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10.48550_arXiv.0801.3321 | ##
(color online) Temperature dependent ZFC (open symbol) and FC (solid symbol) magnetization (M), of LaBiMn\({}_{4/3}\)Co\({}_{2/3}\)O\({}_{6}\) thin film on LaAlO\({}_{3}\) substrates (H = 1000 Oe, applied parallel to film surface H\(\parallel\)\({}_{\rm S}\), and perpendicular to the film surface, H\(\parallel\)\({}_{\rm S}\). The insets show the magnetic hysteresis curves at different temperatures (H\(\parallel\)\({}_{\rm S}\)). Contribution from the substrate was substracted.
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(color online) Variation of the relative permittivity (\(\varepsilon_{\rm r}\)) with temperature for LBMCO compound (left for bulk, and right for thin film). The inset shows the variation of \(\Delta\varepsilon\) % {[\(\varepsilon\) (H)- \(\varepsilon\)/ \(\varepsilon\)] x 100} with magnetic field at 10K for LBMCO/LAO film.
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(color online) Raman spectrum of the LBMCO/LAO in both HH and HV geometry. Inset shows the softening of a Raman mode with temperature in both substrates.
\(\Delta\)(a) (b) (d) (a) (b) (c) (d) (e) (f) (g) (h)
\(\langle\)\(\rangle\) vs \(\langle\)\(\rangle\) for \(\langle\)\(\rangle\) = 0.5, 0.6, 0.7, 0.8, 0.9, 0.1, 0.1, 0.2, 0.3, 0.4, 0.5, 0.6, 0.7, 0.8, 0.9, 0.1, 0.2, 0.
The
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Ranjith, Asish K. Kundu, M. Filippi, B. Kundys, W. Prellier, B. Raveau, J. Laverdiere, M. P. Singh, S. Jandl | 2,650 |
10.48550_arXiv.0711.4745 | ###### Abstract
Thin film coatings have been essential in development of several micro and nano-scale devices. To realize thin film coatings various deposition techniques are employed, each yielding surface morphologies with different characteristics of interest. Therefore, understanding and control of the surface growth is of great interest. In this paper, we devise a novel network-based modeling of the growth dynamics of such thin films and nanostructures. We specifically map dynamic steps taking place during the growth to components (e.g., nodes, links) of a corresponding network. We present initial results showing that this network-based modeling approach to the growth dynamics can simplify our understanding of the fundamental physical dynamics such as shadowing and re-emission effects.
1
H.1Models and PrinciplesMiscellaneous I.6Simulation and ModelingModel Development C.2.1Computer-Communication NetworksNetwork Architecture and Design[network topology, network communications]
## 1 Introduction
Thin film coatings have been the essential components of various devices in industries including microelectronics, optoelectronics, detectors, sensors, micro-electro-mechanical systems (MEMS), and more recently nano-electro-mechanical systems (NEMS). These coatings have thicknesses typically in the nano- to micro-scales and are grown using vacuum deposition techniques. Thin film surface morphology controls many important physical and chemical properties of the films. It is therefore of great interest to understand and control the development of the surface morphology during thin film growth.
Commonly employed deposition techniques are _thermal evaporation, sputter deposition, chemical vapor deposition (CVD)_, and _oblique angle deposition_. Different than others, oblique angle deposition technique is typically used for the growth of nanostructured arrays of rods and springs through a physical self-assembly process. In many applications, it is often desired to have atomically flat thin film surfaces. However, in almost all of the deposition techniques mentioned above, the surface morphology generates a growth front roughness. The formation of growth front is a complex phenomenon and very often occurs far from equilibrium. When atoms are deposited on a surface, atoms do not arrive at the surface at the same time uniformly across the surface. This random fluctuation, or noise, which is inherent in the process, may create the surface roughness. The noise competes with surface smoothening processes, such as surface diffusion (hopping), to form a rough morphology if the experiment is performed at either a sufficiently low temperature or a high growth rate.
A conventional statistical mechanics treatment cannot be used to describe this complex phenomenon. About two decades ago, a dynamic scaling approach was proposed to describe the morphological evolution of a growth front. Since then, numerous modeling and experimental works have been reported based on this dynamic scaling analysis. On the other hand, there has been a significant discrepancy among the predictions of these growth models and the experimental results published. Briefly, theoretical predictions of growth models in dynamic scaling theory basically fall into two categories. One involves various surface smoothing effects, such as surface diffusion. The other category involves the shadowing effect (which originates from the preferential deposition of obliquely incident atoms on higher surface points and always occurs in sputtering and CVD) during growth. However, experimentally reported values of growth exponent (which measures how fast the root-mean-square roughness of the surface evolves as a function of time according to a power-law relation) are far from agreement with the predictions of these growth models. Especially, sputtering and CVD techniques are observed to produce morphologies ranging from very small to very large growth exponent values.
Understanding the thin film and nanostructure growth dynamics under the above-mentioned deposition techniques has been of high importance. There have been several studies revealing fundamental dynamic effects (e.g., _shadowing, re-emission, surface-diffusion, and noise effects_) taking place during the growth process. Studies towards explaining the growth dynamics have been partly successful and only the simulation-based studies were able to include all these effects. In this paper, we devise a novel network-based modeling approach to better understand the growth dynamics. We define a concise mapping between a network and the basic physical operations taking place in the growth process.
We, then, develop qualitative and quantitative understanding of the growth dynamics by studying the corresponding network model. We present our initial results based on previously recorded simulations of the growth process.
The rest of the paper is organized as follows: We start with covering the thin film and nanostructure growth process and the basic physical effects involved in Section 2. We then survey the applications of dynamic network models on various areas in Section 3. Section 4 describes the details of our methodology of mapping growth dynamics to a network. We present initial results of our network-based modeling approach in Section 5, and conclude in Section 6.
## 2 Basics of Thin Film and Nanos-Tructure Growth
Only recently, it has been recognized that in order to better explain the dynamics of surface growth one should take into account the effects of both "shadowing" and "re-emission" processes. As illustrated in Figure 1, particles can approach the surface at oblique angles and be captured by higher surface points (hills) due to the shadowing effect. This leads to the formation of rougher surfaces with columnar structures that can also be engineered to form "nanos-tructures" under extreme shadowing conditions, as in the case of oblique angle deposition that can produce arrays of nanorods and nanosprings. In addition, depending on the detailed deposition process, particles can either stick to or bounce off from their impact points, which is determined by a sticking probability, also named "sticking coefficient" (s). Non-sticking particles are re-emitted and can arrive at other surface points including shadowed valleys. In other words, re-emission has a smoothening effect while shadowing tries to roughen the surface. Both the shadowing and re-emission effects have been proven to be dominant over the surface diffusion and noise, and act as the main drivers of the dynamical surface growth front. The prevailing effects of shadowing and re-emission rely on their "non-local" character: The growth of a given surface point depends on the heights of near and far-away surface locations due to shadowing and existence of re-emitted particles that can travel over long distances.
Due to the complexity of the shadowing and re-emission effects, no growth model has been developed yet within the framework of dynamical scaling theory that take into both these effects and still that can be analytically solved to predict the morphological evolution of thin film or nanostructure deposition. A dynamic growth equation that was proposed by Drotar et al. and developed for plasma and reactive ion etching processes (where in etching surface atoms are removed instead of being incorporated to the surface as in the case of deposition) that take into the re-emission and shadowing effects could only be solved numerically for a limited case of re-emission and shadowing scenarios. Only recently, shadowing and re-emission effects could be fully incorporated into the Monte Carlo lattice simulation approaches.
In brief, conventional growth models, which do not include re-emission effects, in dynamic scaling theory can not explain most of the experimental results reported for dynamic thin film growth. On the other hand, simulation techniques that include re-emission effects along with other important processes such as shadowing, surface diffusion, and noise can successively predict the experimental results but can not always be easily implemented by a widespread of researchers.
## 3 Dynamic Network Models
The study of complex networks pervades various areas of science ranging from sociology to statistical physics. A network in terms of modeling can be defined as a set nodes with links connecting them. Examples of real life complex networks include the Internet, the World Wide Web, metabolic networks, transportation networks, social networks, etc. Recent works, motivated by a large number of natural and artificial systems, such as the ones listed above, have turned the focus onto processes on networks, where the interaction and dynamics between the nodes are facilitated by a complex network. Here, our aim is to construct the network from the apparent dynamics. These systems also typically constitute large scale elements unlike the atomic processes involved during thin film or nanostructure growth.
By using network-based modeling, fundamental understanding of many natural and artificial systems has been attained. In complex networks research, two major types of network models are used for various applications: Small-world and scale-free (power-law) networks. Watts and Strogatz, inspired by a sociological experiment, have proposed a network model known as the small-world (SW) network, which means that, despite their often large size, there is a relatively short path between any two nodes in most networks with some degree of randomness. The SW network was originally constructed as a model to interpolate between regular lattices and completely random networks. Systems and models (with well known behaviors on regular lattices) have been studied on SW networks, such as the Ising model, phase ordering, the Edwards-Wilkinson model, diffusion, and resistor networks.
The other major type of network is based on an observation made in the context of real networks such as the Internet, World Wide Web, scientific collaboration network, and e-mail network. The common characteristic among these networks is that they all exhibit power-law degree (connectivity) distributions. These networks are commonly known as power-law or scale-free networks since their degree distributions are free of scale (i.e., not a function of the number of nodes \(N\)) and follow power-law distributions over many orders of magnitude. This phenomenon has been represented by the probability of having nodes with k degrees as \(P(k)\sim k^{-\gamma}\) where \(\gamma\) is usually between 2 and 3. The origin of the scale-free behavior can be traced back to two mechanisms that are present in many systems, and have a strong impact on the final topology. First, networks are developed by the addition of new nodes that are connected to those already present in the system. This mechanism signifies continuous expansion in real networks.
Surface of a growing thin film (growth front) under shadowing and re-emission effects.
With appropriate mapping to a network model, both of these mechanisms can be qualitatively shown in thin film and nanostructure growth dynamics. If we consider the thin film surface as a set of nodes and re-emissions as the links between them, the first mechanism refers to the understanding that each particle gets "connected" to the grid network by falling on to the film surface. Similarly, the second mechanism refers to that a falling particle will more likely to land on a large-size node thereby contributing to the scale-free topological behavior of the growth dynamics.
## 4 Mapping Growth Dynamics to a Network Model
Interestingly, non-local interactions among the surface points of a growing thin film that originate from shadowing and re-emission effects can lead to non-random preferred trajectories of atoms/molecules before they finally stick and get deposited. For example, during re-emission, the path between two surface points where a particle bounces off from the first and head on to the second can define a "network link" between the two points. If the sticking coefficient is small, then the particle can go through multiple re-emissions that form links among many more other surface points. In addition, due to the shadowing effect, higher surface points act as the locations of first-capture and centers for re-emitting the particles to other places. In this manner, hills on a growing film resembles to the network "nodes" of heavy traffic, where the traffic is composed by the amount of re-emitted particles.
Several issues need to be considered in making a useful and appropriate mapping between the growth dynamics of thin films and nanostructures to a network modeling framework. Let us consider a snapshot of a growing thin film's landscape. In Figure 2(a), let us say that blue color shows currently elevated (i.e. hills) regions of the film and yellow color shows currently not elevated (i.e. valleys) regions of the film. The first mapping issue is to define a "node" in the corresponding network model. That is, what should be the boundary of the corresponding network node on the thin film surface? Intuitively, each blue or yellow region in Figure 2(a) should ideally get mapped to a network node. However, this depends on the resolution of the grid being used for developing a network model. If the grid resolution is too fine, then a blue/yellow region of the film can correspond to multiple nodes as in Figure 2(b). Conversely, if the grid resolution is too coarse, then multiple blue/yellow regions can correspond to one network node as in Figure 2(c). Having finer grid is more likely to capture dynamics of the growth; therefore, we will develop our network models in as fine granularity as possible. For a fine granularity network model, it is always possible to aggregate the data pertaining to neighboring nodes and observe the behavior at coarser granularity. This is illustrated in Figure 3, where the grid network model can be developed at various scales in space.
After fixing the placement of nodes on the thin film, we then map growth dynamics to components of the corresponding grid network model as shown in In general, we argue that we can make an analogy that hills and valleys are nodes of the network system, but hills act as distributing centers, and valleys as gathering centers due to the shadowing and re-emission effects, respectively. The re-emissions of particles can, then, be modeled as a "link" from the re-emission's starting node to the re-emission's ending node. The time it takes for the particle to reach to its new point can be considered as the link's "propagation delay", which implicitly expresses the distance between the starting and the ending nodes of the re-emission. It is even possible to consider the link's "capacity" as the highest possible number of particles that can simultaneously travel from the starting and the ending nodes of the re-emission, which is limited by the physical space corresponding to the link and average size of the re-emitting particles.
Since it is not possible to experimentally track the tra
Some basic processes in the simulation: A particle is sent towards surface with angles \(\theta\) and \(\phi\) based on an angular distribution chosen based on the deposition technique. This particle sticks to the surface with probability \(s_{0}\). If it does not stick, then it is re-emitted after which it may find another surface feature and stick there with probability \(s_{1}\). This re-emission process continues like this for higher-order particles, too. An adatom can diffuse on the surface. Some surface points are shadowed from the incident and re-emission fluxes of particles due to the nearby higher surface features.
Grid network model development in space: Consider two, red and green, particles falling on a growing thin film sample. The red particle makes four re-emissions while the green one makes three re-emissions. We model each re-emission as a “link” between the nodes corresponding to the starting and ending points of the re-emission.
Identification of network “nodes” in a grid network model corresponding to a landscape of a growing thin film.
In these simulations, each incident particle (e.g., atom or molecule) is represented with the dimension of one lattice point. A specific angular distribution for the incident flux of particles is chosen depending on the deposition technique being simulated. At each simulation step, a particle is sent toward a randomly chosen lattice point on the substrate surface. Depending on the value of sticking coefficient \(s\), the particle can bounce off and re-emit to other surface points. At each impact sticking coefficient can have different values represented as \(s_{n}\), where \(n\) is the order of re-emission (\(n=0\) being for the first impact)1. In all the emission and re-emission processes shadowing effect is included, where the particle's trajectory can be cut-off by long surface features on its way to other surface points. After the incident particle is deposited onto the surface, it becomes a so called "adatom". Adatoms can hop on the surface according to some rules of energy, which is a process mimicking the surface diffusion. This simulation steps are repeated for other particles being sent onto the surface. illustrates the basic growth processes included in a typical Monte Carlo simulation approach.
Footnote 1: In this paper we assume a constant sticking coefficient for all subsequent re-emissions.
## 5 Initial Results
In order to explore existence of such a network behavior during thin film and columnar nanostructure growth, we developed 3D Monte Carlo simulations that take into shadowing, re-emission, surface diffusion, and noise effects. These effects simulate the evolution of surface topography and also the simulation environment allows us to record the trajectories of re-emitted atoms. As an example, shows the snapshot top view images of two surfaces simulated for a CVD type of deposition, at two different sticking coefficients. also displays their corresponding particle trajectories projected on the lateral plane. Qualitative network behavior can easily be realized in these simulated morphologies as the trajectories of re-emitted atoms "link" various surface points. It can also be seen that larger sticking coefficients (Figure 5(a) and Figure 5(c)) leads to fewer but longer range re-emissions, which are mainly among the peaks of columnar structures. Therefore, these higher surface points act as the "nodes" of the system. This is due to the shadowing effect where initial particles preferentially head on hills. They also have less chance to arrive down to valleys because of the high sticking probabilities (see also particle A illustrated in Figure 1). On the other hand, at lower sticking coefficients (Figure 5(b) and Figure 5(d)), particles now go through multiple re-emissions and can link many more surface points including the valleys that normally shadowed by higher surface points (e.g. particle B in Figure 1).
Another interesting observation revealed in our Monte Carlo simulations was the dynamic change of network behavior on the trajectories of re-emitted particles. shows top view images and their corresponding particle trajectories obtained from the simulations for a sticking coefficient of \(s=0.9\), but this time at different film thicknesses that is proportional to the growth time. The dynamic change in the network topography can be clearly seen: at initial times, when the hills are smaller and more closely spaced, the re-emitted particles travel from one hill to another one or to a valley. However, as the film gets thicker, and some hills become higher than the others and get more separated, particles travel longer ranges typically among these growing hills. The shorter hills that get shadowed become the valleys of the system. It is expected that this dynamic behavior should be strongly dependent on the values of sticking coefficients and angular distribution of the incident flux of particles, which determine the strength of re-emission and shadowing effects, respectively.
Top view images of simulated thin film surfaces grown under shadowing, re-emission, and noise effects for sticking coefficients (a) \(s=0.9\) and (b) \(s=0.1\). Corresponding projected trajectories of the re-emitted particles are also mapped on the top view morphologies for (c) \(s=0.9\) and (d) \(s=0.1\).
First row: Top view images from the simulated thin film surfaces for a CVD growth with \(s=0.9\) at different film thicknesses \(d\). Bottom row: Corresponding projected trajectories of the re-emitted particles qualitatively show the dynamic change in the network topography.
## Behavior of degree and distance distributions for network models of a CVD thin film growth.
To make some initial observations on the network characteristics based on our network-based models of the growth dynamics, we plotted the degree and distance distributions in for a thin film of size \(512\times 512\) lattice units. We used each lattice unit on the thin film as a node in the corresponding network model and each re-emission as a directed/undirected link between the nodes of the surface locations. We developed the network models for snapshots of the growth where each snapshot being composed of \(10\times 512\times 512\) particles' trajectories. We took four snapshots at different film thickness \(d\). Since the complete growth process is very long this many particles, in some sense, samples the surface morphology. We did this network modeling for two different thin film growths, one with sticking coefficient \(s=0.1\) and the other with \(s=0.9\).
In this manner, Figure 7(a) and (b) shows the degree distribution for the network models of the snapshots when the links are undirected and directed respectively. Overall, the degree distributions exhibit an exponential behavior while becoming power-law as time progresses during the growth. This means that the interrelationship of the surface points become more dominant and some nodes (i.e., columnar structures) on the surface become the main hubs. The degree distributions are quite well characterized even though the growth dynamics are very chaotic. Another interesting observation is that, as time progresses, the degree distribution for the case with high \(s\) converges to the one with low \(s\), which is a non-intuitive result.
Figure 7(c) shows the relationship between the indegree and outdegree by plotting the average outdegree of nodes with a particular indegree value. From this graph also, it seems that the degree distributions converge to a common behavior as time progresses even though sticking coefficients are quite different. Similarly, Figure 7(d) shows the distance distribution of the links in the network, which clearly exhibits a power-law structure. The network model, again, clearly captures the behavior and shows that a higher sticking coefficient yields larger average distance with a pseudo-power-law structure.
## 6 Conclusions
Our initial results on the observation of dynamic network behavior in simulated CVD thin films are very promising and indicate that a novel network modeling approach can be developed for various deposition systems. We showed that particles with non-unity sticking probabilities that are re-emitted and deposited to other parts of the surface can form a network structure constructed by the links among each impact point, which defines nodes of the network. In addition, due to the shadowing effect where obliquely incident particles hit preferentially to the higher surface points, hills of the morphology act as the hubs of the network where most of the particles are re-emitted from these regions. Columnar morphologies formed under high sticking coefficients promote the creation of long-distance network links mainly among the hills, while smoother morphologies of smaller sticking coefficient depositions leads to the formation of shorter range but well-connected links all over the surface points also including valleys. Therefore, this dynamic network behavior during thin film growth strongly depends on the sticking probabilities, presence of obliquely incident particles, and time-dependent morphology of the growing thin film, which leads to the realization of a rich dynamic network system. We believe that this work can lead to an unprecedented understanding of thin film and nanostructure growth, which has been long sought by the researchers. However, in order to fully develop our network concept as a viable modeling approach, more in-depth investigations are necessary.
| 10.48550/arXiv.0711.4745 | Networking Behavior in Thin Film and Nanostructure Growth Dynamics | Murat Yuksel, Tansel Karabacak, Hasan Guclu | 4,698 |
10.48550_arXiv.1511.08868 | ## Capping layer thickness dependence of SOT
The oxygen level can be controlled by the thickness of the SiO\({}_{2}\) capping layer in our layer structure. The sputter-deposited film structure of Pt/CoFeB/MgO/SiO\({}_{2}\) is shown in Fig. 1a, in which the thickness \(t\) of the SiO\({}_{2}\) layer is varied from 0 to 4 nm. For small \(t\), oxygen can easily diffuse through both the SiO\({}_{2}\) and MgO layers, and reach the CoFeB layer. A scanning electron microscope (SEM) image of the patterned Hall bar is shown in We find that \(t\) variation alters the SOT considerably. shows the anomalous Hall resistance (\(R_{\rm H}\)) of the device as a function of the in-plane current (\(I\)) applied to the device. In addition to the current, a small constant magnetic field of 40 mT is applied along the positive current direction to break the symmetry of the device and allow for selective magnetization switching by the in-plane current. Since \(R_{\rm H}\) probes the average \(z\)-component of the CoFeB magnetization, the hysteretic switching of \(R_{\rm H}\) confirms that the current-induced SOT indeed switches the magnetization. The arrows represent the current sweep direction. Interestingly, the resulting switching sequence is clockwise for \(t>1.5\) nm, but counterclockwise for \(t\leq 1.5\) nm. Only the switching sequence for large \(t\) (i.e., low oxygen level) is consistent with the previously reported switching sequence for Pt/Co/AlOx and Pt/CoFe/MgO. Thus, the switching sequence for small \(t\) is abnormal.
The current-induced Oersted field cannot explain the sequence reversal. The switching sequence reversal is not due to the sign reversal of the relation between \(R_{\rm H}\) and the \(z\)-component of the magnetization either, since the purely magnetic-field driven magnetization switching (see Supplementary Figs. 2 and 6) does not exhibit the switching sequence reversal with \(t\). The inset of shows \(I\)s versus \(t\), in which \(I\)s is defined as the current at which \(R_{\rm H}\) changes from a positive to a negative value (note the sign change of \(I_{\rm S}\) around \(t=1.5\) nm). Interestingly, this threshold thickness of 1.5 nm matches the native oxide thickness of Si for passivation. The main panel in shows the ratio between the anisotropy field \(H_{\rm an}\) and \(I_{\rm S}\) as a function of \(t\), which provides a rough magnitude estimation of SOT. Note the abrupt sign reversal of the ratio while its magnitude remains roughly the same before and after the sign reversal. This implies that a new mechanism of SOT is suddenly introduced by the oxidation, generating SOT that is two times stronger and of opposite sign. The abrupt and full sign reversal differs qualitatively from continuous and marginal sign reversal in a previous study.
For independent confirmation of the SOT sign reversal, we perform lock-in measurements of SOT. We apply a small amplitude sinusoidal ac current with a frequency of 13.7 Hz to exert periodic SOT on the magnetization, so that the induced magnetization oscillation around the equilibrium direction generates the second harmonic signal \(V_{\rm 2o}\). Depending on the measurement geometry, \(V_{\rm 2o}\) measures the damping-like or field-like component of SOT, which are two mutually orthogonal vector components of SOT. As current induced magnetization switching is driven mainly by the damping-like SOT, we present here the results for the damping-like SOT only, which can be probed by applying an external dc magnetic field \(H\) along the current direction (tilted 4 degree away from the film plane) to tilt the equilibrium magnetization direction accordingly. The results for the field-like SOT are presented in Supplementary The magnetization switching characteristics have been measured from the first harmonic signals, and asymmetric \(V_{{}_{2o}}\) loops have been observed as shown in For \(t=0\) and 1.2 nm, there is a dip in \(V_{{}_{2o}}\) at a positive field and a peak at a negative field, while the opposite behaviour is observed for \(t=1.8\) and 3 nm. Opposite polarities in \(V_{{}_{2o}}\) prove that the damping-like SOT is pointing in opposite directions for small and large \(t\), confirming the conclusion drawn from the switching sequence in For \(t=1.5\) nm, the \(V_{{}_{2o}}\) signal for positive field contains both a peak and a dip, which may indicate the coexistence of regions with opposite damping-like SOT directions.
## Characterisation of oxidation
In order to verify the oxidation for small \(t\), we have carried out various measurements. shows the oxygen depth profiles obtained by secondary ion mass spectroscopy (SIMS) for devices with \(t=0\) and 2 nm. The depth profile for \(t=0\) nm shows a significantly enhanced oxygen level in the CoFeB layer compared to that of \(t=2\) nm, confirming the oxidation for small \(t\). On the other hand, the two depth profiles are almost identical in the Pt layer, indicating no oxidation of the Pt layer even for small \(t\). This is natural since Pt has excellent resistance to oxidation, which is supported also by the essentially indistinguishable Pt 4\(f\) x-ray photoelectron spectroscopy (XPS) spectra in for the \(t=0\) and 2 nm samples. We also use the x-ray absorption spectroscopy (XAS) to probe the electronic structures of Fe and Co. The Fe and Co \(L_{2,3}\)-edge (\(2p\to 3d\)) XAS spectra in and in Supplementary exhibit spectral features quite similar to those of \(\alpha\)-Fe\({}_{2}\)O\({}_{3}\) and CoO, indicating (Fe,Co)-oxide formation with Fe\({}^{3+}\) and Co\({}^{2+}\), respectively. The XAS data show that the fraction of the oxidized atoms increases as the thickness \(t\) decreases (Figs. 2f and Supplementary Fig. 11). Since Fe\({}^{3+}\) and Co\({}^{2+}\) ions do not have net magnetic moments, the saturation magnetization is expected to decrease with decreasing \(t\), which is indeed the case as confirmed by vibrating sample magnetometry (VSM). The magnetic properties of Fe and Co can also be probed by the x-ray magnetic circular dichroism (XMCD). Both the Fe and Co (Supplementary Fig. 10b) \(L_{2,3}\)-edge XMCD signals become weaker with decreasing \(t\), which is consistent with the VSM measurements, since the XMCD signals arise from the ferromagnetic atoms that remain unionized. The orbital magnetic moments of the ferromagnetic atoms can be evaluated from the XMCD sum rule. For the ferromagnetic Fe atoms, the ratio between the orbital to the spin magnetic moments is about 0.06, which is about 40% larger than the bulk value 0.043 and comparable to the value for epitaxial Fe film at the two-dimensional percolation threshold. Interestingly, as \(t\) decreases, the ratio increases even further to 0.065, indicating an enhancement of the orbital moment of the FM at the interface with the oxide. Considering that orbital moment enhancement typically occurs when ferromagnetic atoms are in an environment with broken symmetry, this suggests that the ferromagnetic atoms are subject to more strongly broken symmetry as \(t\) decreases.
## Sign reversal of SOT by _in-situ_ oxidation
All these measurements support the oxidation of the CoFeB layer for small \(t\). However, we cannot yet rule out the possibility that the oxidation is merely correlated with instead of the cause of the SOT sign reversal. In order to verify that the oxidation of the FM is the key parameter for the sign reversal, we examine a sample with an oxidized CoFeB layer but with large \(t\) (3 nm). To prepare such a sample, we intentionally oxidize the CoFeB layer with O\({}_{2}\) gas during its deposition before depositing the capping layers including 3 nm of SiO\({}_{2}\) as shown in In this case, even for \(t=3\) nm, we observe an abnormal anticlockwise switching loop as shown in Hence what is important for the SOT sign reversal is the FM layer oxidation itself rather than the value of \(t\). Small \(t\) is merely a method to induce the oxidation. This result rules out other \(t\)-dependent changes such as strain from being key parameters of the SOT sign reversal.
## SOT beyond the spin Hall effect
We now discuss the connection between the FM layer oxidation and the SOT sign reversal. The bulk spin Hall effect in the HM layer is an important mechanism of SOT. According to the spin Hall interpretation of SOT, the sign of the damping-like SOT is determined by the bulk spin Hall angle of the HM layer. The FM affects the damping-like SOT through the real part of the spin mixing conductance, which is always positive and cannot change its sign since its being negative implies a negative charge conductance. The spin Hall interpretation is thus inadequate to explain the oxygen-induced sign reversal of the damping-like SOT, since the HM layer is not affected by oxidation (and b). Furthermore, we have verified that the SOT change caused by the oxidation is essentially independent of the Pt thickness (Supplementary Fig. 5). Hence our data necessitate a new source of SOT, other than the bulk spin Hall effect of the HM layer. One can think of two possibilities; one is the oxidized FM layer itself being a SOT source, and the other is the top or bottom interfaces of the oxidized FM layer being a SOT source.
To examine the first possibility, we change the thickness \(d_{\rm CFB}\) of the CoFeB layer for fixed \(t=0\) nm. Up to \(d_{\rm CFB}=2\) nm, the perpendicular magnetic anisotropy is well maintained, and the saturation magnetization (\(M\)s) values are almost constant and do not increase with \(d_{\rm CFB}\), implying an almost \(d_{\rm CFB}\)-independent oxidation level. The change of \(d_{\rm CFB}\) has a negligible effect on the current density, since the resistivity of CoFeB is much greater than that of Pt. While the abnormal anti-clockwise switching sequence is maintained with changing \(d_{\rm CFB}\), \(H_{\rm an}/I_{\rm S}\) and \(H_{\rm L}\) change almost linearly with 1/\(d_{\rm CFB}\). That is, \(H_{\rm L}\)\(\times\)\(d_{\rm CFB}\), which is proportional to the total torque acting on the CoFeB layer, does not increase with \(d_{\rm CFB}\). This implies that the bulk part of the oxidized FM layer is not an SOT source. Hence, one can exclude the first possibility.
To examine the second possibility, we eliminate the MgO layer from the device stack structure. The switching sequence reversal from the normal clockwise to abnormal anti-clockwise direction is still observed, when \(t\) changes from 4 to 1.2 nm as shown in This shows that the interface between the oxidized FM layer and the MgO layer is not the new SOT source. Next we eliminate the Pt layer instead. The current-induced switching itself is not observed nor is the 2\({}^{\rm nd}\) harmonic signal (Supplementary Fig. 13). This leads us to conclude that the interface between the oxidized CoFeB layer and the Pt layer is the new SOT source. We further extend our experiments to devices in which the FM material CoFeB is replaced with Co. As shown in Fig. 5c, the current-induced switching sequence shows normal clockwise behaviour for \(t\) = 3 nm, but reverses to abnormal anti-clockwise behaviour for \(t\) = 0 nm, which is similar to the results from devices with CoFeB as the FM layer. Hence the observed sign reversal phenomenon is not restricted to a specific FM material, but can be universal.
The most plausible mechanism consistent with our experimental data is then the interfacial spin-orbit coupling, some signatures of which have been reported in earlier experiments by varying the degree of oxidation or changing the thickness of the Ta underlayer.
If its contribution to the damping-like SOT is of opposite sign to the spin Hall contribution and becomes larger with oxidation than the spin Hall contribution, the competition between these two contributions can explain the sign reversal of the damping-like SOT upon oxidation. For this, the oxidation should enhance either (i) the interfacial spin-orbit coupling strength or (ii) the efficiency to generate the damping-like torque for a given interfacial spin-orbit coupling strength.
There is a well-known example of (i). The interfacial spin-orbit coupling at the magnetic Gd surface becomes three times stronger upon oxidation, and interestingly reverses its sign. The enhanced strength is attributed to the enhanced internal electric field at the surface. An additional mechanism of (i) may arise from the atomic orbital degree of freedom. When atomic orbitals with angular momentum \(\overline{L}\) are linearly superposed to make a Bloch state with crystal momentum \(\overline{K}\), the quantum interference between orbitals of neighbouring atoms generates an electric dipole moment towards the direction \(\overline{L}\)\(\times\)\(\overline{K}\), which couples with the internal electric field \(\overline{E}\) at the surface to generate a Coulomb energy proportional to \(-\overline{E}\cdot\left(\overline{L}\times\overline{K}\right).\) When \(\overline{E}\) is sufficiently strong and the orbital quenching is weak, this energy tends to align\(\overline{L}\) along the direction \(\overline{K}\)\(\times\)\(\overline{E}\). Such \(\overline{L}\) couples with spin \(\overline{S}\) through the atomic spin-orbit coupling \(\alpha_{SO}\overline{L}\)\(\cdot\)\(\overline{S}\) at the surface, where the coupling constant \(\alpha_{SO}\) is large due to the hybridization between Pt \(5d\) orbitals and ferromagnetic \(3d\) orbitals. Subsequently, the strong atomic spin-orbit coupling \(\alpha_{SO}\overline{L}\)\(\cdot\)\(\overline{S}\) is converted to the strong interfacial spin-orbit coupling \(\alpha_{SO}\)\(\overline{K}\)\(\times\)\(\overline{E}\)\(\cdot\)\(\overline{S}\). It has been suggested that this orbital-based mechanism may enhance the Rashba-type interfacial spin-orbit coupling significantly. A recent first principles calculation for a Pt/Co bilayer confirms that a strong interfacial spin-orbit coupling can indeed arise at the HM/FM interface near the Fermi energy.
Regarding (ii), we are not aware of any theoretical mechanism that predicts efficiency enhancement by oxidation. We remark, however, that the damping-like SOT caused by interfacial spin-orbit coupling has been significantly underestimated in earlier theories. It was later pointed out that due to the Berry phase effect, the efficiency of the interfacial spin-orbit coupling mechanism is actually much higher and comparable to that of the spin Hall mechanism. The Berry phase effect has been confirmed for a bulk inversion symmetry broken material (Ga,Mn)As, but not yet for structural inversion symmetry broken interfaces. Previous observations of the damping-like SOT were attributed to the bulk spin Hall mechanism.
Next we discuss the abruptness of the SOT sign reversal (Figs. 1e & 1f) despite the rather gradual changes of oxidation level (Figs. 2f & Supplementary Fig. 11). Concomitant with the sudden SOT sign reversal, the coercivity (Supplementary Fig. 2) and the temperature dependence of \(R_{\rm H}\) (Supplementary Fig. 14) also change suddenly. We suspect that such sudden changes may be manifestations of SOT instability. A possible origin of the SOT instability is the competition between the orbital ordering \(\overline{L}\!\propto\!\overline{K}\!\times\!\overline{E}\) and the orbital quenching common in transition metals. Although its origin is still unclear, the abrupt SOT reversal is of considerable value for device applications. When the oxidation level is near the threshold value, a tiny change of the oxidation level by electric gating can induce a large change of SOT. This takes the SOT engineering to a whole new level and may even pave the way towards reconfigurable logic devices.
Our results may also be relevant to recent SOT experiments using Ta which is more susceptible to oxidation compared to Pt. Our results indicate that even for the exact same layer structure, very different SOTs can be obtained depending on the detailed device preparation procedure, which may affect the oxygen content in the sample. Furthermore, we hope our work initiates efforts to bridge the gap between metal spintronics and oxide electronics to combine the merits of the both fields.
## Methods
The stacked films were deposited on thermally oxidized Si substrates by magnetron sputtering with a base pressure \(<2\times 10^{-9}\) Torr at room temperature. The structure of the \(t\) series films is substrate/MgO/Pt/Co\({}_{60}\)Fe\({}_{20}\)B\({}_{20}\) (0.8)/MgO/SiO\({}_{2}\) (\(t\)) with \(t=0\sim 4\), and that of the \(d_{\rm{CFB}}\) series films is substrate/MgO/Pt/Co\({}_{60}\)Fe\({}_{20}\)B\({}_{20}\) (\(d_{\rm{CFB}}\))/MgO/SiO\({}_{2}\) (\(t=0\) and 3) with \(d_{\rm{CFB}}\) = 0.8 \(\sim\) 2 (numbers are nominal thicknesses in nanometers). The bottom MgO layer is used to promote perpendicular anisotropy. The other film structures are schematically shown in Figs. 3 and 5. After deposition, except for the oxygen doped CoFeB sample in Fig. 3, all the other films were post-annealed at 300 \({}^{\circ}\)C for 1 hour in a vacuum to obtain perpendicular anisotropy. The multilayers were coated with a ma-N 2401 negative e-beam resist and patterned into 600 nm width Hall bars by electron beam lithography and Ar ion milling as shown in PG remover and acetone were used to lift-off the e-beam resist. Contact pads were defined by photolithography followed by the deposition of Ta (5 nm)/Cu (150 nm)/Ru (5 nm) which are connected to the Hall bars. Before the deposition of the contact pads, Ar ion milling was used to remove the SiO\({}_{2}\) and part of the MgO layer, in order to make low-resistance electrical contacts. All the devices for each batch were processed at the same time to ensure the same fabrication conditions, which was important for this study. Devices were wire-bonded to the sample holder and installed in a physical property measurement system (Quantum Design) for the transport studies.
We performed the measurements of current induced switching and the ac harmonic anomalous Hall voltage loops for the \(t\) and \(d_{\rm{CFB}}\) series devices at 200 K for the data set in Figs. 1and 4, at which temperature all the devices retain desirable perpendicular anisotropy (Supplementary Figs. 1-3). The current induced switching was measured using a combination of Keithley 6221 and 2182A. A pulsed dc current of a duration of 50 us was applied to the nanowires and the Hall voltage was measured simultaneously. An interval of 0.1 s was used for the pulsed dc current to eliminate the accumulated Joule heating effect.
The x-ray absorption spectroscopy (XAS) and x-ray magnetic circular dichroism (XMCD) measurements were performed at the 2A beamline in the Pohang Light Source. All the spectra were measured at 200 K in the total electron yield mode, and the energy resolution was set to be \(\sim\) 300 meV. The reference XAS spectra were obtained from \(\alpha\)-Fe\({}_{2}\)O\({}_{3}\) and CoO bulk crystals at room temperature. An electromagnet with a 0.4 T magnetic field was used for the magnetization along the normal direction of the sample for the XMCD measurements at normal incidence of more than 95% circularly polarized light. The spin and orbital magnetic moment ratio was estimated by using the sum rule, and the metal to the oxide ratio was extracted from the XAS spectra by using the reference spectra of metallic Fe (Co) and \(\alpha\)-Fe\({}_{2}\)O\({}_{3}\) (CoO).
| 10.48550/arXiv.1511.08868 | Spin-orbit torque engineering via oxygen manipulation | Xuepeng Qiu, Kulothungasagaran Narayanapillai, Yang Wu, Praveen Deorani, Dong-Hyuk Yang, Woo-Suk Noh, Jae-Hoon Park, Kyung-Jin Lee, Hyun-Woo Lee, Hyunsoo Yang | 4,817 |
10.48550_arXiv.0705.2645 | ###### Abstract
We investigate graphene and graphene layers on different substrates by monochromatic and white-light confocal Rayleigh scattering microscopy. The image contrast depends sensitively on the dielectric properties of the sample as well as the substrate geometry and can be described quantitatively using the complex refractive index of bulk graphite. For few layers (\(<\)6) the monochromatic contrast increases linearly with thickness: the samples behave as a superposition of single sheets which act as independent two dimensional electron gases. Thus, Rayleigh imaging is a general, simple and quick tool to identify graphene layers, that is readily combined with Raman scattering, which provides structural identification.
Graphene is the prototype two dimensional carbon system. Its electron transport is described by the (relativistic-like) Dirac equation and this allows access to the rich and subtle physics of quantum electrodynamics in a relatively simple condensed matter experiment. The scalability of graphene devices to true nanometer dimensions makes it a promising candidate for future electronics, because of its ballistic transport at room temperature combined with chemical and mechanical stability. Remarkable properties extend to bi-layer and few-layers. More fundamentally, the various forms of graphite, nanotubes, buckyballs can all be viewed as derivatives of graphene.
Graphene samples can be obtained from micro-mechanical cleavage of graphite. Alternative procedures include chemical exfoliation of graphite or epitaxial growth by thermal decomposition of SiC. The latter has the potential of producing large-area lithography compatible films, but is substrate limited. It is hoped that in the near future efficient large area, substrate independent, growth methods will be developed, as it is now the case for nanotubes.
Despite the wide use of the micro-mechanical cleavage, the identification and counting of graphene layers is still a major hurdle. Monolayers are a great minority amongst accompanying thicker flakes. They cannot be seen in an optical microscope on most substrates. Currently, optically visible graphene layers are obtained by placing them on the top of oxidized Si substrates with typically 300 nm SiO\({}_{2}\). Atomic Force Microscopy (AFM) is viable but has a very low throughput. Moreover, the different interaction forces between the AFM probe, graphene and the SiO\({}_{2}\) substrate, lead to an apparent thickness of 0.5-1 nm even for a single layer, much bigger of what expected from the interlayer graphite spacing. Thus, in practice, it is only possible to distinguish between one and two layers by AFM if graphene films contain folds or wrinkles. High resolution transmission electron microscopy is the most direct identification tool, however, it is destructive and very time consuming, being viable only for fundamental studies.
Optical detection relying on light scattering is especially attractive because it can be fast, sensitive and not-destructive. Light interaction with matter can be elastic or inelastic, and this corresponds to Rayleigh and Raman scattering, respectively. Raman scattering has recently emerged as a viable, non-destructive technique for the identification of graphene and its doping. However, Raman scattered photons are a minority compared to those elastically scattered. Here we show that the elastically scattered photons provide another very efficient and quick means to identify single and multi-layer samples and a direct probe of their dielectric constant.
Rayleigh scattering was previously used to monitor size, shape, concentration and optical properties of nano
Schematic experimental set-up for combined Rayleigh and Raman spectroscopy. The inset shows a cross sectional view of the interaction between the optical field and graphene deposited on Si covered with SiO\({}_{2}\).
Rayleigh scattering experiments can be performed using two different strategies. In one, the background signal is minimized by making free-standing samples, as done in the case of carbon nanotubes, or by dark-field configurations. Alternatively, the background intensity is utilized as a reference beam, while the sample signal is detected interferometrically. Here, we combine the second approach with the interferometric modulation of the contributing fields and we show that the presence of a background is essential to enhance the detection of graphene over a certain wavelength range.
Graphene samples are produced by micro-mechanical cleavage of bulk graphite and deposited on a Si substrate covered with 300 nm SiO\({}_{2}\) (IDB Technologies LTD). The sample thickness is independently confirmed by a combination of AFM and Raman spectroscopy. AFM is performed in tapping mode under ambient conditions. Raman spectra are measured at 514 nm using a Renishaw micro-Raman 1000 spectrometer. Rayleigh scattering is performed with an inverted confocal microscope, Fig.1. Either a He-Ne laser (633 nm) or a collimated white-light beam are used as excitation source. Coherent white-light pulses are generated by pumping a photonic crystal fibre with the output of a Ti:Sa oscillator operating at 760 nm. The beam is reflected by a beam splitter and focused by a microscope objective with high numerical aperture (NA\(=0.95\)). However,the objective lens is not totally filled, which results in an effective NA\(\sim\)0.7 thereby increasing the image contrast as discussed at the end of this paper. The scattered light from the sample is collected in backscattering geometry, transmitted by a beam splitter and detected by a photon-counting avalanche photodiode (APD), Fig.1. Alternatively, the reflected light is filtered using a notch filter to remove the laser excitation and sent to a spectrometer. _This allows simultaneous Rayleigh and Raman measurements_, Fig.1,2a. Confocal Rayleigh images are obtained by raster scanning the sample with a piezoelectric scan stage. The acquisition time per pixel varies from few ms in the case of Rayleigh scattering to few minutes for Raman scattering. This empirically indicates that Rayleigh measurements are almost 5 orders of magnitude quicker than Raman measurements. The spatial resolution is \(\sim\) 800 nm.
Fig.2(b) shows an AFM image of monolayer graphene. The AFM cross section gives an apparent height of \(\sim\)0.6 nm. Raman spectroscopy confirms that the sample is a single layer (Fig.2(a)). Fig.2(b) is the corresponding confocal Rayleigh image obtained with monochromatic laser light (633 nm). Fig.3(a) shows an optical micrograph of a sample composed of a varying number of layers. Once the single layer is identified by Raman scattering, we get the total number of layers from the measured AFM height, considering the interlayer spacing of \(\sim\) 0.33 nm: z [nm]\(=\) 0.27 \(+\) 0.33 N. This confirms that the sample is composed of 1, 2, 3 and 6 layers, as for Fig.3 (a). These layers have a slightly different color in the optical microscope (Fig.3 (a)). It appears that the darker color corresponds to the thicker sample. Note, however, that the color of much thicker layers (more than 10 layers) does not follow this trend and can change from blue, to yellow, to grey. The number of layers is further confirmed by the evolution of the 514 nm Raman spectra,(b). Fig.4(a) shows a confocal Rayleigh map for 633nm excitation. The signal intensity of in Fig.4 appears to increase with N.
We now discuss the physical origin of the image contrast (\(\delta\)). This is defined as the difference between substrate and sample intensity, normalized to the substrate intensity. The single layer contrast at 633 nm is \(\sim 0.08\). The contrast is positive, i.e. the detected intensity from graphene is smaller than that of the substrate. The Rayleigh images in Fig.2 (c) and Fig.4 (a) are reversed for convenience, in order to compare them with AFM.
We explain the sign and scaling of the contrast for increasing N in terms of interference from multiple reflections. The inset in Fig.1 shows a schematic of the
(a) Raman spectrum at 514 nm, showing the features of graphene; (b) AFM image of single layer graphene (c) Confocal Rayleigh image obtained by raster scanning the sample with a piezoelectric scan stage.
When the light impinges on a multi-layer, multiple reflections take place. Thus, the detected signal (I) results from the superposition of the reflected field from the air-graphene (\(E_{G}\)), graphene-SiO\({}_{2}\) (\(E_{SiO_{2}}\)), and SiO\({}_{2}\)-Si interfaces (\(E_{Si}\)). The back-ground signal (\(I_{Bg}\)) results from the superposition of the reflected field from the air-SiO\({}_{2}\) interface and the Si substrate.
Before giving a complete quantitative model, it is useful to consider a simplified picture that captures the basic physics and illustrates why a single atomic layer can be visualized optically. The field at the detector is dominated by two contributions: the reflection by the graphene layer, and the reflection from the Si after transmission through graphene and after passing through the SiO\({}_{2}\) layer twice. Thus, the intensity at the detector can be approximated as:
\[I\sim|E_{G}+E_{Si}|^{2}=|E_{G}|^{2}+|E_{Si}|^{2}+2|E_{G}||E_{Si}|\cos\phi \tag{1}\]
This includes the phase change due to the optical path length of the oxide, \(d_{SiO_{2}}\), and that due to the reflection at each boundary, \(\vartheta_{Si}\) and \(\vartheta_{G}\):
\[\phi=\vartheta_{G}-(\vartheta_{Si}+2\pi\ n_{SiO_{2}}2d_{SiO_{2}}/\lambda_{0}) \tag{2}\]
Assuming the field reflected from graphene to be very small, \(|E_{G}|^{2}\simeq 0\), the image contrast \(\delta\) results from interference with the strong field reflected by the silicon:
\[\delta=(I_{Si}-I)/I_{Si}\simeq-2\cdot|E_{G}|/|E_{Si}|\cdot\cos\phi \tag{3}\]
The sign of \(\delta\) depends on the sign of \(\cos\phi\), which is given by Eq. 2. The reflectance, R, is the ratio between the reflected power to the incident power. Assuming the Si reflectance as one, Eq. 3 can be written as:
\[\delta=-2\surd R_{G}\ \cos\phi \tag{4}\]
This is in turn related to the reflection coefficient \(r_{G}\):
\[r_{G}=\surd R_{G}\cdot exp(i\vartheta_{G}) \tag{5}\]
Eq. 4 shows that the main role of the SiO\({}_{2}\) is to act as a spacer: the contrast is defined by the phase variation of the light reflected by the Si. _Thus, the contrast for a given wavelength can be tailored by adjusting the spacer thickness or its refractive index._
In order to investigate the wavelength dependence of the image contrast, we perform Rayleigh spectroscopy
(a) Optical micrograph of multi-layer with 1, 2, 3 and 6 layers; (b) Raman spectra as a function of number of layers.
(a) Three-dimensional confocal Rayleigh map for monochromatic 633 nm excitation. The window size is 49\(\mu\)m x 49\(\mu\)m; (b) Experimental (dots) and theoretical (line) contrast as a function of excitation wavelength.
A grating is used to analyze the detected light. Fig.4 (b) shows that for N=1 the contrast is maximum at \(\sim\) 570 nm. The contrast at 633 nm is \(\sim\) 0.08, in agreement with the monochromatic Rayleigh scattering experiment. The contrast is zero at 750 nm and it is small and negative for \(\lambda\)\(>\) 750 nm. From Eqs. 2 and 4 and assuming \(\vartheta_{Si}=-\pi\), the phase of graphene is \(\vartheta_{G}\simeq-\pi\) as expected for an ultra-thin film. The contrast decreases in the near IR (for \(d_{SiO_{2}}\)= 300 nm) since the wavelength becomes larger than twice the optical path length provided by the SiO\({}_{2}\)-spacer. (b) shows that while the contrast increases for increasing N, the phase remains constant.
We now present a more accurate model, with no assumptions, which describes the light modulation by multiple reflections based on the recurrent matrix method for reflection and transmission of multilayered films. We calculate the total electric and magnetic fields in the various layers, applying the boundary conditions at every interface. The fields at two adjacent boundaries are described by a characteristic matrix. This depends on the complex refractive index and the thickness of the film and the angle of the incident light. By computing the characteristic matrix of every layer and taking into account the numerical aperture of the objective and the filling factor, it is possible to find the reflection coefficient for an arbitrary configuration of spacer and substrate and for any number of graphene layers (G).
\[M_{12}=\bigg{[}\cos\phi_{G}\cos\phi_{2}\bigg{(}1-\frac{n_{Air}}{ n_{3}}\bigg{)}-\] \[\sin\phi_{G}\sin\phi_{2}\bigg{(}\frac{n_{G}}{n_{2}}-\frac{n_{Air} n_{2}}{n_{G}n_{3}}\bigg{)}\bigg{]}\] \[-i\bigg{[}\cos\phi_{G}\sin\phi_{2}\bigg{(}\frac{n_{2}}{n_{3}}- \frac{n_{Air}}{n_{2}}\bigg{)}-\] \[\sin\phi_{G}\cos\phi_{2}\bigg{(}\frac{n_{G}}{n_{3}}-\frac{n_{Air} }{n_{G}}\bigg{)}\bigg{]} \tag{7}\]
\[M_{22}=\bigg{[}\cos\phi_{G}\cos\phi_{2}\bigg{(}1+\frac{n_{Air}}{ n_{3}}\bigg{)}-\] \[\sin\phi_{G}\sin\phi_{2}\bigg{(}\frac{n_{G}}{n_{2}}+\frac{n_{Air} n_{2}}{n_{G}n_{3}}\bigg{)}\bigg{]}\] \[-i\bigg{[}\cos\phi_{G}\sin\phi_{2}\bigg{(}\frac{n_{2}}{n_{3}}+ \frac{n_{Air}}{n_{2}}\bigg{)}+\] \[\sin\phi_{G}\cos\phi_{2}\bigg{(}\frac{n_{G}}{n_{3}}+\frac{n_{Air }}{n_{G}}\bigg{)}\bigg{]} \tag{8}\]
For incidence at an angle \(\theta\), with s-polarization (transverse electric field), the same formula applies with the substitution \(n_{i}\to n_{i}\cos\theta_{i}\), while for p-polarization every ratio changes \(n_{i}/n_{j}\to n_{i}\cos\theta_{j}/n_{j}\cos\theta_{i}\). The phases change in both s and p polarizations to \(\phi_{G}=2\pi n_{G}d_{G}\cos\theta_{G}/\lambda_{0}\) and \(\phi_{2}=2\pi n_{2}d_{2}\cos\theta_{2}/\lambda_{0}\). The angle \(\theta_{i}\) for every layer is obtained from Snell's law: \(\theta_{i}=\arcsin(\sin\theta_{0}/n_{i})\). In case any of the layers is absorbing (as in graphene and Si), we need use an effective index \(n^{\prime}_{i}=f(n_{i},\theta_{0})\) which depends on the incident angle from vacuum \(\theta_{0}\). In this case the corresponding refraction angle is \(\theta_{i}=\arcsin[\sin\theta_{0}/Re(n^{\prime}_{i})]\).
The matrix method requires as input the complex refractive index of the sample. The frequency dependent Si and SiO\({}_{2}\) indexes are taken from Ref.. For graphene, few layers graphene and graphite, this is anisotropic, depending on the polarization of the incident light. For electric field perpendicular to the graphene c-axis (in-plane) we need \(n_{GPerp}\), while for electric field parallel to the c-axis we need \(n_{GParal}\). To get these, we use the experimental refractive index taken from the electron energy loss spectroscopy measurements on graphite of Ref.. For s-polarized light (electric field restricted in the plane) the refractive index to be used is simply \(n_{s}=n_{GPerp}\). For p-polarization, both in-plane and out-of-plane field components exists. Thus we have an angle dependent refractive index \(n_{p}^{-2}=n_{GPerp}^{-2}\cos^{2}\theta_{i}+n_{GParal}^{-2}\sin^{2}\theta_{i}\), where the refracted angle \(\theta_{i}\) has to be calculated self-consistently with Snell's law. In order to account for the numerical aperture in the experiment, we need to integrate the response of all possible incident angles and polarizations with a weight distribution accounting for the Gaussian beam profile used in the experiment \(f(\theta_{0})=e^{-2\sin^{2}\theta_{0}/\sin^{2}\theta_{m}}2\pi\sin\theta_{0}\), where \(\theta_{m}=arcsin(NA)\).
Fig.4(b) shows the calculated contrast for N between 1 and 6 (lines). This is in excellent agreement with the experiments: i) the contrast scales with number of layers; ii) it is maximum at \(\sim\)570 nm; iii) no phase shift is observed in this N range. Thus, for N between 1 and 6, \(\cos\phi(\lambda=570\ nm)=-1\). The contrast of graphene at 570 nm is \(\sim\) 0.1. From Eqs. 4 and 5 we get \(r_{G}\) (\(\lambda\)= 570 nm)= 0.05. Thus, \(R_{G}\) (\(\lambda\)= 570 nm)= 0.003.
It quite remarkable that, without any adjustable parameter, graphene's response can be successfully modeled using graphite's dielectric constant. This implies that the optical properties of graphite do not depend on the thickness, i.e. graphene and graphite have the same optical constants. The electrons within each graphene layer form a two dimensional gas, with little perturbation from the adjacent layers, thus making multi-layer graphene optically equivalent to a superposition of almost non-interacting graphene layers. This is intuitive for s-polarization. However, quite notably this still holdswhen the out-of-plane direction (p-polarization) is considered. This is because, compared to the in-plane case, graphite's response is much smaller, and in addition it gets smeared out by the NA integration. Thus, the maximum contrast (\(\lambda\)= 570 nm) of a N-layer is: \(\delta(N)=0.1\cdot N\). Fig. 6(a) shows that this approximation fails for large N. When valid, the relation between topography and contrast is given by: z[nm]= 0.27 + 3.3\(\delta(N)\).
Fig.5(b) plots the contrast as a function of wavelength and SiO\({}_{2}\) thickness for a single layer. The maximum contrast occurs at the minima of the background reflectivity. This is expected because this is the most sensitive point in terms of phase matching, and small changes become most visible. Thus, the optimal configuration requires the SiO\({}_{2}\) to be tuned as an anti-reflection (AR) coating, i.e. with its optical length a quarter wavelength. The yellow dotted lines trace the quarter-wave condition \(2n_{Si0_{2}}d_{Si0_{2}}/\lambda_{0}=(m+1/2)\), and indeed they closely follow the calculated contrast maxima. A second point of interest are the bright spots around 275 nm. These are due to the absorption peak at the \(\pi\rightarrow\pi*\) transition of graphite. For this excitation, the graphene monolayer not only becomes much more visible, but the contrast change also directly reveals the frequency dependence of the graphene's refractive index. Thus, as for nanotubes, white light Rayleigh scattering is a direct probe of the dielectric function.
For thicker samples (\(N>10\)) the phase change due to the optical path in graphite cannot be neglected. Fig.6(a) shows the calculated contrast for a 50 layer sample as a function of SiO\({}_{2}\) thickness, while Fig.6(b) plots the contrast for a fixed 300nm SiO\({}_{2}\) thickness, but for a variable number of layers. At 633 nm, as N increases, the response first saturates, then decreases and red-shifts, finally becoming negative, as found experimentally ((c)). It is also interesting to note that for small N the variation along the vertical (wavelength) axis is largely between zero and positive (i.e. reflectivity reduction only), while for large number of layers, the variation is from positive to negative (i.e. both reflectivity reduction and enhancement). This points to two different mechanisms. For small N, the effect of the graphene layers is just to change the reflectivity of the air/SiO\({}_{2}\) interface, while they offer no significant optical depth.
(a) Calculated contrast of 50 layers as a function of oxide thickness and excitation wavelength; (b)contrast at 633 nm for 300 nm SiO\({}_{2}\) as a function of N; (c) Experimental contrast at 633 nm for a thick sample.
(a) Maximum contrast at 633nm as a function of N; (b) Calculated contrast of graphene as a function of oxide thickness and excitation wavelength. Dotted lines trace the quarter-wavelength condition.
This change is not a monotonic function of N. While these two effects are different, they both contribute to a shift of the reflectivity resonance condition, and thus explain the increasing opaqueness of thicker graphene layers, when measured for a fixed excitation energy.
It is also interesting to consider the contrast as a function of NA. The calculations show that measurements at a reduced NA would give a stronger contrast, as one could intuitively expect. However, there is a nontrivial implication when varying NA, if one tries to maximize the contrast by using the anti-reflection coating rule for the spacer. The ideal AR coating over a substrate of index \(n_{subst}\) must have an index \(n_{spacer}=\surd n_{subst}\) and quarter wave thickness \(d_{spacer}=(m+1/2)\lambda_{0}/2\surd n_{subst}\). Since \(n_{Si}\sim 4\) at 600 nm, it is natural to think that a spacer of n=2 (e.g. Si\({}_{3}\)N\({}_{4}\)) would be ideal. To explore this, plots the contrast for different NAs as a function of \(n_{spacer}\) at 600nm and for spacer thickness \(d_{spacer}=300nm(n_{Si02}/n_{spacer})\), which serves to maintain the AR condition and thus the maximum response.
Contrary to expectations, the contrast maximizes for different spacer indexes depending on NA. For normal incidence, it is maximum at 1.93 with a huge contrast of 0.6 for a single layer, It also has a strong variation thereafter, and becomes negative. As NA further increases, the peak moves to a smaller index (around 1.5 for NA=0.7), becomes relatively flat, and eventually goes to \(n_{spacer}=1\). Thus, for large NA, it makes little difference what the spacer index is, as long as the quarter-wave condition is satisfied. Indeed, for the ideal AR condition the background reflectivity goes to zero and thus the contrast becomes large, however this condition strongly depends on the incidence angle and is thus easily destroyed at large NAs. For all possible spacer refractive indexes, a reduction in NA results into an increased contrast, however, the magnitude of this increase varies: at n=1.5 going from 0.7 to 0.0 NA changes the contrast by a factor 2, while at n=1.9 one can gain a factor of 6, For maximum visibility, a \(Si_{3}N_{4}\) spacer of thickness 225nm with NA=0.0 would be ideal. However,if high resolution is needed, as for nano-ribbons or, in general, to analyze edges and defects, a compromise between resolution and image contrast is necessary.
A second point to note is that for all NAs the contrast converges to the same value for n=1, i.e. for a suspended graphene layer over the substrate. Indeed, optically visible suspended layers were recently reported (see Fig.1 of Ref.). Maximum visibility is achieved if the quarter-wave condition is satisfied, as indeed in Ref., where the 300 nm SiO\({}_{2}\) spacer is etched to create an air gap between graphene and the Si substrate. Interestingly, in this case any measurement with any NA will yield the same contrast. The same considerations are relevant for the case of a thin free-standing spacer (no substrate). By tuning at the low reflection point (now at half-wavelength) and with an NA=0.0 one could get fair contrasts. However, as soon as NA increases, the resonance condition is destroyed and the contrast becomes much smaller than for the SiO\({}_{2}\)/Si system.
The matrix method can be extended to every film configuration. To prove this, we measure graphene layers on glass. For N=1, the calculated contrast at 633 nm is expected to be\(\sim\)0.01. Note the different sign compared with the Si/SiO\({}_{2}\) substrate. This is due to the different optical properties of glass and Si. Fig.8(a) shows an optical micrograph of a multi-layer and Fig.8(b) the corresponding Rayleigh image at 633 nm. Raman spectroscopy shows that the sample is composed of layers of different thickness: A (7-10 layers), B (3-6 layers), C (1-2 layers).
Maximum calculated contrast as a function of spacer refractive index and objective numerical aperture (NA).
(a) Optical micrograph of flakes on glass. (b) Rayleigh image at 633 nm excitation. The contrast is much higher compared to (a).
Note that the use of UV light could enhance the contrast to \(\sim\) -0.04 at 300 nm excitation ((b)).
In conclusion, we used white light illumination combined with interferometric detection to study the contrast between graphene and Si/SiO\({}_{2}\) substrates. We modeled the light modulation by multiple reflections, showing that: i) the contrast can be tailored by adjusting the SiO\({}_{2}\) thickness. Without oxide, no modulation is possible; ii) the light modulation strongly depends on the graphite thickness. For few layers (\(<6\)) the samples behave as a superposition of single sheets. For thicker samples, both amplitude and phase change with thickness. Thus, Rayleigh spectroscopy provides a simple and quick way to map graphene layers on a substrate. It can also be combined with Raman scattering, which is capable of structural identification.
_Acknowledgements._ The authors acknowledge A.K. Geim for useful discussions. CC acknowledges S. Reich for useful discussions. CC acknowledges support from the Oppenheimer Fund. ACF from the Royal Society and Leverhulme Trust. HH from the School of Graduate Studies G. Galilei (University of Pisa). AH from the German Excellence Initiative via the Nanosystems Initiative Munich (NIM).
| 10.48550/arXiv.0705.2645 | Rayleigh Imaging of Graphene and Graphene Layers | C. Casiraghi, A. Hartschuh, E. Lidorikis, H. Qian, H. Harutyunyan, T. Gokus, K. S. Novoselov, A. C. Ferrari | 3,172 |
10.48550_arXiv.0812.2400 | ###### Abstract
We theoretically study the propagation of sound waves in GaAs/AlAs superlattices focussing on periodic modes in the vicinity of the band gaps. Based on analytical and numerical calculations, we show that these modes are the product of a quickly oscillating function times a slowly varying envelope function. We carefully study the phase of the envelope function compared to the surface of a semi-infinite superlattice. Especially, the dephasing of the superlattice compared to its surface is a key parameter. We exhibit two kind of modes: Surface Avoiding and Surface Loving Modes whose envelope functions have their minima and respectively maxima in the vicinity of the surface. We finally consider the observability of such modes. While Surface avoiding modes have experimentally been observed (Phys. Rev. Lett. 97, 1224301), we show that Surface Loving Modes are likely to be observable and we discuss the achievement of such experiments. The proposed approach could be easily transposed to other types of wave propagation in unidimensional semi-infinite periodic structures as photonic Bragg mirror.
## I Introduction
Propagation of waves in periodic structures have been studied for decades. Indeed, the periodicity leads to band gaps where propagation is forbidden. This property is very general and could be observed in many different fields. Electrons in crystals experience the periodic atomic potential leading to electronic band gaps. Electromagnetic waves cannot propagate in a spatial periodic dielectric constant system if their frequencies fall in the band gap: Photonic crystals are based on this statement. Elastic waves in solids could also endure a periodic Young modulus and/or a periodic density. Studies on such systems are currently numerous and lead to a new field of physics namely Phononics.
In periodic structures, two kinds of vibrations can be pointed out: 1) extended eigenmodes that propagate through the system; their frequencies fall out of the gaps and they satisfy the Floquet-Bloch theorem; 2) localized modes that can be found around isolated defects including surfaces(surface modes belong to this category). Due to their localized character, these modes are not so sensitive to the periodicity of the underlying Crystal and thus their frequency can take any value, in particular within the forbidden band gaps.
Extended acoustical vibrations in superlattices (SL) have been described quantitatively in the frame of the elastic model in the middle of the century. The superlattice, with its long period compared to the underlying crystals periods, results in a much shorter reciprocal Brillouin zone and hence a multiple folding of the initial acoustical branches. Experimentally, observations of these folded acoustical vibrations have first been performed using Raman scattering experiments for which selection rules are now well understood. Since the nineties, time-domain optical experiments have investigated folded acoustical vibrations. They exhibit two types of modes: phonons with finite wave vectors (\(q\neq 0\)) and zone-center modes (\(q=0\)) in the reduced Brillouin zone.
Recently, Trigo et al. studied both theoretically and experimentally these zone-center modes in phononic semi-infinite GaAs/AlAs SL yielding two conclusions. First, modes near the Brillouin zone-center observed by pump-probe experiments have not rigorously a zero wave vector, but instead an almost zero wave vector \(q\approx 0\). Second, these modes present a slowly varying envelope wave function with an amplitude minimum in the vicinity of the surface: they claim these modes "have a tendency to avoid the boundaries, irrespective of the boundary conditions" and for this reason, refer to them as Surface Avoiding Modes (SAM). Note that these surface avoiding modes have also been encountered near photonic and acoustic band gap materials and their nature is very general. In this work, we study in detail acoustical modes in the vicinity of Brillouin zone center, focussing on their avoiding character near the GaAs/AlAs SL surface.
Our work is based on analytical studies corroborated by numerical simulations. Analytical calculations are based on the analogy between the propagation of waves in a periodic structure and the spatial parametric oscillator. Since this analogy is not well-known, we devote Sect. II to prove the fruitfulness of such analogy to describe waves in periodic structures in the vicinity of gaps. Considering the propagation in an infinite SL of longitudinal modes in the vicinity of band gaps, we show that these modes are the product of a fast oscillating function and a slowly varying envelope function. We derive the dispersion diagram in the vicinity of band gaps using the parametric oscillator equations. Then considering the case of a semiinfinite SL in Sect. III, we give the analytical expression of longitudinal waves assuming free boundary conditions and considering the dephasing of the SL compared to its surface. We show the existence of Surface Avoiding and Surface Loving Modes depending if the envelope function is minimum or maximum near the SL surface. We then discuss in detail the avoiding or loving character of modes near the center of the Brillouin zone. In Sect. IV, we discuss on the electron-phonon coupling of these modes and show both Surface Avoiding and Loving Modes are likely to be observable using time-domain optical experiments or Raman and Brillouin scattering experiments. Finally, Sect. V will be devoted to a discussion on our results.
## II Parametric oscillator
### Parametric oscillator analogy
We consider the propagation of a sound wave in an unidimensional infinite SL made of 2 solid materials: respectively named A and B. \(d_{A}\) and \(d_{B}\) are the width of the layers, and \(d=d_{A}+d_{B}\) is the period of the SL. If \(\vec{u}\) is the atomic displacement, in each material sound waves are solutions of:
\[\Delta\vec{u}-\frac{1}{c_{i}^{2}}\frac{\partial^{2}\vec{u}}{\partial t^{2}}= \vec{0} \tag{1}\]
To determinate sound waves in a SL from Eq., one needs to apply at each interface, the continuities of the displacement field and of forces \(d\vec{F}=\vec{\sigma}d\vec{S}\) acting on each elementary interface \(d\vec{S}\), where \(\vec{\sigma}\) is the stress tensor.
In a SL, the displacements can be determined exactly by the transfer matrix method. Though very powerful, and clearly grounded, this method suffers from its numerical nature. It is not easy to get analytical expressions for finite size superlattices.
In this work, we will use the analogy between the propagation of sound waves in periodic media and the parametric oscillator. As it will be shown, this analogy involves severals approximations we will justify. However, it has two benefits. First, it yields analytical expressions of the displacements in closed form, with relatively simple calculations. Second, and especially, it allows to point out and to understand the main physical phenomena near the Bragg reflexions in the SL. We stress that the present treatment is complementary to the transfer matrix method and does not pretend to challenge it for quantitative analysis.
Finally, one must note that the present method could be applied to the propagation of light in photonic crystals or electrons in crystals since their wave propagation equations are similar.
From now on, we consider a harmonic, plane, longitudinal sound wave of angular frequency \(\omega\): \(\vec{u}(z,t)=u(z)e^{-i\omega t}\vec{e_{z}}\) along the \(z\)-axis, perpendicular to the SL layers. Continuity of forces acting on each interface imposes the continuity of \(C_{11}(z)\frac{\partial u(z)}{dz}\) where \(C_{11}\) is the stiffness constant, for a longitudinal sound wave propagating along \(z\). In the following, we will focus on GaAs/AlAs superlattices, because they are the most experimentally used superlattices. Stiffness constant contrast between GaAs and AlAs is very weak: \(\frac{\Delta C_{11}}{C_{11}}\approx 0.014\) so that the continuity of \(C_{11}(z)\frac{\partial u(z)}{dz}\) at each interface reduces to the continuity of \(\frac{\partial u(z)}{dz}\). In such a case, the determination of sound waves in the SL can be directly achieved by the resolution of Eq. considering only the variation of the sound speed since \(u(z)\) becomes a \(\mathcal{C}^{2}\) function. Consequently, since the sound speed in a GaAs/AlAs SL is a periodic function of \(z\), we may expand \(\frac{1}{c^{2}}\) as a Fourier series \(\frac{1}{c^{2}}=a_{0}+\sum_{m=1}^{\infty}a_{m}\cos(\frac{2\pi ms}{d})+b_{m}\sin (\frac{2\pi ms}{d})\), so that sound waves in GaAs/AlAs superlattices are solutions of:
\[\frac{d^{2}u}{dz^{2}}+\omega^{2}a_{0}\left[1+\sum_{m=1}^{\infty}\frac{a_{m}}{a _{0}}\cos(\frac{2\pi mz}{d})+\frac{b_{m}}{a_{0}}\sin(\frac{2\pi mz}{d})\right] u=0 \tag{2}\]
We now introduce \(G=\frac{2\pi}{d}\) the primitive vector of the reciprocal lattice, \(c_{m}=\frac{a_{m}}{a_{0}}\) and \(d_{m}=\frac{b_{m}}{a_{0}}\) the reduced amplitudes of the \(m\)th harmonic, and \(k_{0}(\omega)=\omega\sqrt{a_{0}}\) the zero order wave number i.e. the wave number a wave would have in the absence of any periodic modulation of the sound speed. \(k_{0}(\omega)=\omega\sqrt{a_{0}}\) is the usual dispersion relation for acoustic phonons in an homogeneous medium with mean sound speed \(c_{0}=\frac{1}{\sqrt{a_{0}}}\).
Eq. includes all harmonics of the inverse square sound velocity. In order to understand the effect induced by the different harmonics, let us first consider the effect of an isolated harmonic. Eq., in this case, reduces to a Mathieu equation of the form:
\[\frac{d^{2}u}{dz^{2}}+k_{0}^{2}\left[1+a\cos(k_{e}z)\right]u=0 \tag{4}\]
\(k_{e}\) is the wave number of the periodic excitation. General solutions of Eq. are periodic except in some regions of the plane \((k_{0}/k_{e},a)\): they are then the product of an oscillating function with a linear combination of an increasing and a decreasing exponential. These non periodic solutions correspond to the resonances of the parametric oscillator or band gaps of the SL. presents the stability diagram of Eq. limited to the first four resonances as a function of \(k_{0}/k_{e}\) and \(a\): regions of non-periodic solutions are dashed. This diagram has been calculated numerically using the resolvent of Eq. and the Liouville and Floquet-Bloch theorems.
Regions of non periodic solutions form in the plane \((k_{0}/k_{e},a)\) bands that, when \(a\) tends to \(0\), converge to points \(\frac{k_{0}}{k_{e}}=\frac{n}{2}\) with \(n\in\mbox{I\kern-2.0ptN}^{*}\). i.e. wave numbers \(k_{e}=2k_{0},k_{0},\frac{2}{3}k_{0},\frac{k_{0}}{2},\ldots\)etc. Moreover, for \(a\ll 1\), it can be shown that the band width of the \(n\)th resonance is a decreasing function of \(n\) proportional to \(a^{n}\). As a result, the larger resonance is obtained for \(n=1\) i.e. for \(k_{e}=2k_{0}\), the most famous resonance of the parametric oscillator, corresponding to an excitation twice faster than the zero order wave number.
Hence, going back to Eq., varying the value of \(k_{0}\), each harmonic \(m\) of the excitation at wave number \(mG\), if treated independently, is expected to create band gaps around \(\frac{k_{0}}{mG}=\frac{n}{2}\). Thus, band gaps occur for \(\frac{2k_{0}}{G}=nm=l\in\mbox{I\kern-2.0ptN}^{*}\). schematically presents the band gaps independently created by each of the first fifth harmonics. Excepted the first band gap (\(l=1\)) due exclusively to the first harmonic (\(m=1\)), all other band gaps result from at least two harmonics. However, as a first approximation, since the larger resonance due to the \(m\)th harmonic is for \(n=1\) i.e. around \(\frac{k_{0}}{mG}=\frac{1}{2}\), we consider that the \(l\)th band gap of Eq. is essentially due to the first resonance (\(n=1\)) of the \(l\)th harmonic (\(m=l\)). We call this approximation the Single Harmonic Approximation (SHA). This approximation is equivalent to treating the modulation of the sound speed as a first order perturbation: \(|c_{m}|\ll 1\) and \(|d_{m}|\ll 1\)\(\forall m\in\mbox{I\kern-2.0ptN}^{*}\). Our main motivation for the SHA is the possibility to derive analytical calculations and the possibility to produce good analytical approximations of solutions of Eq. around a given gap. Of course, the validity of SHA will be discussed below in the considered case. Note that values of \(k_{0}\) for which there are no periodic solution, correspond to a range of frequencies \(\omega\) which is the usual band gap of the SL. Indeed, \(k_{0}\) has been defined by \(k_{0}=\omega\sqrt{a_{0}}\) in Sect. II.1.
### Infinite medium: band gap
In order to test the validity of SHA, we solve Eq. near the resonance at \(\omega_{BG}^{}\) defined by \(k_{0}(\omega_{BG}^{})=G\) i.e. the first resonance due to several harmonics (\(m=1\) and \(m=2\)). The expression of \(\omega_{BG}^{}\) reads: \(\omega_{BG}^{}=\frac{G}{\sqrt{a_{0}}}\). The superscript "\(\)" in \(\omega_{BG}^{}\) refers to the second band gap. Using the SHA, this band gap is only produced by the harmonic at \(2G\) in the Fourier series of \(\frac{1}{c^{2}}\). We thus study a reduced equation taking into account only that harmonic:
\[\frac{d^{2}u}{dz^{2}}+k_{0}^{2}\left[1+c_{2}\cos(2Gz)+d_{2}\sin(2Gz)\right]u=0 \tag{5}\]
Where we have simplified the notation of \(k_{0}(\omega)\) using \(k_{0}\). The SHA will be checked a posteriori. Switching to complex number representation and introducing \(p_{2}=\frac{c_{2}-id}{2}\), Eq.
Phase diagram of the Mathieu equation. Parameters inducing non periodic solutions are in dashed regions (“parametric resonances” or “band gaps”).
Schematic illustration of band gaps produced by each harmonic.
to \(A(z)\) and \(B(z)\) defined by:
\[u(z)=A(z)e^{iGz}+B(z)e^{-iGz} \tag{7}\]
Introducing Eq. in Eq. and neglecting fast oscillating terms (at wavevector 3 \(G\)) as well as the second derivatives of \(A(z)\) and \(B(z)\), the identification of coefficients of plane waves at \(G\) and \(-G\) reads:
\[\left[\begin{array}{cc}\frac{dA}{d\xi}\\ \frac{d\xi}{dz}\end{array}\right]=\left[\begin{array}{cc}-\frac{k_{0}^{2}-G ^{2}}{2iG}&-\frac{p_{2}k_{0}^{2}}{2iG}\\ \frac{p_{2}k_{0}^{2}}{2iG}&\frac{k_{0}^{2}-G^{2}}{2iG}\end{array}\right] \left[\begin{array}{c}A\\ B\end{array}\right] \tag{8}\]
whose general solutions are:
\[A(z) = A_{+}e^{i\kappa z}+A_{-}e^{-i\kappa z} \tag{9a}\] \[B(z) = B_{+}e^{i\kappa z}+B_{-}e^{-i\kappa z} \tag{9b}\]
where \(\kappa\) is given by:
\[\kappa^{2} = \left[\frac{(k_{0}^{2}-G^{2})^{2}}{4G^{2}}-\frac{\left|p_{2} \right|^{2}k_{0}^{4}}{4G^{2}}\right] \tag{10}\] \[= \left(\frac{k_{0}^{2}}{2G}\right)^{2}\left(\delta^{2}-\left|p_{2} \right|^{2}\right)\] \[\mbox{and }\delta = \frac{k_{0}^{2}-G^{2}}{k_{0}^{2}} \tag{11}\]
\(\delta\) characterizes the detuning between the excitation wave number \(G\) and the zero order wave number \(k_{0}\).
The system undergoes a bifurcation when \(\kappa^{2}\) changes its sign as a function of \(\omega\):
* Either \(\kappa^{2}<0\). Solutions of Eq. are then hyperbolic: \(A(z)\) and \(B(z)\) are a linear combination of increasing and decreasing exponentials. And therefore, no propagation can occur in the SL. The condition \(\kappa^{2}<0\) defines the band gap of the SL around \(\omega_{BG}^{}\).
* Or \(\kappa^{2}>0\). Solutions of Eq. are sinusoidal and the general solution \(u(z)\) is thus periodic. Note that in this case, the spectrum of \(u(z)\) will be composed of a mixing of \(G+\kappa\) and \(G-\kappa\) plane waves. \(u(z)\) is then the product of a quickly oscillating function at \(G\), times a slowly varying functions at \(\kappa\). Moreover, Eq. establishes a relation between \(\omega\) (through \(k_{0}(\omega)\)) and wave vectors \(G\pm\kappa\) i.e. the dispersion relation of the SL around the considered gap.
Note that the band gap only depends on the squared magnitude of \(p_{2}\): \(\left|p_{2}\right|^{2}=\frac{c_{2}^{2}+6^{2}}{4}\) i.e. on the amplitude of the second harmonics at \(2G\).
Besides, we already pointed out the similarities of sound waves propagation in SL and electron waves propagating in crystals. Usually, when studying the independent electrons in a weak crystalline potential, one applies the first order perturbation theory to determine the width and position of electronic band gap: the expression of the band gap width is analogous to Eq. (9a) and could be derived using the parametric oscillator analogy and the SHA.
### Dispersion diagram: numerical study
In this part, we will numerically check our analytical results of Sect. II.3 and the SHA applying our predictions on a GaAs/AlAs SL with \(d_{A}=d_{GaAs}=5.9\) nm and \(d_{B}=d_{AlAs}=2.35\) nm. Such a SL has been used in an experimental setup by Trigo at al.. For these materials, longitudinal sound speeds are:
\[c_{A}=c_{\mathrm{GaAs}}=4726\ \mathrm{ms}^{-1} \tag{12a}\] \[c_{B}=c_{\mathrm{AlAs}}=5630\ \mathrm{ms}^{-1} \tag{12b}\]
These parameters will be used in all the numerical examples given in this article.
(SHA approx.) and the transfer matrix method(exact solutions) for this SL. is restricted to the two first band gaps respectively due to the harmonics at spatial frequencies \(G\) and \(2G\). We would like to emphasize that in Eq., the first band gap is obtained using coefficient \(p_{1}=\frac{c_{1}-id_{A}}{2}\) (instead of \(p_{2}\)) and the second bang gap using only \(p_{2}=\frac{c_{2}-id_{2}}{2}\) whereas the transfer matrix method considers all harmonics.
According to Fig. 3, our predictions of band gaps with the SHA are in good agreement with the exact calculation of the transfer matrix method. Moreover, dispersion diagrams around the gaps also agree very well. They slowly diverge when moving away from the gaps: the \(A\) and \(B\) functions are varying faster and thus their second derivatives are not negligible any more. In addition, the development in Eq. neglecting harmonics of order higher than 2 and keeping only the main term around wave number \(G\) is no longer appropriate. Finally, the SHA, keeping only one harmonic in Eq. naturally becomes inadequate when moving too far away from the gap.
Below, we will focus on periodic solutions around the second band gap: SL eigenmodes in the vicinity of this gap will thus be well described by solutions of Eq..
## III Sam and SLM
In the previous section, we have studied the general solution of Eq. and derived the dispersion diagram of the SL. Since experiments always involve a finite SL, we now consider the effects of the presence of a free surface at \(z=0\) with definite boundary conditions on SL eigenmodes.
### Semi-infinite medium: surface effects
More precisely, we intend to study _periodic_ solutions of Eq. in the vicinity of the band gap at \(\omega_{BG}^{}\) using free boundary conditions i.e. \(u=u_{0}\) and \(\frac{du}{dz}=0\). In the following, we will use the abbreviation NBPM (Near Bang gap Propagative Mode) in reference to such modes. plots \(\frac{1}{c^{2}}\) as a function of z. The dephasing \(\tau\) of the SL compared to \(z=0\) is defined on As shown below, \(\tau\) will be a key parameter in our study. Coefficients \(a_{0}\), \(c_{m}\) and \(d_{m}\) in Eq.
\[p_{m}=\frac{c_{m}-id_{m}}{2}=\left(\frac{c_{0}^{2}}{c_{A}^{2}}-\frac{c_{0}^{2} }{c_{B}^{2}}\right)\frac{e^{-i2\pi m\gamma}-1}{-i2\pi m}e^{-i2\pi m\alpha} \tag{13d}\]
Where \(\alpha=\tau/d\) is the reduced dephasing and \(\gamma=d_{A}/d\) is the cycle ratio of the SL. The mean sound speed \(c_{0}\) is defined by Eq. (13a).
Since we study NBPM in the vicinity of the second band gap around \(\omega_{BG}^{}\), it is instructive to consider the evolution of \(p_{2}\) as a function of the cycle ratio \(\gamma\) for different values of the reduced dephasing \(\alpha\) as shown in Fig.5.
\(p_{2}\) vanishes twice as the cycle ratio \(\gamma\) varies from 0 to 1. So there are two values of \(\gamma\), independent of \(\alpha\), for which the SHA predicts no gap opening around \(\omega_{BG}^{}\). Note that a gap might actually be opened through the second parametric resonance \(n=2\) produced by the first harmonic \(m=1\) (cf. Fig. 2).
For a given \(\gamma\), the effect of \(\alpha\) is merely to rotate the
(color online) Schema of the SL compared to the surface \(z=0\)
(color online) Plot in the complex plane of \(p_{2}\) (13d), for \(\gamma\in\). The first case considered is \(\alpha=0\) (black). The curve is not a single circle, since the average sound speed \(c_{0}\) depends on \(\gamma\), as seen in Eq.(13a). The reduced dephasing \(\alpha\) of the SL merely amounts to a rotation of the curve around 0. The dot circles correspond to the SL taken as an example in this paper with a cycle ratio \(\gamma=\gamma_{SL}\). For \(\alpha=\alpha_{1}=1/2-\gamma_{SL}/2\) or \(\alpha=\alpha_{1}+1/2=1-\gamma_{SL}/2\) (red) or \(\alpha=\alpha_{2}=3/4-\gamma_{SL}/2\) or \(\alpha=\alpha_{2}+1/2=5/4-\gamma_{SL}/2\) (blue), this brings \(p_{2}(\gamma_{SL})\) on the real axis.
(color online) Dispersion diagram of a GaAs/AlAs SL calculated using the transfer matrix method (dashed black) and predictions of the gaps using the parametric oscillator analogy and the SHA: (red) first gap calculated from the first harmonic of the inverse squared sound speed and (blue) second gap calculated from the second harmonic. We report using green horizontal dotted lines the positions of the \(l\)th band gap \(\omega_{BG}^{(l)}=\frac{lG}{2\sqrt{\alpha_{0}}}\). Parameters corresponding to the SL described in section II.4 have been used.
Note that experimentally, most of the SL have \(\alpha=0\) and that for \(\alpha=0\), \(p_{2}\) as a function of \(\gamma\) never cross the real axis except when \(p_{2}=0\): we will illustrate below to what extend the case \(p_{2}\) non null and real is relevant.
Details of the resolution of Eq. are given in Appendix A. Since we are focussing on periodic modes, i.e. outside the gap, \(\kappa^{2}\) in Eq. is positive.
\[u(z)=2A_{0}\left[(1-\zeta)\cos(Gz+\psi/2)\cos(\kappa z+\phi)\right.\] \[\left.-(1+\zeta)\sin(Gz+\psi/2)\sin(\kappa z+\phi)\right] \tag{15}\]
where \(\zeta\), \(\phi\) and \(A_{0}\) are defined by
\[\zeta = \frac{\delta}{\left|p_{2}\right|}-\sqrt{\left(\frac{\delta}{ \left|p_{2}\right|}\right)^{2}-1} \tag{16}\] \[\tan\phi = -\tan(\psi/2)\frac{G(1-\zeta)+\kappa(1+\zeta)}{G(1+\zeta)+ \kappa(1-\zeta)}\] \[A_{0} = \frac{u_{0}}{2\left[(1-\zeta)\cos\frac{\psi}{2}\cos\phi-(1+\zeta )\sin\frac{\psi}{2}\sin\phi\right]} \tag{18}\]
\(u(z)\) is real and \(u(z,t)\) is a stationary wave, as expected since the surface is a perfect mirror for acoustic waves.
Eq. is linear combination of two terms each involving a quickly oscillating function at \(G\) and a slowly one at \(\kappa\) corresponding respectively to Bloch and envelope wave functions. The parameter \(\zeta\) governs the relative contributions of each term and hence their dominant character which we will discuss in Sec. III.2.2. The parameter \(\phi\) is a dephasing of the envelope function with respect to the SL surface: it will be discussed in Sect. III.2.3. \(A_{0}\) governs the global amplitude of the displacement field. There are in general beatings in the solution \(u(z)\), as emphasized by the analytical expression for its root mean squared envelope \(u_{\rm rms}\), derived in Appendix B:
\[u_{\rm rms}^{2}(z)=2A_{0}^{2}\left[1+\zeta^{2}-2\zeta\cos 2(\kappa z+\phi)\right] \tag{19}\]
have been smoothed out.
### Surface Loving and Avoiding Modes
In this section, we discuss the effect of the reduced dephasing \(\alpha\) of the SL compared to the surface \(z=0\) on NBPM near the second gap around \(\omega_{BG}^{}\). We especially focus on the phase of the envelope wave function at wave vector \(\kappa\). Examining the general case requires a numerical study. However, two limiting cases where \(p_{2}\) is real, are remarkable.
#### iii.2.1 Remarkable cases. Pure SAM and pure SLM
* \(p_{2}\) is real and \(p_{2}>0\) (\(\alpha=\alpha_{2}\) or \(\alpha_{2}+1/2\) on Fig. 5). Therefore from, \(\psi=0\) and from, \(\phi=0[\pi]\).
* Just above the gap, we have \(\delta\approx\left|p_{2}\right|\) (cf Eqs. and), implying \(\zeta\approx 1(\)Eq.). Thus from Eq., the main contribution to \(u(z)\) is proportional to \(\sin(Gz)\sin(\kappa z)\): the envelope of the vibration mode is null at the surface. Following the denomination of MerlinMerlin, we call such a mode a Surface Avoiding Mode (SAM): these SL eigenmodes shy away from the boundaries.
* Just below the gap, \(\delta\approx-\left|p_{2}\right|\) and thus \(\zeta\approx-1\). The main contribution to \(u(z)\) is \(\cos(Gz)\cos(\kappa z)\): The envelope amplitude is maximal at the surface. In view of that, we refer to such SL eigenmode as Surface Loving Mode (SLM).
* \(p_{2}\) is real and \(p_{2}<0\) (\(\alpha=\alpha_{1}\) or \(\alpha_{1}+1/2\) on Fig. 5), \(\psi=\pi\) and thus \(\phi=\pi/2[\pi]\).
* Just above the gap, \(\delta\approx\left|p_{2}\right|\). The main contribution to \(u(z)\) is then \(\sin(Gz+\pi/2)\sin(\kappa z\pm\pi/2)=\pm\cos(Gz)\cos(\kappa z)\). Therefore, the SL eigenmode corresponds to a Surface Loving Mode.
* just below the gap \(\delta\approx-\left|p_{2}\right|\). The main contribution to \(u(z)\) is \(\cos(Gz+\pi/2)\cos(\kappa z\pm\pi/2)=\pm\sin(Gz)\sin(\kappa z)\): Therefore, the SL eigenmode corresponds to a Surface Avoiding Mode.
Note that we have chosen to base the latter discussion on Eq., but Eq. could also be used and leads to the same conclusions.
If the distinction between SAM and SLM is obvious in the cases where \(p_{2}\) is real, in the general case we need a more precise definition: we will speak about SAM or SLM depending if the envelope amplitude at the surface (\(z=0\)) is smaller or higher than its value a quarter period later at \(z=\frac{\lambda_{\kappa}}{4}=\frac{\pi}{2\kappa}\) where \(\lambda_{\kappa}\) is the wavelength associated with the envelope wave function. With this definition, according to Eq., knowing \(\zeta\) and \(\phi\) is enough to determine the SAM or SLM character.
#### iii.2.2 Beatings contrast
Let us focus on \(\zeta\) which, as previously quoted, governs the relative amplitude of the two contribution in Eq. and hence drive the beatings contrast, which is defined in Eq. and plotted in Fig.6 as a function of the frequency \(\nu=\frac{\omega}{2\pi}\). From Fig. 6, the contrast reaches its maximum value of 1 only at the precise band gap edges. In such a case, the envelope function exactly vanishes at its minimum: one could then speak about perfect SAM or SLM. Moving away from the gap, the contrast tends to 0 giving the same weight to both terms in Eq..
These beatings are also present in an infinite SL, just like in a temporal parametric oscillator. Away from the resonance condition, the excitation goes successively in-phase and out-of-phase with the movement of the system, leading to an increase or a decrease in the amplitude of the oscillations.
#### iii.2.3 SAM or SLM?
To get a strong SAM or SLM, a good beating contrast is needed. Then if the envelope has a minimum or a maximum at the surface, it is a SAM or a SLM, respectively. Indeed, from Eq. the phase of the envelope relative to the SL surface depends essentially on \(\phi\) and on the sign of \(\zeta\). The latter is seen in Fig.6 to be negative below the gap and positive above. In this section we focus on \(\phi\), the dephasing of the envelope function.
For a given type of SL (materials and thicknesses), \(\phi\) depends essentially on the frequency \(\nu=\frac{\omega}{2\pi}\) and on the reduced dephasing \(\alpha\) of the SL. As in the whole article, we use here the numerical parameters of the SL described in Sect. II.4 whose dispersion diagram is plotted in
Figure 7a) presents in a 3D plot the variations of \(\phi\) as a function of \(\alpha\) and of the frequency \(\nu\), calculated using Eq. around the second band gap. Figure 7b) reports the value of \(\phi\) as a function of \(\alpha\) for two frequencies above and below the gap: 0.5807 THz and 0.6182 THz. These two frequencies reported in Fig. 6, correspond to a 0.5 beating contrast \(C\) (Eq. (B4)), so that at 0.5807 THz (resp. 0.6182 THz), the displacement \(u(z)\) (Eq.) is dominated by the \(\cos(Gz+\psi/2)\cos(\kappa z+\phi)\) term (resp. \(\sin(Gz+\psi/2)\sin(\kappa z+\phi)\)).
From Fig. 7a) and b), \(\phi\) is a periodic function of \(\alpha\) with period \(\frac{1}{2}\) at fixed frequency \(\nu\): indeed, we are looking at the second band gap due to the harmonic at wavevector \(2G\); by changing \(\alpha\) from 0 to 1, the surface \(z=0\) sweeps two periods of that harmonic.
Beating contrast C of the displacement \(u\), defined in Eq.(B4) as a function of the frequency \(\nu=\frac{\omega}{2\pi}\). \(C\) equals \(\zeta\) above the gap and \(\frac{1}{-\zeta}\) below. Parameters corresponding to the SL described in section II.4 have been used. Dot circles mark a beating contrast of 0.5 for frequencies 0.5807 THz and 0.6182 THz.
(color online) a) Value of \(\phi\) (color blue \(\phi=-\pi/2\), red \(\phi=\pi/2\)) as a function of the frequency \(\nu\) (THz) and the reduced dephasing \(\alpha\). In the gap, \(\phi\) is not defined. b) \(\phi\) as a function of \(\alpha\) at 0.5807 THz (black) and 0.6182 THz (red). Dark curves separate SAM and SLM regions. These curves have been calculated looking for couples (\(\alpha,\nu\)) that equal the envelope wave function amplitude in \(z=0\) and \(z=\frac{\pi}{2\kappa}\), and correspond to \(\phi=\pm\frac{\pi}{4}\) (cf Eq.). Parameters corresponding to the SL described in section II.4 have been used.
(13c) (\(n=2\)). This has also been illustrated in Fig.5. The case \(p_{2}\) real and positive corresponds to \(\alpha=\alpha_{2}=3/4-\gamma/2\simeq 0.39\) or \(\alpha=\alpha_{2}+1/2=5/4-\gamma/2\simeq 0.89\), whereas the case \(p_{2}\) real and negative corresponds to \(\alpha=\alpha_{1}=1/2-\gamma/2\simeq 0.14\) or \(\alpha=\alpha_{1}+1/2=1-\gamma/2\simeq 0.64\).
In Fig. 7a), in addition to the value of \(\phi\), we plot dark curves separating SAM and SLM regions. These curves have been calculated looking for couples \((\alpha,\nu)\) that equal the envelope wave function amplitude in \(z=0\) and \(z=\frac{\pi}{2\kappa}\), and correspond to \(\phi=\pm\frac{\pi}{4}\) (cf Eq.).
Below the gap, Figure 7a) shows that regions of SAM dominate: especially, as the frequency closes with the gap, the SLM regions are reduced. This result is also coherent with Fig. 7b): for 0.5807 THz, the slope of \(\phi\) as a function of \(\alpha\) in the vicinity of \(\alpha=3/4-\gamma/2\simeq 0.39\) is very steep, and would be steeper for a frequency closer to the gap. Above the gap, the same conclusions apply: SAM regions dominate the phase space.
#### iii.2.4 SAM/SLM character
In order to summarize the results of the two previous sections, let us define a "SAM/SLM character" as:
\[\chi=\ln\Bigl{[}u_{\mathrm{rms}}/u_{\mathrm{rms}}\left(\frac{\pi}{2\kappa} \right)\Bigr{]} \tag{20}\]
The highest \(\left|\chi\right|\), the more pronounced is the SAM or SLM character of the mode.
\(\chi\) is plotted as a function of the frequency \(\nu\) and the reduced dephasing \(\alpha\) of the SL in Fig.8. It is clear that the more pronounced SAM or SLM (highest \(\left|\chi\right|\)) are localized close to the gap, and that SAM dominate the phase space.
#### iii.2.5 Numerical Study
To check our analytical predictions based on the SHA, we numerically integrate the differential equation Eq. using a 7th-order Runge-Kutta integrator around the 2nd band gap: we thus take into account all harmonics of the inverse square sound speed. We choose here to represent two different cases closed to the remarkable modes we point out above: \(p_{2}\) real positive for \(\alpha=\alpha_{2}=3/4-\gamma/2\approx 0.39\) and \(p_{2}\) real negative for \(\alpha=\alpha_{1}=1/2-\gamma/2\approx 0.14\). In both cases, we compute solutions of Eq. for 0.5807 THz and 0.6182 THz (0.5 beating contrast) with free boundary conditions at the surface, i.e. \(u=1\) and \(\frac{du}{d(2}=0\); these numerical results are plotted in For \(p_{2}\) real positive (\(\alpha=\alpha_{2}\)), we clearly observe a SAM above the gap and a SLM below, whereas for \(p_{2}\) real negative (\(\alpha=\alpha_{1}\)), it is the contrary. With respect to the SAM or SLM character, the numerical results thus fully agree with our analytical analysis.
However, please note that numerical values of the contrasts \(C\) and the envelope wavelengths measured in differ from the ones predicted by Eq. and Eq.. We attribute these discrepancies to the SHA. Indeed, the SHA allows the prediction of the width of the 2nd band gap with an error of the order of \(\left|p_{1}\right|^{2}\). Thus, the gap edges predicted by the parametric oscillator analogy slightly differ from the one calculated from the exact transfer matrix method (or by a direct integration of Eq.). Due to the almost horizontal dispersion diagram curve near the gap edges, a small variation of frequency \(\omega\) may induce a non negligible variation in \(\kappa\). Consequently, while the parametric oscillator analogy gets the main physics and especially allows us to find out the parameters to see SLM, it remains based on the SHA and thus, some care should be taken in order to predict quantitative values, which will better be derived from an exact approach like the transfer matrix method.
## IV Observation of SLM and SAM
We now address the problem of the experimental observability of NBPM. Trigo et al. claim they have shown the existence of SAM using pump-probe experiments coupled to a theoretical analysis. In the following, we would like to focus in detail on the optical activity of NBPM.
Colvard et al. addressed the phonon-electron coupling in GaAs/AlAs SL. The reduced dephasing of phonons compared to layers of the SL is a key parameter for the strength of this coupling.
(color online) SAM/SLM character \(\chi\) as defined in Eq.. A black curve delimits regions of positive and negative \(\chi\). Parameters corresponding to the SL described in section II.4 have been used.
On the contrary, \(A_{1}\) phonons, in quadrature phase with layers of the SL do create a strong electron-phonon coupling and are thus observable. These conclusions rely on the electron-phonon coupling mechanism: the deformation potential (in GaAs/AlAs SL), whose hamiltonian is proportional to the divergence of vibration modes.
In the case of NBPM around the band gap at \(\omega_{BG}^{}\), their wave vector \(G\pm\kappa\) is slightly different from the primitive vector \(G\) of the reciprocal lattice, thus their relative phase compared to layers of the SL slowly shifts with \(z\) as \(\pm\kappa z\). Thus, such modes could be in phase or out of phase with the SL depending on the position \(z\) in the SL (as long as \(\kappa\neq 0\)). However, though the envelope amplitude is varying, we can assume that _these modes will be observable if they have the \(A_{1}\) symmetry (i.e. in quadrature phase with the layers of the SL) at their maximum envelope amplitude_.
As demonstrated in Appendix C for the remarkable cases of Sect. III.2.1, we find that, out of resonance conditions, observable modes are found below the gap independently on their SAM or SLM nature.
To check our latter analytical prediction, numerical solutions of Eq. below (0.5807 THz) and above (0.6182 THz) the gap obtained in Sect. III.2.5 are compared to the SL in regions of maximum envelope amplitude. reports these results. In both cases \(p_{2}\) real positive (\(\alpha=3/4-\gamma/2\)) and \(p_{2}\) real negative (\(\alpha=1/2-\gamma/2\)), phonons above the gap are in phase with the layers of the SL at their maximum amplitude envelope in agreement with our prediction: they are hardly observable. On the contrary, phonons below the gap are in quadrature with the layers of the SL and will thus induce a strong electron-phonon coupling in agreement with our predictions.
## V Discussion
We have shown that the envelope amplitude of near Brillouin zone center acoustical phonon in the vicinity of the surface can be maximum (SLM) or minimum (SAM) depending on the reduced dephasing of the SL relative to the surface. For a given function \(\frac{1}{c^{2}}(z)\), Eq. is a second order differential equation whose solutions are entirely determined by two boundary conditions we can reduce to the definition of \((u,\frac{du}{dz})\). Knowing one solution \(u_{1}(z)\), \(u_{2}(z)=u_{1}(z+z_{ref})\) is also a solution for the boundary conditions \(u_{2}=u_{1}(z_{ref})\) and \(\frac{du_{2}}{dz}=\frac{du_{1}}{dz}(z_{ref})\), and for the function \(\frac{1}{c^{2}}(z+z_{ref})\). Consequently, if SAM exists, a judicious choice of \(z_{ref}\) allows to obtain SLM: one just needs to cut the SL at \(z_{ref}\) chosen in the region where the envelope of the SAM is maximum. Assuming free boundary conditions, this choice imposes the reduced dephasing \(\alpha\) of the SL, which precisely corresponds to our study. Reciprocally, choosing \(\alpha=0\), the choice imposes the boundary conditions.
We now address two discrepancies between our results
Numerical solutions of Eq. for 0.5807 THz (below the gap) and 0.6182 THz (above the gap) with free boundary conditions \(u=1\) and \(\frac{du}{dz}=0\). The reduced dephasing \(\alpha\) of the SL compared to the surface \(z=0\) is respectively \(\alpha=\alpha_{2}=3/4-\gamma/2\simeq 0.39\) for figure a) and b) and \(\alpha=\alpha_{1}=1/2-\gamma/2\simeq 0.14\) for figure c) and d). For each graph, we plot both a zoom and overall picture of the mode. The overall picture exhibits the Loving or Avoiding character of the modes. The zoom allows the discussion on the phases of the mode compared to the SL when the amplitude of the mode is maximum. Dashed parts represent AlAs layers and blank ones GaAs layers.
The first one concerns conclusions of Ref.: "In the physically important cases of a free and a clamped interface, we emphasize, surface repulsion is a property of all modes with wavevectors close to those of dispersion gaps", suggesting that only SAM exist. We have shown in this work, that in the case of free boundary conditions, both SAM and SLM exist. The discrepancy between our results and the cited work may be explained since calculations of Ref. only consider a SL with a reduced dephasing \(\alpha=0\) relative to the SL surface. In the precise case of free boundary conditions and \(\alpha=0\), our results agree with conclusions of Ref..
The second discrepancy concerns claims of Ref.: NBPM "have a tendency to avoid the boundaries, irrespective of the boundary conditions" suggesting that, _whatever_ the boundary conditions, only SAM exist (note that this work also only considers SL with \(\alpha=0\)). If we have restricted our study to the case of free boundary conditions, of course, the study could easily be extended to the case of any boundary conditions. Actually, as suggested in the begining of the section, we found that fixing the reduced dephasing \(\alpha=0\) and varying the boundary conditions produce the same qualitative conclusions as fixing the boundary conditions and varying the relative phase: both SAM and SLM exists and SAM regions dominate the phase space. We have chosen not to discuss in detail such effects of changing boundary conditions because the case of free boundary conditions is experimentally the most relevant. To illustrate the correctness of our analysis, and in disagreement with statements of Ref., the reader can easily check the existence of a SLM by direct integration of Eq. using the following parameters: \(u=1.5\) and \(\frac{du}{dz}=1\) and \(\nu=0.5807\) THz and the SL defined in Sect. II.4 with \(\alpha=0\) (the same as Ref.).
Let's now consider experimental observations of SAM or SLM. Ref. claims to observe SAM under the gap using a SL with \(\alpha=0\) and free boundary conditions. This observation agrees with our analysis: from Fig. 8, NBPM under the gap for \(\alpha=0\) are SAM and from Sect. IV they are likely to be observable.
In Sect. IV, we show that both SAM and SLM could in principle be observable. We would like to address here technical problems associated with the observation of SLM. Indeed, in Fig. 8, we show that SAM regions dominate the phase space. Thus, a high precision on the value of \(\alpha\) is needed if one wants to observe SLM. This might prove challenging for SL growth. As an illustration, let's consider the precision on the thickness of layers produced by molecular beam epitaxy, which is about 2 monolayers (\(\approx\)0.5 nm). In the case of the SL already studied (AlAs(2.35 nm)/GaAs(5.9 nm)) a precision of \(\pm\)0.25 nm on the value of the dephasing \(\tau\) corresponds to a precision of \(\pm\)0.03 on the value of \(\alpha\). If \(\alpha\) exactly matches \(3/4-\gamma/2=0.39\), the NBPM is a SLM for any value of \(\nu\) below the gap. However, if \(\alpha\) lays in the range \(0.39\pm 0.03\), The frequency \(\nu\) has to be lower than 0.583 THz to get SLM. Thus, because of the lack of precision on the layers thicknesses, observations of SLM would require a protocol that selects NBPM with a low enough frequency (below the boundary SAM/SLM shown in Fig. 8): we think such a selection may be achieved by the presence of a substrate. Indeed, the length of the SL and the presence of a substrate is an important parameter that tunes the lifetime of observable phonons created in the SL by the excitation laser. A precise study of these effects will be detailed in a forthcoming publication.
## VI Conclusion
We have shown using a fruitful analogy with the parametric oscillator that NBPM could either avoid (SAM) or love the surface (SLM) depending on the relative phase of the SL compared to the sample surface. Moreover, we have shown that both modes should in principle be observable using judicious experimental parameters. Whereas SAM have already been observed, experimental evidences of SLM will be difficult in view of the technical challenge to achieve a convenient SL. Despite these difficulties, we think such experiments are possible choosing an appropriate SL length. The experimental distinction between SAM and SLM may be demonstrated using pump-probe experiments with different penetrating length lasers.
Finally, we would like to underline that our study of sound waves in SL could be generalised to any kind of waves. So that electronic or electromagnetic wave functions in electronic SL or Bragg mirrors also show a Surface Avoiding or Surface Loving character depending on the surface termination of the SL.
## Appendix A Resolution of Eq. in a semi-infinite SL
Finding the solutions (9a) and (9b) amounts to diagonalize the square matrix in Eq.. The eigenvalues are \(\pm i\kappa\), defined by. Since we are focusing to the outside of the gap, \(\kappa\) is real and we may choose it positive (Changing \(\kappa\) by \(-\kappa\) is equivalent to permute constant \(A(B)_{+}\) and \(A(B)_{-}\) in Eq.):
\[\kappa=\frac{k_{0}^{2}}{2G}\sqrt{\delta^{2}-|p_{2}|^{2}}, \tag{10}\]
\(|p_{2}|\) denotes the magnitude of \(p_{2}\): \(|p_{2}|=\sqrt{p_{2}p_{2}}\). After diagonalization, the following relations are found between the constants in Eq.:
\[B_{+}=-\zeta e^{-i\psi}A_{+} \tag{11a}\] \[A_{-}=-\zeta e^{+i\psi}B_{-} \tag{11b}\]where \(\psi\) is the argument of \(p_{2}\), so that \(p_{2}=|p_{2}|\,e^{i\psi}\) and
\[\zeta=\frac{\delta}{|p_{2}|}-\sqrt{\left(\frac{\delta}{|p_{2}|}\right)^{2}-1} \tag{11}\]
Let us now impose a real amplitude \(u_{0}\) at the surface, \(u=u_{0}\).
\[A_{+}\left(1-\zeta e^{-i\psi}\right)+B_{-}\left(1-\zeta e^{+i\psi}\right)=u_{0} \tag{12}\]
We may try \(B_{-}=\overline{A_{+}}\), since their respective factors are complex conjugates, and \(u_{0}\) is real. Please note that changing \(u_{0}\) to be complex would merely amount to dephase the vibration in time.
\[u(z)=A_{0}\left\{\cos\left[(G+\kappa)z+\frac{\psi}{2}+\phi\right]-\right.\] \[\left.\zeta\cos\left[(G-\kappa)z+\frac{\psi}{2}-\phi\right]\right\} \tag{13}\]
with
\[A_{0}=\frac{u_{0}}{2\left(\cos\left(\frac{\psi}{2}+\phi\right)-\zeta\cos \left(\frac{\psi}{2}-\phi\right)\right)} \tag{14}\]
The boundary condition \(\frac{du}{dz}=0\) yields to the following expression which allows the determination of the integration constant \(\phi\):
\[\tan\ \phi=-\tan(\psi/2)\frac{(G+\kappa)-\zeta(G-\kappa)}{(G+\kappa)+\zeta(G- \kappa)} \tag{15}\]
Note that Eq. defines \(\phi\) modulo \(\pi\). However, \(u(z)\) is fully defined in Eq. by defining the couple \(A_{0}\) and \(\phi\). We thus use the convention to choose \(A_{0}>0\) in Eq.: \(\phi\) is now defined modulo \(2\pi\). Note that close to the gap, this procedure always yields a value of \(\phi\) in \(]-\frac{\pi}{2},\frac{\pi}{2}]\).
We found one solution of this second order differential equation. This is the only one satisfying the given free boundary conditions. Finally, Eq. can be written in the alternative form, more convenient for the interpretation.
## Appendix B Envelope equation
As seen in Fig.9, the displacement consist in a fast oscillating term at wave vector \(G\), modulated by a lower wave vector envelope. One way to define this envelope is to take the square of \(u\) in Eq.
\[u^{2}(z)= 4A_{0}^{2}\left\{(1-\zeta)^{2}\cos^{2}(Gz+\frac{\psi}{2})\cos^{ 2}(\kappa z+\phi)+\right.\] \[\left.(1+\zeta)^{2}\sin^{2}(Gz+\frac{\psi}{2})\sin^{2}(\kappa z+ \phi)-\right.\] \[\left.(1-\zeta)(1+\zeta)2\cos(Gz+\frac{\psi}{2})\sin(Gz+\frac{ \psi}{2})\right.\] \[\left.\cos(\kappa z+\phi)\sin(\kappa z+\phi)\right\} \tag{16}\]
So, we define:
\[u_{\rm rms}^{2}(z)=\frac{G}{2\pi}\int_{z-\frac{2\pi}{G}}^{z+\frac{2\pi}{G}}u^{ 2}(z^{\prime})dz^{\prime} \tag{17}\]
This procedure amounts to take a local mean, around every position \(z\), on a \(\frac{2\pi}{G}\) range. In Eq., using \(\kappa\ll G\), the \(\cos^{2}(Gz+\frac{\psi}{2})\) and \(\sin^{2}(Gz+\frac{\psi}{2})\) yield \(\frac{1}{2}\) and the last term vanishes.
\[C=\frac{\max(u_{\rm rms})-\min(u_{\rm rms})}{\max(u_{\rm rms})+\min(u_{\rm rms })} \tag{19}\]
## Appendix C Observation of SAM or SLM: analytical study
To check if NBPM around band gap at \(\omega_{BG}^{}\) have the \(A_{1}\) symmetry at their maximum envelope amplitude, we examine their relative phase compared to the layers of the SL. So that, whereas in Sect. III.2 we were interested in the phase \(\phi\) of the envelope compared to the SL surface, we now get interested in the phase \(\frac{\psi}{2}\) of the quickly oscillating function (at spatial frequency G) compared to the layers of the SL.
Eq. gives the analytical expression of NBPM. The layers of the SL can be described using the inverse square sound speed as shown in However, only the first harmonic (at \(G\)) of the SL is necessary to get a picture of the layers of the SL: the coefficient \(p_{1}\) defined by Eq. (13d) determines the phase of layers of the SL.
We thus, only have to compare the relative phase of the phonons (at their maximum envelope amplitude) to the one of the first harmonic of the inverse square sound speed.
We now examine the two remarkable cases mentioned in Sect III.2.1.
Let us start with the cases \(p_{2}\) real and positive obtained for \(\alpha=3/4-\gamma/2\) or \(\alpha=5/4-\gamma/2\): the SAM mode, above the gap, varies like \(\sin(Gz)\sin(\kappa z)\), whereas the SLM, below the gap, varies like \(\cos(Gz)\cos(\kappa z)\). From Eq. (13b) and Eq. (13c), we can see that \(c_{1}=0\) and \(d_{1}\neq 0\) which implies that the first harmonic of the SL varies like \(\sin(Gz)\). Thus, it turns out that the SAMmode is in phase with the SL at its maximum amplitude (\(kz=\pi/2[\pi]\))): the electron-phonon coupling is thus weak which makes it hardly observable. On the contrary the SLM mode is in quadrature with the SL at its maximum amplitude (\(kz=0[\pi]\))): the electron-phonon coupling is high and so, this SLM mode is likely to be observable.
A similar analysis for the cases \(p_{2}\) real and negative (\(\alpha=1/2-\gamma/2\) or \(\alpha=3/2-\gamma/2\)) leads to the conclusion that in that case, the SLM mode, above the gap, will be hardly observable whereas the SAM mode just below the gap, is likely to be observable.
Hence, in both remarkable cases, observable modes are found below the gap.
| 10.48550/arXiv.0812.2400 | Surface Loving and Surface Avoiding modes | Nicolas Combe, Jean Roch Huntzinger, Joseph Morillo | 4,207 |
10.48550_arXiv.1904.08235 | ###### Abstract
Melting experiments require rapid data acquisition due to instabilities of the molten sample and optical drifting due to the high required laser power. In this work, the melting curve of zirconium has been determined for the first time up to 80 GPa and 4000 K using in-situ fast x-ray diffraction (XRD) in a laser-heated diamond anvil cell (LH-DAC). The main method used for melt detection was the direct observance of liquid diffuse scattering (LDS) in the XRD patterns and it has been proven to be a reliable melting diagnostic. The effectiveness of other melting criteria such as the appearance of temperature plateaus with increasing laser power is also discussed.
Moreover, its good strength and ductility at high temperatures and the low thermal neutron cross-section absorption make it an ideal material for use as cladding at nuclear reactors. Alloys of Zr with Cu, Al, Ti and Ni have been demonstrated to exhibit extraordinary glass forming ability, while metallic glass formation in single-element zirconium has also been discovered, with a wide stability in high pressure and temperature conditions.
Zirconium is a d-orbital transition metal with a rich and interesting phase diagram. At ambient conditions it crystallizes to an hcp structure (\(\alpha\)-phase), while at temperatures higher than 1136 K it transforms to a bcc (\(\beta\)-) phase. By increasing pressure at ambient temperature it transforms to another hexagonal, but not close-packed, called the \(\omega\)-phase and then back to \(\beta\)-phase around 35 GPa. Similar transitions also occur in other group IV transition metals, such as Ti and Hf, and it seems that the electronic transfer between the broad sp band and the much narrower d band is the driving force behind those structural transitions.
The high melting point (2128 K) of Zr often classifies it as a refractory metal. Although there are some works in the high temperature behavior of zirconium at high pressures, its melting curve has not yet been investigated and this absence of experimental data has strongly motivated this study. On the other hand, the high pressure melting of transition metals has always been a subject of intense debate, because of the large uncertainties in the temperature measurements and the criteria used to identify the melting, so that different approaches can yield very different results. In most cases shock wave (SW) experiments and molecular dynamics (MD) calculations provide dramatically steeper curves than those obtained with the laser speckle method in a LH-DAC, where the melting is visually detected by observing the movements on the sample surface during heating. Tantalum is a good example of such a controversy, with melting temperatures that differ thousands of K at 100 GPa by applying different experimental techniques. Another more recent example is that of iron, where the speckle method was found to coincide with the onset of dynamic recrystallization rather than melting. The observation of a temperature plateau versus laser power during a laser heating experiment has also been suggested as a melting diagnostic that works fine in the case of nickel, but has not been always reliable. Geballe and Jeanloz have argued that the latent heat of the melting inside the LH-DAC is insignificant compared to the heat provided by the lasers, and proposed that the observed temperature plateaus in the laser heating of metals are mainly associated with discontinuous increases in reflectivity rather than melting. Lately, energy dispersive X-ray absorption spectroscopy (XAS) has been also used as a method of detecting melting in metals under high pressure, by tracking the disappearance of the shoulder of the XANES signal, as well as the flattening of the first few oscillations. One of the most consistent and recent methods for determining melting is with in-situ synchrotron X-ray diffraction (XRD), by the direct observation of the first liquid diffuse scattering (LDS) signal in the XRD patterns, as the temperature is gradually increased.
In this work we investigate the melting curve of the \(\beta\)-phase (cubic bcc) of zirconium at pressures up to 80 GPa. To our knowledge there are not any scientific data concerning the melting curve of zirconium. We apply the synchrotron XRD technique in a LH-DAC, which permits to track any chemical reactions in the sample, such as the formation of carbides due to the reaction of the sample with the diamond anvils, and allows for in-situ temperature measurements using pyrometry. We compare the effectiveness of two different melting criteria: the appearance of the first liquid signal in the XRD patterns by increasing temperature, and the observation of temperature plateaus with increasing laser power.
Several membrane-driven diamond anvil cells with culet sizes ranging from 150 to 350 \(\mu\)m were used to pressurize the samples. Rhenium gaskets, pre-indented to 30 \(\mu\)m and drilled with a Nd:YAG pulsed laser formed the sample chamber. The sample assembly consisted of high purity (99.2%) flattened grain Zr pieces of thicknesses around 10 \(\mu\)m. Having a high purity sample for melting studies is important, since a large amount of impurities could lower the free energy of the system and thus lower the melting point. The sample was embedded between two thin disks of KCl in order to provide thermal and chemical insulation from the diamonds during the laser heating experiment. KCl also serves as a soft pressure transmitting medium and a pressure calibrant. A ruby chip was also installed next to the sample for the estimation of the initial compression of the cell before heating. All the DAC loadings have been carried out in a glove box under an argon atmosphere and the KCl was dried at 100 \({}^{\circ}\)C for several hours to avoid any amount of water that could trigger chemical reactions between the sample and the diamond anvils.
The in-situ synchrotron x-ray diffraction experiments have been carried out at the ID27 beamline of the ESRF, using a monochromatic beam with a wavelength of 0.3738 A. The beam was focused to a spot of 3x2 \(\mu\)m\({}^{2}\). The XRD data were collected by a MAR165 CCD detector calibrated for sample to detector distance with a CeO\({}_{2}\) standard. Typical exposure times were in the order of 5 sec and the 2D images were converted to 1D patterns using the Dioptas software. The diffractograms were fitted by the Le Bail method using the Fullprof software.
The samples were heated simultaneously from both sides by two continuous YAG fiber lasers (wavelength 1.064 \(\mu\)m), providing a maximum combined power of 200 W. The YAG laser is very well absorbed by the surface of opaque metallic samples such as Zr. However, there can be gradients in the temperature inside the bulk of the sample, especially if the area of the melt is small and thus difficult to detect by XRD. Therefore, the experimental procedure has been devised to minimize the temperature gradients in the samples and in the same time provide definitive melting criteria. The alignment of the x-rays, lasers and pyrometry spot were verified after each heating cycle, and the temperature between the two sides was kept very similar as we gradually increased the laser power. The laser spot size was slightly defocused to 20 \(\mu\)m diameter (much larger than the x-ray spot) to obtain a more uniform heating of the sample and reduce the temperature gradients due to the Gaussian shape of the TEM\({}_{00}\) mode of the lasers.
The temperature has been determined by pyrometry measurements using the online system of ID27, with reflective, Schwarzschild objectives which are by construction free of chromatic aberrations. Using this setup, the uncertainty related to the radial temperature gradient is less than 50 K, while the axial component is below 100 K, giving a maximum uncertainty of 150 K. The optical path of the collected black body radiation has been calibrated using a tungsten ribbon lamp, with a reference temperature of 2500 K. The temperature is given by the Planck fit in the wavelength window 600-900 nm. Despite the absence of chromatic abberations in the optics, the temperature uncertainty can be much higher, especially at higher temperatures, because of the wavelength dependency of the emissivity. For this reason we have compared the Planck fit with two-color pyrometry measurements, as discussed in. The pressure was determined before and after heating by the fluorescence ruby scale. The final pressure was estimated by the XRD measurements, correcting for thermal pressure by using the thermal equation of state of KCl.
At the high temperatures provided by the lasers, Zr was always in the \(\beta\)-phase. We were able to record the appearance of liquid Zr for six different pressures between \(\sim\)29 and \(\sim\)80 GPa. Every pressure point corresponds to a different DAC loading. For most pressure points, we have carried multiple temperature runs by laser heating a different, fresh region of the sample. The main criterion for the detection of melting in our experiments was the first observation of a diffuse liquid signal in the diffraction patterns. The liquid signal is mainly located around the peak of \(\beta\)-Zr (Figs.1a&b). This peak is very close to the main reflection of KCl, however the melting curve of KCl is much stiffer than the one of Zr, meaning that the observed melt is indeed Zr. Zr can form glasses at high temperatures, especially at high pressures, so we were careful to take the melting points in this study outside the glass formation line. As a verification, the diffuse signal disappears in the quenched (Fig.1c) data (i.e. data taken right after switching off the lasers,) meaning that it cannot correspond to an amorphous/glass phase. The quenched pattern of Fig.1c also shows that there was no carbide (ZrC) or oxide (ZrO\({}_{2}\)) formation during laser heating due to interaction with the diamond anvils or the sample environment. Chemical interactions such as carbide formation can be a big problem in some laser heating experiments, as in the case of Ta.
In Figs.1a&b we plot the XRD data for a selected pressure of \(\sim\)54 GPa and increasing temperatures. It can be clearly observed that the liquid signal in XRD increases with increasing temperature, indicating that the amount of molten Zr is also increasing. In the same time the main peak of Zr at \(\sim\)9.3 degrees decreases with increasing temperature, showing the reduction of solid amount in the sample. However, solid Zr persists for temperatures much higher than the melting point in our XRD patterns, because of the temperature gradients between the surface and the bulk of the sample, which is scanned thoroughly by the x-rays. For a given laser power, the temperature of each side was measured several times before and after the XRD pattern, from both sides, in order to verify that the temperature is not shifting significantly between measurements. The measured x-ray background is constant at low temperatures and starts increasing gradually after a given temperature, providing the signature of melting. The melting temperature was defined as the average temperature between the last solid-only pattern and the first observed pattern with liquid signal as we increased the temperature. For the example of Fig.1a, the liquid signal starts to appear between 3290 and 3450 K and thus the melting temperature was defined to be (3290+3450)/2=3370 K at 54 GPa. In Fig.2 the two-dimensional XRD data are shown for two cases: a) hot but solid \(\beta\)-Zr (2960 K) and b)solid and liquid mix of Zr well above the melting line (3610 K). The diffuse scattering background is obvious and even at the highest temperature reached in this run (3610 K), some signal from solid Zr persists, although it is reduced significantly, meaning that even at this temperature Zr is only partially melted. KCl remains unmelted, since it has a much higher melting point, however it has started growing single crystals, as it can seen from the spots in the diffraction patterns of Fig.2b.
Fig.3 resumes the experimental pressure-temperature conditions for all the XRD patterns recorded. Zr crystallizes upon heating and these small crystals appear and disappear in the 2D diffraction patterns with increasing temperature due to small movements in the surface of the sample. This effect has been referred to as fast recrystallization and in many cases it can lead to an underestimation of the melting point. In some previous works using optical-based diagnostics, such as the speckle method, the melting was attributed to this "fast recrystallization" and yielded temperatures much lower than the actual melting temperature, especially at higher pressures. In our data the threshold of fast recrystallization and melting seem to differ around 1000 K for most pressure points. In the absence of other melting data for Zr, comparing with a recent work on Ti, another d-orbital transition metal, has shown discrepancies of 500 K between fast recrystallization and melting, while in Fe the difference was found to be close to 1000 K.
Hrubiak et al., on their recent work on Mo, they discussed thoroughly the phenomenon of fast recrystallization by observing the quenched data obtained from different temperatures. They observed the appearance of preferred orientations on the quenched data obtained from a recrystallization temperature (i.e. before melting). This preferred orientation gave place to a fine grain structure (with random orientation) for quenches obtained from temperatures above melting.
(Color online) XRD patterns for a heating run at 54 GPa: a) Full spectra, indicating the main peaks of \(\beta\)-Zr and KCl. b) Zoom of the molten region for different temperatures. c) Quenched data, indicating only Zr and KCl peaks, thus no ZrC was formed upon heating.
(Color online) 2D XRD spectra at a thermal pressure of 54 GPa: (a) solid \(\beta\)-Zr at 2960 K and (b) solid and (mainly) molten Zr at 3610 K.
However, we have observed a fine-grain structure for Zr in the quenched patterns obtained after melting (as shown in the Supplemental Material), in good agreement with the work of Hrubiak et al..
Another method that has been thoroughly used for the estimation of the melting line in various works is the presence of temperature plateaus upon heating. Figs.3a&b show the sample temperature as a function of time, which accounts for increasing laser output. In most of the cases the temperature is increasing rapidly and linearly at low laser power inputs (Fig.3a). In some of the runs the temperature exhibited a plateau, or showed minor fluctuations within error bars near the melting point (Fig3a). However, there are runs where plateaus can be seen well below the detected melting points, that are not associated with fast recrystallization either (Fig.3b). For some experiments the temperature slope during the laser power increase changed several times during a heating cycle (Fig.3b). It is clear that the temperature saturation method as a melting criterion has not a great reproducibility in the case of Zr. There are many reasons why the rate the temperature increases can change in a LH-DAC. The main reason that has been proposed are the discontinuous reflectivity increases in the sample, that cannot however be a reliable indicator of melting since it is not an intrinsic property of materials. The increase in the conductivity of KCl with the increasing heat provided by the lasers could also explain why the temperature does not always increase with laser power. Alternatively, the fact that the two criteria of XRD diffuse scattering and temperature plateaus do not always agree in the case of Zr could also be related to the small differences in the thickness of the sample and KCl layers between the runs, which affect the thermal isolation of the sample, or the heating efficiency inside the DAC. Thus the observation of temperature plateaus for defining melting, although it can work in some cases, does not seem to be a consistent method because it depends on many parameters that cannot be calculated quantitatively and in-situ in a laser heating experiment. The temperature vs laser power data for runs at different pressures are shown in the supplemental material (SM).
All the melting points obtained in this study are displayed in Fig.5. The melting curve presented is corrected for thermal pressure and is fitted using the Simon-Glatzel equation, yielding:
\[T_{m}=T_{0}\times(\frac{P_{m}}{42.58\pm 11.34}+1)^{0.56\pm 0.09} \tag{1}\]
The Simon-Glatzel equation gives satisfactory results for the given pressure range, as is the case for many metals.
(Color online) Experimental points for LH-DAC \(\beta\)-Zr and KCl.
(Color online) Temperature measurement vs time (laser power) for a heating run at a)65 GPa and b) 54 GPa. The melting temperature defined from XRD (including errors) is denoted by the pink lines.
The melting curve can be also calculated using the Lindemann law:
\[T_{m}=T_{0}\times(V/V_{0})^{2/3}exp(2\gamma_{0}/q(1-(V/V_{0})^{q})) \tag{2}\]
In this equation \(\gamma_{0}\)=1.01 is the Gruneisen parameter for Zr taken from Goldak et al and its volume dependence is taken as q=1. As seen in Fig.5, the results obtained from the Lindemann equation are generally in good agreement with the Simon-Glatzel fit for most of our data. However, it appears that the melting slope obtained this way is slightly higher. Errandonea in a work concerning Mo, Ta and W has found that the Lindemann law could overestimate the melting point of bcc transition metals at high pressures. This can be due to the fact that the Lindemann law takes into account only the thermodynamic parameters of the solid phase and completely neglects the liquid, which could lead to inaccuracies.
In absence of other experimental data for Zr, we compare the melting curve with the one of Ti from ref., which is a transition metal with the same electronic configuration. The melting curves in both Zr and Ti have been determined using similar experimental methods in these two works. By the Simon-Glatzel fit it is possible to calculate a melting slope dT\({}_{m}\)/dP = 28 K/GPa for Zr, defining it as a low melting slope material. The low slope for Zr (and also Ti) is because of the partially filled d-electron bands and it is a frequent phenomenon in transition metals. It has been proposed that the loss of long-range order due to melting can induce a change in the liquid density of states (DOS) and therefore a decrease in the melting slope, and materials with filled bands and DOS that do not change significantly upon melting, such as Cu, Al, or noble gases, have systematically much stepper melting curves than the partially filled bcc transition metals. Also in metals with partial filled d-bands, icosahedral short-range order can be favored energetically in the supercooled liquids and melts, and it has already been observed for Zr. These short-range structures can act as impurities that lower the free energy and thus the melting slope. Although we have only observed partial melt in our data and thus it is not possible to perform a full pair distribution function analysis, we cannot rule out the possibility of an icosahedral short-range order in Zr. In both Zr and Ti, the pressure generates an s-d electron transfer and an increase in the concentration of local structures which lowers even further the melting slope.
In conclusion, the high-pressure melting curve of \(\beta\)-Zr has been studied using synchrotron x-ray diffraction in a laser-heated diamond anvil cell. The appearance of liquid signal in the in-situ x-ray patterns has been proven to be a reliable diagnostic, that has in the same time the advantage of detecting any possible unwanted reactions in the sample during heating. On the other hand, the observation of temperature plateaus cannot reproduce a specific pattern for melting, and therefore this method was proven unreliable in the case of Zr. This is, to our knowledge the first set of experimental data on the melting curve of Zr and we expect that it will greatly motivate further experimental and theoretical studies on the subject, using different techniques.
| 10.48550/arXiv.1904.08235 | Melting curve of elemental zirconium | Paraskevas Parisiades, Federico Cova, Gaston Garbarino | 6,067 |
10.48550_arXiv.1308.2517 | ### Single domain and superparamagnetic radii
In the absence of an external field, the critical diameter for single domain configuration is a function of exchange length \(l_{ex}\) as follows
\[d_{SD} = 72kl_{ex}\]
Substituting for \(k\) and \(l_{ex}\) yields
\[d_{SD} = 72\sqrt{\frac{K}{\mu_{o}M_{s}{}^{2}}}\sqrt{\frac{A}{\mu_{o}M_{s}{}^{2}}}\]
For a given particle, though its diameter is below \(d_{SD}\), it need not necessarily be superparamagnetic below a certain transition temperature since the surrounding thermal energy is not sufficient enough to flip the dipole moment randomly inside the domain in the considered observation time.
\[d_{SPM}=2\sqrt{\frac{6k_{b}T}{K}} \tag{3}\]
The \(d_{SD}\) and \(d_{SPM}\) calculated for magnetite and maghemite spherical particles at 300K using equation and are given inTable. 1. The variation of \(d_{SPM}\) with temperature is shown in 0
\begin{table}
\begin{tabular}{|p{56.9pt}|p{56.9pt}|p{56.9pt}|p{56.9pt}|p{56.9pt}|p{56.9pt}|} \hline & First anisotropy constant, \(K\) (kJ/m\({}^{3}\)) & Exchange stiffness constant, \(A\) (pJ/m) & Saturation magnetisation, \(M_{s}\) (kA/m) & Single domain critical diameter, \(d_{SPM}\) (nm) \\ \hline Magnetite & 13.5 & 13.3 & 446 & \(\sim 103\) & \(\sim 24\) \\ \hline Magnetite & 4.6 & 10 & 380 & \(\sim 85\) & \(\sim 35\) \\ \hline \end{tabular}
\end{table}
Table 1: **Anisotropy and crystalline parameters defining SD and SPM critical diameters at 300K**
Single domain critical diameter \(d_{\rm SD}\), superparamagnetic diameter \(d_{\rm SPM}\) as a function of temperature for magnetite and maghemite particles
### 2.2. Relaxometric parameters and complex susceptibility.
The magnetic moment flips between parallel or antiparallel easy axes and the effective relaxation time constant for a magnetic particle suspension is
\[\tau_{\mathit{eff}}=\frac{\tau_{N}\tau_{B}}{\tau_{N+\tau_{B}}} \tag{4}\]
\(1/\tau_{o}\) is attempt frequency characteristic to material, \(V\) the particle volume, \(K_{r}\) geometric rotational shape factor and \(\eta\) carrier medium viscosity.
In an external alternating field, the absolute susceptibility of particle suspension is exclusively determined by the effective relaxation time.
\[\chi=\chi^{-}i\chi^{*}=\frac{\chi_{o}}{1+\omega^{2}\tau_{\mathit{eff}}^{2}}-i \frac{\chi_{o}\omega\tau_{\mathit{eff}}}{1+\omega^{2}\tau_{\mathit{eff}}^{2}} \tag{5}\]
Where \(\chi_{o}\) is the DC susceptibility and \(\omega\) the angular frequency.For equation to be theoretically useful, other approximations for eg.Langevin approximation for \(\chi_{o}\) is essential.
The effective relaxation time \(\tau_{\mathit{eff}}\) for magnetite and maghemite spherical particles at different SD diameters\[M_{DC}=\phi M_{s}\left[\coth(\alpha)-\frac{1}{\alpha}\right] \tag{6}\]
### Langevin magnetisation with relaxometric parameters
For an AC field of strength, \(H_{x}\sin\omega t\), the Langevin variable in equation can be modified with the notions \(\chi^{\prime}\)\(=\)\(\chi_{o}\)_cosot_ and \(\chi^{\prime\prime}\)\(=\)\(\chi_{o}\)_sinot_, to include the real and imaginary susceptibility and frequency components as,
\[M_{AC}=\phi M_{s}\left[\frac{1}{1+\omega^{2}\tau_{eff}^{2}}\left(\coth(\alpha \cos\omega t)-\frac{1}{\alpha\cos\omega t}\right)+\frac{\omega\tau_{eff}}{1+ \omega^{2}\tau_{eff}^{2}}\left(\coth(\alpha\sin\omega t)-\frac{1}{\alpha\sin \omega t}\right)\right] \tag{7}\]
At 0Hz Equation converges to Equation. This equation is useful for predicting volume magnetisation at high temperature and only in the SD-SPM regime and never predicts coercivity or remanence observed in many SD magnetisation experiments.
### Langevin magnetisation with relaxometric and coercivity parameters
The temperature dependent SD magnetic coercivity for a randomly oriented non interacting particle system can be expressed as,
\[H_{c}=H_{co}\left[1-(T/T_{B})^{\frac{1}{2}}\right] \tag{8}\]
Where \(H_{co=}\)_2K/ \(\mu_{o}M_{s}\)_ is the coercivity at \(0K\) according to the Stoner-Wohlfarth theory&\(T_{B}\)=\(K\)_V/ \(k_{b}ln(\tau_{m}/\tau_{o})\),_ is the critical superparamagnetic transition temperature (blocking temperature).
\[H_{c}=H_{co}\left[1-\left(\frac{k_{b}T}{K\nu}\ln(\frac{\tau_{m}}{\tau_{o}}) \right)^{\frac{1}{2}}\right] \tag{9}\]
Equation is valid when \(T\)\(<\)\(T_{B}\) since \(H_{c}\) cannot have negative values in forward magnetisation. When substituted for \(T_{B}\) in equation, the same approximation is followed hence the coercivity \(H_{c}\)\(\geq\) 0. The temperature and frequency dependence of coercivity of magnetite particles at different single domain diameters is plotted in Fig. 3To account for coercive force in magnetisation, equation can be modified by including \(H_{c}\) in \(\alpha\) and is rewritten for forward and backward measurements as
\[\alpha_{\mathit{eff}}=\frac{\pi\mu\cdot M\mathit{d}^{\prime}\left(\mathit{H}_{ \mathit{c}}\doteq Hc\right)}{6kT} \tag{10}\]
\[M_{\mathit{AC}}=\phi M_{\mathit{s}}\left[\frac{1}{1+\omega^{2}\tau_{\mathit{ eff}}^{2}}\left(\coth(\alpha_{\mathit{eff}}\cos\omega t)-\frac{1}{\alpha_{ \mathit{eff}}\cos\omega t}\right)+\frac{\omega\tau_{\mathit{eff}}}{1+\omega^{2 }\tau_{\mathit{eff}}^{2}}\left(\coth(\alpha_{\mathit{eff}}\sin\omega t)-\frac{1 }{\alpha_{\mathit{eff}}\sin\omega t}\right)\right] \tag{11}\]
Equation accounts for the frequency dependent volume magnetisation and volume and temperature dependent coercive force. The equation covers all diameters (SPM and nonSPM) in the complete SD regime.
The equation for instantaneous volume susceptibility can be derived by differentiating equation with respect to effective field either for forward \(H_{\mathit{eff}}=H_{x}+H_{c}\) or backward \(H_{\mathit{eff}}=H_{x}\) - \(H_{c}\) magnetisation measurement as follows
**Coercivity as a function of particle diameter a) at different temperatures and b) at different field frequencies. The zero coercivity corresponds to the superparamagnetic transition which is clearly a function of temperature (blocking temperature) and measurement frequency.**
\[\chi_{inst}=\frac{d}{dH_{\mathcal{A}}}(M_{AC})=\frac{-\phi M_{s}}{1+\omega^{2}r_{ eff}^{2}}\left[\frac{k_{i}}{H_{\mathcal{A}}}\Big{(}\coth^{2}(k_{i})-1\Big{)}+ \frac{\omega r_{eff}k_{2}}{H_{\mathcal{A}}}\Big{(}\coth^{2}(k_{2})-1\Big{)}- \frac{k_{i}+k_{i}\omega r_{eff}}{k_{i}k_{2}H_{\mathcal{A}}}\right] \tag{12}\]
Where \(k_{I}=\alpha_{eff}cosot\) and \(k_{2}=\alpha_{eff}sinot\).
A very useful application of equation is to approximate the DC susceptibility (0Hz) which can be reduced to,
\[\chi_{DC}=\phi M_{s}\left[\frac{1}{\alpha_{eff}H_{sf}}-\frac{\alpha_{eff}}{H_{sf} }\left(\coth^{2}(\alpha_{eff})-1\right)\right] \tag{13}\]
In reality, equation consists of real and imaginary components which can be separately redefined as
\[\chi^{*}=\frac{\phi M_{s}}{1+w^{2}\tau_{eff}^{2}}\left[\frac{1}{\alpha_{eff}H_ {sf}}-\frac{\alpha_{eff}}{H_{sf}}\left(\coth^{2}(\alpha_{eff})-1\right)\right] \tag{14}\]
\[\chi^{*}=\frac{w\tau\phi M_{s}}{1+w^{2}\tau_{eff}^{2}}\left[\frac{1}{\alpha_{ eff}H_{sf}}-\frac{\alpha_{eff}}{H_{sf}}\left(\coth^{2}(\alpha_{eff})-1 \right)\right] \tag{15}\]
The \(\chi^{\prime}\)_and_\(\chi^{\prime\prime}\) plots for SD- SPM and SD- nonSPM particles for magnetite and
The instantaneous susceptibility (full volume susceptibility) plots for two diameters 10% above and below the critical \(\text{d}_{\text{SPM}}\)for a) SD magnetite and b) SD maghemite at different temperatures. The maximal influence of coercive field at low temperature (blue) and above critical \(\text{d}_{\text{SPM}}\)(magenta) is seen as peaks in full susceptibility measurement. As the strength of the applied field increases, the peak susceptibility is seen when the maximum magnetic energy is used to overcome the demagnetising coercive field. Thereafter the superparamagnetic behaviour dominates.
maghemite at different frequencies are given in
Finally the cusp observed in experimental \(\chi^{\prime}\) versus \(T\) plots can be effectively predicted by our model as in
## 3 Conclusion
A new model to interpret superparamagnetic and nonsuperparamagnetic behaviour in single domain magnetic nanoparticles weighted by coercivity influence is presented. Equations for directly computing coercivity weighted stationary or time varying magnetisation and susceptibility for non-interacting nanoparticle samples are derived. All equations are derived for monodisperse particles but in reality most of the particle samples from different vendors are polydisperse.
The\(\chi^{\prime}\)_and\(\chi\)_” plots for SD- SPM and SD- nonSPM particles for magnetite and maghemite at different frequencies.
\(\chi^{\prime}\) versus \(T\) curve for magnetite particle of diameter equals 90% of \(d_{SPM}\)
Direct calculation of magnetisation and susceptibility would be helpful in many biomedical areas where parameters like magnetisation dependent voltage, magnetisation dependent polarisation, magneto optic effect etc. are to be estimated.
| 10.48550/arXiv.1308.2517 | Coercivity weighted Langevin magnetisation; A new approach to interpret superparamagnetic and nonsuperparamagnetic behaviour in single domain magnetic nanoparticles | Dhanesh Kattipparambil Rajan, Jukka Lekkala | 611 |
10.48550_arXiv.1808.06909 | ###### Abstract
Despite most of the applications of anatase nanostructures rely on photo-excited charge processes, yet profound theoretical understanding of fundamental related properties is lacking. Here, by means of _ab-initio_ ground and excited-state calculations we reveal, in an unambiguous way, the role of quantum confinement effect and of the surface orientation, on the electronic and optical properties of anatase nanosheets (NSs). The presence of bound excitons extremely localized along the direction, whose existence has been recently proven also in anatase bulk, explains the different optical behavior found for the two orientations when the NS thickness increases. We suggestalso that the almost two-dimensional nature of these excitons can be related to the improved photo-conversion efficiency observed when an high percentage of facet is present in anatase nanocrystals.
Among semiconducting oxides, TiO\({}_{2}\) is the most widely used for energy and environmental oriented applications. Since 1972 when Fujishima and Honda discovered the phenomenon of photo-catalytic water splitting by shining TiO\({}_{2}\) nanoparticles with ultraviolet (UV) radiation, a plethora of works has appeared in literature dedicated to the study of this manifold and extremely appealing material. In this regard, nanostructured TiO\({}_{2}\) based materials are largely investigated due to the enhancement of the surface area and to the observed improvement of photo-chemical and photo-physical activity with respect to the bulk phase. Great attention is devoted to the study of anatase form which becomes more stable than rutile at the nanoscale and shows superior performances in photo-voltaic (PV) and in photo-catalytic applications. Among the possible morphological shapes of anatase nano-materials, the study of-oriented nanosheets (NSs) is becoming particularly attractive in the last years, thanks to several studies which illustrate different routes of synthesis and thanks to the indication that, when the facets are the dominant ones, the samples are extraordinary photo-reactive. Concerning NSs, some works in the last years have also paid attention to the photo-catalytic activity of anatase nanomaterials with high percentage of these facets. Peng et al. reported excellent photocatalytic activity for samples experimentally obtained via a "_chimie-douce_" plus heat treatment method, showing that a large amount of photoactive sites are present, improving the performances with respect to _Degussa_ samples. It is important to stress that even if the facets seem to be more photo-reactive than ones, experiments do not provide homogeneous results. Indeed, in terms of presence of fivefold Ti atoms at the surface, the facet should be more photoreactive than the one, but the opposite should be true because highly reductive electrons should be generated at the, being the conduction band minimum higher in energy.
As a partial explanation for the conflicting results concerning and facet photoconversion activity one could adduce the fact that the stabilization of the "bulk-cut" surface, by means of a fluorine mediated passivation technique, is a process that has been only recently theoretically predicted and experimentally developed, thus providing a conclusive remark about the photoreactivity issue can appear still a quite cumbersome task. On the other hand, experiments have shown that the surface of anatase TiO\({}_{2}\) is much more reactive than the more stable surface. In particular, in the photocatalytic oxidation process the termination is the main source of active sites. Anyway, the presence of less reactive \(\{101\}\) facets characterizes the vast majority of anatase nanocrystals. According to these findings, it is expected that TiO\({}_{2}\) NSs with exposed \(\{001\}\) facets have an enhanced photocatalytic activity compared to those nanoparticles where larger is the amount of terminations. To further stress the dualism in terms of photoreactivity between the two facets, it is worth pointing out that novel PV technologies based on organic-inorganic halide perovskites highly rely on the chemistry of anatase NSs. It is interesting to observe that the power conversion efficiency (PCE) of mesoscopic CH\({}_{3}\)NH\({}_{3}\)PbI\({}_{3}\)/TiO\({}_{2}\) heterojunction solar cells depends on the TiO\({}_{2}\) NS facet exposed. When the is the dominant one a PCE of the device double than that obtained with the facet is reported.
Surprisingly, despite the large number of scientific works, even on bulk anatase, fundamental properties like the exact value of its electronic gap or the presence of a strongly bound two-dimensional exciton have been only very recently clarified as a result of a combined experimental and theoretical effort. Furthermore the impact of quantum-confinement (QC)effect on the electronic and optical properties of anatase nanostructures, is still under debate. This is due, on one side, to the fact that experiments reach contrasting results, being often influenced by several factors, like synthesis conditions, presence of defects dopants and co-dopants, and on the other hand to the lack of results obtained by means of predictive quantum-mechanical excited-state calculations. Then our goal here is to investigate the role played by QC effect, focusing both on the electronic and optical properties of anatase NSs of increasing thickness and using the NSs to point out also the role of different orientations.
For this reason, we have specifically selected NSs with the simplest surface models, avoiding reconstructed and/or hydroxylated facets, with the idea to leave to future investigations the study of many-body effects in NSs with less ideal surface motifs. By means of Density Functional Theory (DFT) and post-DFT excited-state (namely GW ad Bethe-Salpeter Equation(BSE)) calculations, we demonstrate the mismatch between electronic and optical gap, with the latter mainly associated to the presence of strongly bound bidimensional excitons, confirming the results recently reported by Baldini et al. for the bulk anatase.
Ground-state atomic structures have been relaxed using DFT, as implemented in the VASP package, within the generalized gradient approximation (GGA) of Perdew-Burke-Ernzerhof (PBE). The Blochl all-electron projector augmented wave (PAW) method has been employed. For Ti atoms a 12\(e\) potential has been employed along with a cut-off energy set to 500.0 eV for the plane wave basis. A force convergence criterion of 0.035 eV/A has been chosen, while different samplings Gamma centered of the Brillouin zone have been employed according to the lateral dimension of the system under investigation. Using the relaxed atomic structures we performed further relaxation of the atomic structures using the Quantum Espresso package in a consistent PBE scheme but using norm-conserving pseudopotentials with a plane-wave expansion of 2450 eV of cutoff. Then self-consistent and non self-consistent calculations have been performed to obtain DFT-KS eigenvalues and eigenvector to calculate the quasi-particle (QP) energies in GW approximation and optical excitation energies solving the Bethe-Salpeter equation (BSE) by means of the many-body code YAMBO. For the GW simulations a plasmon-pole approximation for the inverse dielectric matrix has been applied, 136 eV (952 eV) are used for the correlation \(\Sigma_{c}\) (exchange \(\Sigma_{x}\)) part of the self-energy and the sum over the unoccupied states for \(\Sigma_{c}\) and the dielectric matrix is done up to about \(\sim\)50 eV above the VBM. In order to speed up the convergence with respect empty states we adopted the technique described in Ref.. Finally when the QP energies and eigenfunctions are known, the optical properties are calculated by solving the Bethe-Salpeter equation (BSE) where the electron-hole interaction is also taken into account. A _k_-points grid of \(10\times 10\times 1\) (\(10\times 5\times 1\)) has been used in the GW and BSE calculations for [] sheets. A cutoff in the coulomb potential in the direction perpendicular to the sheet, has been used in the excited state calculations to eliminate the spurious interactions along the non periodic direction and to simulate a real isolated nanosheet. Since our goal is mainly to analyze the role of quantum confinement we focus first on four-oriented anatase NSs of increasing thickness. Then in order to capture the role of the surface orientation, we perform simulations also on two-oriented NSs of different thickness. For the we use a (1\(\times\)1) cell with the lattice parameter equal to the bulk one, allowing a full relaxation of the atomic positions. We adopt this choice because we consider the experimental conditions of the delamination process along direction of anatase and we aim to compare the results with the corresponding bulk data. Two asymmetric Ti-O bonds (1.74 and 2.24 A) at surface are formed. In a previous work, where we focused on anatase and leptodecrocite thinnest sheets, we have named this structure Anatase-AS2. Despite the asymmetric Ti-O bond structure has not been detected experimentally, the larger stability of the asymmetric bond type with respect to bulk-terminated symmetric one is widely predicted at the theoretical level. This result can be ascribed to the residual stress amount present in the reconstruction. We focus on-oriented NSs formed by two, four, six, and eight atomic layers with an estimated atomic thickness of 0.36, 0.80, 1.3, and 1.8 nm, respectively. The lateral views of two of these atomic structures are shown in the insets of Fig.1. Concerning the orientation we assemble two NSs with thickness of about 0.95 nm and 1.65 nm, respectively and using a 1\(\times\)1 cell in the surface plane. Similarly to the sheets, we have kept the in-plane lattice parameters frozen to the bulk optimized value, i.e. 3.78 \(\times\) 10.24 A, to focus only on the confinement effects. The QP bandstructures of two and two oriented NSs (blue curves) of different thickness, plotted along the high-symmetry directions X\(\rightarrow\Gamma\rightarrow\)M (where X = (0.5,0,0), \(\Gamma\)= and M= (0.5,0.5,0) in reciprocal lattice units), are shown in Few conduction and valence bands around the Fermi energy, calculated at the DFT-KS level, are also reported (gray lines). First of all we can observe that the self-energy correction to the KS gap increases reducing the sheet thickness and is larger than the value found in anatase bulk which is of the order of 1.4 eV (see ref. for more details). This is consistent with the fact that the dielectric screening reduces decreasing the size of the nanostructure. Moreover, at a given thickness, a small but not negligible _k_-dependence of the QP correction is found and for this reason we avoid the use of a rigid scissor operator to open the unoccupied bands. We finally point out that the indirect gap character typical of anatase bulk remains in all the considered NSs, both at DFT and QP level of approximation. For a comparison with anatase bulk band structure, obtained at the same level of theoretical approximation, we refer the reader to Fig.5 of ref.. Fig.2(a) shows the optical spectra of the anatase nanosheets (for light-polarized \(\perp\) to the _c_-axis) compared to the corresponding optical spectrum of anatase bulk. Self-energy, local-fields, and excitonic effects are included through the solution of the Bethe-Salpeter equation. It is important to underline that due to the depolarization effect the optical spectra for light depolarized perpendicularly to the nanosheet (not reported here) are almost zero when, as in the present study, the single-particle approach is overcome taking into account local-field effects. First of all we note that increasing the sheet thickness, the position of the first optical peak rapidly recovers the bulk-like position. Moreover, from the analysis of the optical spectrum in terms of excitonic eigenvalues and eigenvectors, we know that, while in the bulk the first optical peak (at\(\sim\)3.8 eV) corresponds to a bright exciton (indicated by B in the figure) and any lower dark exciton is present, in the NSs several excitons with weak oscillator strength appear, where the position of the first one is indicated in the figure by D. Fig.2(b) shows the corresponding optical spectra calculated for the two-oriented NSs. In this case a larger QC effect with respect to the other orientation is clearly visible. As we will discuss later on, this different behavior due to quantum-confinement is strictly related to the spatial character of the first exciton.
To further illustrate the role of quantum-confinement in Fig.3 we report the value of the direct and indirect QP electronic gaps together with the energy of the first bright exciton as function of the sheet thickness; since we considered only two sheets of different thickness we did not include the corresponding data for this orientation.
As reported in previous literature (see i.e. refs.) focusing on the size dependence of the electronic and optical gaps in semiconducting nanostructures, the direct QP electronic gap values are fitted with the scaling law \(E_{gap}^{bulk}+C/d^{\alpha}\), where \(E_{gap}^{bulk}\) is the bulk gap value and \(d\) is the NS thickness. Similar scaling laws have been used to fit the indirect QP electronic gaps and the energies of the first bright excitons (B).
Still looking at Fig.3, it is clear that the electronic and optical gaps have a different behavior decreasing the thickness of the nanosheet. Indeed, the electronic QP direct and indirect gaps remain larger than the corresponding bulk value for the considered thicknesses. This can be explained by the fact that the self-energy correction, due to the reduced dielectric screening and localization of the wavefunctions, strongly increases reducing the NS thickness. The scaling exponents able to fit the direct and indirect gaps are 1.05 and 0.88, respectively. Similar exponents (\(\sim\)1) have been obtained in previous works that take into account the many-body self-energy corrections in nanostructures of different dimensionality.
On the other hand, as the NS thickness increases, the optical (excitonic) direct gaps converge more rapidly to the corresponding optical direct gap of the bulk. Again this finding is consistent with previous studies of many-body effects in low-dimensional materialsQuasi-Particle bandstructures (blue lines) of the 4L, and 8L and 3L and 5L NSs. To show the entity and the small _k_-dependence of the QP correction some PBE conduction and valence bands are also reported (gray). The QP corrections were calculated on a regular grid and band interpolation was performed following the method of Ref. using the WanT package.
In other words, for the optical data, the convergence is reached as soon as the bulk-like excitonic wave function is contained in the NS thickness which is at \(d\sim 2\) nm. As already observed in other studies, the scaling exponent able to fit the optical data results larger (for the anatase nanosheets is 1.97) than the value used to fit the electronic QP gaps. It is worth observing that although this value is very similar to \(2\)\(-\) the exponent of the ideal particle-in-a-box model \(-\) the physics of the exciton described here is completely different and this value can be understood only in terms of the cancellation of the induced polarization effect which is present both in the GW and in the BSE kernel that then rapidly cancels out as the size of the nanostructure increases.
We aim now to discuss the spatial character of the first bright exciton (B) of the
Optical spectra of bulk, anatase and for light polarized \(\perp\)\(c\)-axis. First Dark (D) and Bright (B) exciton are also indicated. For NSs the intensities are renormalized to their effective thickness.
in anatase bulk. Indeed, the direct optical gap of single crystal anatase is dominated by a strongly bound exciton, rising over the continuum of indirect interband transitions, which possesses an intermediate character between the Wannier-Mott and Frenkel regimes and displays a peculiar 2D wavefunction in the 3D lattice.
By fitting the exciton wavefunction with a 2D hydrogen model, the exciton Bohr radius results \(\sim 3.2\) nm, while the 90% of the excitonic squared modulus wavefunction is contained within 1.5 nm. The top panel of Fig.4 reports the exciton spatial distribution for the-NS (\(d=1.8\) nm), showing the same high degree of spatial localization, along the c-axis, observed in bulk (see ref.), while the bottom panel reports the corresponding distribution for the sheet of thickness \(d=1.6\) nm.
Direct (squares) and Indirect (diamonds) electronic (QP) gaps; first bright (circles) excitons (BSE) as function of NS thickness. Scaling law fits \(\simeq 1/d^{\alpha}\), where \(d\) is the NS thickness, are similarly reported. The red (blue) and yellow solid lines represent the QP direct (indirect) and optical direct gap in bulk.
It is worth pointing out that this localized behavior is substantially the same placing the hole in different positions and that it holds both in the case the hole is fixed in a bulk (shown in Fig.4) or in a sub-surface position (not shown here). The analysis of the dark states (not shown here) does not provide noticeable differences in the spatial exciton localization plots with respect to the bright exciton case. Furthermore, from the corresponding analysis of the nanosheet we have found that the B-exciton is composed by the mixing of single-particle vertical transitions mainly at \(\Gamma\) (with smaller contribution from points near it) between VBM-1 to CBM+1 and VBM-4 to CBM, while for the D-exciton the involved transition is that from VBM to CBM, still at \(\Gamma\). From the spatial analysis of these states they are mainly localized in the central part of the sheet. Minor contribution from other points near \(\Gamma\) and other bands like VBM-3 to CBM,CBM+1 and VBM-2 to CBM,CBM+1 occurs. This finding is consistent with what observed in bulk, where the excitonic wavefunction of the first bright exciton results formed by the mixing of transitions from VBM,VBM-1 to CBM along the \(\Gamma\)\(\to Z\) direction (where the bands near the gap are almost parallel, see of ref.). Indeed, it is important to recall that all the points along the bulk \(\Gamma\)\(\to Z\) direction are folded in \(\Gamma\) in the nanosheet.
Then from this specific, quasi 2D, spatial distribution of the exciton, mainly induced by the lattice geometry (see ref. for more details), we can deduce some conclusions. Due to the extreme localization of the exciton along the direction, when the hole is created in a bulk position the exciton does not touch the surfaces already for thickness of the NS of the order (or larger) of 1.8 nm. As consequence, as we have shown in Fig.2, a bulk-like behavior of the optical spectrum is rapidly recovered in the-oriented NSs, while a larger QC effect is visible in the NSs of comparable thickness, due to the fact that in this case the excitonic wavefunction extends up to the two surfaces, remaining confined.
Moreover, we suggest that it could contribute to the larger photo-reactivity often reported in anatase nanostructures when a large percentage of facets is present. Indeed recent experiments have shown that i) in the photo-catalytic process the reduction and oxidation reactions preferably occur on/ and facets, respectively ii) the thickness of anatase nanocrystals when a large percentage of facets is present, is thinner in the than in other crystallographic directions. These two facts, in addition to the observed exciton spatial distribution, suggest that when a large percentage of facet is present, the photo-excited hole can easily reach the termination, especially if created not far from it and, at the same time, due to the delocalized nature of the exciton in the plane, there is a non-zero probability to collect instantaneously the electron at other terminations, like the surfaces. Although at the moment this is only a speculation, we point out that recent works in organic and hybrid organic-inorganic PV materials show that the presence of extremely delocalized coherent excitonic bound states, due to the low dielectric screening, can contribute to enhance the photo-conversion efficiency.
To summarize, by means of the Bethe-Salpeter equation solution, we have here investigated the excitonic behavior of anatase nanosheets with majority and minority surface orientation. In particular, for and-oriented NSs, we have focused on quantum-confinement effects and on the role they play on the optical properties of such NSs. While the former show a more marked QC ascribed to the exciton confinement induced by the two \(\{101\}\) delimiting surfaces, in the case of the orientation we instead observe the existence of a threshold thickness value (\(\sim\) 2 nm) above which the bulk optical behavior is recovered. The specific quasi two-dimensional character of the exciton can be related to the difference in terms of photo-reactivity between the two orientations, with relevant consequences in devices exposing the two different facets.
| 10.48550/arXiv.1808.06909 | Role of Quantum-Confinement in Anatase Nanosheets | Daniele Varsano, Giacomo Giorgi, Koichi Yamashita, Maurizia Palummo | 4,099 |
10.48550_arXiv.1406.2048 | ###### Abstract
We have demonstrated that the evolution of the two-dimensional electron gas (2DEG) system at an interface of metal and the model topological insulator (TI) Bi\({}_{2}\)Se\({}_{3}\) can be controlled by choosing an appropriate kind of metal elements and by applying a low temperature evaporation procedure. In particular, we have found that only topological surface states (TSSs) can exist at a Mn/Bi\({}_{2}\)Se\({}_{3}\) interface, which would be useful for implementing an electric contact with surface current channels only. The existence of the TSSs alone at the interface was confirmed by angle-resolved photoemission spectroscopy (ARPES). Based on the ARPES and core-level x-ray photoemission spectroscopy measurements, we propose a cation intercalation model to explain our findings.
*E-mail: Since the discovery of topological insulators (TIs) with topologically protected metallic surface states, much attention has been paid to characterization of the topological surface states (TSSs), unveiling the abundant exotic properties of the TSSs such as the absence of back-scattering, the spin-momentum locking, the robustness against various kinds of surface perturbations. In particular, the studies on the effect of surface adsorbates to the TSSs have found very intriguing phenomena that new Rashba-type spin-split surface states emerge on the surface of Bi\({}_{2}\)Se\({}_{3}\) TI and that they have large amounts of common features irrespective of the electric/magnetic properties of the adsorbates. The successive experimental and theoretical studies have suggested that the newly emerging surface states are actually interface states in a quantum-confined two-dimensional electron gas (2DEG) induced by a surface band-bending effect.
The coexistence of TSSs and 2DEG states is a very exotic example in a viewpoint of surface physics and have potential applications in the field of nanoscale spintronic devices, so the understanding of their origin and properties is essential in this TI physics. However, the newly developed 2DEG has a complex spin-orbit-split electronic structure, which could be an obstacle depending on the way how to implement TI-based devices. For example, when we want to use an electric current through only the TSSs in Bi\({}_{2}\)Se\({}_{3}\) or when we want to make an n-type TI-metal junction whose current channel is only TSSs, those 2DEG states are definitely to be removed. In spite of these practical needs, all reports on the effects of adsorbates or surface impurities on Bi\({}_{2}\)Se\({}_{3}\) surfaces have only shown that the 2DEG states are easily formed by various kinds of adsorbates in an ultra-high vacuum, but have not suggested a possible way to remove the developed 2DEG states without killing the TSSs.
In this study, we focus on this issue and performed systematic angle-resolved photoemission spectroscopy (ARPES) measurements in order to explore whether there is a way to remove the 2DEG states with leaving the TSSs alone at a metal/Bi\({}_{2}\)Se\({}_{3}\) interface. Keeping in mind that the practicality of the surface states described above appears at an interface between metal and TI, we deposited _in situ_ several kinds of metal elements such as Cu, In, and Mn on a surface of Bi\({}_{2}\)Se\({}_{3}\) in a well-controlled manner, which actually corresponds to an early stage of a metal thin film synthesis, and observed an evolution/devolution of the TSSs and 2DEG states as a function of deposition thickness by ARPES and core-level x-ray photoemission spectroscopy (XPS). Our ARPES measurements reveal that in the case of Cu and In, the 2DEG states are developed with a Rashba-type spin-split electronic structure and are saturated up to \(\sim\)1 monolayer (ML), but that in the case of Mn, the 2DEG states are developed up to \(\sim\)0.4 ML, then disappear above \(\sim\)1.0 ML. The disappearance of the 2DEG states and the existence of the TSSs alone were observed on a Bi\({}_{2}\)Se\({}_{3}\) surface covered with up to \(\sim\)10 ML of Mn atoms by ARPES, indicating that only the TSSs exist at the Mn/Bi\({}_{2}\)Se\({}_{3}\) interface prepared by our procedure. In order to explain why Mn deposition makes a difference from Cu or In deposition, we kept track of the XPS spectral changes for Bi 4\(f\), Se 4\(d\), and Cu/In/Mn core levels, and proposed a cation intercalation model where the intercalated cations act as a potential gradient reducer.
Evolution of the TSSs and the 2DEG states on the surface of Bi\({}_{2}\)Se\({}_{3}\) TI as a function of Cu/In deposition thickness. (a) ARPES-measured TSSs obtained from a clean surface of Bi\({}_{2}\)Se\({}_{3}\) along the \(\overline{\Gamma}-\overline{K}\) direction within 15 minutes after _in situ_ cleaving at 40 K. (b)-(d) ARPES images after Cu 0.2, 0.4, and 0.8 ML deposition, respectively. (e)-(h) ARPES images after In 0.2, 0.5, 0.8, and 1.1 ML deposition, respectively.
## Results
Typical features of the TSSs in a fresh surface of Bi\({}_{2}\)Se\({}_{3}\) are presented in Fig. 1(a). The ARPES image was obtained along the \(\overline{\Gamma}-\overline{K}\) direction within 15 minutes after cleaving the sample at 40 K during which we did not observe any aging effect due to the adsorption of residual gas molecules in the vacuum chamber. This is quite different from the p-doped Bi\({}_{2}\)Te\({}_{3}\) case in our previous report. The V-shaped linear dispersive surface bands form two Dirac cones sharing the Dirac point (DP) located at around 200 meV below the Fermi level. Below the DP, a clear M-shaped bulk valence band is seen. On this cleaved surface, we deposited copper or indium atoms _in situ_ by evaporating pure (99.999%) copper or indium metal with a well-controlled manner described below. Figures 1(b)-1(d) show an evolution of the TSSs and the 2DEG states as a function of Cu deposition thickness. At a small amount of Cu deposition (\(\sim\)0.2 ML), newly developed surface states with Rashiba-type spin-splittings are clearly seen together with the TSSs. The DP and the M-shaped valence bands shift to the higher binding energy side by \(\sim\)0.3 eV. In Figs. 1(e)-1(h), a similar surface state evolution is displayed with the increase of In deposition. A prominent difference from the Cu deposition case appears in the size of the Rashba-type spin splittings, but both cases share common features in many aspects. The similar behavior of the surface state evolution has been reported for various kinds of surface deposition or surface adsorption. However, the development of the surface states does not originate from a topological property of TIs but from an interface property of metal/semiconductor. In the case of Bi\({}_{0.9}\)Sb\({}_{0.1}\) TI, our previous ARPES study shows just a small shift of the Fermi level in the electronic structure with response to the surface adsorption.
The origin of these surface states emerging in the surface of Bi\({}_{2}\)X\({}_{3}\) (X=chalcogen) compounds has proved controversial initially, but one of the most persuasive models argues that those are a kind of 2DEG states developed in a 2-dimensional quantum well which is induced by a strong band bending at a metal/semiconductor interface. According to this scenario, the dominant factors for the 2DEG states are the shape of the potential profile as a function of depth from the interface and the induced charge density at the interface, but the kinds and the amount of adsorbates are not important if only they stick on the surface to form a well-defined interface. Thus, in order to remove the 2DEG states at a metal/Bi\({}_{2}\)Se\({}_{3}\) interface, or at least in order to make a different surfaceelectronic structure from the prototypical 2DEG states, metal elements that can intercalate or be interstitial defects are expected to be more effective. In this viewpoint, one different behavior between Cu- and In-depositions can be qualitatively understood. At relatively thick deposition (\(\gtrsim\)0.8 ML), the 2DEG states look more shrunk in the Cu-deposited sample than in the In-deposited sample as shown in Figs. 1(d) and 1(g). This is possibly due to the intercalatability difference between Cu and In atoms for Bi\({}_{2}\)Se\({}_{3}\). Actually, it has been reported that Cu atoms can intercalate into the van der Waals (vdW) gaps in Bi\({}_{2}\)Se\({}_{3}\), while there is no such report for In atoms to our knowlege.
Keeping in mind that the intercalatability of the deposited atoms may be the crucial factor for our purpose, we chose Mn as a deposition material and kept track of the 2DEG states evolution as a function of deposition thickness since it is known that a small amount of Mn
Evolution of the TSSs and the 2DEG states on the surface of Bi\({}_{2}\)Se\({}_{3}\) TI as a function of Mn deposition thickness. (a) ARPES-measured FSs (upper) and the dispersion relation of TSSs (lower) obtained from a clean surface of Bi\({}_{2}\)Se\({}_{3}\) along the \(\overline{\Gamma}-\overline{M}\) direction within one hour after _in situ_ cleaving at 40 K. (b)-(d) The corresponding ARPES images after Mn 0.2, 0.6, and 1.0 ML deposition, respectively. Up to \(\sim\)0.2 ML of Mn, the 2DEG states are developed, but they becomes weaker and the Dirac point moves \(\sim\)0.1 eV upward at Mn \(\sim\)0.6 ML thickness. The 2DEG states almost disappear at \(\sim\)1.0 ML thickness, and only TSSs are seen in the ARPES images.
Rubidium was also reported to intercalate into the gap, but its effect on the 2DEG states lies in another direction to our purpose. Figures 2(a)-2(d) show the ARPES-measured Fermi surface (upper panel) and the corresponding energy dispersion relation (lower panel) along the red dotted line (approximately \(\overline{\Gamma}-\overline{M}\) direction) for a clean surface, 0.2 ML, 0.6 ML, and 1.0 ML Mn-deposited surface, respectively. Interestingly enough, Mn-deposited surfaces show a very different evolution of the 2DEG states. At a small amount of Mn deposition (\(\sim\)0.2 ML), quite a complex 2DEG structure is developed as shown in Fig. 2(b). It looks similar to that of the Cu-deposited surface. However, when the deposited layer is thicker than \(\sim\)0.2 ML, the Fermi surface (FS) size of the 2DEG states becomes smaller, and the DP shifts up by \(\sim\)0.1 eV as is shown in Fig. 2(c). At around Mn 1.0 ML, the 2DEG states almost disappear and only the TSSs remain as can be seen in Fig. 2(d). Based on the facts that the energy position of the DP in Mn 1.0 ML deposition is \(\sim\)0.1 eV deeper than that of the clean surface and that the size of the hexagonal FS of the TSSs is a little larger than that of the clean surface, the surface does not recover to its original condition, but forms another interface that gives a similar environment to the original vacuum/ interface for the TSSs and 2DEG.
In order to figure out why the Mn deposition makes such an intriguing behavior in the 2DEG states evolution, we carried out core-level XPS measurements as is displayed in Since XPS is sensitive to the chemical valency and chemical environment of ions in solids, important information or at least a clue on the reason of the disappearance of the 2DEG states can be obtained by analyzing the XPS spectra. In the left column, the center one, and the right one of Fig. 3, the XPS spectra of each deposited element core level, Bi \(4f\), and Se \(3d\) level are presented with the increase of the deposition thickness, respectively. In the case of In deposition, the Bi \(4f_{5/2,7/2}\) and Se \(3d_{3/2,5/2}\) peaks shift toward lower kinetic energy side by \(\sim\)0.4 eV with the increase of the deposition thickness as shown in Fig. 3(a). The amount of the energy shift corresponds to the band bending shown in Figs. 1(a) and 1(h), so it can be interpreted as a surface potential shift due to the accumulated surface charge induced by the In deposites. The similar core-level shifts are observed in the Cu- and Mn-deposited surface, respectively, as is shown in Figs. 3(b) and 3(c), and the amount of energy shift is also very consistent with that of the band shift in Figs. 1 and 2.
Several interesting features in the core-level spectra of each deposited element, Bi \(4f\), and Se \(3d\) are found in a different dependence on deposition. While In deposition does not make any change in the In 3\(d\), Bi 4\(f\), and Se 3\(d\) spectral line shape, Cu(Mn) deposition induces an evolution in Cu 3\(p\) (Mn 2\(p\)), Bi 4\(f\), and Se 3\(d\) level, respectively. In the Cu 3\(p\) region, two doublets (75 eV and 71 eV) are observed, each of which can be assigned to surface peak and intercalation peak(\(\bigtriangledown\)) as shown in Fig. 3(b). The relative peak weight dependence on deposition thickness strongly supports this assignment, and the similar result has been obtained for the 3\(reported in the Rb deposition/annealing study. Meanwhile, Mn \(2p_{3/2}\) do not make a prominent difference in their shape with the amount of Mn deposition, so it is not clear whether a part of the adsorbed Cu atoms intercalates or not. However, Mn deposition effects on the other core level spectra are not similar to the indium case. For both Cu and Mn deposition cases, a new spectral weight appears between the spin-orbit-split peaks of the Se \(3d\) together with the increase of the peak width. This indicates an occurrence of another kind of Se ions chemically different from those in pristine Bi\({}_{2}\)Se\({}_{3}\). The similar evolutionary behavior is observed in the Bi \(4f\) region (\(\blacktriangledown\)) of Fig. 3(c). The prominent evolution of the Bi \(4f\) peaks in the Mn-deposited surface is quite contrastive to the Cu case where the Bi \(4f\) peaks hardly show the dependence on the Cu deposition thickness. This element-specific response to the deposites indicates that intercalatability and intercalated sites of the deposites into Bi\({}_{2}\)Se\({}_{3}\) bulk are different among In, Cu and Mn. The invariance of the XPS spectra in the In-deposited surface suggests that In atoms do not intercalate into the Bi\({}_{2}\)Se\({}_{3}\) bulk. Meanwhile, a part of the Cu or Mn deposites definitely intercalate into the bulk but their intercalated sites are different. Based on the Bi \(4f\) and Se \(3d\) XPS spectra, the intercalated Cu atoms affect on the Se anions only. Meanwhile, the intercalated Mn atoms affect on the Bi ions as well as the Se ions. This contrastive response suggests that the intercalated Cu atoms reside mainly in the vdW gaps between the quintuple layers (QLs) while the Mn atoms are in the interstitial sites as well as in the vdW gaps. This also explains why the binding energy of the intercalated Cu atom \(3p\) levels is smaller than that of the deposited Cu atoms on the surface. Since the QLs are chemically very stable and electrically close to neutral, the intercalated atoms into the vdW gaps are also very close to charge-neutral, so the binding energy of the atoms is smaller than that of Cu metal which is close to monovalent ion. The zero valence of the intercalated Cu ions into the vdW gaps of Bi\({}_{2}\)Se\({}_{3}\) has been reported in an electron energy loss spectroscopy study. Meanwhile, the interstitial Mn atoms are likely to act as cations as Bi atoms do. If this is the case, the intercalated Mn cations cannot help making moderate the surface band bending effect induced by the deposited Mn adsorbates.
## Discussion
The XPS measurements described above give important information to explain the 2DEG states appearance/disappearance in our ARPES data. Probably, the most simple way to the 2DEG removal is to weaken the adsorbates-induced band bending at the interface. We have evidenced in the XPS study that the Mn adsorbates intercalate into the interstitial sites as well as the vdW gap between the QLs of Bi\({}_{2}\)Se\({}_{3}\) and make the surface potential gradient more gentle, acting as a buffer. Although most of the intercalated Mn atoms stay charge neutral, a part of them gives a few electrons per atom and act as Mn cations, so the surface potential gradient gets smaller as the average intercalation depth gets larger. Figures 4(a) and 4(b) show a schematic configuration for our cation intercalation model. In the case of In deposition, the surface charges induced by the deposited indium ions attract electrons toward the In/Bi\({}_{2}\)Se\({}_{3}\) interface, making a 2DEG system.
Schematics for the surface electronic structure of a metal/Bi\({}_{2}\)Se\({}_{3}\) interface. (a) In the case of In deposition, the atoms are adsorbed on the surface of Bi\({}_{2}\)Se\({}_{3}\), which induces a strong downward band bending at the metal/TI interface. (b) In the case of Mn deposition, the Mn atoms intercalate presumably into the interstitial sites as well as the vdW gaps, which reduces the potential gradient beneath the interface. (c) The ARPES image after Mn 10 ML deposition. Even when the Mn-deposited layers are \(\sim\)10 ML thick, there are only TSSs at the interface.
However, as in the case of Mn deposition, if parts of the deposited Mn atoms intercalate into the interstitial sites of Bi\({}_{2}\)Se\({}_{3}\), and are ionized into Mn cations, the deep well-like surface potential changes to a wide valley with a gentle slope. In this model, the different response to Cu deposition is also naturally explained. As is described above, Cu atoms are also known to intercalate into the vdW gap, but they do not intercalate into the interstitial sites, so almost all of the intercalated Cu atoms stay charge neutral. The lower binding energy of the intercalated Cu 3\(p\) peak in Fig. 3(b) supports this interpretation. Actually, if we compare the Cu and In deposition results, the 2DEG states in the Cu deposition shrink a little more than those in the In deposition (See Fig. 1(d) and (g)). So, the Cu intercalation effects definitely exist, but are not so prominent like the Mn case. This is due to a smaller fraction of the ionized Cu atoms to the intercalated ones than that of Mn atoms.
Finally, we have checked whether the TSSs still exist even in an environment that can be regarded as a real metal/TI interface. In Fig. 4(c), the ARPES image shows dim V-shape TSSs near the Fermi level. This image was obtained from a 10 ML Mn-deposited Bi\({}_{2}\)Se\({}_{3}\) surface. Since the scattering rate of the photoelectrons is very high due to the thick Mn layers, the ARPES-measured TSSs look very blurred, but definitely we can see the existence of the TSSs alone at the interface.
### Experimental methods
The Bi\({}_{2}\)Se\({}_{3}\) single crystals were grown by the melting method. The stoichiometric mixture of Bi(99.999%) and Se(99.9999%) was loaded in a evacuated quartz ampoule. The ampoule was heated up to 850\({}^{\circ}\)C for 48 hours, followed by slow cooling to 500\({}^{\circ}\)C at a rate of 2\({}^{\circ}\)C/h. The furnace was kept at this temperature for 5 additional days for post-annealing before furnace cooling. The ARPES and XPS experiments were performed at the 4A1 beamline of the Pohang Light Source with a Scienta SES-2002 electron spectrometer and \(\hbar\omega\)=25 and 800 eV photons. The total energy (momentum) resolution is \(\sim\)20 meV (\(\sim\)0.01 A\({}^{-1}\)) for ARPES and 0.8 eV for XPS. The crystals were cleaved _in situ_ by the top-post method at 40 K under \(\sim 7.0\times 10^{-11}\) Torr. Surface adsorbates/intercalates were introduced on the sample surfaces by evaporating the pure (99.999%) metal elements with a rate of 0.1 A/minin an ultra-high vacuum of \(\sim\)1.0\(\times\)10\({}^{-10}\) Torr at 50 K of the sample temperature.
| 10.48550/arXiv.1406.2048 | Controlling the 2DEG states evolution at a metal/Bi$_2$Se$_3$ interface | Han-Jin Noh, Jinwon Jeong, En-Jin Cho, Joonbum Park, Jun Sung Kim, Ilyou Kim, Byeong-Gyu Park, Hyeong-Do Kim | 96 |
10.48550_arXiv.1207.3144 | ### Introduction of Model Concepts
According to Fig. 3, we assume that the mean effects of phonon scattering at surfaces and voids is confined to the 100 A ranges, at least for voids of comparable sizes to those simulated. Considering that the average spacing between point defects such as vacancies exceeds 1000 A even when the material contains a high defect concentration of \(10^{15}\)\(cm^{-3}\)The phonon density of states for a periodic system: (a) locations of 5 regions (red boxes) for calculations of density of states (from region 1 closest to the voids to region 5 furthest); and (b) the density of states calculated for the 5 locations shown in (a). Here the values of density of states have error bars of near the width of the variation of the acoustic modes (0-45 meV) and are omitted for clarity. The density of acoustic modes appears to be equivalent but the density of optical modes increases slightly near the void.
For convenience, we further assume that defects are uniformly distributed, which is roughly true for the most homogeneous experimental material, but is exact for simulations with periodic boundary conditions where defects are replicated uniformly in space. On the other hand, when the defect scattering effect is fully confined to a relatively small region, the overall thermal conductivity becomes independent on the particular distribution of sparse defects. This is examined in using heat transport through a prism as an example. For sparse defects, we can embed each defect (circle) in a section (dark box) with a sufficiently large dimension of \(d_{x}\). Because defect effects are confined to a relatively small region, we can assume that only the thermal conductivity inside the section changes to \(\kappa_{d}\) whereas the conductivity outside the section remains to be the defect-free matrix value \(\kappa_{m}\). It is plausible that, for the two cases where the three defects are either uniformly distributed over the prism length \(L\), Fig. 5(a), or concentrated on the left-hand side, Fig. 5(b), the overall thermal resistivities (inverse of thermal conductivity) of the two prisms are the same, both equal to a length weighted average of resistivities of defective sectors and matrix as: \(3d_{x}/L\cdot\kappa_{d}^{-1}+\left(L-3d_{x}\right)/L\cdot\kappa_{m}^{-1}\). Note that our model will be accurate when these assumptions are valid. The model may still capture well the scaling law even for materials with phonon mean free path longer than defect spacing. Model evaluation should be performed by comparing model predictions with either experiments or direct simulations (such as MD simulations as will be shown in the following for the GaN case).
When the defect distribution is uniform, the material can be viewed as composed of equivalent small volumes \(\delta V=\delta_{x}\cdot\delta_{y}\cdot\delta_{z}\) each containing a defect. Because these volumes are assumed to be identical and independent of each other, the overall thermal conductivity of the material equals the thermal conductivity of each individual volume. As a result, only one representative volume needs to be addressed. Here, we consider heat conduction in the x dimension of the volume \(\delta V=\delta_{x}\cdot\delta_{y}\cdot\delta_{z}\) containing a defect in its center as marked by a spherical ball in Fig. 6(a). The presence of this defect changes the thermal conductivity of its surroundings defined by a sub-volume \(dV=d_{x}\cdot d_{y}\cdot d_{z}\). Because the defect concentration is low, we can always choose large values of \(d_{x}\), \(d_{y}\), and \(d_{z}\) (comparable to the defect scattering distance shown in Fig. 3). Hence, the material outside the \(dV=d_{x}\cdot d_{y}\cdot d_{z}\) volume can be viewed as far away from the defect and therefore its thermal conductivity remains to be the value of the (defect-free) matrix material, \(\kappa_{m}\). Despite the non-uniformthermal properties on a fine scale, the smaller scattering volume \(dV\) exhibits an apparent overall thermal conductivity \(\kappa_{d}\). Since \(\kappa_{d}\) is essentially the coarse-grained, average thermal conductivity of the volume \(d_{x}\cdot d_{y}\cdot d_{z}\), it therefore depends strongly on \(d_{x}\), \(d_{y}\), and \(d_{z}\); however, \(\kappa_{d}\) can be viewed as a constant for a given defect size and type once \(d_{x}\), \(d_{y}\), and \(d_{z}\) are given.
Using the model shown in Fig. 6(a), we define two geometric parameters \(\alpha=\left(d_{y}\cdot d_{z}\right)/(\delta_{y}\cdot\delta_{z})\) and \(\beta=d_{x}/\delta_{x}\). Considering that \(\delta_{x}\), \(\delta_{y}\), and \(\delta_{z}\) are the overall material dimensions divided by numbers of defects in the three coordinate directions, and \(d_{x}\), \(d_{y}\), and \(d_{z}\) are given constants physically representing the dimension around the defect where scattering is significant, parameters \(\alpha\) and \(\beta\) prescribe the relative defect densities (areal and lineal fractions) in the cross-sectional and axial dimensions respectively. In particular, \(\alpha\) and \(\beta\) can be termed defect "scattering" densities because \(\alpha=1\) and \(\beta=1\) do not represent 100% defect volume fraction (or site fraction), but rather indicate that the ratio between defect scattering volume and total volume equals one. We can also define a defect volume scattering density \(\rho=\alpha\cdot\beta\).
Comparison of two defect distributions: (a) uniform and (b) nonuniform defect distributions.
_(a)_ average total volume (\(\delta\)x\(\cdot\)\(\delta\)y\(\cdot\)\(\delta\)z) containing one defect
_(b)_ prism (\(\delta\)x\(\cdot\)dy\(\cdot\)dz) containing one defect
_(c)_ slab (\(dx\)\(\cdot\)\(\delta\)y\(\cdot\)\(\delta\)z) containing one defect
density \(\rho\) is proportional to the defect volume fraction \(\xi\), \(\xi=f\cdot\rho\), with the scaling factor \(f\) defined as
\[f=\frac{\Omega}{\Omega_{s}}=\frac{\ell_{x}\cdot\ell_{y}\cdot\ell_{z}}{(\ell_{x}+ \ell_{s})\cdot(\ell_{y}+\ell_{s})\cdot(\ell_{z}+\ell_{s})} \tag{1}\]
Eq. indicates that \(f\) depends on defect size. As will be clear below, this is an important concept illustrating why thermal conductivity is sensitive to defect sizes.
Our distinction of two defect densities \(\alpha\) and \(\beta\) may not appear critical for uniform defect distributions. These two separate measures of defect density, however, are useful when the defect distribution is not uniform. For instance, consider the case of heat conduction through a long nanowire that is almost defect-free along its axial direction (\(\beta\approx 0\)) but is severely damaged or even completely fractured on one cross-section (\(\alpha\approx 1\)). Such a nanowire is expected to have a low thermal conductivity. On the other hand, if the same amount of defects is uniformly distributed along the axis of a long nanowire, defect densities become low in both cross-sectional and axial dimensions (\(\alpha\approx 0\), \(\beta\approx 0\)). Such a nanowire is expected to have a thermal conductivity close to that of the matrix, \(\kappa_{m}\). Clearly, the two parameters \(\alpha\) and \(\beta\) can be used to distinguish the two cases.
### Derivation of Thermal Conductivity as a Function of Defect Density, Size, and Distribution
Analytical expression can be derived for thermal conductivity of a material as a function of defect density, size, and distribution using the coarse-grained conductivity concept just introduced. The free parameters of this model are the matrix and defect volume conductivities, \(\kappa_{m}\) and \(\kappa_{d}\), and the chosen defect volume size \(d_{x}\), \(d_{y}\), and \(d_{z}\). First, the material volume \(\delta_{x}\cdot\delta_{y}\cdot\delta_{z}\) is divided along the \(x\) direction into a central rectangular prism with a cross-section area \(d_{y}\cdot d_{z}\) that contains the defect and the surrounding material that is far away from the defect, as shown in the left frame of Fig. 6(b). The apparent overall thermal conductivity of the central prism can be assumed to be \(\kappa_{d_{y}d_{z}}\), whereas that of the remaining material is \(\kappa_{m}\). Because the heat conducts through the two parts of the material in parallel (on average), the thermal conductivity of the entire material (matrix + defect) can be calculated as an area weighted average:
\[\kappa_{m+d}=\frac{d_{y}\cdot d_{z}}{\delta_{y}\cdot\delta_{z}}\cdot\kappa_{d_{y}d _{z}}+\frac{\delta_{y}\cdot\delta_{z}-d_{y}\cdot d_{z}}{\delta_{y}\delta_{z}} \cdot\kappa_{m}=\alpha\cdot\kappa_{d_{y}d_{z}}+\left(1-\alpha\right)\cdot\kappa _{m} \tag{2}\]
In Eq., \(\kappa_{d_{y}d_{z}}\) can be expanded by observing that the central prism is composed of a central section with a length \(d_{x}\) and a conductivity \(\kappa_{d}\) and two end sections with a total length \(\delta_{x}-d_{x}\) and a conductivity \(\kappa_{m}\) as shown in the right frame of Fig. 6(b). Because heat conducts through the three sections in serial, the overall thermal resistivity is calculated as a length weighted average:
\[\frac{1}{\kappa_{d_{y}d_{z}}}=\frac{d_{x}}{\delta_{x}}\cdot\frac{1}{\kappa_{d} }+\frac{\delta_{x}-d_{x}}{\delta_{x}}\cdot\frac{1}{\kappa_{m}}=\frac{\beta}{ \kappa_{d}}+\frac{1-\beta}{\kappa_{m}} \tag{3}\]
Substituting Eq. into Eq., we have:
\[\kappa_{m+d}=\kappa_{m}-\kappa_{m}\cdot\left[1-\frac{1}{\beta\cdot\left(\kappa _{m}-\kappa_{d}\right)/\kappa_{d}+1}\right]\cdot\alpha=\kappa_{m}-\kappa_{m} \cdot\frac{\left(\kappa_{m}-\kappa_{d}\right)/\kappa_{d}}{\beta\cdot\left( \kappa_{m}-\kappa_{d}\right)/\kappa_{d}+1}\cdot\rho \tag{4}\]
Eq. correctly predicts a small thermal conductivity of \(\kappa_{m+d}\approx 0\) at \(\alpha=1\), \(\beta\approx 0\), and \(\kappa_{d}=0\) (i.e., the cross-section is completely fractured), and a large thermal conductivity of \(\kappa_{m+d}\approx\kappa_{m}\) at \(\alpha\approx 0\), \(\beta\approx 0\) and \(\kappa_{d}\neq 0\).
Eq. can be simplified for common scenario where defect distribution is uniform and defect concentration is low. In such cases, \(\kappa_{d}\neq 0\) (note that the \(\kappa_{d}=0\) assumption used above implies a complete fracture of a cross-section normal to the heat flux direction). When large \(d_{x}\), \(d_{y}\) and \(d_{z}\) values are used to contain the defects, we can always approach the limits \(\kappa_{d}\approx\kappa_{m}\) and \(\beta\cdot\left(\kappa_{m}-\kappa_{d}\right)/\kappa_{d}\approx 0\). Using the relation \(1/\left(1+x\right)\approx 1-x\) for small \(x\), Eq.
\[\kappa_{m+d}\approx\kappa_{m}-\left(\kappa_{m}-\kappa_{d}\right)\cdot\alpha \cdot\beta=\kappa_{m}\left[1-\left(\kappa_{m}-\kappa_{d}\right)/\kappa_{m} \cdot\rho\right]=\kappa_{m}\left(1-\eta\cdot\xi\right) \tag{5}\]
Eq. indicates that thermal conductivity is a linear function of defect volume scattering density \(\rho\). This verifies that for independent uniform defect distributions, distinction between areal and axial defect densities is not necessary. Furthermore, Eq. also indicates that thermal conductivity is a linear function of defect volume fraction \(\xi\), for a given defect size, by way of the volume ratio \(f\), Eq.. Eq. can describe the scaling for the defect distribution shown in Fig. 1(a) if defect density is changed by the defect spacing \(\delta\). It can also be used to describe the scaling for the defect distributions shown in Figs. 1(b) and Fig. 1(c) if the defect arrays remain unchanged (i.e., \(\iota_{x}\), \(\iota_{y}\), \(\iota_{z}\) are kept constant) and the defect density is only changed by the spacing \(\delta\) between arrays. In the latter case, we are studying the scaling of arrays rather than the scaling of individual defect.
Eq. can also be derived based upon Fig. 6(c) as is described in Appendix B.
## IV Molecular dynamics verification
In this section, we verify the model by performing large scale MD simulations (400 processors or more) using the bulk configuration (periodic boundary condition in the \(y-\) direction). Two series of MD simulations are conducted to explore effects of the areal and axial defect densities respectively. For areal effects, we use the configuration shown in Fig. 1(b) where defect spacings in the \(x-\) and \(z-\) directions are small but are kept fixed whereas spacing between the \(x-z\) defect arrays in the \(y-\) direction is large and varied to change defect areal density. For the axial effects, we use the configuration shown in Fig. 1(c) where defect spacings in the \(y-\) and \(z-\) directions are small but are kept fixed whereas spacing between the \(y-z\) defect arrays in the \(x-\) direction is large and varied to change defect axial density. Because only the spacing in the direction with a sparse defect distribution is varied, this study satisfies our model assumption and can be used to test the model. The relatively small defect spacings in the other two directions enable small systems to be used to significantly reduce the computational expenses. In particular, all of our simulations used a small fixed dimension of \(n_{3}=6\) (W \(\approx 19\AA\)) in the \(z-\) direction.
### Effects of Areal Defect Density
To explore the areal defect density effect, our first series of simulations employ a fixed \(x-\) dimension (aligned with the flow of heat) of \(n_{1}=136\) (\(2L\approx 707\AA\)), and various \(y-\) dimensions between \(n_{2}=30-100\) (\(t\approx 166-553\AA\)). As a reference of the dimensions, the smallest and the largest systems contain respectively 195,840 and 652,800 atoms. A heat flux of 0.00035 \(eV/\AA^{2}\cdot ps\) is used to introduce the temperature gradient. The choice of heat flux used in the work is to ensure that temperature difference between heat source and heat sink does not exceed 10 K but is significant enough to enable converged calculations.
Four equal-spaced voids are introduced between heat source and sink (the system contains a pair of heat sources and sinks and therefore eight voids). Notice that the void spacing in the \(x-\) direction does not affect the analysis of defect density in the \(y-\) direction, nor does the particular choice of four voids affect the generality of the results. As has been well established in the past, the use of different \(y-\) dimensions above a certain threshold does not affect the thermal conductivity of a defect-free matrix with periodic boundary conditions in the lateral directions. It does affect the distance between periodic defects in the \(y-\) direction (i.e., the spacing between the \(x-z\) defect arrays) and hence the areal defect density. Clearly, the defect spacings \(\delta_{x}\), \(\delta_{y}\), and \(\delta_{z}\) relate to system dimensions as \(\delta_{x}=L/4\), \(\delta_{y}=t\), and \(\delta_{z}=W\). Since \(L\) and \(W\) are not varied (i.e., we are not exploring the scaling in the \(x-\) and \(z-\) directions), we can simply set the dimensions of the (four) scattering volumes in these directions to their maxima \(d_{x}=L/4\) and \(d_{z}=W\). A large \(d_{y}\) value is desired to contain the defect scattering region in the \(y-\) direction, but the geometry requires that \(d_{y}\leq\delta_{y}\). We can maximize the constant \(d_{y}\) by setting \(d_{y}=\delta_{y,min}=t_{min}\), where \(t_{min}\) is the minimum thickness of all the computational systems used in the series. This ensures that the geometry constraint is satisfied for all the samples. Under these conditions, the defect densities \(\alpha=t_{min}/t\), \(\beta=1\), and \(\rho=t_{min}/t\). Note that eight voids correspond to the removal of 96 atoms. For the smallest system, \(\rho=1\), and the defect volume fraction (site fraction) \(\xi=96/195840\approx 4.9\times 10^{-4}=4.9\times 10^{-2}\%\). This means that the defect volume fraction conversion factor for the series of samples is \(f=\xi/\rho=4.9\times 10^{-2}\%\).
According to Eq. or (B5), thermal conductivity is a linear function of \(\rho\). This relationship can be verified using data from direct simulations which are shown in as a function of defect scattering density \(\rho\) or defect volume fraction \(\xi\) using unfilled circles. It should be noted that the shaded area corresponds to the low defect density regime that cannot be easily calculated via MD using current computers. The thermal conductivity at \(\rho=0\), however, can be estimated using a smaller, defect-free system with periodic boundary conditions. The value thus obtained is shown in with an unfilled star to distinguish it from other data points. The solid line in is a linear function fitted to the simulated data. Note that the dashed line and unfilled diamonds show the effects of the axial defect density obtained in another series of MD simulations, to be discussed in the following section. At this point, the most important result in is that the solid trend line fitted to the defect data _predicts_ closely the defect-free value. This is a strong verification of Eq. or (B5). Also, since data at \(\xi=0\) and large \(\xi\) can both be directly calculated using MD, Eq. or (B5) can be used in an _interpolation_ to reliably predict thermal conductivities in the low defect density regime that is not directly accessible by MD simulations. This is unlike the extrapolation typically used to infer the bulk limit in the presence of size-effects.
Numerically, our calculations indicated that the defect-free thermal conductivity at a sample length of \(n_{1}=136\) (\(2L\approx 707\AA\)) is \(\kappa_{m,354}=38.56\)\(W/K\cdot m\) (see Fig. 7).
Thermal conductivity as a function of the defect density \(\rho\) or \(\xi\). Unfilled circles and solid line correspond to planar void arrays spaced \(88\times 19\) Å\({}^{2}\) in the \(x-z\) planes, and unfilled diamonds and dashed line correspond to planar void arrays spaced \(110\times 19\) Å\({}^{2}\) in the \(y-z\) planes. Spacing between arrays is varied to change defect densities. Note that the unfilled diamonds are not the direct result of MD simulations (see Section IV.2) and hence are not associated with error bars.
The void configurations discussed here are essentially planar void arrays spaced \(88\times 19\) A\({}^{2}\) in the \(x-z\) planes with spacing between arrays being varied between 166 and 553 A. For such particular defect configurations, we obtained a (non-dimensional) scaling coefficient of \(\eta=244.9\) for the effect of defect volume fraction on thermal conductivity.
### Effects of Axial Defect Density
To explore the axial defect density effect, our second simulation series employs a fixed \(y-\) dimension (perpendicular to the flow of heat) of \(n_{2}=20\) (\(t\approx 110\AA\)), and various \(x-\) dimension between \(n_{1}=152\) and 500 (\(2L\approx 790-2600\AA\)). As a reference of the dimensions, the smallest and the largest systems contain respectively 145,920 and 480,000 atoms. A heat flux of \(0.0002eV/\AA^{2}\cdot ps\) is used to introduce the temperature gradient. One void is created in the middle between the heat source and sink (two voids in the system). Unlike the areal case where the change of thermal conductivity due to the change of system thickness \(t\) in the \(y-\) dimension comes only from the change of the areal defect density \(\alpha\) (scales with \(1/t\)), the change of thermal conductivity due to the change of system length \(L\) in the \(x-\) dimension comes from both: (a) the change of the axial defect density \(\beta\) (scales with \(1/L\)) and, (b) the size-dependence of the interfacial scattering. Our scaling model can be used to derive an analytical expression of thermal conductivity as a function of both defect density and sample length, as described by Eq. (C1) in Appendix C. Based upon the relation \(\rho\propto 1/L\), Eq. (C1) indicates that thermal resistivity of defective material with a finite length \(L\) is a linear function of \(1/L\). The validity of the analytical model on the axial defect density can therefore be verified by checking this linear relationship. Furthermore, since the length scaling coefficient \(p\) (refer to Eq. C1) can be determined from a series of defect-free samples with different lengths, the effect of defects can be isolated.
Here, the simulated thermal resistivity results are shown in as a function of \(1/L\) using the unfilled diamonds. For comparison, similar data for defect-free samples obtained previously are included as the filled circles. Note that the lines are calculated from a function fitted to Eq. (C1) as will be described in the following. It can be seen that the linear relations are very well satisfied. Most importantly, the thermal conductivity data for the defective samples is seen to be lower than that of the defect-free samples, and both defective and defect-free samples approach the _same_\(\kappa_{m,\infty}\) limit at \(L\rightarrow\infty\) (also \(\rho\to 0\)). As described in Appendix A, a direct test of how good the linear relationship extends to the infinite sample length is not possible using merely defect-free samples because it is increasingly difficult to obtain accurate thermal conductivities from MD simulations when sample length is increased. However, the convergence of defective and defect-free samples to the same thermal conductivity at the infinite sample length limit is a strong verification of the linear scaling law, Eq. (C1). Eq. (C1) therefore can accurately predict thermal conductivities in the defect density and sample dimension space that cannot be directly assessable by MD.
Now, we compare the results of this section to those of the previous section which examined the influence of areal density. Here, we set \(d_{y}=\delta_{y}=t\), \(d_{z}=\delta_{z}=W\).
Thermal resistivity as a function of inverse of sample length \(1/L\) with and without the defect in the middle of sample.
\(136\times 30\times 6\) cells\({}^{2}\) system. This density is equivalent to two voids in a \(51\times 20\times 6\) cells\({}^{2}\) system. To match the previous defect volume scattering density definition, we hence choose \(d_{x}=51/2\) cells \(=132.6\) A (\(<\delta_{x,min}=L_{min}\), where \(L_{min}\) corresponds to the minimum \(x-\) dimension per defect used in the simulation series). At these values, the defect densities \(\alpha=1\), \(\beta=132.6/L\), and \(\rho=132.6/L\). Using these definitions, we fit Eq. (C1) to our data. This reproduced the bulk limit of the thermal conductivity value of \(\kappa_{m,\infty}=184.97\)\(W/K\cdot m\) obtained previously, and resulted in the determination of a length scaling coefficient of \(p=7.4248\times 10^{-10}\)\(K\cdot m^{2}/W\), and a defect volume scattering density scaling coefficient of \(q=0.0109\)\(K\cdot m/W\), see Eq. (C1). The lines shown in are calculated using these parameters. Note that in this series of simulations, the defects are essentially planar arrays of voids spaced \(110\times 19\) A\({}^{2}\) in the \(y-z\) planes with array spacing being varied between 395 and 1300 A.
Eq. (C1) also allows us to cast the thermal conductivity obtained at one sample length \(L_{1}\) to another \(L_{2}\) at a constant defect density:
\[\frac{1}{\kappa_{m+d}\left(\rho\right)|_{L_{2}}}=\frac{1}{\kappa_{m+d}\left( \rho\right)|_{L_{1}}}+p\cdot\left(\frac{1}{L_{1}}-\frac{1}{L_{2}}\right) \tag{6}\]
Using Eq., thermal conductivity vs. defect density data at a fixed sample length of \(n_{1}=136\) (\(2L\approx 707\) A) is obtained. Fitting Eq. or (B5) to such data resulted in \(\kappa_{m,345}=38.56\)\(W/K\cdot m\) and a scattering coefficient of \(\eta=918.4\) for the effect of defect volume fraction on thermal conductivity. Pertaining to the same sample length as the unfilled circles in Fig. 7, the converted data and the corresponding fitted function are shown as unfilled diamonds and dash line in It can be seen that like the areal defect density effect, reducing the axial defect density also causes the thermal conductivity to approach the value of the defect-free sample. With consideration of Appendix A which discusses the difficulties in directly testing the thermal conductivity at very large sample length, the convergence to the same defect-free sample point from two defect relations shown in is again a strong verification of the analytical model.
For non-zero defect densities, thermal conductivities obtained from the areal and axial series are different, with different coefficients \(\eta\). This means that when defects are not sparse, their scattering regions can overlap, resulting in different effects on thermal conductivities depending on how defects are distributed. For example, the configurations shown in Figs. 1(b) and 1(c) have different scaling coefficients.
## V Effect of defect spatial distribution
A series of MD simulations are performed to study the effect of defect location with respect to a surface using a film configuration with the free boundary condition in the \(y-\) direction. The system has a fixed dimension of \(n_{1}=136\) (\(2L\approx 707\) A), \(n_{2}=20\) (\(t\approx 110\AA\)), and \(n_{3}=6\) (W \(\approx 19\AA\)) for a total of 130,560 atoms. Four voids are created between the heat source and sink, resulting in a void volume fraction of \(\xi=7.35\times 10^{-2}\) %. These defects are uniformly distributed in the \(x-\) direction (aligned with the heat flow), but the position of the row of defects is varied along the \(y-\) direction so that it has different distance \(s\) from the free surface, The calculated thermal conductivities as a function of \(s\) are shown with unfilled diamonds in clearly indicates that thermal conductivity depends on the defect-surface distance. In particular, thermal conductivity increases as defects move towards the surface.
Thermal conductivity of a thin film as a function of void-surface distance predicted by MD simulations and an analytical model.
Analysis can be used to understand the results shown in Eq. or indicates that if the total scattering strength of one type of defects is \(\eta_{1}\) (aggregating the dependence on the defect volume fraction \(\xi\)), then the thermal conductivity \(\kappa_{1}\) is reduced from the matrix value \(\kappa_{0}\) through \(\kappa_{1}=\kappa_{0}\cdot\left(1-\eta_{1}\right)\). Similarly, if there is a second type of defects with the scattering strength \(\eta_{2}\), then the thermal conductivity \(\kappa_{2}\) of the material can be viewed as reduced from the new matrix value \(\kappa_{1}\) through \(\kappa_{2}=\kappa_{1}\cdot\left(1-\eta_{2}\right)=\kappa_{0}\cdot\left(1-\eta _{1}\right)\cdot\left(1-\eta_{2}\right)\). In general, we can assume a multiplication rule of \(\kappa_{N}=\kappa_{0}\cdot\Pi_{i=1}^{N}\left(1-\eta_{i}\right)\) to account for \(N\) types of defects. To relate to the MD results, a two dimensional illustration of the system is shown in Fig. 10, where voids are assumed to lie on the center line of the shaded region a distance \(s\) below the surface, and the total sample thickness is \(t\). Because the interaction between surface and defects needs to be addressed, the thermal conductivity must be considered locally as a function of position y.
It is obvious from that the local scattering strength \(\eta_{l,s}\left(d\right)\) of a surface is a decreasing function of distance \(d\) from the surface.
Illustration of defect population model.
of the surfaces (but no defects) is therefore:
\[\kappa_{l,s}\left(y\right)=\kappa_{m}\cdot\left[1-\eta_{l,s}\left(t/2-y\right) \right]\cdot\left[1-\eta_{l,s}\left(t/2+y\right)\right] \tag{8}\]
Similarly, we can assume that the local scattering strength \(\eta_{l,d}\left(d\right)\) of defects is a decreasing function of distance \(d\) from the defect plane based upon Again, we postulate that the behavior can be well captured by the function:
\[\eta_{l,d}\left(d\right)=\delta_{\kappa,d}\cdot\exp\left(-\nu\cdot d\right) \tag{9}\]
The local thermal conductivity \(\kappa_{l,d}\left(y\right)\) due to the presence of the defects (but no surface) is then
\[\kappa_{l,d}\left(y\right)=\kappa_{m}\cdot\left[1-\eta_{l,d}\left(\left|t/2-s -y\right|\right)\right] \tag{10}\]
Note that Eq. prescribes the "apparent" thermal conductivity in a thin slice parallel to the defect plane at a location of \(y\). For such a slice, Eq. correctly specifies a drop of the thermal conductivity by a fraction of \(\delta_{\kappa,d}\) at the defect plane \(y=t/2-s\), and the recover of the bulk value \(\kappa_{m}\) when \(y\) is far away from the defect plane with the parameter \(\nu\) essentially capturing the scattering distance.
In regions where both surface and defects have significant scattering, the total local thermal conductivity \(\kappa_{l}\left(y\right)\) can be written as:
\[\kappa_{l}\left(y\right)=\kappa_{m}\cdot\left[1-\eta_{l,s}\left(t/2-y\right) \right]\cdot\left[1-\eta_{l,s}\left(t/2+y\right)\right]\cdot\left[1-\eta_{l,d} \left(\left|t/2-s-y\right|\right)\right] \tag{11}\]
The apparent thermal conductivity of the system can then be found by averaging Eq. as
\[\kappa=\frac{1}{t}\int_{y=-\frac{t}{2}}^{y=\frac{t}{2}}\kappa_{l}\left(y \right)dy \tag{12}\]
To apply Eq., parameters \(\kappa_{m}\), \(\mu\), \(\nu\), \(\delta_{\kappa,s}\), and \(\delta_{\kappa,d}\) are needed. For the simulated sample length, the matrix conductivity is \(\kappa_{m}=\kappa_{m,354}=38.56\)\(W/K\cdot m\). The other parameters can be estimated from independent simulations.
\[\kappa_{s}=\frac{2}{t}\int_{y=0}^{y=t/2}\kappa_{l,s}\left(y\right)dy \tag{13}\]Here we only integrate half of the sample thickness due to the symmetry of the problem. A series of MD simulations are carried out to obtain \(\kappa_{s}\) vs. \(t\) data for defect-free thin film samples. In particular, we use a fixed \(x-\) dimension of \(n_{1}=136\) (\(2L\approx 707\AA\)) and various \(y-\) dimensions of \(n_{2}=30,50,70,96\) with free \(y-\) surfaces (\(t\approx 166-530\AA\)). In addition, we perform an additional simulation with the periodic condition in the \(y-\) direction at \(n_{2}=10\) (\(t\rightarrow\infty\)). By fitting Eq. to the \(\kappa_{s}\) vs. \(t\) results obtained from MD simulations, we find \(\delta_{\kappa,s}=0.15135\) and \(\mu=0.00705\) A\({}^{-1}\).
The average thermal conductivity of a periodic (i.e., no surfaces) system containing defects on the center line (i.e., \(s=t/2\)) is expressed as
\[\kappa_{d}=\frac{2}{t}\int_{y=0}^{y=t/2}\kappa_{l,d}\left(y\right)dy \tag{14}\]
Note that the defect scattering densities used in (unfilled circles) satisfy \(\beta=1\) and \(\rho=t_{min}/t\), where \(t_{min}=166\) A (\(n_{2}=30\)) is the minimum system thickness used in the series. Hence, the thermal conductivity vs. defect density (\(\rho\)) data shown in can be converted to the thermal conductivity vs. \(1/t\) data. By fitting Eq. to these conductivity vs. \(1/t\) data, we obtain \(\delta_{\kappa,d}=0.29017\) and \(\nu=0.02759\) A\({}^{-1}\).
Based upon the parameters thus obtained, Eq. is used to calculate thermal conductivity as a function of \(s\), and the results are included as the dots in A good agreement between the analytical prediction and the MD data is clearly shown, thereby verifying the analytical model. The mechanism of the defect spatial effects is now clear. Eq. indicates that when \(s\to 0\), the defect affected region merges with one of the surface affected regions. Consequently, the total scattering affected region is reduced from the case where defect is in the middle of the sample (\(s\to t/2\)). It is this consolidation of the defect and surface scattering that causes an increase in thermal conductivity when defects approach the surface.
## VI Discussion
Thermal conductivity as a function of void density, size, and distribution have not been experimentally measured. However, analytical expressions for thermal conductivity as a function of porosity (i.e., volume fraction \(\xi\)) have been experimentally derived for a variety of porous materials. While different forms of analytical expressions are used to enable good fit of experimental data up to a high porosity of 0.8, all expressions can be reduced accurately to a linear function \(\kappa_{m+d}=\kappa_{m}\cdot(1-\eta\cdot\xi)\) when porosity \(\xi\) is small enough (below 0.05). This is in good agreement with Eq. or (B5), which is also valid for void volume fraction far less than 0.05. However, there is a significant difference between the nanoscale void effects and the macro-scale porosity effects as the scaling parameter \(\eta\) for different porous materials falls between 1.0 and 4.6 whereas \(\eta\) is 244.9 and 918.4 for the two void configurations explored in This means that the thermal conductivity at a given void density cannot be interpolated linearly between a defect-free sample and porous materials, i.e. as void size changes.
The discussion of the spatial effects of defects presented in the preceding section indicates that thermal conductivity increases when defects move closer because their scattering regions overlap resulting in a reduction of total scattering volume. This can account for the difference between voids and pores, which can be more clearly illustrated using Eq.. When a large number of voids are closely packed to form a pore, the pore size \(\ell_{x}\), \(\ell_{y}\), and \(\ell_{z}\) become very large (\(>2\mu m\)) but the scattering dimension \(\ell_{s}\) remains small (assumed to be comparable to that in the void case). This means that the defect volume-to-scattering density conversion factor \(f\) approaches 1 for large pores. If the thermal conductivity \(\kappa_{d}\) of a pore approaches 0, \(\eta\) approaches 1 by definition. This explains why the parameter \(\eta\) for a variety of porous materials falls in the order of 1. On the other hand, if a large pore is split into a large number of voids of sizes around 5 A distributed uniformly in the material, a large number of independent scattering volumes will be created, resulting in a significant reduction of the conversion factor \(f\), a significant increase in the parameter \(\eta\), and hence a significant reduction of thermal conductivity. This accounts well with the previous MD results that vacancies cause a more significant reduction of thermal conductivity than voids given the same defect site fraction. Interestingly, experiments also indicate that at a given porosity, reducing pore sizes causes an increase in \(\eta\), which is consistent with an increase in total scattering volume. Our analytical model and MD data, therefore, are well corroborated by the experimental data on porous materials.
At the lower limit of defect size, point defects have been studied extensively and the low temperature effect of Rayleigh scattering is well-known. However, there have been relatively few experimental studies of actual vacancies - most experimental data is for isotopic substitutions. Che et al. calculated the effects of point vacancy concentrations in the range 0.01-0.16% for carbon. Unfortunately, the power-law-like empirical model they fitted to the data does not have finite derivative at a zero defect concentration. Nevertheless, it is clear that this limit of phonon scattering from voids results in a scaling coefficient significantly larger than the macroscopic porosity case.
Molecular dynamics simulations have been recently used to study thermal transport of nanoporous crystalline and amorphous silicon. The results clearly indicate that thermal conductivity depends on pore size and pore fraction. In particular, thermal conductivity was found to reduce with an increasing interfacial area concentration, which relates well to the effects of the scattering volume discussed above. Our studies, therefore, verify the previous results. In addition to the pore size and pore fraction effects, we further show that thermal conductivity is sensitive to defect population (for example, areal and axial populations have different effects).
## VII Conclusions
An integrated approach combining a physically-motivated analytical model, large scale MD simulations, and extensive experimental thermal conductivity data of porous materials is used to study defect effects on thermal conductivity. Corroborated results lead to an explicit functional expression of thermal conductivity on defect density, size, and spatial population. The following conclusions are of particular interests to both theoretical thermal transport studies and phonon engineering of materials:
* Thermal conductivity depends strongly on total scattering volume of defects. This scattering volume differs from the physical volume of defects. It, however, can be linearly correlated with physical volume of defects when defect type and size are fixed. It is the defect configuration that minimizes this scattering volume but not the physical volume that will increase thermal conductivity.
* When defects are close, their scattering regions overlap, resulting in reduced total scattering volume. As a result, thermal conductivity increases when voids move towards surfaces, or small voids collapse to form large pores.
* For uniform, sparse defect distribution with a given defect size, thermal conductivity is a linear function of defect volume fraction. However, thermal conductivity at a given void density cannot be interpolated between defect-free samples and macroscopically porous materials due to the difference in defect sizes. In general, the dependence of thermal conductivity on defects is not simply through volume fraction, but strongly depends on the size distribution of the voids.
* The analytical model enables thermal conductivities obtained from molecular dynamics simulations to be extrapolated/interpolated reliably to realistic defect density ranges, as well as the defect-free limit.
Finally, we point out that the success of our approach can be related to the explicit incorporation of scattering of short wavelength and short mean free path phonons in direct molecular simulations. These modes are scattered the most given the general behavior of Rayleigh scattering and therefore their contribution to the overall conductivity is the most sensitive to changes in defect density. On the other hand, the longer wavelength modes are less affected by point-like defects and their contribution to the thermal conductivity is given by the scaling analysis that estimates the long sample length limit.
| 10.48550/arXiv.1207.3144 | Effects of nano-void density, size, and spatial population on thermal conductivity: a case study of GaN crystal | Xiaowang Zhou, Reese E. Jones | 2,399 |
10.48550_arXiv.1803.06302 | ###### Abstract
We bring forth a consistent theory for the electron-mediated vibrational intermode coupling that clarifies the microscopic mechanism behind the vibrational relaxation of adsorbates on metal surfaces. Our analysis points out the inability of state-of-the-art nonadiabatic theories to quantitatively reproduce the experimental linewidth of the CO internal stretch mode on Cu and it emphasizes the crucial role of the electron-mediated phonon-phonon coupling in this regard. The results demonstrate a strong electron-mediated coupling between the internal stretch and low-energy CO modes, but also a significant role of surface motion. Our nonadiabatic theory is also able to explain the temperature dependence of the internal stretch phonon linewidth, thus far considered a sign of the direct anharmonic coupling.
_Introduction.-_ Most of the theories describing dynamical processes at surfaces rely on the validity of the adiabatic Born-Oppenheimer approximation. However, in the case of metal surfaces, high concentrations of conducting electrons can, in principle, exchange energy with the adsorbate nuclear degrees of freedom. In fact, there is growing experimental evidence that points out the existence of such nonadiabatic effects. Among them, the significant vibrational linewidths of molecules adsorbed on metal surfaces, which are reported in infrared absorption or pump-probe spectroscopy experiments, are considered to be clear fingerprints of the electron-mediated vibrational relaxation. Besides, the nonadiabatic coupling underlies a variety of surface reactions and adsorbate motions (e.g., vibrations, rotations, and lateral hopping) induced by means of inelastic electron currents and femtosecond laser pulses.
Apart from nonadiabatic coupling, a key ingredient for depicting the aforesaid processes is the intermode vibrational coupling. For example, hot-electron induced nonequilibrium dynamics of CO on Cu, which is studied by time-resolved vibrational sum-frequency generation, reveals the importance of intermode coupling in the CO desorption process. Furthermore, the temperature dependence of the CO internal stretch (IS) mode linewidth studied by infrared spectroscopy was explained in terms of the coupling with the thermally-excited low-energy (LE) CO vibrational modes.
The ensuing theoretical efforts aimed to comprehend these experimental observations are remarkable. In particular, the nonadiabatic relaxation of vibrationally excited adsorbates is most commonly studied either by relaxation-rate calculations based on first-order perturbation theory or by performing molecular dynamics with the corresponding electronic friction. In the former case, the intermode coupling is usually tackled by including an additional damping rate due to direct anharmonic coupling. Despite the valuable qualitative insight gained, these theories are still unable to give precise quantitative estimations of the experimental vibrational relaxation rates and, hence, neither can they clarify which relevant mechanisms are ruling them. An emblematic example is the IS mode of CO on Cu. Even if its relaxation is considered to be mostly nonadiabatic, the corresponding vibrational linewidths calculated recently with _ab initio_ first-order perturbation theory are significantly lower than the experimental values. This discrepancy together with the indisputable evidence for the intermode coupling in CO/Cu points out to an overlooked relaxation mechanism that should incorporate both the nonadiabatic and intermode couplings on the same footing.
In this Letter we demonstrate how nonadiabatic coupling can naturally account for intermode coupling if the former is treated up to second-order in the electron-phonon interaction. This theory that includes the process conjoining nonadiabaticity and intermode transitions - known as electron-mediated phonon-phonon (EMPP) coupling- can correctly describe the mechanisms behind the vibrational relaxation of ordered molecules on metal surfaces. Using the prototypical IS mode of CO on Cu, we show that the present phonon linewidth formula combined with first-principles methodologies is finally able to explain the experimental relaxation rates. Importantly, our results show that the EMPP process dominates over the commonly-used first-order nonadiabatic contribution. Specifically, IS couples strongly via electron-hole (_e-h_) pairs to other LE molecular phononmodes, i.e., to the frustrated rotation (FR) and frustrated translation (FT) modes [see Fig. 1(a)]. Even more surprising, surface motion also plays an important role in the electron-mediated vibrational relaxation. Finally, as another remarkable success, our theory explains the temperature dependence of the IS mode linewidth. This result proves that the temperature dependence is not only triggered by the usual direct phonon-phonon coupling and that the commonly overlooked electron-mediated processes are also a significant factor in this respect.
_State-of-the-art theory.-_ Thus far the vibrational relaxation of molecules at metal surfaces was studied mostly by means of the (first-order) Fermi's golden rule formula (i.e., by only considering the terms \(\propto g^{2}\), where \(g\) is the electron-phonon matrix element). An analogous pathway in many-body perturbation theory (MBPT) is to calculate the phonon self-energy due to electron-phonon coupling up to first order, \(\pi_{\lambda}^{}(\mathbf{q},\omega)\) (\(\lambda\), \(\mathbf{q}\), and \(\omega\) are the index, momentum, and energy of the phonon mode, respectively). The corresponding phonon linewidth is obtained by taking the imaginary part of \(\pi_{\lambda}^{}\), i.e., \(\gamma_{\mathbf{q}\lambda}^{}=-2\mathrm{Im}\,\pi_{\lambda}^{}(\mathbf{q },\omega_{\mathbf{q}\lambda})\). In infrared spectroscopy, light is directly exciting only the long-wavelength phonons (i.e., \(\mathbf{q}\approx 0\)).
\[\pi_{\lambda}^{}(\omega)=\sum_{\mu\mu^{\prime}\mathbf{k}}\left|g_{\lambda}^ {\mu\mu^{\prime}}(\mathbf{k},0)\right|^{2}\frac{f(\varepsilon_{\mu\mathbf{k}}) -f(\varepsilon_{\mu^{\prime}\mathbf{k}})}{\omega+\varepsilon_{\mu\mathbf{k}}- \varepsilon_{\mu^{\prime}\mathbf{k}}+i\eta}, \tag{1}\]
The Fermi-Dirac distribution function is defined as \(f(\varepsilon_{\mu\mathbf{k}})=1/(e^{\beta(\varepsilon_{\mu\mathbf{k}}- \varepsilon_{F})}+1)\), where \(\beta=1/(k_{B}T)\), \(k_{B}\) is the Boltzmann constant, \(T\) is the temperature and \(\varepsilon_{F}\) is the Fermi energy. In the limit \(\eta\to 0^{+}\) one obtains the exact first-order phonon linewidth formula, in which the intraband part (\(\mu=\mu^{\prime}\)) of Eq. vanishes and the only remaining contribution comes from the direct interband excitations (\(\mu\neq\mu^{\prime}\)).
Early parameter-dependent calculations of the IS linewidth based on the Fermi's golden rule reported good agreement with experiments. However, here we show that Eq. is not enough to explain the experimental linewidths. In fact, our _ab initio_ calculations of Eq. for \(\mathrm{c}(2\times 2)\)-\(\mathrm{CO}/\mathrm{Cu}\) based on accurate density functional perturbation theory give \(\gamma_{0\lambda}^{}=26.7\,\mathrm{GHz}\) (see Ref. for computational details), which is far from the experimental values \(\gamma_{\mathrm{exp.}}>50\,\mathrm{GHz}\) [see Fig. 2(a)]. The discrepancy between the phonon linewidth obtained from Eq. and in experiments has also been discussed recently. Note in passing that our value \(\gamma_{0\lambda}^{}\) is in almost perfect agreement with the result obtained within a mixed quantum-classical theory when an infinite electron coherence time \(\tau_{e}\) is used.
As shown in Ref., the agreement with the experimental linewidth can be improved by taking into account interband electronic scattering processes (i.e., electron-impurity, electron-phonon, or electron-electron scattering). Phenomenologically, this can be done by replacing the infinitesimal \(\eta\) in Eq. with a finite broadening \(\Gamma\). By using a physically motivated \(\Gamma=60\,\mathrm{meV}\), we get \(\gamma_{0\lambda}^{}=40\,\mathrm{GHz}\) [second column in Fig. 2(a)]. However, to achieve agreement with experiments one needs to use non-physically large \(\Gamma\) values, i.e., much larger than 100 meV. In a similar way, the agreement with experiments in Ref. is improved by using a finite \(\tau_{e}\) value (note that \(\tau_{e}=2\pi\hbar/\Gamma\)).
For completeness, we have additionally performed _ab initio_ molecular dynamics simulations with electronic friction (AIMDEF) as in Refs. to calculate the energy relaxation of the IS mode. Position-dependent electronic friction coefficients are calculated within the local density friction and independent atom approximations (see Ref. for computational details). By restricting the CO dynamics along the surface normal on the rigid surface, a value \(\gamma_{\mathrm{AIMDEF}}=18.4\,\mathrm{GHz}\) is obtained, which is in quite fair agreement with the result obtained from Eq., and, hence, it also fails to reproduce the experimental linewidths.
All the aforesaid results highlight that additional channels must be behind the large experimental IS relaxation rate. In particular, the non-negligible temperature dependence of the IS mode linewidth reported in infrared spectroscopy and the recent experiments on CO hot-electron induced nonequilibrium dynamics suggest that a key role is played by intermode transitions, for which a more rigorous study is required.
_Electron-mediated phonon-phonon coupling.-_ The usual procedure for treating intermode transitions is to consider the direct anharmonic coupling between two phonon modes. However, for the high-energy (HE) IS mode (\(\omega_{\mathrm{exp.}}=62.54\,\mathrm{THz}=256\,\mathrm{meV}\)), direct anharmonic coupling with the LE phonon modes, such as FR (\(\omega_{\mathrm{exp.}}=8.54\,\mathrm{THz}=35.32\,\mathrm{meV}\)), is very inefficient due to the large difference in energy.
(a) Schematic of the EMPP coupling process of the IS mode of CO on Cu. The IS mode (red) couples to low-energy FR and FT modes, as well as to the modes consisting of the joint motion of CO and Cu surface atoms (blue). (b) Indirect phonon-assisted electronic transitions involved in the EMPP process at \(T=0\,\mathrm{K}\) (phonon emission) and \(T>0\,\mathrm{K}\) (phonon absorption). In each case, both intraband and interband scattering are allowed.
In other words, the HE mode can excite _e-h_ pairs which in turn can undergo further electron-phonon scattering with the LE modes [see Fig. 1(a)].
\[\pi_{\lambda}^{}(\omega)=-\sum_{\mu\mu^{\prime}\mathbf{k}\mathbf{ \lambda^{\prime}}\mathbf{k^{\prime}}}|g_{\lambda}^{\mu\mu}(\mathbf{k},0)|^{2} \left[1-\frac{g_{\lambda}^{\mu^{\prime}\mu^{\prime}}(\mathbf{k^{\prime}},0)}{g_ {\mu\mu}^{\mu\mu}(\mathbf{k},0)}\right]\] \[\times\left|g_{\lambda^{\prime}}^{\mu\mu^{\prime}}(\mathbf{k}, \mathbf{q^{\prime}})\right|^{2}\sum_{s,s^{\prime}=\pm 1}\frac{f(\varepsilon_{\mu \mathbf{k}})-f(\varepsilon_{\mu^{\prime}\mathbf{k^{\prime}}}-s^{\prime}s\omega _{\mathbf{q^{\prime}}\lambda^{\prime}})}{\varepsilon_{\mu\mathbf{k}}-( \varepsilon_{\mu^{\prime}\mathbf{k^{\prime}}}-s^{\prime}s\omega_{\mathbf{q^{ \prime}}\lambda^{\prime}})}\] \[\times\frac{s\left[n_{b}(s\omega_{\mathbf{q^{\prime}}\lambda^{ \prime}})+f(s^{\prime}\varepsilon_{\mu^{\prime}\mathbf{k^{\prime}}})\right]}{ \omega\left[\omega+i\eta+s^{\prime}(\varepsilon_{\mu\mathbf{k}}-\varepsilon_{ \mu^{\prime}\mathbf{k^{\prime}}})+s\omega_{\mathbf{q^{\prime}}\lambda^{\prime }}\right]}, \tag{2}\]
As noted before, in infrared spectroscopy the IS mode is not able to excite direct intraband transitions due to the vanishingly small phonon momentum. Thus, the first non-vanishing intraband contribution is Eq., which breaks down the momentum conservation by coupling the studied (\(\mathbf{q}\approx 0,\lambda\)) mode with other (\(\mathbf{q^{\prime}},\lambda^{\prime}\)) phonon modes via _e-h_ pairs [see Fig. 1(b)]. Such indirect phonon-assisted electronic transitions involved in this process are considered to be an important mechanism in hot-carrier generation via plasmon decay in Cu and other noble metals.
With very few exceptions these kinds of electron-mediated, second-order processes have not been considered in the context of vibrating molecules at metal surfaces. In these few references, effects due to _e-h_ pair dephasing and direct anharmonic coupling followed by _e-h_ pair excitation and vice versa were investigated with parametrized models that only accounted for quasielastic electron scattering and that were fitted to match the experimental data. In contrast to these studies, here we include and calculate from first principles both the elastic and inelastic (i.e., fully dynamical) electron-phonon scattering.
Table 1 (last row) shows the results for the EMPP coupling term of the IS phonon linewidth [\(\gamma_{0\lambda}^{}=-2\mathrm{Im}\,\pi_{\lambda}^{}(\omega_{0\lambda})\)] decomposed into contributions coming from different CO/Cu phonon modes: IS (\(\mathbf{q^{\prime}}>0\)), CO-Cu or external stretch (ES), FR, FT, as well as modes consisting of the joint motion of CO and Cu atoms (CO/Cu). We also provide the calculated phonon mode energies and compare them to available experimental data. As Table 1 shows, a significant damping of the IS mode is caused by the electron-mediated coupling with the FR modes within the molecular overlayer. On the other hand, the smallest contribution comes from the ES and the IS (\(\mathbf{q^{\prime}}>0\)) modes. These results are actually in line with existing femtosecond laser experiments showing the importance of the IS-FR mode coupling in the desorption of CO from Cu and Ru. Similarly, the coupling between the IS and LE modes are thought to promote the surface hopping of CO on Pd induced by inelastic tunneling electrons and of CO on Pt induced by a laser pulse. Hence, also in these cases our theory of EMPP coupling can be useful in elucidating what modes are actually involved in these kinds of surface reactions. Another important conclusion extracted from Table 1 is that a considerable contribution to the IS phonon linewidth comes from the CO/Cu modes. This result contradicts the usual assumption that considers the coupling between the adsorbate IS mode and the modes involving surface motion irrelevant for the IS relaxation process, due to the large energy mismatch. Here we show that such coupling is indeed possible whenever the continuum of surface conducting electrons is present to compensate the energy gap.
Next, we summarize in Fig. 2(a) all the contributions to the phonon linewidth and compare them to the experimental findings. Blue and brown dashed lines represent the total experimental phonon linewidths obtained in infrared absorption spectroscopy and infrared pump-probe spectroscopy, respectively. The latter technique makes it possible to extract the energy relaxation contribution (i.e., inelastic process, red dashed line) from the total linewidth. The remaining contributions to the total phonon linewidth are elastic processes (i.e., dephasing via _e-h_ pairs and anharmonic coupling) and inhomogeneities (e.g., impurities and disorders).
(a) Different contributions to the IS phonon linewidth \(\gamma_{0\lambda}\) coming from direct interband transitions (dark blue), interband transitions with phenomenological broadening \(\Gamma=60\,\)meV (light blue), and the EMPP coupling at \(T=10\,\)K (orange). The experimental total linewidths obtained in Refs. and are shown with blue and brown dashed lines, respectively. The inelastic contribution obtained in Ref. is shown with a red dashed line. Gray shaded area represents all the other experimental values, including experimental error bars. (b) Temperature dependence of the IS phonon linewidth as obtained from the EMPP coupling term (blue dots) and the corresponding experimental values (orange diamonds).
In our theoretical framework, the interband contribution [Eq.] is a purely inelastic process, while the EMPP coupling term [Eq.] contains both inelastic (i.e., \(\varepsilon_{\mu\mathbf{k}}\neq\varepsilon_{\mu^{\prime}\mathbf{k}^{\prime}}\)) and elastic (i.e., \(\varepsilon_{\mu\mathbf{k}}\approx\varepsilon_{\mu^{\prime}\mathbf{k}^{\prime}}\) as in _e-h_ pair dephasing) processes and both are together included in our calculations. All in all, the EMPP coupling term greatly improves upon the state-of-the-art theory, and is essential for understanding the experimental linewidths. Nevertheless, further work is desirable in order to reach the ultimate accuracy in the calculation of the phonon linewidth. Specifically, inclusion of inhomogeneities (e.g., electron-impurity scattering) and electron correlation effects might improve the overall result.
_Temperature dependence.-_ The analysis of temperature effects in the CO IS mode also confirms the relevance of the EMPP coupling mechanism. Temperature enters the usual (first-order) electron-phonon term [Eq.] only through the Fermi-Dirac distribution functions. In other words, the temperature is skewing the electron distribution and thus changing the rate of upward and downward electron transitions. However, the overall sum of these transitions [i.e., \(f(\varepsilon_{\mu\mathbf{k}})-f(\varepsilon_{\mu^{\prime}\mathbf{k}})\)] is practically \(T\)-independent at moderate temperatures (\(k_{B}T\ll\omega_{0\lambda}\)). In our case, the linewidth of the CO IS mode coming from this term varies less than 1 GHz for a temperature range of 40-300 K, which is also in accordance with recent calculations. In contrast, temperature effects coming from the EMPP coupling [Eq.], which enter the phonon linewidth through \(n_{b}(\omega_{\mathbf{q}^{\prime}\lambda^{\prime}})\), are significantly more pronounced, as also suggested in studies of optical phonon modes in metals. Figure 2(b) shows the results for the \(T\)-dependence of the IS phonon linewidth coming from the EMPP coupling [the zero-temperature value is \(\gamma_{0\lambda}^{T=0}=87.1\) GHz as in Fig. 2(a)]. We observe that \(\gamma_{0\lambda}^{}\) increases as \(T^{3}\) at low temperatures and linearly with \(T\) at higher temperatures. The overall increase in the range \(T=0-160\) K is around 22 GHz. Such a result remarks that these low temperatures are enough to significantly enhance the population of the excited LE phonons and to thus cause an increase in the probability of electron-LE phonon scattering events. Table 1 shows that the main contributors to the increase of the linewidth with temperature are the lowest-energy modes, i.e., CO/Cu and FT (see Ref. for more details).
In Ref. the \(T\)-dependence of the IS phonon linewidth is measured between 100-160 K using infrared absorption spectroscopy (orange diamonds). The linear increase we obtain in that temperature range is in both qualitative and quantitative agreement with their findings. Usually, the \(T\)-dependence of the adsorbate HE modes is considered to be the footprint of an underlying pure dephasing mechanism, either coming from elastic scattering with LE modes (direct elastic anharmonic effect) or with _e-h_ pairs. The parametrized model of Ref. assigns the \(T\)-dependence to the former process and, then, the model parameters are adjusted to fit the experimental data. However, our results clearly show that it is the EMPP coupling the mechanism governing the \(T\)-dependence. Note that apart from inelastic nonadiabatic effects, this mechanism also includes the aforementioned _e-h_ pair dephasing when the studied (\(\mathbf{q}\approx 0,\lambda\)) mode is coupled to the different \(\mathbf{q}^{\prime}\) modes within the same \(\lambda\) phonon band.
As a final remark, note that the dynamical models aimed to understand the experiments on CO desorption and surface hopping very often require including a phenomenological \(T\)-dependent relaxation-rate term (i.e., friction coefficient), even if the microscopic justification for such \(T\)-dependent friction is not clear enough. In searching a plausible explanation, Ueba and Persson constructed a parameter-based model in which an effective \(T\)-dependent friction, consisting of \(T\)-independent nonadiabatic coupling followed by direct anharmonic coupling, enters the heat transfer equation. In this respect, our theory of EMPP coupling follows a similar line of reasoning. However, only this kind of theory that is entirely based on first principles and rigorous MBPT can finally provide new and detailed quantitative information on these \(T\)-dependent processes. In fact, we show that nonadiabaticity and anharmonic coupling are inseparable parts of the very same process if the electron-phonon coupling is treated consistently up to second order.
_Conclusion.-_ By combining many-body perturbation theory with density functional theory, we have investigated a new relaxation mechanism that bridges the nona
\begin{table}
\begin{tabular}{c c c c c c} & \multicolumn{5}{c}{\(\lambda^{\prime}\) mode (\(\lambda=\) IS)} \\ & IS & ES & FR & FT & CO/Cu \\ \hline \(\omega_{\mathbf{q}^{\prime}\lambda^{\prime}}\) [THz] & 58.70 - 61.37 & 12.11 - 12.33 & 9.14 - 9.78 & 5.79 - 6.33 & \(<7\) \\ \(\omega_{0\lambda^{\prime}}\) [THz] & 61.37 & 12.33 & 9.78 & 6.05 & \(<7\) \\ Exp. \(\omega_{0\lambda^{\prime}}\) [THz] & 62.54 & 10.34 & 8.54 & 0.96 & \(\lesssim 7\) \\ \(\gamma_{0\lambda}^{}\) [GHz] & 0.48 (1.04) & 1.46 (1.56) & 15.58 (17.73) & 8.61 (12.60) & 20.96 (36.20) \\ \end{tabular}
\end{table}
Table 1: Energy range (THz) of the \(\mathrm{c}(2\times 2)\)-CO/Cu phonon modes in the 1st Brillouin zone (\(\omega_{\mathbf{q}^{\prime}\lambda^{\prime}}\)), phonon energies for \(\mathbf{q}\approx 0\) (\(\omega_{0\lambda^{\prime}}\)), and the corresponding experimental values. The last row shows contributions to the EMPP coupling term of the IS phonon linewidth \(\gamma_{0\lambda}^{}\) (GHz) coming from different \(\lambda^{\prime}\) modes at \(T=10\) K (\(T=160\) K).
We have shown that the state-of-the-art nonadiabatic theory is not able to explain the internal stretch mode phonon linewidth obtained in infrared absorption spectroscopy and that inclusion of the electron-mediated phonon-phonon coupling is essential in this regard. The latter mechanism reveals a significant role of the low-energy modes in the relaxation process, such as CO frustrated rotation, but also the modes consisting of the joint motion of CO and surface atoms. Importantly, this new nonadiabatic mechanism is able to explain the temperature dependence of the internal stretch mode linewidth, which was hitherto ascribed to the direct anharmonic coupling or anharmonic dephasing processes. The proposed mechanism is quite general and can be used to elucidate which modes are involved in those surface reactions in which nonadiabatic effects and intermode coupling are expected to be relevant.
| 10.48550/arXiv.1803.06302 | Electron-mediated phonon-phonon coupling drives the vibrational relaxation of CO on Cu(100) | Dino Novko, Maite Alducin, Joseba Iñaki Juaristi | 1,002 |
10.48550_arXiv.1909.06574 | ###### Abstract
We propose a mathematical description of crystal structure: underlying translational periodicity together with the distinct atomic positions up to the symmetry operations in the unit cell. It is consistent with the international table of crystallography. By the Cauchy-Born hypothesis, such a description can be integrated with the theory of continuum mechanics to calculate a _derived crystal structure_ produced by solid-solid phase transformation. In addition, we generalize the expressions for orientation relationship between the parent lattice and the derived lattice. The derived structure rationalizes the lattice parameters and the general equivalent atomic positions that assist the indexing process of X-ray diffraction analysis for low symmetry martensitic materials undergoing phase transformation. The analysis is demonstrated in a CuAlMn shape memory alloy. From its austenite phase (L\(2_{1}\) face-centered cubic structure), we identify that the derived martensitic structure has the orthorhombic symmetry P\(mmm\) with derived lattice parameters \(a_{\rm dv}=4.36491\AA\), \(b_{\rm dv}=5.40865\AA\) and \(c_{\rm dv}=4.2402\AA\), by which the complicated X-ray Laue diffraction pattern can be well indexed, and the orientation relationship can be verified.
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Footnote †: journal: Journal of LaTeX Templates
## 1 Introduction
Materials undergoing reversible martensitic transformations show great potentials in many emerging applications such as biomedical implants, nano/microactuators and solid state caloric coolings. The underlying functionality of using these materials is the ability to recover a large macroscopic deformation (i.e. 5% - 10%) during the reversible structural transformation. Many applications require these materials to run millions of transformation cycles, their functionality typically degrades quickly in the first couple of hundreds of cycles, even for the most successful commercial alloy - Nitinol. Recent significant advances in developing ultra-low fatigue martensitic materials show that the design of phase-transforming materials can be guided through some kinematic compatibility principles called cofactor conditions, i.e. thesuper compatibility conditions for the existence of stressed-free microstructure during phase transformation. When the cofactor conditions are satisfied, the thermal hysteresis is minimized without compromising the amount of latent heat. Meanwhile the thermomechanical response does not degrade at all even upon tens of million mechanically induced transformation cycles. These discoveries underlie a theory-driven design strategy for phase transforming materials, of which the most crucial step is to precisely determine the crystal structures of the transforming material as austenite (high symmetry structure in high temperature phase) and martensite (low symmetry structure in low temperature phase). However, this step is non-trivial, and often quite tedious.
In principle, the structural parameters of a crystalline solid are determined by X-ray diffraction (XRD) experiments. One of the most common XRD measurement used for structural determination is Rietveld refinement of powder diffraction data obtained with either CuK\(\alpha\) or MoK\(\alpha\) radiation. The testing specimen should be either in powder or bulk form with sufficient randomization of grain orientations illuminated by the monochromatic X-ray beam. However, the as-cast metallic specimen after proper heat treatment produced in laboratory is mostly in bulk form with coarse grain size. For example, the grain size of common Cu-based \(\beta\) alloys is about 200 - 500 \(\mu\)m. The orientation randomness of the specimen from the lab production is insufficient for ordinary XRD powder method, especially for the low symmetry structures. In most cases, the crystal structure of the developed material is unknown, which makes the Rietveld analysis impossible. Structure solution through single crystal x-ray diffraction requires isolated good quality single crystals and is therefore hardly applicable to the bulk samples. The lack of structural knowledge for low symmetry metallic materials highly hinders the material development for desirable properties. Therefore, it is very important to have a unified way for structural determination.
For martensitic materials, most are inherited from a high symmetry phase, austenite, through solid-solid phase transformation. The formation of martensite microstructure is strongly restricted by the crystallographic compatibility to the austenite structure. Therefore, the nature of the phase transformation and the orientation relationships between austenite and martensite can be used to propose a universal structural determination method for the martensite crystal structure transformed from the cubic austenite. The cubic structures, including simple cubic, face-centered cubic and body-centered cubic, have only one structural dimensional parameter, \(a_{0}\) that can be accurately determined by ordinary XRD experiments, from which we can derive the primitive lattice metric - the underlying 3-dimensional periodicities of the martensite lattice - through the crystallographic relationships with the austenite phase and the Cauchy-Born rule for solid-solid phase transformation. Such a derivation does not require any pre-knowledge of the crystal symmetry or lattice parameters. The derived lattice metric can be slightly perturbed from the reference lattice and used to present the unknown martensite structure for advanced structural characterization and analysis such as XRD, EBSD and so on. In this paper, we lay down the fundamental formulation for the derived lattice from a cubic structure through the solid-solid phase transformation. We then used synchrotron Laue x-ray microdiffraction experiment combined with energy scans to demonstrate our method for an unknown Cu-based\(\beta\) alloy. To bridging the discrepancy between the previous mathematical description of lattice and the symmetry calculation of crystal structures by the international union of crystallography, we propose a modified description for the _crystal structure_ consisting of two parts: underlying translational periodicity and the fractional atomic positions in the unit cell in consistent with the general equivalent positions used in the international table of crystallography. Consider the parent and child phases as two discrete vector spaces mapped by a homogeneous linear transformation, the orientation relationship can be expressed by the _lattice correspondence matrix_, by which specific lattice vector/plane parallelisms can be derived.
## 2 Derived lattice from solid-solid phase transformation
### Mathematical representation of lattice and crystal structure
For any three-dimensional Bravais lattice:
\[\mathcal{L}(\mathbf{E})=\{\mathbf{E}\mathbf{n}:\forall\mathbf{n}\in\mathbb{Z}^ {3}\}, \tag{1}\]
Its three linearly independent column vectors are the _lattice vectors_. The symmetry of a Bravais lattice \(\mathcal{L}(\mathbf{E})\) is represented by an orthogonal tensor \(\mathbf{R}\in O\) such that :
\[\mathbf{RE}=\mathbf{E}\mathbf{L} \tag{2}\]
All symmetry operations of the Bravais lattice \(\mathcal{L}(\mathbf{E})\) form a finite group \(\mathcal{P}\), defined as the point group of \(\mathcal{L}(\mathbf{E})\). To express the lattice parameters, we introduce the _lattice metric_ tensor:
\[\mathbf{C}=\mathbf{E}^{T}\mathbf{E}\text{ for a Bravais lattice }\mathcal{L}(\mathbf{E}). \tag{3}\]
The lattice metric tensor is always positive definite and symmetric. The lattice parameters of \(\mathcal{L}(\mathbf{E})\) are a sextuplet \(\mathbf{p}=(p_{1},p_{2},p_{3},p_{4},p_{5},p_{6})\) depending on the lattice metric tensor. \(p_{i}=\sqrt{\mathbf{C}_{ii}}\) (no sum) for \(i=1,2,3\), and \(p_{4},p_{5},p_{6}\in[-1,1]\) satisfy:
\[p_{4}=\frac{\mathbf{C}_{2,3}}{p_{2}p_{3}},\ p_{5}=\frac{\mathbf{C}_{1,3}}{p_{1 }p_{3}},p_{6}=\frac{\mathbf{C}_{1,2}}{p_{1}p_{2}}. \tag{4}\]
One can consider the underlying periodicity of a lattice by either the lattice metric tensor or the lattice parameters, which are both invariant under symmetry operations and rigid body rotations. Table 1 lists the expressions of lattice parameters for all 14 Bravais lattices in 3-dimension written in their primitive basis, i.e. the basis underlies the smallest unit cell defined as \(\mathcal{U}(\mathbf{E})=\{\mathbf{E}\lambda:\forall\ (\lambda\cdot\lambda)\in[0,1)\}\), which only consists of one lattice point. However, crystallography theory does not deal with the primitive lattice basis, because it is not always orthogonal for all types of Bravais lattices. To facilitate the crystallographic calculations, X-ray crystallography uses the _conventional basis_ in their formula and equations. There are 7 out of 14 Bravais lattices in Table 1: fcc, bcc, bct, bco, fco, ico and bcm, whose primitive basis is not consistent with their conventional basis. Therefore, the expression of lattice parameters in primitive basis written in Table 1 for these Bravais lattices is different from what we usually use for crystallographic calculation.
The conventional lattice described by the conventional basis is a sublattice of the original Bravais lattice. A sublattice can be considered as a multilattice defined as:
\[\mathcal{M}(\mathbf{E};\mathbf{w}_{i})=\{\mathbf{E}(\mathbf{n}+\mathbf{w}_{i}): \forall\mathbf{n}\in\mathbb{Z}^{3},\text{ for some }\mathbf{w}_{i}\in\mathbb{R}^{3}\text{ with }|\mathbf{w}_{i}|\in[0,1),i=1,2,...,m\}. \tag{5}\]
Here, the basis \(\mathbf{E}\) is not necessarily the primitive basis. By the definition of, there exist \(m\) lattice points in the unit cell \(\mathcal{U}(\mathbf{E})\). The fractional vectors \(\mathbf{w}_{i},i=1,...,m\) can be interpreted as \(m\) Bravais lattices that are displaced by \(\mathbf{E}\mathbf{w}_{i}\) respective to each other. For example, we can choose \(a_{0}\mathbf{I}\) as the basis of a multilattice to express the face-centered cubic unit cell:
\[\mathcal{M}_{\text{fcc}}(a_{0}\mathbf{I};\mathbf{w}_{1},\mathbf{w}_{2},\mathbf{w}_{3}, \mathbf{w}_{4}),\text{ for }\mathbf{w}_{1}=\begin{bmatrix}0\\ 0\\ 0\end{bmatrix},\text{ }\mathbf{w}_{2}=\begin{bmatrix}\frac{1}{2}\\ \frac{1}{2}\\ 0\end{bmatrix},\text{ }\mathbf{w}_{3}=\begin{bmatrix}0\\ \frac{1}{2}\\ \frac{1}{2}\end{bmatrix},\text{ }\mathbf{w}_{4}=\begin{bmatrix}\frac{1}{2}\\ 0\\ \frac{1}{2}\end{bmatrix}. \tag{6}\]
It describes the same periodicity as what \(\mathcal{L}(\mathbf{E})\) does where \(\mathbf{E}\) is the primitive lattice basis given by
\[\mathbf{E}=\frac{a_{0}}{2}\begin{bmatrix}1&0&1\\ 1&1&0\\ 0&1&1\end{bmatrix}=a_{0}\mathbf{I}\chi,\text{ where }\chi=(\mathbf{w}_{2}, \mathbf{w}_{3},\mathbf{w}_{4}). \tag{7}\]
\begin{table}
\begin{tabular}{c|c|c|c} & Bravais lattice & Lattice parameters & Order of the point group1 \\ \hline
1 & simple cubic (sc) & \(a_{0}\) & 24 \\ \hline
2 & face-centered cubic (fcc) & \(\left(\frac{a_{0}}{\sqrt{2}},\frac{a_{0}}{\sqrt{2}},\frac{a_{0}}{\sqrt{2}}, \frac{1}{2},\frac{1}{2},\frac{1}{2}\right)\) & 24 \\ \hline
3 & body-centered cubic (bcc) & \(\left(\frac{\sqrt{3}a_{0}}{2},\frac{\sqrt{3}a_{0}}{2},\frac{\sqrt{3}a_{0}}{2}, \frac{1}{3},\frac{1}{3},-\frac{1}{3}\right)\) & 24 \\ \hline
4 & hexagonal & \(\left(a,a,c,0,0,\frac{1}{2}\right)\) & 12 \\ \hline
5 & trigonal & \(\left(a,a,a,\cos\alpha,\cos\alpha,\cos\alpha\right)\) & 6 \\ \hline
6 & tetragonal & \(\left(a,a,c,0,0,0\right)\) & 8 \\ \hline
7 & body-centered tetragonal (bct) & \(\left(\alpha,\alpha,\alpha,\frac{-2a^{2}+c^{2}}{4\alpha^{2}},\frac{c^{2}}{4 \alpha^{2}},\frac{c^{2}}{4\alpha^{2}}\right)^{2}\) & 8 \\ \hline
8 & primitive orthorhombic (po) & \(\left(a,b,c,0,0,0\right)\) & 4 \\ \hline
9 & base-centered orthorhombic (bco) & \(\left(\frac{\gamma}{2},\frac{\gamma}{2},c,0,0,\frac{-a^{2}+b^{2}}{\gamma^{2}} \right)^{3}\) & 4 \\ \hline
10 & face-centered orthorhombic (fco) & \(\left(\frac{\gamma}{2},\frac{\alpha}{2},\frac{\beta}{2},\frac{c^{2}}{2\alpha}, \frac{a^{2}}{\gamma^{2}},\frac{b^{2}}{\gamma\alpha}\right)^{3}\) & 4 \\ \hline
11 & body-centered orthorhombic (ico) & \(\left(\frac{\alpha}{2},\frac{\alpha}{2},\frac{\alpha}{2},\frac{\alpha^{2}-b^{2} }{\alpha^{2}},\frac{-a^{2}+c^{2}}{\alpha^{2}},\frac{\alpha^{2}-2a^{2}}{\alpha^{ 2}}\right)^{4}\) & 4 \\ \hline
12 & primitive monoclinic (pm) & \(\left(a,b,c,0,\cos\beta,0\right)\) & 2 \\ \hline
13 & base-centered monoclinic (bcm) & \(\left(\frac{m}{2},\frac{m}{2},c,-\frac{ac\cos\beta}{mc},\frac{ac\cos\beta}{mc}, -\frac{a^{2}+b^{2}}{m^{2}}\right)^{5}\) & 2 \\ \hline
14 & triclinic & \(\left(a,b,c,\cos\alpha,\cos\beta,\cos\gamma\right)\) & 1 \\ \hline \end{tabular}
\end{table}
Table 1: Lattice parameters sextuplets of the 14 Bravais latticesProof.: For any lattice point in \(\mathcal{M}_{\text{fcc}}\), it is expressed as :
\[a_{0}\mathbf{I}(\mathbf{n}+\mathbf{w}_{i})=\mathbf{E}\chi^{-1}(\mathbf{n}+ \mathbf{w}_{i})=\mathbf{E}(\tilde{\mathbf{n}}+\chi^{-1}\mathbf{w}_{i})=\left\{ \begin{array}{l}\mathbf{E}\tilde{\mathbf{n}}\text{ for }i=1\\ \mathbf{E}(\tilde{\mathbf{n}}+^{T})\text{ for }i=2\\ \mathbf{E}(\tilde{\mathbf{n}}+^{T})\text{ for }i=3\\ \mathbf{E}(\tilde{\mathbf{n}}+^{T})\text{ for }i=4\end{array}\right.,\]
Among the 14 Bravais lattices, there are 7 choices of the conventional basis corresponding to the 7 crystal systems. They are the bases of simple cubic, hexagonal, trigonal, primitive tetragonal, primitive orthorhombic, primitive monoclinic (with b-axis as the unique axis) and triclinic. The periodicity of the other non-primitive Bravais lattices can be expressed by the multilattice description using the corresponding conventional basis. Then the integer triplet \(\mathbf{n}=[n_{1},n_{2},n_{2}]\) in is consistent with the notation of Miller indices for crystallographic direction introduced by William H Miller in crystallography. Since the reciprocal basis is derived by taking the inverse of the transpose of the real lattice basis, all the calculations for the reciprocal lattice remains the same as given in, except that the integer triplet \(\mathbf{n}\) represents the index of a crystallographic plane.
The multilattice \(\mathcal{M}(\mathbf{E};\mathbf{w}_{1},...,\mathbf{w}_{m})\) can be used for representing the crystal structure as well, i.e. the isometries with the consideration of both the lattice points and atomic/molecular sites. The symmetry of a crystal structure is defined by its _space group_: skeleton Bravais lattice + site symmetry (point group). In consistency with the symmetry operations defined in the International Tables of Crystallography, the fractional vectors \(\mathbf{w}_{i}\) should be classified into 1) lattice points; 2) sites. For those Bravais lattices whose primitive bases are consistent with their conventional bases, they only have one lattice point, _i.e._\(\) by default. For the rest of Bravais lattices whose primitive bases are different from their conventional bases, they can be expressed mathematically by the multilattice defined in using the conventional basis vectors and the corresponding fractional atomic position vectors in the conventional unit cell. For examples, the lattice points of the conventional body-centered cubic are \(\) and \(\frac{1}{2}\); the lattice points of the conventional face-centered cubic are \(\), \([\frac{1}{2},\frac{1}{2},0]\), \([\frac{1}{2},0,\frac{1}{2}]\) and \([0,\frac{1}{2},\frac{1}{2}]\).
The meaning of _site_ is slightly different from the lattice point. The site of a crystal structure is the spatial position occupied by a real atom/molecule in the unit cell. To distinguish the lattice point from the site, we define a _crystal structure_ as:
\[\mathcal{S}(\mathbf{E},\mathbf{w}_{i};\mathbf{s}_{\alpha})=\{\mathcal{M}( \mathbf{E};\mathbf{w}_{i})+\mathbf{E}\mathbf{s}_{\alpha}:\mathbf{s}_{\alpha} \in\mathbb{R}^{3},|\mathbf{s}_{\alpha}|\in[0,1),\alpha=1,...,\nu,i=1,...,m\}. \tag{8}\]
In the above definition, the fractional vectors \(\mathbf{w}_{i}\), \(i=1,...,m\) are the lattice points. \(\{\mathbf{s}_{1},...,\mathbf{s}_{\nu}\}\) are the sites occupied by atoms in the unit cell framed by the skeleton lattice \(\mathcal{M}(\mathbf{E};\mathbf{w}_{i})\). The site symmetry means that the site is invariant under the point group operations including \(n-\)fold proper rotations, mirrors and reflections. For example, the site \(\mathbf{s}=\epsilon\) where \(|\epsilon|<1\) and its isometries of \(\mathcal{S}(a_{0}\mathbf{I},\mathbf{w}_{1},...,\mathbf{w}_{4};\mathbf{s})\) is invariant under \(m\bar{3}m\) where \(\{\mathbf{w}_{1},...\mathbf{w}_{4}\}\) is given in.
In X-ray experiments, the diffraction intensity strongly depends on the scattering of atoms occupying the certain sites of a crystal structure having special site symmetry, although the necessary condition of diffraction (_i.e._ Bragg condition) is governed by the underlying periodicity. X-ray analysis software generally requires the knowledge of both the lattice parameters for the skeleton lattice (_i.e._ parameters in Table 1), and all sites occupied by different species of atoms. In particular, these sites are classified as the _Wyckoff positions_ used by crystallographers and most materials scientists.
For cubic structures, it is not very hard to make an ansatz for the sites in the conventional unit cell. For example, most AB type alloys, the site is among the special positions such as corners, side centers, face centers and body centers. In some cases, atoms of small radii occupy the tetrahedron interstitial sites such as \([\frac{1}{4},\frac{1}{4},\frac{1}{4}]\) and \([\frac{1}{4},\frac{1}{4},\frac{3}{4}]\). However, for some complex crystal structures in low symmetry, _e.g._ martensite structures, there is no rational way of obtaining the sites for each of the atoms. From the database of the binary phase diagram, we observe that many low symmetry structures of metallic materials are in fact formed through solid-solid phase transformations from a high temperature phase of cubic symmetry. Examples include steel, CuAl alloy, Nitinol, and many other Cu-based \(\beta\) shape memory alloys. Using the mathematical formulation of the crystal structure given in, we can derive the crystal structure of the low symmetry phase from their cubic parent phase through the assigned lattice correspondences.
### Sublattice and rebase
When phase transformation occurs, the average lattice distortion can be calculated as a linear transformation that maps the proper lattice basis of parent lattice to that of the product lattice based on the Cauchy-Born rule:
\[\mathbf{m}_{i}=\mathbf{F}\mathbf{g}_{i} \tag{9}\]
Solving these linear equations, we obtain the deformation gradient \(\mathbf{F}=\mathbf{m}_{i}\otimes\mathbf{g}^{i}\in\mathbb{R}^{3\times 3}\) where \(\mathbf{g}_{i}\otimes\mathbf{g}^{i}=\mathbf{I}\). Chen et al. and Koumatos and Muehlemann showed that an admissible linear transformation should be consistent with the least transformation strain associated with the proper choice of the correspondent lattice vectors in the parent and product lattices. In crystallography community, the lattice correspondence between phases is understood as the orientation relationship, which usually consists of a set of parallel crystallographic planes and directions in both phases. Here, the transformation strain is calculated as \(\sqrt{\mathbf{F}^{T}\mathbf{F}}-\mathbf{I}\) where \(\mathbf{F}\) is the deformation gradient given by the Cauchy-Born rule in.
In real materials, both parent and product phases are the crystal structures mathematically expressed by. Suppose the parent phase is a cubic type structure defined as \(\mathcal{S}(a_{0}\mathbf{I},\mathbf{w}_{1},...,\mathbf{w}_{m};\mathbf{s}_{1},...,\mathbf{s}_{\nu})\). \(m=1\) for sc, \(m=2\) for bcc, and \(m=4\) for fcc. The skeleton of the product phase can be derived by a linear transformation of a _sublattice_ of the parent phase, which underlies a correspondent unit cell of the parent lattice that transforms to the primitive cell of the product lattice. Mathematically, it can be easily proved that a sublattice of Bravais lattice \(\mathcal{L}(\mathbf{E})\) is also a Bravais lattice \(\mathcal{L}(\mathbf{E}\mathbf{L})\) where \(\mathbf{L}\in\mathbb{Z}^{3\times 3}\) and \(\det\mathbf{L}\geq 1\). The three column vectors of \(\mathbf{L}\) are the lattice correspondence vectors for the phase transformation written in the primitive basis \(\mathcal{L}(\mathbf{E})\). Therefore the point group of \(\mathcal{L}(\mathbf{E}\mathbf{L})\) is a subgroup of the point group of \(\mathcal{L}(\mathbf{E})\). From the sublattice \(\mathcal{L}(\mathbf{E}\mathbf{L})\), we can calculate the _reference_ lattice parameters for the product phase from its lattice metric tensor:
\[\bar{\mathbf{C}}=\mathbf{L}^{T}\mathbf{C}\mathbf{L}, \tag{10}\]
Note that the lattice metric \(\bar{\mathbf{C}}\) without any linear perturbation still represents the same symmetry of the parent phase in multilattice setting by putting back all missed lattice points in the frame of the sublattice \(\mathcal{L}(\mathbf{E}\mathbf{L})\) (This will be explained later). As an example, we consider a phase transformation from face centered cubic to tetragonal subjected to the lattice correspondence:
\[[\tfrac{1}{2},\tfrac{1}{2},0]_{\text{fcc}} \rightarrow_{\text{t}}\] \[[-\tfrac{1}{2},\tfrac{1}{2},0]_{\text{fcc}} \rightarrow_{\text{t}}\] \[_{\text{fcc}} \rightarrow_{\text{t}}.\]
The matrix representation of the lattice correspondence is:
\[\mathbf{L}=\begin{pmatrix}\tfrac{1}{2}&-\tfrac{1}{2}&0\\ \tfrac{1}{2}&\tfrac{1}{2}&0\\ 0&0&1\end{pmatrix}. \tag{11}\]
Note that the components of \(\mathbf{L}\) are integers only when they are written in the primitive lattice basis. The components of are in terms of the conventional basis in consistent with the crystallographic expressions mostly used by experimentalists. The component transformation between the primitive and conventional bases follows \([\mathbf{L}]_{p}=\chi[\mathbf{L}]_{c}\) where \(\chi\) is the conversion matrix from the primitive basis of fcc to its conventional basis, defined by. Using, the reference lattice metric for the product lattice before transformation is calculated as:
\[\bar{\mathbf{C}}=[\mathbf{L}]_{p}[\mathbf{E}]_{p}^{T}[\mathbf{E}]_{p}[\mathbf{ L}]_{p}=[\mathbf{L}]_{c}[\mathbf{E}]_{c}^{T}[\mathbf{E}]_{c}[\mathbf{L}]_{c}= \begin{bmatrix}\tfrac{a_{0}^{2}}{2}&0&0\\ 0&\tfrac{a_{0}^{3}}{2}&0\\ 0&0&a_{0}^{2}\end{bmatrix}. \tag{12}\]
Therefore the sublattice parameter sextuples are \((\frac{a_{0}}{\sqrt{2}},\frac{a_{0}}{\sqrt{2}},a_{0},0,0,0)\) before phase transformation.
For materials undergoing solid-solid phase transformation without chemical doping or reaction, the atoms in the parent crystal structure will neither appear nor disappear upon the phase transformation. But there are lattice points of \(\mathcal{L}(\mathbf{E})\) that are not covered by the periodicity of \(\mathcal{L}(\mathbf{E}\mathbf{L})\). Let \(\mathbf{G}=\mathbf{E}\mathbf{L}\) be the sublattice basis where \(\det\mathbf{L}=m\geq 1\). For any lattice point in \(\mathcal{L}(\mathbf{E})\):
\[\mathbf{x}=\mathbf{E}\mathbf{n}=\mathbf{G}(\mathbf{L}^{-1}\mathbf{n})\text{ for some }\mathbf{n}\in\mathbb{Z}^{3}. \tag{13}\]
Since \(\det\mathbf{L}^{-1}=1/m\leq 1\), the coordinates of the lattice point \(\mathbf{x}\) in basis \(\mathbf{G}\) are not always integers. Those points having fractional coordinates are the missed lattice points under periodicity defined by basis \(\mathbf{G}\).
\[\mathcal{M}=\{\mathbf{G}(\bar{\mathbf{n}}+\bar{\mathbf{w}}_{i}):\text{ for all }\bar{\mathbf{n}}\in\mathbb{Z}^{3},|\bar{\mathbf{w}}_{i}|\in[0,1),i=1,...,m,\det \mathbf{L}=m\}. \tag{14}\]
The fractional vectors in the frame of the unit cell \(\mathcal{U}(\mathbf{G})\) can be calculated by:
\[\bar{\mathbf{w}}_{i}=\mathbf{L}^{-1}\mathbf{n}-\lfloor\mathbf{L}^{-1}\mathbf{n }\rfloor,\text{ for all }\mathbf{n}\in\mathbb{Z}^{3}. \tag{15}\]
The above calculation is to transform the coordinates of the missed lattice points in the basis of \(\mathbf{E}\) to the basis of \(\mathbf{G}\), and modulate them back into the unit cell. We define this operation as _rebase_. Using the primitive lattice basis of fcc lattice and the transformation matrix \([\mathbf{L}]_{p}\) with \(\det[\mathbf{L}]_{p}=2\), by equation, the multilattice after rebasing consists of two lattice points \(\) and \([\frac{1}{2},\frac{1}{2},\frac{1}{2}]\). The derived lattice looks like a body centered tetragonal with lattice parameters \((\frac{a_{0}}{\sqrt{2}},\frac{a_{0}}{\sqrt{2}},a_{0},0,0,0)\), it still presents the fcc symmetry as discussed earlier. To break and obtain a lower symmetry, we need to perturb the periodicity of \(\mathcal{M}(\mathbf{G};,[\frac{1}{2},\frac{1}{2},\frac{1}{2}])\) by a linear transformation:
\[\mathbf{P} = \frac{\delta_{1}}{\mathbf{g}_{1}\cdot\mathbf{g}_{1}}\mathbf{g}_{1 }\otimes\mathbf{g}_{1}+\frac{\delta_{2}}{\mathbf{g}_{2}\cdot\mathbf{g}_{2}} \mathbf{g}_{2}\otimes\mathbf{g}_{2}+\frac{\delta_{3}}{\mathbf{g}_{3}\cdot \mathbf{g}_{3}}\mathbf{g}_{3}\otimes\mathbf{g}_{3} \tag{16}\] \[+ \frac{\gamma_{12}}{\sqrt{(\mathbf{g}_{1}\cdot\mathbf{g}_{1})( \mathbf{g}_{2}\cdot\mathbf{g}_{2})}}\mathbf{g}_{1}\otimes\mathbf{g}_{2}+\frac{ \gamma_{13}}{\sqrt{(\mathbf{g}_{1}\cdot\mathbf{g}_{1})(\mathbf{g}_{3}\cdot \mathbf{g}_{3})}}\mathbf{g}_{1}\otimes\mathbf{g}_{3}+\frac{\gamma_{23}}{\sqrt{ (\mathbf{g}_{2}\cdot\mathbf{g}_{2})(\mathbf{g}_{3}\cdot\mathbf{g}_{3})}} \mathbf{g}_{2}\otimes\mathbf{g}_{3}, \tag{17}\]
Physically, these stretch and shear quantities are small values, which underlie the deformation of the reference lattice metric of the parent phase to the metric of the product phase.
For complex parent crystal structures such as alloys and compounds, we can use the _substructure_\(\mathcal{S}(\mathbf{PG},\bar{\mathbf{w}}_{i};\bar{\mathbf{s}}_{\alpha})\) as the derived crystal structure of the product phase. Both lattice points and sites are rebased and transformed by the same calculations using and. The derived lattice parameters and the atomic positions in the unit cell can be used to define the unknown structure of low symmetry from solid-solid phase transformation, which provides the initial conditions for structural refinement in X-ray diffraction analysis.
## 3 Orientation relationship and parallelism
In this section, we work on the Bravais lattice index only.1 That is the integer tuple \(\mathbf{n}=(n_{1},n_{2},n_{3})\in\mathbb{Z}^{3}\) of a Bravais lattice \(\mathcal{L}(\mathbf{E}_{p})\) where \(\mathbf{E}_{p}\) is the primitive lattice basis. Therefore, it is not always referred to the Miller index of crystallographic direction, especially for the lattices whose primitive lattice bases are not consistent with the conventional bases such as bcc, fcc, bct, bco, fco, ico and bcm (see Table 1). Let \(\mathbf{E}_{c}\) denote the conventional lattice basis. The primitive lattice basis is related to the conventional basis by \(\mathbf{E}_{p}=\mathbf{E}_{c}\chi\) where \(\chi\in\mathbb{R}^{3\times 3}\) is the conversion matrix between two bases.
\[\chi_{\mathrm{fcc}}=\begin{bmatrix}\frac{1}{2}&0&\frac{1}{2}\\ \frac{1}{2}&\frac{1}{2}&0\\ 0&\frac{1}{2}&\frac{1}{2}\end{bmatrix}. \tag{18}\]
For bcc lattice,
\[\chi_{\mathrm{bcc}}=\begin{bmatrix}\frac{1}{2}&\frac{1}{2}&-\frac{1}{2}\\ -\frac{1}{2}&\frac{1}{2}&\frac{1}{2}\\ \frac{1}{2}&-\frac{1}{2}&\frac{1}{2}\end{bmatrix}. \tag{19}\]
The presentation of lattice basis varies upon the symmetry and/or lattice invariant transformations, therefore the expressional examples given in and may vary as the change of lattice basis. Using the conversion matrix, the Bravais lattice index is related to the Miller index (\(\mathbf{n}_{c}\)) of crystallography by \(\mathbf{n}=\chi^{-1}\mathbf{n}_{c}\).
\[\mathbf{n}_{\mathrm{s}}=[\mathbf{L}]_{p}^{-1}\mathbf{n}=(\chi[\mathbf{L}]_{p} )^{-1}\mathbf{n}_{c}=([\mathbf{L}]_{c})^{-1}\mathbf{n}_{c}. \tag{20}\]
The equation underlies the transformation rule for the Miller index of crystallography from the Bravais lattice to its derived lattice. Through solid-solid phase transformation, the skeleton periodicity of the derived lattice changes for the energetic reasons, but the index of the product lattice remains the same relationship with the parent lattice.
\[\mathbf{n}_{\mathrm{dv}}\ ||\ ([\mathbf{L}]_{c})^{-1}\mathbf{n}_{c}. \tag{21}\]
For the crystallographic plane, the lattice plane index relationship can be expressed as the parallelism
\[\mathbf{n}_{\mathrm{dv}}^{*}\ ||\ ([\mathbf{L}]_{c})^{T}\mathbf{n}_{c}^{*}, \tag{22}\]
By, and the Cauchy-Born rule in, we generalize the presentation of orientation relationship and the associated transformation strains for solid-solid phase transformation through a unified quantity: the lattice correspondence matrix. This correspondence matrix can be rigorously determined by the StrucTrans algorithm, which minimizes the transformation strains between two lattices of arbitrary symmetries.
## 4 Structural determination of martensite CuAlMn
Among all shape memory alloys, CuAuZn, CuAlNi and CuAlZn alloy systems form the second largest group of material candidates used in research and development. However, their commercialization is highly confined due to the poor fatigue life when they are in a polycrystalline state. Compared to these Cu-based alloy systems, the CuAlMn system is much less developed. For some aluminum compositions, people demonstrated good ductility in this alloy system. In-depth study of the crystallography and the formation of microstructure is highly hindered in this system because of the lack of the structural parameters for martensite. Some discussions about its micromechanical behaviors are based on the lattice parameters of the cubic to monoclinic transformation of its sibling system CuAlNi. In this section, we will show how the derived lattice assists the analysis of the structural determination of martensite structure of CuAlMn alloy.
According to the binary phase diagram of Cu-Al alloy system, the martensite can be induced by suppressing the eutectic transformation through the rapid cooling process. The high temperature \(\beta\) phase directly transforms into an ordered structure \(\beta_{1}\) (DO3, or L2\({}_{1}\)) at \(T_{c}\) (marked as the red curve in Figure 1). The \(\beta_{1}\) phase further undergoes a martensitic transformation at temperature \(M_{s}\) marked as the blue curve in This also works for ternary Cu-based \(\beta\) alloys doped by Mn, except that the \(\beta_{1}\) zone (the yellow region in Figure 1) and the composition dependent \(M_{s}\) curve may vary with the addition of Mn. The symmetry of the product phase formed by martensitic transformation from the \(\beta_{1}\) phase varies with the Al concentration.
The most studied compositions of Cu-Al-Mn are those with Al compositions 14 - 17 at% and Mn composition around 10 at% since the alloys within this compositional range show a better ductility than CuAlNi and CuAlZn in polycrystalline form. The crystal structure of martensite has been characterized by both X-ray powder diffraction and transmission electron microscopy for Al compositions 14, 16 and 17 at%. It was found that the martensite of as-aged samples are 18R (_i.e._ 18 layers modulated monoclinic structure).
Binary phase diagram of Cu-Al alloy from FScopp alloy database 2012.
### Experiment
A mixture of high purity Cu (99.99 wt%), Al (99.999 wt%), and Mn (99.95 wt%) ingots were melted in a quartz tube placed in an evacuated (10\({}^{-5}\) mbar) induction furnace under argon atmosphere. The melt was injected into a cylindrical copper mold and solidified as a rod of diameter 5 mm. It was homogenized at 800\({}^{\circ}\)C for 3 hours under argon atmosphere, then cooled down in the furnace. We cut the rod into thin slices of thickness 1 mm, which were sealed in a vacuum quartz tube, heat treated at 900\({}^{\circ}\)C for 1 hour and quenched in water.
The transformation temperature of the specimen was measured by differential scanning calorimetry (DSC) by TA Instruments Q1000 at a heating and cooling rate of 10\({}^{\circ}\)C/min for three complete thermal cycles in the range from -75\({}^{\circ}\)C to 20\({}^{\circ}\)C. The austenite start/finish \(A_{s}/A_{f}\) temperatures and martensite start/finish \(M_{s}/M_{f}\) temperatures are determined as the onsets of the heat absorption/emission peaks as shown in \(A_{s}=-26^{\circ}\)C, \(A_{f}=-8^{\circ}\)C, \(M_{s}=-48^{\circ}\)C and \(M_{f}=-62^{\circ}\)C. The thermal hysteresis is quite large and measured to be \(\Delta T=\frac{1}{2}(A_{s}+A_{f}-M_{s}-M_{f})=38^{\circ}\)C. Unlike those reported transforming Cu-based \(\beta\) alloys, this one shows the thermal bursts in large magnitude over a wide temperature range during the phase transformation. Evidently this thermal signature was observed in a close-by alloy system with slightly different manganese compositions, but the detailed crystal structure of martensite has not been thoroughly studied for this series. In their work, the total entropy change from the cubic phase (austenite) is calculated based on the DSC measurements, which was found to be highly correlated to the average electron concentration (_i. e. e/a_) of the alloy. Only those with \(e/a>1.46\) showed the jerky thermal behaviors, which was conjectured as a different type of martensitic transformation: bcc to 2H. Their room-temperature powder diffraction measurement showed some residual peaks corresponding to the 2H structure. The martensite finish temperatures of this series are quite low, _i.e._ around \(-60^{\circ}\)C, therefore the room-temperature diffractometry with mixed phases is not sufficient to identify and solve the martensite crystal structure for the 2H phase, nor the report of lattice parameters were reported for this phase in any other CuAlMn system.
### Advanced structural characterization by synchrotron X-ray microdiffraction
To obtain precise information for the crystal structures of austenite and martensite and to show how the derived lattice theory assists the structure determination, we conducted a temperature-varying single crystal synchrotron X-ray Laue microdiffraction experiment combined with monochromatic energy scans at beamline 12.3.2 of the Advanced Light Source, Lawrence Berkeley National Lab. The X-ray beam with energy bandpass from 6 keV to 24 keV was focused down to 1 \(\mu\)m in diameter by a pair of elliptically bent Kirkpatrick-Baez mirrors. The focused high-brightness X-rays illuminated a single grain of the bulk sample, and generated a single crystal Laue pattern. We used the custom-made thermal stage to drive the phase transformation of the bulk sample, which controls the sample temperature from -100\({}^{\circ}\)C to 200\({}^{\circ}\)C with ramping rate of 15\({}^{\circ}\)C/min. The bulk sample was polished in the austenite form at room temperature. An optical microscope attached to the end-station optic box allows to observe in-situ the sample surface reliefs while collecting the Laue patterns at a specified sample position during cooling process.
Figures 3 (a) - (c) show the evolution of Laue patterns as the sample was cooled down through the phase transformation temperature while the corresponding microstructures in (d) - (f) sufficiently reveal that the Laue pattern in (a) purely represents the austenite phase. As the temperature going down, we observed that martensite laths appear and grow as shown in (e) and (f). The Laue patterns (b) and (c) suggest that they are purely in the martensitic phase.
We used the L2\({}_{1}\) structure (space group F\(m\bar{3}m\)) to index the austenite Laue pattern. The crystal structure is depicted in Figure 4(b). The stoichiometric ratio of atoms for L2\({}_{1}\) is supposed to be ABC\({}_{2}\) where A atoms occupy the site 4a at \(\) with site symmetry \(m\bar{3}m\), B atoms occupy the site 4b at \([\frac{1}{2},\frac{1}{2},\frac{1}{2}]\) also with site symmetry \(m\bar{3}m\), and C atoms occupy 8c site at \([\frac{1}{4},\frac{1}{4},\frac{1}{4}]\) with site symmetry \(\bar{4}3m\). In the case of Cu\({}_{67}\)Al\({}_{24}\)Mn\({}_{9}\), we assume that the Al atoms fully occupy the 4a site, the Mn atoms fully occupy the 4b site and the Cu atoms occupy the 8c sites. Using the XMAS software, we successfully indexed the Laue pattern by the proposed L2\({}_{1}\) structure as shown in Figure 4(a). To determine the austenite lattice parameter, we chose four (\(hkl\)) reflections:,, and, and precisely measure their interplanar distances by performing energy scans of the reflections. The refined lattice parameter was measured to be \(a_{0}=5.87897\AA\).
Differential Scanning Calorimetry of CuAl\({}_{24}\)Mn\({}_{9}\).
### Determination of martensitic structure by derived lattice theory
We assume the Bain lattice correspondence for the phase transformation from L2\({}_{1}\) to orthorhombic given in :
\[\mathbf{L}=\begin{bmatrix}\frac{1}{2}&0&-\frac{1}{2}\\ 0&1&0\\ \frac{1}{2}&0&\frac{1}{2}\end{bmatrix} \tag{23}\]
After rebasing the original L2\({}_{1}\) lattice, the sublattice structure \(\mathcal{S}=\{\mathcal{L}(a_{0}\mathbf{L})+a_{0}\mathbf{L}\mathbf{s}_{\alpha}: \text{ for }\alpha=1,...,8\}\) retains its stoichiometry with sites for :
\[\text{Al: }\quad\mathbf{s}_{1}=,\ \mathbf{s}_{2}=[\frac{1}{2}, \frac{1}{2},\frac{1}{2}],\] \[\text{Mn: }\quad\mathbf{s}_{3}=[\frac{1}{2},0,\frac{1}{2}],\ \mathbf{s}_{4}=[0,\frac{1}{2},0], \tag{24}\] \[\text{Cu: }\quad\mathbf{s}_{5}=[\frac{1}{2},\frac{1}{4},0],\ \mathbf{s}_{6}=[0,\frac{3}{4},\frac{1}{2}],\ \mathbf{s}_{7}=[\frac{1}{2},\frac{3}{4},0],\ \mathbf{s}_{8}=[0,\frac{1}{4},\frac{1}{2}].\]
By equation, the sublattice parameters are \((\bar{a},\bar{b},\bar{c},0,0,0)\) where \(\bar{a}=\bar{c}=\frac{a_{0}}{\sqrt{2}}=4.15706\AA\) and \(\bar{b}=a_{0}=5.87897\AA\). In (d), the red and blue boxes underlie the sublattice cells of \(\mathcal{L}(a_{0}\mathbf{L})\).
(a) – (c) Laue diffraction patterns of the bulk sample from high temperature to low temperature corresponding to the microstructures in (d) – (f), in which the red circles denote the surface positions illuminated by the focused X-ray beam.
By observation, the sites given in imply a body centered symmetry, which can be naturally expressed as the space group I\(mmm\) (number in international table: 71). The derived substructure can be fully described by the Wyckoff positions listed in Table 22. Multiplicity of the Wyckoff position means the number of equivalent sites under the site symmetry operations. Using our definition of crystal structure, this derived I\(mmm\) structure plotted in Figure 5(a) can be expressed as :
Footnote 2: Wyckoff position is equivalent to the definition of site in this paper.
\[\mathcal{S}_{\rm Immm}=\{\mathcal{M}(\mathbf{E}_{\rm orth};\mathbf{w}_{1}, \mathbf{w}_{2})+\mathbf{E}_{\rm orth}\mathbf{s}_{\alpha}:\alpha=1,2,3\} \tag{26}\]
where the structural parameters are given by :
\[\mathbf{E}_{\rm orth}=\begin{bmatrix}a_{\rm dv}&0&0\\ 0&b_{\rm dv}&0\\ 0&0&c_{\rm dv}\end{bmatrix},\mathbf{w}_{1}=\begin{bmatrix}0\\ 0\\ 0\end{bmatrix},\mathbf{w}_{2}=\begin{bmatrix}\frac{1}{2}\\ \frac{1}{2}\\ \frac{1}{2}\end{bmatrix}, \tag{27}\]
We use this crystal structure as input for the XMAS Laue indexing algorithm, and get the indexed Laue
Austenite crystal structure of Cu\({}_{67}\)Al\({}_{24}\)Mn\({}_{9}\): (a) Indexed Laue pattern by (b) the L\(2_{1}\) structure. (c) The stereographic projection of the reciprocal lattices of austenite and the transformed martensite with respect to the stage coordinate system (X – Y – Z). (d) The theoretical atomic structure in plane that is aligned horizontally along direction.
The indexing program suggests two martensite variants corresponding to the indexed reflections marked by yellow and orange colors respectively in Figure 5(c). However, the indices of many major reflections are still not found by the crystal structure \(\mathcal{S}_{\text{Immm}}\), which implies the existence of an even lower symmetry structure. To slightly lower the symmetry of \(\mathcal{S}_{\text{Immm}}\), we propose to introduce shuffles as indicated in many Cu-based alloys.
2 & a & \(mmm\) & \(\) \\ \hline
2 & d & \(mmm\) & \([\frac{1}{2},0,\frac{1}{2}]\) \\ \hline
4 & h & \(m2m\) & \([0,\frac{1}{4},\frac{1}{2}]\) \\ \hline \end{tabular}
\end{table}
Table 2: Wyckoff positions of derived \(\text{Immm}\) structure for CuAlMn
Derived orthorhombic structures of CuAl\({}_{24}\)Mn\({}_{9}\) with space group (a) I\(mmm\) and (b) P\(mmm\), by which the synchrotron X-ray Laue diffraction pattern of martensite is indexed in (c) and (d) respectively.
that directly shuffles the sites of structure \(\mathcal{S}_{mmm}\) to :
\[\begin{split}\text{Al:}&\quad\bar{\mathbf{s}}_{1}=[0,0,0 ],\bar{\mathbf{s}}_{2}=[\frac{1}{3},\frac{1}{2},\frac{1}{2}]\\ \text{Mn:}&\quad\bar{\mathbf{s}}_{3}=[\frac{1}{2},0,\frac{1}{2}],\bar{\mathbf{s}}_{4}=[\frac{5}{6},\frac{1}{2},0]\\ \text{Cu:}&\quad\bar{\mathbf{s}}_{5}=[\frac{1}{2}, \frac{1}{4},0],\bar{\mathbf{s}}_{6}=[\frac{5}{6},\frac{1}{4},\frac{1}{2}].\end{split} \tag{28}\]
The above atom sites do not show the body centered symmetry any more, therefore the underlying periodicity reduces to a primitive orthorhombic symmetry. In the international table of crystallography, the highest symmetry corresponding to the primitive orthorhombic symmetry is P_mmm_ (number: 47), of which the crystal structure is shown in Figure 5(b), expressed as :
\[\mathbf{S}_{\text{P}mmm}=\{\mathcal{L}(\mathbf{E}_{\text{orth}})+\mathbf{E}_{ \text{orth}}\bar{\mathbf{s}}_{\alpha}:\alpha=1,...,6\} \tag{29}\]
The site symmetry and corresponding Wyckoff positions are listed in Table 3, with which the chemical stoichiometry of Al, Mn and Cu still maintains \(1:1:2\) in consistent with austenite. Using the crystal structure \(\mathcal{S}_{\text{P}mmm}\) as the input for XMAS Laue indexing algorithm, we get the indexed Laue pattern of martensite in Figure 5(d). All major reflections are indexed by two martensite variants, which implies the possible crystal structure of martensite is likely to be \(\mathcal{S}_{\text{P}mmm}\) given by.
Finally, the orientation relationship between the austenite (F_m_\(\bar{3}m\)) and the derived martensite (P_mmm_) is confirmed by overlapping their stereographic projections calculated from the indexed Laue patterns, shown in (c). The normal vectors of the crystallographic planes, (1\(\bar{1}\)0) and in austenite lattice are parallel to those of the crystallographic planes, and in the lattice of one of the martensite variants. This result also confirms our conjecture of the lattice correspondence for the derived martensite structure by.
In the pure martensite phase, we conduct the monochromatic energy scan in a wide photon energy spectrum (_i.e._ from 8keV to 16keV), and get a list of interplanar distance measures in Table 4 for the indexed crystallographic planes in a reference Laue pattern. In general, the theoretical interplanar distance for a plane \(\mathbf{h}=(hkl)\) can be expressed as \(d_{theo}=|\mathbf{E}^{*}\mathbf{h}|\) where \(\mathbf{E}^{*}=\mathbf{E}^{-T}\).
\begin{table}
\begin{tabular}{c|c|c|c} Multiplicity & Wyckoff letter & Site symmetry & Site with lattice points: \(\) \\ \hline
1 & a & \(mmm\) & \(\) \\ \hline
2 & l & \(2mm\) & \([\frac{1}{3},\frac{1}{2},\frac{1}{2}]\) \\ \hline
1 & d & \(mmm\) & \([\frac{1}{2},0,\frac{1}{2}]\) \\ \hline
2 & k & \(2mmm\) & \([\frac{5}{6},\frac{1}{2},0]\) \\ \hline
2 & o & \(m2m\) & \([\frac{1}{2},\frac{1}{4},0]\) \\ \hline
4 & z & \(..m\) & \([\frac{5}{6},\frac{1}{4},\frac{1}{2}]\) \\ \hline \end{tabular}
\end{table}
Table 3: Wyckoff positions of derived P_mmm_ structure for CuAlMnbasis for the Bravais lattice \(\mathcal{L}(\mathbf{E})\). In the case of orthorhombic lattice, the lattice basis \(\mathbf{E}\) is a diagonal matrix with diagonal elements \((a,b,c)\).
\[(a^{*},b^{*},c^{*})=\arg\min_{(a,b,c)\in\mathbb{R}^{3}}\sum_{\mathbf{h}\in \mathcal{H}}\|d_{theo}(a,b,c;\mathbf{h})-d_{exp}(\mathbf{h})\|^{2}, \tag{30}\]
We use the derived lattice parameters \((a_{\mathrm{dv}},b_{\mathrm{dv}},c_{\mathrm{dv}})\) as the initial condition and get the refined lattice parameters \(a=4.43196\AA\), \(b=5.34533\AA\), \(c=4.26307\AA\). Using the refined lattice parameters, we calculated the interplanar distances for the \((hkl)\) planes shown in Table 4, which agree with the measured values up to \(0.01\%\).
## 5 Conclusion
Using the mathematical description of the crystal structure: Lattice + atom sites in consistent with the international table of crystallography, we underlie a route to calculate the derived crystal structure for the martensitic materials transformed from solid-solid phase transformation. This approach is useful for the structure determination by X-ray diffraction analysis. Not only it provides the heuristic lattice parameters by the small perturbation based on continuum mechanics theory, but it also calculates the atom positions and their site symmetries based on the Cauchy-Born rule from the orientation relationship. We use a typical Cu-based \(\beta\) alloy to demonstrate our theory and showed that the indexing accuracy of X-ray Laue pattern has been improved by a large margin based on our mathematical framework.
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(hkl\) & \(E\) (kev) & \(\lambda\) (Å) & \(2\theta\) (\({}^{\circ}\)) & \(d_{exp}\) (Å) & \(d_{theo}\) (Å) & \(d_{exp}/d_{theo}\) \\ \hline (5\(\bar{4}\bar{1}\)) & 11.0201 & 1.12507 & 101.1893 & 0.728038 & 0.727824 & 1.00029 \\ (6\(\bar{4}\bar{2}\)) & 14.4006 & 0.860965 & 88.1896 & 0.618645 & 0.618648 & 1.00000 \\ (5\(\bar{4}\bar{2}\)) & 13.4255 & 0.923498 & 82.8361 & 0.697982 & 0.697949 & 1.00005 \\ (4\(\bar{3}\bar{2}\)) & 12.0003 & 1.033176 & 73.7746 & 0.860631 & 0.860778 & 0.99983 \\ (3\(\bar{2}\bar{2}\)) & 11.3352 & 1.093798 & 59.2741 & 1.105955 & 1.105480 & 1.00043 \\ (5\(\bar{2}\bar{2}\)) & 12.6406 & 0.980841 & 77.6421 & 0.782307 & 0.782578 & 0.99965 \\ (7\(\bar{2}\bar{1}\)) & 13.1555 & 0.942451 & 101.2271 & 0.609698 & 0.609752 & 0.99991 \\ (5\(\bar{2}\bar{1}\)) & 9.9452 & 1.246674 & 98.0783 & 0.825439 & 0.825409 & 1.00004 \\ (4\(\bar{1}\bar{2}\)) & 12.0503 & 1.028889 & 64.3184 & 0.966504 & 0.966894 & 0.99960 \\ (60\(\bar{1}\)) & 12.3899 & 1.000688 & 86.8573 & 0.727825 & 0.727816 & 1.00001 \\ (50\(\bar{1}\)) & 10.7153 & 1.157076 & 83.5984 & 0.867996 & 0.867833 & 1.00019 \\ (40\(\bar{1}\)) & 9.1151 & 1.360207 & 78.7222 & 1.072370 & 1.072365 & 1.00000 \\ \hline \hline \end{tabular}
\end{table}
Table 4: The results of monochromatic energy scan and the corresponding indices obtained from numerical analysis for martensite phase.
Finally, we show that the precise lattice parameters of the 2H martensite of Cu\({}_{67}\)Al\({}_{24}\)Mn\({}_{9}\) alloy can be determined without pre-knowledge of the structure.
| 10.48550/arXiv.1909.06574 | Derived Crystal Structure of Martensitic Materials by Solid-Solid Phase Transformation | Mostafa Karami, Nobumichi Tamura, Yong Yang, Xian Chen | 233 |
10.48550_arXiv.0708.1457 | ###### Abstract
Multibillion-clone landscape phage display libraries were prepared by the fusion of the phage major coat protein pVIII with foreign random peptides. Phage particles and their proteins specific for cancer and bacterial cells were selected from the landscape libraries and exploited as molecular recognition interfaces in detection, gene- and drug-delivery systems. The biorecognition interfaces were obtained by incorporation of the cell-specific phage fusion proteins into liposomes using intrinsic structural duplicity of the proteins. As a paradigm, we incorporated targeted pVIII proteins into commercially available therapeutic liposomes "Doxil", which acquired a new emergent property--ability to bind target receptors. Targeting of the drug was evidenced by fluorescence-activated cell sorting, microarray, optical and electron microscopy. In contrast to a poorly controllable conjugation targeting, the new landscape phage-based approach relies on very powerful and extremely precise mechanisms of selection, biosynthesis and self assembly, in which phages themselves serve as a source of the final product.
## 1 Introduction
Phage display technology emerged as a synergy of two fundamental concepts: _combinatorial peptide libraries_ and _fusion phage_. The first concept replaced the traditional collections of natural or individually synthesized compounds for libraries of peptides obtained in parallel synthesis as grouped mixtures (reviewed in); the second--allowed displaying foreign peptides on the surface of bacterial viruses (bacteriophages) as part of their minor or major coat proteins(reviewed in). The merge of these two concepts resulted in development of _phage display libraries_--multibillion clone compositions of self-amplified and self-assembled biological particles. In particular, a paradigm of _landscape phage libraries_ evolved, in which the phage is considered not just as a genetic carrier for foreign peptides, like in the traditional phage display approach, but rather as a nanoparticle (nanotube) with emergent physico-chemical characteristics determined by specific phage landscapes formed by thousands of random peptides fused to the major coat protein pVIII. These constructs display the guest peptide on every pVIII subunit increasing the virion's total mass by 10%. Despite of the extra burden, such particles could retain their infectivity and progeny-forming ability.
There is a fast growing interest to the landscape phages as a new type of selectable nanomaterials. The landscape phage can bind organic ligands, proteins and antibodies, induce specific immune responses in animals, or resist stress factors such as chloroform or high temperature. Landscape phages have been shown to serve as substitutes for antibodies against cell-displayed antigens and receptors, diagnostic probes for bacteria and spores, gene- and drug-delivery systems, and biospecific adsorbents. Phage-derived probes inherit the extreme robustness of wild-type phage and allow fabrication of bioselective materials by self-assemblage of phage or its composites on metal, mineral or plastic surfaces. Landscape phages specific for bacterial cells and spores have been exploited as molecular recognition interfaces in detection systems.
In this report we first demonstrate the use of the landscape phage as a bioselective interface in drug delivery systems. The concept of using targeted pharmaceutical nanocarriers to enhance the efficiency of anti-cancer drugs has been proven over the past decade both in pharmaceutical research and clinical setting. Examples of a successful realization of this concept are listed in numerous reviews, for example. In particular, it is commonly accepted that selectivity of drug delivery systems can be increased by their coupling with peptide and protein ligands targeted to differentially expressed receptors (reviewed in). The abundance of these receptors was demonstrated recently by comparative analysis of gene expression in tumor cells and tumor vascular endothelial cells versus adjacent normal tissues, and their targeting is turning into a routine procedure in the most advanced laboratories due to the progress in combinatorial chemistry and phage display (a long lists of different target-specific peptidesidentified by phage display can be found in recent reviews.
A new challenge, within the frame of this concept, is to develop highly selective, stable, active and physiologically acceptable ligands that would navigate the encapsulated drugs to the site of the disease and control unloading of their toxic cargo inside the cancer cells. We have shown earlier that the tumor-specific peptides fused to the major coat protein pVIII can be affinity selected from multibillion clone landscape phage libraries by their ability to bind very specifically cancer cells.
Thus, the major principle of our approach is that targeted drug-loaded nanoparticles recognize the same receptors, cells, tissues and organs that have been used for selection of the precisely targeted landscape phage.
## 2 Landscape phage libraries as a source of biosellective probes
The filamentous bacteriophages Ff (fd, f1 and M13) are long, thin viruses, which consist of a single-stranded circular DNA packed in a cylindrical shell composed of the major coat protein pVIII (98% of the total protein mass), and a few copies of the minor coat proteins capping the ends of the phage particle. Foreign peptides were displayed on the pVIII protein soon after the display on the minor coat protein pIII was pioneered (reviewed in). The pVIII-fusion phages display the guest peptide on every pVIII subunit, increasing the virion's total mass by up to 15%. Such particles were given the name "landscape phage" to emphasize the dramatic change in surface architecture caused by arraying thousands of copies of the guest peptide in a dense, repeating pattern around the tubular capsid, as illustrated by **Fig. 3**.
A _landscape library_ is a diverse population of phages, encompassing billions of clones with different surface structures and biophysical properties. Therefore, the landscape phage is unique micro-fibrous material that can be selected in the affinity binding protocol and obtained by a routine and simple microbiological procedure. Binding peptides, 8-9 amino acid long, comprising up to 20% of the phage mass may be easily prepared by cultivation of the infected bacteria and isolation of the secreted phage particles by precipitation. Landscape phages have been shown to serve as substitutes for antibodies against various antigens and receptors, including live cancer and bacterial cells, in gene-delivery vehicles, and as analytical probes in biosensors. Although phages themselves are very attractive bioselective materials, alternative targeted forms containing phage proteins may be more advantageous in medical applications where nanoparticulate materials are required, such as bioselective drug delivery liposomes described below.
Drug-loaded liposome targeted by the pVIII protein. The hydrophobic helix of pVIII spans the lipid layer and binding peptide is displayed on the surface of the carrier particles.
Filamentous phage. Left: electron micrograph. The aminometinus of pIII proteins are pointed by arrow. Right: Segment of -1% of phage virion with the array of the pVIII proteins shown as electron densities (Micrograph and model courtesy of Gregory Kishchenko)
## 3.
The ability of the major coat protein pVIII to form micelles and liposomes emerges from its intrinsic function as membrane protein judged by its biological,
During infection of the host _Escherichia coli_, the phage coat is dissolved in the bacterial cytoplasmic membrane, while viral DNA enters the cytoplasm. The protein is synthesized in infected cell as a water-soluble precursor, which contains a leader sequence of 23 residues at its N-terminus. When this protein is inserted into the membrane, the leader sequence is cleaved off by a leader peptidase. Later, during the phage assembly, the processed pVIII proteins are transferred from the membrane into a coat of the emerging phage. Thus, the major coat protein can change its conformation to accommodate to various distinctly different forms of the phage and its precursors: phage filament, intermediate particle (I-form), spheroid (S-form), and membrane-bound form. This structural flexibility of the major coat protein is determined by its unique architecture, which is studied in much detail. In virions, mostly a-helical domain of pVIII are arranged in layers with a 5-fold rotational symmetry and approximate 2-fold screw symmetry around the filament axis, as shown on the **Fig. 3**, right. In opposite, in the membrane-bound form of fd coat protein, the 16-A-long amphipathic helix (residues 8-18) rests on the membrane surface, while the 35-A-long trans-membrane (TM) helix (residues 21-45) crosses the membrane at an angle of 26o up to residue Lys40, where the helix tilt changes, as illustrated by **Fig. 4.** The helix tilt accommodates the thickness of the phospholipid bilayer, which is 31 A for
_E. coli_ membrane components. Tyr 21 and Phe 45 at the lipid-water interfaces delimit the TM helix, while a half of N-terminal and the last C-terminal amino acids, including the charged lysine side chains, emerge from the membrane interior. The transmembrane and amphipathic helices are connected by a short turn (Thr 19-Glu 20) that differs from the longer hinge loop (residues 17-26).
## 4.
### 4.1.
In model experiments, landscape phages were converted to a new biorecognition affinity reagent -- "stripped phage". The stripped phage is a composition of disassembled phage coat proteins with dominated (98%) recombinant major coat protein pVIII, which is genetically fused to the foreign target-binding peptides. The stripped phage proteins can form bioselective vesicles decorated by target-binding peptides, which can be used for the targeted drug delivery.
In preliminary experiments with streptavidin- and bacterial binders it was demonstrated by competition ELISA, acoustic wave sensor and transmission electron microscopy that the stripped phage proteins retain the target-binding properties of the selected phage.
The model of the pVIII protein in the lipid environment (adapted from Stopar et al. 2003).
Transmission electron microscopy of landscape phage converted to spheroids.
### Fusion of the stripped phage proteins with Doxil
Using intrinsic mechanism of fusion of the phage proteins with lipid membranes, we incorporated streptavidin-targeted proteins into the commercially available Doxil liposomes. The streptavidin-binding landscape phage was affinity selected from 9-mer landscape library. The phage was converted into spheroids with chloroform and incubated with Doxil to allow fusion of the phage proteins with liposome membrane, as illustrated by **Fig. 1**. As a result of the phage fusion, the liposome acquired a new emergent property--ability to bind streptavidin and streptavidin-conjugated fluorescent molecules, as was evidenced by protein microarrays, fluorescent microscopy and fluorescence-activated cell sorting (FACS).
## 5 Conclusion.
We proved a novel approach for specific targeting of pharmaceutical nanocarriers through their fusion with stripped proteins of the affinity selected landscape phage. The phage specific for the target organ, tissue or cell is selected from the multibillion landscape phage libraries, and is combined with liposomes exploring intrinsic amphiphilic properties of the phage proteins. As a result, the targeting probe--the target-specific peptide fused to the major coat protein--is exposed on the shell of the drug-loaded vesicle. In contrast to sophisticated and poorly controllable conjugation procedures used for coupling of synthetic peptides and antibodies to the targeted vesicles, the new phage-based concept relies on very powerful and extremely precise mechanisms of _selection, biosynthesis and self assembly_. When landscape phage serve as a reservoir of the targeted membrane proteins, one of the most troublesome steps of the conjugation technology is bypassed. Furthermore, it does not require idiosyncratic reactions with any new shell-decorating polymer or targeting ligand and may be easily adapted to a new liposome or micelle composition and a new addressed target. No reengineering of the selected phage is required at all: the phages themselves serve as the source of the final product--coat protein genetically fused to the targeting peptide.
Microarray test for Doxil targeting. Streptavidin-microarrayed slide was treated with unmodified Doxil, Doxil modified with wild-type phage and Doxil targeted with streptavidin binding phage.
Transmission electron microscopy of original Doxil liposomes (A), and complexes of the targeted Doxil liposomes with streptavidin-coated 20 nm gold beads. An evagage size of unmodified Doxil particles – 80 nm.
They are secreted from the cell nearly free of intracellular components; their further purification could be easily accomplished by simple steps that do not differ from one clone to another. The major coat protein constitutes 98% of the total protein mass of the virion -- the purity hardly reachable in normal synthetic and bioengineering procedures. As a normal intestinal parasite, phage itself and its components are not toxic and have been already tested for safety in preclinical and clinical trials. In contrast to immunization procedure, the phage selection protocol may require tiny amounts of a target material (thousands of tumor cells available in biopsy procedure for obtaining the tumor-specific phage ligands, affinity and selectivity of which may be controlled by exploring well developed depletion and affinity maturation procedures.
| 10.48550/arXiv.0708.1457 | Fusion Phage as a Bioselective Nanomaterial : Evolution of the Concept | Valery A. Petrensko, S. -N. Ustino, I-Hsuan Chen | 3,090 |
10.48550_arXiv.1702.05017 | ###### Abstract
Research on black phosphorus (BP) has been experiencing a renaissance over the last few years, after the demonstration that few-layer BP exhibits high carrier mobility and a thickness-dependent band gap. For a long time, bulk BP is also known to be a superconductor under high pressure exceeding 10 GPa. The superconductivity is due to a structural transformation into another allotrope of phosphorous and accompanied by a semiconductor-metal transition. No superconductivity could be achieved for BP itself (that is, in its normal orthorhombic form) despite several attempts reported in the literature. Here we describe successful intercalation of BP by several alkali metals (Li, K, Rb, Cs) and alkali-earth Ca. All the intercalated compounds are found to be superconducting, exhibiting the same (within our experimental accuracy) critical temperature of 3.8\(\pm\)0.1 K and practically identical characteristics in the superconducting state. Such universal superconductivity, independent of the chemical composition, is highly unusual. We attribute it to intrinsic superconductivity of heavily-doped individual phosphorene layers, while the intercalated layers of metal atoms mostly play a role of charge reservoirs.
Bulk BP is the most thermodynamically stable phosphorus allotrope with a moderate direct bandgap (0.3 eV for bulk BP, increasing to 2 eV for monolayer phosphorene). Recent demonstrations of high mobilities, quantum oscillations, thickness-dependent gap and field-effect transistors with high on-off ratios using few-layer BP has led to a strong wave of interest in this material in its quasi-two-dimensional form. Furthermore, the band gap of few-layer BP has been predicted to be tuneable by strain and electric field, whereas surface doping of few-layer BP with potassium was found to result in a metallic state. While the interest in semiconducting BP is focused on its promise for nanoelectronics and nanophotonics, metallic BP was predicted to have sufficient electron-phonon coupling to become a superconductor.
Metallic state and superconductivity in phosphorous were previously achieved in the 1960s (ref.) by applying high pressure. The transition to the metallic state was shown to be associated with the structural transition from the orthorhombic to simple-cubic crystal lattice, which actually means that the latter material was no longer BP but somewhat closer to white phosphorous that also has a cubic lattice. The superconducting transition was found at 4.8K at 10 GPa and the transition temperature, \(T_{\rm c}\), could be further increased to 9.5K at 30 GPa (refs.). The question whether it is possible to induce superconductivity in BP by other means has remained open. Recent experiments on electrostatically-doped thin flakes of BP (ref.) did not find superconductivity, despite being able toachieve carrier concentrations well above 10\({}^{13}\) cm\({}^{-2}\), sufficient to induce superconductivity in, e.g., electrostatically-doped MoS\({}_{2}\) (ref.). There were also several attempts to obtain intercalated compounds of BP by reacting it with alkali metals (K, Cs and Li), iodine, and AsF\({}_{5}\) (ref.) and by electrochemical lithiation. No stable intercalation compounds could be achieved in either of these studies, and no superconducting response was reported or discussed.
To overcome the problem of disintegration of BP crystals when in contact with highly reactive vapours of alkali metals, we have employed an alternative technique of liquid ammonia intercalation. This allowed us to achieve successful intercalation of BP crystals with several alkali and alkali-earth metals: lithium (Li), potassium (K), rubidium (Rb), caesium (Cs) and calcium (Ca). All our intercalation compounds show superconductivity with \(T_{c}\) of 3.8 K, the same value within our experimental accuracy of \(\pm\)0.1 K. We emphasise that \(T_{c}\) does not depend on the intercalating metal, which indicates that the superconductivity is an intrinsic property of electron-doped phosphorene (individual layers of black phosphorus), as described below.
The atomic arrangements expected for pristine and intercalated BP are shown schematically in Fig. 1a,b (refs.). BP consists of weakly-bonded phosphorene layers, within which covalently bonded P atoms form a honeycomb network, similar to graphene, but each layer is puckered with a zigzag-shaped edge along the \(x\) axis and an armchair-shaped edge along the \(y\) axis. To achieve metal intercalation and obtain intercalated compounds M\({}_{x}\)P (here M stands for Li, K, Rb, Cs and Ca), crystals of black phosphorus were immersed in a metal-liquid ammonia solution at a temperature of -78degC in a dry ice/isopropanol (IPA) bath (see Methods for details). The starting solution has a characteristic deep blue colour due to dissociation of metal atoms into solvated cations (M\({}^{+}\)) and solvated electrons (e), ref..
**Structure and composition of pristine and intercalated black phosphorus.****(a, b)** Schematic structure of black phosphorus before (a) and after (b) intercalation. \(\cdot\)P’ stands for phosphorus atoms and \(\cdot\)M’ represents intercalating metals Li, K, Rb, Cs or Ca. The shown positions of metal atoms correspond to the results of first-principle calculations. Those suggested that the adjacent phosphorene layers should slide with respect to each other, effectively changing the stacking order of phosphorene layers and providing a maximum space for the metal ions. **(c, d)** Cross-sectional scanning electron microscopy images of pristine (c) and Cs-intercalated (d) BP. Scale bars, 1 μm. **(e)** Typical EDS spectra for pristine and intercalated BP.
This allowed us to monitor the process visually. Intercalation was also apparent from significant swelling of the crystals, with an expansion along the z axis (for example, a sample with thickness of 0.4 mm expanded to 1.0 mm). The intercalation process started at the surface of the host crystal and, as described below (see also ref.), resulted in a superconducting fraction below 100% (up to 10% if thin BP crystals were used). We estimate that only ~10 mm thick surface layers were fully intercalated.
To determine a chemical composition of our intercalated compounds, we used energy dispersive X-ray spectroscopy (EDS). EDS typically probes a few-micron thick surface layer and is therefore the most appropriate method to determine the stoichiometry of our surface-intercalated samples. EDS spectra for our M\({}_{x}\)P are shown in and the corresponding elemental maps in Supplementary Figure S1. The distribution of intercalated metal atoms is fairly uniform for all M\({}_{x}\)P compounds, and our EDS analysis yielded the average concentrations of 20 at.%, 17 at.%, 18 at.%, and 8 at.%, for K, Rb, Cs and Ca intercalated crystals, respectively (typical error in these measurements was 3 at %). Within this accuracy, the found concentrations correspond to the same composition of monovalent alkali-metal intercalated BP (namely, K\({}_{x}\)P, Rb\({}_{x}\)P, Cs\({}_{x}\)P where x \(\approx\) 0.2) and an approximately twice lower content for divalent Ca (Ca\({}_{x}\)P with x \(\approx\) 0.1). It was not possible to determine the concentration of Li using EDS because the technique is insensitive to elements with atomic numbers <5. The above compositions are stoichiometric (or stage-l, ref.), in agreement with the first-principles calculations for Li, Na and Mg intercalation, which predicted that the metal atoms should occupy certain positions between the puckered hexagonal layers, as shown in
We have also used X-ray diffraction (XRD) in order to probe the interlayer spacing after intercalation. For pristine black phosphorus our XRD spectra exhibited three main characteristic peaks at 20 = 17.1\({}^{\circ}\), 34.4\({}^{\circ}\) and 52.5\({}^{\circ}\), corresponding to, and crystallographic planes, respectively (Supplementary Figure S2a). This gives a layer spacing of 5.18 A, in agreement with other studies. However, after intercalation, while the peaks characteristic of pristine BP have almost disappeared, as expected, no new peaks corresponding to an expanded crystal lattice are visible. We attribute this to the fact that intercalation is limited to ~10um-thick layers near the surface, while X rays probe the entire volume of the crystals. Let us mention that such disappearance of XRD peaks is not unusual and was also reported for BP electrochemically intercalated with Na\({}^{+}\) (ref.). However, in contrast to the latter study, in our experiments the peaks corresponding to pristine BP re-appeared after exposure of the intercalated samples to air for about 100 hours (Supplementary Figure S2b) and all signs of superconductivity disappeared (Supplementary Figure S3). This indicates that, unlike the irreversible electrochemical intercalation, our procedure preserved the original structure of the puckered BP layers.
To detect the superconducting response, we used SQUID magnetometry (Methods). shows the temperature dependence of dc magnetic susceptibility _X = M/H_ for all our intercalated BP compounds under zero-field-cooling (ZFC) and field-cooling (FC) conditions, and provides examples of the magnetization as a function of applied magnetic field \(H\). Here \(M\) is the magnetic moment. Both ZFC and FC curves show a sharp increase in diamagnetic susceptibility at 3.8 K, characteristic of a superconducting transition. To find the transition temperature, we used the derivatives of the ZFC curves, and _T_c for each intercalated sample was determined as the temperature at which _dY_/_dT_ exhibited a sharp increase (see inset in and Supplementary Figure S4a). This yielded _T_c =3.8\(\pm\)0.1 K for all M\({}_{\text{x}}\)P, independent of the metal and whether it is monovalent or divalent. Note that we found no correlations between the observed slight variations in _T_c and the intercalant (the same spread of \(\pm\)0.1 K in _T_c was found for different samples intercalated with the same metal; Supplementary Fig. S4b).
To further characterize the superconductivity in our samples, we measured ac susceptibility, _X_ac, as a function of both \(H\) and \(T\). This allowed us to accurately determine the orientation-dependent critical field, _H_c corresponding to the disappearance of superconductivity, as shown in Fig. 3a, where the dc magnetization and ac susceptibility for a Cs-intercalated BP are shown together (more examples of ac susceptibility are shown in Supplementary Figure S5). These measurements were also used to find the temperature and orientation dependence of _H_c as shown in The _H_c values were determined both from _X_ac(_H_,_T_) (and Supplementary Figure S5) and _M_(_T_,_H_) curves (and Supplementary Figure S6), with good agreement between the values obtained by the two methods. Within our experimental accuracy, both parallel and perpendicular _H_c(_T_) are universal for all M\({}_{\text{x}}\)P, i.e. do not depend on the intercalant. This is similar to their intercalant-independent _T_c.
**Superconductivity in intercalated black phosphorus.****(a)** Temperature dependence of magnetic susceptibility \(X\) for Li, K, Rb, Cs and Ca intercalation. Shown are ZFC (solid symbols) and FC (open symbols) measurements for an in-plane magnetic field of 10 Oe. The arrow indicates _T_c as determined from _dY_/_dT_. The inset shows how _T_c was determined for individual samples: It is defined as the sharp change in _dY_/_dT_ (see also Supplementary Fig. S4). **(b)** Magnetization \(M\) as a function of magnetic field for our intercalated compounds.
Let us note that the superconducting fraction in the measurements shown in Figs. 2-4 was rather small, \(\sim\)1-2%, which we attribute to the dynamics of the intercalation process, as already mentioned above. It is much easier for metal ions to start intercalating between outer phosphorene layers, so that the process proceeds from the surface of each crystal into its bulk. The data in Figs. 2-4 were obtained on relatively large individual crystals, \(\sim\)3\(\times\)2\(\times\)0.3 mm, in order it would be possible to determine their orientation with respect to the applied magnetic field. By intercalating thinner crystals, with larger surface-to-volume ratios (Methods), we were able to increase the superconducting fraction to \(\sim\)10% (Supplementary Figure S7), which confirms the bulk nature of superconductivity.
**Magnetic field dependence of magnetization in M\({}_{\rm P}\).****(a)** dc magnetization, \(M\), and real part of ac susceptibility, \(m^{\prime}\), as a function of in-plane magnetic field, \(H\). For clarity, only data for Cs-intercalated BP are shown. Open symbols are data points measured as the dc field varied from -1kOe to +1kOe and from +1kOe to -1kOe. Solid lines show the same data smoothed using Savitzky-Golay filter. \(H_{\rm p}\) and \(H_{\rm c}\) denote the penetration and critical magnetic field, respectively. Inset: Zoomed-up \(m^{\prime}\) near the critical field in two orientations, with the arrows indicating \(H_{\rm c}^{\parallel}\) and \(H_{\rm c}^{\perp}\). **(b, c)** Temperature dependences of \(M(H)\) and \(m^{\prime}(H)\). Only data for Cs\({}_{\rm s}\)BP are shown. Both types of measurements give the same values of \(T\)-dependent \(H_{\rm p}\) and \(H_{\rm c}\).
There are several notable features in the \(H\) dependence of intercalated BP's magnetization, both dc and ac (Figs. 2b and 3). Firstly, the \(M(H)\) curves are practically identical, apart from somewhat different absolute values of the maximum diamagnetic moment for different intercalating metals in Fig. 2b, which is related to different superconducting fractions. Indeed, all the compounds exhibit the same penetration field, \(H_{\rm p}\) (1.8K) \(\sim\) 180 Oe (determined from dc magnetization curves as shown in Fig. 3a), and the same magnetic field corresponding to the disappearance of the diamagnetic response, \(H_{\rm c}\) (1.8K) \(\sim\) 500 Oe. Secondly, the sharp fall in \(M\) just above \(H_{\rm p}\) is reminiscent of the behaviour of type-I, rather than type II, superconductors. In the former case, the magnetic field penetrates either uniformly at the thermodynamic critical field, \(H_{\rm m}\), destroying superconductivity or, for a finite demagnetization factor, in the form of normal domains. The relatively rapid decrease is also seen in ac susceptibility. However, there is a clear difference from a typical type I superconductor with the Ginzburg-Landau parameter \(\kappa<<\) 1 if we compare our \(M(H)\) curves in with \(M(H)\) for indium that we used as a reference (Supplementary Figure S8). The 'tail' on the \(M\) and \(m^{\prime}\) curves seen at \(H>\)300 Oe corresponds to small but still finite diamagnetic response above \(H_{\rm p}\). This behaviour suggests that the intercalated BP is probably a borderline type I/type II superconductor with \(\kappa\) close to the critical value of \(\approx\)0.7, similar to Nb (ref.). Thirdly and very unusually, the \(M(H)\) curves for all M\({}_{\rm x}\)P compounds display practically the same hysteresis (trapping of the magnetic flux). The latter corresponds to pinning of the magnetic flux by defects or impurities and, therefore, is sensitive not only to a particular chemical compound, but also to structural details of individual samples. The fact that, in our case, different intercalated compounds trap the same amount of flux at \(H=\)0 also points in the direction of type-I superconductivity because pinning of superconducting domains is known to be less sensitive to crystal imperfections.
To relate the appearance of superconductivity with the structural changes in BP which occur as a result of intercalation, we used Raman spectroscopy (Supplementary Figure S9).
**Transition temperature as a function of magnetic field.****(a)** Temperature dependence of ZFC dc susceptibility \(\chi\) at different \(H\) applied parallel to the surface (ab plane). For clarity, only data for Cs-intercalated BP are shown. **(b)** Parallel and perpendicular critical fields as a function of temperature for all M\({}_{\rm x}\)P compounds in this study. Error bars indicate the accuracy of determination of the critical temperature from \(\chi(T)\) curves such as those shown in (a) (open symbols) and of the critical field from m\({}^{\prime}(H)\) (solid symbols). Solid curves are fits to the BSC expression \(H_{\rm c}\propto 1-(T/T_{\rm c})^{2}\).
Raman spectroscopy is sensitive to the properties of a few near-surface layers and, therefore, probes the same part of the crystals as EDS and magnetization measurements. Atomic displacements corresponding to the Raman modes are illustrated in Supplementary Figure S9a. There are three main phonon modes that in our pristine BP result in peaks at 362, 440 and 467 cm-1 (see Supplementary Figure S9b). From comparison of the Raman spectra for pristine and intercalated BP it is clear that intercalation has only a small effect on atomic vibrations of P atoms. The modes are slightly broadened but essentially unchanged, which is consistent with the preserved structure of phosphorene layers and in agreement with the mentioned experiments in which samples exposed to air recovered their initial structural state. At the same time, one can see a clear shift of all three modes to lower frequencies, corresponding to phonon softening within phosphorene layers. Such softening is expected as a result of the doping of phosphorene, which according to the first-principle calculations should result in charge transfer of 0.8e- per P atom for monovalent and 1.5e- for divalent donors.
The most surprising result of our work is the practically identical superconducting characteristics of all the tested intercalated compounds, independent of the mass and atomic radius of the intercalating metal and its valency, i.e. the lack of any isotope-like effect. This is in stark contrast to other superconducting intercalated compounds, of which intercalated graphite is an archetypal example and has obvious structural similarities with BP. In the case of intercalated graphite, its superconducting properties are strongly dependent on the chemical composition, with the critical temperature ranging from 0.02 K for Cs-intercalated graphite to 11.5 K for CaC6 (ref.) and LiC6 not superconducting at all. The mechanism of superconductivity in the intercalated graphite compounds is well understood. In terms of the standard phonon-mediated superconductivity, superconductivity in graphite cannot be achieved within individual electron-doped graphene layers because the frequencies of in-plane graphene phonons are too high to lead to efficient electron-phonon coupling, whereas the planar structure of graphene forbids coupling between the n* electronic states and softer out-of-plane vibrations. Therefore, an incomplete ionization and a partially occupied metal-derived electronic band is a key for achieving superconductivity in intercalated graphite, and the superconductivity cannot be attributed to either the graphene n* band or the metal-derived electronic band alone. Instead, the interaction between these two bands plays a critical role, according to theory, and the transition temperature is strongly dopant dependent.
The situation in black phosphorus is quite different. The experiment clearly shows that the chemical composition of intercalated BP is irrelevant for the occurrence of superconductivity, and one has to find a physical mechanism relying on heavy electron doping of individual phosphorene layers. This is in good agreement with theoretical expectations. Indeed, the puckered structure and sp\({}^{3}\) hybridization of phosphorene layers remove the symmetry constraints on electron-phonon coupling, with more phonon modes able to contribute, increasing the overall coupling constant, \(\lambda\). In addition, electron transfer from intercalant atoms to phosphorene softens its vibrational modes, especially the sublayer breathing modes, further enhancing \(\lambda\) without the need of changing the crystal symmetry of phosphorene layers. In our experiments such phonon softening is clear from the Raman spectra, where a similar frequency shift is observed for all intercalants (Supplementary Figure S9b). Furthermore, recent calculations for Li-intercalated bilayer phosphorene indicate that the electronic bands near the Fermi level of intercalated BP are mainly \(\pi\)*-like bands derived from phosphorene, while the contribution to the density of states from the metal-derived band is very small. This basically means that the superconductivity is intrinsic to individual phosphorene layers rather than the entire compound, in agreement with our experiments. Let us mention that the theoretical explanation is also consistent with recent ARPES measurements of the BP intercalated with K (ref.) or Li (ref.), which found no additional interlayer bands that could be attributed to the alkali metals.
The intercalant-independent \(T_{\rm c}\) implies the same doping level induced within phosphorene layers. The situation is somewhat similar to equal doping and intercalant-independent \(T_{\rm c}\) in K, Rb and Cs-intercalated MoS\({}_{2}\) (refs. 40,41). This is also in agreement with theory. The first-principle calculations found an optimum charge transfer for monovalent alkali atoms (Li and Na) of 0.8 e' per P atom and for divalent Mg of 1.5 e'. It is reasonable to expect the same electron doping for K, Rb and Cs that were not covered in the calculations and a close doping for divalent Ca that is chemically similar to Mg and intercalates in twice smaller concentrations than alkali metals. Furthermore, several studies, both theoretical and experimental found that the stoichiometric composition M\({}_{0.2}\)P found in our studies is the maximum - and probably optimum - alkali content, for which the structure of phosphorene layers is preserved and the intercalation can be reversed. If more metal atoms are forced into the space between phosphorene layers by, e.g., electrochemical intercalation, this leads to irreversible changes including chemical bonding and formation of new alloys. Taken together, the available evidence suggests that the amount of intercalating metal atoms in BP is determined primarily by an electrostatic-like equilibrium between the charged phosphorene layers and metal ions. In our case, this conclusion is supported by the factor-of-2 difference in the concentration of monovalent and divalent intercalants as found experimentally.
In summary, we have achieved successful intercalation of black phosphorus with several alkali and alkali-earth metals using the liquid ammonia method. The intercalated compounds exhibit universal superconducting properties: same \(T_{\rm c}\) = 3.8\(\pm\)0.1K, same critical fields \(H_{\rm c}^{||}\) \(\sim\) 630 Oe and \(H_{\rm c}^{\perp}\) \(\sim\) 440 Oe and even similar flux pinning. The lack of variations strongly suggests a physical mechanism behind the superconductivity, such as doping of phosphorene layers by intercalating atoms. In this respect, intercalated BP is very different from extensively-studied intercalated graphite compounds, despite their structural similarities. In the latter case, both electronic and phonon contributions from intercalated metal layers play a key role in inducing superconductivity. In intercalated BP, the superconductivity can entirely be attributed to electron-doped phosphorene whereas changes in the electron and phonon spectra induced by intercalation play little role, according to the experiment and in qualitative agreement with theory. Nevertheless, the degree to which the superconducting behavior disregards the chemical composition is puzzling and perhaps suggests some more fundamental rules at play behind the observed superconductivity.
## Methods
To achieve intercalation, BP crystals (99.998%, Smart Elements) and a desired metal (99.95% Li, K, Rb, Cs and Ca from Aldrich) were sealed in a quartz tube under the inert atmosphere of a glove box with oxygen and moisture levels less than 0.5 ppm. The reactor tube was then evacuated to \(\approx\)10\({}^{5}\) mbar, connected to a cylinder of pressurised ammonia (99.98% from CK Gas) and placed in a bath of dry ice/ethanol (-78 degC). As gaseous ammonia was allowed into the reactor tube, it condensed onto the reactants (BP and the metal), dissolving the metal and forming a deep-blue solution. The colour depth was used as an indicator of the concentration of dissolved metal. To prevent contamination with oxygen or moisture, the empty space in the reactor tube left after ammonia condensation was filled with argon gas (Zero-Grade from BOC). The tube was kept in the dry-ice/ethanol bath for 48 h, after which the system was warmed up, ammonia and argon evacuated and the intercalated BP recovered inside the glove box. As alkali-metal intercalated compounds are extremely sensitive to oxygen and moisture, they were handled in the inert atmosphere of the glovebox, immersed in paraffin oil or protected by using sealed containers. As a reference, we tested our method by intercalating MoS\({}_{2}\) with different alkalis, which produced superconducting samples with \(T_{\mathrm{c}}\approx\)6.9K, in agreement with literature. In addition, to ensure that the observed superconductivity could not be related to accidental contamination, we used mixtures of BP and alkali metals (without placing them in liquid ammonia) and measured their magnetization. No superconducting response could be detected unless the intercalation reaction was induced.
The chemical composition was determined using EDS (Oxford Instruments X-Max detector integrated with Zeiss' Ultra scanning electron microscope). The metal content was determined at 10 or more randomly selected areas on each sample and the average was taken as the intercalant concentration.
X-ray powder diffraction (XRD) measurements were performed using Bruker D8 Discover diffractometer with Cu Ka radiation (\(\lambda\) = 1.5406 A). Before each measurement, samples were roughly ground and mixed with paraffin oil, and then sealed in an airtight XRD sample holder (Bruker, A100B36/B37) inside the glove-box. XRD spectra were obtained at room temperature in the 2th range of 5-75deg with a step of 0.03deg and a dwell time of 0.5 s/step.
Raman spectroscopy was performed at room temperature using the Renishaw inVia Reflex system with a 532-nm laser. All spectra were recorded using a power of \(\sim\)1 mW with the samples sealed between two thin glass plates.
Magnetization measurements were carried out using Quantum Design MPMS-XL7 SQUID magnetometer. To protect the intercalated samples from degradation they were immersed in paraffin oil and sealed inside polycarbonate capsules in a dry argon atmosphere of the glovebox. Measurements were carried out in a dc magnetic field applied either parallel or perpendicular to the crystal surface. Temperature dependences were measured in zero-field cooling and field-cooling modes. In the former, the samples were cooled in zero magnetic field from \(\sim\)10 to 1.8 K, then the field \(H\) was applied and \(M\) was measured as the temperature increased from 1.8 to, typically, 6 K. In latter mode, _M_(_T_) was measured as the temperature decreased from above 6 to 1.8 K. The ac susceptibility was measured using ac magnetic fields (typically, 1 Oe at 8 Hz) applied parallel to the dc field.
To estimate the superconducting fraction in different samples, we used the initial (Meissner) slope of the _M_(_H_) curves (for a 100% superconducting sample the slope should be -1/4\(\pi\)). This yielded a fraction of \(\sim\)1-2% if relatively large (\(\sim\)3\(\times\)2\(\times\)0.3 mm) crystals were intercalated. To increase the intercalated volume, we used two different methods. First, a crystal of BP was ground into a powder consisting of many smaller platelet-shaped crystals that were then intercalated as described above.
This increased the superconducting fraction by a factor of \(\sim\)3. However, we believe that the smallest BP crystals within this powder were non-superconducting, reacting with minute amounts of oxygen and moisture left even under inert atmosphere. The second and more successful approach was to break up BP crystals into smaller ones during the intercalation process. To this end, the liquid ammonia solution was subjected to mild shaking using a magnetic stirrer. This allowed an increase in the superconducting fraction to \(\sim\) 10%.
# Supplementary Figure S9 | Raman spectra for pristine and intercalated black phosphorus.
(a) Schematic representation of the atomic displacements corresponding to Raman-active modes in BP. (b) Raman spectra for pristine and metal-intercalated black phosphorus (see labels). Dashed lines indicate the positions of the \(A_{g}^{1}\), B\({}_{\text{2g}}\) and \(A_{g}^{2}\) Raman peaks for pristine BP. | 10.48550/arXiv.1702.05017 | Intercalant-independent transition temperature in superconducting black phosphorus | Renyan Zhang, John Waters, Andre K. Geim, Irina V. Grigorieva | 2,417 |
10.48550_arXiv.1404.0278 | 10.48550/arXiv.1404.0278 | Spin photovoltaic cell | F. Bottegoni, M. Celebrano, M. Bollani, P. Biagioni, G. Isella, F. Ciccacci, M. Finazzi | 3,931 |
|
10.48550_arXiv.0811.2967 | ###### Abstract
We present a study of the local strain effects associated with vacancy defects in strontium titanate and report the first calculations of elastic dipole tensors and chemical strains for point defects in perovskites. The combination of local and long-range results will enable determination of x-ray scattering signatures that can be compared with experiments. We find that the oxygen vacancy possesses a special property -- a highly anisotropic elastic dipole tensor which almost vanishes upon averaging over all possible defect orientations. Moreover, through direct comparison with experimental measurements of chemical strain, we place constraints on the possible defects present in oxygen-poor strontium titanate and introduce a conjecture regarding the nature of the predominant defect in strontium-poor stoichiometries in samples grown via pulsed laser deposition. Finally, during the review process, we learned of recent experimental data, from strontium titanate films deposited via molecular-beam epitaxy, that show good agreement with our calculated value of the chemical strain associated with strontium vacancies.
pacs: 61.72.Bb,61.72.Hh,61.72.jd,62.20.D-
## I Introduction
Perovskites in general, and strontium titanate in particular, are some of the most frequently and exhaustively studied materials in solid-state physics and chemistry. This attention has largely derived from their diverse and interesting properties: high piezoelectricity, quantum paraelectricity, ferroelectricity, uniaxial stress, and colossal magnetoresistance. Further, the cubic perovskites manifest intriguing effects of underlying quantum fluctuations, since, although they are comprised of relatively heavy atomic constituents, a number of competing structures are energetically and structurally similar. Characterization of the low-temperature order parameters of these materials remains an open and engaging question.
Strontium titanate is a model perovskite -- commonly available and reflective of many of the above properties of that material family. Specifically, while strontium titanate is a wide band-gap insulator at room temperature, it exhibits semiconductivity at elevated temperatures through doping or non-stoichiometric composition and superconductivity at low temperatures through reduction via addition of oxygen. The structural phase diagram of strontium titanate comprises a high-temperature cubic phase and a low-temperature tetragonal phase, with a transition temperature near 105 K. The cubic perovskite structure is particularly interesting due to the richness of its phase diagram (nonpolar antiferrodistortive to ferroelectric to antiferroelectric phases), to the capacity of these phases to emerge from miniscule deviations from the cubic lattice and its skeleton of octahedral oxygens (often through rigid rotations of such), and to open questions regarding the types (displacive or order-disorder) of the transitions among these phases.
Defects and vacancies play a particularly important role in the chemistry of perovskites and deserve continued study in strontium titanate due to the electronic and superconducting effects of doping as well as their role in the interface region of heterostructures. In the dilute limit, the mechanics of defects are fully determined by an examination of stress-strain effects, in particular the elastic dipole tensor, which motivates this work's emphasis on such a quantity. Our presentation of both short-range displacements around a point defect as well as long-range effects (characterized by the elastic dipole tensor) allows for the calculation of x-ray scattering signatures. These quantities also enable the prediction of defect mechanics, such as the behavior of defects within an externally imposed strain gradient (as present in heterostructures), as well as the ratio of chemical strain to stoichiometric deviation (a direct experimental observable). Finally, through comparison of our predictions of chemical strain with experimental results, we draw a number of important conclusions regarding the nature of point defects in non-stoichiometric strontium titanate.
## II Background
Point defects introduce lattice distortions on both local and long-range scales. While the short-range distortions must be described by a potentially large set of atomic displacements, the long-range _elastic_ distortions may be completely described by a single tensor, the elastic dipole tensor.
The elastic dipole tensor and its relation to elastic effects may be understood by the following simple considerations. Consider to quadratic order the most general expansion of the free energy per unit volume of a crystal in terms of the strain \(\epsilon_{ij}\) (\(i\) and \(j\) refer to coordinate axes) and the number of defects per unit volume \(n_{d}\),
\[f\left(\epsilon_{ij},n_{d}\right) =f_{0}+n_{d}\,E_{d}+\tfrac{1}{2}\,n_{d}^{2}\,E_{dd}\] \[\qquad+\tfrac{1}{2}\sum_{ijkl}C_{ijkl}\,\epsilon_{ij}\,\epsilon_{ kl}-n_{d}\sum_{ij}\epsilon_{ij}\,G_{ij}. \tag{1}\]
The Taylor expansion coefficients \(E_{d}\), \(E_{dd}\), \(C_{ijkl}\), and \(G_{ij}\) are, respectively, the defect formation energy, an average inter-defect interaction energy, the components of the _elastic stiffness tensor_ of the material, and the components of the _elastic dipole tensor_ of the defects.
\[-\frac{\partial f}{\partial\epsilon_{ij}}\equiv\sigma_{ij}=-\sum_{kl}C_{ijkl} \,\epsilon_{kl}+n_{d}\,G_{ij}. \tag{2}\]
To isolate the elastic dipole tensor, we can consider the rate of change of the stress in the crystal per unit concentration of defects, while holding _strain fixed_, that is, under _strain control_. Although challenging experimentally, strain control is quite convenient computationally since it corresponds to performing calculations with fixed lattice vectors.
\[\frac{\partial\,\sigma_{ij}}{\partial\,n_{d}}\bigg{|}_{\mathbf{\epsilon}}=\,G_{ij}, \tag{3}\]
Alternatively, we can also consider the derivative of the strain in the crystal per unit concentration of defects under _stress_ or _load control_ (holding stress fixed). While stress control is computationally more complicated than strain control, it is the most common experimental situation. Under experimental conditions, the crystalline lattice vectors relax such that there is essentially zero stress (under normal laboratory conditions, atmospheric pressure corresponds to a negligible stress). This criterion allows the relation of strain to a newly defined quantity, \(\mathbf{\Lambda}\),
\[\frac{\partial\,\epsilon_{ij}}{\partial\,n_{d}}\bigg{|}_{\mathbf{\sigma}}=\sum_{kl }S_{ijkl}\,G_{kl}\,\equiv\,\Lambda_{ij}, \tag{4}\]
Relying upon the above derivations, the numerical calculation of the elastic dipole tensor \(\mathbf{G}\) is straightforward. We compute the stress induced with the introduction of a single defect in a supercell (maintaining fixed lattice vectors, but allowing relaxation of the atomic coordinates).
\[G_{ij} =\tfrac{1}{n_{d}}\left(\sigma_{ij}^{\,d}-\sigma_{ij}\right) \tag{5}\] \[=V_{\circ}\,\Delta\,\sigma_{ij},\]
Once \(\mathbf{G}\) is known, \(\mathbf{\Lambda}\) may also be computed directly from Equation. As a practical matter, we note that in this approach, the lattice vectors need not be those of a fully relaxed bulk crystal, provided the strain is small and kept fixed.
Experimental works often report the variations in _chemical strain_, the strain due to the presence of defects, with respect to stoichiometric deviations in the crystalline chemical formula. From the above considerations, the chemical strain is \(\mathbf{\epsilon}_{c}\equiv n_{d}\mathbf{\Lambda}\). Deviations in stoichiometry specify the number of defects per chemical unit, \(\delta\), so that, in this context, the concentration of defects per unit volume is \(n_{d}=\delta/V_{c}\), where \(V_{c}\) is the volume of the chemical unit. These two relations then immediately provide the chemical strain as proportional to this _stoichiometric defect deviation_, \(\delta\),
\[\mathbf{\epsilon}_{c}=\left(\frac{\mathbf{\Lambda}}{V_{c}}\right)\delta. \tag{6}\]
Experimentally, one does not generally obtain a full tensor for \(\mathbf{\epsilon}_{c}\) but, instead, an average over all equivalent defect orientations which restore the symmetry of the underlying crystal. In cubic crystals, one measures a scalar \(\epsilon_{c}\) which corresponds to the mean diagonal component of \(\mathbf{\epsilon}_{c}\).
## III Methods
To simulate strontium titanate, we employ a shell-potential model parameterized for strontium titanate. Shell-potential models are formulated as an extension to ionic pair potentials and employed to capture the polarizability of the atomic constituents. The shell model separates each ion into two parts, a core and an outer shell, which possess individual charges that sum to the nominal charge of the ion.
\[U\equiv U_{P}+U_{C}+U_{B}, \tag{7}\]
The polarizability is captured by harmonic springs connecting the core and shell of each ion, so that \(U_{P}\) has the form,
\[U_{P}=\tfrac{1}{2}\sum_{i}k_{i}\left|\Delta r_{i}\right|^{2}, \tag{8}\]
The Coulomb contributions take the form,
\[U_{C}=\tfrac{1}{2}\sideset{}{{}^{\prime}}{\sum}_{i,j}\frac{k_{c}q_{i}q_{j}}{r _{ij}}, \tag{9}\]where \(i\) and \(j\) range over all cores and shells (excluding terms where \(i\) and \(j\) refer to the same ion), \(q_{i}\) and \(q_{j}\) are the corresponding charges, \(r_{ij}\) is the distance between the charge centers, and \(k_{c}\) is Coulomb's constant. We compute this Coulombic interaction using a Particle Mesh Ewald algorithm with all real-space pair-potential terms computed out to a fixed cutoff distance using neighbor tables.
\[U_{B}=\tfrac{1}{2}\sum_{i,j}\left(A_{ij}e^{-r_{ij}/\rho_{ij}}-C_{ij}r_{ij}^{- 6}\right), \tag{10}\]
Here, the first term (Born-Mayer) serves as a repulsive short-range interaction to respect the Pauli exclusion principle, and the second term (Lennard-Jones) models the dispersion or van der Waals interactions. The specific electrostatic and short-range shell-model parameters used in this study were fit to strontium titanate by Akhtar et al., with values as listed in Tables 1 and 2. Finally, we wish to emphasize again, as it is rarely mentioned explicitly in the shell-potential literature, that the pair-potential terms in \(U_{B}\) apply to the _shells only_, and _not_ to the cores.
Shell models have been extensively used for decades as the primary empirical potential for modeling perovskites and other oxides. We tested the correctness of our coded implementation of this potential through comparisons of lattice constants and elastic moduli and find excellent agreement. For instance, using the same shell potential and ground-state structure, we predict a lattice constant for cubic strontium titanate of 3.881 A, which is within 0.3% of the value calculated by Akhtar et al. For the elastic moduli, we find \(C_{11}=306.9\) GPa, \(C_{12}=138.7\) GPa, and \(C_{44}=138.8\) GPa, which are within 1.8%, 1.0%, and 0.7%, respectively, of the values from Akhtar et al. From this, we conclude that our implementation of the potential is correct.
We further note that the static dielectric constant of 216.99, as calculated by Akhtar et al. for the same parameter set as our shell potential, is 28% lower than the experimental value for strontium titanate of 301.00, a level of agreement typical of results from empirical potentials. As Gillan and Stoneham discuss, there exists an important electrostriction effect whereby the tendency of the crystal to screen electric fields impacts the elastic dipole tensor. (Note that this effect scales as \(1-\varepsilon^{-1}\), where \(\varepsilon\) is the static dielectric constant.) Given the high dielectric constant of strontium titanate, this screening is nearly perfect in both the actual experiment and our model case; therefore, we expect that this effect is captured well in our calculations below, despite our relatively large fractional error in the static dielectric constant.
As is well known, strontium titanate has a large number of similar, competing ground-state structures. We should emphasize, at this point, that the main quantities of interest to this study, either local atomic displacements or the elastic dipole and defect-strain tensors [from Equations-], are all defined as defect-induced changes relative to the bulk structure and so are likely quite insensitive to which of the competing structures are used to represent the bulk.
Accordingly, we have carried out what we regard as a thorough, but not exhaustive, search for a _probable_ ground-state structure. Indeed, we have found no alternative structure which relaxes to an energy less than our candidate ground-state structure within our potential. We performed quenches on hundreds of random displacements from the idealized positions of the \(1\times 1\times 1\) primitive unit cell to explore various potential reconstructions for supercells up to \(6\times 6\times 6\). We also considered a number of highly ordered, human-inspired configurations commensurate with the antiferrodistortive disordering that is observed experimentally and predicted theoretically. Among those minima which we explored, we selected the lowest-energy configuration to serve as the bulk crystalline state throughout this study. This configuration possesses a fairly regular pattern, namely, each oxygen octahedron rotates slightly along \(\pm\) (trigonal) directions in an alternating three-dimensional \(2\times 2\times 2\) checkerboard pattern. To aid in visualizing this reconstruction, depicts the atomic structure of reference non-reconstructed cubic strontium titanate, with the oxygen octahedra lattice indicative of perovskite materials.
We investigated five defects in strontium titanate: the oxygen, strontium, and titanium vacancies and the strontium-oxygen and titanium-oxygen divacancies. Since the octahedra rotations break the original crystalline symmetry and generate a set of different symmetry-related reconstructions, each of these five defects can be situated in multiple equivalent sites within each reconstruction.
\begin{table}
\begin{tabular}{c c c c} \hline \hline Interaction & A [eV] & \(\rho\) [Å] & C [eV \(\cdot\) Å\({}^{6}\)] \\ \hline S\({}^{2+}\)\(\Leftrightarrow\) O\({}^{2-}\) & 776.84 & 0.35867 & 0.0 \\ Ti\({}^{4+}\)\(\Leftrightarrow\) O\({}^{2-}\) & 877.20 & 0.38096 & 9.0 \\ O\({}^{2-}\)\(\Leftrightarrow\) O\({}^{2-}\) & 22764.3 & 0.1490 & 43.0 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Short-range shell-model potential parameters for strontium titanate (from Akhtar et al.).
\begin{table}
\begin{tabular}{c r r r} \hline \hline Ion & Shell & Core & Spring Constant \\ & Charge [e] & Charge [e] & [eV \(\cdot\) Å\({}^{-2}\)] \\ \hline Sr\({}^{2+}\) & 1.526 & 0.474 & 11.406 \\ Ti\({}^{4+}\) & \(-35.863\) & 39.863 & 65974.0 \\ O\({}^{2-}\) & \(-2.389\) & 0.389 & 18.41 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Electrostatic shell-model potential parameters for strontium titanate (from Akhtar et al.).
Even well below room temperature, strontium titanate shows local fluctuations among the possible reconstructions. Thus, in addition to raw results for a specific realization of each defect in a given reconstruction, we also report results for each defect averaged over all possible reconstructions for a given orientation of the defect. The strontium and titanium vacancies do not select a specific direction and, thus, this averaging represents the full cubic crystalline symmetry group; their respective tensors therefore are always diagonal with cubic symmetry. The other defects do select specific crystalline directions, which must be specified when reporting the corresponding defect tensors. We thus reconstructionally average these latter defect tensors using either symmetry arguments or explicit calculations with different reconstructions, as appropriate. Finally, with a view to chemical strain measurements in macroscopic samples, we also report final averages over all defect orientations.
In the case of the oxygen vacancy, the oxygen sits between two unique nearest-neighbor titanium atoms, thus uniquely distinguishing the Ti-V\({}_{\text{O}}\)-Ti direction, which we define as, among the three cubic axes. Next, the titanium-oxygen divacancy selects a unique V\({}_{\text{Ti}}\)-V\({}_{\text{O}}\) direction, which we define as, directed from the titanium site toward the oxygen site. Finally, the strontium-oxy-gen divacancy selects a unique V\({}_{\text{Sr}}\)-V\({}_{\text{O}}\) direction, which we define as, directed from the strontium toward the oxygen site. Once the reconstructional averaging is accomplished, the average over defect orientations requires generation of the crystal's response to all different possible orientations (three for the oxygen vacancies, six for the titanium-oxygen divacancy, and twelve for the strontium-oxygen divacancy) and restores full cubic symmetry.
To obtain the results in Section IV, each of these defects was placed within the bulk-reconstructed strontium titanate supercell, with cubic symmetry as experimentally observed for \(T\gtrsim 105\) K, and then relaxed via the technique of preconditioned conjugate gradient minimization (specifically, the Polak-Ribiere method) to find the minimal energy configuration (to within \(\sim\)1 \(\upmu\)eV). Supercell convergence studies examined all five defects in cells containing up to 13 720 atoms and verified that such defects were sufficiently separated so that interactions across periodic supercell boundaries were negligible. The final relaxed atomic configurations in the largest cells (\(14\times 14\times 14\)) provide the local strain fields which we report below for each defect. To determine the long-range strain fields, we compute the elastic dipole tensor \(\mathbf{G}\) through the prescription described in Equation above; namely, we calculate the stress induced by the introduction of a single defect in the supercell, holding the lattice vectors fixed while allowing the atomic coordinates to relax.
## IV Results
For a series of important and fundamental strontium titanate defects, we examine both the elastic dipole and defect-strain tensors, as introduced in Section II, as well as the local strains surrounding each defect. This section first examines the role of the oxygen vacancy as a case study of a defect in strontium titanate. Subsequently, the results for the same set of studies are presented for four other point defects in strontium titanate: strontium and titanium vacancies and strontium-oxygen and titanium-oxygen divacancies. Section V continues with more general implications of our results.
### Oxygen vacancy
As described above, there are three distinct orientations for the oxygen vacancy as defined by the Ti-V\({}_{\text{O}}\)-Ti direction. Moreover, because of the reconstruction, there are in fact two distinct classes of site within each possible orientation. As one moves in the positive sense of direction along the Ti-V\({}_{\text{O}}\)-Ti axis, these sites are distinguished by whether the rotation of the octahedra surrounding the two titanium sites changes from positive to negative or from negative to positive (in the right-hand sense about the \(+\) direction). Below, we report results for the latter type of site.
We first examine the elastic dipole tensor computed according to Equation. To explore convergence, we compute the dipole tensor components in supercells of sizes \(2\times 2\times 2\), \(4\times 4\times 4\),..., \(14\times 14\times 14\), containing between 40 and 13 720 atoms with defect separations
Atomic structure of non-reconstructed cubic strontium titanate: oxygen octahedra (\(\blacklozenge\)) surrounding titanium atoms (\(\blacklozenge\)), with strontium atoms (\(\blacksquare\)) outside the octahedra and equidistant from the titanium sites.
depicts the convergence of the diagonal components of the elastic dipole tensor as a function of inverse linear dimension of the supercell. The linear behavior in the figure for large cells indicates that this quantity converges in the same way as the Coulomb interaction between defects and allows extrapolation of the fully converged value for these components in an infinite supercell. We observe exactly the same linear behavior with inverse linear dimension of cell for the convergence of the off-diagonal components of the dipole tensor.
\[\mathbf{G}_{\mathrm{O}}=\left(\begin{array}{rrr}4.53&-3.11&-3.11\\ -3.11&-2.13&1.06\\ -3.11&1.06&-2.13\end{array}\right)\mathrm{eV}.\]
As described above in Section III, at finite temperatures (\(T\gtrsim 105\) K), local fluctuations in the reconstruction make it appropriate to average this tensor over all reconstructions.
\[\overline{\mathbf{G}}_{\mathrm{O}}=\left(\begin{array}{rrr}4.53&0.00&0.00\\ 0.00&-2.13&0.00\\ 0.00&0.00&-2.13\end{array}\right)\mathrm{eV}.\]
Since the underlying, non-defected, crystal is now cubic, we can readily apply Equation to \(\overline{\mathbf{G}}_{\mathrm{O}}\) to obtain the reconstructionally averaged defect-strain tensor,
\[\overline{\mathbf{\Lambda}}_{\mathrm{O}}=\left(\begin{array}{rrr}16.33&0.00&0.00\\ 0.00&-8.05&0.00\\ 0.00&0.00&-8.05\end{array}\right)\hat{\mathrm{A}}^{3}.\]
The above results indicate that the oxygen vacancy tends to cause the crystal to expand along the Ti-V\({}_{\mathrm{O}}\)-Ti direction and to contract along the two orthogonal directions by an amount which results in negligible net volume change in the crystal. When the above tensor is averaged over all defect orientations (permutations of the three coordinate axes), the result is near perfect cancellation, resulting in a constant tensor (multiple of the identity) with a uniform chemical strain per unit defect concentration of \(+0.07\) A\({}^{3}\). This corresponds to a ratio of chemical strain \(\epsilon_{c}\) to the deviation from stoichiometry \(\delta\) in SrTiO\({}_{3-\delta}\) of \(\epsilon_{c}/\delta=+0.001\), indicating a _very slight_ tendency for the crystal to expand due to the presence of oxygen vacancies.
Following Figure 3, we now examine the local strain of this system after reconstructional averaging. We note that the removal of the oxygen ion, with a nominal charge of \(-2\), leaves an effective local positive environment in the location of the vacancy. We should expect pronounced Coulombic response to this environment in the form of local crystal polarization. Indeed, the nearest neighbors of the oxygen vacancy, the two titanium atoms, now move directly _away_ from the vacancy on the precise vector connecting them, by 0.21 A. The next-nearest neighbors are the eight oxygens comprised of two equidistant rings of four oxygens each, each of which move 0.21 A _toward_ the vacancy (0.19 A along \(\hat{e}_{1}\) with remaining projection of 0.08 A either along \(\hat{e}_{2}\) or \(\hat{e}_{3}\) as dictated by symmetry), and one ring of four strontium atoms, each of which
Local strain pattern for oxygen vacancy in the \(\hat{e}_{1}\hat{e}_{2}\) plane of strontium titanate: titanium atoms (\(\blacklozenge\)), oxygen atoms (\(\blacklozenge\)), oxygen vacancy (\(\blacklozenge\)). Atomic displacements exaggerated by a factor of three for clarity (\(\filleddiamond\)), displayed for significant in-plane displacements (\(>0.1\) Å) only.
Diagonal components of elastic dipole tensor for oxygen vacancy in strontium titanate. Data show linear convergence with inverse linear dimension of supercell size.
The third shell of neighbors, a set of six oxygens, each one lattice constant away from the vacancy along the three lattice directions in the crystal, moves different amounts depending upon the vector; the two oxygens along the Ti-V\({}_{\text{O}}\)-Ti direction move 0.06 A directly _away_ from the vacancy (pushed sterically by the second-shell titanium atoms), while the other four oxygens move by a negligible amount (only 0.01 A). While the fourth shell of neighbors, a set of eight titanium atoms at a distance \(\sqrt{5}/2\,a_{\circ}\) from the vacancy, moves negligibly, the fifth shell of neighbors (sixteen oxygen and eight equidistant strontium atoms) shows significant movement in the strontium atoms of 0.15 A _away_ from the vacancy (0.08 A along \(\hat{e}_{1}\) with remaining projections of 0.09 A along both \(\hat{e}_{2}\) and \(\hat{e}_{3}\)), even though the oxygen atoms move a negligible amount. Finally, in the sixth shell of neighbors, a set of twelve oxygen atoms, eight oxygens (those not in the plane containing the defect and perpendicular to \(\hat{e}_{1}\)) move 0.10 A _toward_ the vacancy (0.02 A along \(\hat{e}_{1}\) with remaining projection of 0.10 A either along \(\hat{e}_{2}\) or \(\hat{e}_{3}\) as dictated by the symmetry), while four others (those in the plane perpendicular to \(\hat{e}_{1}\)) move a negligible amount. All other atoms in the crystal move less than 0.06 A.
Finally, we would like to comment on the relation between local strains and defect tensors. We observe that the direction of motion of the near-neighboring atoms often correlates with the far-field motion described by the defect-strain tensor. In this case, in the first and second shells, we see a general pattern of movement which is away from the defect along \(\hat{e}_{1}\) and toward the defect in the other two directions, consistent with the signs in the long-range defect-strain tensor.
### Strontium vacancy
We now repeat the above procedures to obtain similar results for the strontium vacancy. As described above, the strontium-vacancy site defines no unique direction and reconstructional averaging recovers the full cubic symmetry group. For any realization of the reconstruction there are in fact two distinct types of strontium sites. Each such site sits at the center of a cube with oxygen octahedra at the corners with alternating signs of rotations. The results reported below, prior to reconstructional averaging, correspond to the site in which the rotation at the corner is positive.
For the elastic dipole tensor we find
\[\mathbf{G}_{\text{Sr}}=\left(\begin{array}{rrr}2.08&-0.23&-0.23\\ -0.23&2.08&-0.23\\ -0.23&-0.23&2.08\end{array}\right)\text{eV},\]
Performing the reconstructional average gives
\[\overline{\mathbf{G}}_{\text{Sr}}=\left(\begin{array}{rrr}2.08&0.00&0.00\\ 0.00&2.08&0.00\\ 0.00&0.00&2.08\end{array}\right)\text{eV},\]
with a corresponding defect-strain tensor,
\[\overline{\mathbf{\Lambda}}_{\text{Sr}}=\left(\begin{array}{rrr}1.78&0.00&0.00 \\ 0.00&1.78&0.00\\ 0.00&0.00&1.78\end{array}\right)\text{A}^{3}.\]
Diagonal components of elastic dipole tensor for strontium vacancy in strontium titanate. Data show linear convergence with inverse linear dimension of supercell size.
Local strain pattern for strontium vacancy in the \(\hat{e}_{1}\hat{e}_{2}\) plane of strontium titanate: strontium atoms (\(\blacksquare\)), oxygen atoms (\(\blacklozenge\)), strontium vacancy (\(\blacksquare\)). Atomic displacements exaggerated by a factor of three for clarity (\(\blacktriangle\)), displayed for significant in-plane displacements (\(>0.1\) Å) only.
This result expresses the tendency of the crystal to expand due to the strontium vacancy, by an amount significantly greater than the net effect of the oxygen vacancy. Since the original defect defines no unique direction, no orientational averaging is necessary, and we find a ratio of chemical strain \(\epsilon_{c}\) to the deviation from stoichiometry \(\delta\) in Sr\({}_{1-\delta}\)TiO\({}_{3}\) of \(\epsilon_{c}/\delta=+0.030\), indicating a tendency for the crystal to expand due to the presence of strontium vacancies.
We now examine the local strains around such a strontium vacancy after reconstructional averaging. Twelve oxygen atoms (three in each of four neighboring octahedra) are nearest neighbors to the strontium vacancy; these twelve atoms all move directly _away_ from the vacancy by a distance of 0.15 A. The next-nearest neighbors are the eight titanium atoms in each of the surrounding octahedra, each of which moves 0.09 A directly _toward_ the strontium vacancy. All other atoms move less than 0.08 A.
In this case, as expected, both the defect-strain tensor and the local strain displacements are symmetric after reconstructional averaging. The nearest-neighbor displacement shows an expansion in all directions, similar to the defect-strain tensor. The next-nearest neighbors, which have an opposite charge from that of the nearest neighbors, move in the opposite direction, reinforcing that there is no simple connection between local displacements and far-field strain patterns.
### Titanium vacancy
The titanium-vacancy site also defines no unique direction in the ideal crystal, and the reconstructional averaging restores the full cubic symmetry. For any realization of the reconstruction there are, in fact, two distinct types of titanium site. Each such site sits at the center of an octahedron with either positive or negative signs of the rotations relative to the \(+\) axis. The results reported below, prior to reconstructional averaging, correspond to the site in which the rotation is positive.
We first report our results for the elastic dipole tensor,
\[\mathbf{G}_{\mathrm{Ti}}=\left(\begin{array}{ccc}28.08&-0.70&-0.70\\ -0.70&28.08&-0.70\\ -0.70&-0.70&28.08\end{array}\right)\mathrm{eV},\]
The reconstructional average is then
\[\overline{\mathbf{G}}_{\mathrm{Ti}}=\left(\begin{array}{ccc}28.08&0.00&0.00\\ 0.00&28.08&0.00\\ 0.00&0.00&28.08\end{array}\right)\mathrm{eV},\]
with defect-strain tensor,
\[\overline{\mathbf{\Lambda}}_{\mathrm{Ti}}=\left(\begin{array}{ccc}23.92&0.00 &0.00\\ 0.00&23.92&0.00\\ 0.00&0.00&23.92\end{array}\right)\mathrm{\AA}^{3}.\]
This defect-strain tensor expresses the tendency of the crystal to expand due to the titanium vacancy -- a significantly greater amount even than that of the strontium. Again, since the original defect defines no unique direction, no orientational averaging over defect types is necessary. Finally, we report a ratio of chemical strain \(\epsilon_{c}\) to the deviation from stoichiometry \(\delta\) in SrTi\({}_{1-\delta}\)O\({}_{3}\) of \(\epsilon_{c}/\delta=+0.402\), indicating a significant tendency for
Diagonal components of elastic dipole tensor for titanium vacancy in strontium titanate. Data show linear convergence with inverse linear dimension of supercell size.
Local strain pattern for titanium vacancy in the \(\hat{e}_{1}\hat{e}_{2}\) plane of strontium titanate: titanium atoms (\(\blacklozenge\)), oxygen atoms (\(\blacklozenge\)), titanium vacancy (\(\blacklozenge\)). Atomic displacements exaggerated by a factor of three for clarity (\(\blacktriangle\)), displayed for significant in-plane displacements (\(>0.1\) Å) only.
As depicted in Figure 7, we now describe the local strain effects on the atoms surrounding the titanium vacancy. The nearest neighbors are six surrounding oxygen atoms which move 0.22 A directly _away_ from the titanium vacancy. The next-nearest neighbors are the eight surrounding strontium atoms along the body diagonals from the titanium vacancies; these strontium atoms move the very significant distance of 0.52 A directly _toward_ the titanium vacancy. The third shell is made up of six titanium atoms separated by a lattice constant from the vacancy along all three directions (positive and negative); all of these titanium atoms move 0.09 A directly _away_ from the titanium vacancy. Finally, the fourth shell of atoms is comprised of twenty-four oxygen atoms, arranged as six groups of four oxygens, each in a diamond-shape with its center one lattice coordinate away from the titanium vacancy in each lattice direction. These oxygens each move 0.08 A _away_ from the vacancy (0.08 A along the vector separating the diamond-group from the vacancy, and 0.03 A along either of the two other directions, so as to cause the diamond-group to spread outward). All other atoms in the crystal move less than 0.07 A.
We again observe connections between the reconstructionally averaged local displacements and the far-field defect-strain tensor which have similar symmetry. In this case of the titanium vacancy, the nearest-neighbor atoms are displaced away from the vacancy, showing the same behavior as the defect-strain tensor. Interestingly, however, the next-nearest neighbors, which move toward the titanium vacancy, actually have more than twice the displacement of the nearest neighbors. So here we observe that the far-field strain does not correlate with the largest magnitude displacement, but instead with that of the nearest-neighbor atoms.
### Strontium-oxygen divacancy
As described above in Section III, there are twelve distinct orientations for the strontium-oxygen divacancy as defined by the \(\mathrm{V_{Sr}}\)-\(\mathrm{V_{O}}\) direction. For any realization of the reconstruction there are in fact two distinct types of strontium-oxygen sites. Each such strontium site sits at the center of a cube with oxygen octahedra at the corners with alternating signs of rotations. The results reported below, prior to reconstructional averaging, correspond to the \([1\bar{1}0]\) defect with negative sense of rotation for the octahedron at the \(\) corner.
The elastic dipole tensor for this defect is
\[\mathbf{G}_{\mathrm{SrO}}=\left(\begin{array}{rrr}-4.62&3.00&-2.28\\ 3.00&-4.62&-2.28\\ -2.28&-2.28&6.95\end{array}\right)\mathrm{eV},\]
The reconstructional average is then
\[\overline{\mathbf{G}}_{\mathrm{SrO}}=\left(\begin{array}{rrr}-3.00&1.78&0.00 \\ 1.78&-3.00&0.00\\ 0.00&0.00&4.27\end{array}\right)\mathrm{eV},\]
with defect-strain tensor,
\[\overline{\mathbf{\Lambda}}_{\mathrm{SrO}}=\left(\begin{array}{rrr}-9.37&2.1 4&0.00\\ 2.14&-9.37&0.00\\ 0.00&0.00&17.25\end{array}\right)\mathrm{\AA}^{3}.\]
This defect-strain tensor expresses a slight tendency of the crystal to contract due to the strontium-oxygen divacancy. The lower symmetry of this defect, with its orientation along a diagonal, leads to remaining off-diagonal elements even after reconstructional averaging. However, when the above tensor is averaged over all twelve defect orientations, the result is a constant tensor with a uniform chemical strain per unit concentration of defect of \(-0.50\) A\({}^{3}\). This corresponds to a ratio of chemical strain \(\epsilon_{c}\) to the deviation from stoichiometry \(\delta\) in \(\mathrm{Sr}_{1-\delta}\mathrm{TiO}_{3-\delta}\) of \(\epsilon_{c}/\delta=-0.008\), indicating a tendency for the crystal to contract due to the presence of strontium-oxygen divacancies.
Referring to Figure 9, we now examine the local displacements after reconstructional averaging. The situation with this divacancy is more complicated than that of earlier isolated atomic vacancies, leading us to characterize the atomic displacements with respect to each independent missing atom in the strontium-oxygen divacancy. The first shell is comprised of the eleven remaining oxygen atoms that are nearest neighbors of the strontium vacancy. This strontium vacancy would normally have twelve neighboring oxygen atoms forming three squares in the three planes, each comprised of four atoms centered around the strontium vacancy; however, one of these oxygen atoms is missing to form the divacancy. The four oxygens, which are farthest from the oxygen vacancy but not in the same square as the oxygen vacancy, all move _away_ from the strontium vacancy by 0.14 A (0.10 A along \(\hat{e}_{3}\), and 0.09 A along either \(\hat{e}_{1}\) or \(\hat{e}_{2}\), depending upon symmetry, with 0.004 A along the opposite vector, chosen such that most of each displacement is within the plane of squares to which that oxygen belongs). The four oxygens that are closest to the oxygen vacancy and still not in the same square as the oxygen vacancy also move as a group. These oxygen atoms all move _away_ from the strontium vacancy by 0.11 A (0.05 A along \(\hat{e}_{3}\), and 0.09 A along either \(\hat{e}_{1}\) or \(\hat{e}_{2}\), depending upon symmetry, with 0.05 A along the opposite vector, chosen such that more of each displacement is within the plane of the square to which that oxygen belongs). The two oxygens, in the same square as the oxygen vacancy and closer to such vacancy, are displaced _away_ from the defect, in that plane, by 0.12 A (0.12 A along either \(\hat{e}_{1}\) or \(\hat{e}_{2}\), and 0.04 A along the opposite vector, chosen to ensure that the displacement maximizes its overall movement away from the remaining oxygen in this square). The final oxygen atom is also in the same square as the oxygen vacancy, but is the farthest atom from the vacancy in this square; it is displaced directly _away_ from the defect in that plane by 0.14 A. Also included in the first shell are the remaining oxygen atoms that are nearest neighbors to the missing oxygen vacancy. That oxygen vacancy, part of two oxygen octahedra, has eight neighboring oxygens, four in each octahedron; of those eight oxygens, the four closest to the strontium vacancy have already been considered as part of the nearest neighbors to the missing strontium atom. The remaining four oxygen atoms in the other octahedron, farthest from the strontium vacancy, move the appreciable distance of 0.33 A _toward_ the vector connecting the strontium and oxygen vacancies (0.30 A along \(\hat{e}_{3}\), and 0.11 A along either \(\hat{e}_{1}\) or \(\hat{e}_{2}\), depending upon symmetry, with 0.005 A along the opposite vector, chosen such that most of each displacement in the \(\hat{e}_{1}\hat{e}_{2}\) plane is toward the oxygen vacancy). Finally, this first shell also includes the three strontium atoms (the fourth is itself missing) nearest the oxygen vacancy. The one strontium atom farthest from the strontium vacancy moves directly _toward_ the oxygen vacancy by 0.03 A (0.02 A along both \(\hat{e}_{1}\) and \(\hat{e}_{2}\)). The other two strontium atoms move _away_ from the oxygen vacancy by 0.10 A (0.10 A along either \(\hat{e}_{1}\) or \(\hat{e}_{2}\), depending upon symmetry, with 0.04 A along the opposite vector, chosen such that the displacement maximizes the distances of these strontium atoms from the strontium vacancy).
The second shell is comprised of the eight titanium atoms that are the nearest neighbors to the missing strontium atom. Two of these nearest neighboring titanium atoms, which could alternatively have been categorized as the nearest-neighboring shell of atoms from the oxygen vacancy, move _away_ from the strontium vacancy (as well as the oxygen vacancy) by 0.19 A (0.19 A away from the vacancies along \(\hat{e}_{3}\), and 0.03 A toward the strontium vacancy along both \(\hat{e}_{1}\) and \(\hat{e}_{2}\)). Two other nearest-neighbor titanium atoms, those that are farthest from the oxygen vacancy among these eight titanium atoms, move 0.09 A _toward_ the strontium vacancy (0.06 A along \(\hat{e}_{3}\), and 0.05 A along both \(\hat{e}_{1}\) and \(\hat{e}_{2}\)). The final four of these eight nearest-neighbor titanium atoms move _toward_ the vacancies by 0.11 A (0.05 A along \(\hat{e}_{3}\), and 0.08 A along either \(\hat{e}_{1}\) or \(\hat{e}_{2}\), depending upon symmetry, with 0.05 A along the opposite vector, chosen such that more of each displacement in the \(\hat{e}_{1}\hat{e}_{2}\) plane is toward the oxygen vacancy). All other atoms move less than 0.17 A.
We now investigate correlations between these above reconstructionally averaged local displacements and the far-field defect-strain tensor. We find surprising results in this case of the strontium-oxygen divacancy. While the defect-strain tensor and local displacement pattern both show expansion on the axis outward from the defect, we see a disagreement in sign in the plane between the far-field contraction and the expansion of the nearest neighbors. This serves as a cautionary note that far-field and near-field strain patterns need not be simply related and reinforces the importance of providing both sets of information for further experimental analysis of x-ray scattering signatures.
Diagonal components of elastic dipole tensor for strontium-oxygen divacancy in strontium titanate. Data show linear convergence with inverse linear dimension of supercell size.
Local strain pattern for strontium-oxygen divacancy in the \(\hat{e}_{1}\hat{e}_{2}\) plane of strontium titanate: strontium atoms (\(\blacksquare\)), oxygen atoms (\(\blacklozenge\)), strontium vacancy (\(\blacksquare\)), oxygen vacancy (\(\copyright\)). Atomic displacements exaggerated by a factor of three for clarity (\(\blacktriangle\)), displayed for significant in-plane displacements (\(>0.1\) Å) only.
### Titanium-oxygen divacancy
As described above in Section III, there are six distinct orientations for the titanium-oxygen divacancy as defined by the V\({}_{\rm Ti}\)-V\({}_{\rm O}\) direction. Moreover, because of the reconstruction, there are in fact two distinct classes of sites within each possible orientation, as distinguished by the rotation state of the octahedron in which the titanium sits. Below, we report results for a [\(\bar{1}00\)] defect with the titanium sitting in an octahedron of positive rotation.
The elastic dipole tensor for this defect is
\[{\bf G}_{\rm TiO}=\left(\begin{array}{ccc}14.74&-0.08&-4.76\\ -0.08&18.13&-3.87\\ -4.76&-3.87&21.56\end{array}\right){\rm eV},\]
The reconstructional average gives
\[\overline{\bf G}_{\rm TiO}=\left(\begin{array}{ccc}14.74&0.00&0.00\\ 0.00&19.84&0.00\\ 0.00&0.00&19.84\end{array}\right){\rm eV},\]
with corresponding defect-strain tensor,
\[\overline{\bf\Lambda}_{\rm TiO}=\left(\begin{array}{ccc}3.00&0.00&0.00\\ 0.00&21.68&0.00\\ 0.00&0.00&21.68\end{array}\right)\hat{\rm A}^{3}.\]
The defect-strain tensor expresses the tendency of this defect to expand the crystal in all directions, but primarily along the directions orthogonal to the V\({}_{\rm Ti}\)-V\({}_{\rm O}\) axis. When the above tensor is averaged over all six defect orientations (oxygen sites in octahedra surrounding the central titanium site, recalling that the V\({}_{\rm Ti}\)-V\({}_{\rm O}\) axis is direction dependent), the result is a constant tensor with a uniform chemical strain per unit concentration of defect of \(+15.45\) A\({}^{3}\). Such expansion corresponds to a ratio of chemical strain \(\epsilon_{c}\) to the deviation from stoichiometry \(\delta\) in SrTi\({}_{1-\delta}\)O\({}_{3-\delta}\) of \(\epsilon_{c}/\delta=+0.260\), indicating a significant tendency for the crystal to expand due to the presence of titanium-oxygen divacancy.
Now that our reconstructional averaging has symmetrized the displacement patterns of neighboring atoms, it is instructive to examine this set of near-field atomic strains, shown in The situation with this divacancy is again more complicated than that of earlier isolated atomic vacancies. There are six atoms in the first shell around the divacancy. The five remaining (the sixth is itself missing) oxygen atoms that are nearest neighbors of the titanium vacancy all move _away_ from that vacancy: the one that is furthest from the oxygen vacancy moves 0.20 A directly _away_ from the vacancy, while the other four (which are in the \(\hat{e}_{2}\hat{e}_{3}\) plane) move 0.23 A _away_ from the vacancy (0.13 A along \(\hat{e}_{1}\) toward the oxygen vacancy, with the remaining projection of 0.19 A away from the divacancy either along \(\hat{e}_{2}\) or \(\hat{e}_{3}\) as dictated by the symmetry). The sixth atom in this first shell is the sole titanium atom that is the nearest neighbor to the oxygen vacancy; it moves directly _away_ from the vacancy by 0.13 A. In the second shell, we identify four of the oxygen atoms that are nearest neighbor to the oxygen vacancy (we already counted the other four nearest-neighbor oxygens above, "assigning" them to the titanium vacancy); these oxygens move 0.33 A _toward_ the oxygen vacancy (0.31 A along \(\hat{e}_{1}\)
Local strain pattern for titanium-oxygen divacancy in the \(\hat{e}_{1}\hat{e}_{2}\) plane of strontium titanate: titanium atoms (\(\blacklozenge\)), oxygen atoms (\(\blacklozenge\)), titanium vacancy (\(\blacklozenge\)), oxygen vacancy (\(\varotimes\)). Atomic displacements exaggerated by a factor of three for clarity (\(\blacktriangle\)), displayed for significant in-plane displacements (\(>0.1\) Å) only.
Diagonal components of elastic dipole tensor for titanium-oxygen divacancy in strontium titanate. Data show linear convergence with inverse linear dimension of supercell size.
Also, in this second shell, we can loosely consider the eight nearest-neighbor strontium atoms to the titanium vacancy, all of which move toward the titanium vacancy: four of these strontium atoms are in a plane that includes the oxygen vacancy, and these move 0.22 A _toward_ the titanium vacancy (0.17 A along \(\hat{e}_{1}\), and 0.09 A along both \(\hat{e}_{2}\) and \(\hat{e}_{3}\)); the other four strontium atoms are on the opposite side of the titanium vacancy, and these move significantly _toward_ the vacancy by 0.58 A (0.38 A along \(\hat{e}_{1}\), and 0.31 A along both \(\hat{e}_{2}\) and \(\hat{e}_{3}\)). All other atoms move less than 0.20 A.
We again examine connections between the reconstructionally averaged local displacements and the far-field defect-strain tensor. In this case of the titanium-oxygen divacancy, the nearest-neighbor atoms to both vacancies move outward, showing the same behavior as the defect-strain tensor. These atoms also demonstrate larger movements in those directions orthogonal to the \(\mathrm{V_{Ti}}\)-\(\mathrm{V_{O}}\) axis, which conforms with the far-field tensor. The next-nearest neighbors to both vacancies move inward with significant displacements, with those closest to the titanium vacancy moving by almost three times the amount that the nearest-neighbor atoms move. So here we note once more that the far-field strain does not appear to correlate with the largest magnitude displacement, but instead with that of the nearest-neighbor atoms.
## V Discussion
The above results for the local strain patterns and detailed defect tensors are now available for direct comparison with diffuse x-ray measurements; however, we are not aware of any such x-ray data to date. Nonetheless, the fully averaged (over both reconstructions and orientations) defect-strain tensors relate directly to measurements which are commonly done of chemical strain as a function of defect concentration. Table 3 summaries the results from Section IV for the ratios of chemical strain \(\epsilon_{c}\) to stoichiometric defect deviation \(\delta\) for all of the defects considered in this study.
Oxygen-vacancy concentration is widely thought to serve a crucial role in the properties of perovskites, is readily varied but difficult to control, and is experimentally observed to affect chemical strain. Moreover, cation stoichiometry is also difficult to control and so it is uncertain whether, as oxygen vacancies are introduced into the crystal, such vacancies bind to cation vacancies or form in isolation. In the former case, where the oxygen vacancies eventually bind to preexisting cation vacancies, the reference configuration should be the crystal containing the cation vacancy. Hence, it is the _difference_ between the chemical strain of the oxygen-cation divacancy and that of the isolated cation vacancy that describes the change in the crystal lattice as a function of varying oxygen-vacancy concentration. In the latter case of isolated vacancies, the bulk crystal is in fact the system into which these vacancies are introduced, and thus the chemical strain as a function of oxygen-vacancy concentration is described precisely by that of the isolated oxygen vacancy in our study. Table 4 summarizes the resulting _net_ chemical strain \(\Delta\epsilon_{c}\) versus oxygen-vacancy concentration \(\delta\).
For the oxygen vacancy, we have the intriguing result that the elastic dipole tensor and corresponding defect-strain tensor almost vanish under orientational averaging. Thus, very little net effect on the lattice can be expected from the presence of isolated oxygen vacancies. Moreover, the large anisotropy of the dipole tensor of the oxygen vacancy and the ease of introduction and high mobility of such vacancies should allow for the control of the population and orientation of oxygen vacancies by applying external stress. (For instance, at 1% strain, the orientational energy differences from \(\overline{\mathbf{G}}_{\mathrm{O}}\) are 66 meV, or about 2.6 times room temperature.) Also, such vacancies can be expected to tend to shield internal crystalline stresses that result from materials processing, a fact potentially related to the observed difficulties in controlling the oxygen-vacancy concentration during crystalline growth.
One of the earliest sets of available experimental data on chemical strain due to oxygen vacancies in strontium titanate comes from Yamada and Miller, who unfortunately found a null result. Nonetheless, that null result places bounds which, in conjunction with our results, allow some conclusions to be drawn. Yamada and Miller varied the oxygen-vacancy concentration over a range from nearly zero up to \(3.24\times 10^{19}\) cm\({}^{-3}\) (\(\delta=0.0019\) in \(\mathrm{SrTiO_{3-\delta}}\)), stating that "no volume change upon reduction was assumed," due to the experimental uncertainty of the lattice constant (\(\Delta a=5\times 10^{-4}\) A) in their x-ray diffraction measurements.
\begin{table}
\begin{tabular}{c c} \hline \hline Reference state & \(\Delta\epsilon_{c}/\delta\) \\ \hline Bulk & 0.001 \\ \(\mathrm{V_{Sr}}\) & \(-0.038\) \\ \(\mathrm{V_{Ti}}\) & \(-0.142\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Net ratios of chemical strains to stoichiometric defect deviation for oxygen vacancies, referenced against bulk and isolated cation vacancies as appropriate.
\begin{table}
\begin{tabular}{c c} \hline \hline Defect & \(\epsilon_{c}/\delta\) \\ \hline \(\mathrm{V_{O}}\) & 0.001 \\ \(\mathrm{V_{Sr}}\) & 0.030 \\ \(\mathrm{V_{Ti}}\) & 0.402 \\ \(\mathrm{V_{SrO}}\) & \(-0.008\) \\ \(\mathrm{V_{TiO}}\) & 0.260 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Individual ratios of chemical strain \(\epsilon_{c}\) to stoichiometric defect deviation \(\delta\) for different defects as calculated in Section IV.
From Table 4, it is evident that this bound is consistent with either isolated oxygen vacancies or strontium-oxygen divacancies, but is inconsistent with titanium-oxygen divacancies.
The literature also presents studies of chemical strain due to _cation_ non-stoichiometry. Ohnishi et al. present experimental results on the ratio of chemical strain to deviation from cation stoichiometry for samples grown by pulsed laser deposition. Specifically, they measure the lattice changes for both strontium-rich and strontium-poor strontium titanate. They associate the strontium-rich phase with creation of Ruddlesden-Popper planar faults (extra SrO layers) and the strontium-poor regime with the presence of strontium vacancies, possibly bound into strontium-oxygen divacancies. A least-squares fit to the results of Ohnishi and coworkers provides a value for \(\epsilon_{c}/\delta\) between +0.13 (non-weighted) and +0.14 (weighted by reported experimental uncertainty) for the strontium-rich phase (Sr\({}_{1+\delta}\)TiO\({}_{3}\)) and between +0.5 (non-weighted) and +0.8 (weighted by reported experimental uncertainty) for the strontium-poor phase (Sr\({}_{1-\delta}\)TiO\({}_{3}\)).
Our calculations of titanium-vacancy chemical strains are not directly relevant to the strontium-rich phase because they do not account for Ruddlesden-Popper planar faults. On the other hand, our results for strontium vacancies, under the interpretation of Ohnishi et al., should be directly relevant to their strontium-poor samples. Table 3 gives \(\epsilon_{c}/\delta=+0.030\) and \(\epsilon_{c}/\delta=-0.008\), respectively, for isolated strontium vacancies and bound strontium-oxygen divacancies. However, both of these are an order of magnitude smaller than the observed chemical strain. (We note parenthetically that, while our calculations reflect chemical strains for isolated defects and Ohnishi and coworkers measure strains for relatively high defect densities, Figures 4 and 8 show that trends in our data with increasing defect concentration only tend to reinforce our conclusions.) We remark that our results are consistent with the observation that, within a given structural class, the lattice constants of the titanates are largely insensitive to the nature of the A-site cations. (See data on A\({}^{2+}\)B\({}^{4+}\)O\({}_{3}\) perovskites compiled by Galasso.) From both the results of our calculations and this general observation, it seems implausible that simple A-site vacancies should produce the measured magnitude of chemical strain. Chemical strains of the magnitude measured by Ohnishi and coworkers more plausibly arise from B-site vacancies or defect complexes associated with such vacancies.
We note that, intriguingly, there is an approximate coincidence between the order of magnitude of our calculated ratio of chemical strain to stoichiometric deviation due to titanium vacancies, \(|\epsilon_{c}/\delta|=0.4\), and the observed values (+0.5 and +0.8, depending upon weighting of fit) for the strontium-poor samples. Given the magnitude of the observed lattice expansion and the fact that _only_ titanium vacancies appear capable of producing an effect of this size, we are led to the intriguing conjecture that, perhaps, the strontium-poor samples exhibit defects that include titanium vacancies and thus a structure more complex than initially thought (simple strontium vacancies or strontium-oxygen divacancies). Clearly, more investigation is needed on this point, specifically as to the nature of the point defects in the strontium-poor samples grown by pulsed laser deposition.
The above measurements by Ohnishi and coworkers were performed on strontium titanate samples deposited by the highly energetic process of pulsed laser deposition. We have learned of recent results by Brooks et al., that repeat the above measurements on strontium titanate samples grown by molecular-beam epitaxy, which is a lower-energy deposition process and thus less prone to the creation of point defects. Performing a weighted least-squares fit to the new data of Brooks and coworkers, we find a value for \(\epsilon_{c}/\delta\) of +0.032\(\pm\)0.019 in the dilute limit of the strontium-poor regime. This chemical strain ratio, published _after_ our initial calculated prediction was submitted to this journal, shows good agreement with our value of +0.030, and thereby lends convincing support for our methodology as well as the applicability of empirical shell potentials to calculate reasonable estimates of the chemical strain per stoichiometric defect deviation of vacancies.
## VI Summary and Conclusion
We have calculated both near- and far-field strains for five defects in reconstructed strontium titanate: isolated oxygen, strontium, and titanium vacancies, as well as strontium-oxygen and titanium-oxygen divacancies. Given the propensity of the crystal for local fluctuations in the reconstruction, we report results both for a particular reconstructed state and as averaged over all possible local reconstructions. The reconstructionally averaged near-field strain results are presented and interpreted in terms of the movement of neighboring shells of atoms at increasing distances from the vacancy or divacancy. We report far-field strain results in terms of both elastic dipole tensors, with and without reconstructional averaging, and associated defect-strain tensors, with reconstructional averaging. Anticipating that far-field effects will necessarily involve contributions from an ensemble of defects, we also present results averaged over all possible orientations of the defect within the bulk crystal. From these averaged tensors, we extract the resultant ratio of chemical strain to stoichiometric defect deviation. Finally, the combination of local and long-range results presented herein will enable determination of x-ray scattering signatures for comparison with experimental results and should further motivate future work on defect mechanics, including the influence of externally imposed strain (such as in heterostructures) on vacancy populations.
For the oxygen vacancy, we find a highly anisotropic elastic dipole tensor, with almost perfect cancellation under orientational averaging. This may be correlated with observed difficulties in controlling oxygen concentration and lack of observation of effects of oxygen-vacancy concentration on lattice strain. The anisotropy of this tensor also suggests that oxygen vacancies may provide a mechanism to shield local internal strains and that application of external stress may allow for control of oxygen stoichiometry. From comparison to lattice-strain studies, we identify both isolated oxygen vacancies and bound strontium-oxygen divacancies as consistent with the experimentally observed chemical strain as a function of oxygen-vacancy concentration in strontium titanate.
For cation non-stoichiometry, we find strong indications that the point defects in strontium-poor strontium titanate samples grown by pulsed laser deposition are not simple strontium vacancies or strontium-oxygen divacancies, but likely more complicated defect complexes. Further, we identify indications that titanium vacancies may play a role in these defect complexes. Finally, during the review process, we learned of recent experimental data, from strontium titanate films deposited via molecular-beam epitaxy, that show good agreement with our calculated value of the chemical strain associated with strontium vacancies.
| 10.48550/arXiv.0811.2967 | Elastic effects of vacancies in strontium titanate: Short- and long-range strain fields, elastic dipole tensors, and chemical strain | Daniel A. Freedman, D. Roundy, T. A. Arias | 511 |
10.48550_arXiv.1804.09357 | ###### Abstract
Zinc tin nitride (ZnSnN\({}_{2}\)) is attracting growing interest as a non-toxic and earth-abundant photoabsorber for thin-film photovoltaics. Carrier transport in ZnSnN\({}_{2}\) and consequently cell performance are strongly affected by point defects with deep levels acting as carrier recombination centers. In this study, the point defects in ZnSnN\({}_{2}\) are revisited by careful first-principles modeling based on recent experimental and theoretical findings. It is shown that ZnSnN\({}_{2}\) does not have low-energy defects with deep levels, in contrast to previously reported results. Therefore, ZnSnN\({}_{2}\) is more promising as a photoabsorber material than formerly considered.
ZnSnN\({}_{2}\) has a wurtzite-derived structure with a minimum gap of 1.4 eV in its direct-type band structure, which is very close to the best place of the Shockley-Queisser limit of 1.34 eV. In addition, the electron effective mass of 0.17\(m_{0}\) and its heavy-hole mass of 2.00\(m_{0}\) are comparable or even superior to those of GaN (Fig. S2, Supplemental Material). However, there are several fundamentally and technologically important issues regarding the use of ZnSnN\({}_{2}\) as a light absorption layer that need to be assessed: (i) _Point defect properties_. It is necessary to identify any point defects with deep levels, because they trap electrons and/or holes and cause carrier recombination, leading to loss of cell efficiency. (ii) _Unintentional n-type doping_. The carrier-electron concentration generally unintentionally increases up to 10\({}^{21}\) cm\({}^{-3}\) in ZnSnN\({}_{2}\) as well as other narrow gap nitrides like InN, Zn\({}_{3}\)N\({}_{2}\), and ScN. For solar cell applications, the carrier-electron concentration needs to be lowered to around 10\({}^{16}\)-10\({}^{18}\) cm\({}^{-3}\). Besides, the photoabsorption onset is increased up to 2.4 eV because of the Burstein-Moss (BM) shift, which has raised considerable controversy regarding the fundamental gap of ZnSnN\({}_{2}\). The BM shift is also directly related to the formation of donor-type point defects. Understanding the point-defect properties of ZnSnN\({}_{2}\) is thus imperative to optimize its performance in photovoltaics.
Chen _et al._ investigated native point defects and oxygen impurities in ZnSnN\({}_{2}\) with an ordered orthorhombic structure. They reported that Sn-on-Zn antisite (Sn\({}_{\text{Zn}}\)) and O-on-N (O\({}_{\text{N}}\)) substitutional defects have low formation energies with deep donor levels, and thus do not cause the BM shift in ZnSnN\({}_{2}\). However, given recent experimental and theoretical findings, this conclusion needs to be reviewed. This motivated us to reevaluate point defects in ZnSnN\({}_{2}\) including as-yet-uninvestigated hydrogen impurities. Our results show that all the defects with deep levels are very high in energy and the abundant defects act as shallow donors. Considering these favorable defect properties together with its appropriate band gap and high absorption coefficient, ZnSnN\({}_{2}\) is a promising photoabsorber for thin-film photovoltaics.
Our theoretical investigation of the point defects in ZnSnN\({}_{2}\) was performed for the _Pna2\({}_{1}\)_ orthorhombic structure with 16 atoms in the unit cell, referred to here as the ordered model. However, a certain level of disorder in the cation sublattice appears to be unavoidable because of the very low cation order-disorder transition temperature. At low temperature, the disordered phase strictly retains the local charge neutrality, in which each N atom is necessarily coordinated by two Sn and two Zn atoms. We refer to such a structure as the disordered structure with local charge neutrality (DLCN). It has been reported that the electronic structure of the DLCN is almost identical to that of the ordered structure. Thus, the DLCN should be ideal as a photoabsorber, unless its defects are detrimental to its efficiency. We thus considered both ordered and DLCN models in our defect calculations. Note that cation disorder breaking the local charge neutrality occurs at very high temperature (\(>\) 1750 K). Some authors have used special quasirandom structures (SQS) to examine the behavior of the cation disordered phase. Such a fully random cation disordered model is not appropriate for ZnSnN\({}_{2}\) because its typical growth temperature is much lower than 1750 K. Besides, when the charge neutrality is broken, a high concentration of Zn-on-Sn (Zn\({}_{\text{Sn}}\)) defects is introduced. This situation should be avoided for photovoltaic applications because ZnSn defects give rise to deep levels, as discussed later.
The DLCN model was generated by Monte Carlo simulated annealing of a 128-atom orthorhombic supercell. The DLCN model shown in belongs to space group _Pna2\({}_{1}\)_ and consists of a 32-atom unit cell, consistent with that reported elsewhere. Note that there are infinite possible configurations for the DLCN models when different cell sizes are considered; however, within the 128-atom supercell, we found only one configuration owing to the strong geometrical constraint of the local charge neutrality.
First, we discuss the band gap of ZnSnN\({}_{2}\) because it is a central controversy in the research of this material. Asthe carrier-electron concentration increases, the photoabsorption onset also increases due to the BM shift. The blueshift estimated from the Heyd-Scuseria-Ernzerhof (HSE06) hybrid functional calculations is 0.2-2.3 eV when the carrier-electron concentration is 10\({}^{18}\)-10\({}^{21}\) cm\({}^{-3}\) observed in experiments (Fig. S6, Supplemental Material). Thus, larger experimental band gaps than 1.4 eV could be attributed to the BM shift, as has also been discussed by Lahourcade et al. Second, the band gaps of fully cation disordered systems are ill-defined because the minimum gap decreases as the model size increases. In the case of ZnSnN\({}_{2}\), the density of states tends to continuously develop within the band gap when N atoms are coordinated by three or four Sn or Zn atoms, as discussed later. Note that this does not necessarily indicate metallic behavior because the defect states are discontinuous in real space. Refer to the Supplemental Material for a more detailed discussion considering SQS (Fig. S8).
We now discuss the energetics of point defects in ZnSnN\({}_{2}\). To investigate the site dependency in the DLCN model, we conducted over 300 point-defect calculations using the modified Perdew-Burke-Ernzerhof generalized gradient approximation tuned for solids (PBEsol-GGA). We confirmed that most of the defect species show only negligible site dependencies in energy, while the interstitial defects do relatively large ones probably due to the difference in neighboring cation configurations at the octahedral sites (Fig. S9, Supplemental Material). Thus, we performed HSE06 calculations for a few configurations with lower energies in PBEsol calculations for each defect and show only the results of the lowest energy configurations hereafter. shows the resultant formation energies of the native defects as well as hydrogen and oxygen impurities, which are ubiquitous in nitrides. Since ZnSnN\({}_{2}\) is only slightly stable (\(\Delta H_{f}\) = -0.015 eV/atom) compared to Zn metal, Sn metal, and N\({}_{2}\) molecules using HSE06, the growth condition does not largely affect the formation energies under the equilibrium condition. Therefore, here, the chemical potentials of constituent elements are set at the point where Zn and Sn metals equilibrate with ZnSnN\({}_{2}\). As mentioned above, the standard HSE06 functional used in this study yielded a direct gap of 1.4 eV. In the study by Chen _et al._, the Fock exchange ratio was increased to reproduce a reported experimental gap of 2.0 eV although one of the authors attributed this larger gap to the BM shift and stated that the fundamental gap is about 1.4 eV.
Before discussing the results, let us consider the technical details that can alter conclusions even qualitatively. Chen _et al._ did not consider the finite cell-size effects when calculating defect formation energies and the reciprocal space sampling was performed using only the \(\Gamma\) point. According to our calculated cell-size dependencies, a maximum error of 0.72 eV arises when the 128-atom supercell is used with these settings. In this study, we decreased these errors using the extended Freysoldt-Neugebauer-Van de Walle (FNV) corrections and 2\(\times\)2\(\times\)2 Monkhorst-Pack _k_-point sampling for the 128-atom supercell. Consequently, the errors based on our test calculations are less than 0.06 eV. In the case of defects exhibiting hydrogenic states, a huge supercell including tens of thousands of atoms is usually required to avoid overlap between widespread defect orbitals. When using the 128-atom supercell in ZnSnN\({}_{2}\), the thermodynamic transition levels of such defects are overestimated by a few tenths of eV because of the defect-defect interaction (see Fig. S10, Supplemental Material). Therefore, we discuss the transition levels associated with hydrogenic states only qualitatively.
Figures 2(a-d) reveals that there is little difference between the defect formation energies of the ordered and DLCN models, which means that not only the bulk properties but also the point-defect properties are strongly correlated with its immediate coordination environment. This conclusion would also hold for similar cation disordered systems such as ZnSnP\({}_{2}\) and ZnGeN\({}_{2}\) if the local charge neutrality is preserved. It is also notable that defects and impurities that show low formation energies simultaneously with deep levels do not exist when the Fermi level is located within the band gap. This is in stark contrast to the conclusion of a previous study that the Zn interstitial (Zn\({}_{i}\)), Sn\({}_{2n}\), nitrogen vacancy (\(V_{\rm N}\)), and O\({}_{\rm N}\) have deep defect levels and act as carrier recombination centers. This discrepancy is mainly caused by the aforementioned difference in the treatment of the cell-size corrections and _k_-point sampling.
As shown in Figures 2(e) and (g), Sn\({}_{\rm Zn}^{0}\) exhibits an occupied hydrogenic state, namely perturbed host state, while the localized defect state is not confirmed near the conduction-band minimum (CBM), which is different from the results by Chen _et al._ Therefore, Sn\({}_{\rm Zn}\) is a dominant shallow donor among the native defects. Its formation energy becomes zero at the Fermi level being 0.9 eV above the valence-band maximum
Atomic structures of the ordered and DLCN models of ZnSnN\({}_{2}\) used in this study. These models contain 128 atoms. Both structures belong to space group \(Pna2_{1}\).
(VBM), which hinders _p_-type conversion even with acceptor doping. \(V_{\rm N}\) is a single shallow donor in the entire Fermi level range, but has a higher formation energy than that of Sn\({}_{\rm Zn}\). In contrast, acceptor-type defect ZnSn forms two deep transition levels between the 0, -1, and -2 charge states. Indeed, as seen in Figures 2(f) and (h), Zn\({}_{\rm Sn}^{0}\) shows two deep localized defect states within the band gap. However, the formation energy of ZnSn is rather high when the Fermi level is within the band gap. The other defects, namely, Zn vacancy (\(V_{\rm Zn}\)), Sn vacancy (\(V_{\rm Sn}\)), Sn interstitial (Sn\({}_{i}\)), and N interstitial (Ni\({}_{i}\)), create deep levels but have very high formation energies. It is also noteworthy that Sn\({}_{\rm Zn}^{0}\) does not become a DX center (\(V_{\rm Zn}+\) Sn\({}_{i}\)) in our calculations unlike ZnSnP\({}_{2}\) probably because of the high formation energies of \(V_{\rm Zn}\) and Sn\({}_{i}\) and/or the difference in crystal structures.
Oxygen and hydrogen impurities are energetically favorable at the N sites (O\({}_{\rm N}\)) and interstitial sites (H\({}_{i}\)), respectively. These impurities also act as single shallow donors and exhibit low formation energies even when the Fermi level is at the CBM. Orbital analyses indicated that their donor electrons are located at the perturbed conduction bands. Therefore, these impurities should primarily cause the BM shift. However, in the previous study, the origin of the high carrier-electron concentration was attributed to the defect band originating from Sn\({}_{\rm Zn}\) and O\({}_{\rm N}\), which is different from our results. H\({}_{i}\) forms an N-H bond (1.01 A) between N and Zn atom, as commonly observed for nitrides. We also found that anionic hydrogen H\({}_{i}^{-1}\) is not stabilized even when the Fermi level is located at 1 eV above the CBM in ZnSnN\({}_{2}\) (see Fig. S12, Supplemental Material), unlike in GaN.
Recently, Fioretti _et al._ showed that annealing Zn-rich Zn1\({}_{1+\rm{s}}\)Sn\({}_{1-\rm{N}}\)2 grown in a hydrogen atmosphere decreased its carrier-electron concentration to 4 \(\times\) 10\({}^{16}\) cm\({}^{-3}\). They explained this observation from the viewpoint of hydrogen passivation of acceptors, i.e., ZnSn\({}_{\rm{s}}\) + H\({}_{i}\) in Zn1\({}_{1+\rm{s}}\)Sn\({}_{1-\rm{s}}\)N\({}_{2}\) during growth, which lowers the driving force for the forma
Formation energies of point defects in ZnSnN\({}_{2}\) as a function of the Fermi level. (a, b) Native defects and (c, d) impurities in the (a, c) ordered model and (b, d) DLCN model. The valence band maximum is set to the zero of the Fermi level and the upper limit corresponds to the conduction band minimum. The defect species and sites are indicated by \(X_{Y}\), where \(X\) means a vacancy (\(V\)) or an added element and \(Y\) is an interstitial site (\(i\)) or substutional site. The chemical potentials are set at a point where the Zn and Sn metals equilibrate with ZnSnN\({}_{2}\). Note that shallow donor levels in the vicinity of the conduction-band minimum are designated by open circles (see text in detail). Note also that the Sn vacancy does not appear in Figures (a) and (b) owing to its very high formation energy (see Fig. S12, Supplemental Material). The inset of (c) shows the atomic structure for H\({}_{i}^{+}\). (e, f) Band structures for the Sn\({}_{\rm Zn}^{0}\) and Zn\({}_{\rm Sn}^{0}\) in the 128-atom supercell of the ordered model. Note that Zn\({}_{\rm Sn}^{0}\) shows spin polarization, and only the down-spin channel, which indicates two unoccupied deep localized states, is shown. (see Fig. S14, Supplemental Material, for the up-spin channel) (g, h) Squared wave functions of the highest occupied and the lowest unoccupied bands for Sn\({}_{\rm Zn}^{0}\) and Zn\({}_{\rm Sn}^{0}\), respectively, highlighted by blue color in Figures (e) and (f).
Indeed, our results indicate that complexing with hydrogen is exothermic and drastically decreases the formation energy of acceptor ZnSn (Fig. S13); the binding energy, i.e., the energy change from isolated ZnSn\({}^{-2}\) and H\({}_{+}^{+}\) to (ZnSn + H\({}_{i}\))\({}^{-}\) is -1.43 eV and that from isolated H\({}_{+}^{+}\) and (ZnSn + H\({}_{i}\))\({}^{-}\) to (ZnSn + 2H\({}_{i}\))\({}^{0}\) is -0.73 eV. Therefore, abundant ZnSn antisites are easily introduced by hydrogen passivation as discussed by Fioretti _et al._ We, however, emphasize that ZnSn antisites, which generate deep transition levels and trap minority carrier holes, persist even after removing the passivating hydrogen by post-deposition annealing. Moreover, the determination of the intrinsic band gap would be inhibited by the optical absorption related to the defect band.
Here, we propose an alternative route to achieving ZnSnN\({}_{2}\) with lower carrier-electron concentration. In the above discussion, the chemical potentials were set at a point where the Zn and Sn metals equilibrate with ZnSnN\({}_{2}\). However, recent growth techniques such as reactive sputtering can be used to raise the chemical potential of N (\(\mu_{\rm N}\)) by up to +1 eV/N from that of an inactive N\({}_{2}\) molecule. Consequently, metastable nitrides can be synthesized. A notable example is Cu\({}_{3}\)N, in which \(\mu_{\rm N}\) needs to be at least +1.04 eV higher than that of an N\({}_{2}\) molecule. When \(\mu_{\rm N}\) was increased to this value, we can use more advantageous condition for lowring the carrier-electron concentration. shows the formation energies of native defects and oxygen impurity under \(\Delta\mu_{\rm N}\) = +1 eV/N and Zn-rich (Sn-poor) condition (see Fig. S16 for details). The equilibrium Fermi level is located at the 0.70 eV from the CBM at 600 K which assumes synthesis temperature of ZnSnN\({}_{2}\). In this situation, all the defect concentrations are less than \(10^{14}\) cm\({}^{-3}\) and corresponding carrier concentration is \(n\)=\(1.9\times 10^{14}\) cm\({}^{-3}\). One might expect that \(p\)-type conversion by acceptor doping is plausible in this condition as all the donor-type defects compensating holes are high in energy. Therefore, we performed the extensive calculations for impurities Li, Na, K, Cu, and C using HSE06. (see Fig. S17, Supplemental Material) As a result, however, we found that none of them act as effective acceptor dopants for ZnSnN\({}_{2}\) because of deep acceptor levels or incorporation into interstitial sites.
_Methods._ First-principles calculations were performed using the projector augmented wave (PAW) method as implemented in VASP. The PBEsol-GGA was used to calculate the total energies of various DLCN models and SQS with different sizes, and to check the site and cell-size dependencies of defect formation energies. The HSE06 hybrid functional with standard parameters was used for the calculations of the band structure, density of states, and defect formation energies. The plane-wave cutoff energy was set to 550 eV for the lattice relaxation of the ordered and DLCN models without defects and 400 eV for the other calculations with fixed lattice constants. The residual forces were converged to less than 0.01 and 0.04 eV/A for the calculations without and with defects, respectively. Spin polarization was considered for all the defect calculations using HSE06.
The formation energy of a point defect was calculated as
\[E_{f}[D^{q}] =\left\{E[D^{q}]+E_{\rm corr}[D^{q}]\right\}-E_{P}+\sum n_{i}\mu_ {i}\] \[+q(\epsilon_{\rm VBM}+\Delta\epsilon_{F}), \tag{1}\]
\(n_{i}\) is the number of removed (\(n_{i}>0\)) or added (\(n_{i}<0\)) \(i\)-type atoms and \(\mu_{i}\) is the chemical potential representing the growth conditions. The referenced competing phases used were hexagonal Zn, cubic Sn, wurtzite ZnO, and N\({}_{2}\) and H\({}_{2}\) molecules. \(\epsilon_{\rm VBM}\) is the energy level of the VBM, and \(\Delta\epsilon_{F}\) is the Fermi level (\(\epsilon_{F}\)) with respect to \(\epsilon_{\rm VBM}\). Therefore, \(\epsilon_{F}=\epsilon_{\rm VBM}+\Delta\epsilon_{F}\). \(E_{\rm corr}[D^{q}]\) corresponds to the correction energy for a finite supercell size error associated with spurious electrostatic interactions between charged defects. We used our extended FNV correction scheme in the calculations. More computational details are described in the Supplemental Material.
Both DLCN and SQS models were generated by Monte Carlo simulated annealing using in-house and CLUPAN codes, respectively (see Fig. S5, Supplemental Material, for more details).
_Conclusions._ We theoretically revisited the point defects in ZnSnN\({}_{2}\) by realistic modeling of its disordered phase with the local charge neutrality. Our calculations revealed that the ordered and DLCN models exhibited nearly the same stability, volumes, electronic structures, and even point-defect properties, indicating these properties are determined mainly by the immediate coordination environment. It was also found that low-energy defects with deep levels are absent in ZnSnN\({}_{2}\) and, therefore, there is less carrier recombination caused by point defects than thought previously.
Same as Fig. 2, but the nitrogen chemical potential is increased by 1 eV under the Zn-rich condition (see text for details). Since hydrogen impurity could be reduced by post-growth annealing, we consider only oxygen as an impurity, here.
using non-equilibrium growth techniques was proposed. Our study has unveiled the further potential of ZnSnN\({}_{2}\) as a photoabsorber in thin-film photovoltaics.
_Acknowledgements._ This work was supported by the MEXT Elements Strategy Initiative to Form Core Research Center, Grants-in-Aid for Young Scientists A (Grant No. 15H05541) and Scientific Research A (Grant No. 17H01318) from JSPS, and PRESTO (JPMJPR16N4), and Support Program for Starting Up Innovation Hub MI\({}^{2}\)I from JST, Japan. The computing resources of ACCMS at Kyoto University were used for a part of this work.
| 10.48550/arXiv.1804.09357 | Electrically Benign Defect Behavior in Zinc Tin Nitride Revealed from First Principles | Naoki Tsunoda, Yu Kumagai, Akira Takahashi, Fumiyasu Oba | 6,142 |
10.48550_arXiv.0910.1051 | ###### Abstract
We investigate the crystallization of amorphous arsenic-selenium alloys with 0%, 0.5%, 2%, 6%, 10%, and 19% arsenic by atomic concentration using synchrotron X-ray absorption spectroscopy. We identify crystalline order using the extended X-ray absorption fine structure (EXAFS) spectra and correlate this order to changes in features of the X-ray absorption near edge structure (XANES) spectra. We find supporting evidence that the structure of amorphous selenium is composed of disordered helical chains, and is therefore closer to the trigonal crystalline phase than the monoclinic crystalline phase.
## 1 Introduction
The photoconducting properties and the low lateral diffusion of photoinduced carriers in amorphous selenium (aSe) have recently led it to find use in commercial direct conversion X-ray imaging devices in medical applications. Two open problems with the utility of amorphous selenium as a photoconductor are charge trapping and crystallization. aSe spontaneously crystallizes at room temperature. This process can be delayed by alloying the selenium with between 0.2% and 0.5% arsenic by atomic concentration, but such doping increases the charge trapping.
The local structure of aSe has been the subject of considerable interest over the years, and the relation between aSe and the common crystalline phases of monoclinic and tSe (mSe and tSe, respectively) have been discussed. The current consensus is that aSe is mainly composed of disordered helical chains similar to the ordered hexagonal arrangement of helical chains in tSe, with very little contribution from Se\({}_{8}\) rings (which form the basis for mSe).
Herein we attempt to induce crystallization in arsenic-selenium alloys and use X-ray absorption near edge structure (XANES) and extended X-ray absorption fine structure(EXAFS) spectra recorded at the selenium _K_-edge to probe the local structure. We find evidence supporting the belief that aSe is composed of helical chains rather then Se\({}_{8}\) rings, and we identify XANES features which act as fingerprints identifying the degree of crystallization in the material.
## 2 Experimental Procedure
The samples were prepared by vapour deposition of pure (99.999%) selenium and arsenic-selenium alloys on a room temperature polycarbonate substrate. The different arsenic concentrations in the arsenic-selenium alloys were 0.5%, 2%, 6%, 10% and 19% by atomic composition. The vapour deposition process formed homogenous films between 24 \(\mu\)m and 35 \(\mu\)m thick. This range of thickness was chosen to give an absorption step of roughly 2 at the the selenium _K_-edge.
As reference standards, tSe crystals were grown from vapour produced due to the sublimation of pure selenium kept at a temperature of 200\({}^{\circ}\)C in a closed glass vessel for 7-10 days. The crystals grown by this method have needle like shape that is typical for tSe. mSe crystals were grown by a saturated solution of selenium in methylene iodide (CH\({}_{2}\)I\({}_{2}\)). The tSe crystals were ground into a fine powder (less then 50 \(\mu\)m grain size). The mSe crystals were essentially grown in powder form, and were not further modified since mechanical stress is known to induce conversion to the trigonal phase. These powders were homogeneously mixed with boron nitride. The ratio of selenium to boron nitride was chosen to produce a similar absorption step to that of the arsenic-selenium alloy films.
Some of the film samples were annealed in a convection oven to induce crystallization. The films produced samples with 6% or less arsenic were annealed at 100\({}^{\circ}\)C for 24 hours. The annealing temperature was substantially higher than the highest glass transition temperature for these materials (about 80\({}^{\circ}\)C for the Se:6%As alloy). The films with 10% and 19% arsenic were annealed for about 100 hours at 110\({}^{\circ}\)C (the maximum temperature of the furnace). This temperature was only slightly higher than the highest glass transition temperature (about 106\({}^{\circ}\)C for Se:19%As). Finally, aSe films were measured as prepared, and then remeasured immediately after annealing at 60\({}^{\circ}\)C, 80\({}^{\circ}\)C, and finally 100\({}^{\circ}\)C for three hours each.
The X-ray measurements were performed at the Hard X-ray Microprobe Analysis (HXMA) beamline of the Canadian Light Source (CLS). The measurements were performed with a Si crystal monochromator and an Rh-coated harmonic rejection mirror. The measurements were performed in transmission mode. A liquid helium cryostat was used for the low temperature measurements, the temperature was stabilized between 30 K and 45 K. Since no difference was observed between the room temperature and low temperature XANES measurements, the measurements of the progressively annealed pure selenium were conducted at room temperature. A standard pure selenium film was measured concurrently with all spectra as a reference for calibration. The absorption edge of this reference was calibrated to 12658 eV using the zero-crossing in the second derivative nearest the edge.
## 3 Results and Discussion
The EXAFS of the low temperature selenium \(K\)-edge measurements reveal some long range order in some of the annealed samples. shows that in samples with an arsenic concentration of less than 6% the second nearest neighbour at roughly 3 Ais readily apparent. This medium range order in the EXAFS indicates that the samples with less than 6% arsenic are polycrystalline. There is a weak feature at about 3.3 Ain samples with arsenic concentration greater than 6%, but this feature is also present in aSe. Annealing does increase the intensity of this feature, but it is still much weaker than the medium range features in samples with less than 6% arsenic. The shape of the EXAFS \(|\chi(R)|\) is consistent with literature reports on the EXAFS of tSe, and this suggests that the polycrystalline arsenic-selenium alloys have some degree of helical chain structure consistent with the structure of tSe. To contrast, the EXAFS of mSe has the second nearest neighbour at about 3.6 eV, a feature which is not present in any of the annealed samples here.
The XANES of the low-temperature selenium \(K\)-edge measurements shows the prominent white line (at roughly 12660 eV) and secondary post-edge feature (at roughly 12668 eV) that are common to both trigonal, monoclinic, and amorphous pure selenium. shows that there is a minor tertiary post-edge feature at roughly 12673 eV that occurs in the annealed samples with less than 6% arsenic that does not occur in the other annealed samples. Since the EXAFS spectra reveal that the samples with less than 6% arsenic are polycrystalline, this suggests that this post-edge XANES feature is connected to polycrystalline order.
The EXAFS oscillations, \(k^{2}\chi(k)\) and the magnitude of the Fourier transform, \(|\chi(R)|\) for the low-temperature selenium \(K\)-edge measurements of annealed samples and unannealed tSe, and aSe. Note the nearest neighbour appears at roughly 2.1 Å, the actual bondlength is about 2.34 Å.
The post-edge shape of the mSe XANES resembles that of the amorphous arsenic-selenium alloys, while the the post-edge shape of the tSe XANES resembles that of the polycrystalline arsenic-selenium alloys. The shape of the post-edge XANES has been previously proposed used as a fingerprint to identify the phase of biologically produced selenium in Reference, but this study did not recognize that aSe shares the same XANES features as mSe.
As pure selenium is progressively annealed at higher temperatures, the post-edge XANES feature gradually changes from that of aSe to that of tSe. Since recent studies on the structure of aSe indicate that it does not have a significant amount of Se\({}_{8}\) rings, the similarity between the post-edge XANES features in amorphous and mSe should not be interpreted as an indication of local "monoclinic-like" structure. On the contrary, since it is generally accepted that aSe is composed of disordered helical chains and since the similarity in post-edge XANES features between annealed arsenic-selenium alloys and tSe is concurrant with a similarity in the EXAFS, we propose that the post-edge XANES features can be an indication of local "trigonal-like" structure.
To summarize, we have measured the EXAFS and XANES spectra of several arsenic-selenium alloys after annealing in a convection oven. We have identified that materials with a low concentration of arsenic change to a polycrystalline phase composed of helical chains, similar to tSe. The threshold for this polycrystalline phase change is between 2% and 6% arsenic. We have also identified a post-edge XANES feature that we suggest may be used as a quick fingerprint to identify the degree of "trigonal-like" polycrystallinity in arsenic-selenium alloys.
We acknowledge the support of the Natural Sciences and Engineering Research Council of Canada (NSERC) and the Canada Research Chair program. The research described in this paper was performed at the Canadian Light Source, which is supported by NSERC, NRC, CIHR, and the University of Saskatchewan.
| 10.48550/arXiv.0910.1051 | Identifying structural order in Selenium with Near-Edge Spectroscopy | J. A. McLeod, N. Chen, R. E. Johanson, G. Belev, D. Tonchev, A. Moewes, S. O. Kasap | 5,711 |
10.48550_arXiv.2105.12602 | ## Chemicals.
Poly(ethylene glycol) (PEG, Sigma Aldrich, molecular weight Mn = 6000 g.mol-1), LiClO4 (Sigma Aldrich, 99.99%), oleylamine (OLA, Acros, 80-90%), %), lead chloride (PbCl2, Afla Aesar, 99%), sulfur powder (S, Afla Aesar, 99.5%), oleic acid (OA, Afla Aesar, 90%), trioctylphosphine (TOP, Afla Aesar, 90%),n-hexane (VWR), Ethanol (VWR, >99.9%), Toluene (Carlo Erba, >99.8%).
## Precursor.
Lead oleate Pb(OA)2 0.1 M: 0.9 g of PbO are mixed in a three neck flask with 40 mL of oleic acid. The flask is degassed under vacuum for one hour. The atmosphere is switched to Ar and the temperature raised to 150 \({}^{\circ}\)C for two hours. The final solution is typically clear and yellowish.
## PbS CQD Synthesis.
In a three-neck flask, 300 mg of PbCl2, together with 100 mL of TOP and 7.5 mL of OLA are degassed, first at room temperature and then at 110 \({}^{\circ}\)C for 30 min. Meanwhile, 30 mg of S powder is mixed with 7.5 mL of OLA until full dissolution and an orange clear solution is obtained. Then under nitrogen at 80 \({}^{\circ}\)C, this solution of S is quickly added to the flask. After 2 minutes, the reaction is quickly quenched by addition of 1 mL of OA and 9 mL of hexane. The nanocrystals are precipitated with ethanol and redispersed in 5 mL of toluene. This washing step is repeated one more time and the pellet is this time dispersed in 10 mL of toluene with a drop of OA. The solution is then centrifugated at is, to remove the unstable phase. The supernatant is precipitated with methanol and redispersed in toluene. Finally, the PbS CQD solution in toluene is filtered through a 0.2 um PTFE filter. The obtained solution is used for further characterization and devices' fabrication.
## Sample fabrication.
The interdigitated electrodes are defined by optical lithography on a borosilicate glass substrate from Plan Optik using an MJB4 aligner and the AZ 5214E photoresist. After deposition of 3 nm of Ti and 70 nm of Au in a Plassys MEB 550S electron beam evaporator, followed by a lift-off in acetone, the resulting electrodes define an area of 870 \(\times\) 1400 um\({}^{2}\) and the spacing between two consecutive digits is 10 um. The remaining steps of the fabrication are performed in an N\({}_{2}\)-filled glovebox. A 15-20 nm thick layer of CQDs is spun onto the sample, creating a coating above the metallic patterns. An examination of various cross-sections with a scanning electron microscope reveal that the thickness of the layer is uniform between the interdigitated electrodes, except for small edge effects in their immediate vicinity. Finally, the sample is dipped into a solution of EDT in ethanol for 60 s or into a solution of Na\({}_{2}\)S in ethanol for 10s followed by a rinse in ethanol to replace the native OA ligands by shorter molecules. For the transistor measurements, an additional step is necessary to fabricate and deposit the electrolytic gel onto the layer of CQDs. The latter is obtained by mixing 230 mg of PEG and 50 mg of LiClO\({}_{4}\) at 150 \({}^{\circ}\)C for several hours in the glovebox. The resulting mixture is then cooled down until it solidifies and then reheated at 100 \({}^{\circ}\)C for 10 minutes to soften it prior deposition onto the sample.
## Transistor measurements.
The characterization is performed under ambient atmosphere with an electrolytic gate applied above the CQD layer. Automatic transfer curves are measured using a Keithley 2636B sourcemeter. The source and drain are the interdigitated electrodes and the field effect is applied by contacting an electrical probe to the electrolytic gate. The transistor characterizations are performed after all the other experiments because the latter cannot be done anymore once the electrolytic gate covers the CQDs. From these measurements, the carrier mobility of the samples can be obtained using the slope of the transfer curve:
\[\mu^{FET}=\frac{L}{WCV_{DS}}\frac{\partial I_{DS}}{\partial V_{G}}\bigg{|}_{V_{ DS}},\]
Ins is the drain source current and V\({}_{\text{G}}\) is the gate bias.
## Absorption measurements.
The characterization is performed under ambient atmosphere using Fourier transform photocurrent spectroscopy. The samples are placed under a Hyperion microscope from Bruker equipped with a 10X infrared objective from Olympus, biased at 11 V (10.8 V to be more exact, which is the maximum value that we can reach with our apparatus) and illuminated with the white light originating from a Fourier Transform Infrared (FTIR) spectrometer (Invenio R from Bruker). The resulting photocurrent is amplified using a DLPCA-200 amplifier from Femto, converted into a voltage and plugged back into the electronics of the FTIR via an analog/digital converter. The spectra obtained with this technique are then normalized with the spectrum taken with a calibrated 818-ST2 Ge photodiode from Newport. At this stage we have the responsivity of the sample, from which we can derive the absorbance spectrum in arbitrary units by dividing the data by the wavelength. Note that our structures are not good photodetectors because the thickness (and therefore the absorbance) of the CQDs is very small.
## Photoluminescence measurements.
The characterization is performed under ambient atmosphere. The sample is investigated with an Olympus BX51WI upright microscope equipped with an LCPLN50XIR 50X infrared objective. The laser pump is a HeNe laser at 633 nm, filteredwith a BG40 filter from Thorlabs to remove the parasitic spontaneous emission in the near-infrared. The laser beam is directed through a set of neutral density filters that makes it possible to adjust its intensity, then directed into the microscope via the top port and finally focused onto the sample with the 50X objective. The infrared luminescence, collected by the same objective, is separated from the laser pump with a Thorlabs DMLP950R dichroic mirror and a RG780 long pass filter. The signal is analysed with an Acton SP-2356 spectrograph equipped with a 85 groove/mm monochromator and coupled to a NIRvana 640ST InGaAs camera from Princeton Instruments. The resulting data are then corrected by taking into account the transfer functions of the 50X objective and the 85 groove/mm grating. The dispersion of the other elements of the setup is sufficiently small to neglect their impact on the spectra.
## Why we use two different microscope objectives for the absorption and PL measurements.
We chose a 10X objective for the absorption measurements because the device cannot be illuminated in full with a higher magnification, resulting in a degraded signal-over-noise ratio (but the spectra are the same regardless of the objective used). We chose a 50X objective for the PL measurements for two reasons. First, this magnification ensures that the pumping spot is much smaller than the spacing between two interdigitated electrodes; second, this objective has the largest numerical aperture that we have at hand (0.65), maximizing the solid angle of collection and therefore the PL signal.
| 10.48550/arXiv.2105.12602 | Identification of Two Regimes of Carrier Thermalization in PbS Nanocrystal Assemblies | Augustin Caillas, Stéphan Suffit, Pascal Filloux, Emmanuel Lhuillier, Aloyse Degiron | 4,685 |
10.48550_arXiv.1810.13291 | ###### Abstract
In this paper we study a role of F-centers, hole centers and excitons in energy transfer in Eu-doped BaBrI crystals. Optical absorption spectra, thermally stimulated (TSL) and photostimulated (PSL) luminescence in wide temperature range 7-300 K are studied in undoped and doped with different concentrations of Eu ions BaBrI crystals. Based on experimental and calculated results two possible energy tranfer processes from host to Eu\({}^{2+}\) ions are established.
keywords: energy transfer, scintillators, halides, alkali earth halides, europium +
Footnote †: journal: Radiation Measurements
## 1 Introduction
Mixed alkaline earth halides BaBr\(Y\) (where Y=I, Cl) are recently developed scintillation materials having good potential. The light output for BaBrI:Eu\({}^{2+}\) crystals was estimated as 90,000 photons/MeV, whereas in BaBrCl it was about 52,000 photons/MeV, respectively. Additionally, they are less hygroscopic than LaBr\({}_{3}\) and may have a broad implementation. Theoretically these crystals could reach a higher luminosity, therefore further optimization toward improvements in the scintillation properties of this material should be possible. Thereby, it is necessary to study excitons, electron and hole centers and their role in energy transfer to emission centers.
In this paper we discuss the role of excitons, electron and hole centers in energy transfer process in BaBrI crystals. The results obtained by optical absorption, photostimulated and thermally stimulated luminescence are presented and discussed. The obtained results demonstrate possible energy transfer processes from host to Eu\({}^{2+}\) ions. Explanations for the role of electron and hole traps are presented as well.
## 2 Methodology
Eu-doped crystals and undoped crystals were grown by Bridgman technique reported in. To study transformation of one type of F-centers into another, depending on the preparation conditions of the BaBrI we applied a Czochralski crystal growth technique. A conventional growth setup with a graphite thermal screens and resistivity heater was used. The crystals were grown in an inert (argon) atmosphere in glassy carbon crucible. For all growth experiments we used the seeds cut along from the crystal grown by Stockbarger method. The crystal pulling rate was 0.5 mm/hour, and rotation rate was 5 rpm.
Despite the fact that the quasi-binary phase diagram of BaI\({}_{2-}\) BaBr\({}_{2}\) shows region where solid solution for either BaI\({}_{2}\) or BaBr\({}_{2}\) in BaBrI is not formed, an imbalance induced on purpose in the bromine/iodine ratio can alter the shape of the absorption spectrum from F(Br) to F(I)-centers. As shown below using imbalance between BaBr\({}_{2}\) and BaI\({}_{2}\) in raw the undoped crystal containing mostly F(I) centers can be grown. Further in this paper this sample is called non-stoichiometric crystal.
Optical absorption spectra were measured with a Perkin Elmer Lambda 950 spectrophotometer. The crystals were irradiated either by x-rays from a Pd tube operating at 40 kV and 40 mA regime for not more than one hour or 185 nm light from a low-pressure Hg-lamp. Absorption spectra of x-ray and photo colored crystals contain the same bands. Thermally stimulated (TSL) and photostimulated (PSL) luminescence spectra were measured in vacuum cold-finger cryostat. Before measurements samples were irradiated under 185 nm light from low-pressure mercury lamp at 77 K (for PSL) and at 7 K (for TSL) temperatures. PSL was registered with SDL-1 grating monochromator equipped with Hamamatsu photomodule H10721-04. PSL was excited using 150 W halogen lamp and MDR2 grating monochromator. TSL glow curves were measured in linear heating regime with rate 1 K/min using cryocooler Janis Research CCS-100/204N.
The calculations have been performed in embedded cluster approach implemented in Gaussian 03 computer code. The QM cluster with vacancy and nearest neighbors (four or five barium ions) was surrounded by point charges. We used SDD basis with pseudopotential on barium ions and did not use any basis for vacancies, because it is shown that for the correct description of the F-center there are enough d-functions of the surrounding cations. For optical absorption spectra calculations the time depended DFT (TD DFT) method was used.
## 3 Results and discussion
### F-centers
After irradiation of undoped stoichiometric samples at room temperature bands at about 2.05 and 1.55 eV appear in optical absorption spectrum (Fig. 1, curve 3). Intensity of band peaked at about 2.05 eV is higher than one of 1.55 eV band. In crystals grown using imbalance composition of BaIr\({}_{2}\) and BaBr\({}_{2}\) absorption spectrum is changed. Higher energy band shifts to lower energy and its intensity becomes lower whereas intensity of low energy band is increased (Fig. 1, curve 1). In irradiated Eu-doped samples these bands are not observed.
As shown in the inset of Fig. 1, efficiency of F-centers creation decreases during cooling at temperatures about 100 K where intensity of STE band emission grows up. This anticorrelation behaviour is similar to observed in alkali halides crystals.
In related crystals of BaBrBr, BaBr\({}_{2}\) the absorption bands in irradiated crystals at about 2-2.5 eV are attributed to F(Br)-center. Therefore, the observed bands could be assigned to F-centers. There exist one family of Ba\({}^{2+}\) ions and two families of anion ions. Since two types of halogen ions exist in these crystals, two different types of F centers may be expected. These are related to iodine F(I) and bromine F(Br) vacancies trapped an electron.
Ab initio calculation of F(I) and F(Br) centers were performed. The results are given in Fig.1, curves 2 and 4, respectively. In calculated spectra several bands for each type of F-centers are observed. This is due to the fact that the ground state of F centers has approximately an \(s\)-like wavefunction. Whereas, the triple degeneracy of the excited \(p\)-states is partly lifted by the crystal field. The excited states split into three states due to monoclinic group C\({}_{S}\). In absorption spectrum of F(Br) centers the most intensive band is located at about 2 eV. For F(I) centers lower energy band at about 1.55 eV becomes the most intensive.
The calculated and experimental spectra agree well. In stoichiometric crystals two types of centers: F(Br) and F(I) are found. In non-stoichiometric samples grown by Czochralski method we observed an excess F(I) centers and optical absorption peak shifts to low energy region.
### Photostimulated luminescence
When BaBrI doped with Eu sample is exposed to x-ray or 185 nm irradiation at 77 K photostimulated luminescence is observed. Spectrum of photostimulated luminescence under 2 eV light irradiation is shown in Fig. 2, curve 1. The band peaked at about 3 eV in PSL spectrum is attributed to 5d-4f emission of Eu\({}^{2+}\) ions. The excitation spectrum of this PSL is given in Fig. 2, curve 2. This spectrum shows peak at about 2 eV in the region where absorption of F(Br) centers is observed, see Fig.1 and Fig. 2, curve 5. Similar low temperature PSL has been observed in BaBrCl-Eu crystals. At temperatures higher than 120 K intensity of PSL is dramatically decreased.
Photostimulated luminescence is also observed in irradiated undoped crystals. The photostimulated luminescence spectrum is depicted in curve 3 of PSL peak at about 3.9 eV is attributed to STE exciton emission. Photostimulation spectrum of exciton related PSL shows wide peak at about 2 eV (curve 4, Fig. 2) similar to Eu-doped samples.
### Thermally stimulated luminescence
In addition to photostimulated luminescence, thermally stimulated luminescence properties of BaBrI and BaBrI doped with 0.05 and 1 mol.% were investigated to clarify the recombination mechanism. In figure3, curve 1 the glow curve for the exciton emission around 320 nm for x-ray irradiated at 7 K undoped BaBrI is depicted. Strong peaks around 50 and 60 K and two weaker peaks at 95 K and 140 K are observed.
Optical absorption spectra of nominally undoped BaBrI crystals irradiated at room temperature. Curves 1 and 3 are experimental spectra of irradiated non-stoichiometric and stoichiometric crystals. Curves 2 and 4 are calculated spectra of F(I) and F(Br) centers.
Photostimulated luminescence (curve 1), excitation of PSL (curve 2) of BaBrI-0.05 mol.% Eu crystal; and photostimulated luminescence (curve 3), excitation of PSL (curve 4) and optical absorption (curve 5) spectra of undoped BaBrI. All samples were irradiated at 77 K.
For 50 and 60 K peaks thermal trap depth was estimated using second-order kinetic peak shape. The values are 0.10 and 0.13 eV respectively. Higher energy peaks are more complex and simple deconvolution is not possible.
The glow curves for Eu-doped samples are different to undoped sample. In figure3, curve 2 the glow curves for the Eu\({}^{2+}\) emission around 410 nm for x-ray irradiated at 7 K BaBrI-Eu crystals are shown. Intense wide peaks at 95 and 140 K are found. In 1 mol.% Eu\({}^{2+}\) doped sample intensity of TSL is weaker and two wide peaks at about 140 K and 200 K are observed (Fig.3, curve 3). After irradiation in all samples hyperbolic afterglow is observed.
For all crystals the peaks in the glow curves correspond to beginning of thermally stimulated motion of hole centers, because F-centers are stable up to 600 K. Low temperature peaks at 50 and 60 K could be attributed to simple hole centers of I\({}_{2}^{-}\) or Br\({}_{2}^{-}\). Higher temperature peaks could be related to complex centers called in alkali halides V\({}_{2}\), V\({}_{3}\) centers.
### Energy transfer to Eu\({}^{2+}\) ions
Excitation of Eu\({}^{2+}\) emission in the region of exciton peak can indicate the possibility for energy transfer from exciton to Eu\({}^{2+}\) ions as well as in LaBr\({}_{3}\)-Ce, where resonance exciton energy transfer is dominant. Absorption of 4f-5d transition of Eu\({}^{2+}\) ions and exciton emission spectra overlap in BaBrI crystals (Fig.4). Therefore dipole-dipole energy transfer from exciton to Eu\({}^{2+}\) ion is possible. The decrease in the exciton luminescence intensity accompanying the increase of Eu\({}^{2+}\) concentration is observed. At the level of 0.1 mol.% of Eu\({}^{2+}\) the exciton emission is almost completely suppressed. Considering the uniform Eu\({}^{2+}\) ions distribution in the host lattice, the radius of resonant energy transfer from exciton to Eu\({}^{2+}\) ions, at which exciton emission is suppressed, can be estimated as half the distance between Eu\({}^{2+}\) ions. A half Eu-Eu distance at the level of 0.1 mol.% is about 29 A.
This radius can be also estimated from overlapping emission and absorption spectra integral. The radius \(R\) of dipole-dipole energy transfer defined as the distance at which the probability of donor (exciton in our case) radiative transitions is equal to the probability of transfer to acceptor (Eu\({}^{2+}\) ion), is given by:
\[R=\frac{B}{n^{4}N_{A}}\int\limits_{0}^{\infty}\frac{f_{D}(E)\mu_{A}(E)}{E^{4} }dE \tag{1}\]
Here \(n\) is the refractive index of crystal, the subscript \(A\) denotes the acceptor center and \(D\) denotes the donor center, \(N_{A}\) is concentration of acceptor centers (in cm\({}^{-3}\) ) and \(\mu_{A}(E)\) is their absorption coefficient (in cm\({}^{-1}\)), \(f_{D}(E)\) is the emission spectrum of the donor centers after \(\int\limits_{0}^{\infty}D(E)dE=1\) normalization. Constant \(B=3h^{4}c^{4}/4\pi\) is equal to \(3.7\cdot 10^{-20}\) (eV\({}^{4}\cdot\)cm\({}^{4}\)). From the experimental data we estimate the radius \(R\) as 27.6 A.
Both values obtained for the distance of energy transfer from the host to the Eu\({}^{2+}\) ion are in good agreement. Therefore, efficient energy transfer from host to Eu\({}^{2+}\) ion takes place. It explains the growing light output with concentration of Eu\({}^{2+}\) ions found in, because distance between Eu-Eu pairs is decreased with increasing of Eu\({}^{2+}\) ions concentration.
BaBrI optically excited decay curves were fitted with only one single exponential decay component. However, the decay curve under x-ray excitation was well approximated with three exponential decay components. This fact, together with the presence of TSL, indicates the presence of delay energy transfer processes in the crystals, when hole or electron traps take place in energy transfer from host to emission center.
In irradiated at 77 K samples PSL is observed. In stimulation spectrum of undoped crystal the band attributed to F(Br) centers appears. Taking into account the fact that only hole centers are unstable below room temperature we can propose following model.
Overlapping optical absorption of BaBrI-0.05 mol.% Eu\({}^{2+}\) ions (dotted curve) and STE emission (solid curve) spectra
TSL glow curves monitored in STE band (320 nm) (red curve 1) in undoped BaBrI and 5d-4f Eu\({}^{2+}\) band (420 nm) in BaBrI doped with 0.05 mol.% Eu (blue curve 2) and 1 mol. % Eu (black curve 3)
Here F is F-center and H is H-center, which can be regarded as an X\({}_{2}^{-}\) molecule occupying an X\({}^{-}\) anion site. The anions concerning here can be iodine or bromine. H-centers become movable at relatively low temperatures about 40 K and they can recombine in thermally assisted process with F-centers causing TSL. Also they can form more complex (BrI)\({}^{-}\) hole centers. They are relatively stable at 77 K. Stimulation by light in absorption band of F(Br) centers causes that an electron is liberated from an F-center through conduction band and recombine with a hole center with exciton related luminescence. Afterglow is explained by tunneling recombination between an electron from F-center and hole.
In Eu-doped samples the delayed energy transfer process is more involved. From the one hand, Eu ions should be a hole trap relying on PSL data, where an electron is liberated from F-centers and radiatively recombine with hole. The simple assumption is that Eu\({}^{2+}\) ion captures a hole and becomes Eu\({}^{3+}\) ion. However similarly to BaBrBr and CsBr no Eu\({}^{3+}\) traces are found in irradiated crystals. Furthermore, TSL glow curves demonstrate that Eu\({}^{2+}\) should be an electron trap, because glow peaks correspond to hole traps which in thermally assisted process recombine with an electron on Eu ion. So, we conclude that as the case for BaBrBr-Eu and CsBr-Eu the hole trap contains Eu\({}^{2+}\) ion with a hole nearby.
As the case BaBrBr crystals F-centers and H-centers formation can be spatially correlated and appear near Eu\({}^{2+}\) ion with further stabilization a hole centers by Eu\({}^{2+}\) ion. When an electron released from an F-center recombines with a hole, Eu\({}^{2+}\) emission excited. From the other hand, a hole became thermally unstable can recombine with an electron in F-center and Eu\({}^{2+}\) emission is also excited. This process is more effective in crystals doped with high Eu\({}^{2+}\) concentrations due to smaller distances between Eu ions. Therefore, long time components make lower contribution to luminescence decay.
## 4 Conclusion
STE/Eu\({}^{2+}\) emission/absorption overlap allows to determine dipole-dipole transfer distance. The distance is sufficient to resonant energy transfer. This concludes that the fast energy transport from host to activator responsible for the scintillation of BaBrI-Eu proceeds by STE creation and resonance dipole-dipole transfer. At the same time, delayed energy transfer with participation of electron and hole centers formed by exciton decay takes place. In this process spatially correlated pair of F- and hole-center is produced, and a hole recombines with F-center in thermally assisted process leading to resonance transfer to Eu ion. Contribution of delayed energy transfer process decreases with increase of concentration of Eu\({}^{2+}\) ions due to diminution of distance between the dopant ions.
| 10.48550/arXiv.1810.13291 | Role of electron and hole centers in energy transfer in BaBrI crystals | Roman Shendrik, Alexandra Myasnikova, Alexey Rupasov, Alexey Shalaev | 4,935 |
10.48550_arXiv.1501.03934 | ## Figures
##
STM images of a Pb-covered Si surface before (a) and after Si deposition of 0.032 ML (b-d) at room temperature. A white arrow indicates the \(\sqrt{3}\times\sqrt{3}\) phase in each image. Small yellow triangles indicate white streaks at certain boundary sites. From the deposition coverage, we have estimated that 32\(\pm\)6 Si atoms are deposited on this Pb-covered region.
##
Evolution on a Pb-covered region after Si deposition at room temperature. The Si coverages are 0.012 ML (a), 0.024 ML (b), 0.036 ML (c and d), 0.048 ML (e and f).
Possible scenario for epitaxial growth of covalent materials. Basically all processes are reversible and the transition rates are determined by the related activation energies. There are processes illustrated with only one direction because their reverse transitions occur rarely. Some processes are illustrated with both directions, but solid lines and dashed lines refer to the cases with very likely and less than 100 K.
Growth of a Si atomic wire after Si deposition of \(\sim\) 0.03 ML at 240 K. In the Pb-covered Si region indicated with an arrow, we have estimated 79\(\pm\)9 Si atoms are deposited. The STM image is taken at the temperature of 240 K (a), 250 K (b), 270 K (c), and 300 K (d). The temperature is increased by 10 K each step and no Si deposition is carried out.
| 10.48550/arXiv.1501.03934 | Nucleation and Growth Mechanism of a Covalent Material: Magic Clusters and Chemical Reactions | Yi-Hsien Lee, Kuntal Chatterjee, Chung-Kai Fang, Shih-Hsin Chang, I-Po Hong, Tien-Chih Chang, Ing-Shouh Hwang | 3,153 |
10.48550_arXiv.1202.3168 | ###### Abstract
We calculate the bulk photovoltaic response of the ferroelectrics BaTiO\({}_{3}\) and PbTiO\({}_{3}\) from first principles by applying "shift current" theory to the electronic structure from density functional theory. The first principles results for BaTiO\({}_{3}\) reproduce experimental photocurrent direction and magnitude as a function of light frequency, as well as the dependence of current on light polarization, demonstrating that shift current is the dominant mechanism of the bulk photovoltaic effect in BaTiO\({}_{3}\). Additionally, we analyze the relationship between response and material properties in detail. The photocurrent does not depend simply or strongly on the magnitude of material polarization, as has been previously assumed; instead, electronic states with delocalized, covalent bonding that is highly asymmetric along the current direction are required for strong shift current enhancements. The complexity of the response dependence on both external and material parameters suggests applications not only in solar energy conversion, but to photocatalysis and sensor and switch type devices as well.
Introduction
The bulk photovoltaic effect - or photogalvanic effect - refers to the generation of intrinsic photocurrents that can occur in single-phase materials lacking inversion symmetry. Ferroelectrics (materials that possess intrinsic, switchable polarization) exhibit this effect strongly, producing current in response to unpolarized, direct illumination. Traditionally, photovoltaic materials are heterogeneous, doped structures, relying on the electric field at a \(p-n\) junction to separate photoexcited electrons and holes. By contrast, the bulk photovoltaic effect can be observed even in pure homogeneous samples, as with BaTiO\({}_{3}\). Recently, the effect has been demonstrated in the multiferroic BiFeO\({}_{3}\), with reported efficiencies as high as 10%. Though ferroelectric photovoltaics are currently receiving a great deal of interest, the origins of their photovoltaic properties are considered unresolved. Attention has been focused on interface effects, crystal orientation, and the influence of grain boundaries and defects, while any bulk photovoltaic contributions have been largely ignored. Its mechanism has been proposed to be a combination of nonlinear optical processes, especially the phenomenon termed the "shift current", but this has not been firmly established, and the detailed dependence on material properties, especially in ferroelectrics, is not known. This has also hindered progress towards understanding other photovoltaic effects, as the bulk contribution could not be separated out. While shift current calculations have been performed for some non-ferroelectrics, no experimental comparisons were performed. Here we present the first direct comparison of current computed from first principles with experimentally measured short-circuit photocurrent. Using the shift current theory, we successfully predict short circuit photocurrent direction, magnitude, and spectral features, demonstrating that shift current dominates the bulk photovoltaic response. Additionally, we explore the relationship between material polarization and shift current response, making progress towards identifying the electronic structure properties that influence current strength.
## II Main
We emphasize that nonlinear optical processes can give rise to a truly bulk effect. The results demonstrate that the most important of these is the shift current,which arises from the second-order interaction with monochromatic light. The electrons are excited to coherent superpositions, which allows for net current flow due the asymmetry of the potential. Bulk polarization is not required; only inversion symmetry must be broken. Shift currents have been investigated experimentally, analytically, and computationally, though only for a few nonpolar materials.
\[J_{q} =\sigma_{q}^{rs}E_{r}E_{s}\] \[\sigma_{q}^{rs}(\omega)= \pi e\left(\frac{e}{m\hbar\omega}\right)^{2}\sum_{n^{\prime},n^{ \prime\prime}}\int\mathrm{d}\mathbf{k}\,\left(f[n^{\prime\prime}\mathbf{k}]-f [n^{\prime}\mathbf{k}]\right)\] \[\times\left\langle n^{\prime}\mathbf{k}\right|\hat{P}_{r}\left|n^ {\prime\prime}\mathbf{k}\right\rangle\left\langle n^{\prime\prime}\mathbf{k} \right|\hat{P}_{s}\left|n^{\prime}\mathbf{k}\right\rangle\] \[\times\left(-\frac{\partial\phi_{n^{\prime}n^{\prime\prime}}( \mathbf{k},\mathbf{k})}{\partial k_{q}}-\left[\chi_{n^{\prime\prime}q}( \mathbf{k})-\chi_{n^{\prime}q}(\mathbf{k})\right]\right)\] \[\times\delta\left(\omega_{n^{\prime\prime}}(\mathbf{k})-\omega_{n ^{\prime}}(\mathbf{k})\pm\omega\right) \tag{1}\]
It is worth noting that while the Berry connections introduce a gauge dependence, it is exactly canceled by the gauge dependence of \(\frac{\partial\phi_{n^{\prime}n^{\prime\prime}}(\mathbf{k},\mathbf{k})}{ \partial k_{q}}\), so that the overall expression is gauge invariant.
We may view this expression as the product of two terms with physical meaning
\[\sigma_{q}^{rs}(\omega)=e\sum_{n^{\prime},n^{\prime\prime}}\int\mathrm{d} \mathbf{k}\,I^{rs}(n^{\prime},n^{\prime\prime},\mathbf{k};\omega)R_{q}(n^{ \prime},n^{\prime\prime},\mathbf{k})\]
where
\[I^{rs}(n^{\prime},n^{\prime\prime},\mathbf{k};\omega)= \pi\left(\frac{e}{m\hbar\omega}\right)^{2}\left(f[n^{\prime\prime }\mathbf{k}]-f[n^{\prime}\mathbf{k}]\right)\] \[\times\left\langle n^{\prime}\mathbf{k}\right|\hat{P}_{r}\left|n^ {\prime\prime}\mathbf{k}\right\rangle\left\langle n^{\prime\prime}\mathbf{k} \right|\hat{P}_{s}\left|n^{\prime}\mathbf{k}\right\rangle\] \[\times\delta\left(\omega_{n^{\prime\prime}}(\mathbf{k})-\omega_{n ^{\prime}}(\mathbf{k})\pm\omega\right) \tag{2}\]
is the transition intensity, which is proportional to the imaginary part of the permittivity and describes the strength of the response for this transition, and
\[R_{q}(n^{\prime},n^{\prime\prime},\mathbf{k})=-\frac{\partial\phi_{n^{\prime} n^{\prime\prime}}(\mathbf{k},\mathbf{k})}{\partial k_{q}}-\left[\chi_{n^{\prime \prime}q}(\mathbf{k})-\chi_{n^{\prime}q}(\mathbf{k})\right] \tag{3}\]is the shift vector, which gives the average distance traveled by the coherent carriers during their lifetimes.
\[\bar{R}_{q}(\omega)=\!\sum_{n^{\prime},n^{\prime\prime}}\int\mathrm{d}\mathbf{k} \,R_{q}(n^{\prime},n^{\prime\prime},\mathbf{k})\delta\left(\omega_{n^{\prime \prime}}(\mathbf{k})-\omega_{n^{\prime}}(\mathbf{k})\pm\omega\right). \tag{4}\]
We note that \(\bar{R}\) has units of length over frequency and is not physical, nor is it weighted by intensity; as such the \(\bar{R}\) only provides qualitative information about the aggregate shift vector. For additional information, see the supplemental materials.
## Methods
Wavefunctions were generated using the Quantum Espresso and Abinit plane-wave Density Functional Theory(DFT) codes with the generalized gradient approximation exchange correlation functional. Norm-conserving, designed non-local pseudopotentials were produced using the Opium package. Self-consistent calculations were performed on \(8\times 8\times 8\) k-point grids with energy cutoffs of 50 Ry; the resulting charge densities were used as input for non-self-consistent calculations on finer k-point grids as necessary.
### PbTiO\({}_{3}\), BaTiO\({}_{3}\)
Lead titanate and barium titanate derive from the cubic perovskite structure and become tetragonal in the ferroelectric phase at room temperature with five-atom unit cells. Both exhibit strong, robust polarization; combined with their simplicity this makes them ideal candidates for investigating the structural influence on the shift current response.
The calculations for PbTiO\({}_{3}\)(PTO) and BaTiO\({}_{3}\)(BTO) were performed using experimental room temperature geometries. Shown in are the shift current tensor elements, along with the imaginary component of the permittivity and \(\bar{R}_{Z}\). Only current response in the direction of material polarization (\(Z\)) is shown. The two materials show broadly similar behavior, with the peak of response several eV above the band gap and well outside the visible spectrum, while the shift current at energies near the band gap is small. The shift current for both materials is stronger in response to incident light polarization parallel to the direction of ferroelectric distortion than when normal to it.
The shift current depends weakly on the aggregate transition intensities and shift vectors. In PTO, at 5 eV there is a peak in both the current and intensity, yet the shift vector is relatively small. In fact, despite negative aggregate shift vector at many frequencies, the majority of the current response of PTO to \(xx\) polarized light is nonetheless positive. In BTO, the aggregate shift vector direction is largely positive with a negative shift current response under \(zz\) polarized illumination. This indicates that contributions to response can vary significantly across the Brillouin zone, and suggests that strong correlations between the two components are not present in the current.
The overall current susceptibility \(\sigma\), along with the imaginary component of the permittivity, \(\epsilon^{i}\), and shift vector integrated over the Brillouin zone \(\bar{R}\), are shown for (a) PbTiO\({}_{3}\), (b) BaTiO\({}_{3}\). DFT-computed direct band gaps are marked with vertical lines.
The product of the aggregate transition intensity and shift vector does not determine even the direction of photocurrent; to find the shift current, it is vital to multiply the transition intensity by its associated shift vector, and then sum over bands and k-points.
### Experimental Comparison
For bulk, single-crystal BaTiO\({}_{3}\), experimental spectra are available for energies near the band gap.
\[J_{q}(\omega)=\frac{\sigma_{q}^{rr}(\omega)}{\alpha^{rr(\omega)} }E_{r}^{2}w \tag{5}\] \[J_{q}(\omega)=G_{q}^{rr}Iw \tag{6}\]
This expression applies to samples of sufficient thickness to absorb all incident light. We obtained the light intensity and crystal dimensions from, which were \(\approx 0.35-0.6\) mW/cm\({}^{2}\) and 0.1-0.2 cm, respectively. In Fig. 2, the experimental current response from is compared to the response computed using shift current theory. Despite the uncertainty in experimental parameters, the agreement is striking, in both magnitude and spectrum profile, for both tensor elements. This includes the difference of sign between the majority of the transverse and longitudinal response, which is unusual, as well as the small positive region of the longitudinal response near the band edge. For PbTiO\({}_{3}\), experimental results suitable for quantitative comparison could not be located. However, we note that our calculation for PbTiO\({}_{3}\) correctly predicts that the current direction is toward the positive material polarization for light frequencies near the band gap, as well as that it's relatively insensitive to light polarization, in contrast to BaTiO\({}_{3}\).
We emphasize that these calculations not only reproduce the magnitude of response, but its idiosyncratic features as well. Because this theory reproduces all the salient features found in the experiments, this comparison provides strong evidence that shift current is the correct description of the the bulk photovoltaic effect.
## Polarization Dependence
Presently unknown is the relationship of the bulk photovoltaic effect to the material polarization. Identification of the bulk photovoltaic effect with shift current makes it clear that there is no direct, mechanistic dependence of response on material polarization, as is the case for many mechanisms to which photovoltaic effects in ferroelectrics have been attributed. However, shift current requires broken inversion symmetry, which here derives from the lattice distortion that produces ferroelectric polarization, suggesting that the response may appear to depend on polarization in some fashion. However, Eq.
For BaTiO\({}_{3}\), the experimental current and computed current (this work), for transverse (\(xxZ\)) and longitudinal (\(zzZ\)) electric field orientation, as a function of energy above their respective bandgaps. The solid lines are calculated results for a choice of experimental parameters of 0.5 mW/cm\({}^{2}\) illumination intensity and 0.15 cm sample width. The shaded regions are bounded by the results using experimental parameters in the given range that provide the lowest and highest response.
The presented data suggest that stronger polarization does not necessarily imply greater response; photocurrent densities in BaTiO\({}_{3}\) and PbTiO\({}_{3}\) are of similar magnitude, despite PbTiO\({}_{3}\) possessing more than double the material polarization of BaTiO\({}_{3}\).
To further investigate the connection of photovoltaic effect to material polarization, we studied a systematic family of structures based on PbTiO\({}_{3}\). Starting with the cubic perovskite in the paraelectric structure, we rigidly displaced oxygen ions along a single Cartesian axis by amplitudes ranging from 0.01 to 0.09 lattice vectors, without otherwise altering the geometry. The spectra of shift current and aggregate shift vector are shown in for several displacements. The results indicate a complex relationship between shift current and material polarization. As shows, with soft mode amplitude 0.01, the shift current at 3.2 eV above band gap is negative; with amplitude 0.07, the shift current reverses direction, resulting in a change of -200%. With amplitude 0.01, there is a negative peak at 3.8 eV; with amplitude 0.07, the peak shifts to 4.2 eV and is four times the size, for an increase of over 300%.
The overall current susceptibility and aggregated shift vector \(\bar{R}\) are shown for PbTiO\({}_{3}\) with varying polarization.
Next, we turn our attention to the integrated shift vector \(\bar{R}_{Z}(\omega)\). The changes in shift vector are of special interest, since the symmetry constrains the overall shift current expression via the shift vector. The integrated shift vector spectrum echoes the overall current response, but contains some distinct features. The increase in current from 4-5 eV does not appear to result from increased shift vector length, but from stronger coincidence of high transition intensity and large shift vectors. In fact, the overall shift vector changes little with displacement. However, from 7.5-8.5 eV, the integrated shift vector changes dramatically, suggesting that at some points in the Brillouin zone the oxygen displacement substantially alters the shift vector. Changes to the overall response are thus a combination of changes both to shift vector and associated intensity.
To understand these results, the electronic bands participating in transitions in these frequency ranges were examined directly. For the 4-5 eV range, examples of the transitions and associated Bloch states that dominate are shown in Fig. 4(a) at 0.01 and 0.09 lattice vector displacements. For this transition, the shift vector is 0.6 A at displacement of 0.01, and 1.0 A at 0.09. The valence state is largely composed of oxygen \(p\)-orbitals, while the conduction state is essentially a titanium \(d_{xy}\) state. The states, like the shift vectors, are largely unchanged by the oxygen displacement.
However, the transitions in the higher energy range are notably different. Shown in Fig. 4(b) are examples of the dominant transitions in the 7.5-8.5 eV range. The shift vector is large and positive (32.3 A) for 0.01 lattice vector displacement, and large but negative (-22.7 A) at 0.09 displacement. The participating valence state can be characterized as bonding between the Ti and O atoms collinear with polarization, while the conduction state features Ti-O anti-bonding. These results point not to a simple dependence on material polarization, but to a dependence of shift current on the extent of localization of the initial and final states, which in turn depends on atomic displacement. Transitions between states that do not experience bonding interactions in the direction of ferroelectric polarization manifest short shift vectors and insensitivity to oxygen displacement.
(a) The non-bonding Bloch states of PbTiO\({}_{3}\) are involved in a transition that is insensitive to material polarization, with a shift vector length change from 0.6 Å to 1.0 Å as O sublattice displacement increases from 0.01 to 0.09, and (b) a transition from bonding to antibonding gives a shift vector that is highly sensitive to material polarization, with shift vector length change from 32.4 Å to -22.7 Å for increasing O sublattice displacement.
## Conclusion
The shift current response was calculated for ferroelectrics barium titanate and lead titanate. In the case of barium titanate, the shift current closely matches experiment, successfully predicting magnitude, sign, and spectral profile, including the notable dependence of current direction on polarization. For lead titanate, where reliable quantitative data are unavailable, the direction and its lack of dependence on polarization were reproduced. This strongly suggests that shift current is the dominant mechanism of the bulk photovoltaic effect in these materials.
For the materials analyzed, the strongest responses are at frequencies well into the UV spectrum and outside the spectral range probed in most experiments. Consequently, the potential for large shift current response may not yet be fully realized, a conclusion supported by the very large bulk photovoltaic effect observed in response to X-rays. Furthermore, the strength and direction of the photocurrent subtly depend on the electronic structure of the material, including covalent bonding interactions. This suggests that ferroelectric compounds can vary widely in response profile, and could potentially perform much better than previous results have indicated, encouraging efforts to design materials with large shift current response in the visible spectrum.
| 10.48550/arXiv.1202.3168 | First principles calculations of the Shift Current Bulk Photovoltaic Effect in Ferroelectrics | Steve M. Young, Andrew M. Rappe | 549 |
10.48550_arXiv.1005.1322 | ## 1 Introduction
The gapless energy spectrum of electrons and holes with linear dispersion relation up to a few eV at K points (Dirac points) in graphene at equilibrium leads to unique features in electrical transport and optical properties. Dynamics associated with carrier intraband relaxation and interband recombination have been studied using ultrafast pump-probe spectroscopy of ultra-thin film of graphite, epitaxial graphene and graphene suspensions. A faster component of \(\sim\) 100 fs has been attributed to intraband carrier-carrier scattering and a slower component between 0.4 ps to 5 ps to intraband carrier-phonon scattering. The latter depends on the defect density in the sample - being smaller for more defected sample. In other studies on ultrathin graphitic film of thickness 17 nm and epitaxial graphene grown on SiC, it was shown that photoexcited carriers loose most of their energy within first 500 fs by emitting optical phonons resulting in creation of hot optical phonons which survive up to a few ps and present the main bottleneck to the subsequent cooling of the carriers.
Here, we report the results on relaxation dynamics of photoexcited carriers in reduced graphene oxide (RGO) flakes suspended in water (RGO-suspension) and deposited on indium-tin-oxide coated glass plates (RGO-film). The suspension contains mostly single layer graphene as evident from a single '2D' band at 2686 cm\({}^{-1}\) in the Raman spectra. There are three time constants associated with the decay of the photoexcited carriers: first one \(\sim\) 200 fs, second one between 1 to 2 ps and third one \(\sim\) 25 ps, all of them nearly independent of the pump fluence. The first and third components are attributed to electron-optical phonon and electron-acoustic phonon scattering, which are found to be similar for the suspension and the film. The second time constant, on the other hand, which is attributed to the lifetime of optical phonons in graphene, is smaller for the RGO-film. We estimate the nonlinear absorption coefficient from the degenerate pump-probe signal and compare it with that obtainedfrom open aperture Z-scan measurements for the RGO-suspension.
## 2 Experimental Procedure
RGO-suspension having graphene concentration of \(\sim\) 80\(\upmu\)g/ml was prepared and characterized as reported in Ref.. 150\(\upmu\)l of the suspension taken in a 1 mm optical path length quartz cuvette was used in the degenerate pump-probe and Z-scan experiments. To make film of RGO (thickness \(\sim\) 75 nm), 20 \(\upmu\)l of the suspension was dropped on indium tin oxide (ITO) coated glass plate and allowed to dry for a day. Degenerate pump-probe experiments at 790 nm were performed on the suspension and the film using 80 fs laser pulses from a Ti:sapphire laser amplifier (1kHz, Spitfire, Spectra Physics). The pump fluence was varied between 0.12ml/cm\({}^{2}\) and 1.21 mJ/cm\({}^{2}\) whereas the probe fluence was kept constant at 9.6\(\upmu\)l/cm\({}^{2}\). The pump beam was modulated at a frequency of 383Hz using an optical chopper and the change in probe transmitted intensity due to the presence of the pump was recorded using a Si-PIN diode in a lock-in amplifier detection scheme.
## 3 Results and Discussion
Results from degenerate pump-probe experiments for the RGO-suspension and RGO-film are presented in where \(T\) is the linear transmission of the probe beam in absence of the pump and \(\Delta T(t)\) is the change in probe transmission in presence of the pump as a function of time delay (\(t\)) between the pump and probe pulses. shows that the magnitude of the signal at zero delay is higher for the RGO-suspension as compared to RGO film at the same pump fluence. The magnitude of positive \(\Delta T\) signal at zero delay corresponds to hot thermal carrier distribution which is established immediately after photo-excitation within a time scale comparable to the laser pulse width. This nonthermal carrier distribution relaxes towards equilibrium distribution on a time scale of \(\sim\) 25 ps for both the suspension and the film. It can be seen from that the signal decreases rapidly to almost 15% of its peak value in the first 250 fs and then decreases much slowly afterwards.
The differential transmission signal is fitted with the following function:
\[\frac{\Delta T}{T}\,\,\,=\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,attributed to the lifetime of the optical phonons in graphene as discussed in the next paragraph. In the suspension, the graphene layers are isolated from each other and hence the generated optical phonons survive for a longer time. On the other hand, in case of the film, layer to layer interaction and interaction with the substrate open up new channels for the phonons to release their energy to low energy acoustic phonons.
The number density of photoexcited carriers is a function of pump fluence and determines the interband electron-hole recombination times. We note from that \(\tau_{3}\) is nearly independent of the pump fluence and hence can not be attributed to carrier recombination. Since the defect levels need not be same for the RGO in suspension and the film, \(\tau_{3}\) being almost the same for both of them can be associated with the carrier-acoustic phonon scattering in graphene rather than carrier-defect scattering.
The photoexcited carrier density can be approximately calculated as \(n\sim F\alpha l/h\nu\), where \(F\) is the pump fluence, \(\alpha\) is the linear absorption coefficient and \(h\nu\) is the pump photon energy. For RGO-suspension, \(\alpha\)\(\sim\) 1.0x10\({}^{4}\) cm\({}^{-1}\), and considering, the thickness of the graphene sheet, \(d\)\(\sim\) 1 nm, we obtain \(n\sim\) 5x10\({}^{12}\) cm\({}^{2}\) at \(F=\) 1.2 mJ/cm\({}^{2}\) which corresponds to the initial carrier temperature of \(\sim\) 3030 K. At such level of carrier densities, the carrier mean free path, \(l\) has been theoretically estimated to be \(\sim\) 50 nm, which in turn corresponds to carrier scattering time \(\tau-l/v_{\rm F}\) (Fermi velocity \(v_{\rm F}\sim\) 10\({}^{8}\) cm/s) of the order of 50 fs. For samples having defects, this time constant is expected to be even shorter. In our present experiments, the fastest component, \(\tau_{1}\sim\) 200 fs, is much larger than 50 fs for both the suspension and the film. This leads us to associate \(\tau_{1}\) with electron-optical phonon scattering instead of electron-electron scattering as suggested in earlier studies on epitaxial graphene and ultrathin film of graphite.
Optical excitation by \(\sim\)1.6 eV (790 nm) laser pulse creates a nonthermal carrier population 800 meV above the band extrema. The initial relaxation takes place within \(\tau_{1}\sim\) 200 fs and slows down later, resulting in a bottleneck which can be explained due to two possible reasons; firstly, for the relaxation from conduction band to valence band all the electrons have to cross the vicinity of the K point which has a low density of states and therefore acts as a bottleneck for the dynamics. Secondly, the early relaxation leads to emission of cascade of G-band (\(\sim\)193 meV) optical phonons which survive for \(\tau_{2}\sim\) 1.8 ps in the suspension and \(\sim\) 1.2 ps in the film. These numbers for \(\tau_{2}\) are close to the life time of G-phonons in graphite (\(\sim\) 2.2 ps) measured using time-resolved Raman spectroscopy. Since the efficient channels are now occupied after heating the optical phonons which further split into acoustic phonons, a large time scale for electron-acoustic phonon scattering, \(\tau_{3}\sim\) 25 ps is observed for both the suspension and the film.
The positive sign of \(S=\Delta T/T\) at \(t=0\) is generally referred to as bleaching of the ground state of the system or saturable absorption in the degenerate pump-probe experiments. compares the values of \(S\) for both the suspension and the film at three pump fluences. It is seen that \(S\) for the film is less than half as compared to the suspension. The magnitude of the nonlinear absorption coefficient, \(\beta\) can be approximately determined from \(S\) by using the relation\(\beta=\ln(1+S)(L_{\rm eff}l_{\rm pump})\), where \(L_{\rm eff}=\) (cuvette optical length x graphene concentration) is the effective optical path length of the sample (\(\sim\) 80 nm for the suspension and the film) and \(I_{\rm pump}\) is the pump intensity. The parameter \(\beta\), in turn, is related to the imaginary part of third order nonlinear optical susceptibility, \(\ln\chi^{}=[10^{{}^{\prime}}c\lambda\eta^{2}/96\pi^{2}]\beta\), where \(\ln\chi^{}\) is in esu, \(\beta\) is in cm/W, \(c\) is speed of light in vacuum, \(\lambda\) is the laser wavelength and \(\eta\) is the linear refractive index of the sample (taken as 1.5). At the maximum pump fluence of 1.21 mJ/cm\({}^{2}\), the calculated values of \(\beta\) and \(\ln\chi^{}\) for RGO-suspension are 7.5x10\({}^{-8}\) cm/W and 4.2x10\({}^{-11}\) esu. For the RGO-film, these values are 2.9x10\({}^{8}\) cm/W and
Comparison between the amplitude of three components and decay time constants of photoexcited carriers in RGO-suspension (open circles) and RGO-film (filled circles).
1.6x10\({}^{\text{-}11}\) esu, almost 2.5 times smaller than that for the suspension.
In we show the results from open aperture (OA) Z-scan measurements on the RGO-suspension and water only, taken in a 1 mm optical path length cuvette. A saturable absorption behavior for the RGO can be clearly seen. The data has been fitted (solid line) with the model given in Ref. to deduce nonlinear optical constant, \(\beta\) and the saturation intensity \(I_{\text{s}}\) as \(\beta\) = 4.5x10\({}^{\text{-}8}\) cm/W and \(I_{\text{s}}\) = 100 GW/cm\({}^{2}\). A figure of merit (FOM), defined as \(\text{FOM}=|\text{Im}\chi^{}|/\alpha\) gives \(\text{FOM}=4.5x10^{\text{-}15}\) esu.cm.
## 4 Conclusion
In summary, we have studied the ultrafast response after femtosecond pulse excitation of single layer reduced graphene oxide flakes suspended in water as well as few layer thick film deposited on ITO coated glass plate. Three component relaxation dynamics shows a fast component \(\sim\) 200 fs, second component between 1 and 2 ps and a slower one \(\sim\) 25 ps. The assignment of these components is discussed in our work.
| 10.48550/arXiv.1005.1322 | Femtosecond Photoexcited Carrier Dynamics in Reduced Graphene Oxide Suspensions and Films | Sunil Kumar, N. Kamaraju, K. S. Vasu, A. K. Sood | 1,562 |
10.48550_arXiv.1005.4369 | ###### Abstract
We demonstrate the realization of nearly massless electrons in the most widely used device material, silicon, at the interface with a metal film. Using angle-resolved photoemission, we found that the surface band of a monolayer lead film drives a hole band of the Si inversion layer formed at the interface with the film to have nearly linear dispersion with an effective mass about 20 times lighter than bulk Si and comparable to graphene. The reduction of mass can be accounted for by repulsive interaction between neighboring bands of the metal film and Si substrate. Our result suggests a promising way to take advantage of massless carriers in silicon-based thin-film devices, which can also be applied for various other semiconductor devices.
The ultimate performance of electronic devices is largely governed by the effective mass of charge carriers. Therefore, the never-ending quest for higher performance devices has looked after high speed carriers with light effective mass. This quest may have reached its ultimate goal through the recent findings of massless electrons with their speed close to light in graphene and bismuth compounds. However, for these materials various difficulties exist in promptly realizing practical and mass-producible devices.
In general, the effective mass of electrons is determined by their energy dispersion. Especially, the edge structures of conduction and valence bands near the energy gap (\(E_{g}\)) are important. In the band-edge region, the dispersion is sensitive to the interaction between the bottom conduction and the top valence band as described well by the \(k\cdot p\) theory. This standard theory provides an approximate analytic expression for band dispersion under the interband interaction around high-symmetry points of momentum (\(k\)) space. The interaction energy, \(\pm\sqrt{(E_{g}/2)^{2}+(P\cdot k)^{2}}\) (\(P\), the optical matrix element), gives rise to repulsion between bands and provides the linear component in dispersion. As \(E_{g}\) approaches zero, the dispersion becomes nearly linear \(E(k)\approx P\cdot k\) with a much lighter effective mass of electrons \(m^{*}=\hbar^{2}E_{g}/(2P^{2})\). This effect explains the trend that a semiconductor with a narrower gap has a lighter effective mass and the zero-gap semiconductor of graphene has a negligible effective mass. This theory also tells us that if the energy gap could be controlled, the effective mass would be tuned through the interband interaction. While the energy gap of bulk materials cannot be tuned easily as fixed by their crystalline structure, here we show that a similar effect can be obtained at a semiconductor interface where a proper interface state is formed within the band gap.
We chose an ultrathin (only single layer) Pb film on \(n\)-type Si substrate, which has ideally abrupt interfacial structure and strongly dispersing metallic electron bands within the Si band gap. For this system, the film-substrate interaction, which is exploited in the present work, was already invoked to explain the anomalous superconductivity at the two-dimensional (2D) limit. Figure 1(a) illustrates the atomic structure of the interface with the Pb density of 1.2 monolayer (ML) (7.84 atoms/nm\({}^{2}\)). Pb atoms are densely packed within a single layer upon the bulk-terminated Si surface. Most of Pb atoms sit on top of underlying Si atoms (T1 or T1\({}^{\prime}\) sites) while part of them are slightly displaced (T1\({}^{\prime}\)) by additional Pb atoms in hollow sites (H3), which lead to the formation of a uniform \(\sqrt{7}\times\sqrt{3}\) unit cell (grey lines). Due to the anisotropy of this unit cell and the three-fold symmetry of the substrate, one inevitably obtains triply rotated domains, which should be considered in interpreting experimental data.
(color online). (a) Crystal- and (b) reciprocal- lattice structures for Pb/Si at 1.2 ML with a \(\sqrt{7}\times\sqrt{3}\) symmetry. Grey (dashed) line represents the \(\sqrt{7}\times\sqrt{3}\) (1\(\times\)1) surface unit cell. APRES data collected along (c) long and (d) short arrows in (b) crossing 1\(\times\)1 (\(\bar{\Gamma}\)) and \(\sqrt{7}\times\sqrt{3}\) (\(\bar{\Gamma}\)\Angle-resolved photoemission (ARPES) measurements were conducted in an ultra-high vacuum chamber (6.5 \(\times\) 10\({}^{11}\) torr) equipped with a hemispherical electron analyzer (R4000, VG Scienta) and a high-flux He discharge lamp for 21.2-eV photons. The samples were cryogenically cooled down to 90-100 K for measurements. The overall energy and momentum resolutions were better than 20 meV and 0.02 A\({}^{-1}\). Figure 1(c) shows band dispersions along the high-symmetry direction through the center of Brillouin zone \(\bar{\Gamma}\) [the long arrow in Fig. 1(b)] measured by ARPES. Near \(\bar{\Gamma}\), an intense parabolic band is readily found, which is the well-known direct transition from bulk Si valence bands. Another feature with strong intensity appears near each zone boundary (\(\bar{K}\)) with little dispersion around 0.7 eV, which is the covalent-type bonding state between Pb and Si. In contrast, around \(\bar{K}\), there are parabolic bands dispersing toward the Fermi energy (\(E_{F}\)), which were identified as due to 2D metallic electrons localized within the Pb layer (called S2 hereafter). This metallic surface state induces a huge upward band bending in the \(n\)-type substrate, and forms a \(p\)-type inversion layer (IL) as shown in Fig. 1(e). The strong band bending yields a triangular potential well to confine electrons in the interface IL into 2D quantum well states (QWS). An IL state at interfacial layers is typically not easily probed by very surface sensitive ARPES but, for an ultrathin metal film like single-layer In on Si, a previous ARPES study traced its detailed dispersion. We also found such IL states; two hole-like bands are observed with weak intensity between the bulk and surface bands in Fig. 1(c). A relatively prominent band is identified as the so called heavy-hole (HH) bands but the other branch with a lighter effective mass, light-hole (LH) band, is barely observable as a weak and broad feature on top of the Si bulk band.
Although the LH band is not clear enough here [Fig. 1(c)] due to its weak intensity, we can map its dispersion clearly at the center of the surface Brillouin zone (\(\bar{\Gamma}_{S}\)) away from \(\bar{\Gamma}\) as translated by the surface periodicity. Figure 1(d) shows band dispersions taken through two such \(\bar{\Gamma}_{S}\)'s and the \(\bar{M}\) point [the short arrow in Fig. 1(b)]. There are strongly dispersing bands with dominant intensity, crossing \(E_{F}\) at \(\pm\)0.21 A\({}^{-1}\), which correspond to the S2 state for Pb metallic electrons. In addition, two other bands are identified, S1 folded back near \(E_{F}\) with respect to \(\bar{M}\) and R folded with respect to \(\bar{\Gamma}_{S}\). Surprisingly, their dispersions are \(\Lambda\)-shaped and apparently very linear, especially for R. The detailed dispersion of R, located on \(\bar{\Gamma}_{S}\) and related to the Si LH band below, can be shown more clearly and quantified by the peak positions of momentum distribution curves (MDCs) in Fig. 2(c). The dispersion is indeed linear within the experimental uncertainty as shown in Fig. 2(b) (open circles). Thus, the effective mass value would be extremely small and cannot be quantified by a simple parabolic fit of the dispersion as in the case of graphene and Bi compounds. Instead, the Fermi velocity can straightforwardly be extracted from the slope of the band and is as high as 4.6 \(\pm\) 0.4 \(\times\) 10\({}^{5}\) m/s, which reaches to half of those in graphene and Bi\({}_{0.9}\)Sb\({}_{0.1}\), and is similar to those in Bi\({}_{2}\)Se\({}_{3}\) and Bi\({}_{2}\)Te\({}_{3}\). The effective mass value of this linear band will be discussed further below.
Since the R band with a striking dispersion cannot be found in Si bulk crystals [Fig. 2(b)] and on clean Si surfaces, it must be due to electron states localized at the interface (Si ILs) or within the Pb layer. To identify the origin of each band further, we performed theoretical calculations based on density functional theory (DFT). The calculations were performed using the Vienna _ab-initio_ simulation package within the generalized gradient approximation and the ultrasoft pseudopotential scheme with a plane wave basis. The surface is modeled by a periodic slab geometry with 6 and 12 Si layers whose bottoms are terminated by hydrogen.
Figure 3(a) shows the calculated constant energy contours at \(E_{F}\). There are two kinds of surface-state Fermi contours, point-like crossings and wavy lines repeated following the surface periodicity. Since ARPES detects signals coming from all triply-rotated domains, the expected Fermi contours result in a complex pattern [Fig. 3(b)]. Nevertheless, the agreement between the calculated and experimental Fermi contours is excellent in Figs. 3(c) and 3(d). The point-like crossings and the details of the wavy contours are all precisely reproduced, which correspond to S1 and S2, respectively. The dispersions of S1 and S2 are also reproduced quantitatively well in the calculation [Fig. 3(f)]. These states come from the Pb layer due to in-plane Pb 5\(p\) orbitals. In fact, S1 and S2 bands are doubly-degenerated with contributions from domains \(B\) and \(C\) overlapped exactly [Fig. 3(f) and green and blue ones in Fig. 3(b)]. The contribution from domain \(A\), which pass through two \(\bar{\Gamma}_{S}\) points [Fig. 3(e) and red ones in Fig. 3(b)], is not degenerated. For this domain, the Pb-derived band (S3) is located just above \(E_{F}\) and two Si-derived bands below \(E_{F}\) at \(\bar{\Gamma}_{S}\) [Fig. 3(e)]. The calculated electron density for the latter is distributed within Si layers and their dispersions are consistent with the Si IL states (LH and HH) mentioned above. In
(color online). (a) ARPES data of the R band. Data below the dashed line are those symmetrized with respect to \(\bar{\Gamma}_{S}\) to eliminate the strong neighboring feature S2. (b) Spectral peak positions of the R band (open circles) extracted from MDCs shown in (c). Data points given in closed circles are those symmetrized. The dispersion of a normal Si LH band (black parabola) is compared to show the reduction of curvature. Red and blue lines are the results of \(k\cdot p\) model fit. The error bar in \(k\) is less than \(\pm\)0.02 Å\({}^{-1}\), which brackets the quantified effective mass (orange lines) (see also the supplementary information). Red lines overlaid in (c) show fits by Lorenzian functions.
2(b). The calculation, thus, clearly identifies S1, S2, S3 and R bands as three 5\(p\) states of the Pb layer and a Si-derived state, and reproduces the dispersions of S1, S2, and S3 quantitatively well. This result is consistent with the very recent work in Ref. for the single-domain surface.
One notable feature of the calculation is that the bottom of the S3 band lies on \(E_{F}\), namely, just above the experimentally measured band R with a very small energy separation. Thus, S3 has a chance to interact strongly with R from Si subsurface layers through the interband interaction. As mentioned above, this interband interaction determines the effective mass of the bands according to the \(k\cdot p\) model. Indeed, such an interaction between the band of a Ag monolayer film and the band of a Si substrate was observed recently and affects the dispersion of a S3-like surface state of Ag.
The fitting of our experimental dispersion using the \(k\cdot p\) model with the band-gap size determined by the calculated energy position of S3, 23 meV, reproduces well the linear dispersion of the R band as shown in Fig. 2(b), yielding an extremely light effective mass of 0.0074 \(\pm\) 0.0015\(m_{e}\), where \(m_{e}\) is the electron rest mass. In fact, the \(k\cdot p\) fitting is simple enough to treat the band-gap size as another free parameter and the band gap is reliably determined to be within 18-24 meV in good accord to the theoretical value. The effective mass determined is 20 times lighter than the normal Si LH band (0.15\(m_{e}\)). Note also that, for the HH bands, the interband interaction is forbidden from the band symmetry. Therefore, we conclude that the R band with an extremely linear dispersion is the modified LH band through the interband interaction with S3, while we do not know why the HH band itself is not observed around \(\bar{\Gamma}_{S}\).
The \(k\cdot p\) method has been widely applied to various semiconductors and the smallest effective mass reported is 0.016\(m_{e}\) for InSb. This yields the carrier mobility as high as 30,000 cm\({}^{2}\) V\({}^{-1}\) s\({}^{-1}\), about 2 orders of magnitude higher than bulk Si. The present effective mass for the Si IL, thus, indicates the possibility of an ultrafast carriers at Si interfaces. This method has not been applied to graphene and Bi compounds, for which only the effective mass values measured by transport experiments are available; about (0.002-0.009)\(m_{e}\) in a similar range with the present value. In contrast, the DFT calculation shown above does not reproduce exactly the linear dispersion of the LH band, yielding an effective mass of roughly 0.07\(m_{e}\). We think this discrepancy is due to the limitation of the present slab calculation with only few Si layers in reproducing properly the QWS of the much thicker IL.
The proposed mechanism for the linear dispersion is summarized in Fig. 4(a). The contact of a metal film to a semiconductor substrate plays two key roles; (i) to induce a strong IL near the interface lifting the hole bands of the substrate up close to \(E_{F}\) and (ii) to provide a proper band (S3 here) within the band gap inducing the strong repulsive interaction with hole bands for linear dispersion. In order to verify this mechanism, we took advantage of diverse surface band structures in metal/Si systems. We examined three representative metal overlayers -- Pb, Au, and Ag on Si -- with a common \(\sqrt{3}\times\sqrt{3}\) symmetry but with different surface bands. Figures 4(b)-4(d) describe corresponding band dispersions around \(\bar{\Gamma}\) taken from previous reports in literature. The \(\sqrt{3}\times\sqrt{3}\)-Pb system (with the Pb density of 4/3 ML) has no bands close to Si LH bands while the others (at 1 ML) have adequate bands from the films similar to the present case to allow the interaction with the LH band (band gaps of 40-60 meV). The consequence of such a difference on the dispersion of the LH band is remarkable. In Fig. 4(e), our ARPES data for the Pb system show a normal parabolic dispersion consistent with bulk Si. However, for Au and Ag systems, dispersions are obviously much more linear [Figs. 4(f) and 4(g)].
(color online). Constant energy contours at \(E_{F}\) calculated from DFT for the (a) single-domain and (b) triply-domain surfaces. Contours from each domain (\(A\), \(B\), and \(C\)) are indicated by different colors in (b). (c) ARPES constant energy contours at \(E_{F}\). The DFT contours from (b) are superimposed. The raw data (region without DFT contours) are symmetrized reflecting the fundamental mirror symmetry. (d) Enlarged data around \(\bar{M}\) for the detailed comparison between theory (while lines) and experiment (red circles extracted by MDC peak positions). DFT band dispersions along the arrow in (b) for (e) domain \(A\) and (f) domains \(B\) and \(C\), which are overlapped together in (g). The shaded area represents the projected bulk band. For the best match with the experimental data the calculated surface state energies are rigidly shifted by 120 meV with respect to Si bands.
Our result suggests an unprecedented way to ultrafast electronic devices based on Si. Since this technique directly controls the carrier mass at the interface by a metal overlayer, it does not require any artificial modification or chemical engineering on the Si bulk lattice itself. Furthermore, this mechanism, in principle, can be generally applied to various other semiconductors even with thicker films, provided that proper interface states exist. For example, this may solve the notorious imbalance between the hole and electron mobility of III-V compounds.
This work was supported by KRF through the CRi program. M.H.K. acknowledges the support from KRF (Grant No. 2009-0074825).
| 10.48550/arXiv.1005.4369 | Nearly Massless Electrons in the Silicon Interface with a Metal Film | Keun Su Kim, Sung Chul Jung, Myung Ho Kang, Han Woong Yeom | 132 |
10.48550_arXiv.1410.1765 | ###### Abstract
We show elasto-capillary folding of silicon nitride objects with accurate folding angles between flaps of \((70.6\pm 0.1)^{*}\) and demonstrate the feasibility of such accurate micro-assembly with a final folding angle of 90deg. The folding angle is defined by stop-programmable hinges that are fabricated starting from silicon molds employing accurate three-dimensional corner lithography. This nano-patterning method exploits the conformal deposition and the subsequent timed isotropic etching of a thin film in a 3D shaped silicon template. The technique leaves a residue of the thin film in sharp concave corners which can be used as an inversion mask in subsequent steps. Hinges designed to stop the folding at \(70.6^{*}\) were fabricated batchwise by machining the V-grooves obtained by KOH etching in silicon wafers; 90deg stop-programmable hinges were obtained starting from silicon molds obtained by dry etching on wafers. The presented technique is applicable to any folding angle and opens a new route towards creating structures with increased complexity, which will ultimately lead to a novel method for device fabrication.
## 1 Introduction
### Self-folding of 3D micro-structures
The fabrication of 3D micro-structures has become an important field of interest in the scientific community over the past three decades. Traditional mask-based approaches, such as photo-lithography and its developments (including X-ray lithography, electron-beam lithography and dip-pen nanolithography), have proven to be inadequate for fabricating truly 3D-patterned structures. The main limitations include: an inherent two-dimensionality, size limitations, being time-consuming, and demanding a complex fabrication.
Fabrication examples of 3D structures abound in nature. Salt crystallization and the folding of protein or DNA are processes that engineers dream of reproducing in a laboratory with as much precision and reproducibility as is seen in nature. The process by which disordered components are organized into patterns or structures without human intervention is known as "self-assembly" or, by analogy with the previously mentioned top-down methods, a "bottom-up" approach. Although great proofs-of-concept have been published, such engineering suffers from a too high level of uncertainty, as pointed out by Gracias _et al._ in their excellent review. Therefore, they prefer the use of a more deterministic form of self-assembly known as "self-folding" or "micro-origami". Combining the strengths of both lithography and self-assembly, the final 3D structure is predetermined by the linkages between the different parts that are assembled. The obvious link with origami, the ancient Japanese art of folding paper, provided its name to this technique. While origami-like planar structures are fabricated using standard micro-machining techniques, many methods of self-folding have been investigated, some more common than others. These methods include ultrasonic pulse impact, pneumatics, electroactive swelling, thermal actuation of polymer films, thin-film stress-based assembly (TFSA), magnetic forces, and capillary forces. Surface tension is probably the most common method of self-folding. In the micro/nanometer world, interfacial forces dominate over body forces such as gravity, making them a perfect candidate for the self-folding of micro-structures. Syms was the first to introduce this method by using solder pads which are melted to power assembly before solidification in their final state. More recently, this method has had great nanoscale applications as a result of the work of Gracias _et al._. For a complete overview of self-folding techniques, see the recent reviews.
An elegant macro-scale illustration of self-folding by surface-tension is Bico _et al._, who demonstrated the spontaneous wrapping of thin millimeter-sized polydimethylsiloxane (PDMS) sheets around a water droplet: the so-called elastocapillary folding technique. Using the same concept, our group has demonstrated the fabrication of silicon nitride 3D micro-objects by capillary forces in which the actuating liquid, in our case water, disappears as a result of its spontaneous evaporation. Final closure is assured by the strong cohesion between flaps without the need for solder,and the assembly is carried out under ambient conditions either by simply depositing water on top of the structures or by providing a liquid through a tube at the centre of the objects.
A crucial feature of self-folded objects is to pre-determine their final shape. Some techniques require a locking mechanism, for example, some research on self-folding by magnetic interaction, while another treatment of magnetic self-folding relies on plastic deformation. Using solder assembly, the quantity of melting material determines the final folding angle. Likewise, in TFSA, the final radius of curvature is a function of the stresses in the different layers, with curvatures ranging from a few millimeters down to nanometers. The final shape from using self-folding polymer films can be controlled by designing several small shrinking hinges in series.
Structures folded by elasto-capillary interactions are limited in terms of their final three-dimensional shapes. Folding ceases once the moving flaps encounter an obstacle, typically a nearby flap, and the elasticity of the hinges causes the objects to re-open if not enough stiction is present between the flaps. The work presented here aims at extending the scope of elastocapillary folding of silicon nitride micro-objects by predefining the final assembly.
Made of a thick rigid part and a thin flexible part, these complex hinges are designed in such a way that the final assembly angle is predefined by their shape. After folding through evaporation of water, the two opposite thick parts meet and adhere. The final folding angle therefore depends on the initial angle between the substrate and the thick parts of these smart hinges.
### Corner lithography and self-folding
Corner lithography is a wafer scale nano-patterning technique that offers the opportunity to form structures in sharp concave corners, independently of their orientation in space. The conformal deposition of a material layer over a patterned substrate will result in a greater effective thickness in any sharp concave corner. Isotropic etching therefore yields nano-features as presented in This technique was first developed and used in our laboratory to create silicon nitride nano-wire pyramids. We then extended the scope of this technique by demonstrating the use of the structures formed by corner lithography as a mask for subsequent patterning steps. In the meantime, Yu _et al._ demonstrated the fabrication of nano-ring particles and photonic crystals using corner lithography. More recently, our group has continued the development of this technique and demonstrated the parallel nano-fabrication of fluidic components with cell culturing application, as well as the wafer-scale fabrication of nanoapertures and the machining of silicon nitride 3D fractal structures.
In this paper, we use corner lithography to fabricate the smart hinges presented in Sharp features must be avoided when it comes to bending or folding, since they lead to an extreme concentration of stress. Consequently, corner lithography needs to be performed in rounded molds for our purposes. This situation leads to conditions on the radius of curvature of the mold, as well as on the thicknesses of the subsequently deposited material.
## 2 Fabrication process flow
In this section we present the fabrication steps necessary to build the stop-programmable hinges depicted in The main steps are the same for the fabrication of both
The stop-programmable folding principle. The design of the complex hinges is such that once folded, the flap forms a predefined angle with the planar support. Self-folding of the structure is enabled through evaporation of water and decrease of the liquid–air interface of the meniscus. (a) \(70.6^{\circ}\) stop-programmable hinge. (b): \(90^{\circ}\) stop-programmable hinge. In both cases, the flaps adhere thanks to a sufficiently large stiction area and there is no need for a locking mechanism.
The two following sections extensively describe the process flow, with the first part describing the whole procedure for 70.6\({}^{\circ}\) smart hinges, and the second part pointing out the differences when fabricating 90\({}^{\circ}\) stop-programmable hinges.
### \(70.6^{\circ}\) stop-programmable hinges
The strategy is similar to that presented by our lab in previous publications. Corner lithography is here employed to create a masking layer which will be subsequently used to etch the underlying body layer before being removed. shows the procedure step by step for machining a 70.6\({}^{\circ}\) stop-programmable hinge.
The initial Si (silicon) molds will define the shape of our final object. The opening angle \(\alpha\) of the molds, see and 3, defines the final folding angle, \(\beta\), through the relation \(\beta=\pi-\alpha.\) Silicon has a face-centred cubic structure with a well-defined lattice. The angle between the top \(<\)110\(>\) plane and the {111} planes is exactly 35.3\({}^{\circ}\), as depicted in Etching the molds with KOH on oriented wafers yields well-defined openings with the desired opening angle (\(\alpha\)=109.4\({}^{\circ}\), \(\beta\)=70.6\({}^{\circ}\)) with sharp transitions between the different planes, see (a). Note that the mask pattern, consisting of rectangular openings, needs to be rotated by 54.7\({}^{\circ}\) with respect to the vertical \(<\)111\(>\) planes in order to get rectangular V-grooves with KOH etchant.
The bending of sharp objects is to be avoided since it leads to extreme stress concentrations.
Corner lithography in a rounded corner. (a): When a material layer of thickness \(t\) is conformally deposited over a mold with radius of curvature \(R\) that is greater than \(t\), the thickness at the tip is unchanged. (b): On the other hand, when \(R\) is less than \(t\), a concave corner is created and the effective thickness is \(c=R+(t-R)/\!\sin(\nicefrac{{\alpha}}{{2}})\).
Corner lithography in a sharp corner. (a): When a conformal layer of thickness \(t\) is deposited over a concave corner of opening \(\alpha\), the effective thickness of material at the corner is \(a=\nicefrac{{t}}{{\sin(\nicefrac{{\alpha}}{{2}})}}>t\). (b): After isotropic etching by an amount of \(r\), material with thickness \(b=a-r\) remains in the corner.
(b). Kim _et al._ showed that high temperature oxidation is an efficient method to round off sharp silicon V-grooves. Linear relations between the oxidation time and the final achieved radius of curvature were experimentally found by the authors for oriented wafers but not for wafers. Our own oxidation experiments followed by SEM inspections allowed us to determine a similar relation for a 1150\({}^{\circ}\)C wet oxidation step applied to wafers:
\[R=(124\pm 72)+(94\pm 7)\sqrt{t} \tag{1}\]
A 98 min oxidation step yields a 1 um radius of curvature and is used for our fabrication, (b). It is difficult to use a higher radius of curvature since that would imply a deposition of a thicker material layer, as described in (a). Once SiO\({}_{2}\) (silicon dioxide) is stripped in HF, the molds are ready for corner lithography.
First, two layers are deposited by low-pressure chemical vapor deposition (LPCVD): first, a thick silicon rich nitride (SiRN) layer that is to be structured to form the thick part of the smart hinges, see Fig. 3, followed by a polysilicon (polySi) layer. The total thickness of material here needs to be greater than the initial radius of curvature \(R\) as was emphasized in Typically, we deposit a 1 um SiRN layer and a 150 nm polySi layer in a mold where the radius of curvature \(R\) is 1 um. In any case, the following design criterion should be respected:
\[t_{\text{SiRN-1}}+t_{\text{polySi}}\geq R \tag{2}\]
On top of this stack, a last conformal layer of SiRN is deposited. Provided that the design criterion has been respected, the cross section of the stack should look like (d). There are no constraints on the thickness of the top SiRN layer, but it needs to be perfectly known since this layer will be time-etched in phosphoric acid (\(H_{3}PO_{4}\)) to form SiRN nanowires at the bottom of the grooves as shown in (e). In the case of a \(\alpha=109.4^{\circ}\) opening, the material is 22 % thicker at the tip (see Fig. 2) and over-etching of the SiRN layer is allowed within this limit. We have used a thin SiRN layer of about 100 nm to reduce the etching time.
Thanks to the newly formed nanowires running in the three dimensions, the underlying polySi layer can be partially oxidized ((f)) using SiRN nanowires as an inversion mask. Half of the polySi thickness can be consumed during this step without issues. Once the nanowires are selectively etched, the polySi layer can be retracted starting from the tips where nicely defined access points are now formed, step (g). TMAH as an etchant is a nice option here since its selectivity between SiRN and SiO\({}_{2}\) is high and the etching speed not too fast to be controllable, but KOH could also be used. Careful timing is necessary during the retraction, since the length of the flexible parts of the smart hinges are determined during this step, see Typically,a TMAH solution at 25 wt% at 95\({}^{\circ}\)C attacks polySi at a speed of 1 um min\({}^{-1}\).
The above steps result in a patterned polySi layer on top of the thick SiRN layer, (h). Using the polySi layer as a mask, it is now possible to etch the underlying SiRN layer in HF 50 %. Due to the isotropic nature of the etchant, a retraction equal to the thickness of the material etched will occur under the polySi masking layer.
\[c=\frac{(t-R)}{\sin(\frac{\alpha}{2})} \tag{3}\]
In order to remove the polySi masking layer, an oxidation step should be preferred over a wet etching step since we want to conserve the rounding of the mold. This short oxidation step will consume the polySi material and slightly increase the radius of curvature of the mold. Moreover, the oxidation of the Si substrate will yield a specific shape, known as a bird's beak, at the transition between the Si and the SiRN. This shape is not visible in but will be shown in the results part of the paper.
After removal of the polySi layer, (i), the flexible part of the hinges is deposited by LPCVD. The bending stiffness of the hinges is \(B=\nicefrac{{Et^{3}}}{{12(1-\nu^{2})}}\) for thin plates, where \(E\) is Young's modulus and \(\nu\) is Poisson's ratio). It is highly dependent on the thickness \(t\) of the thin plate, and the thinner is this layer, the more flexible is the hinge. 100 nm thin hinges offer both mechanical solidity and flexibility. On the other hand, the final release of the foldable objects requires the etching of the Si molds in SF\({}_{6}\), which also slightly attacks SiRN. The thinning of the hinges during the release step should therefore be taken into account when depositing the thin SiRN layer. We measured a selectivity of around 1000 between silicon and SiRN on the part of our etching system (Adixen AMS100 Reactive Ion Etcher).
A second lithography step follows to define the overall geometry of the smart hinges. Making holes along the length of the hinges permits reducing their stiffness and facilitates the folding. Lithography in molds up to 10 um deep is relatively straightforward when using an appropriate photoresist. In this specific case, dry etching can be used to remove the SiRN, step (j). An extra lithography step must be performed to protect the SiRN structures by photoresist during the semi-isotropic etching of the silicon. Once released from the silicon substrate, the stop-programmable hinges are ready for assembly. When designing the masks, extra care was taken to assure that all structures were etched free from the substrate at the same time during the last step. Ideally, the mask openings should be of the same size and placed at the same distance from the stop-etching point.
The 70.6\({}^{\circ}\) stop-programmable hinges thus fabricated can be used to self-fold perfectly defined tetrahedrons. Since only one out of three hinges in a tetrahedron pattern lie at the right intersection between the planes in the silicon lattice, as shown in Fig. 5, only one smart hinge can be formed in the way that was just described. The other two hinges are therefore flat junctures made by standard micro-machining.
Fabrication of a stop-programmable hinge, example of a 70.6\({}^{\circ}\) smart hinge. (a): On a oriented silicon wafer, V-grooves are etched. (b) and (c): These grooves are rounded off by means of oxidation and subsequent etching. (d): A SiRN/polySi/SiRN stack of layers is conformally deposited. (e): The top SiRN layer is isotropically wet etched using a time stop such that material remains in the corner. SiRN wires run in all three directions. (f): This remaining SiRN nanowire is used as a protection mask during partial oxidation of the underlying polySi layer. (g): After removal of the SiRN line the polySi layer is retracted. (h): SiO\({}_{2}\) is stripped. (i): Using the polySi layer as a mask, an opening is etched in the SiRN layer. An oxidation and subsequent wet etching step follow to remove the polySi layer (not shown here). (j): After deposition of flexible layer of SiRN, a mask is applied through lithography and the overall geometry of the structure is determined by directive ion etching. (k): A last lithography step follows for protection of the SiRN objects during semi-isotropic etching of Si. Once released from the substrate, the smart hinge is ready to be self-folded.
### \(90^{\circ}\) stop-programmable hinges
The procedure to fabricate \(90^{\circ}\) stop-programmable hinges is nearly the same as for the \(70.6^{\circ}\) hinges described above, except for two important differences: making \(90^{\circ}\) molds in silicon is very difficult using wet etching--our attempts using correctly oriented wafers always resulted in tiny bumps at the bottom edges of the molds--and the vertical SiRN sidewalls obtained in these molds cannot be patterned using directive dry etching. Steps (a) and (i-j) in therefore differ when it comes to the micromachining of \(90^{\circ}\) stop-programmable hinges.
Given that the depth of the molds is small, cryogenic dry etching is a good option for our purpose. Unlike the BOSCH processes, cryogenic etching yields smooth sidewalls, which are crucial for our sensitive corner-lithography technique. Retraction of the mask is also a well known problem in dry etching and might be an issue for corner-lithography. In general, when developing a dry etching step for the purpose of performing corner lithography later on inside the molds, any concave corners other than the ones at the bottom of the molds should be avoided. This includes potential roughness of the masking material and retraction of the mask. Moreover, the final opening angle \(\alpha\) is highly dependent on the etching conditions (type of mask used, loading, gas flows) and requires precise tuning.
After performing the corner lithography, it is necessary to pattern the SiRN features before releasing the structures, (i-j). While reactive ion etching is a perfect option for \(70.6^{\circ}\) stop-programmable hinges, it is impossible to use it in the case of the vertical molds obtained by dry etching. The difficulties arising with upright sidewalls are twofold, as shown in One is the difficulty to etch several \(\upmu\)m of material from the top. And the other is the vertical thick SiRN plate of the complex hinge, see Fig. 11, that makes impossible a proper illumination of the photoresist for the subsequent lithography steps. Therefore an alternative method must be considered.
We suggest here to pattern the vertical sidewalls in two main steps, which are shown in Since it is impossible to illuminate the photoresist when it is masked
Fabrication of a tetrahedron folding pattern, extra steps. (a): Since only one of the hinges lies on the correct intersection of the planes, standard lithography is used to define flat hinges for the other faces. Faces 1⃝2 and 2⃝3 are designed with small appendices on their sides to allow them to lock onto face 1⃝1 while folding. (b): Under-etching of Si by semi-isotropic etching of silicon (SF\({}_{6}\) etchant). Etching is stopped when the hinges are free and the central flap rests on a silicon pillar.
The whole thick SiRN plate must be patterned first by wet etching before depositing the flexible part. In order to avoid an extra masking layer deposition and time-consuming wet etching step, the already patterned polySi layer by means of corner lithography, Fig. 4-(h), can be used for this purpose. A short oxidation step is performed to form a \(\simeq 5\) nm SiO\({}_{2}\) layer on top of the polySi. A lithography follows to define the overall geometry of the foldable objects. SiO\({}_{2}\) is then patterned in three dimensions in wet etchant BHF. After stripping the photoresist, TMAH is used to pattern the underlying polySi using the SiO\({}_{2}\) layer as a mask, Fig. 7-(b). The thick SiRN layer is then accessible and can be selectively etched, independently of the spatial direction.
After the conformal deposition of a thin SiRN layer (150 nm, Fig. 7-(d)) followed by a polySi layer (100 nm), the exact same procedure can be applied again: short oxidation of polySi, second lithography using the same mask, patterning of oxide and polySi etching, Fig. 7-(e). Wet etching of thin SiRN and final stripping of masking polySi layer follow to complete the origami patterns, Fig. 7-(f).
The use of wet etching arranges that the SiRN layers are etched under the polySi masking layers.
Difficulties arising when implementing corner lithography in an upright mold. We consider the case where the process flow depicted in has been followed up to step (i) inclusive. (a): Dry etching is impossible, a wet etching strategy must be considered using a masking layer. A polySi layer is deposited followed by short oxidation. (b): Photoresist is spun over the wafer. The use of a thick photoresist dedicated to high-aspect ratio structures protection is necessary to protect the molds. (c): Directional nature of UV illumination makes impossible the proper patterning of photoresist under the thick SiRN plate. (d): Consequently, the masking layer is not etched away everywhere. (e): SiRN is still present all around the molds at their bottoms.
## 3 Results
### Fabrication results--\(70.6^{\circ}\) stop-programmable hinges
By varying the oxidation time, two molds with different radii of curvature were made (Fig. 4-(b)) and subjected to the exact same fabrication steps until the isotropic etching of the bottom SiRN layer (Fig. 4-(i)). Since the thickness of the stack of the layers in the first case does not exceed \(R\), there is no concave corner at the bottom of the groove. Consequently, the timed-etching step of the top SiRN layer (Fig. 3-(f)) does not yield nanowires and the entire polySi layer is oxidized during the subsequent step (g). Without access points below the nanowires, no retraction can occur. Once the oxide is stripped, a non-patterned thick SiRN and polySi layer stack is obtained, as can be seen in Fig. 8-(a).
Alternative steps to pattern upright sidewalls. (a): The thick SiRN should be patterned first. The polySi masking layer obtained thanks to corner lithography can be used. (b): A short oxidation step follows to form a thin SiO\({}_{2}\) layer. Unlike in Fig. 6, lithography is possible to pattern the masking layer. (c): SiRN is wet etched using the oxidized polySi as a masking layer. PolySi is fully oxidized and stripped. (d): A flexible SiRN layer is deposited by LPCVD. (e): After LPCVD of a polySi layer followed by a short oxidation, the mask used in step (b) is applied a second time to pattern the masking layer. (f) SiRN is wet etched and the masking layer stripped.
In the correct case of Fig. (b), the radius of curvature \(R\) is smaller than the thickness \(t\) of the SiRN layer. The entire etching procedure can proceed and yields a clear opening in the polySi layer through which SiRN can be etched.
Corner lithography is a powerful three-dimensional patterning technique. The trick works for concave corners of any size and spatial configuration. As an illustration, shows the result when starting from V-grooves made by anisotropic etching in KOH. The planes are organized in a well known fashion, and the use of corner lithography leads to retraction in all three dimensions.
The use of a thick photoresist developed for high aspect ratio features is necessary for good protection of the deep V-grooves during the dry etching of the SiRN features, Fig. 4-(j). The result obtained with AZ(r) 9260 photoresist is good, as shown in photographs (a) and (b). Such planar protection is obtained by coating and spinning the resist at 300 rpm for 10 s then 60 s at 2400 rpm. The resist is exposed three times for 10 s at intervals of 10 s and is developed for 7 min. As can be seen in the inset of picture (b),
Illustration of the design criterion described in Figs. (a) and (b) show samples with different radii of curvature for the initial mold after performing corner lithography and partially etching the bottom SiRN layer, Fig. 4-(i). The thickness \(t\) of the first deposited SiRN layer is 1.1 μm. (a): When \(R>t\), the entire surface of the polySi layer was oxidized during step (f), consequently the retraction, step (g), had no effect. (b): For \(R<t\), the polySi layer is opened and the underlying SiRN layer can be etched, Fig. 4-(i).
(a): Overview of a V-groove after etching of the SiRN layer, Fig. 4-(h). (b): Zoom in the extremity of the groove after stripping away the polySi top layer, Fig. 4-(i). Retraction occurs in all concave corners, including the vertical planes. The retraction length is the same in every direction.
SiRN remains on the upright sidewalls after the dry etching step: only the top part was attacked by reactive ion etching. These small SiRN spots are, however, not a problem for our folding structures.
As is visible in pictures (a), (b) and (d), pinholes were present all over the wafer. They originate from nanoscopic defects in the bottom SiRN layer that turn into microscopic features because of the corner lithography: another proof of the extreme sensitivity of the technique. As long as the pinholes appear on the flaps and leave the hinges intact, they do not represent an issue for our folding purposes. However, the quality of the first SiRN layer should be checked at the beginning of the fabrication.
The stop-programmable hinge shown in Fig. 10-(c) is nearly identical to the schematic presented in the process flow, Fig. 4-(k), except for the small bumps visible at the transitions between the thin and the thick parts of the smart hinges.
A nearly released tetrahedral pattern is presented in Fig. 10-(d). A circular protection of photoresist was present on top of it before the release step in SF\({}_{6}\), Fig. 4-(k), hence the circular shape of the Si pillar.
SEM images of final fabrications steps. (a): Tetrahedral pattern of thick photoresist. (b): SiRN template after over-etching in the grooves and stripping of photoresist. The combination of a complex hinge with two flat hinges can be observed, see Fig. 5-(a). (c): Close-up image of a complex hinge after the silicon mold was etched away in dry etching and the photoresist stripped, Fig. 4-(k). (d): Overview of an unfolded tetrahedral structure at the end of the process. Isotropic etching of Si is stopped once the pattern comes to rest on a central pillar, Fig. 5-(b). Note that this structure is not completely released, since Si is still present under the flaps.
### Fabrication results--\(90^{\circ}\) stop-programmable hinges
As was emphasized in Part 2.2, the fabrication for both stop-programmable hinges is identical except for the creation of the molds and for the final etching of the SiRN layers.
Our best results for getting molds using dry etching are presented in We used a mixture of SF\({}_{6}\) (\(200\,\mathrm{sscm}\)) and O\({}_{2}\) (\(15\,\mathrm{sscm}\)) gases at \(-110\,^{\circ}\)C (Adixen AMS100 Reactive Ion Etcher, pressure \(1.6\times 10^{-2}\,\mathrm{mbar}\), RF \(200\,\mathrm{W}\), LF \(20\,\mathrm{W}\) on/off time \(25/75\,\mathrm{ms}\)). The dry etching step yields a rounded mold, picture (a), which reduces the oxidation time necessary to get the final desired radius of curvature. The depth of the molds was checked on six different spots spread over one dummy wafer and was found to be the same as that depicted in within an error of \(5\,\%\). Photograph (b) shows a similar mold on top of which the three layers necessary for corner lithography were deposited, Fig. 4-(d). Since the design criterion (Equation) is respected, the rounding has disappeared after the conformal deposition of the first SiRN layer.
In Fig. 11-(b), the defects of the initial mask can be observed on the vertical sidewalls. In order to avoid undesirable corner lithography starting points at that location, the over-etching of the top SiRN layer (Fig. 4-(e)) was deliberately long. The corners formed due to irregularities of the mask have large opening angles, so the surplus of material is thinner than in the \(90^{\circ}\) corners at the bottom of the grooves. Since in a perpendicular corner the material is in theory \(41\,\%\) thicker (see Fig.2), a \(25\,\%\) over-etch was performed.
The successful retraction after the corner lithography is shown in photograph (a). Unlike the results presented in Fig.9, no retraction occurred on the top or in the corners of the mold. This is a consequence of the long over-etching explained in the previous paragraph. This long over-etch of the top SiRN layer when performing corner lithography removed all unwanted material in the irregularities of the molds, as well as in their corners.
As explained in Part 2.2, dry etching cannot be used in the case of upright molds and a double wet etching strategy must be used. Fig. 12-(b) shows the patterned masking layer, corresponding to step (b) in The oxidized polySi layer following the retraction shown in (a) was further shaped by oxidation and subsequent lithography steps. This way, the material at the bottom of the molds is etched and the geometry of folding patterns is defined simultaneously. Picture (c) shows the resulting structures after SiRN etching, with only the flexible part of the hinges missing, see Fig. 7-(c). The same wet etching procedure is applied a second time after depositing a thin SiRN and polySi layer, resulting in the structures shown in photograph (d), Fig. 7-(f). Misalignment of the mask in the second lithography step on top of the first patterned SiRN explains the staircase-like shape of the final SiRN object.
12-(d).
(a): Cross section of a mold obtained by dry etching of silicon. The corner is not exactly perpendicular (measured to be 89\({}^{\circ}\)) and the process yields a round corner. These parameters can be modified by fine tuning the dry etching step. Note that photoresist is still present on top. (b): Stack of the three layers necessary for corner lithography: thick SiRN (1070 nm), polySi (around 150 nm) and a second layer of SiRN (146 nm).
SEM images of fabrication steps for 90\({}^{\circ}\) complex hinges. (a): Retraction of polySi is visible under SiO\({}_{2}\), Fig. 4-(g). The polySi plate is not exactly straight. These irregularities are exact replicas of the defects of the silicon mold, which have been magnified through corner lithography. (b): Thick SiRN with patterned masking partially oxidized polySi layer on top. (c): Same as (b) right after wet etching of SiRN. Stress in SiO\({}_{2}\) mask causes the curtain-like overhanging thin film. (d): Final structure before release after the second wet etching procedure was applied.
### Folding experiments
Self-folding experiments were carried out by manually depositing water droplets of 5 to 15 nL. An accurate positioning system allowed us to deposit the liquid right at the centre of the templates using a hollow fibre (50 um diameter) connected to a high precision Hamilton glass syringe filled with ultra pure water that was manually actuated. The folding of the structures typically takes around one minute, depending on the size of the structures and the volume of liquid deposited.
Fig. (a) shows a wing folded at the designed angle of 70.6\({}^{\circ}\). Fig. (b) shows a tetrahedron resting on a silicon pillar. The large contact areas between the thick SiRN parts provide a good stability for the 3D objects after drying. The tetrahedral structure was folded from a similar pattern presented in Fig. 5-(b) and Fig. 10-(d). One stop-programmable hinge makes sure that the folding stops at 70.6\({}^{\circ}\) while large appendices designed on the side of the other faces allow them to lock onto the first flap.
A too long retraction of the polySi during the corner lithography, coupled with the wet etching steps necessary to pattern the final objects, led to wrongly shaped complex hinges. The thick parts of the hinges are too small, and the folding did not stop at all (a), or stopped too late (b). However, these results are encouraging and demonstrate that 90\({}^{\circ}\) stop-programmable hinges are feasible. Reducing the TMAH retraction to 2 um, which would lead to 4 um long flexible plates, would most probably be sufficient for a successful assembly.
## 4 Discussion
The technique presented in this paper was first successfully used to fabricate 70.6\({}^{\circ}\) stop-programmable hinges. The same principle was then applied to upright molds, but issues were encountered during the process. Difficulties with the molds (retraction of the mask and roughness) caused our first attempt to fail.
(a): Final structure before folding. Residual stress in the thin SiRN layers provoked a slight bending upward of the flap. (b): Zoom in at a stop-programmable hinge.
We initially used a 1.05 times over-etching in the case of \(70.6^{\circ}\) stop-programmable hinges and increased it to 1.25 for \(90^{\circ}\) complex hinges. Later on during fabrication, difficulties in patterning upright sidewalls forced us to modify the process. Dry etching of SiRN with rotated samples, as well as a single wet etching step, were first unsuccessfully carried out before coming up with the appropriate two-step wet etching patterning strategy described here.
The results presented here are encouraging. As long as the necessary molds can be obtained in Si, the procedure that has been described in this paper is applicable to virtually any folding angle. In principle, the wet etching method for patterning SiRN onto the sidewalls should be applied for any mold opening angle \(\alpha\leq 90^{\circ}\), while dry etching should be preferred when \(\alpha>90^{\circ}\).
For clean water and air at standard conditions, the transition is around \(2\,\mathrm{mm}\). Elastocapillary folding of several mm long silicon-based objects is therefore theoretically possible. It is known to be hard to fabricate features of this size out of the wafer plane by conventional two-dimensional micro-fabrication techniques. Indeed, \(2\,\mathrm{mm}\) is more than four times the standard thickness of a standard silicon wafer. \(90^{\circ}\) stop-programmable hinges are especially interesting since they would permit popping up several mm long features out of the plane of the silicon wafers exactly where the hinges are designed. This technique, combining the strengths of well known standard fabrication techniques with the ease of self-folding, could have many applications, such as 3D sensing, Micro-Opto-Electro-Mechanical Systems (MOEMS), or 3D memory.
Furthermore, we have shown that the use of corner lithography leads to retraction in all three dimensions. For the purpose of self-folding, only the features at the bottom of the molds were used, the others being etched away in Fig. 4-(j). Other features might be of interest for different purposes. For example, unique 3D membranes or channels could be obtained. Several applications have already been developed using corner lithography, such as the wafer scale fabrication of nano-apertures, photonic crystals or micro-cages in which the culture of bovine cells has been demonstrated.
## 5 Conclusion
Starting from a simple micrometer-sized mask, accurate stop-programmable hinges for complex elastocapillary folding were micromachined in three dimensions using corner lithography. In order to limit the stress in the hinges while folding, it was necessary to start with round molds. Corner lithography can be performed only if the combined thicknesses of the two first deposited layers is greater than the radius of curvature of the mold. When this design criterion is respected, material can be accurately etched starting from any concave corner, independently of their spatial orientation.
The definition of the silicon molds at the start of the process determines both the locking angle and the accuracy of the stop-programmable hinges. Selective KOH etching on wafers yields well defined sharp corners that, in turn, can be used to micro-machine smart hinges that stop folding at \((70.6\pm 0.1)^{\circ}\). Micrometer sized three dimensional tetrahedral structures were successfully self-folded using capillary forces, thanks to these hinges.
We demonstrated the feasibility of \(90^{\circ}\) stop-programmable hinges. Such complex hinges can be fabricated by making initial molds with dry etching at the cost of a poorer accuracy \((89\pm 4)^{\circ}\) compared to wet etching. Extra care must be taken when using dry etching to avoid any undesirable sharp corners, and a long over-etching during corner lithography must be performed to remove unwanted material. Moreover, small irregularities in the molds will be magnified by the process and will cause defects in the final shape of the flaps.
Stop-programmable hinges extend the possibilities for implementing the elastocapillary folding of microstructures. Using a simple filling procedure, millimeter-long silicon-based structures can be accurately popped out of the plane. We believe that the accuracy and versatility of the technique will find widespread application in 3D sensing, MOEMS, or 3D electronics, for instance.
| 10.48550/arXiv.1410.1765 | Elastocapillary folding using stop-programmable hinges fabricated by 3D micro-machining | A. Legrain, J. W. Berenschot, N. R. Tas, L. Abelmann | 4,180 |
10.48550_arXiv.1405.5570 | ###### Abstract
Modeling semicoherent metal-metal interfaces has so far been performed using atomistic simulations based on semiempirical interatomic potentials. We demonstrate through more precise ab-initio calculations that key conclusions drawn from previous studies do not conform with the new results which show that single point defects do not delocalize near the interfacial plane, but remain compact. We give a simple qualitative explanation for the difference in predicted results that can be traced back to limited transferability of empirical potentials.
Nanostructured metallic multilayer composites (NMMC) are known to have superior mechanical properties compared to standard coarse grained metals along with the ability to efficiently self-heal radiation damage. The latter is crucial for the material to inhibit creep and swelling in harsh environments. In order to utilize these materials in most effective manner, it is necessary to understand the underlying mechanisms leading to the aforesaid advantages. One route to accomplish that is via experiments where state of the art work has reached to a point where it is possible to engineer bulk nanostructured bi-metal multilayers while having control over structural features at atomic level which can considerably alter the mechanical properties and thermal stability of these materials.
Another way to gain insight into the way these materials behave during subjection to extreme mechanical or radiation environments is by theoretical means via computational methods. Since the time and length scales these processes cover are exceptionally widely spread, starting from attoseconds and picometers for electronic effects and going up to meters and years in continuum mechanics, there is no single equation or model that can currently cover all of this complexity. Therefore only a multi-scale modeling approach allows to eventually predict the properties and design optimal NMMCs for future industrial and energy technology applications.
This study concentrates on the atomic level part of the multi-scale method, where there has been considerable effort to model the structure and behavior of NMMCs in order to better understand the traits leading to the high tolerance to radiation damage. While the effects caused by irradiation are essentially macroscopic, they are still governed by changes in atomic level that can be traced back to single point defects. Therefore efforts have been directed to identifying possible lowest energy structures for the undamaged interfaces as well as describing the point defect properties such as configurations, formation energies, migration barriers and mechanisms near the interfacial plane. Previous studies have shown that the interface does not support conventional point defects, that is vacancies and interstitial atoms, but instead pairs of extended jogs will form. Those delocalized defects have been shown to exhibit low formation energies and interaction through long-range forces. This will also result in more complex migration pathways and recombination mechanisms than in bulk material.
The studies described above were performed using atomistic modeling based on classical molecular dynamics and empirical interatomic potentials which have a crucial role in determining the outcome of the simulations. Fitting empirical models to ab-initio or experimental data always presents the challenge of obtaining good transferability to the problem under study which often explores regions of phase space not used in the fitting process. For metallic systems the most widely utilized model is the embedded-atom method (EAM) developed by Daw and Baskes. Although there are alternatives, arguably having greater accuracy, it is still relevant because of its simplicity and computational scalability while providing relatively accurate description, especially for FCC metals.
There are two EAM potentials available for copper-niobium system which were fitted using two different methods. First by Demkowicz et al (hereafter EAM1) uses modified Morse function for Cu-Nb interaction and has been fitted to dilute enthalpies of mixing and bulk modulus and lattice constant of hypothetical CuNb alloy in B2 structure. The second one by Zhang et al (EAM2) uses more flexible polynomial-like function and is fitted to enthalpies of mixing of special quasi-random structures over the whole composition range with the aim of correctly reproducing experimental thermodynamics for the system.
In this work we show, that EAM1 and EAM2 give markedly different results for both the structure and energetics of point defects near the interface. Then we propose a solution to this discrepancy by relaxing the structures predicted by aforementioned two potentials using density-functional theory (DFT) calculations which essentially do not rely on empirical parameters thereby producing more accurate results. We then propose an explanation why some interatomic potentials might lead to erroneous characterization of the interface and how to possibly prevent this in future works.
All DFT calculations were done using plane-wave pseudopotential code Vienna Ab initio Simulation Package (VASP) with supplied PAW pseudopotentials and GGA-PBE approximation. For niobium the semi core \(p\) states were treated as valence. The cutoff energy for the plane waves was 273.214 eV. Single k-point (\(\Gamma\)-point) was used and smearing was handled by 1st order Methfessel-Paxton scheme with width of \(\sigma=0.01\) eV. Atomistic simulations were performed with classical molecular dynamics code LAMMPS.
The structure of the interface can be described by specifying the orientation of the surface normal to the interfacial plane and two parallel directions, one for each surface, that will be parallel when the interface is formed. It has been shown experimentally that copper-niobium interface forms predominantly in Kurdjumov-Sachs orientation. In general calculating the energies of such structures using DFT is a complex task solely because of the number of atoms needed, and hence the required computational effort, to retain characteristic features and periodicities of the interface. The periodicity of the interface is defined by the locations of misfit dislocation intersections (MDIs), that is the areas where the atoms on each side of the interface overlap. In case of copper and niobium in KS orientation the distances between the MDIs are relatively small enabling this specific interface to be modeled using a quasi-unit cell appropriately sized for DFT. This cell is an approximation and not a true unit cell for the larger system since albeit similar, the local environments around the MDIs are not equivalent.
In order to keep the calculations computationally feasible the number of atoms in the cell perpendicular to the interface must be limited. This results in two choices, either make the simulation box periodic or add vacuum in this direction. Former corresponds to having infinite number of thin alternating copper and niobium layers and latter to single interface and two free surfaces. We opted for having 1.65 nm layer of vacuum between the free surfaces. Unit cell vectors (in nm) for the resulting unit cell are \(a_{x}=(2.3,0,0)\), \(a_{y}=(0.75,1.33,0)\) and \(a_{z}=(0,0,3.0)\) and it consists of 216 Cu and 120 Nb atoms. We checked for possible errors by doubling the number of layers and calculating the structure, which did not change, and formation energy of vacancy, which reduced by 20 meV. While constructing a small unit cell using the method described above gives an appropriate representation of the undamaged interface, there is still the problem of finding ground state configurations of point defects. Therefore we calculated candidate structures using both EAM1 and EAM2 with molecular dynamics and then relaxed these using DFT. The ratio of lattice constants of Cu and Nb calculated using DFT and MD differ by less than 0.1% thereby making this method valid.
In order to a) check whether the vacuum layer or distortion of the true periodicity has any effect on the structure of the interface, b) to get input structures for DFT calculations and c) to assess whether relaxing the box has important effect on the outcome of the results, we first performed different molecular dynamics simulations with both EAM1 and EAM2. First the initial system was quenched from 600 K to 0 K followed by the energy minimization using conjugate gradient method. Next the copper atom with highest potential energy was removed and the process was repeated. A typical final structure as predicted by EAM1 is shown in and has the same 4- and 5-atom rings as found in previous works while using EAM2 a compact single vacancy is formed. Relaxing the simulation box using EAM2 has minuscule effect on the formation energies while using EAM1 results in somewhat smaller energies. In either case the structural features are not affected.
Same process was carried out with single interstitial copper atom which was inserted into the interface after initial energy minimization next to the MDI. Again using EAM1 results in delocalization of the defect (on Fig. 3a) while EAM2 produces clear interstitial which resides between the copper and niobium layers.
Next all four structures were relaxed using DFT.
Constructing small quasi-unit cell for Cu/Nb interface in Kurdjumov-Sachs orientation. Blue spheres represent niobium and yellow spheres copper atoms. (a) The unit cell is “cut out” from larger structure and has approximately same periodicity as the misfit dislocation intersections, that is the areas where copper and niobium atom positions coincide in x- and y-directions. (b) A 3D view of the simulation cell, where each layer consists of either 54 Cu or 40 Nb atoms. Total number of atoms is 336.
The box size was kept constant for consistency and lower computational cost based on the fact that no change in structure and only negligible effect on formation energies was observed in constant pressure molecular dynamics runs using either EAM potential. Resulting structures are depicted on and for the vacancy and interstitial respectively. The final structures are very similar to the ones obtained with EAM2 potential, that is no delocalization happens and compact vacancy or interstitial is formed.
The formation energies for copper vacancies and interstitials at the interface calculated with the two potentials and DFT are listed in Table 1. It must be noted though, that the values cannot be directly compared. The reason for that is the difference in defect energies of pure copper which will carry over to the formation energies of defects near the interface. Similarly, these values cannot be compared to the ones calculated using larger unit cell, firstly because of possible defect-defect interaction in neighboring cells due to periodicity and secondly due to the probable errors introduced by approximating the large cell using a smaller one. Moreover, while the defect delocalization predicted by EAM1 can result in diverse final structures with different energies as reported in Ref., a single point defect has a well-defined formation energy.
Our results demonstrate that different empirical potentials can lead to contrasting results when the structures studied are substantially different from those used for the fitting procedure. The result of the fitting of alloy properties in case of EAM is a single function relating distance between two different species to the energy. Since the data used to fit EAM1 depends only on a small discrete set of distances, the energy function is also well defined only at these points. At the same time the energy of a semicoherent interface contains distances of nearly continuous spectrum which means that when using this method the energy of the interface and thus the formation energies of defects can take nearly arbitrary values limited only by the chosen functional form.
The same method of fitting as described above has been also used to "tune" the potentials to yield different enthalpies of mixing and to show how this would affect the behavior of vacancies and interstitials. While this method is a nice example of the ability of simulations to investigate scenarios which are impossible to achieve in experiments, having too small fitting database (too few distances in case of EAM) might again interfere with other physical quantities which can lead to incorrect conclusions.
A method has been proposed by Ercolessi and Adams in which instead of energies, forces are fitted to reproduce those obtained from first-principles calculations.
Top view of interfacial copper layer with single vacancy at the interface before and after relaxing using DFT. (a) The structure was obtained by relaxing the interface with EAM1 which results in delocalization of the defect and formation of 4- and 5-atom rings. (b) Relaxing in DFT yields single compact vacancy. Lines and colors represent the in-plane bonds and coordination numbers respectively with orange being 6, green 5, cyan 4. The position of vacancy is at the MDI, but is shifted due to layers moving with respect to each other in simulation.
Top view of interfacial copper layer with added interstitial atom before and after relaxing using DFT. (a) The structure was obtained by relaxing the interface with EAM1 potential which results in delocalization of the defect and formation of 4-atom rings. (b) After relaxing in DFT a single interstitial between the interfacial copper and niobium layers will emerge. Lines and colors represent in-plane bonds and coordination numbers respectively, with red being 7, orange 6, green 5 and cyan 4 and blue 3.
However, increasing the fitting database, as it is in the case of FAM2, can also result in significant improvement. shows the forces on atoms for the relaxed DFT structure with one vacancy calculated using both EAM1 and EAM2. The forces predicted by latter differ predominantly at free surfaces and for niobium atoms. It has been shown that it is quite hard or even impossible to accurately reproduce DFT forces for niobium using EAM so this behavior is expected. With EAM1 the difference between DFT and MD forces for the first copper layer (where the vacancy is located) is much larger which leads to reconstruction of the layer and delocalization of the vacancy.
To summarize, we have shown that single compact point defects can exist at semicoherent metal-metal interfaces without any delocalization contrary to results of previous studies. This could necessitate further investigation of defect migration, clustering and recombination. In addition, we have provided an explanation for the discrepancy between earlier atomistic studies and this work which can be attributed to different fitting strategies and intrinsic transferability limitations of empirical potentials.
The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007-2013) under grant agreement n\({}^{\circ}\) 263273. Computational resources were provided by Swedish National Infrastructure for Computing. Work of A. Caro performed at the Center for Materials at Irradiation and Mechanical Extremes, an Energy Frontier Research Center funded by the U.S. Department of Energy (Award No. 2008LANL1026) at Los Alamos National Laboratory.
| 10.48550/arXiv.1405.5570 | First-principles study of point defects at semicoherent interface | Erki Metsanurk, Alfredo Caro, Artur Tamm, Alvo Aabloo, Mattias Klintenberg | 2,004 |
10.48550_arXiv.1908.06876 | # Interconversion of multiferroic domains and domain walls
Ehsan Hassanpour
These two authors contributed equally Department of Materials, ETH Zurich, Vladimir-Prelog-Weg 4, 8093 Zurich, Switzerland
Mads C.
Correspondence email address: Department of Materials, ETH Zurich, Vladimir-Prelog-Weg 4, 8093 Zurich, Switzerland
Amade Boris
Department of Materials, ETH Zurich, Vladimir-Prelog-Weg 4, 8093 Zurich, Switzerland
Yusuke Tokunaga
University of Tokyo, Department of Advanced Materials Science, Kashiwa 277-8561, Japan
Yasujiro Taguchi
RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan
Yoshinori Tokura
RIKEN Center for Emergent Matter Science (CEMS), Wako 351-0198, Japan Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan
Andres Cano
Institut Neel, CNRS & Universite Grenoble Alpes, 38042 Grenoble, France
Thomas Lottermoser
Department of Materials, ETH Zurich, Vladimir-Prelog-Weg 4, 8093 Zurich, Switzerland
Manfred Fiebig
Present address: ]Department of Materials Science, Kashiwa 277-8561, Japan
November 6, 2021
###### Abstract
Materials with long-range order like ferromagnetism or ferroelectricity exhibit uniform, yet differently oriented three-dimensional regions called domains that are separated by two-dimensional topological defects termed domain walls. A change of the ordered state across a domain wall can lead to local non-bulk properties such as enhanced conductance or the promotion of unusual phases. Although highly desirable, controlled transfer of these exciting properties between the bulk and the walls is usually not possible. Here we demonstrate this crossover from three- to two-dimensions for confining multiferroic Dy\({}_{0.7}\)Tb\({}_{0.3}\)FeO\({}_{3}\) domains into multiferroic domain walls at a specified location within a non-multiferroic environment. This process is fully reversible; an applied magnetic or electric field controls the transformation. Aside from the aspect of magnetoelectric functionality, such interconversion can be key to tailoring elusive domain architectures such as in antiferromagnets.
antiferromagnet, weak ferromagnet, domain wall, interconversion pacs: +
Footnote †: These two authors contributed equally
Recently, the interests in ferroic materials with magnetic or electric order evolved from the three-dimensional (3D) bulk domains to the two-dimensional (2D) domain walls. As inherent inhomogeneity, domain walls are a source of specific phenomena that are forbidden in the uniform interior of the corresponding domains. Examples are the occurrence of magnetization, polarization, magnetoelectric coupling, (super-)conductivity, memory effects or a change of melting temperature. Hence, domain walls may be regarded as novel state of a material, virtually a world in its own self and seemingly separated from the surrounding bulk phase.
It would now be very desirable if the dimensional limitation of these exciting phenomena could be overcome. For example, one could consider the confinement of a multifunctional bulk state into domain walls where they could establish a form of rewritable electromagnetic circuits. Reversely, the domain walls may seed the recovery of the original bulk state, acting as its memory.
A class of materials in which the transition from the 3D bulk to the 2D domain walls could be virtually continuous are compounds with phase transitions in which the symmetry of the domains and domain walls coincide cross-wise on either side of the phase boundary. In such materials, domain walls can in principle gradually transform into domains, and vice versa, across a first-order phase transition. This concept was discussed theoretically and systems showing a behaviour consistent with this concept were reported. Conclusive observation of a smooth, deterministic transfer of a domain wall into a domain, however, and, in particular, the reversible interconversion between domains and domain walls have not been presented. Notably, the practical consequences of such interconversion for the functionalities of materials were never debated.
We demonstrate this interconvertibility and apply it to tune a multiferroic state between three and two dimensions in a fully controlled way. Specifically, we confine a multiferroic bulk state into a multiferroic domain wall at a specified location within a non-multiferroic environment. We act on this 2D magnetoelectric object with magnetic or electric fields and evidence the presence of a switchable magnetization and polarization in the wall. We furthermore employ the fields to transfer the multiferroic domain wall back into a multiferroic bulk state. We then discuss the general occurrence and benefits of an ordered state with controllable transfer in between 3D and 2D.
For the reasons given below, we choose the rare-earth orthoferrite Dy\({}_{0.7}\)Tb\({}_{0.3}\)FeO\({}_{3}\) (see Ref. about sample preparation) as our model system. Multiferroicity occurs at \(T_{C}\simeq 2.65\) K, caused by simultaneous antiferromagnetic ordering of the rare-earth and iron spin systems, coupled by exchange striction. An electric polarization \(P_{s}=0.12\)\(\mu\)C/cm\({}^{2}\) coexists with a Dzyaloshinskii-Moriya-type weak ferromagnetic moment \(M_{s}=0.15\)\(\mu_{B}\) per formula unit, both oriented along the \(c\)-axis. Subsequently, at 2.3 K a first-order spin reorientation of the iron sublattice from the multiferroic to antiferromagnetic phase with \(M_{s}\), \(P_{s}=0\) occurs.
\[H=A|\nabla\theta|^{2}+K_{1}\sin^{2}\theta+K_{2}\sin^{4}\theta. \tag{1}\]
As depicted in Fig. 1, the angle \(\theta\) distinguishes the domain states on either side of the \(M1\leftrightarrow M2\) phase transition. \(K_{1}\) and \(K_{2}\) are magnetocrystalline anisotropy parameters. \(K_{1}=0\) and \(K_{1}=-2K_{2}\) set the limits of the coexistence region. The gradient term with \(A\) as exchange constant further determines the order-parameter rotation across the domain walls.
In Dy\({}_{0.7}\)Tb\({}_{0.3}\)FeO\({}_{3}\), \(M1\) and \(M2\) are associated with the multiferroic (weak ferromagnetic) and non-multiferroic (purely antiferromagnetic) phases, respectively, with \(\theta_{\pm M1}=+\pi/2,-\pi/2\) and \(\theta_{\pm M2}=0,\pi\). depicts the evolution according to Eq. of a configuration starting with two well-defined domains outside the coexistence region separated by a single domainwall. By entering the \(M1/M2\) coexistence region, the domain wall widens and transforms into a domain of the other phase. Conversely, the initial domain shrinks and eventually transforms into a domain wall separating the new domains on the opposite side of the coexistence region. Hence, the first-order phase transition between the multiferroic \(M1\) and the antiferromagnetic \(M2\) phase in Dy\({}_{0.7}\)Tb\({}_{0.3}\)FeO\({}_{3}\) should in principle allow us to reversibly convert a multiferroic domain into a multiferroic domain wall in an antiferromagnetic, non-multiferroic environment.
We first verify the theoretically predicted transformation of a domain wall into a domain experimentally. In Dy\({}_{0.7}\)Tb\({}_{0.3}\)FeO\({}_{3}\), we can exploit the magnetization of the multiferroic phase to distinguish the \(\pm M1\) and \(M2\) domain states by spatially resolved real-time imaging experiments, using the magneto-optical Faraday effect as magnetization probe. Furthermore, Dy\({}_{0.7}\)Tb\({}_{0.3}\)FeO\({}_{3}\) exhibits a relatively broad region of phase coexistence. This gives us ample time to image the phase coexistence while tuning the balance between the competing multiferroic and antiferromagnetic phases.
Figure 1c-e shows sequential Faraday images of a \(c\)-oriented Dy\({}_{0.7}\)Tb\({}_{0.3}\)FeO\({}_{3}\) sample cooled across the multiferroic-to-antiferromagnetic transition. The corresponding order-parameter distribution is sketched beneath each image with arrows representing the local value of \(\theta\) in Eq.. We refrain from quantifying \(K_{1}\) and \(K_{2}\) since their absolute values are irrelevant for the further course of this manuscript. In we see a \(\pm M1\) domain pair at \(T>T_{C}\). Because of the opposite direction of magnetization and, hence, Faraday rotation, the domains appear as bright and dark regions. Figures 1d and 1e show the same region about 4 and 8 minutes after cooling the sample to \(T\lesssim T_{C}\). A grey stripe centers at the domain-wall position of that widens with time. Its zero Faraday rotation identifies it as antiferromagnetic region. Below \(T_{C}\), this is the dominating phase and, starting from the original \(\pm M1\) domain wall, we perceive the homogeneous expansion of this wall into a \(M2\) domain engulfing the sample as time progresses. Note that this expansion of the antiferromagnetic phase occurs in a deterministic way, starting uniformly from the center of the domain wall of the multiferroic phase. This is in stark contrast to nucleation on random structural defects as in most first-order phase transitions. It is also in contrast to random, needle-like emergence of domains of the target phase at or within domain walls of the seed phase, or to a coexistence of phases in stripe domains of variable width.
Now we employ this deterministic aspect for confining a multiferroic domain into a multiferroic domain wall at a specified location within a non-multiferroic environment. We begin at \(T>T_{C}\) with a \(-M1\) domain sandwiched between \(+M1\) domains. In a magnetic field we enlarge the \(+M1\) domains until they meet so that the \(-M1\) domain is no longer detectable. After cooling the sample to \(T\lesssim T_{C}\), the antiferromagnetic state emerges and grows with time until it has filled the entire field of view. Note that the \(M2\) state originates from the meeting position of the \(+M1\) domains in Fig. 2c, evidencing the presence of a remaining topological object at this location that seeds the \(M2\) order.
This topological object is scrutinized in As we see, there are two types of meetings between \(+M1\) domains on a \(-M1\) background. They reflect the two types of domain walls a multiferroic domain is expected to exhibit, namely \(+M1\rightarrow-M1\) walls with either clockwise or counterclockwise rotation of spins across the wall. When domain walls of the opposite type meet, the respective spin rotations cancel and the walls annihilate so that the meeting domains coalesce. Alternatively, for a meeting of the same type of domain walls, a \(360^{\circ}\) spin rotation as sketched in occurs. This object, equivalent to a topologically protected antiferromagnetic 1D-skyrmion, prevents the coalescence of the domains as indicated by the brightness dip in Instead, it can seed a \(+M2\) and a \(-M2\) domain as sketched beneath Fig. 2d,e. Note that a magnetic field three times the value of the coercive field is required to destroy the \(360^{\circ}\) wall.
Hence, shows a \(+M2\) and a \(-M2\) domain; according to Fig. 1a, they are separated by a multiferroic domain wall into which the multiferroic bulk state of has been confined. This wall has been placed at the meeting point of the two former \(+M1\) domains in Figs. 2c. For confirmation, we re-heat the sample to \(T\gtrsim T_{C}\); shows that this reconverts the suspected multiferroic domain wall into a multiferroic \(-M1\) domain.
To describe the wall as multiferroic, however, it is not sufficient to show that we can convert it into a multiferroic domain, but rather to demonstrate the coexistence of a polarization and a magnetization for the wall itself. This is done in which shows the response of domain walls generated as in to static \(c\)-oriented magnetic or electric fields. We see that either field initiates the transfer of the wall back into a domain of the multiferroic phase, even though temperature-wise the material still favours the antiferromagnetic phase. This is only possible, if the wall already has a magnetization \(M_{s}\) and a polarization \(P_{s}\) the respective applied field can act on to initiate the transfer. Furthermore, the transfer is energetically beneficial only if \(M_{s}\) or \(P_{s}\) points in the direction of the applied field. Since the field has triggered the transfer in all our experiments the wall magnetization and polarization itself must be switchable and hence set the direction of \(M_{s}\) or \(P_{s}\) of the expanding domain. Determination of the sign of \(M_{s}\) by the magnetic field is directly seen in Figs. 4a,b, whereas shows that the electric field acts on the magnetization via magnetoelectric coupling. (Note that because of the involvement of three order parameters in this coupling, the sign of \(M_{s}\) is not determined by the sign of \(P_{s}(E)\) but rather by the history of the sample, as shows.) We thus see that despite the confinement and the non-multiferroic environment, a wall as in retains the properties of the multiferroic bulk phase.
To conclude, we have experimentally demonstrated the interconversion of a multiferroic state in between three and two dimensions. We have confined the multiferroic bulk phase into a multiferroic domain wall at a predetermined location inside a non-multiferroic environment. The initial multiferroic hallmark properties, namely a magnetization and a polarization that are coupled and switched by an applied field, are retained by the wall. Magnetic or electric fields act on the wall and convert it back into a bulk multiferroic domain with a predetermined direction of magnetization or polarization.
The concept of interconversion can be expanded beyond multiferroics. First, there are plenty of first-order magnetic phase transitions fulfilling the rather basic conditions of Eq. with CaFe\({}_{2}\)O\({}_{4}\) as a very recent example to catch scientific attention. Note that even if the region of phase coexistence may be narrower than in our case, possibly even too narrow for convenient experimental probing during the transition, a system will still be subject to the same dynamics as found in Dy\({}_{0.7}\)Tb\({}_{0.3}\)FeO\({}_{3}\). Even non-magnetic types of order can show this phenomenon. For example, a (static) expansion of domain walls into domains with phase coexistence based on a chemical gradient rather than the first-order nature of a phase transition was observed in the ferroelectric-to-nonferroelectric transition in InMnO\({}_{3}\). Even organic materials may show a transition between phases with neutral and ionic building blocks emerging out of an expanding singularity by progressive charge transfer.
Second, our work expands the possibilities for functionalizing domain walls. Unlike in previous examples, the magnetoelectric properties are not confined to the domain walls but can be expanded into the bulk when needed. The magnetoelectric properties of the walls may be employed to visualize or even tailor domain patterns in materials whose order is normally difficult to access, such as 180\({}^{\circ}\) antiferromagnets. Finally, our work provides a rare opportunity of deterministic nucleation in a first-order phase transition since the nucleating phase is seeded by the order and symmetry in domain walls rather than occurring on random defects.
We thank Dr. Morgan Trassin for deposition of Pt electrodes on the samples. This work was financially supported by SNSF (Grant No. 200021_178825/1) and European Research Council (Advanced Grant 694955-INSEETO). Y. Tokunaga was supported by JSPS Grant-in-Aids for Young Scientists (A) (No. 25707032). Y. Tokunaga, Y. Taguchi and Y. Tokura were supported by the Japan Society for the Promotion of Science (JSAP) through its "Funding Program for World-Leading Innovative R&D on Science and Technology" (FIRST Program). M. F. thanks ETH Zurich and CEMS at RIKEN for support of his research sabbatical.
| 10.48550/arXiv.1908.06876 | Interconversion of multiferroic domains and domain walls | E. Hassanpour, M. C. Weber, A. Bortis, Y. Tokunaga, Y. Taguchi, Y. Tokura, A. Cano, Th. Lottermoser, M. Fiebig | 5,819 |
10.48550_arXiv.1410.1498 | ###### Abstract
Understanding fluctuation-induced breakages in polymers has important implications for basic and applied sciences. Here I present for the first time an analytical treatment of the thermal breakage problem of a semi-flexible polymer model that is asymptotically exact in the low temperature and high friction limits. Specifically, I provide analytical expressions for the breakage propensity and rate, and discuss the generalities of the results and their relevance to biopolymers.
## Introduction.
From man-made materials to biopolymers, semi-flexible polymers are ubiquitous in science and engineering. Better understanding of their stability is therefore of high importance. A semi-flexible polymer can naturally be broken by stretching or bending _via_ external forces, but thermal fluctuations alone will also induce breakage. Fluctuation-induced breakage is particularly relevant in the biological world as many biopolymers are stabilized by hydrophobic interactions and hydrogen bonds between proteins, which are relatively weak compared to covalently bonded synthetic polymers. Indeed, for amyloid fibrils, a kind of polymer implicated in numerous human diseases including Parkinson's and Alzheimer's, it has been advocated that thermal breakage could be a key mechanism underlying amyloid fibril proliferation. Despite the importance of understanding how polymers break, it is surprising that the basic physics remains to be elucidated. For instance, there is currently no concensus on the breakage profile of the polymer: some have advocated that breakage happens predominantly in the middle of the polymer while others have assumed uniform breakage propensity. This confusion is due in part to the fact that most existing results rely on considering simplified polymer models in one dimension or numerical simulations. Here I will provide for the first time analytical expressions of the breakage profile and rate of a highly-rigid polymer in the high friction (highly damped) and low temperature (or high Arrhenius energy) regimes.
## Minimal model.
I will employ a bead-and-stick representation of a semi-flexible polymer in which the sticks are massless and the beads experience isotropic thermal fluctuations (see Fig. 1).
\[\frac{\mathrm{d}\vec{r}_{i}}{\mathrm{d}t}=-\frac{1}{\zeta}\vec{\nabla}_{\vec{ r}_{i}}H+\sqrt{\frac{2k_{B}T}{\zeta}}\vec{\eta}_{i} \tag{1}\]
Also, the tensile and bending rigidities are enforced by two energy potentials \(U\) and \(V\).
\[H=\sum_{k=1}^{M-1}U(\theta_{k})+\sum_{k=1}^{M}V(r_{k,k-1})\, \tag{2}\]
Thermal fluctuations induce tensile and bending strains on the polymer and the polymer is broken if the bond is stretched or the angle is bent beyond certain thresholds. Since I have previously analysed thermal breakage by fluctuations-induced tensile strain for a polymer on a one dimensional track, I will focus here on breakage by bending, and I will comment on how the results are modified if both breakage by stretching and bending are possible in Discussion. Here, I assume that the polymer is broken if one of the angles is greater than some material-dependent threshold angle \(\Theta_{c}\) (Fig. 1(b)), which is expected to be much less than 1 for highly rigid
b) A particular form of energy function (\(U(\theta)=A\theta^{2}/2\)) that enforces bending rigidity (solid blue line) and the resulting quasi-stationary probability distribution \(P(\theta)\propto\mathrm{e}^{-\beta V(\theta)}\), with \(\Theta_{c}=0.1\)rad. c) For a 4-bead polymer, the corresponding energy landscape is two dimensional with 4 minimal-energy breakage points at \((\theta_{1},\theta_{2})=(\pm\Theta_{c},0)\) and \((\pm 0,\Theta_{c})\).
With this definition, we are ready to pose the two questions that will be answered in this paper:
1. What is the rate for one of the angles to be pushed beyond \(\Theta_{c}\) due to fluctuations?
2. What is the breakage propensity as a function of the monomer position in the polymer?
Since the breakage criterion concerns the set of angles \(\theta_{k}\), it is more natural to consider the EOM of the \(\theta_{k}\) instead of \(\vec{r}_{k}\).
\[\frac{\mathrm{d}\theta_{i}}{\mathrm{d}t} = \sum_{k=0}^{M}\vec{\nabla}_{\vec{r}_{k}}\theta_{i}\cdot\frac{ \mathrm{d}\vec{r}_{k}}{\mathrm{d}t} \tag{3}\] \[= \sum_{k=i-1}^{i+1}\vec{\nabla}_{\vec{r}_{k}}\theta_{i}\cdot\left( -\frac{1}{\zeta}\vec{\nabla}_{\vec{r}_{k}}H+\sqrt{\frac{2k_{B}T}{\zeta}}\vec{ \eta}_{k}\right) \tag{4}\]
and the summation index is now restricted to \([i-1,i+1]\) since variations in other beads do not affect \(\theta_{i}\). We now focus on the highly rigid polymer so that \(\Theta_{c}\) is expected to be small. In this limit, the EOM can be rewritten as (see Appendix A):
\[\frac{\mathrm{d}\theta_{k}}{\mathrm{d}t}=\sum_{h=1}^{M}L_{kh}\left(-\frac{U^{ \prime}(\theta_{h})}{\ell\zeta}+\sqrt{\frac{2k_{B}T}{\ell^{2}\zeta}}\xi_{h}\right) \tag{5}\]
In other words, the set of angles may be viewed as under the influence of the potential energy \(U_{tot}(\{\theta\})\equiv\sum_{k}U(\theta_{k})\) and a thermal heat bath with the nondiagonal mobility matrix \(L\).
Footnote 14: The term \(\frac{1}{\zeta}\vec{\nabla}_{\vec{r}_{k}}H\) is the term \(\frac{1}{\zeta}\vec{\nabla}_{\vec{r}_{k}}H\), which is the term \(\frac{1}{\zeta}\vec{\nabla}_{\vec{r}_{k}}H\).
## Two dimensions.
Equipped with the EOM for \(\theta_{k}\), let us now calculate the escape rate and the breakage profile by employing the quasi-static approximation. While the arguments below will be heuristic, the results can be shown to be exact in the asymptotic limit of \(\beta\to\infty\) with \(\beta\equiv 1/k_{B}T\)15; 16. In this approximation scheme, we assume that the probability distribution is normalized to one and the distribution is equilibrated to be at the Gibbs state: \(P(\{\theta\})=N_{M}\mathrm{e}^{-\beta U_{tot}(\{\theta\})}\) (Fig. 1(b) & (c)). Since at low \(T\), the distribution is highly centered around \(\theta_{k}=0\), \(P(\{\theta\})\) is well approximated as \(N_{M}\mathrm{e}^{-\beta A}\sum_{k}\theta_{k}^{2}/2\) where \(A=U^{\prime\prime}\). By integrating this multi-dimensional Gaussian distribution, we find that for a \((M+2)\)-bead polymer, \(N_{M}=(\beta A/2\pi)^{M/2}\). As \(T\to 0\), breakage is highly improbable and is dictated by the configurations where one of the angle is at \(\Theta_{c}\) (say \(\theta_{1}\sim\Theta_{c}\)) while the rest remain close to \(0\). Around this breakage boundary, we can expand the energy landscape in a Taylor series.
Footnote 15: The term \(\frac{1}{\zeta}\vec{\nabla}_{\vec{r}_{k}}H\) is the term \(\frac{1}{\zeta}\vec{\nabla}_{\vec{r}_{k}}H\), which is the term \(\frac{1}{\zeta}\vec{\nabla}_{\vec{r}_{k}}H\).
\[N_{M}\mathrm{e}^{-\beta\left[A\sum_{k>1}\theta_{k}^{2}/2-\triangle E+b(\theta_ {1}-\Theta_{c})+\mathcal{O}((\theta_{1}-\Theta_{c})^{2})\right]} \tag{6}\]
The rate of breakage \(R_{1}\) corresponds to the flux across the breakage boundary \(\theta_{1}=\Theta_{c}\). In one-dimension, this amounts to the negation of the diffusion coefficient multiplied by the derivative of the probability distribution.
\[R_{1} = -\left(\frac{L_{11}}{\beta\ell^{2}\zeta}\right)\left.\frac{ \partial}{\partial\theta_{1}}\right|_{\theta_{1}=\Theta_{c}}\int\mathrm{d} \theta_{2}\cdots\mathrm{d}\theta_{M}P_{1} \tag{7}\] \[= \left(\frac{6}{\beta\ell^{2}\zeta}\right)\sqrt{\frac{\beta A}{2 \pi}}\beta b\mathrm{e}^{-\beta\triangle E}=\frac{3b}{\ell^{2}\zeta}\sqrt{ \frac{2\beta A}{\pi}}\mathrm{e}^{-\beta\triangle E} \tag{8}\]
Since the diagonal elements in the mobility matrix \(L\) are identical, we can conclude that _the breakage propensity is uniform along the polymer_.
\[R^{\mathrm{[2D]}}=2MR_{1}=\frac{6Mb}{\ell^{2}\zeta}\sqrt{\frac{2\beta A}{\pi}} \mathrm{e}^{-\beta\triangle E}\, \tag{9}\]
These analytical predictions are supported by Brownian dynamics simulations that show the convergence of theoretical and simulation results as \(T\to 0\). (Fig. 2(a)).
## Three dimensions.
Going from 2D to 3D introduces a new degree of freedom as the polymer can now rotate around its longitudinal axis. Using the longitudinal axis of the polymer as the \(z\)-axis, \(\theta_{k}\) corresponds to the polar angle while we denote the azimuth angle by \(\phi_{k}\).
\[N_{M}=\left(\int_{0}^{2\pi}\mathrm{d}\phi\int_{0}^{\pi}\mathrm{d}\theta \theta\mathrm{e}^{-\beta A\theta^{2}/2}\right)^{-M}\simeq\left(\frac{\beta A}{ 2\pi}\right)^{M}. \tag{10}\]
Once again, defining \(\mathcal{P}_{1}\) as \(P(\{\theta_{1}\sim\Theta_{c}:\theta_{k>1}\sim 0\})\) which has the same expression as in Eq., the breakage rate \(R_{1}\) can be similarly calculated as
\[R_{1} = -\left(\frac{L_{11}}{\beta\ell^{2}\zeta}\right)\left.\frac{ \partial}{\partial\theta_{1}}\right|_{\theta_{1}=\Theta_{c}}\int\mathrm{d} \phi_{1}\Theta_{c}\prod_{k=2}^{M}(\mathrm{d}\phi_{k}\mathrm{d}\theta_{k}\theta_{ k})\mathcal{P}_{1} \tag{11}\] \[= \frac{6\beta Ab\Theta_{c}}{\ell^{2}\zeta}\mathrm{e}^{-\beta \triangle E}\.\]
Therefore, for a \((M+2)\)-bead polymer in 3D, the breakage propensity is again uniform and the overall escape rate is:
\[R^{\mathrm{[3D]}}=\frac{6M\beta Ab\Theta_{c}}{\ell^{2}\zeta}\mathrm{e}^{-\beta \triangle E}. \tag{12}\]
These analytical predictions are also supported by Brownian dynamics simulation (Fig. 2(b)).
## Discussion.
We have seen that for the minimal semiflexible polymer model considered here, the answers to the two questions posed earlier are analytical tractable in the \(T\to 0\) limit. I will now discuss the generalities and limitations of the results.
_Beyond thermal systems._ Although the model formulation focuses on polymers at thermal equilibrium, the results remain valid as long as the dynamics of the polymer is well approximated by Eq.. In other words, the results also apply to polymers enclosed in a volume that is at "local" equilibrium, or under active fluctuations of the form depicted in Eq..
_Low \(T\) limit vs. high energy barrier limit._ Mathematically, these two limits are not reversible: In the first limit, there is only one small parameter in the equation of motions (Eq.), while in the high energy barrier limit, both the fluctuation strength and the distance from the bottom of the well to the escape boundary become effectively small. Physically, since the units of time and length can be set arbitrarily, the high energy barrier and the low temperature limits are in principle interchangeable with the caveat that the high friction assumption has to remain valid. This leads to the next comment.
_Drag coefficient._ The results presented apply to the high drag regime where the inertia term is neglected. The drag coefficient may be seen as an effective measure of the friction due to _both_ the internal degrees of freedom within the beads (e.g., protein), and bead-solvent interactions. These two effects combined seem to lead to the general validity of considering biomolecular kinetics in the highly damped regime.
_Threshold on bending._ The analysis presented here assumes that the threshold bending angle \(\Theta_{c}\) is small. For biopolymers, I am only aware of one paper on microtubules that provides access to this parameter. Specifically, it was observed that upon bending, a microtubule forms an arc with curvature of around \(1\mu\)m\({}^{-1}\) before breaking. Since the size of the tubulin dimers making up the microtubule is around 10nm, one may deduce that for microtubules, \(\Theta_{c}\) is in the order of 0.01rad.
_Extensile vs. bending breakage._ I have omitted discussion of breakage by extensile fluctuations here because the results have already been derived previously, albeit with the extra assumption that the polymer is on a one-dimensional track. But since the bending fluctuations are orthogonal to the stretching fluctuations. In the low \(T\) limit, breakage by stretching is analytically identical for a polymer in 1D or in higher dimensions. If both extensile and bending breakage events are possible, then the one with a lower Arrhenius activation energy will dominate. If both have the same activation energy, then the relative proportion of bending vs. tensile breakage will be given by the ratio of the corresponding prefactors.
_Other types of potential energy._ Although our analysis assumes that \(U^{\prime}(\theta_{c})\) is non-vanishing here for simplicity. Similar analysis can be extended to other types of energy such as the single-hump energy function usually employed in the discussion of Kramers escape rate. The expression for the breakage rate will naturally be modified accordingly, but the conclusion that the breakage propensity is uniform remains valid.
_Internal structure and end effects of a polymer._ The model polymer considered (Fig. 1(a)) is certainly a drastically simplified version of a real polymer, but in the spirit of a ghost chain in polymer physics, each bead may be seen as a unit of polymer structure with internal dynamics that are decoupled from the bending dynamics considered here. If such decoupling is valid, the prediction of uniform breakage propensity should hold true even along the body of the polymer with internal structures. However, the ends of the polymers may be subject to different kinds of energy function. For instance, in the case of protein amyloid fibrils where each fibril is a bundle of filaments, the filaments at the ends may not be as tightly bound and so the bending rigidity at the ends may differ from that in the middle of the fibril. As a result, the breakage propensity close to the ends of the polymer may differ from that in the main body of the polymer. Another complication is the possibility of having a rugged energy landscape that connects two neighboring beads. In this case, the drag coefficient may need to be modified using the theory of diffusion on rugged landscape.
Convergence of analytical and simulational results as \(T\to 0\). a) The ratio of the theoretical breakage rate (Eqs & ) vs. the breakage rate from Brownian dynamics simulations for a 9-band polymer in 2D (a) and in 3D (b). The inset plots show the respective breakage frequency with respect to the breakage locations at the lowest temperature considered. The energy functions employed and simulation procedure are detailed in the Appendix B.
Summary & outlook.I have considered thermal breakage of a semi-flexible polymer and have obtained analytical results in the highly rigid, highly damped and low temperature limits. My calculations indicate that a semi-flexible polymer satisfying these conditions has uniform breakage propensity and analytical expressions of the breakage rates in 2D and 3D were provided. All analytical results were verified by Brownian dynamics simulations. The generalities of the results and their relevance to biopolymers were discussed. Future work on this problem will include a thorough analysis of the thermalization kinetics of polymerization since both the equilibrium configuration and the breakage kinetics are now known. Other interesting directions are the incorporation of higher order corrections into the calculations, and the investigation of polymer breakage under nonequilibrium and anisotropic fluctuations, such as under shear flows and sonications, which constitute two standard experimental procedures in investigating biopolymer self-assembly.
## Appendix A EOM for the \(\theta_{k}\)
Let us first focus on the following terms in Eq.:
\[\sum_{k=i-1}^{i+1}\vec{\nabla}_{\vec{r}_{k}}\theta_{i}\cdot\vec{ \nabla}_{\vec{r}_{k}}H \tag{10}\] \[=\sum_{k=i-1}^{i+1}\vec{\nabla}_{\vec{r}_{k}}\theta_{i}\cdot\vec{ \nabla}_{\vec{r}_{k}}\left(\sum_{h=1}^{M-1}U(\theta_{h})+\sum_{h=1}^{M}V(r_{h, h-1})\right)\] \[=\sum_{k=i-1}^{i+1}\vec{\nabla}_{\vec{r}_{k}}\theta_{i}\cdot\vec{ \nabla}_{\vec{r}_{k}}\left(\sum_{h=k-1}^{k+1}U(\theta_{h})+\sum_{h=k}^{k+1}V(r _{h,h-1})\right)\.\]
The terms of the form \(\vec{\nabla}_{\vec{r}_{k}}\theta_{i}\cdot\vec{\nabla}_{\vec{r}_{k}}V(r_{h,h-1})\) are of order \(\mathcal{O}(\Theta_{c})\), To see this, consider the particular term \(\vec{\nabla}_{\vec{r}_{k}}\theta_{k}\cdot\vec{\nabla}_{\vec{r}_{k}}V(r_{k,k-1})\) in 2D. Without loss of generality, assume \(\vec{r}_{k,-1}\) is along the \(x\)-axis, \(\vec{\nabla}_{\vec{r}_{k}}\theta_{k}=\ell^{-1}\theta_{k}\hat{\bf x}+\ell^{-1} \hat{\bf y}+\mathcal{O}(\theta_{k}^{2})\) and \(\vec{\nabla}_{\vec{r}_{k}}V(r_{k,k-1})=\hat{\bf x}\). Since \(\theta_{k}<\Theta_{c}\) before breakage, \(\vec{\nabla}_{\vec{r}_{k}}\theta_{k}\cdot\vec{\nabla}_{\vec{r}_{k}}V(r_{k,k-1} )=\mathcal{O}(\Theta_{c})\). Other terms of the same form can similarly be shown to be of the same order, which are small for rigid polymer where \(\Theta_{c}\ll 1\). Physically, what it means is that the fluctuations that contribute to bond extension is orthogonal to the fluctuations contributing to bending in the small angle limit. The same arguments apply in 3D.
In Eq., we are thus left with the following terms:
\[\sum_{k=i-1}^{i+1}\sum_{h=k-1}^{k+1}\vec{\nabla}_{\vec{r}_{k}} \theta_{i}\cdot\vec{\nabla}_{\vec{r}_{k}}U(\theta_{h}) \tag{12}\] \[=\sum_{k=i-1}^{i+1}\sum_{h=k-1}^{k+1}U^{\prime}(\theta_{h})\vec{ \nabla}_{\vec{r}_{k}}\theta_{i}\cdot\vec{\nabla}_{\vec{r}_{k}}\theta_{h}\equiv \sum_{h}L_{ih}U^{\prime}(\theta_{h})\,\]
where \(L\) to \(\mathcal{O}(\Theta_{c})\) is a \(M\times M\) matrix of the form
\[L=\left[\begin{array}{ccccc}6&-4&1&0&0&\cdots\\ -4&6&-4&1&0&\cdots\\ 1&-4&6&-4&1&\cdots\\ &&&\ddots\\ \cdots&0&0&1&-4&6\end{array}\right]. \tag{13}\]
The only remaining terms in Eq. that we have to consider are from the coupled noise terms: \(\vec{\nabla}_{\vec{r}_{k}}\theta_{i}\cdot\vec{\eta}_{k}\equiv G_{ik}\).
\[G=\left[\begin{array}{ccccc}-1&2&-1&0&0&\cdots\\ 0&-1&2&-1&0&\cdots\\ 0&0&-1&2&-1&\cdots\\ &&&\ddots\\ \cdots&0&0&-1&2&-1\end{array}\right]. \tag{14}\]
The matrix \(G\) is nondiagonal because there are \(M\) variables (\(\theta_{k}\)) and \(M+2\) noise terms (\(\eta_{k}\)). One could therefore reduce the number of the noise terms by 2 by introducing a new set of \(M\) noise terms \(\xi_{k}\) with statistics depicted below:
\[\langle\xi_{k}(t)\rangle = 0 \tag{15}\] \[\langle\xi_{k}(t)\xi_{k}(t^{\prime})\rangle = 6\delta(t-t^{\prime})\] \[\langle\xi_{k}(t)\xi_{k\pm 1}(t^{\prime})\rangle = -4\delta(t-t^{\prime})\] \[\langle\xi_{k}(t)\xi_{k\pm 2}(t^{\prime})\rangle = \delta(t-t^{\prime})\] \[\langle\xi_{k}(t)\xi_{h}(t^{\prime})\rangle = 0\ \ \mbox{for}\ h\neq k,k\pm 1,k\pm 2. \tag{19}\]
In other words, the covariance matrix of \(\xi_{k}\) is exactly the matrix \(L\), this explains the EOM in Eq..
## Appendix B Simulation procedure
The simulation is based on numerically integrating the set of stochastic differential equations shown in Eq. as follows:
\[\vec{r}_{i}(t+\triangle t)=\vec{r}_{i}(t)-\frac{\triangle t}{\zeta}\vec{\nabla }_{\vec{r}_{i}}H(\{r(t)\})+\sqrt{\frac{2k_{B}T\triangle t}{\zeta}}\vec{g}_{i} \tag{20}\]
The time increment \(\triangle t\) is set to be \(2\times 10^{-6}\), the drag coefficient \(\zeta\) is one and the energy function \(U\) and \(V\) in \(H\) are of the form:
\[U(\theta)=\frac{A}{2}\theta^{2}\quad,\quad V(r)=\frac{B}{2}(r-1)^{2} \tag{21}\]
The polymer is always initialized in its lowest energy state and the simulation stops if one of the angles \(\theta\) is beyond \(\Theta_{c}=0.1\). Five hundred sample runs are performed at each distinct \(k_{B}T\) for the 2D case (Fig. 2(a)) and one thousand sample runs are performed at each distinct \(k_{B}T\) for the 3D case (Fig. 2(b)).
| 10.48550/arXiv.1410.1498 | Thermal breakage of a semiflexible polymer: Breakage profile and rate | Chiu Fan Lee | 5,542 |
10.48550_arXiv.1207.2916 | ###### Abstract
We use symmetry analysis and first principles calculations to show that the linear magnetoelectric effect can originate from the response of orbital magnetic moments to the polar distortions induced by an applied electric field. Using LiFePO\({}_{4}\) as a model compound we show that spin-orbit coupling partially lifts the quenching of the 3d orbitals and causes small orbital magnetic moments (\(\mu_{(L)}\approx 0.3\mu_{B}\)) parallel to the spins of the Fe\({}^{2+}\) ions. An applied electric field \(\mathbf{E}\) modifies the size of these orbital magnetic moments inducing a net magnetization linear in \(\mathbf{E}\).
Such _magnetoelectric coupling_ manifests in numerous macroscopic phenomena: Two well known examples are so-called type-II multiferroism in which the onset of magnetic order induces a spontaneous polarization, and linear magnetoelectricity, where an applied electric field \(\mathbf{E}\) (magnetic field, \(\mathbf{H}\)), induces a magnetization \(M_{j}=\alpha_{ij}E_{i}\) (polarization, \(P_{i}=\alpha_{ij}H_{j}\)). Although the two phenomena are non-reciprocal, that is many multiferroics do not show a linear magnetoelectric effect and vice-versa, they are believed to share closely-related microscopic mechanisms.
First-principles computations have been particularly informative in resolving quantitatively the various microscopic contributions to magnetoelectric response. The first study extracted the "ionic spin" contribution to \(\alpha\), by calculating the change in spin canting caused by an \(\mathbf{E}\)-induced polar distortion. Subsequently, the methodology to calculate the "electronic spin" component was implemented, through calculating the electric polarization induced by an applied Zeeman \(\mathbf{H}\) field that couples only to the spin component of the magnetization. In this method, the electronic spin response is obtained by "clamping" the ions during the calculation; relaxing the ionic positions in response to the \(\mathbf{H}\) field yields both the ionic and electronic spin components. Interestingly, and perhaps surprisingly, this study showed that the ionic and electronic contributions to \(\alpha\) can have similar magnitudes.
These spin-based contributions to \(\alpha\) have been shown to capture much of the experimental response. For the case when the magnetic field is applied perpendicular to the spins in a collinear antiferromagnet, the magnetoelectric coupling, \(\alpha_{\perp}\) is relativistic in origin, resulting e.g. from the electric-field dependence of the antisymmetric Dzyaloshinskii-Moriya exchange. The calculated zero kelvin polarizations are consistent with experimental values, and the temperature evolution of \(\alpha_{\perp}\) follows that of the antiferromagnetic order parameter. The behavior of \(\alpha_{\parallel}\) - obtained when the magnetic field is applied parallel to the spins - is more complicated. In this case, the Heisenberg exchange interactions between the spins induce an electric polarization at finite temperature which is approximately an order of magnitude larger than that from the anisotropic exchange interactions of relativistic origin responsible for \(\alpha_{\perp}\). It has been shown that responses calculated within this Heisenberg exchange model agree closely with experiment in the region close to \(T_{N}\) (Fig. 1(a)). One experimentally observed feature is lacking, however: While Heisenberg exchange predicts \(\alpha_{\parallel}\to 0\) for \(T\to 0\) K, consistent with the vanishing parallel spin susceptibility at zero kelvin, many magnetoelectrics with collinear antiferromagnetism have non-zero \(\alpha_{\parallel}\) at zero kelvin, and instead follow the temperature dependence sketched in Fig. 1(a) (solid line). An obvious candidate for the discrepancy is the neglect of orbital contributions.
While the neglect of orbital magnetism in the above methods is partially justified by the strong quenching of \(3d\) orbital moments which usually occurs in transition metal oxides, spin-orbit coupling, \(H_{so}=\lambda\mathbf{L}\cdot\mathbf{S}\), can of course reduce the quenching, and allow a non-negligible orbital magnetization. This scenario is likely in the collinear antiferromagnet LiFePO\({}_{4}\) and in LiCoPO\({}_{4}\). Both these compounds have a substantially non-zero \(\alpha_{\parallel}\) as \(T\to 0\) and an anomalously large anisotropy of the magnetic g-tensor.
Calculation of the orbital contribution to the magnetoelectric response is not straightforward, and only a few examples, for limited cases and specific approximations, exist in the literature. An early study of LiCoPO\({}_{4}\) calculated the "electronic orbital" (clamped ion) contribution analytically, by determining the change in g-factor with electric field using perturbation theory within a single-ion Hamiltonian. While giving a non-zero value for \(\alpha_{\parallel}\) at \(T=0\), this method underestimated its magnitude. More recently first-principles finite-electric-field methods were used to calculate the electronic orbital contributionsto the trace of the magnetoelectric tensor - the Chern-Simons term - for Cr\({}_{2}\)O\({}_{3}\) and BiFeO\({}_{3}\). This contribution was shown to be negligible with respect to the spin contribution in both cases. In this letter we explore the remaining "ionic orbital" contribution to the magnetoelectric response by calculating the dependence of the local, on-site orbital magnetic moments on polar lattice distortions using density functional theory. Using magnetoelectric LiFePO\({}_{4}\) as a model compound, we show that this ionic orbital contribution to \(\alpha\) is unexpectedly large and can explain the anomalous low-temperature behavior observed in certain components of \(\alpha\) that were previously not understood.
LiFePO\({}_{4}\) is orthorhombic (space group \(Pnma\)) and its unit cell (see Fig. 1(b)) contains four magnetic sublattices occupied by Fe\({}^{2+}\) (\(S\)=2) ions. Each magnetic ion is surrounded by strongly distorted polar oxygen octahedra for which the only remaining local symmetry is a mirror transformation perpendicular to the crystallographic axis \(\mathbf{b}\) giving local \(C_{s}\) symmetry. At temperatures below T\({}_{N}\approx 50\) K the Fe\({}^{2+}\) magnetic moments order in the antiferromagnetic collinear structure with order parameter \(\mathbf{G}=\mathbf{m}_{1}-\mathbf{m}_{2}+\mathbf{m}_{3}-\mathbf{m}_{4}\) where \(\mathbf{m}_{i}\) is the magnetization of the \(i\)-th sublattice. The spin orientation in the antiferromagnetic state is still slightly controversial. Early elastic neutron scattering and X-ray diffraction data suggested that the magnetic moments are fully oriented along the \(\mathbf{b}\) direction. However, recent neutron scattering measurements provide evidence for a magnetic structure in which \(\mathbf{G}\) is slightly rotated from \(\mathbf{b}\). In this paper, we study only those components allowed with \(\mathbf{G}\parallel\mathbf{b}\); \(G^{a}\neq 0\) or \(G^{c}\neq 0\) would give rise to additional non-zero components of the magnetoelectric tensor that have not yet been reported.
The onset of the antiferromagnetic order breaks inversion symmetry and allows for linear magnetoelectric couplings in the free energy
\[\Phi_{\parallel}=\lambda_{\parallel}G^{b}E^{a}H^{b}\quad\text{and}\quad\Phi_{ \perp}=\lambda_{\perp}G^{b}E^{b}H^{a}, \tag{1}\]
\(\alpha_{\parallel}\) follows the typical form discussed previously and sketched in (a): Decreasing the temperature from \(T_{N}\), \(\alpha_{\parallel}\) rapidly increases and reaches a maximum at \(T_{max}\approx 45\) K. Below \(T_{max}\), \(\alpha_{\parallel}\) decreases until 20 K at which it becomes almost temperature independent with a value of \(\sim 2\) ps/m. Importantly, it does not approach zero as \(T\to 0\) K. \(\alpha_{\perp}\) has the simpler temperature dependence mentioned earlier, increasing with decreasing temperature below \(T_{N}\) to reach a roughly constant value below 25 K (4 ps/m).
We focus on the microscopic couplings which can induce \(\alpha_{\parallel}\). Phenomenologically, exchange-striction couplings between electric polarization and spins are allowed by symmetry and give rise to the term: \(P^{a}\propto(\mathbf{m}_{1}\cdot\mathbf{m}_{3}-\mathbf{m}_{2}\cdot\mathbf{m}_{ 4})\) (see Tab. 1). This coupling results in a temperature behavior of \(\alpha_{\parallel}\) similar to that discussed above for Cr\({}_{2}\)O\({}_{3}\). We note that the local symmetry \(C_{s}\) of the crystal field around each Fe\({}^{2+}\) ion has only one-dimensional irreducible representations and, therefore, the \(d\) orbitals are non degenerate. When the orbital moments are fully quenched the magnetic moment at the \(i\)-th site is proportional to the spin \(\mathbf{m}_{i}=2\mu_{B}\mathbf{S}_{i}\). As discussed above, at \(T=0\) the spins in a uniaxial antiferromagnet are not modified by \(\mathbf{H}_{\parallel}\) weaker than the magnetic field necessary to flop the spins. Therefore, the electric polarization generated at \(T=0\) by the above couplings in response to \(\mathbf{H}_{\parallel}\) is zero.
Next we analyze the orbital contribution to \(\alpha_{\parallel}\). We begin by discussing the orientation and size of orbital moments in zero applied field. From an atomistic perspective, when \(H_{so}=\lambda\mathbf{L}\cdot\mathbf{S}\) is considered the orbital moments are partially unquenched and the magnetic moment at site \(i\) is:
\[m_{i}^{\mu}=\mu_{B}(2S_{i}^{\mu}+L_{i}^{\mu})=\mu_{B}g_{i}^{\mu\nu}S_{i}^{\nu} \tag{2}\]
For an ion with non-degenerate ground state first-order corrections in \(\lambda\) lead to \(g_{\mu\nu}=(2-\lambda\Lambda_{i}^{\mu\nu})\) where \(\Lambda_{i}^{\mu\nu}=\sum_{n}\frac{(\psi_{0}|L^{\mu}|\psi_{n})\langle\psi_{n}| L^{\nu}|\psi_{0}\rangle}{\epsilon_{n}-\epsilon_{0}}\).
(b) The orthorhombic unit cell of LiFePO\({}_{4}\) contains four Fe\({}^{2+}\) magnetic cations (brown spheres) which are coordinated by distorted oxygen (red spheres) octahedra. Li and P ions are represented, respectively, by green and purple spheres. The blue numbers label the magnetic sublattices. The arrows indicate the screw rotation axis parallel to \(\mathbf{b}\) and \(\mathbf{c}\) while the black dot indicates the center of inversion.
Since the magnetic moments are parallel to \({\bf b}\) we consider the components \(\Lambda_{i}^{\mu b}\). The transformations of these components under the generators of the space group (modulo primitive translations) are listed in Tab. 1, where we see that \(\Lambda_{i}^{ab}=\Lambda_{i}^{cb}=0\) and \(\Lambda_{i}^{bb}\equiv\Lambda^{bb}\) at every magnetic sublattice.
The mean values of the orbital parts of the magnetic moments induced by the antiferromagnetic ordering are \(\mu_{(L)i}^{\nu}=-\lambda\Lambda_{i}^{\mu b}\langle S_{i}^{b}\rangle\). For d\({}^{6}\) ions, \(\lambda<0\), therefore the orbital moment is parallel to the spins in every magnetic sublattice and like the spins, gives rise to zero net magnetic moment.
Next we consider the case \(E\neq 0\). Electric-field-induced polar lattice distortions modify the crystal field around each Fe\({}^{2+}\) ion and the energies \(\epsilon_{n}=\epsilon_{n}({\bf E})\).
\[\partial_{E^{\alpha}}\Lambda_{i}^{\mu\nu}=-\sum_{n}\frac{\langle\psi_{0}|L^{ \mu}|\psi_{n}\rangle\langle\psi_{n}|L^{\nu}|\psi_{0}\rangle}{(\Delta\epsilon_{ n})^{2}}\frac{\partial(\Delta\epsilon_{n})}{\partial E^{\alpha}}+\xi_{\rho}^{\mu\nu} \tag{3}\]
\(\xi_{\rho}^{\mu\nu}\) are the remaining terms containing derivatives of wave functions with respect to \(E^{\alpha}\). The transformations of the derivatives \(\partial_{E^{\alpha}}\Lambda_{i}^{\mu b}\) under the space group of LiFePO\({}_{4}\) can be obtained from those of \(\Lambda_{i}^{\mu b}\) and those of \({\bf E}\) in Tab. 1. From these transformations we obtain \(\partial_{E^{\alpha}}\Lambda_{1}^{bb}=\partial_{E^{\alpha}}\Lambda_{3}^{bb}=- \partial_{E^{\alpha}}\Lambda_{2}^{bb}=-\partial_{E^{\alpha}}\Lambda_{4}^{bb} \equiv\partial_{E^{\alpha}}\Lambda^{bb}\). Therefore, the response of the average orbital-induced magnetic moment to an electric field along \({\bf a}\) gives rise to a net magnetization along \({\bf b}\)
\[\mu_{(L)}^{b}=\mu_{B}\partial_{E^{\alpha}}\Lambda^{bb}(\langle S_{1}^{b} \rangle-\langle S_{2}^{b}\rangle+\langle S_{3}^{b}\rangle-\langle S_{4}^{b} \rangle)E^{a} \tag{4}\]
To calculate the strength of the linear magnetoelectric coupling arising from this mechanism, we perform first principles calculations using the Vienna _ab initio_ simulation package (VASP). We use a plane-wave basis set for the expansion of the electronic valence wave function and PAW potentials for the treatment of core electrons. The exchange-correlation potential is described within the local-spin-density approximation plus a rotationally invariant Hubbard-\(U\) (LSDA+\(U\)) with a \(U\) value of 5 eV, and \(J\) values between 0 and 1 eV. Calculations are performed at the experimental unit cell volume of 291 A\({}^{3}\). We first relax the structure in the absence of spin-orbit coupling and then we include spin-orbit coupling to calculate the orbital magnetic moment. We obtain an orbital moment \(\mu_{(L)}=0.306\mu_{B}\) parallel to the spins when we use a \(J\) value of 1eV. We note that the magnitude of the magnetic moment depends on \(J\) and on the PAW sphere radius used as discussed in Ref..
To calculate the ionic orbital response - that is the change in orbital magnetic moments when the ions are displaced by an applied electric field - we adapt the framework introduced in Ref. to obtain the ionic spin response. As in Ref., we shift the equilibrium positions \({\bf r}_{i}\) of the ions by \(\Delta r_{i}^{\mu}=E^{\rho}\sum_{\nu j}\phi_{\mu,\nu j}^{-1}Z_{j,\rho\nu}^{*}\) where \(\phi_{\mu,\nu j}^{-1}\) is the inverse of the force constant matrix after the acoustic modes are traced out and \(Z_{j,\rho\nu}^{*}\) are the Born effective charges, both calculated in the absence of spin-orbit coupling. Since we aim to separate the orbital from the spin contribution, we constrain the orientation of the spins to lie along the \({\bf b}\) direction, which we call the "clamped spin" approximation (note, however that the magnitude of the spin is unconstrained.) After making the \(\Delta r_{i}^{\mu}\)
\begin{table}
\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|} \hline & 1 & 2 & 3 & 4 & \(\Lambda_{i}^{ab}\) & \(\Lambda_{i}^{ab}\) & \(\Lambda_{i}^{cb}\) & \(E^{a}\) & \(E^{b}\) & \(E^{c}\) \\ \hline \(I\) & 4 & 3 & 2 & 1 & \(\Lambda_{I}^{ab}\) & \(\Lambda_{I}^{ab}\) & \(\Lambda_{I}^{ab}\) & \(-E^{a}\) & \(-E^{b}\) & \(-E^{c}\) \\ \(2_{c}\) & 2 & 1 & 4 & 3 & \(\Lambda_{2}^{ab}(i)\) & \(\Lambda_{2}^{bc}(i)\) & \(-\Lambda_{2}^{bc}(i)\) & \(-E^{a}\) & \(-E^{b}\) & \(E^{c}\) \\ \(2_{b}\) & 4 & 3 & 2 & 1 & \(-\Lambda_{2}^{ab}(i)\) & \(\Lambda_{2}^{bc}(i)\) & \(-\Lambda_{2}^{bc}(i)\) & \(-E^{a}\) & \(E^{b}\) & \(-E^{c}\) \\ \hline \end{tabular}
\end{table}
Table 1: Transformation of the four magnetic sublattices (second to fifth column) under the three generators of the space group (modulo a primitive translation) of LiFePO\({}_{4}\): inversion \(I\), two fold screw rotations around the \({\bf c}\) axis \(2_{c}\), and around the \({\bf b}\) axis \(2_{b}\) (see Fig. 1). Columns six to eight show the transformation of three components of the rank 2 axial tensor \(\Lambda_{i}\) at the \(i\)-th magnetic sublattice. Here the subscripts refer to the change of magnetic sublattice, e.g. \(2_{c}(\Lambda_{3}^{cb})=-\Lambda_{2}^{cb}=-\Lambda_{4}^{cb}\). The last three columns show the transformations of \({\bf E}\).
Calculated electric-field dependence of the net orbital magnetic moment per unit cell. \({\bf E}\parallel{\bf a}\) (panel a) results in an orbital magnetization along \({\bf b}\) (\(\alpha_{\parallel}\)) while \({\bf E}\parallel{\bf b}\) (panel b) produces a net orbital magnetic moment along \({\bf a}\) (\(\alpha_{\perp}\)). Blue dots and red squares are calculated values at J=1 eV and J=0 eV respectively, while the straight lines are linear fits to the calculated values. The cartoons on the right panels show the size and orientation of the orbital magnetic moments (gray arrows) of Fe\({}^{2+}\) (brown spheres) when the electric field (yellow arrow) is applied along \({\bf a}\) (panel c) and \({\bf b}\) (panel d). In the cartoon the size of the effect is increased for clarity.
Figure 2(a) shows the evolution of the calculated net orbital magnetic moment \(\mu_{(L)}=\sum_{i=1,4}\mu_{(L),i}\) of one unit cell of LiFePO\({}_{4}\) for an electric field applied along \(\mathbf{a}\) with \(J=1\) eV (blue points) and \(J=0\) eV (red points). (Note that, while the electric field is applied perpendicular to the spins, this corresponds to the parallel component of \(\alpha\), since the magnetoelectric response is off-diagonal). We find that at non-zero electric field the orbital magnetic moments remain parallel to the spins, and consistent with Eq. the change of their size is opposite for odd and even magnetic sublattices giving rise to a net magnetization. The linear fits of the \(E^{a}\) responses of the orbital magnetization at \(J=1\) eV (blue line) and \(J=0\) eV (red line) give \(\alpha_{\parallel}=2.3\) ps/m and \(\alpha_{\parallel}=9.3\) ps/m respectively. The \(\alpha_{\parallel}\) value for \(J=1\) eV is reasonably close to the experimental value of \(\alpha_{\parallel}\sim 2\) ps/m at \(T=0\) K. This value of \(J\) is consistent with Ref. which showed that it is necessary to use \(J>0.6\) eV to obtain the correct magnetic easy axis. To summarize this section, we find that the calculated zero kelvin ionic orbital contribution to \(\alpha_{\parallel}\) has a value which is consistent with the measured value of \(\alpha_{\parallel}\). We suggest, therefore, that the previous discrepancy between the measured zero kelvin magnetoelectric response and the calculated spin-only response can be explained by this contribution. At non-zero temperatures, contributions to \(\alpha_{\parallel}\) that are inactive in the absence of thermal fluctuations, have to be taken into account. These terms comprise the electric field dependence of single-ion anisotropy, which has the same nature as the orbital magnetic moment, as well as the Heisenberg interactions mentioned earlier.
Finally, we investigate the ionic orbital contribution to \(\alpha_{\perp}\), by calculating the effect of an electric field applied along \(\mathbf{b}\). While the spin-only contribution was not inconsistent with experiment in this case, contributions to \(\alpha_{\perp}\) from the electric field dependence of \(\mu_{(L)i}\) have not been previously investigated and might also play a role. First we use similar symmetry arguments as those used for \(\alpha_{\parallel}\) to find constraints on \(\partial_{E^{b}}\Lambda_{i}^{\mu\nu}\). From Tab. 1 we find: \(\partial_{E^{b}}\Lambda_{i}^{ab}=\partial_{E^{b}}\Lambda_{3}^{ab}=-\partial_{E^ {b}}\Lambda_{2}^{ab}=-\partial_{E^{b}}\Lambda_{4}^{ab}\equiv\partial_{E^{b}} \Lambda^{ab}\), \(\partial_{E^{b}}\Lambda_{1}^{cb}=\partial_{E^{b}}\Lambda_{2}^{cb}=-\partial_{E ^{b}}\Lambda_{3}^{cb}=-\partial_{E^{b}}\Lambda_{4}^{cb}\equiv\partial_{E^{b}} \Lambda^{cb}\) and \(\partial_{E^{b}}\Lambda_{i}^{ab}=0\). On one hand, we note that the transformation properties of \(\partial_{E^{b}}\Lambda_{i}^{ab}\) are identical to those of \(\partial_{E^{b}}\Lambda_{i}^{bb}\). This allows for a linear dependence of the orbital magnetization along \(\mathbf{a}\) when the electric field is applied along \(\mathbf{b}\): \(\mu_{a}=4\mu_{B}E^{b}\partial_{E^{b}}\Lambda^{ab}|\langle S^{b}\rangle|\) where \(|\langle S^{b}\rangle|\) is the absolute value of the average spin component along \(\mathbf{b}\). In contrast, the transformation properties of \(\partial_{E^{b}}\Lambda_{i}^{cb}\), together with the spin ordering of LiFePO\({}_{4}\) show that the change in orbital moment along \(\mathbf{c}\) under an applied \(E^{b}\) field have opposite sign for sublattices 1, 4 compared with 2,3, yielding zero net moment in this direction.
To obtain the size of the ionic orbital contribution to \(\alpha_{\perp}\) we perform _ab initio_ calculations using the same method discussed for \(\alpha_{\parallel}\) but with the electric field applied along \(\mathbf{b}\). As before, we adopt the clamped-spin approach, and constrain the spins in the \(\mathbf{b}\) direction. The resulting calculated values of net orbital magnetic moment are shown in Fig. 2(b) as a function of \(E^{b}\). Here blue and red points show results for, respectively, \(J=1\) eV and \(J=0\) eV. Even when the spins are constrained to be parallel to the \(\mathbf{b}\) axes, the applied \(\mathbf{E}_{\parallel}\) induces a canting of the orbital magnetic moments from the \(b\) direction. In agreement with the constraints found for \(\partial_{E^{b}}\Lambda_{i}^{ab}\) the resulting canting is uniform along the \(\mathbf{a}\) axis for all magnetic sublattices giving rise to a net magnetization linear in \(E_{b}\). Furthermore, as predicted using the transformations of \(\partial_{E^{b}}\Lambda_{i}^{cb}\) for finite \(E_{b}\) we observe a tiny staggered canting of the orbital moment along \(c\) which gives rise to zero net magnetization. The solid lines in Fig. 2(b) are linear interpolations of the calculated values and give linear magnetoelectric responses of 1.9 ps/m and 9.7 ps/m for \(J=1\) eV and \(J=0\) respectively. To these values, which contain only the ionic orbital magnetoelectric effect, one should add the spin-only contribution to \(\alpha_{\perp}\), which in contrast to the case of \(\alpha_{\parallel}\) does not vanish at \(T=0\). These include the rotation of easy axis anisotropy, that shares the same origin as the canting of orbital magnetic moment, as well as Dzyaloshinskii-Moriya interaction. Using the approach described in Ref., which includes these contributions but not the orbital moment part, we obtain for \(J=1\) eV a value for \(\alpha_{\perp}\) of 2.6 ps/m with sign opposite to the orbital one. Importantly, these considerations can also be used to describe resonant excitation of waves of oscillating magnetization \(M\parallel a\) with an oscillating electric field of a light wave \(E\parallel b\), resulting in the so-called "electromagnon" peaks in optical absorption. Thus the coupling between the orbital magnetic moment and electric field gives rise to both static and dynamic magnetoelectric effects.
In summary, we have shown that a linear magnetoelectric effect can arise from the dependence of orbital magnetic moments on the polar distortions induced by an applied electric field, the so-called "ionic orbital" contribution to the magnetoelectric response. We presented a symmetry analysis which allows the components of \(\alpha_{\mu\nu}\) for which this effect exists to be determined, and a methodology which can be used to calculate _ab initio_ those components at \(T=0\). We applied the methodology to LiFePO\({}_{4}\) and resolved the previous discrepancy between previous calculations of the spin-only contributions and experiment for \(\alpha_{\parallel}\). Our results show that the orbital contributions to the magnetoelectric response can be comparable in size to the spin contributions of either relativistic or exchange-striction origin in \(3d\) transition metal compounds. As suggested by Eq., the temperature dependence of the magnetoelectric effect caused by orbital magnetism coincides with that of the order pa rameter which, added to the temperature dependence of magnetoelectric effect originated by striction gives rise to a qualitative agreement for various collinear antiferromagnets such as Cr\({}_{2}\)O\({}_{3}\), LiCoPO\({}_{4}\) and TbPO\({}_{4}\).
Furthermore, we note that if such coupling between orbital magnetization and polar distortion is allowed by symmetry, its strength does not depend solely on the strength of the spin-orbit interaction. As shown in Eq., from a single ion perspective, the strength of such an effect is determined by the energy gap between the ground state and the excited states for which \(\langle\psi_{0}|L^{\mu}|\psi_{n}\rangle\neq 0\) and also by the dependence of the energies of ionic orbitals on polar distortions of the crystal field. This suggests that large magnetoelectric effect due to the orbital moment correlates with the enhanced anisotropic g-tensor and the anisotropy of the magnetic susceptibility in the paramagnetic state. In particular, large response of orbital magnetism to an applied electric field might be found in compounds with reasonably small electronic gap, containing magnetic ions with large spin-orbit coupling and with low symmetry polar oxygen coordination.
This work was supported by ETH Zurich and by the European Research Council Advanced Grants program under the FP7, grant number 291151. E. B. thanks FRS-FNRS Belgium for support.
| 10.48550/arXiv.1207.2916 | Linear Magnetoelectric Effect by Orbital Magnetism | Andrea Scaramucci, Eric Bousquet, Michael Fechner, Maxim Mostovoy, Nicola A. Spaldin | 6,245 |
10.48550_arXiv.2204.02881 | ###### Abstract
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data-driven data-driven-driven data-driven data-driven data-driven data-driven-driven data-driven data-driven-driven data-driven data-driven data-driven data-driven-driven data-driven data-driven-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven-driven data-driven data-driven data-driven-driven data-driven data-driven data-driven data-driven data-driven data-driven-driven data-driven data-driven data-driven data-driven-driven data-driven data-driven-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven-driven data-driven data-driven data-driven data-driven-driven data-driven data-driven data-driven-driven data-driven data-driven-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven data-driven-The FAIR Principles, applicable to _any_ type of data, provide unifying guidelines for the effective sharing, discovery, and reuse of digital resources, including data, metadata, protocols, workflows, and software. FAIR data for Materials will enable better science via reproducibility and transparency and provide a path to reward valued data generators. Widespread FAIR data will unleash an era of materials informatics where exploring prior work is nearly instantaneous and drive development of advanced analytics and machine learning for materials.
Realizing the promises of MGI and FAIR, however, requires community agreement and implementation. General FAIR principles are necessary but not sufficient to transform the field of materials, where varied interpretations and definitions of basic composition and property terms hold back effective implementation. Each data type has different forms, vocabularies, and descriptors across material types, from polymeric systems to metals, biomaterials, ceramics, and functional materials.
As depicted in **Figure 1**, making materials data FAIR need not involve heroic efforts but does require attention and deliberate and consistent adoption of available protocols. For example, the use of globally unique, persistent identifiers (UUIDs or PIDs) as long-lasting references for digital resources is "FAIR", while the typical protocol of making data "available upon request" is "not FAIR".
### Materials Data Stakeholders: Barriers and Hopes
In planning the operational and cultural changes required to achieve broadly FAIR materials data, we must consider the agendas, needs, and concerns of five large cadres of stakeholders: _researchers_ who generate data; _developers of hardware and software tools_ used to produce research results; _publishers_ and _repository developers_ that transmit research results; _funders
Concise definitions of the FAIR principles translated to two specific examples of materials research datasets: 1) microstructure images, containing nuances such as grain boundaries, dislocations, inclusions, and/or dispersion of particles; 2) numeric data representing spectral responses for a property value across space, temperature, time, and frequency. The inherently heterogeneous data include image and video data at many length scales; scalar, vector, tensorial, and tabular physical property values; text and numbers defining compositions and processing conditions; and metadata on computational methods, experimental protocols, assumptions, and analyses. Findable and Accessible materials data resources include the Materials Project, OpenKim, Materials Data Facility (MDF). Interoperability is being tackled by the Crystallographic Information Framework (CIF, 20), Simplified Molecular-Input Line-Entry System (SMILES, 21), and OPTIMADE. Reusability, the ultimate goal of FAIR for materials science, and depends on development and consistent use of metadata standards.
We interviewed members of each group in developing our recommendations.
The number one **barrier** to FAIR materials data is fear of productive _time_ lost in archiving, cleaning, annotating, and storing data and associated metadata. Funders and researchers are concerned about lost productivity, publishers about barriers and delays to publication when data sharing is enforced, and consumers about spending time finding data in a new and unfamiliar landscape. Other major concerns identified include: _navigation of licensing, fear of being scooped / fear of losing credit, intellectual property restrictions for materials data_, and _quality control for data housed in repositories_.
Stakeholders simultaneously expressed **great hope** for a data-rich future where _journal articles are linked with FAIR datasets_; ever-growing supplementary information (SI) is replaced with references to _cleanly annotated data in repositories_; _measures of quality and FAIR metrics_ naturally evolve for housed data; and data are _citable, findable, and reusable, and have significantly larger impact_.
Achieving widespread FAIR materials data requires overcoming both sociological and technical challenges. To combat the major fear of "lost time", we need demonstrations of FAIR data enabling success, incentives for sharing FAIR data, and infrastructure to simplify or automate data upload and annotation. Data literacy and best practices need to become part of education and researchers' daily workflow so that making data FAIR is no longer a taxing after-thought nor a fear of lost credit. Sustainability models must be developed and implemented to support hosting large quantities of data and required infrastructure.
## A Roadmap to FAIR Materials Data Infrastructure
We depict in a roadmap (**Figure 2**) both individual and community-level actions to accelerate materials research via FAIR data. The community-level actions are:
* **Incentivize and recognize data literacy and reward best practices in data stewardship.** Track _"data use" citations_ and create a _data citation index_ to reward publishing of FAIR data; create open educational content for FAIR materials data methodologies.
* **Prioritize capture of materials research products beyond datasets:** Archive post-processing methods, trained models, and codes; establish links between materials data repositories and associated models/software.
* **Establish benchmark materials datasets** of high-value and high-profile to drive algorithm development. Establish an award for materials discoveries based on prior data.
* **Define high-impact community data generation tasks in subfields of materials science.** Challenge materials subfields to prioritize specific data products (e.g., microstructural image collections) for transformational change. Engage repositories and communities to catalyze these changes.
* **Promote trustworthy repositories.** Define audit and certification criteria for materials repositories to ensure long-term storage, access, and preservation of data as part of the global materials data infrastructure.
* **Collect and publicize success stories.** Collate compelling examples of data-driven approaches used to advance materials research, curated and promoted by professional organizations and funding agencies.
##
also shows four levels of individual action that can be taken by researchers, research groups, and labs to produce FAIR data and enhance scholarly output. The practices encompassed by these levels - organized in roughly increasing order of complexity - can be adopted one at a time, in various orders, and in any materials research effort.
Roadmap towards FAIR materials data, including four levels of individual action built on a foundation of community actions, all of which create value and motivate change to accelerate the widespread adoption of FAIR in the materials domain. Community networks such as MaRDA (Materials Research Data Alliance) play a vital role in enhancing synergy between individual and community driven actions, including by synchronization and promotion of efforts and defining and updating a formal roadmap with tangible goals and target dates.
Define materials data and metadata at project outset. Consider how the data could be reused by others for tasks unrelated to the originator's work, quantifying and capturing uncertainties is often critical in this step. (R) Use electronic lab notebooks to facilitate data and metadata extraction as well as documenting and publishing data management workflows []. (I) Make data available through a general repository with persistent identifiers (e.g., DOIs) for datasets (e.g., Zenodo, Figshare, Dryad). (F) Include licensing information and _how to cite_ examples in metadata, as supported by Figshare, Dryad, MDF, and nanoHUB. (R)
## Level 2: Materials-Specific Metadata and Complete Submission
Include detailed descriptive metadata, via for example metadata columns in a CSV data file. (R, F) Place data and metadata in materials-specific repository (F, A) with fields designed to handle and share materials relevant terms: e.g., OpenKIM for interatomic models, MDF for heterogenous datasets up to many terabytes in size, Foundry for structured ML-ready datasets, MaterialsMine for polymer nanocomposites and structural metamaterials, or AFLOW or OQMD for DFT calculated data on thermodynamic properties of crystallographic materials.
## Level 3: Enhanced Functionality
Ensure data and metadata are both human and machine readable; employ "tidy" data protocols []. Place data in repositories that support long-term storage and query via standard interfaces interfaces (e.g., APIs) (F, A): e.g., Materials Project, AFLOW, OQMD, MDF.
## Level 4: Community standards, Provenance, and Reusing Data
Use community standards for knowledge representation and standard file formats for data and metadata. Examples include SMILES for molecules and CIF for crystals which can be automatically processed by visualization and machine learning packages. (I) Include metadata that points to other metadata as needed to provide detailed context, ensure software and protocols have well-defined and verified requirements (inputs) and services (outputs). (I) Reuse others' data in your research, e.g., for benchmarking or in analyses to create new data. (R)
Community networks such as the US Materials Research Data Alliance (MaRDA) and materials subgroups in the Research Data Alliance (RDA), working closely with stakeholders, can support the transition to FAIR materials data. Critical actions include providing the coordination and engagement required to develop and maintain protocols, standards and best practices; development and promotion of sustainability models for materials data repositories; regular updates to the roadmap to FAIR materials data and annual scoring of the communities' progress.
New data-driven approaches to materials innovation promise transformational contributions to human health and prosperity, but are hindered by inadequate access to data on materials and material properties. The roadmap presented here highlights policies and practices that the materials community and individuals can adopt to catalyze the creation of a distributed, yet unified, worldwide materials innovation network within which data can be reused and recombined to unleash a new era of accelerated innovation and progress.
| 10.48550/arXiv.2204.02881 | Community Action on FAIR Data will Fuel a Revolution in Materials Research | LC Brinson, LM Bartolo, B Blaiszik, D Elbert, I Foster, A Strachan, PW Voorhees | 6,176 |
10.48550_arXiv.0802.2503 | ###### Abstract
We show that adsorbates on surfaces can form islands even if there no attractive interactions. Instead strong repulsion between adsorbates at short distances can lead to islands, because such islands increase the entropy of the adsorbates that are not part of the islands. We suggest that this mechanism cause the observed island formation in O/Pt, but it may be important for many other systems as well.
pacs: 68.43.Hn, 64.60.De, 68.37.-d
Lateral interactions between adsorbates are extremely important for the kinetics of surface reactions, mainly because they determine the structure of the adlayer. Island formation is invariably assigned to attractive interaction between the adsorbates, but reliable estimates for lateral interactions are hard to obtain so that there is really little know about such interactions. State-of-the-art calculations of the lateral interactions for atoms and small molecules yield reliable repulsive interactions, but attractive interactions, which are generally weaker, are harder to obtain. Moreover, other experimental results may only be consistent with repulsive interactions or attractive interactions that are too small to stabilize islands. We discuss this for oxygen atoms on Pt, which is a particularly well studied system. We show that the islands that are observed for that system may be formed without attractive interactions. On the contrary, strong repulsive interactions between adsorbates at short distance may lead to islands because this lowers the entropy.
Low Energy Electron Diffraction (LEED) patterns of O/Pt indicate island formation with a \(p(2\times 2)\) structure, which is assigned to attractive interaction between oxygen atoms at a distance of \(2a\) with \(a\) the distance between two neighboring fcc hollow adsorption sites. Recently there have been Density-Functional Theory (DFT) calculations of the lateral interactions that showed indeed attraction at that distance. The problem with DFT calculations is whether this interaction can be calculated reliably. We have also done DFT calculations of a large number of adlayer structures of O/Pt, but instead of determining the lateral interaction by straightforward multivariate linear regression we used a cross validation method as well. This is a statistical technique that has been used extensively to determine the interactions between atoms in alloys reliably, and also more recently for lateral interactions. We found that it was not possible to compute an accurate interaction at distance \(2a\). If we nevertheless tried to do that, we found error bars that were almost an order of magnitude larger than the absolute value of the interaction itself. If we varied the set of adlayer structures, from which the lateral interactions were determined, we found a large variation in the value of the interaction; most of the time it was repulsive, but sometimes attractive. So we concluded that a value of this interaction from DFT calculations is not to be trusted.
That current DFT may not be able to say anything about the interaction at the \(2a\) distance does not mean that it can not be attractive. However, the presence of absence of an attractive interaction in O/Pt has also been discussed by Zhdanov and Kasemo while discussing Temperature-Programmed Desorption (TPD) experiments of this system. In these spectra there is no indication that there is an attractive lateral interaction. Their conclusions were that if there is an attractive interaction then it so small that it can not lead to island formation at the temperatures of the LEED experiment. We have refined the kinetic Monte Carlo simulation of the TPD spectra that were used by Zhdanov and Kasemo, and determined the lateral interactions by fitting the simulated TPD spectra to the experimental ones. No attractive interactions were obtained. Moreover, the values we obtained in this way agreed very well with those obtained from DFT calculations when cross validation was used.
These observations lead to the question if it is possible to have island formation without attractive interactions. We will show in this paper that this is indeed possible, because of entropic reasons. Remarkably, we will show that small islands lead to a higher entropy only when there are repulsive interaction, not at the distances between the adsorbates as observed in the island, but at shorter distances.
To study the effect of entropy on the island formation in O/Pt we have modeled the system as a hexagonal grid representing the fcc hollow sites, which are the preferred sites of oxygen atoms. The interactions between the oxygen atoms are modeled using hard-sphere interactions. These are such that two nearest-neighbor sites can not both be occupied at the same time, and neither can two next-nearest neighbor sites. The shortest distance between two oxygen atoms is then \(2a\). Apart from these two hard-sphere interactions there no other interactions in our model. This model has been studied by Koper and Lukkien to model the butterfly in voltammetry. It has a order-disorder phase transition at around \(0.18\,\mathrm{ML}\). At higher coverages the adlayer has the \(p(2\times 2)\) structure also observed in LEED. We are however interested at lower coverages where the adlayer has no long-range order.
It can clearly be seen that the adlayer has some structure. At coverages just below the order-disorder transition the peaks are sharp. The pattern is characteristic for a \(p(2\times 2)\) structure with a nearest distance between the adsorbates of \(2a\). At very low coverages the peaks become diffuse, but are still clearly visible.
To understand how the LEED pattern arises it is convenient to look at another one-dimensional model that has essential the same characteristics as the hard-sphere model for O/Pt. In this model we have \(S\) sites numbered from \(0\) to \(S-1\) with periodic boundary conditions. There are \(A\) adsorbates, and one adsorbate is always adsorbed on site 0. Two neighboring sites can not both be occupied by an adsorbate at the same time. The probability that a site \(n\) is occupied, \(P(n)\), can be determined from the fundamental hypothesis of statistical mechanics that all acceptable configurations are equally likely.
Simulated LEED patterns for the hard-sphere model of O/Pt at coverages 0.16 (a), 0.12 (b), 0.08 (c), and 0.04 (d). The relative intensity of the peaks are 100, 11, 3.7, and 1.3, respectively.
The probability \(P\) is equal to the ratio of the number of configurations with site 2 occupied and the total number of configuration.
\[P=\frac{N(A-2,S-5)}{N(A-1,S-3)} \tag{1}\]
We are assuming that the adsorbates are indistinguishable, so that we have the recursion relation
\[N(n,L)=N(n,L-1)+N(n-1,L-2) \tag{2}\]
(The expressions change somewhat when we assume distinguishable adsorbates, but the probabilities \(P(n)\) remain the same.)
For \(A=2\) we have \(N(0,L)=1\) and \(N(1,L)=L\) from the recursion relation so that \(P=1/(S-3)\). This value is equal to the average occupation of sites 2 to \(S-2\). This is to be expected; the second adsorbate will have equal probability to occupy any of the sites 2 to \(S-2\). For \(A=3\) we have \(N(2,L)=(L-1)(L-2)/2\) if \(L\geq 3\) so that \(P=2/(S-4)\). We see that in this case the probability of occupation is more than double the average occupation, which is the first indication that there is a tendency for clustering.
If site 3 is occupied, then the other adsorbates must be somewhere at sites 5 to \(S-2\).
\[P=\frac{N(A-2,S-6)}{N(A-1,S-3)}. \tag{3}\]
For \(A=2\) we get \(P=P=1/(S-3)\). For \(A=3\) we get \(P=2(S-6)/((S-4)(S-5))\). Again we have a higher probability than the average occupation if \(S\geq 7\), but \(P=0\) if \(S=6\). In any case we have \(P>P\). We see that there is a tendency for the other adsorbates to be as close as possible to the adsorbate that is always at site 0. The tendency lessens if one goes farther from site 0. When \(A=3\) and \(S=6\) we have \(P=0\), because there is a maximum number of adsorbates with alternating sites occupied and vacant.
If the number of adsorbates increases and when we look at the occupation of sites for farther from site 0, then the analytical expressions for the probabilities of occupation become quite complex.
\[P(n) = \frac{1}{N(A-1,S-3)}\] \[\times \sum_{m=0}^{A-1}N(m,n-3)N(n-m-1,S-m-4).\]It is hard to see from this expression how \(P(n)\) varies with \(n\). It seems therefore easier for moderate values of \(S\) to simply do a simulation in which all configurations are generated, and from that determine the occupation of all sites. shows the result for \(S=42\). The tendency, which we mentioned already, of the adsorbates to cluster is apparent from this figure even if there are only a few adsorbates. We also see that we get "islands" with adsorbates separated by a distance that is twice the distance between neighboring sites. The origin of this clustering is the fact that when two subsequent adsorbates are closer together, then the other adsorbates have more sites over which to distribute, and hence a higher entropy.
The same probabilities can be computed for the hard-sphere model of O/Pt (see Fig. 3). This can either be done with a small grid and generating all possible configurations, or with a larger grid and doing a Monte Carlo simulation. We did both using a \(10\times 10\) grid for generating all configurations, and a \(64\times 64\) grid for Monte Carlo simulations. The results were the same.
Occupation of all sites for the one-dimensional model with 42 sites. The number of adsorbates ranges from 1 to 21 being small in front and large at the back. Site 0 on the left is always occupied.
For coverages just below the phase transition there is a clear indication already that a \(p(2\times 2)\) structure is being formed (see Fig. 3a). For lower coverages this structure becomes harder to see in the figure. (The nearest and next-nearest sites are not occupied, of course.) There is however a higher probability than average for the next-next-nearest neighbor site; i.e., the nearest site to be occupied in a \(p(2\times 2)\) structure. This is clearly visible at a coverage of 0.12, also visible at 0.08, but hard to see at 0.04, although the occupation of the next-next-nearest neighbor site is still about 11% higher than average even at this low coverage.
The islands that are formed are not static features of the adlayer. They are also quite small at low coverages, as can be seen from Snapshots of the Monte Carlo simulations with a \(64\times 64\) grid show islands of about 10 to 15 oxygen atoms at a coverage of 0.12, and 5 to 8 atoms at 0.08. At a coverage of 0.04 only rarely more than two atoms are found together.
A remarkable aspect of the mechanism of the island formation here is the fact that there must be repulsive interactions. If we allow two next-nearest neighbor sites to be occupied simultaneously, we still get island formation, but the structure observed in the LEED is then \(p(\sqrt{3}\times\sqrt{3})\). If we also allow nearest-neighbor sites to be occupied, then no island formation takes place anymore. The reason is that there is no entropy gain anymore by moving two adsorbates closer together to increase the number of configurations for the other adsorbates. Without the repulsion the number of available sites for those other adsorbates is always equal to the number of vacant sites, and independent of the way the two adsorbates are positioned.
A better model for O/Pt than the hard-sphere model is one with realistic values for the lateral interactions. Although DFT calculations with cross validation indicate that the next-next-nearest neighbor interaction can not be determined, there are other interactions that can be obtained. We have shown that DFT results can be reproduced with an error of only 2.6 kJ/mol with an adsorption energy of an isolated oxygen atom of \(-396.3\) kJ/mol (with respect to a bare substrate and an oxygen atom in the gas phase), a nearest-neighbor interaction of 19.9 kJ/mol (positive values indicate repulsion), an next-nearest neighbor interaction of 5.5 kJ/mol, and a three-particle interaction of 6.1 kJ/mol that occurs if three atoms are in a row at nearest-neighbor distances. The last interaction can be ignored for the coverages of interest here, because the strong repulsion between oxygen atoms at nearest neighbor positions prevents even two atoms getting at these positions, let alone three. The next-nearest neighbor interaction corresponds to a thermal energy of about 660 K, so there is an appreciable probability to find two oxygen atoms at next-nearest neighbor positions.
To summarize, island formation in adlayer as observed in LEED does not need to be caused by attractive interactions between adsorbates. Island formation can also be favored for entropic reasons, because when some adsorbates get close together then there is more space for other adsorbates. This extra space means that these other adsorbates can form more different configurations and hence have a higher entropy. A remarkable requisite for this mechanism to work is that there must be a strong repulsion between the adsorbates at short distances. The distance between the adsorbates in the islands is then larger than this distance at which there is repulsion. We have shown results of this mechanism for a hard-sphere interaction model and a model with realistic lateral interactions for O/Pt, but there seems to be no reason why the mechanism should not work in other adlayers too. Calculations of lateral interactions seem to suggest that a strong repulsion between adsorbates in nearest-neighbor positions is common. This means that island formation at low coverages should be common as well. Because the mechanism here is purely entropic and there is no energy, the island formation is temperature independent. If there are also energetic contributions, then the mechanism should work especially at higher temperatures, where it may dominate interactions that favor other adlayer structures. (Strictly speaking we have not proven that there is no attractive interaction in O/Pt, but such an interaction is, as shown by Zhdanov and Kasemo, too weak to be relevant.)
Ordering effects due to entropy date back at least to Onsager's hard-rod model for liquid crystals. The depleted volume effect in that model and the spatial effects due to the repulsive interactions here are similar. There is an important difference however. There is a clear distinction between degrees of freedom in the hard-rod model. In that model the orientational entropy decreases when a nematic phase is formed but the positional entropy increases. A similar partitioning of degrees of freedom is found in more recent models. Here this is not the case, and the model is simpler. Oscillations found in the density of a gas near the wall of a microchannel could be explained with a model with similarities to the hard-sphere model here, as could the variation in the distribution of molecules in the channels of a one-dimensional zeolites with variable pore diameter. Once again however, the model here is simpler. Moreover it is the first model on entropic ordering in adlayers.
| 10.48550/arXiv.0802.2503 | Island formation without attractive interactions | A. P. J. Jansen | 4,489 |
10.48550_arXiv.1909.06820 | ## 1 Introduction
Low tritium (T) retention is one of the main advantages of tungsten(W) as the plasma-facing material (PFM) for future magnetic fusion devices, such as the International Thermonuclear Experimental Reactor (ITER) and the China FusionEngineering Test Reactor (CFETR). However, during the long-term operation of fusion devices, hydrogen isotpes can easily diffuse deep into W bulk even from the redeposited layers to the W substrate. The tritium retention may increase significantly due to the interaction of thermal diffusion and plasma exposure. Retention and removal of tritium is still one of the most important issues in fusion devices and attract much attention.
The retention of hydrogen isotopes in W is mainly affected by trap sites, such as dislocations, grain boundaries, vacancies and microvoids in the matrix material. It is necessary to evaluate the release behaviors of hydrogen isotopes trapped in different sites. There have been extensive studies on the thermal desorption of deuterium(D) in W by plasma exposure. The 'ion-induced' vacancies are usually associated with D agglomeration in molecules and bubbles near the implanted surface. And the voids are often formed from vacancies clustering together to form small microvoids. A single vacancy can trap multi-D atoms, so can voids and bubbles. The detrapping energies of D in vacancies are largely scattered, reported to be 1.0\(\sim\)2.2 eV. For example, the lower temperature desorption peaks (500-650 K) likely represent single vacancies with a trap energy of 1.0-1.6 eV, while the high-temperature desorption peaks (\(\sim\)900 K) are likely associated with the formation of microvoids or vacancy clusters in W with a trap energy of 1.7\(\sim\)2.2 eV. Note specially, most of the detrapping energies mentioned above were obtained by fitting numerical calculations based on diffusion-trapping codes to experimental thermal desorption spectra(TDS). A large uncertainty in determination of characteristics of defects in such approach is given by dependence of TDS spectrum simulation on many input parameters, such as characteristic frequencies for trapping or detrapping, depth distribution of traps and trapped D. The direct experimental values of detrapping energies are still scarce. Zibrov et al. made these efforts. The D detrapping energies for single vacancies and vacancy clusters was determined experimentaly to be 1.56 eV and 2.10 eV, respectively.
On the other hand, as expected, hydrogen isotopes will also enter into W by thermal diffusion during the process of continuous plasma bombardment. Compared with plasma exposure W, gas charging W can avoid the influence of surface damage layer, and is more conducive to the understanding of the intrinsic mechanism of deuterium retention in W by thermal diffusion. Moreover, due to the deeper diffusion depth in W, this part of hydrogen isotpes may be hard to remove.
This work is devoted to the study on the D retention behavior of W samples by two treatment processes, respectively: D plasma exposure in which W was exposed to D plasma with 35 eV/D\({}^{+}\) at 393 K to the fluence of 3.8\(\times\)10\({}^{24}\) D/m\({}^{2}\) (the sample was denoted as plasma exposure W); D\({}_{2}\) gas charging in which W was exposed to D\({}_{2}\) gas of 500 kPa at 773 K for 4 hours (the sample denoted as charging W). The amount, desorption temperatures and detrapping energies of deuterium in W exposure to D plasma and D\({}_{2}\) gas were determined experimentally.
## 2 Experimental
Rolled W (purity of \(>\)99.95%) purchased from Advanced Technology & Materials Co., Ltd. was used in this study. The samples with dimensions of 10\(\times\) 10\(\times\)1 mm\({}^{3}\) were all mechanically polished to obtain a mirror-like surface, and then ultrasonically cleaned in ethanol and acetone. Thereafter, high-temperature annealing (1273 K/1 h) was performed in a vacuum of better than 10\({}^{\text{-}5}\) Pa for outgassing. The sample has a typical microstructure with the average grain size of about 4 \(\upmu\)m (details are given in).
The schematic diagram of the gas charging system was shown in D\({}_{2}\) was stored in a ZrCo storage bed, which was heated to drive D\({}_{2}\) into the standard vessel. The pipe system was cleaned three times by 10 kPa D\({}_{2}\) before each experiment. The samples were exposed to D\({}_{2}\) with a pressure of 500 kPa at 773 K for 4 h to obtain D saturated W samples according to the diffusive transport parameters of D\({}_{2}\) through W samples.
After exposure, all samples were cooled rapidly to room temperature in deuterium atmosphere. Note specially, the effect of deuterium desorption during cooling is not considered here, since all the samples have this same process. Subsequently TDS experiments were followed. The 316 L stainless steel pipe which ions passed through was baked for 24 h to remove residual gas in the materials before TDS experiment. The samples were heated linearly up to 1273 K in the quartz tube, with a vacuum of 10-5 Pa. In addition, in order to calculate the activation energy of D\({}_{2}\) desorption peak, the heating rate was set to be 5 K/min, 10 K/min, 15 K/min and 20 K/min respectively for the D saturated W samples. D\({}_{2}\) and HD signals were tracked by quadrupole mass spectrometer (QMS). And the D\({}_{2}\)/H\({}_{2}\) signal was calibrated using two standard leaks after experiments. HD signal could be theoretically calibrated, which was the average of the sum of D\({}_{2}\) and H\({}_{2}\).
Deuterium plasma exposure was performed for W in the linear experimental plasma system (LEPS) at Lanzhou Institute of Chemical Physics,Chinese Academy of Sciences. During the exposure, ion energy was set at 35 eV/D\({}^{+}\), and the exposure temperature was kept at around 393 K. The ion flux which consisted dominantly of D\({}_{3}\)\({}^{+}\) ions was around 2.1\(\times\)10\({}^{21}\) D/m\({}^{2}\) s. Thus, the fluence was 3.8\(\times\)10\({}^{24}\) D/m\({}^{2}\) for exposure periods of 0.5 h. TDS was also used to get data on D trapping and retention in plasma exposure W.
Schematic diagram of the gas charging system. A: scroll pump, B: pressure gauge (\(\sim\)1 bar), C: standard vessel, D: resistive vacuum gauge, E: film vacuum gauge, F: heating apparatus, G: pressure gauge (\(\sim\)10 bar), H: Zr-Co alloy storage bed, 1–11: Swagelok valves.
## 3 Results and discussion
### 3.1 Desorption Characteristics of deuterium
Fig.2 shows the desorption rate of D\({}_{2}\) for plasma exposure W and gas charging W. It is clear that the D\({}_{2}\) desorption peak of gas charging W is about 952 K, much higher than 691 K of plasma exposure W. Moreover, the temperature range of significant D\({}_{2}\) release for gas charging W is extending from 600 to 1173 K; while for the plasma exposure W, this range is 550 to 800 K. Deuterium in gas charging W is more difficult to release than that in plasma exposure W. It suggests that there are different D trap sites in these two kinds of samples.
As the literatures states, the peak at around 690 K in plasma exposure W, may correspond to the D release from single vacancies, while the peak at around950 K in gas charging W may correspond to the D release from vacancy clustersor microvoids. In addition, the spectra for 952 K have a shoulder at the temperature of 700 K that is presumably corresponding to the D release from single vacancies. We will discuss this further in the next session.
It is worth mentioning that although the deuterium injection layer is very shallow(about 4 nanometers from SRIM simulation). However, the amount of deuterium retention by plasma exposure is 1.90\(\times 10^{22}\) D/m\({}^{2}\), nearly one order of magnitude higher than that by gas charging, as shown in Fig.3. The origin of such diferences is not completely clear. It may be due to that the deuterium entering into W by plasma exposure is supersaturated. For example, Gao et al. even found the existence of a 10 nm thick D-supersaturated surface layer (DSSL) with an unexpectedly high D concentration of \(\sim\)10 at.% after irradiation with ion energy of 215 eV at 300 K.
The comparison of deuterium retention for plasma exposure W and gas charging W indicates that the non-equilibrium process of deuterium entering into W by plasma exposure may be the main component of deuterium retention in tungsten, and near-surface vacancies are responsible for this retention.
The measured deuterium content in gas charging W is the order of 10\({}^{21}\) D/m\({}^{2}\)
Comparison of total D retention for plasma exposure W and gas charging W
dislocations, grain boundaries and vacancies) would play a dominant role in the actual amount of deuterium retention in W H loaded over solubility limits should stay somewhere in the bulk accompanying some lattice distortion and also making such as H-induced superabundant The peak of 952 K in gas charging W is also in accordance with the characteristics of the D release from vacancy clusters
## 3.2 Detrapping energies of deuterium
Fig.4 shows the TDS spectra of D\({}_{2}\) in plasma exposure W and gas charging W at different heating rates. It can be seen that the desorption peaks of deuterium in W moves to higher temperatures with the increase of heating rate. For example, when the heating rate is 5 K/min, 10 K/min, 15 K/min and 20 K/min, the desorption peaks of gas-charging W locate at 909, 925, 943 and 952 K, respectively; while the desorption peaks of plasma exposure W locate at 646, 670, 683 and 691 K, respectively. The desorption peak temperatures of deuterium in plasma exposure W were generally lower than that in gas charging W at the same heating rate. This further confirms that deuterium in gas charging W is more difficult to be removed.
The detrapping energy means the energy barrier for D escape from a trap, which is usually defined as the sum of the bingding energy and the activation energy for D Deuterium trapped by various types of defects in tungsten requires different detrapping energies. The literature states that desorption peaks near 900 K correspond to trap energies of \(\sim\)2.1 These traps are typically voids vacancy The peaks at around 600 K correspond to the D release from single vacancies with trap energies ranging from 1.1 to 1.4 eV Thus, it is interesting to determine the detrapping energies of D in gas charging W and plasma exposure W.
The detrapping energies of trapped deuterium can be determined experimentally from the measured peak temperatures at different heating rates, as described by the following equation:
\[\ln(\frac{\beta}{T_{p}^{2}})=\ln(A\frac{R}{E_{dt}})-\frac{E_{dt}}{R}\frac{1}{T_{p}} \tag{1}\]
Since \(\beta\) and \(T_{p}\) are known, the detrapping energy \(E_{dt}\) of deuterium evolution from a trapping site can be calculated from the slope of a ln(\(\beta\)/ \(T_{p}\)\({}^{2}\)) vs (1/ \(T_{p}\)) plot. The results are shown in The fitting results have good linear relationships and satisfy the Arrehenius relationship. The detrapping energies of deuterium in gas charging and plasma exposure samples can be calculated as 2.17 eV and 1.04 eV, respectively.
##
The plot of [-ln(\(\beta\)/ \(T_{p}\)\({}^{2}\))] versus 1/ \(T_{p}\) for the main peaks of plasma exposure W and and gas charging W. The best linear fit and the determined value of Edt is also shown.
The detrapping energy of deuterium in gas charging samples is 2.17 eV, which is close to the value of 2.10 eV that was interpreted by several others as trapping of deuterium in vacancy clusters or voids. The measured deuterium content in gas charging samples is much higher than deuterium solubility in tungsten. It also indicates that vacancy clusters or voids may play a dominant role in the actual amount of deuterium retention in tungsten. Note specially, these high-energy traps in W are usually formed by irradiation with neutron, high energy ions (such as 10 keV/Dions), or by the plasma exposure at high surface temperatures ( 680-950 K ) of W, while they are seldom reported to be found in the gas charging W.
The positron anihilation spectroscopy (PAS) measurements confirmed that monovacancy is usually found to be dominant in the undamaged W. Considering that W was exposed to D\({}_{2}\) with a pressure of 500 kPa at 773 K for 4 h in this work, this results further comfirmed that H isotopes could enhance vacancy formation. Vacancy clustering or voids may also easily take place due to the temperature higher than 550 K. Thus, the high-temperature desorption peak (950 K) in the gas charging sample is likely associated with the formation of microvoids or vacancy clusters in the W with a trap energy of \(\sim\)2.17 eV.
The detrapping energy of the sample irradiated by plasma is 1.04 eV, which is in agreement with the trapping energy of deuterium by a single vacancy as reported by Poon (\(\sim\)1.07 eV). It is worth noting that although the amount of deuterium retention by gas charging is much lower than that by plasma exposure, higher trap energies and physically deeper trap location could all increase the temperature of the release peaks of D in the gas charging sample.
## 4 Conclusions
We carried out a study on the amount, desorption temperatures and detrapping energies of deuterium in tungsten(W) exposure to D plasma and D\({}_{2}\) gas. The results showed that the total D retention in plasma exposure W is \(\sim\)10\({}^{22}\) D/m\({}^{2}\), one order of magnitude higher than that of gas charging W. Whereas, the D\({}_{2}\) desorption peak of gas charging W is around 950 K, higher than that of plasma exposure W. The detrapping energies of deuterium in gas charging W and plasma exposure W were found to be 2.17 eV and 1.04 eV, respectively. Higher trap energy and physically deeper trap location could all increase the temperature of the release peaks of D in the gas charging sample. These results also suggests that deuterium in gas charging W was likely trapped by vacancy clusters while deuterium in plasma exposure W was trapped by a single vacancy.
| 10.48550/arXiv.1909.06820 | Comparison of deuterium retention in tungsten exposed to deuterium plasma and gas | Xiaoqiu Ye, Wei Wang, Yifang Wang, Xiaohong Chen, Jiliang Wu, Yao Xiao, Xuefeng Wang, Jun Yan, Wenzhen Yu, Changan Chen | 2,505 |
10.48550_arXiv.1512.08890 | ###### Abstract
We study the problem of magnetization and heat currents and their associated thermodynamic forces in a magnetic system by focusing on the magnetization transport in ferromagnetic insulators like YIG. The resulting theory is applied to the longitudinal spin Seebeck and spin Peltier effects. By focusing on the specific geometry with one YIG layer and one Pt layer, we obtain the optimal conditions for generating large magnetization currents into Pt or large temperature effects in YIG. The theoretical predictions are compared with experiments from the literature permitting to derive the values of the thermomagnetic coefficients of YIG: the magnetization diffusion length \(l_{M}\sim 0.4\,\mu\)m and the absolute thermomagnetic power coefficient \(\epsilon_{M}\sim 10^{-2}\) TK\({}^{-1}\).
pacs: 75.76.+j, 85.75.-d, 05.70.Ln
## I Introduction
The recent discovery of the longitudinal spin Seebeck effect in ferromagnetic insulators has raised a renewed interest in the non equilibrium thermodynamics of spin or magnetization currents. Experiments have shown that a temperature gradient applied across an electrically insulating magnetic material is able to inject a spin current into an adjacent metal, where the spin polarization is revealed by means of the inverse spin Hall effect (ISHE). Typical experiments have been performed by using ferrimagnets, like the yttrium iron garnet (Y\({}_{3}\)Fe\({}_{5}\)O\({}_{12}\), YIG) as insulating magnetic material and Pt or other noble metals, as conductors. In analogy to thermoelectrics, the reciprocal of the spin Seebeck effect has been called spin Peltier effect. This reciprocal effect has been recently observed by using the spin Hall effect of Pt as spin current injector and observing the thermal effects on YIG. All these experiments show that the magnetization current can propagate along different media using different type of carriers. While spin currents in metals are associated to the unbalance in the spin polarization of conduction electrons, in magnetic insulators the magnetization transport is due to spin waves or magnons. Spin Seebeck and spin Peltier experiments reveal that the magnetization current carried by magnons in the magnetic insulator can be transformed into a spin current carried by electrons and viceversa. The mechanism of this conversion is seen as the interfacial s-d coupling between the localized magnetic moment of the ferromagnet (which is often due to d shell electrons) and the conduction electrons of the metal (which are often s shell electrons).
The thermodynamics of thermo-magneto-electric effects, i.e. spin caloritronics, has been already developed for metals by adding the spin degree of freedom to the thermo-electricity theory. However, spin caloritronics cannot be directly applied to electrical insulating magnetic materials like YIG. Therefore it is necessary to develop a more general theory which could be applied to both conductors and insulators. The formulations of the problem present in the literature often focus on the microscopic origin without paying much attention to the formal thermodynamic theory that is expected as a result. Refs. describe the non equilibrium magnon distribution through an effective magnon temperature different from the lattice temperature. However from an experimental point of view in Ref. it was observed a close correspondence between the spatial dependencies of the exchange magnon and phonon temperatures. The Boltzmann approach for magnon transport was used in Ref., combined by a YIG/Pt interface coupling. Within these approaches the spin accumulation and the magnon accumulation take the role of an effective force able to drive the magnetization current. The use of different quantities between the two sides of a junction requires therefore the introduction of a spin convertance to account for the magnon current induced by spin accumulation and the spin current created by magnon accumulation.
The aim of the present paper is to define the macroscopic non-equilibrium thermodynamics picture for the problems related to magnetization currents that could be used independently of the specific magnetic moment carrier. To this aim we start from the results of the thermodynamic theory of Johnson and Silsbee. The main difference with respect to the classical theories of the thermoelectric effects is that the magnetization current density \(j_{M}\) is not continuous. The magnetic moment can both flow through a magnetization current but also can be locally absorbed and generated by sinks and sources. Here, by limiting the analysis to the scalar case, we state the simplest possible continuity equation for the magnetization. As a result we find that the potential for the magnetization current is the difference \(H^{*}=H-H_{eq}\) between the magnetic field \(H\) and the equilibrium field \(H_{eq}\). The gradient of the potential \(\nabla H^{*}\) is the thermodynamic force to be associated to the magnetization current.
With this definition it is then possible to state the constitutive equations for the joint magnetization and heat transport and to identify the absolute thermomagnetic power coefficient \(\epsilon_{M}\) relating the gradient of the potential of the magnetization current \(\mu_{0}\nabla H^{*}\) with the temperature gradient \(\nabla T\), in analogy with thermoelectricity.
The same coefficient also determines the spin Peltier heat current \(\epsilon_{M}Tj_{M}\) when the system is subjected to a magnetization current.
In the present work we apply the previous arguments to describe the generation of a magnetization current by the spin Seebeck effect and the heat transport caused by the spin Peltier effect. To this end we have to complement the constitutive equations for the thermo-magnetic active material (YIG) with the equations for the spin Hall active layer (Pt). Once the equations for the two materials are written by using the same thermodynamic formalism, one can apply the theory to solve specific problems of magnetization current traversing different layers. The diffusion length for the magnetization current \(l_{M}=(\mu_{0}\sigma_{M}\tau_{M})^{1/2}\) is related to intrinsic properties of each material: the magnetization conductivity \(\sigma_{M}\) and the time constant \(\tau_{M}\), describing how fast the system is able to absorb the magnetic moment in excess. We are also able to show that the passage of the magnetization current from one layer to the other is governed by the ratio between \(l_{M}/\tau_{M}\) of the two layers.
By focusing on the specific geometry with one YIG layer and one Pt layer, we obtain the optimal conditions for generating large magnetization currents into Pt in the case of the spin Seebeck effect and for generating large heat current in YIG in the case of spin Peltier effects. In both cases we find that efficient injection is obtained when the thickness of the injecting layer is larger than the critical thickness \(l_{M}\) as recently experiments confirm. We finally determine the values of the thermomagnetic coefficients of YIG by comparing the theory to recent experiments.
The paper is organized as follows. In Section II we first discuss the thermodynamic properties of an out-of-equilibrium but spatially uniform magnetic system and on that basis we introduce, for non spatially uniform system, the currents and the thermodynamic forces in analogy with the non equilibrium thermodynamics of thermoelectric effects. In Section III we set the constitutive equations for the magnetization and heat transport in both an insulating ferrimagnet and a metal with the spin Hall effect. Section IV is devoted to the solutions of the magnetization current problem. In Section V we focus on the specific longitudinal spin Seebeck geometry and on the spin Peltier effect. Finally some conclusive remarks are drawn in Section VI.
## II Thermodynamics of magnetization currents
### Thermodynamics of uniform magnetic systems
We consider a magnetic system that can be described by a scalar magnetization \(M\). Suitable systems can be ferromagnetic or ferrimagnetic materials where an easy axis is present, due for example to an anisotropic crystal structure, along which all the vector quantities are lying. We take spatially uniform quantities and all extensive quantities as volume densities.
\[H_{eq}=\frac{1}{\mu_{0}}\frac{\partial u}{\partial M}\bigg{|}_{s} \tag{1}\]
In equilibrium the magnetic field \(H\) is equal to the state equation \(H=H_{eq}(M,s)\). When \(H\) is different from its equilibrium value \(H_{eq}\) the system state will try to reach the equilibrium by the action of dissipative processes. In a generic out-of-equilibrium situation the variation of internal energy must take into account that dissipative processes correspond to an entropy production.
\[du=Tds+\mu_{0}HdM-T\sigma_{s}dt \tag{2}\]
When approaching equilibrium, the magnetization \(M\) will change until the equilibrium condition \(H=H_{eq}(M)\) will be reached. The typical situation is sketched in Fig.1 showing two processes connecting the equilibrium states and. The equilibrium path (solid line) corresponds to the slow variation of field \(H\) from \(H_{1}\) to \(H_{2}\) through the equilibrium state equation \(H=H_{eq}(M)\). The out-of-equilibrium path (dashed line) passes through the out-of-equilibrium state (\(1^{\prime}\)) and corresponds to the sudden variation of the field from \(H_{1}\) to \(H_{2}\) and to the subsequent time relaxation. As the initial and final states are always equilibrium states, the final internal energy variation must be the same for any process. This is obtained by assuming that the part of the work going into the internal energy is always the equilibrium one. Then, by inserting \(du=\mu_{0}H_{eq}dM\) (from Eq.) into Eq.
\[\sigma_{s}=\mu_{0}\frac{H-H_{eq}}{T}\frac{dM}{dt}. \tag{3}\]
As expected, the entropy production rate is the product of a generalized force, or affinity, represented by the term \(\mu_{0}(H-H_{eq})/T\), times a generalized flux, or velocity, represented by \(dM/dt\). If the distance from equilibrium is not too large one can consider the linear system approximation and assume the velocity to be proportional to the affinity.
\[\frac{dM}{dt}=\frac{H-H_{eq}}{\tau_{M}}, \tag{4}\]where the temperature \(T\) and \(\mu_{0}\) appearing in the generalized force, have been incorporated into the definition of the time constant. Eq. provides a kinetic equation for the magnetization describing the time relaxation from a generic out-of-equilibrium state by showing that the velocity \(dM/dt\) depends on the distance from equilibrium \(H-H_{eq}\) (see Fig.1).
The interesting physics behind Eq. is that it also expresses the non conservation of the magnetic moment with the presence of sources and sinks, although the total angular momentum for an isolated system is conserved. As a matter of fact in the solid state there is a huge reservoir of angular momentum available (electrons, nuclei, etc) and only a very small part of it is associated to the magnetic moment. As a result, the magnetization can be easily varied by exchanging angular momentum with the reservoir constituted by non magnetic degrees of freedoms. With this in mind, the physical meaning of Eq. is to express how fast the angular momentum from the magnetization subsystem can be exchanged with the reservoir.
Finally, as it happens in many problems involving a non conserved magnetization, also the internal energy is a non conserved quantity. To avoid the problem, we pass to the enthalpy potential \(u_{e}=u-\mu_{0}HM\) which contains the magnetic field \(H\) and the entropy \(s\) as independent variables. Dealing with out-of-equilibrium processes, the potential \(u_{e}\) is also a non equilibrium one which depends on the magnetization \(M\) as an internal variable. From Eq., the enthalpy variation is
\[du_{e}=Tds-\mu_{0}MdH-\mu_{0}(H-H_{eq})dM \tag{5}\]
The expression for the variation of the enthalpy potential, together with the kinetic equation, constitutes the out-of-equilibrium thermodynamics of the system and can be employed to build up the thermodynamics of fluxes and forces.
### Thermodynamics of fluxes and forces
We now pass from the out-of-equilibrium thermodynamics of a spatially uniform magnetic system to the problem of having a non uniform situation involving currents of the extensive variables, entropy and magnetization, and the associated thermodynamic forces. Both the extensive and intensive variable are now allowed to vary as a function of space coordinates \(\mathbf{r}\). In the case of extensive variables the volume densities are intended as moving averages over a small volume \(\Delta V\) around the point \(\mathbf{r}\). As the magnetization is a non conserved quantity, we need to explicitly express the fact that any magnetization change \(dM\) is in part drawn from the reservoir of angular momentum, which is external to the thermodynamic system, and in part exchanged between the surrounding regions of the thermodynamic system itself, giving rise to a current of magnetic moment \(\mathbf{j}_{M}\). The sources and sinks of the magnetic moment are exactly those described in the previous Section by Eq., then we can immediately write a continuity equation for the magnetization by extending Eq., obtaining
\[\frac{\partial M}{\partial t}+\nabla\cdot\mathbf{j}_{M}=\frac{H-H_{eq}(M)}{\tau_{M }}. \tag{6}\]
Next, as it is usually done in the non equilibrium theory of fluxes and forces, we use Eq.
\[ds=\frac{1}{T}du_{e}+\frac{\mu_{0}M}{T}dH+\frac{\mu_{0}(H-H_{eq})}{T}dM. \tag{7}\]
As we aim to define the entropy current as a function of the other currents, we have to look at the previous equation in search for the variations of the extensive variables. Eq. contains the variation of the enthalpy \(du_{e}\) and the magnetization \(dM\) which both have associated currents, while the variation of the magnetic field \(dH\) does not corresponds to any current and has not to be taken into account in the definition of the entropy current.
\[\mathbf{j}_{s}=\frac{1}{T}\mathbf{j}_{u_{e}}+\frac{\mu_{0}(H-H_{eq})}{T}\mathbf{j}_{M} \tag{8}\]
where \(\mathbf{j}_{s}\) is the entropy current and \(\mathbf{j}_{u_{e}}\) is the enthalpy current which obeys the following continuity equation
\[\frac{\partial u_{e}}{\partial t}+\nabla\cdot\mathbf{j}_{u_{e}}=-\mu_{0}M\frac{ \partial H}{\partial t}, \tag{9}\]
Equilibrium path (solid line) and out-of-equilibrium path (dashed line) connecting the equilibrium states and in the \(H\) versus \(M\) diagram of a magnetic material. \(H_{eq}(M)\) is the equilibrium state equation at constant entropy. (\(1^{\prime}\)) is an out-of-equilibrium state obtained by the sudden change of the field from \(H_{1}\) to \(H_{2}\). In the relaxation path from (\(1^{\prime}\)) to the work is \(\mu_{0}HdM\), the internal energy change is \(du=\mu_{0}H_{eq}dM\) and the entropy production is \(T\sigma_{s}dt=\mu_{0}(H-H_{eq})dM\). The relaxation equation is Eq.
The continuity equation for the magnetization is Eq.
\[\frac{\partial s}{\partial t}+\nabla\cdot\mathbf{j}_{s}=\sigma_{s}. \tag{10}\]
As it is done in the classical treatment, one expresses the entropy production rate \(\sigma_{s}\) in terms of a sum of products of each current times its thermodynamic force. By using Eqs.- into Eq.
\[\sigma_{s}=\nabla\left(\frac{1}{T}\right)\cdot\mathbf{j}_{q}+\mathcal{F}_{M}\cdot \mathbf{j}_{M}+\frac{1}{\tau_{M}}\frac{\mu_{0}\left(H-H_{eq}\right)^{2}}{T} \tag{11}\]
where we have defined the thermodynamic force associated to the magnetization current
\[\mathcal{F}_{M}=\frac{1}{T}\mu_{0}\nabla\left(H-H_{eq}\right). \tag{12}\]
In Eq. we see the products of the heat current \(\mathbf{j}_{q}\) times its force \(\nabla(1/T)\), of the magnetization current \(\mathbf{j}_{M}\) times its force \(\mathcal{F}_{M}\) and the last term which is exactly the entropy production associated with the out-of-equilibrium homogeneous processes and not to the fluxes. The last term can be also recognized as entropy production of Eq. where the affinity is \(\mu_{0}(H-H_{eq})/T\) and the magnetization change \(dM/dt\) is \((H-H_{eq})/\tau_{M}\) as given by Eq..
As a main result we have found that the gradient of the distance from equilibrium Eq. is the generalized force associated with the magnetization current \(\mathbf{j}_{M}\). For simplicity we define \(H^{*}=H-H_{eq}\) to specify the distance from equilibrium and we observe that the driving force of the magnetization current appears as soon as the system is brought out-of-equilibrium. In that case the system may find more effective to draw magnetization from the surroundings rather than from the local spin reservoir. The strength of this effect is given by a further parameter, the magnetization conductivity \(\sigma_{M}\), which establishes the relationship between the magnetization current \(\mathbf{j}_{M}\) and the gradient of \(H^{*}\)
\[\mathbf{j}_{M}=\sigma_{M}\mu_{0}\nabla H^{*}. \tag{13}\]
\(H^{*}\) can be different from zero in stationary situation every time the material experiences the accumulation of magnetization (i.e. spin accumulation in the case of metallic conductors). We have to notice that even if \(H^{*}\) has the units of a magnetic field, it is not a magnetic field in the sense of the Maxwell equations of electromagnetism. Its status is analogous to the exchange field or the anisotropy field of ferromagnets whose origins is in the quantum mechanics of the solid. \(H^{*}\) represents the thermodynamic reaction of the system for finding itself in an out-of-equilibrium situation. In the following we refer to \(H^{*}\) as the potential for the magnetization current.
## III Constitutive equations
Having defined the potential \(H^{*}\) associated with the magnetization current, we are ready to write the constitutive equations for the two materials of interest for the spin Seebeck and spin Peltier effects: a magnetic insulating material with a spin Seebeck effect and a metallic conductor with the spin Hall effect.
### Thermomagnetic effects in magnetic insulators
In analogy with the thermoelectric effects, we can write the constitutive equation for the joint transport of magnetization and heat by using the potential associated with the magnetization current derived in the previous Section. The general case which includes the presence of electric current is reported in Appendix A. Here we limit to insulators and we take currents and forces in one dimension (\(\nabla_{x}=\partial/\partial x\)).
\[j_{M} =\sigma_{M}\,\mu_{0}\nabla_{x}H^{*}-\sigma_{M}\epsilon_{M}\, \nabla_{x}T \tag{14}\] \[j_{q} =\epsilon_{M}\sigma_{M}T\mu_{0}\nabla_{x}H^{*}-(\kappa+\epsilon_ {M}^{2}\sigma_{M}T)\nabla_{x}T \tag{15}\]
Since the magnetization is not conserved, the magnetization current is not continuous and we have always to add the continuity equation. In non-equilibrium stationary states we always ask the condition \(\partial M/\partial t=0\) to be true, so Eq.
\[\nabla_{x}\,j_{M}=\frac{H^{*}}{\tau_{M}}. \tag{16}\]
#### iii.0.1 Uniform temperature gradient
If we disregard for the moment the heat currents, the solution of magnetization current problems will correspond to find solutions to the system composed by Eqs. and. Under a uniform temperature gradient, where \(\nabla T\) is a constant, the second term at the right hand side of Eq. is just a magnetization current density source \(j_{MS}=-\sigma_{M}\epsilon_{M}\,\nabla_{x}T\).
\[j_{M}=j_{MS}+\sigma_{M}\,\mu_{0}\nabla_{x}H^{*} \tag{17}\]together with Eq., considering constant coefficients, leads to a differential equation for the potential
\[l_{M}^{2}\nabla_{x}^{2}H^{*}=H^{*} \tag{18}\]
where
\[l_{M}=(\mu_{0}\sigma_{M}\tau_{M})^{1/2} \tag{19}\]
The differential equation has general solutions in the form
\[H^{*}(x)=H_{-}^{*}\exp(-x/l_{M})+H_{+}^{*}\exp(x/l_{M}) \tag{20}\]
By looking at Eqs. and we have that if the conduction process is present in different materials, the solution is made by taking Eq. for each material and finally joining the solutiosn by requesting the continuity in both \(j_{M}\) and \(H^{*}\).
#### ii.1.2 Adiabatic conditions
When the temperature is not externally controlled, we have to formulate the thermal problem by writing the heat diffusion equation. To this aim we need to write the continuity equations for the entropy. In stationary conditions Eq. becomes \(\nabla_{x}j_{s}=\sigma_{s}\) where the term at the left hand side is written by using \(j_{s}=j_{q}/T\) and Eq.
\[j_{q}=\epsilon_{M}Tj_{M}-\kappa\nabla_{x}T \tag{21}\]
After a few passages, we obtain
\[\nabla_{x}j_{q}=\mu_{0}\nabla_{x}H^{*}j_{M}+\frac{\mu_{0}\left(H^{*}\right)^{2 }}{\tau_{M}} \tag{22}\]
Both terms are quadratic in the force and the potential, therefore if we assume small currents and forces we are allowed to neglect them in a first approximation. In this case we obtain the condition \(\nabla_{x}j_{q}=0\) which, in one dimension, corresponds to a constant heat flux traversing the material. Moreover we choose here to study the adiabatic condition corresponding to \(j_{q}=0\) in which the two terms at the right hand side of Eq., the spin Peltier term \(\epsilon_{M}Tj_{M}\) and the heat conduction caused by the temperature profile \(T(x)\), counterbalance each other, giving no net heat flow through the layer. The profile \(T(x)\) will be stable if temperature of the thermal baths at the boundaries of the material are let free to adapt at the temperatures of the two ends. By using the adiabatic condition \(j_{q}=0\) in Eq.
\[\nabla_{x}T=\frac{1}{\hat{\epsilon}_{M}}\mu_{0}\nabla_{x}H^{*} \tag{23}\]
where \(\hat{\epsilon}_{M}\) is the thermomagnetic power coefficient in adiabatic conditions
\[\frac{1}{\hat{\epsilon}_{M}}=\frac{1}{\epsilon_{M}}\frac{\kappa_{M}}{\kappa+ \kappa_{M}} \tag{24}\]
From Eq. we see that the temperature profile depends on the profile of the potential \(H^{*}\). This last one is determined by inserting Eq. into Eq..
\[j_{M}=\hat{\sigma}_{M}\mu_{0}\nabla_{x}H^{*} \tag{25}\]
However now the diffusion length is the adiabatic value \(\hat{l}_{M}=(\mu_{0}\hat{\sigma}_{M}\tau_{M})^{1/2}\) where
\[\hat{\sigma}_{M}=\sigma_{M}\frac{\kappa}{\kappa+\kappa_{M}} \tag{26}\]
### Spin Hall effect in non-magnetic metals
The spin Hall effect is due to the spin orbit interaction for conduction electrons. This effect is particularly relevant for noble metals with high atomic number. Because of the spin orbit interaction, a spin polarized electric current is deflected by an angle which is called the spin Hall angle \(\theta_{SH}\). To include spin Hall effects into the theory of Section III.1 one should first extend the equations for the thermo-magnetic effects to the presence of an additional electric current. This is straightforward and the formal result is reported in Appendix A. However to state the equation for the spin Hall effect, the equations must be further extended for two dimensional flow. The complete constitutive equations are characterized by six force variables, namely: the derivative along \(x\) and \(y\) of the three driving forces for magnetic, electric and heat currents. Here we simplify the problem by just disregarding the thermal effects. For our final aims this is a reasonable approximation, since the contribution arising from the thermomagnetic coefficients of Pt is smaller than the other contributions involved in the full matrix of the thermo-magneto-electric effects. The general constitutive equations for the joint electric and magnetic transport are reported in Appendix B. Here we analyze in more detail the case of a non magnetic conductor with negligible Hall effect. We select the conditions in which the electric current \(j_{e}\) is always along \(y\), and the magnetization current \(j_{M}\) along \(x\). We have then the equations for the spin Hall and the inverse spin Hall effects from Eqs.(B5) and (B6).
\[j_{ey} =-\sigma_{0}\nabla_{y}V_{e}+\sigma_{0}\theta_{SH}\left(\frac{\mu_ {B}}{e}\right)\mu_{0}\nabla_{x}H^{*} \tag{27}\] \[j_{Mx} =\sigma_{0}\theta_{SH}\left(\frac{\mu_{B}}{e}\right)\nabla_{y}V_{e }+\sigma_{M}\mu_{0}\nabla_{x}H^{*} \tag{28}\]
The equations contain the spin Hall effects in the non diagonal terms which couples different directions and different currents. It is worthwhile to notice that the effects are fully described by the spin Hall angle \(\theta_{SH}\) which for metals is a definite negative quantity.
#### iii.2.1 Spin Hall effect
In the spin Hall effect a magnetization current is generated in the parallel direction \(x\) because of an electric current in the perpendicular one \(y\). By eliminating \(\nabla_{y}V_{e}\) by Eq. and Eq.
\[j_{Mx}=-\theta_{SH}\left(\frac{\mu_{B}}{e}\right)j_{ey}+\sigma_{M}^{\prime}\mu _{0}\nabla_{x}H^{*} \tag{29}\]
If the electric current density is uniform, the spin Hall effect corresponds to a magnetization current source \(j_{MS}=-(\mu_{B}/e)\theta_{SH}j_{ey}\). The profile of the magnetization current \(j_{Mx}\) which is actually traversing the layer also depends on the boundary conditions posed by the adjacent layers. Then, to find the profile \(j_{Mx}(x)\), Eq. must be solved together with the continuity equation giving a differential equation for the driving potential \(H^{*}(x)\) which has the same from of Eq. but with \(l_{M}=(\mu_{0}\sigma_{M}^{\prime}\tau_{M})^{1/2}\).
#### iii.2.2 Inverse spin Hall effect
In the configuration corresponding to the inverse spin Hall effect one has a magnetization current in the parallel direction which generates an electric effect perpendicular to it.
\[j_{ey}=-\sigma_{0}^{\prime}\nabla_{y}V_{e}+\theta_{SH}\left(\frac{e}{\mu_{B}} \right)j_{Mx} \tag{30}\]
The magnetization current traversing the layer is not constant and it will be given by the solution of Eq. if the electric current \(j_{ey}\) is constrained or by the solution of Eq. if the electric potential \(\nabla_{y}V_{e}\) is constrained. In both cases the constitutive equation must be solved together with the continuity equation, giving again the differential equation.
## IV Solutions of the magnetization current problem
### Single active material
For an active material both the spin Seebeck effects and the spin Hall effect results in a magnetization current source and the profile of the magnetization current will be due to the boundary conditions. In presence of boundaries blocking the flow of the magnetization current, the magnetic moments accumulate giving rise to the potential \(H^{*}\). The magnetization current close to a boundary is therefore absorbed by the materials itself as the potential \(H^{*}\) is also the driving force for the non conservation of the magnetic moment (Eq.). As it was shown in the previous Section, both spin Seebeck and spin Hall effects are characterized by constitutive equations that have the same functional form. Then we can work out the solution for the profile of the magnetization current independently of the specific effect and considering boundary conditions only. The specific solution will correspond to use as the current source \(j_{MS}\) the expression derived from the spin Seebeck Eq. or to the spin Hall Eq.. We initially consider a single material with generic boundary conditions. The solution of the magnetization current problem with several layers will then be obtained by applying appropriate boundary conditions and joining the solutions for different layers. We take a material from \(x=d_{1}\) to \(x=d_{2}\) with a uniform source of magnetization current \(j_{MS}\). Starting from the formal solution Eq., we derive the magnetization current by Eq. and we fix arbitrary values of the current at both boundaries, i.e. \(j_{M}(d_{1})\) and \(j_{M}(d_{2})\).
\[j_{M}(x)= j_{MS}-(j_{M}(d_{1})-j_{MS})\frac{\sinh((x-d_{2})/l_{M})}{\sinh(t/l _{M})}+\] \[+(j_{M}(d_{2})-j_{MS})\frac{\sinh((x-d_{1})/l_{M})}{\sinh(t/l_{M })} \tag{31}\]
and for the potential is
\[H^{*}(x)= -(j_{M}(d_{1})-j_{MS})\frac{1}{(l_{M}/\tau_{M})}\frac{\cosh((x-d_ {2})/l_{M})}{\sinh(t/l_{M})}+\] \[+(j_{M}(d_{2})-j_{MS})\frac{1}{(l_{M}/\tau_{M})}\frac{\cosh((x-d_ {1})/l_{M})}{\sinh(t/l_{M})}, \tag{32}\]where \(t=d_{2}-d_{1}\). Figs.2 and 3 shows the profiles of the magnetization current and the effective field along the material for different thicknesses \(t/l_{M}\). The spin accumulation close to the boundaries generates, as a reaction, an effective field which counteracts the effect considered (e.g. the spin Seebeck effect) in order to let the current to go to zero at the interface.
### Injection of a magnetization current
We consider the injection of a magnetization current from an active material which is acting as current generator, or current injector, into a passive material which is acting as a conductor. It is known that the quality of the interface plays an important role in the injection of the spin currents. In Ref. the condition of the Pt/YIG interface was intentionally modified by creating a thin amorphous YIG layer varying from 1 to 14 nm and it was shown that the spin Seebeck effect is depressed as the thickness of the amorphous layer increases. The maximum value is obtained with a fully crystalline interface and the typical decay length of the effect with thickness is 2.3 nm. In the present theory this kind of interlayer interface can be taken into account by introducing a third effective layer, with degraded properties, between the two. In the present paper we consider ideal interfaces between injector and conductor which is appropriate for spin Seebeck experiments characterized by crystalline interfaces. To analyze the injection of a magnetization current, we simplify the notation by dropping the \(M\) subscript and employing subscripts describing the role of the material: for the injector and for the conductor. The magnetization current source is that of the active material and is denoted \(j_{MS}\). The connection between the two media is set at \(x=0\). The boundary conditions for the magnetization current is \(j_{1}=j_{2}=j_{0}\) and the boundary condition for the potential is \(H_{1}^{*}=H_{2}^{*}=H_{0}^{*}\). Appendix C reports the formal solutions in the case in which each layer has finite width. These solutions will be employed in the comparison with real experiments performed in bilayers. Here we discuss how the efficiency of the injections is determined by intrinsic parameters.
\[j_{1}(x)=j_{MS}-(j_{MS}-j_{0})\exp(x/l_{1}) \tag{33}\]
and
\[j_{2}(x)=j_{0}\exp(-x/l_{2}) \tag{34}\]
for the currents and
\[H_{1}^{*}(x)=\frac{j_{0}-j_{MS}}{(l_{1}/\tau_{1})}\exp(x/l_{1}) \tag{35}\]
and
\[H_{2}^{*}(x)=-\frac{j_{0}}{(l_{2}/\tau_{2})}\exp(-x/l_{2}). \tag{36}\]
By setting the boundary condition at the interface between the two media \(H_{1}^{*}=H_{2}^{*}\) we find the value of the current at the interface
\[j_{0}=\frac{j_{MS}}{1+r_{12}} \tag{37}\]
If \(r_{12}\ll 1\) the current is efficiently injected, while if \(r_{12}\gg 1\) the magnetization current is not transmitted into the conductor.
Magnetization potential profile \(H^{*}\) for a single active material. Curves are Eq. with \(d_{1}=-t/2\) and \(d_{2}=t/2\), boundary conditions fixed to zero (\(j_{M}(-t/2)=j_{M}(t/2)=0\)) and show different thicknesses \(t/l_{M}\) (same as Fig.2). The curves are normalized to \(H_{0}^{*}=j_{MS}/(l_{M}/\tau_{M})\).
Magnetization current profiles for a single active material. Curves are Eq. with \(d_{1}=-t/2\) and \(d_{2}=t/2\), boundary conditions fixed to zero (\(j_{M}(-t/2)=j_{M}(t/2)=0\)) and show different thicknesses \(t/l_{M}\). The curves are normalized to \(j_{MS}\).
\[r_{12}=\sqrt{\frac{\sigma_{1}}{\sigma_{2}}\frac{\tau_{2}}{\tau_{1}}}. \tag{38}\]
So a junction with an efficient injection from to should have a conductor with a magnetization conductivity much larger than the injector \(\sigma_{2}\gg\sigma_{1}\) and a time constant much smaller \(\tau_{2}\ll\tau_{1}\).
## V Spin Seebeck and Spin Peltier Effects
In this Section we apply the theory previously developed to the spin Seebeck and spin Peltier effects.
### Spin Seebeck effect
The spin Seebeck effect consists in a magnetization current generated by a temperature gradient across a ferromagnetic material. We study the longitudinal spin Seebeck effect (LSSE) where the magnetization current and the temperature gradient are along the same direction. We consider experiments in which the active layer is YIG, the injector, labeled as and the sensor layer is Pt, the conductor, labeled as. The geometry of the experiment is schematically shown in Fig.4. The YIG injector has thickness \(t_{1}=t_{YIG}\) while the Pt conductor has thickness \(t_{2}=t_{Pt}\). The interface is set at \(x=0\).
The temperature gradient is applied along \(x\), the magnetic field is along \(z\), the electric effects (ISHE voltage) are measured along \(y\). We consider a constant temperature gradient \(\nabla_{x}T\), therefore the magnetization current source of YIG is \(j_{MS}=-\sigma_{YIG}\epsilon_{YIG}\,\nabla_{x}T\) given by the equations of Section III.1. The solutions of the magnetization current problem are Eqs. and reported in Appendix C and the magnetization current at the interface is given by Eq. in which \(l_{1}=l_{YIG}\), \(\tau_{1}=\tau_{YIG}\) and \(l_{2}=l_{Pt}\), \(\tau_{2}=\tau_{Pt}\). As the thickness of the Pt layer is generally of the same order of the spin diffusion length (\(t_{Pt}\sim l_{Pt}\sim 10\) nm), we can approximate Eq. for the case of \(t_{2}\sim l_{2}\) and find that the profile of the magnetization current is, at a good approximation, a linear decay from \(j_{0}\) at the interface \(x=0\) to zero at the border \(x=t_{2}\). The average magnetization current in the Pt layer is therefore \(\langle j_{Mx}\rangle_{x}=j_{0}/2\) where \(j_{0}\) is the magnetization current injected at the interface. If the experiments are performed by measuring the ISHE voltage, by taking Eq. with \(j_{ey}=0\), we obtain the relation between the magnetizations current along \(x\) and the electric potential along \(y\). We assume the relation to be valid for the average values along \(x\) over the thickness \(t_{2}\).
\[\langle\nabla_{y}V_{e}\rangle_{x}=\frac{\theta_{SH}}{\sigma_{e}}\left(\frac{e} {\mu_{B}}\right)\langle j_{Mx}\rangle_{x}. \tag{39}\]
The current injected at the interface \(j_{0}\) can therefore be estimated by the gradient of the ISHE voltage \(\nabla_{y}V_{ISHE}=\langle\nabla_{y}V_{e}\rangle_{x}\),
\[j_{0}=2\frac{\sigma_{e}}{\theta_{SH}}\left(\frac{\mu_{B}}{e}\right)\nabla_{y} V_{ISHE}. \tag{40}\]
In experiments, the spin Seebeck coefficient is determined as \(S_{LSSE}=\nabla_{y}V_{ISHE}/\nabla_{x}T\). The magnetization current at the interface can be calculated by Eq. where the spin Hall angle is evaluated as \(\theta_{SH}=-0.1\) from Ref.. In turn, the relation between the spin Seebeck current \(j_{MS}\) and \(j_{0}\) at the interface, given by Eq., will depend on the intrinsic parameters of both layers and their thickness.
\[\epsilon_{YIG}=\frac{1}{\sigma_{YIG}}\left(\frac{j_{MS}}{-\nabla T}\right). \tag{41}\]
In Pt the magnetization diffusion length is known to be \(l_{Pt}=7.3\) nm. The spin conductivity can be estimated by assuming that in a normal metal the scattering acts independently of the spin. Then, by converting the electrical conductivity of Pt \(\sigma_{e}=6.4\cdot 10^{6}\,\Omega^{-1}\)m\({}^{-1}\), into the conductivity for the magnetization current, we obtain \(\mu_{0}\sigma_{Pt}=2.6\cdot 10^{-8}\) m\({}^{2}\)s\({}^{-1}\). The time constant is finally calculated and results \(\tau_{Pt}=l_{Pt}^{2}/(\mu_{0}\sigma_{Pt})\simeq 2\cdot 10^{-9}\) s.
In YIG the estimations of the magnetization diffusion length present in literature, range from micron to millimeter for the transverse experiment (in which current and magnetization are parallel) to much lower value (i.e. \(<1\mu\)m) for the longitudinal effect (in which current and magnetization are perpendicular). From Ref. the LSSE coefficient measured on 1 mm of YIG, \(S_{LSSE}\simeq 4\cdot 10^{-7}\) VK\({}^{-1}\), results to be larger than the one measured on a 4 \(\mu\)m sample\(S_{LSSE}\simeq 2.8\cdot 10^{-7}\) VK\({}^{-1}\), but of the same order of magnitude.
Geometry of the longitudinal spin Seebeck effect.
In a more recent study, the dependence of the spin Seebeck effect on the thickness of YIG was investigated. It has been reported that the typical diffusion length is below \(l_{YIG}=1.5\,\mu\)m. We set in the following \(l_{YIG}=1\,\mu\)m. For the evaluation of the absolute thermomagnetic power coefficient \(\epsilon_{YIG}\) we use the result of Ref. where the thermal conditions were properly taken into account. These experiments were performed by using a YIG layer of 4 \(\mu\)m and a Pt layer of 10 nm.
By using the LSSE coefficient estimated at the saturation magnetization of YIG we obtain \(j_{0}/(-\nabla_{x}T)\simeq 2\cdot 10^{-3}\) As\({}^{-1}\)K\({}^{-1}\)m. The only missing intrinsic parameter is the magnetization conductivity of the YIG, \(\sigma_{YIG}\). To have an order of magnitude we suppose a reasonable injection from YIG into Pt (i.e 50%, with \(j_{0}=0.5\,j_{MS}\)). Then we set \(r_{12}=1\), i.e. \(l_{1}/\tau_{1}=l_{2}/\tau_{2}\). By using the resulting value for the magnetization conductivity of YIG \(\mu_{0}\sigma_{YIG}\sim 4\cdot 10^{-7}\) m\({}^{2}\)s\({}^{-1}\), we finally obtain an order of magnitude for the absolute thermomagnetic power coefficient as \(\epsilon_{YIG}\sim 10^{-2}\) TK\({}^{-1}\). In analogy with the thermoelectric effects where the absolute thermoelectric power coefficient is compared to the classical value \(\epsilon_{e}=-k_{B}/e\simeq-86\cdot 10^{-6}\) VK\({}^{-1}\), the value found here can be compared with the ratio \(k_{B}/\mu_{B}\simeq 1.49\) TK\({}^{-1}\). Furthermore, as the experiments show that \(\nabla_{y}V_{ISHE}\) and therefore \(j_{MS}\), changes sign when the magnetization of the YIG layer is inverted, this means that \(\epsilon_{YIG}\) changes sign when inverting the magnetization \(M\). The value reported before corresponds to the absolute value when the magnetization of YIG is at saturation.
### Spin Peltier effect
In the spin Peltier experiments a magnetization current is generated by the spin Hall effect in a Pt layer, labeled as and is injected into a YIG layer, labeled as. The injection of the magnetization current into the YIG, generates thermal effects. The geometry of the experiment is schematically shown in Fig.5.
The interface is set at \(x=0\), the electric current is along \(y\), the magnetization current is along \(x\) and the magnetic field is along \(z\). The magnetization current source is now \(j_{MS}=-\theta_{SH}\left(\mu_{B}/e\right)j_{ey}\) given by the spin Hall effect in Pt discussed in Section III.2. When the magnetization current diffuses inside YIG, it also generates a heat current because of the spin Peltier effect described in Section III.1.2. The solution of the magnetization conduction problem is mathematically identical to the spin Seebeck one, but with the role of YIG and Pt inverted. For this reason we have employed label for the injector, which is now Pt, and label for the conductor which is now YIG. The solutions of the magnetization current problem are again Eqs. and reported in Appendix C and the magnetization current at the interface is given by Eq.. With respect to the previous spin Seebeck case, the diffusion length of YIG is the adiabatic value \(\hat{l}_{YIG}=(\mu_{0}\hat{\sigma}_{YIG}\tau_{YIG})^{1/2}\). In the spin Peltier experiment the temperature profile in YIG is given by the integration of Eq.
\[T(x)-T=\frac{1}{\hat{\epsilon}_{YIG}}\mu_{0}\left(H_{2}^{*}(x)-H_{2}^{*}\right) \tag{42}\]
The result is shown in Fig.6.
By looking at the magnetization current profile (Fig.7), we see, as in the spin Seebeck experiment, that in order to have a good efficiency, the thickness of each layer should be larger than its diffusion length (\(t_{1}>l_{1}\) and \(t_{2}>l_{2}\)) to permit to the magnetization current to develop. Moreover the efficiency of the injection is regulated by the ratio of intrinsic parameters \(r_{12}=(l_{1}/\tau_{1})/(l_{2}/\tau_{2})\), where is the injector Pt and is the conductor YIG. Again the magnetization current at the interface is large if the ratio \(r_{12}\) is small. However it should be noticed that given the two materials in the junction (i.e. Pt,YIG) we have that \(r_{Pt\to YIG}=1/r_{YIG\to Pt}\).
Geometry of the spin Peltier effect.
Finally from the temperature profile Fig.6 obtained in adiabatic conditions we can reach information about the coefficient of the absolute thermomagnetic power in adiabatic conditions \(\hat{\varepsilon}_{YIG}\).
\[\Delta T_{SH}=\frac{1}{\hat{\varepsilon}_{YIG}}\mu_{0}H_{SH}^{*} \tag{43}\]
From the literature the thermal conductivity of YIG is \(\kappa=6\) W K\({}^{-1}\)m\({}^{-1}\). From Section V A, \(\varepsilon_{YIG}\simeq 10^{-2}\) TK\({}^{-1}\) and the parameter \(\kappa_{YIG}\simeq 10^{-2}\) W K\({}^{-1}\)m\({}^{-1}\). Moreover the potential \(H_{SH}^{*}\) is related to the spin Hall current \(j_{MS}=-(\mu_{B}/e)\theta_{SH}j_{ey}\) injected from Pt. Using the values from\(l_{YIG}/\gamma_{YIG}=3\) ms\({}^{-1}\) and \(\theta_{SH}=-0.1\) we are able to give an order of magnitude estimate of the temperature change, obtaining \(\Delta T_{SH}/j_{ey}=4\cdot 10^{-13}\) K A\({}^{-1}\)m\({}^{2}\).
Experimental values are taken from Ref., where in correspondence to an electric current density of \(3\cdot 10^{10}\) A m\({}^{-2}\) in Pt, the temperature difference measured by a thermocouple in YIG was \(2.5\cdot 10^{-4}\) K, considering that the Joule heating of the electric current in Pt was already subtracted. The parameter \(\Delta T_{SH}\) results \(1.2\cdot 10^{-2}\) K which is of the correct order of magnitude. Consequently by using \(t_{1}=t_{Pt}=5\) nm and \(t_{2}=t_{YIG}=0.2\,\mu\)m in Eqs.(C4) and, we find an adiabatic temperature change of \(T(t_{YIG})-T\simeq 2.5\cdot 10^{-4}\) K with \(l_{YIG}=0.4\,\mu\)m. This value refines the upper limit of \(1\,\mu\)m which was found in Section V A, however the phenomenology of the spin Peltier effect in YIG seems coherent with the absolute thermomagnetic power coefficient derived previously.
## VI Conclusions
In this paper the problem of magnetization and heat currents is investigated through a non equilibrium thermodynamics approach. Based on the constitutive equations of a ferromagnetic insulator and a spin Hall active material we are able to solve the problem of the profiles of the magnetization current and of the potential in the geometry of the longitudinal spin Seebeck and of the spin Peltier effects. By focusing on the specific geometry with one YIG layer and one Pt layer, we obtain the optimal conditions for generating large magnetization currents into Pt in the case of the spin Seebeck effect and for generating large heat current in YIG in the case of spin Peltier effects. In both cases we find that efficient injection is obtained when the thickness of the injecting layer is larger than the diffusion length \(l_{M}\). The theory predictions are compared with experiments and this permits to determine the values of the thermomagnetic coefficients of YIG: the magnetization diffusion length \(l_{M}\sim 0.4\,\mu\)m and the absolute thermomagnetic power coefficient \(\epsilon_{M}\sim 10^{-2}\) TK\({}^{-1}\).
| 10.48550/arXiv.1512.08890 | Non-equilibrium thermodynamics of the spin Seebeck and spin Peltier effects | Vittorio Basso, Elena Ferraro, Alessandro Magni, Alessandro Sola, Michaela Kuepferling, Massimo Pasquale | 2,348 |
10.48550_arXiv.0708.1445 | ###### Abstract
A short overview is given of recent advances in the field of nanosemiconductors, which are suitable as materials for spin polarized transport of charge carriers. On the basis of last theoretical and experimental achievements it is shown that development of diluted and wide forbidden zone semiconductors with controlled disorders as well as their molecular structures is the very prospective way for magnetic semiconductors preparation.
| 10.48550/arXiv.0708.1445 | Nanostructures, Magnetic Semiconductors and Spinelectronics | P. Kervalishvili | 4,623 |
10.48550_arXiv.0807.2719 | ## 1 Introduction
Understanding the rupture mechanisms of solids has become an important goal of fracture physics in order to improve the strength of structures and avoid catastrophic failures. Characterization of rupture properties very often involve the growth of a dominant crack. For instance, there is an extensive literature, both experimental and theoretical, discussing the slow growth dynamics of a single crack along a straight path in brittle or visco-plastic materials. However, in many practical situations, the crack path is not straight. For one thing, the crack path can be slightly destabilized and develop some roughness due to dynamical instabilities or material heterogeneities. This kind of path instability has motivated a large amount of experimental works analyzing the roughness of cracks as well as several models describing roughening mechanisms and non-trivial effects of heterogeneities on the rate of crack growth. Much larger deviations of the crack path from a straight line can be observed during the growth of an array of interacting cracks. Comparatively to the case of roughness, this rather complex situation has been studied essentially theoretically and very little experimentally despite its practical importance especially for heterogeneous materials where multiple cracks are likely to form. Understanding the growth of interacting cracks is also a relevant issue for fault dynamics as well as crack pattern formation during drying processes.
There are several levels of complexity that can play a role in the growth of interacting cracks. First, depending on the geometry of the crack array and the loading, the stress field around a crack will be amplified or shielded because of the existence of other cracks. Then, in \(2d\), the effective contribution of the loading on each crack will usually be a mixture of mode I and mode II 1. It has been argued that crack deviations occur when the shear stress on the crack lips (mode II) is non-zero and that the crack will grow so as to minimize the mode II contribution and maximize the mode I contribution. A noticeable observation related to this property is the preferential merging of cracks joining each other forming a right angle, commonly observed for crack patterns in drying experiments. This occurs naturally because the principal stress along a crack lip is parallel to the crack direction so that a crack approaching the lip with a right angle is propagating mainly in mode I. On the other hand, when two collinear mode I cracks are growing towards each other, they do not merge tip to tip, but instead repel each other. The origin of this effect has been discussed by several authors and numerical simulations have been able to reproduce, at least qualitatively, experimental observations. However, there are not many experimental data to compare with these theoretical predictions, and furthermore, there is very little knowledge about the effect of material heterogeneities on the crack path selection.
In this paper, we study experimentally the trajectories of interacting cracks in an almost two dimensional brittle and heterogeneous material. The array of cracks has been initiated prior to the application of the external stress and the heterogeneity of the material (paper sheets) allows us to test the stability of crack paths to small perturbations. We uncover that for a given geometry of the crack array, a crack can follow statistically two stable trajectories : an attractive one and a repulsive one with respect to the neighboring cracks. The main result of this investigation is the analysis of the geometrical conditions for which cracks are attracted towards another and when they are repelled. The paper is organized as follows. In the next section, we describe the experimental apparatus. In section 3, the extraction procedure of the post-mortem shapes of the crack profiles is described. In sections 4, 5 and 6, we analyze the different types of crack paths and their statistics as a function of the initial crack pattern geometry. In section 7, we finally discuss the results and conclude.
## 2 The experiments
To study the growth of several interacting cracks, we have loaded bi-dimensional brittle samples made of fax paper sheets (Alrey(r)) with a tensile machine. The samples were previously prepared cutting several straight cracks in the paper sheets. In order to observe clearly the interaction between cracks during their growth, we had to find a geometry for which all crack tips are equivalent. Actually, we wanted to prevent the isolated growth at a particular crack tip that will inhibit the growth of the other cracks. One possible geometry that follows this condition consists in an array of cracks as presented in The pattern formed by the two lines of cracks offers the advantage to have some translational invariance in the cracks direction. The stress is applied perpendicularly to the cracks direction and uniformly on the sample borders. Therefore, stress intensification is theoretically equivalent at each crack tip. In this geometry, there are three adjustable parameters : the crack length, the vertical spacing of cracks on a line and the horizontal spacing between the two lines. In this article, the first two parameters are fixed to 1cm and only the distance \(d\) between the two lines of cracks has been varied.
Each sample is prepared with the 11 initial cracks forming the array using a cutter blade. The experimental set-up consists of a tensile machine driven by a motor (Micro Controle UE42) controlled electronically to move step by step (Micro Controle ITL09). The paper sheets are mounted on the tensile machine with both ends attached with glue tape and rolled twice over rigid bars clamped on jaws. The motor controls the displacement of one jaw (400 steps per micrometer) while the other jaw is rigidly fixed to a force gage (Hydrotonics-TC). The tensile machine is placed in a box with a humidity level stabilized at 5% and at room temperature. The samples are loaded by increasing the distance between the jaws on which they are clamped such that the resulting force \(F=\sigma eH\) (\(e=50\mu\)m is the film thickness and \(H=20.95\)cm the sample height) is perpendicular to the initial cracks direction. The loading is enforced at a constant and very slow velocity (between \(0.84\mu\)m.s\({}^{-1}\) and \(2\mu\)m.s\({}^{-1}\)) until the final breakdown of the sample. More details about the experimental set-up can be found in. Actually, once the force threshold \(F_{c}\) of the sample is overcome, we observe a quasi-instantaneous rupture of the sample with a quasi-simultaneous growth of all the cracks. There is no visual evidence of any sub-critical crack growth process before the final breakdown of the sample. During this breakdown, the applied force drops. When it reaches about half its maximum value, we stop loading the sample before breaking it in two pieces. It is a convenient procedure that allows us afterwards to image properly the crack path configurations in each sample.
## 3 Post-mortem analysis of crack trajectories
After each experiment, post-mortem samples are digitized using a high resolution scanner. The obtained images are processed in order to get a binary image in which the crack paths can be distinguished. In figure 2, we show an example of a digitized sample and of the corresponding extracted binary image. This image processing allows to finally create for each crack path the profile \(y(x)\) that describes its shape as a function of the abscissae \(x\) along the axis corresponding to the initial crack direction and centered on the corresponding initial crack (cf. figure 3).
## 4 Two types of crack path
For an horizontal spacing \(d=0\)cm between the two crack lines, the initial crack array collapses into a single crack line. In figure 4, one can see that the crack paths show a repulsion phase before an attraction one. This initial crack pattern always leads to the so-called type B\({}_{0}\) growth behavior. This is the signature of the fact crack tips repulse each other. Indeed, the crack tips move away from the initial crack line before the neighboring tips overtake each other. Then, the paths curve to join the neighboring crack lips tending to form a right angle.
Geometry of the crack array initiated in the paper sample and the direction of the applied stress \(\sigma\).
sort of spiral shape as one can see in
For \(0<d<d_{\rm int}\) (experimentally \(d_{\rm int}\approx 1.8\pm 0.2\)cm), two different kinds of behavior have been identified with respect to the crack path shapes among the fifty crack arrays that have been fractured.
Binary image of a fractured sample with \(d=0\)cm.
On the left, binary image showing the path followed by a crack, and on the right, corresponding extracted crack profile \(y(x)\).
Post-mortem images of a fractured sample with \(d=1.5\)cm : at the top, initial digitized image and, at the bottom, corresponding binary image with extracted crack paths obtained through image processing.
For the first one, that will be referred to as type A, neighboring cracks on different lines attract each other from the beginning of the crack growth up to the end as it is the case in (\(d=1.5\)cm). In contrast, for the type B, the crack paths are a mixture of two paths, a repulsive plus attractive path on one end and an attractive path tip to tip on the other end of each crack. For instance, in (\(d=0.4\)cm), we observe a repulsion phase for the right tip of the bottom cracks followed by an attraction phase by the neighboring crack on the other line while the right tip of the top cracks tends to join the left tip of the bottom cracks in a rather unexpected way. It seems that the symmetry breaking associated to the repulsive phase of crack growth on one end of the crack allows simultaneously the merging of tips on the other end of the crack.
Finally, when \(d>d_{\rm int}\), the two lines of cracks do not interact (cf. figure 7). Actually, one of the two lines fractures preferentially and we get the type B\({}_{0}\) behavior previously observed for \(d=0\)cm since the situation is comparable.
To summarize the results :
* For \(d=0\), only type B\({}_{0}\) is observed.
* For \(0<d<d_{\rm int}\), types A and B are observed.
* For \(d_{\rm int}<d\), only type B\({}_{0}\) is observed.
The statistical proportion between types A and B is a function of the spacing \(d\) between the two initial crack lines.
Close-up of a binary image of a fractured sample with \(d=0\)cm.
Binary image of a fractured sample with \(d=0.4\)cm.
Binary image of a fractured sample with \(d=5\)cm.
This percentage decreases regularly starting from 100 % for \(d=0\) down to zero for \(d=d_{\rm int}\) where it jumps back brutally to 100% (this last case corresponds to rupture on a single crack line as observed in figure 7).
In the case of a homogeneous material, we would have expected the crack trajectories to be reproducible from one experiment to the other. A sheet of paper is actually made of a complex network of cellulose fibers. Scanning electron microscopy on our samples shows fiber diameters between 4 and 50\(\mu\)m with an average of 18\(\mu\)m. Cellulose fibers are themselves a bundle of many microfibrils that have a crystalline structure. The statistical selection between two types of crack path shapes for a given initial crack geometry is probably triggered by this heterogeneity of the fractured material. Indeed, heterogeneity and anisotropy in the initial crack tip toughness and shape due to the heterogeneity of the material might induce small perturbations in the initiation of the crack growth path. We discover that these perturbations are large enough for the crack to explore statistically two very different "meta-stable" crack paths.
Additionally, the crack patterns observed in this section allow us to confirm the fact that, in general, two crack tips repulse each other while a crack tip and a crack lip attract each other.
## 5 Analysis of attractive cracks in type A samples
In this section, we study the average behavior followed by type A cracks that corresponds to experiments in which the cracks have been attracting each other during their whole growth. For each value of the spacing \(d\) between the two crack lines, we extract the average profile of the cracks.
Percentage of samples for which the crack path are of type B (including the case B\({}_{0}\)). Experimentally, there is some uncertainty concerning the distance at which there is no more interactions between the two crack lines. This is schematically shown by the dotted line and lighter color band around \(d=1.8\)cm.
In figure 9(a), we plot for three different samples the averaged profiles of the two crack lines separated by \(d=1.2\)cm. Then, we average the profiles \(\langle y(x)\rangle\) over all the type A samples corresponding to the same value of \(d\) (cf. figure 9(b)). We get the mean type A profile \(\langle\langle y(x)\rangle\rangle\) for each value of \(d\).
It is important to notice that, depending on the position \(x\), the number of profiles on which the averaging is performed is variable since all the cracks do not have the same length in the \(x\) direction. This is the reason why the averaged profiles, \(\langle y(x)\rangle\) and \(\langle\langle y(x)\rangle\rangle\), are not continuous at all points. For the same value of \(d\), cracks of different samples appear to a have very similar mean profiles \(\langle y(x)\rangle\) as one can see in figure 9(a).
(a) Average crack profile \(\langle y(x)\rangle\) for a sample with \(d=1.2\)cm and its extension allowing to define \(\ell_{j}\) and (b) distance \(\ell_{j}\) as a function of the spacing \(d\) between initial crack lines, for experiments with a type A behavior. Each point corresponds to one experiment.
(a) Average crack profile \(\langle y(x)\rangle\) for three samples with \(d=1.2\)cm and (b) average crack profile \(\langle\langle y(x)\rangle\rangle\) for five values of \(d\), for experiments with a type A behavior.
The main difference is that the crack paths extend farther away when the distance \(d\) is increased (cf. figure 9(b)).
Extending intuitively each crack path to join the neighboring crack lip on the other crack line, we are able to define an hypothetical junction position \(\ell_{j}\) measured from the center of the joined crack (cf. figure 10(a)). In figure 10(b), we see that this junction distance \(\ell_{j}\) increases rapidly with \(d\). Actually, the rapid increase of this distance between \(d=1.2\)cm and \(d=1.6\)cm suggests that there might be a divergence of the junction position \(\ell_{j}\) when the distance \(d\) approaches \(d_{\rm int}(\approx 1.8\)cm). A divergence of \(\ell_{j}\) would make sense since when \(d>d_{\rm int}\) the two crack lines do not interact anymore and thus, the cracks of one line cannot cross those of the other line.
## 6 Analysis of repulsive cracks in type B samples
In this section, we study in type B (including B\({}_{0}\) for \(d=0\)) samples the average behavior followed by the repulsive crack paths. In figure 11(a), we plot the average profile of repulsive cracks \(\langle y(x)\rangle\) for two samples with \(d=1\)cm. We note that, for a given distance \(d\), the crack path is reproducible from one sample to the other. In figure 11(b), we plot the average crack profile \(\langle\langle y(x)\rangle\rangle\) as a function of \(d\) and \(x\). Clearly, the smaller \(d\), the larger the repulsion. One way to quantify this is to measure the maximum angle of deviation \(\theta_{max}\) during the repulsion phase. As can be seen in figure 12, this quantity decreases with the distance \(d\). If we extend linearly this curve, we can see that the effects of the repulsion disappear when \(d\) reaches a value of about \(1.9\)cm. Interestingly, we recover a value in the range corresponding to the characteristic distance \(d_{\rm int}\) above which the two crack lines do not interact.
For \(d=0\), we find a maximum deviation angle \(\theta_{max}\simeq 27^{\circ}\).
Paths of repelled cracks (type B). (a) Average crack profile \(\langle y(x)\rangle\) for two samples with \(d=1\)cm, and (b) average crack profile \(\langle\langle y(x)\rangle\rangle\) for four value of \(d\).
has a different approach to explain the deviation of collinear cracks. She analyzes the stability of the crack path to a local perturbation. The crack deviation angle is then \(\theta=\rm{atan}(2\delta y/L)\) where \(\delta y\) is the amplitude of the local deviation and \(L\) is the crack length. To obtain \(\theta_{max}\simeq 27^{\circ}\) would require a perturbation \(\delta y\simeq L/4=0.25\)cm, a rather unphysical value. A large deviation angle could also be obtained assuming that several smaller local deviations of the crack path occur due to the heterogeneous structure of paper. However, if that was the case, one would expect the crack to progressively rotate instead of turning rather abruptly (see figure 11(b)).
## 7 Discussion and conclusion
In this article, we have seen that the interactions between two lines of cracks growing under a very low strain rate in a heterogeneous material lead to a statistical behavior. Indeed, for the same initial sample geometry, we get two types of crack path pattern. The type A behavior, that corresponds to a permanent attraction between the cracks during their growth, appears to present a universal shape that does not depend significantly on the initial crack array geometry. On the contrary, for type B behavior, for which the crack paths present either a repulsion phase followed by an attraction or an attraction phase tip to tip, the path trajectories appear to be dependent on the distance between the two initial crack lines \(d\). Indeed, the repulsion decreases as \(d\) increases and finally totaly disappears as \(d\) tends to a certain distance \(d_{\rm{int}}\). This particular value of \(d\) is also a critical value above which the two lines do not interact anymore. Understanding this value is still an open issue. Also, the value of the crack deviation angle observed in our experiments during the repulsive phase remains unexplained. In particular, it is rather large compared to theoretical predictions.
The statistics observed for the crack path shape is the signature of the meta-stability of two types of crack trajectory for a given geometry.
Maximum angle of repulsion of the crack path during the repulsion phase as a function of the distance \(d\) for type B experiments.
It is likely that the trigger for this statistics comes from the complex structure of paper sheets that makes this material heterogeneous and leads to anisotropy on the initial crack tip shape and toughness. The two types of crack path are two meta-stable paths between which each crack has to choose when it initiates its growth. Only at this initiation stage, may the heterogeneity of the sample strongly influence the crack in its "choice".
It is also important to point out the collective behavior of the cracks located on the same initial crack line. Indeed, inside a given crack line, all the cracks select a specific dynamics with quasi-identical crack path shape. Actually, the initiation of the global rupture of a sample is probably triggered at one particular crack tip. The choice for the crack shape type is made at this particular moment. Then, an unexplained collective effect operates to make all the cracks throughout a line behave the same way.
| 10.48550/arXiv.0807.2719 | Attractive and repulsive cracks in a heterogeneous material | Pierre-Philippe Cortet, Guillaume Huillard, Loïc Vanel, Sergio Ciliberto | 544 |
10.48550_arXiv.0903.0278 | ###### Abstract
Graphane is a two-dimensional system consisting of a single planar layer of fully saturated (sp\({}^{3}\) hybridization) carbon atoms with H atoms attached to them in an alternating pattern (up and down with relation to the plane defined by the carbon atoms). Stable graphane structures were theoretically predicted to exist some years ago and just experimentally realized through hydrogenation of graphene membranes. In this work we have investigated using _ab initio_ and reactive molecular dynamics the role of H frustration (breaking the H atoms up and down alternating pattern) in graphane-like structures. Our results show that H frustration significantly contributes to lattice contraction. The dynamical aspects of converting graphene to graphane is also addressed.
Carbon-based materials are among the most studied ones, however the discovery of new structures seems endless, colossal carbon tubes and graphene being recent examples.
Graphenes are flat monolayers of carbon atoms in a sp\({}^{2}\) hybridization. They have very interesting electronic and mechanical properties, which make them one of most important subject in materials science today.
It has been theoretically predicted that a related structure, called graphane, could exist. Graphane consists of a single planar layer structure with fully saturated (sp\({}^{3}\) hybridization) carbon atoms with H attached to them in an alternating pattern (up and down with relation to the plane defined by the carbon atoms). The two most stable conformations are the so-called chair-like (H atoms alternating on both sides of the plane) and boat-like (H atoms alternating in pairs). A third member of these two-dimensional planar carbon structures, called graphynes, have also been predicted to exist but up to now only molecular fragments have been synthesized.
Indirect experimental evidences of graphane-like structures have been reported. More recently, in a series of very elegant experiments, Elias _et al._ demonstrated the existence of graphane formation from graphene through hydrogenation with cold plasma. They also demonstrated that the process is reversible. These fundamental discoveries open new and important perspectives to the use of graphene-based devices since the electronic gap values in graphanes could be controlled by the degree of hydrogenation.
The Elias _et al._ experiments consisted in exposing graphene membranes to H\({}^{+}\) from cold plasma. The H incorporation into the membranes results in altering the C sp\({}^{2}\) hybridizations to sp\({}^{3}\) ones. The experiments were also done with the membranes over SiO\({}_{2}\) substrates (only one membrane side exposed to H\({}^{+}\) attacks) and produced a material with different properties.
One very interesting result from their TEM (transmission electron microscopy) data is the distribution of the lattice spacing value of the hydrogenated membranes. For an ideal graphane structure theoretical calculations indicate a lattice spacing value larger than the graphene one. Although the TEM data (from) show events with values larger than the graphene one, the majority of counts show smaller ones.
Elias _et al._ discuss in details a series of experimental and material conditions that could explain the contracted lattice values, but from the experimental point of view it
Structural models. (a) graphene; (b) graphane-boatlike; (c) graphane-chairlike. See text for discussions.
Of particular interest it is the role of the ordered and/or disordered arrays of H incorporation into the membranes.
In this work we have investigated using _ab initio_ quantum and classical binding energy bond order (BEBO) molecular dynamics methods how the graphene hydrogenation is reflected into lattice deformations.
We have carried out _ab initio_ total energy calculations in the framework of the density functional theory (DFT), as implemented in the DMol\({}^{3}\) code. DMol\({}^{3}\) is considered state of the art DFT methodology. Exchange and correlation terms were treated within the generalized gradient (GGA) functional by Perdew, Burke, and Ernzerhof. Core electrons were treated in a non-relativistic all electron implementation of the potential. A double numerical quality basis set with polarization function (DNP) were considered, with a real space cutoff of 3.7 A. The tolerances of energy, gradient, and displacement convergence were 0.00027 eV, 0.054 eV/A and 0.005 A respectively.
We have considered finite and infinite (cyclic boundary conditions(CBC)) structures. As we need to investigate many structures of different sizes and at different temperatures (molecular dynamics (MD) simulations) the extensive use of _ab initio_ methods is computationally cost prohibitive. Thus, for the larger structures and for MD simulations, we opted to use a BEBO method like ReaxFF. ReaxFF is a reactive force field developed by Adri van Duin, William Goddard III and co-workers for use in MD simulations. This empirical method allows the simulation of many types of chemical reactions, including bond dissociation (making/breaking bonds).
ReaxFF is similar to standard non-reactive force fields, like MM3, where the system energy is divided into partial energy contributions associated with, amongst others, valence angle bending, bond stretching, and non-bonded van der Waals and Coulomb interactions. However, one main difference it is that ReaxFF can handle bond formation and dissociation using bond order approach. ReaxFF was parameterized against DFT calculations. The average deviation between the ReaxFF predicted heat of formation values and the experimental ones are of 2.8 and 2.9 Kcal/mol for non-conjugated and conjugated systems, respectively.
We have carried out geometry optimizations using gradient conjugated techniques (stopping condition, gradient values less than \(10^{-3}\)). For MD simulations we used scaling methods, like Beredsen thermostat, in order to control temperature.
We started carrying out DMol\({}^{3}\) calculations for the infinite (CBC) structures shown in The cell unit parameters were optimized using the Murnaghan procedure. Sofo, Chaudhari, and Baker in their pioneering graphane work considered compacted (interacting layers) structures. Here, in order to mimic the experimental conditions, we have considered isolated (non-interacting) layers. The results are displayed in table 1. The most stable conformation (chair) has just one type of C-C bonds, while the boat conformation has two types (carbon bonded to H on opposite side of plane of symmetry and those connected to H on the same side of it), with their associated H bonds.
As we can see from Table 1, the lattice parameter values for the graphane structures are larger than the graphene one. We have also considered the case of the minimum unit cell with H atoms parallelly aligned (just one side of the membrane). Our results show that the system is unstable with the tendency of H\({}_{2}\) recombination and/or C-C breaking bonds. These results are in good agreement with previous work and with the available experimental data.
Considering the stochastic nature of the experiments,
\begin{table}
\begin{tabular}{l c c c} \hline
## System** & **Energy(Ha)** & **Lattice(Å)** & **C-C(Å)
\\ \hline Graphene & -304.68 & 2.465 & 1.423 \\ G-chair & -309.41 & 2.540 & 1.537 \\ G-boat & -309.38 & 2.592 & 1.581 \\ & & 2.509 & 1.537 \\ \hline \end{tabular}
\end{table}
Table 1: DMol\({}^{3}\) results for the crystalline structures shown in See text for discussions.
Scheme of the H frustrated domains. See text for discussions.
H frustration is a configuration where the sequence of alternating up and down H atoms is broken (frustrated). This is similar to spin frustration in magnetic materials.
In we show a domain of up and down H atoms. From energetic (stability) point of view, after the first (up or down) H is incorporated into the C layer, the next favorable site is its first inverse neighbor (down or up), and so on. If the system is large enough uncorrelated domains might be formed. As the H coverage is continued it could occur that it is no longer possible the alternating sequence of up and down H atoms.
We have investigated finite fragments with and without H frustrations. We analyzed the associated geometrical changes in order to determine whether the lattice expands or contracts with relation to the ideal graphane structure (table 1). We have carried out DMol\({}^{3}\) (table 2) and ReaxFF (for larger fragments) calculations (table 3, Fig. 4).
DMol\({}^{3}\) and ReaxFF show similar and consistent results. The H frustration produce out of plane distortions which induce in-plane shrinkage. The net result is a decrease of the lattice parameter in relation to the ideal graphane value. This effect is amplified when first-neighbor H atoms are parallelly aligned.
In order to further test these ideas we have also carried out ReaxFF molecular dynamics simulations of graphene hydrogenation on very large membranes, at different temperatures and at different H atom densities.
In we show representative snapshots from the early stages of a simulation at 300 K. The results show that significant percentage of uncorrelated H frustrated domains are formed in the early stages of the hydrogenation process leading to lattice decreased values and extensive membrane corrugations. These results also suggest that large domains of perfect graphane-like structures are unlikely to be formed, H frustrated domains are always present. The number of these domains seems to be sensitive to small variations of temperatures and H gas densities. This can perhaps explain the significant broad lattice parameter distribution values experimentally observed.
We run annealing cycle simulations to verify whether thermal healing occurs. Our preliminary results show that, once formed, H frustrated domain are resilient to thermal annealing. Further studies are necessary to clarify this issue.
\begin{table}
\begin{tabular}{l c c c c c c} \hline
## System
& \(\mathbf{d}_{A-B}\) & \(\mathbf{d}_{B^{\prime}-C}\) & \(\mathbf{d}_{C-D}\) & \(\mathbf{d}_{D^{\prime}-A}\) & \(\mathbf{d}_{A-C}\) & \(\mathbf{d}_{B-D}\) \\ \hline Graphene & 9.804 & 9.799 & 9.804 & 9.799 & 15.569 & 9.974 \\ G-chair & 9.861 & 9.841 & 9.882 & 9.847 & 15.636 & 10.050 \\ G-boat & 9.852 & 9.818 & 9.852 & 9.818 & 15.622 & 9.977 \\ frust-0 & 9.788 & 9.857 & 9.876 & 9.823 & 15.588 & 10.003 \\ frust-1 & 9.740 & 9.802 & 9.866 & 9.786 & 15.465 & 9.990 \\ \hline \end{tabular}
\end{table}
Table 2: Distances between reference points depicted in Frust-1 and Frust-0 refer to parallel (see Fig. 4a) and missing (see Fig. 4b) hydrogen atoms in frustrated domains, respectively.
Schematic draw of a graphene fragment before hydrogenation. The letters are the reference points for the distances displayed in Tables 2 and 3.
A detailed MD analysis including the effect of many layers, substrates and association of inert gas with H atoms will be published elsewhere.
Acknowledgments - This work was supported in part by the Brazilian Agencies CNPq, CAPES and FAPESP. The authors wish to thank Prof. A. van Duin for his very helpful assistance with the ReaxFF code and for his kind hospitality in Pennsylvania.
# System
& \(\mathbf{d}_{A-B}\) & \(\mathbf{d}_{B^{\prime}-C}\) & \(\mathbf{d}_{C-D}\) & \(\mathbf{d}_{D^{\prime}-A}\) & \(\mathbf{d}_{A-C}\) & \(\mathbf{d}_{B-D}\) \\ \hline Graph. & 22.26 & 22.26 & 22.26 & 22.26 & 37.08 & 22.47 \\ G-chair & 22.95 & 22.94 & 22.95 & 22.94 & 38.40 & 23.01 \\ Frust-1 & 22.21 & 22.92 & 22.80 & 22.90 & 37.95 & 22.47 \\ Frust-0 & 22.38 & 22.95 & 22.77 & 22.82 & 38.12 & 22.37 \\ \hline \end{tabular}
\end{table}
Table 3: Distances between the reference points depicted in calculated with ReaxFF. The number in parenthesis indicate the number of frustrated domains in the structure. See text for discussions. | 10.48550/arXiv.0903.0278 | Graphene to Graphane: The Role of H Frustration in Lattice Contraction | Sergio B. Legoas, Pedro A. S. Autreto, Marcelo Z. S. Flores, Douglas S. Galvao | 5,743 |
10.48550_arXiv.0902.4774 | ###### Abstract
The effect of edge-type dislocation wall strain field on the Hall mobility in \(n\)-type epitaxial GaN was theoretically investigated through the deformation potential within the relaxation time approximation. It was found that this channel of scattering can play a considerable role in the low-temperature transport at the certain set of the model parameters. The low temperature experimental data were fitted by including this mechanism of scattering along with ionized impurity and charge dislocation ones.
## I Introduction
As is known GaN films are under extensive examination for many years because of their promising application for the construction of short-wavelength light emitting devices. However, the performance of these devices is limited by defects, both native and impurity types. Native defects, in particular, are threading dislocations with high densities (\(10^{8}-10^{11}\) cm\({}^{-2}\)) which are result from the large lattice mismatch between epilayer and substrate. Dislocations, being charged objects, act as scattering centers (core effect) for carriers affecting the transverse mobility in films. In most studies in the context of the GaN layers this Coulomb scattering has been considered. At the same time, in addition to the core scattering, dislocations can give contribution to the resistivity through deformation and piezoelectric potentials. In GaN layers these potentials associate with the strain field of dislocation arrays which form low-angle grain boundaries or separate dislocations in the specific cases. However, the carrier scattering due to piezoelectric potential has been found as negligibly small within the bulk of GaN.
## II Model
In this paper we theoretically investigate the contribution to the Hall mobility in GaN layers from a wall of dislocations of edge type. The scattering of electrons by dislocation wall (DW) is treated in the framework of the deformation potential approach.
\[\delta U({\bf r})=G\Delta({\bf r}), \tag{1}\]
Let the threading dislocation segments with coordinates \((0,h)\) along the \(\) axis (perpendicular to the interface/layer plane) form a dislocation wall of finite length \(2L\). The dilatation at the point \({\bf r}\geq 2L\) around such defect, then, can be found based on the so-called disclination model of grain boundaries and dislocations for isotropic medium.
\[\Delta({\bf r})=\frac{(1-2\sigma)}{(1-\sigma)}\frac{b}{4\pi p}\Bigl{(}\ln\frac {\sqrt{\rho_{+}^{2}+z^{2}}-z}{\sqrt{\rho_{+}^{2}+(h-z)^{2}}+(h-z)}-\]\[\ln\frac{\sqrt{\rho_{-}^{2}+z^{2}}-z}{\sqrt{\rho_{-}^{2}+(h-z)^{2}+(h-z)}}\Big{)}, \tag{2}\]
It is interesting to note that Eq. is also the exact formula for the dilatation around the high-angle grain boundary as well as the disclination dipole.
The square of the matrix element of electron scattering in momentum states \(\mathbf{k}\) to states \(\mathbf{k}^{{}^{\prime}}\) with the perturbation energy given by Eq.
\[|\langle\mathbf{k}|U(r)|\mathbf{k}^{{}^{\prime}}\rangle|^{2}=\frac{32\pi^{2}A^ {2}}{(q_{z}^{2}+q_{\perp}^{2})^{2}V^{2}}\frac{\sin^{2}(q_{z}h/2)}{q_{z}^{2}}( 1-J_{0}(2q_{\perp}L)), \tag{3}\]
As one can see from Eq. and Eq., the scattering due to perturbation \(\delta U(\mathbf{r})\) is three dimensional, therefore \((\mathbf{k}-\mathbf{k}^{{}^{\prime}})_{z}=q_{z}=\sqrt{q^{2}-q_{\perp}^{2}}\neq 0\) in contrast to the case of an infinitely long dislocation line along the \(\)-axis. Omitting the details of further calculations, we can come to the following equation for the relaxation time due to the strain field of DW
\[\tau_{wall}^{-1}(k)=\frac{32A^{2}L^{2}n_{def}\pi m^{*}}{\hbar^{3}k_{\perp}{}^{ 2}}\Big{(}\text{Si}(hk_{\perp})+\frac{\cos(hk_{\perp})}{hk_{\perp}}-\frac{1}{ \hbar k_{\perp}}\Big{)}\]
\[\Big{(}J_{0}^{2}(2Lk_{\perp})+J_{1}^{2}(2Lk_{\perp})-\frac{1}{2Lk_{\perp}}J_{0 }(2Lk_{\perp})J_{1}(2Lk_{\perp})\Big{)}, \tag{4}\]
When a film is thick (\(k_{\perp}h>>1\), bulk regime) Eq.
\[\tau_{wall}^{-1}(k)=\frac{16A^{2}L^{2}n_{def}\pi^{2}m^{*}}{\hbar^{3}k_{\perp}{ }^{2}}\]
\[\Big{(}J_{0}^{2}(2Lk_{\perp})+J_{1}^{2}(2Lk_{\perp})-\frac{1}{2Lk_{\perp}}J_{0 }(2Lk_{\perp})J_{1}(2Lk_{\perp})\Big{)}, \tag{5}\]
The relaxation time due to Coulomb scattering at charged dislocation lines can be written in the well-known form as
\[\tau_{dis}(k)=\frac{\hbar^{3}\epsilon^{2}c^{2}}{N_{dis}e^{4}f^{2}m^{*}}\frac{( 1+4\lambda_{d}^{2}k_{\perp}^{2})^{3/2}}{\lambda_{d}}, \tag{6}\]
To determine the filling factor \(f\), the procedure from has been used.
Assuming the validity of the non-degenerate statistics, we can evaluate the DW contribution \(\mu_{wall}\) to the total mobility using the well-known formula
\[\mu(k)=\frac{e\hbar^{2}}{m^{*2}k_{B}T}\frac{\int\tau(k)k_{i}^{2}f_{0}d^{3}k}{ \int f_{0}d^{3}k}=\frac{e\langle\tau\rangle}{m^{*}}, \tag{7}\]where \(k_{i}=k_{x(y)}\) is the planar component of the wave vector, and \(f_{0}\) is the Boltzmann distribution function.
## III. NUMERICAL RESULTS
We first analyzied the drift mobility contribution due to grain boundary strain field together with other important mechanisms of scattering at low temperatures (ionized impurities, charged dislocation lines). The component of the scattering due to ionized impurities has been taken in the form of the Brooks-Herring-like formula obtained on the basis of the partial-wave phase-shift method\({}^{19}\). This approach yields the correct results for low \(T\) and high \(n\) where the Born approximation can be false.
\[\frac{1}{\tau_{ii}(k)}=N_{I}v\sigma^{B}(k)H_{0}, \tag{8}\]
We found that the Coulomb dislocation scattering is major at free carriers concentration \(n<10^{17}\)cm\({}^{-3}\) when \(N_{dis}\approx 10^{8}\) cm\({}^{-2}\) (\(T<100\)K), and above this \(n\) the impurity scattering dominates. In our calculations dislocation core scattering always dominates above \(N_{dis}\approx 10^{8}\) cm\({}^{-2}\). The numerically calculated \(\mu_{wall}\) on the basis of the Eqs., is shown in Fig.1 as a function of temperature for some selected model parameters together with other contributions. The deformation-potential constant \(G\) has been taken equal to 4 eV, that corresponds to the typical values for semiconductors. From our analysis we found that \(\mu_{wall}\sim T^{3/2}\) (unlike the case of the separate dislocations where \(\mu\sim T\)), and this contribution can be maximal for some chosen set of the parameters when the concentration of the DW is sufficiently high (\(n_{def}\simeq 10^{10}\)cm\({}^{-2}\)) (see Fig.1).
The low temperature part of the experimental data from\({}^{21}\) for two GaN samples with \(N_{dis}=4\times 10^{8}\) cm\({}^{-2}\) and \(2\times 10^{10}\) cm\({}^{-2}\) on the sapphire substrate has been fitted based on the formula for the Hall mobility \(\mu_{H}=e\langle\tau^{2}\rangle/m^{*}\langle\tau\rangle\).
\[\langle\tau\rangle=\langle\frac{1}{\tau_{wall}^{-1}+\tau_{dis}^{-1}+\tau_{ii}^ {-1}}\rangle. \tag{9}\]
The results are presented in Fig.2. As shown, there is agreement with the experimental data when 30K\(<T<100\)K for both samples. We found that the result of the fit essentially depends on the distance between dislocations in the wall \(p\), as compared to other parameters, and, hence on the angle of misorientation between grains (\(\theta=\arcsin(b/2p)\)).
Our preliminary results show that this effect of the angle variation on the Hall mobility can be noticeable even for a narrow interval of \(\theta\) between \(1^{\circ}\) and \(5^{\circ}\). The second very sensitive parameter in our calculations is the position of the dislocation acceptor level energy referred to the conduction band edge energy.
As noted above, Eq. describes both the dilatation around the low-angle grain boundary and disclination dipole. The concept of disclination dipole has been applied to obtain the deformations around high-angle grain boundaries and linear defects of the rotational type\({}^{22}\). In this connection, the observed in hexagonal GaN layers deformations associated with the 5/7 and 4/8 rings and high-angle grain boundaries (see, for example, Refs.\({}^{23,24}\)) can be considered in the framework of the disclination dipole model. The second point which should be noted concerns the interfacial misfit dislocations\({}^{25}\). Misfit dislocation scattering along with considered in this paper can be given in the framework of the multi-layer model proposed in\({}^{26}\).
## IV. CONCLUSIONS
In this article we have theoretically investigated the possible role of the strain field associated with the dislocation wall on the mobility in hexagonal GaN layers. It has been found that this contribution to the total transverse mobility can be noticeable at low temperatures for given above densities of such defects and deformation constant typical for semiconductors. Our calculations show the core scattering due to charged dislocation lines is dominant mechanism when \(N_{dis}>10^{9}\)cm\({}^{-2}\). This supports findings in previous publications devoted to this material. At lower dislocation densities the ionized impurity scattering and strain field scattering can dominate.
| 10.48550/arXiv.0902.4774 | Electron scattering due to dislocation wall strain field in GaN layers | S. E. Krasavin | 2,114 |
10.48550_arXiv.1211.3809 | ###### Abstract
We study the electrical transport properties of ensembles of bismuth telluride (Bi\({}_{2}\)Te\({}_{3}\)) nanoplates grown by solution based chemical synthesis. Devices consisting of Bi\({}_{2}\)Te\({}_{3}\) nanoplates are fabricated by surface treatment after dropping the solution on the structured gold plates and the temperature dependence of resistance shows a nonmetallic behavior. Symmetric tunneling behavior in _I-V_ was observed in both our experimental results and theoretical calculation of surface conductance based on a simple Hamiltonian, which excludes carrier-carrier interactions. Here, we present two devices: one showing symmetric, the other showing a two-step tunneling behavior. The latter can be understood in terms of disorder.
* E-mail address: (X. Lu), (H.
Bismuth telluride (Bi\({}_{2}\)Te\({}_{3}\)), a semiconductor with an indirect bulk energy band gap of 0.165 eV, is a unique multifunctional material. It is an attractive thermoelectric material with the highest figure of merit (\(ZT=0.68\)) at room temperature in its bulk. It was recently shown that in thin films of Bi\({}_{2}\)Te\({}_{3}\)\(ZT\) can be enhanced about ten times due to line dislocations in topologically protected perfectly conductive one dimensional state. This physical property is under investigation, yet it is a well-known fact that Bi\({}_{2}\)Te\({}_{3}\), like other members of its family (i.e. Bi\({}_{2}\)Se\({}_{3}\) and Sb\({}_{2}\)Te\({}_{3}\)) exhibit exotic properties in low dimension, such topological insulating state with a single Dirac cone. Dirac cone was clearly shown by angle resolved photoemisson spectroscopy (ARPES) studies, which have motivated a strong interest to elaborate dissipationless spin currents at room temperature in this material, in addition to the aforementioned superior thermoelectric properties.
In order to understand this newly discovered feature of Bi\({}_{2}\)Te\({}_{3}\) family and make use of it for spintronics applications, there have been many electrical transport studies. However, most of these observations have demonstrated that bulk transport dominates due to low conductivity of surface states compared to bulk and it appears as metallic behavior in resistance vs. temperature. There are some recent experimental studies which show an insulating behavior in electrical transport of Bi\({}_{2}\)Te\({}_{3}\) family. One of them attributes this insulating behavior to the coupling of top and bottom surfaces, as in conventional semiconductors. Recently, scanning tunneling microscopy (STM) studies showed a high transmission probability of surface states which are topologically protected, making tunneling in transport of Bi\({}_{2}\)Te\({}_{3}\) family a crucial exploratory field. However, tunneling has not been studied in detail for these materials.
It is well established that quantum spin Hall (QSH) state is robust against weak non-magnetic impurities and surface states are not affected, leading to high mobilities. However, recent work on the effect of strong disorder in HgTe/CdTe quantum wells, has shown the existence of Anderson insulator behavior. Depending on the strength of disorder, the size of the samples, and the position of the Fermi level, it is possible to observe the quantization of conductance. It is also possible to observe Anderson insulator behavior in disordered Bi\({}_{2}\)Te\({}_{3}\) family with strong spin-orbit interaction, in which disordered insulating bulk and conducting surface states coexist. This suggests Bi\({}_{2}\)Te\({}_{3}\) family can be promising Anderson insulator candidates. However, so far Bi\({}_{2}\)Te\({}_{3}\) has been reported to show metallic behavior except the case where Bi\({}_{2}\)Te\({}_{3}\) was prepared via a special cleaving after high temperature synthesis in a vacuum quartz tube.
Here, we present the current-voltage (\(I\)-\(V\)) characteristics of an ensemble of Bi\({}_{2}\)Te\({}_{3}\) nanoplates (NPs) for temperatures between 6 K and 300 K which exhibit insulating behaviour. Hexagonal Bi\({}_{2}\)Te\({}_{3}\) NPs with an average size of 250 nm and a thickness about 10 nm were prepared via a solution method. These plates were then dispersed in ethanolor chloroform and deposited onto gold pads of 100 nm thickness. Devices are formed, after the solution dries and leaving aggregates of the nanoplates on and between the gold contacts. In a device containing many NPs a clear asymmetric and two-step tunneling behavior appears at temperatures lower than 200 K, even though a simple theory based on the newly discovered state of matter, which does not account for carrier-carrier interactions, suggests symmetric _I-V_ curves at all temperatures. Based on a recent theory, we discuss the possibility that this behavior may be due to randomly distributed impurities in Bi\({}_{2}\)Te\({}_{3}\) NPs.
## 2 Chemical Synthesis and Structural Characterization
The chemical synthesis of Bi\({}_{2}\)Te\({}_{3}\) NPs is as follows. In a glove box, 0.0386 g Bi(CH\({}_{3}\)COO)\({}_{3}\) was mixed with 4 mL squalane. 0.5 M of Tellurium (Te) in trioctylphosphine (TOP) was prepared by dissolving 0.033 g of Te in 2.6 mL of TOP and stirring for two hours in the glove box. Then, 4 mL of Bi(CH\({}_{3}\)COO)\({}_{3}\) solution in squalane was mixed with 0.7 mL of Te(TOP), followed by rapidly injection of the mixture into a three-neck flask equipped with a condenser under nitrogen protection. The reaction mixture was heated to 250 \({}^{\circ}\)C for 15 minutes. Black precipitate was isolated by centrifugation, followed by washing with chloroform, for five times. The resulting Bi\({}_{2}\)Te\({}_{3}\) NPs were collected and dispersed in hexane.
The inset to Figure 1(d) shows a typical scanning electron microscopy (SEM) image of NPs. The NPs solution then was drop cast onto a 300 nm Si/SiO\({}_{2}\) substrate with lithographically patterned 100 nm thick gold plates. Upon evaporation of the solvent, aggregate of Bi\({}_{2}\)Te\({}_{3}\) NPs formed between gold plates. The substrate was then subjected to a surface treatment with ozone stripping at 140 \({}^{\circ}\)C for 20 minutes in order to eliminate carbon radical residues. We perform X-ray photoemission spectroscopy (XPS) just before (red dashed curve) and after (black solid curve) the latter process to confirm the removal of the chemical residues, as in Figure 1(a) and 1(b), for Bi and Te peaks, respectively. In Figure 1(a), Bi 4f\({}^{2}\) and 4f\({}^{62}\) peaks at 160 eV and 165.3 eV, respectively appear after surface treatment; similarly in Figure 1(b) Te 3d\({}^{52}\) and 3d\({}^{32}\) peaks at 577.2 and 587.6 eV, respectively appear after surface treatment. In Figure 1(c), the X-ray diffraction (XRD) data of Bi\({}_{2}\)Te\({}_{3}\) NPs show that it is single phase, and the peaks are indexed to the rhombohedral Bi\({}_{2}\)Te\({}_{3}\). The calculated lattice parameters are \(a\) = 4.369 A and \(c\) = 30.423 A, slightly smaller than those of JCPDS 08-0021, \(a\) = 4.381 A and \(c\) = 30.483 A. After the surface treatment we measured Raman spectroscopy with excitation of 785 nm, with a power of \(\sim\) 0.2 mW, as seen in Figure 1(d). We observe the optical phonon mode A\({}_{1u}\) which is Raman-inactive in bulk, but infra-red active due to crystal symmetry breaking, as reported for mechanically exfoliated Bi\({}_{2}\)Te\({}_{3}\).
## 3 Electrical Transport Results and Discussion
The samples were annealed at 250 \({}^{\circ}\)C for 30 minutes to obtain electrical continuity between the gold plates and the ensemble of NPs. SEM images of two different junctions with separations of 2 \(\upmu\)m and 5 \(\upmu\)m for devices 1 and 2 are shown in Figure 2(a) and 2(b), respectively. Device 1 contains fewer NPs than device 2. The resistance as a function of temperature is measured by applying a current of 1 \(\upmu\)A. For comparison, we amplified the resistance of device 2 by 35 times. The electrical transport is throughout many conduction paths between source and drain. The resistance of the device with more conduction paths would be lower. This qualitatively explains why device 2 has about 35 times less resistance. Both samples show an insulating behavior as indicated in Figure 2(c), with device 2 exhibiting a relatively rapid increase in resistance at temperatures lower than 20 K which may be due to carrier-carrier interactions. For device 1, we observe symmetric tunneling I-V curves at all temperatures. As we increase the temperature from 6 K up to 300 K, the measured current increases for the same applied voltage, as shown in the _I-V_ characteristics in This is in agreement with the resistance vs. temperature data in Figure 2(c).
This system can be simply thought of Bi\({}_{2}\)Te\({}_{3}\) NPs in between two metal (M) contacts, i.e. M/Bi\({}_{2}\)Te\({}_{3}\)/M. We can make a qualitative comparison between experiment and theory by making use of a two dimensional system described by the Bernevig-Hughes-Zhang (BHZ) model, excluding carrier-carrier interactions.
\[H=\sum_{i,\sigma,a}\varepsilon_{a}c_{ia\sigma}^{+}c_{ia\sigma}-\sum_{ia\omega \omega\beta}t_{aa}c_{i+aa\sigma}^{+}c_{i\beta\sigma}\;\;, \tag{1}\]
The electron hopping integral is a 2\(\times\)2 matrix in this basis
\[t_{a\sigma}=\begin{pmatrix}t_{ss}&t_{sp}e^{i\omega\theta_{a}}\\ t_{sp}e^{-i\omega\theta_{a}}&-t_{pp}\end{pmatrix}, \tag{2}\]
The bands near the \(\Gamma\)-point become inverted if
\[\varepsilon_{s}-\varepsilon_{p}\leq 4(t_{ss}+t_{pp})\;. \tag{3}\]
Our calculations of the _I-V_ characteristics of M/Bi\({}_{2}\)Te\({}_{3}\)/M junctions with \(N\)=1 to 4 monolayer (ML) at room temperature are shown in The _I-V_ curves strongly depend on the number of MLs. The current density for \(N\)=4 is much larger than that ones for \(N\)=1-3, where \(N\) is the number of ML. On the other hand, there is no big difference between _I-V_ curves for \(N\)=2 and \(N\)=3. This effect can be explained in terms of the different density of states for different number of MLs.
Independent of the thickness, there is a symmetric behavior of the _I-V_ curves which stays symmetric as temperaturedecreases. However, it is not possible to make a direct quantitative comparison with experiment since NPs are quite randomly distributed. Ideally, it is expected that only surface states are conductive and the largest contribution to the measured current comes from the surface (as calculated in the simulation). In addition, the current density at the surface must be much larger while in the bulk it is very small. Therefore, the experimental current density is expected to be not uniform within the sample. Nevertheless, the symmetric tunneling behavior in _I-V_ curves in the simulations is similar to our experimental observations in device 1 and there is a qualitative agreement. Extension of this simple model to 3D and inclusion of carrier-carrier interactions would provide more insight into tunneling studies and for understanding the behavior in device 2, as in the following.
For device 2, we measure asymmetry in _I-V_ as shown in which is accompanied by a peak in the junction resistance versus bias voltage appearing around 0.5 V at 6 K as shown in As we increase the temperature gradually to 300 K, the asymmetry in _I-V_ changes into symmetric _I-V_ as in device 1. The inset to clearly reveals this behavior as we plot the calculated resistance (voltage/current) vs. bias voltage. This secondary hump has a resistance value of approximately 10 % (~45 k\(\Omega\)) of the resistance value (~0.45 M\(\Omega\)) around zero applied voltage. At 6 K, a potential barrier \(\sim\) 0.5 V gets activated which weakens as temperatures increases. We extract this potential barrier by Gaussian fitting of the calculated resistance vs.
Although the origin of this feature is not clear, we ascribe this temperature dependent activation potential to the existence of randomized impurities as predicted in Anderson insulators in which surface states are formed due to impurities induced in the system. In the case of strong disorder, at certain current densities, it is possible to observe local conductance extrema. As temperature increases, this effect may be smeared out due to phonon scattering. Alternatively, it is possible that for positive applied voltages hopping mechanism is leading to a decrease in the current density compared to negative applied voltages in which the current density is exponentially dependent on the applied voltage. Although this could be valid for samples with relatively lower resistance (e.g. device 2), it might not be possible to resolve for samples with higher resistance (e.g. device 1). Recently, we observed similar I-V characteristics in Bi\({}_{2}\)Te\({}_{3}\) flakes that are mechanically exfoliated, i.e. two step tunneling behavior at much lower positive applied voltages.
## 4 Conclusion
In conclusion, the _I-V_ characteristics of ensemble of chemically synthesized Bi\({}_{2}\)Te\({}_{3}\) nanoplates have shown a tunneling behavior as determined from _I-V_ curves measured from 6 K up to room temperature. As expected from a simple 2D Hamiltonian considering no carrier-carrier interactions, a symmetric tunneling behavior has been observed, except one showing a two-step tunneling behavior. The latter is believed to be due to randomly distributed impurities,but it is also possible due to unconventional hopping mechanism. Future theoretical elaborations and experimental investigations on disorder and transport on single nanoplates could give insight on this promising thermoelectric and spintronic material for device applications.
| 10.48550/arXiv.1211.3809 | Tunneling behavior of bismuth telluride nanoplates in electrical transport | Mustafa Eginligil, Weiqing Zhang, Alan Kalitsov, Xianmao Lu, Hyunsoo Yang | 3,947 |
10.48550_arXiv.1511.09087 | ## I Introduction
Recently, the quest for new thermoelectric materials beyond Bi\({}_{2}\)Te\({}_{3}\) has gained considerable momentum which is owed to the increasing need of highly performing materials for energy harvesting and conversion. The most promising materials classes identified so far are found among the semiconductors, which allow a careful adjustment of charge carrier type and concentration in order to achieve the optimum balance between thermopower, electrical, and thermal conductivity. Together with the idea to exploit low-dimensional quantum-well structures to increase the thermoelectric performance, this leads to a series of new high performance materials which are formed of heterostructures or quantum dot superlattices. Oxides, such as Na\({}_{x}\)CoO\({}_{2}\)-type or Ca\({}_{3}\)Co\({}_{4}\)O\({}_{9}\), represent another promising materials class with the advantage of non-toxicity and abundance of their components in combination with the chemical and thermal stability rather than record-breaking performance values.
Low-dimensionality is readily realized in naturally layered structures, which are for instance found in a specific class of cobaltates, the delafossites. Among those hexagonal \(AB\)O\({}_{2}\) compounds, PdCoO\({}_{2}\) and PtCoO\({}_{2}\) take a special position, since these are - unlike most of the oxides, which are semiconducting or insulating - very good metallic conductors. Such a material usually disqualifies for thermoelectric applications, but it was discovered early that the conductivity of PdCoO\({}_{2}\) and PtCoO\({}_{2}\) is highly anisotropic. Recent theoretical work reports also a qualitative difference in the thermoelectric properties with respect to transport within the \(a\)-\(b\) plane and along the hexagonal \(c\)-axis. The calculations predict significant negative values of about \(-100\,\mu\)V K\({}^{-1}\) (PdCoO\({}_{2}\)) and \(-250\,\mu\)V K\({}^{-1}\) (PtCoO\({}_{2}\)) for the out-of-plane component of the thermopower at room temperature, in contrast to moderate \(+5\,\mu\)V K\({}^{-1}\) in-plane. Such a large anisotropy in the thermopower can give rise to a considerable laser induced voltage, which can be exploited to design photosensors based on the transverse thermoelectric effect.
Both compounds are stable and have been synthesized as high quality single crystals employing the so-called methathetical reaction. In PtCoO\({}_{2}\) and PdCoO\({}_{2}\), the Co ion is found in the trivalent Co\({}^{3+}\) state, which shows a \(S\!=\!0\) low spin configuration, while Pd and Pt are monovalent. Experiment and first principles theory show consistently that the Fermi level of PdCoO\({}_{2}\) is populated by Pd states, while oxygen states are scarce and Co states are essentially absent. Photoemission spectroscopy confirms the validity of this picture also for PtCoO\({}_{2}\). Thus the high in-plane conductivity arises from the hybridized noble metal 4d and 5s electrons which contribute states at the Fermi level, whereas the CoO\({}_{6}\) octahedra can be thought to form an insulating layer, inhibiting transport along the perpendicular \(z\)-axis. The absence of hybridized Co states at the Fermi level results in a quasi-two-dimensional electronic arrangement. This is mirrored by a Fermi-surface, which shows nearly no dispersion along \(k_{z}\) and thus has a quasi two-dimensional shape. In consequence, the Fermi velocities, i. e., the gradients of \(k\)-resolved band structure at \(E_{\rm F}\), are restricted to the \(x\)-\(y\) plane, while the rather flat Fermi surface along \(k_{z}\) prevents a significant contribution to the out-of-plane conductivity \(\sigma_{xx}\). The predicted cross-section of the Fermi surface in the shape of a closed hexagon was validated by Noh _et al._ based on angular-resolved photoemission spectroscopy (ARPES).
In the present work, we consider PdCoO\({}_{2}\) and PtCoO\({}_{2}\) as an intrinsically layered system serving as a simple prototype of a heterostructure considered for future thermoelectric applications. We determine the dependence of the electronic structure on epitaxial in-plane strain by means of first-principles calculations. Using this as input, we obtain the transport properties in the framework of semi-classical Boltzmann transport theory in the constant relaxation time approximation. We identify a remarkable dependence of the electronic transport of PtCoO\({}_{2}\) on epitaxial strain, which stands in clear contrast to the rather uniform strain response of the elastic properties. We will show that this stunning discrepancy is related to an electronic topological transition, which, in turn, is beneficial in improving the thermoelectric performance of the material. It manifests in a variation of the out-plane thermopower by a factor of three and the out-of-plane conductivity by more than one order of magnitude.
After a short survey of the computational details in Sec. II, we will shortly review the similarities in the structural behavior of expitaxially strained PtCoO\({}_{2}\) and PdCoO\({}_{2}\) in Sec. III.1. The different strain response of the electronic structure is explained in Sec. III.2, while Sec. III.3 is devoted to the immediate consequences for in-plane and out-of-plane conductivity. Thermopower and a prediction for the thermoelectric figure of merit for the strained materials are presented in Sec. III.4.
## II Numerical details
The electronic structure was investigated within the framework of density functional theory (DFT) employing the Vienna ab initio Simulation package (VASP),Kresse and Furthmuller; Kresse and Joubert which uses a plane wave basis set for the description of the valence electrons in combination with the projector augmented wave approach (PAW).Blochl In our calculations, we considered explicitly the \(2s^{2}2p^{4}\) electrons for O, \(3d^{8}4s^{1}\) for Co, \(4p^{6}4d^{9}5s^{1}\) for Pd and \(5d^{7}6s^{1}\) for Pt, choosing a cutoff energy of 500 eV. The exchange-correlation part of the Hamiltonian was represented by the generalized gradient approximation (GGA) of Perdew, Burke and Ernzerhof.Perdew et al.
The response of the system to epitaxial strain exerted along the \(a\)-\(b\)-plane was calculated from the hexagonal 12 atom unit cell within the scalar-relativistic approximation. The hexagonal basis of this cell allows a straightforward manipulation of the lattice parameters according to the epitaxial constraint. For a given value of \(a\), the \(c\) lattice parameter and atomic positions were optimized for minimum energy modelling the epitaxial constraint. In order to obtain accurate eigenvalues, as needed for the transport calculations described below, we transformed the 12 atom unit cell with hexagonal basis to the 4 atom primitive cell with rhombohedral basis using a mesh of 41\(\times\)41\(\times\)41 \(k\)-points. We included the spin-orbit term to the Hamiltonian in our self-consistent treatment, which yields small but noticeable changes to the results. For additional corroboration and high quality Fermi surfaces, parts of the calculations were repeated with the full-potential augmented plane wave method Wien2k.Wien2k Thermoelectric properties at finite temperatures were obtained in the framework of semiclassical Boltzmann transport theory in the constant relaxation time approximation under the constraint of a conserved number of carriers. This step was carried out with the BoltzTraPBlochl code based on the the eigenvalues obtained from our VASP calculations. In order to test the effect of additional correlation on the electronic structure and transport properties, we applied the GGA+U scheme in the rotationally invariant formulation of Dudarev _et al.Dudarev et al._ on the Co \(d\) states for selected cases, using different values of \(U-J\) up to 5 eV. Further technical details can be found in the supplementary material.Dudarev et al.
## III Results and discussion
### Structural properties strained PdCoO\({}_{2}\) and PtCoO\({}_{2}\)
The delafossites crystallize in a hexagonal structure (see Fig. 1), which can be described by the rhombohedral space group 166 with symmetry \(R\overline{3}m\). Pt or Pd are found on the (1a) site at \(\), Co on the (1b) site at \((0,0,1/2)\) and the two oxygen on the (2c) site at \((0,0,\pm u)\), with \(u\) as an internal structural parameter. This results in a naturally layered arrangement, consisting of layers of corner-sharing CoO\({}_{6}\) octahedra and linear O-Pd-O or, respectively, O-Pt-O dumbbells. The Pd/Pt atoms form a hexagonal layer with the triangular faces of the CoO\({}_{6}\) octahedra on top of the triangular facets of the Pd/Pt, such that the edge oxygen of CoO\({}_{6}\) become part of the dumbbells.
We find the ground state of both delafossites at
(color online) Representation of PdCoO\({}_{2}\) and PtCoO\({}_{2}\) delafossites in a unit cell with 12 atoms corresponding to the hexagonal basis (left image, 3 times replicated in \(a\) and \(b\) direction) and a four-atom rhombohedral primitive cell (right).
e., \(a\!=\!2.870\) A and \(c\!=\!17.94\) A for PdCoO\({}_{2}\) and \(a\!=\!2.861\) A and \(c\!=\!17.95\) A for PtCoO\({}_{2}\). Consequently, the atomic volume of the Pd-compound is with \(V\!=\!10.66\) A\({}^{\frac{3}{4}}\)/atom slightly larger than the volume of the Pt-compound (\(V\!=\!10.61\) A\({}^{\frac{3}{4}}\)/atom). These values agree well with previous DFT investigations and experiment (\(a\!=\!2.83\) A for both compounds), with the typical overestimation of the lattice constant by approximately one percent as a consequence of the use of the GGA for the exchange-correlation potential. Both systems show an essentially identical response to epitaxial strain. The lower panel of indicates that epitaxial growth with in-plane strains along \(a\) of up to 5 % might be realistic, since the corresponding deformation energies are still in the range of typical thermal energies. Applying in-plane strain results in a corresponding opposite strain of the out-of-plane lattice parameter \(c\), which is, however, approximately only half as large. This has the consequence that upon straining the system by \(\varepsilon\!=\!\Delta a/a\!=\!\pm 4\) % the equilibrium volume changes significantly by \(\Delta V/V\!=\!\pm 5\) %. The increase of \(c/a\) with decreasing volume is consistent with pressure experiments on PdCoO\({}_{2}\). Epitaxial strain also causes a variation of the internal parameter \(u\), which determines the oxygen position relative to the other ions. At equilibrium conditions, we find \(u\!=\!1.1129\) for PdCoO\({}_{2}\) and \(u\!=\!1.1123\) for PdCoO\({}_{2}\). This corresponds to a Pd-O distance of 2.026 A and a Pt-O distance of 2.015 A in combination with Co-O distances of 1.917 A and 1.919 A, respectively. Four percent of tensile strain causes \(u\) to increase by approximately 2 %. This has an opposite effect on the Co-O distance as compared to the distance between O and the Pt-group metal. For both oxides, the Co-O distance increases by +1.8 %, while the distance between O and the noble metal decreases by approximately \(-0.5\) %. Under \(-4\) % compressive strain, we observe the opposite effect with variations of \(-1.5\) % and +0.6 %, respectively.
### Electronic structure and Fermi surface of strained PtCoO\({}_{2}\) and PdCoO\({}_{2}\)
The band structure of the unstrained delafossites, compared in Fig. 3, suggests at first sight that the electronic features relevant for the transport properties, which appear close to the Fermi level, \(E_{\rm F}\), are similar for both compounds. In both cases only one band (i. e., the 16th band of valence electrons disregarding the semicore states in counting) is crossing \(E_{\rm F}\) which is formed by \(d_{3z^{2}-r^{2}}\) and \(s\)-type orbitals with Pd/Pt character. Its rather steep slope in \(k_{x}\)-\(k_{z}\)-direction, which is responsible for the excellent in-plane conductivity, corresponds to a section of a parabolic structure, which has its minimum at \(\Gamma\) at \(-2.6\) eV for PtCoO\({}_{2}\) and \(-2.3\) eV for PdCoO\({}_{2}\) and is intersected by hybridizing bands with d-character several times below \(E_{\rm F}\). Although most other bands of PtCoO\({}_{2}\) are shifted downward due to the larger bandwidth related to the larger extent of the \(5d\) shell of Pt compared to the \(4d\) electrons of Pd, the 16th band does not change its shape in the immediate vicinity of \(E_{\rm F}\).
(color online) Comparison of total energy (bottom graph, left axis), internal parameter \(u\) (bottom graph, right axis), out-of-plane strain (top graph, left axis) and unit cell volume (top graph, right axis) relative to the respective bulk equilibrium value as a function of the relative in-plane strain for PdCoO\({}_{2}\) (dark blue color) and PtCoO\({}_{2}\) (bright orange color) obtained from our first-principles calculations.
(color online) Band structure of PtCoO\({}_{2}\) (solid orange line) and PdCoO\({}_{2}\) (dashed blue line) in the vicinity of the Fermi level. The numbers denote the bands close to \(E_{\rm F}\), which are relevant for our discussion. The counting starts from the valence band minimum (core and semicore states such as the Pd \(4p^{6}\) are not considered).
This is different for the next lower, completely filled band (i. e., the 15th band), which forms hole pockets in the vicinity of the zone boundary, in particular, near the special points F and L. The next higher unoccupied band (17th band) displays a marked minimum (electron pocket) along \(\Gamma\)-L, which comprises Co \(e_{\rm g}\) states that are absent in the vicinity of \(E_{\rm F}\) otherwise. In PtCoO\({}_{2}\), these features of the 15th and the 17th band come very close to \(E_{\rm F}\) within 100 meV, while in PdCoO\({}_{2}\), they are separated from the Fermi surface by more than 300 meV. As we will show below, these fine details of the electronic structure give rise to qualitatively different transport properties under epitaxial strain which stand in contrast to the rather uniform elastic response of both systems.
The energy minimum along \(\Gamma\)-F in the 17th band might be interpreted as a consequence of the hybridization between the Co \(3d_{e_{\rm g}}\) bands with the the parabolic section of the 16th band (see the Co-resolved band structure of PtCoO\({}_{2}\) in Fig. 4d-4f), which manifests in a sharp peak in the electronic density of states 0.4 eV above the Fermi level (see supporting material and Refs. for the density of states and orbital resolved band-plots), while the Co \(3d_{t_{\rm 2g}}\) and the Pd \(4d_{xz}\) and \(4d_{yz}\) states are located 0.5-2 eV below \(E_{\rm F}\). In comparison, the hybridization between Pt \(5d_{3z^{2}-r^{2}}\) and Co \(3d_{e_{\rm g}}\) states is less pronounced and the band-crossing is encountered in the immediate vicinity of the Fermi surface. A strain-induced change of the Brillouin-zone can influence the position of this crossing and thus show significant impact on the shape of the Fermi surface. This might be exploited to alter transport properties by applying external fields, such as mechanical stress.
The crossing of the 16th band results in a hexagonal rod-like shape of the Fermi surface, which is a pronounced feature of both compounds. Under a moderate compressive epitaxial strain of \(\varepsilon\!=\!-4\,\%\), the extremal features of the 15th and 17th band of PtCoO\({}_{2}\) touch the Fermi surface and changes its topology significantly. This is depicted in and The (blue) needle-type features outside the (orange) hexagonal rod evolve from the rising of the hole pocket in the 15th band, which is best seen at the L-point. At the same time, the 17th band approaches the Fermi level along \(\Gamma\)-F from above resulting in the (red) pills inside the hexagonal rods. According to its Pt-type band-character (not shown), we expect the 15th band to contribute predominantly to the in-plane conductivity. In turn, the 17th band shows a strong Co-character close to its minimum where the electron pocket forms and is consequently expected to increase the out-of-plane conductivity (see Sec. III.3). While the faces of the hexagon formed by the 16th band are essentially dispersionless irrespective of the strain, we find bulb-like extrusions at its rounded corners. The extrusions are also present on the Fermi surface of PdCoO\({}_{2}\) and are again associated to a slight admixture of Co-states in band 16 along \(\Gamma\)-F. In PtCoO\({}_{2}\), these features grow under compressive strain and eventually touch the Brillouin zone boundary, while they become significantly flatter under tensile strain, in turn. This corresponds to a diminished band velocity component in \(z\)-direction. Tensile epitaxial strain is thus a suitable possibility to control the Pd/Pt-character of the 16th band and thus enhance the twodimensional nature of the electronic structure. In reverse, compressive strain can stimulate an electronic topological transition, which arises from the the appearance of additional bands at the Fermi level and the 16th band touching the Brillouin zone boundary. In PdCoO\({}_{2}\), the distance between the Co- and Pd-states along \(\Gamma\)-F is significantly larger, which makes the electronic states at the Fermi level much less susceptible to strain. However, electron doping may provide an alternative way to bring the systems close to this instability and produce a Fermi surface of similar shape.
Based on experimental evidence, PtCoO\({}_{2}\) and PdCoO\({}_{2}\) were considered as weakly correlated oxides. Furthermore, Ong _et al._ found a very good agreement between the first-principles electronic structure and experimental spectroscopy, which justifies the use of conventional GGA for the exchange and correlation part of the Hamiltonian. However, recently, Hicks _et al._ argued that static Coulomb repulsion in terms of a Hubbard model with an effective \(U\)-parameter on the Co \(3d\) states improves the agreement with their de-Haas-van-Alphen data. Therefore, we carried out additional GGA+\(U\) calculations for PdCoO\({}_{2}\) and PtCoO\({}_{2}\) under a systematic variation of the effective Coulomb repulsion \(U_{\rm eff}\!=\!U-J\) on Co \(3d\) from 0 (pure GGA) to 5 eV. The additional term mainly affects the bonding \(3d\) Co \(t_{\rm 2g}\) states below \(E_{\rm F}\) which move further down, while we observe only a minute shift in the position of the relevant Co \(e_{\rm g}\) states, which are close to \(E_{\rm F}\). This separates occupied and unoccupied Co and O states further, but hardly affects the Pd-states at the Fermi surface, which are responsible for metallicity. Thus \(U_{\rm eff}\) has only minor effect on the position of the Pt-dominated 15th band but larger values will eventually inhibit the crossing of the 17th band or shift it to larger strains.
It is worthwhile to note that the occurrence of the electronic topological transition is independent of the calculation method and likewise reproduced with VASP and Wien2k. Nevertheless, concerning the transport calculations we obtain slight differences between the codes. Interestingly, we find a closer agreement of the GGA results obtained from the Wien2k calculations with the VASP calculations for \(U_{\rm eff}\!=\!1\,\)eV, rather than for the pure GGA, although the same exchange-correlation functional (PBE) was used in both cases. We ascribe this to the specific implementation of the Co PAW-potential employed in the VASP calculations, which apparently allows for a somewhat stronger interaction between Co and Pd states resulting in a slightly increased presence of residual Co-states at the Fermi-level compared to the full-potential calculations. For additional details see the supplementary information.
### Electronic conductivity under epitaxial strain
Starting from the electronic eigenvalues obtained with DFT, we determined the transport properties in the framework of semi-classical Boltzmann transport theory using the constant relaxation time approximation. This approach has evolved as a standard tool for the prediction and identification of qualitative trends in a wide variety of materials, including the prediction of oxidic systems, and has been applied successfully to several delafossites in the past. Concerning PdCoO\({}_{2}\) and PtCoO\({}_{2}\), particular emphasis was laid on the explanation of the large anisotropy in conductivity and thermopower (Seebeck coefficient), but also on the description of the large out-of-plane magnetoresistance encountered in PdCoO\({}_{2}\) under the rotation of a large in-plane magnetic field. The relaxation time approximation, with a single energy- and momentum-independent relaxation time, assumes that the combined effect of all scattering processes is such that the electronic system relaxes back to equilibrium exponentially with a single time constant once the perturbation is switched off.
(color online) Fermi surfaces (top and center rows, subfigures a-c) and corresponding band structure (bottom row, subfigures d-f) of strained and unstrained PtCoO\({}_{2}\). The left column (subfigures a and d) refers to \(\varepsilon\!=\!-4\,\%\) compressive epitaxial strain, the right column (subfigures c and f) to \(\varepsilon\!=\!+4\,\%\) tensile strain and the center column (subfigures b and e) to the unstrained case. In the images of the Fermi surfaces the different colors refer to the different bands crossing the Fermi levels (blue: 15th band, orange: 16th band, red: 17th band). The vertical width of the (bright blue) fat bands in the bottom row is proportional to the Co-character of the band. Again, the bands crossing \(E_{\rm F}\) are denoted by their respective number. We find that the 15th and 17th band cross the Fermi energy at compressive strains, whereas for tensile strains, the dispersion of the 16th band at \(E_{\rm F}\) is influenced by the nearby crossing of a parabolic section of the 16th Pt-band and the 17th band which is dominated by Co states.
In any case, the relaxation time approximation allows for solving the Boltzmann equation in a simple way, and hence an accurate determination of transport coefficients using \(10^{4}\) to \(10^{5}\)\(k\)-points, which are necessary for converged results. From the comparison with experiment, on the other hand, one is then able to deduce information on the relevant scattering mechanisms.
A more detailed theoretical description could be based on Boltzmann theory, but would require a first-principles calculation of phonon dispersions as well as electron-phonon scattering matrix elements, which is beyond the scope of the present work. Alternatively, the determination of transport coefficients could be based directly on a Green's function approach and Kubo's linear response theory (see Ref. for a recent discussion).
At room temperature and above, it can be expected that electron-phonon scattering is the dominant relaxation mechanism. For the nearly-free electron model, the scattering operator indeed has been studied in detail. For example, it is well known that for temperatures below the Debye temperature, the momentum and energy relaxation rates are very different, the latter being much shorter than the former. Above the Debye temperature, electron-phonon scattering is essentially elastic, and both rates are very similar. These results rely, in particular, strongly on the fact that the electronic density of states (DOS) is practically independent of energy around the Fermi level. Thus predictions based on an energy-independent \(\tau\) are likely to fail in case sharp features exist close to \(E_{\rm F}\). This was shown recently for the simple metal Li, where the constant relaxation time approximation predicts the wrong sign of the thermopower. For Li, quantitative agreement can be achieved within a variational approach to Boltzmann theory. Since our calculated transport properties are consistent with the available experimental data for PdCoO\({}_{2}\), as shown below, we do not expect significant new insights from an advanced treatment of the Boltzmann equation. However, our results will demonstrate clearly that - according to the anisotropic nature of the lattice structure - one must take into account at least a directional dependence of \(\tau\) in addition to its variation with temperature.
In practice, all transport tensors obtained from Boltzmann transport theory turn out to be essentially diagonal; there are only two distinct entries. The first, marked with the index "xx", corresponding to in-plane transport, the second, marked as "zz" corresponding to transport properties perpendicular to the Co-O and Pd-O/Pt-O layers.
As already anticipated, the significant changes at the Fermi surface of PtCoO\({}_{2}\) should leave a corresponding signature in the strain-dependent conductivity, shown in The variation of the in-plane component, which accounts for the already very good conductivity in this direction, is not substantial (upper panel). According to the steeper slope at \(E_{\rm F}\), we find a larger ratio \(\sigma_{\rm xx}/\tau\) for PtCoO\({}_{2}\) compared to PdCoO\({}_{2}\). Both values are steadily reduced for tensile strains. However, as indicated earlier, the conductivity is highly anisotropic and thus orders of magnitude smaller in the perpendicular direction. According to the changes in the Fermi surface, the strain induced variation of the out-of-plane conductivity is much more significant for bulk PtCoO\({}_{2}\). Expanding the \(a\)-axis by 4% decreases the out-of-plane conductivity by almost one order of magnitude. In turn, the opposite trend is encountered for compressive strain. This significantly increases the conductivity compared to PdCoO\({}_{2}\), where the variations are much smaller and do not exhibit a consistent trend for both strain directions.
The strong impact of the Fermi surface on the conductivity anisotropy in PtCoO\({}_{2}\) is explained by the band resolved in-plane and out-of-plane components of the electrical conductivity tensor of PtCoO\({}_{2}\). We see that for tensile strains only the 16th band contributes since it is still the only one crossing the Fermi surface.
(color online) In-plane and out of plane components of the electrical conductivity tensor, \(\sigma_{\rm xx}\) (upper panel) and \(\sigma_{\rm zz}\) (lower panel), as a function of epitaxial strain. The results are specified relative to the empirical parameter of Boltzmann transport theory, the relaxation time constant \(\tau\). We compare the two different compounds PdCoO\({}_{2}\) (dashed lines and squares) and PtCoO\({}_{2}\) (solid lines and circles) are compared at two different temperatures \(T\!=\!300\) K (open symbols) and \(T\!=\!600\) K (filled symbols). Please note the semi-logarithmic scale in the bottom panel.
The additional bands start contributing to the conductivity tensor below an in-plane lattice constant \(a\!<\!2.80\) A (15th band) and \(a\!<\!2.83\) A (17th band), respectively, see For tensile strains, the conductivity is maintained by the 16th band, alone. Its strong decrease can be related to an increasingly better definition of the hexagonal corners of the Fermi-surface along the \(k_{z}\), as visible from the comparison of the Fermi surface under tensile strain with the equilibrium Fermi surface (Fig. 4a-4c, center row). This corresponds to a diminished band velocity component in \(z\)-direction, which consequently reduces the respective element of the conductivity tensor. In a simple geometric picture, this complies with the decreasing width of the conducting PtO\({}_{2}\)-layer, while the separation between O and Co and thus the width of the insulating layer becomes larger.
Recent experiments obtained for the anisotropy \(\sigma_{\rm xx}/\sigma_{\rm zz}\) of PdCoO\({}_{2}\) values ranging from 150 (Ref.), 280 (Ref.) to 400 (Ref.). This large discrepancy has been noticed earlier and related to the treatment of umklapp processes and the anisotropy of defect scattering cross sections. In particular, the latter aspects point out possible shortcomings of the single relaxation time approximation and semi-classical Boltzmann transport theory. Computing the anisotropy ratio from assuming a single constant \(\tau\), the in-plane conductivity is a factor 30 larger than out-of-plane. This ratio can be somewhat increased by applying static Coulomb correlations within the GGA+\(U\) scheme (see the supporting information for more details) but it never becomes comparable to the experimental results. We take this as another indication that a directional dependence of \(\tau\), which has been neglected so far, should enter the anisotropy ratio.
The diagonal structure of the transport tensors allows us to pragmatically circumvent this problem by introducing two effective, temperature dependent relaxation time constants for in-plane and out-of-plane processes, \(\tau_{\rm xx}\) and \(\tau_{\rm zz}\), respectively, which we obtain by comparison with the experimental conductivities. This effectively reproduces the \(k\)-vector dependence of \(\tau\), which is discarded in Boltzmann theory within the common single relaxation time approximation. As for the anisotropy, experiment offers a considerable span of results also for the in-plane experimental conductivities \(\sigma_{\rm xx}\), ranging from \(14.5\,\mu\Omega^{-1}\)m\({}^{-1}\) (Ref.), over \(32.3\,\mu\Omega^{-1}\)m\({}^{-1}\) (Ref.) and \(38.5\,\mu\Omega^{-1}\)m\({}^{-1}\) (Ref.) to \(50\,\mu\Omega^{-1}\)m\({}^{-1}\) (Ref.). This results in relaxation time constants \(\tau_{\rm xx}(300\,{\rm K})\) of \(46\,{\rm fs}\), \(101\,{\rm fs}\), \(121\,{\rm fs}\) and \(157\,{\rm fs}\), respectively. These comparatively large values are consistent with the previous assessment of Ong _et al._, obtained in a similar fashion by comparing Boltzmann theory and experimental conductivity data. From the residual in-plane resistivities for \(T\to 0\), Hicks _et al._ concluded on an extremely large transport mean free path \(l_{\rm MFP}\!=\!20\,\mu{\rm m}\). This is considerably larger than the earlier estimate \(l_{\rm MFP}\!=\!60\) A of Noh _et al._ based on the peak width obtained from ARPES. These measurements also yield a value for the in-plane carrier velocity of \(4.96\,{\rm eV}\,\dot{\Lambda}\,h^{-1}\) which is consistent with first-principles results. Thus, relaxation time constants obtained from experiment were either significantly smaller, such as the \(7.6\,{\rm fs}\) estimated by Noh _et al._ or larger, based on the de-Haas-van-Alphen measurements of Hicks _et al._ or the analysis of anomalous magnetoresistance. For the out of plane direction, ambient \(\sigma_{\rm zz}\) was measured as \(0.096\,\mu\Omega^{-1}\)m\({}^{-1}\) (Ref.), \(0.097\,\mu\Omega^{-1}\)m\({}^{-1}\) (Ref.) and \(0.115\,\mu\Omega^{-1}\)m\({}^{-1}\) (Ref.), corresponding to \(\tau_{\rm zz}(300\,{\rm K})\!=\!9\,{\rm fs}\) and \(11\,{\rm fs}\), respectively, which yields a more uniform picture than the in-plane-case.
Most experiments were carried out at room temperature and below, only Takatsu _et al._ performed conductivity measurements up to \(500\,{\rm K}\). Thus, only Ref. offers a reasonable possibility to extrapolate the experimental conductivities to \(T\!=\!600\,{\rm K}\). We find then \(\sigma_{\rm xx}\!=\!5\,\mu\Omega^{-1}\)m\({}^{-1}\) and \(\sigma_{\rm zz}\!=\!0.04\,\mu\Omega^{-1}\)m\({}^{-1}\), which corresponds to \(\tau_{\rm xx}(600\,{\rm K})\!=\!16\,{\rm fs}\) and \(\tau_{\rm zz}(600\,{\rm K})\!=\!4\,{\rm fs}\). For PtCoO\({}_{2}\), Rogers _et al._ reported conductivities \(\sigma_{\rm zz}\!=\!33\,\mu\Omega^{-1}\)m\({}^{-1}\) and \(\sigma_{\rm zz}\!=\!0.1\mu\Omega^{-1}\)m\({}^{-1}\). This leads to \(\tau_{\rm xx}(300\,{\rm K})\!=\!70\,{\rm fs}\) and \(\tau_{\rm zz}(300\,{\rm K})\!=\!12\,{\rm fs}\), which is fairly close to the relaxation times obtained for PdCoO\({}_{2}\).
### Thermoelectric performance
The thermopower of PdCoO\({}_{2}\) was first reported by Yagi _et al._ for temperatures above \(500\,{\rm K}\). Since the authors measured on polycristalline materials, the highly anisotropic behavior of this quantity was overseen. Later, Hasegawa _et al._ measured the thermopower at lower
(color online) Band-resolved diagonal components \(\sigma_{\rm xx}\) and \(\sigma_{\rm zz}\) of the electronic conductivity tensor at \(T\!=\!0\) of PtCoO\({}_{2}\) as a function of the in-plane lattice constant \(a\). The values are specified relative to the relaxation time constant \(\tau\). Only three bands crossing \(E_{\rm F}\) contribute to the transport properties.
The authors obtained a positive value of \(S\!=\!2\ldots 4\,\mu\)V K\({}^{-1}\) at room temperature, which is comparable to conventional metals. In a very recent study the in-plane thermopower \(S_{\rm xx}\) and thermal conductivity of PdCoO\({}_{2}\) was measured for temperatures below 300 K.\(S_{\rm xx}\) exhibits a change of sign below \(T\!<\!100\) K, but approaches the expected linear temperature dependence at higher temperatures, reaching \(S_{\rm xx}\!=\!5\,\mu\)V K\({}^{-1}\) at ambient conditions.
For both systems, we confirm the marked anisotropy in the Seebeck coefficient, which is significantly more pronounced for PtCoO\({}_{2}\) than for PdCoO\({}_{2}\) and increases with temperature. It is positive for the in-plane component \(S_{\rm xx}\), which indicates a higher mobility of p-type carriers (holes), while large negative values for \(S_{\rm xx}\) point our the dominance of n-type carriers (electrons) for transport in out-of-plane direction. The estimate for \(S_{\rm xx}\) of unstrained PdCoO\({}_{2}\) from Boltzmann theory at room temperature is approximately 1.6 times larger than the experimental value published in Ref.. This might still be considered a reasonable agreement keeping in mind the simplifications of Boltzmann transport theory and experimental difficulties in obtaining this quantity. Previous calculations by Ong _et al._ report moderate positive values in-plane for \(S_{\rm xx}\), while the out of plane component \(S_{\rm zz}\) provides a large negative contribution. Our values compare well to the results of Ong _et al._, who carried out their calculations with the Wien2k code for the experimental lattice constant, which differs slightly from our setup.
We now turn to the effect of strain, which is varied from -5% compressive to +5% tensile strain. The in-plane thermopower \(S_{\rm xx}\) and its variation under strain remains moderate in absolute numbers. We achieve changes in the range of \(10\ldots 12\,\mu\)V/K at \(T\!=\!600\) K as compared to \(\varepsilon\!=\!0\) for both systems. Nevertheless, as the absolute values are small, maximum strain corresponds to a relative change by a factor of two for the Pt-based oxide. The oscillations observed at lower temperatures are a consequence of the electronic topological transition related to the two additional bands consecutively crossing the Fermi level. For the out-of-plane component \(S_{\rm zz}\), we find much larger changes in absolute numbers, in particular, around room temperature. Here, the tensile strain yields a relative increase of 60 % to 180 %, which corresponds to rather significant changes in absolute numbers of \(-47\,\mu\)V/K and \(-196\,\mu\)V/K for PdCoO\({}_{2}\) than for PtCoO\({}_{2}\), respectively.
In particular, for PtCoO\({}_{2}\), the variation of the out-of-plane thermopower, \(S_{\rm zz}\), with strain bears close similarities with the logarithm of the strain dependence of the conductivity element \(\sigma_{\rm zz}\). Such a relation is motivated by the textbook formula of Mott, which connects the scalar thermopower \(S\) of an isotropic system with the logarithmic derivative of the (scalar) conductivity \(\sigma\) with respect to the chemical potential \(\mu\):
\[S=\frac{\pi^{2}k_{\rm B}^{2}T}{3e}\,\frac{\partial}{\partial\mu}\ln\left[ \sigma(\mu)\right]\,, \tag{1}\]
We therefore conclude that the large strain variations in the thermopower originate from the corresponding changes in the conductivity discussed above. The negative sign of \(S_{\rm zz}\) is related to the decrease of \(\sigma_{\rm zz}\) with increasing chemical potential \(\mu\) at a given strain, while the strong variation of the magnitude of \(|S_{\rm zz}|\) with strain relates inversely to the change in conductivity.
The figure of merit of a material with respect to its thermoelectric performance is given by the dimensionless number \(ZT\!=\!S^{2}\sigma\,T/(\kappa^{\rm el}+\kappa^{\rm ph})\), where \(\kappa^{\rm el}\) and \(\kappa^{\rm ph}\) are the electronic and lattice thermal conductivity, respectively. Apart from \(\kappa^{\rm ph}\), all quantities are accessible within our approach. However, \(\kappa^{\rm ph}\) becomes the dominant contribution, when the electrical conductivity is minute, e.g. in semiconductors or insulators, and will thus be relevant for the out-of-plane transport. The calculation of \(\kappa^{\rm ph}\) from first-principles requires the determination of the anharmonic contributions to lattice dynamics, which is beyond the scope of the present work.
(color online) Diagonal elements \(S_{\rm xx}\) (upper panel) and \(S_{\rm zz}\) (lower panel) of the tensorial thermopower of PdCoO\({}_{2}\) (blue dashed lines and squares) and PtCoO\({}_{2}\) (orange solid lines and circles) as a function of the epitaxial in-plane strain for two different temperatures \(T\!=\!300\) K (open symbols) and \(T\!=\!600\) K (filled symbols). The black star in the upper panel marks the experimental in-plane thermopower of PdCoO\({}_{2}\) at \(T\!=\!300\) K taken from Ref..
For the in-plane case, the authors reported a rather large total thermal conductivity \(\kappa_{xx}^{\rm tot}\) of 250 W K\({}^{-1}\)m\({}^{-1}\), while the total out-of-plane conductivity \(\kappa_{zz}^{\rm tot}\) amounts to approximately 70 W K\({}^{-1}\)m\({}^{-1}\) at ambient conditions. According to the Wiedemann-Franz law, the electronic contribution \(\kappa_{zz}^{\rm el}\) is expected to be at least one order of magnitude smaller, based on the very low out-of-plane conductivity. This yields thus a direct estimate of the lattice thermal conductivity \(\kappa_{zz}^{\rm ph}\).
According to the Wiedemann-Franz law, the electronic conductivities \(\kappa^{\rm el}\) increase with temperature, whereas for the lattice thermal conductivity \(\kappa^{\rm ph}\) a fast decrease is expected according to the Debye-Callaway model. For the in-plane component of \(ZT\), we can neglect \(\kappa^{\rm ph}\) as a first approximation since it is significantly exceeded by \(\kappa^{\rm el}\). The corresponding quantity, which we denote by \(ZT|^{\rm el}\), provides thus an upper limit for the true \(ZT\). According to the large thermal conductivity, the in-plane values turn out to be prohibitively low from the application point of view. Since the strain dependence of \(\sigma\) and \(\kappa^{\rm el}\) essentially cancels out, the strain dependence of \(ZT|^{\rm el}_{\rm xx}\) is dominated by the contribution from \(S^{2}_{\rm xx}\) (cf. Fig. 8). This is also the case for \(ZT|^{\rm el}_{\rm zz}\). However, since the thermal conductivity is dominated by the lattice contribution, we cannot use \(ZT|^{\rm el}_{\rm zz}\) to assess the thermoelectric performance of the material. Instead, we determine the complete \(ZT|_{\rm zz}\) with the help the experimental value \(\kappa_{\rm zz}^{\rm ph}\) of Ref. for \(T\!=\!300\) K and \(\kappa_{\rm zz}^{\rm ph}\!=\!10\) W K\({}^{-1}\)m\({}^{-1}\) for \(T\!=\!600\) K, which can be estimated from the fit to the Debye-Callaway model provided in Ref.. In addition, we use an average \(\tau_{\rm zz}\!=\!10\) fs for \(T\!=\!300\) K, while for \(T\!=\!600\) K, we take \(\tau_{\rm zz}\!=\!4\) fs, as discussed in Sec. III.3. As there is only sufficent data for PdCoO\({}_{2}\), we use the same parameters for both systems. The strain depedece of \(ZT|_{\rm zz}\) is plotted as thick dashed (PdCoO\({}_{2}\)) and solid (PtCoO\({}_{2}\)) lines in the bottom part of Since the dominant contribution to thermal conductivity comes from the lattice, which we assume constant for all \(\varepsilon\), the strain dependence of \(ZT|_{\rm zz}\) rather resembles the power factor (see supplementary information). As a general trend, we see that \(ZT|_{\rm zz}\) increases strongly with temperature, while PtCoO\({}_{2}\) clearly exceeds the performance of PdCoO\({}_{2}\). Thus, only PtCoO\({}_{2}\) can reach a reasonable figure of merit of \(ZT|_{\rm zz}\!=\!0.25\) at compressive epitaxial strains and sufficiently high temperatures, which is a consequence of the electronic topological transition.
## IV Conclusions
Based on first-principles calculations in combination with Boltzmann transport theory in the single relaxation time approximation, we provide a systematic analysis of the electronic structure and anisotropic transport properties of the delafossites PdCoO\({}_{2}\) and PtCoO\({}_{2}\) under epitaxial strain. We demonstrate that despite the large similarities in both systems concerning their structural properties that PtCoO\({}_{2}\) has - unlike PdCoO\({}_{2}\) - the propensity to undergo an electronic topological transition, which might be triggered by a realistic compressive epitaxial strain. In turn, by expanding the in-plane lattice constant, it exhibits a dimensional crossover from a three-dimensional open Fermi surface, which touches the zone boundary in-plane and out of plane, to a nearly perfect twodimensional electronic system, with a closed hexagonal Fermi-surface with perfectly flat sides extending in \(k_{z}\)-direction.
(color online) Thermoelectric figure of merit, \(ZT\), calculated from the diagonal elements of the transport tensors (upper panel, xx elements; lower panel zz elements, symbols and colors as in the preceding figures). Thin lines and data points in the upper and midel panel denote an upper boundary \(ZT|^{\rm el}\) as lattice thermal conductivity is neglected. The bottom panel shows an estimate of the total \(ZT\) using the experimental \(\kappa_{\rm zz}^{\rm ph}\) for PdCoO\({}_{2}\) from Ref. (see text) and the relaxation time constant \(\tau\!=\!10\) fs (same values for both compounds and all strains). The thin lines refer to \(T\!=\!300\) K and the thick lines to \(T\!=\!600\) K, while broken lines denote PdCoO\({}_{2}\) and solid lines PtCoO\({}_{2}\). The black star in the upper panel marks the experimental in-plane \(ZT\) of PdCoO\({}_{2}\) at \(T\!=\!300\) K calculated from the experimental data of Daou and coworkers.
Comparing the in-plane and out-of-plane conductivities from our calculations based on GGA correlation and the GGA+\(U\) approach with experiment, we propose that two different relaxation times must be introduced to describe in-plane and out-of-plane transport appropriately in Boltzmann transport theory. We consider this approach justified in the present case due to the essentially diagonal structure of the transport tensors.
We finally predict that, despite the apparent similarities of both oxides, PtCoO\({}_{2}\) exhibits a much better thermoelectric performance than PdCoO\({}_{2}\) and might thus be a better model system for applications. Our analysis includes the thermoelectric figure of merit \(ZT\), which we obtain by combining theoretical results with recently available experimental data for the lattice thermal conductivity of PdCoO\({}_{2}\). The presence of the topological transition significantly helps in improving this number, since it increases the out-of-plane conductivity, which can be tuned efficiently by an external control parameter, such as epitaxial strain. In this way, we can balance the contributions of electronic and lattice thermal conductivity effectively, while still allowing for sufficiently large absolute values of the thermopower. With this strategy we arrive at a reasonable figure of merit of \(ZT\!=\!0.25\) at \(T\!=\!600\,\)K in out-of-plane direction for a system under compressive in-plane strain. For tensile strains the out-of-plane conductivity becomes very low and \(ZT\) small, since the lattice thermal conductivity dominates. But since the thermopower is almost three times larger than for compressive strains, appropriate doping of carriers may still improve the thermoelectric performance reasonably.
In conclusion, the results for PtCoO\({}_{2}\) as a model system confirm that metallic materials characterized by a quasi-twodimensional Fermi surface may be well suited for thermoelectric applications. However, we emphasize that for a reasonable performance the two-dimensional shape must not be too perfect. In the present case, epitaxial strain was used effectively as an external parameter to obtain a certain level of imperfection that optimizes the relation of electrical and thermal conductivities in the figure of merit.
| 10.48550/arXiv.1511.09087 | Impact of strain-induced electronic topological transition on the thermoelectric properties of PtCoO$_2$ and PdCoO$_2$ | Markus Ernst Gruner, Ulrich Eckern, Rossitza Pentcheva | 324 |
10.48550_arXiv.1406.1763 | ###### Abstract
We investigate by near-forward Raman scattering a presumed reinforcement of the (A-C,B-C)-mixed phonon-polariton of a A\({}_{1,}\)B\({}_{x}\)C zincblende alloy when entering its longitudinal optical (_LO_)-regime near the Brillouin zone centre \(\Gamma\), as predicted within the formalism of the linear dielectric response. A choice system to address such issue is ZnSe\({}_{0.08}\)S\({}_{0.32}\) due to the moderate dispersion of its refractive index in the visible range, a _sine qua non_ condition to bring the phonon-polariton insight near \(\Gamma\). The _LO_-regime is actually accessed by using the 633.0 nm laser excitation, testified by the strong emergence of the (Zn-Se,Zn-S)-mixed phonon-polariton at ultimately small scattering angles.
pacs: 63.20.-e pacs: 63.50.Gh pacs: 78.30.-j Phonons in crystal lattices Disordered crystalline alloys Infrared and Raman spectra
Due to the polarity of the chemical bonding in such a ionic crystal as a zincblende AB semiconductor compound, the long-wavelength (\(\Gamma\)-like, \(q\)\(\sim\)0) transverse optical (_TO_) phonon, corresponding to anti-phase displacement of the intercalated A-like and B-like _fcc_ sublattices (mechanical character), is likely to be accompanied by a macroscopic electric field. The latter is transversal to the direction of propagation, thus identical in nature to that carried by a pure electromagnetic wave, namely a photon. Now, due to the quasi vertical dispersion of a photon at the scale of the Brillouin zone, the electromagnetic character of a _TO_ mode can only emerge very close to \(\Gamma\). The concerned \(q\) values are of the order of one per ten thousands of the Brillouin zone size. At this limit the electromagnetic and mechanical characters combine, conferring on a _TO_ mode the status of a so-called phonon-polariton (_PP_). For certain \(q\) values the _PP_ might acquire a dominant electromagnetic character, thus propagating at lightlike speeds. This stimulates interest in view of ultrafast (photon-like) signal processing at THz (phonon-like) frequencies.
The \(\omega\) vs. \(q\) dispersion of _PP_'s propagating in the bulk of various AB zincblende compounds have been abundantly studied, both experimentally and theoretically. In a nutshell, it can be grasped within four asymptotic behaviors, i.e. two photon-like ones (\(o\)-related) and two phonon-like ones (\(q\)-related). For large \(q\) values, i.e. falling within few percent of the Brillouin zone size, as routinely accessible in a conventional backscattering Raman experiment (schematically operating in a "reflection mode", see below), a transverse electric field cannot propagate at THz (phonon-like) frequencies, because the considered (\(o\),\(q\))-domain falls far away from the natural dispersion of a photon (quasi vertical). In such so-called \(q_{w}\)-regime, a _TO_ mode thus reduces to a purely mechanical oscillator (abbreviated _PM_-_TO_ hereafter, deprived of electric field), whose frequency, noted \(\omega_{TO}\), constitutes the first phonon-like asymptote, i.e. away from \(\Gamma\). The frequency of the non-dispersive longitudinal optical (_LO_) mode, noted \(\omega_{LO}\), larger than \(\omega_{TO}\), defines the second phonon-likeasymptote, near \(\Gamma\) then, taking into account that the _TO_ and _LO_ modes are degenerate strictly at \(\Gamma\). Two remaining photon-like asymptotes determine limit _PP_-behaviors away from the (_PM-TO_)\(-\)_LO_ resonance, as dictated by the static \(\varepsilon_{0}\)(\(\omega\)\(>\)\(\omega_{TO}\)) and high-frequency \(\varepsilon_{\infty}\)(\(\omega\)\(>\)\(\omega_{TO}\)) relative dielectric constants of the crystal. The strong _PP_ coupling occurs when the quasi vertical photon-like asymptotes cross the horizontal _TO_ and _LO_ phonon-like ones. This gives rise to an anticrossing, resulting in two distinct _PP_ branches. The upper branch is phonon-like (_LO_) when \(q\)\(\rightarrow\)\(0\) and photon-like (\(\omega\)=q\(\times\)\(\times\)\(\varepsilon_{\infty}\)\({}^{-1}\) where c represents the speed of light in vacuum) at \(\omega\)\(>\)\(\omega\)\(>\)\(\omega_{TO}\) while the lower branch is photon-like (\(\omega\)=q\(\times\)\(\times\)\(\varepsilon_{0}\)\({}^{-1}\)) at \(\omega\)\(<\)\(\omega_{TO}\) and phonon-like in the \(q_{\infty}\)-regime (_PM_-_TO_). Note that the (_PM_-_TO_) _-_LO_ band is forbidden for the propagation of bulk _PP_'s, only surface _PP_'s can propagate therein.
An interesting question is how such _PP_ picture modifies for a multi-oscillator system such as a AB\({}_{1.\text{a}}\)C\({}_{\text{x}}\) zincblende alloy (A standing for a cation or an anion)? Bao and Liang provided a pioneering theoretical insight into the '_o_ vs. \(q\)' _PP_ - dispersion of various AB\({}_{1.\text{a}}\)C\({}_{\text{x}}\) zincblende alloys. As a starting point they assumed a crude two-mode [1\(\times\)(A-C),1\(\times\)(A-C)] _PM_-_TO_ pattern behind the _PP_'s, as explained within the modified-random-element-isodisplacement (MREI) model. Besides the lower and upper alloy-related branches, assimilating to those of a pure compound as described above, an intermediary (A-C, B-C) - mixed _PP_ was predicted by Bao and Liang. The latter branch is distinct in nature from the former parent-like two in that it exhibits an overall S-like shape governed by two phonon-like asymptotes only, i.e. the higher _PM_-_TO_ frequency in the \(q_{\infty}\)-regime, say the BC\(-\)like one, and the lower _LO_ frequency near \(\Gamma\), the AC-like one then. As such, its dispersion covers the gap between the natural \(A\)\(-\)\(C\) and \(B\)\(-\)\(C\) vibration frequencies, possibly a considerable one depending on the alloy.
We have refined the above _PP_-picture at the occasion of a recent near-forward Raman study (schematically operating in a "transmission mode", see below) of the Zn\({}_{0.67}\)Be\({}_{0.33}\)Se zincblende alloy characterized by a three-mode [1\(\times\)(_Zr_-_Se_),2\(\times\)(_Be_-_Se_)] _PM_-_TO_ pattern in the \(q_{\infty}\)-regime. Such three-mode pattern falls beyond the scope of the MREI scheme. It was explained by introducing the phenomenological percolation model. In brief, this model distinguishes between the vibrations of the short (Be-Se) bonds depending on whether their local environment is more rich of one or the other (Zn-Se or Be-Se) species (1-bond\(\rightarrow\)2-mode behavior). At this occasion, two intermediary (Zn-Se,Be-Se)-mixed _PP_'s were revealed, and not only one as predicted within the MREI scheme. Each intermediary _PP_ relates to a particular BeSe-like _PM_-_TO_ in the \(q_{\infty}\)-regime, and collapses with an S-like shape onto the _LO_ immediately underneath near \(\Gamma\), consistently with the basic MREI trend (see above). In particular, the lower-intermediary _PP_, noted _PP_\({}^{-}\), attracts attention, for two reasons. First, in contrast with the upper-intermediary _PP_, noted _PP_\({}^{+}\), which remains confined within the upper doublet of _PM_-_TO_'s (BeSe-like in this case), _PP_\({}^{-}\) may exhibit a considerable dispersion, as discussed above within the MREI scheme. Second, as soon as entering the _PP_ regime, _PP_\({}^{-}\) becomes much dominant over _PP_\({}^{+}\).
The \(q\)-dependence of the _PP_\({}^{-}\) Raman intensity constitutes _per se_ an interesting issue. Based on our recent near-forward Raman study of Zn\({}_{0.67}\)Be\({}_{0.33}\)Se limited to the early stage of the _PP_ regime (a moderate _PP_\({}^{-}\) red-shift of \(\sim\)15 cm\({}^{-1}\) was detected, representing less than ten percent of the total _PP_\({}^{-}\) dispersion), we already know that the initial \(q\)-induced softening of _PP_\({}^{-}\) goes with a progressive collapse of this mode. Now, as _PP_\({}^{-}\) is supposed to assimilate to a _LO_ near \(\Gamma\), _a priori_ showing up strong and sharp in the Raman spectrum, we anticipate that, after its initial collapse, _PP_\({}^{-}\) should reinforce. This points to a specific _PP_ feature of an alloy, yet unexplored, neither experimentally nor theoretically.
In this work we tackle such issue both theoretically, within the formalism of the linear dielectric response, and experimentally, by applying the near-forward Raman scattering to the ZnSe\({}_{0.68}\)S\({}_{0.32}\) zincblende alloy. Due to large optical gaps of ZnSe (2.7 eV) and ZnS (3.6 eV), ZnSe\({}_{0.68}\)S\({}_{0.32}\) is transparent to visible laser lines, thus well-suited in view of a near-forward Raman study. Besides, the frequency gap between the _PM_\(-\)_TO_'s of its (Zn-Se and Zn-S) constituting bonds is narrow (235-285 cm\({}^{-1}\), see below), with concomitant impact on the magnitude of the _PP_\({}^{-}\) dispersion. This offers an opportunity to explore the _PP_\({}^{-}\) dispersion in a different context than was earlier done with Zn\({}_{0.67}\)Be\({}_{0.33}\)Se, the latter alloy being characterized by a huge _PP_\({}^{-}\) dispersion (250 - 450 cm\({}^{-1}\)). Basically, our aim is to penetrate deep into the _PP_\({}^{-}\) dispersion of ZnSe\({}_{0.68}\)S\({}_{0.32}\) so as to address minimal \(q\) values likely to fall into the _LO_-regime of _PP_\({}^{-}\), near \(\Gamma\).
Generally, the wavevector \(\vec{q}\) of a _TO_ mode (a _PP_ one near \(\Gamma\) or the corresponding _PM_\(-\)_TO_ one in the \(q_{\infty}\)-regime) accessible in a Raman experiment is governed by the conservation rule \(\vec{q}=\vec{k}_{i}\) - \(\vec{k}_{s}\) in which \(\vec{k}_{i}\) and \(\vec{k}_{s}\) are the wavevectors of the incident laser beam and of the scattered light, respectively, both taken inside the crystal, forming an angle \(\theta\). In a standard backscattering geometry \(\vec{k}_{i}\) and \(\vec{k}_{s}\) are (nearly) antiparallel (\(\theta\)-180\({}^{\circ}\)), so that \(q\) is maximum, falling deep into the \(q_{\infty}\)-regime. Minimum \(q\) values, i.e. those likely to address the _PP_-regime, are achieved by taking \(\vec{k}_{i}\) and \(\vec{k}_{s}\) (nearly) parallel (\(\theta\)-0\({}^{\circ}\)), using a near-forward Raman setup.
\[q=\text{c}^{-1}\times\{n^{2}(\omega_{i}x)\times\omega_{i}^{2}+n^{2}(\omega_{o}x) \times\omega_{s}^{2}\]\[-2\times n(\omega_{i}x)\times n(\omega_{n}x)\times\omega_{i}\times\omega_{s} \times cos\theta)^{0.5} \tag{1}\]
If we refer to pure ZnSe and pure ZnS, the refractive index of ZnSe1\({}_{x}\)S\({}_{x}\) is expected to decrease with the frequency in the visible range. Accordingly one cannot achieve \(q\)=0 (\(\Gamma\)) in practice. Indeed the minimum \(q\) value, accessed in a perfect forward scattering experiment (\(\omega_{i}\)-0\({}^{\circ}\)), i.e. \(q_{min}\) - |n(\(\omega_{i}\))\(\times\)\(\omega_{i}\) - n(\(\omega_{s}\))\(\times\)\(\omega_{s}\)|, remains finite because the difference in frequencies is augmented by the difference in refractive indexes. Optimum conditions are thus reached by minimizing the dispersion of the refractive index around the used laser excitation. The alloy composition (ZnSe0.68S0.32) as well as the laser excitation (\(\omega_{i}\)) used, were selected in this spirit, as discussed below.
The ZnSe0.68S0.32 sample considered in this work was grown from the melt as a single crystal (cylinder, ~3 mm-high and ~8mm in diameter) by using the high-pressure Bridgman method (see detail, e.g. in Ref.). The near-forward and backward Raman spectra are taken along the-growth axis, corresponding to a nominal (_TO_-allowed, _LO_-forbidden) geometry, after optical polishing of the opposite faces to optical quality until quasi parallelism was achieved.
The selected alloy corresponds to the highest achievable S incorporation by the Bridgman method. This provides optimal conditions with respect to the dispersion of the refractive index, since the latter is larger for ZnSe than for ZnS in the visible range. In fact, the wavelength (\(\lambda\)) dependence of the ZnSe0.68S0.32 refractive index (\(n\)) measured by ellipsometry in the \(\lambda\)-range 450 - 770 nm (not shown), can be accurately fitted to the Cauchy dispersion formula
\[n\ (\lambda)=X+Y\times\lambda^{2}\times 10^{4}+Z\times\lambda^{-4}\times 10^{9} \tag{2}\]
The minimum dispersion of the ZnSe0.68S0.32 refractive index is further achieved by shifting the Raman analysis to the less energetic end of the visible spectral range. We accordingly use the 633.0 nm line from a He-Ne laser to record the ZnSe0.68S0.32 near-forward Raman spectra, instead of the 514.5 nm and 488.0 nm Ar\(+\) ones earlier used with Zn0.68Fe0.33e, notwithstanding its inferior Raman efficiency. Along the same line, the Stokes experiment (\(\omega_{i}\)>\(\omega_{s}\)) was preferred against the anti-Stokes one (\(\omega_{i}\)<\(\omega_{s}\)), not to mention that the Stokes process is, further, more efficient.
Preliminary insight as to whether there is any chance to access experimentally the presumed _LO_-like _PP_\({}^{-}\) reinforcement with ZnSe0.68S0.32 in a near forward Raman scattering experiment using the 633.0 nm laser excitation is achieved by calculating the related multi-_PP_ near-forward Raman cross section (_RCS_) in its (\(q\),\(\theta\))-dependence. For doing so we use the generic _RSC_ expression established in Ref., which we reproduce hereafter in its specific form applying to multi-_PP_'s propagating in the bulk of an alloy containing \(p\) oscillators (in reference to the multi _PM_-_TO_' s of the \(q_{x}\)-regime behind the _PP_'s),
\[\begin{array}{l}\mathrm{RCS}\ (\omega,q.x)\rightarrow\mathrm{Im}\ \{-\mathrm{e}_{x}(\omega,x)\!\!-\!\!q^{2}c^{2}\omega^{-2}\!\!\!\}^{-1}\\ \times\ [1\!+\!\sum\!C_{p}(x)L_{p}(\omega,x)]^{2}\\ +\sum\,C_{p}^{\,\Raman frequencies of (~210, ~285, ~303) cm\({}^{-1}\), correspondingly.
The theoretical _q_-dependent near-forward multi-_PP_ Raman lineshapes derived for ZnSe\({}_{0.68}\)S\({}_{0.32}\) via **Eq.** using the above [_p,o\({}_{p}\)(x),f\({}_{p}\)(x)_ ] input parameters are shown in Fig 1, taking a uniform phonon damping of 1 cm\({}^{-1}\), for reasonable resolution. In fact, instead of \(q\) we conveniently use the dimensionless parameter _y=q_x_x_x_o\({}_{1}\) taking arbitrarily _o\({}_{1}\)_, the _PM_-_TO_ frequency of the pure ZnSe crystal in the _q\({}_{\infty}\)_-regime, for the change of variable. As anticipated, in the theoretical _RCS_(_y_) curves the initial _PP_\({}^{-}\)collapse at large \(y\) values is relayed by a _LO_-like _PP_\({}^{-}\)reinforcement at moderate-to-small \(y\) values. Interestingly, a moderate penetration into the _PP_\({}^{-}\)dispersion curve, not exceeding one third of the total dispersion (covering the _TO\({}_{Zn-Se}\)_-_ PM- _TO\({}_{Zn-S,I}\)_ frequency gap, see Fig 1), seems sufficient to access the _LO_-like reinforcement regime. The crucial question is whether the latter can be accessed experimentally, or not. Basically this depends on the dispersion of the refractive index around the used laser excitation, as already discussed. For a direct insight we indicate in front of a given theoretical _PP_\({}^{-}\)peak in Fig. 1, corresponding to a certain \(\omega=\left|\,\omega_{i}-\omega_{s}\,\right|\) frequency at a certain \(y\) value, the relevant _th_ angle for the used 633.0 nm laser excitation, as directly inferred from **Eq.** via **Eq.**. As shown in the minimum accessible \(y\) value with the latter laser line, corresponding to \(\theta=0^{\circ}\), hopefully falls into the _PP_\({}^{-}\)reinforcement regime. This is encouraging in view of a possible detection by near-forward Raman scattering using the 633.0 nm excitation.
We turn to experiment. For reference purpose, we show at the top of (thick spectrum) the nominally pure-_TO_ Raman spectrum taken in a backscattering geometry along the-growth crystal axis using the most energetic laser line at hand, namely the 488.0 nm one. This line is more absorbed than the 633.0 nm one in our sample, which minimizes the parasitical near-forward Raman signal due to reflection of the incident laser at the back surface of the (quasi) transparent sample.
(Colour on-line) Theoretical _RSC(y)_ multi-_PP_ near-forward Raman lineshapes of the three-oscillator ZnSe\({}_{0.68}\)S\({}_{0.32}\) alloy. A uniform phonon damping \(\gamma_{p}\)(_x_) of 1 cm\({}^{-1}\) (_p_=1-3) is taken. The _th_ angles for the relevantpeaks addressed in a near-forward Raman scattering experiment using the 633.0 nm laser excitation are indicated. The theoretical (bold) and experimental (underlined, in reference to Fig. 2) _th_–limits are emphasized. For clarity, a simple _TO_-labeling is used for the _PM_-_TO_’s of the reference backscattering signal (bold-red).
Intensity-normalized ZnSe\({}_{0.68}\)S\({}_{0.32}\) near-forward Raman spectra taken with the 633.0 nm laser excitation. The nominal/near-forward (_PP_ labeling) and parasitical/backward (a simple _TO_ labeling is used for the _PM_-_TO_’s, marked by dashed lines) Raman signals are distinguished. The ZnSe\({}_{0.68}\)S\({}_{0.32}\) backward Raman spectrum taken with the 488.0 nm laser excitation is added (top-thick curve), for reference purpose. The theoretical (thin lines) backward (_θ_\(\sim\)180\({}^{\circ}\)) and near-forward (_θ_\(\sim\)0.3\({}^{\circ}\)) Zn-S Raman signals are superimposed onto the corresponding experimental ones, for comparison.
We mention that the emergence of theoretically forbidden optic modes or acoustic bands is a common feature of alloys, due to a partial breaking of the wavector conservation rule by the alloy disorder. As for the minor \(LO_{Zn\text{-}S}\)\({}^{-}\) mode of the Zn-S doublet, this remains screened by the related \(PM\)-\(TO\)'s nearby (see Fig. 1). Satisfactory contour modeling of the nominal three-mode \(PM\)-\(TO\) pattern (top thin curve in Fig. 2) is achieved both in the Zn-Se (not shown) and Zn-S (thin line) spectral ranges by injecting our above selection of input parameters into the asymptotic form of Eq. valid in the \(q_{\infty}\)-regime (backscattering-like), in fact reduced to its second term, taking the same \(\gamma_{p}(x)\) damping of 24 cm\({}^{-1}\) for the two Zn-S sub-modes.
Now we change the Raman setup from backscattering to near-forward scattering. A representative series of near-forward Raman spectra taken at different \(\theta\) angles with respect to the-growth axis of the ZnSe\({}_{0.68}\)S\({}_{0.32}\) ingot using the 633.0 nm laser line are shown in (thin spectra). The intensities, normalized to the \(\theta\)-insensitive A-band, are directly comparable. In each spectrum the near-forward Raman signal (\(PP\)-labeling in Figs. 1 and 2) is partially obscured by the \(\theta\)-insensitive backscattering one (\(TO\) -labeling in abbreviation of \(PM\)-\(TO\)) generated after reflection of the laser beam at the top surface (detector side) of the sample. As \(PP\)\({}^{+}\) is located within the Zn-S doublet of parasitical \(PM\)-\(TO\)'s (see Fig. 1), it remains screened at any angle, only \(PP\)\({}^{-}\), falling outside the doublet, is visible. Its (\(q\),\(\theta\))-dependence is discussed hereafter.
The relevant \(\theta\) angle per spectrum (see Fig. 2) is determined by adjusting the frequency of the theoretical \(PP\)\({}^{-}\) peak obtained by Eq. to the experimental frequency. As long as \(\theta\) is larger than \(\sim\)4.0\({}^{\circ}\), the near-forward Raman signal of ZnSe\({}_{0.68}\)S\({}_{0.32}\) remains stable, similar to the reference backscattering one (\(\theta\)-180\({}^{\circ}\)). Below this critical angle, the multi-\(TO\)'s enter their \(PP\)-regime, signed by the red-shift of \(PP\)\({}^{-}\) (see arrows in Fig. 2). Note that the minimum achievable \(\theta\) angle remains finite (\(\sim\)0.3\({}^{\circ}\)), meaning that the near-forward Raman study cannot be fully developed in practice. This, we attribute to slight \(\overline{k_{i}}\) -disorientations inside the crystal due to inherent defects in an alloy. Nevertheless, the accessible \(\theta\)-domain suffices to reach the \(LO\)-like reinforcement regime (\(\theta\) :1.0\({}^{\circ}\)\(\rightarrow\)0.3\({}^{\circ}\)) beyond the initial collapse regime (\(\theta\) :4.0\({}^{\circ}\)\(\rightarrow\)1.3\({}^{\circ}\), see Figs. 1 and 2), testified by the development of \(PP\)\({}^{-}\) into a sharp and intense feature at near-normal incidences. Fair contour modeling of the peak \(PP\)\({}^{-}\)at the minimum achievable \(\theta\) angle (\(\sim\)0.3\({}^{\circ}\)) is obtained by using the same input parameters as specified above, taking a reduced damping of 12cm\({}^{-1}\). We emphasize that the backscattering (\(PM\)-\(TO\)'s abbreviated \(TO\)'s, upper thin curve in Fig. 2) and near-forward (\(PP\)\({}^{-}\), lower thin curve) theoretical signals are directly comparable. Incidentally, we have checked that a similar near-forward Raman study with Zn\({}_{0.67}\)Be\({}_{0.33}\)Se (see Ref.) using the 633.0 nm laser excitation falls short of engaging the \(PP\)\({}^{-}\) reinforcement regime, due to an unfavorable dispersion of the refractive index.
Last, we discuss briefly the Zn-Se signal. At first sight it remains \(\theta\)-insensitive, pinned by a Fano interference with the \(A\)-band nearby (a characteristic antiresonance is marked by an asterisk in Fig. 2), as earlier discussed for Zn\({}_{0.67}\)Be\({}_{0.33}\)Se. The dramatic weakening of the near-forward Zn-Se signal with respect to the backscattering one (divided by \(\sim\)5, see Fig. 2) is attributed to a reinforcement of the Fano interference that occurs when the Zn-Se mode engages its \(PP\) regime and starts to red-shift towards the \(\theta\)-insensitive \(A\)-band. Now, a careful examination reveals that the Zn-Se line develops an increasing asymmetry on its low-frequency side when \(\theta\) reduces, eventually disappearing at near-normal incidences. In the process, the Raman intensity reduces dramatically. A comparison with the reference \(A\)-band is explicit with this respect. The discussion of the appearance and then disappearance of such asymmetrical broadening under \(\theta\) reduction, presumably related to some fine structuring of the ZnSe-like \(PP\), falls beyond the scope of this work.
For sake of completeness, we provide in a comparison between the theoretical (thin curves) and
Comparison between the theoretical and experimental ‘\(o\) vs. \(q\)’ multi-\(PP\) dispersions of ZnSe\({}_{0.68}\)S\({}_{0.32}\), as obtained via Eq. (thick lines) and by near-forward Raman scattering (symbols with error bars representing the Raman linewidths at half height), respectively. No experimental data is reported for \(PP\)\({}^{+}\) due to a screening at any \(\theta\) angle. The relevant ‘\(\theta\) vs. y’ correspondences per Raman spectrum, in reference to Eq., are added (refer to the oblique-hatched lines), for sake of completeness. The phonon asymptotes in the \(q_{\infty}\)-regime (\(TO\) labeling in abbreviation of \(PM\)-\(TO\)’s) and near \(\Gamma\) (\(LO\) labeling), together with the photon ones (thin lines) far-below (a) and far beyond (b) the phonon resonance are also specified, for reference purpose.
\(q\)' multi-_PP_ dispersions of ZnSe\({}_{0.68}\)S\({}_{0.32}\), as obtained via Eq. - in reference to Fig. 1, and by near-forward Raman scattering - in reference to Fig. 2, respectively.
Summarizing, we perform a near-forward Raman study of the three-mode [1\(\times\)(_Zn\(-\)Se_), 2\(\times\)(_Zn\(-\)S_)] ZnSe\({}_{0.68}\)S\({}_{0.32}\) alloy in search of the presumed _LO_-like reinforcement of the (Zn-Se,Zn-S)-mixed _PP_ - near \(\Gamma\). The laser excitation as well as the alloy compositions were selected so as to minimize the dispersion of the refractive index, a prerequisite to penetrate deep into the _PP_ - dispersion in view to address the vicinity of \(\Gamma\). The _LO_-regime is successfully accessed, as evidenced by the development of _PP_ - into a giant feature at ultimately small scattering angles, solving the raised issue in the positive sense. The discussion is supported by a contour modeling of the ZnSe\({}_{0.68}\)S\({}_{0.32}\) multi-_PP_ near-forward Raman lineshapes in their (_q,\(\theta\)_)-dependence within the formalism of the linear dielectric response.
***
We would like to thank P. Franchetti and J.-P. Decruppe for technical assistance in the Raman measurements, C. Jobart for sample preparation, and A.V. Postnikov for useful discussions and careful reading of the manuscript. This work has been supported by the "Fonds Europeen s de DEvelopment Regional" of Region Lorraine (FEDER project N\({}^{\circ}\). 34619).
| 10.48550/arXiv.1406.1763 | Near-forward Raman study of a phonon-polariton reinforcement regime in the Zn(Se,S) alloy | Rami Hajj Hussein, Olivier Pagès, Franciszek Firszt, Agnieszka Marasek, Wojtek Paszkowicz, Alain Maillard, Laurent Broch | 5,231 |
10.48550_arXiv.0907.1859 | # Figures
Figure 4 | 10.48550/arXiv.0907.1859 | Evolution of graphene growth on Cu and Ni studied by carbon isotope labeling | Xuesong Li, Weiwei Cai, Luigi Colombo, Rodney S. Ruoff | 2,840 |
10.48550_arXiv.0708.3769 | ###### Abstract
Ultraviolet and visible radiation is observed from the contacts of a scanning tunneling microscope with Si and wafers. This luminescence relies on the presence of hot electrons in silicon, which are supplied, at positive bias on \(n\)- and \(p\)-type samples, through the injection from the tip, or, at negative bias on \(p\)-samples, by Zener tunneling. Measured spectra reveal a contribution of direct optical transitions in Si bulk. The necessary holes well below the valence band edge are injected from the tip or generated by Auger processes.
This emission has been observed and analysed from metals, adsorbed molecules and semiconductors. While on metals the light emission is excited in the tunneling gap region, the fairly intense luminescence from direct semiconductors has been shown to occur inside their bulk. Recently, tunneling-induced luminescence from Si, an indirect-band material, has been observed and attributed to inelastic transitions between Si dangling bond states and states specific to W tips. Very similar isochronat spectra were reported from \(n\)- and \(p\)-type material independently of the polarity of the tip-to-sample bias. Besides, the radiation from a STM contact with Si has been reported and related to a localised plasmon which was suggested to arise on this Si surface. Emission from the bulk of Si owing to the diffusing carriers was excluded. On the other hand, luminescence of bulk silicon is a known phenomenon, which is currently being used, e.g., in studies of metal-oxide-semiconductor (MOS) devices. A number of theoretical analyses have addressed the role of direct and phonon-assisted processes and of impurities.
Here we undertake an experimental investigation into the STM-induced luminescence from \(n\)- and \(p\)-type Si. For the first time, detailed luminescence spectra are reported. We included both and orientations in our study, in order to vary the Si surface electronic structure, but the data suggests that this structure does not significantly affect the spectra. As will be shown further, our results are consistent with the model of radiative transitions within the bulk Si. In general, the luminescence of a STM contact has much in common with that of planar MOS structures. However, while the luminescence studies in MOS devices are limited to a low bias regime due to the oxide breakdown at elevated electric fields, this problem is less severe in STM experiments. As a result, we can observe the emission of ultraviolet photons with energies exceeding 4 eV.
The experiments were carried out in a custom built STM under ultra high vacuum conditions at ambient temperature. Si samples (B doped \(p\)-Si, 1.5 \(\Omega\)cm; P doped \(n\)-Si, 56.5 \(\Omega\)cm; P doped \(n\)-Si, 10 \(\Omega\)cm) were prepared by resistive heating. NaOH etched and _in vacuo_ heated tungsten tips were used. The photon detection setup was similar to the one reported in Ref.. A solid angle of 0.6 sr centered at \(\angle=30^{\circ}\) with respect to the surface was detected. Light emission spectra were recorded on clean, reconstructed (7\(\times\)7) Si and (2\(\times\)1) Si surfaces. In all cases, they were corrected for the wavelength dependency of the detection sensitivity. The spectra were not noticeably affected by scanning the tip in a constant current mode. Since tip changes or surface damage are conveniently diagnosed during scanning we used this mode for spectroscopy and made sure that tip and surface remained unaltered.
Series of luminescence spectra from \(p\)-Si recorded at different sample voltages \(V\) are presented in For negative \(V\), clear peaks are observed at photon energies \(h\nu=4.5\) eV and 5.4 eV. An additional structure is discernible at \(h\nu=3.4\) eV for \(V=-4.5.\ldots-8\) V. For positive \(V\), the peaks at \(h\nu=4.5\) eV and 5.4 eV were observed too, while at \(h\nu=3.4\) eV, enhanced intensity is hardly discernible except for the \(V=6\) V data. Spectra from \(n\)-Si exhibit pronounced peaks at \(h\nu=3.4\) eV and 4.5 eV, but at 5.4 eV the intensity is only weakly perturbed. All the spectra show an intensity increase at low photon energies which is consistent with the luminescence of almost thermalized carriers observed also on Si MOS capacitors. No radiation was detected from the \(n\)-Si samples for \(V<0\), unless we previously touched the Si surface with the W tip.
The observed peaks match the energies of direct interband transitions in Si as determined from reflectance and ellipsometry measurements. Three sets of transitions can be identified: (a) \(\Gamma_{25^{\prime}}\rightarrow\Gamma_{15}\)and \(L_{3^{\prime}}\to L_{1}\) at \(h\nu\sim 3.4\) eV, (b) \(\Gamma_{25^{\prime}}\rightarrow\Gamma_{2},\Delta_{5^{\prime}}\rightarrow\Delta_{1 },X_{4^{\prime}}\to X_{1}\), \(\Sigma_{5^{\prime}}\rightarrow\Sigma_{1}\) at \(h\nu\sim 4.5\) eV, and (c) \(L_{3^{\prime}}\to L_{3}\) at \(h\nu\sim 5.4\) eV. Some of these transitions, e.g. \(L_{3^{\prime}}\to L_{3}\), involve holes with energies much lower than that of the valence band edge \(E_{V}\).
In this work we did not intend to perform detailed electrical or optical simulations considering the exact geometry of a STM contact. Instead, we restrict ourselves to a qualitative analysis and use models of a one-dimensional (planar) MOS tunnel structure. Such models allow to estimate the band bending in the semiconductor, the electron and hole components of the tunneling current through the insulator, the Zener current within silicon etc. and thus capture the essential physics. Note that the hole tunneling in a vacuum gap, oppositely to an oxide, occurs through the upper barrier formed by this gap, as there is no "insulator valence band". Like in a regular MOS structure, the charge states of depletion, inversion and accumulation can be supported in a STM contact. An important difference between the planar and real topologies is that the depletion layer width for the given band bending value will be smaller than in planar case as can be verified from a calculation for a junction between a semiconductor and a metal sphere. When the sphere radius is large, the flat situation is imitated.
Below we use the energy diagrams of the tip-vacuum-semiconductor system, as if this system were planar, (Figs. 3,4) to illustrate the processes responsible for light generation.
At positive sample bias, \(V>0\) (Figs. 3a,b), electrons (**1**) are injected into Si with energies up to 9 eV above the conduction band edge of the quasineutral area. At such energies, Auger ionization processes are the predominant relaxation mechanism with a quantum yield around one. The probability for the resulting hole \(\mathbf{1^{\prime}_{p}}\) to be created significantly below \(E_{V}\) is high (e.g., for a 4.5 eV electron, the probability of generating a hole with an energy under \(E_{V}-2\) eV exceeds 0.2). A second injected electron **2** can then radiatively recombine with \(\mathbf{1^{\prime}_{p}}\).
Luminescence spectra recorded at positive bias \(V\) from (a) \(n\)-Si at \(I=9.2\) nA and (b) \(n\)-Si at \(I=10.7\) nA.
Luminescence spectra recorded from \(p\)-Si at tunneling current \(I=26.7\) nA for (a) negative and (b) positive sample voltage \(V\). ”Noisy” curves represent raw data acquired with 1340 channels, smoother curves are obtained by averaging. Photon energies \(h\nu=3.4\), 4.5 and 5.4 eV are marked with vertical dashed lines. Dash-dotted lines represent the condition \(h\nu=eV\).
The above scenario is valid for both _p_- and _n_-type silicon. However, the energy of the injected electron **1** with respect to \(E_{C}\) and, therefore, the threshold for exciting a transition involving a specific conduction band state for a given bias voltage depends on doping. Experimentally, such a variation is indeed observed. displays isochromat spectra for the most intense spectral feature in the photon energy range (\(4.28\pm 0.15\)) eV. The observed shift \(\Delta=0.76\) eV between spectra from _n_- and _p_-type Si agrees with the difference in bulk Fermi energies. This is strong indication that most of the photons are generated in the bulk material. For hypothetical near-surface transitions, \(\Delta\) should have reflected the band bending which is rather large and different for _n_- and _p_- Si.
At negative sample voltage, as already mentioned, the light was observed only for _p_-Si. The absence of luminescence from _n_-Si at \(V<0\) is easy to understand. The junction operates at forward bias and the band bending is small. No hot electrons appear in the conduction band of silicon in such a case.
The origin of luminescence from _p_-Si at \(V<0\) can be explained as follows (Fig. 4a,b). The junction is reversely biased and strong band bending occurs. As a consequence, Zener tunneling of electrons from the valence band is possible; subsequently, these electrons are moving within the conduction band gaining high energy. Thus hot electrons **2** and, via Auger transitions, hot holes \(\mathbf{1^{\prime}_{p}}\) are generated. An alternative process leading to the appearance of hot holes is tunneling between the tip and the states in Si underneath the valence band edge (\(\mathbf{3_{p}}\) in Fig. 4a, inset to Fig. 4b). The final step is radiative recombination of **2** with \(\mathbf{1^{\prime}_{p}}\) or with \(\mathbf{3_{p}}\).
There might be doubts about the validity of a Zener transport mechanism in our relatively low doped _p_-silicon samples. Indeed, this mechanism is known from the textbooks to come into the play at fairly high fields for which high dopant concentrations, say 10\({}^{18}\)-10\({}^{19}\) cm\({}^{-3}\), are required. However, one should not forget that these values refer to the planar geometry, and the same fields in the depleted region of a semiconductor may be attained in a STM contact topology even for much more moderate doping levels.
(dashed lines) displays the calculated ratio of conduction and valence band current densities \(j_{c}/j_{v}\), where the acceptor concentration was taken larger than
Band diagrams and relevant processes for STM-induced light emission at positive substrate bias \(V\). (a) _p_-Si. Hot electrons **1** and **2** are injected from the tip. Electron **1** undergoes an Auger transition creating an electron \(\mathbf{1^{\prime}}\) and a hot hole \(\mathbf{1^{\prime}_{p}}\). Electron **2** radiatively recombines with \(\mathbf{1^{\prime}_{p}}\) which may have drifted in the accumulation zone. (b) _n_-Si. Similar processes occur as in _p_-Si. Dotted line indicates the position of quasi Fermi level for holes. (c) Isochromat spectra for photon energies (\(4.28\pm 0.15\)) eV from _n_- (dots) and _p_- (squares) Si. Spectra have been scaled to identical maximum intensity. The solid line connects _p_-Si data points. A shift \(\Delta=0.76\) eV yields the dashed line.
(a) Band diagram and relevant processes for STM-induced light emission from _p_-Si at negative bias \(V\). Zener tunneling of electrons **1** and **2** occurs. **1** undergoes an Auger transition creating \(\mathbf{1^{\prime}}\) and \(\mathbf{1^{\prime}_{p}}\). **2** radiatively recombines either with \(\mathbf{1^{\prime}_{p}}\) or with a hot hole \(\mathbf{3_{p}}\), which tunneled from the tip. (b) Band bending for a planar W-vacuum-Si junction along with the ratio \(j_{c}/j_{v}\) of tunnel components. Calculations were performed for 1 nm (thin curves) and 2 nm (thick) vacuum gap; dopant concentration is \(5\cdot 10^{18}\) cm\({}^{-3}\). An energy distribution of the valence band current (inset, \(V=-6.5\) V) shows that hole tunneling some eV below \(E_{V}\) is significant.
Qualitatively, at high negative bias, most hot holes are found to be provided by Zener tunneling followed by Auger processes, while hole injection from the tip becomes more significant for lower voltages. Note also that for a positive bias \(V>0\), Fig. 3, the role of valence band tunneling in supplying hot holes is always minor, since \(j_{c}\gg j_{v}\).
It is not useless to mention that ordinary MOS structures on highly doped \(p\)-Si emit no light for \(V<0\), although Zener tunneling certainly occurs there. This may be due to the intense non-radiative scattering of hot electrons on defects in the presence of a large impurity concentration. In a STM contact, due to the difference in a topology, less highly doped samples can be used, which favors the observation of luminescence.
In summary, we have reported ultraviolet and visible light from Si and surfaces in a scanning tunneling microscope. While electron injection causes luminescence from \(n\)- and \(p\)-type samples, no emission is detected at negative sample bias from \(n\)-Si in contrast to previous reports. We have presented the first detailed luminescence spectra revealing a great contribution of direct transitions of hot electrons and holes. The luminescence from \(p\)-Si at reverse bias involves Zener tunneling and injection of holes well below the valence band edge.
Our results, in particular the similarity of the data from and surfaces, the spectral features observed, and the polarity dependence of the emission are not consistent with the interpretation of Si light emission in terms of localised plasmons. They also are at variance with the interpretation of Refs. in terms of an inelastic transition between a specific W tip state and Si dangling bond states. It is important to note, however, that the dopant density of the Si samples used in Refs. was substantially higher than in our present work.
We thank P. Johansson, University of Orebro, for many discussions and for calculations of the electromagnetic response of a tunneling gap between a W tip and a Si surface. One of the authors (MIV) thanks the A. von Humboldt foundation for a support of his stay at the TU Braunschweig where part of this work has been done.
| 10.48550/arXiv.0708.3769 | Ultraviolet light emission from Si in a scanning tunneling microscope | Patrick Schmidt, Richard Berndt, Mikhail I. Vexler | 3,311 |
10.48550_arXiv.2110.01486 | ###### Abstract
Understanding how a spin current flows across metal-semiconductor interfaces at pico- and femtosecond timescales has implications for ultrafast spintronics, data processing and storage applications. However, the possibility to directly access the propagation of spin currents on such time scales has been hampered by the simultaneous lack of both ultrafast element specific magnetic sensitive probes and tailored metal-semiconductor interfaces. Here, by means of free electron laser-based element sensitive Kerr spectroscopy, we report direct experimental evidence of spin currents across a Ni/Si interface in the form of different magnetodynamics at the Ni M\({}_{2,3}\) and Si L\({}_{2,3}\) absorption edges. This further allows us to calculate the propagation velocity of the spin current in silicon, which is on the order of 0.2 nm/fs.
## I Introduction
Independent control of charge and spin carriers is the main goal in the field of spintronics. Spin-based electronics has been compared favourably to present electronics in terms of switching energy and speed; spin currents are believed to flow nearly dissipationless and spin systems show coherence times that are larger with respect to charge confinement. While the flow of charges is readily manipulated by voltages, the control of spin currents still represents a challenge. In this context, the injection of the so-called superdiffusive spin currents (SCs) within metals represented an important achievement. Yet, in order to integrate spintronic devices with present semiconductor technology, it is required to demonstrate the injection of spins through metal/semiconductor interfaces. SCs are generated inside ferromagnetic metals by the use of ultrashort infrared (IR) pulses. The IR pulse absorption creates a spin-preserving out-of-equilibrium hot electron distribution. Since the excited carrier lifetime and velocity are, consistently in all ferromagnetic metals, much larger for spin majority electrons than for spin minority, within few hundreds of femtoseconds a diffusion process of strongly spin-polarized excited carriers is set, leading to an ultrashort spin current pulse. Nowadays the injection of SCs from nickel into metals like Au, Fe or Co-Pt multilayer is well established. Conversely, the spin injection into semiconductors presents new challenges, as carriers need to overcome the band gap to enter the semiconductor conduction band. Nonetheless, the process is still possible and a recent theoretical work proposed the spin injection from nickel to silicon to be i) charge-less, i.e., independent from the charge carrier flow; ii) up to 80% spin polarized, and finally iii) ultrashort. To the best of our knowledge, there is still no experimental proof of ultrafast spin injection into silicon. Silicon, besides its role in semiconductor technology, allows long lived spin currents due to its small spin-orbit interaction, the reduced mean nuclear spin and the crystal inversion symmetry. Also, the inspection of mechanism of SCs at metal/semiconductor interfaces is vital for further advancement in spintronics.
In this work, we provide experimental evidence of the metal-to-semiconductor spin injection in a Ni/Si interface and put forth an insight into the mechanism by comparing ultrafast demagnetization measurements performed using the time resolved resonant magneto-optical Kerr effect (RMOKE) at Si L\({}_{2,3}\) and Ni M\({}_{2,3}\) absorption edges. Thanks to the site selectivity, RMOKE allowed us to decouple the magnetodynamical response of the Si and Ni layers. We observed a slower demagnetization response at the Si edge compared to the Ni edge and interpret this difference as an evidence of a transient spin current from the ferromagnetic layer into silicon.
## II Experimental
We investigated a Ni/Si\({}_{3}\)N\({}_{4}\)/Si interface whose structure is schematically represented in Fig. 1(a).
The sample was synthesized at the VUV-Photoemission beamline (Elettra Sincrotrone Trieste) according to the recipe of Ref.s. Section VII.1 reports on the spectroscopic characterization of the sample. The Ni/Si interface has been realized by depositing a 7 nm layer of Ni on top of a p-doped (B dopant, 0.05 \(\Omega\cdot\)cm resistivity) Si-7x7 surface reconstructed substrate. Nitride passivation of the Si surface was required in order to avoid the formation of unwanted metallic silicides and diminish the migration of the metallic ions into the substrate. Finally, a Ag capping layer was deposited in order to avoid the oxidation of the Ni layer. The Ni/Si\({}_{3}\)N\({}_{4}\)/Si nanostructure was characterized by high resolution transmission electron microscopy (HRTEM). A representative bright-field HRTEM image of the Ni/Si\({}_{3}\)N\({}_{4}\)/Si heterostructure is shown in Fig. 1(a). Further details on HRTEM measurements are provided in section VII.2.
The room-temperature transient magnetic response of our sample was measured via the longitudinal RMOKE in a pump-probe scheme, which allows one to detect the effects on the sample magnetization \(\mathbf{M}\) induced by an IR pump beam. All the RMOKE measurements were carried out at the MagneDyn end-station at the externally seeded EUV free-electron laser (FEL) FERMI at Elettra Sincrotrone Trieste.
A schematic illustration of the experimental scattering configuration is shown in Fig. 1(b). The sample was excited by a \(\sim\)70 fs pump pulse at 1.55 eV with 25 Hz repetition rate decimated with respect to the FEL 50 Hz repetition rate for achieving the standard pump-on/off data acquisition mode. The IR pump pulse was generated from the same laser used to seed the FERMI FEL and had a root-mean-square timing jitter with respect to the FEL pulses of \(\sim\)7 fs. The angle of incidence of the incoming IR pump pulse is 43.5\({}^{\circ}\). The probe consists of \(\sim\)50 fs FEL light pulses whose energy is tuned to the absorption edges of Ni and Si. The electric field of the linearly polarized incoming light lays in the scattering plane, while the angle of incidence was set to 45\({}^{\circ}\). The spot size and fluence of the IR pump pulses at the sample were \(\sim\)60\(\times\)60 \(\upmu\)m\({}^{2}\) and 120 mJ/cm\({}^{2}\) for the measurements at the Ni M\({}_{2,3}\) edge and \(\sim\)140\(\times\)110 \(\upmu\)m\({}^{2}\) and 30 mJ/cm\({}^{2}\) for the Si L\({}_{2,3}\) edge. Instead, the FEL spot size and fluences were \(\sim\)54\(\times\)60 \(\upmu\)m\({}^{2}\) and 8.0 mJ/cm\({}^{2}\) for the Ni M\({}_{2,3}\) edge and \(\sim\)61\(\times\)60 \(\upmu\)m\({}^{2}\) and 2.3 mJ/cm\({}^{2}\) for the Si L\({}_{2,3}\) edge, respectively. Considering the reflectivity and absorbance coefficients of the overall sample stack at 1.55 eV and limiting the absorption up to the first 100 nm of the Si substrate, the fraction of the total absorbed intensity of the incoming optical pump pulse released in the Ni layer is 95%. The remaining energy is absorbed by the Ag capping layer (1%) and the Si substrate (4%).
The RMOKE analysis of the FEL light polarization angle \(\theta\) as a function of the pump-probe time delay (\(t-t_{0}\)) is carried out with a Wollaston-like polarimeter, that collects the reflected FEL pulses. The polarimeter decomposes the polarization of the beam in two orthogonal components with I\({}_{1}\) and I\({}_{2}\) intensities.
\[\theta=\frac{1}{2}\frac{I_{1}-I_{2}}{I_{1}+I_{2}}\,. \tag{1}\]
At time \(t_{0}\) the IR laser pulses (1.55 eV, gold arrow) hit the sample. The FEL pulses, tuned at the Si L\({}_{2,3}\) edge (102.5 eV, blue arrow) and at the Ni M\({}_{2,3}\) edge (67 eV, red arrow), probe the sample status at later timescales. The penetration depth of the pump and of the probe pulses is represented by the length of the respective arrows. The fraction of the total energy of the IR laser absorbed by each layer is reported above the pump arrow in percent. An HRTEM image of the heterostructure is superimposed to the lower part of the schematic (see section VII.2). b) Scheme of the FEL RMOKE set-up at MagneDyn. The pump and probe pulses are in a quasi collinear configuration. RMOKE was probed in longitudinal configuration at an angle of incidence of 45\({}^{\circ}\). The linear polarization \(\mathbf{E}\) of the FEL pulses reflected from the sample is rotated form the scattering plane (dashed line) of the Kerr angle \(\theta\).
Element sensitivity of the Ni and of the Si layer is achieved by resonantly tuning the FEL radiation at the Ni M\({}_{2,3}\) edge and the Si L\({}_{2,3}\) edge. The applied magnetic field **B**, whose direction is parallel to the k-vector of the incoming FEL radiation, orients the magnetization of Ni along the line of intersection between the sample surface and the plane of incidence.
Finally, the transient relative change of the sample magnetization magnitude M is defined as
\[\frac{\Delta M}{M}(t)=\frac{\theta(t)^{+}-\theta(t)^{-}}{\theta_{sat}^{+}- \theta_{sat}^{-}}\,, \tag{2}\]
## III Results
Fig. 2(a) displays the Kerr rotation collected at the Ni M\({}_{2,3}\) edge (67 eV) as a function of the applied magnetic field taken before the pump arrival (empty circles) and 500 fs after the pump absorption (filled circles). The silicon Kerr rotation collected with the FEL photon energy resonantly tuned to the Si L\({}_{2,3}\) edge (102.5 eV) and taken before and at a 300 fs time delay after the pump arrival are reported in Fig. 2(b).
The Kerr rotation displays a hysteresis shape for both the Ni and the Si edges. The difference of the respective saturation values \(\theta_{sat\mathrm{Ni}}^{\pm}\) and \(\theta_{sat\mathrm{Si}}^{\pm}\) originates from the different magneto-optical constants at the two absorption edges. The measured coercitivity field for Ni is \(\sim\)50 mT that confirms the ferromagnetic state of the Ni layer. Noteworthy, the Si Kerr signal measured at negative time delay reveals a non-vanishing magnetization at equilibrium in the silicon which is then partially reduced after the arrival of the pump pulse. This experimental finding will be further discussed below.
The reduced values of \(\theta_{sat\mathrm{Ni}}^{\pm}\) and \(\theta_{sat\mathrm{Si}}^{\pm}\) after the pump arrival prove the photoinduced partial demagnetization of both layers. It is interesting to note that the shapes of the demagnetized hysteresis are preserved indicating no detectable damaging of the sample caused by the pump pulse and the subsequent thermal heating.
Fig. 2(c) displays the characteristic dynamics of the ultrafast relative change of the layer magnetization \(\Delta\)M/M measured at fixed FEL photon energies resonantly tuned to the Ni (red marks) and the Si (blue marks) edges with a saturating external field of 550 mT. The relative difference between the two dynamics is also reported (grey pentagons).
\[f(t)=\frac{\Delta M}{M}\Theta(t)\Big{(}1-e^{-t/\tau_{m}}\Big{)}e^{-t/\tau_{r}}\,, \tag{3}\]
Results of the fitting are shown in Fig. 2(c) and summarized in Table 1 together with typical demagnetization and recovery times of Ni found in earlier experiments.
## IV Discussion
The diagram shown in summarizes our experimental findings and sketches the undergoing phenomena at the interface. The left panel shows the sample magnetization in the Ni film (\(\mathbf{M}_{\mathrm{Ni}}\)) and in the Si substrate (\(\mathbf{M}_{\mathrm{Si}}\)) before and after the pump arrival. The right panel represents the relative change of the Ni and Si magnetizations \(\Delta\)M/M and their relative difference as a function of the time delay. The initial state consists of a magnetized Ni film and a magnetized Si substrate as revealed by the measured RMOKE hysteresis reported in Fig. 2(a) and (b). Unlike the Ni film case, the origin of the static magnetized state of the Si substrate is not trivial and it deserves a thorough explanation.
\(\mathbf{M}_{\mathrm{Si}}\) originates from a proximity effect of the spin polarized electron distribution in the Ni film into the semiconductor. In fact, the magnetic Ni film acts as a reservoir of spin-polarized thermal electrons. These low-energy electrons diffuse at equilibrium in the metallic layer and, due to the different exchange interaction with the spin-polarized electron background, they will be scattered differently depending on their spin. As a result, thermal electrons that impinge the metal/semiconductor interface are also spin-polarized and if their energy is above the Schottky barrier they can diffuse as a thermionic current (\(j_{th.i.}\)) inside the Si substrate. This proximity effect is also called the spin-polarized thermionic effect at the metal-semiconductor interface.
\begin{table}
\begin{tabular}{c c c} \hline & \(\tau_{m}\) _(fs)_ & \(\tau_{r}\) _(ps)_ \\ \hline Ni & **100 \(\pm\) 12** & **20.3 \(\pm\) 5.6** \\ & 140 \(\pm\) 10\({}^{}\) & — \\ & 208 \(\pm\) 33\({}^{}\) & 22 \(\pm\) 17\({}^{}\) \\ \hline Si & **255 \(\pm\) 86** & \(>\) **100** \\ \hline \end{tabular}
\end{table}
Table 1: The fitting results for the demagnetization \(\tau_{m}\) and recovery \(\tau_{r}\) times obtained by the double-exponential function on Ni and on Si are reported. The results for Ni are compared with literature.
3, left panel). Accordingly, the proximal layer of the Si substrate retains a net collinear depth dependent magnetization because of the spin-polarized thermal electrons populating the semiconductor conduction band in the proximity of the Ni/Si interface. However, depth dependent investigation of the magnetization in Si is out of the scope of the present work.
Now we turn to the discussion of the magnetodynamics observed at the Ni and Si edges. The optical absorption of an ultrafast pulse by the Ni film is accompanied by a sudden increase of the electron temperature and a consequent reduction of the spin-polarization of the exchange-split Ni bands. In turn, while the Ni film demagnetizes in \(\sim\)100 fs, the consequent reduction of the spin-polarization of the Ni bands reflects in a reduction of the spin-polarized thermionic current \(j_{th.i.}\) in Si. Accordingly, \(\mathbf{M}_{\mathrm{Si}}\) diminishes as revealed by the magnetodynamics at the Si edge. The most striking feature of Fig. 2(c) is that within the first \(\sim\)1 ps the Ni and Si (\(\Delta\)M/M)(t) exhibit different dynamical responses to the laser excitation. Specifically, \(\mathbf{M}_{\mathrm{Ni}}\) reacts faster then \(\mathbf{M}_{\mathrm{Si}}\) and the resulting demagnetization rate of change in Ni is 2.5 times as fast as its Si counterpart. The difference (\(\Delta\)M/M)\({}_{\mathrm{Si}}\)-(\(\Delta\)M/M)\({}_{\mathrm{Ni}}\), that we indicate as (\(\Delta\)M/M)\({}_{j_{s}}\), is plotted in Fig. 2(c) (grey pentagons) and it represents the transient spin current. In fact, as previously mentioned, since in thermal equilibrium the Ni spin minority-majority unbalance is at the origin of the spin-polarized thermionic effect in Si, the magnetodynamics measured at the two distinct absorption edges are expected to be identical and the difference between the two dynamics to be zero. However, as the Ni layer absorbs the femtosecond IR pump pulse, a spin-polarized current pulse \(j_{s}\) is set and superdiffuses into the Si substrate through the interface. The propagation of \(j_{s}\) is illustrated in the left panel of The spin polarization of \(j_{s}\) reflects that of the spin majority in Ni. Consistently, as \(j_{s}\) propagates into the Si substrate, it contributes as a transient increase of the Si magnetization, \(\mathbf{M}_{\mathrm{Si}}\). The combined effect between the spin polarized \(j_{s}\) and \(j_{th.i.}\) currents on the total transient magnetization of the Si substrate results in a longer demagnetization time \(\tau_{m}\) of (\(\Delta\)M/M)\({}_{\mathrm{Si}}\) with respect to the Ni layer case.
RMOKE magnetic hysteresis in degrees of Kerr rotation at the Ni M\({}_{2,3}\) edge (panel a) and at the Si L\({}_{2,3}\) edge (panel b). The empty and filled circles curves represent the unpumped (label \(up\)) and pumped (label \(pp\)) hysteresis measured at a delay time of 0.5 ps (for the Ni edge) and 0.3 ps (for the Si edge) (colored star markers in panel c). Panel c) relative change of the site resolved magnetization M (Ni - red dots and Si - blue dots) as a function of the time delay measured in saturation with an applied magnetic field of 550 mT. The Ni demagnetization curve was rescaled to account for the different pump fluence applied. The solid lines represent the fitting results, from which we extract the two characteristic times for demagnetization (\(\tau_{m}\) ) and recovery (\(\tau_{r}\)). The difference of the two magnetization dynamics, defined as (\(\Delta\)M/M)\({}_{j_{s}}\), is also shown (gray pentagons).
At a longer time scale, following the propagation of the spin current pulse inside the Si substrate, the observed \(j_{s}\) contribution becomes irrelevant.
Finally, having established the presence of a spin current injected in the Si substrate across a Ni/Si interface, further quantitative information could also be extracted. (\(\Delta\)M/M)\({}_{j_{s}}\) displays a maximum at \(\sim\)150 fs after the pump arrival, followed by an exponential decay time \(\tau=248\ \pm\ 128\) fs, as obtained by an exponential fitting of the trace (see also the grey guide for the eye in Fig. 2(c)). Considering the spin current pulse decay time \(\tau=\Gamma/v\), where \(\Gamma\) is the optical attenuation length at the Si \(L_{2,3}\) edge, which in the present case is \(\sim\)55 nm, the calculated velocity \(v\) of the spin pulse propagating in the Si substrate results to be \(\sim\)0.2 nm/fs. This experimental finding matches the theoretical predictions made on an ideal Ni/Si system (see Ref.).
## V Conclusions
In summary we experimentally observed magnetodynamics at the Ni M\({}_{2,3}\) and Si L\({}_{2,3}\) edges in a Ni/Si interface by making use of FEL-based element sensitive Kerr effect. The results indicate the onset of transient superdiffusive spin current triggered by an optical pump pulse from the ferromagnetic Ni layer into the Si substrate. The analysis of the magnetodynamics at the Ni and Si edges reveals the slower demagnetization lifetime in Si when compared to Ni, which can be ascribed to the concomitant mechanisms of the thermionic proximity effect at the Ni/Si interface and the onset of a transient spin current triggered by an optical pump pulse. Finally, the measured spin pulse propagation velocity into Si substrate reported here (\(\sim\)0.2 nm/fs) is benchmarking the theoretical values reported in literature giving further support to these models.
| 10.48550/arXiv.2110.01486 | All-optical spin injection in silicon revealed by element specific time-resolved Kerr effect | Simone Laterza, Antonio Caretta, Richa Bhardwaj, Roberto Flammini, Paolo Moras, MatteoJugovac, Piu Rajak, Mahabul Islam, Regina Ciancio, Valentina Bonanni, Barbara Casarin, Alberto Simoncig, Marco Zangrando, Primoz Rebernik Ribic, Giuseppe Penco, Giovanni De Ninno, LucaGiannessi, Alexander Demidovich, Miltcho Danailov, Fulvio Parmigiani, Marco Malvestuto | 1,393 |
10.48550_arXiv.0810.3057 | ###### Abstract
Graphene field-effect transistors with Co contacts as source and drain electrodes show anomalous distorted transfer characteristics. The anomaly appears only in short-channel devices (shorter than approximately 3 \(\upmu\)m) and originates from a contact-induced effect. Band alteration of a graphene channel by the contacts is discussed as a possible mechanism for the anomalous characteristics observed.
Author to whom correspondence should be addressed; present address: WPI-Advanced Institute for Materials Research, Tohoku University, Sendai 980-8577, Japan; electronic mail: one-atomic carbon sheet with a honeycomb structure, has been attracting significant attention due to its unique physical properties, such as the massless Dirac fermion system. This material shows an extraordinarily high carrier mobility of more than 200,000 cm\({}^{2}\) V-1 s-1 (ref. 2), and is considered to be a major candidate for a future high-speed transistor material. In addition, graphene has shown its ability to transport charge carriers with spin coherence even at room temperature, and is regarded as a pivotal material in the emerging field of molecular spin electronics. In order to construct such electronic devices, metallic materials should make a contact with the graphene layers. The effect of metal contacts can be detected using the structure of a field-effect transistor (FET) and measuring the transfer characteristics (drain current, \(I_{\text{D}}\), vs. gate voltage, \(V_{\text{G}}\), characteristics). For instance, the difference between the drain currents of graphene FETs at exactly opposite charge densities (at the same carrier densities with opposite charge polarities) has been explained by a metal-contact effect. Charge transfer from metal to graphene leads to a \(p\)-\(p\), \(n\)-\(n\) or \(p\)-\(n\) junction in graphene, depending on the polarity of carriers in the bulk of the graphene sheet. An additional resistance arises as a result of the density step created along the graphene channel, which causes asymmetry.
In this letter, we report the effect of metal contacts on the transfer characteristics of graphene FETs. In particular, the choice of metal and the gap between the metal contacts (source and drain electrodes) have been examined by employing a FET structure. It was found that graphene FETs with Co contacts and short channels display anomalous distorted transfer characteristics, indicating that the anomaly originates from Co contacts.
Graphene layers were formed onto a highly-doped Si substrate with a 300 nm thick thermal oxide layer using conventional mechanical exfoliation. The starting graphite crystal used was Super Graphite(r) from Kaneka Corporation. The thicknesses of the graphene layers were determined to be approximately 1 nm by atomic force microscopic observations in tapping mode. These layers were determined to be one-atom thick from the optical contrast. Metal electrodes (Co and Au) were fabricated onto the graphene layers by electron beam lithography and liftoff techniques. For the Au electrodes, 5 nm thick Cr was deposited as an adhesive layer prior to Au deposition. The electrodes fabricated in this study had a total thickness of 50 nm. The FET characteristics were measured in low vacuum at room temperature.
The graphene FET device structure is schematically displayed in Fig. 1(a), and the transfer characteristics are shown in Figs. 1(b) and 1(c) for Cr/Au and Co source/drain electrodes, respectively. Cr/Au is a conventionally used metallic material for electronic devices, and Co is a popular material for spin-electronic devices as a source of spin-polarized current. Although the graphene FET with Cr/Au contacts exhibits conventional transfer characteristics, as widely reported previously, that with the Co contacts displays anomalous distorted characteristics, especially in the negatively gated region. The distortion disappeared when the gap between the metal contacts (channel length) was lengthened, which indicates a contact-related effect. The shorter channel results in lower channel resistance, and the resistance originating from the contacts should have a more dominant effect on the two-terminal resistance. In fact, the resistances at the \(I_{\text{D}}\) minima are not proportional to the ratio of channel length to channel width, and thus the contact-related effects contribute to the device resistance.
Distorted transfer characteristics can be observed in a double-gated device under a specific relation between the two gate voltages. In such a device, the graphene channel is divided into two parts: a double-gated region and a bottom-gated region. By applying a gate voltage from the top gate electrode, the Fermi level of the top-gated region is shifted, compared to the rest of the channel. The resulting transfer characteristics display two local minima when the shift is sufficiently large. If such a large shift occurs due to the charge transfer from the metal electrodes, devices without a top gate also appear to display transfer characteristics with two local minima. However, this is not the case for the Co-contacted devices. In the wide-ranging transfer characteristics of the 3 \(\upmu\)m channel device, decreases in the current compared with ordinary transfer characteristics can be seen at gate voltages of -70 and +30 V in addition to the minimum at a gate voltage of +2 V. The minimum at +2 V is considered to be the charge neutrality point (the so-called Dirac point); the other two anomalous points are therefore a consequence of the metal contacts. Such additional decreases can also be distinguished in the 2 \(\upmu\)m channel device [Fig. 1(c)], although the appearance of the anomaly around 30 V is somewhat weaker than that of the 3 \(\upmu\)m channel device. The difference in the strength of the anomaly might be caused by contamination of the graphene surface, which would vary from device to device and cause the difference in the strength of the Co-graphene interaction.
Generally accepted values of work functions of polycrystalline Au and Co are 5.1 and 5.0 eV, respectively. These values are higher than that of graphene [4.6 eV (ref.)] and there should exist an electron transfer from graphene to the metals. However, the two metallic materials have very similar work functions; thus the simple ionic charge transfer effect alone cannot account for the observed difference in the shape of transfer characteristics. A recent first-principle calculation at the level of density functional theory indicates a strong chemical interaction between Co and graphene, where conical points at \(K\) disappear, and instead a mixed metal-graphene character appears in the electronic structure. From examination of Fig. 3, two peaks centered at -40 and +20 V can be distinguished. The Fermi energy shifts from the Dirac point at +2 V are determined to be -0.20 and +0.13 eV, respectively, using the relationship
\[\Delta E_{F}=sgn(n)h\nu_{F}\sqrt{\pi\left|\pi\right|}\,,\,\text{where}\,\,\,h\, \,\,\text{is the Dirac constant,}\,\,\,\nu_{F}\,\,\,\text{is the Fermi velocity}\] (around \(10^{6}\) m s-1), and \(\,\,n\,\,\) is the two-dimensional charge density. The gate voltage changes \(\,\,n\,\,\) through the relation \(\,\,n=\varepsilon_{r}\varepsilon_{0}\left(V_{G}-V_{G0}\right)/\sigma\,\), where \(\,\,\varepsilon_{r}\,\,\) is the relative permittivity of the gate dielectric (3.8 for SiO\({}_{2}\)), \(\,\varepsilon_{0}\,\,\) is the electric constant, \(\,V_{G0}\,\,\) is the gate voltage corresponding to the Dirac point, and \(\,\,d\,\,\) is the thickness of the gate dielectric (300 nm in this study). Similar peaks have been reported in scanning tunneling spectroscopic data of a graphene layer formed on Ru, where two peaks were observed at -0.4 and +0.2 V. The peaks were not observed with the bare Ru surface, and were therefore considered to originate from the metal contact. A band structure for the graphene/Ni system has been calculated using a discrete variational X\(\alpha\) method, and a gap of approximately 1.0 eV was expected to open at the \(K\) points. Such a gap opening affects the transfer characteristics and should be detected as an increase in \(\,I_{D}\,\,\)at gate voltages corresponding to the edges of the gap. However, the transfer characteristics in display a decrease in \(\,I_{D}\,\,\)around -70 V, which cannot be explained by a simple gap opening. This decrease is possibly explained by contact-induced states formed by the hybridization of graphene \(\pi\) and Co \(d\) bands, which may form some resonant states and cause the negative \(\left.dI_{D}\right/d\left|V_{G}-V_{GO}\right|\) values.
Another possible mechanism is the diffusion of Co atoms into/onto graphene channels. In a single charge tunneling device of a single CdTe nanorod with Cr/Pd contacts, a chemical transformation was found to occur by the diffusion of Pd atoms 20-30 nm into the nanorod. However, the robust honeycomb lattice structure of graphene and the possibly strong chemical interactions at Co/graphene interfaces should prevent Co atoms from diffusing a long distance into and onto graphene channels.
In summary, the effect of metallic electrode materials contacting graphene channel layers was studied using FET structures. Cr/Au and Co contacts were investigated, and it was found that graphene FETs with Co contacts and short channels exhibit distorted transfer characteristics that have two peaks at -0.20 and +0.13 eV, in addition to the common minimum at the Dirac point. The anomalous distortion can be considered to be the result of band alteration of the graphene channel underneath the Co contacts. The present study ascertained the metal-induced alteration of the FET characteristics of graphene. These results indicate particularly crucial issues for the development of future graphene microelectronics that consist of short-channel devices.
We acknowledge Dr. Mutsuaki Murakami of Kaneka Corporation for providing the Super Graphite(r) Crystal. This work was supported by a grant from the Foundation Advanced Technology Institute.
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(Color online) (a) Schematic diagram of a graphene FET. (b) Transfer characteristics of a graphene FET with Cr/Au electrodes. The channel length was 1.5 \(\upmu\)m. (c) Transfer characteristics of a graphene FET with Co electrodes. The channel length was 2.0 \(\upmu\)m. An anomalous distortion is clearly seen in the characteristics.
Wide-ranging transfer characteristics of Co-contacted graphene FETs with a channel length of 3 \(\upmu\)m. The device is identical to that shown in the left panel of Fig. 2(b). In order to avoid an accidental breakdown of the device, the upper limit of the gate voltage was set to 50 V. The gate voltage corresponding to the charge neutrality was determined to be identical to that showing the smallest current.
Transfer characteristics of longer-channel graphene FETs with (a) Cr/Au and (b) Co contacts. The distortion is barely observed in the 5 \(\upmu\)m channel device, even with Co contacts. | 10.48550/arXiv.0810.3057 | Transfer Characteristics in Graphene Field-Effect Transistors with Co Contacts | Ryo Nouchi, Masashi Shiraishi, Yoshishige Suzuki | 1,097 |
10.48550_arXiv.0805.3712 | ###### Abstract
Graphene has recently emerged as an interesting material for electronics due to extremely high carrier mobility in bulk graphene and the demonstration of all-semiconducting sub-10nm graphene nanoribbons. Aggressive device scaling requires integration of ultrathin high-x dielectrics in order to achieve high on-state current and ideal subthreshold swing without substantial gate leakage. Deposition of metal oxides including high-x dielectrics on graphene has not been systematically investigated thus far. Uniform oxide deposition on graphene by atomic layer deposition (ALD) is expected to be difficult due to the lack of dangling bonds in the graphene plane, as in the case of carbon nanotube. As a result, previously reported topgated graphene devices used very thick (\(\geq\)15nm) ALD dielectrics on top of negative tone resist or functional layer to prevent gate leakage. Functionalization is likely needed for uniform ALD on graphene.
Here we show that ALD of metal oxides gives no direct deposition on defect-free pristine graphene. On the edges and defect sites however, dangling bonds or functional groups can react with ALD precursors to afford active oxide growth. This leads to an interesting and simple way to decorate and visualize defects in graphene. By non-covalent functionalization of graphene with carboxylate terminated perylene molecules, one can coat graphene with densely packed polar groups for uniform ALD of high-\(\kappa\) dielectrics. Uniform high-\(\kappa\) coverage is achieved on large pieces of graphene sheets with size of greater than 5\(\mu\)m. This method opens the possibility of integrating ultrathin high-\(\kappa\) dielectrics in future graphene electronics.
Our graphene sheets were obtained by standard peel-off method on 300nm SiO\({}_{2}\)/Si substrate described in Ref.. We first identified few-layer (\(\leq\)5 layers) graphene sheets under an optical microscope. Then the chip was annealed at 600\({}^{\circ}\)C in vacuum with 1Torr argon atmosphere to clean the substrate and graphene sheets, followed by atomic force microscopy (AFM) imaging of the few layer graphene sheets. Then the chip was soaked in 3,4,9,10 perylene tetracarboxylic acid (PTCA) solution for -30mins, thoroughly rinsed and blown dry (see Supporting Information). The chip was immediately moved into the ALD chamber to prevent contamination by molecules in the air. We then deposited Al\({}_{2}\)O\({}_{3}\) at \(\sim\)100 \({}^{\circ}\)C using trimethylaluminum and water as precursors. In the same run we also included a SiO\({}_{2}\)/Si chip covered by PMMA except some pre-patterned squares to measure Al\({}_{2}\)O\({}_{3}\) thickness (See Supporting Information).
We first carried out study on pristine graphene. and b show AFM images of the same area before and after ALD of -2nm Al\({}_{2}\)O\({}_{3}\). Before ALD, the height of the triangular graphene piece at the bottom and the large piece on the left was \(\sim\)1.7nm and \(\sim\)2.0nm, respectively. Near the edge of the 2.0nm graphene, there was also a narrow \(\sim\)1.0nm high stripe. After ALD, \(\sim\)2.0nm Al\({}_{2}\)O\({}_{3}\) was coated on SiO\({}_{2}\), the apparent topography height of the three graphene sheets was obviously reduced, to a similar level as the ALD coated SiO\({}_{2}\). The height difference before and after ALD indicates no Al\({}_{2}\)O\({}_{3}\) coating on pristine graphene sheets.
Since pristine graphene does not have any dangling bonds or surface groups to react with precursors, no ALD occurs on the graphene plane. Interestingly, we observed quasi continuous bright lines of Al\({}_{2}\)O\({}_{3}\) preferentially grown on the edges of graphene sheets, suggesting dangling bonds on the edges or possible termination by -OH or other reactive species. We also observed some bright dots in the middle of graphene sheets, likely corresponding to defects such as pentagon-hexagon pairs or vacancies known to exist in graphite. Thus, our ALD method could be used as a novel way to probe and visualize defects in graphene, which is much simpler and more efficient than current STM and TEM measurements. The peeled off graphene sheets could be defect free for a few micrometers as evidenced by AFM images after ALD (see Supporting Information). However, defects do exist in the graphene planes and should be considered in other studies.
In order to afford uniform ALD coating on pristine graphene, functionalization of graphene is needed to induce uniform surface groups as active ALD nucleation sites.
ALD of Al\({}_{2}\)O\({}_{3}\) on pristine graphene. (a) AFM image of graphene on SiO\({}_{2}\) before ALD. The height of the triangular shaped graphene is \(\sim\)1.7nm as shown in the height profile along the dashed line cut. Scale bar is 200nm. (b) AFM image of the same area as (a) after \(\sim\)2nm Al\({}_{2}\)O\({}_{3}\) ALD deposition. The height of the triangular shaped graphene becomes \(\sim\)0.3nm as shown in the height profile along the dashed line cut. Scale bar is 200nm. (c) and (d) are schematics of graphene on SiO\({}_{2}\) before and after ALD. The Al\({}_{2}\)O\({}_{3}\) grows preferentially on graphene edge and defect sites.
While PTCA non-covalently partitions and adheres to graphene, likely via \(\pi\)-\(\pi\) stacking and hydrophobic forces, basic conditions deprotonate the terminal acid groups, yielding a tetra-anionic state which is repulsed from the oxidized silica surface. Following adsorption onto graphitic surfaces, the carboxylate functional groups on the perylene moiety serve as uniformly distributed sites for nucleation of ALD.
Before ALD, the graphene under the line cut was \(\sim\)1.6nm high, while after ALD, the height increased to \(\sim\)3.0nm as shown in the height profile in The relative height increase of Al\({}_{2}\)O\({}_{3}\) coated graphene was partly attributed to the thickness of PTCA layer, which was usually \(\sim\)0.5-0.8nm as observed by AFM after PTCA coating step (see Supporting Information). The actual Al\({}_{2}\)O\({}_{3}\) on graphene was \(\sim\)2.8\(\pm\)0.2nm thick in Complete and uniform PTCA packing is evidenced by Al\({}_{2}\)O\({}_{3}\) coated over the whole piece of graphene, which is several micrometers in size. The mean roughness of Al\({}_{2}\)O\({}_{3}\) film on graphene is \(\sim\)0.33nm over 2.5\(\mu\)m x 2.5\(\mu\)m area as measured with several pieces (see Supporting Information). Previous high vacuum STM studies have confirmed two modes of epitaxial packing of the PTCA precursor, perylene-tetracarboxylic dianhydride, along the lattice lines of highly oriented pyrolytic graphite. As the partitioning of PTCA from methanol to the graphene interface should be highly favorable, it is likely that similar epitaxial packing is occurring in the solution phase, yielding dense, uniform coating of graphene sheets. This self-assembly process of PTCA on graphene is responsible for uniform Al\({}_{2}\)O\({}_{3}\) film coating. Note that non-covalent functionalization for ALD is not expected to degrade the electrical properties of graphene, similar to the case of single-walled carbon nanotubes.
In summary, we found that ALD of metal oxide cannot directly be deposited on pristine graphene without surface functionalization due to the lack of dangling bonds and surface functional groups. ALD could grow actively on edge and defect sites of graphene, which could be used as a simple and effective probe to defects. We used carboxylate terminated perylene molecules to functionalize graphene with densely packed -OH groups and achieved uniform ultrathin Al\({}_{2}\)O\({}_{3}\) deposition on graphene over a few micrometers. The non-covalent functionalization method is not destructive to graphene and could be used for depositing ultrathin high-\(\kappa\) dielectrics for future graphene electronics.
# Height increase after PTCA coating on graphene.
We observed ~0.5-0.8nm height increase after the PTCA treatment described in main text. This indicated that probably only a monolayer of closed packed PTCA is absorbed on graphene. | 10.48550/arXiv.0805.3712 | Atomic Layer Deposition of Metal Oxides on Pristine and Functionalized Graphene | Xinran Wang, Scott Tabakman, Hongjie Dai | 5,290 |
10.48550_arXiv.1309.1405 | 10.48550/arXiv.1309.1405 | Direct Optical Coupling to an Unoccupied Dirac Surface State in the Topological Insulator Bi$_2$Se$_3$ | Jonathan A. Sobota, Shuolong Yang, Alexander F. Kemper, J. J. Lee, Felix T. Schmitt, Wei Li, Robert G. Moore, James G. Analytis, Ian R. Fisher, Patrick S. Kirchmann, Thomas P. Devereaux, Zhi-Xun Shen | 4,157 |
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10.48550_arXiv.1201.6013 | ## I Introduction
The spinel cobalt oxide Co\({}_{3}\)O\({}_{4}\) is a magnetic semiconductor and widely used catalyst for a variety of reactions. Recently, this material has attracted further interest as a promising catalyst for energy and environment-related applications such as low-temperature CO oxidation, water splitting, and the oxygen reduction reaction. Surfaces have a key role in these applications, and a detailed understanding of the physical and chemical properties of Co\({}_{3}\)O\({}_{4}\) surfaces is important for the design of Co\({}_{3}\)O\({}_{4}\)-based functional materials with improved performance. Experimental atomic-scale investigations of Co\({}_{3}\)O\({}_{4}\) surfaces are relatively scarce however. To help obtaining a better fundamental understanding of the surface properties of Co\({}_{3}\)O\({}_{4}\), in this work we present a first principles Density Functional Theory (DFT) study of the Co\({}_{3}\)O\({}_{4}\) surface, which is the predominant one on Co\({}_{3}\)O\({}_{4}\) nano-rods, and is believed to be mainly responsible for the oxidation reactivity of this material.
Co\({}_{3}\)O\({}_{4}\) crystallizes in the cubic normal spinel structure with magnetic Co\({}^{2+}\) ions in tetrahedral sites and non-magnetic Co\({}^{3+}\) ions in octahedral sites. The surface is a Type III polar surface according to Tasker's criterion. It has two different terminations, usually denoted as the A and B terminations (see Fig. 1): the-A termination exposes both Co\({}^{2+}\) and Co\({}^{3+}\) ions, whereas the-B termination has only Co\({}^{3+}\) ions. As Co\({}_{3}\)O\({}_{4}\) is basically ionic, the unit cell of the-A termination \(-\) exposing two Co\({}^{2+},\) two Co\({}^{3+},\) and four O\({}^{2-}\) ions \(-\) has formal charge \(+2\), whereas the same unit cell on the-B termination exposes two Co\({}^{3+}\) and four O\({}^{2-}\) ions, and therefore has formal charge -2. Thus a slab can be viewed as a stack of charged layers as sketched in While in principle such a system has a polarization which increases linearly with slab thickness and eventually diverges, in reality polarity compensation mechanisms exist which prevent the "polar catastrophe" and stabilize the surface (also see Fig. 2)A number of first principles studies of Co\({}_{3}\)O\({}_{4}\) have been already reported, but some basic properties, including the polarity compensation mechanism, have not been examined in detail and/or are not yet well understood. An objective of this work is thus to investigate how polarity is compensated on the two different surface terminations of Co\({}_{3}\)O\({}_{4}\). Since experiments do not show evidence of surface reconstruction on either termination, we will restrict to undefected and unreconstructed-A and-B terminations obtained by simply relaxing the bulk truncated structures, and will study the compensation mechanism by focusing on the surface electronic structure. We will also examine the surface magnetic structure as recent experiments on Co\({}_{3}\)O\({}_{4}\) nanostructures have revealed interesting features which cannot be fully explained simply on the basis of the magnetic properties of bulk Co\({}_{3}\)O\({}_{4}\).
Following our recent investigation of bulk Co\({}_{3}\)O\({}_{4}\), the present study of the Co\({}_{3}\)O\({}_{4}\) surface is based on DFT calculations within the generalized gradient approximation (GGA) augmented with an on-site Coulomb repulsion U term in the 3d shell of the cobalt ions. The GGA+U approach reduces significantly the delocalization error arising from the incomplete cancellation of the Coulomb self-interaction in pure GGA calculations, and gives a value of the band gap for bulk Co\({}_{3}\)O\({}_{4}\) (1.96 eV) in reasonable agreement with experiment (\(\sim\)1.6 eV). The U repulsion terms in Ref. were determined from first principles using linear response. The resulting values, U = 4.4 and 6.7 eV for the Co\({}^{2+}\) and Co\({}^{3+}\) ions, respectively, reflect the different oxidation states and local electronic structure of the two ions. For surfaces, however, it is difficult to pre-identify the oxidation states of the surface Co ions. Moreover, the use of multiple U values renders the calculation of surface energies and other thermodynamics quantities more involved. Therefore in this work we use a single U value for all Co ions in our models, namely U = 5.9 eV, which corresponds to the weighted average of the two computed U values for the bulk. The bulk properties computed using this U for all Co ions are very similar to those reported in Ref.. For example, the band gap is 1.96eV using two U values, and 1.92eV using their weighted average.
This paper is organized as follows. After a brief description of the computational methods in Sec. II, in Sec. III we first present our results on the surface structural, magnetic, and electronic properties. Next, based on analysis of the Wannier functions, the polarity compensation and surface charge are discussed, and the critical thickness for polarity compensation is evaluated. Conclusions are given in Sec. IV.
## II Methods and Models
Calculations were performed within the plane wave-pseudopotential scheme as implemented in the Quantum Espresso package. Spin polarization was always included and exchange and correlation were described using the gradient corrected Perdew-Burke-Ernzerhof (PBE) functional with on-site Coulomb repulsion U term on the Co 3d states. As mentioned in the Introduction, we used a single U value for all Co ions, namely U = 5.9 eV, which corresponds to the weighted average of the two computed U values for the bulk. For comparison, pure PBE calculations have been also performed; however, unless otherwise specified, only PBE+U results are reported in the following. Ultrasoft pseudopotentials were used and the valence electrons included O 2s, 2p and Co 3d, 4s states. Plane wave energy cutoffs of 35 Ry for the smooth part of the wavefunction and 350 Ry for the augmented density were found sufficient to ensure a good convergence of the computed properties.
Surfaces were modeled using a periodic slab geometry, with consecutive slabs separated by a vacuum layer 15 A wide. We adopted the PBE+U lattice constant from our previous work which is 2% larger than the experimental one. (Pure PBE calculations were performed with the corresponding optimized lattice constant.) To study the properties of a single A or B termination, we considered symmetric slabs with odd number of layers, for which the total dipole moment is zero. Although non stoichiometric, these models provide useful information in the thick sample limit, when the effect of the nonstoichiometry becomes negligible. We performed tests on slabs with different number of layers, from 5 up to 11 layers, and found that a well converged description could be achieved with 9-layer models. On the other hand, to achieve perfect stoichiometry, one should consider slabs with even number of layers, which expose the A and B terminations on the two different sides, and have a dipole moment perpendicular to the slab. We also performed tests to compare the results obtained with symmetric and non-symmetric slabs and found that the surface properties (e.g. the surface electronic structures of the different terminations, see Sect. 3C) obtained with 9-layer models agree well with those from symmetric slabs of 8 or 10 layers. Results reported in the following thus refer to calculations on 9-layer models, unless otherwise specified. Structural optimizations were carried out by relaxing all atomic positions until all forces were smaller than 1\(\times\)10-3 a.u.
For most calculations the rectangular surface cell depicted in was used, and the sampling of the surface Brillouin zone was performed using a 3\(\times\)4 k-point grid. Comparisons to calculations using a 4\(\times\)6 k-point grid show surface energy differences of \(\sim\) 1meV/A\({}^{2}\). Maximally-localized Wannier functions (MLWFs) were obtained using the \(\Gamma\) point only on models with a surface supercell twice the size of the rectangular cell in Test calculations showed that the results for the two setups were in satisfactory agreement. The MLWFs were calculated with the algorithm developed by Sharma et al..
## III Results and Discussion
### Energetics and structure
####.1.1.1 A.1.
Experimental studies on Co\({}_{3}\)O\({}_{4}\) epitaxial films grown on MgAl\({}_{2}\)O\({}_{4}\) single crystal substrates found that the surfaces of the as-grown films are relatively disordered and have an oblique low-energy electron diffraction (LEED) pattern characteristic of the-B termination, whereas the annealed surfaces show a sharp rectangular LEED pattern indicating a well ordered-A termination. These findings indicate that the-A termination is more stable than the B one under Ultra High Vacuum (UHV) conditions. However, the occurrence of the-B termination on the as-grown films suggests the existence of kinetic limitations, so that the actual exposed termination may depend on the synthetic method and the post-treatment of the samples.
In order to study the properties of a single termination, it is convenient to consider symmetric, non-stoichiometric slabs, and express their surface formation energies in terms of the chemical potentials of Co (\(\mu_{\rm Co}\)) and oxygen (\(\mu_{\rm O}\)). Since \(3\mu_{\rm Co}+4\mu_{\rm O}=\mu_{\rm Co3O4}\) under equilibrium conditions, \(\mu_{\rm Co3O4}\) being the chemical potential of bulk Co\({}_{3}\)O\({}_{4}\), it is possible to eliminate the dependence on \(\mu_{\rm Co}\), and express the surface energy only in terms of the oxygen chemical potential \(\mu_{\rm O}\) or, equivalently, \(\mu_{\rm O}\)'\(\equiv\mu_{\rm O}\) - \(\gamma_{2}\) E\({}_{\rm tot}\)(O\({}_{2}\)), where E\({}_{\rm tot}\)(O\({}_{2}\)) is the total energy of an O\({}_{2}\) molecule. The oxygen potential \(\mu_{\rm O}\)' satisfies the condition \(1/4\) H\({}_{\rm f}\leq\mu_{\rm O}\)'\(\leq 0\), where H\({}_{\rm f}\) is the heat of formation of bulk Co\({}_{3}\)O\({}_{4}\) and the lower and upper limits correspond to O-poor and O-rich conditions, respectively. Values for H\({}_{\rm f}\) are given in Ref..
The computed surface energies for slab models with 5, 7 and 9 layers in the O-rich limit (\(\mu_{\rm O}\)' =0) are listed in Table 1, whereas shows the surface energies in the full range of \(\mu_{\rm O}\)' for the 9-layer slabs. For the sake of comparison with previous GGA calculations, in results obtained at both the pure PBE and PBE+U levels are presented. We can see a significant difference between the results of the two approaches. According to the pure PBE calculations the-B termination has lower surface energy except at very low \(\mu_{\rm O}\)', in agreement with previous published results. By contrast, the PBE+U calculations predict the A- termination to be more stable in a wide range of the oxygen chemical potential, consistent with the experimental results of Ref.. This difference between the PBE and PBE+U results can be understood on the basis of the computed surface electronic structures, reported in Sect. 3C. Briefly, the B termination is found to have delocalized metallic surface states, for which the energy penalty from the Hubbard U term is larger, thus making the surface energy of the B termination higher. The PBE functional is known to overestimate the O\({}_{2}\) binding energy: our computed value is 130 Kcal/mol, against 118 kcal/mol from experiment.
\begin{table}
\begin{tabular}{c c c} \hline & \multicolumn{2}{c}{Surface Energy (eV/Å\({}^{2}\))} \\ & A termination & B termination \\ \hline
5-layer & 0.081 & 0.080 \\
7-layer & 0.085 & 0.081 \\
9-layer & 0.082 & 0.080 \\ \hline \end{tabular}
\end{table}
Table 1: Surface energies of Co\({}_{3}\)O\({}_{4}\), computed at the PBE+U level and in the O-rich limit, for symmetric slabs of different thicknesses.
### A.2.
The A-terminated Co\({}_{3}\)O\({}_{4}\) surface exposes all types of ions present in the bulk, namely Co\({}^{2+}\), Co\({}^{3+}\) and O\({}^{2-}\) ions. (We identify the surface ions with the oxidation state they have in the bulk, even though their actual oxidation state may be different at the surface.) The Co\({}^{2+}\) (Co\({}^{3+}\)) ions are 3-fold (4-fold) coordinated and form bonds with two surface oxygen ions and one (two) oxygen(s) in the second layer; they will be denoted Co-3f (Co-4f) in the following. All surface oxygens are equivalent and 3-fold coordinated to one Co-3f and one Co-4f surface ion as well as to one 6-fold Co\({}^{3+}\) in the second layer (see Fig. 1). Calculated atomic relaxations on the-A termination are listed in Table 2. While all surface atoms undergo an inward relaxation, this relaxation is larger for the Co than for the oxygen ions, and therefore the surface becomes slightly buckled. The reflection symmetry of the surface remains during relaxation, so that on the relaxed-A surface there is one type of 3-fold and one type of 4-fold Co ion as well as one type of oxygen ion. As shown in Table 2, all surface Co-O bonds are shorter after relaxation.
\begin{table}
\begin{tabular}{l c c c c c} \hline & \multicolumn{3}{c}{Atomic displacement (Å)} & \multicolumn{3}{c}{Bond expansion} \\ & \(\Delta\)x & \(\Delta\)y & \(\Delta\)z & Label & \(\Delta\) \\ \hline Co3f1 & 0.17 & 0.00 & -0.22 & Co3f2-O1 & -5.9\% \\ Co3f2 & -0.17 & 0.00 & -0.22 & Co4f1-O1 & -0.2\% \\ Co4f1 & 0.00 & 0.00 & -0.19 & Co4f1-O3 & -0.2\% \\ Co4f2 & 0.00 & 0.00 & -0.19 & Co3f1-O3 & -5.9\% \\ O1 & 0.00 & -0.06 & -0.05 & & \\ O2 & 0.00 & 0.06 & -0.05 & & \\ O3 & 0.00 & 0.08 & -0.05 & & \\ O4 & 0.00 & -0.08 & -0.05 & & \\ \hline \end{tabular}
\end{table}
Table 2: Atomic displacements from bulk-like positions on the relaxed-A surface. Displacements along the, [1\(\,\)0] and directions are denoted as (\(\Delta\)x, \(\Delta\)y, \(\Delta\)z). Atoms are labeled as in Fig.1.
The less dense-B surface exposes only Co\({}^{3+}\) and O\({}^{2-}\) ions. All Co ions are equivalent and 4-fold coordinated to two surface and two second layer oxygens. There are two different types of surface oxygen ions: one is 2-fold (O-2f) coordinated to one surface Co ion and one 4-fold Co\({}^{2+}\) ion in the second layer; the other is 3-fold (O-3f) coordinated to one surface Co and two Co\({}^{3+}\) ions in the second layer (see Fig. 1). Table 3 shows the computed atomic relaxations for the-B termination. The surface 2-fold and 3-fold oxygen ions behave differently upon relaxation: O-2f ions relax outwards and the bond with Co ions weakens, whereas O-3f ions relax inwards and their bonds to Co ion become stronger upon relaxation.
### Surface magnetization
In bulk Co\({}_{3}\)O\({}_{4}\), only the Co\({}^{2+}\) ions at tetrahedral sites have a magnetic moment, whereas the Co\({}^{3+}\) ions at octahedral sites are non-magnetic. At the surface, the bulk symmetry is broken and the ionic coordinations are reduced, and therefore the magnetic properties of the surface cobalt ions can differ from those in the bulk. We computed the magnetic moments of the different surface ions on the-A and-B surfaces using a Lowdin charge analysis.
\begin{table}
\begin{tabular}{l c c c c} \hline & \multicolumn{2}{c}{Atomic displacement (Å)} & \multicolumn{2}{c}{Bond expansion} \\ & \(\Delta\)x & \(\Delta\)y & \(\Delta\)z & Label & \(\Delta\) \\ \hline Co4f & -0.05 & 0.08 & -0.08 & Co1-O2f & 2\% \\ O2f & -0.05 & -0.04 & 0.08 & Co1-O3f & -3\% \\ O3f & 0.00 & -0.02 & -0.14 & & \\ \hline \end{tabular}
\end{table}
Table 3: Atomic displacements from bulk-like positions on the relaxed-B surface. Displacements along the, [1\(\,\)\(\,\)\(\,\)\(\,\)\(\,\)\(\,\)\(\,\)\(\,\)\(\,\)\(\,\)\(\,\)\(\,\,\)\(\,\,\)\(\,\,\)\(\,\,\)\(\,\,\)\(\,\,\)\(\,\,\)\(\,\,\)\(\,\,\)\(\,\,\,\)\(\,\,\)\(\,\,\,\)\(\,\,\)\(\,\,\,\)\(\,\,\,\)\(\,\,\,\)\(\,\,\,\)\(\,\,\,\)\(\,\,\,\)\(\,\,\,\)\(\,\,\,\)\(\,\,\,\)\(\,\,\,\)\(\,\,\,\)\(\,\,\,\,\)\(\,\,\,\)\(\,\,\,\)\(\,\,\,\)\(\,\,\,\)\(\,\,\,\)\(\,\,\,\)\(\,\,\,\)\(\,\,\,\,\)\(\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\,\,\)\(\,\,\,\,\,\,\)\(\,\,\,\,\,\)\(\,\,\,\,\,\,\)\(\,\,\,\,\,\,\)\(\,\,\,\,\,\,\)\(\,\,\,\,\,\,\)\(\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\)\(\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\)\(all surface Co ions have similar magnetic moments, which are also similar to the computed magnetic moment, 2.59\(\upmu_{\text{B}}\), of the Co\({}^{2+}\) ions in bulk Co\({}_{3}\)O\({}_{4}\). Contour plots of the surface spin density for both terminations are shown in We can see that on the-A surface the oxygen ions are essentially non-magnetic, whereas on the-B termination a slight spin polarization is present on the O-2f ions. The ionic magnetic moments in the second layer are already the same as in the bulk.
To determine the ground state surface magnetic configuration, we need to analyze the couplings between the different magnetic moments. In contrast to the bulk, where magnetic couplings are due to weak superexchange interactions (two metal ions separated by two oxygen ions), on the surface the presence of magnetic Co\({}^{3+}\) ions gives rise to normal superexchange interactions (two metal ions separated by one oxygen ion). There are normal superexchange interactions between surface Co ions, as well as between surface ions and the magnetic Co\({}^{2+}\) ions in the next layer (Fig.1). For the A termination, there are three different superexchange interactions. The coupling between surface neighboring Co-4f ions (J1 in Fig. 1) is via an intermediary oxygen ion in the second layer, with a Co-O-Co angle of 90\({}^{\circ}\). According to the Goodenough-Kanamori-Anderson (GKA) rules, the exchange interaction between them is ferromagnetic. The other two superexchange interactions are associated with angles of about 120\({}^{\circ}\), for which the GKA rules do not make well defined predictions. The ground state ordering obtained by calculating the surface energies of different magnetic configurations is given in Table 5.
\begin{table}
\begin{tabular}{c c c c} \hline \multicolumn{2}{c}{**A termination**} & \multicolumn{2}{c}{**B Termination**} \\ \hline Ion type & Magnetic moment & Ion type & Magnetic moment \\ \hline Co3f (Co\({}^{2+}\) in bulk) & 2.64 & Co4f (Co\({}^{3+}\) in bulk) & 2.56 \\ Co4f (Co\({}^{3+}\) in bulk) & 2.52 & O2f & 0.08 \\ O & 0.02 & O3f & 0.02 \\ \hline \end{tabular}
\end{table}
Table 4: Magnetic moments (\(\upmu_{\text{B}}\)) of surface ions determined through Lowdin charge analysis On the-B termination, the distance between the surface magnetic Co-4f ions is quite large, and therefore the coupling between them can be considered weak. The only normal superexchange interaction is the one between surface Co-4f and Co\({}^{2+}\) ions in the second layer, which is also associated with a Co-O-Co angle of about 120\({}^{\circ}\). From total energy differences between different magnetic configurations, it appears that this coupling is antiferromagnetic (see Table 5).
Based on the results in Table 5, the expected surface ground state magnetic configurations for the A and B terminations are schematically illustrated in The surface region comprises the first and second layers, and is characterized by normal superexchange couplings, whereas below the second layer only weak antiferromagnetic superexchange interactions are present, as in bulk Co\({}_{3}\)O\({}_{4}\). The presence of a ferrimagnetic surface region on the A termination is interesting. It can provide the mechanism to understand a number of experimental observations on Co\({}_{3}\)O\({}_{4}\) nanostructures, notably: (i) the decoupling of magnetic core and shell contributions ; (ii) the ferrimagnetic behavior of porous nanostructures; (iii) the exchange anisotropy phenomena observed in Co\({}_{3}\)O\({}_{4}\) nanowires.
\begin{table}
\begin{tabular}{c c c c} \hline & A Termination & & B Termination \\ \hline & 0.0 & & 0.0 \\ & 2.8 & & 6.5 \\ & 3.3 & & \\ \hline \end{tabular}
\end{table}
Table 5: Surface energies (meV/Å\({}^{2}\)) of various magnetic configurations relative to the lowest energy state, taken as zero. Co4f ions are schematically indicated by underlined arrows, Co3f ions and Co\({}^{2+}\) ions in the second layer are indicated by arrows without underlines.
## C.
Surface electronic states in the bulk band gap are of great interest because they can strongly influence the physical and chemical properties of semiconductor materials. For Co\({}_{3}\)O\({}_{4}\), evidence of surface states in the band gap has been recently found in STM and STS studies on nanowires. In this section we characterize the surface states on both Co\({}_{3}\)O\({}_{4}\) terminations, by studying their energies and spatial distributions, i.e. on what ions these states are primarily localized, and how fast they decay when moving from the surface toward the bulk. The calculations were performed on symmetric slab models of 9 layers, for which spin densities are also symmetric, and spin up and spin down states are degenerate in energy. For this reason, we do not distinguish between spin up and spin down in the following; instead, all results include the sum over the two spin directions.
By comparison with the projected bulk structure (shaded area in Fig. 6), it is evident that on both surface terminations several surface state bands are present in the lower half of the bulk band gap. Partially occupied bands are present, indicating a metallic state. In Figure 7, we plot the Layer-Resolved Density of States (LRDOS) for surface models of A, B termination and a 4-layer bulk model. The DOS curves for the inner layers have a clear bulk-like character, as shown by the similarity between the bulk DOS and the DOS for the 4\({}^{\rm th}\) and 5\({}^{\rm th}\) layers of both surface models. At the surface new states appear close to the top of the valence band, while in the second layer, just below the surface, the tail of these states is still present, more prominent for the B termination, but starting from the third layer the DOS is already bulk-like.
To clarify the character of the surface states, in we show the partial densities of states, obtained by projecting the surface LRDOS onto the different surface oxygen and cobalt ionsseparately. On the-A termination, surface states originate predominantly from surface O 2p states, and may be described as oxygen dangling bond-like states. On the-B termination, both Co and oxygen contribute to the surface states which look more delocalized and metallic-like in character in comparison to those on the A termination. Partially metallic surface states are known to occur on other transition-metal oxide polar surfaces as well, notably on the Zn-terminated ZnO (000 I) surface, suggesting that partial metallization may be a quite common phenomenon on surfaces of transition metal oxides.
Work functions for the two surface terminations were computed at both PBE and PBE+U levels. The results, reported in Table 6, clearly show a larger work function for the B termination relative to the A case, which can be attributed to the different surface dipoles on the two surfaces. We can also notice that PBE+U predicts a larger value of the work function in comparison to PBE, which may be attributed to the stabilization of the Co d states at the Fermi energy caused by the U term.
## 4 Compensating charges and bonding properties from the analysis of Wannier functions
### Compensating charges
A simple way to determine the value of the compensating charge for each termination is by calculating the total charge Q\({}_{l}\) in each layer of the slab. This can done very effectively and precisely by counting the number of Wannier centers (WCs) associated with each ion in that layer.
\begin{table}
\begin{tabular}{c c c} \hline & A Termination & B Termination \\ \hline PBE & 3.96 & 4.59 \\ PBE+U & 5.28 & 5.97 \\ \hline \end{tabular}
\end{table}
Table 6: Computed work functions (eV) from PBE and PBE+U calculationsa total charge Q\({}_{I}\) = +1, instead of the value +2 found for the same layer in the bulk (see Figure 2). Similarly, for the-B termination, the total charge of the top layer is Q\({}_{I}\) = -1, instead of the value -2 for the same layer in the bulk. Below the second layer, the charge of each layer is the same, +2 or -2, as in the bulk. As expected, the compensating charges are \(\Delta\)Q= -1 and +1/cell for the A and B termination, respectively.
The same result can be also obtained by using a result of the modern theory of polarization which shows that the compensating (or external) surface charge density \(\sigma_{ext}\) is equal to the component of the bulk polarization, P\({}_{bulk}\), normal to the surface
\[\sigma_{ext}=P_{bulk}\cdot\stackrel{{\wedge}}{{n}}\;\;\;.\;cannot be described by a single Wannier function or Kohn-Sham state. Similarly, on the B termination the compensating hole is shared between two Co\({}^{3+}\) ions which are thus are partially oxidized. On the B termination, one MLWF has relatively large spread, indicating that this termination has a metallic character.
### _D.3. Non-symmetric stoichiometric slab models_
So far, our results were obtained from calculations on symmetric, non-stoichiometric slab models appropriate for the study of the surface properties of thick samples, on which charge compensation occurs naturally. In the case of thin films and nanostructures, however, the polarity may remain uncompensated below a critical thickness and possibly affect the properties and reactivity of these systems. It is therefore interesting to determine what is this critical thickness for Co\({}_{3}\)O\({}_{4}\). To this end we considered non-symmetric, stoichiometric slab models with different (even) number of layers and calculated the formation energy E\({}_{\mathrm{form}}\) (total energy difference between the slab and an equal number of bulk Co\({}_{3}\)O\({}_{4}\) units ) and the electrostatic potential energy drop along the slab \(\Delta\)V as a function of the number of layers. The results show that both E\({}_{\mathrm{form}}\) and \(\Delta\)V become approximately constant when the number of layers is larger than 4, implying that the critical thickness is 4 layers.
## **IV** Summary and Conclusion
We have presented an accurate and comprehensive computational study of the structural, electronic and magnetic properties of the polar Co\({}_{3}\)O\({}_{4}\) surface by the GGA+U method. We found the atomic relaxations give rise to a surface buckling of \(\sim\) 0.2 A on both surface terminations. Surface energy calculations indicate that the-A termination is more stable in a wide range of the oxygen chemical potential, in agreement with surface science experiments.
The Co\({}^{3+}\) ions do not have a magnetic moment in the bulk but become magnetic at the surface which leads to interesting surface magnetic properties, as found also in recent experiments on Co\({}_{3}\)O\({}_{4}\) nanostructures. From band structure and density of states calculations, we found that surface electronic states are present in the bulk band gap for both terminations, consistent with STM experiments on Co\({}_{3}\)O\({}_{4}\) nanowires. The B termination is found to have a more pronounced metallic character compared to the-A surface. It has also a larger work function, which could play an important role in the study of surface redox reactions. Maximally localized Wannier functions clearly show that charge compensation takes place on the top layer of both terminations. They also reveal that the surface is more covalent with respect to the bulk. Calculations on asymmetric models predict a critical thickness for polarity compensation of 4 layers. We hope that these predictions can be tested experimentally in the near future.
#
Charge densities of typical covalent MLWFs on the-A (left) and-B (right) termination.
Formation energy and electrostatic potential energy drop (eV) for stoichiometric slab models as a function of the number of layers in the slab. | 10.48550/arXiv.1201.6013 | Electronic states and magnetic structure at the Co3O4 (110) surface: a first principles study | Jia Chen, Annabella Selloni | 4,691 |
10.48550_arXiv.1412.3228 | ### 6- Conclusion and discussion
This short note demonstrates on a simple numerical experiment how filtering material properties is able to efficiently improve the spectral methods applied to heterogeneous materials, essentially by reducing the required spatial resolution and/or reducing spurious oscillations. This approach only modifies the input parameters of these methods and can be used in addition to improved algorithms or modified discrete Green operators. In contrast with Voigt and Reuss filters, the 2-layer filter, proposed in this note, is efficient in both cases of a stiff or compliant inclusion within a matrix.
The choice of the filter radius should be at least the optimal value found on (B and D), almost 0.5 (approximately the size of a pixel, or voxel in 3D), which significantly decreases the minimum required spatial resolution \(N_{1\%}\) but without any significant effect on the local fields. Increasing the filter radius reduces spurious oscillations but increases the minimum required resolution \(N_{1\%}\). Hence, the choice will result from a compromise between the computation time (closely related to the resolution) and the quality of the local fields.
These conclusions are discussed in the view of extrapolated material properties obtained from truncated Fourier series with the Fourier coefficients evaluated from FFT applied to the fields \(\widetilde{c}(x_{\alpha})\). Without any filter, important oscillations are observed and a part of this oscillations seems to correspond to spurious oscillations observed on figure 2-A. For a radius of 0.55, oscillations are reduced but still visible, and for a radius of 1.0, oscillations are almost invisible in line with the reduced spurious oscillations on figure 2-E and F. On the other hand, increasing the filter radius increases the 'interphase' thickness and decreases the agreement with the real microstructure leading to increase the required spatial resolution. Finally, minimizing the gap between the exact microstructure and extrapolated ones lead to an optimal value of 0.55, consistent with the optimal value deduced from numerical simulations. To conclude this discussion, the Fourier extrapolation of material properties could explain our results in terms of spurious oscillations and evolution of the minimum required resolution \(N_{1\%}\).
Number of iterations at convergence (spatial resolution=65) as a function of the elastic contrast (A) and of the material reference (B). The parameter \(\alpha\) multiplies the optimal choice given in for isotropic elastic phases. Below this optimal value, the algorithm does not converge. Solid, dashed and dashed-dot lines, associated to 0.55, 1.0 and 1.5 filter radius, are almost superimposed.
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‘Exact’ material property (A). Extrapolation with truncated Fourier series associated to a 65x65 grid of the material properties without any filter (B) and with filter radii of 0.55 (C) and 1.0 (D).
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* F. Willot, B. Abdallah, and Y.P. Pellegrini. Fourier-based schemes with modified Green operator for computing the electrical response of heterogeneous media with accurate local fields. _International Journal for Numerical Methods in Engineering_, 98:518-533, 2014.
* J. Zeman, J. Vondrejc, J. Novak, and I. Marek. Accelerating a FFT-based solver for numerical homogenization of periodic media by conjugate gradients. _Journal of Computational Physics_, 229:8065-8071, October 2010. | 10.48550/arXiv.1412.3228 | Filtering material properties to improve FFT-based methods for numerical homogenization | Lionel Gélébart, Franck Ouaki | 1,657 |
10.48550_arXiv.2206.08285 | ## Antimony telluroiodide (SbTeI) is predicted to be a promising material in many technological applications based on theoretical simulations, however the bulk structure solution remains elusive. We consolidate SbTeI belonging to the base-centered monoclinic lattice with a space group C 2/m by combining single crystal X-ray diffraction and X-ray photoemission spectroscopy techniques. The atomic arrangement of the reported crystal structure is remarkable with one-dimensional double-chains forming two-dimensional blocks. In this structure, the Sb\({}^{3+}\) ion is surrounded by Te\({}^{2-}\) and \(\Gamma\), which is distinguishable by an incomplete polyhedron resulting in the 5s\({}^{2}\) (Sb) lone pair electrons in the valence band. Manipulation of this material with pressure to induce novel structures and properties is highly anticipated.
*_Corresponding author:
## I.
Antimony and bismuth chalcohalides show great promise for photovoltaic applications, where they also feature flexible electronics and novel piezoelectric devices. In contrast to common Pb and Sn, compounds containing trivalent Sb/Bi have ecological advantage and demonstrate good stability parameters in air. Recently, SbTeI material was highlighted as relevant absorbent of the solar spectrum with high conversion efficiency evaluated from the theoretical absorption coefficients. In addition, it shows low thermal conductance of the sheet which is favorable for 2D thermoelectric materials from the monolayer. Furthermore, Rashba effect in a single-layer antimony telluroiodide was predicted for potential spintronics applications. Theoretical calculations of the low formation energy and real phonon modes imply that the system is stable. It should be noted that the Rashba spin splitting in the spin-orbit coupling (SOC) band structure of single-layer SbTeI is significantly larger than that of a number of two-dimensional systems, including surfaces and interfaces.
A lot of research has been done recently with the relatively close compound BiTeI, which has many unique properties, including the Rashba effect.. However, structurally, SbTeI is much different from BiTeI which is clearly layered 2D material and of significantly higher symmetry - trigonal system, space group \(P3m1\) (No. 156). As a semiconductor, SbTeI has more in common with 1D ferroelectric SbSI-type compounds, most of which are characterized by orthorhombic crystal symmetry comprising of infinite double-chains. Yet, in this work, we show that SbTeI may be an intermediate example between 1D - 2D systems, as it holds somewhat both layer and chain arrangement.
In the literature, there are only a few studies dealing with crystal structure of SbTeI and their results are quite different. This causes confusion, for example, when performing theoretical_ab-initio_ calculations. Initially, it was announced that the structure may adopt the orthorhombic or even the lowest triclinic symmetry. Investigating more precisely, A. G. Papazoglou and P. J. Rentzeperis found that the symmetry should be monoclinic with the space group \(C\) 2/_m_. On the other hand, theoretical simulations indicate that bulk SbTeI with the trigonal symmetry (analogous to the BiTeI crystal) exhibits energy 19 meV/atom lower than that of monoclinic SbTeI. Therefore, seeing the growing interest in these ternary materials as well as many uncertainties, we have undertaken growth and structural analysis of pure single crystals of SbTeI. In this study, we also examine their surface chemical composition and internal electronic properties.
## II.
SbTeI compound was prepared by heating a high purity mixture of antimony (99.999%), tellurium (99.999%), and iodine (\(\geq\)99.99%) purchased from "Sigma-Aldrich". Stoichiometric amounts of Sb, Te, and I powders were sealed in an evacuated quartz tube and placed in a rotating furnace. The temperature was slowly raised up to 670 K and held at this level for 24 h. Then, the tube was fixed and kept stable in turn temperature, which was gradually lowered to the room temperature. The resulting crystals were black-gray in color and had a mirror-like surface (see Fig. 1a).
A specimen of SbTeI was used for the X-ray crystallographic analysis. The X-ray intensity data were measured on a Bruker D8 VENTURE PHOTON II system equipped with a Microfocus Incoatec Ims 3.0 (Ag Ka, \(\lambda\) = 0.56086 A) and a multilayer optic monochromator. A total of 872 frames were collected. The total exposure time was 0.48 h. The frames were integrated with the Bruker SAINT software package using a narrow-frame algorithm. The integration of the data using a monoclinic unit cell yielded a total of 2132 reflections to a maximum \(\theta\) angle of 22.12\({}^{\circ}\) (0.74 A resolution), of which 604 were independent (average redundancy 3.530, completeness = 100.0 %, Rint= 4.36 %, Rsig= 3.97 %) and 542 (89.74%) were greater than 2\(\sigma\)(F\({}^{2}\)). The final parameters listed in the Tables 1-4 are based upon the refinement of the XYZ-centroids of 2513 reflections above 20 \(\sigma\)(I) with 4.476\({}^{\circ}<20\)\(<\) 44.15\({}^{\circ}\). The data were corrected for absorption effects using the Multi-Scan method (SADABS). The ratio of minimum to maximum apparent transmission was 0.462. The structure was solved and refined using the Bruker SHELXTL Software Package, using the space group \(C\) 2/\(m\) (No. 12, unique axis \(b\)), with \(Z=4\) for the formula unit, SbTeI. The final anisotropic full-matrix least-squares refinement on F\({}^{2}\) with 19 variables converged at R\({}_{1}=\) 3.27% for the observed data and wR\({}_{2}=8.72\) % for all data. The goodness-of-fit was 1.20. The largest peak in the final difference electron density synthesis was 1.792 e\({}^{\text{-}}\)/A\({}^{3}\) and the largest hole was -1.711 e\({}^{\text{-}}\)/A\({}^{3}\) with an RMS deviation of 0.387 e\({}^{\text{-}}\)/A\({}^{3}\). On the basis of the final model, the calculated density was 6.009 g/cm\({}^{3}\) and F, 624 e\({}^{\text{-}}\).
X-ray photoelectron spectroscopy (XPS) was used to investigate the surface composition and chemical state of the elements within a SbTeI. X-ray photoelectron spectra were acquired at room temperature by using a PHI Versaprobe 5000 spectrometer. The photoelectrons were excited using monochromatized 1486.6 eV Al radiation, 25 W beam power, 100 \(\upmu\)m beam size and 45\({}^{\circ}\) measurement angle. Sample charging was compensated using dual neutralization system consisting of low energy electron beam and ion beam. The random C 1\(s\) line with binding energy fixed at 284.6 eV was used for correction of the charging effects. After background subtraction, a non-linear least squares curve fitting routine with a Gaussian/Lorentzian product function was used for the analysis of XPS spectra.
Theoretical simulations of different structures and compositions within the ternary Sb-Te-1 system were performed using the crystal structure prediction method implemented in the USPEX code. The USPEX was used in coupling with the first-principles calculations within the framework of density functional theory (DFT) using the Vienna _ab initio_ simulation package VASP. For the determinations of the total energies, optimized lattice structures and their corresponding electronic structures, the Perdew, Burke and Ernzerhof generalized gradient exchange-correlation functional (PBE-GGA) was selected. The plane-wave kinetic energy cutoff was set to 500 eV and a Brillouin zone sampling resolution was 2\(\pi\)\(\times\) 0.05 A-1. All structures were relaxed at ambient pressure and 0 K, and the enthalpy was used as fitness function. In addition, for the structural candidates searched by USPEX, we further performed DFT analysis using the WIEN2k software. Here, the calculations were carried out on a mixed basis set of augmented plane waves (APW) and linearized augmented plane waves (LAPW). The exchange and correlation potentials have been treated according to the PBE-GGA scheme. Wave functions in the interstitial regions were expanded in plane waves, with the plane wave cutoff chosen so that \(R_{\rm MT}k_{\rm max}=7\) (where \(R_{\rm MT}\) represents the smallest atomic sphere radius and \(k_{\rm max}\) is the magnitude of the largest wave vector). The energy separation between core and valance states was -6.0 Ry. For the structure optimization, we required that the forces are smaller than 1 mRy/a.u. When performing self-consistent calculations for fixed geometry, the iteration halted when the difference in the eigenvalues was less than 0.0001 between steps of convergence criterion.
## III.
Our single crystal X-ray diffraction (XRD) measurements indicate that SbTeI adopts a monoclinic lattice (space group \(C\)\(2/m\)) with the unit cell parameters \(a=13.698\) A, \(b=4.2289\) A, \(c=9.191\) A, \(\beta=\)128.626\({}^{\circ}\) (see more details provided in Table 2 and supplementary material). This coincides with the results presented in Ref., where the unit cell parameters \(a=13.7008\) A, \(b=4.2418\) A, \(c=9.2005\) A, and \(\beta=\)128.631\({}^{\circ}\) were reported. This also confirms that SbTeI crystals synthesized at ambient pressure are indeed monoclinic rather than orthorhombic or trigonal, in contrast to the triclinic symmetry (\(P1,P\overline{1}\)) announced in Ref..
As it was briefly mentioned in the introduction, the double-chains of SbTeI extending through the shorted \(b\)-axis of the unit cell are very similar to those in SbSI-type structure. However, double-chains of SbTeI are uniquely situated forming plate-like blocks (see Fig. 1b). These blocks are likely held together by a weak van der Waals connection. Such an arrangement has strong anisotropic properties and a rather brittle nature, which can be easily observed by mechanical pressing of the sample. The calculated universal anisotropy index, \(A^{U}\), gives the value of 1.85 indicating strong anisotropy as well. For comparison, BiTeI has \(A^{U}\)\(=0.37\) and \(A^{U}\)\(=0\) represents locally isotropic crystals. As for the atomic bonds inside the chains, considerable covalent character can be pondered in the Sb-Te bond because it is shortest in length and closer to the sum of the covalent radii. Overall, a mixed covalent-ionic-metallic character prevails the structure where metallicity is gained from tellurium. Mossbauer spectroscopy showed that Sb systems containing tellurium, in most cases, have negative quadrupolar splitting, indicating a tendency to metallic character. On the contrary, sulfur (as well as selenium in similar systems) usually tends to ionic character.
The metallicity of bonds as well as forbidden gap value are closely related to the stereochemical activity of the lone pair which is often considered as a key factor for the efficient carrier transport in optoelectronic materials. On the other hand, the example of SbSI showed that the lone pair of antimony plays an active role in the stabilization of ferroelectricity. The monoclinic SbTeI variation indeed has the lone pair of antimony. shows that there is one missing atom at one edge of the octahedron for SbTeI. Thus, the octahedron is incomplete and takes square pyramidal shape: tellurium is at the apex of a pyramid whose base is one of the Te\({}_{2}\)I\({}_{2}\) faces of the triangular prism. Although lone pair does not participate in bonding, its properties are connected to the local atomic arrangement of antimony. In fact, the stereochemical activity of the lone pair is sensitive to the local environment because it affects the loss of sphericity of the 5_s_' electron distribution around the Sb atom. Usually, small coordination polyhedrons with strong covalent bonds have active lone pairs and their stereochemical activity increases with the distortion of the local environment. The structural modifications then correspond to changes in the nature of bonds from more or less ionic to covalent or metallic. On the basis of chemical bonding, the stereochemical activity of the lone pair for SbTeI is considered to be low in activity. However, it is likely that chemical or physical pressure-induced changes would cause electronic transformations because the lone pair electrons are directed to occupy the space between plate-like blocks. Such an effect on the structure can be quite significant, as in the case of layered Bi\({}_{2}\)O\({}_{2}\)S, where the high-pressure enforces the lone pair electrons to disappear and triggers 2D-to-3D structural transition.
In general, to balance the charge, SbTeI with 1:1:1 stoichiometry should be formed from a trivalent cation with divalent and monovalent anions, respectively. Here, we used XPS methods to define the chemical state and related peculiarities of SbTeI. shows the peaks of Sb 3\(d\),I 3\(d\), Te 3\(d\) and valence band (VB) spectrum collected at room temperature. The Sb 3\(d\) spin-orbit doublet is situated at the binding energies (BEs) of 530.3 eV and 539.7 eV, for Sb 3\(d_{5/2}\) and Sb 3\(d_{3/2}\), respectively. The fitting of the experimental data indicates that each line of the Sb 3\(d\) spin-orbit doublet has an additional minor component approximately spaced by 1.4 eV (see Fig. 2a). All four lines are well matching with Sb 3\(d\) in sonochemically prepared SbSI. For the antimony chalcohalides the lines divided into two groups indicate separate states of surface and bulk. Although the surface tends to oxidize, both BEs represent Sb\({}^{3+}\) species. No other antimony species contributions corresponding to Sb\({}^{4+}\) or Sb\({}^{5+}\), which appear at slightly higher BEs, are seen. The oxidation of the surface is also evident from I 3\(d\) and Te 3\(d\) spectra given in Figs. 2b and 2c, respectively. We observe that I 3\(d\) region has well separated spin-orbit doublet, \(\Delta\) = 11.5 eV. The position of I 3\(d_{5/2}\) is at BE = 618.8 eV and of I 3\(d_{3/2}\) - at BE = 630.3 eV. The chemical shift is -0.7 eV to lower energy values (pure I 3\(d_{5/2}\) it is at BE = 619.5 eV and for I 3\(d_{3/2}\) at BE = 631.0). The main lines have two adjacent components spaced 1.3 eV apart. The main Te 3\(d_{5/2}\) and 3\(d_{3/2}\) peaks were located at BEs of 573.1 and 583.5 eV, respectively. This indicates metallic tellurium similar, as for example, in PtTe\({}_{2}\) crystals. In addition, rather strong peaks at BEs of 576.32 and 586.72 eV were observed as well. The later peaks can be assigned to the Te-O bonds which, as mentioned before, are often found on the surfaces of chalcogenide materials due to the rapid surface oxidation after exposure to the ambient atmosphere.
In the given region, three bands can be singled out. The shape of this structure is comparable to the VBs reported for the SbSI-type crystals and corresponds well to the DFT simulated total density of states (T-DOS) (see Fig. 3a).
However, the well separated bands allow their composition to be identified. According to the simulations, the lowest BE band is mainly the anion-\(p\)-dominated consisting of Sb 5\(p\), I 5\(p\) and Te 5\(p\) hybridized orbitals. The partial densities of states (P-DOS) reveal only very small portion of the Sb 5\(s\) involved in this band. The second band should be related to the lone pair of antimony, because it has most pronounced 5\(s\) states here. Overall, the second and third bands originate from Sb 5\(s\), I 5\(s\) and Te 5\(s\) hybridized orbitals. In addition, as P-DOS indicates, the conduction band (CB) composed of Sb 5\(p\), I 5\(p\) and Te 5\(p\) hybridized orbitals where cation has the largest contribution.
Sb-Te-I system is known to have only one ternary compound SbTeI. Our structure search for 1:1:1 composition using the evolutionary algorithm implemented in USPEX software predicts a series of candidates to match the experiment. The lowest enthalpy structures were taken into account and their corresponding energy-volume curves are presented in (with more details provided in the supplementary material). Here, the volume is given in terms of the reduced volume, \(V/V_{0}\), emphasizing that considered systems have no points of contact at a given range. The resulting \(E=f(V)\) curves shown in correspond to fitting of the calculated points using a Birch-Murnaghan expression:
\[E(V)=E_{0}+\frac{9}{8}V_{0}B_{0}\left[\left(\frac{\nu_{0}}{\nu}\right)^{\frac{ 2}{3}}-1\right]^{2}+\frac{9}{16}B_{0}(B^{\prime}-4)V_{0}\left[\left(\frac{\nu_ {0}}{\nu}\right)^{\frac{2}{3}}-1\right]^{3}, \tag{1}\]
The derived value of the bulk modulus characterizesSbTeI as soft material, \(B_{0}\approx 38\) GPa. This value is also in a good agreement with the data for ternary bismuth tellurohalides.
A series of energy calculations with fixed symmetry determine that orthorhombic _Pnam_ is energetically unfavorable but two other phases (trigonal \(P3m1\) and monoclinic _Cm_) justified as suitable and are more stable than experimentally observed _C_\(2/m\). In all cases, SbTeI is found as semiconductor where the forbidden gap is surrounded by intertwined \(p\) orbitals but, interestingly, for the lowest energy phases, it has clearly layered 2D structure with octahedron being complete (see supplementary material). The absence of both the inversion symmetry and out-of-plane mirror symmetry makes this compound much attractive. However, to obtain bulk SbTeI experimentally in \(P3m1\) or \(Cm\) form is likely to be difficult process requiring very special high-pressure-temperature conditions or even impossible. Despite the technical challenges, other forms such as Janus monolayer are intriguing as they can induce novel electronic and piezoelectric properties as well. Furthermore, the application of external pressure, which is an alternative thermodynamic parameter, could modulate the atomic and electronic structures by changing the bond distances. In this regard, SbTeI may have a transition from the monoclinic - lone-pair phase to the phase of completed polyhedron.
## IV.
Single crystal XRD measurements indicate that SbTeI obtained by using standard synthesis methods at ambient pressure are of base-centered monoclinic lattice with a space group _C_\(2/m\). The structure consists of double-chains that lie in plate-like blocks. Therefore, these crystals are characterized by strong anisotropy, while XPS shows their surface in the air is being oxidized. Such atomic layout is considerably different from BiTeI compound that settle in trigonal \(P3m1\) at ambient. Despite the fact that bismuth and antimony are both trivalent cations, their ternary compounds have several differences that may be detected in this fashion, such as SbTeI being made up of unfinished octahedra, having a structure that is closer to the 1D arrangement, and having a larger band gap. The valence band spectra observed for SbTeI indicated three bands that are dominated by \(p\) and \(s\) orbital hybridization while the conduction band is composed only of hybridized \(p\) orbitals. Such hybridization should result in strong optical absorption near band gap energies, which is important for a solar absorber material. Another important factor is the lone pair which plays a key role in the SbTeI structure and could be further manipulated with pressure to generate novel structures and electrical, and optical properties.
# Te-Sb-I
(\(\times\) 2) & 92.76 & & & \\ \hline \end{tabular}
\end{table}
Table 4: Bond lengths (Å) and angles (\({}^{\circ}\)). | 10.48550/arXiv.2206.08285 | On the structure of SbTeI | Raimundas Sereika, Raimundas Žaltauskas, Šarūnas Varnagiris, Marius Urbonavičius, Fuyang Liu, Yang Ding, Darius Milčius | 5,522 |
10.48550_arXiv.1605.05378 | ###### Abstract
The dynamics of frictional interfaces play an important role in many physical systems spanning a broad range of scales. It is well-known that frictional interfaces separating two dissimilar materials couple interfacial slip and normal stress variations, a coupling that has major implications on their stability, failure mechanism and rupture directionality. In contrast, interfaces separating identical materials are traditionally assumed not to feature such a coupling due to symmetry considerations. We show, combining theory and experiments, that interfaces which separate bodies made of macroscopically identical materials, but lack geometrical reflection symmetry, generically feature such a coupling. We discuss two applications of this novel feature. First, we show that it accounts for a distinct, and previously unexplained, experimentally observed weakening effect in frictional cracks. Second, we demonstrate that it can destabilize frictional sliding which is otherwise stable. The emerging framework is expected to find applications in a broad range of systems.
## I Introduction
Understanding frictional sliding is a long-standing challenge with important practical and theoretical implications. It is relevant in diverse physical systems spanning a broad range of scales, from the nano-scale to the planetary-scale. A complete analytic treatment of sliding frictional interfaces is generally a formidable task. Two major factors are responsible for the complexity of the problem. First, the friction law, i.e. the constitutive relation that describes the shear traction at the frictional interface, poses experimental challenges and depends on the slip rate and slip history in a highly nonlinear fashion. The second factor is the elastodynamics of the sliding bodies, i.e. the time-dependent long-range stress transfer mechanisms between different points along the interface. It is particularly challenging when the two bodies are made of different materials and in the generic case in which spontaneously-generated interfacial rupture fronts dynamically propagate along the interface.
A significant simplification in relation to the second factor is obtained when the system possesses reflection symmetry across the interface, i.e. when the two materials are identical, the geometry is symmetric, and the loading configuration is antisymmetric (here and elsewhere we consider macroscopic geometry. Differences in small-scale roughness typically exist and are effectively incorporated into the interfacial constitutive relation). A prototypical example of such a situation is that of two semi-infinite half-spaces made of identical elastic materials, a situation that was extensively studied in the literature (see, for example,). The main simplification comes from the fact that such a symmetry precludes a coupling between tangential slip and variations in the normal stress. The lack of such symmetry has important implications on the stability of sliding, the failure mechanism and rupture directionality. Physically, this happens because sliding can enhance (reduce) the normal stress, which in turn can inhibit (facilitate) frictional sliding.
The origin of the absence of reflection symmetry is traditionally assumed to be constitutive in nature, i.e. sliding of dissimilar materials is usually considered. This is known as the bi-material effect. Sliding along such bi-material interfaces has been quite extensively studied in the literature and this material contrast is thought to have important implications for frictional dynamics. The purpose of this paper is to explore the possibility of asymmetry of a geometric origin, i.e.
**Examples of physical systems featuring frictional interfaces separating bodies made of identical materials without geometrical reflection symmetry**: (a) A thin block sliding over a thicker block. (b) A block of finite height \(H\) sliding atop a semi-infinite bulk. Sliding occurs in the \(x\)-direction. (c) An idealized schematic geometry of tectonic subduction motion.
Examples of such geometries are depicted in sliding of two blocks with different thickness in the direction orthogonal to sliding, an experimental setup that was used in various recent works and will be theoretically addressed below (panel a); sliding of a block of finite height \(H\) over a semi-infinite bulk, a simple example to be analyzed in depth in this work (panel b); finally, an idealized sketch of tectonic subduction motion is shown (panel c), a situation in which one lithospheric plate is subducted beneath another one and is responsible for most of the large magnitude earthquakes ("megathrust") occurring on the Earth's crust [51; 52; 53; 54; 55; 56; 57; 58; 59; 60; 61; 62; 63; 64; 65; 66; 67; 68; 69; 70; 71; 72; 73; 74; 75; 76; 77; 78; 79; 80; 81; 82; 83; 84; 85; 86; 87; 88; 89; 90; 91; 92; 93; 94; 95; 96; 97; 98; 99; 100; 101; 102; 103; 104; 105; 106; 107; 108; 109; 110; 111; 112; 113; 114; 115; 116; 117; 118; 119; 120; 121; 122; 123; 124; 125; 126; 127; 128; 129; 130; 131; 132; 133; 134; 135; 136; 137; 138; 139; 140; 141; 142; 143; 144; 145; 146; 147; 148; 149; 150; 151; 152; 153; 154; 155; 156; 157; 158; 159; 160; 161; 162; 163; 164; 165; 166; 167; 168; 169; 170; 171; 172; 173; 174; 175; 176; 177; 178; 179; 180; 181; 182; 183; 184; 185; 186; 187; 188; 189; 190; 191; 192; 193; 194; 195; 196; 197; 198; 199; 199; 198; 199; 199; 190; 191; 193; 195; 196; 197; 198; 199; 199; 199; 198; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 199; 1999; 199; 199; 199; 199; 199; 1999; 1999; 199; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 19999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 19999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 19999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 19999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 19999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 19999; 1999; 1999; 1999; 1999; 1999; 1999; 19999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 19999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 19999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 19999; 1999; 1999; 1999; 19999; 1999; 1999; 1999; 19999; 19999; 1999; 19999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 1999; 19999; 1999; 1999; 19999; 1999; 19999; 1999; 1999; 1999; 1999; 1999; 1999; 19999; 1999; 1999; 19999; 19999; 19999; 19999; 1999; 19999; 1999; 19999; 19999; 1999; 19999; 19999; 19999; 19999; 199coupling is precluded by symmetry. To see this, note that in this case the off-diagonal elements of \(\mathbf{M}^{\mbox{\tiny}}\) and \(\mathbf{M}^{\mbox{\tiny}}\) are identical, and thus \(\mathbf{M}^{\mbox{\tiny}}\!-\!\mathbf{M}^{\mbox{\tiny}}\), as well as its inverse, is diagonal. This immediately implies \(G_{y}\!=\!0\). In what follows, we study two important frictional problems in which the sliding bodies are made of _identical materials_, i.e. \(\rho^{\mbox{\tiny}}\!=\!\rho^{\mbox{\tiny}}\!\equiv\!\rho\), \(\mu^{\mbox{\tiny}}\!=\!\mu^{\mbox{\tiny}}\!\equiv\!\mu\) and \(\nu^{\mbox{\tiny}}\!=\!\nu^{\mbox{\tiny}}\!\equiv\!\nu\), yet reflection symmetry relative to the interface is absent due to asymmetry in the _geometry_ of the bodies, leading to \(G_{y}\!\neq\!0\). These problems highlight the importance of geometrical asymmetry to frictional sliding and its relation to the conventional bi-material effect.
## III "thin-on-thick" systems and the propagation of frictional cracks
The first problem that we examine, which is directly motivated by recent experimental observations, is depicted in In this system, a thin block of width \(W\!=\!5.5\) mm is pushed along its length (the \(x\)-axis) on top of a significantly thicker block (here 30 mm). This "thin-on-thick" experimental setup was used in various studies, where a transparent glassy polymer (poly(methyl-methacrylate), PMMA) was used. The transparent material allows a direct real-time visualization and quantification of a fundamental interfacial quantity: the real contact area, \(A_{r}\). The latter is the sum over isolated micro-contacts formed due to the small scale roughness of macroscopic surfaces.
\(A_{r}\) is typically orders of magnitude smaller than the nominal contact area, \(A_{n}\). Their ratio, \(A\!\equiv\!A_{r}/A_{n}\!\ll\!1\), plays a critical role in interfacial dynamics because the frictional resistance/stress is proportional to \(A\), \(\sigma_{xy}\!\propto\!A\), i.e. the larger the real contact area the larger the frictional resistance. \(A\) itself depends on the normal stress and also on the slip history of the interface according to
\[\sigma_{xy}\propto A\propto\sigma_{yy}(1+\psi)\, \tag{4}\]
Frictional sliding leads to reduction of \(\psi\), i.e. to a reduction of the contact area. In the absence of sliding, \(\psi\) (and hence \(A\)) grows logarithmically with time, a process known as frictional aging.
In it was found that sliding is mediated by a succession of crack-like rupture fronts propagating along the frictional interface and that these fronts are surprisingly well described by the classical theory of shear cracks propagating along an interface separating identical materials, Linear Elastic Fracture Mechanics (LEFM). The variation of \(A\) along a few of these fronts is shown here in It is seen that the rupture fronts involve a significant overall reduction of the contact area, which weakens the interface (i.e. reduces \(\psi\)) and facilitates sliding. We would like to focus our attention on a distinct feature of these curves: As observed in Fig. 2a, fronts which travel at 90% of the Rayleigh wave-speed \(c_{R}\), here \(c_{R}\!\simeq\!1237\) m/s (for plane-stress conditions), or slower (not shown), feature a monotonic decrease of \(A\). However, in fronts propagating even closer to \(c_{R}\), \(A\) features a non-monotonic behavior, i.e. \(A\) undershoots the asymptotic value \(A_{\infty}\) (i.e. \(A\) as \(x\!\to\!-\infty\)) and then rapidly increases, at a rate way too high to be explained by slow frictional aging. This non-monotonic behavior remained unexplained in, where it was stated that "the non-monotonic behavior of \(A\)... suggests interesting dynamics as \(c\!\to\!c_{R}\)...".
In we show the spatial profiles of the slip velocity \(v\), corresponding to the contact area profiles shown in These profiles were calculated from the experimental data using the simplest cohesive zone model which is consistent with the measurements of the fracture energy and cohesive zone size (see for more details). This model, while generally used to describe identical materials, is motivated by the empirical observation that the strain fields are, to first order, quite similar in the thin-on-thin and thin-on-thick setups. This approximation would, of course, have to be modified in cases of strong material contrast, where the fields on both sides of the interface differ strongly.
We denote the maximum slip velocity in these profiles by \(v_{m}\). Next, in order to quantify the non-monotonic effect, we define the magnitude of the undershoot \(\Delta A\) as the difference between the asymptotic value \(A_{\infty}\) and the minimum of the profile over the range \(-5\) mm \(<\!x\!<\!0\), which is the typical spatial range for which \(\Delta A\!>\!0\) is observed in the thin-on-thick setup, see \(\Delta A/A_{\infty}\) is plotted vs. \(v_{m}\) in (red symbols), demonstrating that the former is quasi-linear (i.e. predominantly linear) in the latter. Note that the spread in the data does not allow to identify any systematic deviations from linearity. We stress that the effect is not only qualitatively novel, i.e. the existence of a non-monotonic contact area behavior \(\Delta A/A_{\infty}\!>\!0\), but it is also quantitatively important. As shows, the local _reduction_ in the real contact area \(\Delta A/A_{\infty}\) can reach nearly 25%. This is a large quantitative effect, compared to other documented frictional effects, implying the existence of significant local frictional _weakening_ which can significantly influence interfacial dynamics.
What is the source of this non-monotonicity, why does it scale quasi-linearly with the slip velocity and why does it appear only at sonic propagation velocities? Such behavior has recently been observed in when investigating the frictional motion of bi-material interfaces in a geometrically symmetric system. There, a very large local reduction of \(A\) was observed at sonic propagation velocities, but it entirely disappeared when the upper and lower blocks were made of the same material. We propose that the same happens in our case, only here it is due to geometric asymmetry. That is, we suggest that the non-monotonicity of \(A\) stems from the absence of geometrical reflection symmetry of the two blocks, i.e. from the difference in their thickness. If true, then the fast non-monotonic variation of \(A\) is not an intrinsically frictional phenomenon, i.e. a result of the dynamics of the state of the interface \(\psi\), but rather an elastodynamic effect emerging from the coupling between slip and normal stress variations, solely induced by geometrical effects. In terms of Eq., we propose that \(\psi\) is monotonic and that the non-monotonicity of \(A\) results from a non-monotonic behavior of \(\sigma_{yy}\).
Our strategy in testing and exploring this idea is two-fold. First, our idea can be directly tested by a definitive experiment. That is, we expect that when the width of the lower block equals that of the upper one, i.e. in a "thin-on-thin" setup, the non-monotonicity in \(A\) disappears altogether even in the limit \(c\!\rightarrow\!c_{R}\). We performed this experiment, as in, and present a representative example (for \(c\!=\!0.993c_{R}\)) in the inset of (blue line). The curve is indeed monotonic. Moreover, note that the asymptotic value \(A_{\infty}\) is the same as that in the "thin-on-thick" setup (cf. Fig. 1a, for the same propagation velocity), even though the latter exhibits a large undershoot. In (main panel) we added \(\Delta A/A_{\infty}\) of many rupture fronts in the "thin-on-thin" setup (blue symbols). \(\Delta A/A_{\infty}\) is indeed very close to zero (small negative values simply correspond to monotonic behavior), i.e. all of the \(A\) profiles in the "thin-on-thin" setup are monotonic. This direct experimental evidence provides unquestionable support of our basic idea that the non-monotonic behavior corresponding to the "thin-on-thick" setup data in emerge from the absence of geometrical reflection symmetry.
Next, our aim is to develop a theoretical understanding of the origin of non-monotonicity. The challenge is to explain both the fact that it emerges at asymptotic propagation velocities (\(c\!\rightarrow\!c_{R}\)) and the quasi-linear relation between \(\Delta A/A_{\infty}\) and \(v_{m}\).
**Experimental results**. (a) Snapshots of the spatial profile of the contact area \(A\) of rupture fronts in the “thin-on-thick” setup (see and for additional details). These fronts propagate to the right at velocities \(c\) indicated in the legend of panel b (\(0.900c_{R}\!<\!c\!<\!0.993c_{R}\), see), where \(x\!=\!0\) corresponds to the tip of each rupture front. The contact area is normalized by its value \(A_{0}\) before the passage of the front. (b) The slip velocity profiles corresponding to the snapshots in panel a (see for details). (c) \(\Delta A/A_{\infty}\), where \(\Delta A\) is the magnitude of the real contact area undershoot and \(A_{\infty}\) is the asymptotic value (see inset), vs. the maximal slip velocity \(v_{m}\) (see panel b) for both the “thin-on-thick” setup (red symbols) and the geometrically symmetric “thin-on-thin” setup (blue symbols). Different symbols correspond to different experiments and their size roughly corresponds to the measurement error. The red line is the best linear fit for the red symbols. (inset) The contact area profile for \(c\!=\!0.993c_{R}\) in the “thin-on-thick” setup (red line, already appearing in panel a) and in the geometrically symmetric “thin-on-thin” setup (blue line).
We approach the problem by breaking it into two steps. First, we perform a simplified analysis, invoking physically-motivated approximations, which allow us to reduce the mathematical complexity of the problem and gain analytic insight into it. The major simplification is to consider the corresponding quasi-static problem instead of the full elastodynamic one. The physical rationale for this is clear: the absence of geometrical reflection symmetry should manifest its generic implications also in the framework of static elasticity and hence the simplified analysis is expected to reveal the origin of the non-monotonicity of the real contact area. Then, in the second step, we use the static results in an effective dynamic calculation, to be explained below.
The main outcome of the first step is that the static 3D problem can be approximately mapped onto a 2D problem involving two elastically _dissimilar_ materials. That is, we show that the _geometric_ asymmetry can be approximately mapped onto an effective _constitutive_ asymmetry, i.e. an effective _material contrast_. To see how this emerges, we assume that both blocks are infinite in the \(y\)-direction and that the thicker (lower) block is also infinite in the \(z\) direction. That is, the lower block is replaced by a semi-infinite 3D half-space, which allows us to use the well-known interfacial Green's function. More specifically, the 3D real-space Green's function matrix \(\mathbf{\hat{M}}^{\mbox{\tiny 3D}}(\mathbf{r}-\mathbf{r}^{\prime})\) allows us to express the interfacial displacements at a point \(\mathbf{r}\!=\!(x,y\!=\!0,z\!=\!0)\) on the symmetry line, \(\mathbf{u}(\mathbf{r})\!=\!(u_{x},u_{y})\), induced by a point force applied by the upper block at \(\mathbf{r}^{\prime}\!=\!(x^{\prime},y\!=\!0,z^{\prime})\), \(\mathbf{F}(\mathbf{r}^{\prime})\!=\!(F_{x},F_{y})\). Note that the latter is assumed not to contain an out-of-plane component, i.e. \(F_{z}\!=\!0\), which in principle could emerge from frustrated Poisson expansion at the interface. It is reasonable, though, to neglect it to leading order.
We physically expect shear tractions to be uniform across the thickness \(W\), hence they are taken to be constant for \(|z|\!\leq\!\frac{W}{2}\) (and of course to vanish for \(|z|\!>\!\frac{W}{2}\)).
\[\begin{pmatrix}u_{x}\\ u_{y}\end{pmatrix}=\mathbf{M}^{\mbox{\tiny eff}}(k)\begin{pmatrix}\sigma_{xy}\\ \sigma_{yy}\end{pmatrix}\;, \tag{5}\]
where the effective 2D response matrix \(\mathbf{M}^{\mbox{\tiny eff}}(k)\) of the thicker (lower) block is given by the Fourier transform of \(\mathbf{\hat{M}}^{\mbox{\tiny 3D}}\) over the strip \(|z|\!\leq\!\frac{W}{2}\),
\[\mathbf{M}^{\mbox{\tiny eff}}(k)\!=\!\int_{-\infty}^{\infty}\!\!\!\!\!dx^{\prime} \int_{-\frac{W}{2}}^{\frac{W}{2}}dz^{\prime}\,e^{ik(x-x^{\prime})}\mathbf{\hat{M} }^{\mbox{\tiny 3D}}(x\!-\!x^{\prime},z^{\prime})\;. \tag{6}\]
The integration can be carried out, resulting in
\[\mathbf{M}^{\mbox{\tiny eff}}\!\simeq\!\frac{1}{\mu k}\!\begin{pmatrix}(1\!-\!\nu) B(q)&-\frac{i}{2}(1\!-\!2\nu)\!\left(1\!-\!e^{-\frac{|q|}{2}}\!\right)\\ \frac{i}{2}(1\!-\!2\nu)\left(1\!-\!e^{-\frac{|q|}{2}}\right)&(1\!-\!\nu)B(q) \end{pmatrix}\!,\]
The outcome of the analysis, which is presented in full detail in, is that \(\mathbf{M}^{\mbox{\tiny eff}}(k)\) appears to identify with the 2D response matrix of Eq., if one defines the effective elastic moduli of the lower (thicker) block as
\[\begin{split}\mu^{\mbox{\tiny eff}}(q)&\simeq\frac{\mu}{2(1 -\nu)B(q)-(1-2\nu)\left(1-e^{-\frac{|q|}{2}}\right)}\;,\\ \nu^{\mbox{\tiny eff}}(q)&\simeq\frac{(1-\nu)B(q)-(1-2\nu) \left(1-e^{-\frac{|q|}{2}}\right)}{2(1-\nu)B(q)-(1-2\nu)\left(1-e^{-\frac{|q|} {2}}\right)}\;.\end{split} \tag{7}\]
These are plotted in
The mapping of the 3D problem onto an effective 2D problem is formally valid as long as the interfacial stresses (and hence displacements) in Eq. are approximately localized in Fourier \(k\)-space. Otherwise, Eq. will not identify with Eq. due to the extra \(k\)-dependence of \(\mu^{\mbox{\tiny eff}}(kW)\) and \(\nu^{\mbox{\tiny eff}}(kW)\), which is a result of the 3D nature of the original problem. We note in passing that in the limit \(q\!=\!kW\!\gg\!1\), \(\mu^{\mbox{\tiny eff}}\!\to\!\mu\) and \(\nu^{\mbox{\tiny eff}}\!\to\!\nu\), which corresponds to 2D plane-strain conditions. This is expected for small wavelengths, for which the thinner block also appears infinitely thick, and hence is a consistency check on our calculation. The important observation, though, as is clearly seen in Fig. 3a, is that for the thicker block \(\mu^{\mbox{\tiny eff}}(k)\!>\!\mu\) for all experimentally relevant \(k\)'s. This suggests that the thicker block is effectively stiffer than the thinner one, as hypothesized in where the thicker block was assumed to correspond to plane-strain conditions in numerical simulations. That is, the main physical insight gained from the performed analysis is that geometric asymmetry gives rise to an effective material contrast.
With this physical insight in hand, we aim now at addressing the non-monotonicity of \(A\) discussed in panels a and c of The 3D static analysis presented above may not yield quantitatively accurate predictions when strongly elastodynamic 2D interfacial rupture fronts are considered. Yet, we believe that the insight embodied in the relations \(\mu^{\mbox{\tiny eff}}(k)\!>\!\mu\) and \(\nu^{\mbox{\tiny eff}}(k)\!>\!\nu\) is physically robust and hence try to explore their quantitative implications in relation to the experimental observations in the dynamic regime.
To accomplish this, we consider the 2D dynamic transfer function \(G_{y}(c,k)\) in Eq. and take it to approximately describe the experimental system when the effective moduli \(\mu^{\mbox{\tiny eff}}(k)\!>\!\mu\) and \(\nu^{\mbox{\tiny eff}}(k)\!>\!\nu\) are used for the thicker (lower) block and plane-stress conditions are assumed for the thinner (upper) block. Note that it is justified to treat the heights of the two blocks as infinite since the experimental rupture fronts are so fast that they do not interact with the upper and lower boundaries before traversing the whole system. Therefore, Eq.
\[\Delta\sigma_{yy}(c,k,v)=-c^{-1}\mu\,G_{y}\left[c,k;\mu^{\mbox{\tiny eff}}(k), \nu^{\mbox{\tiny eff}}(k)\right]v\;, \tag{8}\]
The 2D infinite-system dynamic transfer function \(G_{y}(\cdot)\) in Eq. was calculated by Weertman for sliding of dissimilar materials quite some time ago. We reiterate that the basic idea here is to use a known result for dissimilar materials to represent a system composed of identical materials with geometric asymmetry, utilizing the effective moduli derived in Eq., \(\mu^{\text{eff}}(k)>\mu\) and \(\nu^{\text{eff}}(k)>\nu\). In the presence of any contrast between the shear moduli of the materials, \(G_{y}(c)\) is finite and increases significantly at elastodynamic velocities (in fact, it diverges when \(c\) approaches the shear wave-speed \(c_{s}\) of the more compliant material), as shown in Thus, we expect rupture fronts that propagate at near-sonic velocities to be accompanied by a significant reduction in the local normal stress as implied by Eq., reducing locally the real contact area. In turn, this reduces the interfacial strength, which facilitates sliding. This is consistent with the experimental observations of Fig. 2a, where the non-monotonicity of \(A\) becomes substantial at asymptotic propagation velocities (\(c\to c_{R}\)). This normal stress reduction is also remarkably similar to the recent observations of in bi-material systems, a similarity that further strengthens the analogy between geometric asymmetry and material asymmetry.
The connection between geometric and material asymmetries is yet further strengthened when the directionality of rupture is considered. The sub-Rayleigh (\(c<c_{R}\)) rupture fronts, shown in Fig. 2b, propagate from left to right, in the direction of sliding of the _thinner_ (upper) block (see also Fig. 1a). Sub-Rayleigh rupture fronts that are accompanied by normal stress reduction are known to propagate in the direction of sliding of the more _compliant_ material in a bi-material setup, the so-called "preferred direction". This is fully consistent with our result that the thinner (upper) block is effectively softer than the thicker (lower) block (or alternatively, that the thicker block is effectively stiffer than the thinner one).
The quasi-linearity of \(\Delta A\) with the (maximal) slip velocity, observed in Fig. 2c, naturally emerges from Eq.. To see this, note that \(\Delta A\propto\Delta\sigma_{yy}\) according to Eq. (recall that \(\psi\) in that equation is expected to be monotonic) and that \(c\) remains close to \(c_{R}\) to within a few percent. In this regime (\(c\!\simeq\!c_{R}\)), \(G_{y}\) does not change appreciably as a function of \(c\), while the maximal \(v\) varies quite substantially (cf. Fig. 2a). That means that while \(c\!\simeq\!c_{R}\) is required for the existence of the weakening effect, its variability is mainly determined by \(v\). Put together, we obtain \(\Delta A\propto v\). To obtain some estimate of the proportionality factor between \(\Delta A\) and \(v\) along this line of reasoning, we interpret \(\Delta\sigma_{yy}\) in Eq. to be a function of \(v\) alone, with \(c\!=\!c_{R}\) and \(k\!\sim\!\mathcal{O}(W^{-1})\), where \(\mu^{\text{eff}}(k)\!>\!\mu\) (cf. Fig. 3a).
The results for \(\Delta\sigma_{yy}(v)\) with \(kW\!=\!3,4,5\), normalized by the experimentally applied normal stress \(\sigma_{0}\), are shown in The slope of the \(kW\!=\!5\) line is very close to the slope of the linear fit in Fig. 2c, which was added to for comparison (gray dashed line). Note that the experimental line features a finite \(v\) intercept, which is absent in the theoretical one. This is expected since the undershoot, \(\Delta A\), is generally susceptible to variations both of \(\sigma_{yy}\) and the fracture of contacts (variations of \(\psi\) in Eq.). For low values of \(v\), variations of \(\sigma_{yy}\) should be small, and the spatial profile of \(A\) is therefore dominated by variations of \(\psi\). \(A(x)\) should therefore be monotonic in space, similar to the spatial profile in the "thin-on-thin" setup, thus rendering any undershoots (i.e. \(\Delta A\)) to be unmeasurable.
**Analytical results**. (a) The effective shear modulus \(\mu^{\text{eff}}\) of the thicker block, in units of \(\mu\), vs. the dimensionless wavenumber \(q\!=\!kW\), cf. Eq.. (inset) The variation of the effective Poisson’s ratio \(\nu^{\text{eff}}(q)\) with \(q\!=\!kW\). In both we used \(\nu\!=\!0.33\), which is relevant for PMMA. (b) The response function \(G_{y}\), quantifying the _effective_ bi-material contrast according to \(\mu^{\text{eff}}(q)\) and \(\nu^{\text{eff}}(q)\) (for the thicker block, the thinner one is represented by plane-stress conditions), corresponding to selected values of \(q\!=\!kW\). The corresponding values of the elastic moduli \(\mu^{\text{eff}}(q)\!>\!\mu\) and \(\nu^{\text{eff}}(q)\!>\!\nu\) are marked in panel a and its inset using the same color code. (c) \(\Delta\sigma_{yy}\) given in Eq., normalized by the experimentally applied normal stress \(\sigma_{0}\!=\!4.5\) MPa, vs. the slip velocity \(v\), where the propagation velocity was set to \(c\!=\!c_{R}\!\simeq\!1237\) m/s.
This quantitative agreement should be taken with some caution in light of the various approximations invoked above. Yet, the existence of a characteristic wavenumber \(kW=5\) is not unreasonable as the typical scale of the velocity peaks (see Fig. 2b), the spatial scale of the undershoot in the contact area (see Fig. 2a) and \(W\) are all in the mm-scale. Furthermore, the _relative magnitudes_ of the slopes in provide a testable prediction for how the slope decreases with increasing \(W\). This should be experimentally tested in the future. Finally, as \(W\) increases and approaches the width of the lower block, the non-monotonicity is predicted to disappear, as demonstrated experimentally in (blue symbols).
The results presented in this section demonstrate that global geometric features of the sliding bodies in a frictional problem, here a difference in their thickness, affect the frictional resistance to sliding and in fact makes it easier for interfacial rupture fronts that mediate sliding to propagate. In fact, the effect of geometric asymmetry is maximal at the extreme rupture velocities that are the norm in frictional sliding. This reduction in frictional dissipation applies to any engineering or tribological system involving identical materials and geometric asymmetry. As such, it implies that the design and friction control of any real-life tribological application must take into account not only the interfacial properties, but also the relative size of the sliding bodies. In the next section we show that the same concept applies to another class of important sliding friction problems, where a different form of geometric asymmetry controls the dynamic response of the system.
## IV Stability of frictional sliding
We now focus on a different, yet conceptually related, physical situation in which geometric asymmetry plays a crucial role as well. While in Sect. III geometric asymmetry was associated with a difference in the thickness of the sliding bodies, here its origin is a difference in their height. Moreover, while in Sect. III we addressed the propagation of spatially-localized interfacial cracks, here the focus will be on the stability of homogeneous sliding. Yet, in both cases a geometry-induced coupling between interfacial slip and normal stress variations, encapsulated in the function \(G_{y}\) in Eq., is the dominant physical player.
We consider an elastic block of a height \(H^{}\) sliding atop a block of height \(H^{}\!=\!\eta H^{}\) (with a dimensionless positive \(\eta\), \(0\!<\!\eta\!<\!\infty\)), both made of the same material under plane-strain conditions, as depicted in Note that \(\eta\!=\!1\) corresponds to a symmetric system. The blocks initially slide at a fixed velocity and all of the fields are assumed to reach steady state. A homogeneous compressive normal stress \(\sigma_{yy}^{}\!=\!-\sigma_{0}\) is imposed at both \(y\!=\!H^{}\) and \(y\!=\!-H^{}\). In addition, a constant velocity \(\dot{u}_{x}^{}\!=\!v\) in the positive \(x\)-direction is imposed at \(y\!=\!H^{}\) and \(\dot{u}_{x}^{}\!=\!0\) at \(y\!=\!-H^{}\). In this problem, unlike the problem considered in Sect. III, the interfacial dynamics are coupled to the boundaries at \(y\!=\!H^{},-H^{}\), and hence the heights \(H_{1,2}\) are expected to play a central role here.
To fully define the problem, one needs to specify the frictional boundary condition at the interface. Friction is commonly modeled as a linear relation between the interfacial normal stress and the interfacial shear stress (frictional resistance/stress), i.e.
\[\sigma_{xy}=-f(\cdot)\sigma_{yy}\, \tag{9}\]
Our major goal here is to understand the destabilizing effect associated with geometric asymmetry, i.e. \(\eta\!\neq\!1\), which to the best of knowledge has not been studied before. Consequently, in order to isolate the geometric effect, we will focus below on situations in which friction is intrinsically stabilizing such that any instability, if exists, is associated with the absence of geometrical reflection symmetry.
To achieve this, we proceed in two steps. First, in Sec. IV.1, we present a simplified analysis involving a simple velocity-dependent friction law and strong geometrical asymmetry. This will allow us to gain much insight into the role of geometric asymmetry in frictional sliding and to clearly identify the physical origin of instability. Then, in Sec. IV.2, we present a significantly generalized analysis for a realistic friction law, including an internal state variable and an interfacial memory length, and for any level of geometric asymmetry. The emerging results strengthen the findings of Sec. IV.1 and extend them.
### Simplified analysis: Velocity-dependent friction and large geometric asymmetry
As a primer, we use here a simple friction law where \(f(\cdot)\) in Eq. depends only on the interfacial slip velocity \(v\!\equiv\!\dot{\epsilon}_{x}\), i.e. \(f(v)\). We focus on velocity-strengthening interfaces, \(f^{\prime}(v)\!>\!0\), because in this case sliding is unconditionally stable for symmetric systems and thus the origin of any emerging instability must be associated with the absence of geometrical reflection symmetry. Moreover, steady state velocity-strengthening friction has been recently shown to be a generic feature of dry interfaces over some velocity range. Finally, to simplify the analysis further we consider the case in which the lower block is much higher than the upper one, \(\eta\!\gg\!1\). That is, we take the limit \(H^{}\!\rightarrow\!\infty\), such that \(H^{}\!\equiv\!H\) is the only lengthscale in the problem.
Under what conditions is homogeneous sliding stable? This question, which is of fundamental importance in a broad range of frictional problems (see, for example), is first investigated in the context of the simplified problem defined above. As the interface is characterized by velocity-strengtheningfriction, \(f^{\prime}(v)>0\), friction itself tends to _stabilize_ sliding. Consequently, the only possible destabilizing piece of physics can be the geometric-asymmetry-induced coupling between interfacial slip and normal stress variations, encapsulated in the function \(G_{y}\) (cf. Eq.), which also played a crucial role in Sect. III. Can geometric asymmetry destabilize velocity-strengthening frictional interfaces in much the same way as material asymmetry (the bi-material effect) can?
To address the stability question, we perturb Eq.
\[\mu\,G_{x}(c,k)+i\,\mu\,f\,G_{y}(c,k)+i\,c\,\sigma_{0}\,\delta\!f/\delta v=0\, \tag{10}\]
In the simple velocity-dependent friction case considered here, we have \(\delta\!f/\delta v\!=\!f^{\prime}(v)\) (more general interfacial constitutive laws are considered in Sect. IV.2). Perturbations with \(\Im[c]>0\) are unstable and will grow exponentially, while perturbations with \(\Im[c]<0\) are stable (remember that \(k>0\)).
\[G_{y}\!=\!\frac{c_{s}^{2}}{c^{2}}\left(\frac{2(\alpha_{s}^{2}+1)}{1\!+\!\tanh (kH\alpha_{d})}-\frac{2(\alpha_{s}^{2}+1)}{1\!+\!\tanh(kH\alpha_{s})}\right)\, \tag{11}\]
The limit \(H\!\to\!\infty\) amounts to a symmetric system, in which case \(\eta\!\to\!1\), and indeed \(G_{y}\) vanishes in this limit. We can thus expect the system to be unconditionally stable for \(H\!\to\!\infty\). \(G_{y}\) also vanishes in the limit \(H\!\to\!0\).
\[G_{x}\!=\!\frac{c_{s}^{2}}{c^{2}}\!\left(\frac{\left(\alpha_{s}^{2}+1\right)^{ 2}\alpha_{s}^{-1}}{1\!+\!\tanh(kH\alpha_{s})}\!-\!\frac{4\alpha_{d}}{1\!+\! \tanh(kH\alpha_{d})}\right). \tag{12}\]
Equipped with the results for the dynamic response functions \(G_{i}(c,k)\), the implicit equation for the spectrum, Eq., can be in principle solved, at least numerically. The equation admits a few solution branches, and in general its analysis is far from trivial. However, since the purpose of the present discussion is not a complete analysis of Eq., but rather a demonstration of the qualitative effect of the absence of geometrical reflection symmetry, we focus here on a particular branch of solutions which is shown in It is observed that for a range of parameters, and for a finite range of wavenumbers, the solutions are unstable (\(\Im[c]>0\)). This is a direct numerical evidence that geometric asymmetry can destabilize systems which are otherwise stable (remember that \(f^{\prime}(v)>0\)).
It seems natural at this point to ask under what conditions this instability is observed. What are the conditions on the various system parameters such that there will be a range of \(k\)'s for which \(\Im[c(k)]>0\)? As a prelude, we perform a dimensional analysis. Clearly, the only length-scale in the problem is \(H\) and indeed the wavenumber \(k\) only appears in the dimensionless combination \(kH\). Thus, large (small) \(k\) is equivalent to large (small) \(H\) and since \(G_{y}\) vanishes in both limits \(H\!\to\!0\) and \(H\!\to\!\infty\), we expect to find unstable modes only in a finite range \(k_{min}\!<\!k\!<\!k_{max}\), if any.
Another dimensionless combination is \(\gamma\equiv\mu/(\sigma_{0}c_{s}f^{\prime}(v))\), which is the ratio of the elastodynamic quantity \(\mu/c_{s}\) -- proportional to the so-called radiation damping factor for sliding -- and the response of the frictional stress to variations in the sliding velocity. As such, \(\gamma\) quantifies the importance of elastodynamics, which tends to destabilize sliding when geometrical asymmetry is present, relative to velocity-strengthening friction, which generically stabilizes sliding. We thus expect large \(\gamma\) to promote instability, if \(G_{y}\neq 0\). In addition, as \(G_{y}\) is the only possible source of instability in the problem, the appearance of \(fG_{y}\) is associated with destabilization (because \(f\) and \(G_{y}\) enter the spectrum in Eq. only through the combination \(fG_{y}\)). Finally, the ratio of the two wave-speeds \(\beta\equiv c_{s}/c_{d}=\sqrt{(1-2\nu)/(2-2\nu)}\) is also a dimensionless parameter of the system which depends only on the bulk Poisson's ratio.
**Linear stability: Simplified analysis.** Imaginary (a) and real (b) parts of solutions to the linear stability spectrum in Eq.. \(\Im(c)>0\) implies an instability and note that only one solution branch is discussed (other solution branches exist as well, but are not discussed here). The solid lines show numerical solutions to Eq. and the dashed lines show the approximate analytic solutions obtained by a linear expansion around \(c=c_{R}\). The parameters used are \(f=0.9\) and \(\beta=0.3\), where \(\gamma\!\equiv\!\mu/(\sigma_{0}c_{s}f^{\prime}(v))\) is varied according to the legend. (c) The instability threshold \(\chi_{c}\), i.e. for \(\chi\!\equiv\!\gamma f\!<\!\chi_{c}\) sliding is stable for all \(k\), vs. \(\beta\!\equiv\!c_{s}/c_{d}\). The open symbols show direct numerical results and the solid line is the prediction in Eq..
To obtain analytic insight into the instability presented in Fig. 4a, note that solutions in this instability branch are located near the Rayleigh wave-speed, as shown in (note that here \(c_{R}\!\simeq\!0.95c_{s}\)). Consequently, we expand Eq. to linear order around \(c\!=\!c_{R}+\delta c\), obtaining an explicit expression for \(\delta c(kH)\).
\[\gamma fG_{y}(c_{R},k)\approx-c_{R}/c_{s}. \tag{13}\]
This approximate stability criterion explains the existence of an instability and in fact gives reasonable quantitative estimates for its onset.
To see this, note that since \(G_{y}(c_{R},k)\) (which is negative, cf. Eq. and) vanishes for both \(k\!=\!0\) and \(k\!=\!\infty\), and attains a global minimum for \(k\) of order \(H^{-1}\), Eq. admits solutions only for certain values of the product \(\chi\!\equiv\!\gamma f\). When \(\chi\) is smaller than a critical value \(\chi_{c}\), no solutions exist and this branch of solutions is stable for all wavenumbers. Note that this criterion has exactly the expected structure: the instability is indeed governed by \(G_{y}\), and large \(\gamma\) or \(f\) promote instability, which only happens at a finite range of wavenumbers. These predictions are quantitatively verified in In addition, the real and imaginary parts of the approximate solution for \(\delta c(kH)\) are added to Figs. 4a-b (dashed lines), demonstrating reasonable quantitative agreement with the full numerical solution for various parameters.
The results presented in this section demonstrate the destabilizing role that the absence of geometrical reflection symmetry may play in frictional dynamics. In the next section, we significantly extend the analysis to include more realistic friction laws and any geometric contrast.
### Generalized analysis: State dependence, memory length and arbitrary geometric asymmetry
The analysis presented in the previous section adopted two simplifying assumptions, i.e. that the frictional response depends only on the instantaneous slip velocity \(v\) and that the lower block is much higher than the upper one, \(\eta\!\to\!\infty\). Frictional interfaces, however, are known to depend also on the state of the interface, not just on the slip velocity, and obviously the sliding bodies can feature any geometric asymmetry, i.e. the system can attain any value of \(\eta\). Consequently, our goal here is to relax these simplifying assumptions and to present a significantly generalized analysis applicable to a broad range of realistic frictional systems.
It is experimentally well-established that the response of frictional interfaces depends, in addition to the slip velocity \(v\), on the state of the interface through the (normalized) real contact area \(A(\phi)\!\propto\!\sigma_{yy}(1+\psi(\phi))\), as discussed in relation to Eq.. The auxiliary internal state variable \(\phi\), which represents the age/maturity of the contact and is of time dimensions, carries memory of the history of the interface. This implies that irrespective of the exact functional form of \(\psi(\phi)\) (with \(d\psi/d\phi\!>\!0\)) the frictional response \(f(\cdot)\) in Eq. depends on both \(v\) and \(\phi\), i.e. we have \(f(v,\phi)\).
\[\partial_{\!v\!}f\equiv\frac{\partial f(v,\phi)}{\partial v}\qquad\text{and} \qquad d_{v\!}f\equiv\frac{df(v,\phi_{0}(v))}{dv}\, \tag{14}\]
It is also well-established that after a rapid variation in \(v\), accompanied by an instantaneous frictional response characterized by \(\partial_{\!v\!}f\), a new steady state is established over a characteristic slip distance \(D\), which can be regarded as an interfacial memory length. This generic behaviour is described by the following evolution equation for \(\phi\)
\[\dot{\phi}=g\Big{(}\frac{v\,\phi}{D}\Big{)}\, \tag{15}\]
While several functions \(g(\cdot)\) were proposed and extensively studied in the literature, the only property that affects the linear stability is \(g^{\prime}\). Note that if \(g\!>\!0\) (corresponding to \(v\!=\!0\)), the equation describes frictional aging (\(\phi\) increases linearly with time under quiescent conditions) and that \(g\!=\!0\) corresponds to steady state, \(\dot{\phi}\!=\!0\), implying \(\phi_{0}(v)\!=\!D/v\). The latter describes contact rejuvenation, where the typical contact lifetime is inversely proportional to \(v\).
The physics incorporated in the distinction between \(\partial_{\!u\!}f\) and \(d_{u\!}f\), and in the memory length \(D\) -- within the so-called rate-and-state friction constitutive framework -- imply the existence of two dimensionless parameters that are absent in the simplified analysis of Sect. IV.1
\[\Delta\equiv\frac{d_{u\!}f}{\partial_{\!v\!}f},\qquad\qquad\xi\equiv\frac{Dc_ {s}}{Hv|g^{\prime}|}. \tag{16}\]
Frictional interfaces generically feature \(\partial_{\!u\!}f>0\), which is termed the "direct effect" (associated with thermally activated rheology). As in Sect. IV.1, we are interested in \(d_{u\!}f\!>\!0\) (i.e. in steady state velocity-strengthening friction), which implies a positive \(\Delta\). In fact, \(\Delta\) varies in the range \(0\!<\!\Delta\!<\!1\), while \(\xi\) can attain any positive value.
Within this generalized framework, \(\delta f/\delta v\) of Eq.
\[\frac{\delta f}{\delta v}=\partial_{\!u\!}f\left(1+\frac{\Delta-1}{1-i\,\xi \frac{c}{c_{s}}kH}\right). \tag{17}\]
In the limit \(\Delta\!\to\!1\), i.e. when there is no distinction between \(\partial_{\!u\!}f\) and \(d_{u\!}f\) (\(\partial_{\!v\!}f\!\to\!d_{u\!}f\)), and when \(\xi\!\to\!0\), i.e. when the memory length \(D\) becomes vanishingly small, we obtain \(\delta f/\delta v\!\to\!d_{u\!}f\). This recovers the result of Sect. IV.1 where \(d_{u\!}f\) simply identifies with \(f^{\prime}(v)\).
To understand the effect of \(\Delta\) and \(\xi\) on frictional stability, we need to solve Eq. using Eq.. As we also want to consider arbitrary values of the height ratio \(\eta\), we should first derive expressions for the interfacial elastodynamic transfer function \(G_{x,y}\) for any \(\eta\).
\[\begin{split} G_{x}&=\frac{c_{s}^{2}\left(1+\alpha_{ s}^{2}\right)^{2}\Big{(}\tanh(\eta kH\alpha_{d})+\tanh(kH\alpha_{d})\Big{)}-4 \alpha_{d}\alpha_{s}\Big{(}\tanh(\eta kH\alpha_{s})+\tanh(kH\alpha_{s})\Big{)}} {c^{2}\,\alpha_{s}\Big{(}\tanh(\eta kH\alpha_{d})+\tanh(kH\alpha_{d})\Big{)} \Big{(}\tanh(\eta kH\alpha_{s})+\tanh(kH\alpha_{s})\Big{)}}\,\\ G_{y}&=\frac{2c_{s}^{2}\left(1+\alpha_{s}^{2}\right) }{c^{2}}\frac{\tanh(kH\alpha_{s})\tanh(\eta kH\alpha_{d})-\tanh(kH\alpha_{d}) \tanh(\eta kH\alpha_{s})}{\Big{(}\tanh(\eta kH\alpha_{d})+\tanh(kH\alpha_{d}) \Big{)}\Big{(}\tanh(\eta kH\alpha_{s})+\tanh(kH\alpha_{s})\Big{)}}\,\end{split} \tag{18}\]
Note that Eqs.- are obtained from Eq. by taking the \(\eta\!\to\!\infty\) limit, which amounts to setting \(\tanh(\eta kH\alpha_{i})\) to unity (since both \(k\) and \(\Re[\alpha_{i}]\) are positive). In addition, as expected, \(G_{y}\) vanishes for symmetric systems, i.e. for \(\eta\!=\!1\).
We are now ready to study the effect of the geometric dimensionless parameter \(\eta\), and of the constitutive dimensionless parameters \(\Delta\) and \(\xi\), on the linear stability of frictional interfaces. That is, we aim at solving the implicit linear stability spectrum in Eq., with Eqs.-. The ultimate goal of such a generalized linear stability analysis is to derive the stability phase-diagram in the \(\gamma\) (here \(\partial_{\eta}f\) replaces \(f^{\prime}(v)\) in the definition of \(\gamma\) in Sect. IV.1), \(f\), \(\beta\), \(\eta\), \(\Delta\) and \(\xi\) parameter space, where the stability boundary is a complex hypersurface in this multi-dimensional space.
As it is obviously impossible to visualize this high-dimensional stability boundary and in order to gain clear physical insight, we analyze this hypersurface by studying its sections along various parameter directions. A first step was done in Sect. IV.1, where the analysis was performed for fixed values of geometric asymmetry \(\eta\), frictional resistance \(f\) and wave-speed ratio \(\beta\), while \(\gamma\) varied. As a simple velocity-dependent friction model was adopted there, we also had \(\Delta\!=\!1\). As observed in and analyzed theoretically in relation to Eq., an instability emerges when \(\gamma\) becomes sufficiently large (here somewhere between \(\gamma\!=\!2\) and \(\gamma\!=\!3\)). As \(\gamma\!=\!\mu/(\sigma_{0}c_{s}\partial_{t}f)\) quantifies the importance of elastodynamics relative to instantaneous velocity-strengthening friction, the instability emerges when elastodynamics becomes more dominant in the presence of large geometric asymmetry, \(\eta\!=\!\infty\).
Our next step is to isolate the geometric asymmetry effect embodied in \(\eta\). We therefore use the parameters of Fig. 4a-b, together with \(\gamma\!=\!3\), and vary \(\eta\) over a very broad range, essentially from \(\eta\!=\!1\) (corresponding to a symmetric system) to \(\eta\!=\!\infty\). \(\Im[c(kH)/c_{s}]\), obtained by numerically solving Eqs., and, is shown in It is observed that for symmetric systems, \(\eta\!=\!1\), sliding is stable for all wave-numbers. As \(\eta\) is increased, \(\Im[c(kH)/c_{s}]\) approaches the x-axis until they first intersect when \(\eta\!\simeq\!3.3\) at \(kH\!\sim\!{\cal O}\), signaling the onset of instability. This result provides direct evidence for the destabilizing role played by geometric asymmetry in frictional sliding. As \(\eta\) is further increased, the system becomes more unstable in the sense of an increased range of unstable wave-numbers and a larger growth rate. Obviously, the result in the \(\eta\!=\!\infty\) limit identifies with that of In fact, the \(\eta\!=\!\infty\) analysis well captures the salient features of the instability spectrum for \(\eta\) values moderately above the critical value \(\eta\!\simeq\!3.3\).
**Linear stability: Generalized analysis.**\(\Im[c/c_{s}]\) (i.e. the rate of exponential growth/decay of perturbations, \(\Im[c]\!>\!0\) corresponds to instability) vs. \(kH\) for a broad range of physical parameters. In all panels the parameters are the same as in with \(\gamma\!=\!3\). (a) The dependence of \(\Im[c(kH)/c_{s}]\) on \(\eta\) for \(\Delta\!=\!1\). The curve \(\eta\!=\!\infty\) identifies with the blue curve of The case \(\eta\!=\!1\) corresponds to a symmetric system and is thus stable for all \(k\). (b) The dependence of \(\Im[c(kH)/c_{s}]\) on \(\Delta\) for \(\xi\!=\!1\) and \(\eta\!=\!\infty\). (c) The dependence of \(\Im[c(kH)/c_{s}]\) on \(\xi\) for \(\Delta\!=\!0.5\) and \(\eta\!=\!\infty\).
Next, we would like to understand the effect of \(\Delta\), i.e. of a difference between the instantaneous response \(\partial_{u}\!f\) and the steady state response \(d_{u}\!f\), on the sliding stability in the presence of geometric asymmetry. For that aim, we plot in \(\Im[c(kH)/c_{s}]\) for various values of \(\Delta\), spanning the whole range \(0<\Delta<1\), and fixed \(\xi\!=\!1\) and \(\eta\!=\!\infty\). It is observed that as \(d_{u}\!f\) decreases relative to \(\partial_{u}\!f\), i.e. as \(\Delta\) decreases, sliding becomes less stable, resulting in a broader range of unstable wavenumbers and a larger instability growth rate. This result demonstrates the stabilizing role played by steady state velocity-strengthening friction in frictional sliding. We note, though, that the qualitative properties of the instability spectrum are rather well captured by the \(\Delta\!=\!1\) analysis (i.e. for velocity-dependent friction, where no distinction is made between \(d_{u}\!f\) and \(\partial_{v}\!f\)). We stress that while \(\Delta\) affects the properties of instability, the origin of instability is still geometric asymmetry (i.e. sufficiently large \(\eta\)).
Finally, we explore the effect of varying the interfacial memory length \(D\), corresponding to varying \(\xi\), on frictional stability in the presence of geometric asymmetry. We plot in \(\Im[c(kH)/c_{s}]\) for a broad range of \(\xi\) values, and fixed \(\eta\!=\!\infty\) and \(\Delta\!=\!0.5\). It is observed that increasing \(D\) (i.e. \(\xi\)) tends to stabilize sliding (i.e. shrink the instability range and growth rate) as it makes the real contact area less sensitive to slip velocity perturbations. We also stress here that while \(\xi\) affects the range and growth rate of instability, its origin is geometric asymmetry (i.e. sufficiently large \(\eta\)).
The results presented in this section provide a rather comprehensive physical picture of the implications of geometric asymmetry on the stability of frictional sliding, and of the interplay between geometric asymmetry and generic constitutive properties of frictional interfaces, most notably the effect of the state of the interface and of an interfacial memory length. The results significantly extend those presented in Sect. IV.1, yet they show that the simplified analysis properly captured the destabilizing geometric asymmetry effect. We stress again that additional solutions to Eq. (with Eqs.-) exist. These additional solution branches, along with a more detailed analysis of the multi-dimensional stability phase-diagram, will be presented in a follow-up report.
The results presented in this section regarding the stability of homogeneous sliding in the presence of geometric asymmetry may have far reaching implications for the dynamics of frictional interfaces in a variety of frictional systems. Under homogeneous loading applied to the top of long enough sliding bodies, as assumed in the analysis, we predict that no homogeneous steady state will be established experimentally under certain conditions that were carefully quantified. Instead, the interface separating geometrically asymmetric bodies will experience inhomogeneous slip related to the most unstable mode identified in the analysis. This will lead to spatiotemporal stick-slip-like motion, accompanied by distinct acoustic signature as in squeaking door hinges.
In frictional systems where the loading configuration promotes inhomogeneous slip, the obtained results may still be relevant. Inhomogeneous slip in slowly driven frictional interfaces typically takes the form of an expanding creep patch. The conditions under which an expanding creep patch spontaneously generates rapid/unstable slip, an important process known as nucleation, may be related to the minimal unstable wavelength in the stability analysis presented in this section for geometrically asymmetric systems. In particular, the minimal unstable wavelength may determine the size at which the expanding creep patch loses stability.
Finally, when rapid slip develops, it is typically mediated by the propagation of rupture modes. Which mode is actually realized in a given experimental system may be affected by the stability analysis presented here. In particular, extended crack-like rupture modes leave behind them a homogeneous sliding state, which may be precluded under certain conditions predicted by our analysis. Instead, localized pulse-like rupture modes may develop. Consequently, the results presented in this section may affect rupture modes selection, a basic open problem in the field of friction. Additional theoretical and experimental research should be carried out in order to fully explore these potential implications.
## V Concluding remarks
In this paper, combining experiments and theory, we showed that frictional interfaces which separate bodies made of identical materials, but lack geometric reflection symmetry about the interface, generically feature coupling between interfacial slip and normal stress variations. This geometric asymmetry effect is shown to account for a sizable, and previously unexplained, normal-stress-induced weakening effect in frictional cracks. New experiments support the theoretical predictions. We then showed that geometric asymmetry can destabilize homogeneous sliding with velocity-strengthening friction which is otherwise stable. These analyses demonstrate that the effect of geometric asymmetry resembles, sometimes qualitatively and sometimes semi-quantitatively, that of material asymmetry (the bi-material effect).
Since no system is perfectly symmetric, we expect the geometrically-induced coupling between interfacial slip and normal stress variations to generically exist in a broad range of man-made and natural frictional systems. Consequently, it should be incorporated into various theoretical approaches, into engineering models and employed in interpreting experimental observations. The implications in geophysical contexts, such as in subduction zone sliding (cf. Fig. 1c), call for further investigation.
_Acknowledgements_ E.B. acknowledges support of the Israel Science Foundation (grant 295/16), the William Z. and Eda Bess Novick Young Scientist Fund, COST Action MP1303 and of the Harold Perlman Family. J.F.
acknowledge support of the James S. McDonnell Fund (grant 220020221). J.F. and I.S. acknowledge support of the European Research Council (grant 267256) and the Israel Science Foundation (grants 76/11 and 1523/15).
| 10.48550/arXiv.1605.05378 | Frictional sliding without geometrical reflection symmetry | Michael Aldam, Yohai Bar-Sinai, Ilya Svetlizky, Efim A. Brener, Jay Fineberg, Eran Bouchbinder | 1,263 |
10.48550_arXiv.1205.1486 | ###### Abstract
Here we present a model to study the micro-plastic regime of a stress-strain curve. In this model an explicit dislocation population represents the mobile dislocation content and an internal shear-stress field represents a mean-field description of the immobile dislocation content. The mobile dislocations are constrained to a simple dipolar mat geometry and modelled via a dislocation dynamics algorithm, whilst the shear-stress field is chosen to be a sinusoidal function of distance along the mat direction. The sinusoidal function, defined by a periodic length and a shear-stress amplitude, is interpreted to represent a pre-existing micro-structure. These model parameters, along with the mobile dislocation density, are found to admit a diversity of micro-plastic behaviour involving intermittent plasticity in the form of a scale-free avalanche phenomenon, with an exponent and scaling-collapse for the strain burst magnitude distribution that is in agreement with mean-field theory and similar to that seen in experiment and more complex dislocation dynamics simulations.
In 1964, using state-of-the-art torsion experiments, Tinder and co-workers were able to achieve a strain resolution of \(10^{-8}\) for sub mm-sized samples to study the micro-plastic regime of highly pure poly-crystal Cu samples, followed by tests on Zn single crystals some years later. Their plastic-strain versus stress curves contained plateaus of stress which were attributed to the occurrence of discrete dislocation glide activity over a length scale comparable to the dislocation spacing of an assumed three-dimensional network. Indeed the authors write in ref., "The results suggest that an important fraction of the total strain, in the initial stages of deformation, involved motion of a few favourably situated dislocation segments through distances large enough to form new interactions with other elements of the three dimensional network. If this were so, then most elements of the network must have been relatively immobile, making little or no contribution to the strain." That most dislocations remain immobile remains a contemporary viewpoint.
Another more recently pursued route is to probe discrete dislocation activity via the stress-strain curve of micron sized focused ion beam (FIB) milled single crystals. Here, advantage is taken of a nano-indentation platform equipped with a flat punch tip to compress micro-/nano-crystals. In addition to the sub-nanometer displacement resolution of the system, which notably has a lower strain resolution than the above mentioned torsion experiments, the sample size is decreased to the micron range and below. As a result, the strain associated with the discrete dislocation activity is increased to an easily detectable magnitude. Whilst this more contemporary work has been primarily motivated by the "smaller-is-stronger" size-effect paradigm, an extensive analysis of the statistics of the discrete dislocation activity has revealed power-law behavior in the distribution of strain-burst magnitudes giving an exponent of \(\approx 1.6-2.2\). Similar exponents can also be found in bulk samples via detailed analysis of load displacement signals, where structural evolution is also seen to occur. The very recent work of Dahmen and co-workers suggest that the variation in literature exponent values could be due to the different stress intervals used to bin the strain burst magnitude data, where a total integration over the stress interval to material value should, within mean field theory, give an exponent equal to precisely two. Such exponents are indicative of crackling or Barkhausen noise, and more generally of avalanche phenomena, indicating that dislocation mediated plastic deformation belongs to a universality class that encompasses many natural phenomenon over a variety of different length and timescales. Indeed, similar power-law exponents can also be found for metallic glasses in which the underlying plastic deformation is fundamentally different to crystalline metals.
Another class of experiments revealing the intermittent nature of dislocation dynamics is the acoustic emission monitoring of ice. Such experiments measure the acoustic energy released by intermittent dislocation activity during constant stress deformation (in the tertiary creep regime). Indeed, via single sensor acoustic emission signals, such dislocation activity can be well characterised in time revealing power-law behaviour with exponents of 1.6 to 1.8, which is very similar to that seen via the micro-compression stress-strain curves. Furthermore, via multiple sensor monitoring, time and space clustering of dislocation avalanches could be observed. It was found that avalanche epi-centres were correlated in space according to a non-integer power-law exponent indicating scale-free clustering and at short enough times such clusters were correlated in time indicating collective activity.
That such scale-invariant dislocation activity occurs, is a signature of an underlying complex dislocation based micro-structure. An entity whose properties and evolution under an applied stress play a central role in the more general subjects of material strength and strain hardening. Due to the complex dynamics and evolution of the dislocation structure, computer simulation based approaches have helped greatly over the past decades to clarify the underlying dislocation based mechanisms responsible for such structural evolution. One such method is the so-called dislocation dynamics (DD) approach. Early work involved two dimensional arrays of straight edge dislocations interacting via elasticity using single- and multi-slip geometries. These works demonstrate that under an external stress, dislocation patterning emerges analogous to what is seen in static transmission electron microscopy experiments. Other and more recent works have developed these numerical techniques in terms of efficiency, strain boundary-conditions and obstacle/composite geometries to study more complex patterning such as the emergence of granular cell structures and the study of grain boundary network evolution, strain hardening, and material strength in both bulk and confined volume systems. Analogous methodologies and investigations have also been carried out in three dimensions.
Intermittent dislocation activity, in the form of dislocation avalanches, has also been studied using the DD simulation technique. Such simulations have produced power-lawexponents of the distribution of strain burst magnitudes similar to that seen in experiments, indicating some degree of scale-free behaviour, where dislocations are arranged in meta-stable cell and/or wall structures, with only a minor fraction of the dislocation population moving intermittently, thereby creating discrete strain jumps. The so-called self organised criticality (SOC) view of the dislocation network state offers one theoretical platform for the understanding of the observed universality, in which the material system organises itself into a configuration that is critical, resulting in scale-free behaviour upon transiting to a new realisation of the critical state. Originally developed to describe sand-pile dynamics, the approach is somewhat at odds with the historical viewpoint that dislocation structure evolution is primarily driven by equilibrium driving forces such as that embodied in Low Energy Structures (LES) theory and in a wide range of strain hardening theories. The finding that the occurrence of avalanche phenomenon is insensitive to the nature of the forming immobile dislocation network, the slip geometry, the deformation mode, and the details of the dislocation dynamics and spatial dimension is a central hallmark of SOC, which is robust against the details of the underlying physical model.
This motivates investigating plastic flow with simpler models that explicitly do not take into account the fine details of individually interacting sessile and mobile dislocations, naturally shifting the focus of plastic flow from complex dislocation structural evolution to the interaction of a minute mobile dislocation population with a simplified description of the sessile dislocation population. Such an approach is analogous to the study of dislocations in the presence of pinning potentials and more generally to coarse grained models of plasticity that study the depinning transition (see ref. and references therein, and also refs.). Dislocation field theories are also well able to study intermittency, however unlike dislocation dynamics based methods which are able to simulate only small plastic strains, these models are able to incorporate dislocation transport over non-neglible distances and therefore able to simulate significant structural evolution.
In this work, a simple model is therefore proposed in which a dipolar dislocation mat is embedded into an internal static sinusoidal stress field defined by a wave-length and a shear-stress amplitude. The explicit dislocation population is modelled via a standard DD algorithm. It is found that, similar to very complex and detailed 3D-DD simulations, the resulting stress-strain curves evolve in a discrete manner that reflects an underlying intermittent plasticity originating from irreversible changes in dislocation configuration, and that the distribution of the corresponding strain-burst magnitudes reveal both extremal value statistics and scale-free avalanche behaviour. Thus, although the complex details of microscopic dislocation mechanisms and structure are omitted, the simple model is still able to capture the fundamental properties of intermittent micro-plastic flow.
In the following section, the model is formally introduced and the DD technique described, and in sec. III various loading modes to produce a stress-strain curve are presented. Sec. IV presents the DD simulations for a wide range of model parameters to investigate their effect on deformation behaviour. The statistical analysis of the stress at which intermittent activity occurs, as well as the distribution of related strain-bursts, are presented in sec. V. Finally, sec. VI discusses the context for the model, where it is argued that its applicable range is restricted to the micro-plastic region of the stress-strain curve -- a regime where significant structural evolution and work hardening are, to a large extent, absent.
## II Description of model
The proposed model consists of two parallel slip planes, in the \(x-z\) plane, separated by a distance \(h\) along the \(y\) direction. Each slip plane is populated by infinitely long straight edge dislocations where one slip plane contains \(N_{+}\) dislocations with Burgers vector \(\vec{b}=(b,0,0)\) and the other \(N_{-}\) dislocations with Burgers vector \(\vec{b}=(-b,0,0)\). Equal numbers of each type of dislocation are considered to ensure no net Burgers vector content: \(N_{+}=N_{-}=N/2\). Such a structure is traditionally known as a simple dipolar mat.
\[\tau_{\rm Internal}(x)=\tau_{0}\cos\left(\frac{2\pi x}{\lambda}\right), \tag{1}\]
displays a schematic of the model system in which the line direction of each straight dislocation is perpendicular to the plane of the figure. The dislocation density is defined by the total number of dislocations divided by the area of the system. The area of the system is defined by its length along \(x\), defined as \(d\), and its spatial extent along \(y\), defined as \(2h\). Detailed discussion on the origin of the internal sinusoidal stress field is deferred to sec. VI, however, its existence should be viewed as a simplified representation of a pre-existing (and unchanging) immobile dislocation population. The explicit dislocations, schematically shown in fig. 1, are therefore to be viewed as the mobile dislocation population of the system, and are quantified by their density \(\rho_{\rm m}\).
In addition to the force arising from the internal shear-stress field, each dislocation will experience a force of elastic origin from all other dislocations within the system.
\[f_{x,ij}=\frac{Gb_{x,i}b_{x,j}}{2\pi(1-\nu)}\frac{\Delta x(\Delta x^{2}-\Delta y ^{2})}{(\Delta x^{2}+\Delta y^{2})^{2}}. \tag{2}\]
Here \(G\) is the isotropic shear modulus and \(\nu\) Possion's ratio of the isotropic elastic medium, \(b_{x,i}\) (\(b_{x,j}\)) is the Burgers vector in the x-direction of the \(i\)th (\(j\)th) dislocation, and \((\Delta x,\Delta y)\) is the two dimensional vector defining the dislocations' spatial separation. Presently a model isotropic Cu system is implemented, in which the shear modulus is taken as \(G=42\) GPa, the Possion's ratio as \(\mu=0.43\), and the Burgers vector magnitude as \(b=2.55\) A.
For the present work, periodic boundary conditions along \(d\), the dipolar mat direction, and open boundary conditions along \(h\), are assumed. Due to the long range nature of eqn. 2, the correct treatment of periodicity invovles the summation of all dislocation image contributions to the force per unit dislocation length on a given dislocation.
Schematic of the two-dimensional dislocation dipolar mat system consisting of edge dislocations populating the two slip planes and the external sinusoidal stress field visualised in blue.
by
\[f_{x,ij} = -\frac{Gb_{x,i}b_{x,j}}{2(1-\nu)}\sin\left(\frac{2\pi\Delta x}{d} \right)\times \tag{3}\] \[\frac{\left[d\left(\cos\left(\frac{2\pi\Delta x}{d}\right)-\cosh \left(\frac{2\pi\Delta y}{d}\right)\right)+2\pi\Delta y\sin\left(\frac{2\pi \Delta y}{d}\right)\right]}{d^{2}\left(\cos\left(\frac{2\pi\Delta x}{d}\right)- \cosh\left(\frac{2\pi\Delta y}{d}\right)\right)^{2}}.\]
A simple derivation of this equation is detailed in appendix A.
The temporal evolution of a particular dislocation configuration is characterised by the choice of an empirical mobility law. Due to the actual discreteness of the lattice at the atomic scale, a dislocation segment must overcome an energy barrier associated with the local shearing of atoms in order to move an atomic distance. This so-called Peierls energy barrier and the associated Peierls stress, the stress at which the dislocation can begin to move (defined at a given temperature), results in the dislocation moving quasi-statically from atomic lattice site to atomic lattice site. At the meso-scopic scale this results in overdamped motion where the dislocation's velocity is proportional to the force acting on the dislocation -- the present mobility law. The material specific proportionality constant is referred to as a damping parameter and is dependent on dislocation type, geometry, and on temperature.
The equation of motion along the \(x\) direction for the \(i\)th dislocation is then given by
\[\frac{\delta x_{i}}{\delta t}=\frac{F_{x,i}}{B}, \tag{4}\]
In eqn. 4, \(F_{x,i}\), is the total force per unit dislocation length acting on the dislocation,
\[F_{x,i}=\left[\tau_{\text{Internal}}(x_{i})-\tau_{\text{External}}\right]b_{x,i }+\sum_{j\neq i}f_{x,ij}. \tag{5}\]
Here \(\tau_{\text{External}}\) is an externally applied homogeneous shear-stress field and \(\tau_{\text{Internal}}\) is the static sinusoidal shear-stress field defined in eqn. 1.
The numerical solution of eqn. 4 constitutes the Dislocation Dynamics (DD) algorithm presently used in which an appropriate finite time-step, \(\delta t\), is used to integrate the equations of motion.
\[\delta\varepsilon=\frac{1}{2dh}\sum_{i}^{N}b_{x,i}\delta x_{i}. \tag{6}\]It is again emphasised that over the time-scale of \(\delta t\) all atomic scale aspects are averaged over and inertial effects are ignored. Since the edge dislocations are infinitely long and straight such dynamics falls into the class of two dimensional DD modelling.
The dislocation density is given by \(\rho_{\mathrm{m}}=\frac{N}{2dh}\). For the present work, the periodic length \(d\) is chosen to define the distance \(h\) between the two populated slip systems via \(2d/N\). This sets the mean distance between dislocations along the \(x-\) and \(y-\)direction to be the same, and defines the dislocation density as \(\left(N/2d\right)^{2}\). Thus, \(h=1/\sqrt{\rho_{\mathrm{m}}}\), and choosing a value of the dislocation density will fix the scale of interaction between the two parallel slip planes of the dipolar mat geometry -- see appendix A. The motivation for such a restriction is to give the dislocation density a greater bulk-like relevance, where in the bulk limit it represents an isotropic density of dislocations. Other definitions of \(h\) are, of course, possible.
## III Loading and the calculation of stress-strain curves
To simulate a loading experiment and thus a stress-strain curve, a sample must be produced and a loading mode chosen. Presently, sample preparation involves the chosen number of dislocations being initially placed at random positions within the dipolar mat geometry, and the structure relaxed via eqn. 4 to minimise the force on each dislocation to within a chosen tolerance. This is performed at \(\tau_{\mathrm{External}}=0\). In the present work, no attempt is therefore made to determine a low (or lowest) energy dislocation structure, an approach that is compatible with the fact that the explicit dislocations within the model are only those of the mobile variety. This aspect is found to be crucial to the properties of the model and will be discussed in more detail in sec. VI.
There exist a number of ways in which a deformation simulation can be done. The first such loading mode is referred to as "stress-relaxed" and involves incrementing the external shear-stress by a value \(\delta\tau\) (\(\tau_{\mathrm{External}}\rightarrow\tau_{\mathrm{External}}+\delta\tau\)), and relaxing the structure until the the sum of the dislocation force magnitudes, or equivalently, the dislocation velocity magnitudes, varies by less than the fraction \(10^{-8}\).
During the relaxation, the associated plastic shear-strain increment may be calculated as the sum of eqn. 6 over all \(\delta t\) time steps of the relaxation. Once convergence is obtained this shear-strain increment is added to the total plastic shear-strain and the cycle is repeated until the desired stress-strain curve is obtained. displays a typical shear-stress versus plastic shear-strain curve during the initial stages of loading. Discrete dislocation activity, via strain bursts, is clearly evident and is separated by continuous an-elastic regions. By adding the elastic shear-strain, \(\tau/G\), to the plastic shear-strain, the total shear-strain is obtained. displays the corresponding shear-stress versus total shear-strain curve for a small range of stresses and an approximately linear stress-strain curve with intermittent stress plateaus is seen. When displaying a similar curve for the full range of stresses, as seen in fig. 2a, only a straight line is resolvable indicating that such strain bursts are well within the micro-plastic regime of deformation. At larger stresses a plastic flow regime is entered which will be investigated in more detail in subsequent sections.
Experimentally, two distinct deformation modes can be used: displacement controlled and load controlled. Here we consider an inherently force controlled testing device. In such a case, displacement controlled testing is done by adjusting the applied load via a feedback loop such that the displacement rate is held at a fixed value throughout the loading, whereas for load controlled experiments, the applied load simply increases at a chosen rate.
Stress plateaus indicating strain burst activity is clearly evident. b) Displays, for the same simulation, the corresponding stress versus total (elastic plus plastic) strain for a small range of shear-stresses.
To obtain a stress-strain curve with a constant shear-stress rate, a numerical value for the applied stress rate, \(\dot{\tau}_{\text{External}}\), is chosen. This then defines a stress increment \(\delta\tau=\dot{\tau}_{\text{External}}\delta t\), where \(\delta t\) is the time-step used to evolve the dislocation network according to eqn. 4. Thus at every simulation iteration, the stress is increased by \(\delta\tau\) and the configuration evolves in time by an amount \(\delta t\). To implement a constant shear-strain rate loading mode, \(\dot{\varepsilon}\), a numerical value is chosen and the appropriate \(\delta\tau\) stress increment, to achieve such a strain rate, is performed every simulation step.
\[\dot{\tau}_{\text{External}}=G\left(\dot{\varepsilon}-\dot{\varepsilon}_{ \text{plastic}}\right) \tag{8}\]
or
\[\delta\tau=G\left(\dot{\varepsilon}-\dot{\varepsilon}_{\text{plastic}}\right) \delta t, \tag{9}\]
6. Thus the stress increment \(\delta\tau\) is determined by the correction needed to achieve the required constant strain rate for the next simulation iteration. In the above, due to the simplified geometry of the model, a pure shear modulus, \(G\), rather than a Young's modulus is used.
To investigate how such loading modes affect the discrete dislocation activity seen in fig. 2, the appropriate stress-rate and strain-rate must be chosen. This is done by first choosing \(\dot{\varepsilon}\), from which \(\dot{\tau}_{\text{External}}\) is obtained via \(\dot{\varepsilon}/G\) to ensure that in the elastic/an-elastic regime both deformation modes have the same total strain rate. displays a single strain burst in all three considered loading modes. For the constant strain rate and stress rate modes two strain rates are considered: \(0.1\) s\({}^{-1}\) and \(1.0\) s\({}^{-1}\). For the "stress-relaxed" loading mode a sharp plateau is evident in with an identically zero gradient. In this region, the constant stress rate mode also exhibits a plateau but with a (non-zero) positive gradient since, during the evolution of the strain burst, the stress is rising at the chosen rate. Also, the stress at which the strain burst initiates is somewhat higher (and increasing with increasing strain rate) than that in the stress relaxed mode indicating a strain-rate effect. Inthis regard, the "stress-relaxed" deformation mode can be considered as the zero stress-rate limit of the constant stress rate loading mode in which the dislocation configuration always has time to relax before the next stress increment. For the constant strain-rate loading mode, the onset of the strain burst occurs at similar stresses to that of the constant stress rate, however as the strain-burst evolves, the stress decreases to maintain the chosen strain-rate. displays the same strain burst with the stress now as a function of the total strain. The greatest effect is seen in the constant strain-rate mode where due to the drop in stress during the strain-burst, there is a rapid drop in elastic strain. displays the plastic strain rate as a function of plastic strain for the burst shown in (for a constant strain rate of 1.0 s\({}^{-1}\)). The elastic strain rate response as dictated by eqn. 9 is also shown and correspondingly reduces to compensate for the rise in the plastic strain rate. Data for a constant strain rate of 0.1 s\({}^{-1}\) differs little from the 1.0 s\({}^{-1}\) data, a result (along with the high strain-rates of fig. 4a) due to the driven zero temperature nature of the simulation.
It is noted that for the constant strain rate loading mode, the loading system responds instantaneously (to within \(\delta t\)) to any discrete plastic event (hence leading to the backwards curvature seen in fig. 3b). To model an instrumentally realistic device, a more complex differential equation than that of eqn. 9 would need to be developed, which takes into
Close up of a dislocation burst for the three considered loading modes and at different strain rates. a) Stress versus plastic strain and b) stress versus total (elastic plus plastic) strain.
now shows the average dislocation velocity during the "stress relaxed" deformation mode for four consecutive applied shear stresses in which the first constant stress relaxation undergoes the irreversible plastic strain rate seen in For this stress value, an initial relaxation of the dislocation velocities is seen, followed by a rapid acceleration of the dislocations corresponding to the emerging instability and the eventual irreversible plastic event of This continues until a maximum velocity is reached at which the dislocations then begin to decelerate until the convergence criterion is met. Three further constant stress increments result in an immediate relaxation of the dislocation velocities corresponding to a reversible (anelastic) relaxation of the dislocation configuration.
## IV Results
In this section the "stress-relaxed" loading mode is used to investigate the influence of the model parameters on the stress-strain curve.
Stress-strain behaviour as a function of micro-structural parameters: \(\tau_{0}\), \(\lambda\), and the mobile dislocation density, \(\rho_{\rm m}\).
The influence of the sinusoidal shear-stress field amplitude is first investigated. displays the resulting shear-stress versus plastic shear-strain curves for three different choices of \(\tau_{0}\); namely 100MPa, 10MPa and 1MPa for a system with \(d=200\)\(\mu\)m and \(\lambda=2\)\(\mu\)m, containing \(N=N_{+}+N_{-}=20+20\) dislocations. This gives a mobile dislocation density of \(\rho_{\rm m}=1\times 10^{10}\)m\({}^{-2}\). The figure demonstrates that the choice of \(\tau_{0}\) strongly controls the stress at which macroscopic plastic flow occurs. In fig. 5a, the vertical axis is plotted as a logarithmic scale to reveal the fine structure of the micro-plastic region. In all cases discrete strain bursts are evident. displays the corresponding shear-stress versus total shear-strain curves. The small plastic shear-strain values evident in and the sharp yield transition in 5b emphasise that the shear-stress versus plastic shear-strain data prior to flow is clearly in the micro-plastic regime of the stress-strain curve.
In figs. 6a-b, \(\rho_{\rm m}\) is now varied from \(1\times 10^{10}\)m\({}^{-2}\) to \(2.56\times 10^{12}\)m\({}^{-2}\) by changing the number of dislocations for a fixed \(d=100\)\(\mu\)m and \(\lambda=2\)\(\mu\)m. Here \(\tau_{0}\) is 50 MPa. Figs. 6a-b demonstrate that with increasing mobile dislocation density the flow stress decreases from \(\tau_{0}\). Thus at low enough dislocation densities (as in fig. 5), \(\tau_{0}\) defines the flow stress of the system, a natural result since a well isolated dislocation will be primarily affected by
the sinusoidal shear-stress field. Figs. 6a-b also demonstrate that the micro-plastic regime broadens with increasing dislocation density and that the first discrete dislocation event occurs at a decreasing applied homogeneous shear-stress (see arrowed regions in fig. 6b). Figs. 6c-d show similar data for \(\tau_{0}=10\) MPa, for the same values of dislocation density. Although a similar trend is seen, the deviation away from the maximum flow stress of \(\tau_{0}=10\) is a proportionally stronger function of the increasing dislocation density.
Stress versus plastic strain for different mobile dislocation densities for (upper panels) \(\tau_{0}=50\) MPa and (lower panels) \(\tau_{0}=10\) MPa. The right panels display the same data with strain plotted using a logarithmic scale to emphasise the initial micro-plastic strain region where the arrows (in b) indicate the first occurrence of a strain response.
The remaining micro-structural variable that can be varied is the characteristic length scale of the sinusoidal stress field, \(\lambda\). displays stress versus plastic strain curves as \(\lambda\) increases from \(\lambda=0.5\mu\)m to \(\lambda=20\mu\)m, with \(\tau_{0}=10\) MPa and \(50\) MPa, \(d=800\mu\)m and \(\rho_{\rm m}=1\times 10^{10}\)m\({}^{-2}\). Here a larger \(d\) was used to obtain a better statistical sample of the dislocation environment at the length scale of \(\lambda\). The chosen dislocation density is similar to that of in which \(\tau_{0}\) largely determined the flow stress. Inspection of reveals that with increasing \(\lambda\) the micro-plastic regime is broadened due to increasing strain burst magnitudes with increasing \(\lambda\). For small strain (figs. 7a and 7c), it also appears that the flow stress regime occurs at a reduced stress (for increasing \(\lambda\)), however at larger strains (figs. 7b and 7d) flow stresses approximately equal to \(\tau_{0}\) are eventually reached. It is also noted that with increasing \(\lambda\) the strain burst magnitude increases. With decreasing \(\tau_{0}\), the deviation of the curves away from \(\tau_{0}\) is a stronger function of increasing \(\lambda\) resulting in a somewhat broader micro-plastic region.
The central properties of the present model are reflected in figures 5, 6 and 7. If the dislocation density is low enough or \(\tau_{0}\) sufficiently high, then \(\tau_{0}\) will primarily determine the shear-stress at which the extended plastic flow regime begins. In this limit, the internal shear-stress that each dislocation experiences is due to the sinusoidal stress field and when \(\tau_{\rm applied}\) approaches \(\tau_{0}\), the plastic flow regime is immediately encountered with a negligible micro-plastic region. By increasing the mobile dislocation density or decreasing \(\tau_{0}\) the micro-plastic regime broadens and also the stress at which extended plastic flow occurs decreases due to the increasing influence of the elastic interaction between the dislocations. This latter feature is also reflected in the decreasing stress at which the first strain burst is seen. By increasing \(\lambda\), the strain burst magnitude increases which broadens the micro-plastic regime.
### Stain burst behaviour as a function of periodic length scale
The stress versus plastic strain behaviour for three different values of \(d\), using \(\tau_{0}=10\) MPa and \(\rho_{\rm m}\) equal to \(1\times 10^{10}\)m\({}^{-2}\) is presently investigated. For all three simulations \(\lambda=2\)\(\mu\)m. displays the resulting stress-strain curves demonstrating that there is no strong overall \(d\) dependence. Indeed, the stress at which the first strain burst occurs differs little for the three samples: 3.4 MPa for \(d=400\mu\)m, 3.2 MPa for \(d=800\mu\)m and 3.0 MPa for \(d=1600\mu\)m. It is seen, however, that with increasing \(d\) the scale of the discrete plastic strain bursts becomes finer, leading to small-scale differences in the curves. Inspection of the minimum magnitude of the plastic strain bursts observed in this figure, and in figures 5 and 6, reveals it to be approximately \(\left|b\right|\lambda/2dh\), indicating that the strain bursts correspond to dislocation motion over a distance equal to at least \(\lambda\). It is worth noting that the existence of such _minimum_ discrete strain bursts is an artifact of the periodicity used to approximate a system of infinite extent.
Stress versus plastic strain for a range of \(\lambda\) choices, where for each simulation \(\tau_{0}=50\) MPa (upper panels) and \(\tau_{0}=10\) MPa (lower panels), \(d=800\mu\)m and a mobile dislocation density \(1\times 10^{10}\)m\({}^{-2}\). The left panels show the initial strain regime and the right panels show a larger strain range.
## V Statistical properties of intermittent flow
### First burst behaviour
To better understand the relationship between the mobile dislocation density and \(\lambda\), a series of simulations is performed to investigate the statistics of the shear-stress at which the first strain burst occurs. For computational efficiency these simulations were done using the constant stress-rate loading mode with \(d=800\)\(\mu\)m and \(\tau_{0}=50\) MPa. displays such shear-stress values as a function of \(\lambda\) for three different mobile dislocation densities. For each \(\lambda\) and dislocation density value, the results of many different simulations are shown to indicate the degree of statistical variation. Inspection reveals that by increasing the dislocation density, the first-burst stress value becomes increasingly dependent on the value of \(\lambda\), decreasing with decreasing \(\lambda\). For all three considered dislocation densities, the degree of scatter increases with decreasing \(\lambda\).
The state of the dislocation configuration prior to loading provides an understanding of the results of \(N\) dislocations are introduced into the dipolar mat by randomly
b) A close up of the stress-stain curves indicating that with increasing periodicity the minimum discrete strain bursts reduce in strain magnitude.
For large enough \(\lambda\) this relaxation process will result in similar numbers of dislocations for each \(\lambda\) unit, producing a dislocation configuration that is increasingly ordered. However, as \(\lambda\) decreases, there will exist an increasingly varying amount of dislocations for each \(\lambda\) unit, and the original disorder associated with the random initial positions of the dislocations is increasingly preserved. These two trends constitute the origin of the increased scatter and general decrease seen in with respect to decreasing \(\lambda\), since the first-burst stress value is directly related to the lowest critical shear-stress required for an irreversible configurational change to occur. That is, the first-burst stress probes the extremal values of the dislocation environment and a greater variation in dislocation environment will naturally lead to a greater variation in the first-burst stress, which also increases scatter and results in an on-average reduced stress magnitude. Dislocation interaction enhances this effect by generally increasing the degree of disorder in the low-lambda limit.
The above result is, in fact, dependent on the system size (the periodicity length, \(d\)) being finite. As the system size grows at a fixed value of \(\lambda\) and dislocation density, the chances for an extremal dislocation configuration to occur increases, resulting in a increased scatter and reduction of the first-burst shear-stress. It is therefore expected that as the system size increases, the second-burst stress is reduced to a greater variation in dislocation environment. This is the case for the case of a dislocation density, where the dislocation density is increased, and the second-burst stress is reduced to a greater variation in dislocation environment. This is the case for the case of a dislocation density, where the dislocation density is increased, and the second-burst stress is reduced to a greater variation in dislocation environment. 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This is the case for the case of a dislocation density, where the dislocation density is increased, and the second-burst stress is reduced to a greater variation in dislocation environment. This is the case for the case of a dislocation density, where the dislocation density is increased, and the second-burst stress is reduced to a greater variation in dislocation environment. This is the case for the case of a dislocation density, where the dislocation density is increased, and the second-burst stress is reduced to a greater variation in dislocation environment. This is the case for the case of a dislocation density, where the dislocation density is increased, and the second-burst stress is reduced to a greater variation in dislocation environment. This is the case for the case of a dislocation density, where the dislocation density is increased, and the second-burst stress is reduced to a greater variation in dislocation environment. This is the case for the case of a dislocation density, where the dislocation density is increased, and the second-burst stress is reduced to a greater variation in dislocation environment. This is the case for the case of a dislocation density, where the dislocation density is increased, and the second-burst stress is reduced to a greater variation in dislocation environment. This is the case for the case of a dislocation density, where the dislocation density is increased, and the second-burst stress is reduced to a greater variation in dislocation environment. This is the case for the case of a dislocation density, where the dislocation density is increased, and the second-burst stress is reduced to a greater variation in dislocation environment. This is the case for the case of a dislocation density, where the dislocation density is increased, and the second-burst stress is reduced to a greater variation in dislocation environment. This is the case for the case of a dislocation density, where the dislocation density is increased, and the second-burst stress is reduced to a greater variation in dislocation environment. This is the case for the case of a dislocation density, where the dislocation density is increased, and the second-burst stress is reduced to a greater variation in dislocation environment. This is the case for the case of a dislocation density, where the dislocation density is increased, and the second-burst stress is reduced to a greater variation in dislocation environment. This is the case for the case of a dislocation density, where the dislocation density is increased, and the second-burst stress is reduced to a greater variation in dislocation environment. This is the case for the case of a dislocation density, where the dislocation density is increased, and the second-burst stress is reduced to a greater variation in dislocation environment. This is the case for the case of a dislocation density, where the dislocation density is increased, and the second-burst stress is reduced to a greater variation in dislocation environment. This is the case for the case of a dislocation density, where the dislocation density is increased, and the second-burst stress is reduced to a greater variation in dislocation environment. This is the case for the case of a dislocation density, where the second-burst stress is reduced to a greater variation in the case of a dislocation density, where the second-burst stress is reduced to a greater variation in dislocation environment. This is the case for the case of a dislocation density, where the second-burst stress is reduced to a greater variation in dislocation environment. This is the case for the case of a dislocation density, where the dislocation density is increased, and the second-burst stress is reduced to a greater variation in the case of a dislocation density, where the dislocation density is increased, and the second-burst stress is reduced to a greater variation in the case of a dislocation density, where the dislocation density is increased, and the second-burst stress is reduced to a greater variation in the case of a dislocation density, where the dislocation density is increased, and the second-burst stress is reduced to a greater variation in the case of a dislocation density, where the dislocation density is increased, and the second-burst stress is reduced to a greater variation in the case of a dislocation density, where the dislocation density is increased, and the second-burst stress is reduced to a greater variation in the case of a dislocation density, where the dislocation density is increased, and the second-burst stress is reduced to a greater variation in the case of a dislocation density, where the dislocation density is increased, and the second-burst stress is reduced to a greater variation in dislocation environment. This is the case for the case of a dislocation increases, the saturation in first burst stresses as a function of increasing \(\lambda\) will be shifted to larger values of \(\lambda\). Such a size effect must also be reflected in the choice of \(\tau_{0}\) which, along with the dislocation interaction, determines the degree of configurational disorder in the initial state. For a given value of \(\lambda\), a smaller value of \(\tau_{0}\) should increase the disorder due to the increasing influence of the dislocation interaction on the initial configuration. displays a similar figure of first burst stresses as a function \(\lambda\) using a value of \(\tau_{0}=10\) MPa and, indeed, demonstrates a shift to the right of the saturation region.
Although these results are an artifact of the system size, they reveal that through proper choice of model parameters, the intermittent plasticity seen in the past sections is driven by the extreme value statistics associated with the initial dislocation configuration -- a behaviour that is to be equally expected in real crystals.
### Distribution of strain burst magnitudes
The statistical properties of the strain burst magnitudes are now investigated for a system with \(\tau_{0}=10\) MPa, \(\lambda=2\)\(\mu\)m, and \(\rho_{\text{m}}\) equal to \(1\times 10^{10}\)m\({}^{-2}\). Three systems with \(d=400\), \(800\) and \(1600\)\(\mu\)m are first considered. The stress-strain curves for the \(d=400\mu\)m and \(d=800\mu\)m systems span up to the flow stress regime at approximately \(\tau_{0}=10\) MPa, whereas for the \(d=1600\mu\)m data is only available up to approximately \(9.4\) MPa. plots the magnitude of the strain burst as a function of the shear-stress at which it occurred, for all three samples. In the first instance, the plastic strain magnitudes for each stress step (as calculated via eqn. 6) were analysed and it was found that irreversible plasticity via discrete strain bursts begins at approximately \(3\) MPa with magnitudes equal to \(|b|\,\lambda/2dh\) (indicated by the corresponding coloured arrows). With increasing stress, larger strain bursts occur with magnitudes equal to multiples of the basic strain burst unit for each sample. displays the entire range of strain bursts using a vertical log-scale, and it is seen that the spread in strain bursts increases rapidly as the shear-stress approaches the flow regime at approximately \(\tau_{0}=10\) MPa.
Plotting the strain-burst magnitude distribution derived from the data of fig. 10, reveals a power law behaviour which is similar for each of the three considered systems. However from these graphs (not shown), a reliable exponent could not be obtained and to investigate this aspect further improved statistics is required. shows strain-burst magnitude distributions from a much larger number of events, plotted using a log-log scale, for three different choices of parameters: 1) \(\lambda=2\)\(\mu\)m and \(\tau_{0}=10\) MPa (same as that of fig. 10), 2) \(\lambda=10\)\(\mu\)m and \(\tau_{0}=10\) MPa, and 3) \(\lambda=2\)\(\mu\)m and \(\tau_{0}=50\) MPa. For all simulations \(d=800\)\(\mu\)m and \(\rho_{\rm m}=1\times 10^{10}\) m\({}^{2}\). For 1), eleven deformation simulations were performed, and for 2) and 3) sixteen independent simulations were performed for each system. All distributions were obtained by logarithmic binning of identified irreversible plastic strain events. Inspection of this figure reveals good power law behaviour over a number of decades. This is particularly the case for the parameter set 1), in which more statistics could be obtained from each individual curve. Indeed, for this parameter-set, shows that the first-burst stresses are low and exhibit strong scatter indicating significant intermittent plasticity. For this parameter set, a power law fit with an exponent of \(-1.96\pm 0.03\) was found. Visual inspection of the remaining two parameter sets reveals an approximately similar exponent. For parameter set 2), finite size effects for the largest strain burst magnitudes are apparent. This is to be expected since at \(\lambda=10\)\(\mu\)m, there exists just 80 \(\lambda\) lengths within the \(d=800\)\(\mu\)m system, whereas for parameter sets 1) and 3), where \(\lambda=2\), there exist 400 \(\lambda\) lengths.
The above mentioned data of was obtained by logarithmic binning of all discrete
The arrows indicate the (minimum) fundamental strain bursts for each sample, indicating that larger strain bursts occur as multiples of these values. b) Vertical log plot for the entire applied stress range demonstrating that as the flow stress regime is approached individual strain bursts can become large.
In the recent mean-field theory (MFT) work of Dahmen and co-workers this is referred to as the stress integrated distribution, \(D_{\rm int}(S)\) in which \(S\) is the strain burst magnitude, whose integrand is given by the stress dependent distribution function,
\[D(\tau,S)\sim S^{\kappa}f\left(S(\tau_{\rm c}-\tau)^{1/\sigma}\right), \tag{10}\]
From MFT theory \(\kappa=3/2\) and \(\sigma=1/2\). Performing the stress integral gives \(D_{\rm int}(S)\sim S^{-(\kappa+\sigma)}=S^{-2}\) where the last step uses the MFT exponents. The data of is therefore in excellent agreement with that of MFT.
\[C(\tau,S)=\int_{S}^{\infty}\,dSD(\tau,S), \tag{11}\]
Fig. 11b, plots this quantity for the case of \(\lambda=2\)\(\mu\)m and \(\tau_{0}=10\) MPa and gives an exponent equal to \(-0.99\pm 0.02\). also plots \(C(\tau,S)\) for a selection of stress intervals revealing strong cut-off effects due to the tunable criticality of the MFT model. Following ref., these curves are found to collapse when appropriately scaled via the function \(f=(\tau_{\rm c}-\tau)/\tau_{\rm c}\) onto the universal mean field result, as shown in fig. 11c, giving quantitative evidence that scale free avalanche behavoir does indeed occur for the current model. Here \(\tau_{\rm c}\) is taken as \(\tau_{0}\). Similar behaviour is seen for the case of \(\lambda=10\)\(\mu\)m and \(\tau_{0}=10\) but with \(f=(\tau_{c}-\tau)/\tau_{c}-c\) in which the \(c=0.19\) is an adjustment parameter to correct for finite size effects. For the case of \(\lambda=2\)\(\mu\)m and \(\tau_{0}=50\) the statistics was not of a sufficient quality to perform the procedure.
When compared to more realistic two and three dimensional simulations that include dislocation reactions and much larger dislocation numbers, it might seem remarkable that such a simple model is able to exhibit scale-free behaviour. That this is the case, is a central feature of the SOC phenomenon in which the universal features of the critical dislocation configuration are robust against the details of the underlying physical model. Moreover, due to this simplicity, extremely good power law behaviour is obtained over many decades for a minimal computational effort.
## VI Discussion
The simulations of sec. IV demonstrate that two dominant factors control the characteristic stress scale of the stress-strain curve. These are the choice of \(\tau_{0}\) and the mobile dislocation density. Figs. 5 and 6 demonstrate that \(\tau_{0}\) sets an upper stress limit for extended plastic flow to occur.
Data is shown as a log-log plot of the histogram of strain bursts in units of the fundamental strain burst magnitude. b) Corresponding stress dependent and stress integrated complementary cumulative distribution for the case of \(\lambda=2\)\(\mu\)m and \(\tau_{0}=10\) MPa and c) via an appropriate scaling procedure where \(f=(\tau_{0}-\tau)/\tau_{0}\) all stress dependent complementary cumulative distributions collapse on the universal curve given by mean field theory — see ref..
By increasing the mobile dislocation content, the characteristic shear-stress scale reduces due to the increasing role of the internal stress fields arising from the elastic interaction between dislocations. At the same time a broadening of the micro-plastic regime is observed, which very much is a general property of micro-yielding, where a greater number of mobile edge dislocations correlates with larger measurable plastic strain. Figs. 7 to 9 demonstrate the effect of \(\lambda\) (and \(d\)) on the stress-strain curve is somewhat subtler. The choice of \(\lambda\) influences the initial configuration of the mobile dislocation population, thereby affecting the statistics of the stress at which the first strain-burst occurs and also the way in which the extended plastic flow regime is reached. shows, however, that the statistics of the strain burst magnitude is insensitive to all three discussed parameters, reflecting the universality of SOC with respect to micro structural details.
How shall one quantitatively interpret the imposed sinusoidal stress field? In a real material that is nominally free of dislocation structures, some type of length scale will naturally emerge as a function of macroscopic plastic strain due to the evolution of a growing and interacting dislocation structure that eventually leads to the phenomenon of patterning. Although patterning is a term predominantly referring to the effects of latter stage II and III hardening regimes, the emergence of micro-structure length scales is expected to occur at all stages of plasticity ranging from slip, dipole and eventually cellular patterns. In fact, a micro-structural length-scale can equally well be defined for an undeformed as-grown material, where the mean dislocation spacing can be used to describe the initially present internal stress fluctuations -- a view point that is central to the early work of Tinder and co-workers. From this perspective the imposed sinusoidal stress field, can be viewed as the simplest realisation of the internal stress field arising from such a structure.
In the present model this stress field is time independent, implying it is constructed by that part of the dislocation population that is immobile, with the explicit dislocations and their dynamics, arising from the (much smaller) mobile component of the dislocation population. Thus a typical loading simulation can be seen as the deformation of a model material that has a particular sample preparation or deformation history characterised by \(\tau_{0}\) and \(\lambda\), and a mobile dislocation density that is only a small part of the total dislocation density.
Much past work exists concerning the emergence of internal stress and length scales as a function of deformation history. In early work on the theory of cell formation, two relationships have emerged in which the total evolving dislocation density, \(\rho_{\rm total}\), plays a central role.
\[\lambda_{\rm material}\propto\frac{1}{\sqrt{\rho_{\rm total}}}, \tag{13}\]
In the above, \(\tau_{\rm material}\) is the evolving flow stress of the material and \(\lambda_{\rm material}\) is an evolving internal length scale that can be referred to as a cell size. The first expression has its earliest origins in a Taylor hardening picture in which the total dislocation density is seen as an immobile forest dislocation population. The second equation has its theoretical origins in the early work of Holt Holt who derived it for a dipolar population of screw-dislocations, showing that a uniform arrangement was unstable to fluctuations with one such length scale dominating, characterised by eqn. 13. This length scale, which could be related to a fixed self-screening distance of the dislocation network, was postulated to reflect an emerging cell size. The approach was based on an energy minimisation principle, however due to dislocation reactions, the more modern viewpoint is that the dynamics of cell formation lies in a statistical process involving dislocation reactions and that the screening length, and therefore cell-size, is an evolving variable Herring.
Thus, the model parameter \(\lambda\) has a direct counterpart in cell formation theory, \(\lambda_{\rm material}\), which can represent quite generally, a mean-free path for a mobile dislocation, a dipolar screening length or a well evolved cell length scale. Moreover, since an unloading/loading cycle will generally return a system to the flow stress before unloading, and that the present simulations have shown that the flow stress is partly controlled by \(\tau_{0}\), \(\tau_{0}\) should be in some way related to \(\tau_{\rm material}\). From this perspective, \(\tau_{0}\) and \(\lambda\) are parameters that are not entirely independent from each other. In fact, eqns. 12 and 13 express that the cell size decreases inversely as a function of flow stress, a well known experimental observation that is refered to as "similitude" Herring.
Although similitudity is generally confirmed by experiment, some experimental work does present a more complicated picture. Early tensile/TEM work on tapered Cu single crystals finds an initially broad distribution of cell sizes that narrows and shifts to small lengths with increasing flow stress Klimov. This result suggests that a single structural length-scale might not always be a good statistical description of the evolving micro-structure. Indeed, more modern viewpoints, in which dislocation structure evolution is a non-equilibrium process, tend to suggest a distribution of emerging length-scales leading to a scale-free fractal-like structure. Although such micro-structures have been quantitatively established by TEM investigations of latter-stage hardened single crystals of Cu, their existence is not universal, depending strongly on material type and deformation history. The current work does not address this aspect. More general forms of an inhomogeneous internal stress field that capture such scale-free micro-structural features can be envisioned; a direction which will be investigated in future work.
Whilst the dipolar mat geometry in an external field offers a platform with which to study the depinning transition and more generally the transition to extended plastic flow, when comparing to experiment, careful consideration has to be given to its regime of applicability. To do this, a typical simulation of secs. IV and V is now broadly summarised. Upon choosing numerical values for all model parameters, the \(N\) dislocations are introduced to the system via a distribution of random positions. This unstable configuration is then relaxed to a local minimum energy in which the forces on each dislocation are below a small threshold value. The deformation simulation is then begun using one of the three loading modes of sec. III. As the stress increases, intermittent plasticity increasingly occurs until a stress is reached at which extended and overlapping strain events occur, which in the previous sections has loosely been referred to as the plastic flow regime.
It is important to emphasise that no attempt has been made to obtain the global energy minimum of the starting configuration. Such an initial state turns out to play a crucial role in the observed properties of the model, since many high energy configurations will exist, and it is these that dominate the early stages of plasticity. As a deformation simulation proceeds, such high energy configurations structurally transform eventually leading to a plastic flow regime and often to the homogenisation of the dislocation configuration. In other words, the extended plastic flow regime should be considered to be outside the applicability regime of the present model when a comparison to experiment is made, or equivalently, the present model is only suitable for the study of the micro plastic regime of the stress-strain curve.
The rational behind the use of an initial high-energy dislocation configuration originates from the assumption that the explicit dislocation population of the model represents only the mobile dislocation network, which constitutes only a small part of the true population. Thus,in the same way as \(\tau_{0}\) and \(\lambda\) characterise the sample preparation or deformation history of the model material, so does the initial high energy (explicit) mobile dislocation content. This is quite compatible from the perspective of SOC in which the dislocation structure reaches a critical configuration that is far from equilibrium, and that structural rearrangements correspond to the system transforming from one SOC state to another. By construction, that part mediating the structural transformation will be the current mobile dislocation content. The central simplification of the present model, is that it separates the mobile and immobile populations, associating the former to an explicit mobile dislocation content that represents the non-equilibrium component of the network, and relegating the latter to an effective static internal stress field. That this internal stress field is unchanging and that the same explicit mobile dislocation population exists as a function of strain for the entire deformation simulation, is of course different from a real material, where the structure evolves with strain, and at any particular non-negligible strain interval, quite different dislocations might constitute the mobile dislocation population. This again emphasises that the present model should only be applied to the micro plastic regime, where significant structural evolution is minimal.
Experimental evidence for a lack of structural evolution in the micro-plastic regime is best seen in low amplitude cyclic deformation experiments of FCC metals, in which the plastic strain per cycle can be as low as \(\simeq 10^{-5}\) leading to significant changes in load stress and internal length scale only after the occurrence of several tens-to-hundreds of thousands of cycles. It is further noted, that documented experimental studies on the micro-plasticity at room temperature primarily report on movements of edge or non-screw type dislocations, whereas a clear increase in dislocation density or the formation of dislocation structures as a result of multiplication remains absent. In the bulk case there are exceptions to this trend where in the case of a work-hardened Al-Mg alloy which exhibits dynamic strain ageing, emerging structural length-scales were already detected in the micro-plastic regime using high-resolution extensometry methods.
The results of sec. V.2 demonstrate that the developed model exhibits power-law behaviour in the distribution of strain burst magnitudes, and thus the scale-free avalanche phenomenon seen in experiments, either via the stress-strain curve of micro-compression tests or via in situ acoustic emission experiments, and in simulation, via two or three dimensional dislocation dynamics simulations in which the entire network is represented by an explicit dislocation population and individual dislocation reaction mechanisms are taken into account. With an exponent of approximately -1.96, the model gives a value that is somewhat higher than that seen in simple metals and ice, but more comparable to that seen in LiF crystals. That such a simple model can admit scale-free behaviour, is connected to the dependence of the intermittent plasticity on the extremal configurations of the explicit dislocation population. This was directly seen in the statistics of the first-burst shear-stress and also the distribution of strain burst magnitudes, where with a large enough increase of \(\lambda\) (say from 2 \(\mu\)m to 10 \(\mu\)m) the first burst statistics changes from being dominated by extreme value statistics to that being dominated by the statistics of the most probable corresponding to an increased presence of cut-off effects in the statistics of strain burst magnitudes. This is a natural result of the observation that quantities that depend on extreme value statistics can exhibit power-law behaviour in their distributions, emphasising a connection with SOC that is related to only the mobile dislocation population being in a non-equilibrium state and not to the characteristics of the present simplified immobile dislocation network -- a manifestation of a scenario referred to as "nearly critical" or "robust critical".
## VII Concluding remarks
A simplified two dimensional dislocation modelling framework has been introduced in which the explicit interacting dislocation population, constrained to a simple dipolar mat geometry, represents only the mobile dislocation density component of the total dislocation density, and the much larger immobile dislocation population is described by a static internal sinusoidal shear-stress field defined by an internal shear-stress amplitude and wavelength. These model parameters, along with the initial non-equilibrium explicit mobile dislocation content characterise either the deformation or sample preparation history of the model material. Because of the static nature of the internal field and the lack of dislocation-dislocation reactions, upon loading, the present model is restricted to the micro-plastic region of the stress-strain curve, and therefore to a deformation regime for a given material that involves negligible structural evolution. Despite the simplicity of the model and the restriction to the micro-plastic regime, the deformation behaviour exhibits a rich variety of properties as a function of the model parameters. In particular, intermittent plasticity is observed whose strain burst magnitude distribution exhibits scale-free avalanche behaviour.
| 10.48550/arXiv.1205.1486 | Micro-plasticity and intermittent dislocation activity in a simplified micro structural model | P. M. Derlet, R. Maaß | 1,702 |
10.48550_arXiv.1010.0049 | ## 2.
## 2.1 Transient homogeneous nucleation
In case of transformation by homogeneous nucleation we use the phantom nuclei concept. The critical volume of each of the spherical nuclei with radius \(R_{c}\)is \(v_{\tau}=\frac{4}{3}\pi R_{c}^{3}(\mbox{in}\,L^{3}\,\mbox{units})\) for homogeneous nucleation. We shall indicate progress of time by an iteration index \(i\). We introduce \(V_{ur}\), \(V_{ac}\) and \(V_{ex}\) respectively as untransformed, actual transformed and extended volumes. At \(t=0\) (corresponding to \(i=0\)), these are \(V_{ur}=V_{0}\), \(V_{ac}=0\) and \(V_{ex}=0\), where \(V_{0}\)is the initial volume of the parent phase. We recast equation, for an iteration interval \((i\geq 1)\) as \(N(i)=N_{s}\exp\bigl{(}\tau/i\bigr{)}\) (where \(N_{s}\) and \(N\)_(i)_ are the number of nuclei born per unit time per unit volume corresponding to the steady and the transient nucleation rates) number of nuclei is made available in random distribution of nuclei by their proportional distribution in transformed and untransformed volumes. The value of \(N_{s}\)is kept fixed for all computations. The expression of extended volume in the \(i^{\rm th}\) iteration can be written in a manner analogous to our earlier work and is computed by the following equation.
\[V_{ex}(i)=V_{ex}(i-1)+N(i)v_{\tau}V_{ur}(i-1)+N(i)v_{\tau}V_{ac}(i-1)+V_{gr}(i) \tag{3}\]
The term \(N\)_(i)_ in the above takes care of the transient effects.
\[V_{gr}(i)=V_{gr}^{ur}(i)+V_{gr}^{ac}(i) \tag{4a}\]
\(V_{gr}^{ur}(i)\)is the increment in growth of nuclei formed on untransformed volume and \(V_{gr}^{ac}(i)\)is the increment in growth of the phantom nuclei.
\[V_{gr}^{ur}(i)=\sum_{z=1}^{i-1}N(z)\Bigl{[}v_{(i-z)}-v_{(i-(z+1))}\Bigr{]}.V_{ ur}(z-1) \tag{4b}\]
where \(i>1\)and
\[V_{gr}^{ac}(i)=\sum_{z=1}^{i-1}N(z)\Bigl{[}v_{(i-z)}-v_{(i-(z+1))}\Bigr{]}.V_{ ac}(z-1) \tag{4c}\]
Here \(v_{i}=\frac{4}{3}\pi(R_{c}+i\Delta r)^{3}\) and growth rate \(\gamma=\frac{\Delta r}{\Delta t}\). Thus \(\Delta r\), is the increment in the radius of each growing nuclei in every iteration interval \(\Delta i\). \(V_{ac}(i)\)is calculated using the following equation.
\[V_{ac}(i)=V_{ac}(i-1)+N(i).V_{ur}(i-1)v_{\tau}+V_{gr}(i)\{1-X(i-1)\} \tag{5}\]and the fraction transformed is obtained from \(X(i)=V_{ac}(i)/V_{0}\). Please note that the last term in Eq. represents the contribution of \(V_{gr}(i)\) to actual transformed volume. This is proportional to the available untransformed volume.
## 2.2 Transient heterogeneous nucleation
Under this condition the nuclei are formed only in the untransformed volume on nucleants in the parent phase. The corresponding nucleation rate is defined as the number of nuclei per unit time per unit untransformed volume. As mentioned earlier, we assume heterogeneous nucleation is facilitated by the presence of randomly distributed planar nucleants in the volume of the parent phase. Nucleation occurs on the surface of the nucleants with \(\theta\) (\(180^{\circ}<\theta<0^{\circ}\)) as the contact angle between the product phase and the surface. Each nuclear provides a site for only one nucleation event.
\[v_{\tau}^{het}=v_{\tau}.S(\theta) \tag{6}\]
The available number of nucleants and corresponding nucleation events are limited. They may get exhausted before the completion of the transformation. However, in our computations we assume that the number of nucleants is sufficiently large and the phase transformation process is completed before heterogeneous nucleation events are exhausted.
The expression for \(V_{ex}\)(i) for heterogeneous transient nucleation takes the form of
\[V_{ex}(i)=V_{ex}(i-1)+N(i).v_{\tau}^{het}V_{ur}(i-1)+V_{gr}(i) \tag{7}\]
Further, the growth of nuclei is computed assuming their growth to retain the shape till the end of transformation. If we denote \(i_{het}\)as the iteration when all nucleants have been utilized, then for the condition \(i<i_{het}\) and \(i>1\),
\[V_{gr}(i)=\sum_{z=1}^{i-1}N(z)\Big{[}v_{(i-z)}^{het}-v_{(i-(z+1))}^{het}\Big{]} V_{ur}(z-1) \tag{8a}\]
The corresponding \(V_{ac}\) is written as following.
\[V_{ac}(i)=V_{ac}(i-1)+N(i).V_{ur}(i-1)v_{\tau}^{het}+V_{gr}(i)\{1-X(i-1)\} \tag{8b}\]
**3. Results**As mentioned earlier, the incubation time takes into account the time lag before \(X\) reaches measurable values. In normal experimental situations, the incubation time \(t_{inc}\) is defined as the as the time scale between \(t_{0}\) and \(t_{1\%}\), where \(t_{0}\) is the time to reach the annealing temperature and \(t_{1\%}\)is the time to reach \(1\%\) crystallized volume fraction. On the other hand, at high values of the fraction transformed, the nucleation rate starts to decrease as the nucleation sites become filled with nuclei. Thus, the ends of the Avrami plots deviate from the expected constant nucleation rate straight-line plot. We, therefore, compute and analyze the fraction-transformed data in the range \(X=0.01-0.95\), assuming \(t_{inc}=t_{1\%}\). The error due to very late stage transformation is avoided by analyzing data only up to \(X=0.95\). Further, it is important to mention that \(X\left(t\right)\)is itself a functional of the parameter determined directly in the experiment, for example, electrical resistivity. An uncertainty in determination of the experimental observable will cause a new uncertainty in \(X\left(t\right)\). The Avrami approach does not concern such experimental noise.
The Avrami exponent \(n\)is determined from the slope of the plots of ln[- ln(1 - \(x\))] versus \(\ln(t-t_{inc})\)assuming increasing transient nucleation times. As described in the Introduction, increasing values of \(\tau\) may be assumed to correspond to experimental annealing temperatures at increasing distance from the peak transformation temperature. On the other hand different rates of heating to a given isothermal annealing temperature will also result in different transient nucleation times.
As mentioned earlier, to define the system we need to calculate its characteristic length and time. In comparison to the steady state nucleation case, here we define the characteristic time by\(t_{c}=\xi/\gamma\) where \(\xi=\gamma/I_{s}\). The use of \(I_{s}\)in the calculation of \(\xi\)is a necessary assumption, since otherwise we are not able to define the system by a unique characteristic length. In unit iteration interval (\(\Delta i=1\)) the use of corresponding \(N_{s}\)and \(\Delta r\)values define \(\Delta t=1(T)\)for each \(\Delta i=1\). All computations are done well within the condition \(\Delta t\ll t_{c}\), so that the finite time interval error is negligible. Since we are using the finite difference numerical technique for these computations, therefore, there are no random errors associated with the data generated.
### Homogeneous nucleation
As noted earlier, time dependent nucleation rates violate the basic KJMA assumptions. As a first step, we therefore study the effect of increasing transient nucleation time \(\tau\)on the Avrami exponent. Other conditions of the KJMA model are strictly followed. Thus, the transformation proceeds by homogeneous nucleation. The nuclei are of negligible size and grow in an isotropic manner at a constant rate.
b) Avrami exponents \(\left(n\right)\) with increase in \(X\)under homogeneous nucleation conditions. These are line plots (lines as guide to eye) prepared by smoothing the fluctuations in data points.
Average Avrami exponent \(\left(n\right)\) versus dimensionless transient nucleation time \(\left(=\tau/t_{c}\right)\) for systems defined by three different \(\xi\)values (satisfying the condition \(R_{c}\ll\xi\)).
The slopes of the best linear fits to \(\ln(\ln\left(1/\left(1-X\right)\right)\)versus \(\ln\left(t\right)\) plots give the Avrami exponent\(\left(n\right)\)values. The correlation coefficient of the fits always satisfies the condition \(\left(R\right)>0.99\). For the sake of comparison across systems with different\(\xi\), we represent the systems by their dimensionless transient nucleation times\(\left(=\tau/t_{c}\right)\). As expected, we observe that increasing transient nucleation time's delays the time origin of transformation. In we show the change in local Avrami exponents with increase in the fraction of the product phase \(\left(X\right)\) for the systems considered in Throughout the transformation \(n\left(X\right)\) is always > 4. But, at the start of the transformation \(n\left(X\right)\)is at a high value and then it decreases, initially at a rapid rate and later more gradually.
We observe that the \(n\)values for systems with same\(\xi\)but different \(\tau\)characteristics fall on a smooth non-linear curve, which fits a third-order polynomial function. However, there is no universality, since one single third order polynomial function does not fit the three different\(\xi\)curves.
The aforementioned results correspond to the negligible nuclei size. It is already well known that including finite nuclei size can factor in bulk nanocrystallization kinetics for polymorphic transformations. We now try to investigate the effect of finite nuclei size on the Avrami exponent scaling relations with respect to transient nucleation time. shows Avrami exponent \(n\)versus \(\xi_{c}\left(=R_{c}/\xi\right)\)plots. Each plot is for system with different transient nucleation time. For effective comparison the transient nucleation times are divided by the corresponding characteristic time to yield dimensionless values\(\left(\tau/t_{c}\right)\). We observe that for fixed value of the reduced transient nucleation time \(\left(\tau/t_{c}\right)\)the Avrami exponent follows a linear scaling relation as \(\xi_{c}\to 0\). In we plot \(n\) versus reduced transient nucleation time \(\left(\tau/t_{c}\right)\). Plots are given for systems defined by different \(R_{c}\) values. The plot for \(R_{c}=0.1L\) is similar to that of \(0.01L\). However, for \(R_{c}=1L\) the plot becomes completely random in the earlier portion or shorter transient nucleation times relative to \(t_{c}\).
## 3.2 Heterogeneous nucleation
In our previous work, we established that the Avrami exponents show a linear scaling relation with decrease in \(R_{c}\) for a fixed \(\theta\) value. Therefore, it is necessary that we first establish the relations for change in \(n\) with \(\theta\)keeping \(R_{c}\) and \(\xi\) fixed under constant nucleation rate conditions. We stress the importance of using \(\xi\) instead of \(\zeta=\left(V_{0}/N_{t}\right)^{1/D}\) where \(N_{t}\) is the total number of potential heterogeneous nucleation sites in system. This is because in all of our computations the number \(N_{t}\)is chosen such that it does not get exhausted and growth only stage is never encountered. To avoid any unaccounted effect, we fix the value of \(\zeta\) for all the computations given here.
In we plot \(n\) versus \(\theta\)for systems at different \(\xi\)values. All plots are for systems with negligible critical nuclei (\(R_{c}=0.01L\)). For systems with different \(\theta\) conditions having the same \(\xi\)value, \(n\) values follow a sigmoidal fit. Systems with the largest \(\xi\)value and smallest\(\xi_{c}\), show least variation. On comparing the three plots, we observe with increase in \(\xi_{c}\), the variation in \(n\)values increases. shows \(n\)versus \(\theta\)plots for systems at different \(\xi\)values and same\(R_{c}=1L\). The corresponding\(\xi_{c}\)values are therefore larger by a factor of 100. The trend found in is no more followed by systems with such large \(\xi_{c}\)values. As \(\xi_{c}\)increases from 0.0563 to 0.067 the fit to plots changes from sigmoid to Gaussian. Finally, for \(\xi_{c}=0.01\)we are only able to describe it
Plots showing different aspects of transformation under heterogeneous but constant nucleation rate conditions. a) \(n\)versus \(\theta\)plots for systems at different \(\xi\)values under \(R_{c}\ll\xi\)conditions. Lines in different plots represent the best sigmoidal fits to the computed data. Correlation coefficients for all fits are >0.99. b) \(n\)versus \(\theta\)plots for systems at different \(\xi\)values under \(R_{c}<\xi\)conditions. Line in the plot \(\xi_{c}=0.0563\) represents the best sigmoidal fits to the computed data. Fourth order polynomial regression fits are obtained for other two \(\xi_{c}\)values considered. Correlation coefficients for all fits are >0.99. c) Linear scaling relations shown by \(n\)against \(\xi_{c}\)plots at different \(\theta\)values. Lines in different plots represent the best linear fits to the computed data. Correlation coefficients for all linear fits are equal to 1.
In Fig. 4c, keeping \(\theta\)constant, we plot \(n\)against\(\xi_{c}\). The Avrami exponents scale linearly with \(\xi_{c}\)when \(\theta\)is held constant. For a given\(\theta\), as we decrease\(\xi_{c}\), the \(n\)value tends to a fixed value for the negligible nuclei size condition.
Now we present results for heterogeneous transient nucleation conditions. In Fig. 5, we consider systems with negligible nuclei radius conditions (\(R_{c}\ll\xi\)).
Different facets of transformation kinetics under heterogeneous transient nucleation rate and \(R_{c}\ll\xi\)conditions: a) Local Avrami exponent versus fraction transformed \(\left(X\right)\) plots for systems (with contact angle \(\theta\)= 30\({}^{\circ}\)) at different dimensionless \(\tau/t_{c}\) values. These are line plots (lines as guide to eye) prepared by smoothing the fluctuations in data points. b) \(n\)versus \(\theta\)plots for systems at different dimensionless \(\tau/t_{c}\) values. Lines in different plots represent the best sigmoidal fits to the computed data. Correlation coefficients for all fits are >0.99. c) \(n\)versus \(\tau/t_{c}\) plots for systems at different \(\theta\)contact angle values. Lines in different plots represent the best sigmoidal fits to the computed data. Correlation coefficients for all fits are >0.99.
As mentioned earlier, each plot is obtained by taking the derivative of the corresponding \(\ln(\ln\left(1/\left(1-X\right)\right)\)versus \(\ln\left(t\right)\)plot against \(X\). All plots in the figure are at \(\theta=30^{\circ}\), the minimum contact angle considered for heterogeneous nucleation cases in this study. We observe that the plots shift to higher range of \(n\)values with increase in \(\tau/t_{c}\) value. Otherwise, the nature of the curves does not change. As in systems transforming by homogeneous nucleation conditions, we take the slope of the best linear fit to \(\ln(\ln\left(1/\left(1-X\right)\right)\)versus \(\ln\left(t\right)\)plot as the average Avrami exponent\(\left(n\right)\). The correlation coefficients of such fit always satisfy \(R>0.99\). The average \(n\)values obtained in this manner are plotted against \(\theta\)in These plots constructed at increasing \(\tau/t_{c}\) values are described by sigmoidal function fits. The other aspect of these plots is given by \(n\)against \(\tau/t_{c}\) plots at different \(\theta\)values in Perfect sigmoidal function fits describe the plots at all \(\theta\)conditions.
That is, now the nuclei size is finite in comparison to the characteristic length of the system\(\left(R_{c}<\xi\right)\). shows the local Avrami exponent \(n\left(X\right)\)versus \(X\)plots for this particular case. We observe two types of plots, depending on the \(\tau/t_{c}\) values. While the latter portion of plots and are similar, the initial trends are different. Plots and are similar; however the average \(n\)obtained is different. depicts average Avrami exponent\(\left(n\right)\)versus the contact angle \(\left(\theta\right)\)plots at different \(\tau/t_{c}\) values but the same \(\xi_{c}\left(=0.0178\right)\). All plots show perfect sigmoidal fits. The next shows the variation of average Avrami exponent \(\left(n\right)\)with\(\tau/t_{c}\). Each plot is at different \(\xi_{c}\)and its value affects the range of \(n\)value variation the most. Again all plots show perfect sigmoidal fits. Finally, shows the variation of \(n\)with\(\xi_{c}\)at a fixed contact angle\(\left(\theta=30^{\circ}\right)\). Plots given are for different values of \(\tau/t_{c}\). All plots show linear change with \(\xi_{c}\).
## 4 Discussion4. 1 Variation of the local Avrami exponent
Our calculations suggest for transformations initiated by homogeneous transient nucleation, the \(n\big{(}X\big{)}\)value changes with time. The characteristic effect of transient nucleation is to increase the initial \(n\big{(}X\big{)}\)value to > 4, depending on \(\tau\). In the later stages it decreases and ultimately enters a steady state but always with a value > 4. This is similar to intermediate stages of experimental local Avrami exponent versus \(X\)plots for crystallization kinetics of some metallic glass systems. Recently Zheng et al conducted molecular dynamics (MD) simulations to study melt nucleation and growth process at atomistic scales in copper. After the incubation period, Avrami exponents gradually increased to values >4.
On the other hand heterogeneous nucleation leads to an exponent value < 4. The variation of the local Avrami exponent \(n\big{(}X\big{)}\)value versus \(X\) for transformations initiated by heterogeneous nucleation at fixed contact angle values has been shown in reference. On comparing and Fig. 5a, we realize that the initially high value of \(n\big{(}X\big{)}\), due to the magnitude of \(\tau\)value, is diminished when the effect of heterogeneous nucleation is included. Also, in contrast to Fig. 1b, there is no steady \(n\)value achieved in the later stages of the transformation. On further addition of the finite nuclei effect, the initial \(n\big{(}X\big{)}\)value variation with \(X\)is affected. The later stage transformation kinetics trend in remains similar to that seen in This has to be analyzed from the perspective of the negative deviation in the \(n\)values when a system is subjected to a finite nuclei effect. As reported in reference, \(n\)is at a low value at the start of the transformation and then increases to ultimately reach a steady value.
4.2 Scaling relations for average Avrami exponent\(\big{(}n\big{)}\)
In Fig. 2, we observe a near collapse of data points corresponding to systems with different \(\xi\)values when the variation of \(n\)with \(\tau\big{/}t_{c}\) is considered. No such data collapse is observed when the effect of finite nuclei size is included (compare and Fig. 3b). Finally, in Fig. 5c, we consider the effect of transient nucleation and heterogeneous nucleation together. As noted earlier, \(n\)versus \(\tau\big{/}t_{c}\)plots at different contact angles follow the similar sigmoidal functions. This suggests that for a given transient nucleation time we observe similar collapse of data points corresponding to systems defined by different \(\xi\)values. Again, the data collapse observed in is violated in where, besides transient heterogeneous nucleation, systems also include the effect of nuclei of finite sizes.
We now compare the results of \(n\)versus \(\theta\)plots across systems with different features. The extent of deviation of the average value of \(n\)from the universal value 4 is determined by the contact angle \(\theta\), lesser the contact angle more the deviation. On evaluating and 4b, we find that for increasing \(\xi_{c}\)value the plots deviate from the sigmoidal function. Finally, systems with large \(\xi_{c}\)values may even show Gaussian or anomalous function dependence. Since, there is no transient nucleation, therefore, in all observations 3 < \(n\) < 4. However, for systems with negligible \(\xi_{c}\)values, transient nucleation does not affect the sigmoidal function dependence of \(n\)versus \(\theta\)plots. Thus, the plots in are similar in nature to those in In contrast to Fig. 4b, in we again observe sigmoidal dependence of \(n\)versus \(\theta\)plots for systems at different transient nucleation times initiated by nuclei of finite size\(\left(R_{c}<\xi\right)\).
We now discuss the effect on the linear scaling relations found in reference between the \(n\)values of different systems, as their \(\xi_{c}\)values tend to zero. When such plots are constructed at different transient nucleation times, they still give linear scaling relations. Further the slopes of these plots are also same, although with different intercepts. In contrast to this, in and 6d, heterogeneous nucleation is also one of the factors. Although linear scaling relations are still followed, their slopes change for different systems.
## Conclusions
We have investigated the effect of transient nucleation, transient heterogeneous nucleation as applied to bulk materials and materials with nano grain (or finite nuclei) sizes in the KJMA formalism. To delineate the effect of various factors we first consider separately transient and heterogeneous nucleation cases before taking them together. The local Avrami exponent change with fraction transformed has been described for each such factor. Non-linear (sigmoidal) scaling relations have been found for deviations from the universal Avrami exponent for transformations initiated by transient (homogeneous)nucleation. Limited universality is followed in terms of systems defined by different characteristic lengths. However, such universality is violated in different forms, as heterogeneous nucleation conditions are also included. Finally, even when transformation is by transient heterogeneous nucleation, we find that linear scaling relations are still followed between \(\,n\) and \(\xi_{c}\,\), although there is no universality.
| 10.48550/arXiv.1010.0049 | Avrami exponent under transient and heterogeneous nucleation transformation conditions | I. Sinha, R. K. Mandal | 4,758 |
10.48550_arXiv.1512.06119 | ## I Introduction
Transparent conducting oxides (TCO) are a unique class of material that combines two features, electrical conductivity and optical transparency, which are not typically found in the same material. A good TCO material should have a carrier concentration of the order of \(10^{20}\)\(cm^{-3}\), a resistivity of the order of \(10^{-3}\)\(\Omega\) cm, an optical transmission above 80% in the visible range of the electromagnetic spectra. Tin oxide (\(SnO_{2}\)) thin films with such unique features have been extensively used in optoelectronic applications for flat panel displays, smart windows, solar cells, and electromagnetic interference shielding windows. Recently there is an increased interest to introduce magnetic functionality in tin oxide semiconductors due to their promising applications in spintronics. The tin oxide semiconductor can be made ferromagnetic by doping with transition-metal (TM) ions. The first report of high Curie temperature ferromagnetism in tin oxide thin films was by Ogale et. al., who reported a Curie temperature \(T_{c}=650\) K in pulsed laser deposited rutile (\(Sn_{1-x}Co_{x}\))\(O_{2}\) thin films with x = 5-27%, and an amazingly giant magnetic moment of (7.5\(\pm\)0.5)\(\mu_{B}\) per Co ion. High Curie temperature ferromagnetism was latterly reported for (\(Sn_{1-x}Ni_{x}\))\(O_{2}\) with x = 5%, (\(Sn_{1-x}V_{x}\))\(O_{2}\) with x = 7%, (\(Sn_{1-x}Cr_{x}\))\(O_{2}\) with x = 5%, and (\(Sn_{1-x}Fe_{x}\))\(O_{2}\) with x = 14% & x = 0.5-5%. Gopinadhan et. al. investigated (\(Sn_{1-x}Mn_{x}\))\(O_{2}\) (with x = 10%) thin films deposited by spray pyrolysis method and found ferromagnetic behavior above room temperature with low magnetic moment of 0.18\(\pm\)0.04 \(\mu_{B}\) per Mn ion. Fitzgerald et. al. studied 5% Mn-doped \(SnO_{2}\) bulk ceramic samples and reported a Curie temperature of \(T_{c}=340\) K with magnetic moment of 0.11 \(\mu_{B}\) per ordered Mn ion. On the contrary, Duan et. al. reported an antiferromagnetic superexchange interaction in Mn-doped \(SnO_{2}\) nanocrystalline powders and Kimura et. al. observed paramagnetic behavior of Mn-doped \(SnO_{2}\) thin films. Apart from this some other experiments were also carried out by various research groups on \(SnO_{2}\) based dilute magnetic semiconductors (DMS) and reported interesting results regarding the absence or presence of ferromagnetism. DMS based on \(SnO_{2}\) could be useful for a variety of applications requiring combined optical and magnetic functionality. Several devices such as spin transistors, spin light-emitting diodes, very high-density nonvolatile semiconductor memory, and optical emitters with polarized output have been proposed using \(Sn_{1-x}(TM)_{x}O_{2}\) materials.
The origin of n-type conductivity in tin oxide semiconductor is a subject of controversy. Thin films of pure, but non-stoichiometric, \(SnO_{2}\) mostly show free-electron concentrations in the \(n\sim 10^{20}\)\(cm^{-3}\) range. The prevalence of such high electron concentrations has historically been explained by oxygen vacancies (\(V_{O}\)) and tin interstitials (\(Sn_{i}\)). Recent density functional theory (DFT) calculations and experiments, however, have provided evidence that usual suspects such as \(V_{O}\) and \(Sn_{i}\) are actually not responsible for n-type conductivity in majority of the cases. These calculations indicate that the oxygen vacancies are a deep donor, whereas tin interstitials are too mobile to be stable at room temperature. As discussed further in this paper, hydrogen is an ubiquitous impurity that can lead to n-type doping.
Swift heavy ion (SHI) irradiation, in which an energetic ion beam is allowed to pass through a material, is a very effective method to induce structural/microstructural modifications in materials and has been used to tailor the properties of various materials including insulators, metals, semiconductors, and polymers. High electronic energy released by the SHI beams in a very short span of time produces significant excitation in the crystal lattice, causing changes in the structural, optical, electrical and magnetic behaviour of the materials. The SHI irradiation is known to generate controlled defects(points/clusters and columnar) and structural disorder. The parameters of SHI beam such as mass and energy of the ion and the properties of the target materials such as conductivity and microstructure, play crucial role in defect generation and engineering. The ion energy and its mass, decide the magnitudes of the electronic as well as nuclear energy losses. The other ion beam parameter, fluence, dictates the number of defects (created). The number of ion tracks per \(cm^{2}\) is the same as that of fluence (ions/\(cm^{2}\)) because each ion creates an ion-track. Modifications produced by SHI are quite different from that of low energy ions. There is a threshold of electronic energy loss, beyond which the creation of columnar defect or the latent track occurs in the materials. This threshold energy depends on the electron-phonon coupling (g) and conductivity of the target material. It can be up to about a few hundred \(eV/\AA\) for polymers and other insulators and it can be a few \(keV/\AA\) for metals. There are certain materials like Ag, Cu, Ge, Si etc., in which track formation is not possible at any energy with monoatomic ion beams. Investigations of defects produced by swift heavy ion irradiation in oxide semiconductors such as \(SnO_{2}\) and \(In_{2}O_{3}\) have become an important area of research in view of multifunctional magneto-optoelectronic devices. It would be interesting to carry out a detailed study on modifications induced by 120 MeV gold ions on conducting tin oxide (\(n\geq 10^{20}cm^{-3}\)) thin films.
## II Experimental Details
Thin films of \(SnO_{2}\) and \(Sn_{0.9}Mn_{0.1}O_{2}\) were deposited by spray pyrolyzing a mixture of aqueous solutions of tin (II) chloride di-hydrate (\(SnCl_{2}\). 2\(H_{2}O\)) and manganese (II) acetate tetra-hydrate ((\(CH_{3}COO)_{2}Mn.4H_{2}O\)) on glass/quartz substrates at \(450^{o}C\). An amount of 11.281 gm of \(SnCl_{2}.2H_{2}O\) (Sigma Aldrich purity \(>99.99\%\)) was dissolved in 5 ml of concentrated hydrochloric acid by heating at \(90^{o}C\) for 15 min. The addition of hydrochloric acid rendered the solution transparent, mostly, due to the breakdown of the intermediate polymer molecules. The transparent solution subsequently diluted with ethyl alcohol formed the precursor. To achieve Mn doping, \((CH_{3}COO)_{2}Mn.4H_{2}O\) was dissolved in ethyl alcohol and added to the precursor solution. The amount of \((CH_{3}COO)_{2}Mn.4H_{2}O\) to be added depends on the desired doping concentration. The overall amount of spray solution in each case was made together 50 ml. The repeated experiments of each deposition showed that the films could be reproduced easily. Pyrex glass and fused quartz slides (10 mm \(\times\) 10 mm \(\times\) 1.1 mm), cleaned with organic solvents, were used as substrates for various studies. During deposition, the solution flow rate was maintained at 0.2 ml/min by the nebulizer (droplet size 0.5-10 \(\mu\)m). The distance between the spray nozzle and the substrate as well as the spray time was maintained at 3.0 cm and 15 min, respectively. One set of as-deposited \(SnO_{2}\) films were annealed in air for 4 h at \(850^{o}C\).
The as-deposited \(Sn_{0.9}Mn_{0.1}O_{2}\) films were irradiated with 120 MeV \(Au^{9+}\) ions at six different fluences (\(1\times 10^{11}\), \(3\times 10^{11}\), \(1\times 10^{12}\), \(3\times 10^{12}\), \(1\times 10^{13}\) and \(3\times 10^{13}\) ions/\(cm^{2}\)) in a 15 MV pelletron accelerator at Inter University Accelerator Centre (IUAC), New Delhi. During irradiation the samples were attached to a massive sample holder using a double sided tape. The irradiation was performed in the direction nearly perpendicular to the sample surface. Ion beam was focused to a spot of 1 mm diameter and then scanned over 1 \(cm^{2}\) area using a magnetic scanner. The vacuum of \(10^{-6}\) Torr was maintained during irradiation experiments. During irradiation the current of the ion beam was kept at 0.5 PnA. The fluence values were measured by collecting the charges falling on the sample mounted on an electrically insulated sample holder placed in secondary electron suppressed geometry. Ladder current was integrated with a digital current integrator and the charged pulses were counted using scalar counter.
The gross structure and phase purity of all films were examined by X-ray diffraction (XRD) technique using a Bruker AXS, Germany X-ray diffractometer (Model D8 Advanced) operated at 40 kV and 60 mA. In the present study, XRD data of unirradiated and irradiated thin films were recorded in the scanning angle (\(2\theta\)) range \(20^{o}-60^{o}\) using \(Cu-K_{\alpha}\) radiations (\(\lambda=1.5405\)\(\AA\)). All the diffraction patterns were collected under a slow scan with a \(0.01^{o}\) step size and a counting velocity of \(0.3^{o}\) per minute. The experimental peak positions were compared with the data from the database Joint Committee on Powder Diffraction Standards (JCPDS) and Miller indices were assigned to these peaks. Transmission Electron Microscopy (TEM) measurements were carried out on a Tecnai \(20^{2}\)G microscope with an accelerating voltage of 200 kV. All the images were digitally recorded with a slow scan charge-coupled device camera (image size 688 \(\times\) 516 pixels), and image processing was carried out using the digital micrograph software. The TEM data were used for the study of grain size distribution and the crystalline character of the prepared samples. These TEM micrographs were also used to identify secondary phases present, if any, in the \(Sn_{0.9}Mn_{0.1}O_{2}\) matrix. Surface morphology of the samples was analyzed by means of Scanning Electron Microscopy (SEM) using a MIRA II LMH, TESCAN: field emission scanning electron microscope with a maximum resolution of 1.5 nm at 30 kV. Atomic Force Microscopy (AFM) was performed with Multi Mode SPM (Digital Instrument Nanoscope E) in AFM mode to examine the microstructural evolution and root mean square (rms) surface roughness of the sample before and after irradiation. Hall measurements were conducted at room temperature to estimate the film resistivity (\(\rho\)), donor concentration (n) and carrier mobility (\(\mu\)) by using the van der Pauw geometry employing Keithley's Hall effect card and switching the main frame system. A specially designed Hall probe on a printed circuit board (PCB) was used to fix the sample of the size 10 mm \(\times\) 10 mm. Silver paste was employed at the four contacts. The electrical resistivity and the sheet resistance of the films were also determined using the four-point probe method with spring-loaded and equally spaced pins. The probe was connected to a Keithley-voltmeter-constant-current source system and direct current and voltage were measured by slightly touching the tips of the probe on the surface of the films. Multiple reading of current and the corresponding voltage were recorded in order to get average values. Optical absorption measurement was performed at room temperature within a wavelength range of 200-800 nm using a Cary 5000 UV-Vis spectrophotometer having spectral resolution of 0.05 nm in the UV-Vis range. As a reference, 100% baseline signals were displayed before each measurement. Magnetic measurements were carried out as a function of temperature (5 to 300 K) and magnetic field (0 to \(\pm\)2 T) using a 'EverCool 7 Tesla' SQUID magnetometer. Measurements were carried out on small size films placed in a clear plastic drinking straw. The data reported here were corrected for the background signal from the sample holder (clear plastic drinking straw) independent of magnetic field and temperature. Thickness of the deposited films was estimated by an Ambios surface profilometer and was approximately 500 nm.
## III Results and Discussion
The electronic energy loss \(S_{e}\), nuclear energy loss \(S_{n}\) and range \(R_{p}\) of 120 MeV \(Au^{9+}\) ions in the \(Sn_{0.9}Mn_{0.1}O_{2}\) thin film are 27.79 \(keV/nm\), 0.485 \(keV/nm\) and 8.33 \(\mu m\), respectively. These values were calculated through the standard simulation program SRIM. From the calculated values, it is clearly evident that the \(S_{e}\) is about two orders of magnitude more than that of \(S_{n}\) and the range is much larger than the film's thickness (\(\sim\) 500 nm). The variation of energy loss (both electronic and nuclear) for 120 MeV gold ions incident on \(Sn_{0.9}Mn_{0.1}O_{2}\) thin film with the film thickness from the surface is shown in Near the surface of the film, \(S_{e}\) exceeds \(S_{n}\) by two orders of magnitude and is almost constant throughout the film thickness, as shown in the inset of This reveals that the morphological and structural changes in \(Sn_{0.9}Mn_{0.1}O_{2}\) thin film on irradiation by 120 MeV \(Au^{9+}\) ions are due to electronic excitations. The values of \(S_{n}\), \(S_{e}\) and \(R_{p}\) for 120 MeV \(Au^{9+}\)-ions-irradiated \(Sn_{0.9}Mn_{0.1}O_{2}\) thin film (thickness \(\sim\) 500 nm) on glass substrate (thickness \(\sim\) 1.1 mm) are mentioned in the Table 1 along with the threshold value of electronic energy loss (\(S_{eth}\)), calculated later in the paper.
### Threshold energy loss
To create latent tracks or to induce amorphization in crystalline materials a certain threshold value of \(S_{e}\) is required and according to Szenes' "thermal spike model" this threshold value \(S_{eth}\) depends on the material parameters such as electron-phonon coupling efficiency g, density \(\rho\), melting temperature \(T_{m}\), average specific heat C, irradiation temperature \(T_{irr}\), and initial width of the thermal spike a. The Szenes' equation for the threshold energy can be expressed as follows:
\[S_{eth}=\frac{\pi\rho C(T_{m}-T_{irr})a^{2}}{g} \tag{1}\]
The value of a, for semiconductors, depends on their energy band gap \(E_{g}\), while for insulators it is almost constant. This ion induced thermal spike width a in semiconductors can be best approximated by the expression \(a=b+c(E_{g})^{\frac{-1}{2}}\) where b and c are constants. Using an \(E_{g}\) value of 4.3 eV (see Section III.H) for \(Sn_{0.9}Mn_{0.1}O_{2}\), the value of a was determined from the plot of \((E_{g})^{\frac{-1}{2}}\) versus a as given in reference-59, and was found to be \(\sim\) 6.2 nm. The electron-phonon coupling efficiency (g) can be defined as the ability of electrons to transfer their energy to the lattice. According to Szenes, the value of g depends on electron concentration (n). It is found that, for conductive materials (n \(\geq\) 10\({}^{20}\)\(cm^{-3}\)), the value of g is \(\sim\) 0.092, while for insulators it is \(\sim\) 0.4. In a later section-III.F, we have reported the results of electrical measurements. From these measurements, we can say as-deposited \(Sn_{0.9}Mn_{0.1}O_{2}\) samples are in conducting state with n = \(1.93\times 10^{20}\)\(cm^{-3}\). By substituting the values of \(\rho\) = 6.99 \(g/cm^{3}\), \(T_{m}\) = 1898 K, \(T_{irr}\) = 300 K, and C = 0.349 J/(gK) in equation-1, one can obtain \(S_{eth}\) = 31.92 \(keV/nm\) for \(Sn_{0.9}Mn_{0.1}O_{2}\). For the present case of 120 MeV \(Au^{9+}\) ions irradiating the \(Sn_{0.9}Mn_{0.1}O_{2}\) film, \(S_{e}\) is 27.79 keV/nm (see
Variation of \(S_{e}\) and \(S_{n}\) of 120 MeV \(Au^{9+}\) ions incident on \(Sn_{0.9}Mn_{0.1}O_{2}\) thin film as a function of film thickness. The inset shows the almost constant value of \(S_{e}\) for even 500 nm depth from the surface of the thin film.
Table-I), which is less than the threshold value (\(S_{eth}\) = 31.92 \(keV/nm\)) required to produce tracks/melted zones. Therefore we expect that only point defects or clusters of point defects will be produced in \(Sn_{0.9}Mn_{0.1}O_{2}\) film after irradiation. On the other hand, for glass substrate, the value of \(S_{eth}\) is 3.033 keV/nm (calculated by using \(\rho\) = 2.53 \(g/cm^{3}\), a = 4.5 nm, C = 0.88 J/(gK), \(T_{m}\) = 1673 K, \(T_{irr}\) = 300 K, g = 0.4), which is much smaller than the available \(S_{e}\) of 14.88 keV/nm induced by 120 MeV \(Au^{9+}\) ions. Hence, track formation is only possible in the glass substrate but not in the \(Sn_{0.9}Mn_{0.1}O_{2}\) film. This discontinuity in track formation can induce stress at the film-substrate interface.
It is important to understand that the way in which the huge energy (\(S_{e}\sim\) 27.79 keV/nm) is deposited to the system will modify the materials. The swift heavy ion initially interacts with the atomic electrons (electronic subsystem) of the target material and transfers its energy to electrons (valence and core electrons) in a time less than \(10^{-16}\) s and the timescale of electron-ion interaction is very small, initially the lattice (atomic subsystem) of target material does not respond in that timescale. The transfer of the excess heat energy of the excited electronic subsystem to the atomic subsystem may lead to an increase in the local temperature of the target material and therefore a high energy region is developed in the close vicinity of the ion path. The amount of energy transferred (locally) depends on the coupling between the electronic and atomic subsystems. This coupling is known as electron-phonon coupling (g) and it can be defined as the ability of electrons to transfer their energy to the lattice. According to Szenes, the value of g depends on the carrier concentration (n) in the target material. It is found that, for conductive materials (\(n\geq 10^{20}cm^{-3}\)), the value of g is \(\sim\) 0.092, while for insulators it is \(\sim\) 0.4. The amorphized latent tracks are created only above a certain threshold value of \(S_{e}\), which inversely depends on the electron-phonon coupling efficiency (g). Therefore, conductive targets will require very high beam energy to create latent tracks/melted zones. But if \(S_{e}<S_{eth}\), then only point defects or cluster of point defects will create in target material after irradiation. A point to note here is that in insulator if \(S_{e}<S_{eth}\) then point defects will create locally around the ion path whereas in conductive material these defects will create uniformly through out the material. In this way, we can conjecture that the SHI irradiation can create tracks or point defects depending upon the initial state of the material. But if the system already contains such defects, then they can be annealed through the irradiation.
### Equilibrium substrate/film temperature
The input power density can be calculated by using the following equation:-
\[P=\frac{(Irradiation\ Energy\times Beam\ Current)}{Scanning\ Area}\]
In the present case the maximum input power density is
\[P=\frac{(120\ MeV\times 4.5\ nA)}{1\ cm^{2}}=0.54\frac{W}{cm^{2}}\]
The power of the ion beam deposited in the target can be removed mainly by thermal radiation and directed to the heat sink constituted by the vacuum chamber.
\[P=\epsilon\sigma(T^{4}-T_{s}^{4}), \tag{2}\]
\(T_{s}\) is the surrounding target chamber temperature (\(\sim\) 300 K), and T is the equilibrium substrate/film
\begin{table}
\begin{tabular}{||c|c|c||} \hline \hline Parameters & \(Sn_{0.9}Mn_{0.1}O_{2}\) & Glass substrate \\ \hline \hline Nuclear energy loss & 0.485 & 0.216 \\ (\(S_{n}\) in keV \(nm^{-1}\)) & & \\ \hline Electronic energy loss & 27.79 & 14.88 \\ (\(S_{e}\) in keV \(nm^{-1}\)) & & \\ \hline Threshold electronic energy loss (\(S_{eth}\) in keV \(nm^{-1}\)) & 31.92 & 3.033 \\ \hline Range (\(\mu\)m) & 8.33 & 15.28 \\ \hline Track formation & Not possible & Possible \\ \hline \hline \end{tabular}
\end{table}
Table 1: The values of \(S_{n}\), \(S_{e}\), \(R_{p}\) for 120 MeV \(Au^{9+}\)-ion-irradiated \(Sn_{0.9}Mn_{0.1}O_{2}\) thin film deposited on a glass substrate estimated by SRIM. The value of \(S_{eth}\) and possibility of track formation are also mentioned.
According to Kumar et. al., the value of effective emittance (\(\epsilon\)) for glass substrate is 0.2. By using the values of input power density (P = 0.54 \(W/cm^{2}\)) and effective emittance (\(\epsilon\) = 0.2) in equation-2, one can obtain the equilibrium temperature T = 834 K for \(Sn_{0.9}Mn_{0.1}O_{2}\) film/glass substrate. This calculated equilibrium temperature is higher than the crystallization temperature of \(SnO_{2}\) and expected to be develop within the grains of material during irradiation and facilitate (i) increase of grain size (judged by AFM), (ii) removal of micro-strain (evidenced by XRD measurements), and (iii) migration of point defects (explained by XRD and electrical measurements).
The increase rate of the glass substrate temperature during irradiation can be deduced by using the following equation:
\[\frac{dT}{dt}=\frac{[P-\epsilon\sigma(T^{4}-T_{s}^{4})]}{\rho wC}, \tag{3}\]
Eq. can be solved by separation of variable technique as follows:
\[dt=\frac{dT}{a(T^{4}-b)}, \tag{4}\]
The differential Eq. can be numerically solved (by Wolfram language) as follows:
t = \(c_{1}\) - 93.002 ln (834.212 - T) + 93.002 ln (T + 834.212) + 186.004 \(tan^{-1}\) (1.199 \(\times\) 10\({}^{-3}\)T)
Apply initial condition: T = 300 K at t = 0.
t = -93.002 ln (834.212 - T) + 93.002 ln (T + 834.212)
\[+\ 186.004\ \tan^{-1}\ (1.199\ \times\ 10^{-3}\ T) - 134.234 \tag{5}\]
\begin{table}
\begin{tabular}{||c|c|c|c||} \hline \hline Beam fluence (\(ions/cm^{2}\)) & Beam current (nA) & Irradiation period (sec) & Sample temperature in irradiation period (K) \\ \hline \hline \(1\times 10^{11}\) & 4.5 & 32 & Low temperature transient condition \\ \hline \(3\times 10^{11}\) & 4.5 & 96 & Low temperature transient condition \\ \hline \(1\times 10^{12}\) & 4.5 & 320 & Low temperature transient condition \\ \hline \(3\times 10^{12}\) & 4.5 & 960 & \(\sim\) 834 K (not entire time) \\ \hline \(1\times 10^{13}\) & 4.5 & 3200 & \(\sim\) 834 K (almost entire time) \\ \hline \(3\times 10^{13}\) & 4.5 & 9600 & \(\sim\) 834 K (almost entire time) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Summary of the irradiation conditions: beam fluence, beam current, irradiation period, and sample temperature in irradiation period.
The transient substrate/film temperature calculated by using an input power density of 0.54 \(\frac{W}{cm^{2}}\).
In this calculation, the glass substrate thickness (w), effective emittance (\(\epsilon\)), target material density (\(\rho\)), target heat capacity (C), and surrounding target temperature (\(T_{s}\)) are assumed to be 0.11 cm, 0.2, 2.53 \(g/cm^{3}\), 0.88 J/gK, and 300 K, respectively. The irradiation fluences in the present experiment were in the range of \(1\times 10^{11}\) - \(3\times 10^{13}\)\(ions/cm^{2}\), which means that the irradiation periods were in the range of 32-9600 sec when 1 \(cm^{2}\) sample was used (see Table 2). The results of Eq. are shown in It is clear from that the glass substrate temperature reaches its equilibrium value (\(\sim\) 834 K) in several hundred seconds. In the case of high fluence (\(\geq 3\times 10^{12}\)\(ions/cm^{2}\)), irradiation period is much longer than the amount of time needed to achieve equilibrium temperature. Therefore, in this case, the samples spend their almost full irradiation period at equilibrium temperature. This equilibrium temperature can influence the physical properties of samples.
### Brief review on the properties of point defects in \(SnO_{2}\)
Point defects are usually electrically active and introduce levels in the energy band gap of semiconductor. These levels involve transitions between different charge states of the same defect. Transition levels can be derived directly from the calculated formation energies. In semiconductor there may exist two types of transition levels: shallow and deep. This distinction is on the basis of their position in the energy band with respect to valence or conduction band. For a defect to contribute in n-type conductivity, it must be stable in a positive charge state and the transition level from the positive to neutral charge state \(\epsilon(+/0)\) should occur close to or above the conduction band minimum (CBM). Shallow donors are defects in which the transition level from a positive to the neutral charge state \(\epsilon(+/0)\) is near or above the CBM. Theoretical studies based on density functional theory have contributed to deeper understanding of the role of point defects on the unintentional n-type conductivity in \(SnO_{2}\). For a long time it has been envisaged that the unintentional n-type conductivity in \(SnO_{2}\) is caused by the presence of oxygen vacancies (\(V_{O}\)) and tin interstitials (\(Sn_{i}\)). However, recent first principle calculations have demonstrated that this attribution to native point defects cannot be correct. It has been shown that oxygen vacancies are very deep rather than shallow donors and therefore cannot contribute to n-type conductivity. According to Singh et. al., the \(\epsilon_{V_{O}}(2+/0)\) transitional level is located at 1.24 eV above the VBM in GGA and at 1.39 eV in GGA + U; the extrapolated value is 1.80 eV. This result distinctly shows that \(V_{O}^{0}\) is a deep donor with ionization energy of 1.80 eV. For \(V_{O}^{0}\), the three nearest-neighbors Sn atoms are displaced inward by 2.5%, whereas for \(V_{O}^{+}\) and \(V_{O}^{2+}\), the relaxations are outward by 5.6% and 10% of the equilibrium Sn-O bond length, respectively. In addition, it was found that tin interstitials (\(Sn_{i}^{4+}\)) and tin antisites (\(Sn_{O}^{4+}\)) are also unlikely causes of the unintentional n-type conductivity in as-deposited \(SnO_{2}\) films. The tin interstitial is a shallow donor, but it is not thermally stable. It has high formation energy about 12 eV in n-type \(SnO_{2}\) and migrates through the interstitial channels along the direction with an energy barrier of only 0.13 eV. This very low energy barrier implies that the Sn interstitials are highly mobile even below room temperature. Tin antisites are also shallow donors but their high formation energies make them unlikely to exist in measurable concentrations under equilibrium conditions. The dissociation barrier of \(Sn_{O}^{4+}\) into \(Sn_{i}^{4+}\) and \(V_{O}^{0}\) is very high in n-type \(SnO_{2}\). Therefore, \(Sn_{O}^{4+}\) is expected to be stable at temperature of up to 500 K. This suggest that \(Sn_{O}^{4+}\) may play a role in n-type conductivity under non-equilibrium conditions such as ion beam irradiation. On the other hand Sn vacancies (\(V_{Sn}^{4-}\)) are deep acceptors and have lowest formation energies among the native point defects under n-type conditions. They can therefore occur as a compensating defects in n-type \(SnO_{2}\). In contrast, oxygen antisites (\(O_{Sn}^{2-}\)) have the highest formation energies among the acceptor type native defects. Hence, it is very unlikely that oxygen antisites would be present in equilibrium. However, \(O_{Sn}^{2-}\) could potentially be created under non-equilibrium conditions in n-type \(SnO_{2}\). Oxygen antisites are also deep acceptors and show large off site displacements, in which the oxygen atom forms a chemical bond with one of the oxygen nearest neighbours. Oxygen interstitials (\(O_{i}\)) can exist either as electrically inactive split interstitials in p-type samples or as a deep acceptors at the octahedral sites in n-type samples. In both forms oxygen interstitials have high formation energies and are therefore unlikely to be present in measurable concentrations under equilibrium conditions. A key conclusion from the recent DFT studies is that native point defects cannot explain the unintentional n-type conductivity. One has to consider the role of impurities that are most likely to be present in different growth and processing environments and act as shallow donors. Hydrogen is indeed a especially ambidextrous impurity in this respect, since it is extremely difficult to detect experimentally. In both forms, substitutional and interstitial, hydrogen has been predicted to act as a shallow donor in n-type \(SnO_{2}\). But by means of density functional calculations, it is found that interstitial hydrogen (\(H_{O}^{+}\)) is highly mobile and can easily diffuse out of the \(SnO_{2}\) samples. The migration barrier of interstitial hydrogen is only 0.57 eV. This low barrier value makes it difficult to explain the stability of the n-type conductivity at relatively high temperature. On the other hand substitutional hydrogen (\(H_{O}^{+}\)) species is much more thermally stable than interstitial hydrogen and can explain stability of n-type conductivity at high temperature and its variation with partial pressure of oxygen in annealing environments. Both the substitutional and interstitial forms of hydrogen have low formation energiesin \(SnO_{2}\), indicating that they can occur in significant concentrations. Hydrogen is by no means the only possible shallow donor impurity in tin oxide, but it is a very likely candidate for an impurity that can be unintentionally incorporated and can explain observed unintentional n-type conductivity. Several groups have reported on the incorporation of hydrogen in tin oxide and many have claimed that hydrogen substitutes for oxygen. Table 3 lists the details of all native point defects and hydrogen impurities, based on several reported results. These summarized results will help us later to interpret experimental data in this research paper.
### X-ray diffraction studies
From the analysis of x-ray diffraction patterns, it is evident that only the peaks corresponding to the rutile-type \(SnO_{2}\) phase (space group \(P4_{2}/mnm\)) are detected in all samples. No additional reflection peaks related to impurities, such as unreacted manganese metal, oxides or any other tin manganese phases are detected. The lack of any impurity phases shows that the Mn is evenly distributed throughout the \(SnO_{2}\). The lattice parameters (a and c) and cell volume (V) are calculated by using the, and reflection planes in the XRD patterns. The complete structural analyses of the samples are given in the Table 4. The observed contraction in cell volume from 72.1681 \(\AA^{3}\) to 71.8616 \(\AA^{3}\) on Mn doping, indicating the replacement of \(Sn^{4+}\) by \(Mn^{3+}\). The lattice parameter a is observed to decreases from 4.7491 \(\AA\) to 4.7413 \(\AA\), while the lattice parameter c decreases from 3.1998 \(\AA\) to 3.1967 \(\AA\). These results are in agreement with the fact that the difference in the ionic radius of \(Mn^{3+}\) (0.58 \(\AA\)) and \(Sn^{4+}\) (0.69 \(\AA\)) is rather small, leading to very small changes in the lattice constants on doping with Mn. For Mn, there are three oxidation states namely \(Mn^{4+}\) (0.53 \(\AA\)),
\begin{table}
\begin{tabular}{||c|c|c|c|c|c|c||} \hline \hline Defect & Character & Formation & Migration & Formation & Formation & Effect on \\ & & energy & energy/temperature & probability in pristine samples & probability in irradiated samples & lattice \\ \hline \hline \(V_{O}^{2+}\) & Donor & Stable in & Very high & - & - & Expansion \\ & & p-type & & & & \\ \hline \(V_{O}^{+}\) & Donor & Unstable for & - & - & - & - \\ & & all \(E_{F}\) & & & & \\ \hline \(V_{O}^{2}\) & Neutral & Modest & Very high & Modest & Very high & Contraction \\ & & (Deep \(\sim\) 1.8 eV) & (\(\approx 900K\)) & & & \\ \hline \(Sn_{i}^{4+}\) & Donor & Very high \(\sim\) 12 eV & Low & Very low & Very low & Expansion \\ & & (Shallow) & \(\sim\) 0.43 eV & & & \\ \hline \(Sn_{i}^{3+}\) & Donor & Unstable & - & - & - & - \\ \hline \(Sn_{i}^{2+}\) & Donor & Unstable & - & - & - & - \\ \hline \(Sn_{O}^{4+}\) & Donor & Very high & High & Very low & High & Expansion \\ & & (Shallow) & (\(\approx 500K\)) & & & \\ \hline \(O_{i}^{2-}\) & Acceptor & High & Modest & Low & High & Expansion \\ & & (Deep) & (\(<500K\)) & & & \\ \hline \(V_{Sn}^{4-}\) & Acceptor & Modest in n-type & High & Modest & Very high & Expansion \\ & & (Deep) & (\(\approx 500K\)) & (Very low \(T>500K\)) & & \\ \hline \(O_{Sn}^{2-}\) & Acceptor & Very high & High & Very low & High & Contraction \\ & & (Deep) & (\(\approx 500K\)) & & & \\ \hline \(H_{i}^{+}\) & Donor & Low & Low & Very low at & - & - \\ & & (Shallow) & \(\sim\) 0.57 eV & room temperature & & \\ \hline \(H_{O}^{+}\) & Donor & Low & Very high & Very high & - & Expansion \\ & & (Shallow) & \(\sim\) 2.2 eV & & & \\ \hline \hline \end{tabular}
\end{table}
Table 3: Properties of native or intrinsic defects in n-type \(SnO_{2}\) (oxygen poor condition).
(0.58 \(\AA\)) and \(Mn^{2+}\) (0.82 \(\AA\)). The ionic radius of each ion in \(\AA\) is given in brackets. But the magnitude of cell volume change indicates that the valence of the Mn ion is most probably +3 instead of +2 or +4. This speculation also finds support from the studies performed on manganese oxides showing that in its oxides, \(Mn^{3+}\) prefers to be in an octahedral coordination, just like \(Sn^{4+}\) in the rutile-type structure of tin oxide (\(SnO_{2}\)). It is clear from that Mn substitution effects the intensity of \(SnO_{2}\) peaks. The intensity of the scattered x-ray is related to structure factor and this factor is determined by the presence of bound electrons in an atom. Since manganese has less bound electrons than tin, the substitution of tin by manganese in the tin oxide lattice should yield a structure factor, which is almost half the value for tin. If there is any change in the occupation site of manganese, i.e., substitutional (\(Mn_{Sn}^{3+}\)) to interstitial (\(Mn_{i}\)), the same may be reflected as an increase in the structure factor.
As far as we know, the migration temperatures of all possible native defects of \(SnO_{2}\) are less than 700\({}^{\circ}\)C (see Table-3). Therefore, samples which have been annealed at a temperature of 850\({}^{\circ}\)C, they may be free of defects. In this article, we have assumed annealed-samples as defect-free samples (see Fig. 5). After analysis of XRD patterns, we found that the unit cell volume of as-deposited \(SnO_{2}\) thin film is slightly more than that of defect free annealed \(SnO_{2}\) thin film (see Fig. 5). This increase in cell volume may be due to presence of expansion-type point defects such as \(H_{O}^{+}\), \(V_{Sn}^{4-}\), \(O_{i}^{2-}\), \(Sn_{O}^{4+}\), \(Sn_{i}^{4+}\). Among all these expansion-type point defects only \(H_{O}^{+}\) and \(V_{Sn}^{4-}\) may be responsible for the observed increase in cell volume, because formation energies of other expansion-type defects such as \(O_{i}^{2-}\), \(Sn_{O}^{4+}\), \(Sn_{i}^{4+}\) are very high as compared to formation energies of \(H_{O}^{+}\) and \(V_{Sn}^{4-}\). It is impossible to create such defects of high formation energies through thermodynamic/equilibrium process. The direct and unambiguous method for introducing such defects in lattice is by high-energy ion beam irradiation. We know that tin vacancies are acceptor type point defects and play a lead role in conductivity killing (see Table-3).
Changes in normalized intensity of respective samples.
Deviations in cell volume of all samples with respect to defect free sample.
X-ray diffraction patterns of unirradiated, irradiated and annealed thin films.
Thus, we can conclude that the slight expansion observed in cell volume of pure \(SnO_{2}\) thin film is only due to \(H_{O}^{+}\) point defects. According to Singh et. al., \(V_{O}^{0}\) defect is a contraction type, but when hydrogen does occupy the oxygen vacancy site (\(V_{O}^{0}\)) then resultant defect \(H_{O}^{+}\) becomes expansion type, because after hydrogen substitution the resultant Sn-H bond length (2.14 \(\AA\)) is slightly more than Sn-O bond length (2.07 \(\AA\)). In a later section-III.F, we have reported the results of electrical measurements. From these measurements we can say carrier concentration of as prepared pure \(SnO_{2}\) thin film is very high. According to Singh et. al. & Janotti et. al., \(H_{O}^{+}\) is also stables enough to explain this observed high unintentional n-type conductivity.
The swift heavy ion irradiation is a very effective technique to create point defects in the target material. Both expansion (\(V_{Sn}^{4-}\), \(O_{i}^{2-}\), \(Sn_{O}^{4+}\)) and contraction (\(V_{O}^{0}\), \(O_{Sn}^{2-}\)) type of native defects can be created with the help of ion beam. These defects are created in accordance with its respective formation energy; those having less formation energy are created in large number and vice-versa. SHI irradiation can also anneal-out the defects via two ways: self-annealing and beam heating. Self-annealing is a process through which pre-existing defects of target are annealed-out. Self-annealing of pre-existing defects depends upon the ion fluence of irradiation. Almost all pre-existing defects may anneal-out at high fluence irradiation. In this process the energy transfer through ion-electron interaction makes the target abnormally excited. On the other hand, the ion beam heating process is based on the heat produced by the transfer of beam power. The power of the ion beam warms up the target to a certain equilibrium temperature in specific period of time (see Section - III.B). In the case of high fluence, irradiation period is much longer than the amount of time needed to achieve equilibrium temperature. Therefore, in this case those defects cannot be generated permanently by irradiation whose migration temperature is less than equilibrium temperature. The cell volume estimated for pristine, \(1\times 10^{11}\), \(3\times 10^{11}\), \(1\times 10^{12}\), \(3\times 10^{12}\), \(1\times 10^{13}\), \(3\times 10^{13}\)\(ions/cm^{2}\) irradiated films are 71.8616, 72.3000, 72.4320, 72.6942, 72.8267, 71.5438 and 71.4129 \(\AA^{3}\), respectively.
\begin{table}
\begin{tabular}{||c|c|c|c|c|c||} \hline \hline Samples & \multicolumn{2}{c|}{Lattice parameters} & Cell volume & \multicolumn{1}{c|}{Defects present} & \multicolumn{1}{c||}{Defects} \\ & a = b (\(\AA\)) & c (\(\AA\)) & (\(\AA^{3}\)) & in the samples & annealed/migrated \\ \hline \hline \(SnO_{2}\) & 4.7491 & 3.1998 & 72.1681 & \(H_{O}^{+}\) & \(V_{Sn}^{4-}\) \\ \hline \(Sn_{0.9}Mn_{0.1}O_{2}\) & 4.7413 & 3.1967 & 71.8616 & \(H_{O}^{+}\) & \(V_{Sn}^{4-}\) \\ (Pristine) & & & & & \\ \hline \(1\times 10^{11}\)\(ions/cm^{2}\) & 4.7521 & 3.2016 & 72.3000 & \(H_{O}^{+}\) (Partially Present) & \(H_{O}^{+}\) (Partially Annealed), \\ & & & & \(V_{Sn}^{4-}>V_{O}^{0}>O_{i}^{2-}>Sn_{O}^{4+}>O_{Sn}^{2-}\) & \(Sn_{i}^{4+}\) \\ \hline \(3\times 10^{11}\)\(ions/cm^{2}\) & 4.7551 & 3.2034 & 72.4320 & \(H_{O}^{+}\) (Partially Present) & \(H_{O}^{+}\) (Partially Annealed), \\ & & & & \(V_{Sn}^{4-}>V_{O}^{0}>O_{i}^{2-}>Sn_{O}^{4+}>O_{Sn}^{2-}\) & \(Sn_{i}^{4+}\) \\ \hline \(1\times 10^{12}\)\(ions/cm^{2}\) & 4.7611 & 3.2069 & 72.6942 & \(H_{O}^{+}\) (Partially Present) & \(H_{O}^{+}\) (Partially Annealed), \\ & & & & \(V_{Sn}^{4-}>V_{O}^{0}>O_{i}^{2-}>Sn_{O}^{4+}>O_{Sn}^{2-}\) & \(Sn_{i}^{4+}\) \\ \hline \(3\times 10^{12}\)\(ions/cm^{2}\) & 4.7641 & 3.2087 & 72.8267 & \(H_{O}^{+}\) (Partially Present) & \(H_{O}^{+}\) (Partially Annealed), \\ & & & & \(V_{Sn}^{4-}>V_{O}^{0}>O_{i}^{2-}>Sn_{O}^{4+}>O_{Sn}^{2-}\) & \(Sn_{i}^{4+}\) \\ \hline \(1\times 10^{13}\)\(ions/cm^{2}\) & 4.7354 & 3.1905 & 71.5438 & \(V_{O}^{0}\) & \(H_{O}^{+},Sn_{i}^{4+},O_{i}^{2-}\), \\ & & & & & \(Sn_{O}^{4+},O_{Sn}^{2-},V_{Sn}^{4-}\) \\ \hline Annealed \(850^{o}C\) & 4.7431 & 3.1935 & 71.8442 & - & \(H_{O}^{+},Sn_{i}^{4+},O_{i}^{2-},Sn_{O}^{4+}\), \\ (\(SnO_{2}\)) & & & & & \(O_{Sn}^{2-},V_{Sn}^{4-},V_{O}^{0}\) \\ \hline Annealed \(850^{o}C\) & 4.7437 & 3.1933 & 71.8578 & - & \(H_{O}^{+},Sn_{i}^{4+},O_{i}^{2-},Sn_{O}^{4+}\), \\ (\(3\times 10^{13}\)) & & & & & \(O_{Sn}^{2-},V_{Sn}^{4-},V_{O}^{0}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Summary of various results obtained through analysis of XRD patterns.
But here, instead of decrease in cell volume, an increase has been observed upto \(3\times 10^{12}\)\(ions/cm^{2}\) (see Fig. 5). In spite of self-annealing, ion beam can also generate defects in accordance with its formation energy. As we know, the expansion-type \(V_{Sn}^{4-}\) defects have lowest formation energy among the all native point defects (\(V_{Sn}^{4-}\), \(V_{O}^{0}\), \(O_{i}^{2-}\), \(Sn_{O}^{4+}\), \(O_{Sn}^{2-}\)) under n-type conditions (see Table 3). Therefore, after irradiation, they may be present in large numbers as compared to other native defects and can explain the observed increase in cell volume. After the irradiation fluence of \(3\times 10^{12}\)\(ions/cm^{2}\), we have observed a decrease in the cell volumes of samples (see Fig. 5). This decrease in the cell volumes can be explained in terms of ion beam heating phenomena. In case of high fluence (such as \(1\times 10^{13}\) or \(3\times 10^{13}\)\(ions/cm^{2}\)), material spends almost entire time during irradiation at equilibrium temperature (\(\sim\) 834 K). So possibly in such case, those point defects cannot be created permanently through irradiation whose migration temperature is less than equilibrium temperature. Among the all possible defects (\(V_{Sn}^{4-}\), \(V_{O}^{0}\), \(O_{i}^{2-}\), \(Sn_{O}^{4+}\), \(O_{Sn}^{2-}\)) only contraction-type \(V_{O}^{0}\) defect has greater migration temperature (\(\sim\) 900 K) than equilibrium substrate/film temperature (\(\sim\) 834 K). Therefore, only contraction-type \(V_{O}^{0}\) defects may be present in high fluence irradiated samples and can explain the observed contraction in cell volume. We have also analyzed the effect of annealing on the cell volumes of irradiated and pure \(SnO_{2}\) samples. It should be noted that the cell volumes of both annealed films (irradiated at \(3\times 10^{13}\)\(ions/cm^{2}\) and pure \(SnO_{2}\)) are almost equal. This observation confirms the creation of point defects in irradiated materials. If we keep annealing temperature of irradiated samples higher than the migration temperature of all possible defects (approximately 700\({}^{o}\)) then all defects present in the irradiated samples becomes migrate and consequently samples acquire defect free state.
### Texture coefficient, crystallite size, micro and macro-strain
The preferential orientation of the crystallites in the films was studied by calculating the texture coefficient C(hkl) of each XRD peak using the equation:
\[C(hkl)=\frac{N(I(hkl)/I_{o}(hkl))}{\sum(I(hkl)/I_{o}(hkl))} \tag{6}\]
C(hkl) is unity for each XRD peak in the case of a randomly oriented film and values of C(hkl) greater than unity indicate preferred orientation of the crystallites in that particular direction. The degree of preferred orientation \(\sigma\) of the film as a whole can be evaluated by estimating the standard deviation of all the calculated C(hkl) values:
\[\sigma=\sqrt{\frac{\sum[C(hkl)-C_{o}(hkl)]^{2}}{N}} \tag{7}\]
The zero value of \(\sigma\) indicates that the crystallites in the film are oriented randomly. The higher value of \(\sigma\) indicates that the crystallites in the film are oriented preferentially. The texture coefficient C(hkl) of all the XRD peaks along with the value of \(\sigma\) for each film is given in Table 5. It can be seen that the plane has a high texture coefficient for all the films. The thin film attained maximum preferential orientation (\(\sigma=0.367\)) when the fluence value is \(3\times 10^{13}\)\(ions/cm^{2}\). However, it should be highlighted that none of the films possess a significant preferential orientation since the value of \(\sigma\) is less than unity for all the films.
In polycrystalline materials, two types of lattice strain can be encountered: uniform strain (macro-strain) and non-uniform strain (micro-strain). Uniform strain causes the unit cell to contract/expand in an isotropic way. This simply leads to a change in the lattice parameters and shift of the x-ray diffraction peaks. There is no peak broadening associated with this type of lattice strain. Moreover, non-uniform strain leads to systematic shifts of atoms from their ideal positions and to peak broadening. In the case of nano-materials where crystallite size also plays a role in peak broadening the two effects strain and size broadening overlap each other. In such cases the micro-strain or crystallite size can only be determined by separating the two effects. However, the peak width derived from micro-strain varies as \(\tan\theta\), whereas crystallite size varies as \(1/\cos\theta\). This difference in behaviour as a function of \(2\theta\) enables one to discriminate between the strain and size effects on peak broadening. Williamson-Hall (W-H) analysis is a simplified integral breadth method where micro-strain-induced and crystallite size-induced broadening are de-convoluted by considering the peak width as a function of \(2\theta\). Williamson and Hall assumed that both strain and size broadened profiles are Lorentzian. Based on this assumption a mathematical relation was established between the integral breadth (\(\beta\)), the lattice micro-strain (\(\varepsilon\)) and volume weighted average crystallite size (D) as follows:
\[\beta\cos\theta=\frac{k\lambda}{D}+4\varepsilon\sin\theta \tag{8}\]
Where k is a constant equal to 0.94, \(\lambda\) is the wavelength of the radiation (1.5405 \(\AA\) for \(CuK_{\alpha}\) radiation), and \(\theta\) is the peak position. The instrumental resolution in the scattering angle \(2\theta\), \(\beta_{inst}\), was determined by means of a standard crystalline silicon sample and approximated by \(\beta_{inst}\) = 9 \(\times\) 10\({}^{-6}\) (2\(\theta\))\({}^{2}\) - 0.0005 (2\(\theta\)) + 0.0623
Finally, the integral breadth \(\beta\) without instrumental contribution was obtained according to the relation:
\[\beta=\beta_{measured}-\beta_{instrumental} \tag{9}\]
Eq. 8 represents the general form of a straight line \(y=mx+c\). The plot between \(\beta\cos\theta\) and \(4\sin\theta\) gives a straight line having slope \(\varepsilon\) and intercept \(k\lambda/D\). The values of crystallite size and micro-strain can be obtained from the inverse of intercept and the slope of the straight line, respectively. The Williamson-Hall plots for all the samples are given in and the results extracted from these plots are listed in Table 5. Although W-H analysis is an averaging method apart from TEM imaging, it still holds a dominant position in crystallite size determination. It is clear from intercept values that SHI irradiation does not influence the crystallite size much in the low fluence regime, but causes a significant increase at the fluence of \(1\times 10^{13}\) and \(3\times 10^{13}\)\(ions/cm^{2}\). The crystallite size of the target changes from 29 nm to 45 nm after irradiation at the fluence of \(3\times 10^{13}\)\(ions/cm^{2}\). Such significant change in the crystallite size at higher fluence may be due to ion-beam heating effect. In ion beam heating process the power of ion beam warms up the target to a certain equilibrium temperature in specific period of time. In the case of high fluence (\(>1\times 10^{12}\)\(ions/cm^{2}\)) material spends almost entire time during irradiation at equilibrium temperature of \(\sim\) 834 K and feels annealing like atmosphere for whole irradiation period (see section-III.B).
Furthermore, we have calculated macro-strains of all the samples along a and c directions with the help of the following relations:-
\[\varepsilon_{a}^{{}^{\prime}}=\frac{(a-a_{o})}{a_{o}} \tag{10}\]
\[\varepsilon_{c}^{{}^{\prime}}=\frac{(c-c_{o})}{c_{o}} \tag{11}\]
Where \(a_{o}\) = 4.7431 \(\AA\) and \(c_{o}\) = 3.1935 \(\AA\) are the lattice parameters of the defect free sample (annealed \(SnO_{2}\) thin film). The obtained values of \(\varepsilon_{a}^{{}^{\prime}}\) and \(\varepsilon_{c}^{{}^{\prime}}\) for all the samples are listed in Table 5.
It is important to note that the Williamson-Hall method probes the micro-strain induced by dislocation inside the crystalline grains, i.e., opposite to a macro-strain that causes a shift in the Bragg peak position. By inspection of the W-H plots, it appears that the value of micro-strain for pristine film is \(12.7\times 10^{-4}\) and is reduced to \(11.4\times 10^{-4}\), \(10.8\times 10^{-4}\), \(8.71\times 10^{-4}\), \(6.24\times 10^{-4}\)
\begin{table}
\begin{tabular}{||c|c|c|c|c|c|c|c|c||} \hline \hline & \multicolumn{2}{c|}{Texture coefficient C(hkl)} & \multicolumn{2}{c|}{Crystallite} & Non-Uniform & \multicolumn{2}{c||}{Uniform Strain} \\ \cline{2-9} Samples & & & & & \(\sigma\) & Size (nm) & Strain & a - direction & c - direction \\ \hline \hline \(SnO_{2}\) & 0.792 & 0.865 & 1.442 & 0.901 & 0.258 & 27 & \(12.5\times 10^{-4}\) & \(1.265\times 10^{-3}\) & \(1.973\times 10^{-3}\) \\ \hline \(Sn_{0.9}Mn_{0.1}O_{2}\) & 0.814 & 0.783 & 1.504 & 0.899 & 0.294 & 29 & \(12.7\times 10^{-4}\) & \(-0.379\times 10^{-3}\) & \(1.002\times 10^{-3}\) \\ (Pristine) & & & & & & & & & \\ \hline \(1\times 10^{11}\)\(ions/cm^{2}\) & 0.844 & 0.826 & 1.498 & 0.833 & 0.287 & 26 & \(11.4\times 10^{-4}\) & \(1.897\times 10^{-3}\) & \(2.536\times 10^{-3}\) \\ \hline \(3\times 10^{11}\)\(ions/cm^{2}\) & 0.827 & 0.824 & 1.476 & 0.873 & 0.276 & 26 & \(10.8\times 10^{-4}\) & \(2.529\times 10^{-3}\) & \(3.100\times 10^{-3}\) \\ \hline \(1\times 10^{12}\)\(ions/cm^{2}\) & 0.820 & 0.812 & 1.515 & 0.853 & 0.298 & 29 & \(8.71\times 10^{-4}\) & \(3.795\times 10^{-3}\) & \(4.196\times 10^{-3}\) \\ \hline \(3\times 10^{12}\)\(ions/cm^{2}\) & 0.850 & 0.816 & 1.480 & 0.855 & 0.277 & 34 & \(6.24\times 10^{-4}\) & \(4.427\times 10^{-3}\) & \(4.759\times 10^{-3}\) \\ \hline \(1\times 10^{13}\)\(ions/cm^{2}\) & 0.743 & 0.813 & 1.473 & 0.972 & 0.285 & 39 & \(3.77\times 10^{-4}\) & \(-1.623\times 10^{-3}\) & \(-0.939\times 10^{-3}\) \\ \hline \(3\times 10^{13}\)\(ions/cm^{2}\) & 0.739 & 0.719 & 1.622 & 0.920 & 0.367 & 45 & \(1.15\times 10^{-4}\) & \(-2.256\times 10^{-3}\) & \(-1.503\times 10^{-3}\) \\ \hline Annealed \(850^{o}C\) & 0.828 & 0.795 & 1.460 & 0.916 & 0.269 & 121 & \(2.51\times 10^{-4}\) & - & - \\ \((SnO_{2})\) & & & & & & & & \\ \hline Annealed \(850^{o}C\) & 0.849 & 0.839 & 1.424 & 0.887 & 0.246 & 139 & \(0.49\times 10^{-4}\) & \(0.126\times 10^{-3}\) & \(-0.062\times 10^{-3}\) \\ \((3\times 10^{13})\) & & & & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 5: The Texture coefficient C(hkl), the degree of preferential orientation \(\sigma\), crystallite size, non-uniform and uniform strain for all the samples.
\(3.77\times 10^{-4}\), \(1.15\times 10^{-4}\) for films irradiated with the fluence of \(1\times 10^{11}\), \(3\times 10^{11}\), \(1\times 10^{12}\), \(3\times 10^{12}\), \(1\times 10^{13}\), and \(3\times 10^{13}\)\(ions/cm^{2}\), respectively. The reduction of microstrain after irradiation is indicative of the improvement in crystallinity due to self-annealing of pre-existing irregular lattice defects such as dislocations, faults etc. Practically, it has been found that the higher the ion fluence, more is the heat generated, and hence smaller the microstrain. On the other hand the macro-strain of the target is increased after irradiation (see Table 5). The increase in macro-strain may be due to the formation of fresh regular expansion/contraction type point defects. It can be clearly seen from the Table 5, the macro-strain is of tensile type upto \(3\times 10^{12}\)\(ions/cm^{2}\) fluence and after that it is of compressive type. That is because before \(1\times 10^{13}\)\(ions/cm^{2}\) fluence the density of expansion type defects is more as compare to contraction type defects and it is vice versa for \(1\times 10^{13}\) and \(3\times 10^{13}\)\(ions/cm^{2}\) fluence (already discussed in the XRD section). The micro-strain for the unirradiated pure \(SnO_{2}\) sample is measured as 12.5 \(\times\)\(10^{-4}\), which changes to 2.51 \(\times\)\(10^{-4}\) after annealing at 850\({}^{o}\) (see Table 5). From comparing the micro-strain of pure \(SnO_{2}\) samples annealed at 850\({}^{o}\)C with samples irradiated at \(3\times 10^{13}\)\(ions/cm^{2}\) fluence, we find that high fluence irradiation more efficiently releases micro-strain from the samples. The macro-strain was also found to be negligibly small for the annealed samples (pure \(SnO_{2}\) and irradiated at \(3\times 10^{13}\)\(ions/cm^{2}\) fluence). The reduction in micro- and macro-strain values after thermal annealing may be due to the migration of lattice defects at 850\({}^{o}\)C. From W-H plots, it is clear that the grain growth has occurred on thermal annealing at 850\({}^{o}\)C.
### Electrical properties
The electrical properties of the films were estimated by resistivity and Hall effect measurements made at room temperature. The room temperature results are presented for all measured films in Table 6. The pure \(SnO_{2}\) film shows the best combination of electrical properties as follows: resistivity (\(\rho\)) of 2.41 \(\times\)\(10^{-3}\)\(\Omega\) cm, carrier concentration (n) of 2.735 \(\times\)\(10^{20}\)\(cm^{-3}\), and mobility (\(\mu\)) of 9.497 \(cm^{2}V^{-1}s^{-1}\). The lower resistivity in the as-deposited \(SnO_{2}\) film may be due to the presence of substitutional hydrogen (\(H_{O}^{+}\)). First principle calculations have provided evidence that usual suspects such as oxygen vacancy \(V_{O}\) and tin interstitial \(Sn_{i}\) are actually not responsible for n-type conductivity in majority of the cases. These calculations indicate that the oxygen vacancies are a deep donor, whereas tin interstitials are too mobile to be stable at room temperature. Recent first principle calculations have drawn attention on the role of donor impurities in unintentional n-type conductivity. Hydrogen is indeed a especially ambidextrous impurity in this respect, since it is extremely difficult to detect experimentally. By means of density functional calculations it has been shown that hydrogen can substitute on an oxygen site and has a low formation energy and act as a shallow donor. Hydrogen is by no means the only possible shallow donor impurity in tin oxide, but it is a very likely candidate for an impurity that can be unintentionally incorporated and can explain observed unintentional n-type conductivity. Tin vacancies have the lowest formation energy among the native point defects under n-type condition. Therefore they may exist as compensating defects in n-type \(SnO_{2}\) sample (see Section III.C). But if we make thin films at a specific temperature somewhere in between migration temperature of tin vacancy and substitutional hydrogen, we find that they are highly conducting (\(n\geq 10^{20}\)\(cm^{-3}\)). In the present investigation, we have optimised substrate temperature for the deposition of \(SnO_{2}\) thin films at 450\({}^{o}\) C. This substrate temperature is much higher than the migration temperature of tin vacancy. Thus we can say that the n-type conductivity in \(SnO_{2}\) samples may be caused by \(H_{O}^{+}\) defects. On the other hand, the resistivity of the thermally annealed films shows an insulating behavior. In the thermally annealed film, the transformation towards stoichiometry leads to an increase in the resistivity as expected for a metal-oxide semiconductor. Thermal annealing of as-deposited \(SnO_{2}\) films was carried out beyond the migration temperature of \(H_{O}^{+}\) defects (\(\sim\) 900K) to confirm the role of \(H_{O}^{+}\) defects on the film resistivity. The electrical resistivity of 3.63 \(\times\)\(10^{-3}\)\(\Omega\) cm obtained from the pristine \(Sn_{0.9}Mn_{0.1}O_{2}\) films is increased to 49.6 \(\Omega\) cm for the films irradiated with \(3\times 10^{12}\)\(ions/cm^{2}\) fluence (see Fig. 7). The value of carrier concentration for pristine sample is 1.927 \(\times\)\(10^{20}\)\(cm^{-3}\) and is reduced to 9.658 \(\times\)\(10^{18}\), 3.283 \(\times\)\(10^{18}\), 7.492 \(\times\)\(10^{16}\), 1.831 \(\times\)\(10^{16}\)\(cm^{-3}\) for films irradiated with the fluence
Williamson-Hall plots of all samples.
8). On the other hand, the mobility of \(Sn_{0.9}Mn_{0.1}O_{2}\) films is decreased from the original value of 8.929 \(cm^{2}V^{-1}s^{-1}\) to 6.875 \(cm^{2}V^{-1}s^{-1}\) after irradiation with fluence of 3 \(\times\) 10\({}^{12}\)\(ions/cm^{2}\) (see Fig. 8). Above a typical fluence of 3 \(\times\) 10\({}^{12}\)\(ions/cm^{2}\), electrical measurements show that the sheet resistance of irradiated film is of the order of 10 M\(\Omega/\square\) (see Fig. 7). SHI irradiation is very effective method to keep the sample at very high temperature through electron-phonon coupling in a short span of time. Due to this self annealing effect of irradiation, the concentration of pre-existing defects (\(H_{G}^{+}\)) decreases with increasing ion fluence. Along with this self annealing of pre-existing defects, some fresh electron donor (\(Sn_{O}^{4+}\)) and electron killer (\(V_{Sn}^{4-}\), \(O_{i}^{2-}\), \(O_{Sn}^{2-}\)) type defects can be formed through irradiation in the target.
These fresh defects are created in accordance with its respective formation energy; those having less formation energy are created in large number and vice-versa. The variation in conductivity of irradiated samples may be caused by these mixed effects. The point defects newly created by ion beam irradiation can also be annealed out of the sample at high fluence (\(\geq\)\(1\times 10^{13}\)\(ions/cm^{2}\)). In the case of high fluence, target sample spends almost entire time during irradiation at equilibrium temperature of \(\sim\) 834 K (see Section - III.B). Therefore in that case all those newly created defects may anneal out of the sample whose migration temperature is less than the equilibrium temperature. Among the all possible defects only \(V_{O}^{0}\) defect has a higher migration temperature (\(\sim\) 900 K) than the equilibrium temperature (\(\sim\) 834 K). Therefore other newly created defects (\(V_{Sn}^{4-},O_{i}^{2-},Sn_{O}^{4+},O_{Sn}^{2-}\)) except \(V_{O}^{0}\) may anneal out of the sample during high fluence irradiation. With the help of these arguments one can understand the insulating behaviour of high fluence irradiated samples.
The temperature dependence of electrical resistivity in the range 30-200\({}^{\circ}C\) indicates that the pure and Mn doped \(SnO_{2}\) (pristine) films are degenerate semiconductors.
Changes in the free carrier concentration n and electron mobility \(\mu_{c}\) of respective samples. Here x-axis is not scaled for better clarity.
Changes in the electrical resistivity \(\rho\) and sheet resistance \(R_{s}\) of respective samples. Here x-axis is not scaled for better clarity.
Degeneracy temperature as a function of carrier concentration.
Here, We have tried to identify the main scattering mechanisms that influence the mobility of pure \(SnO_{2}\) films. There are many scattering mechanisms such as grain-boundary scattering, domain scattering, surface scattering, interface scattering, phonon scattering (lattice vibration), neutral, and ionized impurity scattering which may influence the mobility of the films. The interaction between the scattering centres and the carriers determines the actual value of the mobility of the carriers in the samples. In the interpretation of the mobility obtained for pure \(SnO_{2}\) films, one has to deal with the problem of mixed scattering of carriers. To solve this problem, one has to identify the main scattering mechanism and then determine their contributions. The pure \(SnO_{2}\) films prepared here are polycrystalline. They are composed of grains joined together by grain boundaries, which are transitional regions between different orientations of neighboring grains. These boundaries between grains play a significant role in the scattering of charge carriers in polycrystalline thin films. The grain boundary scattering has an effect on the total mobility only if the grain size is approximately of the same order as the mean free path of the charge carriers (\(D\sim\lambda\)). The mean free path for the degenerate samples can be calculated from known mobility (\(\mu\)) and carrier concentration (n) using the following expression:
\[\lambda=(3\pi^{2})^{\frac{1}{3}}(\frac{\hbar\mu}{e})n^{\frac{1}{3}}, \tag{13}\]
The mean free path value calculated for the pure \(SnO_{2}\) film is 1.257 nm which is considerable shorter than crystallite size (D \(\sim\) 27 nm) calculated using XRD data. Moreover, the effect of crystallite interfaces is weaker in semiconductors, with n \(\geq\)\(10^{20}\)\(cm^{-3}\), observed here, as a consequence of the narrower depletion layer width at the interface between two grains. Based on above discussion it is concluded that grain boundary scattering is not a dominant mechanism.
The mobility of the free carrier is not affected by surface scattering unless the mean free path is comparable to the film thickness. Mean free path value calculated for the pure \(SnO_{2}\) film is 1.257 nm, which is much smaller than the film thickness (\(\sim\) 510 nm). Hence, surface scattering can be ruled out as the primary mechanism. Scattering by acoustical phonons apparently plays a subordinate role in the pure \(SnO_{2}\) films because no remarkable temperature dependence have been observed between 30 and 200\({}^{\circ}\)C. Moreover, neutral impurity scattering can be neglected because the neutral defect concentration is negligible in the pure \(SnO_{2}\) films. Electron-electron scattering, as suggested to be important in Ref. 133, can also be neglected as it does not change the total electron momentum and thus not the mobility. In high crystalline \(SnO_{2}\) films, scattering by dislocations and precipitation is expected to be of little importance.
Another scattering mechanism popular in unintentionally doped semiconductors is the ionized impurity scattering.
\begin{table}
\begin{tabular}{||c|c|c|c|c|c|c|c|} \hline \hline & Film & Sheet & & Carrier & Degeneracy & Carrier mobility \\ & thickness & resistance & Resistivity & concentration & temperature & \(\mu\) (\(cm^{2}V^{-1}s^{-1}\)) \\ Samples & (nm) & \(R_{s}\) & \(\rho\) (\(\Omega\) cm) & n (\(cm^{-3}\)) & \(T_{D}\) (K) & Observed & Calculated \\ \hline \hline \(SnO_{2}\) & 510 & 47 \(\Omega/\square\) & 2.41\(\times 10^{-3}\) & \(2.735\times 10^{20}\) & 5767 & 9.497 & 10.346 \\ \hline \(Sn_{0.9}Mn_{0.1}O_{2}\) & 550 & 66 \(\Omega/\square\) & 3.63\(\times 10^{-3}\) & \(1.927\times 10^{20}\) & 4566 & 8.929 & - \\ (Pristine) & & & & & & & \\ \hline \(1\times 10^{11}\)\(ions/cm^{2}\) & 500 & 1.81 \(K\Omega/\square\) & 9.03\(\times 10^{-2}\) & \(9.658\times 10^{18}\) & 621 & 7.163 & - \\ \hline \(3\times 10^{11}\)\(ions/cm^{2}\) & 523 & 5.53 \(K\Omega/\square\) & 0.289 & \(3.283\times 10^{18}\) & 302 & 6.597 & - \\ \hline \(1\times 10^{12}\)\(ions/cm^{2}\) & 555 & 0.223 \(M\Omega/\square\) & 12.4 & \(7.492\times 10^{16}\) & 24 & 6.724 & - \\ \hline \(3\times 10^{12}\)\(ions/cm^{2}\) & 530 & 0.936 \(M\Omega/\square\) & 49.6 & \(1.831\times 10^{16}\) & 10 & 6.875 & - \\ \hline \(1\times 10^{13}\)\(ions/cm^{2}\) & 515 & \(\approx\)10 \(M\Omega/\square\) & - & - & - & - & - \\ \hline \(3\times 10^{13}\)\(ions/cm^{2}\) & 505 & \(\approx\)10 \(M\Omega/\square\) & - & - & - & - & - \\ \hline Annealed 850\({}^{\circ}C\) & 510 & \(\approx\)10 \(M\Omega/\square\) & - & - & - & - & - \\ \((SnO_{2})\) & & & & & & & \\ \hline Annealed 850\({}^{\circ}C\) & 505 & \(\approx\)10 \(M\Omega/\square\) & - & - & - & - & - \\ \((3\times 10^{13}\)\(ions/cm^{2})\) & & & & & & & \\ \hline \hline \end{tabular}
\end{table}
Table 6: Electrical parameters for all the samples.
in the degenerate case, given by
\[\tau_{i}=\frac{(2m^{*})^{\frac{1}{2}}(\epsilon_{o}\epsilon_{r})^{2}(E_{F})^{\frac{ 3}{2}}}{\pi e^{4}N_{i}f(x)}, \tag{14}\]
with \(N_{i}\) the carrier concentration of ionized impurities and f(x) given by
\[f(x)=ln(1+x)-\frac{x}{1+x}, \tag{15}\]
with
\[x=\frac{8m^{*}E_{F}R_{S}^{2}}{\hbar^{2}}, \tag{16}\]
The screening radius \(R_{S}\) is given by
\[R_{S}=(\frac{\hbar}{2e})(\frac{\epsilon_{o}\epsilon_{r}}{m^{*}})^{\frac{1}{2}} (\frac{\pi}{3N_{i}})^{\frac{1}{6}}, \tag{17}\]
The mobility (\(\mu\)) is defined as
\[\mu=\frac{e\tau}{m^{*}}, \tag{18}\]
Substitution of the \(\tau_{i}\) expression [Eq.] in Eq.
\[\mu_{i}=\frac{(\frac{2}{m^{*}})^{\frac{1}{2}}(\epsilon_{o}\epsilon_{r})^{2}(E _{F})^{\frac{3}{2}}}{\pi e^{3}N_{i}f(x)}, \tag{19}\]
Since all the \(H_{O}^{+}\) defects present in the pure \(SnO_{2}\) films will be fully ionized at room temperature, impurity ion concentration will be equal to the free carrier concentration. Thus taking \(N_{i}=\) n, \(m^{*}=0.31\)m, \(\epsilon_{r}=13.5\) and using Eq. in Eq.
\[\mu_{i}=\frac{2.4232\times 10^{-4}}{f(x)}, \tag{20}\]
with
\[x=1.7942\times 10^{-9}n^{\frac{1}{3}}, \tag{21}\]
The calculated mobility and measured mobility values for pure \(SnO_{2}\) thin films are 10.346 and 9.497 \(cm^{2}V^{-1}s^{-1}\) respectively, both are comparable to each other. This clearly indicates that the main scattering mechanism reducing the intra-grain mobility of the electrons in pure \(SnO_{2}\) films is the ionized impurity scattering. Ionized impurity scattering with singly ionized \(H_{O}^{+}\) donors best describes the mobility of pure \(SnO_{2}\) samples. This finding supports our assumption that \(H_{O}^{+}\) defect is source of conductivity in pure \(SnO_{2}\) sample.
A Coulomb interactions between the electrons and the ionized impurities cause collisions and the scattering of the electrons. The amount of deflection depends on the speed of the electron and its proximity to the ion. The electrons will not experience the Coulombic force if they do not spend much time near the ionized impurities. If the electrons are moving fast, the amount of time spent in the location of the ionized impurity will be very short. This will affect the mobility of the electrons. Temperature of the material greatly affects ionized impurity scattering. At low temperatures, the electrons are moving slowly because thermal velocity is low. If the thermal velocity of the electrons is small, the amount of time the electrons stay within the vicinity of an ionized impurity increases. This increases the cross section of scattering. Another factor affecting ionized impurity scattering is the concentration of ionized impurities. The more ionized impurities in the material; the more scattering events occur. With the higher probability of scattering event, the mean time between collisions decrease, which decreases mobility. The mobility of electrons in non-degenerate semiconductor increases with temperature and is independent of the electron concentration, whereas the mobility in a highly degenerate semiconductor is nearly independent of temperature and increases with electron concentration.
### Microstructural properties
Structural characterization through TEM is a direct way of visualizing the estimated grain size. It also gives authentic information regarding the grain size distribution, crystalline nature, and other structural information. The transmission electron micrographs and the corresponding selected area electron diffraction (SAED) patterns for \(SnO_{2}\) and \(Sn_{0.9}Mn_{0.1}O_{2}\) samples are shown in These TEM micrographs and SAED patterns have been analyzed using the IMAGE-J software. The TEM images of the \(SnO_{2}\) and \(Sn_{0.9}Mn_{0.1}O_{2}\) samples show the presence of interconnected nono-sized spheroidal grains. The crystallite size observed by TEM (\(\sim 25\) nm for \(SnO_{2}\) and \(\sim 30\) nm for \(Sn_{0.9}Mn_{0.1}O_{2}\)) is in good agreement with that estimated by x-ray line broadening (\(\sim 27\) nm for \(SnO_{2}\) and \(\sim 29\) nm for \(Sn_{0.9}Mn_{0.1}O_{2}\)). The SAED patterns shown in Figs. 10(b) & 10(d) taken from \(SnO_{2}\) and \(Sn_{0.9}Mn_{0.1}O_{2}\) samples show several sharp rings, which are indexed to the,, and planes of the rutile crystalline structure of \(SnO_{2}\). The electron diffraction pattern has been examined carefully for rings and spots of secondary phases, and it has been found that all the rings and spots belong to the tetragonal rutile structure of \(SnO_{2}\) only. We have observed that there is no formation of any structural core-shell system.
The SEM images of \(SnO_{2}\) and \(Sn_{0.9}Mn_{0.1}O_{2}\) thin films deposited by chemical spray pyrolysis technique at substrate temperature of 450\({}^{o}\)C are shown in These SEM images reveal significant aggregation of the nano-grains. The small grains coagulated to form big clusters that reflect like the pomegranate structure. This can be explained by the fact that smaller primary Scanning electron micrographs of the (a) pure \(SnO_{2}\) and (b) pristine \(Sn_{0.9}Mn_{0.1}O_{2}\) thin films.
Transmission electron micrographs [(a) and (c)] of the pure \(SnO_{2}\) and pristine \(Sn_{0.9}Mn_{0.1}O_{2}\) thin films, respectively. Corresponding selected area electron diffraction (SAED) patterns for the pure \(SnO_{2}\) and pristine \(Sn_{0.9}Mn_{0.1}O_{2}\) thin films are shown in (b) and (d), respectively. Transmission electron micrographs of both thin films showing several nanocubes or nanospheres.
The aggregation of nano-grains is very important for the ferromagnetism in nano-crystalline dilute magnetic semiconductors, which can generate lattice defects and increase the domain size. In our samples, the aggregation should be promising for ferromagnetism. The aggregation makes it difficult to determinate the grain size accurately. The size that is estimated from the few spherical grains is about 25 nm. This is slightly smaller than that obtained from XRD measurements.
Fig. 12(a) shows the AFM image of \(Sn_{0.9}Mn_{0.1}O_{2}\) film irradiated with \(3\times 10^{13}\)\(ions/cm^{2}\) fluence and then annealed at \(850^{o}C\) for 4 hr. The grains of this film are nearly spherical in shape with size in the range of 100 to 150 nm. The root mean square (rms) roughness is 16.5 nm. Fig. 12(b) shows the AFM image of the \(Sn_{0.9}Mn_{0.1}O_{2}\) film annealed at \(850^{o}C\) and then irradiated with \(3\times 10^{13}\)\(ions/cm^{2}\) fluence. Morphology of this film is very much different from that shown in Fig. 12(a). At this condition nanorod/nanoribbon like structures with size in the range of 50-100 nm and rms roughness of 6.4 nm are grown in lateral direction on the surface. Morphology of irradiated film depends on the conductivity of target material. If the target material is a conductor (\(n\geq 10^{20}cm^{-3}\) for as-deposited \(Sn_{0.9}Mn_{0.1}O_{2}\) film) then conduction electrons rapidly spread the energy of incident ion throughout the material and we find similar pristine/target sample like morphology with some dimensional change after irradiation. But if target material is an insulator (such as \(Sn_{0.9}Mn_{0.1}O_{2}\) sample annealed at \(850^{o}\)C) then projectile ions can not spread their energy throughout the target material and therefore they create high energy region in the close vicinity of their paths. This cylindrical region around the path of the ion may become fluid if maximum temperature reached at the centre is higher than the melting temperature of the material. The structure of grains is changed (spherical to ribbon/rod like) only when melting is observed and the nature/size of the new structure corresponded closely to the maximum molten region. The typical amorphous nanoribbon/nanorod like structures may arise on the surface due to rapid resolidification of molten material. Melt recoil pressure, surface tension, diffusion and evaporation dynamics may also effect the resolidification process of molten material and consequently the structure of grains.
### Band-gap studies
The presumed band structure of \(SnO_{2}\) is shown in with the parabolic conduction and valence bands.
\[E_{v}^{0}(k)=\frac{-\hbar^{2}k^{2}}{2m_{v}^{*}}, \tag{22}\]
and
\[E_{c}^{0}(k)=E_{g0}+\frac{\hbar^{2}k^{2}}{2m_{c}^{*}}, \tag{23}\]
\(E_{g0}\) is the intrinsic band gap of the semiconductor, k = \(\frac{2\pi}{\lambda}\) is the wave number, and superscript 0 denotes unperturbed bands.
Atomic force microscopy images of \(Sn_{0.9}Mn_{0.1}O_{2}\) surfaces: (a) annealed at \(850^{o}\)C after irradiation with \(3\times 10^{13}\) ions/\(cm^{2}\), (b) irradiated with \(3\times 10^{13}\) ions/\(cm^{2}\) after annealing at \(850^{o}\)C.
\(n_{c}^{1/3}a_{0}^{*}\sim 0.25\) (\(a_{0}^{*}\) is the effective Bohr radius), the low-energy levels of conduction band are filled up by the conduction electrons. Then photons with energy greater than the intrinsic band gap only get absorbed.
\[E_{g}=E_{c}^{0}(k_{F})-E_{v}^{0}(k_{F}), \tag{24}\]
Alternatively, we may write
\[E_{g}=E_{g0}+\Delta E_{g}^{BM}, \tag{25}\]
where the Burstein-Moss (BM) shift is given by
\[\Delta E_{g}^{BM}=[E_{c}^{0}(k_{F})-E_{v}^{0}(k_{F})]-E_{g0},\]
\[\Delta E_{g}^{BM}=\frac{\hbar^{2}}{2m_{vc}^{*}}(3\pi^{2}n)^{2/3}, \tag{26}\]
with the reduced effective mass
\[\frac{1}{m_{vc}^{*}}=\frac{1}{m_{v}^{*}}+\frac{1}{m_{c}^{*}}, \tag{27}\]
Neglecting the electron-electron and electron-defect interactions, the band gap is
\[E_{g}=E_{g0}+\frac{\hbar^{2}}{2m_{vc}^{*}}(3\pi^{2}n)^{2/3} \tag{28}\]
To determine the absorption band-edge of \(SnO_{2}\) films, we use the theory developed for optical transitions in semiconductor, in which the absorption coefficient (\(\alpha\)) of \(SnO_{2}\) is a parabolic function of the incident energy \(h\nu\) and the optical band gap \(E_{g}\),
\[\alpha(h\nu)=A(h\nu-E_{g})^{1/2} \tag{29}\]
Using this relationship, the band edge of \(SnO_{2}\) is evaluated in the standard manner from a plot of \((\alpha h\nu)^{2}\) versus photon energy (\(h\nu\)). The extrapolation of the linear portion of the \((\alpha h\nu)^{2}\) vs. \(h\nu\) plot to \(\alpha\) = 0 gives the band gap value of the films. shows such plots for all the samples and the linear fits obtained for these plots are also depicted in the same figure.
The Burstein-Moss effect, i.e., the widening of the band gap with increasing n, is exhibited when the free electron density (n) far exceeds the Mott critical density (\(n_{c}\)), whose magnitude can be estimated by Mott's criterion.
\[n_{c}^{1/3}a_{0}^{*}\sim 0.25 \tag{30}\]
The effective Bohr radius \(a_{0}^{*}\) is given by:
\[a_{0}^{*}=\frac{h^{2}\epsilon_{0}\epsilon_{r}}{\pi e^{2}m_{c}^{*}} \tag{31}\]
Using these numbers, one can obtain \(a_{0}^{*}\sim 2.31\) nm, and the critical density \(n_{c}\) is calculated as \(1.3\times 10^{18}\)\(cm^{-3}\). Above this Mott critical density, the material is said to be degenerate.
For degenerate semiconductor (such as pure \(SnO_{2}\), \(Sn_{0.9}Mn_{0.1}O_{2}\) and \(1\times 10^{11}\)\(ions/cm^{2}\)), a plot of \(E_{g}\) versus \(n^{2/3}\) gives \(E_{g0}\) as well as \(m_{vc}^{*}\).
\((\alpha h\nu)^{2}\) vs \(h\nu\) plots for the un-irradiated, irradiated and annealed samples. The direct energy band gap \(E_{g}\) is obtained from the extrapolation to \(\alpha\) = 0.
(a) Schematic band structure of stoichiometric \(SnO_{2}\) showing the parabolic conduction and valence bands separated by the band gap \(E_{g0}\) and (b) the Burstein-Moss shift \(\Delta E_{g}^{BM}\) due to filling up of the lowest states in the conduction band. Shaded areas denote occupied regions. Band gaps, Fermi wave number, and dispersion relations are indicated.
From this plot, the intrinsic band gap \(E_{g0}\) and reduced effective mass \(m^{*}_{vc}\) come out to be 3.99 eV and 0.298m, respectively.
\[m^{*}_{v}=\frac{m^{*}_{c}.m^{*}_{vc}}{m^{*}_{c}-m^{*}_{vc}} \tag{32}\]
Taking the conduction-band effective mass as \(m^{*}_{c}=0.31\)m, \(m^{*}_{v}\) works out to be 7.69m for the present study. It may be pointed out that the reduced effective mass of \(SnO_{2}\) obtained in the present work (\(m^{*}_{vc}=0.298\)m) is comparable than that estimated for \(SnO_{2}\) by Sanon et. al. (\(m^{*}_{vc}=0.237\)m) using \(m^{*}_{c}=0.31\)m and \(m^{*}_{v}=1.0\)m. A positive small value of valence-band effective mass (\(m^{*}_{v}=1.0\)m) can be obtained by including the many-body interactions in the valence-band and conduction bands. The band-gap narrowing due to electron-electron and electron-defect scattering in \(SnO_{2}\) films is quite significant and cannot be neglected. The narrowing when added to experimental widening yields the actual Burstein-Moss widening and a positive small value of valence-band effective mass \(m^{*}_{v}=1.0\)m is thereby obtained.
The band gap obtained for annealed \(SnO_{2}\) sample can be assumed as exact intrinsic band gap. Because the samples which have been annealed at a temperature of 850\({}^{\circ}\)C, they may be free of defects. On comparing the band gap of annealed \(SnO_{2}\) sample (\(\sim 3.97\) eV) with that estimated from (\(\sim 3.99\) eV), we find that both are nearly same. For degenerate samples such as \(SnO_{2}\), \(Sn_{0.9}Mn_{0.1}O_{2}\) and \(1\times 10^{11}\)\(ions/cm^{2}\), it can be seen that the band gap is changing according to Burstein-Moss effect whereas for other irradiated samples (SHI: \(3\times 10^{11}\) to \(3\times 10^{13}\)\(ions/cm^{2}\)) we can notice a significant but small shift in band gap from intrinsic band gap (\(E_{g0}\) = 3.97 eV). This noticeable shift in the band gap upon irradiation (SHI: \(3\times 10^{11}\) to \(3\times 10^{13}\)\(ions/cm^{2}\)) may be due to creation of intermediate energy levels.
### Magnetic properties
For proper investigation and understanding of the magnetic properties of pristine and irradiated \(Sn_{0.9}Mn_{0.1}O_{2}\) films, measurements of magnetization as a function of temperature [M(T)] and magnetic field [M(H)] were carried out over a temperature range of 5-300 K and field range of 0 to \(\pm\) 2 T using a SQUID magnetometer. The magnetic field was applied parallel to the film plane. Figs. 16 and 17 show the magnetization versus applied magnetic field (M-H) curves measured at 5 and 300 K for the pristine and irradiated (SHI: \(1\times 10^{11}\) and \(3\times 10^{11}\)\(ions/cm^{2}\)) \(Sn_{0.9}Mn_{0.1}O_{2}\) films with n = \(1.9\times 10^{20}\), \(9.7\times 10^{18}\) and \(3.3\times 10^{18}\) electrons \(cm^{-3}\), respectively. The inset in Figs. 16 and 17 shows a zoom of the region of low magnetic fields that evidences the presence of a hysteresis. The saturation magnetization (\(M_{S}\)) is estimated to be \(14.503\times 10^{-2}\), \(10.013\times 10^{-2}\) and \(3.745\times 10^{-2}\) emu/g at 5 K and \(13.793\times 10^{-2}\), \(8.265\times 10^{-2}\) and \(1.843\times 10^{-2}\) emu/g at 300 K and the coercivity (\(H_{C}\)) is 89, 45 and 29 Oe at 5 K and 83, 35 and 24 Oe at 300 K for the pristine and irradiated (SHI: \(1\times 10^{11}\) and \(3\times 10^{11}\)\(ions/cm^{2}\)) \(Sn_{0.9}Mn_{0.1}O_{2}\) samples by the M-H curves, respectively. The samples with higher carrier concentration (Pristine: \(Sn_{0.9}Mn_{0.1}O_{2}\)) show ferromagnetic characteristics with higher saturation magnetization and coercivity. On the other hand, thin films irradiated with higher fluence (\(\geq 1\times 10^{12}\)\(ions/cm^{2}\)) are diamagnetic at 5 K, as shown in the inset of The diamagnetic films have \(n\leq 7.5\times 10^{16}\) electrons \(cm^{-3}\), which is much lower than that observed in ferromagnetic pristine films. We have observed that the films are ferromagnetic for the higher and intermediate range of n, \(1.9\times 10^{20}\) - \(3.3\times 10^{18}\) electrons \(cm^{-3}\), and diamagnetic for lower range, \(n\leq 7.5\times 10^{16}\) electrons \(cm^{-3}\). Thus, lowering the n from \(1.9\times 10^{20}\) to \(7.5\times 10^{16}\) electrons \(cm^{-3}\) results in both electrical and magnetic phase transitions, a transition from a ferromagnetic semiconducting state to a diamagnetic insulating state. The diamagnetic background of the pure \(SnO_{2}\) film and substrate has been subtracted from all of the magnetization data shown here. The correlation between the magnetic and electrical properties in the present \(Sn_{0.9}Mn_{0.1}O_{2}\) system formulates it to be a potential aspirant for the spintronics oriented devices wherein the communication between the spin and charge is highly desired.
For the pristine film, the M-H curve at 5 K differed by less than 6% from that measured at room temperature (300 K), from which we guess that the Curie temperature (\(T_{C}\)) of pristine sample is well above room temperature. But, in the case of \(1\times 10^{11}\) and \(3\times 10^{11}\)\(ions/cm^{2}\) irradiated films, the M-H curves at 5 K differed by 16%
Band gap as a function of carrier concentration for the degenerate samples (\(SnO_{2}\), \(Sn_{0.9}Mn_{0.1}O_{2}\) and SHI: \(1\times 10^{11}\)\(ions/cm^{2}\)). The linear-fit of the experimental data gives \(E_{g0}\) = 3.99 eV and \(m^{*}_{vc}\) = 0.298m.
This clearly indicates that the Curie temperature of the irradiated film is less as compared to pristine film. We also recorded the M vs T curves of these samples in a field of 0.5 T. shows the normalized \(M_{S}(T)/M_{S}(5K)\) data from 5-300 K, for a \(Sn_{0.9}Mn_{0.1}O_{2}\) film irradiated at \(1\times 10^{11}~{}ions/cm^{2}\) fluence. The absence of any sharp drop in the normalized \(M_{S}(T)\) plot suggests that the film is ferromagnetic with a Curie temperature exceeding 300 K.
The appearance of room temperature ferromagnetism in pristine and irradiated \(Sn_{0.9}Mn_{0.1}O_{2}\) samples cannot be due to the presence of secondary phases, because the metallic manganese and nearly all of the possible manganese-based binary and ternary oxide phases (MnO, \(MnO_{2}\) and \(Mn_{2}O_{3}\)) are antiferromagnetic with a Neel temperature that is much less than 300 K. However, \(SnMn_{2}O_{4}\) and \(Mn_{3}O_{4}\) phases are exceptions; they are ferromagnetic with Curie temperatures of 46 K and 53 K, respectively. In the present work, the electron and x-ray diffraction analyses have not revealed any manganese oxide phases, although x-ray diffraction technique is not sensitive enough to detect secondary phases, if present at a very minute level. Even if these ferromagnetic \(SnMn_{2}O_{4}\) and \(Mn_{3}O_{4}\) phases are present, these cannot responsible for the ferromagnetic behavior observed at room temperature in pristine and irradiated \(Sn_{0.9}Mn_{0.1}O_{2}\) thin films.
## IV Conclusions
High crystalline thin films of \(SnO_{2}\) and \(Sn_{0.9}Mn_{0.1}O_{2}\) have been deposited by the spray pyrolysis technique and then deposited \(Sn_{0.9}Mn_{0.1}O_{2}\) films have been irradiated with 120 MeV \(Au^{9+}\) ions to study the modification of structural, microstructural, electrical and magnetic properties. The threshold value of electronic energy loss (\(S_{eth}\)) for \(Sn_{0.9}Mn_{0.1}O_{2}\) has been calculated according to Szenes' thermal spike model. The electronic energy loss (\(S_{e}\)) of 120 MeV \(Au^{9+}\) ions in \(Sn_{0.9}Mn_{0.1}O_{2}\)
Field-dependent magnetization of the pristine and irradiated (SHI: \(1\times 10^{11}\) and \(3\times 10^{11}~{}ions/cm^{2}\)) \(Sn_{0.9}Mn_{0.1}O_{2}\) thin films measured at 5K. The inset shows the low-field part in an enlarged scale that evidences the presence of a hysteresis in pristine \(Sn_{0.9}Mn_{0.1}O_{2}\) sample.
Field-dependent magnetization of the pristine and irradiated (SHI: \(1\times 10^{11}\) and \(3\times 10^{11}~{}ions/cm^{2}\)) \(Sn_{0.9}Mn_{0.1}O_{2}\) thin films measured at 300K. The inset shows the low-field part in an enlarged scale that evidences the presence of a hysteresis in pristine \(Sn_{0.9}Mn_{0.1}O_{2}\) sample.
Normalized \(M_{S}(T)\) plot with H = 5000 Oe for \(Sn_{0.9}Mn_{0.1}O_{2}\) sample irradiated at \(1\times 10^{11}~{}ions/cm^{2}\) fluence. The inset shows the 5K field-dependent magnetization of \(Sn_{0.9}Mn_{0.1}O_{2}\) sample irradiated at \(1\times 10^{12}~{}ions/cm^{2}\) fluence.
Therefore, we expect that only point defects or clusters of point defects will be produced in \(Sn_{0.9}Mn_{0.1}O_{2}\) thin films after irradiation. Efforts have been made to calculate equilibrium substrate/film temperature using the Stefan's equation and the calculated temperature (834 K) has been found to be quite higher than the crystallization temperature of tin oxide. This equilibrium temperature is expected to be develop within the grains of \(Sn_{0.9}Mn_{0.1}O_{2}\) during irradiation period and facilitate (i) increase of grain size, (ii) removal of micro-strain, and (iii) migration of point defects. The increase rate of the substrate/film temperature during irradiation has been determined by the difference between the input power density and the heat dissipation via radiation, divided by the heat capacity of the substrate. Through this it has been concluded that, in the case of high fluence, irradiation period is much longer than the amount of time needed to achieve equilibrium temperature. Therefore, in this case, the samples spend their almost full irradiation period at equilibrium temperature. This equilibrium temperature can influence the physical properties of high fluence irradiated samples. X-ray and electron diffraction patterns analysis reveals that all the unirradiated and irradiated thin films are pure crystalline with tetragonal rutile phase of tin oxide which belongs to the space group P4\({}_{2}\)/mmm (number 136). The Williamson-Hall (W-H) method has been used to evaluate the crystallite size and the microstrain of all the samples. The average crystallite size of \(SnO_{2}/Sn_{0.9}Mn_{0.1}O_{2}\) nanoparticles estimated from W-H analysis and TEM/SEM analysis is highly inter-correlated. Electrical measurement shows that as-deposited \(SnO_{2}\) and \(Sn_{0.9}Mn_{0.1}O_{2}\) films are in conducting state with n = \(2.735\times 10^{20}\)\(cm^{-3}\) and \(1.927\times 10^{20}\)\(cm^{-3}\), respectively. The results of electrical measurements suggest that \(H_{O}^{+}\) defects are responsible for the conductivity in as-deposited thin films. Through electrical investigation it has also been found that the main scattering mechanism reducing the intra-grain mobility of the electrons in pure \(SnO_{2}\) films is the ionized impurity scattering. Ionized impurity scattering with singly ionized \(H_{O}^{+}\) donor best describes the mobility of pure \(SnO_{2}\) samples. Measurement of resistivity, mobility, and carrier density as a function of ion fluence (\(1\times 10^{11}\) to \(3\times 10^{13}\)\(ions/cm^{2}\)) promulgates that increasing fluence results in degradation in electrical properties of \(Sn_{0.9}Mn_{0.1}O_{2}\) film. Typical TEM micrographs of as-deposited \(SnO_{2}\) and \(Sn_{0.9}Mn_{0.1}O_{2}\) thin films show well isolated and highly crystallized spherical shaped crystallites. Electron diffraction patterns taken from several crystallites confirm the \(SnO_{2}\) structure in both \(SnO_{2}\) and \(Sn_{0.9}Mn_{0.1}O_{2}\) samples and no evidence for secondary phases are observed. The SEM micrographs of as-deposited \(SnO_{2}\) and \(Sn_{0.9}Mn_{0.1}O_{2}\) samples reveal significant aggregation of the nano-particles. Our AFM study demonstrates that the morphologies of irradiated films are linked with carrier concentration of target materials. If the target material is a conductor (\(n\geq 10^{20}cm^{-3}\) for as-deposited \(Sn_{0.9}Mn_{0.1}O_{2}\) film) then conduction electrons rapidly spread the energy of incident ion throughout the material and we find similar pristine/target sample like morphology with some dimensional change after irradiation. But if target material is an insulator (such as \(Sn_{0.9}Mn_{0.1}O_{2}\) sample annealed at 850\({}^{o}\)C) then projectile ions can not spread their energy throughout the target material and therefore they create high energy region in the close vicinity of their paths. This cylindrical region around the path of the ion may become fluid if maximum temperature reached at the centre is higher than the melting temperature of the material. The structure of grains is changed (spherical to rod like) only when melting is observed and the nature/size of the new structure corresponded closely to the maximum molten region. In the present case, amorphous nanoribbon/nanorod like structures may arise on the surface of the irradiated films due to rapid resolidification of the molten material. The optical band gap (\(E_{g}\)) of the films has been determined from the spectral dependence of the absorption coefficient \(\alpha\) by the application of conventional extrapolation method (Tauc plot). For degenerate samples such as \(SnO_{2}\), \(Sn_{0.9}Mn_{0.1}O_{2}\) and \(1\times 10^{11}\)\(ions/cm^{2}\), it can be seen that the band gap is changing according to Burstein-Moss effect whereas for other irradiated samples (SH: \(3\times 10^{11}\) to \(3\times 10^{13}\)\(ions/cm^{2}\)) we can notice a significant but small shift in band gap from intrinsic band gap (\(E_{g}\) = 3.97 eV). This noticeable shift in the band gap upon irradiation (SHI: \(3\times 10^{11}\) to \(3\times 10^{13}\)\(ions/cm^{2}\)) may be due to creation of intermediate energy levels. Intrinsic room temperature ferromagnetism has been observed in all degenerate films (such as pristine and irradiated (SHI: \(1\times 10^{11}\) and \(3\times 10^{11}\)\(ions/cm^{2}\)) \(Sn_{0.9}Mn_{0.1}O_{2}\) thin films). The variation of \(M_{S}\) values suggest that room temperature ferromagnetism could be enhanced by increasing carrier concentration of the films. No evidence of any impurity phases are detected in \(Sn_{0.9}Mn_{0.1}O_{2}\) suggesting that the emerging ferromagnetism in this system is most likely related to the properties of host \(SnO_{2}\) system.
| 10.48550/arXiv.1512.06119 | Swift Heavy Ion Irradiation Induced Modifications in Structural, Microstructural, Electrical and Magnetic Properties of Mn Doped $SnO_{2}$ Thin Films | Sushant Gupta, Fouran Singh, B. Das | 2,232 |
10.48550_arXiv.1807.06183 | ## Introduction
Several perovskite relaxor ferroelectrics in the PMN-PT family,
(PbMg\({}_{1/3}\)Nb\({}_{2/3}\)O\({}_{3}\))\({}_{1\cdot\text{x}}\)(PbTiO\({}_{3}\))\({}_{\text{x}}\) have been observed to show low-frequency polarization noise of unknown origins so prominent that it shows up in pyroelectric current experiments not intended to look for noise. Since this noise is far above the fluctuation-dissipation noise level required by thermodynamics, it can limit the materials' use as sensitive electromechanical transducers. Unlike the ordinary Barkhausen noise often observed in ferroelectrics, it usually persists for days after changes in electric field and shows no apparent connection to the net rate of polarization change but instead is dependent on the polarization itself. Unlike the thermal polarization noise found in some ferroelectrics, it is not limited to periods in which a phase transition is taking place. The effect is far more dramatic than the relatively subtle violations of the fluctuation-dissipation relation found in non-equilibrium spinglasses. Its origin presents an interesting puzzle.
It has previously been noted that the noise is present whenever a low-x sample (which we shall call here PMNPTx, where x is the percent rather than the fraction), is polarized and absent when the sample is not polarized. Below about x=0.20, bulk samples of these materials exhibit inversion symmetry and thus lack a net spontaneous piezoelectric coefficient, acquiring one only when their symmetry has been broken via an applied field or, at low temperature, via being polarized by prior application of a field. Thus they are microphonic, i.e. generate voltages in response to long-range coherent strain only when polarized. Since the noise appears only when the sample is microphonic, extraneous mechanical vibrations are the obvious first suspect for the source, but these have been consistently ruled out by simple tests to reduce or increase such vibrations, e.g. turning off noisy pumps or stomping around near the cryostat. In a previous paper Zhangfocused on Barkhausen noise, we presented preliminary speculation that this extra low-frequency noise was internally generated, i.e. that the material is mechanically creaking due to very slow thermal equilibration.
Here we present evidence, based largely on sensitivity to thermal and field histories and on the persistence of the noise in the non-ferroelectric state, confirming that speculation. We also find that the anomalous low-frequency noise is almost absent in higher-x materials lacking the bulk cubic so-called "X phase" and the broken-symmetry "skin effect" surface layer that often accompanies the X phase. We shall discuss possible implications of the low frequency noise for the origin of the poorly understood skin effect and the metastability of the X phase.
## Materials and Methods
The samples of PMNPTx are from the same batch previously described in our work on the kinetics and thermodynamics of forming the ferroelectric (FE) phase. They were grown by a modified Bridgman technique and supplied by TRS Technologies (State College, PA). shows an empirical phase diagram of a PMNPT12 sample from this batch, illustrating the ergodic paraelectric (PE), non-ergodic relaxor paraelectric (RXF), and the FE regimes, as well as the hysteresis between RXF and FE. The melting temperature of the FE phase is 302K. The PMNPT20 sample shows similar behavior with somewhat higher characteristic temperatures, with a FE melting at 341K. The noise studies here included the FE regime, the metastable RXF regime, the stable RXF regime, and the PE regime. The PMNPT32 sample goes directly from a PE phase to an FE domain phase at roughly 430 K, but its FE phase depolarizes at \(\sim\)411 K. The noise studies here focus on its transition range.
As background, throughout the temperature range of our measurements these materials are largely filled with polar nanodomains (PNDs), on a scale of \(\sim\)10nm, each locally piezoelectric but with quasi-random orientations. Only the FE phase has bulk long-range polar order. The bulk of the RXF phase is referred to as "phase X", with overall non-polar cubic symmetry. Within about 100\(\upmu\)m of the surface, a lower-symmetry skin-effect region is observed. A comprehensive discussion of the evidence for phase X and the skin effect may be found in reference, with detailed results on phases of powder samples in ref.
The samples were configured as parallel-plate capacitors oriented with the applied **E** along a axis, an easy polarization axis for the ferroelectric state. Contacts were made via evaporated Ag layers of roughly 200nm thickness on top of adhesion-enhancing \(\sim\)10 nm thick evaporated Cr layers. The PMNPT12 sample started at \(\sim\)0.4mm thick, becoming slightly thinner for later measurements after repolishing. Its contact area is a 1 mm\({}^{2}\) disk out of a crystal area \(\sim\)2 mm by \(\sim\)3 mm. The PMNPT20 sample is 0.48mm thick with contacts 1.11 mm by 0.75mm out of a total area \(\sim\)4.22mm by \(\sim\)0.75mm. The PMNPT32 sample is 0.44mm thick with a total contact area 2.8mm\({}^{2}\) out of an irregular shape several times larger.
The measurement circuitry, described elsewhere, allows the sample voltage to be fixed with ac and dc biases while the polarization current I\({}_{\text{P}}\)(t) is measured at a rate of \(\sim\)10 samples/sec. Using a low-pass filter (usually set at 20 Hz) allows simultaneous measurement of the systematic polarization current and low-frequency (LF) noise in I\({}_{\text{P}}\)(t) along with the complex dielectric response function, \(\varepsilon\)'-ie", measured at 100 Hz (using a 28.3 mV rms ac drive) on an unfiltered channel. High-frequency (HF) noise was measured via a channel with a high-pass filter set at 1.0 Hz to remove the main large deterministic signal followed by an anti-alias low-pass filter at 2.1 kHz with 5kHz sampling, in runs where the ac susceptibility was not measured. In order to reduce electromagnetic pickup the nitrogen-flow transfer-line cryostat was mounted inside a double-wall mu-metal shield. The shield was supported on a sand pile to reduce vibrational pickup.
## Results
As before, we found large low-frequency noise when the low-x samples were polarized. The noise was insensitive to very large changes in acoustic input, including turning the nearby vacuum pump on or off and even tapping softly on the transfer line. shows Ip(t) measured at 250K in an applied dc voltage, as the PMNPT12 sample converts from the metastable RXF phase to the FE phase over several hundred seconds. The noise in Ip(t) grows steadily as the polarization creeps up in the RXF phase, then grows further as the polarization increases in the fairly abrupt transition to the FE phase. After the transition the noise gradually decreases slightly. When the applied voltage is removed after conversion to the FE state, causing little change in the polarization, the noise magnitude remains approximately constant. As the sample is subsequently heated the noise magnitude changes as a function of T, reaching a minimum at about 280K, rising as the sample starts to depolarize, then falling to very low levels after the sample abruptly depolarizes on the transition back to the relaxor phase. The noise level and its dependence on polarization during this sort of protocol were very similar to those observed in PMN some years ago.
The large noise during the melting transition may be viewed as a type of thermal Barkhausen noise. The persistent large noise as the polarized sample sits under fixed conditions before warming does not resemble any conventional Barkhausen noise.
LF noise again is found in the polarized sample, but its level is significantly lower than in the similar PMNPT12 or PMN samples, roughly three orders of magnitude in spectral density. (We have seen a similar reduction in each 20%
shows data for the PMNPT32 sample, which forms an aligned FE state almost immediately when a field of 550V/cm is applied. The sample shows thermal Barkhausen noise (toward the left side of the figure) as it is cooled at E=0 through the transition temperatures expected for the two FE phase changes at this composition. The resulting state is FE but lacks overall polarization due to since the domains are not systematically aligned. After field application, while sitting at fixed T in the polarized FE state, the noise is substantially lower than in even the PMNPT20 sample, i.e. barely above instrumental background except for an occasional Barkhausen spike. Since the contact area is larger than for the other samples, the current noise, which is additive over area, is comparatively even lower per contact area.
To test whether the low-frequency noise inherently involved the FE phase we looked for it above the equilibrium FE transition line, in the PE state of PMNPT12. The LF noise again appeared, as shown in Fig. 5, when the sample was polarized via an applied voltage. We cannot separately check the voltage and polarization dependences in this regime, in which the polarization relaxes back to very nearly zero after the applied field is removed, unlike in the lower-temperature regime, which exhibits large remnant FE polarization. The presence of the LF noise in this regime, above the peak in \(\varepsilon\)'(T), shows that it does not require either any long-range FE order or even the major slow polarization response of the non-ergodic RXF regime. Nevertheless, the asymmetry (more upward spikes) evident in the plots indicates that despite being in the PE regime, a small amount of the sample polarization is occurring after long delays via occasional steps.
Immediately after the voltage is applied in this regime, there is a transient period in which the LF noise gradually builds up. That effect is not simply due to an increase in the microphonic sensitivity, since the polarization itself does not show a significant delayed build-up. On longer time scales, as shown in Fig. 6, the LF noise sometimes starts to decrease again, consistent with it coming from very slow relaxation toward equilibrium.
Since the noise appears to come from some microphonic sensitivity to slow strain relaxations in response to thermal and field history, we checked whether it could be reduced by annealing the sample at high temperature, 773 K, to reduce internal strain. On the initial cool-down after such annealing, the LF noise in the PMNPT12 was indeed substantially reduced, although not eliminated. The insert of shows the most dramatic effect of this treatment, that on the first post-anneal warming after cooling there is a temperature range around 200 K - 250 K in which the noise magnitude is much lower than it had been on similar warming prior to the annealing. This result is consistent with annealing reducing the non-equilibrium strain, but could also be consistent with other explanations, since the contacts had to be re-applied after the annealing. After further thermal cycling and field application, the LF noise returned to approximately the pre-annealing magnitude, although with a slightly shifted temperature of the minimum on warming. also shows a slow decrease in noise magnitude during a long period at150K before warming.
To account for why the low-frequency noise becomes evident only when the samples have net polarization, the low-frequency strain changes must be correlated over regions containing many PND. That would give a piezoelectric voltage that is a coherent sum when the sample is polarized but a much smaller incoherent sum in the net unpolarized condition. One would then expect there to be some non-equilibrium polarization noise even in unpolarized samples, although not easy to pick up against background instrumental noise at low frequencies.
The PMNPT20 sample, which shows relatively little LF noise, gave non-equilibrium noise in the HF channel even while sitting at 370 K with E=0 after cooling from 600 K anneal. This temperature is approximately where the 100 Hz susceptibility peaks. Typical spectra are shown in The noise magnitude gradually decreased with a typical time-scale of hours. The approximate absolute magnitudes and general time course were reproducible. The susceptibility also showed aging under these conditions, but the magnitude of the susceptibility aging was much too small to account for the size of the noise aging. Other experiments, not shown, found spectral density a little higher than the 9 hour results as the sample passed through the range 365 K to 375 K on rapid warming of the sample from 350 K. Although this extra noise, like ordinary thermal Barkhausen noise, is caused by changing temperature, in this material it is found well above the temperature at which long-range FE order melts.
## Discussion
Polarized PMN and low-x PMNx show low-frequency current noise far above the value expected in equilibrium while held at fixed E and T. The magnitude depends strongly on sample thermal and field history. Although after temperature cycling and field changes the noise persists for periods of days or longer, it does at least sometimes gradually reduce, as expected for a non-equilibrium effect. Even non-polarized PMNPT20 in the PE state can show significant non-equilibrium current noise in a frequency range comparable to the typical dielectric relaxation rate. The dependences of the excess noise on field and temperature and on their histories are consistent with a picture of non-equilibrium strain relaxation. Even when a sample appears to be near equilibrium, i.e. with its polarization very close to the long-term expectation for a given average field and temperature, the internal pattern of the PND's, interacting both by strain and electric fields, appears usually to be far from equilibrium.
Although unpolarized samples do not show the large excess low-frequency noise, they can show a slowly decreasing non-equilibrium noise at higher frequencies. We believe that the explanation for the distinction is that the long-range correlated relaxations are slow and show up very little unless there is systematic piezoelectricity, while faster strain relaxations with only short-range correlations show up about equally regardless of whether the piezo coefficients of different PND have the same sign. The virtual absence of the excess low-frequency noise in unpolarized low-x samples shows that it comes from strain changes that are coherent over distances large compared to the size of polar nanodomains.
That the LF noise is most prominent in the low-x samples, for which the X-phase is most stable, suggests a connection to that phase, which is out of thermal equilibrium below the melting line shown on Since our samples are only several times the thickness of the temperature-dependent strained skin layer, the existence of major temperature-dependent strains is not a surprise. What static measurements had not shown, however, is that such strains remain out of equilibrium for very long times, with their slow approach to equilibrium creating dramatic polarization fluctuations, even in the PE phase or in the FE phase.
The existence of this noise may provide a clue to one unsolved question about the skin layer, i.e. the origin of the \(\sim\)100\(\upmu\)m characteristic length scale, some 10\({}^{4}\) times the typical PND length scale. The noise indicates that the strain is far from static, so the skin depth may not be an equilibrium property but rather a scale set by the slow kinetics of long-range strain changes in phase X under typical experimental conditions. The symmetry-broken skin-effect region would be, if this interpretation is correct, closer to equilibrium than is the X phase, with the growth of the skin-effect region inhibited by the slow kinetics of large-scale strain changes in the X phase. It would be interesting to explore whether the low-frequency non-equilibrium noise effects are consistent with some of the simpler pictures of the relaxor state, e.g. reference.
The large long-lasting non-equilibrium noise presents obvious difficulties for use of the low-x relaxor materials as sensitive low-noise strain detectors. Its magnitude does depend on field and temperature history and does decrease (sometimes very gradually) under constant E-T conditions, so the difficulties should not be insurmountable.
| 10.48550/arXiv.1807.06183 | Non-equilibrium Strain Relaxation Noise in the Relaxor Ferroelectric (PbMg1/3Nb2/3O3)100-x( PbTiO3)x | Xinyang Zhang, Thomas J. Kennedy, Eugene V. Colla, M. B. Weissman, D. D. Viehland | 1,499 |
10.48550_arXiv.1309.0601 | ## 1 Introduction
The room temperature liquid metal has attracted particular attention in recent years due to its fluidity, high electronic conductivity, and large thermal conductivity at normal atmospheric conditions. Usually, liquid metal refers to the metals that have low melting points close to room temperature. This property endows the liquid metal a unique feature of easy processing around room temperature. In the light of this merit, the liquid metal has been fund with many important applications such as stretchable electronics, printable thermocouple, thermometer, cooling media, and so on. Recently, it has been gradually extended to direct writing electronics. In a word, the liquid metal could be a promising material for electronic circuits in the following decades.
However, it is currently not clear as how the reliability of the liquid metal circuits will be when high currents are passing through them. The early studies have shown that an integrated rigid circuit may fail due to the so-called electromigration phenomenon when it carries a high current density. This is because the electromigration phenomenon can cause voids and hillocks in the conductor, and therefore, result in an open circuit or a short circuit. Although this problem has been studied in the rigid metal films such as aluminum thin films and copper thin films in the field of integrated circuits, little is known about the same issue in the newly emerging liquid metal films. Because such metal has fluidity, the electromigration may cause even more serious problems in the liquid metal circuits. Clarification on the reliability of the liquid metal circuits is therefore critical because it determines the appropriate working of an electronic device thus made. Thus, it is highly desirable to carry out an in-depth investigation on the electromigration phenomenon in the liquid metal thin films.
In this paper, we discovered for the first time the failure phenomenon of liquid gallium thin films. It was experimentally disclosed that a liquid metal thin film breaks up when it carries a high enough current density. The phenomenon was interpreted as a result of electromigration in the liquid metal. This finding is expected to be significant for future studies and applications of liquid metal based printed electronics.
## 2 Experimental set up
The experimental studies on the electromigration phenomenon were carried out in the liquid gallium thin films written on glass and silicon substrates. The liquid gallium used in the experiments has a purity of 99.99%. The glass substrates have dimensions of 76 mm \(\times\) 25 mm \(\times\) 1 mm. The samples were prepared by the so-called "direct writing" method: cover the substrate by a mask with desired geometric shape, heat the substrate up to 50 - 60 \({}^{\circ}\)C, and then paint the liquid gallium onto the substrate with a brush or glass rod. Because the liquid gallium (and its alloys) need oxides to wet the substrates (glass and silicon), the liquid gallium must be repeatedly written at higher temperatures (50 - 60 \({}^{\circ}\)C) to increase its oxide contents. With increasing wettability, the liquid gallium finally sticks to the substrate uniformly.
For the convenience of experimental observation, the thin film samples were prepared with special geometric shapes (see Fig. 1A). The width of the thin film'sends is 7.07 mm and the width of the thin film's middle is 2.32 mm. Consequently, the current density **j** in the middle is more than three times larger than that in the ends. This ensures that the thin films break-up in the middle for the convenient observation under an optical microscope. The thickness of the thin film is 0.012 mm, obtained from the SEM image of the thin film's cross section as shown in Due to the insulation of the glass substrate, the sample was coated with an ultrathin layer of gold by sputter deposition for viewing with the SEM.
In the measurements, the electric current was supplied by a current source with a rated current of 10 A. A thermocouple was buried in the liquid metal close to thin film's tapering part to monitor temperature. Because gallium's melting point is T\({}_{\mathrm{m}}\)=29.7646 \({}^{\circ}\)C (302.9146 K), we carried out all the measurements at the temperatures above T\({}_{\mathrm{m}}\) to ensure that our results are from liquid gallium, but not from the rigid solid gallium (see Fig. 3).
## 3 Results
To obtain a visual observation on the high electric current induced break-up phenomenon in liquid gallium thin film, we took the photographs of the thin films under an optical microscope as shown in Without applying a current to the sample, the middle of the thin film is shown in (before broken). With an increasing current applied to the sample, the middle of the thin film started to break as shown in As the applied current density increased up to j=114.9 A/mm
A. The geometric shape of the tested liquid gallium thin films. B. The SEM image of the thin film’s cross section (the sample is coated with gold).
(current 3.2 A), the thin film broke up and a crack can be clearly seen on it as indicated in The inset of shows the details of part of the crack. The white region is glass and the black region is liquid gallium. One can also refer to the SEM image of another sample as shown in to see the detail of the crack. The grayscales of the selected regions in A, B, and C are analyzed as shown D, E, and F, respectively. It indicates that the grayscale becomes light as the thin film started to break. The counts of the dark points is 1100 before the break, but reduced to 650 after the break.
The break-up phenomenon as observed in is further confirmed by monitoring the corresponding electric current density **j**(t), temperature T(t), and resistance R(t) in the thin film as the functions of time t (see Fig. 3):
The electric current density **j**(t) in the thin film is started to increase at t=23.0 s. As **j**(t) increased to 114.9 A/mm\({}^{2}\) (current 3.2 A) at t = 26.5 s, it dropped drastically under a constant voltage applied to the sample. Finally, it reached zero at t=27.5 s. This means that the thin film was broken up by the electromigration effect and the circuit became open.
The temperature T(t) in the middle of the thin film is shown to be around 38.0 \({}^{\circ}\)C in the interval t \(<\) 23.0 s (before break). This ensures that our measurements are performed in liquid gallium but not in solid gallium. With the increasing electric current **j**(t), the temperature then increased to its maximum value T(t)=44.9 \({}^{\circ}\)C at t=26.5 s due to the Joule heat released.
The thin film's resistance R(t) between the tapering parts as shown in increased from 0.4 \(\Omega\) at t=23.0 s to 0.6 \(\Omega\) at t=26.5 s, and then jumped up to 97451.1 \(\Omega\) at t=27.5 s (see the inset of Fig. 3). This indicates that the thin film was broken within one second. Due to the 2D character and the oxide contents, the resistance of the liquid metal thin film is much higher that of the bulk material.
Time evolution of the electric current, temperature, and resistance of the liquid metal thin film in the break process. An electronic current was applied to the thin films at t=23.0 s to induce the
The thin film broke at 27.5 s where the current density increased up to 114.9 A/mm\({}^{3}\) (current 3.2 A). The temperature is shown to be around 38 \(\,\mathrm{\SIUnitSymbolCelsius}\) measured close to the thin film's middle where the break process occurred. This ensured that our measurements are performed in liquid gallium but not in solid gallium. In the break process, the temperature increased from 38 \(\,\mathrm{\SIUnitSymbolCelsius}\) to 45 \(\,\mathrm{\SIUnitSymbolCelsius}\) due to the Joule heat released.
The contents of the residue in the crack of the broken thin film were analyzed using EDS (Energy Dispersive Spectroscopy) spectrum technique. To avoid the interference from the oxygen in a glass substrate, we prepared a thin film sample on a silicon substrate. The EDS spectrum was measured by choosing a region in the crack of the broken thin film (see Fig. 4). The elements analysis shows that the remained contains 55.31 Atommic% oxygen, and 44.69 Atomic% gallium. This indicates that the remained are mainly oxides.
## 4 Discussion
EDS spectrum of the liquid gallium thin film on a silicon substrate. The elements analysis indicates that the remained contains 55.31 Atommic% oxygen, and 44.69 Atomic% gallium.
The observed break-up phenomenon in the liquid gallium thin film can be explained using the electromigration theory. It is known that the electromigration phenomenon cause the failure of a solid circuit. It is also known that the electromigration phenomenon can induce a liquid metal flow in a micro-channel. On considering that the electromagnetic properties of the liquid metal are similar to those in its solid state, we shall now assume that the electromigration phenomenon will also cause the failure of a conductive liquid circuit.
In the electromigration process, the ion cores are subject to the following two opposing forces as shown in Electron wind force from the momentum exchange between the conducting electrons and ion cores, \(\mathbf{F}_{\mathrm{w}}\!=\!Z_{\mathrm{w}}\mathbf{e}\mathbf{E}\), where \(Z_{\mathrm{w}}\) is the effective valence for the wind force, \(\mathbf{e}\) is the elementary charge, and \(\mathbf{E}\) the electric field. Direct electrostatic force from the applied electric field, \(\mathbf{F}_{\mathrm{d}}\!=\!Z_{\mathrm{d}}\mathbf{e}\mathbf{E}\), where \(Z_{\mathrm{d}}\) is the effective valence for the electrostatic force. Therefore, the resultant force exerts on the ion cores is the addition of the above two forces, i.e., \(\mathbf{F}_{\mathrm{em}}\!=\!\mathbf{F}_{\mathrm{w}}+\mathbf{F}_{\mathrm{d}}= \!Z^{*}\mathbf{e}\mathbf{E}\!\!=\!Z^{*}\mathbf{e}\mathbf{j}\), where \(Z^{*}\!\!=\!\!Z_{\mathrm{w}}\!\!+\!Z_{\mathrm{d}}\) is the effective valence for the resultant force, \(\mathbf{j}\) is current density, and \(\rho\) the resistivity.
In addition to the resultant force \(\mathrm{F}_{\mathrm{em}}\), the ion cores in the liquid metal are still subject to other forces. All these will contribute to the ion core flow \(\mathbf{J}\) in the liquid metal.
\[\frac{\partial n}{\partial t}+\nabla\cdot\mathbf{J}=0\,, \tag{1}\]
The schematic diagram of electromigration in which both the electrons and ion cores move to the opposite direction of the applied electric field.
The meanings of various terms in **J** are as follows: \({\bf J}_{em}=\frac{Dn}{kT}Z^{*}e\rho{\bf j}\) is induced by the electric current density **j**, where D=\(\mu\)kT is diffusivity (Einstein relation for liquid), \(\mu\) is mobility, k is Boltzmann's constant, and T is absolute temperature. \({\bf J}_{n}=-D\nabla n\) is induced by the density gradient \(\nabla n\), \({\bf J}_{T}=-\frac{Dn}{kT}\frac{Q}{T}\nabla T\) is induced by the temperature gradient \(\nabla T\) (Q is the heat of thermal diffusion), \({\bf J}_{p}=\frac{Dn}{kT}\Omega\nabla p\) is induced by the pressure gradient \(\nabla p\) (\(\Omega\) is the atomic volume).
Eq. shows that, if the flux density moving into a region becomes larger than the flux density moving out from the region \(\partial n/\partial t>0\), then \(\nabla\cdot{\bf J}<0\) and material may pile up in this region. However, if the flux density moving into the region becomes smaller than the flux density moving out from the region \(\partial n/\partial t<0\), then \(\nabla\cdot{\bf J}>0\) and voids may form in the region. These will cause the circuit failure. Only if \(\nabla\cdot{\bf J}=0\) (or \(\partial n/\partial t=0\) ), then there is no failure occurs.
There is another possibility that the break-up phenomenon in the liquid metal thin film may be caused by a temperature distribution. However, our early measurements have shown that the thin film stays unaffected even as the temperature in the thin film's middle increased up to 375 \({}^{\circ}\)C. This ruled out the possibility of thermal break-up in our liquid metal thin film. In the following discussion, therefore, we shall focus on the effects of the electromigration, but ignore the others, i.e., \({\bf J}_{em}\neq 0\), \({\bf J}_{T}\approx{\bf J}_{p}\approx{\bf J}_{n}\approx 0\).
Another measuring procedure is also used to confirm that the break-up phenomenon in the liquid metal thin film is induced by the electric current **j** (or electron wind force \({\bf F}_{w}\)), but not by the applied electric field \({\bf E}\) (or direct electrostatic force \({\bf F}_{d}\)). More specifically, apply an electric field \({\bf E}\) (or a voltage U) to the sample, but let the circuit open to ensure that no electric current is observed (**j**=0) in the circuit. Our results have shown that no break-up phenomenon was observed even if the voltage was increased to a higher value. This can be well understood by referring to Eq.. If **j**=0, then \(\nabla\cdot{\bf J}_{e}=0\). Consequently, we have \(\partial n/\partial t=0\). Therefore, the ion core density is a constant and no electromigration phenomenon will occur.
The EDS spectrum results in may suggest that the electron wind force \(\mathbf{F}_{w}\) is small in the few layers of atoms absorbed on the substrate. It is known that gallium and its alloys need oxides to wet the substrates such as glass and silicon, the thin layer between the liquid metal and substrate are mainly oxides (Ga\({}_{2}\)O\({}_{3}\)) that are wide bandgap semiconductors. This layer should not be subject to the electron wind force \(\mathbf{F}_{w}\) because there is no transporting electron within the oxides. In the electromigration process, therefore, the oxides should stay behind because there is no electron flow inside it, but the liquid gallium on the top were pushed away by the electron flow while the oxides underneath stay behind.
Due to its fluidity, therefore, a liquid metal thin film is more vulnerable to the electromigration phenomenon than a solid thin film. Thus, the electromigration phenomenon should have a considerable impact on the reliability of the liquid metal film in printed electronics, electronic circuit, 3-D electronic fabrication or related microtechnology etc., which must be handled with caution.
Finally, we would like to mention that, in addition to Eq., the liquid gallium should obey two more modified equations of fluid mechanics:
_Equation of motion (Navier-Stokes equation)_.
\[n\bigg{[}\frac{\partial\mathbf{v}}{\partial t}+\left(\mathbf{v}\cdot\nabla \right)\mathbf{v}\bigg{]}=-\nabla p+\eta\Delta\mathbf{v}+\bigg{(}\frac{1}{3} \eta+\zeta\bigg{)}\nabla\big{(}\nabla\cdot\mathbf{v}\big{)}+nZ^{\ast}e\mathbf{ j}\,, \tag{2}\]
_Energy equation_.
\[\frac{d}{dt}\bigg{(}n\varepsilon+\frac{1}{2}n\mathbf{v}^{2}\bigg{)}=\nabla \cdot\big{[}\big{(}-p\mathbf{I}+\boldsymbol{\tau}\big{)}\cdot\mathbf{v}\big{]} +\nabla\cdot\big{(}\kappa\nabla T\big{)}+\rho j^{2}\,, \tag{3}\]
The term \(\nabla\cdot\big{(}\kappa\nabla T\big{)}\) represents the heat flowed out from the region and \(\kappa\) is the thermal conductivity. The term \(\mathbf{\beta}^{2}=\mathbf{E}\cdot\mathbf{j}\) is the Joule heat released, which includes the energy increased in the electromigration process.
Furthermore, the liquid metal must also obey Maxwell's equations, which are not given here for brief.
A complete characterization on the above equations is beyond the scope of this study. Overall, the present findings also raised quite a few important fundamental issues worth of pursuing in the coming time.
## 5 Conclusions
In summary, we have discovered for the first time that liquid metal is susceptible to electromigration effect. A liquid metal thin film can be broken up by electron wind force when it carries a high current density. These findings open a new direction for the study of the electromigration phenomenon in conductive fluids which had never been reported before. The electromigration phenomenon also adds a barrier for the large scale applications of the liquid metal in electronics. Further studies need to be carried out to resolve the failure problem in liquid metal film caused by the electromigration phenomenon.
| 10.48550/arXiv.1309.0601 | Break-up phenomena of liquid metal thin film induced by high electric current | Rongchao Ma, Cangran Guo, Yixin Zhou, Jing Liu | 3,467 |
10.48550_arXiv.1102.1703 | ## I Introduction
Recent years have seen a flurry of activity stemming from the recognition that the unusual ground-state correlations of dipolar spin ice should support excitations that behave as magnetic charges. These magnetic charges, or deconfined monopoles, are formed through the fractionalization of a single spin-flip (dipole) excitation. By virtue of the underlying correlations of the dipolar spin ice 'vacuum', the monopoles experience a Coulomb force between them and are found to flow under the influence of the magnetic field. Several experimental works and theoretical analyses of previous experimental work have since verified the existence of these monopole excitations. Furthermore, lithographically patterned, artificial spin ice systems have been created and also shown to exhibit monopole defects. In particular, the slow relaxation that is observed in the ac magnetic susceptibility of spin ice has recently been attributed to the freezing out of dynamics of deconfined magnetic monopoles.
Here, we present detailed low temperature ac susceptibility measurements, \(\chi(f)\), on single crystal Ho\({}_{2}\)Ti\({}_{2}\)O\({}_{7}\). These measurements represent an important step in the study of monopole physics of dipolar spin ice for several reasons. (i) Ho\({}_{2}\)Ti\({}_{2}\)O\({}_{7}\) has received less attention, at least from thermodynamic measurements, than Dy\({}_{2}\)Ti\({}_{2}\)O\({}_{7}\). Ho\({}_{2}\)Ti\({}_{2}\)O\({}_{7}\) are susceptibility measurements thus far have been limited to temperature scans of \(\chi\) in higher temperature and frequency regimes. (ii) Our experiments represent the first low temperature ac susceptibility measurements performed on single crystal spin ice in zero static field along the direction. (iii) In previous work on Dy\({}_{2}\)Ti\({}_{2}\)O\({}_{7}\), there has not been a clear accounting of the demagnetization correction, which turns out to be rather important for quantitatively determining the intrinsic time constant of the system as a function of temperature and matching to theoretical predictions. The demagnetization correction also has qualitative effects on the shape of the frequency spectra. (iv) This work extends measurements to much lower frequencies than previous work on any variety of spin ice, therefore delving deep into the Coulomb phase. It is our hope that it will provide an important benchmark by which to test theories of monopole physics in dipolar spin ice.
In the pyrochlore materials Ho\({}_{2}\)Ti\({}_{2}\)O\({}_{7}\) (HTO) and Dy\({}_{2}\)Ti\({}_{2}\)O\({}_{7}\) (DTO), the magnetic ions Ho\({}^{3+}\) and Dy\({}^{3+}\) occupy a lattice of corner sharing tetrahedra. The crystal field acting on those ions creates a strong Ising anisotropy, with the easy axis, known as the local axis, pointing directly in or out of the tetrahedra. For ferromagnetic nearest neighbor interactions, the spins are highly frustrated and possess a macroscopically degenerate set of ground states, consisting of two spins pointing into and two spins pointing out of each tetrahedron. This situation is directly analogous to the proton bonds in water ice, hence the name'spin ice'. In both spin and water ice, the macroscopic degeneracy of ground states gives rise to an extensive residual entropy \(S_{0}=R/2\ln(3/2)\), known as Pauling's entropy. In HTO and DTO, the nearest neighbor (n.n.) ferromagnetic interaction is a result of dipole-dipole interactions, thus these systems are often referred to as dipolar spin ice. Considering the antiferromagnetic n.n. exchange interaction \(J\) and the Ising anisotropy of the spins, the relevant effective n.n.
\[J_{\rm eff}=m^{2}(5D-J)/3, \tag{1}\]
\(m\) is the magnetic moment of the Dy\({}^{3+}\) or Ho\({}^{3+}\) ions and \(D\) is the strength of the n.n. dipolar interaction. The further neighbor dipolar interactions are very well screened in this system. Nonetheless, they are found, theoretically, to select a unique ground state for the system and should, in principle, result in a sharp, first-order phase transition. However, no such transition has been found in experiment, possibly because of the difficulties of reaching equilibrium at temperatures well below 0.5 K.
The unusual disordered ice-like state of dipolar spin ice, with its two-in, two-out tetrahedra, can be thought of as a divergence-free \(\nabla\cdot\vec{H}=0\) vacuum, with fluctuating magnetic field lines. The simplest excitation out of this vacuum is the flipping of one spin. This single spin flip affects two neighboring tetrahedra resulting in one with 3-in, 1-out and the other with 1-in, 3-out. This removes the divergence-free character of the system and leads to two magnetic charges, or monopoles, of opposite sign, centered on those tetrahedra. In the simple nearest neighbor model of spin ice, these monopole excitations are, once created, free to roam about on the lattice without any energy cost. In dipolar spin ice, however, the longer range interactions give rise to a Coulomb interaction, \(\sim 1/r\), between magnetic monopoles.
This has been associated recently with the slow relaxation observed in the ac magnetic susceptibility of spin ice. At temperatures ranging from about 2 K to 6 K, where the monopole density is high but there are very few double defects (4-out or 4-in), DTO ac susceptibility data from Ref.
\[\tau=\tau_{0}\exp(E_{A}/T)=\tau_{0}\exp(2J_{\rm eff}/T), \tag{2}\]
As the temperature is lowered and the concentration of monopoles decreases, the relaxation becomes slower than predicted by Eq. 2. A reasonably good fit of experimental data is obtained by performing Monte Carlo simulations of a gas of monopole charges on the diamond lattice in the grand canonical ensemble with a largely temperature independent chemical potential \(\mu\).
Despite this strong verification of the monopole hypothesis, there remains some discrepancy between theory and experiment at lower temperatures (below \(\sim 1\) K) in that the experiments find slower relaxation than is predicted by theory. It is, thus, important to further explore the low temperature freezing of spin ice systems. Ensuring an accurate correction of the demagnetization effect and taking measurements to lower frequencies, as performed here, will permit careful comparison of experimental results with the theory of monopole defects. In contrast to the predictions of theory, our results show a temperature activated regime where the dynamics are described well by a surprisingly simple Arrhenius law with \(6J_{\rm eff}\) activation energy at low temperatures. This study, performed on Ho\({}_{2}\)Ti\({}_{2}\)O\({}_{7}\), as opposed to the more commonly studied Dy\({}_{2}\)Ti\({}_{2}\)O\({}_{7}\), also permits the observation of monopole physics with a different set of Hamiltonian parameters, namely the spin flip rate, effective n.n. interaction, \(J_{\rm eff}\), and the monopole charge, \(Q\).
## II Experiment
The single crystal Ho\({}_{2}\)Ti\({}_{2}\)O\({}_{7}\) samples studied here were prepared at McMaster University. The samples are from the same crystal growth employed in the neutron scattering studies reported in Ref.. They are grown by a floating-zone image furnace technique, the details of which are provided in Ref.. Two sample geometries were measured, with dimensions \(1.1\times 1.1\times 2.6\) mm\({}^{3}\) and \(0.6\times 0.6\times 3\) mm\({}^{3}\) respectively. In each case, the longest side of the crystal was the crystal direction, also the direction of the applied ac magnetic field.
Susceptibility, \(\chi\), measurements were performed using a magnetometer based on a superconducting quantum interference device (SQUID), mounted on a dilution re
(color online) Frequency scans of \(\chi^{\prime}\) and \(\chi^{\prime\prime}\) of Ho\({}_{2}\)Ti\({}_{2}\)O\({}_{7}\) with the magnetic field aligned along the crystal orientation. The data shown here has been corrected for demagnetization and was taken on a needle shaped sample (sample 1). Slow relaxation is observed, characterized by a suppression of \(\chi^{\prime}\) at higher frequencies and a broad peak in \(\chi^{\prime\prime}\).
The second-order gradiometer sensing coils are contained in the bore of an excitation coil consisting of 375 turns of NbTi wire wound on a phenolic form with which an applied ac magnetic field, not exceeding 20 mOe, is generated. The gradiometer is coupled to the SQUID by means of a superconducting flux transformer. Further refinement of the magnetometer balance is achieved with a trim coil coupled to one branch of the gradiometer, in parallel with the excitation coil. The SQUID is contained within a small lead shield, and the entire gradiometer within another, larger lead shield. Further shielding is provided by a cryogenic \(\mu\)-metal shield surrounding the vacuum can of the cryostat and by a room-temperature \(\mu\)-metal shield surrounding the liquid helium dewar. The SQUID and SQUID controller, with 100 kHz modulation frequency, were obtained from the company _ezSQUID_. A lock-in amplifier provides an ac source, and reads the feedback output from the SQUID controller as the resulting signal.
The samples were mounted on a sapphire rod using General Electric varnish, the long edge aligned with the direction of the applied magnetic field with an estimated accuracy of \(\pm 2\) degrees. The sapphire rod is clamped into the copper base of the sample holder and this sample holder is in turn heat sunk to the mixing chamber of a dilution refrigerator.
Frequency scans were taken by controlling the cryostat at a given temperature in the range of 500 to 1300 mK, and sweeping the frequency in the range of 1 mHz to 500 Hz. It was ensured that, for all measurements presented here, the apparatus and samples were in thermal equilibrium. Reproducibility, and the absence of further thermal relaxation, was verified by taking multiple scans separated by several hours. The lower temperature limit of our data corresponds roughly to the point at which the maximum in \(\chi^{\prime\prime}\) reaches our lowest measurement frequency of 1 mHz. This temperature limit was chosen since much less information will be gained below that point without employing prohibitively low frequencies of measurement. No problems of thermal equilibration were encountered that would have otherwise increased the base temperature of our measurement. The excitation power was also varied by more than an order of magnitude to rule out heating of the sample or apparatus or other nonlinear effects. Temperature scans of \(\chi\) at fixed frequency were also obtained for four different measurement frequencies: 0.1, 1.2, 10 and 40 Hz.
## III Results
Results of ac susceptibility frequency scans, shown in Fig. 1, are qualitatively consistent with previous results taken on the spin ice material Dy\({}_{2}\)Ti\({}_{2}\)O\({}_{7}\). Slow relaxation is observed, characterized by broad absorption spectra \(\chi^{\prime\prime}(f)\) and suppression or blocking of the in phase susceptibility \(\chi^{\prime}\) at higher frequencies. Rapid freezing of the magnetic moments can be seen through the sharply dropping peak frequency, \(f_{\rm Max}\), of the absorption spectrum \(\chi^{\prime\prime}(f)\) as the temperature is lowered. We see that by 500 mK, \(f_{\rm Max}\) is already below our frequency window at less than 1 mHz. At higher temperatures approaching 1.3 K, the peak position is beginning to plateau at around 100 Hz. The magnitude of the susceptibility monotonically increases with decreasing temperature. However, judging by the peak height of \(\chi^{\prime\prime}\), the susceptibility appears to be leveling off at the lowest temperatures studied.
(color online) Superimposed absorption spectra obtained by plotting \(\chi^{\prime\prime}/\chi^{\prime\prime}_{\rm Max}\) against \(f/f_{\rm Max}\). (a) The absorption spectra taken on the less elongated sample (sample 2) of dimensions \(1.1\times 1.1\times 2.6\) mm\({}^{3}\), _before_ correcting for the demagnetization effect. The raw data, plotted in this way, exhibits a narrowing of the absorption spectrum with decreasing temperature. (b) Data fully corrected for the demagnetization effect, taken on a needle-shaped sample (sample 1). The spectra change qualitatively, with a clear broadening of the low-frequency tail as the temperature is reduced. However, the high-frequency tail continues to narrow with reducing temperature. Above 1 K, the low-frequency tail also shows slight narrowing with reducing temperature, although the effect is rather subtle. Arrows indicate increasing temperature.
In order to make comparison of the spectra more amenable, we superimpose the spectra in This is done by plotting data normalized by the peak susceptibility and frequency, so plotting \(\chi^{\prime\prime}/\chi^{\prime\prime}_{\rm Max}\) against \(f/f_{\rm Max}\). Interestingly, if this is done before correcting for demagnetization with the less elongated sample, one finds a subtle narrowing of the absorption spectrum with lower temperatures, as shown in Fig. 2(a). This effect was previously noticed by Snyder _et al._ in measuring the susceptibility of Dy\({}_{2}\)Ti\({}_{2}\)O\({}_{7}\).
Because of a large magnetic moment on the Ho\({}^{3+}\) sites, the susceptibility of Ho\({}_{2}\)Ti\({}_{2}\)O\({}_{7}\) is quite large and the demagnetization effect is crucially important to consider. In fact, without correcting for demagnetization, one can obtain qualitatively different spectra due to mixing of \(\chi^{\prime}\) and \(\chi^{\prime\prime}\). If a demagnetization correction is performed, the narrowing behavior becomes less obvious, as seen in Fig. 2(b). The low-frequency tails of \(\chi^{\prime\prime}(f)\) are found to broaden with decreasing temperature, whereas the high-frequency tails still narrow significantly with decreasing \(T\). Even in the demagnetization corrected data, there is some hint of the low-frequency side of the spectra beginning to broaden very slightly above 1 K. Thus, the relaxation in spin ice is found to be noticeably different from the behavior of a standard glass or spin glass, where the absorption spectra show clear and largely symmetric broadening with reduced temperature.
Quantitatively, the broadening of the absorption spectra on a log scale can be parametrized by the half width at half maximum, either on the high frequency side (HWHM\({}_{+}\)) or low frequency side (HWHM\({}_{-}\)), both of which are plotted against temperature in the inset of HWHM\({}_{-}\) is roughly constant at around 0.9 decades from 1.3 K down to 0.85 K (though showing barely discernible narrowing with lower \(T\)). Below that point, it shows broadening, to almost 1.2 decades by 650 mK. HWHM\({}_{+}\) shows a stronger and opposite temperature dependence, increasing from \(\sim 1\) decade at the lowest temperatures studied here to 1.5 decades at 1.0 K. These numbers can be compared to the 0.7 decades HWHM and symmetric absorption spectrum that results from a single energy barrier to relaxation. The asymmetry of the spectra indicate that the relaxation is not described by a single characteristic Debye form. Simple empirical forms such as the Cole-Cole, Davidson-Cole and Havriliak-Negami functions are also not good fits to the data at low \(T\), unlike what was found above 1.8 K in a previous analysis of the susceptibility of Dy\({}_{2}\)Ti\({}_{2}\)O\({}_{7}\).
At low temperatures, the peak absorption frequency appears to approach an Arrhenius law, \(f_{\rm Max}=f_{0}\exp(-E_{A}/T)\), thus resulting in a straight line when \(\log f_{\rm Max}\) is plotted against \(1/T\), as seen in Fitting the data below 1 K gives a best fit Arrhenius law with activation energy \(E_{A}=10.70\) K and \(f_{0}=5.74\times 10^{5}\) Hz. The resulting reduced chi-square statistic, \(X^{2}\), is 0.12, implying that it is a very good fit to the data. The value of \(E_{A}\) so obtained is tantalizingly close to \(6J_{\rm eff}=10.80\) K for Ho\({}_{2}\)Ti\({}_{2}\)O\({}_{7}\). In fact, an Arrhenius law with exactly \(E_{A}=6J_{\rm eff}\) is also a more than adequate fit with \(f_{0}=6.57\times 10^{5}\) Hz and reduced \(X^{2}=0.18\). The low values of \(X^{2}\) suggest that the fit is somewhat under-constrained. Taking a 95% confidence interval allows for \(E_{A}=10.70\pm 0.15\) K. Above 1 K, the temperature dependence of the frequency \(f_{\rm Max}\) becomes shallower than the Arrhenius-law fit, likely moving towards the quasi-plateau regime that was seen in DTO. At the lowest measured temperatures, an onset of Arrhenius behavior was seen in DTO and in a later work a \(\sim 6J_{\rm eff}\) Arrhenius law was shown to fit the lower end of that same experimental data. However, insufficient low temperature data has been obtained on DTO to be reasonably certain that the Arrhenius behavior develops as is seen here in HTO.
Detailed temperature scans at four different frequencies of measurement, 0.1 Hz, 1.2 Hz, 10 Hz, and 48 Hz, are shown in Once again, the hallmarks of slow relaxation are observed, with suppression of \(\chi^{\prime}\) at low temperatures and a peak in \(\chi^{\prime\prime}(T)\) near the inflection point of \(\chi^{\prime}(T)\). The peak in \(\chi^{\prime\prime}\) shifts to higher temperatures and broadens as the frequency of measurement, \(f\), is increased. The peak position of the temperature scans, or \(T_{\rm Max}(f)\), may also be extracted in order to characterize the relevant time scale as a function of temperature.
(color online) Frequency versus inverse temperature taken from this work and fits with different methods. Green diamonds are obtained from the maxima \(T_{\rm Max}(f)\) of temperature scans of \(\chi^{\prime\prime}\). The corresponding best fit Arrhenius law is shown as the green line. The blue circles are obtained from the maxima of the absorption spectra, \(f_{\rm Max}(T)\). The best fit Arrhenius law, shown in magenta, gives \(E_{A}/k_{B}=10.70\) K, which is very close to \(6J_{\rm eff}\) for Ho\({}_{2}\)Ti\({}_{2}\)O\({}_{7}\). In fact, an Arrhenius law with \(E_{A}/k_{B}=6J_{\rm eff}\) also fits the data to within the uncertainty, as shown in blue. The fits are plotted as solid lines in the range of the fitted data and extended with dashed lines. The \(2J_{\rm eff}\) Arrhenius law that should be expected in the plateau region is shown in red.
It is found that the peak positions of the frequency scans and of the temperature scans lead to different functional forms, so that \(f_{\rm Max}(T)\neq f(T_{\rm Max})\). This is most easily demonstrated with surface plots of \(\chi^{\prime}(f,T)\) and \(\chi^{\prime\prime}(f,T)\), shown in The maxima in \(\chi^{\prime\prime}\) taken along the frequency axis and along the temperature axis are shown in unfilled and filled circles respectively and do not sit on the same curve. While there is a very significant difference between these parametrizations at higher temperature, the two curves appear to be tending toward a single function at lower temperatures. In previous work,\(f_{\rm Max}(T)\) was used to compare with theory. Ideally one would like to have theoretical calculations that can reproduce the entire \(\chi(f,T)\) surfaces that are shown in Figure 5, but for now, \(f_{\rm Max}(T)\) will serve as a good representative parameterization of our data.
The demagnetization _correction_ shifts the peak frequency significantly lower, especially at lower temperatures where the static susceptibility becomes quite large. In other words, the demagnetization _effect_, increases the measured relaxation times relative to the relaxation times that would be measured from the true, bulk susceptibility. This is a result of \(\chi^{\prime}\) feeding into the measurement of \(\chi^{\prime\prime}\). This reduces the apparent absorption spectrum \(\chi^{\prime\prime}_{A}\) at low frequencies where \(\chi^{\prime}\) is large, thereby pushing the peak frequency \(f_{\rm Max}\) to higher frequency. For more details on the demagnetization effect and correction, see the Appendix.
## IV Comparison with previous work
Multiple experimental techniques have been used to study the relaxation of the three well established spin ice materials: Dy\({}_{2}\)Ti\({}_{2}\)O\({}_{7}\) (DTO), Ho\({}_{2}\)Ti\({}_{2}\)O\({}_{7}\) (HTO) and Ho\({}_{2}\)Sn\({}_{2}\)O\({}_{7}\) (HSO). In ac susceptibility, neutron spin echo, neutron scattering, and \(\mu\)SR relaxation, the following general features are observed.
Above \(\sim 15\) K, one observes a high temperature regime where the transition rate of the individual spins is changing rapidly in \(T\). The energy scale for this regime appears to be set by the energy gap to the next excited crystal field energy level at \(\sim 210\) K for DTO and \(\sim 290\) K for HTO. The Ho\({}^{3+}\) (or Dy\({}^{3+}\)) moments have a truly Ising doublet ground state with matrix elements of the local \(J_{x}\) and \(J_{y}\) angular moment operators identically zero. Transitions within this doublet are thus forbidden transitions. Spin flips must therefore occur through either quantum tunneling (a slow process) or by passing through the excited crystal field energy levels (a much faster process). Relaxation in the high temperature regime can thus be described by an Arrhenius law with an activation energy of \(\sim 300\) K. As the temperature is lowered it becomes difficult to populate the excited crystal field states reducing the transition rate.
Below \(\sim 15\) K, the spins are very Ising-like and the materials enter a quasi-plateau regime where the tunneling rate levels off. The vast majority of spin flips are now occurring via tunneling processes. Note however, the tunneling is also accompanied via a change in energy resulting from monopole excitations, thus there remains a temperature dependence to the relaxation. This plateau should correspond to the high temperature tail of a \(2J_{\rm eff}\) Arrhenius behavior that results from the thermal excitation of independent (deconfined) monopole defects. The monopole density in the plateau regime is high enough that there is significant screening of the magnetic charges to make the nearest neighbor spin ice model largely applicable.
Approaching \(J_{\rm eff}\) (around 1.5 to 2 K), the system enters a regime exhibiting rapid freezing out of dynamics. Here the time scales begin to increase dramatically as the system condenses into a disordered 2-in-2-out configuration. Equivalently, it is now becoming thermally difficult to form monopole defects. The monopole density decreases which in turn reduces screening of the magnetic charges, making full treatment of the dipolar spin ice model necessary to describe the dynamics.
While different techniques and the different spin ice materials give quantitatively different time scales, they all agree on the qualitative behavior and on three clear regimes of relaxation.
(color online) Temperature scans of \(\chi^{\prime}\) and \(\chi^{\prime\prime}\) of Ho\({}_{2}\)Ti\({}_{2}\)O\({}_{7}\) with the probe field aligned along, for four different measurement frequencies: 0.1 Hz, 1.2 Hz, 10 Hz and 48 Hz. This data has been taken on a needle-shaped sample (sample 1) and has been corrected for the demagnetization effect. Again, slow relaxation is observed, with a maxima in \(\chi^{\prime}\) or \(\chi^{\prime\prime}\) moving to lower temperatures with lower frequency of measurement.
Despite this significant difference in relaxation times, the same qualitative shape of \(f(T)\) described above is observed. Neutron scattering measurements by Zhou _et al._ also found a quantum tunnelling regime on the time scale of \(\sim 0.01\) ns in the dynamic spin ice material Pr\({}_{2}\)Sn\({}_{2}\)O\({}_{7}\). Ehlers _et al._ have found, using neutron spin echo experiments, the same three regimes of relaxation. Susceptibility measurements of DTO have found the time scale of the plateau to be on the order of \(\sim 1\) ms, while \(\mu\)SR measurements of DTO have found the above qualitative shape, but with the time scale of the plateau relaxation on the order of 0.1 \(\mu\)s. This mismatch of timescales in the dynamics as determined by different measurement techniques remains an open question in the study of spin ice. The answer may lie in the wavevectors that are accessed by these various measurements; while neutron scattering accesses specific wavevectors, bulk susceptibility addresses \(k=0\) and \(\mu\)SR is a local measurement, thus has contributions from all wavevectors.
Interestingly, measurements of DTO by Orendac _et al._ using the magnetocaloric effect have accessed a frequency and temperature range lower than that of any ac susceptibility measurements. Those measurements also seem to access a different time scale, showing both a higher rate of relaxation and a shallower temperature dependence of the freezing than what is observed in ac susceptibility. Henceforth, we concentrate on comparing our data with that of other bulk susceptibility measurements of spin ice.
Even between ac susceptibility measurements, a direct comparison of data can be difficult to achieve. For many of the measurements published on spin ice, a correction of the demagnetization effect has not been performed, or is not incorporated into the presented data. Many of the previous results were obtained on powder samples, with which it is difficult to compare aligned single crystal measurements. Of course different spin ice materials have different parameters in their respective Hamiltonians, which makes their quantitative behavior different.
Finally, comparison is further hindered by the complex dynamical behavior of spin ice and the various measurement schemes and parametrizations that can be used. One of the most obvious ways to parametrize the data is with the maximum of the absorption spectra at fixed temperature, or \(f_{\rm Max}(T)\). This parameter has been employed in several experimental works and in theory. Alternatively, temperature scans at constant frequency may be employed and two maxima in \(\chi^{\prime\prime}\) can be resolved at \(T_{\rm Low}(f)\) and \(T_{\rm High}(f)\). The higher temperature maximum, \(T_{\rm High}\), appears to be present only for frequencies above a certain threshold (near 200 Hz for DTO). While \(T_{\rm High}\) has not been observed in HTO in zero field due to an unsuitable range of temperature and frequency, it has been observed, by Ehlers _et al._ under an applied field of 1 T. In temperatures scans, one may also extract maxima in \(\chi^{\prime}\). In Figure 6, we show a compilation of several of the previous results on DTO, HTO and HSO compared with our results on HTO.
While qualitatively similar behavior is observed in ac susceptibility for all the materials studied, quantitative differences remain. There is very little difference in the data below 1 K between our results on HTO and results on DTO from Refs., despite the differences in Hamiltonian, specifically the effective n. n. exchange energy scale \(J_{\rm eff}\), and in demagnetization effects. We can draw from the physics of monopole excitations discussed in Refs. to suggest what relations between the different relaxation rates should be expected.
First, the tunneling rates will be different in the two materials studied. Tunneling rates for Ising moments like those in spin ice are functions of several different parameters. The symmetry and strength of the crystal field, hence the energy of excited states above the Ising doublet, play an important role, giving the energy barrier through which the moments must tunnel. Components of the exchange and dipolar interactions transverse to the local Ising direction result in mixing with the excited crystal field energy levels, permitting tunneling to occur. Furthermore, nuclear hyperfine interaction strengths, very different between Dy\({}^{3+}\) and Ho\({}^{3+}\), can play a role in assisting or blocking spin flips.
(color online) Surface plot of \(\chi^{\prime}\) (top) and \(\chi^{\prime\prime}\) (bottom) with temperature \(T\) on the \(y\)-axis and measurement frequency \(f\) on the \(x\)-axis. Also shown, are the positions of the maxima in \(\chi^{\prime\prime}\) either taken from temperature scans (black, filled circles) or frequency scans (red, open circles).
However, spin ice has so far been treated with a single spin flip rate \(f_{0}\). In the regime where tunneling is dominant (from temperatures in the quasi-plateau regime and below), increasing the spin flip rate would be equivalent to stretching the frequency axis. It is not clear at this time what the difference in tunneling rates is between the various spin ice materials, though our results suggest that they are at least in the same order of magnitude.
Within the nearest-neighbor spin ice (NNSI) model, the only other parameter governing the dynamics of the system would be \(J_{\rm eff}\), or the energy barrier that must be overcome in order to excite monopole defects. Changing \(J_{\rm eff}\) would stretch the temperature axis, either elongating or shortening the plateau regime. In Dy\({}_{2}\)Ti\({}_{2}\)O\({}_{7}\), \(D_{mn}=2.35\) K and \(J_{nn}=-1.24\) K giving \(J_{\rm eff}=1.11\) K. In Ho\({}_{2}\)Ti\({}_{2}\)O\({}_{7}\), the dipolar interaction is essentially the same, whereas \(J_{nn}=-0.52\) K, giving a much higher value of \(J_{\rm eff}=1.83\) K. Because of the higher \(J_{\rm eff}\) in HTO, it enters the freezing, 2-in-2-out regime, at higher temperatures than DTO.
When one goes beyond the NNSI model into the dipolar spin ice model, the monopole defects begin to interact with a Coulomb law, \(V(r_{ij})=(\mu_{0}/4\pi)Q_{i}Q_{j}/r_{ij}\), as long as they occupy distant sites on the diamond lattice. Thus a third parameter, the magnetic charge \(Q\), starts to play an important role at lower temperatures as the monopole density is reduced and there is less screening of the magnetic charges.
\[Q=\pm\frac{2\mu}{a_{d}}=\pm\left(\frac{32\pi D_{nn}a}{3\mu_{0}}\right)^{1/2} \tag{3}\]
\(Q\) is not directly related to \(J_{\rm eff}\), but only the strength of the dipolar interaction \(D_{nn}\). If the NNSI model were to hold, a plot of \(f_{\rm Max}\tau_{0}\) versus \(T/J_{\rm eff}\) would in principle give the same result for all spin ice samples, from the quasi-plateau regime and below. Introduction of this third parameter, the monopole charge \(Q\), should make such a simple scaling of relaxation curves impossible.
Surprisingly, despite all the differences between samples and measurements, our frequency scan maxima \(f_{\rm Max}\) are quite close to those of Snyder _et al._ on a polycrystalline sample of DTO. While overlapping temperatures are not available, our results do seem to be approaching those of Matsuhira _et al._, also on polycrystalline DTO, at higher temperatures.
We may more directly compare our temperature scans with the results of Ref. where HTO has also been studied and a demagnetization correction has been applied. Shown in the inset of Figure 6, are our results at \(f=10\) Hz on a single crystal oriented along and those of Matsuhira _et al._ at the same frequency on a powder sample of HTO. There is a clear difference between the curves, suggesting that the magnetic field orientation is an important parameter. The relaxation times determined from \(T_{\rm Low}(f)\) are found to have a steeper temperature dependence for the polycrystalline sample than for the-oriented sample. This would seem to suggest that energy barriers to the movement of monopole defects may be lower along the and symmetry related directions (in other words, along the so-called\(\alpha\) or \(\beta\) chains) than in other crystal directions. In a perfect gas of interacting charges all directions are equivalent, but the magnetic monopoles in spin ice are confined to travel only in certain paths which could give rise to anisotropic behavior.
Shi _et al._ have studied the relaxation of two DTO single crystals, one oriented along the direction and the other oriented along the direction. They note that there is a difference between \(T_{f}(f)\) (the maximum in \(\chi^{\prime}(T)\)) between these two scenarios. However, it is not clear that their data has been corrected for demagnetization. The two samples have different geometries and the different orientations will give rise to different magnitudes of susceptibility since the magnetic field will be differently aligned with the different basis spins. This change in magnitude will heavily influence the apparent susceptibility \(\chi_{A}(f,T)\).
## V Conclusions
To conclude, we have performed a careful study of the frequency dependent spin freezing in a single crystal of the dipolar spin ice material Ho\({}_{2}\)Ti\({}_{2}\)O\({}_{7}\) at low temperatures. The results are qualitatively consistent with research by other groups on Ho\({}_{2}\)Ti\({}_{2}\)O\({}_{7}\), on the related materials Dy\({}_{2}\)Ti\({}_{2}\)O\({}_{7}\) and Ho\({}_{2}\)Sn\({}_{2}\)O\({}_{7}\) and also with theoretical work that relates such dynamics to the motion of magnetic monopole excitations. The results presented here represent an exploration much deeper into the Coulomb phase of spin ice, with much lower temperatures and frequencies of measurement than were employed for ac susceptibility measurements previously.
One of the most striking aspects of our results is the observation of a seemingly temperature activated regime below 1 K where the dynamics are well fit by an Arrhenius law, consistent with a \(6J_{\rm eff}\) activation energy. It has also been noted that an Arrhenius law with an activation energy close to \(6J_{\rm eff}\) is consistent with the low temperature limit of measurements on Dy\({}_{2}\)Ti\({}_{2}\)O\({}_{7}\). While very few points contribute to that conclusion for DTO, it nonetheless supports our observation of such an effect in HTO. A more thorough investigation of the low temperature behavior of DTO with full demagnetization correction would be valuable to verify or disprove the existence of the simple law that we have observed.
It is not clear what could give rise to such a simple temperature activated regime in dipolar spin ice, which seems at odds with the current theoretical treatment of magnetic monopoles in dipolar spin ice. Such theory instead predicts ever increasing activation energies as monopole density and thus screening is reduced. Furthermore, a one-parameter (\(J_{\rm eff}\)) fit to this low temperature regime surprisingly suggests that the monopole charge \(Q\) might be an irrelevant parameter. There remains the possibility that the accessible temperature range is simply narrow enough that a more complex \(f(1/T)\) function only appears to follow a linear behavior. The Arrhenius law holds well over almost 4 orders of magnitude in frequency, though applies only over a factor of 2 in temperature.
Our measurements have also illustrated the complex nature of the spectra. While qualitatively glassy relaxation is seen, the precise behavior of the susceptibility spectra is quite different from that of spin glasses. In particular, as the temperature is lowered, the high frequency tail is found to narrow appreciably where the low frequency tail broadens. Clearly illustrated here is the extreme importance of the demagnetization correction for such materials and its impact on the shape of the absorption spectra, for example.
When compared to experiments on Dy\({}_{2}\)Ti\({}_{2}\)O\({}_{7}\), theory currently shows a persistent mismatch at lower temperatures, with the experiments showing longer relaxation times than would be expected theoretically. While those experiments have not taken into account the demagnetization effect, this will not help explain the discrepancy and should in fact make it worse as correcting for demagnetization will lower the peak frequencies.
In order to resolve such discrepancies, it has been suggested that higher order terms, such as next nearest neighbor exchange, \(J_{2}\), may need to be added to the currently accepted Hamiltonian of the dipolar spin ices. Such interactions were already found to be quite important for matching theory to experiment in DTO. It would also be useful to consider the possibility that the low temperature susceptibility of Ho\({}_{2}\)Ti\({}_{2}\)O\({}_{7}\) is affected by bound pairs of monopoles. Such very slowly relaxing pairs of monopoles have recently been predicted to occur after thermally quenching spin ice. It is not clear, however, whether experimentally inaccessible cooling rates are required to form these pairs or whether they are practically unavoidable in the real systems.
It is hoped that the very exciting theory of monopole excitations in spin ice can begin to explain some of the peculiar effects that are seen in these susceptibility measurements. Equivalently, these results, having been performed with careful demagnetization correction on single crystal samples and to comparatively low frequencies and temperatures, should serve as an important benchmark for testing the theory of monopole physics in spin ice.
| 10.48550/arXiv.1102.1703 | Dynamics of the Magnetic Susceptibility Deep in the Coulomb Phase of the Dipolar Spin Ice Material Ho2Ti2O7 | J. A. Quilliam, L. R. Yaraskavitch, H. A. Dabkowska, B. D. Gaulin, J. B. Kycia | 5,272 |
10.48550_arXiv.1409.7869 | 10.48550/arXiv.1409.7869 | Universal Ratio of Intrinsic Resistivities of Spin Helix in B20 (Fe-Co)Si Magnets | S. X. Huang, Jian Kang, Fei Chen, Jiadong Zang, G. J. Shu, F. C. Chou, S. V. Grigoriev, V. A. Dyadkin, C. L. Chien | 3,841 |
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10.48550_arXiv.1505.00033 | ###### Abstract
Multiferroics permit the magnetic control of the electric polarization and electric control of the magnetization. These static magnetoelectric (ME) effects are of enormous interest: The ability to read and write a magnetic state current-free by an electric voltage would provide a huge technological advantage. Dynamic or optical ME effects are equally interesting because they give rise to unidirectional light propagation as recently observed in low-temperature multiferroics. This phenomenon, if realized at room temperature, would allow the development of optical diodes which transmit unpolarized light in one, but not in the opposite direction. Here, we report strong unidirectional transmission in the room-temperature multiferroic BiFeO\({}_{3}\) over the gigahertz-terahertz frequency range. Supporting theory attributes the observed unidirectional transmission to the spin-current driven dynamic ME effect. These findings are an important step toward the realization of optical diodes, supplemented by the ability to switch the transmission direction with a magnetic or electric field.
BiFeO\({}_{3}\) is by the far most studied compound among multiferroic and magnetoelectric materials. While experimental studies have already reported about the first realizations of the ME memory function using BiFeO\({}_{3}\) based devices, the origin of the ME effect is still under debate due to the complexity of the material. Because of the low symmetry of iron sites and iron-iron bonds, the magnetic ordering can induce local polarization via each of the three canonical terms - the spin-current, exchange-striction and single-ion mechanisms. While the spin-current term has been identified as the leading contribution to the magnetically induced ferroelectric polarization in various studies, the spin-driven atomic displacements and the electrically induced shift of the spin-wave (magnon) resonances were interpreted based on the exchange-striction and single-ion mechanisms, respectively.
In the magnetically ordered phase below \(T_{N}\)=640 K, BiFeO\({}_{3}\) possesses an exceptionally large spin-driven polarization, if not the largest among all known multiferroic materials. Nevertheless, its systematic study has long been hindered by the huge lattice ferroelectric polarization (\(\mathbf{P}_{0}\)) developing along one of the cubic \(\langle 111\rangle\) directions at the Curie temperature \(T_{C}\)=1100 K and by the lack of single-domain ferroelectric crystals. Owing to the coupling between \(\mathbf{P}_{0}\) and the spin-driven polarization, in zero magnetic field they both point along the same axis. A recent systematic study of the static ME effect revealed additional spin-driven polarization orthogonal to.
The optical ME effect of the magnon modes in multiferroics, which gives rise to the unidirectional transmission in the gigahertz-terahertz frequency range, has recently become a hot topic in materials science.
\[\Delta\alpha_{k}(\omega)=\alpha_{+k}(\omega)-\alpha_{-k}(\omega) \approx\frac{2\omega}{c}\Im\{\chi^{m\varepsilon}_{\gamma\delta}(\omega)-\chi^ {em}_{\delta\gamma}(\omega)\}. \tag{1}\]
The dynamic ME susceptibility tensors \(\hat{\chi}^{me}(\omega)\) and \(\hat{\chi}^{em}(\omega)\) respectively describe the magnetization generated by the oscillating electric field of light, \(\Delta M^{\omega}_{\gamma}=(\varepsilon_{0}/\mu_{0})^{1/2}\chi^{me}_{\gamma \delta}(\omega)E^{\omega}_{\delta}\), and the electric polarization induced by its oscillating magnetic field, \(\Delta P^{\omega}_{\delta}=(\varepsilon_{0}\mu_{0})^{1/2}\chi^{em}_{\delta \gamma}(\omega)H^{\omega}_{\gamma}\). Here \(\varepsilon_{0}\) and \(\mu_{0}\) are the vacuum permittivity and permeability, respectively, while \(\gamma\) and \(\delta\) stand for the Cartesian coordinates. Since the two cross-coupling tensors are connected by the time-reversal operation [\(\ldots\)]\({}^{\prime}\) according to \([\chi^{me}_{\gamma\delta}(\omega)]^{\prime}=-\chi^{em}_{\delta\gamma}(\omega)\), the directional dichroism becomes \(\Delta\alpha_{k}(\omega)=\frac{2\omega}{c}\Im\{\chi^{me}_{\gamma\delta}( \omega)-[\chi^{me}_{\gamma\delta}(\omega)]^{\prime}\}\). In other words, the directional dichroism emerges for simultaneously electric- and magnetic-dipole active excitations and its magnitude is determined by the time-reversal odd parts of the off-diagonal \(\chi^{me}_{\gamma\delta}(\omega)\) tensor elements.
In the cycloidal spin state of BiFeO\({}_{3}\), several low-frequency collective modes have been observed by spectroscopic methods including light absorption and Raman spectroscopy. Though the electric-field-induced shift of the resonance frequencies observed in the Raman study indicates the ME nature of these magnon modes, the optical ME effect has not been investigated in BiFeO\({}_{3}\). Here, we performed absorption measurements in the gigahertz-terahertz spectral range on single-domain ferroelectric BiFeO\({}_{3}\) crystals with \(\mathbf{P}_{0}\) along between room temperature and \(T\)=4 K in magnetic fields up to \(\mu_{0}H\)=17 T. We found that some ofthe magnon modes exhibit strong unidirectional transmission. We identified the minimal set of spin-driven-polarization terms and quantitatively reproduced both the spectral shape and the field dependence of the directional dichroism solely by the spin-current mechanism.
The experimental configurations are schematically illustrated in Absorption spectra were obtained for light beams propagating along with two orthogonal linear polarizations, \(\mathbf{E}^{\omega}\)\(\parallel\)[1\(\overline{1}\)0] and \(\mathbf{E}^{\omega}\)\(\parallel\). Static magnetic fields (\(\pm H\)) were applied perpendicular to the light propagation direction along either or [1\(\overline{1}\)0].
In simple magnets, such as ferromagnets, the sign change of the magnetization corresponds to the time reversal operation. Thus, it is equivalent to the reversal of the light propagation direction. Owing to experimental limitations, in such cases, the absorption change upon the magnetic field induced reversal of the magnetization, \(\Delta\alpha_{H}\)=\(\alpha_{+H,+k}\)-\(\alpha_{-H,+k}\), is typically detected instead of the absorption change associated with the reversal of the light propagation direction, \(\Delta\alpha_{k}\)=\(\alpha_{+H,+k}\)-\(\alpha_{+H,-k}\). Though the relation \(\Delta\alpha_{k}=\Delta\alpha_{H}\) does not necessarily hold for complex spin structures, such as BiFeO\({}_{3}\), \(\Delta\alpha_{k}\) and \(\Delta\alpha_{H}\) spectra obtained from our calculations are equal within 1-2 % for the experimental configurations studied here.
for two orientations of the magnetic field and two light polarizations. The absorption coefficient at several magnon resonances depends on the sign of the magnetic field.
\(\mid\)**Experimental configurations used to detect unidirectional transmission in BiFeO\({}_{3}\).****a,** Pseudocubic unit cell of BiFeO\({}_{3}\) showing the positions of Bi, Fe and O ions. The lattice ferroelectric polarization, \(\mathbf{P}_{0}\)\(\parallel\), is schematically indicated on the Fe site. **b,** Illustration of the three equivalent directions of the cycloidal ordering vector \(\mathbf{q}_{i}\) on the Fe sublattice. The frame of reference is common to all panels. **c,** In magnetic fields (\(\pm H\)) applied along [1\(\overline{1}\)0], cycloidal domains with \(\mathbf{q}_{2}\) and \(\mathbf{q}_{3}\) are equally favoured, while the domain with \(\mathbf{q}_{1}\) is suppressed\({}^{28,29}\). **d,** In magnetic fields (\(\pm H\)) applied along, only the cycloidal domain with \(\mathbf{q}_{1}\) is stable\({}^{28,29}\). **e & f,** The propagation direction (**k**) and the two orthogonal polarizations of light beams traveling in the material.
\(\mid\)**Optical diode function in multiferroics.** Ferro-type ordering of the local electric dipoles (red arrows) and magnetic moments (green arrows) produces a ferroelectric polarization \(P\) and a spontaneous magnetization \(M\), respectively. Light interacts with both ferroic order parameters, hence, upon illumination \(P\) and \(M\) oscillate coherently with the electromagnetic field around their equilibria. The light induced polarization has contributions from both the usual dielectric permittivity and the optical ME effect \(\chi^{em}(\omega)\). While the first contribution is independent of the light propagation direction, the polarization induced via the optical ME effect has opposite sign for counter-propagating light beams. This can give rise to either a constructive or a destructive interference between the two terms. Similarly, the magnetization dynamics is governed by the interference between the magnetization induced via the magnetic permeability and the optical ME effect \(\chi^{me}(\omega)\). Consequently, the transmitted intensity depends on the propagation direction (intense and pale yellow beams) even for unpolarized light and can be exploited to produce optical diodes transmitting light in one, but not in the opposite direction. The transmitting direction can be reversed by switching the sign of either \(P\) via an electric voltage (\(V\)) or \(M\) by an external magnetic field (\(H\)).
With increasing magnetic field this resonance becomes almost transparent for \(+H\), while its absorption increases for \(-H\). We also measured the absorption spectra with both light polarizations for \(\mathbf{H}\|\) and could not detect any difference between \(\pm H\).
In order to reproduce the observed directional dichroism on a microscopic basis, we adopt the spin model of Refs., which successfully describes the magnetic field dependence of the magnon resonances (see the Methods section). Similarly to the static ME effect, all the three basic mechanisms--the spin-current, exchange-striction and single-ion mechanism--can in principle contribute to the optical ME effect. By including all symmetry-allowed spin-driven polarization terms, we calculated the optical ME susceptibilities \(\hat{\chi}^{me}(\omega)\) and \(\hat{\chi}^{em}(\omega)\), the dielectric permittivity \(\hat{\varepsilon}(\omega)\) and the magnetic permeability \(\hat{\mu}(\omega)\). Next, we numerically solved the Maxwell equations by including these response functions in the constitutive relations and calculated the transmission of linearly polarized incoming beams for both backward and forward propagation. The same calculation was done for both field directions, \(\pm H\). As already mentioned, we found \(\Delta\alpha_{k}\approx\Delta\alpha_{H}\) irrespective of the magnitude of \(H\).
To identify the spin-driven polarization terms relevant to the optical ME effect, we performed a systematic fitting of the measured \(\Delta\alpha_{H}(\omega)\) by treating the magnitude of the different terms as free parameters.
\(|\)**Absorption spectra of BiFeO\({}_{3}\) in the range of magnon resonances.****a-d** Magnetic field dependent part of the absorption spectra measured at \(\mathrm{T}=2.5\,\mathrm{K}\) in four different configurations, i.e. for the two orientations of the magnetic field (\(\mathbf{H}\)) and the two orthogonal polarizations schematically shown in The light propagation direction is common to all experimental configurations, \(\mathbf{k}\|\). Absorption spectra measured in different magnetic fields are shifted vertically in proportion to the magnitude of the field, and spectra recorded in \(+H\) and \(-H\) are plotted with red and black lines, respectively. Spectra shown in panel **a** & **c** represent absorption form the \(\mathbf{q}_{i}\) cycloidal domain stabilized by \(\mathbf{H}\|[1\bar{1}0]\), while spectra in panel **b** & **d** have contributions from \(\mathbf{q}_{2}\) and \(\mathbf{q}_{3}\) domains favoured by \(\mathbf{H}\|\).
by the following two types of spin-current terms
\[P_{\alpha}^{SC}=\frac{1}{N}\sum_{\langle i,j\rangle}\{\lambda_{\alpha}^{}[{\bf e }_{i,j}\times({\bf S}_{i}\times{\bf S}_{j})]_{\alpha}+(-1)^{n_{i}}\lambda_{ \alpha}^{}[{\bf S}_{i}\times{\bf S}_{j}]_{\alpha}\} \tag{2}\]
The dynamic ME effect generated by the spin-current terms is described by the coupling constants \(\lambda_{\alpha}^{}\) and \(\lambda_{\alpha}^{}\), where \(\alpha\)=\(x^{\prime},y^{\prime},z^{\prime}\) stands for the three coordinates along the axes \({\bf x}^{\prime}\|{\bf q}_{i}\), \({\bf y}^{\prime}\|({\bf P}_{0}\times{\bf q}_{i})\) and \({\bf z}^{\prime}\|{\bf P}_{0}\) (see Fig. 2b).
The best fit was obtained with three independent parameters: \(\lambda_{x^{\prime}}^{}\)=0, \(\lambda_{y^{\prime}}^{}\)=\(-2\lambda_{z^{\prime}}^{}\)\(\approx\)57.0\(\pm\)3.1 nC/cm\({}^{2}\), \(\lambda_{x^{\prime}}^{}\)=\(\lambda_{y^{\prime}}^{}\)\(\approx\)34.5\(\pm\)2.4 nC/cm\({}^{2}\), \(\lambda_{z^{\prime}}^{}\)\(\approx\)11.8\(\pm\)2.9 nC/cm\({}^{2}\). The population of the two cycloidal domains with \({\bf q}_{2}\) and \({\bf q}_{3}\) propagation vectors was kept equal. We note that this limited set of parameters provides only a semi-quantitative description of the mean absorption spectra, \(\overline{\alpha}(\omega)\)=\(\alpha_{+H,+k}(\omega)\)+\(\alpha_{-H,+k}(\omega)\).
We found that additional terms did not further improve the quality of the fit. Hence, the optical ME effect in BiFeO\({}_{3}\) is dominated by two types of spin-current polarizations, while the exchange-striction and single-ion polarization terms do not significantly contribute to it. This stems from the general nature of the spin dynamics in BiFeO\({}_{3}\). Due to the very weak on-site anisotropy acting on the \(S=5/2\) iron spins, each magnon mode corresponds to pure precessions of the spins, where the oscillating component of the spin on site \(i\), \(\delta{\bf S}_{i}^{\omega}\), is perpendicular to its equilibrium direction, \({\bf S}_{i}^{0}\). This is in contrast to the spin stretching modes observed in highly anisotropic magnets. Since neighbouring spins are nearly collinear in the cycloidal state with extremely long (62 nm) pitch, a dynamic polarization is efficiently induced via spin-current terms such as \(\delta{\bf P}_{i}^{\omega}\propto{\bf S}_{i}^{0}\times\delta{\bf S}_{i+1}^{ \omega}\). In contrast, the dynamic polarization generated by exchange-striction terms such as \(\delta{\bf P}_{i}^{\omega}\propto{\bf S}_{i}^{0}\cdot\delta{\bf S}_{i+1}^{ \omega}\) is nearly zero.
Despite its success in quantitatively describing the directional dichroism spectra observed for \({\bf H}\|[1\overline{1}0]\), our model may not be complete. When light propagates along, we predict that directional dichroism should be absent for a magnetic field along [\(\eta\eta\kappa\)]. While this is in agreement with \(\Delta\alpha_{H}\)=0 found for \({\bf H}\|\), it cannot account for the finite directional dichroism discerned in Figs. 3c and 3d for \({\bf H}\|\). This discrepancy may come from additional anisotropy terms, neglected in the microscopic spin Hamiltonian adopted from Refs., which further reduce the symmetry of the magnetic state and allow the weak directional dichroism observed for \({\bf H}\|\).
Finally, we turn to the temperature dependence of the directional dichroism presented in for \({\bf H}\|{\bf E}^{\omega}\|[1\overline{1}0]\). With increasing temperature the magnon modes soften and both the mean absorption and the directional dichroism are reduced. Nevertheless, the modes \(\Psi_{1}^{}\) and \(\Phi_{2}^{}\) still exhibit considerable directional dichroism, \(\Delta\alpha_{H}\)\(\approx\)5 cm\({}^{-1}\) at room temperature. At low temperatures, almost perfect unidirectional transmission was observed for the lowest-energy mode \(\Psi_{0}\) with orthogonal light polarization (\({\bf E}^{\omega}\|\)). Though we expect the same at room temperature, \(\Psi_{0}\) is out of our limited spectral window at high temperatures.
The emergence of strong optical ME effect and the corresponding unidirectional transmission require the simultaneous breaking of the space- and time-inversion symmetries by the coexistence of ferroelectricity and (anti)ferromagnetism. While these optical ME phenomena have been investigated recently in various materials hosting multiferroicity at low temperatures, here we studied the unidirectional transmission in the spin excitation spectrum of BiFeO\({}_{3}\), the unique multiferroic compound offering a real potential for room temperature applications up to date.
**Directional dichroism spectra of BiFeO\({}_{3}\) in the range of magnon resonances.****a & b,** Magnetic field dependence of the directional dichroism spectra measured as \(\Delta\alpha_{H}(\omega)\) at \(T\)=2.5 K with the two orthogonal polarizations \({\bf E}^{\omega}\|[1\overline{1}0]\) and \({\bf E}^{\omega}\|\), respectively. Spectra obtained in different magnetic fields are shifted vertically in proportion to the magnitude of the field, which was applied along [1\overline{1}0]\). The field values (common to each panel) are indicated with labels on the top of the spectra in panel **a.****c & d,** Directional dichroism spectra predicted by our model for the case of panels **a & b**, respectively. The calculated mode frequencies are indicated by dashed lines. For the assignment of the different modes see Refs..
Based on the current progress achieved in the electric control of the magnetization in BiFeO\({}_{3}\), we expect that the magnetic switching of the transmission direction, demonstrated here, can be complemented by the electric control of the optical ME effect. Because these functionalities exist at room temperature, they can pave the way for the development of optical diodes with electric and/or magnetic control.
## Methods
## Absorption measurements in the terahertz frequency range.
The terahertz spectroscopy system consists of a Martin-Puplett type interferometer with a Si bolometer operating at 300 mK and a mercury lamp. At high temperatures the spectral window of the measurement was limited by the strong radiation load on the detector. The light is directed to the sample using light pipes. The sample is located in the He exchange gas filled sample chamber, which is placed into the cold bore of a 17 T superconducting solenoid.
The measurement sequence was started by applying high magnetic fields (\(\geq\)12 T) at 4 K for tens of minutes. For \(\mathbf{H}\parallel\) and \([1\overline{1}0]\), this procedure respectively populates a single magnetic domain with \(\mathbf{q}_{1}\) and two domains with \(\mathbf{q}_{2}\) and \(\mathbf{q}_{3}\). Next, spectra were measured in different \(\pm H\) fields. We did not see any change in the magnetic domain population when the \(-17\) T field was applied after +17 T.
The zero field absorption spectrum was subtracted from the spectra measured in finite fields. This procedure cancels out diffraction and interference effects caused by the sample. The differential absorption coefficient \(\alpha(H)-\alpha=-\ln[I(H)/I]d^{-1}\), where \(I\) and \(I(H)\) are light intensity spectra in zero and \(H\) field. The lower envelope of the whole set of differential absorption spectra measured in different fields was used to calculate the zero field spectrum. Magnetic field dependent absorbtion spectra were evaluated as a sum of the zero field spectrum and the corresponding differential spectra. Note that by this method only the field-dependent part of this absorption is recovered and field-independent features are not captured. While this can cause an ambiguity of the mean absorption, the directional dichroism spectrum \(\Delta\alpha_{H}=\alpha(+H)-\alpha(-H)\) are free of such uncertainties.
## Theoretical calculations.
The cycloid of BiFeO\({}_{3}\) is controlled by two Dzyaloshinskii-Moriya (DM) interactions and an easy-axis anisotropy \(K\) along the ferroelectric polarization \(\mathbf{P}_{0}\). Whereas the DM interaction \(D_{1}\) perpendicular to \(\mathbf{P}_{0}\) is responsible for the formation of the long-period (62 nm) cycloids, the DM interaction \(D_{2}\) along \(\mathbf{P}_{0}\) is responsible for a small cycloidal tilt.
In a magnetic field \(\mathbf{H}\), the spin state and the magnon excitations of BiFeO\({}_{3}\) were evaluated from the microscopic Hamiltonian
\[\mathcal{H}=-J_{1}\sum_{\langle i,j\rangle}\mathbf{S}_{i}\cdot \mathbf{S}_{j}-J_{2}\sum_{\langle i,j\rangle^{\prime}}\mathbf{S}_{i}\cdot \mathbf{S}_{j}\] \[+D_{1}\sum_{\langle i,j\rangle}(\mathbf{z}^{\prime}\times\mathbf{ e}_{i,j})\cdot(\mathbf{S}_{i}\times\mathbf{S}_{j})\] \[+D_{2}\sum_{\langle i,j\rangle}(-1)^{n_{z}}\,\mathbf{z}^{\prime} \cdot(\mathbf{S}_{i}\times\mathbf{S}_{j})\] \[-K\sum_{i}(\mathbf{z}^{\prime}\cdot\mathbf{S}_{i})^{2}-2\mu_{ \mathrm{B}}\mathbf{H}\cdot\sum_{i}\mathbf{S}_{i}, \tag{3}\]
\(J_{1}\) and \(J_{2}\) are the nearest and the second nearest neighbour interactions, respectively. The \(D_{1}\) sum, first proposed by Katsura and coworkers, is uniform over the lattice, while the \(D_{2}\) sum alternates sign from one hexagonal layer to the next.
The nearest- and next-nearest neighbor exchange interactions \(J_{1}\)=-5.32 meV and \(J_{2}\)=-0.24 meV were obtained from recent inelastic neutron scattering measurements. \(D_{1}\)\(\approx\)0.18 meV is determined from the cycloidal pitch. \(D_{2}\)=0.085 meV and \(K\)=0.0051 meV were obtained by fitting the four magnon modes observed in zero field. The spin state of BiFeO\({}_{3}\) is solved by using a trial spin state that contains harmonics of the fundamental ordering wavevector \(\mathbf{Q}\). We then minimize the energy \(\langle\mathcal{H}\rangle\) over the variational parameters of that state. A \(1/S\) expansion about the classical limit is used to evaluate the magnon mode frequencies at \(\mathbf{Q}\) as a function of field. We calculated the optical ME susceptibilities, \(\hat{\chi}^{me}(\omega)\) and \(\hat{\chi}^{em}(\omega)\), the dielectric permittivity, \(\hat{\epsilon}(\omega)\), and the magnetic permeability, \(\hat{\mu}(\omega)\), using the Kubo formula.
To determine the directional dichroism, the Maxwell
\(|\) **Temperature dependence of the directional dichroism in BiFeO\({}_{3}\)**. Directional dichroism spectra measured in \(\mu_{0}H\)=\(\pm\)12 T at \(T\)=4, 150 and 300 K. The magnetic field was applied along \([1\overline{1}0]\) and \(\mathbf{E}^{\ast}\)\(\parallel\)\([1\overline{1}0]\). Modes \(\Psi_{1}^{}\) and \(\Phi_{2}^{}\) soften and get weaker with increasing temperature, but are still clearly observable even at 300 K. The \(\Psi_{1}^{}\) is visible until 150 K while the \(\Phi_{1}^{}\) cannot be detected reliably already at 150 K.
Polarization rotation of the transmitted beam was found negligible. We also evaluated the directional dichroism spectra using the approximate formula in Eq., which is valid when the polarization rotation of light can be neglected. Eq. and the numerical solution of the Maxwell equations provide nearly equivalent \(\Delta\alpha_{k}\) spectra if parameters \(\hat{\chi}^{me}(\omega)\), \(\hat{\chi}^{em}(\omega)\), \(\hat{\varepsilon}(\omega)\) and \(\hat{\mu}(\omega)\) realistic to BiFeO\({}_{3}\) are used. Fits to the directional dichroism spectra indicate that the polarization induced by the spin current associated with \(D_{1}\) and \(D_{2}\) can be written in the form given by Eq., where \(\lambda^{}\) and \(\lambda^{}\) are the dynamic ME couplings for the two types of terms, respectively.
| 10.48550/arXiv.1505.00033 | Optical diode effect in the room-temperature multiferroic BiFeO$_3 | I. Kezsmarki, U. Nagel, S. Bordacs, R. S. Fishman, J. H. Lee, H. T. Yi, S-W. Cheong, T. Room | 5,836 |
10.48550_arXiv.1406.3095 | ##
(a) Optical reflection image of WSe\({}_{2}\) flakes on a Si substrate covered by a 300nm SiO\({}_{2}\) layer. A monolayer sample (middle) is outlined by dashed blue lines. (b) Photoluminescence (PL) spectrum of monolayer WSe\({}_{2}\) excited by a cw HeNe laser at 1.96eV. (c) Emission spectrum of monolayer WSe\({}_{2}\) under the excitation of femtosecond infrared pulses centered at 1.07eV. It consists of two features corresponding to the second-harmonic generation (SHG) (blue) and two-photon PL (red). The latter is magnified by 15 times. The PL excited by the cw HeNe laser (green) is included for comparison. (d) Excitation power dependence of the integrated SHG and two-photon PL.
##
(a) Linear absorption (red line, right axis) and 2PPL excitation spectrum (blue symbols, left axis) measured on monolayer WSe\({}_{2}\) at room temperature. Each data point of the 2PPL excitation spectrum corresponds to an integrated PL (1.664 - 1.687eV) normalized by the reference SHG signal from a z-cut quartz crystal according to Eq.. The uncertainty corresponds to the spectral resolution, determined by the bandwidth of the excitation pulse. A and B correspond to the fundamental exciton resonances arisen from transitions from the two highest energy spin-orbit split-off valance bands and the lowest energy conduction bands at the K(K') point of the Brillouin zone. A' and A' denote the \(2s\) state and a broad \(p\)-peak observed in one-photon and two-photon absorption, respectively. (b) Second-order numerical derivative of the linear absorption spectrum of (a). Black dashed line denotes the band edge energy of 2.02eV determined from the fit described in the text.
Exciton excitation spectrum of monolayer WSe\({}_{2}\) determined experimentally in this work (left panel) is compared with the 2D hydrogenic model (right panel). Red lines denote the one-photon active states and the blue box is the unresolved two-photon active states. The exciton binding energy and the bottom of the continuum in the 2D hydrogenic model are chosen to match the values obtained from experiment.
(a) Representative PL spectra of monolayer WSe\({}_{2}\) excited by femtosecond IR pulses centered at 0.85-1.1eV. The spectra were normalized by the SHG intensity from a z-cut single crystal quartz plate recorded under identical experimental conditions. (b) Experimental 2PPL excitation spectrum (symbols) and fit (green line) including contributions from both excitons and band-to-band transitions (dotted red lines) as described in the text.
| 10.48550/arXiv.1406.3095 | Tightly bound excitons in monolayer WSe2 | Keliang He, Nardeep Kumar, Liang Zhao, Zefang Wang, Kin Fai Mak, Hui Zhao, Jie Shan | 4,385 |
10.48550_arXiv.2107.01478 | ## ABSTRACT
Thin film quasicrystal coatings have unique properties such as very high electrical and thermal resistivities and very low surface energy. A nano quasicrystalline thin film of icosahedral Al-Ga-Pd-Mn alloy, has produced by flash evaporation followed by annealing. The icosahedral phase of Al\({}_{65}\)Ga\({}_{5}\)Pd\({}_{17}\)Mn\({}_{13}\) alloy has been used as a precursor material. The X-ray diffraction and transmission electron microscopy confirmed the formation of icosahedral phase in the thin film. The Energy dispersive X-ray analysis investigations suggest the presence of Ga (\(\sim\) 5 at %) in the alloy. Icosahedral Al-Ga-Pd-Mn thin film provides a new opportunity to investigate the various characteristics including surface characteristics. The formation of icosahedral thin film in Al-Ga-Pd-Mn quaternary alloy by present technique has been studied for the first time. These films can be utilized as conversion coatings for Al substrates or incorporated into a full coating system containing an organic primer and a topcoat.
# Fig.5
The SADPs of annealed version of Al\({}_{65}\)Ga\({}_{5}\)Pd\({}_{17}\)Mn\({}_{13}\) thin film, (a) two -fold (b) three fold (c) mirror plane and (d) corresponding bright field microstructure respectively.
Fig.1Fig.2Fig.3Fig.4Fig.5 | 10.48550/arXiv.2107.01478 | Thin film of Al-Ga-Pd-Mn quasicrystalline alloy | Thakur Prasad Yadav | 2,654 |
10.48550_arXiv.1506.04069 | 10.48550/arXiv.1506.04069 | Comments on the paper: Optical reflectance, optical refractive index and optical conductivity measurements of nonlinear optics for L-aspartic acid nickel chloride single crystal | Bikshandarkoil R. Srinivasan, Suvidha G. Naik, Kiran T. Dhavskar | 334 |
|
10.48550_arXiv.0709.0793 | ###### Abstract
With density functional theory, studied are the local magnetic moments in Fe-Al alloys depending on concentration and Fe nearest environment. At zero temperature, the system can be in different states: ferromagnetic, antiferromagnetic and spin-spiral waves (SSW) which has a minimum energy. Both SSW and negative moment of Fe atoms with many Al atoms around them agree with experiments. Magnetization curves taken from literature are analysed. Assumption on percolation character of size distribution of magnetic clusters describes well the experimental superparamagnetic behaviour above 150 K.
pacs: 75.20.Hr; 75.50.Bb; 71.15.Ap
## 1 Introduction
The alloys Fe-Al attracts the attention of researchers as a perspective material in an extreme technology. They possess the properties such as good refractoriness, oxidizing and corrosion resistance, relatively low density, good ductility at room temperature. Intensive study of magnetic properties was firstly initiated by a development of non-destructive control methods, as the magnitudes of all the above properties correlate with magnetic characteristics in these alloys. Afterwards, unusual behavior of magnetic properties generated a separate interest to their study. Mainly the attention is focused on the concentration range from 25 to 50 at.% of Al for quasiordered and from 40 to 60 at.% of Al for disordered alloys. Reliably enough, it was established that in higher and lower concentration ranges the alloys are in the ferromagnetic and paramagnetic states, correspondingly, and in the intermediate region more complicated states are realized. To the beginning of the 80-s a majority of researchers have been convinced that at low temperature the magnetic state is a spin glass in this region. But a row of experimental data that had been then considered as an evidence for the spin glass, had, as a matter of fact, another nature, which has been revealed later. For example, the thermomagnetic hysteresis in these alloys is a consequence of a solely magnetic hysteresis. Recently, the neutron powder diffraction has shown that at low temperature the magnetic order is governed by spin-density waves. At high temperature the alloys are superparamagnetic.
A raw of discrepancies in studies of magnetic properties of these alloys has given an impetus to our paper. Here, a theoretical analysis of the superparamagnetic behavior in the experimental data available is conducted, collinear and spiral magnetic structures are studied with the help of the density-functional theory.
## 2 Superparamagnetic behavior
Magnetization curves of the alloy with 34 at.% Al as a function of the applied magnetic field and temperature have been received in the Ref.. Higher than the blocking temperature of 150 K the magnetization curves join each other, which is typical for the paramagnetic behavior. Besides, the magnetization increases rather quickly at low magnitude of the parameter \(h/T\approx 10^{3}A/m/K\), but does not reach the saturation up to \(5\times 10^{4}A/m/K\), which is an evidence for different by size magnetic clusters that do not interact. Note that the system is structurally homogeneous, so the clusters are governed by magnetic interactions. Using the Arrhenius formula for the relaxation time \(t\) one can estimate the upper limit for the number of atoms in clusters. It is an order of \(n_{max}\approx 10^{4}\) atoms (the characteristic time of magnetic measurement at which the detection of largest clusters is possible equals \(t=t_{0}\times exp\{-n_{max}E_{a}/KT\}\approx 10^{-2}\) s, where \(E_{a}\approx 7\times 10^{-25}J/at\) is energy of the magnetocrystalline anisotropy in iron, \(t_{0}\approx 10^{-6}\) c is the spin-lattice relaxation time, \(T=150K\) is a blocking temperature).
Assuming that the alloy contains clusters of two types, we succeeded in describing the magnetization curves with clusters of 6 nm diameter (6500 atoms in the cluster) and 3 nm diameter (600 atoms in the cluster). 20 % of all atoms belong to the 6 nm clusters and 80 % of all atoms belong to the 3 nm clusters. The average magnetic moment of an Fe atom is \(0.33\mu_{B}\). This description has however essential shortcomings. First, there is no physical or chemical mechanism which could be responsible for just these cluster sizes (as authors of Ref. assure, the sample was homogeneous). Second, with such size distribution of clusters, magnetization at weak fields should be proportional to h/T, which is not corroborated by experiment (see insert in Fig. 1).
More naturally looks the assumption about a continuous size distribution as this is in the case of hierarchy of the cluster size distribution in disordered percolation task. In this case, density of the number of clusters consisting of \(n\) Fe atoms divided by total number of lattice sites is equal to \(w_{n}=x(\tau-2)n_{min}^{\tau-2}n^{-\tau}\), where \(\tau\) is a critical exponent, \(n_{min}\) is a minimal cluster size and \(x\) is concentration of Fe atoms. As the magnetic moment of a cluster is large, one can use classical concepts and calculate the magnetization per iron atom (see Ref.):
\[M=m_{av}(\tau-2)n_{min}^{\tau-2}\times\] \[\int_{n_{min}}^{\infty}n^{1-\tau}[cth(nm_{av}h/kT)-kT/nm_{av}h]\ dn\]
Here \(m_{av}\) is average magnetic moment of an Fe atom in a cluster.
Scaling relations allow us to write \(\tau=\delta^{-1}+2\), where \(\delta\) determines the magnetization behavior
The magnetization curves fitting the experimental data from Ref. in assumption about a continuous cluster size distribution.
\((h/T)^{1/\delta}\) at \(h/T\to 0\). In the insert to Fig. 1, shown is a least-squares adjustment of the experimental data with \(\delta=1.49\). From the above interrelation between coefficients we receive \(\tau=2.67\). Further, using this \(\tau\), we have conducted fitting in the whole range of the parameter \(h/T\). The best agreement with experiment have been achieved at \(n_{min}=62\) atoms and \(m_{av}=0.44\mu_{B}\).
We must note that the coefficient \(\tau\) obtained does not coincide by magnitude with the coefficient in the classical percolation theory (\(\tau=2.2\)). To our opinion, the reasons are the following: first, the sample in the experiment was quasiordered, that is, had a strong short order; second, it did not reach the percolation threshold; third, the interactions between the atomic magnetic moments, that govern the geometry of magnetic clusters, are connected with the chemical configurations of the atoms disposition in a very complicated way.
## 3 Dependence of Fe magnetic moments on the closest atomic environment
To understand the peculiarities of the magnetic interaction in the Fe-Al alloys we have conducted first-principles calculations of the periodical systems \(Fe_{38}Al_{16}\) (29.6 at.% Al), \(Fe_{11}Al_{5}\) (31.3), \(Fe_{34}Al_{20}\) (37.0), \(Fe_{10}Al_{6}\) (37.5) and \(Fe_{9}Al_{7}\) (43.8). These systems have been chosen so that to cover that interesting intermidiate concentration region from 29 to 44 at.% Al and to receive many different chemical configurations of the iron nearest environment. The calculations have been conducted by FP LAPW method with the WIEN 2k program package. The detailed description of the models and approximations used is given in.
One of the main results consists in the following: there are two solutions with collinear magnetic moments found for all the concentrations studied. One of them has Fe local magnetic moments all of a direction, the other has both positive and negative moments depending on environment. Namely, the magnetic moments at Fe atoms with 6 and more Al atoms in nearest environment direct oppositely to those of the rest iron atoms. In the following, we call the first as the solution of a ferromagnet type (FM), and the second as that of an antiferromagnet type (AFM). The AFM states are slightly lower by energy than the FM ones in \(Fe_{11}Al_{5}\), \(Fe_{34}Al_{20}\) and \(Fe_{10}Al_{6}\); in \(Fe_{38}Al_{16}\) and \(Fe_{9}Al_{7}\) the FM state is more preferable. The Fe average magnetic moment in the AFM solutions as a function of Al concentration agrees rather well with experimental data. shows the iron magnetic moments in all the systems studied, in the aggregate, depending on nearest environment. One can see that direction and magnitude of the magnetic moments in AFM solutions are rather accurately determined by chemical composition of the iron nearest environment, and only small variations can be imputed to different structure, concentration or environment in more distant coordination spheres.
Similar results have been also obtained earlier for disordered alloys using a two-band Hubbard model. This behavior of local magnetic moments gives grounds for usage of the Jaccarino-Walker model for interpretation of experiments. The main idea of the Jaccarino- Walker model asserts that the local magnetic moment at a transition-metal atom is determined by chemical composition of the nearest environment and only weakly depends on the overall concentration. Using this model authors of Refs. have described combination of magnetic and Mossbauer experimental data in disordered and partly disordered Fe-Al alloys. Surely, the models describing the magnetic order in terms of closest environment cannot pretend to be very precise in the transition-metal alloys.
The local magnetic moment of Fe atoms as a function of number of Al atoms in the nearest environment.
Average magnetic moment of Fe atoms as a function of Al concentration. Circles denote the first-principles results, squares are for experimental data from Ref.. Triangle shows the moment obtained from analysis of superparamagnetic behavior.
## Spiral magnetic structures
Recently, neutron diffraction studies have shown that quasiordered Fe-Al alloys have spin-density waves with direction at temperature lower than 100 - 150 K. Such study, as authors of Ref. themselves admit, cannot distinguish the spin-density wave with a collinear structure (SDW) and the spin-spiral wave (SSW). SDW in Cr have been rather long ago known and well studied. A nesting in the Fermi surface is considered the most justified mechanism for appearance of SDW in Cr. This mechanism looks impossible for the Fe-Al alloys. First, the Fermi surface of iron does not have nesting; second, the alloys have a disorder in the atomic disposition, which makes the influence of the Fermi surface on the SDW formation very problematic. That is why we think that the oscillations in the experiment come from SSW and consider the conditions of their appearance in the systems \(Fe_{9}Al_{7}\) and \(Fe_{10}Al_{6}\). The calculation is conducted with use of a non-collinear-magnetic version of WIEN2k package. The SSW in \(Fe_{9}Al_{7}\) has been considered earlier in. They have received that the SSW with direction and the wave vector \(q=0.4a_{0}^{-1}\) possesses the minimum energy (here \(a_{0}\) is a bcc lattice parameter). We have received the same result for \(Fe_{9}Al_{7}\). For \(Fe_{10}Al_{6}\) the minimal by energy SSW direction coincides the experimental one. The difference between the collinear and the SSW solutions is less than 7 mRy/cell which is a small value and allows transitions from SSW to a collinear state at small energy of an external excitation (magnetic field or temperature). We must note that wavelength of the SSW received in both our and Ref. calculations is \(4a_{0}\), while the experimental value observed at these concentrations is \(7a_{0}\).
We did not take into account the spin-orbit interaction in our calculations, so it cannot be responsible for the appearance of the SSW as this usually occurs in the magnetics on the basis of rare-earth elements and actinides. To our opinion, the main reason of the SSW appearance as the ground state is a competition of the two collinear magnetic states FM and AFM that are close by energy.
## 5 Conclusions
At temperatures higher than 150 K the alloy with 34 at.% of Al is a typical superparamagnetic. The best theoretical description of experimental magnetization curves is obtained with assumption that cluster distribution by size obeys a scaling law with minimum size clusters as \(\approx\) 60 magnetic atoms, and with the average local magnetic moment of Fe atom is \(m_{av}=0.44\mu_{B}\).
Our study has shown a potential possibility of existence of few types of magnetic order in Fe-Al alloys: collinear structures (FM and AFM) and spin-spiral waves (SSW). The energy of SSW is lower than those of the FM and AFM structures. The difference in energy between these states does not exceed 7 mRy/cell. This allows the system to transform from one magnetic structure to another at weak external influence (magnetic field or temperature). The character of the thermal or field transition from SSW to a collinear state is, however, unclear in actual disordered alloys: is it a kind of phase transition or the transition occurs through a row of continuous reconstructions of the electron structure in local regions?
The average Fe magnetic moments theoretically calculated in structures with AFM ordering are close to the experimental data from direct magnetization measurements and from the analysis of the superparamagnetic behavior.
| 10.48550/arXiv.0709.0793 | The formation and ordering of local magnetic moments in Fe-Al alloys | A. K. Arzhnikov, L. V. Dobysheva, M. A. Timirgazin | 1,345 |
10.48550_arXiv.2307.14478 | 10.48550/arXiv.2307.14478 | An In Situ Study of the Role of Pressure on Fe Recrystallization and Grain Growth during Thermomechanical Processing | Darren C. Pagan, Lukas A. Kissell, Matthew L. Whitaker | 4,013 |
|
10.48550_arXiv.1309.0306 | ###### Abstract
In this study, the temperature dependence of the spin Hall angle of palladium (Pd) was experimentally investigated by spin pumping. A Ni\({}_{80}\)Fe\({}_{20}\)/Pd bilayer thin film was prepared, and a pure spin current was dynamically injected into the Pd layer. This caused the conversion of the spin current to a charge current owing to the inverse spin Hall effect. It was found that the spin Hall angle varies as a function of temperature, whereby the value of the spin Hall angle increases to ca. 0.02 at 123 K.
In recent years, _spincurrentronics_ has been attracting significant attention for potential future applications because of the low energy consumption of spin current. Spin current can be generated by inducing spins into spin sink materials via spin pumping. The spin current can then be detected by utilizing the inverse spin Hall effect (ISHE), which can convert a pure spin current to a charge current by spin-orbit interaction. Many studies have applied this method in order to evaluate the spin Hall angle (\(\theta_{\rm SHE}\)), to inject spins into condensed matter, and estimate the spin diffusion length and spin relaxation time, but most of these studies have been conducted at room temperature (RT). The recent success of dynamical spin injection and transport in Si at room temperature allows the establishment of spin-based logic by using a dynamical method since Si is one of the most suitable materials for beyond CMOS spin devices. In the spin logic using the dynamical method, readout of a spin current is realized by the ISHE. In the case of Si, Pt is not suitable for the "spin-charge converter", because a Pt-silicide alloy can be easily formed, which impedes the conversion. Hence, Pd is a better material for the converter. For future investigations of Si-based spin logic using the dynamical method, the temperature dependence of the logic device performance is also quite important and the temperature dependence of \(\theta_{\rm SHE}\), as a good index of spin-charge conversion, should be investigated. Herein, we report the temperature dependence of the spin Hall angle of palladium (Pd), which is a commonly used spin sink material exhibiting the ISHE. The spin Hall angle of Pd was qualitatively evaluated by changing the temperature from 130 K to RT. This evaluation yields an important spin-related physical parameter that can be investigated in future studies by applying the spin pumping method.
Figure 1(a) shows the schematic illustration of a Ni\({}_{80}\)Fe\({}_{20}\)(Py)/Pd bilayer sample. A 25-nm-thick permalloy (Py) film and a 5-nm-thick Pd film were prepared on an oxidized silicon substrate by electron beam evaporation. Both Py and Pd layers were rectangular with an area of 2 \(\times\) 1 mm\({}^{2}\).
The sample system was placed at the center of a TE\({}_{011}\) microwave cavity in an electron spin resonance (ESR) instrument with a frequency (\(f\)) of 9.12 GHz. An external magnetic field \(H\) was applied to the Py/Pd bilayer at an angle of \(\theta_{\mathrm{H}}\), as shown in Fig. 1(a). The ferromagnetic resonance (FMR) condition was determined using the following equation:
\[(\frac{\omega}{\gamma})^{2}=H_{FMR}(H_{FMR}+4\pi M_{s})\,, \tag{1}\]
Figure 1(b) shows the FMR spectra of Py with and without Pd. The linewidth of the spectrum corresponding to Py/Pd is larger than that corresponding to Py alone, which is attributed to the modulation of the Gilbert damping constant (\(\alpha\)) owing to successful spin pumping into the Pd layer. The FMR spectra and output dc voltages of the Py/Pd bilayer at RT resulting from the ISHE in the Pd are shown in Figs. 1(c) and 1(d), respectively. The FMR intensities and resonance fields were nearly identical for all values of \(\theta_{\mathrm{H}}\), and an electromotive force was induced when \(\theta_{\mathrm{H}}\) was set to 0 and 180\({}^{\circ}\), while the signal was flat at \(\theta_{\mathrm{H}}=90^{\circ}\). Because this is consistent with the symmetry of the ISHE (\(J_{c}\propto J_{s}\times\sigma\)), the observed electromotive force was ascribed to the ISHE of Pd, which was due to a pure spin current generated dynamically by spin pumping at the Py/Pd interface. A theoretical fitting was then performed in order to separate the ISHE and anomalous Hall effect (AHE) signals by using the following equation.
\[V=V_{ISHE}\,\frac{\Gamma^{2}}{\left(H-H_{FMR}\right)^{2}+\Gamma^{2}}+V_{AHE} \,\frac{-2\Gamma(H-H_{FMR})}{\left(H-H_{FMR}\right)^{2}+\Gamma^{2}}+aH+b\,, \tag{2}\]
The value of \(H_{\mathrm{FMR}}\) was experimentally determined to be 89.51 mT at 0 and 180\({}^{\circ}\) at RT. The variables \(\Gamma\), \(a\), and \(b\) are the fitting parameters. As shown in Fig. 1(e), a theoretical fitting using eq.effectively reproduced the experimental results, and \(V_{ISHE}\) was estimated to be 7.66 \(\upmu\)V at RT.
The electromotive force from the Pd layer \(V_{\rm ISHE}\) was proportional to the microwave power. As shown in the inset of Fig. 2, the power dependence of \(V_{ISHE}\) was in good agreement with the theoretical prediction, indicating that the generation of the observed electromotive force is attributed to the ISHE in Pd. More importantly, the unsaturated FMR spectra enabled the estimation of the spin Hall angle of Pd. A pure spin current was injected at the Py/Pd interface via spin pumping under the resonance condition, and the generated spins diffused into the Pd layer and were converted to a charge current (\(j_{c}\)), as shown in Fig. 1(a). The spin current density \(j_{s}\) was theoretically calculated as\(j_{s}=\frac{g_{r}^{\uparrow\downarrow}\gamma^{2}h^{2}h[4\pi M_{s}\gamma+\sqrt{ \left(4\pi M_{s}\right)^{2}\gamma^{2}+4\omega^{2}}\ ]}{8\pi\alpha^{2}[\left(4\pi M_{s}\right)^{2}\gamma^{2}+4\omega^{2}\ ]}\), where \(h\) is the microwave magnetic field, and it was set to 0.16 mT at a microwave power of 200 mW. The variable \(g_{r}^{\uparrow\downarrow}\)and the constant \(h\) are the real part of the mixing conductance and the Dirac constant, respectively. Note that \(g_{r}^{\uparrow\downarrow}\) is given by \(g_{r}^{\uparrow\downarrow}=\frac{2\sqrt{3}\pi M_{s}\gamma d_{P_{y}}}{g\mu_{B}\omega}\left(W_{P_{y}/Pd}-W_{P_{y}}\right)\), where \(g,\mu_{B}\), \(d_{\rm Py}\), \(W_{\rm Py/Pd}\), and \(W_{\rm Py}\) are the g-factor, Bohr magneton, thickness of the Py layer, FMR spectral width of the Py/Pd film, and FMR spectral width of the Py film, respectively. In this study, \(d_{\rm Py}\), \(W_{\rm Py/Pd}\), and \(W_{\rm Py}\) were 25 nm, 3.03 mT, and 2.49 mT, respectively. The electromotive force due to the ISHE can then be expressed in the simplest form as follows:
\[V_{ISHE}=\frac{w\theta_{SHE}\lambda_{Pd}\ \tanh(d_{Pd}\ /2\lambda_{Pd})}{d_{P_{y}}\sigma_{P_{y}}+d_{Pd}\sigma_{Pd}}(\frac{2e}{h})j_{s}\,, \tag{3}\]
1(a), \(d_{\rm Py}\) and \(\sigma_{\rm Py}\) are the thickness (25 nm) and electric conductivity (4.48 \(\times\) 10\({}^{6}\)\(\Omega^{-1}\) m\({}^{-1}\)) of the Py layer at RT, and \(d_{\rm Pd}\) and \(\sigma_{\rm Pd}\) are the thickness (5 nm) and electric conductivity (3.50 \(\times\) 10\({}^{6}\)\(\Omega^{-1}\) m\({}^{-1}\)) of the Pd layer, respectively. From the above calculations, the spin Hall angle \(\theta_{\rm SHE}\)was estimated to be 0.011 at RT, which is reasonably consistent with that obtained in a previous study.To estimate the spin Hall angle at various temperatures, the temperature evolution of the linewidth (\(W\)) of a simple Py film, resonance field (\(H_{\rm FMR}\)), and saturation magnetization (\(4\pi M_{\rm s}\)) were measured and evaluated as shown in Fig. 3(a), and the results were physically reasonable. Here, note that \(\sigma_{\rm Py}\), \(\sigma_{\rm Pd}\), \(V_{\rm ISHE}\), and \(\lambda_{\rm Pd}\) changed with the temperature, and thus, the temperature dependences of \(\sigma_{\rm Py}\) and \(\sigma_{\rm Pd}\) were also evaluated [Fig. 3(b)], and the estimated values of \(V_{\rm ISHE}\) obtained from eq. at different temperatures are shown in the inset of Fig. 3(c). The spin diffusion length of Pd, denoted as \(\lambda_{\rm Pd}\), has been reported to be 9 nm at RT and 25 nm at 4.2 K, so \(\lambda_{\rm Pd}\) can be estimated at each temperature by interpolating the values at RT and at 4.2 K (In fact, \(\lambda_{\rm Pd}\) is estimated to be 9.12 nm at 133 K by our calculation), since the spin lifetime is inversely proportional to temperature in this temperature region. The change in \(\lambda_{\rm Pd}\) yields little contribution to the change in \(\theta_{\rm SHE}\). Thus, \(\theta_{\rm SHE}\) of Pd can be estimated by solving eq. and using the conductivities of Pd and Py and the estimated spin diffusion length of Pd at each temperature. The results of this evaluation are shown in Fig. 3(d). It was found that \(\theta_{\rm SHE}\) increased to 0.020 monotonically as the temperature decreased to 130 K. It is surprising that the temperature evolution of \(\theta_{\rm SHE}\) of Pd exhibits opposite behavior compared with that of Pt. Whereas Pt and Pd belong to the same group in the periodic table, the electron configurations are different, which may induce such difference. In fact, their electron configuration of the d-orbital allows a different sign of \(\theta_{\rm SHE}\). The origin of this characteristic temperature dependence of \(\theta_{\rm SHE}\) of Pd is beyond the scope of this study, but will be investigated in the near future.
In conclusion, in this study, the temperature dependence of the spin Hall angle of Pd was estimated by spin pumping. This approach enables the quantitative estimation of the temperature dependence of the spin transport properties of Si, graphene, and other materials by combining spin pumping and ISHE.
# Figure captions
Figure 1(a) Schematic illustration of the Py/Pd bilayer sample used in this study. \(H\) represents an external magnetic field and \(\theta_{\rm H}\) the angle of the external magnetic field. The dimensions of the sample are shown in the figure. (b) FMR spectra of the Py layer (red line) and the Py/Pd layer on the SiO\({}_{2}\) substrate (black line). An increase in the line width, \(W\), can be observed, which is attributed to the modulation of the Gilbert damping constant \(\alpha\), indicating successful spin injection into the Pd layer by spin pumping. (c) Magnetic field angle (\(\theta_{\rm H}\)) dependence of the FMR signal _dI(H)/dH_ for the Py/Pd bilayer sample at room temperature (RT). (d) Magnetic field angle (\(\theta_{\rm H}\)) dependence of the electromotive force measured from the Py/Pd bilayer sample at RT. (e) External magnetic field dependence of the electromotive force \(V_{\rm ISHE}\) for the Py/Pd bilayer film at 200 mW and at RT. Open circles represent the experimental data. The solid curve shows the fitting result, which was calculated using eq.. Orange and blue dashed lines show the contribution from ISHE and AHE, respectively.
Microwave power dependence of the electromotive force in the Pd film. A magnetic field was applied parallel to the film plane (\(\theta_{\rm z}=0^{\circ}\)). The output voltage increased with increasing microwave power. The inset shows the power dependence of the contribution of ISHE.
Figure 3(a). (upper panel) Temperature dependence of the FMR linewidth of the single Py film and the Pd/Py film. The inset shows the difference in the linewidth between the Py and Pd/Py films as a function of temperature. (lower panel) Ferromagnetic resonance field (\(H_{\rm FMR}\)) and saturation magnetization (\(4\pi M\)s). Saturation magnetization decreased with increasing temperature. An increase in the resonance field with increasing temperature can also be observed owing to the relation shown in eq.. (b) Temperature dependence of the resistivity of the Pd and Pylayers. Red and blue circles indicate the experimental data for Pd and Py, respectively. Red and blue lines show the linear fittings. (c) Temperature dependence of the output electromotive force measured for the Py/Pd sample. The insets show the temperature dependence of the contribution from ISHE (\(V_{\rm ISHE}\)) and an expanded graph of the electromotive force. (d) Temperature dependence of the spin Hall angle \(\theta_{\rm SHE}\) of Pd. The spin Hall angle decreased with increasing temperature.
Figure 1(a).
Figure 1(b).
Figure 1(c) and (d)
Figure 1(e)
Figure 3(a).
Figure 3(b) and (c)
Figure 3(d) | 10.48550/arXiv.1309.0306 | Temperature Dependence of Spin Hall Angle of Palladium | Zhenyao Tang, Yuta Kitamura, Eiji Shikoh, Yuichiro Ando, Teruya Shinjo, Masashi Shiraishi | 2,155 |
10.48550_arXiv.1408.5671 | ## 1 Introduction
Transient electric field (TEF) generally exists in femtosecond laser-matter interactions due to thermionic and/or multi-photon emission of electrons. The strength and evolution of the TEF is critical in laser ablation mechanism studies, and the formation of early stage plasmas after intense laser irradiation. Under moderate laser excitation conditions, it is also a nontrivial influencing factor for photocathode optimizations and time-resolved electron scattering studies, such as ultrafast electron diffraction (UED) and time-resolved angle-resolved photoemission spectroscopy (TR-ARPES). In UED studies, a crystalline sample is first excited by an ultrashort optical pump pulse and then interrogated by an electron probe pulse delivered at a specific delay time. Transient structural information is majorly obtained from the time-dependent evolutions of the diffraction angle and intensity extracted from electron diffraction patterns. However, the existence of the TEF on the sample surface may distort the trajectory of the probe electrons and make the interpretation of diffraction patterns complicated. For example, in the studies of semiconductors by reflection UED and metals by transmission UED, the deflection angles induced by the TEFs were comparable with the changes of the diffraction angle originated from structural dynamics. In the transmission geometry, it has been demonstrated that the structural dynamics and the TEF effect can be distinguished by simultaneously tracking the radii and the centroids of the diffraction rings of polycrystalline crystals. However, their separation in reflection UED is still indistinct. In such case, the TEF is nearly normal to the propagation direction of the reflective probe electrons,which may induce additional and notable deflections to the probe electrons. Furthermore, the field gradient perpendicular to the sample surface may bring non-uniform distortions to different diffraction spots. As a consequence, the convolution of structural dynamics and such TEF is complex. To ultimately separate the TEF effect from structural dynamics in reflection UED, it is crucial to have a better understanding on the origin and evolution of the TEF and its influence on the probe electrons. Moreover, the TEFs induced by femtosecond pump laser pulses may influence the angular and energy resolution of photoelectrons in TR-ARPES studies, which also leads to the necessity of its investigation.
Previously, studies on light induced electron emissions has been focused on the quantum yield or energy spectroscopy of photoelectrons, while the temporal evolution of TEFs is sparsely understood. Recently, ultrafast electron deflection and shadowgraph have provided a direct monitor to transient electromagnetic fields, combining the intrinsic field sensitivity of electrons with the ultrahigh temporal resolution provided by a laser-pump electron-probe configuration.
In this contribution, the TEFs generated by femtosecond laser pulse irradiation of a 25-nm thick aluminum film have been investigated by picosecond electron deflection. Under laser intensities on the order of \(10^{10}\) W/cm\({}^{2}\), it is shown that the TEFs at 120 \(\upmu\)m above the metallic surface last more than one nanosecond with a maximum strength on the level of \(10^{4}\) V/m. The experimental results were explained by a "three-layer"analytic model, which indicates that the observed TEFs were mainly attributed to the thermionic emission of electrons with an initial velocity of 1.4 \(\upmu\)m/ps and a charge density of approximately 10\({}^{7}\) e\({}^{\text{-}}\)/mm\({}^{2}\). Based on the dynamics of the TEFs revealed in this study, we further evaluated their influence on UED and TR-ARPES.
## 2 Ultrafast electron deflection configuration
The laser-pump electron-probe experimental configuration, as shown in Figure 1, includes a Ti:sapphire laser system (1 kHz, 800 nm, 70 fs, 1 mJ/pulse ), a photoelectron gun driven by ultraviolet pulses, a magnetic lens, a sample holder attached to a 5-axial manipulator, an imaging system, and an ultrahigh vacuum chamber. The main laser beam was split into two parts: 90% was used as the pump and directed to a linear translation stage to precisely control its relative time difference (delay time) with respect to the probe beam. The pump beam was focused to a diameter of 0.8 mm (1/e\({}^{2}\)) and normally impinged onto a freestanding 25 nm thick aluminum sample, which was a paradigm for UED experiments and prepared according to a routine procedure. The pump intensities of interest was varied from 2.9 to 7.1\(\times\)10\({}^{10}\) W/cm\({}^{2}\) (2\(\sim\)5 mJ/cm\({}^{2}\) fluence), in the same range as generally applied in time-resolved diffraction studies and well below the \(\sim\)10 mJ/cm\({}^{2}\) damage threshold of aluminum. Under the pump intensities used in this study, the temporal evolutions of the observed deflection angles are repeatable even after a large number of laser shots on the sample. We also inspected the sample after the experiments by an optical microscopy and no observable damage was found on its surface. The remaining 10% of the main beam was converted to 266 nm ultraviolet light through a frequency tripler and directed to the photocathode of the electron gun, a 30-nm silver layer coated on sapphire disc, to generate ultrashort electron pulses. The probe electrons were accelerated to 59 keV, collimated and focused to a diameter of \(\sim\)200 \(\upmu\)m (1/e\({}^{2}\)) by the magnetic lens with their centroid at 120 \(\upmu\)m above the sample surface. After passing through the sample area, the deflected probe electrons were recorded by the two dimensional imaging system containing a phosphor screen, a multi-channel plate (MCP) and a charge-coupled device (CCD) camera. Each electron deflection image was acquired with 1-s CCD exposure time to accumulate 10\({}^{3}\) electron pulses and the signal-to-noise ratio was further improved by averaging more than 15 independent measurements of the electron deflection pattern at each delay time. The \(\sim\)6 ps temporal resolution of the current setup is mainly limited by the travelling time of the 59 keV probe electrons through the laser-aluminum interaction area, which has a dimension similar to that of the pump beam. The time zero was defined as the onset of the observable deflections of the probe electrons.
The schematic illustration of ultrafast electron deflection. (a), the experimental
(b), a detailed view of the sample area. Z axis denotes the sample surface normal direction, while X and Y axis are parallel to the sample surface. The probe electron beam travels along the Y axis before entering the TEF area and its centroid position at the Z axis is Z\({}_{0}\) = 120 \(\mu\)m. \(\Delta\alpha\) is the deflection angle of the probe electron beam. The positive and negative deflection angles represent that the probe electron beam is deflected toward and away from the sample surface, respectively.
## 3 Electron deflection data analysis
The time-dependent evolution of the TEF was represented by the corresponding deflection angle of the probe electrons at each delay time. In order to calculate the electron deflection angle, we first integrated the 2D deflection pattern (see Figure 2(a)) along the X and Z directions to obtain two 1D intensity profiles, which were fitted by Gaussian functions to derive the peak positions as the centroid coordinates of the probe electrons. Then, the absolute change of the centroid position at each delay time, \(\Delta R_{i}\)(t) (i = \(x,z\)), was obtained by subtracting the averaged value before time zero. Because the TEF is mainly perpendicular to the sample surface, only the 1D intensity distribution along the Z-axis is considered in the study presented here. Taking account of the distance between the sample and the detector (\(L\) = 0.46 \(m\)) and the small angle approximation, the deflection angle of the probe electrons is \(\Delta\alpha_{z}\)(t) \(\approx\arctan\left[\Delta\alpha_{z}(t)\right]\) = \(-\dfrac{\Delta R_{z}\)(t) \(L\), where the minus sign represents that the positive direction of the TEF parallels the positive direction of Z-axis. The time-dependent evolution of the electron deflection angle actually represented the averaged TEFstrength at 120 \(\mu\)m above the sample surface, where the centroid of the probe electrons locates. Meanwhile, the width and amplitude derived from the Gaussian fitting of the intensity profile along the Z-axis were normalized by their corresponding averaged values before time zero to deduce their time-dependent evolution.
## 4. Results and discussions:
### 4.1.
Upon femtosecond laser excitation of aluminum, the optical energy is rapidly deposited into conduction electrons because their heat capacity is several orders of magnitude smaller than that of the lattice. Some energetic electrons overcome the \(\sim\) 3.9 eV work function of the nanosized thin aluminum, and escape from the sample surface via thermionic and/or multi-photon emission. The evolution of the emitted electrons and the positive ion layer eventually determine the formation and decay of the TEF.
The TEF represented by the deflection of the probe electrons' centroid is the average strength sensed during their interrogation of the TEF. The relation between the electron deflection angle, \(\Delta\alpha_{z}(t)\), and the averaged TEF strength detected by the probe electrons at each delay time, \(\overline{E_{z}}(z_{0},t)\), is described by the following equation:
\[\Delta\alpha_{z}(t)=-\frac{\Delta R_{z}(t)}{L}=-\frac{\Delta V_{z}(t)}{V_{e}} =\frac{\overline{E_{z}}(z_{0},t)qD}{mV_{e}^{2}}=\frac{\overline{E_{z}}(z_{0},t )qD}{m_{e}V_{e}^{2}/\sqrt{1-V_{e}^{2}/c^{2}}} \tag{1}\]
The time-dependent deflections of the probe electrons and the corresponding TEF strengths were shown in The typical error bar is mainly contributed by the pointing jitter of the probe electron beam, while the pointing jitter of the pump laser beam is negligible due to a much larger irradiation diameter. The maximum averaged electric field appears at t=158 ps and according to Eq., its strength range from 32 to 53 kV/m with the pump intensities varying from 2.9 to 7.1 \(\times\) 10\({}^{10}\) W/cm\({}^{2}\). However, the temporal evolution of the TEF remains the same under different pump intensities. The evolution of the TEF consists of three steps: (i). The negative deflection of the probe electrons reaches its minimum within 33 ps after laser irradiation. Because the centroid of the emitted electrons moves toward to that of the probe beam, the direction of the TEF at Z\({}_{0}\)=120 \(\upmu\)m is along the negative Z direction. Therefore, as shown in Figure 2(b), the probe electrons are deflected along the positive Z-axis by the TEF immediately after laser irradiation. Meanwhile, as shown in Figure 2(e), the deflection angle grows as a function of delay time and reaches its minimum at t=33 ps when the centroid of the probe electrons is probably deflected farthest away from the sample surface. (ii). The deflection of the probe electrons evolves from its minimum to zero at \(\sim\)60 ps and reaches the positive maximum deflection at around 158 ps. Due to Coulomb repulsion inside the emitted electrons and the attractive force from the positive surface ion layer, a significant number of the emitted electrons decelerate from their initial emitting velocities and fall back into the sample while the remaining electrons effectively escape from the sample. Given that the initial emitting velocity of the laser-excited electrons is on the order of 1 \(\upmu\)m/ps, the "fallen back" electrons are always below the centroid position of the probe electrons and become the dominate contribution to the negative TEF strength at the centroid positon. Therefore, accompany with a large amount of the emitted electrons below Z\({}_{0}\)=120 \(\upmu\)m returning back to the sample, the magnitude of the deflection decreases after t=33 ps. At \(\sim\)60 ps, the deflection angle of the probe electrons is zero, which means the direction of the averaged TEF at Z\({}_{0}\)=120 \(\upmu\)m will change from the negative to the positive Z direction. After 60 ps, the attractive force from the surface ion layer dominates the positive deflection of the probe electrons. Meanwhile, after the "effectively emitted" electrons pass the entire probing area, both the ion layer and the "effectively emitted" electrons contribute to the positive deflection of the probe electrons. Therefore, the probe electrons reach their positive maximum deflection at t=158 ps. (iii). The positive deflection of the probe electrons follows a decay process. Due to the expansion and moving away of the emitted electrons, and the accompany decreasing of the charge density, the TEF strength continues to descend for at least one nanosecond.
The horizontal dash and solid lines are marked for the sample position and the centroid of the probe electrons at the Z direction before deflection, respectively. **(e):** Time-dependent evolution of the electron deflection angles at various pump intensities. The negative and positive \(\Delta\alpha_{z}\) indicates that the transient centroid of the probe electrons located above and below the horizontal solid line, respectively.
### The evolution of the TEFs explained by a "three-layer" model
The evolution of the TEFs observed above results from the complex nonlinear many-body interactions between the emitted electrons and the positive surface charges, both evolving fast with time.
Deflection of the probe electron beam as a function of delay time. **(a)-(d):** Snapshots of probe electrons at a pump intensity of \(7.1\times 10^{10}\)W/cm\({}^{2}\). The horizontal dash and solid lines are marked for the sample position and the centroid of the probe electrons at the Z direction before deflection, respectively. **(e):** Time-dependent evolution of the electron deflection angles at various pump intensities. The negative and positive \(\Delta\alpha_{z}\) indicates that the transient centroid of the probe electrons located above and below the horizontal solid line, respectively.
This model is aimed at reproducing the main features of our TEF measurements, together with an evaluation of the key parameters of the emitted electrons and the remaining positive ions. The "three-layer" model, which is illustrated in Figure 3, describes three types of charges that contribute to the TEFs: the positive surface charges that are due to the emission of electrons upon laser excitation, the emitted electrons that will return to the sample (fallen back electrons) and the emitted electrons that will effectively escape from the sample (effectively emitted electrons). The charge density and the longitudinal (perpendicular to the sample surface) and transversal (parallel to the sample surface) dimensions of these three types of charge layers evolve with time and determine the observed TEFs. The "three-layer" model is developed from the two-disk models reported before, which either neglected the thickness of the emitted electron layer or described the two kinds of electrons with different behaviors by a single Gaussian distribution. In the "three-layer" model, the total-emitted electrons are represented by the combination of two different Gaussian distributions, which correspond to the different behaviors of emitted electrons. It is expected to have an improved description of the actual behaviors of the total-emitted electrons.
The initial transversal dimensions of the positive charges and the emitted electrons are assumed to be equal to the pump laser spot with a diameter of \(D\). The positive charge layer contracts at an average speed \(V_{w1}\) due to the degrading of mirror charge effects and neutralization. The emitted electrons expand at an average speed \(V_{w2}\) and \(V_{w3}\) due to the transversal and longitudinal Coulomb interaction, respectively. The longitudinal distribution of the positive charges is assumed to be a Delta function, which means that all the positive charges are confined at the surface of the sample, z=0 \(\mu\)m.
The distribution function of the total-emitted electrons, \(\rho(z,t)\), is defined by the following relation:
\[\rho(z,t)=(1-\alpha)\rho_{E}(z,t)+\alpha\rho_{F}(z,t) \tag{2}\]
Schematic illustration of the “three-layer” model describing the evolution of emitted electrons and positive surface charges. The electron distribution function shown at the left side is for demonstration purpose only. In actual case, the effectively emitted electrons only account for a small portion of the total emitted electrons. Therefore, the effectively emitted electrons appears as a small shoulder in the distribution of the total-emitted electrons.
\(\rho_{\varepsilon}(z,t)\) and \(\rho_{ F}(z,t)\) are two normalized Gaussian distributions representing the "effectively emitted" and the "fallen back" electrons along the Z direction (longitudinal), respectively. The time-dependent widths and peak positions of these two Gaussian functions have the same initial values at time zero. The "effectively emitted" electrons are assumed to move away from the sample surface with an initial emitting velocity \(v_{0}\). The "fallen back" electrons decelerate with a rate \(a\) from the same initial velocity \(v_{0}\)to zero, then accelerate toward the sample surface with the same rate and eventually neutralize the positive ion layer. The propagating velocities are the center-of-mass (CoM) velocities of the effectively emitted" and the "fallen back" electrons. As a result of the velocity distribution inside the emitted electrons, the width of the electron distribution function also evolves with time. The symbols used in the model are summarized in Table 1.
\begin{table}
\begin{tabular}{c c} \hline \hline
## Symbol** & **Description
\\ \hline \(\rho(z,t)\) & Distribution function representing the total-emitted electrons \\ \(\rho_{ E}(z,t)\) & Gaussian distribution function representing the “effectively emitted” electrons \\ \(\rho_{ F}(z,t)\) & Gaussian distribution function representing the “fallen back” electrons \\ \(\alpha\) & Ratio of the “fallen back” electrons to the total-emitted electrons \\ \(a\) & Decelerating rate of the fallen back electrons \\ \(v_{0}\) & Initial CoM velocity of the emitted electrons \\ & Initial transversal diameter of the ions and the emitted electrons (pump laser diameter) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Symbols used in the “three-layer” model.
For simplicity, the averaged TEF strength was represented by the strength at the centroid position of the probe electrons, which is the contribution of both the positive surface ion layer and the negative electrons described below:
\[E_{z}(z_{0},t) = \frac{\sigma_{0}}{2\varepsilon_{0}}\cdot\left\{\left[1-\frac{z_{0} }{\sqrt{{z_{0}}^{2}+(D/2-{v_{w1}}t)^{2}}}\right]\cdot\left[1-\int_{-\infty}^{0 }\rho\left(z,t\right)dz\right]\right.\] \[\left.-\int_{0}^{z_{0}}\rho\left(z,t\right)\cdot\left[1-\frac{z_{0} -z}{\sqrt{\left(z_{0}-z\right)^{2}+(D/2+{v_{w2}}t)^{2}}}\right]dz\right.\] \[\left.+\int_{z_{0}}^{z_{0}}\rho\left(z,t\right)\cdot\left[1-\frac{z -z_{0}}{\sqrt{\left(z_{0}-z\right)^{2}+(D/2+{v_{w2}}t)^{2}}}\right]dz\right\}\]
The experimental data were acquired from the shifting of the probe electrons centroid, which originally locates at 120 \(\upmu\)m above the sample surface, therefore, the value of \(z_{0}\) is 120 \(\upmu\)m in Eq.. Together with Eq., the deflections of the probe electrons at all four pump intensities were well fitted as depicted in Among all the seven fitting parameters (\(\alpha\,,v_{0}\,,\sigma_{0}\,,a\,,V_{w1}\,,V_{w2}\,,V_{w3}\)), the temporal evolutions of the transient electric field are most sensitive to \(\alpha\,,v_{0}\)and \(\sigma_{0}\), which are listed in Table 2 for detailed discussion. In addition, all fitting parameters were set as free variables with no constrains. Because the photon energy of the pump laser is 1.55 eV, much lower than the 3.9 eV work function of the nanosized thin aluminum, electrons are expected to be induced by thermionic and/or multi-photon emission instead of single-photon process. We performed the pump intensity dependence experiments to further distinguish these two emission mechanisms. According to Fowler-DuBridge theory that described the electron emission from a solid surface, the electron yield of the n-th order photoemission is proportional to \(I^{n}\), where \(I\) is the pump laser intensity. However, the amount of the total-emitted electron charges is found to depend linearly on the pump intensity, as depicted in This linear relation indicated that thermionic emission is the dominant mechanism within the pump intensities applied here and the contribution of multiphoton emission is insignificant, which consists with the previous theoretical prediction.
Linear dependence of the amount of the total-emitted electron charges on pump intensities.
The fitting results also suggest that, the temporal evolution of the electron distribution functions changes slightly with the increasing of the laser intensity, while the "fallen back" ratio \(\alpha\) and the initial charge density \(\sigma_{0}\) increase. Therefore, according to Eq. and Eq., \(\alpha\) and \(\sigma_{0}\) only modulate the amplitude of the charge distribution function and the strength of TEF, respectively. This agrees with the experimental results that, the temporal evolutions of the electron deflections are similar at all pump intensities increasing from 2.9 to 7.1\(\times\)10\({}^{10}\) W/cm\({}^{2}\), while the deflection magnitude grows accordingly. The initial CoM emitting velocity of the electrons was fitted to be about 1.4 \(\mu\)m/ps. 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The Coulomb repulsion along Coulomblongitudinal direction causes the return of more than 75% of the total-emitted electrons back to the sample together with the attraction force from the positive surface charges.
### Influence of TEF on time-resolved electron scattering studies.
Pump-probe technique is the primary tool for the studies of transient phenomenon, especially on the picosecond to femtosecond time scale. The pump and probe sources can vary from different combinations of optical, THz, X-ray, and electron pulses. Among all these combinations, ultrafast electron diffraction, which is based on laser-pump electron-probe, has been an effective method that provides direct access to structure evolution with atomic spatial-temporal resolutions. In UED studies, structural dynamics are generally obtained from the time-dependent evolution of electron diffraction angles, line widths, and integrated intensities. However, both structural dynamics and transient electric fields can affect the behavior of the probe electron beam, which gives rise to difficulties in the interpretation of electron diffraction data.
In this study, the TEF at 120 \(\upmu\)m above the metallic surface was found to be on the order of 10\({}^{4}\) kV/m under moderate pump intensities, which are generally applied in UED studies. Its influences on the deflection angle, width, and peak intensity of the probe electron beam profile are depicted in and some characteristic parameters are given in Table 3. For the first tens of picoseconds after laser irradiation, the deflection angle of the probe electrons is on the order of tens micro radians, which is comparable to the typical changes of the diffraction angle induced by structural dynamics in reflection UED. With the fitting parameters obtained under the pump fluence of \(7.1\times 10^{10}\) W/cm\({}^{2}\), we further calculated the TEF gradient along the Z direction according to the "three-layer" model and evaluated its influence on the broadening of the probe beam profile. The results imply that, the maximum beam width reached at t=92 ps is 2.2 times of that before time zero, which is in good agreement with the 2.3-times experimental value. This good agreement suggests that, applying Eq. for the averaged TEF is reasonable and the "three-layer" model is self-consistent. In addition, the broadening of the beam width also induces the attenuation of the peak intensity. Both the width and peak intensity recover toward their original values before time zero along with the decay of the TEF.
In general, the results presented here indicate that, TEFs widely exist in UED studies, and in a reflection configuration, it can affect the position, width and peak intensities of the probe electron beam profile, which may cause misinterpretations to the structural dynamics extracted from diffraction patterns. In future UED studies, the pump laser induced TEFs should be evaluated _in situ_ for a closer understanding of the structural dynamics and a better resolution. Meanwhile, the strong electric field above the sample surface may also contribute to the transient structure change, which has not been considered in the previous UED studies. Therefore, further efforts are necessary to access the role of TEF effects on transient structure changes.
The TEF effects may also be an important issue in TR-APERS studies emerged recently, which provide a temporal, angular and energy resolution of photoelectrons on the order of sub-picoseconds, tenth of a degree and milli-electronvolts, respectively. In these TR-ARPES studies, samples are pumped by a femtosecond laser pulse and probed by ultraviolet (UV) photons to reveal the time-dependent photoemission spectroscopy at the first few picoseconds after laser excitation.
Time-dependent evolution of the (**a**) deflection angle, (**b**) width, and (**c**) peak intensity of the probe electron beam at a pump intensity of 7.1\(\times\)\(10^{\text{\textdegree}}\)W/cm\({}^{2}\). The evolution of the deflection angle was fitted by the “three-layer” model. The peak intensity was normalized to the value before time zero and smoothed with Fast-Fourier-Transformation filtering.
\begin{table}
\begin{tabular}{c c c} \hline & **Tmax (ps)** & **Maximum Value** \\ \hline \(\Delta\alpha\) & 158 & 0.38 mrad \\ W/ W\({}^{0}\) & 92 & 2.3 \\ \(I_{p}\) / \(I_{p}^{0}\) & 92 & 0.47 \\ \hline \end{tabular}
\end{table}
Table 3: Typical parameters of the dynamical position presented in the deflection angle, width and peak intensity of the probe beam profile.
However, even at a low fluence/intensity, the femtosecond pump laser pulse can rapidly heat up the electron system and may generate photoelectrons. This additional effect, which could also be an influencing factor to the energy resolution of TR-ARPES, has rarely been assessed.
We estimated the averaged electric field strength on aluminum surface by the "three-layer" model under the lowest pump intensity used here. As presented in Figure 6, it indicated that the averaged electric field strength within several micrometers above the sample surface is on the order of 100 kV/m for the first few picoseconds. Its modulation to the photoelectrons is on the order of 10\({}^{2}\) meV, which may influence the understanding of TR-ARPES results and limit the improvement of the energy resolution to better than milli-electronvolts. Although the samples of interest in TR-ARPES studies are mainly superconductors or topological insulators, the study presented here may bring into attention that, it is insufficient to only consider the effect of UV induced photoelectrons, and the TEF induced by the pump laser pulse should also be evaluated to optimize the performance of TR-ARPES.
However, limited by the experimental configuration of this study and the simplified "three-layer" analytical model, we can only obtain some finite insights into the transient electric field on the metallic surface. Inspired by proton radiography, in our further efforts, we will experimentally investigate the spatial-temporal evolution of the TEFs by ultrafast electron radiography. A better understanding of the TEFs may help to improve the resolution and accuracy of time-resolved studies that involved with electrons.
## 5 Conclusion:
We used ultrashort electron pulse to directly monitor the femtosecond laser induced
The TEF strength along the Z direction for the first few picoseconds, which is predicted by the “three-layer” model and the fitting parameters under the lowest pump intensity, 2.9\(\times\)10\({}^{10}\) W/cm\({}^{2}\).
Its strength is on the scale of 10\({}^{4}\) V/m at 120 \(\upmu\)m above the sample surface under the pump intensities on the order of 10\({}^{10}\) W/cm\({}^{2}\). The experimental results were explained by a "three-layer" analytic model, and the observed TEFs were attributed to the thermionic emission of electrons with an initial velocity of \(\sim\)1.4 \(\upmu\)m/ps and a charge density of approximately 10\({}^{7}\) e\({}^{\text{-}}\)/mm\({}^{2}\). The study presented here also indicate that, besides deflection, the probe electron beam width, peak intensity and energy dispersion can also been modulated by the transient electric field. Therefore, for time-resolved electron scattering studies, such as ultrafast electron diffraction and time-resolved angle-resolved photoemission spectroscopy, the transient surface electric field should be considered and evaluated _in situ_ for improved resolution and accuracy.
| 10.48550/arXiv.1408.5671 | Investigation of transient surface electric field induced by femtosecond laser irradiation of aluminum | Run-Ze Li, Pengfei Zhu, Long Chen, Tong Xu, Jie Chen, Jianming Cao, Zheng-Ming Sheng, Jie Zhang | 6,032 |
10.48550_arXiv.1306.2395 | ## I Introduction
Almost all materials become stiffer when compressed, as a result of the constituent atoms being squeezed together. It therefore comes as something of a shock that some materials - among them amorphous silica, ZrW\({}_{2}\)O\({}_{8}\) and Zn(CN)\({}_{2}\) - actually become _softer_ under compression. Formally, the stiffness is defined through the zero-pressure bulk modulus \(B_{0}=V_{0}(\partial V_{0}/\partial P)_{T}^{-1}\), and the change in stiffness is defined through the differential \(B_{0}^{\prime}=\partial B_{0}/\partial P\), which is a positive quantity for almost all materials. However, in these cited examples it is found that \(B_{0}^{\prime}\) has a negative value. There is as yet no theoretical explanation for this effect, which can be called "pressure-induced softening", but in a simulation study of pressure-induced softening in amorphous silica we drew attention to the role of fluctuations involving whole-body rotations of SiO\({}_{4}\) tetrahedra. Given that the same fluctuations are implicated in the similarly counter-intuitive phenomenon of negative thermal expansion (NTE), and given that the few materials in which pressure-induced softening has been identified also show NTE, we suggest that most NTE materials will show pressure-induced softening, and in this paper we demonstrate the plausibility of this hypothesis.
The basis for linking pressure-induced softening with negative thermal expansion can be understood by considering the relatively simple example of Zn(CN)\({}_{2}\). Its perfect structure has linear Zn-C-N-Zn linkages of bonds along the crystallographic \(\langle 1,1,1\rangle\) directions. Uniform compression of the perfect structure will force compression of these bonds, which are stiff and will become stiffer on further compression. Hence at a temperature of 0 K, or in a static lattice energy calculation, we might expect to find a positive value of \(B_{0}^{\prime}\). However, on heating thermal fluctuations will cause instantaneous buckling of the Zn-C-N-Zn linkages - a process aided by the fact that the rigid-unit-mode flexibility of the structure allows for localised distortions - so that an external compressive force can be accommodated with relatively low energy cost by further buckling without the need to compress the individual bonds. If we now consider the case of stretching the structure (application of negative pressure), the stretch will first be accommodated by reducing the buckling of the linkages of bonds, but when the buckling has been stretched out the second process is to stretch the individual bonds. This will cost a lot more energy, and the volume change per unit of stretch force will reduce. This means that the bulk modulus will increase on stretching, and hence we have a negative value of \(B_{0}^{\prime}\).
Because the fluctuations that buckle the linkages give rise to a reduction in crystal volume in many framework, and because their amplitude increases with temperature, we have the possibility - perhaps in some cases inevitability - for negative thermal expansion. Thus we might expect pressure-induced softening to be linked to NTE. Put another way, we might expect that many NTE materials will also show pressure-induced softening. In the case of Zn(CN)\({}_{2}\), experimentally it is found to have large NTE, and a negative value of \(B_{0}^{\prime}\), i.e. pressure-induced softening. Our recent simulation study of Zn(CN)\({}_{2}\) is consistent with the experimental data and shows that \(B_{0}^{\prime}\) has a dependence on temperature of the form described above.
In this paper we test the proposal of a direct link between NTE and pressure-induced softening by performing simulation experiments on the full suite of zeolites with cubic lattice symmetry. Zeolites are low-density framework structures built from corner-linked SiO\({}_{4}\) tetrahedra, many of which are found naturally with ionic substitution on the tetrahedral site (e.g. Al for Si) with associated charge-balancing cations (e.g. Na) found in the large pores in the structure. It has long been recognised that some zeolites show NTE, although there has not yet been a systematic study of the set of the cubic zeolites. Here we make two predictions, first that most cubic zeolites will show NTE, and second, based on the preceding discussion, that those that do have NTE will also show pressure-induced softening. This is much easier tested by molecular dynamics simulation than experiment, and for siliceous zeolites we have some good force fields derived from quantum mechanical calculations and tested in many independent studies. There are 13 candidate zeolites with crystal structures of cubic symmetry and fully connected SiO\({}_{4}\) tetrahedra, all of which are investigated in the current work. For reference for the rest of this paper, we note that all zeolites are assigned a three-letter name, sometimes which relates to a historical name (e.g. ANA for analcime, FAU for faujisite).
The thermodynamic theory to link NTE with pressure-induced softening is derived in Section II. Section III gives the computational method and Section IV presents the main results. Conclusions are drawn in Section V.
## II Thermodynamic background
The Helmholtz free energy of an insulating crystal in the classical high-temperature approximation is written as
\[F=\Phi+\sum_{s}\ln{\left(\frac{\hbar\omega_{s}}{\tau}\right)}, \tag{1}\]
The second term involves the sum over all wave vectors \(\mathbf{k}\) on all branches of the phonon dispersion curves \(j\), with \(\omega_{s}\) as the frequency of each phonon mode denoted by \(s=\left\{j,\mathbf{k}\right\}\). \(\tau=k_{\mathrm{B}}T\) is the temperature in the unit of energy.
At equilibrium, the pressure \(p\) is obtained as the derivative of the free energy with respect to the crystal volume \(V\):
\[p=-\frac{\partial F}{\partial V}=-\frac{\partial\Phi}{\partial V}+\frac{3N \tau}{V}\overline{\gamma}, \tag{2}\]
The overall Gruneisen parameter \(\overline{\gamma}\) is defined as the sum over all the mode Gruneisen parameters \(\gamma_{s}=-\left(V/\omega_{s}\right)\left(\partial\omega_{s}/\partial V\right)\):
\[\overline{\gamma}=\sum_{s}\gamma_{s}/\left(3N\right) \tag{3}\]
The bulk modulus of the material can be calculated using
\[B=-V\frac{\partial p}{\partial V}=V\frac{\partial^{2}\Phi}{\partial V^{2}}+ \frac{3N\tau}{V}\overline{\gamma}+\frac{3NB\tau}{V}\frac{\partial\overline{ \gamma}}{\partial p}, \tag{4}\]
where we have used
\[\frac{\partial\overline{\gamma}}{\partial V}=\frac{\partial\overline{\gamma}} {\partial p}\frac{\partial p}{\partial V}=-\frac{B}{V}\frac{\partial\overline{ \gamma}}{\partial p}. \tag{5}\]
Thus,
\[B=\frac{V\partial^{2}\Phi/\partial V^{2}+\left(3N\tau/V\right)\overline{ \gamma}}{1-\left(3N\tau/V\right)\left(\partial\overline{\gamma}/\partial p \right)}. \tag{6}\]
From Equation 6, we can obtain the first derivative of the bulk modulus with respect to pressure as
\[B^{\prime}=\frac{\partial B}{\partial p}=\frac{\partial B}{\partial V}\frac{ \partial V}{\partial p}\approx-\frac{V}{B}\left(\frac{\partial^{2}\Phi}{ \partial V^{2}}+\frac{V\partial^{3}\Phi}{\partial V^{3}}\right)+\frac{3N\tau} {V}\frac{\overline{\gamma}}{B}+\frac{6N\tau}{V}\frac{\partial\overline{\gamma }}{\partial p}+\frac{3N\tau B}{V}\frac{\partial^{2}\overline{\gamma}}{\partial p ^{2}}. \tag{7}\]
In Equation 7, we have used the approximation
\[\left|\frac{3N\tau}{V}\frac{\partial\overline{\gamma}}{\partial p}\right|\ll 1 \tag{8}\]
According to this, from Equation 6, one has
\[B\approx V\frac{\partial^{2}\Phi}{\partial V^{2}}+\frac{3N\tau}{V}\overline{ \gamma}. \tag{9}\]
Accordingly, we can rewrite Equation 7 in the more compact form
\[B^{\prime}=B^{\prime}\left.\right|_{T=0}+\frac{3N\tau}{V}\left[\frac{ \overline{\gamma}}{B}+2\frac{\partial\overline{\gamma}}{\partial p}+B\frac{ \partial^{2}\overline{\gamma}}{\partial p^{2}}\right], \tag{10}\]
where the first term on the right-hand side, namely
\[B^{\prime}\left.\right|_{T=0}=-\frac{V}{B}\left(\frac{\partial^{2}\Phi}{ \partial V^{2}}+\frac{V\partial^{3}\Phi}{\partial V^{3}}\right) \tag{11}\]
In our case, we will calculate this term for all the cubic NTE zeolites in harmonic lattice dynamics using a force field with the Buckingham potential.
Thus, according to Equation 10, if \(B^{\prime}\left.\right|_{T=0}\) is positive, given that all the rest terms are negative, \(B^{\prime}\) may become negative when the temperature is high enough.
\[\overline{\gamma}<0 \tag{12}\]and we will see in Section V, that for all the cubic NTE zeolites,
\[\frac{\partial\overline{\gamma}}{\partial p} <0\] \[\frac{\partial^{2}\overline{\gamma}}{\partial p^{2}} <0. \tag{13}\]
Note that, if \(\overline{\gamma}\), \(\partial\overline{\gamma}/\partial p\) and \(\partial^{2}\overline{\gamma}/\partial p^{2}\) are in the same order of magnitude, the term containing \(\partial^{2}\overline{\gamma}/\partial p^{2}\) will contribute dominantly to a negative \(B^{\prime}\) due to its large coefficient involving the bulk modulus \(B\).
## III Computational methods
The molecular dynamics (MD) simulations were carried out using DL_POLY.
\[\phi_{ij}(r_{ij})=A_{ij}\exp(-r_{ij}/\rho_{ij})-C_{ij}r_{ij}^{-6}, \tag{14}\]
The point charges on theSi and O atoms are (in the units of electron charge) 2.4 and \(-1.2\) respectively. The long-range Coulomb energy was calculated using the Ewald method with precision of \(10^{-4}\). Typical simulations, lasting around 30 ps in the production stage with a 10 ps equilibration stage, were performed using time steps of 0.001 ps and the velocity Verlet scheme. Long-time stability of the structures at high temperature were tested up to 200 ps. Simulations were performed using the Nose-Hoover constant-pressure constant-temperature ensemble, with relaxation times of 1.0 ps for both thermostat and barostat. The first suite of simulations were performed at constant pressure to search for NTE, followed by a large number of simulations over a range of pressure at a fixed temperature for a large number of different temperature values. Typical sample sizes were \(6\times 6\times 6\) unit cells.
The calculations of the density of states and the Gruneisen parameters for the studied zeolites were carried out in harmonic lattice dynamics using GULP. The same potential model as in the MD was used in these calculations.
## IV Results
### Search for negative thermal expansion in cubic zeolites
Only ANA has positive thermal expansion throughout the temperature range. BSV shows almost zero thermal expansion. The other 11 zeolites have NTE. Three of these undergo phase transitions, and it is their high-temperature phases that show NTE. shows the simulated volume-pressure relationships at a temperature of 300 K. Inevitably most examples show a pressure-induced phase transition. RWY, SOD and PAU undergo phase transitions at very low pressure (\(<0.2\) GPa). Thus, from the original pool of 13 zeolites we have 8 useful candidate materials where the transition pressure is not too low and in which we can explore the link between pressure-induced softening and NTE.
Calculated V-P plots from 0.0 to 3.0 GPa at 300 K for the 11 NTE zeolites. RWY, SOD and PAU undergo phase transitions at very low pressure (\(<0.2\) GPa).
Calculated V-T plots from 100 to 1000 K at ambient pressure for all the cubic zeolites. ANA shows positive thermal expansion. BSV shows almost zero thermal expansion. AST, MTN and SOD show negative thermal expansion only in their high-temperature phases.
We found that almost all the modes with negative Gruneisen parameters are rigid unit modes (RUMs) in these cubic zeolites. Calculations using the CRUSH code reveal that these RUMs correspond to the rotations and the translations of the SiO\({}_{4}\) tetrahedra, with the translational modes occupy the lowest energy band. The big difference between the positive-thermal-expansion ANA and other NTE cubic zeolites is that the low-frequency translational RUMs of ANA do not have large negative Gruneisen parameters, while the medium and high-frequency non-RUMs in ANA have relatively large positive Gruneisen parameters. The low-frequency translational RUMs contribute the most to the NTE of the material, which is similar to the finding in our work on Zn(CN)\({}_{2}\). Detailed discussions of the origins of NTE of these cubic zeolites will be given elsewhere.
### Pressure-induced softening in NTE-zeolites
From sequences of pressure-sweep simulations over a range of fixed temperatures we have obtained values of \(V_{0}\), \(B_{0}\) and \(B^{\prime}_{0}\) by fitting isothermal data using the equation of states (EoS). We have explored several EoS formalisms, including 3rd and 4th-order Birch-Murnaghan (BM) EoS, Vinet (Universal) EoS, and Keane EoS. It was found that the 3rd and 4th-order BM EoSs have the greater stability. The Vinet and Keans EoS gave similar results to those of the BM EoS. The 3rd-order BM gave excellent results for most of the studied zeolites, but significant improvements were obtained using the 4th-order BM for AST and MTN. The fitted values of \(B_{0}\) and \(B^{\prime}_{0}\) at 300 K are given in Table 1.
As an example, shows the fitting to isotherms of FAU at different temperatures using the 3rd-order BM EoS. shows the temperature-dependence of the fitted \(B_{0}\) and \(B^{\prime}_{0}\) values. In this case, the fitted value of \(B^{\prime}_{0}\) is negative at all temperatures; in a convex-parabola trend seen in the isotherms shows that the volume contracts more rapidly at higher pressure, and this gives the negative value of \(B^{\prime}_{0}\). In this regard, a 3rd-order BM EoS was adequate to describe this trend in the isotherm. Plots of the fitted isotherms, equilibrium volume \(V_{0}\), \(B_{0}\) and \(B^{\prime}_{0}\) for all the systems are given in the supplemental material.
In principle, at low temperature the values of \(B_{0}\) and \(B^{\prime}_{0}\) measured experimentally may differ from those obtained from a classical MD simulation due to quantum effects.
\begin{table}
\begin{tabular}{c|c|c|c|c|c} \hline \hline Zeolite & \(\alpha_{\rm V}\) (MK\({}^{-1}\)) & \(T_{\rm c}\) (K) & \(P_{\rm c}\) (GPa) & \(B_{0}\) & \(B^{\prime}_{0}\) \\ \hline AST & \(-13.8\) & 250 & 0.5 & 16 & \(-28^{f}\) \\ FAU & \(-8.0^{a}\) & – & 2.2 & 51.1\({}^{d}\) & \(-2.6\) \\ KFI & \(-10.0\) & – & 1.6 & 65.7 & \(-4.4\) \\ RHO & \(-8.3^{b}\) & – & 0.5 & 59 & \(-4\) \\ LTA & \(-11.3\) & – & 1.5 & 61.8\({}^{e}\) & \(-13.0\) \\ MTN & \(-19.3^{c}\) & 200 & 1.5 & 27 & \(-3^{f}\) \\ MEP & \(-20.8\) & – & 0.6 & 58 & \(-30\) \\ TSC & \(-8.2\) & – & 1.75 & 52.4 & \(-4.9\) \\ RWY & \(-35.2\) & – & \(<0.1\) & – & – \\ SOD & \(13.4\) & 600 & \(<0.1\) & – & – \\ PAU & \(-11.2\) & – & \(<0.2\) & – & – \\ BSV & \(-0.9\) & – & – & 56.2 & \(-0.03\) \\ ANA & \(19.7\) & 800 & 1.2 & 10.3 & \(0.3\) \\ \hline \hline \end{tabular} \({}^{a}\)Experimental value averaged over 25-573 K is \(-12.6\) MK\({}^{-1}\).
\({}^{b}\)Averaged value over 0-500 K calculated from lattice dynamics in quasi-harmonic approximation is \(-13\) MK\({}^{-1}\).
\({}^{c}\)Experiment shows that MTN goes through a phase transition at about 370 K and only its high temperature phase shows NTE. The experimental value averaged over 463-1002 K is \(-5.0(0.7)\) MK\({}^{-1}\).
\({}^{d}\)Experimental value is 38 GPa.
\({}^{e}\)DFT calculated value is 46 GPa.
\({}^{f}\)The fitted values of \(B^{\prime}_{0}\) of AST and MTN using the 4th-order BM EoS are \(-108\) and \(-37\), respectively.
\end{table}
Table 1: Calculated coefficients of thermal expansion \(\alpha_{\rm V}\), phase-transition temperature \(T_{\rm c}\) at ambient pressure, and phase-transition pressure \(P_{\rm c}\) at 300 K. Because \(\alpha_{\rm V}\) varies with temperature, the averaged values over the temperature range are given. For AST and MTN which only have NTE in their high temperature phases, the values are averaged over 300-1025 K. The bulk modulus at 0 pressure \(B_{0}\) and its first derivative with respect to pressure \(B^{\prime}_{0}\) of each NTE zeolite is obtained from fitting the 300 K isotherm with the 3rd-order BM EoS. These values are not given for RWY, SOD and PAU because they quickly go through phase transitions on compression.
Fitted isotherms of FAU at different temperatures from 1 to 600 K with a 10 K interval from 10 to 100 K and an 100 K increment from 100 to 600 K, using the 3rd-order BM EoS. The convex-parabola trend of all the isotherms indicate negative \(B^{\prime}_{0}\) and pressure-induced softening of the material. The value of volume in the plot is for the supercell used in MD.
Taking the example of FAU, we have obtained the set of mode Gruneisen parameters from the phonon frequencies calculated using expanded and contracted (\(\pm 0.1\%\)) unit-cell volumes, and have colored the vibrational density of states according to the value of Gruneisen parameter, as shown in (plots of the other NTE zeolites are available in). It is clear that modes with the most negative Gruneisen parameters -- highlighted by red to light violet -- are around 1 THz (48 K) and span to the lowest frequency. This suggests that even at low temperature \(\sim 50\) K, these modes will not be 'frozen' out and can still be excited and contribute to NTE and pressure-induced softening of the material. In such a case, the classical MD results at low temperature would not have too much difference from the real quantum picture.
As shown in Table 1, all the cubic zeolites that have NTE and are stable under compression show pressure-induced softening (negative \(B^{\prime}\)). It is interesting to note that BSV having almost zero thermal expansion shows almost zero \(B^{\prime}\), and ANA with positive coefficient of thermal expansion shows positive \(B^{\prime}\).
To show the consistency between the theory in Section II and the MD results, we calculate \(B^{\prime}\) using Equation 10 and give the values of various terms in Table 2. The values of \(\overline{\gamma}\) were obtained from the phonon frequencies calculated using expanded and contracted \(\pm 0.05\%\) unit-cell volumes at zero pressure. The term \(\partial\overline{\gamma}/\partial p\) was calculated as \(\partial\overline{\gamma}/\partial p\approx\left(\overline{\gamma}_{p}- \overline{\gamma}_{p=0}\right)/p\) for a small \(p\) around zero pressure. Combined with the values of \(\tau/V_{\rm at}=N\tau/V\), one can see the validity of Equation 8. The volume variation for the calculation of \(\overline{\gamma}\) at each pressure is much smaller relative to the volume reduction at that pressure. \(\partial^{2}\overline{\gamma}/\partial p^{2}\) is calculated as \(\partial^{2}\overline{\gamma}/\partial p^{2}\approx\left(\overline{\gamma}_{p _{2}}-2\overline{\gamma}_{p_{1}}+\overline{\gamma}_{p=0}\right)/p_{1}^{2}\) for small pressures \(p_{2}>p_{1}>0\). As we mentioned in Section II, since this term has a coefficient of \(B\), it will contribute the most to the negative \(B^{\prime}\) in Equation 10.
The values in the last two columns of Table 2 show a reasonable degree of consistency. The main differences between the two is mainly due to the anharmonic processes that are missing in the calculations of \(\overline{\gamma}\), \(\partial\overline{\gamma}/\partial p\) and \(\partial^{2}\overline{\gamma}/\partial p^{2}\) using harmonic lattice dynamics.
### Correlation between the pressure-induced softening and NTE
Negative values of \(\overline{\gamma}\), \(\partial\gamma/\partial p\) and \(\partial^{2}\gamma/\partial p^{2}\) can be satisfied by assuming a pressure-induced strain (\(e\)) dependence for the frequency of the NTE modes, namely
\[\omega^{2}=\omega_{0}^{2}+Ce>0, \tag{15}\]
Clearly, the frequency of the mode will decrease with more negative strain \(e\), i.e.
\[\gamma=-\frac{1}{2\omega^{2}}\frac{\partial\omega^{2}}{\partial e}=-\frac{C}{ \omega_{0}^{2}+Ce}<0 \tag{16}\]
From Equation 16, we have
\[\frac{\partial\gamma}{\partial p}=\frac{\partial e}{\partial p}\frac{\partial \gamma}{\partial e}=-\frac{C^{2}}{B\left(\omega_{0}^{2}+Ce\right)^{2}}<0 \tag{17}\]
Fitted \(B_{0}\) and \(B_{0}^{\prime}\) of FAU changing with temperature. Obviously, the average value of \(B_{0}^{\prime}\) throughout the temperature is negative, indicating pressure-induced softening of the material.
Calculated vibrational density of states of FAU using harmonic lattice dynamics, colored according to the values of Grüneisen parameters (see text for details). Red highlights the most negative Grüneisen parameters (\(\leq-9\)), while blue are for the positive values \(\geq 1\).
and
\[\frac{\partial^{2}\gamma}{\partial p^{2}}\approx\left(\frac{\partial e}{\partial p} \right)^{2}\frac{\partial^{2}\gamma}{\partial e^{2}}=-\frac{2C^{3}}{B^{2}\left( \omega_{0}^{2}+Ce\right)^{3}}<0 \tag{18}\]
Thus, a material with whose mode frequency is reasonably represent by Equation 15 will show NTE with negative \(\partial\gamma/\partial p\) as well as negative \(\partial^{2}\gamma/\partial p^{2}\), and therefore is likely to have negative \(B^{\prime}\) on heating according to Equation 10.
One way in which pressure, elasticity and thermal fluctuations are linked is in the pressure-dependence of the coefficient of thermal expansion, \(\alpha\), as shown by
\[\frac{1}{\alpha}\frac{\partial\alpha}{\partial p}=\frac{1}{\overline{\gamma}} \frac{\partial\overline{\gamma}}{\partial p^{2}}\left(\text{GPa}^{-2}\right) \left.\begin{array}{c}B^{\prime}\\ \end{array}\right|_{T=0} \tag{19}\]
For all the NTE zeolites (whose \(\alpha<0\)) listed in Table 2, both terms on the right-hand side of the equation would be positive, resulting in negative \(\partial\alpha/\partial p\), i.e. the coefficient of thermal expansion becomes more negative under pressure. This pressure-enhancement of NTE has been confirmed by our MD results. For example, shows both the simulation and the experimental data for FAU. Plots of this kind for the other zeolites are given in the supplemental material.
## V Conclusions
The main conclusion from this study is that all the cubic zeolites that show NTE and are stable under pressure have negative \(B^{\prime}\). This is a result from simulations and will need experimental verification, but the results are so overwhelmingly positive from simulation that we are confident they reflect the underlying physical processes. This lends strong support to our proposal that many NTE materials are likely to have pressure-induced softening.
The origin of the pressure-induced softening is rooted in the dependence of the frequencies of the NTE phonon modes on strain. With a simple form of frequency having positive dependence on pressure-induced strain, the phonon modes would not only have negative Gruneisen parameters but also have negative first and second derivatives of the Gruneisen parameter with respect to pressure, resulting in NTE as well as pressure-induced softening of the material.
With an increasing number of NTE materials being discovered, we suggest that there should be an increased focus on experimental searches for pressure-induced softening in these materials.
\begin{table}
\begin{tabular}{c|c|c|c|c|c|c|c} \hline \hline Zeolite & \(\tau/V_{\text{at}}\) (GPa) & \(\overline{\gamma}\) & \(\partial\overline{\gamma}/\partial p\) (GPa\({}^{-1}\)) & \(\left.\begin{array}{c}\partial^{2}\overline{\gamma}/\partial p^{2}\ (\text{GPa}^{-2}) \\ \end{array}\right|\) & \(\left.\begin{array}{c}B^{\prime}\\ \end{array}\right|_{T=0}\) & \(\left.\begin{array}{c}{}^{a}B^{\prime}\\ \end{array}\right|\) & \(\left.\begin{array}{c}{}^{b}B^{\prime}\\ \end{array}\right.\) \\ \hline FAU & \(0.16\) & \(-0.40\) & \(-0.27\) & \(-0.20\) & \(2.4\) & \(-2.7\) & \(-2.6\) \\ KFI & \(0.18\) & \(-0.62\) & \(-0.35\) & \(-0.18\) & \(2.6\) & \(-4.1\) & \(-4.4\) \\ RHO & \(0.18\) & \(-0.44\) & \(-0.23\) & \(-0.16\) & \(-0.1\) & \(-5.4\) & \(-4.0\) \\ LTA & \(0.18\) & \(-0.85\) & \(-0.60\) & \(-0.90\) & \(3.0\) & \(-27.1\) & \(-13.0\) \\ MEP & \(0.21\) & \(-1.65\) & \(-1.85\) & \(-5.22\) & \(-3.1\) & \(-192.1\) & \(-30.0\) \\ TSC & \(0.16\) & \(-0.41\) & \(-0.24\) & \(-0.10\) & \(0.5\) & \(-2.2\) & \(-4.9\) \\ \hline \hline \end{tabular} \({}^{a}\)Calculated using Equation 10.
\({}^{b}\)MD results at 300 K.
\end{table}
Table 2: Calculated various terms in Equation 10 for all the cubic NTE zeolites, except for AST and MTN which only show NTE in their high-temperature phases. The second column of the table lists the values of \(\tau/V_{\text{at}}=N\tau/V\) at 300 K, with \(\tau/V_{\text{at}}\) the average volume per atom in the system. The third column lists the values of the overall Grüneisen parameters, \(\overline{\gamma}\). The sixth column lists the values of \(B^{\prime}\) at zero temperature obtained from harmonic lattice dynamics. The seventh column lists the calculated \(B^{\prime}\) using Equation 10. The last column gives the values of \(B^{\prime}\) from molecular dynamics at 300 K.
Temperature dependence of the cell parameter of FAU at different pressures from MD simulations. The results agree well with the experimental data. The inset shows that the coefficient of thermal expansion (\(\alpha\) at 300 K) of the material decreases on compression, i.e. the pressure-enhanced NTE.
| 10.48550/arXiv.1306.2395 | Pressure-induced softening as a common feature of framework structures that have negative thermal expansion | Hong Fang, Martin T. Dove | 6,077 |
10.48550_arXiv.0708.3589 | ###### Abstract
A method to calculate NMR J-coupling constants from first principles in extended systems is presented. It is based on density functional theory and is formulated within a planewave-pseudopotential framework. The all-electron properties are recovered using the projector augmented wave approach. The method is validated by comparison with existing quantum chemical calculations of solution-state systems and with experimental data. The approach has been applied to verify measured J-coupling in a silicophosphate structure, Si\({}_{5}\)O(PO\({}_{4}\))\({}_{6}\).
## I Introduction
Nuclear Magnetic Resonance (NMR) allows information to be relayed through magnetic nuclei in a non-destructive and powerful approach to structural elucidation. It is a fundamental tool in a broad range of scientific disciplines and is a cornerstone of modern spectroscopy. NMR spectra yield a wealth of information, the most commonly reported property being the chemical shift. This parameter relates an externally applied magnetic field to the resulting change in the local electronic environment of the magnetic nuclei, thereby providing key insight into the underlying atomic structure.
NMR J-coupling or indirect nuclear spin-spin coupling is an indirect interaction of the nuclear magnetic moments mediated by the bonding electrons. It is manifested as the fine-structure in NMR spectra, providing a direct measure of bond-strength and a map of the connectivities of the system. The J-coupling mechanism is an essential component of many NMR experiments.
In solution-state, J-coupling measurements can often be obtained from one dimensional spectra where the multiplet splitting in the peaks is clearly resolved. However, in the solid-state this is not the case as these splittings are typically obscured by the broadenings from anisotropic interactions. Fortunately this technical challenge has not prevented the determination of J-coupling in the solid-state, as recent work employing spin-echo Magic Angle Spinning (MAS) techniques has resulted in accurate measurements of J-coupling in both inorganic materials and molecular crystals. In combination with the advances in solid-state experiments, there has also been an increased interest from the biomolecular community as J-coupling has been found to be a direct measure of hydrogen bond strength. Both of these factors have provided a strong impetus to develop first principles approaches to compute the NMR J-coupling constants in order to support experimental work, particularly for solid-state systems.
For finite systems, NMR parameters, including both chemical shifts and J-couplings, can be routinely calculated using traditional quantum chemistry approaches based on localised orbitals. Such calculations have been widely applied to assign the solution-state NMR spectra of molecular systems and establish key conformational and structural trends. In particular NMR J-couplings have been used to quantify hydrogen bonding in biological systems. In order to apply these techniques to solid-state NMR, it is necessary to devise finite clusters of atoms which model the local environment around a site of interest in the true extended structure. While this has led to successful studies of NMR chemical shifts in systems such as molecular crystals, supra-molecular assemblies and organo-metallic compounds, it is clear that there are advantages in an approach that inherently takes account of the long-range electrostatic effects in extended systems.
This observation has led to the recent development of the Gauge Including Projector Augmented Wave (GIPAW) method which enables NMR parameters to be calculated at all-electron accuracy within the planewave-pseudopotential formalism of density functional theory (DFT). The technique has been applied, in combination with experimental NMR spectroscopy, to systems such as minerals, glasses and molecular crystals.
In this paper we introduce a theory to compute NMR J-Couplings in extended systems using periodic boundary conditions and supercells with the planewave-pseudopotential approach. Like the GIPAW approach to calculating NMR chemical shifts, our method is formulated within the planewave-pseudopotential framework using density functional perturbation theory (DFPT). We use the projector-augmented-wave (PAW) reconstruction technique to calculate J-couplings with all-electron accuracy.
In the following section we discuss the physical mechanism of the indirect spin-spin interaction, the basis of the PAW approach and the supercell technique. In Sections III and IV we show how the J-coupling tensor maybe be calculated using PAW and DFPT. The method has been implemented in a parallel plane-wave electronic structurecode and we discuss details of the implementation and provide validation results in Section V.
## II NMR J-Coupling
We consider the interaction of two nuclei, K and L, with magnetic moments, \(\mathbf{\mu}_{\mathrm{K}}\) and \(\mathbf{\mu}_{\mathrm{L}}\), mediated by the electrons. The first complete analysis of this indirect coupling was provided by Ramsey who decomposed the interaction into four distinct physical mechanisms; two involving the interaction of the nuclear spins through the electron spin and two through the electron charge. In the absence of spin-orbit coupling i.e, for relatively light elements, the charge and spin interactions can be treated separately.
\[\mathbf{B}^{}_{\mathrm{in}}(\mathbf{R}_{\mathrm{L}}) = \frac{\mu_{0}}{4\pi}\int\left[\frac{3(\mathbf{m}^{}(\mathbf{r} )\cdot\mathbf{r}_{\mathrm{L}})\mathbf{r}_{\mathrm{L}}-\mathbf{m}^{}( \mathbf{r})|\mathbf{r}_{\mathrm{L}}|^{2}}{|\mathbf{r}_{\mathrm{L}}|^{5}}\right] \,\mathrm{d}^{3}\mathbf{r} \tag{1}\] \[+ \frac{\mu_{0}}{4\pi}\frac{8\pi}{3}\int\mathbf{m}^{}(\mathbf{r} )\delta(\mathbf{r}_{\mathrm{L}})\,\mathrm{d}^{3}\mathbf{r}\] \[+ \frac{\mu_{0}}{4\pi}\int\mathbf{j}^{}(\mathbf{r})\times\frac{ \mathbf{r}_{\mathrm{L}}}{|\mathbf{r}_{\mathrm{L}}|^{3}}\,\mathrm{d}^{3}\mathbf{ r}.\]
\(\mathbf{r}_{\mathrm{L}}=\mathbf{R}_{\mathrm{L}}-\mathbf{r}\), where \(\mathbf{R}_{\mathrm{L}}\) is the position of nucleus L, \(\mu_{0}\) is the permeability of a vacuum and \(\delta\) is the Dirac delta function.
\(\mathbf{\mu}_{\mathrm{K}}\) interacts with the electron spin through a magnetic field generated by a Fermi-contact term, which is due to the finite probability of the presence of an electron at the nucleus, and a spin-dipolar interaction. Both of these terms give rise to a first order spin magnetisation density, \(\mathbf{m}^{}(\mathbf{r})\). This magnetisation density then induces a magnetic field at the receiving nucleus by the same mechanisms, which in this case are given respectively by the first and second terms of Eqn. 1. The interaction between \(\mathbf{\mu}_{\mathrm{K}}\) and the electronic charge gives rise to an induced current density. To first-order this is given by \(\mathbf{j}^{}(\mathbf{r})\) and can be divided into a paramagnetic and a diamagnetic contribution.
The J-coupling tensor, \(\overrightarrow{\mathbf{J}}_{\mathrm{LK}}\), between L and K, can be related to the induced field by
\[\mathbf{B}^{}_{\mathrm{in}}(\mathbf{R}_{\mathrm{L}})=\frac{2\pi}{\hbar \gamma_{\mathrm{L}}\gamma_{\mathrm{K}}}\overleftrightarrow{\mathbf{J}}_{ \mathrm{LK}}\cdot\mathbf{\mu}_{\mathrm{K}}, \tag{2}\]
Although the physical interpretation of J-coupling is simplified by considering the interaction in terms of a responding and a perturbing nucleus, it is a symmetric coupling and either atom L or K can be considered as the perturbing site. Experimental interest is focused primarily on the isotropic coupling constant, \(\mathrm{J}^{n}_{\mathrm{LK}}\), which is obtained from the trace of \(\overleftrightarrow{\mathbf{J}}_{\mathrm{LK}}\) and is measured in Hz. The superscript, \(n\), denotes the order of the coupling in terms of the number of bonds separating the coupled nuclei. In a typical NMR experiment J-coupling can be measured across a maximum of three bonds. In this paper we concentrate solely on obtaining the isotropic or scalar value.
To calculate \(\overleftrightarrow{\mathbf{J}}_{\mathrm{KL}}\) we obtain \(\mathbf{m}^{}\) and \(\mathbf{j}^{}\) within density functional perturbation theory using a planewave expansion for the wavefunctions with periodic boundary conditions and pseudopotentials to represent the ionic cores. The use of pseudopotentials generates a complication as the J-coupling tensor depends critically on the wavefunction in the regions close to the perturbing and receiving nuclei, precisely the regions where the pseudo-wavefunctions have a non-physical form. To compensate for this we perform an all-electron reconstruction of the valence wavefunctions in the core region using Blochl's projector augmented wave (PAW) scheme.
Within this scheme, the expectation value of an operator \(O\), applied to the all-electron wavefunctions, \(|\psi\rangle\), is expressed in terms of the pseudised wavefunctions, \(|\widetilde{\psi}\rangle\), as: \(\langle\psi|O|\psi\rangle=\langle\widetilde{\psi}|\widetilde{O}|\widetilde{ \psi}\rangle\).
\[\widetilde{O} = O+\sum_{\mathbf{R},n,m}|\widetilde{p}_{\mathbf{R},n}\rangle \left[\langle\phi_{\mathbf{R},n}|O|\phi_{\mathbf{R},m}\rangle\right. \tag{3}\] \[\left.-\langle\widetilde{\phi}_{\mathbf{R},n}|O|\widetilde{\phi }_{\mathbf{R},m}\rangle\right]|\widetilde{p}_{\mathbf{R},m}|.\]
\(\mathbf{R}\) labels the atomic site, or augmentation region, and \(n\) and \(m\) are composite indexes which account for the angular momentum channels and the number of projectors. \(|\phi_{\mathbf{R},n}\rangle\) are the all-electron partial waves obtained as eigenstates of an atomic calculation within \(r_{\mathrm{c}}\), the pseudopotential core radius and \(|\widetilde{\phi}_{\mathbf{R},n}\rangle\) are the corresponding pseudo partial waves. \(|\widetilde{p}_{\mathbf{R},n}\rangle\) are the localised projectors which weight the superposition of partial waves where \(\langle\widetilde{p}_{\mathbf{R},n}|\widetilde{\phi}_{\mathbf{R}^{\prime},m} \rangle=\delta_{\mathbf{R}\mathbf{R}^{\prime}}\delta_{nm}\). The PAW method has been used to calculate several all-electron properties from pseudopotential calculations including: EPR hyperfine parameters, electric field gradient tensors and Electron Energy Loss Spectroscopy.
To calculate J-couplings in the solid-state using periodic boundary conditions, the perturbing nucleus can be viewed similar to a defect in a defect calculation. This allows us to use the standard technique of constructing supercells from the unit cell which are large enough to inhibit the interaction between the periodic defects or perturbations. This corresponds to extending the system-size to facilitate the decay of the induced magnetisation and current densities within the simulation cell. Figure. 1 is a schematic of a \(2\times 2\times 2\) supercell constructed from eight unit cells. The perturbing atom now lies at the corner of a much larger cell which decreases the interaction between the perturbation and its periodic image. This approach works very well for localised properties such as J-coupling. To calculate the J-coupling for molecules, we use a vacuum supercell technique. In both cases, the J-couplings must be converged with respect to the cell-size.
## III The spin magnetisation density
We now obtain the contribution to the J-coupling tensor which arises from the interaction of the nuclear spins mediated by the electron spin. We first obtain an expression for the pseudo-Hamiltonian in the presence of a perturbing nuclear spin and show how it can be used to obtain the induced magnetisation density. We then use the magnetisation density to calculate the magnetic field induced at the receiving nucleus.
### Pseudo-Hamiltonian
The all-electron Hamiltonian for a system containing N magnetic moments which interact through the electron spin, \(\mathbf{S}\), is expanded to first order in the magnetic moment of the perturbing site, \(\mathbf{\mu}_{\mathrm{K}}\), to give
\[\mathrm{H}=\frac{1}{2}\mathbf{p}^{2}+\mathrm{V}^{}(\mathbf{r})+\mathrm{V}^ {}(\mathbf{r})+\mathrm{H}_{\mathrm{SD}}+\mathrm{H}_{\mathrm{FC}}. \tag{4}\]
where
\[\mathrm{H}_{\mathrm{SD}}=g\beta\mathbf{S}\cdot\mathbf{B}_{\mathrm{K}}^{\mathrm{ SD}}, \tag{5}\]
and
\[\mathrm{H}_{\mathrm{FC}}=g\beta\mathbf{S}\cdot\mathbf{B}_{\mathrm{K}}^{\mathrm{ FC}}. \tag{6}\]
\(\mathbf{B}_{\mathrm{K}}^{\mathrm{SD}}\) is the magnetic field generated by a spin-dipole interaction,
\[\mathbf{B}_{\mathrm{K}}^{\mathrm{SD}}=\frac{\mu_{0}}{4\pi}\frac{3\mathbf{r}_{ \mathrm{K}}(\mathbf{r}_{\mathrm{K}}\cdot\mathbf{\mu}_{\mathrm{K}})-r_{\mathrm{K}} ^{2}\mathbf{\mu}_{\mathrm{K}}\mathcal{I}}{|r_{\mathrm{K}}|^{5}}, \tag{7}\]
and \(\mathbf{B}_{\mathrm{K}}^{\mathrm{FC}}\) is the Fermi contact interaction,
\[\mathbf{B}_{\mathrm{K}}^{\mathrm{FC}}=\frac{8\pi}{3}\delta(\mathbf{r}_{ \mathrm{K}})\mathbf{\mu}_{\mathrm{K}}. \tag{8}\]
We have defined \(\mathbf{r}_{\mathrm{K}}=\mathbf{R}_{\mathrm{K}}-\mathbf{r}\), where \(\mathbf{R}_{\mathrm{K}}\) is the position of nucleus K and \(\mathcal{I}\) is the identity matrix. Here \(\mathrm{V}^{}(\mathbf{r})\) is the ground-state all-electron local potential and \(\mathrm{V}^{}(\mathbf{r})\) is the corresponding first order variation. The latter term is due to the change in magnetisation density induced by \(\mathbf{\mu}_{\mathrm{K}}\). The perturbation does not give rise to a first order change in the charge density and so there is no change in the Hartree potential for linear order. This can be understood by considering the effect of time reversal; the charge density is even under time inversion, while the spin-magnetisation and magnetic field change sign. As a result the perturbation does not induce a first order change in the charge density, however there is a first order induced spin-magnetisation density. \(\mathrm{V}^{}(\mathbf{r})\) therefore accounts for the first-order variation of the exchange-correlation term which we label as \(\mathrm{H}_{\mathrm{xc}}^{}\).
We now use the PAW transformation (Eqn. 3) to obtain the pseudo-Hamiltonian. As Eqn. 3 does not contain any field dependence it is sufficient to apply it to each term of Eqn. 4 individually.
\[\widetilde{\mathrm{H}}^{}=\frac{1}{2}\mathbf{p}^{2}+\mathrm{V}_{\mathrm{loc} }(\mathbf{r})+\sum_{\mathbf{R}}\mathrm{V}_{\mathrm{nl}}^{\mathbf{R}}, \tag{9}\]
\(a_{n,m}^{\mathbf{R}}\) are the strengths of the nonlocal potential in each channel at each ionic site.
\[\widetilde{\mathrm{H}}^{}=\widetilde{\mathrm{H}}_{\mathrm{xc}}^{}+ \widetilde{\mathrm{H}}_{\mathrm{SD}}+\widetilde{\mathrm{H}}_{\mathrm{FC}}. \tag{10}\]
\(\widetilde{\mathrm{H}}_{\mathrm{SD}}\) describes the spin-dipolar interaction induced by \(\mathbf{\mu}_{\mathrm{K}}\) and is given by
\[\widetilde{\mathrm{H}}_{\mathrm{SD}}=g\beta\mathbf{S}\cdot\mathbf{B}_{\mathrm{ K}}^{\mathrm{SD}}+g\beta\mathbf{S}\cdot\Delta\mathbf{B}_{\mathrm{K}}^{ \mathrm{SD}}. \tag{11}\]
The first term on the right-hand side is the all-electron operator and the second term is the augmentation to this,
\[\Delta\mathbf{B}_{\mathrm{K}}^{\mathrm{SD}} = \sum_{n,m}|\widetilde{p}_{\mathbf{R},n}\rangle\left[\ \langle \phi_{\mathbf{R},n}|\mathbf{B}_{\mathrm{K}}^{\mathrm{SD}}|\phi_{\mathbf{R},m}\rangle\right.\] \[\left.-\langle\widetilde{\phi}_{\mathbf{R},n}|\mathbf{B}_{ \mathrm{K}}^{\mathrm{SD}}|\widetilde{\phi}_{\mathbf{R},m}\rangle\ \right] \langle\widetilde{p}_{\mathbf{R},m}|,\]
In Eqn. 11 we have only included the augmentation of the spin-dipolar operator at the site of the perturbing atom. This on-site approximation is fully justified given the localised nature of this operator.
\(\widetilde{\mathrm{H}}_{\mathrm{FC}}\) is the Fermi-contact operator and can be constructed in a similar manner to the spin-dipole operator giving an all-electron and an augmentation contribution. However, as the Fermi-contact operator contains a Dirac delta-function and is therefore localised within the augmentation region, \(\widetilde{\mathrm{H}}_{\mathrm{FC}}\) can be simplified considerably.
\[\widetilde{\mathrm{H}}_{\mathrm{FC}}=g\beta\mathbf{S}\cdot\sum_{n,m}| \widetilde{p}_{\mathbf{R},n}\rangle\langle\phi_{\mathbf{R},n}|\mathbf{B}_{ \mathrm{K}}^{\mathrm{FC}}|\phi_{\mathbf{R},m}\rangle\langle\widetilde{p}_{ \mathbf{R},m}|, \tag{13}\]
Schematic of the supercell technique. The unit cell is on the left, a 2\(\times\)2\(\times\)2 supercell of the unit cell is on the right. This supercell doubles the distance between the perturbing atom (black) and its periodic image in the next cell.
This form is more suitable for a practical calculation as it avoids an explicit representation of the delta-function.
### Magnetisation Density
To construct the magnetisation density, we define \({\bf m}_{i}^{}({\bf r})\) to be the linear response to the magnetic field, \({\bf B}_{i}\) induced along the direction \({\bf u}_{i}\) by the spin-dipolar and Fermi-contact interactions. The total magnetisation density is obtained as \({\bf m}^{}=\sum_{i=x,y,z}{\bf m}_{i}^{}({\bf r})\), the sum over the cartesian directions. By choosing \({\bf u}_{i}\) as the spin quantisation axis, \({\rm H}^{}\) is diagonal in the spin-up and spin-down basis. The eigenstates of \({\rm H}^{}+{\rm H}^{}\) are also eigenstates of \({\bf u}_{i}\cdot{\bf S}\) and so the magnetisation density is parallel to \({\bf u}_{i}\) giving;
\[{\bf m}_{i}^{}({\bf r})=[{\bf u}_{i}\cdot{\bf m}_{i}^{}({\bf r})]{\bf u} _{i}=m_{i}^{}({\bf r}){\bf u}_{i}. \tag{14}\]
Here
\[m_{i}^{}({\bf r})=g\beta\left[n_{i,\uparrow}^{}({\bf r})-n_{i,\downarrow }^{}({\bf r})\right]=2g\beta n_{i,\uparrow}^{}({\bf r}), \tag{15}\]
The simplification of the magnetisation density in this way is a consequence of time reversal symmetry, namely the absence of a first-order charge density. This means that the spin-up and spin-down ground-state wavefunctions are equivalent, so that \(|\widetilde{\psi}_{\uparrow\sigma}^{}\rangle=|\widetilde{\psi}_{\downarrow \sigma}^{}\rangle\). Also, the linear variation of the wavefunctions induced by the spin magnetisation are related through \(|\widetilde{\psi}_{\uparrow\sigma}^{}\rangle=-|\widetilde{\psi}_{\downarrow \sigma}^{}\rangle\).
Within PAW, Eqn. 15 becomes
\[m_{i}^{}({\bf r}) = 4g\beta{\rm Re}\sum_{o}\langle\widetilde{\psi}_{\uparrow\sigma}^ {}|{\bf r}\rangle\langle{\bf r}|\widetilde{\psi}_{\uparrow\sigma}^{}\rangle \tag{16}\] \[+\sum_{{\bf R},n,m}\langle\widetilde{\psi}_{\uparrow\sigma}^{} |\widetilde{p}_{{\bf R},n}\rangle\Big{[}\langle\phi_{{\bf R},n}|{\bf r} \rangle\langle{\bf r}|\phi_{{\bf R},m}\rangle\] \[-\langle\widetilde{\phi}_{{\bf R},n}|{\bf r}\rangle\langle{\bf r }|\widetilde{\phi}_{{\bf R},m}\rangle\Big{]}\langle\widetilde{p}_{{\bf R},n}| \widetilde{\psi}_{\sigma\uparrow}^{}\rangle.\]
Re signifies taking the real component, \(|\widetilde{\psi}_{\uparrow\sigma}^{}\rangle\) are the eigenstates of the unperturbed Hamiltonian, \({\rm H}^{},|\widetilde{\psi}_{\uparrow\sigma}^{}\rangle\) are the perturbed pseudowavefunctions and \(o\) indexes the occupied bands. The first term on the right hand side of Eqn. 16 is the pseudo-magnetisation density \(\widetilde{m}_{i}^{}\), and the second term is the corresponding augmentation. For simplicity, we drop the spin indexing on the ground-state wavefunctions from now on as the spin-dependence enters only through the perturbation.
To calculate \(|\widetilde{\psi}_{\uparrow\sigma}^{}\rangle\) we employ a Green's function method where
\[|\widetilde{\psi}_{\uparrow\sigma}^{}\rangle={\cal G}(\epsilon)|\psi_{o}^{( 0)}\rangle=\sum_{e}\frac{|\psi_{e}^{}\rangle\langle\psi_{e}^{}|}{ \epsilon_{o}-\epsilon_{e}}\widetilde{\rm H}_{i}^{}|\psi_{o}^{}\rangle. \tag{17}\]
\(\widetilde{\rm H}_{i}^{}\) is the first order Hamiltonian given by Eqn. 10 with the spin quantised along the \({\bf u}_{i}\) direction. \({\cal G}(\epsilon)\) is the Green's function, \(\epsilon_{o}\) and \(\epsilon_{e}\) are the eigenvalues of the occupied and empty bands. Rather than explicitly sum over the empty states, we project onto the occupied bands by multiplying Eqn. 17 through by \((\epsilon_{o}-\widetilde{\rm H}^{})\). We define \({\cal P}=\sum_{e}|\widetilde{\psi}_{e}^{}\rangle\langle\widetilde{\psi}_{ e}^{}|=1-\sum_{o}|\widetilde{\psi}_{\sigma}^{}\rangle\langle\widetilde{\psi}_{ \sigma}^{}|\) and rewrite Eqn. 17 as
\[(\epsilon_{o}-\widetilde{\rm H}^{})|\widetilde{\psi}_{\uparrow\sigma}^{ }\rangle={\cal P}\widetilde{\rm H}_{i}^{}|\widetilde{\psi}_{\sigma}^{}\rangle. \tag{18}\]
This is then solved using a conjugate gradient minimisation scheme, with an additional self-consistency condition to account for the dependence of \({\rm H}_{\rm xc}^{}\) on the spin-density. For a more detailed account of this type of approach, see Ref..
### Induced Magnetic Field
The induced magnetic field at atom L, and subsequently the J-coupling between L and K, due to the spin magnetisation is obtained by combining Eqns. 1 and 16 to give
\[{\bf B}_{{\bf m}^{}}^{}({\bf R}_{\rm L})=\widetilde{\bf B}_{\rm SD}^{} ({\bf R}_{\rm L})+\Delta{\bf B}_{\rm SD}^{}({\bf R}_{\rm L})+\Delta{\bf B}_ {\rm FC}^{}({\bf R}_{\rm L}), \tag{19}\]
1 to yield three separate terms. The first term is the pseudo spin-dipolar contribution and the second term is the spin-dipole augmentation. The notation used here implicitly assumes that a rotation over the spin-axis has been performed.
In practise, the pseudo-spin dipole term can be constructed from the Fourier transform of Eqn. 1,
\[{\bf B}_{\rm SD}^{}({\bf G})=-\frac{\mu_{0}}{3}\left[\frac{3(\widetilde{ \bf m}^{}({\bf G})\cdot{\bf G}){\bf G}-\widetilde{\bf m}^{}({\bf G})|{\bf G }|^{2}}{G^{2}}\right], \tag{20}\]
The \(G=0\) term is neglected as the cell size is large compared with the strength of the perturbation which is small.
The induced magnetic field at atom L is then recovered by performing a slow inverse Fourier transform at the position of each responding nucleus.
\[\Delta{\bf B}_{\rm SD}^{}({\bf R}_{\rm L})=g\beta\frac{\mu_{0}}{2\pi}{\rm Re }\sum_{n,m}\langle\widetilde{\psi}_{o}^{}|\Delta{\bf B}_{\rm L}^{\rm SD}| \widetilde{\psi}_{\uparrow\sigma}^{}\rangle. \tag{21}\]
\(\Delta{\bf B}_{\rm L}^{\rm SD}\) is defined in Eqn. 12 but now the subscript indicates the responding rather than the perturbing nucleus. To evaluate this term and Eqn. 12, we note that \(|\phi_{n}\rangle\) can be decomposed into the product of a radial (\(|{\rm R}_{nl}\rangle\)) and an angular (\(|Y_{lm}\rangle\)) term. \(B_{\rm L}^{\rm SD}\) can also be rewritten as the product of a radial and angular component such that the computation of the augmentation term involves the on-site calculation of \(\langle R_{nl}|\frac{1}{r_{\rm L}^{\rm L}}|R_{n^{\prime}l^{\prime}}\rangle\langle Y _{n}|\frac{3\mathbf{r}_{\rm L}\mathbf{r}_{\rm L}^{\rm T}}{r_{\rm L}^{\rm L}}- \mathcal{I}|Y_{m}\rangle\). The latter quantity reduces to an integral over spherical harmonics given by the Gaunt coefficients.
The Fermi-contact contribution, \(\Delta\mathbf{B}_{\rm FC}^{}\), is obtained by following the same argument used in constructing \(\widetilde{\rm H}_{\rm FC}\) (Eqn. 13) and is given by
\[\Delta\mathbf{B}_{\rm FC}^{}(\mathbf{R}_{\rm L})=\] \[\frac{4\mu_{0}}{3}\mathrm{Re}\sum_{n,m}\langle\widetilde{\psi}_{ \rm o}^{}|\widetilde{p}_{\mathbf{R},n}\rangle\langle\phi_{\mathbf{R},n}| \delta(\mathbf{r}_{\rm L})|\phi_{\mathbf{R},m}\rangle\langle\widetilde{p}_{ \mathbf{R},m}|\widetilde{\psi}_{\uparrow\circ}^{}\rangle, \tag{22}\]
## IV Current density
We now obtain the contribution to the J-coupling tensor arising from the interaction of the nuclear spins mediated by the electron charge current. The derivation of the current density is similar to that of the magnetisation density and much of the notation is conserved throughout. We first obtain an expression for the pseudo-Hamiltonian in the presence of a perturbing nuclear spin and show how it can be used to obtain the induced current density. We then use this current density to calculate the magnetic field induced at the receiving nucleus.
### Pseudo-Hamiltonian
The all-electron Hamiltonian for a system of N magnetic nuclei which interact through the nuclear vector potential, \(\mathbf{A}(\mathbf{r})=\sum_{\rm N}\mathbf{A}_{\rm N}(\mathbf{r})=\frac{\mu_{ 0}}{4\pi}\sum_{\rm N}\frac{\mu_{\rm N}\times\mathbf{r}_{\rm N}}{|\mathbf{r}_{ \rm N}|^{3}}\), can be expanded to first order in \(\mathbf{\mu}_{\rm K}\) to give,
\[\mathrm{H}_{\rm K}=\frac{1}{2}\mathbf{p}^{2}+\mathrm{V}^{}(\mathbf{r})+ \mathrm{H}_{\mathbf{A}_{\rm K}}, \tag{23}\]
where
\[\mathrm{H}_{\mathbf{A}_{\rm K}}=\frac{\mu_{0}}{4\pi}\mathbf{p}\cdot\mathbf{A}_ {\rm K}(\mathbf{r}), \tag{24}\]
The perturbation does not induce a first order change in either the charge or magnetisation densities and so unlike Eqn. 4, there is no linear variation to the self-consistent potential in Eqn. 23. We have used the symmetric gauge for the vector potential and taken the natural choice of gauge-origin; namely that for the Nth nuclear spin the gauge origin is the Nth atomic site giving \(\mathbf{A}_{\rm N}(\mathbf{r})=1/2\mathbf{B}(\mathbf{r})\times\mathbf{r}_{\rm N}\). This gauge-choice preserves the translational invariance of the system and is much simpler than in the otherwise analogous case of NMR chemical shielding. In the latter situation, due to the use of finite basis sets a rigid translation of the system in the uniform external magnetic field introduces an additional phase factor. For the planewave-pseudopotential approach the problem was solved by Pickard and Mauri with the development of the Gauge-Including Projector Augmented-Wave (GIPAW) approach, an extension which is unnecessary here.
To obtain the pseudo-Hamiltonian we now apply the PAW transformation of Eqn. 3 to Eqn. 23. The zeroth order term is again given by Eqn. 9 and the first order term by
\[\widetilde{\mathrm{H}}_{\mathbf{A}_{\rm K}} = \frac{\mu_{0}}{4\pi}\mathbf{\mu}_{\rm K}\cdot\frac{\mathbf{r}_{\rm K }\times\mathbf{p}}{|\mathbf{r}_{\rm K}|^{3}}+ \tag{25}\] \[-\left\langle\widetilde{\phi}_{\mathbf{R},n}|\frac{\mathbf{L}_{ \rm K}}{|\mathbf{r}_{\rm K}|^{3}}|\widetilde{\phi}_{\mathbf{R},m}\right\rangle \left|\widetilde{p}_{\mathbf{R},m}\right|.\]
The first term on the right-hand side is the all-electron component and the second term is the augmentation which is constructed at the site of the perturbing nucleus. \(\mathbf{L}_{\rm K}=\mathbf{r}_{\rm K}\times\mathbf{p}\) is the angular momentum operator centred on the perturbing atomic site.
### Current Density
The current density operator, \(\mathbf{J}(\mathbf{r})\), is given by the sum of a paramagnetic and a diamagnetic term,
\[\mathbf{J}(\mathbf{r})=\mathbf{J}^{\rm p}(\mathbf{r})+\mathbf{J}^{\rm d}( \mathbf{r}), \tag{26}\]
where the paramagnetic term is given by
\[\mathbf{J}^{\rm p}(\mathbf{r})=-\left[\mathbf{p}|\mathbf{r}\rangle\langle \mathbf{r}|+|\mathbf{r}\rangle\langle\mathbf{r}|\mathbf{p}\right]/2, \tag{27}\]
and the diamagnetic term is
\[\mathbf{J}^{\rm d}(\mathbf{r})=-\mathbf{A}(\mathbf{r})|\mathbf{r}\rangle \langle\mathbf{r}|. \tag{28}\]
If we consider the current due only to the perturbing nucleus, and with our atomic choice of gauge origin, the diamagnetic term can be written
\[\mathbf{J}^{\rm d}_{\rm K}(\mathbf{r})=-\frac{\mu_{0}}{4\pi}\frac{\mathbf{\mu}_{ \rm K}\times\mathbf{r}_{\rm K}}{r_{\rm K}^{3}}|\mathbf{r}\rangle\langle \mathbf{r}|. \tag{29}\]
By applying Eqn. 3 to both Eqns. 27 and 29, we obtain the pseudo-current density operator within PAW
\[\widetilde{\mathbf{J}}(\mathbf{r})=\mathbf{J}^{\rm p}(\mathbf{r})+\mathbf{J}^{ \rm d}_{\rm K}(\mathbf{r})+\left[\Delta\mathbf{J}^{\rm p}(\mathbf{r})+\Delta \mathbf{J}^{\rm d}_{\rm K}(\mathbf{r})\right], \tag{30}\]
where the paramagnetic augmentation operator is
\[\Delta\mathbf{J}^{\rm p}(\mathbf{r}) = \sum_{\mathbf{R},n,m}|\widetilde{p}_{\mathbf{R},n}\rangle\left[ \langle\phi_{\mathbf{R},n}|\mathbf{J}^{\rm p}|\phi_{\mathbf{R},m}\rangle\right. \tag{31}\] \[\left.-\langle\widetilde{\phi}_{\mathbf{R},n}|\mathbf{J}^{\rm p}| \widetilde{\phi}_{\mathbf{R},m}\rangle\right]\langle\widetilde{p}_{\mathbf{R},m}|,\]and the corresponding diamagnetic operator is
\[\Delta\mathbf{J}_{\mathrm{K}}^{\mathrm{d}}(\mathbf{r}) = \sum_{\mathbf{R},n,m}|\widetilde{p}_{\mathbf{R},n}\rangle\left[ \langle\phi_{\mathbf{R},n}|\mathbf{J}_{\mathrm{K}}^{\mathrm{d}}|\phi_{\mathbf{R },m}\rangle\right. \tag{32}\] \[\left.-\langle\widetilde{\phi}_{\mathbf{R},n}|\mathbf{J}_{\mathrm{ K}}^{\mathrm{d}}|\widetilde{\phi}_{\mathbf{R},m}\rangle\ |\widetilde{\phi}_{ \mathbf{R},m}|.\right.\]
Arranging terms in \(\mathbf{J}(\mathbf{r})\) to zeroth and linear order in \(\mathbf{\mu}_{\mathrm{K}}\) gives
\[\mathbf{J}^{}(\mathbf{r})=\mathbf{J}^{\mathrm{p}}(\mathbf{r})+\Delta \mathbf{J}^{\mathrm{p}}(\mathbf{r}), \tag{33}\]
and
\[\mathbf{J}^{}(\mathbf{r})=\mathbf{J}_{\mathrm{K}}^{\mathrm{d}}(\mathbf{r}) +\Delta\mathbf{J}_{\mathrm{K}}^{\mathrm{d}}(\mathbf{r}). \tag{34}\]
Using Eqns. 33 and 34 we are now able to obtain the first-order induced current density, which within density functional perturbation theory is given by
\[\mathbf{j}^{}(\mathbf{r}) = 2\sum_{o}2\operatorname{Re}\langle\widetilde{\psi}_{o}^{}| \widetilde{\mathbf{J}}^{}|\widetilde{\psi}_{o}^{}\rangle+\langle \widetilde{\psi}_{o}^{}|\widetilde{\mathbf{J}}^{}|\widetilde{\psi}_{o}^ {}\rangle, \tag{35}\] \[= \mathbf{j}_{p}^{}(\mathbf{r})+\mathbf{j}_{\mathrm{d}}^{}( \mathbf{r}).\]
Here \(|\widetilde{\psi}_{o}^{}\rangle\) is the unperturbed wavefunction, \(|\widetilde{\psi}_{o}^{}\rangle\) is the perturbed wavefunction and \(o\) indexes the occupied bands. The first term on the right hand side is the paramagnetic contribution to the induced current and the second term is the diamagnetic contribution. The first order wavefunction is again obtained using Eqn. 18 where \(\widetilde{\mathrm{H}}^{}\) is now given by Eqn. 25.
### Induced Magnetic Field
We can now combine Eqns. 1 and 35 to calculate the J-coupling between nuclei L and K arising from the magnetic field induced by the orbital current.
\[\mathbf{B}_{\mathbf{j}^{}}^{}(\mathbf{R}_{\mathrm{L}}) = \widetilde{\mathbf{B}}_{\mathrm{p}}^{}(\mathbf{R}_{\mathrm{L}} )+\widetilde{\mathbf{B}}_{\mathrm{d}}^{}(\mathbf{R}_{\mathrm{L}})+\Delta \mathbf{B}_{\mathrm{p}}^{}(\mathbf{R}_{\mathrm{L}}) \tag{36}\] \[+\Delta\mathbf{B}_{\mathrm{d}}^{}(\mathbf{R}_{\mathrm{L}}),\]
To calculate the pseudised contributions to the current density we obtain the Fourier transform of \(\widetilde{\mathbf{B}}_{\mathrm{p}}^{}(\mathbf{R}_{\mathrm{L}})\) and \(\widetilde{\mathbf{B}}_{\mathrm{d}}^{}(\mathbf{R}_{\mathrm{L}})\) giving
\[\mathbf{B}^{}(\mathbf{G})=\mu_{0}\frac{i\mathbf{G}\times\mathbf{j}_{\mathrm{ p}/\mathrm{d}}^{}(\mathbf{G})}{G^{2}}, \tag{37}\]
To obtain the induced field at the atom site L we perform a slow Fourier transform of Eqn. 37. We again note that the G=0 contribution to \(\mathbf{B}^{}(\mathbf{G})\) is neglected as the contribution is expected to be small.
\[\Delta\mathbf{B}_{\mathrm{p}}^{}(\mathbf{R}_{\mathrm{L}}) = \frac{\mu_{0}}{4\pi}\sum_{n,m}\langle\widetilde{\psi}_{o}^{}| \widetilde{p}_{\mathbf{R},n}\rangle\left[\langle\phi_{\mathbf{R},n}|\frac{ \mathbf{L}_{\mathrm{L}}}{\mathbf{r}_{\mathrm{L}}^{3}}|\phi_{\mathbf{R},m}\rangle\right. \tag{38}\] \[-\left.\langle\widetilde{\phi}_{\mathbf{R},n}|\frac{\mathbf{L}_{ \mathrm{L}}}{\mathbf{r}_{\mathrm{L}}^{3}}|\widetilde{\phi}_{\mathbf{R},m} \rangle\right]\langle\widetilde{p}_{\mathbf{R},m}|\widetilde{\psi}_{o}^{}\rangle,\]
The augmentation to the diamagnetic current is given by
\[\Delta\mathbf{B}_{\mathrm{d}}^{} = \frac{\mu_{0}}{4\pi}\sum_{\mathbf{R},n,m}\langle\widetilde{\psi} _{o}^{}|\widetilde{p}_{\mathbf{R},n}\rangle\] \[\left[\langle\phi_{\mathbf{R},n}|\frac{r_{\mathrm{L}}r_{\mathrm{ K}}-\mathbf{r}_{\mathrm{L}}r_{\mathrm{K}}^{\mathrm{T}}}{r_{\mathrm{L}}^{3}r_{ \mathrm{K}}^{3}}|\phi_{m}\rangle\right.\] \[- \left.\langle\widetilde{\phi}_{\mathbf{R},n}|\frac{r_{\mathrm{L}}r_ {\mathrm{K}}-\mathbf{r}_{\mathrm{L}}r_{\mathrm{K}}^{\mathrm{T}}}{r_{\mathrm{L}}^{3}r_ {\mathrm{K}}^{3}}|\widetilde{\phi}_{m}\rangle\right]\langle\widetilde{p}_{ \mathbf{R},m}|\widetilde{\psi}_{o}^{}\rangle.\]
This is much more difficult to evaluate than any other term as an on-site approximation cannot be used due to the presence of the \(\mathbf{r}_{\mathrm{K}}\) position vector within the augmentation summation. However, previous quantum chemical studies have shown that the overall contribution to the J-coupling from the diamagnetic term is very small compared with any of the other three contributions. In light of this, we have neglected this term in our current implementation.
## V Results
We have implemented our theory into a parallelised plane-wave electronic structure code. The ground-state wavefunctions and Hamiltonian are obtained self-consistently after which the isotropic J-coupling constant is calculated using the outlined approach. In our implementation we use norm-conserving Troullier-Martins pseudopotentials. For the all-electron reconstruction we used two projectors per angular momentum channel.
In the following sections we compare our approach to existing quantum chemistry approaches which use localised basis sets and with experiment.
### Molecules
To validate our method we have calculated isotropic coupling constants for a range of small molecules and compared them with experiment. There are several studies of calculated J-couplings for small molecules reported in the literature, using a variety of theoretical approaches, see Ref. and references therein. We compare our results to calculations presented by Lantto _et al_ which were obtained within DFT using the BLYP functional and with the Multi-configurational Self-Consistent Field (MCSCF) approach. For consistency we use the molecular geometries reported in their work.
To obtain the isotropic J-coupling we use a supercell of size 1728 A\({}^{3}\) for each molecule with the exception of benzene which required a larger cell-size of 3375 A\({}^{3}\). The exchange-correlation was approximated by GGA-PBE and an energy cut-off of 80 Ry was imposed on the planewave expansion. All calculations sample the Brillouin zone at the gamma-point and used norm-conserving Trouillier-Martins pseudopotentials. The calculated J-coupling against experiment for the molecules are shown in alongside the results of a linear regression for each set of data. These results are presented as reduced spin-coupling constants, which are given by \(\overleftrightarrow{\mathbf{K}}_{\mathrm{LK}}=\frac{2\pi^{2}\overrightarrow{ \mathbf{J}}_{\mathrm{LK}}}{\hbar\gamma_{\mathrm{LK}}\gamma_{\mathrm{K}}}\), and so are independent of nuclear species. The graph indicates an excellent overall agreement with both experiment and the other approaches. The accuracy of the planewave PBE calculations is comparable with the all-electron BLYP results, with regression coefficients of 0.92 and 1.08 respectively. The correlation coefficients for the BLYP, MCSCF and PW data are 0.97, 0.97 and 0.99 respectively, suggesting a smaller random error in the planewave approach. Unsurprisingly, the most accurate couplings are given by MCSCF, which provides a more comprehensive description of electron correlation but is computationally more demanding than DFT.
In Table 1 we present the J-coupling values calculated for benzene. The results compare favourably with both the existing approaches and with experiment. The MCSCF approach systematically overestimates the J-coupling for both JCH and JCC compared with experiment. This is due to the use of a restricted basis set which was necessary given the size of the system, for further details see Ref..
The decomposition of the J-coupling into the four components serves as an illustration of the relative strengths of each contribution and the trends over several bonds. Lantto _et al_ have only presented this separation for JCC. It is clear that the Fermi-contact is the dominant mechanism in the coupling and that the diamagnetic component is consistently the smallest and is often negligible.
### Crystals
Due to the difficulties encountered measuring J-coupling in solid-state systems there are very few values to be found in the literature that are suitable for validation of our approach. Recently Coelho _et al_. provided an estimate for the two bond coupling between \({}^{29}\)Si and \({}^{31}\)P pairs in the silicophosphate Si\({}_{5}\)O(PO\({}_{4}\))\({}_{6}\). Subsequently this was followed by a more accurate determination which identified the four Si-O-P couplings. We have calculated NMR chemical shifts and \(J_{\mathrm{P-O-Si}}^{2}\) for Si\({}_{5}\)O(PO\({}_{4}\))\({}_{6}\) to validate our approach.
The structure of Si\({}_{5}\)O(PO\({}_{4}\))\({}_{6}\) is trigonal (a=7.869A, c=24.138A, 36 atoms per primitive cell) and contains
\begin{table}
\begin{tabular}{l l c c c c c c} \hline \hline \(J_{\mathrm{KL}}^{1}\) & Method & D & P & FC & SD & Total & Expt[Hz] \\ \hline J\({}_{\mathrm{CC}}^{1}\) & PW(PBE) & 0.2 & \(-\)6.9 & 58.2 & 1.9 & 53.4 & 55.8 \\ & BLYP & 0.2 & \(-\)6.8 & 63.8 & 1.0 & 58.2 & & \\ & MCSCF & 0.4 & \(-\)6.6 & 75.1 & 1.5 & 70.9 & & \\ J\({}_{\mathrm{CC}}^{2}\) & PW(PBE) & 0.0 & 0.0 & \(-\)1.2 & \(-\)0.3 & \(-\)1.5 & \(-\)2.5 \\ & BLYP & 0.0 & 0.1 & 0.0 & \(-\)0.5 & \(-\)0.4 & & \\ & MCSCF & -0.2 & 0.0 & \(-\)3.7 & \(-\)1.1 & \(-\)5.0 & & \\ J\({}_{\mathrm{CC}}^{3}\) & PW(PBE) & 0.0 & 0.6 & 7.3 & 1.1 & 9.0 & 10.1 \\ & BLYP & 0.0 & 0.5 & 8.4 & 1.5 & 10.4 & & \\ & MCSCF & 0.1 & 0.4 & 16.8 & 1.8 & 19.1 & & \\ J\({}_{\mathrm{CH}}^{1}\) & PW(PBE) & 0.5 & 0.9 & 132.5 & \(-\)0.2 & 133.7 & 158.3 \\ & BLYP & & & & & 155.2 & & \\ & MCSCF & & & & 185.1 & & \\ J\({}_{\mathrm{CH}}^{2}\) & PW(PBE) & \(-\)0.1 & 0.2 & 5.1 & 0.1 & 5.3 & 1.0 \\ & BLYP & & & & & 1.1 & \\ & MCSCF & & & & & \(-\)9.8 & & \\ J\({}_{\mathrm{CH}}^{3}\) & PW(PBE) & \(-\)0.2 & \(-\)1.1 & 6.0 & 0.0 & 4.7 & 7.6 \\ & BLYP & & & & & 7.4 & \\ & MCSCF & & & & 12.9 & & \\ J\({}_{\mathrm{CH}}^{4}\) & PW(PBE) & \(-\)0.1 & 0.3 & \(-\)0.4 & 0.0 & \(-\)0.2 & \(-\)1.2 \\ & BLYP & & & & & \(-\)1.3 & & \\ & MCSCF & & & & & \(-\)6.1 & \\ \hline \hline \end{tabular}
\end{table}
Table 1: J-coupling[Hz] in benzene. PW(PBE) labels the current planewave approach. The BLYP, MCSCF and experimental values were taken from Ref.. D, P, FC and SD label the diamagnetic, paramagnetic, Fermi-contact and spin-dipolar contributions respectively. All values are in Hz.
J-couplings calculated for a set of molecules. Both the MCSCF and BLYP results were taken from Ref.. All of the experimental values were also taken from this paper. The PW(PBE) results are from the present work. The lines are obtained from a linear regression of the calculated values with experiment. All values are quoted in \(10^{19}\)T\({}^{2}\)J\({}^{-1}\), the unit of the reduced coupling constant.
Two of these Si sites are 6-fold coordinated, Si\({}_{1}\) and Si\({}_{2}\), and the third site, Si\({}_{3}\), is four-fold coordinated. Si\({}_{1}\) is bonded to six equivalent oxygen atoms, Si\({}_{2}\) is bonded to six oxygen atoms which are comprised of two distinct sites. Si\({}_{3}\) is bonded to three equivalent oxygen atoms and one oxygen from an Si\({}_{3}\)-O tetrahedron. Thus there is one \({}^{31}\)P chemical shift, three \({}^{29}\)Si chemical shifts and four unique \({}^{2}\)J\({}_{\rm P-O-Si}\) couplings.
We obtained the structure from the Chemical Database Service at Daresbury. Prior to calculating the NMR parameters, we performed a full geometry optimisation on the structure, using a planewave cut-off of 70 Ryd and norm-conserving pseudopotentials. The GGA-PBE exchange-correlation functional was used and a Monkhorst-Pack k-point grid with a maximum of 0.1 A\({}^{-1}\) between sampling points. We calculated the NMR chemical shifts using the GIPAW approach with the same parameters used for the geometry optimisation. The J-coupling between P and Si was obtained using the approach outlined above. A slightly higher maximum planewave energy (80Ryd) was required to give J-couplings converged to within 0.1Hz.
We tested the convergence of the induced magnetization density and current density with respect to supercell size. The results of these calculations for three cell sizes are presented in table 2. From this we can see that both the induced magnetization and current densities have decayed substantially within the single unit cell. The largest of these calculations (144 atoms) was parallelised over 16 dual-core AMD processors and took 45 hours to run. The groundstate calculation took approximately 14 hours and the J-coupling terms; Fermi-contact, spin-dipolar and orbital, required 3.5 hours, 15.5 hours and 11.4 hours respectively.
The results for the 2\(\times\)2\(\times\)1 cells are presented in comparison with experiment in Table 3. From Table 3 it is clear that the calculated J-couplings are in excellent agreement with experiment and fully reproduce the surprisingly large spread in the J-coupling values. Our calculations verify the novel experimental work and also identify the sign of the couplings which are not determined by the experimental spin-echo based approaches.
The NMR chemical shifts are also in good agreement with experiment, particularly for \({}^{29}\)Si. For both \({}^{29}\)Si and \({}^{31}\)P the difference between the calculated and experimental values is a very small fraction of the total shift range. We note that our assignment of the three Si sites in Si\({}_{5}\)O(PO\({}_{4}\))\({}_{6}\) agrees with the assignment based on experimental intensities as discussed by Coelho _et al_. In Table 4 we present the decomposition of the siliconphosphate J-coupling into their constituent terms. As with benzene, the Fermi-contact is found to be consistently the largest component while the diamagnetic and spin-dipolar contributions are very small.
## VI Conclusions
We have developed an all-electron approach for calculating NMR J-coupling constants using planewaves and pseudopotentials within DFT. Our method is applicable to both solution and solid state systems using supercell techniques. We have validated our theory against existing quantum chemical approaches and experiment for molecules. We have calculated the J-coupling between Si and P in a silicophosphate polymorph, for which we have determined the sign of the coupling.
Given the recent experimental interest in J-coupling, we expect that our approach will prove useful in determining both the range and strength of coupling in systems not yet investigated and whether or not such couplings can feasibly be determined by experiment. By combining J-coupling calculations with computations of other NMR parameters, there now exists a comprehensive set of computational tools to complement experimental understanding and design.
| 10.48550/arXiv.0708.3589 | A First Principles Theory of Nuclear Magnetic Resonance J-Coupling in solid-state systems | Sian A. Joyce, Jonathan R. Yates, Chris J. Pickard, Francesco Mauri | 4,652 |
10.48550_arXiv.1911.04637 | ## IV Conclusion
The method introduced in Ref.
\begin{table}
\begin{tabular}{c|c c c c} \hline \hline & \(N_{\mathrm{stdPDEP}}=1000\) & \(N_{\mathrm{stdPDEP}}=2000\) & Fit & \(N_{\mathrm{kinPDEP}}=400\) \\ \hline Si VBM & 5.70 & 5.55 & 5.45 & 5.53 \\ Si CBM & 7.03 & 6.91 & 6.79 & 6.82 \\ \(a-\mathrm{Si_{3}N_{4}}\) VBM & 7.14 & 7.01 & 7.01 & 6.99 \\ \(a-\mathrm{Si_{3}N_{4}}\) CBM & 11.99 & 11.87 & 11.83 & 11.83 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Quasiparticle energies of valence band maximum (VBM) and conduction band minimum (CBM) of bulk silicon and amorphous Si\({}_{3}\)N\({}_{4}\) computed with standard eigenpotentials and by combining standard and kinetic eigenpotentials. Columns \(N_{\mathrm{stdPDEP}}=1000\) and \(N_{\mathrm{stdPDEP}}=2000\) are calculations done with 1000 and 2000 stdPDEPs; column Fit is extrapolated results; column \(N_{\mathrm{kinPDEP}}=400\) is calculation with up to 400 kinPDEPs and extrapolated. (See text)of large dielectric matrices, thus leading to substantial computational savings. Building on the strategy proposed in Ref. and implemented in the WEST code, here we proposed an approximation of the spectral decomposition of dielectric matrices that further improve the efficiency of \(G_{0}W_{0}\) calculations. In particular we built sets of eigenpotentials used as a basis to expand the Green function and the screened Coulomb interaction by solving two separate Sternheimer equations: one using the Hamiltonian of the system, to obtain the eigenvectors corresponding to the lowest eigenvalues of the response function, and one equation using just the kinetic energy operator to obtain the eigenpotentials corresponding to higher eigenvalues. We showed that without compromising much accuracy, this approximation reduces the cost of \(G_{0}W_{0}\) calculations by 10%-50%, depending on the system, with the most savings observed for the largest systems studied here.
\begin{table}
\begin{tabular}{c c|c c c} \hline \hline MethodEnergy & VBO & CBO & \(E_{g}^{\rm Si}\) & \(E_{g}^{\rm Si_{3}N_{4}}\) \\ \hline & LDOS & 0.83 & 1.49 & 0.67 & 3.19 \\ PBE & Potential & 0.89 & 1.63 & 0.76 & 3.19 \\ & Ref\({}^{\rm a}\) & 0.8 & 1.5 & 0.7 & 3.17 \\ \hline & LDOS & 1.41 & 1.88 & 1.29 & 4.77 \\ \(G_{0}W_{0}\) & Potential & 1.46 & 2.02 & 1.29 & 4.77 \\ & Ref\({}^{\rm a}\) & 1.5 & 1.9 & 1.3 & 4.87 \\ \hline & Expt & 1.5-1.78\({}^{\rm b}\)1.82-2.83\({}^{\rm c}\)1.17\({}^{\rm d}\)4.5-5.5\({}^{\rm e}\) & & & \\ \hline \hline \multicolumn{5}{l}{\({}^{\rm a}\) Ref.;} \\ \multicolumn{5}{l}{\({}^{\rm b}\) Ref.–54;} \\ \multicolumn{5}{l}{\({}^{\rm c}\) Estimated by the other three experimental values;} \\ \multicolumn{5}{l}{\({}^{\rm d}\) Ref.;} \\ \multicolumn{5}{l}{\({}^{\rm e}\) Ref.–58.} \\ \end{tabular}
\end{table}
Table 5: Band gaps of bulk Si, \(a-{\rm Si_{3}N_{4}}\), and band offsets (VBO&CBO) of the interface.(see Fig. 4) All values are in eV.
| 10.48550/arXiv.1911.04637 | Improving the efficiency of $G_0W_0$ calculations with approximate spectral decompositions of dielectric matrices | Han Yang, Marco Govoni, Giulia Galli | 2,945 |
10.48550_arXiv.0709.0020 | ## III Practical considerations
### Description of the experiment
An _ab initio_ EELS calculation requires not only a description of the sample as input. It also requires knowledge of the conditions in which the experiment was done. We briefly discuss some experimental parameters specific to EELS. An EELS experiment usually involvesan integration over the DDCS defined in Eq.. Typically, the probe has a certain angular width, characterized by the convergence semi-angle \(\alpha\), allowing a whole set of \(\mathbf{k_{i}}\) incoming plane waves. Similarly, the detector integrates the signal over a certain range of outgoing beam directions, characterized by the collection semi-angle \(\beta\). Both are usually of the order of mrad.
\[\frac{\partial\sigma(E)}{\partial E}=\int\limits_{\alpha;\beta}\frac{\partial^{ 2}\sigma}{\partial\Omega\partial E}\left(\mathbf{q},E\right)d^{3}q \tag{8}\]
. To cast the orientation dependence of the EELS spectrum in a more explicit form, we can rewrite Eq. using the CST :
\[\frac{\partial\sigma\left(E\right)}{\partial E}=\sum\limits_{i,j=1}^{3}\sigma_{ ij}(E)\int\limits_{\alpha;\beta}q_{i}^{\prime}\,q_{j}^{\prime}\,\,\frac{4a_{0} ^{-2}\gamma^{2}}{[q^{2}-(E/\hbar c)^{2}]^{2}}\frac{k^{\prime}}{k}d^{3}q \tag{9}\]
As the CST depends only on the sample, only integrals of functions of \(\mathbf{q}\) need to be calculated. These integrals are approximated by a sum over a finite set of impulse transfer vectors \(\mathbf{q}\). This is considered in more detail in the Appendix.
Other experimental parameters included in our calculations are the electron beam energy, the sample to beam orientation, and the position of the EELS detector in the scattering plane. FEFF8 calculations always include core hole broadening. Additional broadening can be applied.
### Computational details
Our calculations are incorporated in a generalization of the _ab initio_ real space multiple scattering code FEFF8, which we call FEFF8.5. We made modifications to FEFF8 to obtain the full-cross section tensor \(\sigma_{ij}\) from the program, and added a new module that calculates the net cross section of Eq. for a given \(\mathbf{q}\) from the cross section tensor, and carries out the integrals over \(\mathbf{q}\) as described in Sec. III.1. We have taken care to retain all other features of the code, including the use of advanced cards such as TDLDA in the calculations which account for corrections to the independent electron approximation ; the use of Debye-Waller factors to approximately account for temperature effects ; etc.
The calculation of the cross section tensor for EELS is analogous to the case of XAS calculations. For the near edge region or energy loss near edge structure (ELNES)the electron equivalent of XANES), the full multiple scattering technique (FMS) is used, in which all scattering paths within a sphere of limited radius are summed implicitly by matrix inversion. For the extended region or extended energy loss fine structure (EXELFS) - the electron equivalent of EXAFS, the path expansion approach is taken, in which the scattering from a selected number of paths of limited length is summed explicitly. This is done for each of the six independent components of the sigma tensor. Combining the ELNES and EXELFS calculations, one can calculate spectra over hundreds of eV, far beyond the limitations of most band-structure codes.
Note that the calculation of the probe and the sample are treated separately so that it is sufficient to calculate the properties of the sample once for the simulation of many experimental situations. We remark that our EELS code can also calculate mixed dynamic form factors (MDFF), which are off diagonal in \(\mathbf{q}\), but we postpone the discussion of these to a future publication.
## IV Applications
### The C K edge of graphite
The strongly anisotropic nature of graphite makes its EELS spectra highly susceptible to orientation effects and therefore to relativistic contributions to the cross section that we discussed in Sec. II. We demonstrate our method on the C K edge, which has an energy threshold of 285 eV.
#### iv.1.1 Cross-section tensor
In this subsection, we discuss the components of the CST, and show that calculation of its diagonal components is not generally sufficient to calculate the EELS spectrum. To see this, we show the different components of the cross section tensor calculated in two different coordinate systems. System 1 is symmetrical : its \(z\)-axis is perpendicular to the graphene sheets of the sample, \(x\) and y are in-plane. System 2 is non-symmetrical : it is obtained from system 1 by a rotation of 35 around the \(x\)-axis of system 1. We work in a Carthesian representation and refer to the components \(i\),\(j\) of \(\sigma\) as \(x\),\(y\),\(z\).
In Fig. 1, we see that in symmetric coordinates the \(\sigma_{zz}\) spectrum contains the so-called \(\pi\)-transitions, while \(\sigma_{xx}\) and \(\sigma_{yy}\) are identical and contain the \(\sigma\)-transitions. All off-diagonal components are zero, as can be explained by symmetry, i.e., equivalence of \(x\) and -\(x\), \(y\) and -\(y\), and \(z\) and -\(z\).
Additionally, the decrease in symmetry allows \(y\),\(z\) cross terms to exist. The \(x\), -\(x\) symmetry has been preserved, suppressing \(xz\), \(zx\), \(xy\) and \(yx\) components. A more general rotation of the coordinates would make all off-diagonal elements nonzero.
Finally, shows the resulting ELNES spectrum. In system 1, calculation of the direct components of \(\sigma\) is sufficient. To calculate the same spectrum in system 2, however, the off diagonal components (\(yz\) and \(zy\) in this example) make a very important contribution.
Components of the CST of graphite in symmetric coordinates.
Components of the CST of graphite in nonsymmetric coordinates, tilted 35 around the symmetric \(x\)-axis.
#### iii.1.2 The Magic Angle
The magic angle is defined as that value the collection angle \(\beta_{m}\) of for which the EELS spectrum is independent of sample to beam orientation, given a certain convergence angle \(\alpha\). In the dipole approximation, one can prove easily that such an angle exists at which the integrals in Eq. lose their orientation dependence. The magic angle depends only on beam energy and energy loss (but it is approximately constant over the near edge region) and not on any material property. The magic angle has played a key role in recent developments of EELS theory - arguably exactly for that reason. It is of practical importance to experimentators wishing to eliminate the complications of orientation dependence from their investigations, and turns out to be sensitive to the details of scattering theory.
\[\theta_{E}=\frac{\omega}{E_{0}}\frac{E_{0}+m_{e}c^{2}}{E_{0}+2m_{e}c^{2}} \tag{10}\]
In Fig. 4, the magic angle is experimentally found to be \(\beta_{m}=0.68\) mrad for given experimental conditions. This is close to \(\theta_{E}=0.59\) mrad.
We now turn to theoretical calculations using FEFF8. We calculate spectra at different sample to beam orientations, which we characterize by a single tilt angle between the electron beam and the crystal \(c\)-axis. This tilt corresponds to a rotation of \(q^{\prime}_{i}\,q^{\prime}_{j}\) in Eq. 9.
C K edge of graphite calculated in coordinate systems 1 and 2 (see text). The beam is perpendicular to the graphene sheets, the beam energy is 300 keV, \(\alpha=10\) mrad, \(\beta=0\) mrad.
At the magic angle, the partial \(i=j\) cross sections of Eq. 9 are individually rotation invariant, and therefore we may equivalently study the \(\frac{\pi}{\sigma}\) ratio of the spectrum. This function is important as it is related to the \(sp^{2}/sp^{3}\)-ratio which is often used to characterize carbon samples. In symmetric coordinates, it is given by the \(\pi\) term or \(i=j=z\) term of Eq., divided by the \(\sigma\) term or \(i=j=x\) plus \(i=j=y\) term,
\[[\frac{\pi}{\sigma}]:=\frac{\sigma_{zz}}{\sigma_{xx}+\sigma_{yy}} \tag{11}\]
We calculate this quantity at a fixed energy loss of \(E=294eV\) as a function of collection angle \(\beta\), shown in for a nonrelativistic calculation, and in for a relativistic calculation. Three different sample to beam orientations are shown in each Figure. At the magic angle, the spectrum and its \(\frac{\pi}{\sigma}\) ratio are independent of orientation.
The nonrelativistic simulation in gives \(\beta_{m}=4\theta_{E}\), as has been reported in the literature for many years, but is inconsistent with experiment.
C K edge of graphite measured at 300 keV beam energy for 2 orientations, and different collection angles. The two orientations overlap between the camera lengths of 80 cm and 100 cm, corresponding to a magic value of the collection angle of about 0.68 mrad. The convergence angle is close to 0 mrad.
7 and 8, where relativistic calculations of the spectrum are shown at both values of the collection angle. The nonrelativistic calculation would yield identical spectra at \(\beta=2.4\) mrad, in disagreement with experiment.
Our present results agree very well with calculations reported in, which were calculated using the DFT code WIEN2K.
\(\frac{\pi}{\sigma}\) ratio (see text) of the graphite C K edge at 10 eV above threshold for a 300 keV beam and three sample to beam orientations. The magic angle is at the intersection of the three curves. Nonrelativistic calculation. The convergence angle is 0 mrad.
\(\frac{\pi}{\sigma}\) ratio (see text) of the graphite C K edge at 10 eV above threshold for a 300 keV beam and three sample to beam orientations. The magic angle is at the intersection of the three curves. Relativistic calculation. The convergence angle is 0 mrad.
## V Conclusions
We have presented relativistic calculations of electron energy loss spectroscopy using the real space Green's function code FEFF. The calculations correctly accounts for \(\mathbf{q}\)-dependence and microscope settings such as collection and convergence angle. We have demonstrated our method on the C \(K\) edges of graphite, where we calculate the correct magic angle.
| 10.48550/arXiv.0709.0020 | Real Space Multiple Scattering Calculations of Relativistic Electron Energy Loss Spectra | K. Jorissen, J. J. Rehr | 5,939 |
10.48550_arXiv.1010.4888 | ### Summary
In summary, we provided a simple recipe of preparation of graphene films with adsorbed biological molecules. As an example of our technique, we demonstrated images of individual unstained TMV on graphene support recorded in TEM.
| 10.48550/arXiv.1010.4888 | Graphene as a transparent conductive support for studying biological molecules by transmission electron microscopy | R. R. Nair, P. Blake, J. R. Blake, R. Zan, S. Anissimova, U. Bangert, A. P. Golovanov, S. V. Morozov, T. Latychevskaia, A. K. Geim, K. S. Novoselov | 6,071 |
10.48550_arXiv.0807.2771 | ## 1 Introduction
Plastic deformation of pure body-centered cubic (BCC) metals is primarily controlled by the glide of 1/2{111} screw dislocations that have high lattice friction (Peierls) stress owing to their non-planar cores (for reviews see). In the Part I of this series we presented the results of atomistic studies of the core structure and glide of 1/2{111} screw dislocations at 0 K in molybdenum and tungsten that employed the recently developed Bond Order Potentials. The principal conclusion is that the Peierls stress, identified with the critical resolved shear (CRSS) that acts parallel to the slip direction in the maximum resolved shear stress plane (MRSSP) at which the dislocation starts to move, is a function of both the orientation of the MRSSP and the magnitude of the shear stress perpendicular to the slip direction. The orientation of the MRSSP is determined by the angle \(\chi\), defined in of Part I, and the shear stress perpendicular to the slip direction, \(\tau\), defined by equation in Part I. The glide plane is always of the {110} type although not necessarily the {110} plane with the highest shear stress in the slip direction. Comparison with earlier atomistic studies that employed central-force potentials suggests that this conclusion is general and applies qualitatively to all BCC metals and possibly also to other materials with this crystal structure. However, quantitativelythe dependence of the CRSS on \(\chi\) and \(\tau\) differs rather significantly from material to material; this is visible in Figs. 4, 7 and 8 of Part I that display such dependencies for molybdenum and tungsten, respectively.
The calculations presented in Part I were all done for the slip direction and the angle \(\chi\) was measured from the \((\overline{1}01)\) plane. Owing to the three-fold screw axis symmetry of the direction the results would be the same if we chose to measure \(\chi\) from any of the other two \(\{110\}\) planes of the zone, \((01\overline{1})\) or \((1\overline{1}0)\). Furthermore, in any BCC crystal there are twelve independent systems comprised of \(\langle 111\rangle\) slip directions and \(\{110\}\) slip planes. They are all crystallographically equivalent and thus the results found in Part I apply equally to all of them. Hence, when we investigate the plastic yielding of a BCC single crystal all these systems have to be considered. Such analysis for loading by a general stress tensor is presented in Section 2 and a specific example of a combined loading by shear stresses parallel and perpendicular to the direction is the topic of Section 3. In the latter the dependencies of the CRSS on \(\chi\) and \(\tau\), determined in Part I, have been employed. These numerically evaluated dependencies could, of course, be used when analyzing any type of loading. However, it is much more efficient and astute to formulate an analytical yield criterion that applies to any \(\{110\}\langle 111\rangle\) system and reproduces with sufficient accuracy the numerical results. This approach is analogous to that first advanced by Taylor and later developed to include capabilities that account for finite shape change and lattice rotations and applied in studies of multislip hardening and strain localization.
These continuum models all assume the Schmid-type plastic behavior, which means that the only stress component that affects the yielding is the shear stress acting in the slip plane parallel to the slip direction. This is well-established for slip confined to \(\{111\}\) planes in FCC metals and basal planes in HCP metals. Since the plastic flow mediated by dislocations is always driven by the shear stress in the direction of slip, the plastic flow obeying the Schmid law is called associated. However, if also other stress components, called non-glide stresses, affect the yielding and plastic flow but they do not drive the dislocation glide in the slip plane, the Schmid law does not apply and such flow is called non-associated; more exact definition is presented in Section 8. The latter applies in BCC metals and the primary objective of this paper is to formulate the yield criterion for the non-associated flow that includes both the shear stresses parallel and perpendicular to the slip direction that affect the dislocation glide. This is done following the suggestion by Qin and Bassani made when formulating the yield criterion for L\(1_{2}\) intermetallic compounds deformed in the anomalous regime of the temperature dependence of the yield stress. The yield criterion formulated in Section 4 of this paper, which reproduces both the \(\chi\) and \(\tau\) dependence of the CRSS, is written as a linear combination of four shear stresses, the Schmid stress and three non-glide stresses, one parallel and two perpendicular to the slip direction. This approach was employed earlier in the special case of loading by pure shear stress parallel to the slip direction.
As the next step we employ the constructed yield criterion to determine the yield surface and compare this to a hypothetical yield surface that would be obtained if the Schmid law were valid. Furthermore, we use the yield criterion to determine active slip systems that operate during uniaxial loading for all orientations of the loading axis within the standard stereographic triangle. This analysis reveals a very notable contrast between tension and compression as well as a significantly different slip geometry in molybdenum and tungsten. Results of this study are then compared with recent experimental investigation of tension-compression asymmetries in molybdenum. The remarkable agreement between the present calculations and experimental data demonstrates that the tension-compression asymmetry, commonly associated with the twinning-antitwinning asymmetry of shearing, is only partly related to this crystallographic characteristic. The major contribution originates in the effect of shear stresses perpendicular to the slip direction. In Section 8 we utilize the Taylor homogenization model to demonstrate that this asymmetry persists even in random polycrystals and in Section 9 we show on the example of ductile cavitation that the non-glide stresses also significantly affect critical plastic phenomena.
## 2 Plastic flow in a BCC single crystal subject to external loading
The atomistic calculations presented in the Part I of this series of papers were all done for the 1/2 screw dislocation. The angle \(\chi\), determining the orientation of the MRSSP in which the shear stress \(\sigma\) parallel to the direction is applied, was measured from the \((\overline{1}01)\) plane. Owing to the crystal symmetry only \(-30^{\circ}\leq\chi\leq+30^{\circ}\) need to be considered. The shear stress \(\tau\) perpendicular to the slip direction is applied by Eq. of Part I. Hence, the \((\overline{1}01)\) plane is the \(\{110\}\) plane with the highest shear stress parallel to the slip direction in between the three \(\{110\}\) planes of the zone and the \((\overline{1}01)\) system was chosen as a _reference system_ in the atomistic studies presented in Part I. However, in any BCC crystal there are twelve crystallographically equivalent \(\{110\}\)\(\{111\}\) systems and thus the dependencies of the CRSS on \(\chi\) and \(\tau\), found in Part I, apply equally to all of them. In addition, for \(\chi\neq 0\) the CRSS depends on the sense of shearing and changing the sign of \(\sigma\) is equivalent to changing the sign of \(\chi\) while keeping the sign of \(\sigma\) fixed. This corresponds to the so-called twinning-antitwinning asymmetry and an alternative way to capture this effect is to regard positive and negative slip directions as distinct. In this case only positive \(\sigma\), and thus positive CRSS, need to be considered. The ensuing twenty four \(\{110\}\)\(\{111\}\) reference systems are summarized in Table 1 and denoted by \(\alpha\); the system used in atomistic studies corresponds to \(\alpha\)\(=2\). All these systems have to be included on equal footing when analyzing the plastic response of a single crystal to the loading represented by an applied stress tensor.
In such analysis we have to find first the orientations of the MRSSPs for the eight distinct slip directions and determine the shear stresses parallel and perpendicular to these slip directions in the MRSSPs associated with them. As explained above, only positive shear stresses parallel to the slip directions are considered and thus we exclude all reference systems for which these stresses are negative. This means that at this point only four slip directions remain in the subsequent analysis. In the next step we find for each of these four slip directions the \(\{110\}\) plane of its zone that has the highest shear stress parallel to the slip direction. Combination of this \(\{110\}\) plane with the concurrent slip direction represents a reference system that is always one of the systems listed in Table 1. The orientation of the MRSSP is for a system \(\alpha\) characterized by the angle \(\chi_{\alpha}\) that it makes with the corresponding \(\{110\}\) reference plane. The range of this angle is \(-30^{\circ}\leq\chi_{\alpha}\leq+30^{\circ}\) and it is bounded by two \(\{112\}\) planes. For \(\chi_{\alpha}<0\) the nearest \(\{112\}\) plane is sheared in the _twinning_ sense while for \(\chi_{\alpha}>0\) the nearest \(\{112\}\) plane is sheared in the _antitwinning_ sense. This definition conforms to that used for the reference system (\(\overline{1}01\)) (\(\alpha\!=\!2\)) in Part I.
Finally, for each reference system included we transform the externally applied stress tensor into the right-handed coordinate system with the \(z\)-axis parallel to the corresponding \(\langle 111\rangle\) direction, \(y\)-axis normal to the MRSSP and the \(x\)-axis in the MRSSP. This is congruent with the coordinate system defined for the reference system (\(\overline{1}01\)) in conjunction with the application of \(\tau\) by Eq. 2 in Part I. In general, for this orientation all components of the applied stress tensor are nonzero. However, as shown in Part I, the only stress components affecting the glide of screw dislocations are the resolved shear stresses \(\sigma_{a}\) and \(\tau_{a}\), parallel and perpendicular to the slip direction of the reference system \(\alpha\), respectively.
\[\boldsymbol{\Sigma}^{\alpha}(\chi_{\alpha})\!=\!\begin{bmatrix}-\tau_{\alpha}&0& 0\\ 0&\tau_{\alpha}&\sigma_{\alpha}\\ 0&\sigma_{\alpha}&0\end{bmatrix}. \tag{1}\]
This tensor determines for the reference system \(\alpha\) the values of \(\sigma_{\alpha}\) and \(\tau_{\alpha}\) that arise for a given mode of loading when the MRSSP makes the angle \(\chi_{\alpha}\) with the \(\{110\}\) reference plane of the system \(\alpha\). As the applied loading evolves, the shear stresses \(\sigma_{\alpha}\) and \(\tau_{\alpha}\) develop accordingly and the tensor \(\boldsymbol{\Sigma}^{\alpha}(\chi_{\alpha})\) defines a unique dependence of \(\tau_{\alpha}\) on \(\sigma_{\alpha}\). In the following we call their dependence the loading path \(\sigma_{\alpha}-\tau_{\alpha}\).
The atomistic studies of the glide of the 1/2 screw dislocation suggest that for a given \(\chi\) there is a unique relation between the CRSS and \(\tau\), independent of the manner (i.e.
\begin{table}
\begin{tabular}{c c c c c|c c c c} \hline \(\alpha\) & ref. system & \(\mathbf{m}^{\alpha}\) & \(\mathbf{n}^{\alpha}\) & \(\mathbf{n}_{1}^{\alpha}\) & \(\alpha\) & ref.
1 & (01\(\overline{1}\)) & & [01\(\overline{1}\)] & [\(\overline{1}10\)] & [13 & (01\(\overline{1}\))[\(\overline{1}\)\(\overline{history) the corresponding shear stresses \(\sigma\) and \(\tau\) were attained. For several values of \(\chi\) this CRSS vs \(\tau\) dependence was established in the studies presented in Part I. We then consider that a reference system \(\alpha\) becomes activated for slip when \(\sigma_{\alpha}\) that varies along a loading path \(\sigma_{\alpha}-\tau_{\alpha}\) reaches the CRSS corresponding to the actual value of \(\tau_{\alpha}\). In order to demonstrate such transfer of the atomistic results to the analysis of the plastic deformation of a single crystal, we discuss in the following section a combined loading by the shear stresses parallel and perpendicular to the slip direction. The same method will be employed subsequently to analyze uniaxial loading of single crystals.
Plastic yielding in a single crystal loaded by combined shear stresses parallel and perpendicular to the slip direction
In this section we consider that the applied stress tensor comprises a combination of pure shear stresses \(\sigma\) and \(\tau\), parallel and perpendicular to the slip direction. In this case the loading path along which a given combination of \(\sigma\) and \(\tau\) is attained is not unique. In fact it can be achieved in a limitless number of ways1. However, since the CRSS depends only on the value of \(\tau\) and not on the way the corresponding combination of \(\sigma\) and \(\tau\) was produced, we will utilize paths specified by constant ratios of \(\tau/\sigma\). For a given loading path the procedure outlined in the previous section determines the four reference systems \(\alpha\) and the related pairs of stresses \(\tau_{\alpha}\) and \(\text{CRSS}_{\alpha}\) for which the plastic yielding would commence on these systems. The system with the lowest \(\text{CRSS}_{\alpha}\) is the primary system on which the plastic deformation starts first.
Footnote 1: In contrast, the loading path is unique in the usual experimental situations such as loading by tension/compression when it is linear.
We now consider the shear stress parallel to the direction applied such that the (\(\overline{\text{T}}\)01) plane is the MRSSP (\(\chi=0\)), combined with the applied shear stress perpendicular to this direction; in this case the reference system is \(\alpha=2\). This is the same loading as that considered in Section 4.3 of Part I for the 1/2 screw dislocation and the corresponding CRSS vs \(\tau\) dependence is shown for Mo and W in of Part I. Obviously, in the proximity of \(\tau=0\) the (\(\overline{\text{T}}\)01) (\(\alpha=2\)) is the primary slip system. However, as the magnitude of \(\tau\) increases the shear stresses, both parallel and perpendicular to the slip direction, evolve also in other slip systems and which of the available systems is primary has to be investigated as described in the previous section. An example corresponding to the loading path \(\tau_{2}/\sigma_{2}=1.5\) for the system \(\alpha=2\) is shown in where the squares are the data for molybdenum copied from of Part I and represent the dependence of the CRSS on \(\tau\) for the loading considered; the straight line passing through the origin depicts the loading path. If only 1/2 dislocations were gliding the (\(\overline{\text{T}}\)01) system would become operative at the point where this line intersects the CRSS vs \(\tau\) dependence, i.e. at the point marked \(\mathbf{B}\). However, while the loading in the (\(\overline{\text{T}}\)01) reference system proceeds along the path shown in another reference system, (\(0\,\overline{\text{T}}\)1)[\(\overline{\text{T}}\)11] (\(\alpha=5\)) in which the orientation of the MRSSP corresponds to \(\chi_{5}=-7.3^{\circ}\), is subjected to the shear stresses \(\sigma_{5}\) and \(\tau_{5}\) that evolve along the loading path \(\tau_{5}/\sigma_{5}\approx-0.2\). This path is shown as a straight line passing through the origin in It was determined employing the procedure outlined in the previous section using the stress tensor corresponding to the loading path of Note that the slopes of the two loading paths for the two reference systems are different. As already emphasized, the CRSS vs \(\tau\) dependence is the same for every system \(\alpha\) and thus the squares in have the same meaning as in but were now obtained for the MRSSP with angle \(\chi=-9^{\circ}\) that is closest to \(\chi_{{}_{5}}\) in between the angles \(\chi\) for which the CRSS vs \(\tau\) dependence was calculated atomistically in Part I2. The system (\(0\,\overline{1}1\))[\(\overline{1}11\)] becomes operative at the point where the loading path in intersects the CRSS vs \(\tau\) dependence, i.e. at the point marked \(\mathbf{A}\) in this figure. However, in the (\(\overline{1}01\)) system this stress corresponds to the loading at the point marked \(\mathbf{A}\) in Fig. 1a, where the corresponding \(\sigma_{{}_{2}}\) is well below the CRSS. Consequently, the (\(0\,\overline{1}1\))[\(\overline{1}11\)] system will become operative _before_ the (\(\overline{1}01\)) system and it is thus the primary system for the loading path considered. The same analysis as described above for molybdenum can also be performed for tungsten. The only difference is that in this case one uses the CRSS vs \(\tau\) dependencies calculated for tungsten.
Footnote 2: This dependence was not shown in Part I but can be found in.
Evolution of loading in two different {110}\(\backslash\)111) systems, represented by straight lines passing through the origin, induced by the shear stresses parallel and perpendicular to the slip direction applied in the (\(\overline{1}01\)) system: (a) \(\alpha\)= 2, \(\tau_{{}_{2}}/\sigma_{{}_{2}}\) = 2.5, (b) \(\alpha\)= 5, \(\tau_{{}_{5}}/\sigma_{{}_{5}}\) = \(-\)0.2. Squares correspond to the atomistic data obtained for an isolated 1/2 dislocation in Part I. The points marked \(\mathbf{A}\) and \(\mathbf{B}\) correspond to the same applied loading in both cases.
In order to develop a complete description of the yielding of a single crystal on the basis of the atomistic study of the glide of 1/2\(\langle\)111\(\rangle\) screw dislocations the procedure described above has to be repeated for a number of different loading paths \(\tau_{{}_{2}}/\sigma_{{}_{2}}=\eta_{{}_{2}}\) where \(\rightarrow\infty<\eta_{{}_{2}}<+\infty\). For each of these loading paths such calculations yield four reference systems \(\alpha\), each associated with a distinct slip direction, and for every system \(\alpha\) a pair of stresses CRSS\({}_{\alpha}\) and \(\tau_{{}_{\alpha}}\) at which the plastic deformation commences on this system can be determined. It is seen from the results of Part I that for \(\left|\tau_{{}_{\alpha}}/C_{{}_{44}}\right|\leq 0.02\) the dependence of CRSS\({}_{\alpha}\) on \(\tau_{{}_{\alpha}}\) is close to linear (see Figs. 7 and 8 of Part I). This is the range of \(\tau_{{}_{\alpha}}\) in which CRSS\({}_{\alpha}\) is always larger than \(\tau_{{}_{\alpha}}\) and in the following we consider only this regime of applied stresses3. In this case the CRSS\({}_{\alpha}\) evaluated for various loading paths follows in the plot of CRSS vs \(\tau\) a straight line. For molybdenum such lines are drawn in for the reference systems \(\alpha=5\), \(19\), \(2\), \(21\), \(8\) and for tungsten in for the reference systems \(\alpha=19\), \(2\), \(13\), \(17\). The inner envelopes of these lines, drawn in Figs. 2a, b as the solid polygons, encompass the regions of the elastic behavior of the material. Hence, they represent projections of the yield surfaces (defined more precisely in Section 5) onto the CRSS vs \(\tau\) plot. For any loading path characterized by a ratio \(\tau_{{}_{2}}/\sigma_{{}_{2}}\) the yielding occurs when this path reaches the yield surface, i.e. the corresponding polygon in The \(\{110\}\langle 111\rangle\) system associated with the side of the polygon that is intersected by the loading path is the primary slip system for this combination of \(\sigma_{{}_{2}}\) and \(\tau_{{}_{2}}\).
Footnote 3: The problem becomes more complex when \(\left|\tau/C_{{}_{\alpha}}\right|>0.02\) because the CRSS then depends on \(\tau_{{}_{\alpha}}\) non-linearly. For negative \(\tau\) from this region the slip may occur on a \(\{110\}\) plane that is inclined with respect to the reference \(\{110\}\) plane by \(\pm 60^{\circ}\). As discussed in Part I, this may be the origin of the anomalous slip that occurs at very low temperatures and takes place on the slip systems for which the resolved shear stress is substantially lower than that on the most highly stressed \(\{110\}\langle 111\rangle\) system.
Critical lines marking the onset of slip on slip systems in molybdenum (a) and tungsten (b), projected into the CRSS-\(\tau\) plot for the MRSSP (\(\overline{1}01\)) (\(\chi=0\)). The inner envelope of these lines (bold) defines a yield polygon for real single crystals. Squares correspond to the atomistic data obtained for an isolated 1/2 dislocation in Part I.
In we show the projection of the yield surface calculated for tungsten using the procedure outlined above. Since the dependence of the CRSS on \(\tau\) is now different from that for molybdenum, also the \(\{110\}\langle 111\rangle\) systems that comprise the yield polygon are distinct from those in for molybdenum.
## 4 Analytical yield criterion for single crystals
The dependence of the CRSS on the angle \(\chi\) and the shear stress \(\tau\) was in the previous section obtained directly from the results of atomistic calculations. However, for \(\left|\tau\,/\,C_{44}\right|\leq 0.02\), i.e. when for a given \(\chi\) the CRSS depends to a good approximation linearly on \(\tau\), we can formulate an analytical yield criterion that reproduces very closely the atomistic data. In developing such criterion we follow the proposal of Qin and Bassani who introduced the yield criterion for non-associated flow in Ni\({}_{3}\)Al that captured the observed orientation dependence and tension-compression asymmetry of the yield stress in the regime of the anomalous increase of the flow stress with temperature (see e.g.). This criterion included two shear stresses, one parallel and one perpendicular to the slip direction. In our case, in order to capture both the \(\chi\) and \(\tau\) dependence of the CRSS, such criterion needs to comprise two shear stresses parallel and two shear stresses perpendicular to the slip direction, both resolved in two different \(\{110\}\) planes of the zone of the slip direction.
\[\sigma^{(\tau 01)}+a_{1}\sigma^{(\tau 01)}+a_{2}\tau^{(\tau 01)}+a_{3}\tau^{(0\tau 1)}=\tau^{*}_{cr}\, \tag{2}\]
The first term in Eq. is the stress that drives the dislocation motion in the \((\overline{1}01)\) glide plane and does the work through dislocation glide. It is commonly called the Schmid stress. In contrast, the stresses \(\sigma^{(\tau 01)}\), \(\tau^{(\tau 01)}\) and \(\tau^{(\tau 0\tau)}\) affect the dislocation core but do not do any work when the dislocation glides in the \((\overline{1}01)\) plane. These stresses are commonly called non-glide stresses. The second term includes the shear stress parallel to the slip direction in another \(\{110\}\) plane and reproduces the twinning-antitwinning asymmetry of the CRSS. For such loading a yield criterion employing just these two terms was developed earlier. The third and fourth terms incorporate the effect of the shear stress perpendicular to the slip direction. The coefficients \(a_{1}\), \(a_{2}\), \(a_{3}\), as well as \(\tau^{*}_{cr}\),are all adjustable parameters that are ascertained by fitting the CRSS vs \(\chi\) and CRSS vs \(\tau\) dependencies determined by atomistic calculations.
First, \(a_{1}\) and \(\tau^{*}_{cr}\) are fitted to reproduce the CRSS vs \(\chi\) dependence for loading by pure shear stress parallel to the slip direction. In this case Eq. reduces to \(\sigma^{(\tau 01)}+a_{1}\sigma^{(0\tau 1)}=\tau^{*}_{cr}\). The two shear stresses entering this relation can be written in terms of the CRSS applied in the MRSSP and the angle \(\chi\) as \(\sigma^{(\tau 01)}=\text{CRSS}\,\cos\chi\) and \(\sigma^{(\tau 01)}=\text{CRSS}\,\cos(\chi+\pi\,/\,3)\). For a given orientation of the MRSSP, i.e.
\[\text{CRSS}(\chi)=\frac{\tau^{*}_{cr}}{\cos\chi+a_{1}\cos(\chi+\pi\,/\,3)}\,. \tag{3}\]\(a_{1}\) and \(\tau_{\sigma}^{*}\) can then be ascertained by the least squares fitting of this relation to the CRSS vs \(\chi\) dependence found in atomistic studies (of Part I). For molybdenum \(a_{1}\) and \(\tau_{\sigma}^{*}\) were found in this way in and it can be seen in these papers that Eq. reproduces the atomistically determined CRSS vs \(\chi\) dependence very closely for all values of \(\chi\).
In the second step, keeping \(a_{1}\) and \(\tau_{\sigma}^{*}\) fixed, the parameters \(a_{2}\) and \(a_{3}\) have been determined by fitting the CRSS vs \(\tau\) dependence found in the atomistic calculations that employed the combination of the shear stresses perpendicular and parallel to the slip direction (Section 4.3 of Part I). In this case \(\tau^{}=\tau\sin 2\chi\) and \(\tau^{}=\tau\cos(2\chi+\pi/6)\).
\[\text{CRSS}(\chi,\tau)=\frac{\tau_{\sigma}^{*}-\tau[a_{2}\sin 2\chi+a_{3}\cos (2\chi+\pi/6)]}{\cos\chi+a_{1}\cos(\chi+\pi/3)}\,. \tag{4}\]
The coefficients \(a_{2}\) and \(a_{3}\) can again be ascertained by the least squares fitting of this relation to the CRSS vs \(\tau\) dependencies calculated for various angles \(\chi\). For \(\left|\tau/\,C_{44}\right|\leq 0.02\), which is the range of \(\tau\) for which the yield criterion has been developed, we found that the best fit is obtained when considering only three orientations of the MRSSP, namely \(\chi=0\) and \(\chi_{\approx}\pm 9^{\circ}\). In addition, for each of these orientations, only the data for \(\tau/C_{44}=\pm 0.01\) need to be taken into account since only two points specify the slope of the straight line that approximates the CRSS vs \(\tau\) dependence in the region of \(\tau\) specified above. The coefficients \(a_{1}\), \(a_{2}\), \(a_{3}\) and \(\tau_{\sigma}^{*}\) entering the yield criterion for molybdenum and tungsten, which were determined as described above, are listed in Table 2.
The yield criterion applies, of course, to any \(\{110\}\langle 111\rangle\) reference system.
\[\mathbf{m}^{\alpha}\mathbf{\Sigma}^{app}\mathbf{n}^{\alpha}+a_{1}\mathbf{m}^{ \alpha}\mathbf{\Sigma}^{app}\mathbf{n}_{1}^{\alpha}+a_{2}(\mathbf{n}^{\alpha} \times\mathbf{m}^{\alpha})\mathbf{\Sigma}^{app}\mathbf{n}^{\alpha}+a_{3}( \mathbf{n}_{1}^{\alpha}\times\mathbf{m}^{\alpha})\mathbf{\Sigma}^{app}\mathbf{ n}_{1}^{\alpha}=\tau_{\sigma}^{*}\,, \tag{5}\]
This criterion determines a critical applied stress tensor, \(\mathbf{\Sigma}_{c}^{app}\), at which the yielding on the system \(\alpha\) commences. Recall that an angle is positive if measured relative to the reference plane in the sense shown in of Part I. For example, if the reference system is \((\overline{1}01)\), i.e. \(\alpha\)= 2, the three unit vectors are \(\mathbf{m}^{2}=\frac{1}{\overline{\overline{\overline{\overline{\overline{ \overline{\overline{\overline{\overline{\overline{\overline{\overline{\overline{ \overline{\overline{\overline{ \overline \overline { \overline \overline \overline { \overline \overline \overline \overline{ \nu \nu \nu \to 12 and 13 to 24 are conjugate to each other in that the pairs of systems \(\alpha\) and \(\alpha\)+12 have identical normals of the reference planes \(\mathbf{n}^{\alpha}\), opposite slip directions \(\mathbf{m}^{\alpha}\), and complementary vectors \(\mathbf{n}_{{}_{1}}^{\alpha}\).
In the criterion the last three terms represent the effect of non-glide stresses and the first term is the Schmid stress acting in the slip plane of the system \(\alpha\). For any applied loading given by the stress tensor \(\mathbf{\Sigma}^{\mathit{app}}\), one can assess the activity of individual reference systems summarized in Table 1 by evaluating the left side of equation, which we mark \(\tau^{*\alpha}\), for \(\alpha\) ranging from 1 to 24. Following the criterion the slip on a system \(\alpha\) is considered activated when \(\tau^{*\alpha}\) reaches \(\tau^{*}_{{}_{\mathit{cr}}}\) and thus when the applied stress tensor attains the critical value \(\mathbf{\Sigma}^{\mathit{app}}_{{}_{\mathit{cr}}}\). In this framework the plastic deformation at 0 K starts when one of the 24 values of \(\tau^{*\alpha}\) reaches \(\tau^{*}_{{}_{\mathit{cr}}}\). The corresponding \(\alpha\) then identifies the slip system that will be activated first and, in the following, this reference system is called the primary system.
## 5 The yield surface
In order to introduce the notion of the yield surface we define the so-called principal space that is a three-dimensional space with orthogonal axes along which we measure the three principal stresses of any applied stress tensor. Hence, any stress state is represented in this space by a point the coordinates of which are the corresponding principal stresses \(\sigma_{{}_{I}}\), \(\sigma_{{}_{II}}\) and \(\sigma_{{}_{III}}\). The yield surface, determined by the yield criterion, is a convex hyperplane in the principal space that encompasses the region of stress states for which the deformation behavior of the material is purely elastic. When the stress state corresponding to an applied loading reaches the yield surface, the yield criterion is satisfied and the material starts to deform plastically.
For a given yield criterion the yield surface can be constructed as follows. In order to explore all possible orientations of the applied loading we write \(\sigma_{{}_{I}}=\kappa\sin\theta\cos\phi\), \(\sigma_{{}_{II}}=\kappa\sin\theta\sin\phi\) and \(\sigma_{{}_{III}}=\kappa\cos\theta\), where \(\theta\in\left\langle 0,\pi\right\rangle\) and \(\phi\in\left\langle 0,2\pi\right\rangle\) are spherical angles. Every combination of angles \(\theta\) and \(\phi\) corresponds to a well-defined direction in the principal space and for each such combination we evaluate \(\sigma_{{}_{I}}\), \(\sigma_{{}_{III}}\), \(\sigma_{{}_{III}}\) and determine \(\kappa\) such that the yield criterion is satisfied. For this value of \(\kappa\), which is a function of \(\theta\) and \(\phi\), the principal stresses determine the critical stress state for which the yielding commences and thus the point on the yield surface. When this calculation is repeated for all angles \(\theta\) and \(\phi\), one obtains the full three-dimensional yield surface. In the framework of crystal plasticity the procedure described above has to be repeated for every available slip system and the inner envelope of all calculated critical stress states then represents the yield surface.
When employing the yield criterion we first calculate for every combination of angles \(\theta\) and \(\phi\) the principal stresses \(\sigma_{{}_{I}}\), \(\sigma_{{}_{II}}\), \(\sigma_{{}_{III}}\) and thus the stress tensor \(\mathbf{\Sigma}^{\mathit{app}}(\theta,\phi)\). We then evaluate for all slip systems \(\alpha\) from Table 1 the left side of Eq. using the tensor \(\mathbf{\Sigma}^{\mathit{app}}(\theta,\phi)\). If we mark this value \(\tau^{*\alpha}(\theta,\phi)\) then for these angles \(\theta\) and \(\phi\) the yielding will commence on the slip system \(\alpha\) when \(\kappa\) is equal to \(\kappa^{\alpha}_{{}_{\mathit{c}}}(\theta,\phi)=\tau^{*}_{{}_{\mathit{cr}}}/ \tau^{*\alpha}(\theta,\phi)\). For the slip system \(\alpha\) the stresses \(\sigma^{\alpha}_{{}_{I}}=\kappa^{\alpha}_{{}_{\mathit{c}}}(\theta,\phi)\sin \theta\cos\phi\), \(\sigma^{\alpha}_{{}_{II}}=\kappa^{\alpha}_{{}_{\mathit{c}}}(\theta,\phi)\sin \theta\sin\phi\) and \(\sigma^{\alpha}_{{}_{III}}=\kappa^{\alpha}(\theta,\phi)\cos\theta\) determine the critical stress state for which this system becomes activated for slip. The yield surface is then the inner envelope of the critical stresses \(\sigma^{\alpha}_{{}_{I}}\), \(\sigma^{\alpha}_{{}_{II}}\), \(\sigma^{\alpha}_{{}_{III}}\) calculated for all angles \(\theta\) and \(\phi\) and all slip systems \(\alpha\) from Table 1. In general, different slip systems operate in different parts of the yield surface.
Although the stress tensor \(\boldsymbol{\Sigma}^{app}\) entering the yield criterion may contain nonzero hydrostatic stress, this does not affect the magnitude of \(\tau^{*\alpha}\) and thus the onset of yielding. Hence, it is a common practice to represent the yield surface as a two-dimensional yield locus that is obtained as a cross-section of the three-dimensional yield surface by the so-called deviatoric plane whose normal is parallel to the direction in which \(\sigma_{I}=\sigma_{H}=\sigma_{III}\). The three-dimensional yield surface is then uniquely characterized by its yield locus in the deviatoric plane. The yield loci for the yield surfaces determined by the procedure described above are shown in for molybdenum (dashed polygon) and for tungsten (solid polygon). For comparison, we also plot in this figure the yield locus that is predicted by the Schmid law (dotted polygon), i.e. when \(a_{1}=a_{2}=a_{3}=0\) in. In this case, the yield criterion reduces to that of Tresca and the yield locus is a regular hexagon. The remarkable difference between the yield loci obtained from the yield criterion and that corresponding to the Schmid law demonstrates the breakdown of the Schmid law in both molybdenum and tungsten. In molybdenum, this is caused by a combination of the twinning-antitwinning asymmetry of shearing parallel to the slip direction and the effect of the shear stresses perpendicular to the slip direction, while in tungsten only the latter plays role.
## 6 Application of the yield criterion to uniaxial loading
In this section we discuss tensile and compressive loadings along the directions for which the most highly stressed \(\{110\}\langle 111\rangle\) system is \((\overline{1}01)\). In Section 4.2 of Part I several such cases were studied atomistically and orientations of the corresponding loading axes were summarized in of Part I. Assuming a unit uniaxial applied stress (\(+1\) for tension and \(-1\) for compression) we evaluate first for each loading axis studied the corresponding applied stress tensor \(\boldsymbol{\Sigma}^{app}\), defined in Section 4 in connection with equation. In order to consider all possible slip systems that can operate for a given loading axis we identify the four relevant reference systems employing the method described in Section 2.
The yield loci in the deviatoric plane generated by the yield criterion for molybdenum (dashed polygon) and for tungsten (solid polygon). For comparison we also show as the dotted polygon the yield locus that is obtained from the Schmid law (\(a_{1}\)=\(a_{2}\)=\(a_{3}\)=0). In this projection, each edge of the yield locus is shared by four slip systems and the corners by eight slip systems.
As the next step we determine for these systems the corresponding vectors \(\mathbf{m}^{\alpha}\), \(\mathbf{n}^{\alpha}\) and \(\mathbf{n}_{1}^{\alpha}\), defined in Section 4. Finally, we evaluate the left side of Eq. and mark it \(\tau_{\iota/c}^{\ast\alpha}\). According to the yield criterion the uniaxial tensile/compressive stress for which a system \(\alpha\) becomes activated if \(\sigma_{\iota/c}^{\alpha}=\tau_{\iota cr}^{\ast}/\tau_{\iota/c}^{\ast\alpha}\). The actual _uniaxial_ yield stress that induces plastic flow in the crystal corresponds to the minimum among these stresses, i.e. \(\sigma_{\iota/c}=\min\limits_{\alpha}\sigma_{\iota/c}^{\alpha}\), and the corresponding system \(\alpha\) is then the primary system.
Following this procedure we find that in tension (positive \(\tau\)) the (\(\overline{1}01\)) system, which has the highest Schmid stress, is the primary slip system for any orientation of the loading axis within the stereographic triangle for both molybdenum and tungsten. However, in compression (negative \(\tau\)) the primary slip systems vary considerably with the orientation of the loading axis. Regions of different slip systems for different orientations of compressive axes within the stereographic triangle are depicted in for molybdenum and in for tungsten. Since the sense of shearing is reversed relative to tension and we consider that shear stresses parallel to the slip directions are always positive (see Section 2), the slip system with the highest Schmid stress is not (\(\overline{1}01\)) but its conjugate (\(\overline{1}01\))[\(\overline{1}\)\(\overline{1}\)\(\overline{1}\)\(\overline{1}\)].
Consider first the distribution of primary slip systems for molybdenum in In the central region of the triangle, the most operative slip system is, indeed, (\(\overline{1}01\))[\(\overline{1}\)\(\overline{1}\)\(\overline{1}\)\(\overline{1}\)\(\overline{1}\)]. As the loading axis moves towards the [\(011\)]\(-\)[\(\overline{1}1\)] edge, the (\(\overline{1}10\))[\(\overline{1}\)\(\overline{1}\)\(\overline{1}\)\(\overline{1}\)] system, which dominates near the [\(\overline{1}\)\(\overline{1}\)\(1\)] corner, becomes increasingly more prominent. There is a region in which the two systems can operate simultaneously, i.e. the uniaxial yield stress of the secondary system (marked as II) is within 2% of that of the primary system (denoted as I).
Primary slip systems in _compression_ predicted from the yield criterion for molybdenum (a) and tungsten (b). In the regions of simultaneous activity of two slip systems, the primary slip system with lower yield stress is marked as I and the secondary, whose yield stress is at most 2% higher than I, as II.
A similar situation arises on the other side of the triangle. For orientations close to the corner the slip system (\(101\))[\(1\,\overline{1}\,\overline{1}\,\overline{1}\,\overline{1}\,\overline{1}\,\overline{1}\)) dominates but there is again an intermediate region where the slip systems (\(\overline{1}01\))[\(\overline{1}\,\overline{1}\,\overline{1}\,\overline{1}\)) and (\(101\))[\(1\,\overline{1}\,\overline{1}\,\overline{1}\)] can be equally active. However, the slip directions are now different and involve \(1/2\)[\(111\)] and \(1/2\)[\(\overline{1}11\)] dislocations that move simultaneously. This so-called multislip mechanism of plastic deformation has been frequently observed in low-temperature experiments not only in molybdenum but also in other pure BCC refractory metals.
In the case of tungsten, the distribution of primary slip systems, shown in Fig. 4b, is rather different and more complex than for molybdenum. The striking difference between the two metals is that while in molybdenum the (\(\overline{1}01\))[\(\overline{1}\,\overline{1}\,\overline{1}\)] system dominates the central region of the stereographic triangle in which \(\chi\)\(>\)\(0\), in tungsten it is the (\(\overline{1}10\))[\(\overline{1}\,\overline{1}\,\overline{1}\)] system that is dominant in this region. Only near the corner the slip systems (\(\overline{1}01\))[\(\overline{1}\,\overline{1}\,\overline{1}\)] and (\(\overline{1}10\))[\(\overline{1}\,\overline{1}\,\overline{1}\)] may operate simultaneously. It is important to emphasize that the Schmid factor for the (\(\overline{1}10\))[\(\overline{1}\,\overline{1}\,\overline{1}\)] system is only a half of that for the (\(\overline{1}01\))[\(\overline{1}\,\overline{1}\,\overline{1}\)] system. The region of negative \(\chi\) is dominated by the (\(\overline{1}10\))[\(11\,\overline{1}\)] system surrounded by regions of simultaneous operation of the (\(\overline{1}10\))[\(11\,\overline{1}\)] system and (\(\overline{1}10\))[\(\overline{1}\,\overline{1}\,\overline{1}\)] or (\(\overline{1}\,\overline{1}0\))[\(\overline{1}\,\overline{1}\)], respectively. Similarly as in molybdenum, this results in multislip deformation that takes place as a consequence of cooperative motion of two different \(1/2\)/\(111\)) dislocations. As seen in Fig. 4b, a variety of slip systems, several of them leading to the multislip, are found near the and [\(\overline{1}11\)] corners of the stereographic triangle.
The reason for the significant difference between molybdenum and tungsten when loaded in compression is that in molybdenum the difference between orientations corresponding to positive and negative \(\chi\) is to a great extent governed by the twinning-antitwinning asymmetry encountered in pure shear along the slip direction of the most highly stressed (\(\overline{1}01\))[\(\overline{1}\,\overline{1}\,\overline{1}\)] system. In contrast, as shown in Part I, this asymmetry plays virtually no role in tungsten and the choice of the most favored slip system is affected solely by the shear stress perpendicular to the slip direction.
Unfortunately experiments currently accessible are not sufficient to test the predicted variety of slip systems in molybdenum and tungsten loaded in tension/compression at very low temperatures. Especially, no low temperature experiments are available for tungsten that tends to be brittle under these conditions, for which the most interesting finding is that the (\(\overline{1}10\))[\(\overline{1}\,\overline{1}\,\overline{1}\)] system dominates for most orientations for which \(\chi\)\(>\)\(0\) although its Schmid factor is only a half of the most highly stressed (\(\overline{1}01\))[\(\overline{1}\,\overline{1}\,\overline{1}\)] system. However, this deformation mode is reminiscent of the anomalous slip observed in a number of high-purity BCC metals at very low temperatures and it is feasible that this type of slip has the same origin as found in tungsten. Notwithstanding, in recent years, Seeger and Hollang performed a series of detailed experiments on ultra-high purity molybdenum single crystals with the main objective to investigate the tension-compression asymmetry of the yield stress. In the following section we compare our predictions with these experimental results.
## 7 Yield stress asymmetry in tension and compression
In the experiments of Hollang et al. on molybdenum interesting tension-compression asymmetries of the uniaxial yield stress were found at temperatures close to \(123\,\)K. For orientations of the tensile/compressive axes [\(\overline{1}\)49] (\(\chi\) = 0) and [\(\overline{5}\)79] (\(\chi\) = +21\({}^{\circ}\)) the uniaxial yield stress in compression was appreciably higher than in tension. While for the [\(\overline{5}\)79] axis this could be explained by the twinning-antitwinning asymmetry of shearing, these arguments fail for \(\chi\) = 0 when no such asymmetry exists. Moreover, for the [\(\overline{1}\)22] loading axis (\(\chi\) = +29\({}^{\circ}\)) the uniaxial yield stress in tension was found to be higher than in compression. Thus the tension-compression asymmetry changes character between \(\chi\) = +21\({}^{\circ}\) and \(\chi\) = +29\({}^{\circ}\) which can in no way be attributed to the twinning-antitwinning asymmetry. In the following, we employ the proposed yield criterion for molybdenum to predict the tension-compression asymmetries and compare the results with the data obtained in.
\[SD=\frac{\sigma_{t}-\sigma_{c}}{\left(\sigma_{t}+\sigma_{c}\right)/2}\,. \tag{6}\]
For any orientation of the loading axis these yield stresses can be determined on the basis of the yield criterion as described in the previous section. Performing these calculations for a variety of orientations of the tensile/compressive axes we obtain a map of the strength differential \(SD\) within the standard stereographic triangle. This is displayed in by shading the interior of the standard triangle such that different shadings correspond to different values of \(SD\).
In the case of molybdenum, the yield criterion predicts that for most orientations of the loading axis the uniaxial yield stress in compression is larger than in tension and thus \(SD\) < 0. However, for the axes approaching the-[\(\overline{1}\)11] edge, \(\sigma_{t}\) increases relative to \(\sigma_{c}\) and becomes equal to \(\sigma_{c}\) for orientations along the solid curve shown in For orientations close to the corner, \(\sigma_{t}\) > \(\sigma_{c}\) and \(SD\) becomes positive.
Distribution of the strength differential \(SD\) calculated from the yield criterion for molybdenum (a) and tungsten (b). Note the existence of a small region of positive \(SD\) in molybdenum and its lack in tungsten.
For comparison, the maximum negative value of the strength differential, corresponding to the loading axis in the middle of the-[\(\overline{1}\)11] edge, is \(SD\approx-0.45\).
The above calculations of \(SD\) correspond to loading at 0 K since the effect of temperature has not been introduced at this stage. Nevertheless, a direct comparison between the calculated and experimentally observed strength differential can be made since both \(\sigma_{t}\) and \(\sigma_{c}\) vary with temperature in the same way and thus \(SD\) is essentially temperature-independent. Such comparison is presented in Table 3 and it shows that the variation of the tension-compression asymmetry predicted by the yield criterion agrees qualitatively very well with the measurements. Table 3 also contains the strength differential calculated from the restricted yield criterion (\(a_{2}=a_{3}=0\)) that includes the effect of the twinning-antitwinning asymmetry but not the effect of shear stresses perpendicular to the slip direction. This demonstrates most emphatically that the observed tension-compression asymmetry of the yield stress is not just the consequence of the twinning-antitwinning asymmetry of shearing in the slip direction. If this were the sole reason, as has often been assumed, \(SD\) would be antisymmetric with respect to \(\chi\) and thus would necessarily vanish at \(\chi=0\). Hence, the effect of shear stresses perpendicular to the slip direction is a major contribution to the tension-compression asymmetry.
For completeness, we present in the distribution of the strength differential \(SD\) determined from the yield criterion for tungsten. In this case \(SD<0\) for all orientations of the loading axes and thus the yield stress in compression is always higher than in tension. This feature of the deformation behavior can be attributed to even stronger effect of the shear stresses perpendicular to the slip direction in tungsten than in molybdenum, which is apparent from the comparison of the coefficients \(a_{2}\) and \(a_{3}\) for the two metals, presented in Table 2.
## 8 Random polycrystals
The effects of non-glide stresses on the slip behavior of BCC metals have been clearly established from atomistic studies of screw dislocations and predictions of yielding in single crystals.
\begin{table}
\begin{tabular}{l c c c} \hline \hline & \multicolumn{3}{c}{\(SD\)} \\ \cline{2-4} & experiment & restricted criterion & full criterion \\ \hline \([\overline{1}49]\), \(\chi\)=0 & \(-0.06\) & 0 & -0.28 \\ \([\overline{5}79]\), \(\chi\)=+21\({}^{\circ}\) & \(-0.04\) & \(+0.16\) & -0.21 \\ \([\overline{1}22]\), \(\chi\)=+29\({}^{\circ}\) & \(+0.07\) & \(+0.21\) & 0.0 \\ \hline \hline \end{tabular}
\end{table}
Table 3: The strength differential \(SD\) for molybdenum determined experimentally in Ref. and from the yield criterion in the restricted (\(a_{2}=a_{3}=0\)) and full form, respectively. In the former case only the twinning-antitwinning asymmetry of shearing can be responsible for the tension-compression asymmetry.
Similar effects are also predicted for BCC single crystals. We will now illustrate that the effects of non-glide stresses persist in yielding of BCC polycrystals, which also requires non-associated flow descriptions. This has been demonstrated in studies of forming limits associated with necking in thin sheets. In this section, we outline a framework for an isotropic non-associated flow theory appropriate for random BCC polycrystals and in the following section cavitation instabilities are analyzed.
A classical non-associated flow relation for the plastic strain rate, \(\dot{\epsilon}_{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{ \mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{ \mathit{ \mathit{ }}}}}}}}}}}}}}}}^{ \mathrm{p}}\), is defined by4
Footnote 4: Since cavitation instabilities involve proportional stressing, we will not distinguish co-rotational time derivatives of stress from ordinary time derivatives.
\[\dot{\epsilon}_{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{ \mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{ \mathit{\mathit{\mathit{\mathit{ \mathit{ }}}}}}}}}}}}}}}}}}}^{ \mathrm{p}}=\frac{\lambda}{E_{\mathrm{t}}^{\mathrm{p}}}\Bigg{(}\frac{\partial F }{\partial\sigma_{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{ \mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{ \mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{ \mathit{ \mathit{ } \mathit{ \mathit{ } \mathit{ }}}}}}}}}}}}}}}}}}}\,\Bigg{)}\frac{ \partial G}{\partial\sigma_{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{ \mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathit{ \mathit{\mathit{\mathit{\mathit{\mathit{\mathit{\mathitmathit{ \mathitmathit{ \mathitmathitmathitmathit{ \mathitmathitmathit{ \mathitmathitmathit{ \mathitmathitmathitmathit{ \mathitmathitmathit{ \mathitmathitmathit{ \mathitmathitmathit{ \mathitmathitmathit{ \mathitmathitmathit{ \mathitmathitmathit{ \mathitmathitmathit{ \mathitmathitmathitmathit{ \mathitmathitmathitmathit{ \mathitmathitmathit{ \mathitmathitmathitmathit{ \mathitmathitmathitmathit{ \mathitmathitmathitmathit{ \mathitmathitmathitmathitmathit{ \mathitmathitmathitmathit{ \mathitmathitmathitmathitmathitmathit{\the functional form of \(G\) if the flow function is positive and homogenous in stress.
In what follows, we consider isotropic behavior at the macroscopic scale and utilize recent results of a Taylor type calculation for random polycrystals, which assumes that the strain in each crystal is the same as the macroscopic strain (see e.g.). The yield and flow surfaces predicted from the Taylor type calculation, based upon the yield criterion with the \(a_{i}\) derived from atomistic simulations for molybdenum, are plotted as circles in for plane states of stress. A family of similar surfaces is obtained as the magnitudes of the non-glide stress coefficients, \(a_{i}\) in, are varied. The flow surfaces, on the other hand, have little dependence on the non-glide stresses and, therefore, are indistinguishable from the yield surface for associated flow behavior. Finally, we note that we have considered both rate-dependent and rate-independent flow descriptions and, in the latter case, both with smooth yield and flow surfaces and with the effects of corners on the flow surface. Here we limit our attention to smooth yield surfaces and adopt a flow rule of the form of.
Simple isotropic functions that accurately describe computed yield and flow surfaces for random BCC polycrystals obtained from Taylor calculations, such as those shown in Fig. 6, are given by:
\[F=\sqrt{3}\left[\left(J_{2}\right)^{3/2}+bJ_{3}\right]^{1/3} \tag{12a}\] \[G=\sqrt{3J_{2}}\ \, \tag{12b}\]
In this simple isotropic theory, the parameter \(b\) entering the yield function (12a) is the non-associated flow parameter and the only measure of the effects of non-glide stresses. Note that the yield function \(F\) reduces to the classical von Mises function (12b) for \(b\)=0. These functions are plotted as the continuous curves in Fig. 6, with \(b=-0.72\) the
Yield and flow surfaces predicted from a Taylor model for BCC Mo (circles) and best fits: \(F\) with \(b\)=–0.72 in (12a) and \(G\) (12b).
With \(J_{2}\) and \(J_{3}\) computed for uniaxial stress states, the strength differential is given in terms of \(b\) and using (12a) it can be written as
\[SD=2\frac{\left(1-2b/3\sqrt{3}\right)^{1/3}-\ \ \left(1+2b/3\sqrt{3}\right)^{1/3} \approx-0.257\ b\ . \tag{13}\]
It is worth noting that the best fit to the flow surface in with a function of the form of (12a) gives \(b\approx 0\), which is the von Mises function (12b).
In closing this section, we note that non-associated flow models also have been adopted to describe frictional materials, e.g. granular and geological materials, and to account for pressure-sensitive yield in metals. With these material behaviors in mind, the effects of non-associated flow on strain localization have been investigated, while predictions have been mostly restricted to shear bands in plane strain tension. Kuroda has studied sheet necking and has shown that the effect of non-associated flow arising from pressure sensitivity on forming limits is small. In contrast, we have found a significant effect of non-associated flow on sheet forming limits from calculations that utilize the isotropic yield and flow functions (12a) and (12b), respectively, that approximate well the Taylor surfaces for random BCC polycrystals.
## 9 Cavitation instabilities
The fact that non-glide stress effects persist at the level of polycrystals and can significantly alter the failure criteria is readily demonstrated in the problem of cavitation instabilities. These instabilities arise when the energy available from the strained material surrounding the cavity is sufficient to drive continued expansion (see e.g.). The problem addressed here is defined as follows: a spherically symmetric stress (i.e. hydrostatic tension) is applied at infinity and conditions are sought for the critical value of this stress when a spherical cavity grows without bounds, i.e. when the radius of the cavity approaches infinity in the deformed configuration. The existence of cavitation instabilities in elastic-plastic solids was recognized by Bishop et al. and Hill also determined limit states for the equivalent problems of cavities subjected to internal pressure. Huang et al. studied cavitation instabilities under axisymmetric states of remote stress in elastic-plastic solids.
In this analysis, we assume a rigid-plastic rate-independent material behavior for simplicity (i.e. the elastic strains are neglected). Furthermore, we assume that the loading is monotonic and the material strain hardens, in which case the plastic tangent modulus is always positive. Let \(R\) and \(r\) denote the radial coordinates of material points, measured from the center of a spherical cavity, in the undeformed and deformed configurations, respectively.
\[\lambda_{r}=\left(\frac{R}{r}\right)^{2},\ \ \ \lambda_{\phi}\ =\ \lambda_{\phi}=\frac{r}{R} \tag{14}\]and the corresponding radial strain rate is
\[\dot{\varepsilon}_{{}_{rr}}=\frac{\dot{\lambda}_{{}_{r}}}{\lambda_{{}_{r}}}=-2\frac {\dot{r}}{r}. \tag{15}\]
Since hydrostatic tension is applied at infinity, \(\dot{r}>0\) and, therefore, \(\dot{\varepsilon}_{{}_{rr}}<0\).
\[\dot{\varepsilon}_{{}_{rr}}=\frac{3}{2E_{{}_{t}}^{{}^{p}}}\dot{F}\frac{S_{{}_{ rr}}}{G}. \tag{16}\]
From the above expression and the fact that the loading is monotonic it is straightforward to show that \(\dot{F}>0\), and \(s_{{}_{rr}}<0\).
\[\sigma_{{}_{e}}=\frac{3}{2}\Big{|}s_{{}_{rr}}\Bigg{(}\frac{3\sqrt{3}-2b}{3 \sqrt{3}+2b}\Bigg{)}^{\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\! \!\!\[\sigma_{\rm crit}=2\Bigg{(}\frac{3\sqrt{3}+2b}{3\sqrt{3}-2b}\Bigg{)}^{1/3}\int \limits_{1}^{\infty}F_{\rm cr}\Bigg{[}-\frac{2}{3}\ln\left(1-\eta^{-3}\right) \Bigg{]}\eta^{-1}d\eta\ \, \tag{23}\]
Here a class of power-law hardening materials is considered with
\[F_{\rm cr}\left(\varepsilon_{\rm o}^{\rm p}\right)=\left\{\begin{array}{ll} \sigma_{\rm o}\varepsilon_{\rm e}^{\rm p}/\varepsilon_{\rm o}&\mbox{if }\varepsilon_{\rm e}^{\rm p}\leq\varepsilon_{\rm o}\\ \sigma_{\rm o}\Bigg{(}\frac{\varepsilon_{\rm e}^{\rm p}}{\varepsilon_{\rm o}} \Bigg{)}^{N}&\mbox{if }\varepsilon_{\rm e}^{\rm p}\geq\varepsilon_{\rm o}\end{array}\right.\, \tag{24}\]
For this power-law hardening behavior, the estimate for the cavitation limit stress becomes
\[\sigma_{\rm crit}=2\Bigg{(}\frac{3\sqrt{3}+2b}{3\sqrt{3}-2b}\Bigg{)}^{1/3} \left\{\begin{array}{ll}\int\limits_{1}^{\infty}\frac{\sigma_{\rm o}}{ \left(\varepsilon_{\rm o}\right)^{N}}\Bigg{[}-\frac{2}{3}\ln\left(1-\eta^{-3} \right)\Bigg{]}^{N}\eta^{-1}d\eta\ \ +\\ \int\limits_{\eta_{\rm o}}^{\infty}-\frac{2}{3}\frac{\sigma_{\rm o}}{ \varepsilon_{\rm o}}\ln\left(1-\eta^{-3}\right)\eta^{-1}d\eta\end{array}\right\}\, \tag{25}\]
where
\[\eta_{\rm o}=\left(1-e^{-3\varepsilon_{\rm o}}F\right)^{-1/3}. \tag{26}\]
The cavitation limit versus the hardening exponents \(N\) is plotted in for various values of the strength differential, _SD_, which is given in terms of the coefficient \(b\) in. As in the case of associated flow, the limit stress increases with increasing hardening exponent even for non-associated flow. In addition, there is a significant dependence on _SD_, as seen in for \(N=0.1\). Recall that for a random polycrystal with the single crystal yield criterion derived from the atomistic results for molybdenum \(SD\approx 0.2\). For that value of the strength differential there is almost 20% reduction in the cavitation limit.
Cavitation limits versus hardening exponent \(N\) for various values of strength differential _SD_.
## 10 Conclusions
Based on atomistic studies of the glide of an isolated 1/2 screw dislocation, presented in the Part I of this series of papers, we constructed the analytical yield criteria for Mo and W that reflect the effect of non-glide stresses on the CRSS (Peierls stress) for glide of these dislocations. In these criteria, formulated following the original development in, a linear combination of two shear stresses parallel to and two shear stresses perpendicular to the slip direction has to reach a critical value. Only the shear stress parallel to the slip direction acting in the {110} type glide plane, called the Schmid stress, does work as the dislocation glides. The other three shear stresses affect the magnitude of the CRSS by modifying the structure of the dislocation core and their effect causes the breakdown of the Schmid law. In general, when plastic deformation is affected by components of the applied stress tensor other than the Schmid stress, such behavior is defined as a non-associated plastic flow. In the setting of continuum mechanics it means that the yield function and the flow potential are not equal. The physical origin of this non-associated flow behavior is the complex response of the non-planar cores of 1/2\(\langle\)111\(\rangle\) screw dislocations to a general state of stress.
The yield criterion formulated in this paper contains only four adjustable parameters that are readily determined using the results of the atomistic calculations presented in Part I. Specifically, the dependence of the CRSS (Peierls stress) on the orientation of the MRSSP, i.e. the angle \(\chi\), is used to parameterize the dependence of the CRSS on the shear stress parallel to the slip direction in a {110} plane other than the glide plane. This reproduces the twinning-antitwinning asymmetry of the dislocation glide and determines parameters \(a_{1}\) and \(\tau_{cr}^{*}\) in and. Subsequently, the atomistically calculated dependence of the CRSS on the shear stress perpendicular to the slip direction, \(\tau\), is accurately reproduced via parameters \(a_{2}\) and \(a_{3}\) in and.
Since all {110}\(\langle\)111\(\rangle\) slip systems are equivalent, the tensorial representation of the criterion can be directly employed in crystal plasticity studies. For example, it can be utilized to determine the first system to operate under a given applied loading and to assess the slip activity of each system during continued plastic deformation. We have shown for tensile loading alongvarious axes within the standard stereographic triangle that both Mo and W deform predominantly by slip on the most highly stressed (\(\overline{1}01\)) system. On the other hand, in compression the slip activity depends both on the material and on the orientation of the loading axis. For a broad range of orientations the most operative slip system in Mo is still the most highly stressed (\(\overline{1}01\))[\(\overline{1}\)\(\overline{1}\)\(\overline{1}\)\(\overline{1}\)\(\overline{1}\)] system but in W it is the (\(\overline{1}\)\(10\))[\(\overline{1}\)\(\overline{1}\)\(\overline{1}\)\(\overline{1}\)] system. To the best of our knowledge, there are no experimental data to verify or refute the latter prediction. For orientations of loading axes close to the corners and sides of the standard stereographic triangle the calculations predict a simultaneous operation of two or more slip systems in both Mo and W, i.e. the multislip that has been frequently observed. Moreover, in W slip is predicted to occur on a low-stressed (\(\overline{1}\)\(10\))[\(\overline{1}\)\(\overline{1}\)\(\overline{1}\)] system for the loading axes corresponding to \(\chi\)\(>\)\(0\), while in Mo the same slip mode is observed only for orientations close to \(\chi\)\(=\)\(30^{\circ}\). In both metals, this is reminiscent of the anomalous slip observed in many high-purity BCC metals at low temperature.
A direct comparison between experimental observations and predictions based on the constructed yield criterion can be made for the tension-compression asymmetry of the yield stress that was measured in molybdenum in. For Mo the yield criterion predicts that the critical compressive stress is larger than the critical tensile stress for a broad range of orientations of the loading axis within the standard stereographic triangle. The character of this asymmetry reverses only for orientations close to the corner where the yield stress in tension becomes higher than that in compression. This finding is in an excellent qualitative agreement with the observations in. An interesting feature of this asymmetry is that it is significant for the [\(\overline{1}\)\(49\)] loading axis when the (\(\overline{1}\)\(0\)\(1\)) plane is the MRSSP and \(\chi\)\(=\)\(0\). If only the shear stress parallel to the slip direction played role the asymmetry would necessarily vanish for this orientation since such stress can induce an asymmetry, known as the twinning-antitwinning asymmetry, only for orientations of loading with \(\chi\)\(\neq\)\(0\). This disparity was noted in Ref. and attributed to "a modification of the Peierls potential by stress components other than the resolved shear stress". In the setting of the non-associated plastic flow model, we identify this contribution as an effect of shear stresses perpendicular to the slip direction. For other orientations of the loading axis the asymmetry is the result of combination of twinning-antitwinning asymmetry and effects of stresses perpendicular to the slip direction.
For W the yield criterion predicts that the critical compressive stress is larger than the critical tensile stress for any orientation of the loading axis within the standard stereographic triangle. Since the twinning-antitwinning asymmetry is negligible in W the predicted tension-compression asymmetry is entirely due to the shear stresses perpendicular to the slip direction. Unfortunately, no experimental observations that would test this finding are currently available.
This paper concludes with predictions of the effects of non-glide stresses at the polycrystalline level, in which individual grains undergo multiple slip. Using the slip system yield criteria derived from the atomistic studies and a Taylor homogenization procedure for a random polycrystal we have demonstrated that the effects of non-planar cores of screw dislocations persist at the polycrystal level. The polycrystalline constitutive behavior also corresponds to non-associated flow in which the yield and flow functions depend only on deviatoric components of stress. Several earlier continuum analyses have found that the effects of non-glide stresses significantly affect critical phenomena at macroscopic scales such as shear localization in single crystals. In this paper, we demonstrated this fact on the problem of cavitation in a ductile plastic solid. The dependence of the cavitation limit on the strength differential (\(SD\)), which is a convenient measure of the extent of non-associated flow, is shown to scale roughly in proportion to \(SD\) (see Fig. 7). For the strength differential derived from the yield criterion for molybdenum there is almost 20% reduction in the cavitation limit. In another related study, Racherla and Bassani have shown that non-associated flow plays an important role also in forming limits of biaxially stretched sheets. Overall, these findings demonstrate that the effects of non-planar core structure of screw dislocations in BCC metals and related complex influence of applied loading on dislocation glide persist in macroscopic deformation of both single and polycrystals.
| 10.48550/arXiv.0807.2771 | Multiscale modeling of plastic deformation of molybdenum and tungsten: II. Yield criterion for single crystals based on atomistic studies of glide of 1/2<111> screw dislocations | R. Gröger, V. Racherla, J. L. Bassani, V. Vitek | 1,623 |
10.48550_arXiv.1610.08220 | ###### Abstract
In the present work we grow self-organized TiO\({}_{2}\) nanotube arrays with a defined and controlled regular _spacing_ between individual nanotubes. These defined intertube gaps allow to build up hierarchical 1D-branched structures, conformally coated on the nanotube walls using a layer by layer nanoparticle TiO\({}_{2}\) decoration of the individual tubes, i.e. having not only a high control over the TiO\({}_{2}\) nanotube host structure but also on the harvesting layers. After optimizing the intertube spacing, we build host-guest arrays that show a drastically enhanced performance in photocatalytic H\({}_{2}\) generation, compared to any arrangement of conventional TiO\({}_{2}\) nanotubes or conventional TiO\({}_{2}\) nanoparticle layers. We show this beneficial effect to be due to a combination of increased large surface area (mainly provided by the nanoparticle layers) with a fast transport of the harvested charge within the passivated 1D nanotubes. We anticipate that this type of hierarchical structures based on TiO\({}_{2}\) nanotubes with adjustable spacing will find even wider application, as they provide an unprecedented controllable combination of surface area and carrier transport.
Since the groundbreaking work of Fujishima and Honda in 1972, photocatalytic H\({}_{2}\) generation by water splitting based on TiO\({}_{2}\) materials has received immense research interest. Over the past decades, not only TiO\({}_{2}\) but also hundreds of other semiconductors have been investigated for their H\({}_{2}\) generation performance (e.g. ranging from Si to group II-VI, III-V compound semiconductors and to various transition metal oxides, and so on). In spite of some intrinsic deficits of TiO\({}_{2}\), such as a wide band gap and a sluggish electron transfer kinetics, it still remains the most studied photocatalytic material as it has an almost unique (photo)corrosion resistance and comparably low cost. The photoelectrochemical principle to hydrogen generation is that suitable light irradiation promotes electrons from the semiconductor (TiO\({}_{2}\)) valence band to the conduction band, thus generating electron hole pairs that then can react at the semiconductor surface with redox couples in the surroundings - hydrogen (H\({}_{2}\)) generation occurs from ejected conduction band electrons to H\({}_{2}\)O or H\({}^{+}\).
In order to maximize the efficiency of the H\({}_{2}\) photocatalytic reactions on TiO\({}_{2}\), typically TiO\({}_{2}\) nanoparticles, either free floating in the solution or compacted to electrodes, are used. Nanoscale photocatalysts provide not only a high specific surface area but also a short diffusion path for the excited carriers to reach the particle surface (minimizing recombination). More recently, one-dimensional (1D) structures such as nanotubes, nanowires and nanorods have attracted significant interest as they can provide, except for a large surface area, a _directional_ carrier transporter (faster), and in many cases intrinsically support an orthogonal electron-hole separation. These features further aid in minimizing the recombination of charge carriers and thus minimize the efficiency loss.
In the past years, particularly highly ordered TiO\({}_{2}\) nanotube layers (NTs) grown by self-organizing electrochemical anodization of titanium have attracted wide interest. For these tubes, which are vertically aligned on the substrate, geometry and structure can be easily controlled by the anodization parameters and by a post formation thermal annealing. In some cases, such nanotube layers were reported to outperform comparable nanoparticle layersin their photocatalytic performance. This was mainly ascribed to a more efficient charge separation and transport which, in fact, is not surprising considering that the electron mobility of nanoparticle layers has been reported to be orders of magnitude lower than that of nanotubes. Nevertheless, in terms of specific surface area, such NTs provide clearly lower values (30-50 m\({}^{2}\) g-1) than comparable nanoparticle layers where values of 100 m\({}^{2}\) g-1 (and more) can be reached.
In order to combine the benefits of NTs (electron transport properties) and nanoparticles (large surface area), hierarchical structures have been explored where TiO\({}_{2}\) particles are decorated onto the TiO\({}_{2}\) NTs. However, up to now, optimized hierarchical structures require the ability to space the 1D conducting material in desired distances on a substrate, e.g. as in the case of nanowires, nanorods or nanotubes grown by template method. However, using classic anodic TiO\({}_{2}\) nanotube layers does not allow this, as conventional anodization leads to a hexagonally close-packed arrangement of nanotubes, and this geometry allows only a limited decoration with defined secondary nanoparticle layers.
In the present work we introduce the use of _spaced_, self-ordered nanotube layers with controllable regular intertube gaps to build defined hierarchical structures, as shown in In contrast to the classic (hexagonal close-packed) NTs in Fig. 1c, spaced nanotubes can be precisely coated, layer by layer with nanoparticles, up to an optimized number of layers without clogging the tube's mouth. These structures when used for photocatalytic H\({}_{2}\) production can exhibit a strongly enhanced efficiency compared with any conventional TiO\({}_{2}\) NTs or compared to a plain TiO\({}_{2}\) nanoparticle layer.
The formation of self-organized nanotubular layers with defined intertube gaps takes place in some electrolytes under anodization conditions (outlined in the SI) that establish a controlled spacing, in fact, independent control of tube diameter and intertube distance, can be established by control over the water content in the electrolyte and the applied voltage. After a set of preliminary experiments (see SI) where we evaluated the photocurrent response of nanotubes that have different spacing, we selected nanotubes with spacing in the range of 150-200 nm in order to build up hierarchical structures (see Fig. S1, S2).
The spaced TiO\({}_{2}\) NTs were grown by anodization of Ti in hot triethylene glycol (TEG) based electrolyte, while the reference nanotubes (R-NTs) are grown in ethylene glycol based electrolyte, see (for more details, see SI). After anodization, these spaced TiO\({}_{2}\) NTs (S-NTs) uniformly cover the surface, and have a 220 nm diameter with an average intertube space (wall-to-wall) of approx. 150-200 nm. Such a geometry was found to be ideal for a highly conformal layer by layer decoration with TiO\({}_{2}\) nanoparticles (NPs), using a TiCl\({}_{4}\)-hydrolysis approach, as shown in and it allows a controlled decoration with up to 5 layers. Each decorated layer is \(\approx\)17 nm thick and consists of \(\approx\)5-10 nm diameter TiO\({}_{2}\) nanoparticles that homogeneously coat both the interior and exterior surfaces of the nanotube walls, as evident from the SEM images in and e. After annealing the decorated NTs at 450\({}^{\circ}\)C in air, high resolution TEM images (Fig. 1f,g) show that the TiO\({}_{2}\) NPs are interconnected and fully crystallized (anatase crystallographic planes, 3.46 A lattice spacing) can be identified. Moreover, XRD confirms the anatase phase of the NTs (see Fig. 2.c), while XRD and TEM results together indicate that not only the tubes but also the decorated particles were converted to anatase.
In order to assess the photocatalytic H\({}_{2}\) evolution performance, layers were decorated with a Pt co-catalyst (1 nm Pt thick, SI) and illuminated with continuous UV light (325 nm) in an ethanol-water mixture (Fig. 1h,i). shows a comparison of the H\({}_{2}\) amounts evolved from spaced NTs with and without nanoparticle decoration, reference NTs and a nanoparticle (P) film. The H\({}_{2}\) evolution amount from plain S-NTs is already slightly higher than R-NTs, however both are clearly lower than for the defined TiO\({}_{2}\) nanoparticle film on FTO substrate (the nanoparticle film is shown in Figure S4).
The detailed evaluation of the spaced NTs hierarchical structures, with layer by layer TiO\({}_{2}\) nanoparticle decoration, is shown in the SEM images of With every additional layer, not only the inner diameter of NTs decreases but also the outer diameter expands. More importantly, the photocatalytic performance increases and maximizes after three layers, then decreases; higher amounts of NP loadings led to a drop in H\({}_{2}\) generation as NTs open geometry decreases, thus reducing the surface area as well as light penetration.
However, for S-NTs, after the optimal three times layer by layer TiO\({}_{2}\) NP decoration, a seven times improvement of the photocatalytic H\({}_{2}\) generation is obtained and this hierarchical structure clearly outperforms the nanoparticle layer. If the same multiple decoration treatment is attempted with classic NTs (R-NTs), not only does the second decoration already result in an inhomogeneous morphology but also the nanotubes are blocked with TiO\({}_{2}\) NPs (Fig. S5). For these R-NTs, the generated H\({}_{2}\) amount is \(\approx\)16 \(\upmu\)mol h-1 cm-2, which is half of the hierarchical S-NT structure produced under similar conditions.
When investigating the effect of the length of nanotubes for the three times decorated spaced NTs, a maximum H\({}_{2}\) production rate is registered for a 7 \(\upmu\)m tube length (Fig. 1i, Fig. S6). This is in line with literature, where a thickness of 6-7 \(\upmu\)m is found to be an optimum between light absorption and electron diffusion length (previous reports show for NP UV light penetration (325 nm) of 1-3 \(\upmu\)m, and diffusion length in TiO\({}_{2}\) NTs of several 10 \(\upmu\)m while in particle layers this is only some few \(\upmu\)m). Additionally, a 1nm thick Pt decoration is optimal in view of photocatalytic H\({}_{2}\) generation (see Fig. S7, S8). For these optimized S-NTs, the H\({}_{2}\) generation is linear over extended time, indicating that the structures are stable, see Fig. S9 (moreover, SEM images after extended H\({}_{2}\) generation times did not show any significant difference - data not shown).
Usually, as-formed S-NTs are amorphous (only titanium peaks are detected) and by 450 \({}^{\circ}\)C air annealing, crystallization induces anatase formation (XRD patterns in Fig 2.c). S-NTs were layer by layer coated with TiO\({}_{2}\) nanoparticles and air annealed again at 450 \({}^{\circ}\)C (for 10 min) to crystallize the nanoparticles and the hierarchical structures crystallography remains unchanged after the three layers decoration.
In order to characterize the main difference between the hierarchical structure and conventional tubes, several aspects were investigated which include their photocurrent spectra, their relative active surface area, the relative electron transport time from intensity modulated photocurrent spectroscopy (IMPS) measurements and their relative ability to photocatalytically generate OH radicals.
From photocurrent spectra measurements, most apparent is that the nanoparticle layer on FTO shows a very low photon to current conversion efficiency compared with the nanotube layers - this is indicative of the high recombination rate of electron-hole pairs in the nanoparticle structure. Clearly, S-NTs show a significantly higher photocurrent magnitude than R-NTs, with the highest values for the three times loading, and nanoparticle decorated S-NTs show also a shift of the wavelength of the maximum IPCE.
In order to determine the surface area of the structures, dye loading measurements for R-NTs, TiO\({}_{2}\) nanoparticle films, spaced NTs and three times loading spaced NTs (S-NTs-T3) were performed and resulted in 41, 101, 24 and 131 nmol cm\({}^{-2}\), respectively. This shows that the NPs decoration of S-NTs strongly increases their specific surface area, up to 5 times (from dye loadings of 24 to 131 nmol cm\({}^{-2}\)), thus enhancing the active surface area. In fact, the average particle size in the nanoparticle layers (10-15 nm) is larger than the particles obtained from the TiCl\({}_{4}\) layers (5-10 nm), therefore the hierarchical structure shows a higher surface area than the nanoparticle layer.
To examine the electronic properties of the structures, IMPS measurements were performed for these selected structures, see Clearly, electron transport in the TiO\({}_{2}\) particle layer is significantly slower than that for either bare nanotubes - reference or spaced. While both types of non-decorated nanotubes (hexagonally-packed and spaced) have a similar electron transport rates, one layer NP decoration of the spaced NTs improves the electron transport time by a decade and the optimal three times decoration shows only slightly faster transport time compared to one layer, while additional NP layers deposition leads to no further improvement. These findings can be ascribed to the fact that the TiCl\({}_{4}\) treatment not only increases the specific surface area but also passivates the defects on TiO\({}_{2}\) NTs, thus enhancing the electronic properties. A classic photocatalytic test for the formation of OH\({}^{\bullet}\) radicals, i.e. using fluorescence from terephthalic acid under UV light (\(\lambda\)=325 nm) illumination - see Fig. 3d, confirms that under the same conditions in spaced NTs (S-NTs) twice the rate of OH\({}^{\bullet}\) radicals can be formed than with classical hexagonally packed NTs (R-NTs) - this confirms that not only a strongly improved H\({}_{2}\) formation rate can be achieved using hierarchical NTs but also that in classic photocatalytic reactions (photodegradation, photosynthesis) hierarchical structures are able to generate strongly enhanced amounts of active species.
In summary, the present work shows the fabrication of a photocatalytic platform consisting of hierarchical TiO\({}_{2}\) nanostructures. The approach is based on using highly ordered spaced TiO\({}_{2}\) NTs as a scaffold for a controlled layer by layer TiO\({}_{2}\) nanoparticle deposition. After optimizing the nanoparticle decoration, hierarchical TiO\({}_{2}\) nanostructures show a significant enhancement of the photocatalytic performance in comparison to the hexagonally close-packed TiO\({}_{2}\) NTs and conventional TiO\({}_{2}\) nanoparticle films. This improvement is ascribed to the combination of fast electron transfer due to the 1D structure and to the high surface area owing to nanoparticle decoration. This also opens a practical route for the deposition of other materials in the spaced TiO\({}_{2}\) NTs, thus using to full advantage such morphology for the development of other state-of-the art applications.
Dr. Lei Wang is acknowledged for helping in the evaluation of the diffraction data. Xuemei Zhou is acknowledged for XPS measurements. Prof. Kiyoung Lee and Jeong Eun Yoo are acknowledged for valuable discussions. The authors would also like to acknowledge the ERC,the DFG, the Erlangen DFG cluster of excellence EAM, and the DFG funCOS for financial support.
# Figure S9.
Photocatalytic H\({}_{2}\) evolution rate measured for spaced TiO\({}_{2}\) NTs after decoration with 3 layers of nanoparticles and 1 nm thick Pt. | 10.48550/arXiv.1610.08220 | TiO2 nanotubes with laterally spaced ordering enable optimized hierarchical structures with significantly enhanced photocatalytic H2 generation | Nhat Truong Nguyen, Selda Ozkan, Imgon Hwang, Anca Mazare, Patrik Schmuki | 5,861 |
10.48550_arXiv.2210.09379 | ###### Abstract
Metallic nanogranular films display a complex dynamical response to a constant bias, showing up as atypical resistive switching mechanism which could be used to create electrical components for neuromorphic applications. To model such a phenomenon we use a multiscale approach blending together an ab initio treatment of the electric current at the nanoscale, a molecular dynamical approach dictating structural rearrangements, and a finite-element solution of the heat equation for heat propagation in the sample. We also consider structural changes due to electromigration which are modelled on the basis of experimental observations on similar systems. Within such an approach, we manage to describe some distinctive features of the resistive switching occurring in nanogranular film and provide a physical interpretation at the microscopic level.
## I Introduction
The capability of engineering systems down to the atomic level reached in the last decades has open the door to the possibility of an unprecedented tailoring of material properties. Newly synthesized nanostructured materials can in fact also present features bearing the potential of unforeseen technological applications. This is the case, for instance, of cluster-assembled gold thin films, which present a rich dynamical non-Ohmic response to an external bias that could be exploited, it has been suggested, in the fabrication of unconventional computing hardware components. Even under a constant bias, the electrical resistance of such systems can suddenly change over macroscopic scales giving rise to what is usually called resistive switching events. The microscopic mechanisms behind them are still far from being identified and fully characterized. Partially, this is due to a lack of specialized theoretical tools that allow to carry out a quantitative analysis of the phenomenon. In two previous works, the first steps towards the definition of such tools have been moved, and a further one is presented here.
More specifically, we propose to model the electrical response of such systems with a multiscale approach. At the macroscopic scale a set of classical equations, Fourier heat equation and Kirchhoff's circuit laws coupled via Joule heating, are numerically solved on a uniform grid; they allow to estimate the film electrical resistance and temperature. At the microscopic scale, molecular dynamics simulations and an _ab initio_ tool previously developed are used to estimate the value of electrical conductivity that enters into the classical equations. This allows us to treat such a quantity in its general form as a temperature-dependent tensor field, reflecting the anisotropic and non-homogeneous nature of the sample at small scales. In fact, we shall argue that for such systems electrical conductivity sensibly depends on the current as well and propose a model based on observations of electromigration effects in gold nanowires.
In Section II we will present our theoretical device in detail; after discussing some motivations, we shall introduce the macroscopic equations that dictate the evolution of the observables of interest; then we will discuss how theory and experiments can help us in estimating the highly nontrivial values that a temperature- and current-dependent inhomogeneous electrical conductivity can assume in those systems. In Section III we will present the results of a simulation of the entire procedure. Specific features of the experiments will emerge from the few ingredients of the model, allowing us to suggest an explanation about their microscopic nature. Conclusions and outlook will be presented at the end.
## II Multiscale modeling
### Experiments and modeling, an overview
Purely metallic clusters of a few nanometers of diameter can be produced in gas phase and gently deposited on a substrate to form, layer after layer, a "nanogranular film" (or _ng-film_), i.e. a large (\(\sim\)mm\(\times\)mm), uniform agglomerate of clusters that retain their individuality to a high degree. By growing a ng-film between two electrodes, one can probe its response to an external electrical bias. Besides the usual insulating/metal transition occurring around the percolation threshold (a few nanometers), gold ng-films have been reported to present a complex non-Ohmic response even when the film has grown far from that threshold (tens of nanometers). In the simplest situation of a constant bias, the response of the film is highly dynamical. More specifically, upon application of a sufficiently high bias, the resistance of a ng-film, which starts with a low, constant value, suddenly increases to a much higher value (two orders of magnitude); after this initial phase, called of "sample activation", resistive switching events show up, appearingin the form of sudden "jumps" of the sample resistance; while jumps occur over macroscopic scales (seconds), the jump itself is characterized by a much shorter time scale (fraction of second); they can occur in either directions: to higher as well as to lower values; recurrent resistance values can be observed over long observational periods; the higher the voltage of the external bias the more frequent the jumps are; sample activation is permanent: after the bias is switched off, an activated sample will show right away resistive switching events the next time a bias is switched on.
What causes such a form of resistive switching is far from clear. Different mechanisms at different scales may occur; at the atomic scale, local charging/discharging phenomena may happen; at the cluster scale, the structure of single clusters may vary (defect migration, local melting or crystallization,...); at a larger scale, the configuration of groups of clusters may change over time (coalescence, aggregation, sliding, densification...). A cyclic mechanism of electric current bridges forming/breaking at the cluster interfaces has been suggested but not yet supported with sufficient experimental evidence or a convincing quantitative theoretical analysis.
From the theoretical side, indeed, the phenomenon poses some challenges, specifically for its dynamical character. In a static situation, the resistance of a ng-film can be estimated using a variety of approaches. If the ng-film is simply regarded as a thin film, one can use Matthiessen's rule and conceptualize its resistivity as that of bulk Au plus the sum of different contributions; for most of them (grain boundaries, surface effects,...) there are well-developed models are routinely used for polycrystalline thin films, while some other (inter-stitial voids, random grain orientation,...) may require an _ad hoc_ treatment. Regarded as a granular system, the static resistance of ng-films can be estimated adapting analytical and numerical tools from percolation theory, or, for regimes far from the percolation threshold, those developed for ordered grain arrays based on Green's function formalism, which well capture inter-grain quantum effects. If regarded as a inhomogeneous medium, its resistance can be estimated using effective medium approximations which can also accurately describe quantum effects. In Ref. we presented a procedure based on an atomistic modeling of the film via molecular dynamics and a characterization of the electric transport via ab initio techniques which, although computationally expensive, provided us with a flexible tool to estimate the resistance of ng-films.
Although all of the mentioned approaches can, in principle, be made time-dependent, not all of them are flexible enough be model any possible microscopic mechanism. Consider, for instance, the approach we presented in Ref.. To model the film in a dynamical situation we used a time-dependent effective medium approximation with which we were able to prove that local resistance changes can lead to global ones if they are triggered by the temperature; as the sample heats up due to Joule effect, resistive switching events at the sample scale appear in correspondence of some sort of phase transition, during which certain local structural arrangements become more likely then others. The model, however, relies on certain assumptions that prevent us to study other possible situations of interest. For instance, the underlying assumption of homogeneity of charge and heat diffusion prevents us to explore what happens if the current tends to circulate only on a few preferred percolation paths. Or, the independence of local resistance variations that are randomly decided prevents us to model the complex interplay between current and temperature at the cluster scale required by the above mentioned cyclic mechanism put forth in Ref..
In the present work we introduce a new approach designed to overcome these limitations. Using molecular dynamics simulations to model ng-films freed us from the constrains of dealing with clusters as fundamental brick of the modeling. Our system was indeed not just a network of pristine clusters, but a rich structure of randomly oriented grains, amorphous interfaces, interstitial voids, etc. This is a natural framework for describing local melting, crystallization and all sort of unbiased structural changes can arise at the cluster scale. However, molecular dynamics modeling comes with its own limits and one cannot directly introduce an electric current and see its effects on the atomic structure, simply for the lack of electrons in those simulations. Even bypassing the current and simply looking at its effects on an atomic structure due to Joule heating is a procedure that presents its conceptual difficulties: for structures as such where charge transport is often in its ballistic regime, it might be hard to establish which atoms are heated up by Joule effect (cf Ref. and references therein); and even if the Joule heat was appropriately distributed, its propagation through neighboring atoms, in great part due to electronic degree of freedom, would not be correctly captured by molecular dynamics simulations, in which electrons are not included. A more sensible way to deal with Joule heating in atomistic simulations is to couple them to a resolution of an appropriate heat diffusion model. And this is the strategy we present here.
### The non-homogeneous, anisotropic thermistor problem
When a bias is applied to a ng-film, such system can respond in a non-linear way. This is probably the result of structural changes happening at the cluster level and triggered by local variations of temperature and current. Key observables are therefore the temperature and vector current field, \(T(\mathbf{r},t)\) and \(\tilde{I}(\mathbf{r},t)\), respectively. At a macroscpic level they can be described via what is sometimes called "the thermistor problem", namely Kirchhoff's circuit law and the Fourier heat equation with Joule heating as source term, complemented by appropriate initial and boundary conditions.
\[\rho c\frac{\partial T}{\partial t}=\vec{\nabla}\cdot\left(\overleftarrow{\kappa} \cdot\vec{\nabla}T\right)+\vec{I}\cdot\overleftarrow{\sigma}\cdot\vec{I}, \tag{1}\]
\[\vec{\nabla}\cdot\vec{I}=0 \tag{2}\]
The source term in Eqn. is obtained combining \(\vec{I}\cdot\vec{E}\) and \(\vec{I}=\overleftarrow{\sigma}\cdot\vec{E}\), \(\vec{E}\) being the electric field. Just like its homogeneous, isotropic counterpart, Eqn., which determines \(T({\bf r},t)\), requires an initial value and some boundary condition, which can be Dirichelet, Neumann, or a mixture of the two; on the other hand, Eqn., which determines \(\vec{I}({\bf r},t)\), only requires the boundary conditions that are dictated by the request that the electric potential has the bias difference at the electrodes. Details on the numerical resolution of such a set of equations is provided in the Appendix.
The complex microscopic structure of a ng-film enters, therefore, via the quantities, \(\rho\), \(c\), \(\overleftarrow{\kappa}\), and \(\overleftarrow{\sigma}\). By approaching the numerical resolution of the equation with a finite element method, we discretize the space and partition it into a mesh. Those observables are therefore directly related to the mass, heat capacity, thermal and electrical conductance of each mesh cell. To simplify the problem, we consider a mesh large enough for \(\rho\) to be constant (i.e. space and time independent). The existence of such a scale is justified by the fact that during the experiments no appreciable changes to the topography of ng-film samples has been reported: even though, when a bias is applied, connections between clusters may form or deteriorate, the density of clusters inside the film seems to remain constant. Similarly, we also consider constant the specific heat capacity, which is primarily connected to the cell mass and not to the arrangement of clusters inside a cell.1 What can sensibly depend on the cluster arrangement inside a cell are the cell electrical and thermal conductance. For cells containing a small number of clusters (say, a few units), it is reasonable to expect that even a single junction, that for some reason breaks, can lead to a sizeable change of electrical/thermal conductance in that direction. We therefore consider \(\overleftarrow{\sigma}=\overleftarrow{\sigma}({\bf r},t)\) and \(\overleftarrow{\kappa}=\overleftarrow{\kappa}({\bf r},t)\).
Footnote 1: While both \(\rho\) and \(c\) can be estimated using molecular dynamics simulations, \(\rho\) can also be estimated using Au bulk density and a plausible porosity (cf), while \(c\) can be estimated using the Dulong-Petit law, again adjusted to account for the porosity of the medium.
\[\overleftarrow{\kappa}=LT\overleftarrow{\sigma}, \tag{3}\]
\(T\) being the temperature field and \(L\) Lorenz number.
It follows that the key quantity, connecting the macroscopic level of the model to the lower one, is the electrical conductivity \(\overleftarrow{\sigma}({\bf r},t)\). Such an observable indeed encodes whatever changes happens at a "microscopic" scale, which in this context means "at the scale of a single cell". Contrary to other common multiscale approaches, we do not opt to calculate such an information at each timestep of the resolution of the macroscopic equation, but rather use theoretical tools and available experiments to build, once for all, a map that connects \(\overleftarrow{\sigma}\) with \(T\) and \(\vec{I}\) and use that in the resolution of the closed set of equations,,.
### Temperature-induced conductivity variations
As we go from the macroscopic level of the model to the microscopic one, we consider the cell resistance, \(R\), from which \(\overleftarrow{\sigma}({\bf r},t)\) is readily obtained.2 For the equations characterizing the macroscopic level to make sense, each cell must includes (at least) a few gold clusters. This means that, in estimating the cell resistance, we can neglect the contribution of phonons, quantum tunneling, etc. and focus only on structural changes at the atomic level which, as argued in Ref., are the main responsible for large resistance variations. As explained in Ref., in doing this the only relevant quantum effect to take into account is the complex interplay between ballistic and diffusive electronic transport regimes between crystal and amorphous regions inside the system. Molecular dynamics (MD) simulations and the _ab initio_ tool presented in Ref. (an atomically-resolved equivalent resistor network, or AR-ERN) seem therefore adequate tools to estimate a cell resistance.
Footnote 2: The symbol \(R\) should carry multiple indices: one for the cell, one for the timestep and one for the direction on which the tensor is projected (in our case \(i=x,y,z\)), but we shall avoid such a complicated notation unless necessary.
Next, starting from the large scale simulation of a chunck of a ng-film, as described in Ref., we carve out of the entire system a few samples of the size of the cell of the macroscopic level (\(\sim 15\) nm), as shown in Fig. 1, left panel. For the reasons mentioned in Section II.1, we warm up the samples globally, rather then locally, and what we observe in general is a change of the structure of the samples even when the highest temperature is considerably lower than gold melting point (\(T_{melt}\sim 900~{}K\)). As one would expect, those changes depend on the specific function \(T(t)\) one uses. However, when it comes to the electrical resistance of the sample, we do observe that, after a brief period of adjustment, if the sample is brought to a certain temperature \(T=\tilde{T}\), its resistance will roughly be \(R=\tilde{R}\), no matter how \(\tilde{T}\) is achieved. This allows us to establish that, even though some hysteresis effects are present, it is indeed possible to describe \(R\) as function of \(T\). This is well exemplified by Fig. 2, in which we report the behavior of three samples under a complex temperature evolution. By randomly switching the sample temperature between room temperature (300 K) and a certain higher temperature (430 or 600 K), all three samples tend to respond in the same way: after a period during which the structure relaxes a bit, to each temperature switching correspond a resistance switching and a relation between specific values of \(T\) and \(R\) emerges, allowing us to consider a function \(R(T)\) indeed. On our limited set of samples considered, the exact form of \(R(T)\) sensibly depends on the sample, but we do recognize a trend that we encode in the following function:
\[R(T)=R_{0}(1+\vartheta(T-\tilde{T})\Delta) \tag{4}\]
In words, Eqn. encodes a sudden change of the cell resistance that might occur when the cell temperature is brought above a certain threshold. Such a behavior seems to be linked to cycles of amorphization/re-crystallization happening in key regions as cluster interfaces, as shown in Fig. 3, but further investigations would be required to make a general statement.
Based on our simulations, we give the rough estimates of \(\tilde{T}\sim 400\div 600K\) and \(\Delta\sim 0\div 0.2\). Providing more accurate ones would require the study of \(R(T)\) for many different cells under many different \(T(t)\) profiles, which, for its computational cost, is currently the scopes of the present work. A much more accurate estimate of \(R_{0}\), however, can be already given; by considering a large set of samples carved out of the system of Ref., which is indeed at room temperature, we recognise that the \(R_{0}\) roughly follows a Poissonian distribution peaked around 110 \(\Omega\), as shown in Fig. 1, right panel.
Technical details about the MD simulations and the resistance estimates are provided in the Appendix.
Evolution of the resistance of three carved out samples in connection with temperature variations. For each sample, the resistance has been monitored while changing the temperature. Blue shadowed regions correspond to the time intervals in which the temperature is 300K while red ones correspond to \(T=430\) K (upper panels) and \(T=600\) K (lower panels).
To study as a cell resistance varies with the temperature we randomly carve samples out of a large scale MD simulation of a chunk of ng-film (left panel). As cubic samples have the tendency to deform when isolated from the surrounding system, we consider cylindrically shaped samples which present higher stability. We use periodic boundary conditions while cylinders are surrounded by vacuum so to effectively isolate periodic images of the cylinders. To represent cells of size 15nm we consider cylinders of length 15nm and section diameter 15nm (right panel). In calculating the resistance, electrodes are placed on the two opposite flat faces. By taking many samples, one can estimate the random distribution of the cell resistance for a given ng-film (right panel).
### Current-induced conductivity variations
As the dynamical observables of the macroscopic model are \(T\) and \(\vec{I}\), we could speculate about a dependency of \(\vec{\sigma}\) not only on \(T\) but on \(\vec{I}\) as well. In fact, we believe that in our case this is not only possible but actually necessary. When a current high enough is sustained over time, the structure of the conductor may deteriorate as effect of electromigration (EM), an effect that has been widely studied and reported also for poly-crystalline Au nanostructures.
A back-of-the-envelop calculation shows that such an effect is relevant for our case, too. In Ref., Au nanowires are reported to completely break after having transmitted high enough currents. According to the authors, the highest value they can withstand before undergoing failure can be written as \(I_{f}=6\times 10^{3}A_{c}^{0.425}\) A, where \(A_{c}\) stands for the nanowire cross-section. Such an expression has been confirmed for cross-sections ranging from \(5\times 10^{5}\) nm\({}^{2}\) down to \(8\times 10^{3}\) nm\({}^{2}\). In the case of the ng-films of Ref., the cross-section is much higher (a typical ng-films has length and width equal to 1 mm and a thickness of a few tens of nm) and the intricate structure at the cluster scale makes the section of a ng-film quite different from that of a nanowire (and more akin to a section of a random stack of nanowires). Nonetheless we can use the same formula to get an idea of the order of magnitude of the failure current for those systems, which turns out to be \(I_{f}=0.28\) A, a value that is not that far from the typical current in their experiments, \(I\sim 20\)V/\(100\Omega=0.2\) A. We therefore argue that such an effect may be present and must be taken into account.
Because of the interplay between electrons and atoms, it is hard to simulate the effects of electromigration on a generic structure with standard computational tools, since they are designed to treat the dynamics of either electrons or atoms. Nonetheless, to our purposes, a simple characterization based on available experiments is enough to give us some interesting effects in the macroscopic model. More specifically, we consider the work of Ref., in which the degradation of Au junctions under the effects of different current regimes is systematically studied, and propose a simple way to reproduce their results within our macroscopic model.
When a Au junction is exposed to high biases for a long period, the material can develop some discontinuity in its structure that affects the flow of electrons. Such an effect can be quantified by measuring the sample resistance against the bias applied in a quasi-static experiment during which the bias is slowly increased over time. It is then possible to identify three regimes. For low enough voltages, the sample acts in a Ohm-like manner, as one would expect from a standard conductor in normal working conditions. For moderate voltages, electromigration effects kick in and the junction starts shrinking; the beginning of this phase is marked by a sudden fall of the junction conductance, in complete analogy with the activation of a ng-film. For high enough voltages, the junction reaches a failure point after which it no longer can sustain a current: the junction is effectively broken.
An experiment as such can be, not only qualitatively, but also quantitatively reproduced in our modeling if an appropriate \(R(I)\) is considered. For start, we neglect temperature variations and consider only the circuit equation, Eq., with a potential difference at the electrodes that linearly increases in time. At each time step the current \(I\) of each cell is calculated. If such a value overcomes a certain ("activation") threshold \(I_{a}\), which is randomly assigned for each cell from a given range, we say that the current shrinks the effective cross-section \(A_{c}\) that the it experiences inside the cell. Such a shrinking is quantified by a certain parameter \(\beta\), such that \(A_{c}\to A^{\prime}_{c}:=\beta^{-1}A_{c}\). According to Ref., this implies an increase of the cell resistance \(R\to R^{\prime}:=\beta R\). At the next time step, the new values \(R^{\prime}\) and \(A^{\prime}_{c}\) will be considered, alongside some new values for the current activation and failure, for each cell. Note that we allow \(A_{c}\) to be reduced down to 0.16 nm\({}^{2}\) (corresponding to a minimal junction with area \(a^{2}\), being \(a\) the lattice constant of gold) and we use \(\beta=5\) to produce all the results in this work. If the cell current overcomes the failure threshold \(I_{f}\), which is also initially assigned to each cell at random from a certain range, then the cell is considered "failed": a junction along the percolation path inside the cell breaks and no current is allowed to flow through that cell anymore.
Non-hysterical behaviour of the resistance as a function of the temperature. As one can see from the highlighted regions, the rise in temperature leads to an amorphization of a certain region, while the temperature fall favours a recrystallization of the same region. The process seems reversible and justifies our assumption that for each \(T\) a single value of \(R\) can be assigned to the system.
With this we are able to qualitatively reproduce the curves in in which a polycrystalline Au nanowire 100 nm wide and 200 nm long (7 and 14 ERN cells, respectively) is first activated under 0.15 V to finally reach the electrical failure at 0.4 V. The curve obtained with our model is shown in at low voltages, \(V<0.05\) V, the sample displays a low resistance value (\(R<\)100 \(\Omega\)). At \(V=0.1\) V some electromigration events occur (yellowish and bluish elements in the first panel in Figure 4) and, as a consequence, the sample becomes more resistive but still in the low resistance range (\(<1\) k\(\Omega\)). At \(V=0.2\) V EM events have brought the sample to a high resistive state (\(10-100\) k\(\Omega\)) and some gaps can be identified as not colored grid elements in second to forth panel. Further increasing of the voltage produce the failure of an important portion of the grid cells ending up with a short-circuited gold sample. Data shown in are obtained with \(I_{a}\) and \(I_{d}\) uniformly distributed between 19 \(\mu\)A to 23 \(\mu\)A and 32 \(\mu\)A and 38 \(\mu\)A, respectively. The values for \(I_{a}\) at \(t=0\) are chosen so that average current flowing through the sample just before the activation is in the order of magnitude of the measured one, i.e. 7 mA. The upper limit for the \(I_{d}\) range is set to be twice the lower limit for the \(I_{a}\) one.
In a ng-film, where junctions are interfaces between adjacent clusters, we hypothesize that a similar phenomenon happens. More specifically, when the voltage is sufficiently high, electromigration leads to a reduction of the cluster-cluster interface, up until the two clusters actually separate. We can therefore use the model we have outlined above also for ng-films. The only additional feature we introduce is that, within each cell, we allow junctions to shrink/break in each direction independently, to account for the higher degree of inhomogeneity characterizing small length scales of a ng-film. Indeed, in our picture, if in a cell two clusters get separated along, say, the main direction of the current, there might still be within the same cell a path connecting other clusters in some other direction. It follows that the resistance of the cell along the direction of the bias, say \(x\), is infinite, \(R_{x}=\infty\), but \(R_{y}\) and \(R_{z}\) are not, until, at least, the transversal current is not as strong.
Finally, we would like to emphasise that the modeling here proposed neglects one important piece of information: the timescale over which those effects are supposed to emerge (notice that has no indication of time). As pointed out in literature, the damage of even failure of a nanojunction may occur over a period of hours. On the one hand, we do not have enough information to appropriately characterise the phenomenon in time; on the other hand, such a timescale is quite far from that of the phenomena we are interested in (of the order of seconds, at most). For now we make a strong assumption, to take electromigration effects as instantaneous. Later one, we shall come back on it and and offer the reader, on the basis of some concrete results, an explanation on why such an assumption is in fact good enough for our purposes.
## III Results
### Model Parameters
The model just described can be computationally demanding when the evolution of a ng-film is simulated in the experimental conditions. Nevertheless, interesting insights can be gained already by considering smaller length and timescales. By running several simulations of hundreds timesteps on samples of a few thousands of cells under different conditions we were able to see some distinctive effects that can help understand the unusual behavior of ng-films.
Here we present a specific simulation that well captures some interesting, general trends observed. We consider a uniform mesh of \(80\times 80\times 4\) cubic cells of side \(d=15\) nm. A specific ng-film sample is identified by the set of parameters that characterize the cells. In our case, at \(t=0\) each cell is characterized by: an initial resistance tensor \(R^{}_{a;i,j,k}\) (\(a=x,y,z\), \(i=1,...,80\), \(j=1,...,80\), and \(k=1,...,4\)) randomly picked between 0 and 250 \(\Omega\); a temperature threshold randomly picked between 300 and 340 K, and a resistance variation for the thermally-induced structural changes following 4 with \(\Delta\) between 0 and 0.2; two current thresholds randomly picked between \(0.9\mu A-1.08\mu A\) and \(2.0\mu A-2.4\mu A\) a shrinking factor \(\beta=5\).
Replicating the experiment on the effects of electromigration on gold nanojunctions of Ref. using our model for \(R(I)\). The values of the parameters have been chosen to match that experiment and then used in our simulation of a ng-film.
\(0.016kg^{2}*K^{-1}*m^{-1}*s^{-2}\) and \(L=2.44\times 10^{-8}V^{2}K^{-2}\). To ensure convergence of the finite element scheme a time step of \(\tau=1\) ps has been adopted. We allow heat to escape from the sample through all the boudaries of the sample by setting the temperature for those equal to their adjacent elements, i.e. \(\frac{\partial T}{\partial a}|_{boundary}=0\) (\(a=x,y,z\)) and the temperature for all the elements is initially set to \(T=300\) K. Finally, as bias we consider an applied voltage along the \(x\) direction that is ramped up from 0 to 0.014 V, which produces an electric field close to the experimental one, during the first 30 timesteps and kept constant after that. A simulation as such can require up to a few hours on a dual-core laptop.
Some of the above parameters differ from our own estimates presented in the above sections. They have been tuned in order to explore an interesting region of the parameter space within a reasonable simulation time (for instance, temperature thresholds are much lower of those mentioned in Section II.3). As we shall argue, this prevents us to interpret our results in an absolute, quantitative manner, but not from getting some relative, qualitative insights.
### Sample evolution
In the left panel of the evolution of the sample resistance over time is reported. We distinctly recognize two phases, connected with the external voltage. When the voltage is raised, the resistance rises, too, ultimately reaching a value that is orders of magnitudes higher than the initial one. Once the voltage is constant, the resistance stabilizes, although, on a smaller scale, some variations are still present.
As we analyze the change of resistance of each single cell, we recognize that the initial phase is dominated by electromigration effects. They lead to an extensive deterioration of the sample, as shown in the top row of Nearly a third of the cells experienced a shrinkage of an interface along the bias direction, and in most cases (\(\sim 80\%\)) the shrinkage turned into a broken connection. As an effect of the current trying to find its way through such an inhomogeneous medium, also interfaces along the transverse direction were highly affected by electromigration. Nearly 12% of cells experienced interface shrinkage along one or the other transverse direction and 65% of them completely lost the connection. Nevertheless, only 1.5% of cells were left completely insulating, i.e. with no current flowing in any direction. The voltage was chosen to be not high enough to actually break the sample, so at some point the system stabilizes and electromigration effects become negligible.
After that first phase, thermal effects dominates. On small time scales they lead to some noisy variations of the film resistance.
From the right panel of we can see the evolution of the sample temperature. As one would expect from the boundary conditions, it always rises. Also, consistently with the two regimes for the sample resistance, such a rising happens at two different rates: a high one in the beginning, when the sample resistance is relatively low, and a low one, when the resistance is high. Such a behavior is direct consequence of the fact that the increase in temperature is inversely proportional to \(R^{3}\), a relation one can obtain by combining the power dissipated by Joule effect, \(P=I^{2}/R\), and Ohm's first law, \(I=V/R\). In the local temperature map plotted in the second row of Fig. 6, we also see that spots with higher resistance tend to become hotter. This is due to the fact that, as the spot electrical conductance lowers, so does its thermal one (the two are linked by Eqn.); and because of the slow heat dissipation rate, the spot overheats.
We notice that the rise in temperature in the second phase is, rather interestingly, somehow at odds with the trend of the resistance. In a simplified picture, one would expect that, as the sample temperature goes to higher values, cell resistances, which are subject to Eq. 4, would, on average, tend to rise as well; and, in turn, the sample resistance would rise, too. But this is not what happens. Although cell resistances do increase on average, the sample resistance, which is calculated using the circuit law, presents a rather different trend.
In the last row of a comparison between the current before and after the sample deterioration is presented. Before the deterioration occurs, the current basically flows uniformly from one electrode to the other. But once the deterioration is over, clear preferred paths emerge. Although only few cells are completely insulating (\(\sigma_{x}=\sigma_{y}=\sigma_{z}=0\)), the fact that a third of the cells does not let the current flow along the bias direction (only \(\sigma_{x}=0\)) shapes the current in a very specific way.
### Connection with experiments
To provide a physical interpretation of the simulation and compare it with experiments we need to remind the reader that the time scales of the simulations are not realistic, in two ways. First, as discussed in Sec. II.4, the way \(R(I)\) has been modeled on electromigration experiments completely ignores the typical time scales of the phenomenon, for which too little information is given to know. It follows that the time scale of the first phase is somehow artificial, for it should almost certainly happen on macroscopic scales. Second, although we got some realistic estimates for the parameters of the function \(R(T)\), we deliberately altered them in order to probe interesting regions of the parameter space within a reasonable simulation time. Less dramatically than for the first phase, in the second phase the time scale, although not quite realistic, might well be, under certain conditions of external voltage and sample temperature, not that far from a realistic one.
Bearing that in mind, we see from the above analysis that, even though the two effects are treated, as we did, on the same timescale, they are distinct and consequential, with thermal effects being relevant only after electromigration has deteriorated the sample. In fact, this is representative of a more general trend we observed in our simulations. If electromigration effects are artificially switched off, thermally triggered local resistivity variations have little effects on the overall sample resistance. In other words, local resistance variations triggered by the temperature can have a macroscopic relevance only if the sample has previously been deteriorated by electromigration effects. It seems therefore reasonable to draw a direct comparison between the deterioration phase that we observe in the simulations and the sample "activation" observed in the experiments: in both cases the sample resistance rises by orders of magnitudes, it represent a necessary phase to go through for the emergence of resistive switching events at the macroscopic scale, and it is permanent.
As mentioned, in the second phase, when electromigration plays no longer a role, thermally activated local resistance variations can coordinate to lead to macroscopically appreciable ones. Those variations are actually also in a quantitative agreement with experiments: in our simulation the sample goes from an average resistance of 9200 \(\Omega\) to 8900 \(\Omega\), which is indeed strikingly close to the resistance values explored by one of the sample discussed in Ref., ranging from 9900 \(\Omega\) to 9300 \(\Omega\).3 Time scale is indicative of a microscopic phenomenon, meaning that what here appears as a smooth variation, it would appears as a rapid transition on macroscopic scales, in agreement with the "sudden" jumps reported in experiments.
Footnote 3: We recall the reader that, even though our sample is much smaller than that of the experiments, the proportions are kept in such a way the resistance is the same.
A piece of explanation of the reason why resistive switching events emerge on deteriorated samples comes from the analysis of the current field. As mentioned, on the scales considered, the deterioration leads to the emergence of a pattern of current, which does not flow uniformly from one electrode to the others but via some preferred paths. And, just like in the submonolayer regime when only a few percolation paths are available, also here it is true that even slight changes on a path can have macroscopic effects. If, for instance, the switching of just a few cells to a high resistive state makes a path no longer favourable, such an event can lead to a sensible resistance variation of the entire sample if the current has only limited options to run from one side to the other.
Under different experimental conditions, jumps in real samples present different features (cf panels (a) and (c) of in Ref.) that may be sign of different microscopic origins. From current available information it is hard, if not impossible, to say whether the jumps we observe in our simulations are caused by the same mechanisms. Only a synergetic effort of theory and experiment in characterizing features such as recurrent values, plateauing duration, etc. in connection with the experimental setup and ng-film specs will enable a more precise characterisation of the relevant microscopical mechanism(s).
## IV Conclusions and Outlook
In this work we presented a way to model the electric response of large-scale complex nanostructures with local structural variations. The problem has been tackled at different scales with different tools. At the macroscopic level, a set of classical equations (the so-called thermistor problem) have been used to get the evolution in time of the system temperature and resistance in presence of local variations of conductivity. We then considered molecular dynamics simulations to characterize such variations when arising from thermally activated microscopic structural rearrangements.
Temperature (blue line) and resistance (gold line) of a simulated ng-film. In the left panel, electrodes voltage difference is also plotted (magenta line) for reference. In the right panel, an inset plot shows a zoom-in of the resistance after the activation phase. Dashed line in the inset represents the resistance value averaged over 20 previous and 20 following timesteps.
Furthermore, electromigration effects, that were modeled on relevant experiments, were also included. Altogether, the model allowed us to simulate the emergence of resistive switching events in nanogranular gold films.
We identified electromigration as responsible of an extensive deterioration of the sample occurring in experiments during the so called activation phase. Such a deterioration forces the current to flow only on a few preferred paths and amplifies the effect of local variations of resistance which can indeed have macroscopic relevance. Furthermore, in the measurement of the sample resistance over time, local resistance variations appear to coordinate and lead to large variations that, on macroscopic timescales, can be identified as resistive switching event.
Such encouraging results certainly motivate to extend our work in several directions: running larger scale (both of time and length) simulations in search of other key features of resistive switching in ng-films; exploring larger regions of the parameter space (also covering possible-but-yet-to-explore experimental conditions); refine the
Conductivity along the bias direction (first row), temperature (second row), and current field (third row) inside the simulated ng-film at three different time steps. The first time step correspond to the beginning of the simulation when the sample is at room temperature, its conductivity is high and the current flows rather uniformly from one electrode to the other; the second plotted step corresponds to the end of the deterioration phase induced by electromigration effects; the last plotted step is the last simulated one, which shows that changes occurring the second phase are much less visible. Electrodes are placed at the left and right sides of a square; the section is taken at half the ng-film thickness; the arrows in the third row represent the project of the current over that section; to better visualize the current field, both the length and the color of the arrows represent its intensity.
## Appendix
### Numerical resolution of the thermistor problem
Once the map \(T,\vec{I}\rightarrow\overleftrightarrow{\sigma}\) has been characterize, the set of equations,, supplied with appropriate initial and boundary conditions represent a well-defined problem that can be solved numerically. We use a regular mesh of cubic cells, \(\Delta x\times\Delta y\times\Delta z=d^{3}\). This implies that the current can only flow in three directions through the mesh element (along \(x\), \(y\), or \(z\)), i.e. \(\sigma_{ab}\rightarrow\sigma_{a}\) (\(a=x,y,z\)).
\[\rho c\frac{\partial T}{\partial t}=\partial_{a}\kappa_{a}\partial_{a}T+{I_{a} }^{2}/\sigma_{a} \tag{5}\]
As pointed out in Ref., one has to slightly change the algorithm for solving the heat equation in dealing with an inhomogeneous thermal conductivity. Using the anisotropic generalization of their Eqn.'s and, we have that an explicit (in time) finite element version of the above equation is
\[\rho_{i,j,k}c_{p_{i,j,k}}\frac{T^{(n+1)}_{i,j,k}-T^{(n)}_{i,j,k}}{ \Delta t}=2\left(\kappa^{(n)}_{x;i,j,k}\frac{T^{(n)}_{x;i+1/2,j,k}-2T^{(n)}_{ i,j,k}+T^{(n)}_{x;i-1/2,j,k}}{\Delta x^{2}}+\right.\\ +\left.\kappa^{(n)}_{y;i,j,k}\frac{T^{(n)}_{y;i,j+1/2,k}-2T^{(n)}_ {i,j,k}+T^{(n)}_{y;i,j-1/2,k}}{\Delta y^{2}}+\kappa^{(n)}_{z;i,j,k}\frac{T^{(n )}_{z;i,j,k+1/2}-2T^{(n)}_{i,j,k}+T^{(n)}_{z;i,j,k-1/2}}{\Delta z^{2}}\right)+\\ +\left.{I^{(n)}_{x;i,j,k}}^{2}/\sigma^{(n)}_{x;i,j,k}+{I^{(n)}_{y ;i,j,k}}^{2}/\sigma^{(n)}_{y;i,j,k}+{I^{(n)}_{z;i,j,k}}^{2}/\sigma^{(n)}_{z;i,j,k}\right. \tag{6}\]
with
\[T^{(n)}_{x;i\pm 2,j,k}=\frac{\kappa^{(n)}_{x;i,j,k}T^{(n)}_{i,j,k} +\kappa^{(n)}_{x;i\pm 1,j,k}T^{(n)}_{i,j,k}}{\kappa^{(n)}_{x;i,j,k}+\kappa^{(n)}_{y ;i,j,k}+\kappa^{(n)}_{y;i,j,k}}\\ T^{(n)}_{y;i,j\pm 2,k}=\frac{\kappa^{(n)}_{x;i,j,k}+\kappa^{(n)}_{y ;i,j,k}+\kappa^{(n)}_{y;i,j,k}}{\kappa^{(n)}_{x;i,j,k}+\kappa^{(n)}_{x;i,j,k \pm 1}}. \tag{7}\]
With our cubic mesh, the electrical conductivity reduces to
\[\sigma^{(n)}_{a;i,j,k}=1/(d\:R^{(n)}_{a;i,j,k}) \tag{8}\]
while the thermal conductivity, for which we use Eq., is
\[\kappa^{(n)}_{a;i,j,k}=L\:T^{(n)}_{i,j,k}/(d\:R^{(n)}_{a;i,j,k}). \tag{9}\]
\(R\) being the cell resistance.
The initial value is imposed by assigning a value to the entire temperature field. In experimental conditions, one would start from a sample at environment temperature: \(T^{}_{i,j,k}=T_{environment}\) for all \(i,j,k\).
\[T^{(n)}_{i_{min}/max,j,k} =T_{environment},\] \[T^{(n)}_{i,j_{min}/max,k} =T_{environment}, \tag{10}\] \[T^{(n)}_{i,j_{k}min/max} =T_{environment}\]
For Neumann boundary conditions, one imposes
\[T^{(n)}_{i_{min},j,k} =T^{(n)}_{i_{min}+1,j,k}\] \[T^{(n)}_{i_{min}}{}^{2},j,k} =T^{(n)}_{i_{max}-1,j,k}\] \[T^{(n)}_{i,j_{min},k} =T^{(n)}_{i,j_{min}+1,k}\] \[T^{(n)}_{i,j_{max},k} =T^{(n)}_{i,j_{max}-1,k} \tag{11}\] \[T^{(n)}_{i,j_{min}} =T^{(n)}_{i,j_{min}+1}\] \[T^{(n)}_{i,j,k_{max}} =T^{(n)}_{i,j,k_{max}-1}\]
Mixed boundary conditions, or even more complicated ones, can be used to model the presence of the electrodes, the substrate and the heat dissipation of the upper surface.
About the circuit equation. Eqn., one can use the scheme we presented in Ref., in which we actually use the integrated version of the stated circuit equation (sum of cell incoming currents =0) which can be obtained using Gauss's Divergence Theorem to the continuity equation assuming no accumulation/depletion of charges (\(\partial_{t}q=0\)). See that paper for further information about the numerical resolution. In that framework, the current is rewritten in terms of the difference of potentials at the nodes and the cell resistances, \(\vec{I}=-\overleftrightarrow{\sigma}\cdot\vec{\nabla}v\) which becomes \(I_{x;i,j,k}=-(v_{i+1,j,k}-v_{i,j,k})/(d^{2}R_{a;i,j,k})\). It should be mentioned that there is an underlying as sumption of staticity of the current, even when the jumps occur, which is here deemed as reasonable for the electronic dynamics is characterized by scales time quite far from those here considered.
### Molecular dynamics simulation and resistance estimates of carved samples
MD simulations of the three systems presented in Section II.3 have been performed using the LAMMPS code, integrating the equations of motion by the velocity-Verlet algorithm. The Au-Au interactions were sampled using the embedded-atom potential of Ref.. By means of a Nose-Hoover thermostat (relaxation time equal to 100 fs) we are able to dynamically modulate the temperature, \(T\), of each of these gold systems (or "junctions") while keeping track of the fcc and non-fcc regions evolution. We opt for a two-level switching scheme, meaning that we will modulate the junction temperature between two fixed values, \(T_{room}=300\) K and \(T_{hi}\). We expect the micro-structural changes to be more pronounced for higher temperatures. Unfortunately, we have no hint on the local or averaged temperature of the experimental samples in. However, other studies have treated the relation between EM and temperature in gold nanojunctions: maximum temperatures in the range of 450K and 850 K have been reported at moderate current densities, i.e. \(1.5\cdot 10^{12}\) and \(0.3\cdot 10^{12}\) A/m\({}^{2}\), respectively. Consistently, we set \(T_{hi}\) to 430 K and 600 K, the later representing a maximum temperature well below the melting point of the simulated Au junctions (\(\sim 900\) K) and below the reported maximum temperatures. Every junction is initially left to evolve for 1.0 ns at \(T=300\) K. Next, the thermostat temperature is switched from \(T=300\) K to \(T_{hi}\) and the junction is left to evolve again at its new temperature for a period \(\tau_{hi}\). Finally, the thermostat is switched back to \(T=300\) K, this time for a period \(\tau_{room}\). This operation is repeated varying randomly the length of \(\tau_{hi}\) and \(\tau_{low}\), as shown in We use such a scheme rather than a periodic switching scheme to avoid inducing any kind of resonances or other spurious effects. Atoms belonging to the edges of each junction are kept in their initial positions during the simulation at finite temperature. This is done in order to prevent the whole structure to completely relax to a single fcc domain and mimic the presence of the macroscopic granular film.
Every carved sample contains two kinds of atomic arrangements: cubic (fcc) and non-cubic (non-fcc) gold, colored in orange and red respectively in a Polyhedral Template Matching (PTM) Analysis performed with Ovito software allows to distinguish between those Au atoms sitting in fcc sites (forming fcc regions) and those which are not (forming non-fcc regions). In we show a section of a carved sample where atoms are colored according to the PTM analysis. We observe non-fcc regions (red colored atoms) separating different fcc regions (orange colored atoms) within a single junction. As explained in, fcc domains are less resistive than non-fcc ones, thus, less defected junctions tend to conduct better. Moreover, the specific distribution of the fcc and non-fcc regions determines the resulting \(R\) value for the junction.
The color maps displayed in the right-upper panels of show the fcc and non-fcc distribution averaged over the ERN cells at two different time steps corresponding to \(T=300\) K and \(T=600\) K. We observe that a temperature increase produces two effects: (i) a redistribution of the structural defects and (ii) a broadening of non-fcc regions. In these features are highlighted with green and blue arrows, respectively. As a consequence the ratio between fcc and non-fcc cells increases and these variations modulate the electrical response of the junction itself.
The total resistance of each junction is calculated by means of the AR-ERN of Ref., every \(2\cdot 10^{4}\) MD time-steps. As shown in Fig. 7, the total resistance is effectively modulated with the sample temperature in a reversible manner. The same MD protocol has been used for \(T_{hi}=430\) K and 600 K and three different gold junctions, all displaying similar behavior as shown in the total junction resistance is modulated by the sample temperature. The observed jumps in the \(R\) value, \(\Delta R\), reach maximum values in the order of a 25% of its initial value. Importantly, not all junctions perform equally: the junction corresponding to the upper panel shows very little \(R\) variations for the \(T_{hi}=430\) K case (\(\Delta R<5\%\)), while the modulation is much more pronounced for the \(T_{hi}=600\) K case (\(\Delta R\sim 25\%\)). The same feature is observed in the middle panel. Instead the junction corresponding to the lower panel responds similar for the two thermal modulation conditions providing a maximum \(\Delta R\) equal to 7% and 10% for \(T_{hi}=430\) K and \(T_{hi}=600\) K, respectively.
| 10.48550/arXiv.2210.09379 | Multiscale modeling of resistive switching in gold nanogranular films | Miquel López-Suárez, Claudio Melis, Luciano Colombo, Walter Tarantino | 2,890 |
10.48550_arXiv.1203.0466 | ## 1 Introduction
Let \(\mathcal{L}_{n}\) be the \((2+n)\) order operator:
\[\mathcal{L}_{n}=\partial_{t}^{(n)}(\partial_{tt}-c_{n}^{2}\partial_{xx})\ +\ a_{n-1}\ \partial_{t}^{(n-1)}(\partial_{tt}-c_{n-1}^{2}\partial_{xx})\ +\.. \tag{1}\]
\[...a_{0}\ (\partial_{tt}-c_{0}^{2}\partial_{xx})\]
According to the value of \(n\), describes several physical phenomena. As example, when \(n=1\), \(\mathcal{L}_{n}\) can be found in dynamic of relaxing gases, in magnetohydrodynamics, in hereditary electromagnetism (see and references therein) and in isotropic viscoelasticity where, models the evolution of the Standard Linear Solid (S.L.S.) (see, f.i.).
In all these models \(c_{k}\)'s represent the characterized speeds depending on the materials properties of the medium and in many physical problems it results \(c_{0}^{2}\leq c_{1}^{2}..\leq c_{n-1}^{2}\leq c_{n}^{2}\) as it's typical of _wave hierarchies_..
When \(n=1\), the operator is stricty-hyperbolic and it has been widely analized in. It's fundamental solution \(\mathcal{E}_{1}\) has been explicitly determined and singular perturbation problems, together with asymptotic properties, have been estimated.
Aim of the paper is to draw generalizations of the wide analysis related to S.L.S. to the case of with \(n\) arbitrary. For this, a _conditioned_ equivalence between and an integro - differential operator \(\mathcal{M}\) related to an appropriate memory function \(g_{n}(t)\), is considered. Owing to the hypotheses of fading memory, every function \(g_{n}(t)\) can be approximated by Dirichlet polynomials with appropriate restrictions on thecofficients of this expansion. These limitations need that the differential operator \({\cal L}_{n}\) is typical of _wave hierarchies_.
By this equivalence, whatever \(n\) may be, the fundamental solution \({\cal E}_{n}\) of (1.1) is explictly achieved and it is estimated in terms of \({\cal E}_{1}\). So the maximum properties and asymptotic estimates established for the Standard Linear Solid can be applied to operator \({\cal L}_{n}\) defined in (1.1). Moreover, boundary layer problems, typical of dissipative media, can be rigorously estimated..
These results are applied to _the Rouse model_ and _the reptation model_ which describe different aspects of polymer chains and have met with reasonable success.-.
## 2 Differential Constitutive Equation
Let \({\cal B}\) a linear, isotropic, homogeneous system and let \(\underline{u}(\underline{x},t)\) the displacement field from an undeformed reference configuration \({\cal B}_{0}\).
\[\rho_{0}\;u_{tt}=\sigma+f,\quad\varepsilon=u, \tag{2.1}\]
When the viscoelastic behavior of \({\cal B}\) is of _rate-type_, the well known stress-strain constitutive relation is
\[\sum_{k=0}^{n}a_{k}\ \partial_{t}^{k}\sigma=\sum_{k=0}^{n}\alpha_{k}\ \partial_{t}^{k}\varepsilon \tag{2.2}\]
Then, by (2.1), (2.2), the displacement field \(\underline{u}(\underline{x},t)\) is solution of the higher order equation like:
\[{\cal L}_{n}v\equiv\sum_{k=0}^{n}a_{k}\ \partial_{t}^{k}(v_{tt}-c_{k}v_{xx})=F \tag{2.3}\]
where:
\[c_{k}=\alpha_{k}/\rho_{0}a_{k},\quad F=(1/\rho_{0})\ \sum_{k=0}^{n}a_{k}\ \partial_{t}^{k}f. \tag{2.4}\]
The constitutive relation (2.2) includes various classical mechanical models, as Maxwell and Kelvin - Voigt models.
\[0<c_{0}<c_{1},\quad\eta=a_{1}/a_{0}>0 \tag{2.5}\]one has the case of the _Standard-Linear Solid_ (S.L.S.) which is modelled by the strictly-hyperbolic third order equation:
\[\mathcal{L}_{1}v\equiv\eta\ \partial_{t}(v_{tt}-c_{1}v_{xx})+\ v_{tt}-c_{0}\ v_{xx}=(1/a_{0})F. \tag{6}\]
The fundamental solution \(\mathcal{E}_{1}\) of this operator has been obtained in, also when \(\underline{\textbf{x}}\in R^{2}\) or \(\underline{\textbf{x}}\in R^{3}\). Further, numerous basic properties of \(\mathcal{E}_{1}\) have been rigorously estimated and the wave behavior of S.L.S. is now acquired.
When \(n>1\) and all the \(c_{k}\)' s are positive, then waves of different orders appear and their roles must be clarified in order to see how each set is modified by the presence of the other. Obviously, wave or dispersive behavior depend on the requirements of the coefficients \(a_{k}\) and \(c_{k}\) due to physical properties of the system \(\mathcal{B}\).
For this, we will analyze the restrictions imposed on the constants \(a_{k}\) and \(c_{k}\) by usual hypotheses of fading memory for \(\mathcal{B}\).
## 3 Integral Constitutive Equation
When the strain amplitudes are not too large, the behavior of most viscoelastic media is fairly well modelled by linear hereditary equations like:
\[\varepsilon(t)=J[\sigma(t)+\int_{-\infty}^{t}\dot{J}(t-\tau)\sigma(\tau)d \tau], \tag{7}\]
Usually, according to _fading memory_ hypotheses, \(\dot{J}(t)\) is a positive fast decreasing function. For instance, several real materials as polymers, rubbers, bitumines, have satisfactory representations by means of chains of S.L.S. elements in series or parallel.
\[J_{n}(t)=J_{n}[1+\sum_{k=1}^{n}\frac{B_{k}}{\beta_{k}}(1-e^{-\beta_{k}t})], \tag{8}\]
Then, if one puts:
\[c^{2}=[\rho_{0}J_{n}]^{-1},\ \ F_{*}=c^{2}[J_{n}f+\int_{-\infty}^{0}\dot{J} _{n}(t-\tau)\sigma_{x}(\tau)d\tau], \tag{9}\]
by,,, one deduces:\[{\cal M}u\equiv c^{2}u_{xx}-u_{tt}-\int_{0}^{t}g(t-\tau)u_{\tau\tau}d\tau=-F_{*}(x,t), \tag{3.4}\]
with
\[g\ =\ g_{n}(t)=\sum_{k=1}^{n}B_{k}e^{-\beta_{k}t}\ =\dot{J}_{n}(t)/J_{n}. \tag{3.5}\]
In this memory function, \(n\) is quite arbitrary and constants \(B_{k}\) and frequencies \(\beta_{k}\) are such that:
\[0<\beta_{1}<\beta_{2}...<\beta_{n};\ \ \ \ B_{k}>0\ \ \ \ \ \forall k=1,2....n. \tag{3.6}\]
These hypotheses assure that:
\[g(t)>0,\ \ \dot{g}<0,\ \ \ddot{g}>0,\ \ \forall t\geq 0 \tag{3.7}\]
We observe that the representation (3.5) of the memory function \(g\) is not restrictive because well-known Muntz and Schawart'z theorems imply that whatever \(C^{0}(R^{+})\) function can be uniformly represented by means of Dirichlet polynomials. Moreover, as \(n\) is arbitrary, the constants \(B_{k},\beta_{k}\) can be determined in order to fit the experimental curves for \(g(t)\) to any prefixed degree of approximation.
By (3.2), (3.6) one has
\[J_{n}(\infty)=J_{n}[1+\sum_{k=1}^{n}\ \frac{B_{k}}{\beta_{k}}]>J_{n}. \tag{3.8}\]
## 4.
The initial data related to (2.3) and (3.4) let be null and let
\[P(s)=\sum_{k=0}^{n}\ \mu_{k}\ s^{k}\ \ \ \,\ Q(s)=\sum_{k=0}^{n}\lambda_{k}\ s^{k} \tag{4.1}\]
with
\[\mu_{k}=a_{k}/a_{n}\ \ \ \ \,\ \lambda_{k}=a_{k}c_{k}/a_{n}c_{n}\ \ \ \ (k=0,..n) \tag{4.2}\]
Further, let \[G(s)=\sum_{k=1}^{n}\frac{B_{k}}{s+\beta_{k}}\]
Then, if one applies the Laplace transformation to (2.3) and (3.4), it results:
\[\hat{v}_{xx}-\frac{s^{2}}{c_{n}}\ \ \frac{P(s)}{Q(s)}\hat{v}=\ -\frac{1}{a_{n}c_{n} }\ \ \frac{\hat{F}}{Q(s)}\]
\[\hat{u}_{xx}-\frac{s^{2}}{c^{2}}\ \ [1+G(s)]\hat{u}=\ -\frac{\hat{F}_{*}}{c^{2}}\]
By comparing (4.4), (4.5) one deduces
\[\frac{P(s)}{Q(s)}=\frac{c_{n}}{c^{2}}\ \ [1+G(s)]\]
and the polinomial identity implies \(c_{n}=c^{2}\) and
\[\left\{\begin{array}{l}\lambda_{0}=\beta_{1}\beta_{2}..\beta_{n}\\.....................................\\ \lambda_{n-2}=\beta_{1}\beta_{2}+\beta_{1}\beta_{3}+..\beta_{n-1}\beta_{n}\\ \lambda_{n-1}=\beta_{1}+....+\beta_{n}\end{array}\right.\]
So, owing to (3.6), all the \(\lambda_{k}\)'s are positive. Further, as for \(\mu_{k}\), one has:
\[\left\{\begin{array}{l}\mu_{0}=\lambda_{0}+B_{1}(\beta_{2}..\beta_{n})+...B_ {n}(\beta_{1}..\beta_{n-1})\\.....................................\\ \mu_{n-2}=\lambda_{n-2}+B_{1}(\beta_{2}+..+\beta_{n})+...B_{n}(\beta_{1}+..+ \beta_{n-1})\\ \mu_{n-1}=\lambda_{n-1}+B_{1}+....+B_{n}\end{array}\right.\]
As consequence:
\[0<c_{k}<c_{n}=c^{2}\ \ \ \ \ (k=0,...n-1)\]
At last, by (4.7),(4.8), it follows: \(\frac{\lambda_{0}}{\mu_{0}}\,<\,\frac{\lambda_{1}}{\mu_{1}}\,<\,\frac{\lambda_ {n-1}}{\mu n-1}\) and so
\[0<c_{0}<c_{1}....\ <c_{n}.\]
So, the following property holds:
**Property 4.1.**_Hypotheses of fading memory_ (3.5) (3.6) _imply that the differential operator_ (2.3) _is typical of wave hierarchies._Vice versa, the inverse transformation of (4.7),(4.8) requires carefulness.
When the differential equation (2.3) is prefixed, in order to obtain the dual hereditary equation (3.4) with a memory function \(g(t)\) satisfying (3.5), (3.6), appropriate restrictions on the constants \(a_{k},c_{k}\) must be imposed.
At first, (4.3), (4.6) imply
\[\frac{P(s)}{Q(s)}\;=\;B_{0}+\sum_{k=1}^{n}\frac{B_{k}}{s+\gamma_{k}} \tag{4.11}\]
## Example 4.1
- When \(n=1,\) one has: \(c^{2}=c_{1},B_{0}=1,\) and
\[\beta_{1}=\frac{a_{0}c_{0}}{a_{1}c_{1}}>0\quad,B_{1}=\frac{a_{0}}{a_{1}}(1- \frac{c_{0}}{c_{1}})>0 \tag{4.12}\]
## Example 4.2
- When \(n=2,\) one has: \(c^{2}=c_{2},B_{0}=1,\) and \(\beta_{1},\beta_{2}\) are real iff:
\[\omega^{2}=(a_{1}c_{1})^{2}-4(a_{0}c_{0})(a_{2}c_{2})>0 \tag{4.13}\]
Then, it results:
\[\beta_{1}=\frac{1}{2a_{2}c_{2}}(a_{1}c_{1}-\omega),\quad\beta_{2}=\frac{1}{2a _{2}c_{2}}(a_{1}c_{1}+\omega) \tag{4.14}\]
Further
\[B_{i}=\frac{(-1)^{i-1}}{\omega}[a_{0}(c_{2}-c_{0})-a_{1}\beta_{i}(c_{2}-c_{1}) ].\quad(i=1,2) \tag{4.15}\]
Thus, it is \(B_{1}>0,B_{2}>0\) iff
\[\beta_{1}<\frac{a_{o}}{a_{1}}\;\;\frac{c_{2}-c_{0}}{c_{2}-c_{1}}<\beta_{2}. \tag{4.16}\]
Therefore, the fourth-order operator
\[a_{2}(u_{tt}-c_{2}u_{xx})_{tt}+a_{1}(u_{tt}-c_{1}u_{xx})_{t}+a_{0}(u_{tt}-c_{0 }u_{xx}) \tag{4.17}\]
## 5. ESTIMATES FOR THE HEREDITARY MODEL
Let \({\cal B}_{n}\) the viscoelastic model characterized by the memory function \(g_{n}\) in (3.5); the case \(n=1\) corresponds to the L.S.L. \({\cal B}_{1}\).
In, the fundamental solution \(E_{n}\) of the operator \({\cal M}\) in (3.4) has been explicitly determined, whatever \(n\) may be. If \(\eta\) is the step - function and \(I_{0}\) is the modified Bessel function of first kind, it results:
\[E_{n}=E_{n}(\beta_{1}..\beta_{n},B_{1}..B_{n})=\frac{1}{2c}\eta(t-r)(A_{1}+A_{ 2})\]
with
\[A_{1}=e^{-g_{0}t/2}I_{0}(\frac{g_{0}}{2}\ \sqrt{t^{2}-r^{2}})\]
\[A_{2}=\frac{1}{\pi}\int_{0}^{\pi}d\theta\int_{r}^{t}e^{-g_{0}z}H(z,t-w)du,\]
and \(g_{0}=g,r=|x|/c,\ \ 2z=w-cos\theta\ (w^{2}-r^{2})^{1/2}.\) Futher, if
\[\phi_{k}(z,t)=e^{-\beta_{k}t}\sqrt{B_{k}\beta_{k}z/t}\ \ I_{1}(2\sqrt{B_{k}\beta_{k}z\ t}),\]
one has:
\[H(z,t)=\sum\ \phi_{k}+\sum_{k_{1},K_{2}}\phi_{k_{1}}*\phi_{k_{2}}+...\]
Moreover the fundamental solution \(E_{n}\) related to \({\cal B}_{n}\) and defined in (5.1)-(5.5), can be rigorously estimated in terms of the fundamental solution \(E_{1}\) related to an appropriate S.L.S. \({\cal B}_{1}^{*}\) defined by
\[g_{1}=b\ e^{-\beta_{1}t}\ \ \ with\ \ b=\beta_{1}\sum_{1}^{n}\frac{B_{k}}{\beta_{k}}.\]
In fact, if \(\Gamma\) is the open forward characteristic cone \(\{(t,x):t>0\ |x|<ct\},\) and \(\chi_{n}=\prod_{k=2}^{n}\ (\frac{B_{k}}{\beta_{1}})^{2},\) then the following theorem holds:
## Theorem 5.1
- _If the memory function is given by (3.5) (3.6), then the fundamental solution \(E_{n}\) of \({\cal M}\) is a never negative \(\ C^{\infty}(\Gamma)\) function and it satisfies the estimate_:
\[0<E_{n}(\beta_{1}..\beta_{n},B_{1}..B_{n})<\chi_{n}\ \ E_{1}(\beta_{1},b),\]everywhere in \(\Gamma\) and whatever \(n\) may be._
## Remark 5.1
- The model \({\cal B}_{1}^{*}\) defined by (5.6) is physically meaningful.
In fact the memory function \(g_{1}\) is related just to the obliviator because \(\tau_{1}=\beta_{1}^{-1}\) is the longest characteristic time.
Furthermore \({\cal B}_{n}\) and \({\cal B}_{1}^{*}\) verify the same hypotheses of fading memory and by (3.5) (5.6), it results:
\[\int_{0}^{\infty}g_{n}(t)dt =\int_{0}^{\infty}g_{1}(t)dt =\sum_{1}^{n}\frac{B_{k}}{\beta_{k}}. \tag{5.8}\]
Moreover, the integral (5.8) affects the asymptotic analysis of hereditary equation.
## Remark 5.2.
- By known properties of asymptotic behaviour of convolutions, the constitutive relation (3.1) implies:
\[\lim_{t\rightarrow\infty}\varepsilon(x,t)=J_{n}\ [1+\int_{0}^{\infty}g_{n}(t) dt]\ \lim_{t\rightarrow\infty}\sigma(x,t) \tag{5.9}\]
Then \({\cal B}_{n}\) and the model \({\cal B}_{1}^{*}\) exhibit the same asymptotic behaviour (5.9)
Further, by (3.5), (3.8) it results:
\[\frac{J_{n}(\infty)}{J_{n}}=1+\sum_{1}^{n}\frac{B_{k}}{\beta_{k}}=\frac{J_ {1}(\infty)}{J_{1}} \tag{5.10}\]
and so, (5.10) implies \(J_{n}(\infty)\sigma(\infty)=\varepsilon(\infty).\) Consequently, when \(t\) is large, the behaviour of \({\cal B}_{n}\) is typical of an elastic material with modulus \(J_{n}(\infty).\)
## 6. ESTIMATES RELATED TO WAVE HIERARCHIES
When the operator \({\cal L}_{n}\) is reduced to the hereditary operator \({\cal M}\) of (3.4), then estimates of Theorem 5.1 can be applied to wave hierarchies. Obviously, the equivalence is conditioned by inverse transformation of (4.7)- (4.8) together with (3.6) (see n.4).
Let \({\cal L}_{1}^{*}\) the operator (2.6) related to the S.L.S. \({\cal B}_{1}^{*}\) characterized by
\[\eta=\frac{1}{\beta_{1}+b},\quad a_{0}=\frac{\beta_{1}+b}{\beta_{1}},\quad c_{ 0}=\frac{c^{2}\beta_{1}}{\beta_{1}+b}\quad c_{1}=c^{2}. \tag{6.1}\]
Now, let \({\cal L}_{n}\) the differential operator given by (2.3) whatever \(n\) may be, and let \({\cal P}_{n}\) a prefixed boundary value problem related to \({\cal L}_{n}\). The meaningful aspects of qualitative analysis of the solution of \({\cal P}_{n}\) can be obtained by means of Theorem 5.1 and by the known properties of \({\cal L}_{1}^{*}\).
So maximum properties, asymptotic behaviour, boundary layer estimates, etc. for the solution of \({\cal P}_{n}\) are deduced by analogous properties related to \({\cal L}_{1}^{*}\).
Moreover, owing to the equivalence between \({\cal L}_{n}\) and \({\cal M}\), it is possible to have explictly the fundamental solution of \({\cal L}_{n}\) for all \(n\). In fact it suffices to apply the explicit formula (5.1)-(5.3).
As an example the case of _polymeric materials_ can be considered.
## Example 6.1
- Polymeric materials are very flexible and are easily formed into fibres, thin films, additives for oils, etc. So theirs applications to concrete problems are numerous.. According to theories of linear viscoelasticity, two models, that descibe different aspects of polymer chains, have met a reasonable success: _the Rouse model_ and _the reptation model_.
In both cases the memory function \(g(t)\) assumes a form like (3.2)(3.3). In fact in _the reptation model_, the stress relaxation function is:
\[g(t)=k\sum_{h=0}^{n}\frac{1}{(2h+1)^{2}}\ \ e^{-(2h+1)^{2}\frac{t}{\tau_{d}}}, \tag{6.2}\]
When the viscoelastic behaviour is represented by _the Rouse model_, memory function \(g(t)\) is given by :
\[g(t)=k_{1}\sum_{h=1}^{n}e^{2h^{2}\frac{t}{\tau_{1}}}, \tag{6.3}\]
So, if one considers the first two steps in _the reptation model_, it results: \(B_{1}=k,\ \ B_{2}=B_{1}/9,\ \ \beta_{1}=1/\tau_{d},\ \ \beta_{2}=9\beta_{1}.\) Consequently the operator (4.17) is characterized by constants:
\[\left\{\begin{array}{ll}c_{0}=c^{2}\ \frac{81}{81+82k\tau_{d}}&c_{1}=c^{2}\ \frac{9}{9+k\tau_{d}}&c_{2}=c^{2}\\ \\ a_{0}=1+\frac{82}{81}\,k\tau_{d}&a_{1}=\frac{10\tau_{d}^{2}}{9}(\frac{1}{\tau_ {d}}+\frac{k}{9})&a_{2}=\frac{\tau_{d}^{2}}{9}.\end{array}\right. \tag{6.4}\]
Analogously, in _the Rouse model_, beeing \(B_{1}=B_{2}=k_{1},\ \ \beta_{1}=2/\tau_{1},\ \ \beta_{2}=4\beta_{1}\), one has:
\[\left\{\begin{array}{ll}c_{0}=c^{2}\frac{8}{8+5k_{1}\tau_{1}}&c_{1}=c^{2} \frac{5}{5+k_{1}\tau_{1}}&c_{2}=c^{2}\\ \\ a_{0}=1+\frac{5k_{1}}{8}\tau_{1}&a_{1}=\frac{\tau_{1}^{2}}{16}(2k_{1}+\frac{1 0}{\tau_{1}})&a_{2}=\frac{\tau_{1}^{2}}{16}.\end{array}\right. \tag{6.5}\]
The _wave hierarchies_ defined by (6.4) or (6.5) are governed by the operator \({\cal L}_{1}^{*}\) of the Standard Linear Solid defined, respectively, by:\[\left\{\begin{array}{ll}c_{0}=c^{2}\ \frac{81}{81+82k\tau_{d}}&c_{1}=c^{2}\quad a _{0}=1+\frac{82}{81}k\tau_{d}\quad\eta=\frac{81\tau_{d}}{81+82k\tau_{d}},\\ \\ c_{0}=c^{2}\frac{8}{8+5k_{1}\tau_{1}}&c_{1}=c^{2}\quad a_{0}=1+\frac{5}{8}k_{1} \tau_{1}\quad\eta=\frac{4\tau_{1}}{8+5k_{1}\tau_{1}}.\end{array}\right. \tag{6.6}\]
These results have been confirmed also in for entangled polymers with chain stretch.
| 10.48550/arXiv.1203.0466 | Wave hierarchies in viscoelasticity | M. De Angelis, P. Massarotti, P. Renno | 393 |
10.48550_arXiv.0909.4622 | ## Abstract:
Fluorescent carbon nanoparticle (CNP) having 2-6 nm in size with quantum yield of about ~3% were synthesized via nitric acid oxidation of carbon soot and this approach can be used for milligram scale synthesis of these water soluble particles. These CNPs are nano-crystalline with predominantly graphitic structure and shows green fluorescence under UV exposure. While nitric acid oxidation induces nitrogen and oxygen incorporation into soot particle that afforded water solubility and light emitting property; the isolation of small particles from a mixture of different size particles improved the fluorescence quantum yield. These CNP shows encouraging cell imaging application. They enter into cell without any further functionalization and fluorescence property of these particles can be used for fluorescence based cell imaging application.
## Key Words:
Carbon nanoparticles; Fluorescence, Cell labeling.
\({}^{\textbf{(a)}}\)**Present Address: School of Physics, University of the Witwatersrand, Private Bag 3, Wits 2050, South Africa. \({}^{\textbf{\ast}}\)_Corresponding author(s)_: S.C. Ray and Nikhil R.
## 1 Introduction
The emergence of fluorescence carbon nanoparticle (CNP) shows high potential in biological labeling, bioimaging and other different optoelectronic device applications. These carbon nanoparticles are biocompatible and chemically inert, which has advantages over conventional cadmium based quantum dots. However, these fluorescent carbon nanoparticle are poorly studied compared to other carbon-based materials such as carbon nanotubes and fullerenes. In addition, the understanding of the origin of fluorescence in carbon nanoparticle is far from sufficient. For example, information on the microstructure and surface ligands remains unclear and details of the organic passivation is not sufficient to aid understanding of the surface states beneficial for light emission.
Common routes in making fluorescent carbon nanoparticle includes high energy ion beam radiation based creation of point defect in diamond particle followed by annealing, laser ablation of graphite followed by oxidation and functionalization, thermal decomposition of organic compound, electrooxidation of graphite and oxidation of candle soot with nitric acid. A wide range of fluorescent carbon particle of different colors can be prepared by those approaches; e.g. octadecylamine functionalized diamond nanoparticle showed blue fluorescence, nitrogen doped diamond showed red fluorescence [1a] and candle soot derived particle or thermal decomposition method or laser ablation method produced particles with multiple colors. However, quantum yield of most of these particle are too low (<1%) with few exceptions. In addition the synthetic methods are cumbersome and inefficient. For example, in the high energy ion beam radiation based method, it is difficult to introduce a large number of point defects into ultra-fine nano-carbon particles (<10 nm) for bright luminescence. Thermal decomposition based methods produce low yield of soluble and fluorescent particle with a significant fraction of insoluble product. Soot based synthesis produce particle mixture of different colors and isolation of different colored particles by gel electrophoresis is a difficult task. Recent report showed that surface passivation can lead to a significant increase in fluorescence quantum yield (4-15%) however exact mechanism is not yet clear. Thus simple, efficient and large scale synthesis of fluorescent carbon nanoparticle and their isolation, purification and functionalization are very challenging.
Among all these synthetic methods, soot based approach is simple and straightforward. However, quantum yield of fluorescent carbon nanoparticle is too low (<0.1%) to any useful application. Herein, we report an improved soot based method of synthesizing fluorescent carbon nanoparticle (CNP) of 2-6 nm in size with quantum yield of ~3%. There are three distinct improvements in our modified method. First, we developed a simple separation method of small size and fluorescent carbon particle from heterogeneous particle mixture. The method is applicable for milligram scale synthesis of these particles. Second, small particles are more fluorescent than larger one and thus isolation of small particle improves the quantum yield from <0.1% to ~3%. Third, we observed that these small carbon particles enter into cell without any further functionalization and fluorescence property of particle can be used for fluorescence based cell imaging application.
## Experimental Procedures
### Synthesis of carbon particle
25 mg carbon soot (collected from burning candle) was mixed with 15 mL of 5M nitric acid in a 25 mL three necked flask. It was then refluxed at 100\({}^{\circ}\)C for 12 hours with magnetic stirring. After that the black solution was cooled and centrifuged at 3000 rpm for 10 minutes to separate out unreacted carbon soot. The light brownish yellow supernatant was collected that shows green fluorescence under UV exposure. The aqueous supernatant was mixed with acetone (water:acetone volume ratio was 1:3) and centrifuged at 14000 rpm for 10 minutes. The black precipitate was collected and dissolved in 5-10 mL water. The colorless and non-fluorescent supernatant was discarded. This step of purification separates excess nitric acid from the carbon nanoparticles. This concentrated aqueous solution having almost neutral pH was taken for further use. The same synthesis technique was also performed for 6 hours reflux and 18 hours reflux. The supernatant obtained from 6 hours reflux was pale yellow and for 18 hour reflux was dark yellow. We weighted the unreacted carbon soot which was removed as precipitate, in order to find out the yield of soluble carbon nanoparticles. The weight was ~22.5 mg for 6 hours reflux time (yield ~10%) and ~20 mg for 12 and 18 hours reflux time (yield ~20%). It shows that yield increases as reflux time increases from 6 to 12 hours but after that no significant increases was observed. This solution has particles having size ranges from 20-350 nm and called as synthesized carbon particle (CP).
### Size separation of carbon particles
Size separation was performed in a solvent mixture with the combination of high speed centrifuge based separation. As synthesized carbon particles (CP) are soluble in water, ethanol and acetone but insoluble in chloroform. The aqueous and ethanolic particle solution does not precipitate even at 16000 rpm centrifugation. So we chose a solvent mixture of water-ethanol-chloroform (single phase without any phase separation) for the size separation of particle where water-ethanol helps to solubilize the particle but chloroform decreases their solubility. Next, we followed a step-by-step separation using different centrifugation speed from 4000-16000 rpm.
In a typical process, the aqueous solution of carbon nanoparticle was mixed with chloroform and ethanol maintaining water:chloroform:ethanol volume ratio of 1:1:3. Next, 2 mL of this solution was centrifuged with 4000 rpm for 10 minutes. The precipitate was collected and dissolved in 2 mL fresh water. The supernatant was then again centrifuged at 5000 rpm for 10 minutes and the precipitate was collected and redissolved in 2 mL of water. The same procedure was applied at 6000 rpm and 8000 rpm. The supernatant obtained after 8000 rpm was not getting precipitated even at 16000 rpm. This solution was collected and evaporated to dryness to remove ethanol and chloroform and finally dissolved in water. This solution has particle size of 2-6 nm and named as carbon nanoparticle (CNP) that were used for characterization and application. The yield of CNP from CP is ~ 50%, which means 25 mg soot can produce ~2-3 mg of CNP. In this calculation we assumed that carbon dioxide formation is negligible during nitric acid oxidation.
### Cell labeling and cytotoxicity assay
For cell labeling experiment, Ehrlich ascites carcinoma cells (EAC) were collected from peritoneal cavity of adult female mice after 7 days of inoculation. The suspension of cells was prepared with a concentration \(\sim 10^{7}\) cell / mL. Next, 1.0 mL of this suspension was mixed with 10-100 \(\upmu\)L of aqueous CNP solution and incubated for 30 minutes. Next, labelled cells were separated from free CNP by centrifuging at 2000 rpm for 3 minutes. The precipitated cells were suspended in phosphate buffer solution. This type of precipitation-resuspended was repeated for 2 more times and finally cells were suspended in phosphate buffer solution. A drop of this suspension was placed in a glass slide for imaging experiment. Fluorescence image was captured using Olympus IX71 fluorescence microscope with DP70 digital camera.
For MTT and Trypan blue assay, HepG2 cells were trypsinized and resuspended in culture medium. The cells were seeded to flat bottom microplate with 0.5 mL full medium and kept for overnight at 37\({}^{\circ}\)C and 5% CO\({}_{2}\). The CNP solution of different amounts was loaded to each well and each concentration has 3 duplications. After incubation for 24 hours, 50\(\upmu\)L of MTT solution (5mg/mL) was added to each well 4 hours before the end of the incubation. The medium was discarded and produced formazan was dissolved with DMSO. The plates were read with absorbance at 550nm. The optical density is directly correlated with cell quantity and cell viability was calculated by assuming 100% viability in the control set without any CNP. In case of Trypan blue assay, 0.4% of Trypan blue solution was used instead of MTT and after 5 minutes stained cells are counted to determine the cell viability.
## Instrumentation
Diluted CNP solution was dropped onto copper grids to prepare specimens for transmission electron microscopic (TEM) observation which was performed in a FEI Tecnai G2 F20 microscope with a field-emission gun operating at 200 kV. The microstructures of the CNPs were examined using JEOL JSM-6700F scanning electron microscope and Veeco di CP-II atomic force microscope respectively. The fluorescence (photoluminescence) spectra were measured on a Hitachi F4500 fluorescence spectrophotometer at different excitation energy ranging from 325 nm to 600 nm. A Nicolet 6700 Fourier transform infrared (FTIR) spectrophotometer was used to analyze chemical bonds on the surface of CNP. A Perkin-Elmer PH1-1600 X-ray photoelectron spectrometer (XPS) was used for compositional analysis and chemical bond determination of CNPs. Dynamic light scattering (DLS) study was performed using model BI-200SM instrument (BrookhavenInstrument Corporation), after filtering the sample solution with Milipore syringe filter (0.2 micron pore size). Micro Raman studies was performed using an ISA Lab Raman system equipped with 514.5 nm LASER with a 100xobjective giving a spot size about 1 mm with a spectral resolution better than 2 cm\({}^{-1}\). The quantum yield was measured by comparing the integrated photoluminescence intensities and absorbance values of the CNP with the reference fluorescein dye (QY = 95 %).
## Results and discussion
Carbon soot is black in color and completely insoluble in water even after ultra-sonnication. This is because they are large in size and hydrophobic in nature. When this soot is refluxed with nitric acid, light brown colored supernatant solution is obtained along with an insoluble black precipitate. Brownish yellow supernatant indicates a part of carbon particle becomes small and water soluble during the refluxing processes. This soluble particle exhibits green fluorescence when irradiates with UV light, while the precipitate part shows no fluorescence. We have studied the fluorescence spectra at different excitation energy ranging from 325 nm - 600 nm and found that the highest fluorescence intensity was achieved at the excitation wavelength of 450 nm, which shows emission maximum at 520 nm. shows the photoluminescence spectra of as synthesized CP with different excitation wavelength. We have tested the quality and yield of CP as a function of reflux time and found that 12 hour is the optimum time. If the reflux time is less the yield is low and for longer reflux time the yield does not increase appreciably. (See experimental section and supporting Figure S1a).
The nature of fluorescence spectra suggested that there are different types of particles with different colors. Earlier work also showed that these particles can be separated by gel electrophoresis technique. We have performed size separation of CP in order to separate CP of different fluorescent property. The size separation has been performed in a solvent mixture with the combination of high speed centrifuge based separation. We have identified mixture of water-ethanol-chloroform as a single phase solvent for the effective size separation of CP, where water-ethanol helps to solubilize the CP but chloroform decreases their solubility. Next we have followed a step-by-step separation using different centrifugation speed from 4000-16000 rpm.
It shows that smallest particle (CNP) that does not precipitate even at 16000 rpm, shows the highest fluorescence. All the other size particles show very weak fluorescence and there is very little blue shift in fluorescence with decreasing particle size. Fluorescence study of various phosphate buffer solution of CNP shows that fluorescent intensity does not change appreciably on solution pH from 7-9. (See in supporting Figure S1b). The comparative absorption spectra of as synthesized carbon particle (CP) and smallest particle (CNP) are shown in No appreciable change is observed except that CNP shows a stronger absorbance in visible wavelength along with weak band at \(\sim\) 350 nm.
Transmission electron microscopic (TEM) study shows that as synthesized carbon particle (CP) have broad size distribution from 20-350 nm but CNP have small and narrow particle size distribution from 2-6 nm (and 2b). Both CP and CNP has well graphitization, the interlayer spacing between graphitic sheets is d\({}_{}\)=0.33 nm (obtained from HRTEM: shown as inset in Figure 2b), which is very close to that of the ideal graphite. Similar type of carbon nanoparticle (\(1.5-2.5\) nm) is reported earlier following an aqueous route with the help of silica sphere as carrier. Comparison of TEM and fluorescence data shows that CNP of 2-6 nm size has higher fluorescence intensity compared to CP of larger overall size. This type of size dependent fluorescence QY is observed for carbon particle produced via laser ablation technique, thermal decomposition and particle obtained from candle soot.
We found that our CP and CNP has very strong tendency of aggregation during TEM grid preparation or SEM slide preparation. The aggregation is so high that we face difficulty in finding significant amount of isolated small carbon particle (CNP) under TEM either from as synthesized carbon particle (CP) solution or from CNP solution. A similar type of aggregation is observed by Iijima et al. where small carbon particles are found to aggregate into \(\sim\) 80 nm size nano-horn structures. We have done some control SEM experiment to study this aggregation processes. Particle solution has been deposited on Si-substrate by a single or successive multiple drops (after the evaporation of first drop next drop was added). We have always found that multiple drop sample shows larger particle than the single drop. We have estimated the particle size of CNP as 12-15 nmfor single drop sample, but size increases when the sample was prepared from three drops. This observation further indicates that the CNP agglomerate easily in solid form but remain isolated in water. AFM study of CNP also showed the existence of small particles as well as particle aggregates even if a very dilute solution was used for this study (supporting Figure S2). We have also estimated the particle size of CP and CNP from the dynamic light scattering (DLS) measurement and result shows that the hydrodynamic diameters is broad and ranges from 20 to 350 nm for CP but for CNP it shows a narrow distribution with average diameter of \(\sim\) 12.5 nm (supporting Figure S3). The broad size distribution and larger sizes in CP mask the presence of small carbon particle (CNP) in their size distribution. This broad DLS size distribution of CP and narrow size distribution of CNP corroborate the TEM observation. The increased average size of CNP in DLS study (in comparison to TEM which shows 2-6 nm) is because DLS consider overall hydrodynamic diameter that includes particle as well as adsorbed molecules and ions.
The soot contains mainly elemental carbon and oxygen having 96 atomic % and 4 atomic % respectively, whereas CNP shows that the C, O and N are 59 atomic %, 37 atomic % and 4 atomic % respectively as estimated from XPS compositional analysis. (Figure 3a-f), These data shows that CNP is mainly composed of graphitic carbon (\(sp^{2}\)) and oxygen/nitrogen bonded carbon, whereas starting soot is mainly composed of diamond like carbon (\(sp^{3}\)) with oxygen bonded carbon. This composition variation is well matched with bonding structures obtained from FTIR spectrum (see supporting Figure S4 and identification of different bonding configurations).
Spectrum of CNP shows high photoluminescence background with compared to soot. The two signature peaks for carbon i.e. D band and G band are clearly seen for CNP and soot where D band corresponds to disordered structure in crystalline of sp\({}^{2}\) cluster and G band corresponds to the in-plane stretching vibration mode E\({}_{2g}\) of single crystal graphite. The intensity ratio (I\({}_{\text{D}}\)/I\({}_{\text{G}}\)), which is often used to correlate the structural purity to the graphite, also indicates that the CNP are composed of mainly nano-crystalline graphite.
was calculated as 2.2 nm. Similar graphitic structure is also obtained from x-ray diffraction patterns of CNP (supporting Figure S5).
It is well known that nitric acid oxidation produces OH and CO\({}_{2}\)H groups on the carbon nano-particle surfaces that made them hydrophilic and negatively charged particle. In addition this oxidation can also induce small extent of nitration into graphitic carbon. Our experimental data suggest that refluxing step with nitric acid has made two fold chemical modifications to the soot. First, it induces partial oxidation of carbons and introduces functional groups such as OH, CO\({}_{2}\)H and NO\({}_{2}\). Second, it induces doping of nitrogen and oxygen into the carbon particle. Introduction of functional groups induce water solubility and surface charge to the CNP. In addition it helps to break the large aggregated soot particle into small carbon particles. This oxidation step can also be considered as chemical route of incorporating nitrogen and oxygen into the carbon particle as observed from the chemical composition analysis.
The yield of soluble carbon particle depends of oxidation property of nitric acid in refluxing condition whereas the fluorescent quantum yield of carbon particle seems to depend on the efficiency of nitrogen and oxygen incorporation. Smaller particle size and dominant graphitic structure of the raw soot made this oxidation step easier. However, the efficiency of converting soot into soluble carbon particle is still low (yield \(\sim\)20%), as observed from large part of insoluble soot. This suggests that this type of chemical oxidation is not efficient enough for complete conversion into water soluble particle. Longer time refluxing with nitric acid increases the yield of soluble particle to some extent but does not increase the yield of fluorescent carbon particle. This suggests that further oxidation might have other adverse effect such as further oxidation of carbon particle that reduces the conjugated double bond structure in the carbon particle.
Incorporation of nitrogen and oxygen defect via nitric acid oxidation might have a role in producing fluorescent centre into carbon particle. Such defect structures in the fluorescence property of diamond is well established. As soot has some percentage of diamond like carbon (as observed from XPS data), it might happen that during oxidation step nitrogen and oxygen defects are formed into diamond structure. However, their presence in CNP is too low to determine with our presented analytical method and further study is neededto confirm this possibility. An alternative explanation of fluorescence may be that chemical oxidation and doping step introduces more conjugated double bond system into the carbon particle and thereby introduce the fluorescence property. Nevertheless, the advantage of this type of chemical processes of making fluorescent carbon particle is that it is simple and requires less adverse conditions as compared to ion beam radiation method. However, as synthesized carbon particles have heterogeneous size distribution and small size particle are more fluorescent than larger one. Thus successful isolation of small particle is essential for the enhancement of fluorescence quantum yield.
Water soluble fluorescent carbon nanoparticle is ideal cell imaging probe with minimum cytotoxicity. However, functionalization is an important step for cellular and sub-cellular targeting. Interestingly, we found that small carbon particles (CNP) enter into cell without any further functionalization and using the fluorescence property of CNP it is possible track the CNP. CNP solution has been mixed with cell culture media along with cell, incubated for 30 minutes and washed cells were then imaged under bright filed, UV and blue excitations. Cells become bright blue-green under UV excitations and yellow under blue excitation, but they are colorless in the control sample where no CNP was used. This suggests that CNP enter into the cells and labeled cell can be imaged using conventional fluorescence microscope. MTT and Trypan blue assay of cell viability study suggest that CNP has no cytotoxicity. We exposed the cell with CNP of 0.1-1 mg/mL (which is about 100-1000 times higher than it required for imaging application) and for 24 hours. The cell survival rate in <0.5 mg/mL was between 90-100%, suggesting a minimum cell death. However, at higher concentration some percentage of cell death is observed. This result concludes that CNP can be used in high concentration for imaging or other biomedical applications.
## Conclusion
Fluorescent carbon nanoparticle of 2-6 nm in diameter was obtained after nitric acid oxidation of soot-particles. Surface oxidation and subsequent nitrogen and oxygen doping afforded light-emitting property of carbon particle. It is to be noted that the light emitted by these carbon particles depends on the wavelength of light used for excitation. We isolated different size particles and found that emission quantum yield is size-dependent, i.e. smaller the size better is their photoluminescence efficiency. Our approach can be used for milligram scale synthesis of these water soluble particles. The fluorescence property of these particles is useful for cell imaging application. These CNP enter into cell without any further functionalization and fluorescence property of the particle can be used to track their position in cell using conventional fluorescence microscope. The discovery of fluorescent carbon nanoparticles will no doubt lead to more research in this filed, particularly these particles have the potential in biomedical application where cadmium-based quantum dot shows toxic effects. However, synthetic methods of these particles need to be much more advanced so that large quantity of these particles with different emission color can be easily prepared.
The authors would like to thank Dr. Nihar **R.** Jana of National Brain Research Centre (NBRC), Gurgaon, India for providing cellular imaging facility and cytotoxicity study. Authors would like to thank DST (SR/S5/NM-47/2005), Government of India for providing financial support. AS thanks to CSIR, India for providing fellowship.
# XPS peak assignment:
XPS C-1\(s\), N-1\(s\) and O-1s spectra and deconvolution of these spectra into different peaks was executed by curve fitting using Gaussian functions. C-1\(s\) peak of CNP is consisted of four Gaussian peaks centered at 281.8 eV, 284.4 eV, 286.2 eV and 288.6 eV. The peak at 284.4 eV indicates that carbon is mostly in the form of graphite and assigned to _sp\({}^{2}\)_ aromatic hydrocarbons. The peak at 286.2 eV can be assigned to C atoms surrounded by N and H atoms i.e. C-N / C=N and C-H bonds and the peak at 288.6 eV is the -C=O carbonyl groups. The peak at lower energy 281.8 eV is presumably the C-H peak. In case of soot the peaks in C-1s are consisted of _sp\({}^{3}\)_ C-C (285.0 eV), C-O/C-H (285.6 eV), carbonyl groups (288.4 eV) and the peak at 292.3 eV can be attributed to CO\({}_{2}\) and/or C-C=O bonds. In case of N 1s, the peak centered at 399 eV and 401.1 eV are graphitic structured C-N and C=N bond respectively, whereas the peak at 405.4 eV is assigned to some oxidized N-species like N-O and/or N=O bonds. Meanwhile, in case of CNP, the O1\(s\) peak consisted of three Gaussian peaks centered at 529.7 eV, 531.3 eV and 532.7 eV, which are associated with C-O/N-O and C=O/N=O respectively. In case of soot, the O1s peak is consisted of four Gaussian peaks and are associated with C-O (531.8 eV), -C=O (533.9 eV) and COOH (535.1 eV and 540.0 eV) respectively.Figure S1. (a) Fluorescence spectra of carbon particle (CP) produced with different reflux time. (b) Fluorescence spectra of small size carbon particle of 2-6 nm (CNP) at phosphate buffer of different pH.
Figure S2. AFM image of CNP taken at different position and magnification.
Figure S3. Particle size distribution of (a) aqueous solution of as synthesized carbon particle (CP) and (b) smallest size carbon particle of 2-6 nm (CNP) as observed from dynamic light scattering study.
Figure S4. FTIR spectrum of CNP showing the presence of different functional groups. The peaks at \(\sim\)1100 cm-1 and 1265 cm-1 are ascribed as C-O-C and C=C bonds respectively. A few small peaks are observed in between 1400-1650 cm-1 and at 1735 cm-1 ascribed to C-H, conjugated C-N and C=N, N-O bonds of stretching modes respectively. The broad band observed at 1600-1700 cm-1 can be assigned to the C-OH and C=O bonds. A small band at 1835 is observed and is ascribed as associated with C=N (sp2 C-N) stretching vibration. The band observed around 2090 cm-1 is ascribed to C-H bond. The band 2335 and 2360 cm-1 can be attributed to the nitrile (-CN) group. In addition a broad IR band \(\sim\) 3100-3650 cm-1 appeared due to C-OH and COOH bonds.
Figure S5. The wide-angle region of the x-ray diffraction (XRD, Shimadzu, XRD-6000) patterns of CNP exhibit a high-intensity diffraction peak at \(2\theta=24.4^{0}\) and one additional peat at \(2\theta=43.7^{0}\) that are ascribed to and diffraction of graphitic carbon respectively. Si peak is arises because CNPs are deposited on Si substrate using solution drop process and dried at room temperature for this measurement. | 10.48550/arXiv.0909.4622 | Fluorescent Carbon Nanoparticle: Synthesis, Characterization and Bio-imaging Application | S. C. Ray, Arindam Saha, Nikhil R. Jana, Rupa Sarkar | 6,069 |
10.48550_arXiv.1406.1113 | ###### Abstract
We have measured the interlayer and in-plane (needle axis) thermal diffusivity of 6,13-bis(triisopropylsilylethynyl) pentacene (TIPS-Pn). The needle axis value is comparable to the phonon thermal diffusivities of quasi-one dimensional organic metals with excellent \(\pi\)-orbital overlap, and its value suggests that a significant fraction of heat is carried by optical phonons. Furthermore, the interlayer (**c**-axis) thermal diffusivity is at least an order of magnitude larger, and this unusual anisotropy implies very strong dispersion of optical modes in the interlayer direction, presumably due to interactions between the silyl-containing side groups. Similar values for both in-plane and interlayer diffusivities have been observed for several other functionalized pentacene semiconductors with related structures.
Organic semiconducting materials are being studied for a variety of room temperature applications, especially where low-cost conformal materials that can be coated on complex structures are desired. Although most of the emphasis has been on studies of electronic and optical properties, it is also important to know the thermal conductivities (\(\kappa\)) for different functions. For example, for micron-channel thin-film transistors, e.g. to drive pixels in flexible displays, one requires a relatively large thermal conductivity to minimize Joule heating; to keep heating less than \(\sim 10\) degrees, one needs \(\kappa>\kappa_{0}\), where \(\kappa_{0}\sim\)10 mW/cm\(\cdot\)K. On the other hand, low thermal conductivities, e.g. \(\kappa<\kappa_{0}\), are needed for applications as room-temperature thermoelectric power generators.
Because small molecule, crystalline organic semiconductors can have high charge carrier mobilities and offer versatile processing possibilities, they can be advantageous for some applications. For thermoelectric applications, for which materials must be chemically doped to have sufficient conductivities, this will require substitutional doping with structurally similar molecules so as not to overly increase scattering due to disorder. If this can be achieved, the high mobility may allow very low doping levels to be used, increasing the Seebeck coefficient without significantly increasing the thermal conductivity. For example, layered crystals of rubrene has been found to have in-plane and interplane thermal conductivities comparable to that of disordered polymers.
In this regard, 6,13-bis(triisopropylsilylethylnyl) pentacene (TIPS-Pn), for which self-assembled crystalline films can be cast from solution and for which hole mobilities \(\mu>10\) cm\({}^{2}\)/V\(\cdot\)s have been achieved, is a promising material. In this work, we discuss measurements of the room temperature thermal conductivities of TIPS-Pn and several other pentacene based materials with related structures, shown in Because these crystals are undoped, the electronic density is negligible and \(\kappa\) is overwhelmingly due to phonons. All of these materials form layered crystals, with the pentacene (or substituted pentacene) backbones lying approximately in the **ab**-plane (where one hopes for good \(\pi\)-orbital overlap between molecules), and the silyl or germyl side groups extending between layers. For example the "brick-layer" **ab**-plane structure of TIPS-Pn is shown in The interlayer (**c**-axis) and in-plane phonon thermal conductivities are therefore expected to be different and we measured them separately. Because the crystals are small (in-plane dimensions typically 0.5-10 mm and interlayer direction less than 0.6 mm), we used ac-calorimetry techniques, which yield the thermal diffusivity, D = \(\kappa\)/c\(\rho\), where c is the specific heat and \(\rho\) the mass density, rather than conventional techniques.
The interlayer measurement technique is described in detail in Ref.. Visible/NIR light, chopped at frequency \(\omega\), illuminates and heats the top surface of the opaque sample, giving a typical dc-temperature rise of the sample of a few degrees for a sample in vacuum. The temperature oscillations on the bottom surface are measured with a flattened, 25 \(\mu\)m diameter chromel-constantan thermocouple glued to the surface with silver paint. The oscillating thermocouple signal, V\({}_{\omega}\), is measured with a 2-phase lock-in amplifier.
\[T_{\omega}=2P_{0}A\Phi(\omega)/(\pi\omega C)\] (1a) \[\text{with}\ \ \Phi(\omega)\approx[1+(\omega\tau_{\text{meas}})^{2}]^{1/2}\] (1b).
Here P\({}_{0}\) is the intensity of the light, A the area of the sample, and C its heat capacity.
\[\tau_{\text{meas}}\approx\tau_{2}\equiv\text{d}^{2}/\surd 90D_{\text{c}}, \tag{2}\]where d is the thickness of the sample, \(\tau_{2}\) is the internal thermal time constant describing heat flow through the thickness of the sample (i.e. along the **c**-axis for our samples) and D\({}_{\rm c}\) = \(\kappa_{\rm c}\)/cp is the transverse, interlayer thermal diffusivity. For small values of this intrinsic \(\tau_{2}\), however, one also needs to consider the time response of the thermometer, which for our technique is dominated by the interface thermal resistance between the silver paint and sample. We've measured this interface time constant to vary between \(\sim\) 1 and 10 ms for silver paint contact areas \(\sim\) 1 mm\({}^{2}\) on different materials. Because the materials studied here are slightly soluble in the butyl acetate base of the paint, we expect the interface time constants to be closer to 1 ms, but for measured time constants less than a few ms, we only conclude a lower limit for D\({}_{\rm c}\), i.e. D\({}_{\rm c}\)\(>\) d\({}^{2}\)/\(\surd\)90\(\tau_{\rm meas}\). In fact, this was the case for all the samples discussed in this paper. (In contrast, a d \(\approx\) 90 \(\mu\)m crystal of rubrene had \(\tau_{\rm meas}\)\(\approx\) 17 ms with an interface time constant \(<\) 6 ms.)
(a-f) Molecular structures of crystals studied:
(a) bis(trisopropylsilylethylvinyl) pentacene (TIPS-Pn);
(b) bis(trisopropylgermylethylenyl) pentacene (TIPGe-Pn);
(c) bis(trisopropylsilylethylvinyl) octafluoropentacene (F\({}_{8}\)-TIPS-Pn);
(d) bis(trisopropylsilylethynyl) perfluoropentacene (F\({}_{2}\)-TIPS-Pn);
(e) bis(cyclopropyl-diisopropylsilylethynyl) pentacene (CP-DIPS-Pn);
(f) tetraethyl-bis(trisopropylsilylethynyl)-tetraoxadicyclopent[b,m] pentacene (EtTP-5).
(g) Schematic of the **ab**-plane structure of TIPS-Pn, with the heavy bars representing the pentacene backbones. The dashed arrow shows the growth axis of needle-shaped crystals.
For longitudinal, in-plane thermal conductivity (\(\kappa_{\rm long}\)) measurements, we use the technique of Hatta, _etal_. A movable screen is placed between the chopped light source and sample, and the temperature oscillations are measured (with a thermocouple again glued to the sample with silver paint) on the back side of the sample in the screened portion.
\[{\rm dlnT}_{\omega}({\rm x})\,/{\rm dx}=\,-\,(\omega/2{\rm D}_{\rm long})^{1/2}, \tag{3}\]
In our setup, the screen is attached to a micrometer (precision \(\pm\) 3 \(\mu\)m) which measures the distance, \({\rm x}_{0}\)-\({\rm x}\), where the offset \({\rm x}_{0}\) depends on the relative positions of the sample and screen. For these measurements, the frequency is fixed so that the thermal response of the thermometer does not affect results. To be sure that we are in the correct frequency limit and the edge of the screen is not too close to the thermometer (i.e. overlapping with the silver paint spot) we check that we obtain the same values for the "frequency normalized" slopes, \({\rm f}^{1/2}\,{\rm dlnT}_{\omega}/{\rm dx}\) at different frequencies, where \({\rm f}\equiv\omega/2\pi\),as shown in Because Eqtn. assumes that both the illuminated and screened portions of the sample are much longer than the longitudinal diffusion length, \(\left({\rm D}_{\rm long}/\omega\right)^{1/2}\), which is typically \(\sim 0.5\) mm for our samples and frequencies, data was taken on needle shaped samples \(\sim 1\) cm long.
In Figure 2, we plot \({\rm f}^{1/2}\,{\rm lnV}_{\omega}\) as a function of position behind a screen for a TIPS-Pn crystal along its needle axis at a few frequencies. From Eqtn., the slope is inversely proportional to \({\rm D}_{\rm long}^{1/2}\). For large \({\rm x}_{0}\)-\({\rm x}\), the edge of the screen overlaps the silver paint holding the thermocouple, and the signal begins to saturate, whereas for small \({\rm x}_{0}\)-\({\rm x}\), the edge of the screen is far from the thermocouple, so \({\rm V}_{\omega}\) is small and approaches the thermocouple offset voltage and noise level. For intermediate distances, the same slope, \({\rm f}^{1/2}\,{\rm d}\,{\rm lnV}_{\omega}/{\rm dx}\), is measured for different frequencies, yielding \({\rm D}_{\rm long}=1.0\pm 0.1\) mm\({}^{2}\)/s. (Similar values were found for other crystals.) The specific heat of TIPS-Pn, measured on a pellet with differential scanning
Spatial dependence of \({\rm V}_{\omega}\) for an 8 mm long TIPS-Pn crystal at several frequencies. The region of parallel, linear variations of \({\rm f}^{1/2}\,{\rm ln(V}_{\omega})\), shown by the solid lines, is outlined by the dotted lines, as discussed in the text.
Using c = 1.48 J/gK and density \(\rho\) = 1.1 g/cm\({}^{3}\), we find \(\kappa_{\rm long}=16\pm 2\) mW/cm\(\cdot\)K.
This value of \(\kappa_{\rm long}\) is several times larger than the room temperature value for rubrene, but comparable to the phonon thermal conductivity of quasi-one dimensional organic conductors containing segregated molecular stacks with excellent \(\pi\)-orbital overlap. In fact, its value suggests that, in addition to acoustic phonons, a significant fraction of the heat is carried by low energy, propagating optical phonons. The room temperature specific heat is over thirty times the value associated with acoustic modes (\(c_{\rm acoustic}=3\)R/M = 0.039 J/g\(\cdot\)K, where R is the gas constant and the molecular weight M = 638 g/mole), indicating that most of the room temperature specific heat is due to excitation of optical modes.
\[\kappa=(\rho/3)\Sigma_{\rm j}v_{\rm j}j_{\rm j},\quad\]
If the acoustic phonon velocity is between 1 and 3 km/s (typical for molecular solids) and we assume that only the acoustic modes are propagating, Eqtn. implies that \(\lambda>400\) A\({}^{\circ}\), i.e. \(>50\) in-plane lattice constants. This value is much larger than expected, since acoustic modes are expected to have significant scattering both from librations of the side groups and large thermal motion of the TIPS-Pn molecule along the long-axis of the pentacene backbone (\(\sim\)transverse to the needle axis). If the phonon mean-free paths are in fact relatively small, much of the heat must be carried by sufficiently dispersive, low frequency (\(<\) k\({}_{\rm B}\)T) optical modes.
In Ref., we showed results for a d=335 \(\mu\)m crystal, with \(\tau_{\rm meas}\) = 1.3 ms; much thinner crystals, e.g. d \(\approx\) 100 \(\mu\)m, had similar time constants. These values of \(\tau_{\rm meas}\) and their lack of correlation with thickness imply that we are limited by the thermal interface resistance of the silver paint, and from the 610 \(\mu\)m crystal (the thickest available), we deduce D\({}_{\rm c}\)\(>\) 14 mm\({}^{2}\)/s and \(\kappa_{\rm c}\)\(>\) 225 mW/cm\({}^{2}\cdot\)K.
These values are much larger than typically found in a van-der-Waals bonded molecular crystal (e.g. for rubrene we measured D\({}_{\rm c}\)\(\sim\) 0.05 mm\({}^{2}\)/s, giving \(\kappa_{\rm c}\)\(\sim\) 0.7, while for pentacene \(\kappa_{\rm c}\)\(=\) 5.1 mW/cm\({}^{2}\cdot\)K), and are comparable to the values for materials with extended bonding (e.g. Al\({}_{2}\)O\({}_{3}\) has D \(\sim\) 10 mm\({}^{2}\)/s and \(\kappa\)\(\sim\) 300 mW/cm\(\cdot\)K). Also surprising is the anisotropy, \(\kappa_{\rm c}\)\(>\) 14 \(\kappa_{\rm long}\); because the in-plane (\(\pi\)-\(\pi\)) interactions were assumed to be stronger than the inter-plane (hydrocarbon) interactions, we expected the longitudinal thermal conductivity to be greater than the transverse, as we observed for rubrene, \(\kappa_{\rm c}\)\(\sim\)\(\kappa_{\rm long}\)/6. If one assumes that thermal transport is dominated by intermolecular thermal resistances and that these were _equal_ for **c**-axis and in-plane interactions, then the thermal conductivities in different directions would be proportional to the packing densities in their transverse planes, making \(\kappa_{\rm c}\)\(\sim\) 2 \(\kappa_{\rm long}\) because a \(\sim\) b \(\sim\) c/2. The much larger anisotropy we measure therefore indicates that the intermolecular thermal resistances along **c** are _smaller_ than those in the plane, implying stronger phonon interactions between the TIPS side groups than between the in-plane pentacene backbones. For example, the isopropyl groups of molecules in neighboring layers are only separated by \(\sim\) 0.4 nm. Because of the slight ionicity of the C-H bond, there will be significant Coulomb coupling of vibrations (e.g. librations) in these groups, increasing the **c**-axis dispersion of these low-frequency phonons. (If instead, one assumes that most of the heat is carried by acoustic phonons, which seems to be the case for rubrene, then the same analysis used for \(\kappa_{\rm long}\) would give an unreasonably large interlayer phonon mean-free-path \(\lambda_{\rm c}\)\(>\) 300c!)
To check these results for TIPS-Pn, we measured the transverse and longitudinal diffusivities of the related materials listed in shows the results of the frequency dependence of fV\({}_{\rm o}\) for some crystals, along with the values of d and \(\tau_{\rm meas}\). F\({}_{2}\)-TIPS-Pn crystals are needles with a brick-layer structure similar to that of TIPS-Pn. TIPGe-Pn and EtTP-5 crystals, on the other hand, grow as thick (d \(>\) 200 \(\mu\)m) flakes (in-plane dimensions \(<\) 3 mm). Molecules in EtTP-5 are coplanar but insulating substituents keep the aromatic faces \(\sim\) 10 A\({}^{\rm o}\) apart along. In TIPGe-Pn, the orientation of the pentacene backbones alternate in the **ab**-plane so that the aromatic surface of each molecule faces insulating substituents of adjacent molecules. In both materials, therefore, there is poor \(\pi\)-orbital overlap in the **ab**-plane, but the silyl and germyl side groups still extend along the interlayer **c**-axis. For all materials, a few crystals were measured with representative results shown. In all cases, the responses are very fast so only lower limits for D\({}_{\rm c}\) (given in the caption) were determined. As for TIPS-Pn, the large diffusivities are unusual for van-der-Waals bonded molecular crystals and suggest strong phonon interactions between the side groups,giving significant dispersion to low-frequency optical phonons which consequently carry most of the heat.
All of these have brick-layer structures similar to that of TIPS-Pn. Because the cyclopropyl group makes the side groups slightly more rigid in CP-DIPS-Pn than in the other crystals, the molecules are more tightly packed; e.g. the unit cell of CP-DIPS-Pn is 6% smaller than that of TIPS-Pn. The measured slopes and calculated longitudinal diffusivities for the materials are given in the caption. As for TIPS-Pn, slopes were measured at a few frequencies for each crystal and the uncertainties given in the table reflect the variations in slopes. The fluorinated crystals may have slightly steeper slopes, and therefore lower thermal diffusivities, than TIPS-Pn, but the differences are less than the uncertainties in the measurement. The slopes for CP-DIPS-Pn had more scatter, probably reflecting the somewhat shorter crystals available, but were always considerably steeper than for the other compounds. The resulting lower diffusivity for CP-DIPS-Pn is surprising because one would expect the more rigid side-groups to reduce scattering of acoustic phonons. It again suggests that much of the heat is carried by molecular vibrations and that the greater rigidity of the side-groups reduces the dispersion of the relevant low-energy modes.
The large thermal diffusivities indicate that, for those of these materials with brick-layer structures and high electronic mobility, Joule heating of micro-electronic components should not pose a problem. While the longitudinal diffusivities are slightly larger than desired for thermoelectric applications, they are not excessive. However, thermoelectric devices would need to be constructed so that the very high transverse diffusivities do not create thermal shunts.
In summary, we have measured the longitudinal (needle-axis) and transverse, interlayer thermal conductivities of crystals of TIPS-Pn by ac-calorimetry. We have found that the longitudinal value is higher than that of rubrene and pentacene and comparable to quasi-one dimensional conductors with excellent \(\pi\)-orbital overlap. The transverse thermal diffusivity is at least an order of magnitude _larger_ than the longitudinal. These values and inverted anisotropy of \(\kappa\) indicate that molecular vibrations, presumably concentrated on the silyl-containing side groups, have sufficient intermolecular interactions and dispersion to carry most of the heat. Similar values for both the in-plane and interlayer thermal diffusivities were found for several other materials with related structures.
This research was supported in part by the National Science Foundation, grant DMR-1262261. The synthesis and crystal engineering of organic semiconductors was supported by the Office of Naval Research, grant N00014-11-1-0328.
| 10.48550/arXiv.1406.1113 | Thermal Diffusivities of Functionalized Pentacene Semiconductors | H. Zhang, Y. Yao, Marcia M. payne, Karl J. Thorley, J. E. Anthony, J. W. Brill | 3,307 |
10.48550_arXiv.0708.2330 | 10.48550/arXiv.0708.2330 | Light dressed-excitons in an incoherent-electron sea: Evidence for Mollow-triplet and Autler-Townes doublet | Jean Berney, Marcia T. Portella-Oberli, Benoit Deveaud | 4,877 |
|
10.48550_arXiv.1606.05760 | ## A. LEED
All of the patterns were collected with the samples in the same position and using the same electron energy. Figures 1(a) and (b) each display bright and sharp 1x1 LEED patterns, indicating that both _in situ_ cleaving and IBA lead to well-ordered crystalline surfaces. The pattern collected following _ex situ_ cleaving was very dim, however, indicating a disordered surface.
In addition to providing the symmetry of the surface unit cell, LEED can also be used to determine the azimuthal orientation of the sample, although the pattern by itself is sometimes insufficient. For example, with Bi\({}_{2}\)Se\({}_{3}\) the orientation cannot be distinguished from the orientation, as the symmetry of the LEED pattern is 6-fold while the symmetry of the crystal surface is 3-fold. Thus, LEIS spectra are used to identify the specific orientation, as explained below.
## B. XPS
Spectra for _in situ_ cleaving were collected from a Bi\({}_{2}\)Se\({}_{3.12}\) sample that was cleaved under vacuum in the load lock and then transferred into the XPS main chamber. The absence of an O 1s signal and the presence of only a single core level component in the Bi 4f and Se 3d XPS spectra demonstrate that the samples following _in situ_ cleaving were without contamination.
_Ex situ_ cleaving was performed using carbon tape for one Bi\({}_{2}\)Se\({}_{3.12}\) sample and two Bi\({}_{2}\)Se\({}_{3.0}\) samples. For each measurement, the three samples were cleaved, exposed to air for 5 min, 1 hour or 1 day, and then inserted into the chamber together. All three samples gave the same results, so only the data for Bi\({}_{2}\)Se\({}_{3.12}\) is shown here. The data collected following _ex situ_ cleaving display oxygen contamination. The Se and Bi core levels show shifted components consistent with Se-O and Bi-O bonds, similar to what was reported in ref., and there is a clearly visible O 1s core level. Note that the C 1s core level is not visible even after air exposure for 1 day, as it overlaps with the Se L\({}_{2}\)M\({}_{23}\)M\({}_{45}\) Auger line making it difficult to discern. The signals for the Se-O and Bi-O components and the O 1s level all increase with air exposure, indicating that the surfaces oxidize gradually over time. The Bi-Bi metallic bonds that were sometimes observed following _ex situ_ cleaving in refs. were not detected. This is not unexpected, however, considering that the probability for an _ex situ_ cleaved sample to be Bi-rich is small and there were only three trials in which XPS spectra were collected here.
## C.
LEIS involves the bombardment of a surface with keV ions and the collection and energy analysis of the projectiles that scatter back from the sample. Ions in this energy range can be modeled with the binary collision approximation (BCA), which assumes that the projectiles make a series of isolated binary collisions with unbound surface atoms located at the lattice sites. This is a reasonable assumption as the scattering cross sections are generally smaller than the spacings between atoms and the bonding energy of the surface atoms is much smaller than the projectile's kinetic energy. If a projectile backscatters after colliding with only a single surface atom, then the energy of the scattered projectile is primarily a function of the projectile/target mass ratio and the scattering angle. Thus, LEIS spectra display a single scattering peak (SSP) for each element on the surface that is directly visible to both the incident ion beam and the detector.
The intensity of the single scattering yield as a function of angle also depends on the particular atomic arrangement in the outermost two or three layers. The yield for a particular SSP depends primarily on the number of such atoms that are visible, but is also affected by focusing that occurs when the projectile experiences a grazing collision with a surface atom, which involves very little energy loss, and subsequently undergoes a hard collision with another atom that leads to backscattering at the SSP energy. Thus, the yield can change greatly as the specific orientation between the incident ions, the single crystal sample and the detector is adjusted. It is the thus angular dependence of the scattering yield that is measured when using LEIS to determine a surface structure, and there are a number of ways in which this can be accomplished.
For example, the orientation can be set in such a way as to make some near-surface atoms visible to the incoming ion beam, while keeping others hidden. An example of this is illustrated schematically in Fig. 3(a), which shows a side view of the nominal bulk terminated Bi\({}_{2}\)Se\({}_{3}\) surface along the azimuthal direction. The figure depicts an incident ion beam directed along the surface normal and the "shadow cones" formed by interaction with atoms in the first three atomic layers. Such shadow cones are constructed by mapping out the possible trajectoriesof the incoming ions. For Bi\({}_{2}\)Se\({}_{3}\), this means that the normally incident beam can only directly impact atoms in the first three atomic layers, as the fourth, fifth and sixth layer atoms are shadowed by the first, second and third layer atoms, respectively. Such a geometry in which the incident beam is aimed along a low index crystal direction is referred to as "single alignment".
A "double alignment" orientation occurs when the detector is also aligned along a low-index crystal direction. As an example, placing the detector along the bond angle between second layer Bi and first layer Se, which is at \(\sim\)33\({}^{\circ}\) from the surface plane in the bulk-terminated material, leads to a geometry in which the second layer atoms block projectiles scattered from the third layer from directly reaching the detector, as illustrated in Fig. 3(a). Similarly, projectiles that impact the second layer are blocked from reaching the detector by the first layer atoms. There are "blocking cones" illustrated in the figure, which are similar to shadow cones but are formed by the possible trajectories of ions that are initially scattered from a deeper layer atom. Thus, in this double alignment orientation, only the outermost atomic layer contributes to the SSP. Note that since the diameter of shadow and blocking cones are on the order of A's for low energy ions, a scattering angle that is a few degrees off from the actual bond angle is sufficient to maintain the double alignment orientation. Ion scattering spectra collected in such a double alignment orientation are thus an ideal tool for determining the composition of the outermost atomic layer, i.e., the surface termination.
A single alignment orientation occurs by
The spectra shown in Figs. 4(b) and (d) were collected by rotating the samples azimuthally 60\({}^{\circ}\) from the double alignment orientation so that the exit direction is now along the projection. The difference between these two orientations is used here to distinguish the and azimuthal directions, as the LEED pattern alone cannot do this. LEIS spectra are collected for each orientation, and the one that shows only a single SSP is identified as the direction.
The feature at 6.3 \(\upmu\)s in is the Se SSP and the feature at 4.8 \(\upmu\)s is the Bi SSP, which is consistent with the expectation that the particles scattered from heavier target atoms will have a larger kinetic energy and thus reach the MCP detector more quickly. The large backgrounds that extend from about 9 \(\upmu\)s to flight times just beyond the Bi SSP's are due to multiple scattering trajectories. This type of background is present in all of the TOF spectra. Note that the actual background increases at longer flight times due to the cascade of multiply scattered particles, but the transmission function of the detector goes down quickly with lower impact energies because of the decreasing MCP efficiency so that the background in the experimental data approaches zero at the longest flight times.
The TOF spectra collected in the double alignment orientation provide the terminations of the surfaces, as only the outermost atomic layer contributes to the SSP. Figure 4(a) shows a spectrum collected from an _in situ_ cleaved surface. This spectrum shows a clear Se SSP riding on the background, and no Bi SSP, which indicates a Se-termination. Figure 4(c) was collected from an IBA prepared surface, and also indicates a Se-termination. The single alignment spectra in Figs. 4(b) and 4(d) show both Se and Bi SSP's, indicating the presence of Bi in the second or perhaps third layer.
_Ex situ_ cleaved samples have a less reproducible behavior, sometimes showing Se-termination and other times being Bi-rich. Spectra collected after _ex situ_ cleaving that display each of these possibilities are shown in The _ex situ_ cleaved samples were exposed to air for less than 5 min before being placed into the load lock chamber. The spectra shown in Figs. 5(a) and (b) are similar to those collected from _in situ_ cleaved and IBA surfaces, suggesting that this sample is Se-terminated with Bi in the second or third layer. Figures 5(c) and (d) show a Bi SSP in both double and single alignment, indicating that there is Bi in the outermost surface layer. The noise level in these spectra is rather large, however, making it difficult to discern a Se SSP. Thus, it's not clear from the LEIS spectra whether the surface is terminated with an ordered Bi layer or if it is covered by islands of Bi metal that obscure much of the Se signal. Because the LEED pattern associated with _ex situ_ cleaved samples is weak, the latter possibility is more likely, considering that Bi is a strong electron scatterer that would be less affected by adsorbates than an ordered Se-termination would be. Thus, this surface is designated as Bi-rich rather than Bi-terminated.
The data in Figs. 5(a) through (d) also show evidence for adsorbed contaminants on the _ex situ_ cleaved surfaces. First, the spectra all have a shoulder-like feature between 2.2 us and 4.5 us, which is highlighted in Fig. 5(a). This shoulder results from direct recoiling of light adsorbed atoms, such as hydrogen, carbon or oxygen. Direct recoiling occurs when an atom is removed from the surface in a collision that converts most of the projectile's kinetic energy to the kinetic energy of the recoiled atom. Direct recoiling is most pronounced for large projectile/target mass ratios, and such fast recoiled atomic particles have sufficient kinetic energy to induce a signal in the MCP. Note that direct recoiling is different than sputtering. In sputtering, the emitted atoms result from a collision cascade and have less than 10 eV of kinetic energy, so they would not be counted by the MCP detector. Additional evidence for the presence of adsorbates is given by the Se SSPs, which are smaller relative to the multiple scattering background than for an _in situ_ or IBA-prepared surface, which is likely caused by surface contamination partially covering the outermost Se atoms. Surface contamination would also explain why the LEED pattern for a sample cleaved _ex situ_ is always very dim. It is also possible that the contamination could be a factor leading to the non-reproducibility of the surface termination following _ex situ_ cleaving, as discussed below.
The _ex situ_ cleaved Se-terminated surface is further studied by subjecting it to a mild annealing in vacuum. shows TOF spectra collected following _in situ_ cleaving, and then after annealing at 130\({}^{\circ}\)C and 290\({}^{\circ}\)C for 30 minutes each. The data show that annealing largely reduces the feature associated with direct recoiling and increases the size of the Se SSP. The LEED pattern after annealing (not shown) becomes as sharp and bright as that following _in situ_ cleaving. These LEIS and LEED results thus suggest that the _ex situ_ cleaved surface is similar to the Se-terminated surface formed by _in situ_ cleaving, but is covered by contaminants that can be removed by a light anneal.
Different stoichiometries of bismuth selenide are used to investigate if excess Bi or Se affects the surface termination. Although all of the TOF spectra shown in Figs. 4 and 5 were collected from Bi\({}_{2}\)Se\({}_{3.12}\), they are indistinguishable from spectra collected from stoichiometric Bi\({}_{2}\)Se\({}_{3}\). Table 1 shows the number of times that each type of surface was produced when using different stoichiometries and preparation methods. As the table shows, _in situ_ cleaved and IBA-prepared surfaces were always Se-terminated for all of the stoichiometries tested. Out of 17 trials of _ex situ_ cleaving, 13 were Se-terminated and 4 were Bi-rich. The probability for obtaining a Bi-rich surface, averaged over all of the stoichiometries, is thus around 22%. Bi\({}_{2}\)Se\({}_{3.12}\) and Bi\({}_{2}\)Se\({}_{3.0}\) have each produced Bi-rich and Se-terminated surfaces following _ex situ_ cleaving. Thus, the termination does not appear be to sensitive to the bulk stoichiometry, as far as this limited numbers of trials can demonstrate.
## C.
ICISS is a variation of LEIS that provides information about the atomic structure of the outermost few layers of a single crystal solid by monitoring the yield of projectiles that are singly scattered at a large angle as the sample is rotated. The basic idea is that the projection of the shadow cones formed by atoms in the outermost layer changes with respect to other surface and near-surface atoms. The flux of projectiles is zero within a shadow cone, but is peaked at the edges of the cone. Thus, the yield of projectiles singly scattered from the surface and near-surface atoms changes in response to this change in flux as the sample rotates. The advantage to the large scattering angle employed for ICISS is that the effects of the shadow and blocking cones can be considered separately and in terms of two-atom pairs within the atomic structure. With a smaller scattering angle, particular trajectories are more likely to be influenced by multiple cones, making the analysis more complex. The ESA is used for ICISS, rather than TOF, as it can be set to the energy of a particular SSP and that signal then monitored as the incident angle is adjusted. In doing so, it is implicitly assumed that neutralization does not affect the shape of the angular yields.
The peak at 0.8 keV is the Se SSP and the peak at 1.8 keV is the Bi SSP. The background is considerably different than in the TOF spectra because the ESA measures only the ion yield and the transmission function does not change with scattered energy as with the TOF detector. This spectrum is shown because it is used to identify the Se and Bi SSP peak positions that are used for ICISS polar angle scans.
The features in these scans can be qualitatively understood by considering how the shadow and blocking cones associated with one atom interact with a second atom in the crystal structure as the sample rotates. The schematic diagram in shows four such two-atom pairs and helps to illustrate the angles at which three shadow cones and one blocking cone act to sharply increase or decrease the SSP yield, assuming a Se-terminated structure. There will be no backscattered yield when the incident beam is just at or barely above the surface plane, as each surface atom is inside the shadow cone of its neighbor.
A feature that results from the shadow cone of a first layer atom passing through a neighboring first layer atom is called a surface flux peak (SFP). The peak at 12\({}^{\circ}\) in the Se SSP angular scan in Fig. 8(b) is the SFP for the Se-terminated structure. Additional features arise in the ICISS scans as the shadow cones around the first layer atoms interact with atoms in deeper layers. For example, the peak at 55\({}^{\circ}\) for the Bi SSP in Fig. 8(a) occurs when the shadow cone of the outermost Se atom interacts with a second layer Bi atom, as illustrated by trajectory 2s in In a similar way, the broad feature at approximately 60\({}^{\circ}\) in Fig. 8(b) has contributions from the enhancement that occurs when the shadow cone of a second layer Bi atom interacts with third layer Se, as illustrated by trajectory 3s. When the scattering angle is less than 180\({}^{\circ}\), blocking can also contribute features to an ICISS scan. The decrease of at 86\({}^{\circ}\) in Fig. 8(b) is due to the blocking by first layer Se atoms of projectiles scattered from the third layer, as illustrated by trajectory 4b in A detailed analysis of ICISS data from Se-terminated surfaces at several azimuths compared with simulations will be published separately.
The peak positions in the ICISS angular scans are strongly related to the crystal structure, so that the good agreement between the polar scans from the IBA-prepared and _in situ_ cleaved surfaces confirms that the near-surface atomic structures are the same. Furthermore, this analysis of the ICISS data indicates that the top three layers are ordered as Se-Bi-Se, which is consistent with an intact QL at the surface.
To get additional confirmation that IBA and _in situ_ cleaved surfaces are both Se-terminated,ICISS simulations were performed using Se-terminated and single layer Bi-terminated models, with the results shown in The simulations are confined within the (\(\overline{1}\)20) plane, as illustrated in Fig. 9, which greatly reduces the computation time. This simplification works well because the distance between (\(\overline{1}\)20) planes, 2.07 A, is larger than the sizes of blocking and shadow cones so that single scattering trajectories along the azimuth in different (\(\overline{1}\)20) planes are completely independent. Both models have four atomic layers with four atoms in each layer, and a periodic boundary condition is applied along the direction. The Bi-terminated model is assumed to consist of Bi-Se-Bi-Se, with the structural parameters all the same as the Se-terminated model. The experimental data shown as a solid line in are the same as shown in Fig. 8(a). The experimental and simulated data both display peaks at around 55\({}^{\circ}\), indicating 2\({}^{\text{nd}}\) or 3\({}^{\text{rd}}\) layer Bi. The simulation for a Bi-termination has a SFP at 9\({}^{\circ}\), however, which is missing in the experimental data. Thus, the simulations strongly rule out the possibility that the IBA-prepared surface is Bi-terminated.
## IV.
The fact that _in situ_ cleaving and IBA both produce a well-ordered Se-terminated surface suggests that this is the stable configuration, in accordance with the expectations for such a layered material. When Bi\({}_{2}\)Se\({}_{3}\) is cleaved _ex situ_, a contaminant-covered Se-terminated surface is sometimes produced, while at other times a contaminant-covered Bi-rich surface results. Thus, it is also reasonable to conclude that cleaving always produces a Se-terminated surface, but changes can occur after cleaving. Note that this was the conclusion of ref., although that paper reported that the resulting surface was Bi-terminated and not simply Bi-rich. Post-cleavage changes to the surface composition and structure for some _ex situ_ cleaved samples could result from a surface chemical reaction with atmospheric contaminants, and such contaminants are likely to preferentially adsorb at surface defects.
Defects in these materials are typically Se vacancies, as Se dimers and tetramers are stable gas phase species that can desorb from the surface. Se vacancies are often present in the as-grown materials and are generally thought to be the reason for the natural n-type doping of Bi\({}_{2}\)Se\({}_{3}\).
The stress applied to the sample during cleaving can also produce defects. The propensity of defects to form during cleaving depends on many factors, such as the crystal quality and the method used to cleave. After a single crystal is grown, a razor blade is commonly used to break the crystal along the natural cleavage plane, and then cleaving with tape or paste is used to produce a flat surface. Cleaving induces strain in the surface, which might then alter the surface atomic structure and the TSS. For example, mechanical strain in TI's was predicted by DFT to affect the electronic properties by changing the \(\Gamma\) point band gap, which was later verified experimentally using the strain existing at the surface grain boundaries of a Bi\({}_{2}\)Se\({}_{3}\) film. Another indirect example of the creation of defects comes from studies of cleaved Si surfaces, which showed that stress-induced microstructures with two types of terraces, triangular terraces and parallel-steps terraces, can coexist on the same surface. Because the process and the force applied on the samples during cleaving are not well controlled, it is not surprising that the density of defects is not reproducible from sample to sample.
Different cleaving methods can also lead to different defect concentrations. Prior to usage, each Bi\({}_{2}\)Se\({}_{3}\) piece is usually cleaved with tape several times in an attempt to produce a flat surface and reduce the number of defects. Reference claims, however, that cleaving with scotch tape can still induce many defects, while using a mechanical cleaver can produce a nearly atomically flat surface. Triangular defects with half height QL step edges induced by scotch tape cleaving were also shown in Ref..
The sample size is also important. If the sample is larger, the center will be relatively less affected by the stress during cleaving and thus be more defect free. The center of larger samples is also better protected from contamination during handling. These considerations may explain why the samples used in Refs. were very flat and inert, as the Bi metal was purified before usage, the samples were more than 1 cm in diameter and they were prepared with a mechanical cleaver. In contrast, most of the experiments with Bi\({}_{2}\)Se\({}_{3}\) reported in the literature used samples smaller than 5 mm and were cleaved with tape.
Chemical reactions of atmospheric contaminants with surface defects are a likely cause of the surface structural changes that occur with some _ex situ_ cleaved samples. For example, in synthesizing the materials for the present study, it was found that Bi\({}_{2}\)Se\({}_{3}\) is very sensitive to oxygen contamination. When samples are cleaved in UHV or in a glove box filled with an inert gas, the newly cleaved surfaces have no contact with atmospheric gases. If a chemical reaction was responsible for the termination change, this would explain why these remainedSe-terminated. The detection of oxygen on _ex situ_ cleaved surfaces, as well as data widely found in the literature, indicates the possibility of a surface reaction with air. In addition, scanning photoelectron microscopy shows that step edges oxidize much faster than the basal planes for Bi\({}_{2}\)Te\({}_{2}\)Se, leading to the conclusion that step edge densities will have a profound effect on the rate of oxidation.
Based on the above analysis, it is proposed that the Bi-rich surface is formed by chemical reaction of adsorbed contaminants with a Se-terminated surface that has a high defect density. Atmospheric gases, such as water or oxygen, can react with the defects to somehow decrease the surface concentration of Se. For example, oxygen might break a Bi-Se bond and release volatile Se to leave a Bi-rich surface.
If there are too many defects, the Bi-rich surface might lead to an actual Bi-termination as found in our previous LEIS study. The as-grown samples used in Ref. were small, on the order of 3 mm in diameter. Smaller samples will have more surface defects and smaller surface basal planes, as mentioned above. This explanation is also supported by the fact that the TOF spectra collected after _in situ_ cleaving of the small samples used in Ref. usually had obvious recoiling shoulders. Although the TOF spectra that were published in Ref. do not have a noticeable recoiling feature, this doesn't mean there was absolutely no contamination. Instead, this might indicate that only a small concentration of contaminants is actually needed to alter the surface structure, which is consistent with the rapid oxidation widely observed in the literature. Thus, even small amounts of contamination from background gases in the UHVchamber, or from the sample holder itself, might react with a high density of surface defects to alter the termination.
For the present experiments, large samples that are around 10 mm in diameter were grown. In addition, the Bi and Se shot were flashed twice with high purity argon before being sealed in the high vacuum ampule to minimize the intrinsic contamination and density of defects. These improvements helped to reduce the contamination level and the number of defects so that the samples used here are much more likely to remain Se-terminated after cleaving.
## V.
_In situ_ cleaved and IBA-prepared surfaces are both well-ordered and terminated as Se-Bi-Se, which is the expected structure assuming that the surfaces cleave along the van der Waals gap to reveal an intact QL. Surfaces cleaved _ex situ_, however, are covered with a submonolayer of contaminants and can be either Se-terminated or Bi-rich. This contamination may be involved in a chemical reaction that is ultimately responsible for the non-reproducibility of the surface termination. It is proposed that defects on the surface would increase the adsorption of contaminants and thus the propensity for a termination change. Samples that were cleaved _ex situ_ can be returned to a Se-termination by an IBA process, whether they had been Se-terminated or Bi-rich. Thus, IBA would be the preferred method for producing a high quality surface for UHV studies. Investigations of the surface defects produced by cleaving and studies of the chemical reactions of atmospheric contaminants with the defects are needed to fully understand the chemistry of _ex situ_ cleaved surfaces, how such reactions can sometimes lead to a Bi-rich surface and how to develop robust methods of sample preparation.
## VI.
The authors would like to thank Dr. Zhiyong Wang and Prof. Ian Fisher for instructing us about the growth of Bi\({}_{2}\)Se\({}_{3}\) single crystals, and Dr. M.A. Karolewski for help with Kalypso. This material is based on work supported by, or in part by, the U.S. Army Research Laboratory and the U.S. Army Research Office under Grant No. 63852-PH-H.
#
The polar angle dependence of the intensities of (a) the Bi SSP and (b) the Se SSP for Bi\({}_{2}\)Se\({}_{3.12}\) surfaces prepared _in situ_ (dashed line) and by IBA (solid line) collected along the azimuth using a scattering angle of 161\({}^{\circ}\). The intensities are adjusted to match at a 90\({}^{\circ}\) polar angle.
A side view of the (\(\overline{1}20\)) plane containing the azimuth. The arrows show four primary trajectories that contribute to the experimental ICISS polar scans along the azimuth for a scattering angle of 161\({}^{\circ}\). The letter \(s\) refers to a shadowing alignment and the \(b\) to a blocking alignment.
The polar angle scan of the Bi SSP intensity along the azimuth using a scattering angle of 161\({}^{\circ}\). The graph shows experimental data (solid line) and simulations using models with Se-termination (short dashed line) and Bi single layer termination (long dashed line). Spectra are adjusted to make the intensity at the polar angle of around 90\({}^{\circ}\) match.
\begin{table}
\begin{tabular}{|c|l|c|c|c|c|} \hline & & **Bi\({}_{2}\)Se\({}_{2.8}\)** & **Bi\({}_{2}\)Se\({}_{3.0}\)** & **Bi\({}_{2}\)Se\({}_{3.12}\)** & **Bi\({}_{2}\)Se\({}_{3.2}\)** \\ \hline \multirow{2}{*}{_In Situ_} & \multirow{2}{*}{Se-termination} & \multirow{2}{*}{0} & \multirow{2}{*}{9} & \multirow{2}{*}{10} & \multirow{2}{*}{2} \\ \cline{1-1} \cline{5-5} & & & & & \\ \cline{1-1} \cline{5-5} & \multirow{2}{*}{Bi-rich} & \multirow{2}{*}{0} & \multirow{2}{*}{1} & \multirow{2}{*}{3} & \multirow{2}{*}{0} \\ \cline{1-1} \cline{5-5} & & & & & \\ \cline{1-1} \cline{5-5} & \multirow{2}{*}{Se-termination} & \multirow{2}{*}{2} & \multirow{2}{*}{7} & \multirow{2}{*}{4} & \multirow{2}{*}{1} \\ \hline \multirow{2}{*}{**IBA**} & \multirow{2}{*}{Se-termination} & \multirow{2}{*}{0} & \multirow{2}{*}{1} & \multirow{2}{*}{5} & \multirow{2}{*}{0} \\ \hline \end{tabular}
\end{table}
Table 1: The number of times that each type of surface termination resulted when using Bi\({}_{2}\)Se\({}_{3}\) with different stoichiometries and employing the various surface preparation methods. | 10.48550/arXiv.1606.05760 | Termination of Single Crystal Bi2Se3 Surfaces Prepared by Various Methods | Weimin Zhou, Haoshan Zhu, Jory A. Yarmoff | 2,876 |
10.48550_arXiv.0804.0378 | ###### Abstract
We have realized and measured a GaAs nanocavity in a slab photonic crystal based on the design by Kuramochi et al. [Appl. Phys.Lett., **88**, 041112,]. We measure a quality factor Q=700,000, which proves that ultra-high Q nanocavities are also feasible in GaAs. We show that, due to larger two-photon absorption (TPA) in GaAs, nonlinearities appear at the microwatt-level and will be more functional in gallium arsenide than in silicon nanocavities. (c)Optical Society of America
The achievement of quality factors of Q \(\approx\) 10\({}^{6}\) in micron-sized nanocavities carved in two-dimensional photonic crystals (PhC) opens perspectives for all-optical signal processing. The unique property of PhC cavities is their ultra small volume (on the order of 0.1 \(\mu\)\(m^{3}\)) which, combined with a high Q factor, dramatically enhances light-matter interaction. This is highly desirable for fundamental investigations in cavity quantum electrodynamics (QED) in condensed matter, where a single emitter, e.g. a quantum dot (QD), is strongly-coupled to a single optical mode. As emission linewidth of a single QD can be as small as a few \(\mu\)eV at low temperature, reaching a comparable linewidth for the cavity resonance is clearly desirable. Besides QED and applications to micro-lasers and non-conventional laser sources such as single photon emitters for quantum key distribution, the strong enhancement of light-matter interaction is fundamental in order to miniaturize a broad class of devices such as optical sensors and optical modulators. Furthermore, high Q-values are compatible with broadband operation when a so-called CROW type waveguide is built from multiple identical cavities. For slow light purposes or for nonlinearity enhancement, the intrinsic Q factor determines the maximum amount of delay/enhancement which can be achieved.The state-of-the-art in high-Q PhC resonators has been achieved by NTT and Kyoto's teams in silicon-based structures. There is a widespread consensus that silicon processing technology is superior to III-V semiconductor and therefore that there is little chance that PhCs based on III-Vs approach the state-of-the-art. On the other hand, III-Vs offer unique opportunities in photonics, in particular emission/amplification of light.
In this letter, we show that GaAs can reach Q values similar to those of silicon, namely Q\(>\)700,000. This result opens new perspectives for realisations combining the features of III-V materials with the attractive properties of PhC, including ultra-low-power nonlinear optics. Microwatt-level nonlinear operation can therefore be envisioned from the \(\approx\) 5\(\mu\)W threshold power value obtained in our previous results () for the lower quality factor Q\(\approx\)246,000. With the linear bandwidth set at \(\approx\)1 GHz by the cavity, nonlinear processing of microwatt optical signal in the 1 MHz - 1 GHz window can be achieved. From the scaling laws of the various effects (Kerr, free carrier plasma, two-photon absorption, thermo-optic), we pinpoint the more favorable capabilities of GaAs in this respect.
Our strategy was to consider the design that ensured the highest theoretical Q factor and which has also been implemented successfully in Si structures. At the time we designed the cavity, this was the design proposed in ref.. Our underlying idea is that this structure must be also the most robust against fabrication tolerances. Three dimensional finite-difference-time domain (FDTD3D) modeling predicted a Q factor of about 100 millions. We used a 186-nm-thick GaAs membrane, and a basic lattice pitch a=420 nm. As shown in Fig.1a, the access waveguide is designed as W1.07 (W1 refers to the single missing row waveguide along the \(\Gamma\)K direction of the photonic crystal) while the width of structure supporting the cavity is _Wx_, with \(x\) variable but \(x<1\). The hole shifts defining the cavity are 9 nm, 6 nm and 3 nm. The waveguide-cavity separation varies between 7 and 9 rows.
We used a compact and efficient 100kV e-beam writer nB3 (NanoBeam Ltd., Cambridge, UK) to define the patterns in the top resist layer, the rest being unchanged. The good results obtained validate the qualities of this tool. Inductively-Coupled Reactive-Ion-Etching was used to perform GaAs/GaInP vertical etching. As for measurements, we used a tunable laser source (Tunics from Nettest) operated in the fine scanning mode by applying an external voltage. The wavelength shift is monitored with a low-finesse Fabry-Perot (FP) interferometer with 28.6-cm-spaced mirrors (free-spectral range is 526 MHz). We have fabricated nine cavities with slightly modified parameters (controlling the coupling strength to the waveguide) and measured 4 of them (n\({}^{\circ}\) 3, 4, 5 and 7). Cavities n\({}^{\circ}\)5 and n\({}^{\circ}\)7 have been designed for maximizing Q with purposely weak coupling (x=0.98 and spacing is 9 rows), the two other cavities (n\({}^{\circ}\)3 and n\({}^{\circ}\)4)have non optimal parameters and were designed with a stronger waveguide to cavity coupling. Measurements have been performed at different power levels, in order to identify the linear regime whereby the Q factor saturates as power is further reduced. Each of these measurements has been repeated 10 times. The uncertainty is deduced from the fitting procedure and also corresponds to the fluctuation across measurements.
The signal detected from the top of the cavity provides a peak at resonance, while the transmitted waveguide signal displays a corresponding dip whose depth is indicative of coupling conditions. A largely sub-critical coupling was observed (the minimum/max. transmission ratio at resonance being \(T_{m}/T_{M}\approx 90\%\)). The value for the loaded Q factor (Lorentzian fit, averaged over several measurements, Fig. 2a) was \(Q\)=700,000\(\pm\)30,000 (\(\Delta\nu\)=280 MHz, \(\Delta\lambda\)=2.1pm). The estimated optical power coupled into the waveguide is here about 10 nW. The fraction of power \(P_{c}/P_{wg}\) actually coupled into the cavity is \(P_{c}/P_{wg}=2(\sqrt{T_{m}/T_{M}}-T_{m}/T_{M})\approx 0.1\), which makes 1 nW. We had to stick to such small values in order to prevent broadening induced by nonlinear absorption. We have also measured Q\(\approx\) 700,000 in cavity n\({}^{\circ}\)7. In spite of the progress made in PhC microcavities, the gap between theoretical expectation and experimental measurements of the Q factor is large, here is almost two order of magnitude. This holds also for silicon. This suggests that the limit of processing capability is close, even for state-of-the-art e-beam systems. The important point of this work is that the handicap of III-V processing with respect to silicon has almost vanished. We are convinced that the residual gap separating us from the world record is rather related to residual structural disorder (e-beam lithography and plasma etching). We think that other factors, such as residual absorption in the material, are still negligible in this spectral range. We also believe that our process (high density plasma) generates a relatively low number of surface defects so that an additional source of losses is avoided.
A Q scaling from our previous work indicates that for such high values of the Q factor, we enter in a nonlinear regime at the microwatt level in terms of power flow in the waveguide. This is confirmed in Fig. 3, where we report the dependence of the lineshape of another cavity than that of Fig.2 as a function of the power coupled in the waveguide. The onset of nonlinearity (here Two Photon Absorption) appears as the power coupled to the waveguide is on the order of 0.3\(\mu\)W. However, as underlined by this work and previous ones, both thermal and electronic nonlinearities are involved: Kerr effect, two photon absorption (TPA) and the resulting index changes due to free-carrier plasma and thermo-optic effect. For thermo-optic effects, at first sight, silicon is a better heat conductor. We show below that this appearance is largely offset by the intrinsic lower operation point of GaAs, giving to this latter material a clear niche for ultra-low power optical manipulations. Let us detail how the nonlinear operation can be performed to further evaluate GaAs vs. Si. Firstly, in transient operation, the thermo-optic effect has a time-dependent spatial extent \(x\), given by the thermal diffusivity \(C_{p}\), obeying the \(x\approx\left(C_{p}\tau\right)^{1/2}\) scaling (where \(C_{p}\) is 0.31 and 0.78 \(m^{2}/s\) for GaAs and Si respectively).
(a) Schematic description of the GaAs side-coupled cavity system, (b) Experimental setup, (c) SEM image of the cavity.
Spectra of the resonance of the cavity n\({}^{\circ}\)5. The power coupled into the waveguide is about 10 nW. a) Detected signal from the camera as a function of the detuning (markers) and Lorentzian fit (solid line). Transmission of the Fabry Pérot interferometer used as reference (dashed line). b) Transmitted signal through the waveguide. The cavity resonance is pointed by the arrow.
This implies the virtual impossibility to implement an extra thermal sink at these high Q/low nonlinear threshold values. Secondly, the TPA damping threshold (at which the induced damping halves Q) is evaluated in terms of power coupled through the cavity \(P_{c}\) as: \(P_{c}^{th}\)\(=\)(\(4\pi^{2}V_{TPA}\))/(\(\lambda^{2}Q^{2}\beta\)) \(\propto\beta^{-1}\) Here the TPA constant is \(\beta=10\) cm/GW for GaAs and the nonlinear effective volume is about 0.4 \(\mu m^{3}\). This amounts to approximately 100 nW for a mean quality factor Q = 7 10\({}^{5}\). This is in quite good agreement with data measured on cavity n\({}^{\circ}\)3 in where the estimated power at which the Q-factor drops is \(P_{WG}^{th}\approx 2\mu W\) in the waveguide and therefore much lower in the cavity, due to the weak coupling. We have also measured the power for cavity n\({}^{\circ}\)5 and found that Q drops to 500,000 as power in the waveguide is raised to 1 \(\mu W\), i.e. 100 nW in the cavity. Thirdly, the impact of thermo-optic effects derives from the index shift \(\Delta\)n which is proportional to the thermo-optical coefficient \(n_{T}\), the thermal resistance and the amount of power absorbed, which turns out to be governed by the ratio \(n_{T}/(\beta\,C_{p})\). This is a key point: obviously, a stronger TPA coefficient and a lower power threshold weakens the thermal burden. Of particular interest here is the fact that the \(\approx\)4 times weaker \(n_{T}/C_{p}\) ratio of silicon is well offset by the \(>\)10 times larger TPA coefficient \(\beta\) of GaAs.
Fourthly, the carriers generated by TPA induce a negative index shift counteracting the Kerr effect. This is governed by the carrier recombination time \(\tau_{rec}\), which can be considered to be fast (\(\tau_{rec}\) is in 10-100 ps time scale). The amount of energy stored in the cavity \(W\) at which the plasma-induced index change takes over the Kerr effect is \(W_{th}\approx n_{2}m^{*}/(\beta\tau_{rec})\), with \(m^{*}\) the effective electron mass and \(n_{2}\) the Kerr coefficient. With the values 1.6 and 0.45 \(cm^{2}/GW\) for \(n_{2}\) in GaAs and Si respectively, the crossover energy ratio is: \(W_{th,Si}/W_{th,GaAs}\approx\)15\(\times\tau_{rec,GaAs}/\tau_{rec,Si}\). Moreover, the net ratio is almost entirely compensated when considering typical values for \(\tau_{rec}\): 100 ps for Si and 10 ps for GaAs, resulting in similar crossover powers. Usually, Kerr effect is sought for optical manipulation and TPA seen as an hindrance. Taking an opposite approach, i.e. exploiting nonlinear cavity damping, it becomes advantageous to use GaAs : as it operates at a much lower power, it features less index shift from the Kerr effect than silicon.
In conclusion, we showed that an ultra-high Q nanocavity akin to those elaborated in silicon is also feasible in GaAs, making GaAs devices with Q\(\approx 10^{6}\) fully plausible. Additionally, we stressed the possibility to operate at the microwatt level for nonlinear operation, through nonlinear damping based on two-photon-absorption. Importantly, we substantiated the fact that thermal effects inflict less severe penalties when operating nanocavities based on GaAs as compared to those based on Si.
We acknowledge the support of the SESAME action of Conseil General Ile de France for key equipments used in this work. The TRT staff ackowledges the financial support of the European Commission through the IST Project "QPphoton".
| 10.48550/arXiv.0804.0378 | GaAs photonic crystal cavity with ultra-high Q: microwatt nonlinearity at 1.55 $μ$m | Sylvain Combrie', Alfredo De Rossi, Quynh Vy Tran, Henri Benisty | 4,391 |
10.48550_arXiv.2110.00321 | ## 1 Introduction
The crystal structure of hybrid halide-perovskites is a topic of study that has surfaced several times in the last four decades. X-ray powder diffraction experiments of Weber _et.al._ on Methylammonium(MA)-Pb\(X_{3}\) with halogens \(X=\) {I, Br, Cl}, have established a high temperature cubic phase for all \(X\). A perovskite structure is formed by Pb\(X_{6}\) corner-sharing octahedra enclosing the MA molecules. In later years, the low temperature phases and librational modes of the MA molecule at various temperatures were studied. The development of high efficiency solar cells based on these perovskites sparked a rival of interest in its structure characterization. It was shown by high-quality powder neutron-diffraction experiments that the low-temperature orthorhombic phase of MAPbI\({}_{3}\) actually belongs to the \(Pnma\) space-group. This is the space group that was also determined for MAPbBr\({}_{3}\) and MAPbCl\({}_{3}\). The potential for opto-electronic applications raised new questions that are all (in)directly related to the atomic structure: the effect of MA rotation on charge dynamics, dynamic or permanent deformations of the Pb\(X_{6}\) octahedra, the extent of electron-phonon coupling and the Rashba effect, to name a few. Considerable progress has been made, but consensus has not always been achieved. This is in part the result of differences in interpretation of the local microscopic structure. The disorder, be it static or dynamic, of the molecular C-N axes is a well-known problem for diffraction techniques, that makes it difficult to determine their precise orientation. First-principles (FP) methods such as density functional theory (DFT) have shown to be very useful for the determination of crystal structure by augmenting the experimentally resolved inorganic framework with the ordering of the molecules. However, even though commonly used density functional approximations have the required chemical accuracy, their computational complexity prohibits the large length & time scale molecular dynamics (MD) calculations necessary to resolve the free energy landscape and thereby the finite temperature crystal structure. We will use the _on-the-fly_ Machine-Learning Force Field (MLFF) method, which makes it possible to explore the full diversity of atomic structures while going through the entropy-driven phase transformations in hybrid perovskites. This method substantially reduces the computational cost while retaining near-FP accuracy. Recently, it has been shown to be capable to resolve the orthorhombic-tetragonal (Ort-Tet) and tetragonal-cubic (Tet-Cub) phase transitions in MAPbI\({}_{3}\) and the inorganic halide perovskites CsPb\(X_{3}\) in good agreement with experiment. Furthermore, it can be systematically extended to describe mixed MA\({}_{x}\)FA\({}_{1-x}\)PbI\({}_{3}\) perovskites under isothermal-isobaric conditions.
The starting point of our search for the low-temperature (\(\sim\)100 K) orthorhombic structure of MAPb\(X_{3}\) are two seemingly similar, but distinctly different structures: sA and sB. They have the same lattice vectors and inorganic coordinates, but a different molecular ordering pattern as sketched in We have labeled the lattice vectors such that the molecules lie in the \(ab\)-plane. sB is created out of sA by an in-plane rotation of half of the molecules by 180\({}^{\circ}\) as indicated by the curved arrows. Note that in both arrangements the neighboring molecules in the \(c\)-direction (not shown in the figure) are anti-parallel. These structures have been prepared in a \(2\times 2\times 2\) supercell, such that it accommodates the \((\sqrt{2}a_{\rm p},\sqrt{2}a_{\rm p},2a_{\rm p})\) basis of \(X\) = I, Br as well as the \((2a_{\rm p},2a_{\rm p},2a_{\rm p})\) basis of \(X\) = Cl, where \(a_{\rm p}\) is the pseudo-cubic lattice constant of the parent \(Pm\overline{3}m\) cell. The molecules in the often referenced experimental (_Pnma_) structures for \(X\) = I, Br and Cl of Refs and are arranged as in the sB, sA and sA configuration, respectively. Other experimental works did not distinguish between these two arrangements, since refinement of the model structure by permuting N and C with respect to the measured diffraction spectra does not lead to significant improvements of the fit. To date even for the extensively studied MAPbI\({}_{3}\) perovskite, different studies report opposite arrangements: sA and sB.
Initial low-temperature crystal structures of MAPb\(X_{3}\), with molecules in the sA and sB arrangement. Lattice vectors and Pb,\(X\) atom coordinates are adapted from Refs. and. Only one of the two layers of molecules in the \(ab\)-plane is shown. Molecules in the second layer (into the paper) are anti-parallel to the first. sB is created by rotating half of the molecules of sA by 180\({}^{\circ}\) in the ab-plane. The distortions of the PbCl\({}_{6}\) octahedra are indicated in the top right figure by the long (green) and short (orange) Pb-Cl bonds.
This pattern shows a closer resemblance to the 'head-tails' groundstate of a point-dipole model.
In this work, we will analyze the '_ibrational pathways_' of the MA molecules and Pb\(X_{6}\) octahedra in MAPb\(X_{3}\), and use them to identify the most representative low-temperature orthorhombic structure. We sample structures of the crystal by slowly heating up two plausible low-temperature structures (sA and sB) in isothermal-isobaric MD simulations. Structures on the explored pathways through the structural phase space are thermodynamically linked to the starting configuration and result in marked differences in lattice and order parameters that are compared to temperature dependent diffraction studies.
## 2 Computational Details
The DFT calculations are performed with the projector augmented wave method as implemented in the VASP code using the meta-gradient corrected SCAN density functional approximation (DFA), which has shown good performance when compared to high quality many-body perturbation theory reference calculations. A plane-wave basis with a cutoff of 350 eV, Gaussian smearing with a width of 10 meV and 4 (I,Br) or 8 (Cl) k-points of the \(\Gamma\)-centered 2\(\times\)2\(\times\)2 Monkhorst-Pack grid are set, which suffice to obtain the required accuracy of the calculations. The computed lattice parameters as function of temperature should (qualitatively) agree with experiment over the whole temperature range. Therefore, by not limiting the study to a 0 K DFT based relaxation of the internal energy, biases related to the chosen DFA can be detected. Before starting the MD simulations the starting structures of were shortly relaxed by a conjugate gradient algorithm.
MLFFs are trained during MD simulations with VASP, based on calculated total energies, forces and stress tensors for automatically (on-the-fly) selected structures in the isothermal-isobaric ensemble. This approach is described in detail in Refs.. In short, a Bayesian error estimation of the predicted forces is used to select either DFT or MLFF forces to propagate the structure in time (\(t_{n}\to t_{n+1}\)). Whenever the predicted errors exceed the threshold, a new reference structure is picked up, a DFT calculation is performed and the coefficients of the MLFF are re-optimized. In Figs. 3(c,f) a 'density-of-states' like function of the temperature (note, equivalent to simulation time) shows when in the training MD most DFT calculations were performed. It is calculated by: \(\rho_{\rm FP}(T)=\sum_{i=0}^{\rm N_{\rm ref}}\delta_{i}(T-T_{i})\), where \(\delta(T)\) is a Lorentzian function. This function is normalized to the total number of DFT reference structures picked up in training, \(\rm N_{\rm ref}=\int\rho_{\rm FP}(T)dT\). The automatically picked up reference structures form a minimal training database (containing total energies, forces, stress tensors and atomic coordinates) that is well spread over the available structural phase space. We have shared this database via the 4TU.DataBase repository to encourage development of ML potentials based on minimalistic datasets.
A variant of the GAP-SOAP method is used as a descriptor of the local atomic configuration around each atom. Within a cutoff of 7 A a two-body radial probability distribution \(\rho_{i}^{}(r)\) is build, as well as three-body angular distribution \(\rho_{i}^{}(r,s,\theta)\) within a cutoff of 4 A. The atomic coordinates are smeared in the distributions by placing Gaussians with a width of 0.5 A. The obtained distributions are projected on a finite basis set of spherical Bessel functions multiplied with spherical harmonics. The bessel functions are of the order 6 and 7 for the radial and angular part, respectively. Only the angular part has a maximal angular momentum of \(l_{max}=6\). The coefficients of the projections are gathered in the descriptor vector \({\bf X}_{i}\). A kernel-based regression method is applied to map the descriptor to a local atomic energy. The similarity between two local configurations is calculated by a polynomial kernel function: \(K({\bf X}_{i},{\bf X}_{i_{B}})=\nicefrac{{1}}{{2}}({\bf X}_{i}^{}\cdot{\bf X }_{i_{B}}^{})+\nicefrac{{1}}{{2}}({\bf X}_{i}^{}\cdot{\bf X}_{i_{B}}^{(3 )})^{4}\).
## 2 Results & Discussion
To introduce the librational pathways of MAPb\(X_{3}\) we will illustrate them by weighted sums of pair distribution functions (PDFs) in The PDF for the atom types \(\alpha\) and \(\beta\) is defined as
\[g_{\alpha,\beta}(r)=\frac{1}{4\pi r^{2}dr}\frac{\rm V}{\rm N_{\alpha}N_{\beta}} \left\langle\sum_{i\in\alpha}\sum_{\begin{subarray}{c}j\in\beta\\ (i\neq j)\end{subarray}}\delta(|{\bf r}-{\bf r}_{ij}|)\right\rangle,\]
In \(g_{\rm inorganic}(r)\), the pairs of framework components Pb-\(X\),\(X\)-\(X\),Pb-Pb are included, and only the C-N pairs of the MA molecules are included in \(g_{\rm C,N}(r)\). For all halides \(X\), we see that \(g_{\rm inorganic}(r)\) retains most of its structure throughout the whole temperature range, and that \(g_{\rm C,N}(r)\) shows a transition whereby part of the order is lost. The intra-molecular part (\(r\approx 1.5\) A) remains intact, but the inter-molecular pairs show dual peaks merging into a single broad peak centered around the nearest-neighbor distances of the consecutive cubic unit cells, ie. \(\sqrt{1}a_{p},\sqrt{2}a_{p},\sqrt{3}a_{p}\). This is the result of the unfreezing of the molecules whereby they reorient (\(C_{4}\)) rapidly. Around room temperature, neighboring MA molecules are still dynamically correlated to their neighbors. The differences in the DSF between the halides \(X\) are, among other things, related to different Pb-\(X\) bond lengths and to the relative orientation of the molecules in the low temperature phase. As we will show hereafter, the thermodynamically stable molecular configurations at low temperature (sA or sB) depends on the halide type \(X\).
On-the-fly heating MD at \(\frac{2}{3}\frac{\rm K}{\rm ps}\) of MAPbBr\({}_{3}\) starting from the sA and sB arrangement. The \(a,b\) (a,d) and \(c\) (b,e) lattice parameters (blue/red/green) of the orthorhombic system and their running averages (thick lines, window size 5 K). Experimental reference data from Ref.: lattice parameters (symbols) and transition temperatures (thick vertical lines). (c,f) The density \(\rho_{\rm FP}(T)\) of performed DFT calculations as function of temperature (black line).
Combined pair distribution functions of the inorganic (Pb,\(X\)) components compared to \(g_{\rm C,N}(r)\) in the MAPb\(X_{3}\) (\(X\)= I, Br and Cl) perovskites for increasing temperatures with steps of 20 K. The crystals librate from the initial structures I-sB, Br-sA and Cl-sA. Vertical dashed lines indicate typical bond lengths (left) and neighbor distances based on the lattice constant \(a\) in the cubic phase (right).
The starting structures are slowly heated at a rate of \(\frac{2}{3}\frac{\mathrm{K}}{\mathrm{ps}}\) using DFT based MD with a Langevin thermostat and a time step of 2 fs. The PDFs at different temperatures have been obtained by partitioning the resulting trajectory in parts of equal length. In the \(NpT\) ensemble all lattice degrees of freedom are allowed to change as shown for MAPbBr\({}_{3}\) in The two plots correspond to the heating trajectories starting from the sA and sB structures. Thermal fluctuations in structural parameters are smoothed by applying running averages. To accelerate the MD a MLFF is trained on-the-fly as described in Refs.. The algorithm switches between MLFF and DFT forces based on the predicted error of the MLFF. Structural reference configurations to train the MLFF are automatically picked up and by construction lie outside the already 'learned' part of the phase space. This can be seen by the sharp increase in the density of first-principles calculations (\(\rho_{\mathrm{FP}}(T)\)) shown in Figs. 3(c,f). The on-the-fly algorithm decides to do a large number of DFT calculations in the region between 150 and 170 K, ie. when the system undergoes the Ort-Tet phase transition. The total number of DFT reference structures picked up in training, N\({}_{\mathrm{ref}}\) = \(\int_{80}^{280}\rho_{\mathrm{FP}}(T)dT\), is N\({}_{\mathrm{ref}}^{\mathrm{sA}}\) = 933 and N\({}_{\mathrm{ref}}^{\mathrm{sB}}\) = 1074. This transition temperature is expected to be retarded, because the system is out of thermal equilibrium as a result of the still considerable heating rate. Even though, the agreement with the exp. lattice parameters shown by the symbols in is remarkable. We show that the structural transformation and the related librational pathways of the molecules and octahedra (see Supplementary Movies) are accurately described in the on-the-fly heating MD. This opens up the possibility to explore many different perovskites, because only \(\sim\)1.000 out of the total of 150.000 MD steps per heating run were DFT calculations. This reduces the compute time from years to days.
### Librational pathway analysis
This means that the lattice parameters alone provides insufficient information to select either the sA or the sB as the thermodynamically stable low-temperature structure. Therefore, we analyze the structural motif presented by the atomic coordinates in the six heating trajectories. First, we extract solely the orientation of the molecular C-N axes in time. The total electrostatic energy (\(H_{\mathrm{lr}}\)) corresponding to the dipole moments of the molecules is calculated. In short, this point-dipole model assumes a fixed dipole moment on all molecules, no screening and includes all dipole-dipole interactions up to the third nearest-neighbor. In Figure 4(a) \(H_{\mathrm{lr}}\) is plotted as function of temperature for the three halides starting from the sA (black lines) and sB (red lines) configurations. The sB configuration is clearly lower in energy. With increasing temperature the molecules flip/re-order and the stable arrangement is broken down, eventually leading to a disordered state with \(H_{\mathrm{lr}}=0\).
Energies during the heating trajectories of the MAPb\(X_{3}\) perovskites. (a) Electrostatic energy of the molecular dipoles in the point-dipole approximation (\(H_{\mathrm{lr}}\)) and (b) DFT internal energy differences (\(\Delta E\)). (running averages over 25 K)
However, for \(X\) = Br and Cl the pattern remains largely frozen-in until the phase transition. The sA configuration can only be stable at low-T if either a potential energy contribution arising from the inorganic framework or the entropy compensates this internal energy difference.
The DFT/MLFF calculated internal energy (\(E\)) is shown in Figure 4(b) as the energy difference (\(\Delta E=E_{\mathrm{sA}}-E_{\mathrm{sB}}\)) between the heating trajectory starting with the sA and the sB structure. At low temperature and ambient pressure, the volume and entropy contributions to the Gibbs free energy are small, and a sizable positive/negative \(\Delta E\) would indicate that the sB/sA configuration is favored, respectively. For MAPbCl\({}_{3}\) it is clear that the sA structure is favored even though its electrostatic energy in the dipole model was unfavored. Above \(\sim 175\) K the difference between the two initial configurations has been lifted by thermally induced structural rearrangements. For MAPbI\({}_{3}\) and MAPbBr\({}_{3}\) the situation is less clear, at low temperature \(\Delta E\) is positive, however it is of the size of the fluctuations. Even for the fully DFT relaxed (0 K) structures \(\Delta E\) values are small: 33, 3 and \(-72\) meV/formula-unit for I, Br and Cl, respectively. Increasing the precision of the calculation, by doubling the k-point grid density and applying the Tetrahedron method results in 33, \(-6\) and \(-75\) meV/f.u., respectively. This indicates that, especially for MAPbBr\({}_{3}\), we cannot distinguish sA from sB based on the internal energy alone.
The changes of the structural motif as a function of the temperature are compared in and in the Supplementary Movies. Order parameters describing the inter-octahedral (**O**) and inter-molecular (**M**) order are shown for all heating trajectories in Fig. 5(a). The **M/O** order parameters are based on the dot products of \(X\)-Pb-\(X\)/C-N connection vectors located on nearest neighboring sites. A detailed description can be found in Refs.. Looking at **M** for I-sA, we see that the molecular ordering pattern rapidly changes starting from \(\mathbf{M}_{120\text{ K}}=(0.9,0.7,0.7)\) and transforming to the same order observed for I-sB \(\mathbf{M}_{160\text{ K}}=(0.9,0.3,0.3)\), in agreement with the previously seen change in \(H_{\mathrm{lr}}\). At the same time the inter-octahedral order parameter for the inorganic framework, **O**, changes only little.
Heating trajectories of MAPb\(X_{3}\) starting in the sA and sB configuration. The (a) structural order parameters for the octahedra (**O**) and molecules (**M**). The red line shows the classified perovskite crystal based on **O**. (b) The pseudo-cubic lattice constants refined in the classified phase. (running averages over 25 K) The solid circles represent (a) the **O** values for the experimental low-temperature structures and (b) the experimental lattice parameters. Note that the exp. Br **O** lies outside the scale of the graph (a) at 11 K.
This shows that the sA and sB molecular order are more energetically competitive in orthorhombic \(X\) = Br than in I, and that an additional measure is required to determine the stable arrangement. The \(\bf O\) values for the experimental low temperature structures indicated by the circles in Fig. 5(a) provide this measure. Based on their comparison to the simulation the Br-sA is the stable low-temperature orthorhombic structure.
To automatically classify the instantaneous crystallographic phase of all structures within the MD trajectory, we applied a new approach based on the \(\bf O\) order parameter. This allows, for example, to still assign the orthorhombic phase to a structure that is strained in a box with \(a\approx b\approx c\). Whenever the variance of the components of \(\bf O\) is below a threshold it is classified as Cub, and otherwise it is Ort or Tet. We can then differentiate between the last two by counting the number (1\(\rightarrow\) Ort, 2\(\rightarrow\) Tet) of components larger than their mean value. The red line in Fig. 5(a) shows the result of this phase classification. Note that no Tet phase in MAPbCl\({}_{3}\) is recognized. This could be caused by a very small temperature window in which the Tet phase is stable, or that the \(c/a\) ratio is too small to be noticed in the supercell.
Lattice parameters as function of temperature are shown in Figure 5(b). The parameters are refined in the unit cell corresponding to the classified phase and converted to pseudo-cubic lattice parameters (\(a_{p}\)). Experimentally obtained parameters are shown by the circles. Surprisingly, the plot for MAPbBr\({}_{3}\) and MAPbCl\({}_{3}\) show good to very good agreement with experiment. Especially for MAPbI\({}_{3}\) we notice the effect of our limited MD setup, suppressing the Tet phase on the \(T\)-axes. From our previous study we know that this can be improved with a lower heating rate and by applying a larger \(4\times 4\times 4\) supercell. Still, it is noteworthy that, under the same computational settings, the Ort-Tet phase transition becomes more retarded going from Cl, Br to I.
Combining all the above presented analysis leads to the favored initial configurations, I: sB, Br: sA, Cl: sA.
### Training system size dependence
The PDFs in are plotted beyond half of the simulation box width, since the \(2\times 2\times 2\) supercell has an average width of \(2a_{\rm p}\). They are computed in \(4\times 4\times 4\) supercells, which are created by replicating the original \(2\times 2\times 2\) cell, and enables us to sample the PDF on the \([0,2a_{\rm p}]\) domain. These results are compared to a training run performed with a \(4\times 4\times 4\) supercell. Specifically, we have made a test for MAPbBr\({}_{3}\) starting in the sA configuration, and applied the same on-the-fly training directly on the \(4\times 4\times 4\) supercell. In this supercell, the k-points of the k-grid (applied with the \(2\times 2\times 2\) supercell) all fold-down on the Gamma point. For computational tractability we slightly lowered the plane-wave cut-off from 350 to 300 eV and retrained the \(2\times 2\times 2\) cell with the same cut-off for a fair comparison. The smaller plane-wave basis results in higher Pulay stress, which slightly affects the volume, but does not qualitatively change the crystal structure.
shows \(g_{\rm inorganic}(r)\) and \(g_{\rm C,N}(r)\) for the
Influence of supercell dimension. Pair distribution functions of the inorganic (Pb,Br) components (top) and the C,N pairs (bottom) in MAPbBr\({}_{3}\) obtained during on-the-fly heating MD in a \(2\times 2\times 2\) and \(4\times 4\times 4\) supercell. Increasing temperatures shown with steps of 40 K.
The \(g_{\rm inorganic}(r)\) of the two systems are almost identical. The main deviations in \(g_{\rm C,N}(r)\) are the result of a different temperature at which the system switches from the Ort to Tet phase. This is to be expected for simulations with finite system size. The MD trajectory is chaotic and the transition does not occur at the same temperature even when the initial conditions are the same. This very good agreement indicates that the applied \(2\times 2\times 2\) supercell is large enough to capture the crystal symmetry. This is in agreement with a fully _ab-initio_ MD study of the system size dependence of MAPbI\({}_{3}\) going up to the \(6\times 6\times 6\) supercell.
The accuracy of the MLFF model over the entire collected dataset of structures, which includes three different perovskite phases is very high. The DFT reference energy (U\({}_{\rm DFT}\)) and predicted MLFF energy (U\({}_{\rm MLFF}\)) for the test systems are plotted in Figure 7, and show a clear linear relation over a large energy range. Note that both point clouds nicely overlap, whereby the larger variations are, as would be expected, observed for the smaller supercell. The overall root-mean-square (rms) error on the energy is only 1.7 and 0.88 meV/atom for the MLFF trained on the \(2\times 2\times 2\) and \(4\times 4\times 4\) supercells, respectively. This is small and of the same order of magnitude of state-of-the-art ML potentials (kernel-regression, neural networks, etc.). Furthermore, for the two system sizes the errors in the force are 0.081 and 0.077 eV/A and in the stress 1.1 and 0.46 kB. These error estimates are typical for all MLFFs presented in this work. For instance, the rms errors in the energy of the six MLFFs (\(X\)=1,Br,Cl/sA or sB) are all in the 1.2-1.8 meV/atom range.
### Octahedra distortions: dynamic or permanent?
The central question remains, why is the sA configuration more stable in MAPbCl\({}_{3}\) and to a less extent in MAPbBr\({}_{3}\) as compared to sB? Chi._et.al._ have already shown that the PbCl\({}_{6}\) octahedra are distorted. This _polar_ distortion is highlighted in by the green and orange lines indicating the difference between two Pb-Cl bond lengths (3.02 and 2.73 A) in the crystallographic \(a\)-direction, and is in good agreement with our simulations. However, we find no noticeable distortion of the PbI\({}_{6}\) and PbBr\({}_{6}\) octahedra above 80 K, as shown in Again, in agreement with experiments of Refs., however, opposite to the findings of Refs.
(a) Distributions of Pb-\(X\) bond lengths as function of temperature. (b) Mean values (\(\mu\)) and the standard deviations (\(\pm\sigma\) errorbars) as function of temperature for the Pb-Cl bond lengths. The experimental values from Refs. and are shown by the circles and squares, respectively.
DFT internal energies versus predicted MLFF energies for the structures in the MAPbBr\({}_{3}\) training-set obtained during on-the-fly heating MD in a \(2\times 2\times 2\) and \(4\times 4\times 4\) supercell.
In Fig. 8(a) the distribution of the Pb-\(X\) bond lengths as function of temperature are shown. The heating trajectories were cut in parts of equal length and all bond lengths in the \(a\)-direction were added to the distribution. The low-temperature distribution for Pb-Cl has two peaks. This distortion is not observed when starting from the Cl-sB structure, nor when starting from the Br-sB or I-sA structures. The distribution is well described by a combination of two Gaussian distribution functions \(\mathcal{N}(\mu_{1},\sigma_{1})+\mathcal{N}(\mu_{2},\sigma_{2})\). For Cl-sA, the mean values (\(\mu\)) and the standard deviations (\(\sigma\)) as function of temperature are shown in Fig. 8(b). These values have been obtained from a separate heating trajectory with the finished MLFF on a \(4\times 4\times 4\) MAPbCl\({}_{3}\) supercell. The so obtained distributions agree with Fig. 8(a) and improve statistical accuracy. Experimentally determined bond lengths from Refs. and have been added to Fig. 8(b) and agree within the standard deviation. In the simulations a single Gaussian suffices above \(\sim\)175 K, ie. \(|\mu_{1}-\mu_{2}|<(\sigma_{1}+\sigma_{2})\). At this temperature, the octahedral polar distortion is no longer observed on _time average_ and the crystal is in the cubic phase. As for I and Br, _instantaneous_ distortions of the octahedra do occur at these elevated temperatures.
The following scenario now becomes plausible, the sA ordered molecules are stabilized in the MAPbCl\({}_{3}\) orthorhombic phase by an anti-ferroelectric stripes pattern of dipolar octahedra in the \(a\)-direction. As argued in Refs. the volume of the perovskite has to be sufficiently small to induce these distortions, whereby the 'hard' MA deforms the'soft' octahedra. However, the stripes pattern cannot be the only stabilization mechanism, because no distortion is observed in Br-sA.
We would like to note that our 'ensemble average' view on the structural model results in PDFs for MAPbBr\({}_{3}\) that qualitatively agree with those obtained from X-ray diffraction experiments of Ref.. In Figure 9(a) \(g_{\rm inorganic}(r)\) has been plotted on the same length scale as in Ref., whereby the vertical dashed lines indicate the experimental peak positions. Starting from sA, we also do not observe any relevant structural change in the 150-280 K temperature range apart from thermal broadening of the peaks. However, our approach classifies tetragonal and cubic structures within this range, and does not indicate that an orthorhombic structure would be a better fit throughout this temperature range. Starting from sB results in structural changes (indicated by *) in disagreement with the PCF of Ref.. This is another indication that the sB pattern is not the stable low temperature structure. The different structural interpretation of the crystal at elevated temperatures becomes apparent in the H-\(X\) bonding as shown by \(g_{\rm H,X}(r)\) in Fig. 9(b).
Pair distribution functions of the (a) inorganic components in MAPbBr\({}_{3}\) and (b) \(g_{X,\rm H}(r)\) for increasing temperatures with steps of 20 K. Vertical dashed lines in (a) show the peak positions in the X-ray PDF of Ref. and (b) the first peaks in the calculated \(g_{X,\rm H}(r)\).
This finding is different from the 2.5 A peak observed in the neutron PDFs of both MAPbBr\({}_{3}\) and MAPbCl\({}_{3}\).
### Ma \(C_{3}\) dynamics
We would like to note that training of a very accurate MLFF for the hybrid perovskites is not fully completed by the here performed single heating run. Precise values for phase transition temperatures, \(c/a\) ratios, etc., were not the aim of this work, but can be obtained with an accurate MLFF which enables long MD trajectories on large supercells. Limits to the accuracy can be seen in the \(C_{3}\)_torsion/rotation_ degree of freedom of the molecule, for example. shows the NH\({}_{3}\) versus the CH\({}_{3}\) group dihedral angle (\(\phi_{\rm MA}\)) of all MAs in MAPbBr\({}_{3}\) in \(NPT\) ensembles with the finished MLFF. _Torsion_ (\(\phi_{\rm MA}<60^{\rm o}\)) unfreezes at \(\sim 25\) K and also _rotations_ (\(\phi_{\rm MA}>60^{\rm o}\)) occur around our starting temperature (80 K) of the on-the-fly training.
(b) shows that \(\phi_{\rm MA,i}(t)\) of a single molecule shows step-like behavior, occasionally jumping between planes separated by 120\({}^{\rm o}\), and is superimposed by a fast oscillation. For each of the eight molecules in the supercell a probability distribution of \(\phi_{\rm MA}\) as in Fig. 10(c) was made. We then calculate the \(C_{3}\) rotational energy barrier in the two MLFFs of MAPbBr\({}_{3}\) (sA and sB) by a Boltzmann inversion of the distribution. A barrier was only assigned at a temperature \(T\) when the number of 120\({}^{\rm o}\) rotations in the \(\sim 100\) ps long MD trajectories exceeded the number of molecules in the supercell. The error bars in Fig. 10(d) correspond to \(\pm\sigma\), the standard deviation of the eight obtained barriers. The barrier is, within our statistical accuracy, temperature independent and, surprisingly, different between sA and sB. It is tempting to conclude that the sA structure for Br affords a more facile \(C_{3}\) rotational degree of freedom compared to the sB structure within the orthorhombic phase. However, the difference should not persist in the high temperature cubic phase, in which the MAs are orientationally disordered. This should be a warning that training is not yet completed and the MLFF still shows a bias depending on the initial conditions.
We are able to measure the barriers of a single molecule in vacuum in the same manner. The absence a surrounding Pb-Br framework does not destabilize the molecule. Barriers obtained in this way are slightly lower than the DFT value of 105 meV for MA in vacuum. This value was calculated as the difference in internal energies of the optimized \(\phi_{\rm MA}=0^{\rm o}\) and \(60^{\rm o}\) molecule, for which all internal degrees of freedom were relaxed under the constraint of \(\phi_{\rm MA}\). Using these two structures the barriers for the Br-sA and Br-sB MLFFs are 78 and 89 meV, respectively.
Since both Br-sA and Br-sB show \(C_{3}\) rotations at temperatures below the Ort-Tet phase transition temperature, we can conclude that the unfreezing of this motion does not drive it. This is in agreement with the findings of the computational study of Kieslich _et.al._.
3-fold dynamics of the NH\({}_{3}\) versus the CH\({}_{3}\) group. (a) Sketch of the angle \(\phi_{\rm MA}\) in the MA molecule. (b) \(\phi_{\rm MA,i}\) as function of time for each of the eight molecules at 270 K. (c) Dihedral potential obtained by a Boltzmann inversion of the distribution (P) at 270 K. (d) Barrier heights as function of temperature.
This, however, does not preclude the possibility that entropy related to \(C_{3}\) dynamics is involved in this phase transition. Based on the observed initial condition dependence (sA or sB) of the barrier, we _speculate_ that Br-sA is entropically stabilized by the \(C_{3}\) torsion/rotation degree of freedom of the molecule, and thereby determines the orthorhombic structure below the phase transition temperature.
## 4 Conclusions
In conclusion, we have shown that _on-the-fly_ machine-learning force fields are a very powerful tool in determining atomic structure in dynamic, entropically stabilized solids. Already during the training-by-heating MD important structural characteristics are qualitatively correct and even quantitatively useful. As a prime example, the low-temperature ordering pattern of MA molecules in MAPb\(X_{3}\) perovskites, which is not uniquely resolved by diffraction experiments, was studied. We determined the most likely structure by slow-heating DFT-based molecular dynamics and analyzing the _librational pathways_. By comparing this analysis with reported temperature dependent lattice parameters and refined structures, we show that the ordering of the molecules (sA or sB) in orthorhombic phases of MAPbBr\({}_{3}\) and MAPbCl\({}_{3}\) is similar (sA), while in MAPbI\({}_{3}\) they are differently ordered (sB). This is unexpected, since the sA pattern is energetically unfavorable when considering solely the intrinsic dipole moment of the MA molecules. The sA order induces a permanent structural distortion of the PbCl\({}_{6}\) octahedra at low temperature, resulting in an anti-ferroelectric stripes pattern in the crystallographic \(a\)-direction. In the higher temperature cubic phase this distortion is no longer observed in the ensemble average, instead instantaneous dynamic distortions appear. No permanent distortion is observed in the PbBr, nor PbI, octahedra even down to the lowest simulated temperature of 80 K. We have presented indications that the sA order in low temperature, orthorhombic MAPbBr\({}_{3}\) is stabilized by an entropic contribution to the free energy related to the \(C_{3}\) dynamics of the MA molecules. We hope that this paper will stimulate combined experimental and MLFF studies of the structure of many other complex Dynamic Solids.
| 10.48550/arXiv.2110.00321 | Exploring librational pathways with on-the-fly machine-learning force fields: Methylammonium molecules in MAPbX$_3$ (X=I, Br, Cl) perovskites | Menno Bokdam, Jonathan Lahnsteiner, D. D. Sarma | 5,460 |
10.48550_arXiv.1308.0412 | ## I Introduction
Skyrmions were originally introduced by the British particle physicist Tony Skyrme to describe localized, particle-like configurations in the field of pion particles. Since that seminal paper, skyrmions have been developed in many fields, including classical liquids, liquid crystals, Bose-Einstein condensates, quantum Hall magnets, and two-dimensional isotropic magnetic materials. They also have been theoretically predicted to exist in magnets with the DMI. Recently, many experiments have proved their existence in helical magnets, such as MnSi and Fe\({}_{1-x}\)Co\({}_{x}\)Si. However, most of the skyrmions occurring in helimagnets were induced by an external magnetic field and only at low temperatures, which limits the technological application of skyrmions. Efforts have been made to resolve part of these problems. For example, Yu et al. have obtained a skyrmion crystal near room-temperature in the helimagnet FeGe with a high helical transition temperature (280 K), but assisted by an external magnetic field. Heinze et al. observed a spontaneous atomic-scale magnetic ground-state skyrmion lattice in a monolayer Fe film, but still at a low temperature (about 11 K). Despite all these efforts, spontaneous skyrmion-like magnetic ground states at or above room temperature have not been reported. Skyrmions existing in helimagnets are always associated with the chiral magnetic interactions, the so-called DMI, which favors canted spin configuration. We note that the magnetostatic interaction favors spin canting and divergence-free spin alignment to reduce the total energy, an effect similar to that of the DMI. For example, confined magnetic thin films can exhibit variant topological spin textures such as vortex-like or meron-like states. Therefore, spontaneous ground state of skyrmion-like configurations may be obtainable in ordinary magnets of nanostructures without the DMI.
Topologically nontrivial magnetic nanostructures have attracted long-standing attention due to their peculiar spin configurations and dynamic behaviors for promising technological applications, and in particular to the rich physics involved. Extensive theoretical and experimental research has revealed their dynamics depending on their topological charges and spin textures. The topological structure of skyrmions distinguishes from magnetic vortex and bubble. Therefore, we expect a novel dynamical behavior in skyrmions.
In this work, we report direct observation of a spontaneous ground state of skyrmion-like configuration in Co/Ru/Co nanodisks without the DMI by micromagnetic simulations. We observe a novel gyration of the guiding center (\(R_{x}\),\(R_{y}\)) of skyrmions with a star-like trajectory in a pulsed magnetic field and a hexagonal trajectory after the field is switched off, which is different from the pentagram trajectory of the mean position (X,Y) of magnetic bubbles. Moreover, the gyration of skyrmion in the bottom nanolayer without the presence of field can be stimulated by the motion of skyrmion in the top nanolayer. This indirect control of skyrmion motion offers the potential applications in information-signal processing and spin devices.
## II Micromagnetic methods
The hexagonal-close-packed (HCP) Co/Ru/Co nanodisks with high Curie temperature and large uniaxial anisotropy are used to study formation and gyrotropic motion of skyrmions without the DMI by the three-dimensional object oriented micromagnetic framework (OOMMF) code. The material parameters of hexagonal-close-packed (HCP) cobalt chosen include saturation magnetization M\({}_{s}\)=1.4\(\times\)10\({}^{6}\) A/m, exchange stiffness A=3\(\times\)10\({}^{-11}\) J/m and uniaxial anisotropy constant K\({}_{u}\)=5.2\(\times\)10\({}^{5}\) J/m\({}^{3}\) with the direction perpendicular to the nanodisk plane. HCP Co/Ru/Co multilayers can be prepared by electron-beam evaporation, magnetron sputtering deposition or ultrahigh vacuum deposition. Interfacial coupling coefficients of the adjacent surfaces for different thicknesses of Ru were from Ref.. The thickness of Co was tuned from 5 to 25 nm; that of Ru, from 1 to 20 nm. The diameter of a Co/Ru/Co nanodisk in the phase diagram and gyration was chosen to be 200 nm. To study the diameter dependence of the skyrmion radius, the diameter of Co (20 nm)/Ru (2 nm)/Co (20 nm) was from 160 to 210 nm. The cell size was 4\(\times\)4\(\times\)1 nm\({}^{3}\) for simulating magnetic ground states, which is smaller than the exchange length of cobalt (about 4.94 nm). At phase boundaries, the cell size was reduced to 2\(\times\)2\(\times\)1 nm\({}^{3}\) to test stability of the obtained states. The dimensionless damping \(\alpha\) was chosen to be 0.25 for rapid convergence. Different initial magnetic states [vortex-like (with same or opposite chirality), in-plane-like and out-of-plane-like initial states] were used to get the most stable ground state. As for gyration simulation, the cell size was 2\(\times\)2\(\times\)2 nm\({}^{3}\) and \(\alpha\) was 0.02. A pulsed magnetic field of 10-ns width and 50-mT magnitude along the +x direction was applied to the top or bottom nanolayer.
## III Formation of skyrmions
Figure 1(a) is a sketch of a single Co (20 nm)/Ru (2 nm)/Co (20 nm) nanodisk with a diameter of 200 nm. Figure 1(b) represents the micromagnetic simulation result of the nanodisk with out-of-plane-like initial state. The equilibrium states of the top and bottom nanolayers are typical skyrmion-like magnetic configurations. The magnetization **M** is down (along -z axis) in the centers and up (along +z axis) on the boundaries and it rotates gradually from -z axis to +z axis at the intermediate regions of the nanolayers. The magnetic chirality of the top nanolayer is right-handed, while that of the bottom one is left-handed. To elucidate the equilibrium state's nature, we calculate the skyrmion number using the following formula:
\[S=\frac{1}{4\pi}\iint qdxdy,\quad q\equiv\frac{1}{2}\epsilon_{\mu\nu}(\partial _{\mu}\mathbf{m}\times\partial_{\nu}\mathbf{m})\cdot\mathbf{m}, \tag{1}\]
S is found to be approximately -1, showing a signature of skyrmion-like state. Similar magnetic spin textures have been found in a patterned Co/Ru/Co nanodisk array with the diameter the same as that of the single Co/Ru/Co nanodisk. The distance between centers of two nearest neighboring nanodisks was 250 nm, as shown in The result suggests that a stray field between two nearest neighboring nanodisks (250 nm apart) has little influence on the skyrmion spin textures.
The formation of magnetic stable states is the consequence of minimizing the Gibbs free energy of magnetic systems. Figure 3(a) illustrates time dependences of the total energy, exchange energy, uniaxial anisotropy energy, demagnetization energy, and antiferromagnetic coupling energy for the case in The exchange energy, uniaxial anisotropy energy, and demagnetization energy are two orders of magnitude higher than the interfacial antiferromagnetic coupling energy and, thus, are vital to the emergence of a skyrmion spin texture. At the beginning, the out-of-plane-like initial state has a very high value of total energy due to the significant demagnetization energy. To lower the total energy, the demagnetization energy decreases rapidly with time, whereas the exchange energy and uniaxial anisotropy energy both increase significantly. A balance is reached and the total energy is almost unchanged after 0.5 ns. Concurrently, the skyrmion number S drops to about -1 for the top and bottom nanolayers, as shown in the inset in Fig. 3(b),
(Color online) (a) Sketch of a Co/Ru/Co nanodisk. (b) Micromagnetic simulation result for a Co (20 nm)/Ru (2 nm)/Co (20 nm) nanodisk. Arrows and colors correspond to the directions of the local magnetization and the magnitude of the out-of-plane magnetization component (\(M_{z}\)) at every point, respectively. Spin textures in both the top and the bottom nanolayers are skyrmions.
As elaborated in many articles, the DMI is crucial to the formation of magnetic skyrmion-like states, which favors canting spins.
\[H_{DMI}=\iint D\mathbf{M}\cdot(\nabla\times\mathbf{M})dxdy, \tag{2}\]
But in this work, skyrmions are spontaneously formed without the DMI. As discussed above, the competition among the exchange energy, demagnetization energy, and uniaxial anisotropy energy plays a significant role in the emergence of skyrmions. To quantify the competition effect, we define a quantity to mimic the DMI:
\[\Psi=\iint\mathbf{M}\cdot(\nabla\times\mathbf{M})dxdy. \tag{3}\]
From Eq., we calculate \(\Psi\) as a function of time on both the top and the bottom Co nanolayers [see Fig. 3(b)]. Notably, when energies compete drastically with each other before 0.5 ns in Fig. 3(a), The \(\Psi\) for both Co nanolayers significantly changes. Then all energies reach equilibrium and \(\Psi\) remains relatively unchanged after 0.8 ns. This indicates that the competition among the three energies can produce an effect similar to that of the DMI to form skyrmions.
\(t_{Ru}\) is changed from 1 to 20 nm, and \(t_{Co}\) from 5 to 25 nm, with a fixed diameter of 200 nm. The phase diagram shows four regions: the vortex-like state, skyrmion-like state, multidomain state and mixed state (more than one stable or metastable state). Stable skyrmions can exist only in a small fraction of the phase diagram with \(t_{Co}\) appropriately chosen and \(t_{Ru}\) smaller than 4 nm. This is obviously a delicate balance between different energies, especially the interlayer magnetostatic interaction. If \(t_{Co}\) goes beyond 25 nm, regardless of \(t_{Ru}\), the multidomain state appears to minimize the demagnetization energy. If the separation layer of Ru gets very thick where the interlayer magnetostatic interaction between two Co nanolayers becomes diluted, the intralayer demagnetization factor starts to play a more essential role, constraining the local magnetization in plane to form the vortex state. Though we have been emphasizing the importance of interlayer magnetostatic interaction by controlling the \(t_{Ru}\), it is still not sufficient to reduce just the \(t_{Ru}\) for the skyrmion state. Moreover, the interfacial antiferromagnetic or ferromagnetic coupling becomes more effective once \(t_{Co}\) is smaller than 8 nm.
## III Stability of skyrmion
The normalized magnetization curve of the nanodisk and the corresponding skyrmion number as a function of the field are shown in Fig. 5(a). The insets in Fig. 5(a) illustrate the profiles of magnetic stable states of the top nanolayer at different fields. The magnetization increases almost linearly with the magnetic field until saturation at 0.64 T, where skyrmions are suppressed completely.
(Color online) Equilibrium magnetic state of a Co (20 nm)/Ru (2 nm)/Co (20 nm) nanodisk array evolving from out-of-plane-like initial state. All the nanodisks have typical skyrmion-like magnetic configurations.
To investigate the spatial distribution of magnetization, we show the \(r\) dependence of \(\theta\) in Figs. 5(b). \(\theta\) is defined as the averaged angle between the local magnetization and +z axis in the range of (r, r+dr) as described in Fig. 1(a). The behavior of \(\theta\) at zero magnetic field is typically skyrmion-like and rules out the possible existence of cylindrical bubble domains. Besides, the distribution of magnetization does not match the Belavin-Polyakov (BP) solution as demonstrated in With an increasing magnetic field, \(\theta\) values in the center and at the edge of the nanodisk align rigidly till the critical saturation field (0.64 T) is reached. \(\theta\) at the edge remains always tilted from a 0 value, possibly due to the stray field at the edge, which results in the absence of a small fraction of the order parameter sphere when the skyrmion is mapped from real space to an order parameter sphere as illustrated in In contrast to the rigidity of magnetization in the center and at the edge, \(\theta\) in the intermediate region decreases gradually with the field, which has a variation tendency similar to that reported in Ref.. indicates that skyrmions remain in a similar configuration under the external field even up to 0.44 T, which is much higher than the reported one. Obviously, the skyrmion obtained in the nanodisk is quite robust.
(Color online) Phase diagram of spin textures derived from micromagnetic simulations. The spin textures as functions of the thickness of Co and of Ru illustrate four regions: the vortex-like state, skyrmion-like state, multidomain state, and mixed state. Phase boundaries are marked by white lines.
(Color online) (a) Normalized magnetization curve (M-B curve) of the Co (20 nm)/Ru (2 nm)/Co (20 nm) nanodisk and the corresponding skyrmion number of the top (”Top-S”) and the bottom (”Bottom-S”) nanolayers as a function of the field. (b) Averaged angle between the local magnetization and the +z axis in the range of (r, r+dr) under different magnetic fields.
(Color online) (a) Time dependence of different energies for a Co (20 nm)/Ru (2 nm)/Co (20 nm) nanodisk with a diameter of 200 nm. (b) Competition among the exchange energy, demagnetization energy and uniaxial anisotropy energy has an effect similar to the DMI (here represented by \(\Psi\)). Inset: the skyrmion number as a function of time.
## III Dynamics of skyrmions
We also investigated the gyrotropic motion of the guiding center of skyrmions. The guiding center (\(R_{x}\),\(R_{y}\)) is essential for the dynamics of a skyrmion, which is defined by the moments of the topological density:
\[R_{x}=\frac{\iint xqdxdy}{\iint qdxdy},\quad R_{y}=\frac{\iint yqdxdy}{\iint qdxdy}, \tag{4}\]
First, a pulsed magnetic field of 10-ns width and 50-mT magnitude is applied along the +x direction on the top nanolayer, as demonstrated in Fig. 6(a). The guiding center gyrates towards its new equlibrium position along the field, which is different from the motion of the vortex. The trajectory of (\(R_{x}\),\(R_{y}\)) of the top nanolayer is initially like a star (before 2.4 ns), then an elliptical orbit, and, finally, is damped around its new equilibrium position (33.5 nm, 0 nm), represented by the filled black circle. Stimulated by the gyrotropic motion of the top skyrmion through strong interlayer magneto-static interaction, the skyrmions in the bottom nanolayer without the presence of a field almost synchronously gyrate, as shown in Fig. 6(b). This indirect control of skyrmion motion may be used in information-signal processing and spin devices. Once the field is turned off, (\(R_{x}\),\(R_{y}\)) in both nanolayers begin to gyrate back to the origin, as shown in Figs. 6(c) and 6(d). Initally, the trajectories of the two guiding centers of skyrmions are hexagons (before 15.24 ns), which is similar to hypocycloid involute. The period of this hexagon-like motion is approximately T=1 ns (i.e., frequency \(f\)=1 GHz). The average velocity in the first period is about 200 m/s. The corresponding FFT spectra of the novel trajectories illustrate two eigenfrequencies of 0.96 and 4.98 GHz with a ratio of about 1:5 as illustrated in These unique trajectories of (\(R_{x}\),\(R_{y}\)) have never been observed in other topological magnetic systems (vortices or bubbles), which usually have a circular or elliptical orbit. Note that a pentagram trajectory resulting from the deformation of circular domain wall in magnetic bubble was reported previously, which is about the mean position (X,Y) defined by the moments of the magnetization. However, hexagon-like trajectories in this work are due to the distribution variance of the topological density of the whole system and about the guiding centers (\(R_{x}\),\(R_{y}\)) defined by the moments of the topological density. Moreover, the motion of skyrmion can not be correctly described by the mean position (X,Y). Therefore, our hexagon-like trajectories of (\(R_{x}\),\(R_{y}\)) in skyrmion is sharply different from the pentagram trajectory of (X,Y) in magnetic bubble. These may be understood by their differences in topological charges and spin textures as well as the strong magnetostatic interaction of the double-skyrmion system (detailed analysis will be discussed in future work). After 14.24 ns, gyration orbits change to circular ones.
The influence of skyrmion chirality on the guiding center of skyrmion gyration is illustrated in Figures 7(a) and 7(e) illustrate sketches of right-handed and left-handed magnetization.
(Color online) (a, b) Trajectories of the guiding centers of skyrmions in both the top and the bottom nanolayers when a pulsed magnetic field is applied to the top nanolayer. (c), d) Gyrotropic motion of the guiding centers of skyrmions after the applied magnetic field is switched off. Colors are used to indicate time-dependent positions of (\(R_{x}\),\(R_{y}\)). The filled black circle represents the new equilibrium position under the magnetic field.
The motion of opposite chiral skyrmions when a pulsed magnetic field along +x direction is applied on the top or bottom nanolayer are shown in Figs. 7(b) and 7(f). The guiding center of skyrmion gyrates towards new equilibrium position along the field applied on the top nanolayer, while against the field when applying the field on the bottom nanolayer. Figures 7(c) and 7(g) are the corresponding gyrations after the applied field is switched off. The trajectories are similar. Both of the trajectories are hexagons before 15.24 ns and the corresponding fast Fourier transform (FFT) spectra are demonstrated in Figs. 7(d) and 7(h). Both of the hexagonal trajectories have two eigenfrequencies with approximately the same values, about 0.96 and 4.98 GHz. The ratio of the two frequencies is about 1:5.
## IV Discussion
A small fraction of a region is missing from the unit sphere. This missing region corresponds to about 0.5% of the sphere area. Nevertheless, the mapped region can still give rise to a skyrmion number (about -0.995) close to -1.0 for a fully mapped skyrmion sphere.
\(\theta\) as a function of x (x=r/R) is illustrated in Fig. 9, where R is the skyrmion radius in which \(\theta\) goes to \(\pi/2\). Diameters of Co/Ru/Co nanodisks are from 160 to 210 nm. All curves from different nanodisk diameters collapse to one. The skyrmion radius is found to be from 51 to 70 nm, corresponding to a diameter of from 160 to 210 nm. The solid (red) line is the BP solution for the nonlinear SO \(\sigma\) model. Skyrmions obtained in the Co/Ru/Co nanodisks do not match the BP solution. In these nanodisks, the competition among exchange energy, uniaxial anisotropy energy, and demagnetization energy results in the formation of a skyrmion spin configuration instead of the single exchange energy considered in the BP solution.
## V Conclusion
In conclusion, we have revealed the existence of a spontaneous magnetic skyrmion ground state in Co/Ru/Co nanodisks without chiral DMI.
(Color online) Mapping from the skyrmion configuration to the unit sphere.
(Color online) Influence of skyrmion chiralities on gyration of the guiding center of skyrmions. (a, e) schematics of skyrmions with right-handed and left-handed chiralities. (b, f) motion of opposite chiral skyrmions when a pulsed magnetic field is applied along the +x direction to the top or bottom nanolayer, respectively. (c, g) gyrotropic motion of skyrmions after the applied field is switched off. (d),(h) Corresponding FFT spectra of the hexagonal trajectories. (R\({}_{xt}\), R\({}_{yt}\)) and (R\({}_{ab}\), R\({}_{yb}\)) are the positions of the guiding center of skyrmions in the top and the bottom nanolayer, respectively.
(Color online) Variation of \(\theta\) as a function of x (x=r/R) for different diameters of Co (20 nm)/Ru (2 nm)/Co (20 nm) nanodisks. The solid (red) line is the BP solution of the nonlinear SO \(\sigma\)-model.
This nontrivial topological spin texture is very stable even against an external magnetic field up to 0.44 T, higher than the highest reported value, about 0.3 T. We observe a novel gyration of the guiding centers (\(R_{x}\),\(R_{y}\)) with a star-like trajectory in a magnetic field and a hexagonal trajectory after the field is switched off. We also observe synchronous gyrotropic motions of two skyrmions in two Co nanolayers (one with a field, the other one without). These unique gyrotropic motions of the skyrmion are distinguished from the magnetic vortex or bubble and contribute to the understanding of the dynamical properties of skyrmions. Our study suggests a new method to the generation of skyrmion textures in various confined magnetic systems and paves the new way to the design of spintronic and magnetic storage devices.
| 10.48550/arXiv.1308.0412 | Skyrmion ground state and gyration of skyrmions in magnetic nanodisks without the Dzyaloshinsky-Moriya interaction | Yingying Dai, Han Wang, Peng Tao, Teng Yang, Weijun Ren, Zhidong Zhang | 4,966 |
10.48550_arXiv.1308.4357 | ###### Abstract
We report a comprehensive study of the binary systems of the platinum group metals with the transition metals, using high-throughput first-principles calculations. These computations predict stability of new compounds in 37 binary systems where no compounds have been reported in the literature experimentally, and a few dozen of as yet unreported compounds in additional systems. Our calculations also identify stable structures at compound compositions that have been previously reported without detailed structural data and indicate that some experimentally reported compounds may actually be unstable at low temperatures. With these results we construct enhanced structure maps for the binary alloys of platinum group metals. These are much more complete, systematic and predictive than those based on empirical results alone.
## I Introduction
The platinum group metals (PGMs), osmium, iridium, ruthenium, rhodium, platinum and palladium, are immensely important in numerous technologies, but the experimental and computational data on their binary alloys still contains many gaps. Interest in PGMs is driven by their essential role in a wide variety of industrial applications, which is at odds with their high cost. The primary application of PGMs is in catalysis, where they are core ingredients in the chemical, petroleum and automotive industries. They also extensively appear as alloying components in aeronautics and electronics applications. The use of platinum alloys in the jewelry industry also accounts for a sizeable fraction of its worldwide consumption, about 30% over the last decade. The importance and high cost of PGMs motivate numerous efforts directed at more effective usage, or at the development of less-expensive alloy substitutes. Despite these efforts, there are still sizeable gaps in the knowledge about the basic properties of PGMs and their alloys; many of the possible alloy compositions have not yet been studied and there is a considerable difficulty in application of thermodynamic experiments because they often require high temperatures or pressures and very long equilibration processes.
The possibility of predicting the existence of ordered structures in alloy systems from their starting components is a major challenge of current materials research. Empirical methods use experimental data to construct structure maps and make predictions based on clustering of simple physical parameters. Their usefulness depends on the availability of reliable data over the entire parameter space of components and stoichiometries. Advances in first-principles methods for the calculation of materials properties open the possibility to complement the experimental data by computational results. Indeed many recent studies present such calculations of PGM alloy structures. However, most of these studies consider a limited number of structures, at just a few stoichiometries of a single binary system or a few systems. Some cluster expansion studies of specific binary systems include a larger set of structures, but limited to a single lattice type (usually, fcc). Realizing the potential of first-principles calculations to complement the lacking, or only partial, empirical data requires high-throughput computational screening of large sets of materials, with structures spanning all lattice types and including, in addition, a considerable number of off-lattice structures. Such large scale screenings can be used to construct low-temperatures binary phase diagrams. They provide insights into trends in alloy properties and indicate the possible existence of hitherto unobserved compounds. A few previous studies implemented this approach to binary systems of specific metals, hafnium, rhenium, rhodium, ruthenium and technetium.
The capability to identify new phases is key to tuning the catalytic properties of PGM alloys and their utilization in new applications, or as reduced-cost or higher-activity substitutes in current applications. Even predicted phases that are difficult to access kinetically in the bulk may be exhibited in nanophase alloys and could be used to increase the efficiency or the lifetimes of PGM catalysts. Given the potential payoff of uncovering such phases, we have undertaken a thorough examination of PGM binary phases with the transition metals, using the first-principles high-throughput (HT) framework AFLOW. We find new potentially stable PGM phases in many binary systems and, comparing experimental data with our predictions, we construct enhanced Pettifor-type maps that demonstrate new ordering trends and compound forming possibilities in these alloys.
## II Methods
Computations of the low-temperature stability of the PGM-transition metal systems were carried out using the HT framework AFLOW. For each of the 153 binary systems studied, we calculated the energies of more than 250 structures, including all the crystal structures reported for the system in the phase diagram literatureand additional structures from the AFLOWLIB database of prototypes and hypothetical hcp-, bcc- and fcc-derivative superstructures. A complete list of structures examined for each binary system can be found on the on-line repository, www.aflowlib.org. The low temperature phase diagram of a system is constructed as the minimum formation enthalpy convex hull from these candidate structures, identifying the ordering trends in each alloy system and indicating possible existence of previously unknown compounds. It should be noted that there is no guarantee that the true groundstates of a system will be found among the common experimentally observed structures or among small-unit-cell derivative structures. However, even if it is impossible to rule out the existence of additional unexpected groundstates, this protocol (searching many enumerated derivative structures and exhaustively exploring experimentally reported ones) is expected to give a reasonable balance between high-throughput speed and scientific accuracy to determine miscibility (or lack thereof) in these alloys. In Ref., it was shown that the probability of reproducing the correct ground state, if well defined and not ambiguous, is \(\eta_{\mathcal{C}}\sim 96.7\%\) ["reliability of the method," Eq.].
The calculations of the structure energies were performed with the VASP software with projector augmented waves pseudopotentials and the exchange-correlation functionals parameterized by Perdew, Burke and Ernzerhof for the generalized gradient approximation. The energies were calculated at zero temperature and pressure, with spin polarization and without zero-point motion or lattice vibrations. All crystal structures were fully relaxed (cell volume and shape and the basis atom coordinates inside the cell). Numerical convergence to about 1 meV/atom was ensured by a high energy cutoff (30% higher than the maximum cutoff of both potentials) and a 6000 \(\mathbf{k}\)-point, or higher, Monkhorst-Pack mesh.
The presented work comprises 38,954 calculations, performed by using 1.82 million CPU/hours on 2013 Intel Xeon E5 cores at 2.2GHz. It was carried out by extending the pre-existing AFLOWLIB structure database with additional calculations characterizing PGM alloys. Detailed information about all the examined structures can be found on the on-line repository, www.aflowlib.org, including input/output files, calculation parameters, geometry of the structures, energies and formation energies. In addition, the reader can prepare phase diagrams (as in figs. 5 to 12) linked to the appropriate structure URL locations.
The analysis of formation enthalpy is, by itself, insufficient to compare alloy stability at different concentrations and their resilience toward high-temperature disorder. The formation enthalpy, \(\Delta H(A_{x}B_{1-x})\equiv H(A_{x}B_{1-x})-xH(A)-(1-x)H(B)\), represents the ordering-strength of a mixture \(A_{x}B_{1-x}\) against decomposition into its pure constituents at the appropriate proportion \(xA\) and \((1-x)B\) (\(\Delta H\) is negative for compound forming systems). However, it does not contain information about its resilience against disorder, which is captured by the entropy of the system. To quantify this resilience we define the _entropic temperature_
\[T_{s}\equiv\max_{i}\left[\frac{\Delta H(A_{x_{i}}B_{1-x_{i}})}{k_{B}\left[x_{i }\log(x_{i})+(1-x_{i})\log(1-x_{i})\right]}\right], \tag{1}\]
This definition assumes an ideal scenario where the entropy is \(S\left[\{x_{i}\}\right]=-k_{B}\sum_{i}x_{i}\log(x_{i})\). This first approximation should be considered as indicative of a trend (see of Ref. and below), which might be modified somewhat by a system specific thorough analysis of the disorder. \(T_{s}\) is a concentration-maximized formation enthalpy weighted by the inverse of its entropic contribution. It represents the deviation of a system convex-hull from the purely entropic free-energy hull, \(-TS(x)\), and hence the ability of its ordered phases to resist the deterioration into a temperature-driven, entropically-promoted, disordered binary mixture.
## III High-throughput results
We examined the 153 binary systems containing a PGM and a transition metal, including the PGM-PGM pairs, (see Fig. 1). An exhaustive comparison of experimental and computational groundstates is given in tables 1 to 6. Convex hulls for systems which exhibit compounds are shown in the Appendix (figs. 5 to 12). These results uncover 37 alloy systems reported as non-compound forming in the experimental literature, but predicted computationally to have low-temperature stable compounds. Dozens of new compounds are also predicted in systems known to be compound forming.
The top panel of gives a broad overview of the comparison of experiment and computation. Green circles (dark gray) indicate systems where experiment and computation agree that the system is compound forming. Light gray circles indicate agreement that the system is not compound-forming. The elements along the axes of this diagram are listed according to their Pettifor \(\chi\) parameter, leading, as expected, to compound-forming and non-compound forming systems separating rather cleanly into different broad regions of the diagram. Most of the compound-forming systems congregate in a large cluster on the left half of the diagram, and in a second smaller cluster at the lower right corner.
The systems for which computation predicts compounds but experiment does not report any are marked by red squares. As is clear in the top panel of Fig. 1, these systems, which harbor potential new phases, occur near the boundary between the compound-forming and non-compound-forming regions of the diagram. They also fill in several isolated spots where experiment reports no compounds in the compound-forming region (e.g., Pd-W, Ag-Pd), and bridge the gap between the large cluster of compound-forming systems, on the left side of the panel, and the small island of such systems at its center. The computations also predict ordered structures in most systems reported only with disordered phases (yellow circles in top panel of Fig. 1). Two disordered phases, \(\sigma\) and \(\chi\), turn up in the experimental literature on PGM alloys. In the HT search, we included all ordered realizations of these phases (the prototypes Al\({}_{12}\)Mg\({}_{17}\) and Re\({}_{24}\)Ti\({}_{5}\) are ordered versions of the \(\chi\) phase and the \(\sigma\) phase has 32 ordered realizations, denoted by \(\sigma_{XXXXX}\) where \(X=A,B\)). In most of these systems we find one of these corresponding ordered structures to be stable. The only exception is the Cr-Ru system, where the lowest lying ordered phase is found just 4 meV/atom above the elements tie-line (yellow square in Fig. 1). These results thus identify the low temperature ordered compounds that underly the reported disordered phases. The calculated compound-forming regions are considerably more extensive than reported by the available experimental data, identifying potential new systems for materials engineering.
The bottom panel of ranks systems by their estimated entropic temperature \(T_{s}\). Essentially, the (top panel) map, incorporating the computational data, corresponds to what would be observed at low temperatures, assuming thermodynamic equilibrium, whereas a map with only experimental data reports systems as compound-forming when reaching thermodynamic equilibrium is presumably easier. That is not to say, however, that the predicted phases will necessarily be difficult to synthesize--some of the systems where the \(T_{s}\) value is small have been experimentally observed to be compound-forming (e.g., Cr-Pd, Au-Pd, Ag-Pt, Hg-Rh and Co-Pt). \(T_{s}\) decreases gradually as we move from the centers of the compound-forming clusters towards their edges. Most systems with low \(T_{s}\) are adjacent to the remaining non-compound-forming region. This leads to a qualitative picture of compound stability against disorder which is correlated with the position of a system within the compound forming cluster, and with larger clusters centered at systems with more stable structures.
It is instructive to note that many obscure and large unit cell structures that are reported in the experimental literature are recovered in the HT search. For example, compounds of prototypes such as Mg\({}_{44}\)Rh\({}_{7}\), Ru\({}_{25}\)Y\({}_{44}\), Ir\({}_{4}\)Sc\({}_{11}\), Rh\({}_{13}\)Sc\({}_{57}\) from the experimental literature, nearly always turn up as ground states, or very close to the convex hull, in the HT search as well. This is strong evidence that the first-principles HT approach is robust and has the necessary accuracy to extend the PGM data where experimental results are sparse or difficult to obtain. Also of interest is the appearance of some rare prototypes in systems similar to those in which they were identified experimentally. For example, the prototype Pd\({}_{3}\)Ti\({}_{2}\), reported only in the Pd-Ti system, also emerges as a calculated groundstate in the closely related systems Hf-Pd and Pt-Ti.
Top panel: Compound-forming vs. non-compound-forming systems as determined by experiment and computation. Circles indicate agreement between experiment and computation, green for compound-forming systems, gray for non-compound forming systems. Yellow circles indicate systems reported in experiment to have disordered phases, for which low-energy compounds were found in this work. Ru-Cr is the only system (yellow square) experimentally reported to include a disordered phase where no low-temperature stable compounds were found. Red squares mark systems for which low-temperature compounds are found in computation but no compounds are reported in experiment. Bottom panel: \(T_{s}\) for the binary systems in this work. Colors: from red (lowest \(T_{s}\)) to blue (highest \(T_{s}\)).
In the systems we examined, there are nearly 50 phases reported in the experimental phase diagrams for which the crystal structure of the phase is not known. In one half of these cases, the HT calculations identify stable structures for these unknown phases. For the other half of these unknown structures, our calculations find no stable compounds at the reported concentration, but stable compounds at other concentrations. The reported phases (sans structural information) may, therefore, be due to phases that decompose at low temperatures or may merely represent samples that were kinetically inhibited and unable to settle into their stable phases during the time frame of the experiments.
The prototype database included in this study comprise both experimentally-reported structures as well as hypothetical structures constructed combinatorially from derivative supercells of fcc, bcc, and hcp lattices.
\begin{table}
\begin{tabular}{|c|c c c||c|c|c|c|} \hline \hline & \multicolumn{2}{c||}{Compounds} & \(\Delta H\) & \multicolumn{2}{c|}{Compounds} & \(\Delta H\) \\ & Exper. & Calc. & meV/at. & Exper. & Calc. & meV/at. \\ \hline \hline Y & Os\({}_{2}\)Y(C14) & Os\({}_{2}\)Y(C14) & -304 & W & & Os\({}_{3}\)W(D0\({}_{19}\)) & -56 \\ & OsY\({}_{3}\)(D0\({}_{11}\)) & OsY\({}_{3}\)(D0\({}_{11}\)) & -239 & & Os\({}_{0.3}\)W\({}_{0.7}\)(\(\sigma\)) & & \\ \hline Sc & Os\({}_{2}\)Sc(C14) & Os\({}_{2}\)Sc(C14) & -390 & Cr & Cr\({}_{2}\)Os(A15) & & \(\langle 18\rangle\) \\ & & OsSc\({}_{2}\)(fcc\({}_{4}^{}\)) & -400 & & & CrOs\({}_{3}\)(D0\({}_{19}\)) & -22 \\ & Os\({}_{4}\)Sc\({}_{11}\)(Ir\({}_{4}\)Sc\({}_{11}\)) & Os\({}_{4}\)Sc\({}_{11}\)(Ir\({}_{4}\)Sc\({}_{11}\)) & -372 & & & CrOs\({}_{5}\)(Hf\({}_{5}\)Sc\({}^{*}\)) & -19 \\ & Os\({}_{87}\)Sc\({}_{44}\)(Mg\({}_{44}\)Rh\({}_{7}\)) & Os\({}_{7}\)Sc\({}_{44}\)(Mg\({}_{44}\)Rh\({}_{7}\)) & -197 & Tc & - & Os\({}_{3}\)Tc(D0\({}_{19}\)) & -71 \\ \hline Zr & Os\({}_{2}\)Zr(C14) & Os\({}_{2}\)Zr(C14) & -388 & & & OsTc(B19) & -83 \\ & OsZr(B2) & OsZr(B2) & -524 & & & OsTc\({}_{3}\)(D0\({}_{19}\)) & -57 \\ & Os\({}_{4}\)Zr\({}_{11}\)(Ir\({}_{4}\)Sc\({}_{11}\)) & Os\({}_{4}\)Zr\({}_{11}\)(Ir\({}_{4}\)Sc\({}_{11}\)) & -29 & Re & - & Os\({}_{3}\)Re(D0\({}_{19}\)) & -78 \\ & Os\({}_{17}\)Zr\({}_{54}\)(Hf\({}_{5}\)Os\({}_{17}\)) & \(\langle 8\rangle\) & & & OsRe(B19) & -89 \\ & OsZr\({}_{4}\)(D1\({}_{a}\)) & -220 & & & OsRe\({}_{2}\)(Sc\({}_{2}\)Zr\({}^{*}\)) & -68 \\ \hline Hf & Hf\({}_{54}\)Os\({}_{17}\)(Hf\({}_{54}\)Os\({}_{17}\)) & & \(\langle 20\rangle\) & & & OsRe\({}_{3}\)(Re\({}_{3}\)Ru\({}^{*}\)) & -56 \\ & Hf\({}_{2}\)Os(NiTi\({}_{2}\)) & & \(\langle 44\rangle\) & Mn & - & MnOs(B19) & -42 \\ & HfOs(B2) & HfOs(B2) & -709 & & & MnOs\({}_{3}\)(D0\({}_{19}\)) & -36 \\ & HfOs\({}_{2}\)(C14) & & \(\langle 66\rangle\) & Fe & - & - & \\ \hline Ti & OsTi(B2) & OsTi(B2) & -714 & Os & reference & & \\ & OsTi\({}_{2}\)(C49) & -515 & Ru & - & Os\({}_{3}\)Ru(D0\({}_{a}\)) & -9 \\ & OsTi\({}_{3}\)(Mo\({}_{3}\)Ti\({}^{*}\)) & -403 & & & OsRu(B19) & -15 \\ \hline Nb & & Nb\({}_{3}\)Os(HPD\({}_{5}\)) & -200 & & & OsRu\({}_{3}\)(D0\({}_{a}\)) & -11 \\ & Nb\({}_{3}\)Os(A15) & Nb\({}_{3}\)Os(A15) & -275 & & & OsRu\({}_{3}\)(Hf\({}_{5}\)Sc\({}^{*}\)) & -9 \\ & Nb\({}_{0.6}\)Os\({}_{0.4}\)(\(\sigma\)) & Nb\({}_{20}\)Os\({}_{10}\)(\(\sigma_{BAAB}\)) & -274 & Co & - & - \\ & Nb\({}_{0.4}\)Os\({}_{0.6}\)(\(\chi\)) & Nb\({}_{12}\)Os\({}_{17}\)(Al\({}_{12}\)Mg\({}_{17}\)) & -247 & Ir & - & Ir\({}_{5}\)Os(Pt\({}_{8}\)Ti) & -8 \\ & NbOs\({}_{3}\)(D0\({}_{24}\)) & -115 & & & IrOs\({}_{5}\)(Hf\({}_{5}\)Sc\({}^{*}\)) & -7 \\ \hline Ta & & Os\({}_{2}\)Ta(Ga\({}_{2}\)Hf) & -205 & Rh & - & OsRh(RhRu\({}^{*}\)) & -8 \\ & Os\({}_{0.5}\)Ta\({}_{0.5}\)(\(\chi\)) & Os\({}_{12}\)Ta\({}_{17}\)(Al\({}_{12}\)Mg\({}_{17}\)) & -313 & Ni & - & - & \\ & Os\({}_{0.3}\)Ta\({}_{0.7}\)(\(\sigma\)) & Os\({}_{10}\)Ta\({}_{20}\)(\(\sigma_{ABBAB}\)) & -335 & Pt & - & - & \\ & OsTa\({}_{3}\)(A15) & -330 & Pd & - & - & \\ \hline V & & Os\({}_{3}\)V(Re\({}_{3}\)Ru\({}^{*}\)) & -150 & Au & - & - & \\ & & Os\({}_{3}\)V\({}_{5}\)(Ga\({}_{3}\)Pt\({}_{5}\)) & -350 & Ag & - & - & \\ & & OsV\({}_{2}\)(C11\({}_{b}\)) & -354 & Cu & - & - & \\ & OsV\({}_{3}\)(A15) & OsV\({}_{3}\)(D0\({}_{3}\)) & -361\(\langle 21\rangle\) & Hg & - & - & \\ & OsV\({}_{5}\)(Mo\({}_{3}\)Ti\({}^{*}\)) & -253 & Cd & - & - & \\ \hline Mo & Mo\({}_{3}\)Os(A15) & & \(\langle 29\rangle\) & Zn & - & - & \\ & Mo\({}_{0.65}\)Os\({}_{0.35}\)(\(\sigma\)) & & & & & & \\ & MoOs\({}_{3}\)(D0\({}_{19}\)) & -52 & & & & & \\ \hline \end{tabular}
\end{table}
Table 1: Compounds observed in experiments (“Exper.”) or predicted by _ab initio_ calculations (“Calc.”) in **Osmium** binary alloys (structure prototype in parentheses, multiple entries denote different reported structures, in the experiments, or degenerate structures, in the calculations). “-” denotes no compounds. The superscript “\(\star\)” denotes unobserved prototypes found in calculations. \(\Delta H\) are the formation enthalpies from the present study. The energy difference between reported and calculated structures or between the reported structure (unstable in the calculation) and a two-phase tie-line is indicated in brackets “\(\langle\cdot\rangle\)”.
In this work, we find compounds with 5 of these new structures, for which no prototype is known and no _Strukturbericht_ designation have been given. These new prototypes are marked by a \(\dagger\) in tables 1 to VI and their crystallographic parameters are given in Table 7. We also find a few other compounds with unobserved prototypes (marked by a \(\star\) in tables 1 to VI) previously uncovered in related HT studies.
## IV Structure maps
Empirical structure maps present available experimental data in ways that highlight similarities in materials behavior in alloy systems. Their arrangement principles usually depend on simple parameters, e.g., atomic number, atomic radius, electronegativity, ionization energy, melting temperature or enthalpy.
\begin{table}
\begin{tabular}{|c|c c|c||c|c||c|c|c|} \hline \hline & \multicolumn{2}{c|}{Compounds} & \(\Delta H\) & \multicolumn{4}{c|}{Compounds} & \(\Delta H\) \\ & Exper. & Calc. & meV/at. & & Exper. & Calc. & meV/at. \\ \hline \hline Y & Ru\({}_{2}\)Y(C14) & Ru\({}_{2}\)Y(C14) & -313 & Mo & Mo\({}_{0.6}\)Ru\({}_{0.4}\)(\(\sigma\)) & Mo\({}_{14}\)Ru\({}_{16}\)(\(\sigma_{AABAB}\)) & -116 \\ & Ru\({}_{2}\)Y\({}_{3}\)(Er\({}_{3}\)Ru\({}_{2}\)) & & & W & Ru\({}_{0.4}\)W\({}_{0.6}\)(\(\sigma\)) & Ru\({}_{3}\)W(D0\({}_{19}\)) & -65 \\ & Ru\({}_{25}\)Y\({}_{44}\)(Ru\({}_{25}\)Y\({}_{44}\)) & Ru\({}_{25}\)Y\({}_{44}\)(Ru\({}_{25}\)Y\({}_{44}\)) & -342 & Cr & Cr\({}_{0.7}\)Ru\({}_{0.3}\)(\(\sigma\)) & - & \\ & Ru\({}_{2}\)Y\({}_{5}\)(C\({}_{2}\)Mn\({}_{5}\)) & Ru\({}_{2}\)Y\({}_{5}\)(C\({}_{2}\)Mn\({}_{5}\)) & -334 & Tc & - & Ru\({}_{3}\)Tc(D0\({}_{19}\)) & -63 \\ & RuV\({}_{3}\)(D0\({}_{11}\)) & RuV\({}_{3}\)(D0\({}_{11}\)) & -307 & & & RuTc(B19) & -73 \\ \hline Sc & Ru\({}_{2}\)Sc(C14) & Ru\({}_{2}\)Sc(C14) & -389 & & & & RuTc\({}_{3}\)(D0\({}_{19}\)) & -47 \\ & RuSc(B2) & RuSc(B2) & -540 & & & & RuTc\({}_{5}\)(RuTc\({}_{5}\)\({}^{*}\)) & -32 \\ & Ru\({}_{3}\)Sc\({}_{5}\)(D8\({}_{8}\)) & & & Re & - & Re\({}_{3}\)Ru(Re\({}_{3}\)Ru\({}^{*}\)) & -53 \\ & RuSc\({}_{2}\)(NiTi\({}_{2}\)) & RuSc\({}_{2}\)(C11\({}_{8}\)) & -484 & & & & ReRu(B19) & -86 \\ & Ru\({}_{4}\)Sc\({}_{11}\)(Ir4Sc11) & Ru\({}_{4}\)Sc\({}_{11}\)(Ir4Sc11) & -405 & & & & ReRu\({}_{3}\)(D0\({}_{19}\)) & -80 \\ & Ru\({}_{13}\)Sc\({}_{57}\)(Rh\({}_{13}\)Sc57) & & & & Mn & - & Mn\({}_{24}\)Ru\({}_{5}\)(Re\({}_{24}\)Ti\({}_{5}\)) & -18 \\ & Ru\({}_{7}\)Sc\({}_{44}\)(Mg\({}_{44}\)Rh\({}_{7}\)) & Ru7Sc44(Mg\({}_{44}\)Rh\({}_{7}\)) & -226 & Fe & - & - & \\ \hline Zr & RuZr(B2) & RuZr(B2) & -644 & Os & - & Os\({}_{3}\)Ru(D0\({}_{4}\)) & -9 \\ \hline Hf & HfRu(B2) & HfRu(B2) & -819 & & & OsRu(B19) & -15 \\ & HfRu\({}_{2}\)(Unkn.) & & & & & OsRu\({}_{3}\)(D0\({}_{4}\)) & -11 \\ \hline Ti & RuTi(B2) & RuTi(B2) & -763 & & & & OsRu\({}_{5}\)(Hf\({}_{5}\)Sc\({}^{*}\)) & -9 \\ & & RuTi\({}_{2}\)(C49) & -532 & Ru & Reference & & \\ & RuTi\({}_{3}\)(Mo\({}_{3}\)Ti\({}^{*}\)) & -401 & Co & - & & & \\ \hline Nb & & Nb\({}_{8}\)Ru(Pt\({}_{8}\)Ti) & -117 & Ir & - & Ir\({}_{8}\)Ru(Pt\({}_{8}\)Ti) & -20 \\ & & Nb\({}_{5}\)Ru(Nb\({}_{5}\)Ru\({}^{*}\)) & -172 & & & Ir\({}_{3}\)Ru(In\({}_{12}\)) & -34 \\ & & Nb\({}_{3}\)Ru(L0\({}_{6}\)) & -222 & & & IrRu(B19) & -49 \\ & Nb\({}_{5}\)Ru\({}_{3}\)(Ga\({}_{3}\)Pt\({}_{5}\)) & -249 & & & IrRu\({}_{2}\)(Ir\({}_{2}\)Tc\({}^{*}\)) & -54 \\ & NbRu(Unkn.) & & & & IrRu\({}_{3}\)(D0\({}_{19}\)) & -53 \\ & Nb\({}_{3}\)Ru\({}_{5}\)(Ga\({}_{3}\)Pt\({}_{5}\)) & -240 & & & IrRu\({}_{5}\)(Hf\({}_{5}\)Sc\({}^{*}\)) & -37 \\ & NbRu\({}_{3}\)(L1\({}_{2}\)) & & & Rh & - & Rh\({}_{8}\)Ru(Pt\({}_{8}\)Ti) & -2 \\ \hline Ta & Ru\({}_{5}\)Ta\({}_{3}\)(Unkn.) & Ru\({}_{5}\)Ta\({}_{3}\)(Ga\({}_{3}\)Pt\({}_{5}\)) & -332 & & & RhRu(RhRu\({}^{*}\)) & -8 \\ & RuTa(Unkn.) & & & & & RhRu\({}_{2}\)(RhRu\({}^{*}\)) & -6 \\ & & Ru\({}_{3}\)Ta\({}_{5}\)(Ga\({}_{3}\)Pt\({}_{5}\)) & -313 & & & & RhRu\({}_{5}\)(Rh\({}_{8}\)Ru\({}^{*}\)) & -3 \\ & & RuTa\({}_{3}\)(fcc\({}_{AB3}^{}\)) & -281 & Ni & - & - & \\ & RuTa\({}_{5}\)(Nb\({}_{5}\)Ru\({}^{*}\)) & -207 & Pt & - & PtRu(CdTi) & -33 \\ \hline V & & Ru\({}_{3}\)V(Re\({}_{5}\)Ru\({}^{*}\)) & -145 & Pd & - & - & \\ & Ru\({}_{2}\)V(C37) & -192 & Au & - & - & \\ & RuV(B11) & & & Ag & - & - & \\ & Ru\({}_{3}\)V\({}_{5}\)(Ga\({}_{3}\)Pt\({}_{5}\)) & -313 & Cu & - & - & \\ & RuV\({}_{2}\)(C11\({}_{b}\)) & -321 & Hg & - & - & \\ & RuV\({}_{3}\)(Mo\({}_{3}\)Ti\({}^{*}\)) & -296 & Cd & - & - & \\ & RuV\({}_{4}\)(D1\({}_{a}\)) & -262 & Zn & & RuZn\({}_{3}\)(L1\({}_{2}\)) & -150 \\ & RuV\({}_{5}\)(Nb\({}_{5}\)Ru\({}^{*}\)) & -230 & & RuZn\({}_{6}\)(RuZn\({}_{6}\)) & RuZn\({}_{6}\)(RuZn\({}_{6}\)) & -132 \\ & RuV\({}_{8}\)(Pt\({}_{8}\)Ti) & -154 & & & & \\ \hline \end{tabular}
\end{table}
Table 2: Compounds in **Ruthenium** binary alloys. (Unkn.) denotes an unknown structure. All other symbols are as in Table 1.
\begin{table}
\begin{tabular}{|c|c c c|c||c|c|c|} \hline \hline & \multicolumn{2}{c|}{Compounds} & \(\Delta H\) & \multicolumn{2}{c|}{Compounds} & \multicolumn{1}{c|}{\(\Delta H\)} \\ & Exper. & Calc. & meV/at. & & Exper. & Calc. & meV/at. \\ \hline \hline Y & Ir\({}_{3}\)Y(PuNi\({}_{3}\)) & & & W & & Ir\({}_{8}\)W(Pt\({}_{8}\)Ti) & -157 \\ & Ir\({}_{2}\)Y(C15) & Ir\({}_{2}\)Y(C15) & -803 & & Ir\({}_{3}\)W(D0\({}_{19}\)) & Ir\({}_{3}\)W(D0\({}_{19}\)) & -350 \\ & IrY(B2) & IrY(B2) & -787 & & Ir\({}_{2}\)W(C37) & -352 \\ & Ir\({}_{2}\)Y\({}_{3}\)(Rh\({}_{2}\)Y\({}_{3}\)) & & & IrW(B19) & -300 \\ & Ir\({}_{3}\)Y\({}_{5}\)(Pu\({}_{8}\)Rh\({}_{3}\)) & Ir\({}_{3}\)Y\({}_{5}\)(Pu\({}_{8}\)Rh\({}_{3}\)) & -772 & & Cr Cr\({}_{3}\)Ir(A15) & \\ & Ir\({}_{2}\)Y\({}_{5}\)(C\({}_{2}\)Mn\({}_{5}\)) & Ir\({}_{2}\)Y\({}_{5}\)(C\({}_{2}\)Mn\({}_{5}\)) & -640 & & CrIr(B19) & -239 \\ & IrY\({}_{3}\)(D0\({}_{11}\)) & Ir\({}_{3}\)Y\({}_{3}\)(D0\({}_{11}\)) & -564 & & CrIr\({}_{2}\)(C37) & -233 \\ \hline Sc & & Ir\({}_{7}\)Sc(CuPt\({}_{7}\)) & -352 & & & CrIr\({}_{3}\)D(D\({}_{19}\)) & -228 \\ & Ir\({}_{3}\)Sc(L\({}_{12}\)) & & & & Ir\({}_{3}\)Tc (Pt\({}_{8}\)Ti) & -89 \\ & Ir\({}_{2}\)Sc(C15) & Ir\({}_{2}\)Sc(C14) & -783 & & & Ir\({}_{2}\)Tc(Ir\({}_{2}\)Tc\({}^{*}\)) & -224 \\ & IrSc(B2) & IrSc(B2) & -1032 & & & IrTc(B19) & -287 \\ & IrSc\({}_{2}\)(NiTi\({}_{2}\)) & & & & IrTc\({}_{3}\)(D0\({}_{19}\)) & -217 \\ & Ir\({}_{4}\)Sc\({}_{11}\)(Ir\({}_{4}\)Sc\({}_{11}\)) & Ir\({}_{4}\)Sc\({}_{11}\)(Ir\({}_{4}\)Sc\({}_{11}\)) & -686 & & Re & - & Ir\({}_{3}\)Re(Pt\({}_{8}\)Ti) & -94 \\ & Ir\({}_{13}\)Sc\({}_{57}\)(Rh\({}_{13}\)Sc\({}_{57}\)) & & & & Ir\({}_{2}\)Re(Ir\({}_{2}\)Tc\({}^{*}\)) & -227 \\ & Ir\({}_{5}\)Sc\({}_{44}\)(Mg\({}_{44}\)Rh\({}_{7}\)) & Ir\({}_{7}\)Sc\({}_{44}\)(Mg\({}_{44}\)Rh\({}_{7}\)) & -369 & & IrRe(B19) & -274 \\ \hline Zr & Ir\({}_{3}\)Zr(L\({}_{12}\)) & Ir\({}_{3}\)Zr(L\({}_{12}\)) & -709 & & & Ir\({}_{3}\)Dr(L\({}_{12}\)) & -173 \\ & Ir\({}_{2}\)Zr(C15) & Ir\({}_{2}\)Zr(Ga\({}_{2}\)Hf) & -766 & Mn & & Ir\({}_{3}\)Dr(L\({}_{10}\)) & IrMn(B19) & -204 \\ & IrZr(NiTi) & IrZr(NiTi) & -830 & & & IrMn(L\({}_{10}\)) & IrMn(L\({}_{12}\)) & -175 \\ & Ir\({}_{3}\)Zr\({}_{5}\)(Ir\({}_{3}\)Zr\({}_{5}\)) & Ir\({}_{2}\)Zr\({}_{5}\)(Ir\({}_{3}\)Zr\({}_{5}\)) & -732 & & & IrMn\({}_{2}\)(C37) & -175 \\ & IrZr\({}_{2}\)(C16) & IrZr\({}_{2}\)(C37) & -668 & & IrMn\({}_{3}\)(L\({}_{12}\)) & IrMn\({}_{3}\)(L\({}_{60}\)) & -156 \\ & IrZr\({}_{3}\)(SV\({}_{3}\)) & IrZr\({}_{3}\)(SV\({}_{3}\)) & -519 & Fe & & Fe\({}_{3}\)Ir(L\({}_{60}\)) & -44 \\ \hline Hf & Hf\({}_{2}\)Ir(NiTi\({}_{2}\)) & Hf\({}_{2}\)Ir(C37) & -750 & & Fe\({}_{0.6}\)Ir\({}_{0.4}\)(Mg) & FeIr(NbP) & -57 \\ & Hf\({}_{5}\)Ir\({}_{3}\)(D8\({}_{8}\) / Ir\({}_{5}\)Zr\({}_{3}\)) & Hf\({}_{5}\)Ir\({}_{3}\)(Ir\({}_{5}\)Zr\({}_{3}\)) & -814 & & FeIr\({}_{3}\)(D0\({}_{22}\)) & -63 \\ & HfIr (Unkn.) & HfIr(B27) & -949 & Os & - & Ir\({}_{3}\)Os(Pt\({}_{8}\)Ti) & -8 \\ & HfIr\({}_{2}\)(Ga\({}_{2}\)Hf) & -872 & & & IrOs\({}_{5}\)(Hf\({}_{5}\)Sc\({}^{*}\)) & -7 \\ & HfIf\({}_{3}\)(L1\({}_{2}\)) & HfIf\({}_{3}\)(L12) & -800 & Ru & - & Ir\({}_{3}\)Ru(Pt\({}_{8}\)Ti) & -20 \\ & IrNb(L10\({}_{7}\) & IrTa) & IrNb(L10) & -542 & & Ir\({}_{3}\)Ru(L12) & -34 \\ & Ir\({}_{0.37}\)Nb\({}_{0.63}\)(\(\sigma\)) & Ir\({}_{2}\)Nb\({}_{5}\) (\(\sigma\)\({}_{ABBAB}\)) & -484 & & IrRu(B19) & -49 \\ & IrNb\({}_{3}\)(A15) & IrNb\({}_{3}\)(A15) & -433 & & & IrRu\({}_{2}\)(Ir\({}_{2}\)Tc\({}^{*}\)) & -54 \\ \hline Ta & Ir\({}_{3}\)Ta(L12) & Ir\({}_{3}\)Ta(Co\({}_{3}\)V) & -688 & & & \\ & Ir\({}_{2}\)Ta(Ga\({}_{2}\)Hf) & -659 & Pt & - & - & \\ & IrTa(L10 / IrTa) & IrTa(L10) & -594 & Pd & - & & Ir\({}_{3}\)Rh(fcc\({}_{ABB}^{}\)) & -15 \\ & Ir\({}_{0.25}\)Ta\({}_{0.75}\)(\(\sigma\)) & Ir\({}_{10}\)Ta\({}_{20}\)(\(\sigma_{ABBAB}\)) & -528 & & Ir\({}_{2}\)Rh(Pd\({}_{2}\)Ti) & -20 \\ & IrTa(A15) & -479 & Ag & - & - & IrRh(fcc\({}_{ABB}^{}\)) & -21 \\ \hline V & Ir\({}_{3}\)V(L12) & Ir\({}_{3}\)V(D0\({}_{19}\)) & -505 & Cu & - & - & \\ & IrV(IrV / L10) & IrV(L10) & -500\({}^{5}\) & & & CrIr\({}_{3}\)V(D0\({}_{19}\)) & -38 \\ & IrV\({}_{3}\)(A15) & IrV\({}_{3}\)(A15) & -497 & & IrRh\({}_{2}\)(Pd\({}_{2}\)Ti) & -18 \\ & IrV\({}_{8}\)(Pt\({}_{8}\)Ti) & -225 & & & IrW(B19) & -300 \\ \hline Mo & Ir\({}_{3}\)Mo(D0\({}_{19}\)) & Ir\({}_{3}\)Mo(D0\({}_{19}\)) & -332 & & & IrZn\({}_{2}\)(C49) & -238 \\ & Ir\({}_{2}\)Mo(C37) & -337 & & & IrZn\({}_{3}\)(NbPd\({}_{3}\)) & -224 \\ & IrMo(B19) & IrMo(B19) & -321 & Ir\({}_{2}\)Zn\({}_{11}\)(Ir\({}_{2}\)Zn\({}_{11}\)) & Ir\({}_{2}\)Zn\({}_{11}\)(Ir\({}_{2}\)Zn\({}_{11}\)) & -192 \\ & IrMo\({}_{3}\)(A15) & & & & & \\ \hline \end{tabular} \
\begin{table}
\begin{tabular}{|c|c c|c||c|c|c|c|} \hline \hline & \multicolumn{2}{c|}{Compounds} & \(\Delta H\) & \multicolumn{3}{c|}{Compounds} & \(\Delta H\) \\ & Exper. & Calc. & meV/at. & & Exper. & Calc. & meV/at. \\ \hline \hline Y & Rh\({}_{3}\)Y(CeNi\({}_{3}\)) & Rh\({}_{3}\)Y(CeNi\({}_{3}\)) & -569 & W & Rh\({}_{0.5}\)W\({}_{0.2}\)(Mg) & Rh\({}_{8}\)W(Pt\({}_{8}\)Ti) & -140 \\ & Rh\({}_{2}\)Y(C15) & Rh\({}_{2}\)Y(C15) & -742 & & Rh\({}_{3}\)W(D0\({}_{19}\)) & Rh\({}_{3}\)W(D0\({}_{19}\)) & -274 \\ & RhY(B2) & RhY(B2) & -863 & & & Rh\({}_{2}\)W(C37) & -264 \\ & Rh\({}_{2}\)Y\({}_{3}\)(Rh\({}_{2}\)Y\({}_{3}\)) & & \(\langle 8\rangle\) & Cr & Cr\({}_{3}\)Rh(A15) & & \(\langle 103\rangle\) \\ & Rh\({}_{3}\)Y\({}_{5}\)(Unkn.) & Rh\({}_{3}\)Y\({}_{5}\)(Pu\({}_{8}\)Rh\({}_{3}\)) & -727 & & & CrRh\({}_{2}\)(C37) & -117 \\ & Rh\({}_{3}\)Y\({}_{7}\)(Fe\({}_{3}\)Th\({}_{7}\)) & Rh\({}_{3}\)Y\({}_{7}\)(Fe\({}_{3}\)Th\({}_{7}\)) & -606 & & CrRh\({}_{3}\)(L1\({}_{2}\)) & CrRh\({}_{3}\)(L1\({}_{2}\)) & -128 \\ & RhY\({}_{3}\)(D0\({}_{11}\)) & RhY\({}_{3}\)(D0\({}_{11}\)) & -517 & & & CrRh\({}_{7}\)(CuPt\({}_{7}\)) & -65 \\ \hline Sc & \multicolumn{2}{c|}{Rh\({}_{7}\)Sc(CuPt\({}_{7}\))} & -348 & Tc & - & Rh\({}_{2}\)Tc(Ir\({}_{2}\)Tc\({}^{*}\)) & -157 \\ & Rh\({}_{3}\)Sc(L1\({}_{2}\)) & Rh\({}_{3}\)Sc(L1\({}_{2}\)) & -620 & & & RhTc(B19) & -175 \\ & RhSc(B2) & RhSc(B2) & -1035 & & & RhTc\({}_{3}\)(D0\({}_{19}\)) & -158 \\ & Ir\({}_{4}\)Sc\({}_{11}\)(Ir\({}_{4}\)Sc\({}_{11}\)) & -582 & Re & - & Re\({}_{3}\)Rh(D0\({}_{19}\)) & -163 \\ & Rh\({}_{12}\)Sc\({}_{57}\)(Rh\({}_{13}\)Sc\({}_{57}\)) & Rh\({}_{13}\)Sc\({}_{57}\)(Rh\({}_{13}\)Sc\({}_{57}\)) & -424 & & & ReRh(B19) & -184 \\ & Ir\({}_{7}\)Sc\({}_{44}\)(Mg\({}_{44}\)Rh\({}_{7}\)) & -319 & & & & ReRh\({}_{2}\)(Ir\({}_{2}\)Tc\({}^{*}\)) & -173 \\ \hline Zr & Rh\({}_{2}\)Zr(L1\({}_{2}\)) & Rh\({}_{3}\)Zr(L1\({}_{2}\)) & -687 & Mn & Mn\({}_{3}\)Rh(L1\({}_{2}\)) & & \(\langle 153\rangle\) \\ & Rh\({}_{5}\)Zr\({}_{3}\)(Pu\({}_{3}\)Pd\({}_{5}\)) & Rh\({}_{5}\)Zr\({}_{3}\)(Pu\({}_{3}\)Pd\({}_{5}\)) & -811 & & MnRh(B2) & MnRh(B2) & -190 \\ & Rh\({}_{4}\)Zr\({}_{3}\)(Unkn.) & & & & MnRh\({}_{3}\)(L1\({}_{2}\)) & -126 \\ & Rh\({}_{2}\)Zr(NiTi) & RhZr(B33) & -790 & & & MnRh\({}_{7}\)(CuPt\({}_{7}\)) & -66 \\ & RhZr\({}_{2}\)(NTTi\({}_{2}\) / C16) & RhZr\({}_{2}\)(C11\({}_{1}\)) & -568 & Fe & & Fe\({}_{3}\)Rh(bcc\({}^{}_{AB3}\)) & -49 \\ & RhZr\({}_{2}\)(D0\({}_{0}\)) & IrZr\({}_{2}\)(SV\({}_{3}\)) & -428\({}^{\dagger}\) & & & Fe\({}_{2}\)Rh(Fe\({}_{2}\)Rh\({}^{1}\)) & -57 \\ \hline Hf & Hf\({}_{2}\)Rh(NiTi\({}_{2}\)) & Hf\({}_{2}\)Rh(CuZr\({}_{2}\)) & -633 & FeRh(B2) & & \\ & HfRh(B2) & HfRh(B27) & -898 & & FeRh\({}_{3}\)(D0\({}_{24}\)) & -56 \\ & Hf\({}_{3}\)Rh\({}_{4}\)(Unkn.) & & & Os & - & OsRh(RhRu\({}^{*}\)) & -8 \\ & Hf\({}_{3}\)Rh\({}_{5}\)(Ge\({}_{3}\)Rh\({}_{5}\)) & Hf\({}_{3}\)Rh\({}_{5}\)(Ge\({}_{3}\)Rh\({}_{5}\)) & -928 & Ru & - & RhRh(Pt\({}_{5}\)Ti) & -2 \\ & HfRh\({}_{3}\)(L1\({}_{2}\)) & HfRh\({}_{3}\)(L1\({}_{2}\)) & -762 & & & RhRu(RhRu\({}^{*}\)) & -8 \\ \hline Ti & \multicolumn{2}{c|}{Rh\({}_{7}\)Ti(CuPt\({}_{7}\))} & -330 & & & RhRuRu\({}_{2}\)(RhRu\({}^{*}\)) & -6 \\ & Rh\({}_{5}\)Ti(Unkn.) & & & & RhRhCu\({}_{5}\)(RhRu\({}_{5}\)) & -3 \\ & Rh\({}_{3}\)Ti(L1\({}_{2}\)) & Rh\({}_{3}\)Ti(L1\({}_{2}\)) & -631 & Co & - & - & \\ & Rh\({}_{5}\)Ti\({}_{3}\)(Ge\({}_{3}\)Rh\({}_{5}\)) & Rh\({}_{5}\)Ti\({}_{3}\)(Ge\({}_{3}\)Rh\({}_{5}\)) & -790 & Ir & - & Ir\({}_{3}\)Rh(fcc\({}^{}_{AB3}\)) & -15 \\ & RhTi (Unkn.) & RhTi(L1\({}_{0}\)) & -749 & & & Ir\({}_{2}\)Rh(Pd\({}_{2}\)Ti) & -20 \\ & RhTi\({}_{2}\)(CuZr\({}_{2}\)) & RhTi\({}_{2}\)(C11\({}_{8}\)) & -629 & & & IrRh(fcc\({}^{}_{AB2}\)) & -21 \\ \hline Nb & \multicolumn{2}{c|}{Nb\({}_{8}\)Rh(Pt\({}_{8}\)Ti)} & -131 & & & IrRh\({}_{2}\)(Pd\({}_{2}\)Ti) & -18 \\ & Nb\({}_{3}\)Rh(A15) & Nb\({}_{3}\)Rh(A15) & -288 & Rh & reference & & \\ & Nb\({}_{0.7}\)Rh\({}_{0.3}(\sigma)\) & Nb\({}_{20}\)Rh\({}_{10}\)(\(\sigma_{BAAABA}\)) & -342 & Ni & - & - & \\ & NbRh(L1\({}_{0}\) / IrTa) & NbTh(L1\({}_{0}\)) & -436 & Pt & - & PtRh(NbP) & -25 \\ & NbRh\({}_{3}\)(L1\({}_{2}\) / Co\({}_{3}\)V) & NbRh\({}_{3}\)(Co\({}_{3}\)V) & -548 & & & PtRh\({}_{2}\)(Pd\({}_{2}\)Ti) & -21 \\ \hline Ta & Rh\({}_{3}\)Ta(L1\({}_{2}\)) & Rh\({}_{3}\)Ta(L1\({}_{2}\)) & -611 & & & PtRh\({}_{3}\)(D0\({}_{22}\)) & -18 \\ & Rh\({}_{2}\)Ta(C37) & Rh\({}_{2}\)Ta(Ga\({}_{2}\)Hf) & -597\(\langle 13\rangle\) & Pd & - & - & \\ & RhTa(IFTa) & & \(\langle 11\rangle\) & Au & - & - & \\ & Rh\({}_{0.3}\)Ta\({}_{0.7}(\sigma)\) & RhTa\({}_{3}\)(A15) & -333 & Ag & - & - & \\ & RhTa\({}_{5}\)(RuTc\({}_{5}^{*}\)) & -233 & Cu & - & Cu?Rh(CuPt\({}_{7}\)) & -4 \\ & RhTa\({}_{8}\)(Pt\({}_{8}\)Ti) & -159 & Hg & “H\({}_{5}\)Rh” & Hg\({}_{4}\)Rh(
\begin{table}
\begin{tabular}{|c|c c|c||c|c|c|c|} \hline \hline & \multicolumn{2}{c|}{Compounds} & \(\Delta H\) & \multicolumn{2}{c|}{Compounds} & \(\Delta H\) \\ & Exper. & Calc. & meV/at. & & Exper. & Calc. & \(\Delta H\) \\ \hline \hline V & P\({}_{15}\)Y (Ukink.) & P\({}_{15}\)Y(D\({}_{2d}\)) & -677 & Te & - & P\({}_{15}\)Te(be) & -158 \\ & P\({}_{15}\)Y(L\({}_{12}\)) & P\({}_{15}\)Y(L\({}_{12}\)) & -983 & & P\({}_{15}\)Te(HeTeTeTe) & -184 \\ & P\({}_{15}\)Y(C\({}_{15}\)) & P\({}_{15}\)Y(C\({}_{15}\)) & -1095 & & P\({}_{15}\)Te(De) & -267 \\ & P\({}_{14}\)Y\({}_{3}\) (Ukink.) & & & Re & P\({}_{15}\)Re(Uakn.) & P\({}_{15}\)Re(be) & -128 \\ & P\({}_{17}\)Y(B27) & P\({}_{17}\)Y(B33) & -1252 & & P\({}_{15}\)Re(Sb) & -231 \\ & P\({}_{14}\)Y\({}_{5}\)(Ga\({}_{28}\)Sm) & & & Mn & Mn\({}_{29}\)Pt(L\({}_{12}\)) & Mn\({}_{29}\)Pt(D\({}_{10}\)) & -174(140 \\ & P\({}_{13}\)Y\({}_{38}\)(Ma\({}_{5}\)Si\({}_{3}\)) & & & MnPt(Li\({}_{0}\)) & & \\ & P\({}_{17}\)Y(Cl\({}_{2}\)Pb) & P\({}_{17}\)Y(Cl\({}_{2}\)Pb) & -936 & & Mn\({}_{3}\)Pt\({}_{2}\)(Ga\({}_{29}\)Pt\({}_{5}\)) & -363 \\ & P\({}_{18}\)Y(Te\({}_{3}\)Ti\({}_{7}\)) & & & & Mn\({}_{29}\)Pt(La\({}_{29}\)Br) & -365 \\ & P\({}_{19}\)Y(Ba\({}_{0}\)Di\({}_{11}\)) & -709 & & Mn\({}_{29}\)Pt(La\({}_{12}\)) & -363 \\ \hline Sc & \multicolumn{2}{c|}{P\({}_{16}\)Sr(Pt\({}_{8}\)Ti)} & -482 & \multicolumn{2}{c|}{MnPt\({}_{8}\)(Pt\({}_{8}\)Ti)} & -172 \\ & P\({}_{19}\)Sr(Li\({}_{2}\)) & P\({}_{19}\)Sr(Li\({}_{2}\)) & -1050 & Fe & P\({}_{19}\)Pt(Li\({}_{2}\)) & \(\begin{array}{c}\text{}\\ -244\end{array}\) & \multicolumn{2}{c|}{\(\begin{array}{c}\text{}\\ -200\end{array}\)} \\ & P\({}_{12}\)Sr(Ge\({}_{28}\)Hf) & -1143 & \multicolumn{2}{c|}{FePt(Li\({}_{0}\))} & \(\begin{array}{c}\text{}\\ -212\end{array}\)} \\ & P\({}_{18}\)Sr(Be\({}_{2}\)) & P\({}_{18}\)Sr(Be\({}_{2}\)) & -1232 & \multicolumn{2}{c|}{FePt\({}_{2}\)(Ga\({}_{29}\)Hf)} & -220 \\ & P\({}_{15}\)Ca(Cl\({}_{2}\)Pb) & P\({}_{15}\)Ca\({}_{2}\)Cl\({}_{2}\)Pb) & -982 & & Pe\({}_{19}\)Pt(Li\({}_{2}\)) & \(\begin{array}{c}\text{}\\ -212\end{array}\)} & -203 \\ & P\({}_{14}\)Sr(Be\({}_{2}\)Rh\({}_{12}\)Scr) & \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} & \multicolumn{2}{c|}{} \\ \hline Zr & P\({}_{18}\)Zr(Pt\({}_{8}\)Ti) & P\({}_{18}\)Zr(Pt\({}_{8}\)Ti) & -496 & Ru & - & P\({}_{18}\)Ru(Cl\({}_{1}\)) & -33 \\ & P\({}_{19}\)Zr(Di\({}_{2}\)O\({}_{24}\)) & -1031 & Co & Co\({}_{3}\)Pt(Di\({}_{10}\)) & Co\({}_{3}\)Pt(Di\({}_{10}\)) & -97 \\ & P\({}_{12}\)Zr(Cl\({}_{12}\)) & & & CoPt(Li\({}_{10}\)) & CoPt\({}_{12}\)(Co\({}_{2}\)Zr\({}_{2}\)) & -106 \\ & P\({}_{11}\)Zr(Pt\({}_{11}\)Zr) & & & & CoPt\({}_{3}\)(Li\({}_{2}\)) & & \\ & P\({}_{12}\)Zr(Ti\({}_{1}\)) & P\({}_{12}\)Hf(Na\({}_{3}\)) & -1087 & & CoPt\({}_{3}\)(Li\({}_{2}\)) & & \\ & P\({}_{13}\)Zr\({}_{2}\)(Br\({}_{2}\)Sn\({}_{8}\)) & & & & CoPt\({}_{3}\)(Hf\({}_{2}\)) & -55 \\ & P\({}_{12}\)Zr(Sn\({}_{2}\)) & P\({}_{12}\)Zr\({}_{2}\)(Cl\({}_{16}\)) & -759 & Ir & - & - & - \\ \hline Hf & H\({}_{2}\)Pt(SnTi) & H\({}_{2}\)Pt(SnTi) & H\({}_{2}\)Pt(SnTi) & -786 & Rh & - & P\({}_{18}\)Rh(NbP) & -25 \\ & H\({}_{2}\)Pt(B2/B3/Tl) & H\({}_{2}\)Pt(B3/Tl) & -1155 & & & P\({}_{18}\)Rh\({}_{2}\)(Pd\({}_{2}\)Ti) & -21 \\ & H\({}_{2}\)Pt(J\({}_{2}\)/Do\({}_{24}\)) & H\({}_{2}\)Pt(B3/D\({}_{24}\)) & -1100 & & & P\({}_{18}\)Rh\({}_{2}\)(Do\({}_{2}\)) & -18 \\ & H\({}_{2}\)Pt(B3/Tl) & -528 & Ni & Ni\({}_{2}\)Pt(Ukink.) & Ni\({}_{2}\)Pt(Do\({}_{2}\)) & -76 \\ \hline Ti & P\({}_{8}\)Ti(Pt\({}_{8}\)Ti) & P\({}_{8}\)Ti(Pt\({}_{8}\)Ti) & -433 & & NiPt(Li\({}_{0}\)) & NiPt(Li\({}_{0}\)) & -99 \\ & P\({}_{15}\)Ti(Pt\({}_{8}\)Pt\({}_{8}\)) & -617 & & & NiPt2(CoZr\({}_{2}\)) & -75 \\ & P\({}_{16}\)Ti(Pt\({}_{8}\)Pt\({}_{8}\)) & -804 & & & NiPt3(Di\({}_{0}\)Do\({}_{2}\)) & -61 \\ & P\({}_{17}\)Ti(Co\({}_{8}\)Pt\({}_{8}\)) & P\({}_{17}\)Ti(Pt\({}_{8}\)) & -804 & & NiPt3(Di\({}_{0}\)Do\({}_{2}\)) & -61 \\ & P\({}_{17}\)Ti(Pt\({}_{8}\)) & P\({}_{17}\)Ti(Pt\({}_{8}\)) & -912 & Pt & reference & & \\ & P\({}_{19}\)Ti\({}_{2}\)(Pd\({}_{3}\)Ti) & -931 & & & & & \\ & P\({}_{17}\)Ti(B1) & P\({}_{17}\)Ti(\({}_{8}\)Ti) & -938 & P4 & - & P\({}_{4}\)PtPt(CuZr\({}_{5}\)) & -14 \\ & P\({}_{16}\)Zr(Al5) & P\({}_{17}\)Ti(Al5) & -648 & & P\({}_{4}\)Pd\({}_{3}\)Pt(Cd\({}_{5}\)Pt\({}_{5}\)) & -25 \\ \hline Nb & Nb\({}_{2}\)Pt(A15) & Nb\({}_{2}\)Pt(A15) & -415 & & & P\({}_{4}\)PtPt(Li\({}_{1}\)) & -36 \\ & Nb\({}_{2}\)Pt(B19) & NbPt(Li\({}_{10}\)) & -460 & & & P\({}_{4}\)PtPt(CoZr\({}_{5}\)) & -15 \\ & NbPt(2d\({}_{0}\)Pt\({}_{2}\)) & NbPt(2d\({}_{0}\)Pt\({}_{2}\)) & -721 & An & - & & \\ & NbPt3(L60/NbPt3) & NbPt3(Nbr\({}_{3}\)/Do\({}_{0}\)) & -678 & Ag & &
\begin{table}
\begin{tabular}{|c|c c|c||c|c|c|c|} \hline \hline & Compounds & \(\Delta\mu\) & & & & Compounds & \(\Delta\mu\) \\ & Expr., & Calc. & meV/st. & & Expr. & Calc. & meV/st. \\ \hline \hline V & Pd\({}_{3}\)(CuPt\({}_{2}\)) & Pd\({}_{3}\)Y(CuPt\({}_{2}\)) & -442 & Re. & & PdRh\({}_{2}\)(D0\({}_{43}\)) & -56 \\ & Pd\({}_{3}\)Y(Li\({}_{2}\)) & Pd\({}_{3}\)Y(Li\({}_{2}\)) & -863 & Mn\({}_{2}\)Pd\({}_{3}\)(Li\({}_{2}\)) & & \\ & Pd\({}_{2}\)Y (Ukink). & & & Mn\({}_{3}\)Pd\({}_{5}\)(Ga\({}_{2}\)Pt\({}_{5}\)) & Mn\({}_{3}\)Pd\({}_{5}\)(Ga\({}_{2}\)Pt\({}_{5}\)) & -250 \\ & Pd\({}_{3}\)Y\({}_{2}\) (Ukink). & & & Mn\({}_{2}\)Pd\({}_{2}\)(C37) & -252 \\ & Pd\({}_{4}\)Y\({}_{3}\)(Pd\({}_{4}\)Pt\({}_{3}\)) & Pd\({}_{4}\)Y\({}_{3}\)(Pd\({}_{4}\)Pt\({}_{3}\)) & -923 & MnPd\({}_{3}\)(D0\({}_{23}\)) & MnPd\({}_{3}\)(Li\({}_{2}\)) & -234 \\ & Pd\({}_{3}\)Y(Ukink). & & Pav\(\lambda\)(B33) & -913 & & MnPd\({}_{3}\)(D0\({}_{23}\)) & -175 \\ & Pd\({}_{2}\)Y\({}_{3}\)(Ni\({}_{2}\)Br\({}_{3}\)) & & & & MnPd\({}_{4}\)(Pt\({}_{5}\)) & -125 \\ & Pd\({}_{3}\)Y\({}_{2}\)(C37) & -622 & Fe\({}_{3}\)Pt\({}_{5}\) & Po\({}_{5}\)Pd\({}_{4}\)\({}_{1,04}\)(Li\({}_{0}\)) & & \\ & Pd\({}_{3}\)Y(D0\({}_{23}\)) & -116 & Pd\({}_{3}\)Y(D0\({}_{23}\)) & -475 & FePd\({}_{2}\)(Cu\({}_{2}\)) & -116 \\ & Pd\({}_{3}\)Z(D0\({}_{43}\)) & Pd\({}_{3}\)Y(Li\({}_{2}\)) & -411 & & FePd\({}_{3}\)((H2\({}_{2}\))) & -112 \\ & Pd\({}_{3}\)Ge(Li\({}_{2}\)) & Pd\({}_{3}\)Ge(Li\({}_{2}\)) & -555 & & FePd\({}_{5}\)(H\({}_{2}\)Pt\({}_{5}\)) & -81 \\ & Pd\({}_{4}\)ge (Ukink). & Pd\({}_{3}\)Sc(C37) & -898 & Os. & - & - \\ & Pd\({}_{3}\)(H2) & Pd\({}_{3}\)Sr(E02) & -906 & Ru. & - & - \\ & Pd\({}_{3}\)Sr(Ni\({}_{2}\)) & Pd\({}_{3}\)Sr(Ni\({}_{2}\)) & -660 & Co. & - CuPd\({}_{3}\)(wt\(\cdot\)\({}_{2}\) & \(\varepsilon\)=4/2.8) & -10 \\ & Pd\({}_{3}\)Sr(Ni\({}_{2}\)) & Pd\({}_{3}\)Sr(Ni\({}_{2}\)) & -600 & Ir & - & - \\ & Pd\({}_{3}\)Fe({}_{4}\) (Ukink). & & & Ir & - & \\ \hline
2\({}_{x}\) & & Pd\({}_{3}\)Pd\({}_{2}\)(Pt\({}_{5}\)) & -424 & Rb. & - & - \\ & Pd\({}_{3}\)Pd\({}_{2}\)(Pt\({}_{5}\)) & -591 & Ni. & - & NiPd\({}_{3}\)(wt\(\cdot\)\({}_{2}\) & \(\varepsilon\)=2.7) & -6 \\ & Pd\({}_{3}\)Zr(D0\({}_{24}\)) & Pd\({}_{3}\)Zr(D0\({}_{24}\)) & -485 & Pt. & - & Pd\({}_{3}\)(FePt\({}_{5}\)) & -14 \\ & Pd\({}_{3}\)Zr(C11\({}_{3}\)) & & & & Pd\({}_{3}\)Pt(CuPt\({}_{5}\)) & -25 \\ & Pd\({}_{4}\)Zr3(Pt\({}_{4}\)Pt\({}_{3}\)) & & & & Pd\({}_{3}\)(Li\({}_{2}\)) & -36 \\ & Pd\({}_{3}\)Zr(Ukink). & PdZr(N33) & -645 & & PdPd\({}_{3}\)(Li\({}_{2}\)) & -26 \\ & Pd\({}_{3}\)Zr(Os\({}_{8}\)) & & & & Pd\({}_{3}\)Pt(CuPt\({}_{5}\)) & -15 \\ & Pd\({}_{3}\)Zr(Os\({}_{8}\)) & & & & Pd\({}_{3}\)Pt(CuPt\({}_{5}\)) & -15 \\ & Pd\({}_{3}\)Zr(Os\({}_{8}\)) & & & -487 & Pd\({}_{3}\)Pd(O\({}_{23}\)) & -63 \\ & Pd\({}_{3}\)Zr(Os\({}_{8}\)) & & & & Ag\({}_{2}\)Pd(Mg\({}_{2}\)Pd\({}_{5}\)) & -63 \\ & Pd\({}_{3}\)Pd\({}_{4}\) (Ukink). & & & Ag\({}_{2}\)Pd(Mg\({}_{2}\)Pd\({}_{5}\)) & -59 \\ & Pd\({}_{3}\)Pd\({}_{4}\) (Ukink). & & & Ag\({}_{2}\)Pd(Mg\({}_{2}\)Pd\({}_{5}\)) & -59 \\ & Pd\({}_{3}\)Pd\({}_{3}\)Pd\({}_{5}\) & & & -466 & & \({}_{4}\)Pd(Li\({}_{2}\)) & -59 \\ & Pd\({}_{3}\)Pt(D0\({}_{23}\)) & Pd\({}_{3}\)Pt(Pd\({}_{2}\)) & -632 & & AgPd\({}_{3}\)(CaPt\({}_{5}\)) & -31 \\ & Pd\({}_{3}\)Pt(Mg\({}_{4}\)Ti\({}_{3}\)) & Pd\({}_{5}\)Ti(Pd\({}_{4}\)Ti\({}_{3}\)) & -615 & Cu. & Cu\({}_{5}\)Pd(CuPt\({}_{5}\)) & \\ & Pd\({}_{3}\)Ti\({}_{3}\)Pt\({}_{3}\)Pt\({}_{3}\) & Pd\({}_{3}\)Ti\({}_{2}\)(Pd\({}_{3}\)Pt\({}_{3}\)) & -602 & Cu\({}_{5}\)Pd(Li\({}_{2}\)) & -672 \\ & Pd\({}_{3}\)Pt(Ni\({}_{2}\)Pd\({}_{3}\)Pt\({}_{3}\)) & Pd\({}_{3}\)Ti\({}_{2}\)(Pd\({}_{3}\)Pt\({}_{3}\)) & -602 & Cu\({}_{5}\)Pd(Li\({}_{2}\)) & -72 \\ & Pd\({}_{3}\)Pt(Ni\({}_{2}\)Pd\({}_{3}\)Pt\({}_{3}\)) & & & Cu\({}_{5}\)Pd(Li\({}_{2}\) / Sn\({}_{3}\)) & Cu\({}_{5}\)Pd(Li\({}_{2}\)) & -1072,5 \\ & Pd\({}_{3}\)Pt(O\({}_{23}\)) & Pd\({}_{3}\)Pt(O\({}_{23}\)) & -451 & CuPd(Li\({}_{2}\)) & -451 \\ & Pd\({}_{3}\)Pt\({}_{3}\)Pt\({}_{3}\)(Al\({}_{16}\)) & & & CuPd(Ukink). & CuPd(Li\({}_{2}\)) & -125 \\ \hline Nb & & Pd\({}_{3}\)Pd\({}_{3}\)(Pd\({}_{3}\)Pt\({}_{3}\)) & -167 & & CuPd(Li\({}_{2}\)) & -71 \\ & Nb\({}_{3}\)Pd\({}_{3}\)(Pd\({}_{3}\)Pt\({}_{3}\)) & -167 & & CuPd(Li\({}_{2}\)) & -71 \\ & Nb\({}_{3}\)Pd\({}_{3}\)(Pd\({}_{3}\)Pt\({}_{3}\)) & -167 & & CuPd(Li\({}_{2}\)) & -71 \\ & Nb\({}_{3}\)Pd\({}_{3}\)(Pd\({}_{3}\)Pt\({}_{3}\)) & -169 & CuPd(CsPt\({}_{3}\)) & CuPd(Cl\({}_{3}\)Pt\({}_{3}\)) & -37 \\ & Nb\({}_{3}\)Pd\({}_{3}\)(Pd\({}_{3}\)) & -167 & & CuPd(Li\({}_{2}\)) & -71 \\ & Pd\({}_{3}\)Pt(Ni\({}_{2}\)) & & & Pd\({}_{3}\)Pt(Ni\({}_, Miedema formation enthalpy, Zunger pseudo-potential radii maps, and Pettifor maps. These empirical rules and structure maps have helped direct a few successful searches for previously unobserved compounds. However, they offer a limited response to the challenge of identifying new compounds because they rely on the existence of consistent and reliable experimental input for systems spanning most of the relevant parameter space. In many cases, reliable information is missing in a large portion of this space, e.g. less than 50% of the binary systems have been satisfactorily characterized. This leaves considerable gaps in the empirical structure maps and reduces their predictive usefulness. The advance of HT computational methods makes it possible to fill these gaps in the experimental data with complementary _ab initio_ data by efficiently covering extensive lists of candidate structure types. This development was envisioned by Pettifor a decade ago, and here we present its realization for PGM alloys.
The elements along the map axes are ordered according to Pettifor's chemical scale (\(\chi\) parameter). Circles indicate agreement between computation and experiment, regarding the existence of 1:1 compounds, or lack thereof. If the circle contains a label (Strukturbericht or prototype) this denotes the structure that is stable in the given system at this stoichiometry. Rectangles denote disagreement between experiments and computation about the 1:1 compounds, in systems reported as compound forming (blue rectangles) or as non-compound forming (red and gray rectangles). In the lower left part of the map, there is a region of non-compound forming systems, whereas the upper part of the map is mostly composed of compound-forming systems. In the upper part of the map, experiment and computation agree, preserving a large cluster of B2 structures, or differ slightly on the structure reported to have the lowest formation enthalpy at 1:1 (blue rectangles). For example, the 1:1 phases of Hf-Pd and Pd-Zr are unknown according to the phase diagram literature, but we find the stable phases with B33 structure, right next to Hf-Pt in the diagram, which is reported as a B33 structure. Similarly, stable L1\({}_{0}\) structures are identified in the Ir-Ti and Rh-Ti systems, adjacent to a reported cluster of this structure. Two additional L1\({}_{0}\) structures are identified in the Cd-Pd and Pd-Zn systems, instead of the reported CuTi structures, extending a small known cluster of this structure at the bottom right corner of the map. These are examples of the capability of HT _ab initio_ results to complement the empirical Pettifor maps, and extend their regions of predictive input, in a way consistent with the experimental data.
In the middle of the map, in a rough transition zone between compound-forming and non-compound-forming regions, computation finds quite a few cases where stable compounds are predicted in systems where none have been reported experimentally (pink rectangles). Most prominent here is a large cluster of B19 compounds. Nine systems marked by light gray rectangles are reported in experiments as having no compounds, but our calculations find stable compounds at stoichiometries other than 1:1.
At the stoichiometries of 1:2 and 2:1, shows significant additions of the calculations to the experimental data on compound-formation. Again, the systems where computation finds stable compounds in experimentally non-compound-forming systems are found at the border between the compound-forming region (dark gray circles and white labeled circles) and the non-compound-forming region (light gray circles), or fill isolated gaps within the compound-forming regions. The calculations augment islands of structurally-similar regions, yielding a more consistent structure map. For example, calculation finds the CuZr\({}_{2}\) structure for Nb-Pd, extending the island of this structure already present in the experimental results (left panel, upper right). The calculations significantly extend the Hg\({}_{2}\)Pt island in the lower right of the B\({}_{2}\)A panel, from a single experimental entry to 6 systems (in Hg-Pt itself, the calculation finds this structure slightly unstable at \(T=0K\), 25meV/atom above the stability tie-line). A cluster of \(\sigma\) phases in the left panel shows that this reported disordered phase has underlying ordered realizations at low temperatures. Three completely new islands, for the C37, Ga\({}_{2}\)Hf and IrTc\({}_{2}\) structures, appear near the upper center of the A\({}_{2}\)B panel. Another new cluster, of the Pd\({}_{2}\)Ti structure, appears at the lower center of both panels. In general, the clusters of blue rectangles, show that the calculations augment the experimental results in a consistent manner.
\begin{table}
\begin{tabular}{||c|c|c|c|c|c||} \hline \hline Formula & IzZn & Nb\({}_{2}\)Pd & Fe\({}_{2}\)Rh & Ag\({}_{2}\)Pt\({}_{2}\) & Ag\({}_{2}\)Pd\({}_{3}\) \\ \hline Lattice & Monoclinic & Orthorhombic & Orthorhombic & Rhombohedral & Monoclinic \\ \hline Space Group & \(C2/m\) No. 12 & \(Cmmm\) No. 65 & \(Cmmm\) No. 65 & \(R3m\) No. 166 & \(C2/m\) No. 12 \\ \hline Pearson symbol & mS8 & oS8 & oS12 & hR5 & mS10 \\ \hline Bravais lattice type & MCLC & ORCC & ORCC & RHL & MCLC \\ Lattice variation & MCLC\({}_{1}\) & ORCC & ORCC & RRL\({}_{1}\) & MCLC\({}_{3}\) \\ \hline Conv. Cell: \(a,b,c\) (A) & 1.94, 3.83, 1.12 & 1.26, 1.78, 3.56 & 1.78, 5.35, 1.26 & 1.12, 1.12, 13.75 & 3.55, 1.59, 1.94 \\ \(\alpha,\beta,\gamma\) (dec) & 72.98, 90, 90 & 90, 90, 90 & 90, 90, 90, 90 & 90, 90, 120 & 65.9, 90, 90 \\ \hline Wyckoff & Ir \(\frac{1}{2}\),\(\frac{1}{2}\),\(-0.292\) (4i) & Nb1 0 0, 0, \(\frac{1}{4}\) (4k) & Fe1 \(\frac{1}{2}\),\(\frac{1}{6}\),\(0\) (4g) & Ag1 0, 0, \(\frac{1}{2}\) (2c) & As \(\frac{2}{2}\),\(\frac{1}{2}\),\(\frac{1}{2}\),\(\frac{1}{2}\),\(\frac{1}{2}\),\(\frac{1}{2}\),\(\frac{1}{2}\) (4i) \\ positions & Zn \(\frac{1}{2}\),\(\frac{1}{2}\),\(-0.208\) (4i) & Nb2 \(\frac{1}{2}\),\(\frac{1}{2}\) (2c) & Fe2 0, 0, \(\frac{1}{2}\) (2d) & Ag2 0, 0, 0 (1a) & Pd1 0, 0, \(\frac{1}{2}\) (2c) \\ & & Pd \(\frac{1}{2}\), 0, 0 (2b) & Fe3 \(\frac{1}{2}\),\(0\),\(0\) (2b) & Pt 0, 0, \(\frac{1}{2}\) (2c) & Pd2 \(\frac{1}{2}\),\(\frac{1}{2}\),\(\frac{1}{2}\) (4i) \\ & & Rh \(\frac{1}{2}\), 0, \(\frac{1}{2}\) (4h) & Rh \\ \hline AFLOW label & 123 & 72 & bS3 & fS8 & fS55 \\ \hline \hline \end{tabular}
\end{table}
Table 7: Geometry of new prototypes marked by \(\dagger\) in tables III to VI.
The structure map for 1:3 phases is shown in Similarly to the 1:1 and 1:2 maps, the calculation extends structural islands of the experimental data, most new phases in non-compound-forming systems occur in systems at the boundary between compound-forming and non-compound-forming regions, and there is significant agreement between the experimentally reported phases (or lack thereof) and calculated phases. In the upper part of the right panel, the L1\({}_{2}\) and D0\({}_{24}\) clusters are preserved with slight modifications at their boundaries (at Pt-Ti, the PuAl\({}_{3}\) structure is only 3 meV/atom lower than the experimental structure D0\({}_{24}\), a difference too small to be significant). The D0\({}_{19}\) cluster is significantly expanded. In the left panel, the calculations introduce a new D0\({}_{19}\) island near the center of the diagram. New small regions of the D0\({}_{22}\) structures emerge at the right bottom of both panels. Adjacent D0\({}_{23}\) and CdPt\({}_{3}^{*}\) islands appear in the left and right panels, respectively. The experimental D0\({}_{e}\) structure for RhZr\({}_{3}\) may actually be SV\({}_{3}\), since in the calculation the D0\({}_{e}\) structure relaxed into the SV\({}_{3}\) structure, creating a small SV\({}_{3}\) island at the top of the left panel.
The structure maps of figs. 2 to 4 give a bird's eye view of the exhaustive HT search for new structures. Consistently with the empirical maps, they show significant separation of different structures into regions where the constituent elements have a similar Pettifor \(\chi\) number. The HT data significantly enhances the empirical maps, extends the regions of some structures, fills in apparent gaps and indicates previously unsuspected structure clusters. Moreover, the HT data contains more detail than is apparent in the structure maps. Even when calculation and experiment agree that a system is compound-forming (green [dark gray] circles in Fig. 1), the calculations often find additional stable compounds, beyond those known in experiment. When the reported structures are found to be unstable in the calculation, they are usually just slightly less stable than the calculated groundstate, or just slightly above the convex hull in a two phase region. Such cases and numerous additional predictions of marginally stable structures harbor further opportunities for materials engineering and applications.
## V Conclusions
In this study, the low temperature phase diagrams of all binary PGM-transition metal systems are constructed by HT _ab initio_ calculations. The picture of PGM alloys emerging from this study is much more complete than that depicted by current experimental data, with dozens of stable structures that have not been previously reported. We predict ordering in 37 systems reported to be phase-separating and in five systems where only disordered phases are reported. In addition, in the known ordering systems, we find many cases in which more phases are predicted to be stable than reported in the experimental phase diagrams. These _ab initio_ results complement the ordering tendencies implied by the empirical Pettifor maps.
A Pettifor-type structure map for 1:1 stoichiometry compounds in PGM binary systems. Circles indicate agreement between experiment and computation: white circles with Strukturbericht or prototype labels denote 1:1 compounds, dark circles indicate a compound-forming system with no compounds at 1:1, light circles denote non-compound forming systems. Blue rectangles denote compound-forming systems where the reported and computed stable structures differ at 1:1 stoichiometry. The top label in the rectangle is the reported structure, the bottom label is the structure we find to be stable in this work. A dash “\(-\)” indicates the absence of a stable structure. Unidentified suspected structures are denoted by a question mark “?”. Pink rectangles indicate systems reported as non-compound forming, with a dash at the top of the rectangle, but we find a stable 1:1 phase, identified at the bottom of the rectangle. Light gray rectangles indicate systems reported as non-compound forming where a structure is predicted at a stoichiometry different from 1:1. A dark gray rectangle indicates a system reported with a disordered compound where no stable structures are found in the calculation.
These maps demonstrate that the integration of the empirical and computational data produces enhanced maps that should provide a more comprehensive foundation for rational materials design. The theoretical predictions presented here will hopefully serve as a motivation for their experimental validation and be a guide for future studies of these important systems.
The maps in Figs. 2-4 include a large number of light blue rectangles, pointing to experiment-theory mismatches on structures at simple compositions in binary systems known to be compound forming. This may raise reservations that the level of theory employed, DFT-PBE, may not be as good as commonly accepted for transition metal alloys. A more careful look, however, shows that many of these mismatches, e.g.
A Pettifor-type structure map for 1:2 stoichiometry compounds in PGM binary systems. The symbols are as in Fig. 2, with the map stoichiometry changed respectively from 1:1 to 1:2 or 2:1.
The calculation thus reveals the stable structure and closes the gap in the experimental data. In most other cases, e.g. RhZr, PtV, Ir\({}_{3}\)V, Rh\({}_{2}\)Ta, Cu\({}_{3}\)Pt, the energy difference between the reported structure and the calculated structure or two-phase tie-line is rather small and is congruent with the adjacent structure clusters in the maps. Similar improved consistency with reported structure clusters also appears in cases where the discrepancies are considerable, e.g. CdPd and PdZn. In addition, as discussed in Sec. III, the calculations reproduce many complex large unit cell structures that are reported in the experimental literature. Moreover, it is important to remember that experiments are performed at room temperature or higher, while our calculations are carried out at zero temperature. Many phase discrepancies may therefore be due to vibrational promotion, or the tendency of structures to gain symmetries by loosing their internal Peierls instabilities or Jahn-Teller distortions.
A Pettifor-type structure map for 1:3 stoichiometry compounds in PGM binary systems.
retical treatment, but a demonstration of its usefulness is bridging gaps in the experimental data and extending it towards unknown phase transitions at lower temperatures. The ultimate test of this issue rests with experimental validation of at least some of our predictions, which would hopefully be motivated by this work.
To help accelerate this process of experimental validation, discovery and development of materials we are in the process of setting up a public domain REST-API that will allow the scientific community to download information from the www.aflowlib.org repository. It would ultimately enable researchers to generate alloy information remotely on their own personal computers. Extension of the database to nano-alloys and nano-sintered systems is planned within the size-pressure approximation (i.e. Fig.2 of Ref.), to study trends of solubility and size-dependent disorder-order transitions and segregation in nano-catalysts, and nano-crystals.
A few of our predictions correspond to phases where the driving force for ordering is small (i.e., the formation enthalpy is small and it may be difficult to reach thermal equilibrium), however, it should be noted that some experimentally reported phases have similarly small formation enthalpies. Some of these predicted phases could be more easily realized as nano-structured phases, where the thermodynamics for their formation may be more favorable. Our results should serve as the foundation for finite temperature simulations to identify phases that are kinetically accessible. Rapid thermodynamical modelling and descriptor-based screening of systems predicted to harbor new phases should be used to pinpoint those with the greatest potential for applications. Such simulations would be an invaluable extension to this work, however, the necessary tools to accomplish them on a similarly large scale are not yet mature.
| 10.48550/arXiv.1308.4357 | A high-throughput ab initio review of platinum-group alloy systems | Gus L. W. Hart, Stefano Curtarolo, Thaddeus B. Massalski, Ohad Levy | 31 |
10.48550_arXiv.1104.3434 | ## Institute of Solid State Chemistry, Ural Branch of the Russian Academy of Sciences, 620990, Ekaterinburg, Russia
Footnote *: Corresponding author.
_E-mail (Igor R. Shein).
## Abstract
First-principles FLAPW-GGA band structure calculations were employed to examine the structural, electronic properties and the chemical bonding picture for four ZrCuSiAs-like Th-based quaternary pnictide oxides ThCuPO, ThCuAsO, ThAgPO, and ThAgAsO. These compounds were found to be semimetals and may be viewed as "intermediate" systems between two main isostructural groups of superconducting and semiconducting 1111 phases. The Th _5f_ states participate actively in the formation of valence bands and the Th _5f_ states for Th_MPn_O phases are itinerant and partially occupied. We found also that the bonding picture in Th_MPn_O phases can be classified as a high-anisotropic mixture of ionic and covalent contributions: inside [Th\({}_{2}\)O\({}_{2}\)] and [\(M_{2}Pn_{2}\)] blocks, mixed covalent-ionic bonds take place, whereas between the adjacent [Th\({}_{2}\)O\({}_{2}\)]/[\(M_{2}Pn_{2}\)] blocks, ionic bonds emerge owing to [Th\({}_{2}\)O\({}_{2}\)] \(\rightarrow\) [\(M_{2}Pn_{2}\)] charge transfer.
_PACS_ : 71.18.+y, 71.15.Mb,74.25.Jb
_Keywords:_ ThCuPO; ThCuAsO; ThAgPO; ThAgAsO; Structural, Electronic properties, Inter-atomic bonding, first principles calculations
## 1.
The recent discovery of superconductivity in the layered Fe-based pnictide oxides with an unconventional pairing mechanism near the spin-density wave order aroused tremendous interest and in the three last years inspired a great research activity in the field of condensed matter physics and materials science, reviews.
One of the most extensively studied groups of these materials consists of the so-called 1111 phases, namely, quaternary pnictide oxides \(Ln\)Fe\(Pn\)O, where \(Ln\) are early rare earth metals such as La, Ce, Sm, Dy, Gd _etc._, and \(Pn\) are pnictogens. These \(Ln\)Fe\(Pn\)O phases adopt a layered ZrCuSiAs-type structure, where \([Ln_{2}\)O\({}_{2}]^{\delta^{+}}\) blocks are sandwiched between \([\)Fe\({}_{2}Pn_{2}]^{\delta^{-}}\) blocks. Let us note that: (i). among more than 70 synthesized ZrCuSiAs-type 1111 layered phases, only a few (basically, Fe-containing) were involved till now in the search of superconducting materials, and their electronic properties were examined; (ii). non-doped phases \(Ln\)Fe\(Pn\)O are located on the border of magnetic instability and commonly exhibit temperature-dependent structural and magnetic phase transitions with the formation of antiferromagnetic spin ordering, and (iii). superconductivity emerges as a result of hole or electron doping of the parent compounds, see.
The atomic substitutions inside building blocks of 1111 phases exert a profound influence on their properties, and, in particular, on their superconductivity. Therefore chemical substitution (doping effects) is one of the main strategies for improvement of the properties of these systems. A large number of investigations of the doping effects on the above ZrCuSiAs-like \(Ln\)Fe\(Pn\)O phases were performed today, and the most extensively studied situations are \(d^{n<10}\) metals substitutions at the Fe site, P substitutions at the As site, and F substitutions at the oxygen site.
Besides, a set of doped 1111 superconductors were obtained recently using quite an unusual type of substitutions: Th\({}^{4+}\to Ln^{3+}\). So, superconductivity through Th substitution at the lanthanide site was found for GdFeAsO (\(T_{\rm C}\sim\) 56K), NdFeAsO (\(T_{\rm C}\sim\) 38K), LaFeAsO (\(T_{\rm C}\sim\) 30K), and SmFeAsO (\(T_{\rm C}\sim\) 50K). However, as distinct from other \(sp\) and \(d\) impurities, the influence of Th on the electronic structure of 1111 phases remains actually unstudied.
Besides the above thorium-doped 1111 phases, a set of "pure" ZrCuSiAs-like Th-based pnictide oxides Th_MPn_O, where \(M\) are Cu or Ag and \(Pn\) are P or As, were synthesized, see.
In this paper, in order to get a insight into the electronic properties and the peculiarities of inter-atomic bonding in thorium-containing 1111 materials, a first-principles study of four thorium-based pnictide oxides: synthesized ThCuPO, ThCuAsO, ThAgPO and hypothetical ThAgAsO was performed. This choice allows us to compare the above properties of these related phases as a function of: (i) the pnictogen type (P _versus_ As) and (ii) the \(d\) metal type (Cu _versus_ Ag). As a result, the structural parameters, electronic bands, total and site-projected \(l\)- decomposed densities of states, and the peculiarities of the inter-atomic interactions for the above Th_MPn_O phases have been obtained and analyzed for the first time.
## 2.
The examined pnictide oxides Th_MPn_O adopt a tetragonal layered structure of the ZrCuSiAs-type, space group \(P4/nmm\), \(\#\) 129, Z = 2. The atomic positions are Th: 2\(c\) (\(\surd_{4}\),\(\surd_{4}\),\(z_{\rm Th}\)); \(M\): 2\(b\) (\(\surd_{4}\),\(\surd_{4}\),\(\surd_{2}\)); \(Pn\): 2\(c\) (\(\surd_{4}\),\(\surd_{4}\),\(z_{Pn}\)); and O: 2\(a\) (\(\surd_{4}\),\(\surd_{4}\),0); here, \(z_{\rm Th}\) and \(z_{Pn}\) are the so-called internal coordinates. These structures (see Fig. 1) consist of alternating fluorite-type [\(M_{2}Pn_{2}\)] and anti-fluorite-type [Th\({}_{2}\)O\({}_{2}\)] blocks. In turn, the [\(M_{2}Pn_{2}\)] blocks consist of square nets of \(M\) atoms, which are tetrahedrally coordinated by four pnictogens. For [Th\({}_{2}\)O\({}_{2}\)] blocks,oxygen atoms are tetrahedrally coordinated by four thorium atoms, and Th is coordinated by four O to form square pyramids.
The calculations of all mentioned Th_MPn_O phases were carried out by means of the full-potential method with mixed basis APW+lo (LAPW) implemented in the WIEN2k suite of programs. The generalized gradient correction (GGA) to exchange-correlation potential in PBE form was used. The _muffn-tin_ spheres radii were chosen 2.5 a.u. for Th, 1.6 a.u. for O, and 2.0 a.u. both for Cu/Ag and P/As. The starting configurations were: [Rn](\(6d^{2}\)\(7s^{2}\)\(7p^{0}\)\(5f^{0}\)) for thorium, [Ne](\(3s^{2}\)\(3p^{3}\)) for P, [Ar](\(3d^{10}\)\(4s^{2}\)\(4p^{3}\)) for As, [Ar](\(3d^{10}\)\(4s^{1}\)) for Cu, [Kr](\(4d^{10}\)\(5s^{1}\)) for Ag, and [He](\(2s^{2}\)\(2p^{4}\)) for oxygen. The maximal value for partial waves used inside atomic spheres was \(l=12\) and the maximal value for partial waves used in the computation of _muffn-tin_ matrix elements was \(l=4\). The plane-wave expansion with \(R_{\rm MT}\times K_{\rm MAX}\) was equal to 7, and \(k\) sampling with a 10\(\times\)10\(\times\)10 \(k\)-points mesh in the Brillouin zone was used. Relativistic effects were taken into account within the scalar-relativistic treatment including spin-orbit coupling (SOC). The self-consistent calculations were considered to converge when the difference in the total energy of the crystal did not exceed 0.001 mRy and the difference in the atomic forces did not exceed 1 mRy/a.u. as calculated at consecutive steps.
The hybridization effects were analyzed using the densities of states (DOSs), which were obtained by a modified tetrahedron method. The ionic bonding was considered using Bader analysis. In this approach, each atom of a crystal is surrounded by an effective surface that runs through minima of the charge density, and the total charge of an atom (the so-called Bader charge, Q\({}^{\rm B}\)) is determined by integration within this region. In addition, some peculiarities of the inter-atomic bonding picture were visualized.
## 3 Results and discussion
As the initial step, the structure of the Th-containing 1111 phases was fully optimized and the equilibrium values of lattice constants and internal coordinates were found. The obtained results are presented in Table 1 in comparison with available experimental data.
It is seen that both \(a\) and \(c\) parameters increase as going from Th_M_PO to Th_M_AsO and from ThCu_Pn_O to ThAg_Pn_O. This result can be easily explained by considering the atomic radii of \(M\) and _Pn_: R(P) = 1.30 A \(<\) R(As) =1.48 A and R(Cu)=1.28 A \(<\) R(Ag) =1.44 A.
On the other hand, in the P \(\rightarrow\) As or Cu\(\rightarrow\) Ag replacements the effect of _anisotropic deformation_ of the crystal structure was found. Indeed, as going from ThCuPO to ThCuAsO, the \(a\) and \(c\) parameters increase 1.019 and 1.016 times respectively, whereas as going from ThCuPO to ThAgPO, they increase 1.022 and 1.060 times, respectively. The _anisotropic deformation_ of the crystal structure owing to atomic substitutions was also found for a set of related layered phases, see for example. Possibly, this effect (related to strong _anisotropy of inter-atomic bonds_, see also below) has a universal character for all similar layered phases.
### 3.2 Electronic properties
Figures 2-4 and 6 show the band structures and total and atomic-resolved _l_-projected densities of states (DOSs) in ThCuPO, ThCuAsO, ThAgPO, and ThAgAsO phases as calculated for equilibrium geometries.
First of all, since the effect of spin-orbit coupling (SOC) on the electronic structure of materials with light actinides is widely discussed now, we will compare the results for the band structure and DOSs as obtained in our FLAPW calculations without SOC (non-SOC) and within spin-orbit coupling (using ThCuPO as an example), see
It is seen that SOC results mainly in a shift and splitting of semi-core Th 6\(p\) states in the region from -22 eV to -14 eV below the \(\rm E_{F}\). Besides, small splitting of valence bands takes place in the region from -3 eV to -1 eV. However, the common picture of valence bands (and also of DOSs distributions) for ThCuO as obtained in our calculations without SOC and within spin-orbit coupling varies insignificantly. Further we present the results obtained within SOC.
From Figs. 3 and 4 it is seen that all of these phases will be metallic-like, but with very low densities of states at the Fermi level: N(\(\rm E_{F}\))\(\sim\)0.05-0.17 states/(eV\(\cdot\)cell). Therefore these Th-based 1111 phases may be classified as _semimetals_ and may be viewed as "intermediate" systems between the group of the above mentioned superconducting 1111 phases and the group of ZrCuSiAs-like semiconductors such as _LnZnPn_O, see.
Naturally, the topology of the Fermi surface (FS) for Th\(MPn\)O phases should differ completely from the 2D-like FSs for Fe_Pn_-based 1111 superconducting materials, which consist of electron cylinders around the tetragonal \(M\) point, and hole cylinders around the \(I\) point, see.
Let us discuss the common features of the electronic structure of the examined Th-based 1111 phases. Their valence spectra contain three main occupied bands, labeled in as peaks A, B, and C. The lowest band (peak A lying in the region from about -12 eV to -10 eV) arises mainly from _Pn s_ states and is separated from the near-Fermi bands by a gap. These bands (peaks B and C) are located in the energy range from - 7 eV to \(\rm E_{F}\) and are of a mixed type. Indeed, as can be seen from Fig. 6, where the site-projected _l_-decomposed densities of states for ThCuPO are depicted, the peak B contains hybridized Cu\(3d\), O \(2p\), P \(3p\), and Th (\(p\),\(d\)\(f\)) states, which are responsible for the formation of the covalent component of bonds Cu-P (inside [Cu\({}_{2}\)P\({}_{2}\)] blocks) and Th-O (inside [Th\({}_{2}\)O\({}_{2}\)] blocks). The insert in shows also that Th (\(p\),\(d\)\(f\)) states participate actively in the formation of the valence band, and in its topmost part the contributions from Th \(6d\) and Th \(5f\) states are comparable. Thus, like in metallic thorium and a series of thorium compounds with light \(sp\) atoms (H, B, C, N,O), the Th \(5f\) states for 1111 phases are itinerant and partially occupied. This fact means that the charge state of thorium atoms in 1111-phases differs from that of the purely ionic Th\({}^{4+}\), see also below. The bottom of the empty conductivity band for the examined phases is formed mostly by the Th \(6d\) and Th \(5f\) states; the contributions of other atoms are negligible.
Quantitative differences between the examined Th_MPn_O phases may be seen from the data presented in Figs. 3, 4 and Table 2, where the band structure parameters (bandwidths) of these materials are summarized as dependent on P \(\rightarrow\) As or Cu \(\rightarrow\) Ag replacements.
### 3.3 Inter-atomic bonding
Let us discuss the bonding picture in the examined Th_MPn_O phases. To describe the _ionic bonding_ for these materials, we will start with a purely ionic picture with usual oxidation numbers of atoms: Th\({}^{4+}\), \(M^{1+}\), O\({}^{2-}\), and \(Pn^{3-}\). So, the charge states of the blocks are [Th\({}_{2}\)O\({}_{2}\)]\({}^{4+}\) and [\(M_{2}Pn_{2}\)]\({}^{4-}\), _i.e._ the charge transfer (\(4e\)) occurs from [Th\({}_{2}\)O\({}_{2}\)] to [\(M_{2}Pn_{2}\)] blocks. Besides, inside [Th\({}_{2}\)O\({}_{2}\)] and [\(M_{2}Pn_{2}\)] blocks, the ionic bonding takes place between ions with opposite charges: Th - O and \(M\) - \(Pn\).
To estimate numerically the amount of electrons redistributed between various atoms and between adjacent [Th\({}_{2}\)O\({}_{2}\)]\({}^{\delta+}\)/[\(M_{2}Pn_{2}\)]\({}^{\delta-}\) blocks, we carried out a Bader analysis, and the obtained effective atomic charges \(\Delta\)Q are presented in Table 3. These results show that the inter-atomic and inter-blocks charge transfer is considerably smaller than it was predicted in the idealizedionic model. Namely, the transfer \(\Delta\)Q([Th\({}_{2}\)O\({}_{2}\)]\(\rightarrow\) [\(M_{2}\)_Pn\({}_{2}\)_]) is about 2.08-1.94 \(e\); these values are smaller for Cu-containing phases than for their Ag-containing counterparts and are smaller for arsenides as compared with the related phosphides. These results confirm that all Th_MPn_O phases are partially ionic compounds - owing to the presence of appreciable covalent contributions.
Indeed, the character of _covalent Th-O and M-Pn bonding_ in Th_MPn_O phases (owing to hybridization of the states Th (_p,d,f_)- O 2\(p\) and \(M\)\(d\)\(-\)_Pn \(p\)_, respectively) _inside_ [Th\({}_{2}\)O\({}_{2}\)] and [\(M_{2}\)_Pn\({}_{2}\)_] blocks may be understood from site-projected DOSs calculations, see above. These bonds are also well visible on the 3D charge density distribution picture, see On the other hand, no direct inter-atomic bonds _between_ the adjacent blocks are present.
Thus, summarizing the above results, the picture of chemical bonding for Th_MPn_O phases may be described in the following way. Inside [Th\({}_{2}\)O\({}_{2}\)] blocks mixed covalent-ionic bonds Th-O take place (owing to hybridization of Th (_p,d,f_)- O 2\(p\) states and Th \(\rightarrow\) O charge transfer), while the estimated effective charges of Cu and Ag atoms in 1111-compounds are small. Therefoer it should be expected that the covalent component of _M-Pn_ bonding is predominant _inside_ [\(M_{2}\)_Pn\({}_{2}\)_] blocks. Between the adjacent [Th\({}_{2}\)O\({}_{2}\)]/[\(M_{2}\)_Pn\({}_{2}\)_] blocks, ionic bonds emerge owing to [Th\({}_{2}\)O\({}_{2}\)] \(\rightarrow\) [\(M_{2}\)_Pn\({}_{2}\)_] charge transfer. Generally, the bonding in Th_MPn_O phases can be classified as a high-anisotropic mixture of ionic and covalent contributions.
## 4 Conclusions
In summary, by means of the FLAPW-GGA approach, we studied the structural, electronic, and chemical bonding picture for four ZrCuSiAs-like Th-based quaternary pnictide oxides ThCuPO, ThCuAsO, ThAgPO, and ThAgAsO.
Our results show that the replacements of \(d\) metal atoms (Cu \(\leftrightarrow\) Ag) and pnictogen atoms (P \(\leftrightarrow\) As) lead to _anisotropic deformations_ of the crystal structure; this effect is related to strong anisotropy of inter-atomic bonds.
Our studies showed that the examined Th_MPn_O phases may be classified as _semimetals_ and may be viewed as "intermediate" systems between the main groups of superconducting and semiconducting ZrCuSiAs-like phases. The Fermi surfaces for these Th-based 1111 phases (consisting of isolated closed hole-like and electronic-like pockets) are completely different from the FSs for Fe_Pn_-based 1111 superconducting materials, which consist of a set of 2D-like electron and hole cylinders. The data obtained show also that Th 5\(f\) states participate actively in the formation of valence bands. In other words, the Th 5\(f\) states for 1111 phases are itinerant and partially occupied.
Finally, we found that the bonding in Th_MPn_O phases can be classified as a high-anisotropic mixture of ionic and covalent contributions, where mixed covalent-ionic bonds take place inside [Th\({}_{2}\)O\({}_{2}\)] and [\(M_{2}Pn_{2}\)] blocks, whereas between the adjacent [Th\({}_{2}\)O\({}_{2}\)]/[\(M_{2}Pn_{2}\)] blocks, ionic bonds emerge owing to [Th\({}_{2}\)O\({}_{2}\)] \(\rightarrow\) [\(M_{2}Pn_{2}\)] charge transfer.
#
Band structures of ThCuPO, ThAgPO, ThCuAsO, and ThAgAsO.
Total densities of electronic states for ThCuPO, ThAgPO, ThCuAsO, and ThAgAsO.
(_Color online_) Fermi surfaces for ThAgAsO, ThAgPO, ThCuAsO, and ThCuPO.
(_Color online_) Total and partial atomic- and \(l\)-projected densities of states for ThCuPO. _Insert_ – the \(l\)-projected DOS of thorium in the near-Fermi region (1 – Th-\(s\) states, 2 – Th-\(p\) states, 3 – Th-6\(d\) states, and 4 –Th-5\(f\) states). | 10.48550/arXiv.1104.3434 | Electronic band structure and inter-atomic bonding in layered 1111-like Th-based pnictide oxides ThCuPO, ThCuAsO, ThAgPO, and ThAgAsO from first principles calculations | V. V. Bannikov, I. R. Shein, A. L. Ivanovskii | 1,872 |
10.48550_arXiv.1808.05847 | ###### Abstract
Ferroelectric materials are interesting candidates for future photovoltaic applications due to their potential to overcome the fundamental limits of conventional single bandgap semiconductor-based solar cells. Although a more efficient charge separation and above bandgap photovoltages are advantageous in these materials, tailoring their photovoltaic response using ferroelectric functionalities remains puzzling. Here we address this issue by reporting a clear hysteretic character of the photovoltaic effect as a function of electric field and its dependence on the poling history. Furthermore, we obtain insight into light induced nonequilibrium charge carrier dynamics in Bi\({}_{2}\)FeCrO\({}_{6}\) films involving not only charge generation, but also recombination processes. At the ferroelectric remanence, light is able to electrically depolarize the films with remanent and transient effects as evidenced by electrical and piezoresponse force microscopy (PFM) measurements. The hysteretic nature of the photovoltaic response and its nonlinear character at larger light intensities can be used to optimize the photovoltaic performance of future ferroelectric-based solar cells.
## Introduction
The non-centrosymmetric structure of large bandgap polar materials induces an internal electric field comparable to that existing in the p-n junction region of semiconductor based solar cells. Subsequently, electrically polar materials with photovoltaic (PV) properties have gained renewed attention with regard to photovoltaics and other attractive multi-functionalities. Although the photovoltaic effect in non-centrosymmetric crystals has been known for long, this field has attracted increasing attention following the discovery of photovoltaic effects in the multiferroic BiFeO\({}_{3}\)(BFO). Several ferroelectric (FE) materials in the thin film form such as BiMnO\({}_{3}\) (BMO), La and Ni-doped Pb(Zr,Ti)O\({}_{3}\) (PLZT), BaTiO\({}_{3}\) (BTO) and bulk [KNbO\({}_{4}\)]\({}_{1-\{\rm{BaNi}}}\)/[BaNi\({}_{12}\)/Nb\({}_{12}\)O\({}_{3-\delta}\)] (KBNNO) have been studied over the last few years, but none of them were able to generate a remarkable efficiency, mainly because of their bandgap being larger than 2.5 eV. In contrast, the multiferroic BFO and related materials showing moderate energy bandgaps manifest interesting PV properties. Moreover, the bandgap of BFO can be varied by doping and by modulating the preparation conditions. Indeed, Cr doping of BFO results in a double perovskite Bi\({}_{2}\)FeCrO\({}_{6}\) (BFOO) structure with different amounts of Fe-Cr cationic disorder, which allows bandgap engineering down to 1.9 eV. Nechache _et al._ claimed that in perfectly ordered BFCO the bandgap can be even lower (1.4 eV) and succeeded to fabricate BFCO based solar cells with efficiencies of 3.3% and 8.1% using a single layer and a tandem-like configuration, respectively. These efficiencies are expected to improve further because, at least theoretically, solar cells integrating ferroelectric materials could exceed the Shockley-Queisser limit. Moreover, the increase of efficiency in FE solar cells also requires a deep insight into their dependence on the FE state, and the charge generation-evolution mechanisms. Here we investigate these issues by using careful photovoltaic response measurements _versus_ electric field and demonstrate how the PV effect can be tuned by varying the applied voltage and related poling history. We also study the light intensity dependence of the PV effect aiming at the fundamental understanding of the ferroelectric based photovoltaic performance.
## Results and discussion
In order to perform the electrical measurements, a 100 nm thick transparent ITO layer was sputtered at room temperature on the BFCO film, providing both good electrical conductivityand high optical transparency (Fig. 1, left). The 1 mm\({}^{2}\) area of the electrode was then brought into contact with a copper wire using a conductive epoxy (Fig. 1, right). shows the \(\theta\) -2\(\theta\) X-ray diffractogram recorded for Bi\({}_{2}\)FeCrO\({}_{6}\) deposited on the Nb:STO substrate (NSTO). In addition to the 00\(h\) peaks of the NSTO substrate, we observe the 001 peaks of the BFO films. The position of the peaks suggests an epitaxial growth of the pseudo-cubic BFCO phase with the 001 axis parallel to the growth direction. No secondary phases could be observed in the resolution limit of the X-ray diffraction technique. The out-of-plane (OP) lattice parameter \(c\) of the BFCO layer was determined to be 0.3965 nm, close to the lattice parameter of bulk Bi\({}_{2}\)FeCrO\({}_{6}\) (0.3930 nm). The epitaxial growth of our films was unambiguously demonstrated by \(\Phi\) scan measurements.
Given the low doping level (\(<\)1% of Nb) of NSTO substrates, the same epitaxial quality is obtained when BFCO films are grown on STO and NSTO. The optical properties of the as-grown BFCO films were also investigated by UV-Vis-NIR spectrophotometry measurements. shows the normalized absorption spectra recorded on the STO substrate and on a BFCO film grown on STO. The film strongly absorbs in a large range from 200 to 800 nm. illustrates the corresponding Tauc plot, where the optical absorption coefficient (\(a\)) relates to bandgap (\(E_{g}\)) _via_ Planck's constant (\(h\)) and the frequency of the incident photon (\(\nu\)) as: \(\alpha=(h\nu-E_{g})^{1/2}\).
Assuming a direct allowed bandgap for the BFCO film, the bandgap of 1.66 \(\pm\) 0.04 eV is evaluated. This value is consistent with our previous work. Note however that a shoulder is also observed on the Tauc plot below this energy. Nechache _et al._ attributed this shoulder to the absorption edge (1.4 eV) of perfectly ordered BFCO (_i.e._ showing a perfect alternation between Fe and Cr cations) while the second absorption edge (1.66 eV) was attributed to BFCO presenting Fe-Cr disorder.
The electrical measurements were first performed in darkness by sweeping voltage between \(\pm\)8 V to test ferroelectricity. At +6 V, a clear FE peak originating from the polarization reversal is observed ((inset)) even though the curve shows a large level of leakage current. Although the expected peak is not visible for negative voltages due to the overwhelming ohmic conductivity, the FE coercive field of \(\sim\)64 MV m\({}^{-1}\) extracted from the volt-ampere characteristic is in a good agreement with the previous study and the polarization switching can also be evidenced by PFM measurements. In order to study the effect of light on the BFCO sample, a current-voltage measurement has been performed in the dark and under 365 nm wavelength excitation light emitted by a light emitting diode (LED) with 30 nm spectral linewidth.
Schematic representation of the device used for the experiment (left) and a microscopy image (right) of the electrical contact indicating the active sample area.
Absorption spectrum of BFCO deposited on STO (a) and the corresponding Tauc plot (b) indicating two absorption edges at 1.4 and 1.66 eV.
XRD \(\theta\)–2\(\theta\) pattern of a BFCO thin film grown on the NSTO substrate.
After that, the sample was exposed to 365 nm and 153 mW cm\({}^{-2}\) of radiant power, and the current was recorded in the depolarized state during voltage sweep starting from zero to +8 V. Afterwards, the voltage was swept from +8 V to -8 V (marked as positive to negative (P \(\rightarrow\) N) sweep) followed by the reverse negative to positive (N \(\rightarrow\) P) sweep (from \(-\)8 V to +8 V). The FE-history dependence of the photovoltaic effect becomes clearly evident with an important short-circuit photocurrent (\(I_{\text{sc}}\)) evolution. \(I_{\text{sc}}\) can be effectively tuned between 0.9 and 2.2 mA depending on the poling sweep. It also becomes evident that the presence of FE domains (depolarized state) clearly affects the photovoltaic effect. It has to be noted that the role of domain walls in the PV effect is a long debated issue. In that respect, our results clearly demonstrate that the polydomain (depolarized) state can indeed be advantageous for the PV effect under specific conditions (if compared with the random FE ground state or here with the state under the PN electric field sweep). However, a proper electrical pooling (or here NP sweep) can show even better photovoltaic response than that of the depolarized state. The insight into the PV hysteresis is therefore clearly needed to optimize the performance. A similar hysteretic behavior is also detected for the open circuit voltage (\(V_{\text{oc}}\)) with an average value of 0.54 V. To obtain more insight into the photovoltaic memory effect, the open circuit voltage was measured as a function of time during the periodic illumination of the sample. Prior to measurements, the sample was polarized positively by sweeping voltage from \(-\)8 V \(\rightarrow\) +8 V \(\rightarrow\) 0 V. The sample was then first illuminated for 4 minutes and then the light was switched off for 8 minutes. Subsequently, the LED was switched ON and OFF three more times.
The curve features show that at FE saturation the photovoltaic response is in fact a combination of the optically reversible (transient) and irreversible (remanent) effects. With the sample initially polarized positively, the first pulse of light induces a jump in photovoltage reaching a value of 0.56 V. This is followed by an interesting behavior since upon switching off the light, the voltage value did not drop down to the initial starting point, but to a value of 50 mV giving rise to a remanent voltage value. All the subsequent light pulses show the same reversible effect. This behavior can be attributed to the light induced change in the polarization state featuring both reversible and irreversible components as was recently shown by Makhort _et al._. In an attempt to elucidate the observed phenomena, PFM measurements on the film surface free of the ITO electrode layer have been carried out.
The ferroelectric nature of the film is confirmed by the presence of the electrically switchable polarization states. Poling images were obtained by applying +8 V (large square) followed by \(-\)8 V (small square).
\(I\)–\(V\) loop of the BFCO film recorded in darkness.
Current–voltage measurements on a BFCO sample, in darkness and under 365 nm light irradiation.
Transient and remanent photovoltaic effect for the BFCO sample.
6). In agreement with Fig. 6, the further light exposure does not modify the PFM images and the initial contrast can only be recovered electrically. The observed behavior can be explained as follows. For a sample initially polarized positively, the illumination generates carriers which distribute along the previously forced polarization direction. The generated photo-carriers reduce the surface's charges, and consequently change the internal electric field of the material depolarizing the sample. Upon turning off the light, the carrier trapping makes this decrease partly persisting (remannent effect), leaving the sample in a different polarization state. All the subsequent pulses of illumination reveal only a reversible effect; these pulses are in fact of the same power and the corresponding trapping centers are already occupied. The hypothesis of carriers' trapping effect can be confirmed by measuring the photovoltage _versus_ the light intensity. The curve exhibits the nonlinear behavior with three different regions: a fast charge generation, an intermediate saturation and a slow relaxation. The latter can be possibly attributed to the charge trapping processes involved in the remanent effect as shown in It is noteworthy that the possible effect of the light-induced increase of temperature in the sample during illumination can be discarded. The measured temperature change using a thermal camera exhibited only a 1 K temperature increase compared to the unexcited sample. The sample was then heated in a cryostat and it was found that such heating induces a negligible contribution to the sample pyrocurrent and voltage change compared to the light induced effects.
## Experimental
Epitaxial Bi\({}_{2}\)FeCrO\({}_{6}\) (BFOO) films studied in this work were grown on SrTiO\({}_{3}\) (STO) or Nb-doped (0.5%):SrTiO\({}_{3}\) (NSTO) by Pulsed Laser Deposition (PLD) using a KrF (248 nm) laser and a home-made target. The laser fluence was about 2 J cm\({}^{-2}\) and the repetition rate was 2 Hz. The deposition was carried out at 750 \({}^{\circ}\)C under an oxygen pressure of 10\({}^{-2}\) mbar. The as-grown film was cooled down at 5 \({}^{\circ}\)C min\({}^{-1}\) rate under the same atmosphere. A film thickness of 98 nm was measured using X-Ray Reflectivity (XRR). The crystalline structure was investigated by X-ray diffraction (XRD) using a SmartLab Rigaku diffractometer equipped with a Cu source and a Ge (220 \(\times\) 2) crystal delivering the monochromatic CuK\({}_{\alpha\beta}\) radiation (0.154056 nm). The high angle measurements allowed identifying the crystalline structure, calculating the size of the crystallites, and checking the phase purity and epitaxial quality of the films. The optical measurements of the films were investigated in the 200-1200 nm range using a Perkin-Elmer Lambda 950 spectrophotometer working in the ultraviolet-visible-near infrared (UV-Vis-NIR) range. The ferroelectric properties were investigated by the Piezoresponse Force Microscopy (PFM) technique using a Bruker Icon QNM microscope. The tip is a 0.01-0.025 Ohm cm antimony (n) doped Si covered with conductive diamond. A 4 V AC voltage was applied to the tip at 20 kHz. In this configuration, the current-voltage measurements were performed using a precision LCR Meter (Agilent E4980A).
## Conclusions
In this work, the successful preparation of epitaxial BFO films on Nb:STO substrates allowed the observation of ferroelectric and photovoltaic effects demonstrating a clear link between the two.
Relative change in voltage induced by 365 nm illumination plotted _versus_ light intensity.
PFM poling images before and after exposure to 365 nm light.
Moreover, the photoelectric characterization reveals a nonlinear character of the photovoltaic response _versus_ light intensity involving change generation, saturation and recombination processes. The latter effects can be used for all-optical information storage exploiting light induced remanent changes in polarization. More generally, the results reported here are not often encountered, thereby stimulating a new path of research towards electrically optimized ferroelectric-based solar cells.
| 10.48550/arXiv.1808.05847 | Tuning photovoltaic response in Bi2FeCrO6 films by ferroelectric poling | A. Quattropani, A. S. Makhort, M. V. Rastei, G. Versini, G. Schmerber, S. Barre, A. Dinia, A. Slaoui, J. -L. Rehspringer, T. Fix, S. Colis, B. Kundys | 812 |
10.48550_arXiv.0808.1464 | ###### Abstract
We study the effects of growth temperature, Ga:As ratio and post-growth annealing procedure on the Curie temperature, \(T_{C}\), of (Ga,Mn)As layers grown by molecular beam epitaxy. We achieve the highest \(T_{C}\) values for growth temperatures very close to the 2D-3D phase boundary. The increase in \(T_{C}\), due to the removal of interstitial Mn by post growth annealing, is counteracted by a second process which reduces \(T_{C}\) and which is more effective at higher annealing temperatures. Our results show that it is necessary to optimize the growth parameters and post growth annealing procedure to obtain the highest \(T_{C}\).
For this material to be useful in device applications it will be necessary to increase the Curie temperature (T\({}_{\rm C}\)) above room temperature. Theory predicts that T\({}_{\rm C}\) is proportional to the magnetic moment density which depends upon the density of substitutional Mn ions, x\({}_{\rm S}\). This trend has been confirmed experimentally for samples with x\({}_{\rm S}\)\(<\)6.8% grown by molecular beam epitaxy (MBE) with T\({}_{\rm C}\) reaching 173K\({}^{\rm 2}\). x\({}_{\rm S}\)\(\approx\)6.8% is achieved for total Mn concentration x\({}_{\rm total}\)\(\approx\)9% with the additional \(\approx\)2.2% incorporated as interstitial Mn (Mn\({}_{\rm I}\)) which can be removed by post-growth annealing. Recent attempts to grow (Ga,Mn)As with larger x\({}_{\rm total}\) have failed to achieve T\({}_{\rm C}\) in excess of the previous record and have produced conflicting results with T\({}_{\rm C}\) decreasing, saturating or increasing with increasing x\({}_{\rm total}\). In Ref [] it was found that T\({}_{\rm C}\) saturated at 165K for x\({}_{\rm total}\)\(>\)10%, leading the authors to suggest that the Zener model may not be applicable in the heavily alloyed regime. However, in another study, Olejnik et al reported T\({}_{\rm C}\)=180K for x\({}_{\rm total}\)=11%, obtained by etching and annealing the sample. The range of different results obtained by different groups for x\({}_{\rm total}\)\(>\)10% indicates that more research is required to understand how growth parameters and post growth annealing procedures affect the achievable T\({}_{\rm C}\) when x\({}_{\rm total}\)\(>\)10%. Here we present a detailed study of the epitaxial growth of (Ga,Mn)As layers with x\({}_{\rm total}\)\(\approx\)12% and T\({}_{\rm C}\) up to 185K. We show that the T\({}_{\rm C}\) depends sensitively on the growth temperature and the post growth annealing procedures.
25nm (Ga,Mn)As layers were grown by low temperature (\(\sim\)200 \({}^{\circ}\)C) MBE on low temperature grown GaAs buffer layers on semi-insulating GaAs substrates, using a Veeco Mod Gen III MBE system. The As flux was provided by a Veeco Mk5 valved cracker, set to produce As\({}_{2}\). On a separate test sample, the growth rate was calibrated using RHEED oscillations and the valve setting for As stoichiometry at 580\({}^{\circ}\)C was found. The stoichiometric point was found by observing the transition from the As-rich 2x4 to the Ga-rich 4x2 surface reconstruction. The re-evaporation rate of As is higher at 580\({}^{\circ}\)C than at \(\sim\)200 \({}^{\circ}\)C, which was found to correspond to \(\sim\)10% higher As incorporation, allowing a low temperature stoichiometric point to be calculated. Relative adjustments were then made by varying the Beam Equivalent Pressure (BEP).
Excess As or Ga has been shown to lead to the formation of As\({}_{\rm Ga}\) antisites or 3D growth of GaAs respectively. In agreement with previous studies we find that T\({}_{\rm C}\) is highly sensitive to the Ga:As ratio. We obtain the highest T\({}_{\rm C}\) for a (Ga+Mn):As ratio of 1:1.1 (determined using the method described above), and find that a small deviation from this value leads to a pronounced reduction in the maximum obtainable T\({}_{\rm C}\).
Measurement of the substrate temperature T\({}_{\rm g}\) was performed using a band edge spectrometer (Bandit from K-Space) under reflection geometry. Typical measurements of T\({}_{\rm g}\) during growth are shown in Fig. 1(a) for a series of films with x\({}_{\rm total}\)=12%. The growth temperatures were maintained within \(\pm\)1\({}^{\circ}\)C of the nominal value during the growth of the 25nm (Ga,Mn)As layers (4.5 minutes) by using a low temperature buffer layer and suitable power ramps. Typically T\({}_{\rm g}\) remained constant or increased at \(\leq\) 0.5\({}^{\circ}\)/min while 2D growth was maintained and then decreased at a similar rate after the 2D-3D phase boundary was crossed, presumably due to a change in surface emissivity.
Each wafer was cut into samples with dimensions 5mm x 4mm which were then annealed at temperatures below the growth temperature (see below). This is an established method for the removal of Mn\({}_{\rm I}\) which are detrimental to the ferromagnetism. T\({}_{\rm C}\) was determined from measurements of the remnant magnetization in a Quantum Design MPMS SQUID Magnetometer. Each sample was cooled in a field of 0.1T before measuring remnance as a function of increasing temperature in zero applied field.
Figure 1(b) shows T\({}_{\rm C}\) for fully annealed samples taken at different distances from the centre of three wafers, with T\({}_{\rm g}\) at the start of (Ga,Mn)As growth of 204\({}^{\rm o}\)C (wafer 1), 202\({}^{\rm o}\)C (wafer 2), and 200\({}^{\rm o}\)C (wafer 3). The distance at which the maximum T\({}_{\rm C}\) is obtained moves from the edge of the wafer towards the centre as the growth temperature is decreased. Monitoring the RHEED pattern at different points across the wafer indicates that the temperature across the wafer during growth decreases by approximately 2\({}^{\rm o}\)C from the centre to the edge. These results show that the maximum T\({}_{\rm C}\) is very sensitive to the growth temperature, and that the optimum conditions lie very close to the 2D-3D phase boundary which moves closer to the centre of each wafer as the measured substrate temperature is decreased. It is therefore crucial to maintain temperature stability to the accuracy illustrated in Fig. 1(a) in order to obtain the highest possible T\({}_{\rm C}\).
The anneal time t\({}_{\rm a}\) is obtained using the model described in Ref. such that Dt\({}_{\rm a}\) is the same as for a sample annealed at 180\({}^{\rm o}\)C for 48 hours, where D=D\({}_{0}\) e\({}^{\rm{-Q/kT}}\) is the diffusivity of Mn\({}_{\rm I}\) and Q=1.4eV is the activation energy. We found that these anneal times allowed T\({}_{\rm C}\) to reach a saturation value and further annealing did not significantly increase or decrease T\({}_{\rm C}\). It is clear that annealing at higher temperature results in a lower T\({}_{\rm C}\), suggesting that there is a second process, detrimental to ferromagnetism, which is dependent on temperature. As shown in the inset to Fig. 2, annealing a sample at 220\({}^{\rm o}\)C after initially annealing at 170\({}^{\rm o}\)C results in a decrease of T\({}_{\rm C}\) to a value similar to samples annealed at 220\({}^{\rm o}\)C initially. Subsequent annealing at 180\({}^{\rm o}\)C does not result in an increase of T\({}_{\rm C}\) indicating that the second process is not reversible. Further characterization of the samples will be required to elucidate the mechanism responsible for the decrease of T\({}_{\rm C}\) at higher annealing temperatures. Possible mechanisms might involve the loss of substitutional Mn to form MnAs precipitates, or to interstitial sites. Other mechanisms might involve the formation of native defects such as As\({}_{\rm Ga}\) and Ga\({}_{\rm As}\) antisites which would compensate carriers.
In addition to the wafers described above we have grown wafers with x\({}_{\rm total}\)=10% and 11%. In the absence of moment compensation, T\({}_{\rm C}\) should increase roughly linearly with ferromagnetic Mn moment density. shows Curie temperature versus the moment density measured at 2K in zero applied field, for annealed samples showing the highest T\({}_{\rm C}\) for a given x\({}_{\rm total}\) in this and our previous study. For x\({}_{\rm total}\)\(\geq\) 10% the T\({}_{\rm C}\) continues to increase, but with a slightly sub-linear trend. However, we should point out that we consistently achieve higher \(T_{C}\) than previous reports for \(x_{\rm total}\geq 10\%\), and it is clear from that the annealing procedure has not yet been fully optimized for these samples since \(T_{C}\) shows no sign of saturating as the anneal temperature is reduced.
With increasing \(x_{\rm total}\), the growth temperature must be reduced in order to maintain 2D growth, but this increases the probability of forming \(As_{\rm Ga}\) antisites and other compensating defects. Hence, precise control over the growth parameters becomes more important for incorporating Mn onto the substitutional sites and for achieving the corresponding increase in \(T_{C}\). We have shown that \(T_{C}\) is extremely sensitive to the growth temperature and (Ga+Mn):As ratio for MBE grown (Ga,Mn)As layers with \(x_{\rm total}\geq 10\%\). Additionally, we have found that the post-growth annealing temperature determines the maximum achievable \(T_{C}\) due to a temperature dependent mechanism in addition to the out diffusion of Mn\({}_{\rm I}\). These sensitivities may explain why some previous efforts to improve \(T_{C}\) in material with \(x_{\rm total}\geq 10\%\) have not been successful.
This work was done in close collaboration with the group of Vit Novak and Tomas Jungwirth (Prague) who have recently also obtained similar high \(T_{C}\) values. We acknowledge funding from EU grant IST-015728 and EPSRC grants GR/S81407/01.
## (a) Temperature at the centre of three wafers measured by band edge spectrometry as a function of time during the growth of the (Ga,Mn)As layers. The start and end points of the growth of the (Ga,Mn)As layers are indicated by vertical lines. The arrows indicate the approximate point where the 2D-3D RHEED transition occurs. (b) Curie temperature as a function of the distance from the centre of the wafer for wafers 1 (squares) and 2 (circles) annealed at 180\({}^{\circ}\)C for 48 hours and wafer 3 (triangles) annealed at 170\({}^{\circ}\)C for 116 hours.
## Curie temperature after annealing for 2.6hrs (220\({}^{\circ}\)C), 5.2hrs (210\({}^{\circ}\)C), 13hrs (200\({}^{\circ}\)C), 18hrs (193\({}^{\circ}\)C), 48hrs (180\({}^{\circ}\)C), 116hrs (170\({}^{\circ}\)C) and 280hrs (160\({}^{\circ}\)C) for a series of samples taken from wafer 2. Samples were taken from 4 different distances from the centre of the wafer. Inset: TC for a sample taken 7.5mm from the centre annealed at 170\({}^{\circ}\)C for 116 hrs then at 220\({}^{\circ}\)C for 1 hour, 2 hour and 2 hour intervals, then at 180\({}^{\circ}\)C for 48 hours.
## Curie temperature versus the spontaneous magnetization measured at 2K for annealed samples showing the highest Tc for a given xtotal. Open squares correspond to the samples with xtotal\(\geq\)10% discussed in this paper while closed squares correspond to our previously reported results for xtotal\(<\)10% \({}^{2}\) which were grown on a different MBE system (Mod Gen II). For comparison, open circlescorrespond to samples grown on the Mod Gen III for x\({}_{\rm total}\)\(<\)10% showing that we can achieve similar results on either MBE system.
## Abstract
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# 4 Discussion
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10.48550/arXiv.0808.1464 | Achieving High Curie Temperature in (Ga,Mn)As | M Wang, R P Campion, A W Rushforth, K W Edmonds, C T Foxon, B L Gallagher | 2,229 |
10.48550_arXiv.1301.5369 | ###### Abstract
An empirical multiorbital (\(spd\)) tight binding (TB) model including magnetism and spin-orbit coupling is applied to calculations of magnetic anisotropy energy (MAE) in CoPt \(L1_{0}\) structure. A realistic Slater-Koster parametrisation for single-element transition metals is adapted for the ordered binary alloy. Spin magnetic moment and density of states are calculated using a full-potential linearized augmented plane-wave (LAPW) _ab initio_ method and our TB code with different variants of the interatomic parameters. Detailed mutual comparison of this data allows for determination of a subset of the compound TB parameters tuning of which improves the agreement of the TB and LAPW results. MAE calculated as a function of band filling using the refined parameters is in broad agreement with _ab initio_ data for all valence states and in quantitative agreement with _ab initio_ and experimental data for the natural band filling. Our work provides a practical basis for further studies of relativistic magnetotransport anisotropies by means of local Green's function formalism which is directly compatible with our TB approach.
## I Introduction
Ordered CoPt alloys have been studied widely as they hold potential for applications in high density magnetic recording due to the combination of exchange and spin-orbit interactions giving rise to large magnetic anisotropies. Tunnelling magnetoresistance,Tunnelling tunnelling anisotropic magnetoresistance,Tunnelling or spin pumpingTunnelling have been demonstrated in CoPt based devices. Our general objective is to develop an efficient numerical model allowing us to study ground state and in future studies also transport properties of spintronic devices based on CoPt or other intermetallic compound with large magnetic anisotropy energy (MAE).
MAE has been calculated in CoPt \(L1_{0}\) structure using _ab inito_ methods.Kucera; Kucera and Kucera The stability of various bulk CoPt structuresKucera has been studied recently also by _ab initio_ methods. On the other hand, quantum transport in micro-devices is typically described within the Green's function formalism assuming an expansion of the electronic states on a local basis set which facilitates partitioning of the system. The tight binding (TB) description of electronic structure provides a good foundation for subsequent simulations of magnetoresistance in tunnelling or ohmic regime as it assumes a local basis set.Kucera and Kucera; Kucera
TB schemes applied in modelling of magnetotrasport phenomena range from empirical or semi-empirical (charge self-consistent) modelsKucera; Kucera and Kucera; Kucera to tight binding linearized muffin-tin orbital (LMTO) modelKucera and Kucera; Kucera combining density functional theory (DFT) with the TB approach. The state-of-the-art magnetic DFT-based TB schemesKucera and Kucera; Kucera and Kucera; Kucera and Kucera; Kucera; Kucera and Kucera allow for simulations of complex systems (impurities, structural defects, surfaces) that are beyond practical capability of _ab initio_ calculations. In this work, we employ an empirical two-center Slater-Koster TB modelSlater and Koster following the Harrison approachHarrison recently further developed by Shi and PapaconstantopoulosShi and Papaconstantopoulos to investigate MAE of a bulk ordered transition metal alloy and compare the results to density functional theory (DFT) calculations performed using the full-potential linearized augmented plane-wave (LAPW) program package WIEN2k. Thereby we test the transferability of the TB parametersShi and Papaconstantopoulos obtained by fitting to _ab initio_ band structures of single-element solids of the two atoms forming our ordered alloy. Rather than finding a new full set of TB parameters by fitting to the DFT band structure of the compound, we determine a minimal subset of interatomic parameters which influence the spin magnetic moments and projected density of states (DOS) in a transparent way and tune these parameters to improve the agreement of TB and DFT results. MAE calculated in a narrow range of refined values of the interatomic parameters is in quantitative agreement with DFT. We believe that our model and parameters provide good basis for future simulations of magnetotransport in CoPt and other multilayer structures.
Our paper is organized as follows: The TB model including the exchange and spin-orbit interaction is introduced in Sec. II; TB parameters according to Shi and Papaconstantopoulos with our modifications are discussed in Sec. III; First comparison of experimental MAE,Shi and Papaconstantopoulos_ab initio_, and TB results is presented in Sec. IV; Sec. V describes the refinement of TB parameters towards agreement of spin magnetic moment and DOS with _ab initio_ results, it also presents corresponding MAE in comparison with DFT; Our work is briefly summarised in Sec. VI.
## II Tight binding model
Our TB model is a variant of the Linear Combination of Atomic Orbitals method for a periodic crystal wherethe basis has the form of Bloch sums of atomic like wavefunctions:
\[\langle\mathbf{r}|a\alpha\mathbf{k}\rangle=\frac{1}{\sqrt{N}}\sum_{n=0}^{N-1}e^{i\mathbf{k} \cdot(\mathbf{R}_{n}+\mathbf{p}_{n})}\phi_{a\alpha}(\mathbf{r}-\mathbf{R}_{n}-\mathbf{p}_{a}), \tag{1}\]
We consider an orthonormal set of wavefunctions \(\phi_{a\alpha}(\mathbf{r})\) constructed from the atomic orbitals (with the angular part expressed in terms of cubic harmonics given in Eq.) using Lowdin's orthonormalization procedure so the overlap matrix \(\langle b\beta\mathbf{k}|a\alpha\mathbf{k}\rangle=\delta_{b\beta,\alpha a}\). Our single-electron Hamiltonian has three components as follows:
\[H=H_{band}+H_{Stoner}+H_{SO}. \tag{2}\]
The first component \(H_{band}\) contains the kinetic energy and a superposition of atomic potentials centered at each site:
\[V(\mathbf{r})=\sum_{na}V_{a}^{at}(\mathbf{r}-\mathbf{R}_{n}-\mathbf{p}_{a}), \tag{3}\]
The potentials \(V_{a}^{at}(\mathbf{r})\) centred at each site are spherically symmetric so the wavefunctions \(\phi_{a\alpha}\) can be specified by the usual angular momentum quantum numbers.
The matrix elements of the non-magnetic Hamiltonian term \(H_{band}\) can be written in terms of on-site energies \(\varepsilon_{a\alpha}\) and hopping integrals \(E_{b\beta,a\alpha}(\mathbf{\rho}_{n})\) which depend, in the two-center approximation proposed by Slater and Koster, only on the intersite position vector \(\mathbf{\rho}_{n}=\mathbf{R}_{n}+\mathbf{p}_{a}-\mathbf{p}_{b}\) as follows:
\[\langle b\beta\mathbf{k}|H_{band}|a\alpha\mathbf{k}\rangle =\delta_{b\beta,a\alpha}\varepsilon_{a\alpha}+\sum_{n}e^{i\mathbf{k} \cdot\mathbf{\rho}_{n}}E_{b\beta,a\alpha}(\mathbf{\rho}_{n}),\] \[E_{b\beta,a\alpha}(\mathbf{\rho}_{n}) \equiv\int d\mathbf{r}\phi_{b\beta}^{*}(\mathbf{r})H_{n}(\mathbf{r})\phi_{a \alpha}(\mathbf{r}-\mathbf{\rho}_{n}), \tag{4}\]
Following the TB scheme, we parametrize the Hamiltonian matrix instead of performing the integration of Eq.. The hopping integrals \(E_{b\beta,a\alpha}(\mathbf{\rho}_{n})\) can be expressed in terms of Slater-Koster parameters (\(V_{ss\sigma},\,V_{sp\sigma}\), \(V_{sd\sigma}\), \(V_{pp\sigma}\), \(V_{pp\sigma}\), \(V_{p\sigma\mu}\), \(V_{dd\sigma}\), \(V_{dd\sigma}\), \(V_{dd\delta}\), etc.) obtained usually by fitting to _ab initio_ calculations. See Eqs. and in the Appendix for the so called Slater-Koster tables listing the \(s\), \(p\), and \(d\) hopping integrals used in this work.
The on-site energies \(\varepsilon_{a\alpha}\) could be approximated by the atomic values as in the original Harrison approach. Instead, we follow Shi and Papaconstantopoulos who keep the on-site \(k\)-independent matrix diagonal but use \(\varepsilon_{a\alpha}\) obtained by fitting to _ab initio_ data. Hence, both the on-site energies and Slater-Koster parameters are inputs of the model as we discuss in more detail in Sec. III.
Since the Hamiltonian has the same periodicity as the basis functions in Eq. it is diagonal in the \(k\)-vector. The dimension of our Hamiltonian matrix is given by the number of atoms in the unit cell and valence orbitals considered for each atom. We use \(s\), \(p\), and \(d\) orbitals to model CoPt and there are only two atoms in the unit cell of the \(L1_{0}\) structure. The sum in Eq. runs over a limited set of neighbouring sites due to localization of functions \(\phi_{a\alpha}\) around site \(a\).
The Hamiltonian described so far is non-magnetic. Now we double our Hilbert space by including the spin degree of freedom \(|a\alpha\mathbf{k}\rangle\rightarrow|a\alpha\mathbf{\zeta}\mathbf{k}\rangle\) and add a \(\mathbf{k}\)-dependent term \(H_{Stoner}\) to account for the ferromagnetism in our system:
\[\langle b\beta\zeta\mathbf{k}|H_{Stoner}|a\alpha\mathbf{\zeta}\mathbf{k}\rangle =\frac{1}{2}\delta_{b\beta,a\alpha}I_{a\alpha}\hat{\mathbf{m}}\cdot \mathbf{\sigma}_{\zeta\xi}, \tag{5}\] \[+\frac{1}{2}\sum_{n}e^{i\mathbf{k}\cdot\mathbf{\rho}_{n}}I_{b\beta,a \alpha}(\mathbf{\rho}_{n})\hat{\mathbf{m}}\cdot\mathbf{\sigma}_{\zeta\xi},\]
We derive our Stoner parameters from the exchange-split on-site energies and hopping integrals: \(I_{a\alpha}=\varepsilon_{a\alpha\uparrow}-\varepsilon_{a\alpha\downarrow}\), \(I_{b\beta,a\alpha}=E_{b\beta\uparrow,a\alpha\uparrow}-E_{b\beta\downarrow,a \alpha\downarrow}\), using on-site and Slater-Koster parameters fitted independently for the spin-up and spin-down states in the absence of spin-orbit coupling. Using the averages of spin-up/down parameters (non-magnetic structure with \(\varepsilon_{a\alpha}\) and \(E_{b\beta,a\alpha}\)) and their differences (Stoner parameters) allows us to rotate the magnetization direction according to Eq. when the spin-orbit coupling is considered.
In order to account for the MAE we add the spin-orbit coupling term \(H_{SO}\) in its atomic \(\mathbf{k}\)-independent form to the on-site terms of our Hamiltonian:
\[\langle b\beta\zeta\mathbf{k}|H_{SO}|a\alpha\mathbf{\zeta}\mathbf{k}\rangle=\delta_{b,a} \lambda_{a\beta,a\alpha}\mathbf{L}_{\beta\alpha}\cdot\mathbf{S}_{\zeta\xi} \tag{6}\]
The matrix elements of \(H_{SO}\) in the basis of cubic harmonics are given in Eqs. and in the Appendix.
Finally, we do not introduce explicitly any Hamiltonian term controlling the charge transfer between sites occupied by different atoms. Prior to our calculation we shift all on-site energies of Co with respect to Pt so that the Fermi energies calculated for pure Co and pure Pt are equal. We check that the local charges on Pt sites and on Co sites in CoPt \(L1_{0}\) structure are in agreement with the LAPW results within the error-bar caused by charge located outside of atomic spheres used by the _ab initio_ method.
## III Parametrization
As mentioned above, our model relies on input parameters that are obtained by fitting band structures and total energies to _ab initio_ results. Extensive parameter sets are available for bulk single-element metals. They assume non-orthogonal basis set, interaction to higher order neighbours, and reproduce _ab initio_ data with great accuracy. Our aim is to study TB models of more complex systems such as ordered binary alloys (this work) and heterostructures (future work) suitable for exploring the physics of relativistic equilibrium and potentially also transport phenomena in these systems. Therefore we prefer smaller, more transferable sets of parameters assuming interactions only up to third nearest neighbours.
We use a parametrisation by Shi and Papaconstantopoulos which further develops the Harrison approach. Harrison expressed the two-centre Slater-Koster parameters \(V_{\alpha\beta\gamma}(\rho)\) as functions of the interatomic distance \(\rho=|\mathbf{\rho}|\), an effective radius of the \(d\) orbital \(r_{d}\) which is characteristic to each transition metal, and constants \(\eta_{\alpha\beta\gamma}\) which are universal for all elements and lattice structures:
\[V_{\alpha\beta\gamma}(\rho) =\eta_{\alpha\beta\gamma}\frac{\hbar^{2}}{m\rho^{2}},\] \[V_{\alpha d\gamma}(\rho) =\eta_{\alpha d\gamma}\frac{\hbar^{2}r_{d}^{3/2}}{m\rho^{7/2}},\] \[V_{dd\gamma}(\rho) =\eta_{dd\gamma}\frac{\hbar^{2}r_{d}^{3}}{m\rho^{5}}, \tag{7}\]
Values of \(\eta_{\alpha\beta\gamma}\) are listed in Ref.. In case of transition metals, Harrison used only the \(s\) and \(d\) orbitals so there were only two parameters specific to each element, the \(d\)-band width given by \(r_{d}\) and the on-site energy of the \(d\) orbitals with respect to the \(s\) orbitals.
Shi and Papaconstantopoulos significantly improved the ability of the Harrison parametrisation to produce accurate numerical results for the band structure while keeping the form and universality of the Slater-Koster parameters given in Eq.. This is accomplished by: 1) Replacing the atomic energies by on-site energies fitted to Augmented Plain Wave (APW) calculations; 2) Including the \(p\) orbitals into the basis set; 3) Modifying of the \(sp\) Slater-Koster parameters by introducing a dimensionless parameter \(\gamma_{s}\) as follows:
\[V_{\alpha\beta\gamma}(\rho)=\eta_{\alpha\beta\gamma}\frac{\gamma_{s}\hbar^{2}} {m\rho^{2}}; \tag{8}\]
4) Obtaining new prefactors \(\eta_{\alpha\beta\gamma}\) and radii \(r_{d}\) by simultaneously fitting the APW energy bands of 12 transition metals at the equilibrium lattice constants of the particular element. The new parameters reproduced APW energy bands and density of states (DOS) remarkably well, not only for the 12 elements originally fitted, but also for the rest of the transition metals, the alkaline earth and the noble metals as shown in Ref.. This parametrisation assumes an orthogonal basis set and interaction to second (fcc) or third (bcc) nearest neighbours. The on-site and Slater-Koster parameters are exchange-split in case of the ferromagnetic metals.
In this work we build on the results of Shi and Papaconstantopoulos and test the transferability of their TB parameters to an ordered binary alloy. We use the parameters for single-element bulk metals with the following modifications: 1) The interatomic Slater-Koster parameters between Co and Pt atoms are set to a geometric average of the elemental values: \(V_{\alpha\beta\gamma}^{Co,Pt}=\sqrt{V_{\alpha\beta\gamma}^{Co}V_{\alpha\beta \gamma}^{Pt}}\) following the work of Ballhausen and Gray. Using the geometric average is also in line with the LMTO method; 2) The exchange-splittings of the on-site and Slater-Koster parameters enter our model through the on-site (\(I_{\alpha\alpha}\)) and hopping (\(I_{\alpha a,b\beta}\)) Stoner parameters, respectively. We increase \(I_{Co,d}\) and introduce non-zero \(I_{Pt,d}\) to reproduce the LAPW spin magnetic moment and DOS of CoPt more accurately; 3) We vary the on-site energy of unoccupied \(p\) orbitals (\(\varepsilon_{Pt,p}\)), which extend the original Harrison's set of parameters and are specific to a particular single-element crystal, in order to further improve the agreement with spin magnetic moment and DOS obtained by LAPW and explore the dependence of MAE in the CoPt compound on this parameter.
Finally, we add atomic spin-orbit coupling parameters obtained by numerical Hartree-Fock calculations based on the Dirac-Coulomb Hamiltonian to the Pt sites. The \(5d\) orbital has an atomic value \(\lambda_{Pt,d}=0.0445\) Ry and the \(3d\) orbital of Co has \(\lambda_{Co,d}=0.0063\) Ry. We neglect the spin-orbit coupling in \(6p\)-orbitals of Pt and \(4p\)-orbitals of Co to keep the number of input parameters low.
## IV Comparison of tight binding and _ab initio_ results
We check the validity of our extensions of the Harrison TB model and the transferability of Shi and Papaconstantopoulos' parametrization to CoPt ordered alloy by comparing the band structure, spin magnetic moment, DOS, and MAE calculated using our TB code and a well established _ab initio_ code, the full-potential relativistic LAPW package Wien2K.
In CoPt \(L1_{0}\) structure, fcc lattice sites are occupied by alternating layers of Co and Pt atoms and the lattice constant perpendicular to layers \(c\) is smaller than the in-plane lattice constant \(a\). Throughout this work we use the experimental lattice constants: \(a=7.19\) a.u, \(c=7.01\) a.u.
The Slater-Koster parameters for single-element fcc crystals were fitted assuming interaction to second nearest neighbours. However, we intend to account for MAE caused by hybridisation of magnetic \(3d\) orbitals on Co and spin-orbit coupled \(5d\) orbitals on Pt. The coupling between the Co and Pt sites in the \(L1_{0}\) lattice inour two-center approximation is formed by 8 second (third) nearest neighbour hopping integrals. Therefore, to enhance the ability of our model to capture the Co-Pt hybridisation, we add the third nearest neighbours to the sum in Eq..
At this stage we do not modify the TB parameters derived for single-element metals except taking geometric average of Slater-Koster parameters between Co and Pt. We denote this default set of TB parameters as the "atomic parameters".
There is a good agreement of the bands especially in the vicinity of the Fermi energy. TB bands assuming second nearest neighbours match LAPW bands also in the region of \(s\) and \(p\) states. However, the bottom of the \(s\)-band is very far from the Fermi energy and the \(p\)-band has only an auxiliary role in our model so we can conclude that summation to both second and third neighbours has potential for successful simulation of the MAE in CoPt \(L1_{0}\) structure.
Now we turn our attention to the spin magnetic moment, spin-resolved DOS and MAE calculated using second nearest neighbours together with the "atomic parameters" and compare these TB results to LAPW with LSDA and spin-polarised generalized gradient approximation (GGA). It is known that GGA and LDA are suitable for \(3d\) and \(5d\) transition metals, respectively, so it is not clear which approximation is more appropriate for the CoPt compound. Therefore, we try to view the difference between TB and LSDA results in the context of the difference between calculations using LSDA and GGA approximations.
Fig. 2(a) shows the spin magnetic moment on Co and Pt sites per formula unit (f.u.) for energies ranging through the whole valence band for magnetization along the axis. (Our axis is set perpendicular to the alternating Co and Pt atomic planes and the nearest in-plane neighbour lies on the axis.) The Wien2K code places all atoms of the unit cell in non-overlapping spheres leaving some charge in the interstitial region. We set the radii of the atomic spheres to \(r_{Co}=2.2\) a.u. and \(r_{Pt}=2.4\) a.u. for all LAPW calculations causing our spin magnetic moment in the interstitial region to be less than 5% of the total spin magnetic moment. GGA and LSDA give very similar values in case of Pt and deviate slightly in case of Co in the middle of the \(d\)-band. TB spin magnetic moment on Pt is lower than the LAPW prediction in the whole energy range, whereas TB values for Co match the GGA very well except around the Fermi level where Co is predicted to be less spin-polarized by TB than by LSDA and GGA. The total spin magnetic moment is 2.27 \(\mu_{B}\)/f.u. using GGA and 2.21 \(\mu_{B}\)/f.u. using LSDA which is in good agreement with the experimental value 2.4 \(\mu_{B}\)/f.u. TB prediction of the total spin magnetic moment is 1.92 \(\mu_{B}\)/f.u. Very similar results are obtained for magnetization along the axis so we do not plot them.
Fig. 2(b) shows DOS projected on spin-up/down states again for magnetization along the axis and summation to second nearest neighbours. LSDA and GGA are in excellent agreement in case of spin-down but differ slightly in case of spin-up. The DOS calculated by TB has unexpected peaks close to the bottom of the \(d\)-band. A separate calculation of the Pt-component of DOS and the fact that these peaks for spin-up/down are not mutually shifted in energy suggests that they correspond to Pt-states. We hypothesize at this stage that an increased coupling between Co and Pt sites could remove these unrealistic peaks. In general, the spin-up part of the valence band calculated by TB also seems to be less shifted in energy with respect to the spin-down part as compared to the LAPW reference.
(Color online) Comparison of TB (continuous red lines) and LAPW-LSDA (dashed black lines) band structures of CoPt \(L_{1}0\) structure with magnetization along the axis. Second (a) and third (b) nearest neighbours together with “atomic parameters” are assumed in TB whereas LAPW bands are the same in both plots.
2(a) and motivates us to increase the Stoner parameters both for Co and Pt atoms and add the third nearest neighbours to enhance the coupling between Co and Pt in the next section.
Fig. 2(c) compares MAE calculated again using TB with "atomic parameters" and second nearest neighbours and LAPW with LSDA and GGA approximations. Note that in the TB case MAE amounts to the difference of total energies for two magnetization directions: \(\mathrm{MAE}=E_{tot}(M_{110})-E_{tot}(M_{001})\), whereas in case of LAPW we use the force theorem following the work of Shick. (We use about 50000 and 200000 \(k\)-points in LAPW and TB integration, respectively.) MAE is a more subtle quantity than the spin magnetic moment or DOS so we calculate it for a wide range of band filling (b.f.) using the rigid band approximation and compare the trends of MAE(b.f.) rather than individual values.
We observe a very good agreement of the LSDA and GGA data and a broad agreement of the TB and _ab initio_ curves. We consider this a remarkable success of the TB model which employs parameters fitted for pure single-element metals. The amplitude of MAE oscillations decreases towards the edges of the valence band so the relative error of the TB result at the Fermi energy becomes quite large. Both LAPW values \(\mathrm{MAE}_{LSDA}=1.26\) meV/f.u. and \(\mathrm{MAE}_{GGA}=0.85\) meV/f.u. are in good agreement with the measured value \(\mathrm{MAE}_{exp.}=1.0\) meV/f.u., whereas our "atomic parameters" TB prediction based on straight forward transfer of Shi and Papaconstantopoulos' parameters is an order of magnitude lower: \(\mathrm{MAE}_{TB}=0.13\) meV/f.u.
## V Parameter Refinement and MAE
The good agreement of key features in the spin magnetic moment, DOS, and MAE calculated by LAPW and TB motivates us to improve our TB model. The shortcomings of the initial simulations described in the previous section provide useful guidance how to proceed. As concluded in the discussion of Fig. 2(b), we add the third nearest neighbours to enhance the coupling between Co and Pt sites and we increase the on-site Stoner parameters both for Co and Pt atoms to match the larger spin polarization obtained by LAPW.
The results are summarised in which presents the same quantities as computed in three different ways: TB with "atomic parameters" and third nearest neighbours (a,3nn); TB with "atomic parameters" enhanced by \(I_{Co,d}=1.08I_{Co,d}^{atomic}=0.134\) Ry, \(I_{Pt,d}=0.15I_{Co,d}\), and third nearest neighbours (a,I,3nn); LAPW with LSDA (same as in Fig. 2, GGA is not included to maintain legibility of the plots).
Fig. 3(a) can be contrasted with its counterpart, Fig. 2(a).
(Color online) Comparison of TB results assuming “atomic parameters” and second nearest neighbours (a,2nn) to LAPW using LSDA and GGA approximations for quantities: (a) Spin magnetic moment on Co (thick lines) and Pt (thin lines) sites per formula unit with magnetization along the axis (spin in the interstitial region of LAPW is not captured); (b) DOS per formula unit projected on spin-up/down states with magnetization along the axis, the legend of plot (a) applies including the line style used for Pt and color coding; (c) \(\mathrm{MAE}=E_{tot}(M_{110})-E_{tot}(M_{001})\) per formula unit for a range of band fillings where the natural band filling is 19 valence electrons.
However, we managed to tune the Stoner parameters \(I_{Pt,d}\) and \(I_{Co,d}\) to achieve quantitative agreement of the spin magnetic moment on Pt with the LSDA values throughout the whole valence band. The spin magnetic moment on Co cannot be brought to a full quantitative agreement with LSDA by tuning only parameters \(I_{Pt,d}\) and \(I_{Co,d}\) and remains slightly lower than the LSDA or GGA reference at the Fermi energy. We address this deficit in the final refinement of our TB model.
DOS in Fig. 3(b) shows significant improvement over Fig. 2(b) so the summation to third nearest neighbours seems to be more suitable for modelling the coupling between Co and Pt. Moreover, the enhancement of the Stoner parameter increased the mutual shift of spin-up/down DOS to show closer match with LSDA, as expected.
In Fig. 3(c) we focus on band fillings closer to the natural band filling (b.f. = 19 valence electrons). Including the third nearest neighbours does not cause any significant change in the overall MAE dependence on the band filling. Slightly more pronounced change corresponds to increasing the Stoner parameters but the main features of MAE are still in agreement with LSDA and GGA data. Note that the MAE at the natural band filling becomes negative due to the enhanced exchange interaction.
We can conclude that our first TB parameter refinement attempt presented in leads to a better agreement with LAPW in a broad range of valence band energies, however, increasing \(I_{Pt,d}\) and \(I_{Co,d}\) and adding third nearest neighbours does not reproduce accurately the net spin magnetic moment predicted by LAPW at the Fermi energy and the MAE at the natural band filling.
As we mentioned in Secs. III and IV the unoccupied \(p\)-states were added to the Harrison TB model by Shi and Papaconstantopoulos only to produce more realistic warping of the \(d\)-band in single-element crystals close to Fermi energy. Therefore, the position of the unoccupied \(p\)-states in the CoPt compound is the next natural subject to scrutiny.
We note that Pt offers more room for variation of the \(p\)-state on-site energy as its \(s\)-state is much lower in energy than the \(s\)-state of Co: \((\varepsilon_{Pt,p}-\varepsilon_{Pt,d})/(\varepsilon_{Pt,s}-\varepsilon_{Pt,d} )=2.2\) whereas \((\varepsilon_{Co,p}-\varepsilon_{Co,d})/(\varepsilon_{Co,s}-\varepsilon_{Co,d} )=1.02\). Therefore, we can bring \(\varepsilon_{Pt,p}\) closer to \(\varepsilon_{Pt,s}\) (and to \(\varepsilon_{Pt,d}\) further below) without changing the order of the Pt on-site energies. Such shift should increase the hybridisation of the \(p\)-states on Pt with the exchange-split \(d\)-states increasing the deficient net spin magnetic moment.
The LSDA data are the same as in Figs. 2 and 3.
As expected, the greater proximity of \(p\) and \(d\)-states
(Color online) Comparison of TB results assuming third nearest neighbours and “atomic parameters” (a,3nn) or “atomic parameters” with enhanced Stoner parameter (a,I,3nn) to LAPW using LSDA approximation for quantities: (a) Spin magnetic moment on Co (thick lines) and Pt (thin lines) sites with magnetization along the axis; (b) DOS projected on spin-up/down states with magnetization along the axis, the legend of plot (a) applies including the line style used for Pt and color coding; (c) MAE = \(E_{tot}(M_{110})-E_{tot}(M_{001})\) for a range of band fillings where the natural band filling is 19 valence electrons.
4(a). At the same time, the spin magnetic moment of Co at Fermi energy scales with the shift of on-site energy \(\varepsilon_{Pt,p}\) due to hybridisation with Pt and finally reaches the LAPW values when the shift mentioned above is in the range \(\varepsilon_{Pt,p}-\varepsilon_{Pt,d}\approx 0.54-0.69\) Ry.
The agreement of DOS calculated by LAPW and TB with enhanced Stoner parameters shown in Fig. 3(b) is satisfactory. We include Fig. 4(b) to demonstrate that shifting \(\varepsilon_{Pt,p}\) causes only minor deviation from DOS obtained by LSDA at the very bottom of the valence band, whereas it further improves the agreement closer to the Fermi energy.
Fig. 4(c) shows the main result of our work. The general trend of the MAE dependence on the band filling is very robust against small variations of the TB input parameters and is in broad agreement with MAE calculated by LAPW. On the other hand, the value of MAE for the natural band filling turns out to be very sensitive to the input parameters. Remarkably, the MAE for the natural band filling is in quantitative agreement with the LSDA, GGA, and experimental values MAE \(\approx 1\) meV/f.u. when the on-site energy of the \(p\)-states on Pt is in the range determined by comparing DOS and spin magnetic moment to LAPW: \(\varepsilon_{Pt,p}-\varepsilon_{Pt,d}\approx 0.54-0.69\) Ry.
We have also explored the sensitivity of the above quantities to the variation of the Slater-Koster parameters. However, replacing the geometric average by an arithmetic average to obtain \(V_{\alpha\beta\gamma}^{Co,Pt}\) or increasing the relative magnitude of the Slater-Koster parameters to enhance the Co-Pt hybridisation does not change the spin magnetic moment or DOS in a transparent way that would improve our physical understanding of the electronic structure or the overall agreement with the LAPW results.
## VI Summary
We have carried out systematic modeling of electronic structure and relativistic magnetic characteristics of bulk CoPt \(L1_{0}\) structure using TB and _ab initio_ methods. An empirical multiorbital TB model following the Harrison approach with a parametrisation devised by Shi and Papaconstantopoulos was applied. We extended the model by adding an atomic spin-orbit coupling term to the on-site Hamiltonian blocks in order to account for the magnetocrystalline anisotropy. We have focused on the MAE as a function of the band filling so that we could compare general trends of TB and LAPW rather than singe values for the natural band filling which are available in literature.
We started our calculations by checking the validity of the model assuming the "atomic parameters" (parameters optimized for single-element fcc crystals with geometric averaging of Slater-Koster parameters between Co
(Color online) Comparison of TB results assuming third nearest neighbours, “atomic parameters” with enhanced Stoner parameter, and on-site energies \(\varepsilon_{Pt,p}-\varepsilon_{Pt,d}=0.69\) Ry (a,I,p1,3nn) or \(\varepsilon_{Pt,p}-\varepsilon_{Pt,d}=0.54\) Ry (a,I,p2,3nn) to LSDA approximation for quantities: (a) Spin magnetic moment on Co (thick lines) and Pt (thin lines) sites with magnetization along the axis; (b) DOS projected on spin-up/down states with magnetization along the axis, the legend of plot (a) applies including the line style used for Pt and color coding; (c) MAE \(=E_{tot}(M_{110})-E_{tot}(M_{001})\) for a range of band fillings where the natural band filling is 19 valence electrons.
The broad agreement observed throughout the valence band with slightly deficient TB net spin magnetic moment and smaller mutual shift of the projected DOS stimulated further development of the parametrisation focusing on the Stoner parameters and on-site energies of the virtual \(p\)-states.
We continued by adding the interaction to third nearest neighbours to enhance the hybridisation between magnetic and spin-orbit coupled sites and varied three TB parameters (\(I_{Pt,d}\), \(I_{Co,d}\), and \(\varepsilon_{Pt,p}\)) which extend the original Harrison parametrisation and their values were likely to require corrections after the transfer to an ordered binary alloy. We compared the new set of results to _ab initio_ predictions again. In other works, this process is typically replaced by simultaneous fitting of all TB parameters to APW band structures and total energies, however, in our work we seek better physical insight into the spin-orbit coupling phenomena and transferability between different structures rather than precise agreement of TB and _ab initio_ results.
We found that the net spin magnetic moment increases with small enhancement of the Stoner parameters and with shifting of the Pt \(p\)-state on-site energy towards the \(d\)-states as expected. Remarkably, the MAE obtained for a narrow range of these parameters, where the net spin magnetic moment and DOS reached the best agreement with the _ab initio_ predictions, is in quantitative agreement with _ab initio_ results. Such success motivates future investigations of the transferability of the model in other compounds or multilayers. At the same time our TB model is well suited for incorporation of the equilibrium Green's function framework to calculate relativistic magnetotransport phenomena in structures containing the magnetic compounds or multilayers.
| 10.48550/arXiv.1301.5369 | Comparative study of tight-binding and ab initio electronic structure calculations focused on magnetic anisotropy in ordered CoPt alloy | J. Zemen, J. Mašek, J. Kučera, J. A. Mol, P. Motloch, T. Jungwirth | 2,142 |
10.48550_arXiv.1306.4127 | ## 1 Bhelevine
Inmecnenie zlektorvorodnykh svoist' razdinnykh polimmerov svoobodannikh khnikh khnikh khnikh khnikh vvel'enemy vikh ob'em, naprimer, razdinnykh imbedevichnoy (nabolintsel', pastichnikatorov, stadilinatorov i t.I.) ihevstno davho. \(\,\) S' davhoi sotropny, znemst' v dostatonno improkikh predelakh zlektorvorodnykh svoist'va polimmernogo materiala mokho s pomorili'v vipova i nispol'vovannia poty ihnogo vidk samogo polimmera. Pri atom nanobiee viskovinni elektorvorodnykh svoist'vami oblast'i polimmery s sistenoi polimmery sistenoi polimmery sistenoi polimmery sistenoi polimmery. (it'o nispoliee viskovinni polimmery, polinstorhen, polinstentiei i t.I.). Oven' taats dlia svalieningia elektorvorodnykh svoist's polimmerov, soderkashikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh 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khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikhhnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikh khnikhhnikh khnikh khnikh khnikh khnikh khnikh10\({}^{16}\)Om.chn). Ho, c droptu storonyi, korobnio zhsevstu, otob polchnstellich (Pil), soderkazhnii v sostave srohnakh makromolekvi ikshichonitel'no IIC N (sichnoyiuchn) predstavliaiushii soboiu PikN, polvernutrisi operatsii 100%-destadokliornovaniia, odgladiei znahitel'no Bol'shei elektroprovodnost'iu (plei)noe oblemnoe soprintshenie10\({}^{8}\)-10\({}^{10}\)Om.chn) kota sobsoliuchno negrindelen pri atom s teknologicheskoi tochki irennia.
Vuntiviva zin obstotneistva, obluo reiheno uptem machinogo terminoseckogo elektrokhodirovaniia iskhodiuchogo PikN bulipattsi otvskat optimum zlektrokhodiushki i teknologicheskikh svoistr sobolimemorv vhniklorda ihniienta.
S iobel'no v lannoiu rabote nskhodiue ogranznyi PikN polvertaliye kiminneskoi modifikatsii metodom termoniaa noscelestiveno (bezn sichnoyiuchnii pri atom kakh-ihodiushchnistnikatorok, shesiia)nykh dobakov, noniiarmuisteno iznuteniia i t.d.).
Dikcitivne obrazilyi nastichno elektrokhorovannogo PikN metodom termolina v rastvore polchnaini sleduciushchnii odrazom. Piazale poluciushchnii 4% pastvore PikN (marikh C-70) v rastvoretiele (sietrodernie PikN sichnist'lii pri repremensivannii pri komnatiuoi temgraviture v ruehene 12 akov do polchneniia komponentnogo programnogo rastvora. V dal'neiheniem pastvore i ropensiia v roponritsi i polvertalisia termolina pri T=190\({}^{0}\)C v ruehene 20-48minut.
Doriia, neobkhodiimaa dlia polchneniia obraziia plenki zanivanal' na elektjunnogo polchnokiv i polvertaliasu svilke pri T=5\({}^{0}\)C v terminiknady i reihene 48 akov.
Dalee poluchenniye obrazilyi polvertaliye vnazial'nomu, teknologicheskomu i urrancheniveliushchnii nekhodiuchnoyi (progranichost, ihet, elektrosty chntia s polchnokki, proniost', iakshchnist'liiemost' i t.d.).
Limerennia obraziuiu PikN-plenok po pokazannim elektrokhorovhodiushchnist'lii na prilobre E6-3, oblasniannom v s Oostrovannoi kol'tsevoi iammeritel'noi chalekii, a takzhe s risolivovanim mikoronaomnemterra PikN.
Vakhno otmetnist, otop pri nisol'ovanii kornitratii opredeleniia znachini prozolnoyiushchnogo (poverniknost') soprintshenie obraziov soprintshenie obraziov soprintshenie obraziov soprintshenie obraziushchnist'lii elektrokhodiushchnist'lii na prilobre E6-3, oblasniannom v s Oostrovannoi kol'tsevoi iammeritel'noi chalekii, a takzhe s risolivovanim mikoronaomnemterra PikN.
Vakhno otmetnist, otop pri nisol'ovanii kornitratii opredeleniia znachini prozolnoyiushchnogo (poverniknost') soprintshenie obraziov soprintshenie obraziov soprintshenie obraziushchnist'lii elektrokhodiushchnist'lii na prilobre E6-3, oblasniannom v s Oostrovannoi kol'tsevoi iammeritel'noi chalekii, a takzhe s risolivovanim mikoronaomnemterra PikN.
Vakhno otmetnist, otop pri nisol'ovanii kornitratii opredeleniia znachini prozolnoyiushchnogo (poverniknost') soprintshenie obraziov soprintshenie obraziov soprintshenie obraziushchnist'lii elektrokhodiushchnist'lii elektrokhodiushchnist'lii na prilobre E6-3, oblasniannom v s Oostrovannoi kol'tsevoi iammeritel'noi chalekii, a takzhe s risolivovanim mikoronaomnemterra PikN.
Vakhno otmetnist, otop pri nisol'ovanii kornitratii opredeleniia znachini prozolnoyiushchnogo (poverniknost') soprintshenie obraziov soprintshenie obraziov soprintshenie obraziov soprintshenie obraziushchnist'lii elektrokhodiushchnist'lii elektrokhodiushchnist'lii na prilobre E6-3, oblasniannom v s Oostrovannoi kol'tsevoi iammeritel'noi chalekii, a takzhe s risolivovanim mikoronaomnemterra PikN.
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Seludver obsobo otmethy, tno nekobliv i3 CHII v CBI nabholidnics.
distsi | 10.48550/arXiv.1306.4127 | Some features of anomalous conductivity of vinylene and vinyl chloride copolymer films | V. I. Kryshtob, V. F. Mironov, L. A. Apresyan, D. V. Vlasov, S. I. Rasmagin, T. V. Vlasova | 1,507 |
10.48550_arXiv.1209.3660 | ###### Abstract
Density functional theory (DFT) has become the computational method of choice for modeling and characterization of carbon dioxide adsorbents, a broad family of materials which at present are urgently sought after for environmental applications. The description of polar carbon dioxide (CO\({}_{2}\)) molecules in low-coordinated environments like surfaces and porous materials, however, may be challenging for local and semilocal DFT approximations. Here, we present a thorough computational study in which the accuracy of DFT methods in describing the interactions of CO\({}_{2}\) with model alkali-earth-metal (AEM, Ca and Li) decorated carbon structures, namely anthracene (C\({}_{14}\)H\({}_{10}\)) molecules, is assessed. We find that gas-adsorption energies and equilibrium structures obtained with standard (i.e. LDA and GGA), hybrid (i.e. PBE0 and B3LYP) and van der Waals exchange-correlation functionals of DFT dramatically differ from results obtained with second-order Moller-Plesset perturbation theory (MP2), an accurate computational quantum chemistry method. The major disagreements found can be mostly rationalized in terms of electron correlation errors that lead to wrong charge-transfer and electrostatic Coulomb interactions between CO\({}_{2}\) and AEM-decorated anthracene molecules. Nevertheless, we show that when the concentration of AEM atoms in anthracene is tuned to resemble as closely as possible to the electronic structure of AEM-decorated graphene (i.e. an extended two-dimensional material), hybrid exchange-correlation DFT and MP2 methods quantitatively provide similar results.
pacs: 68.43.Bc, 73.63.Fg, 81.05.U-, 88.05.Np
## I Introduction
The concentration of carbon dioxide (CO\({}_{2}\)) in the atmosphere has increased by about 30% in the last 50 years and is likely to double over the next few decades as a consequence of fossil-fuel burning for energy generation. This excess of CO\({}_{2}\) greenhouse gas may have dramatic negative repercussions on Earth's air quality and climate evolution. Besides exploitation of renewable energy resources, carbon capture and sequestration (CCS) implemented in fossil-fuel energy plants and ambient air have been envisaged as promising cost-effective routes to mitigate CO\({}_{2}\) emissions. To this end, membranes and solid sorbents (e.g. activated carbons -AC-, hydrotalcites, and coordination polymers -i.e. zeolitic imidazolate and metal organic frameworks-) are widely considered as the pillars of next-generation CCS technologies because of a encouraging compromise between large gas-uptake capacities, robust thermodynamic stability, fast adsorption-desorption kinetics, and affordable production costs. A key aspect for the success of these materials is to find the optimal chemistry and pore topologies to work under specific thermodynamic conditions. Unfortunately, due to the tremendous variety of possible compositions and structures, systematic experimental searches of this kind generally turn out to be cumbersome. In this context, first-principles and classical atomistic simulation approaches emerge as invaluable tools for effective and economical screening of candidate carbon adsorbents.
Density functional theory (DFT) performed with the local-density (LDA) and generalized-gradient (GGA) approximations of the electronic exchange-correlation energy, has become the _ab initio_ method of choice for modeling and characterization of CCS materials.
MP2 adsorption energy results obtained for a small CO\({}_{2}\)/Ca-benzene system and expressed as a function of the intermolecular distance \(z\) and AO basis set. Ca, C, H, and O atoms are represented with large blue, yellow, small blue and red spheres, and the dashed line is a guide to the eye.
In view of the ubiquity of DFT methods to CCS science, it is therefore crucial to start filling this knowledge gap while putting a special emphasis on the underlying physics. Computational benchmark studies on CO\({}_{2}\)-sorbent interactions, however, are technically intricate and conceptually difficult since most CCS materials have structural motifs that are large in size. In particular, genuinely accurate but computationally very intensive quantum chemistry approaches like MP2 and CCSD(T) can deal efficiently only with small systems composed of up to few tens of atoms, whereas DFT can be used for much larger systems (i.e. extended -periodically replicated in space- systems composed of up to \(1,000-10,000\) atoms). Consequently, computational accuracy tests need to be performed in scaled-down systems resembling to the structure and composition of the material of interest (e.g. the case of organic C\({}_{n}\)H\({}_{m}\) molecules to graphene). Generalization of so-reached conclusions to realistic systems however may turn out to be fallacious since intrinsic DFT limitations (e.g. exchange self-interaction and electron correlation errors) can be crucial depending, for instance, on the level of quantum confinement imposed by the topology of the system.
In this article, we present the results of a thorough computational study performed on a model system composed of a CO\({}_{2}\) molecule and alkali-earth-metal (AEM) decorated anthracene (e.g. X-C\({}_{14}\)H\({}_{10}\) with X=Ca, Li), that consists of standard DFT (i.e. LDA and GGA), hybrid DFT (i.e. PBE0 and B3LYP), van der Waals DFT (vdW), and MP2 adsorption energy, E\({}_{\text{ads}}\), and geometry optimization calculations. It is worth noticing that anthracene is structurally and chemically analogous to the organic bridging ligands found in metal- and covalent-organic frameworks, -MOF and COF-, thus our model conforms to a good representation of a promising class of CCS materials. We find that standard, hybrid and vdW functionals of DFT dramatically fail at reproducing CO\({}_{2}\)/AEM-C\({}_{14}\)H\({}_{10}\) interactions as evidenced by E\({}_{\text{ads}}\) discrepancies of \(\sim 1-2\) eV found with respect to MP2 calculations. This failure is mainly due to electron correlation errors that lead to inaccurate electron charge transfers and exaggerated electrostatic Coulomb interactions between CO\({}_{2}\) and X-C\({}_{14}\)H\({}_{10}\) molecules. In the second part of our study we analyse whether our initial conclusions can be generalized or not to extended carbon-based materials, another encouraging family of CO\({}_{2}\) solvents. For this, we tune the concentration of calcium atoms in anthracene so that the partial density of electronic states (pDOS) of the model system resembles as closely as possible to the pDOS of Ca-decorated graphene. In this case we find that DFT and MP2 methods qualitatively provide similar results, with hybrid DFT and MP2 in almost quantitative agreement.
## II Computational methods
Standard DFT calculations were done using the plane wave code VASP while hybrid DFT and MP2 results were obtained with the atomic orbitals (AO) code NWCHEM. Numerical consistency between the two codes was checked at the DFT-PBE energy level. The value of all technical parameters were set in order to guarantee convergence of the total energy to less than 1 meV/atom. Optimized structures were determined by imposing an atomic force tolerance of 0.01 eV/A and verified as minima on the potential energy surface by vibrational frequency analysis. Basis-set superposition errors (BSSE) in hybrid DFT and MP2 energy calculations were corrected using the counter-poise recipe. Indeed, only results obtained in the complete-basis-set (CBS) limit can be regarded as totally BSSE free however reaching that limit in our calculations turned out to be computationally prohibitive due to the size of the systems and large number of cases considered. Nevertheless, we checked in a reduced Ca-benzene system that MP2 binding energy results obtained with large Dunning-like AO basis sets (i.e. triple zeta cc-pVTZ and quadruple zeta cc-pVQZ) and in the CBS limit differed at most by 20 meV (see Fig. 1), thus we assumed the MP2/cc-pVTZ method to be accurate enough for present purposes (i.e. as it will be shown later, the reported discrepancies are in the order of \(1-2\) eV) and regarded it as "gold standard". It is worth noting that MP2 results obtained with medium and large Pople-like AO basis sets (i.e. 6-311G++(2d,2p) and 6-311G(2df,2pd)) are also in notable agreement with MP2/cc-pVTZ results (i.e. E\({}_{\text{ads}}\) differences of \(20-30\) meV in the worse case) whereas MP2/6-31G++ estimations (not shown in the figure) turn out to be not so accurate.
Front (F) and top (T) views of equilibrium CO\({}_{2}\)-adsorption structures in Ca- and Li-anthracene as obtained with standard and hybrid DFT and MP2 methods. Li atoms are represented with purple spheres.
## III Results
### AEM-decorated anthracene systems
Carbon dioxide adsorption energy and equilibrium geometry results obtained for AEM-anthracene systems are shown in Figs. 2 and 3. Since the number of reaction coordinates in CO\({}_{2}\)/AEM-anthracene systems (i.e. intermolecular distances and molecular bond and torsional angles) is considerably large, rather than parameterizing E\({}_{\rm ads}\) curves as a function of just few of them, first we performed DFT and MP2 atomic relaxations and then calculated all DFT and MP2 energies in the resulting equilibrium structures. In doing this, we disregard local minimum conformations, as customarily done in computational materials studies, and gain valuable insight into the energy landscape provided by each potential. It is important to note that due to the size of the systems considered we were able to perform tight MP2 atomic relaxations only at the 6-31G++ and 6-311G++(2d,2p) levels. Nevertheless, MP2/6-311G++(2d,2p) and benchmark MP2/cc-pVTZ equilibrium structures are very likely to be equivalent since results obtained with both methods are in fairly good agreement (see Figs. 1 and 3), and MP2/6-31G++ and MP2/6-311G++(2d,2p) equilibrium geometries are already very similar (the former case not shown here).
First, we note that equilibrium DFT and MP2 structures obtained for both Ca- and Li-anthracene systems are perceptibly different (see Fig. 2). Of particular concern is the Ca-anthracene system where, depending on the geometry optimization method used, the plane containing the CO\({}_{2}\) molecule orientates perpendicularly (DFT) or parallel (MP2) to anthracene. Also, we observe important differences between DFT-PBE and DFT-B3LYP optimized geometries, the Ca atom being displaced towards an outside carbon ring in the hybrid case (as it also occurs in the MP2-optimized system). Structural differences among Li-anthracene systems are similar to those already explained except that the gas molecule always binds on top of the Li atom and no off-center AEM shift appears in MP2 optimizations.
Concerning E\({}_{\rm ads}\) results (see Fig. 3), let us concentrate first on the Ca-C\({}_{14}\)H\({}_{10}\) case. As one can see, adsorption energies calculated with the same evaluation and geometry optimization method (highlighted with large grey circles in the figure) are very different. In particular, DFT methods always predict thermodynamically favorable CO\({}_{2}\)-binding to Ca-anthracene, with DFT-LDA and DFT-PBE0 providing the largest and smallest E\({}_{\rm ads}\) values, whereas MP2/6-311G++(2d,2p) calculations show the opposite. Moreover, with MP2/6-311G(2df,2pd) and MP2/cc-pVTZ methods large and positive adsorption energies of \(\sim 0.6-1.0\) eV are obtained for DFT-optimized geometries, in stark contrast to DFT E\({}_{\rm ads}\) re
CO\({}_{2}\)-adsorption energy results obtained for Ca- (a) and Li-anthracene (b), using different optimization and evaluation methods. Cases in which optimization and evaluation methods coincide are highlighted with large grey circles. PS, PM, and PL notation stands for 6-31G++, 6-311G++(2d,2p), and 6-311G(2df,2pd) Pople AO basis sets, respectively.
\(\sim-1.0-0.0\) eV). Adsorption energy disagreements in Li-C\({}_{14}\)H\({}_{10}\) systems are not as dramatic as just described, although the performance of standard DFT methods still remains a cause for concern. Specifically, DFT-LDA and DFT-PBE predict unfavorable CO\({}_{2}\)-binding whereas hybrid DFT and MP2 methods predict the opposite. Also, computed MP2 energies in standard DFT-optimized structures are negative and noticeably larger than DFT \(\mid\) E\({}_{\text{ads}}\mid\) values. Interestingly, the series of binding energies calculated for hybrid DFT and MP2/6-311G++(2d,2p) geometries are in remarkably good agreement in spite of the evident structural differences involved (see Fig. 2).
Since the agreement between MP2 and hybrid DFT results in Ca-C\({}_{14}\)H\({}_{10}\) is only marginally better than achieved with LDA or GGA functionals, common self-interaction exchange errors alone cannot be at the root of standard DFT failure. Consequently, DFT difficulties at fully grasping electron correlations, which in the studied complexes account for the 44 % to 57 % of the total binding energy, must be the major factor behind the discrepancy. In fact, upon gas-adsorption important dispersive dipole-dipole and dipole-quadrupole forces appear in the systems as a consequence of CO\({}_{2}\) inversion symmetry breaking (i.e. the polar molecule is bent) which cannot be reproduced by either local, semilocal or hybrid DFT approximations. Moreover, the non-linearity of the gas molecule also indicates the presence of large electron transfers which are well-known to pose a challenge for description to standard and hybrid DFT methods. But, which of these DFT shortcomings, i.e. omission of non-local interactions, charge-transfer errors or a mix of both, is the predominant factor behind E\({}_{\text{ads}}\) inaccuracies? In order to get insight into this question, we performed frontier molecular orbital and charge distribution analysis on isolated and joint Ca-C\({}_{14}\)H\({}_{10}\) and CO\({}_{2}\) complexes. We found that the energy difference between the HOMO of Ca-anthracene and the LUMO of the gas molecule, \(\Delta E_{\text{front}}\), varied from \(0.7-0.9\) eV to \(-1.4\) eV when either calculated with DFT (standard and hybrid) or MP2/cc-pVTZ methods. Positive and large \(\Delta E_{\text{front}}\) values do imply large reactivity and charge transfers from Ca-C\({}_{14}\)H\({}_{10}\) to CO\({}_{2}\) molecules, \(\Delta Q\). In fact, this is consistent with results shown in and also with our charge distribution analysis performed: \(\Delta Q\) values obtained with DFT amount to \(\sim 1\)\(e^{-}\), about a 50 % larger than computed with MP2/cc-pVTZ. Standard and hybrid DFT approximations, therefore, provide overly charged donor (+) and acceptor (-) species which in the joint complexes are artificially stabilized by the action of exaggerated electrostatic interactions. In view of this finding, and also of the large size of the binding energies reported, we tentatively identify the inability of DFT methods to correctly describe charge-transfer interactions (and not the omission of non-local interactions) as the principal cause behind its failure. Calculations done on Li-anthracene systems appear to support this hypothesis since a mild improvement on the agreement between hybrid DFT and MP2 methods is obtained (not seen in the standard cases) which is accompanied by smaller \(\Delta E_{\text{front}}\) discrepancies (i.e. of just few tenths of an eV). Also, the amount of electronic charge transferred from Li-C\({}_{14}\)H\({}_{10}\) to CO\({}_{2}\) does not change significantly when either calculated with hybrid DFT or MP2 methods (i.e. 0.6 and 0.4 \(e^{-}\), respectively).
In order to fully quantify the effect of neglecting non-local interactions and their role on standard and hybrid DFT E\({}_{\text{ads}}\) inaccuracies, we conducted additional energy calculations using two different DFT van der Waals approaches implemented in VASP. The first of these methods corresponds to that developed by Grime, also known as DFT-D2, in which a simple pair-wise dispersion potential is added to the conventional Kohn-Sham DFT energy. The second approach, referred to as DFT-vdW here, is based on Dion _et al._'s proposal for which a non-local correlation functional is explicitly constructed. We considered three different equilibrium configurations (i.e. those obtained in DFT-PBE, DFT-B3LYP and MP2/cc-pVTZ geometry optimizations) and calculated the corresponding DFT-D2 and DFT-vdW adsorption energies. In Table 1, we report the results of these calculations. As one can see, the overall effect of considering non-local interactions is to decrease E\({}_{\text{ads}}\) values by about few tenths of an eV, increasing so slightly the discrepancies with respect to the MP2/cc-pVTZ method. Also, we find that van der Waals corrections obtained with both DFT-D2 and DFT-vdW methods are fairly similar. For instance, \(\Delta E_{\text{D2}}\) and \(\Delta E_{\text{vdW}}\) differences (taken with respect to DFT-PBE values, see Table 1) computed in the DFT-B3LYP case amount to \(-0.08\) and \(-0.13\) eV, respectively. We also found that the equilibrium CO\({}_{2}\)/Ca-C\({}_{14}\)H\({}_{10}\) geometry determined with the DFT-D2 method is practically identical to that found with DFT-PBE. Therefore, our above assumption on the causes behind unsatisfactory description of specific CCS processes by DFT methods (i.e. charge-transfer interaction errors) turns out to be rigorously demonstrated.
Overall, the results presented in this section show the key importance of charge-transfer interactions in AEM-based CCS nanoparticles which, as we mentioned in the Introduction, can be reasonably generalized to similar covalent organic structures. Consequently, MP2 or other efficient computational schemes embodying also many-electron correlations (see for instance Ref.) must be employed for the rational design and characterization of these complexes.
### Emulating extended AEM-decorated carbon surfaces
In view of the composition and structure of anthracene, it appears tempting to generalize our previous conclusions to extended carbon-based materials (e.g. AC and nanostructures). Nevertheless, we show in that Ca-anthracene and Ca-graphene are quite distinct systems in terms of electronic structure since partially occupied \(s\) and \(d\) electron orbitals are missing in the former. It is important to note that equivalent partial density of states (pDOS) dissimilarities are also found when larger C\({}_{n}\)H\({}_{m}\) molecules are considered (e.g. the case of Ca-coronene). In consequence of these differences, hybridization between \(sd\)-sorbent and \(p\)-CO\({}_{2}\) orbitals leading to strong gas attraction will be more limited in Ca-C\({}_{14}\)H\({}_{10}\) complexes than in Ca-decorated graphene.
C\({}_{14}\)H\({}_{10}\) are already compatible with those relevant for CO\({}_{2}\)-binding to Ca-graphene (see Fig. 4) thus, in spite of the obvious chemical and structural deformations introduced (e.g. now the anthracene molecule is significantly bent, see Fig. 5), we performed additional benchmark calculations for this system.
In Figs. 5 and 6 we show the resulting optimized geometries and E\({}_{\text{ads}}\) values. As can be appreciated standard DFT, hybrid DFT and MP2 calculations are now in qualitative good agreement: all methods predict equivalent equilibrium structures and thermodynamically favorable CO\({}_{2}\)-binding. Moreover, adsorption energy differences between hybrid DFT and MP2 methods amount to less than 0.4 eV in most studied geometries with hybrid DFT systematically providing the smaller values. Therefore, the agreement between hybrid DFT and MP2 approaches can be regarded in this case as almost quantitative. On the other hand, standard DFT approximations tend to significantly overestimate E\({}_{\text{ads}}\) (i.e. by about \(\sim 1.0-2.0\) eV with respect to the MP2/cc-pVTZ method). It must be stressed that charge-transfer interactions in Ca-overdecorated organic complexes are very intense and play also a dominant role. Indeed, the amount of electron charge transferred from 3Ca-C\({}_{14}\)H\({}_{10}\) to CO\({}_{2}\) is \(\Delta Q=1.3\)\(e^{-}\) according to the MP2/cc-pVTZ method and also hybrid DFT approximations. In contrast, standard DFT methods predict exceedingly large \(\Delta Q\) values of \(\sim 1.6-1.8\)\(e^{-}\). We identify therefore self-interaction exchange errors, which now appear as a consequence of populating spatially localized \(d\)-Ca orbitals, as the principal cause for the overestimation of CO\({}_{2}\)-binding by standard DFT approaches.
In the light of the results presented in this section, we conclude that standard DFT modeling of extended carbon-based CCS materials can be expected to be correct only at the qualitative level. On the other hand, hybrid DFT approximations conform to a well-balanced representation of the relevant interactions in AEM-decorated carbon surfaces thus we propose using them when pursuing accurate description of these systems.
## IV Conclusions
We have performed a thorough computational study in which the failure of standard, hybrid and van der Waals DFT methods at describing the interactions between X-anthracene (X = Li and Ca) and CO\({}_{2}\) molecules is demonstrated. The origins of this deficiency mainly resides on the inability of standard and hybrid DFT approximations to correctly describe charge-transfer interactions. This finding has major implications in modeling and characterization of coordination polymer frameworks (e.g. MOF and COF) with applications in carbon capture and sequestration. As an effective strategy to get rid of these computational shortcomings, we propose using MP2 or other effective computational approaches incorporating many-electron correlations. Moreover, based on the similarities in electronic structure found between Cagraphene and 3Ca-C\({}_{14}\)H\({}_{10}\) systems (and in spite of their obvious chemical and structural differences) and the tests performed, we argue that standard DFT modeling of extended carbon-based materials may be expected to be correct at the qualitative level. On the other hand, hybrid DFT approximations will provide quantitative information on these systems. The conclusions presented in this work suggest revision of an important number of computational studies that are relevant to CCS materials engineering.
| 10.48550/arXiv.1209.3660 | Accuracy of Density Functional Theory in Prediction of Carbon Dioxide Adsorbent Materials | Claudio Cazorla, Stephen A. Shevlin | 6,047 |
10.48550_arXiv.1908.05807 | ###### Abstract
We have studied the longitudinal spin Seebeck effect (LSSE) in the layered ferromagnetic insulators CrSiTe\({}_{3}\) and CrGeTe\({}_{3}\) covered by Pt films in the measurement configuration where spin current traverses the ferromagnetic Cr layers. The LSSE response is clearly observed in the ferromagnetic phase and, in contrast to a standard LSSE magnet Y\({}_{3}\)Fe\({}_{5}\)O\({}_{12}\), persists above the critical temperatures in both CrSiTe\({}_{3}\)/Pt and CrGeTe\({}_{3}\)/Pt samples. With the help of a numerical calculation, we attribute the LSSE signals observed in the paramagnetic regime to exchange-dominated interlayer transport of in-plane paramagnetic moments reinforced by short-range ferromagnetic correlations and strong Zeeman effects.
By injecting this spin current into a paramagnetic metal it can be measured as a voltage through the inverse spin-Hall effect. Because of the simple bilayer structure needed to generate a thermoelectric voltage, LSSE devices have a potential use as thermoelectric conversion elements. From the point of basic physics, the LSSE is sensitive to spin correlations, and thus can be exploited as a probe to study the dynamical spin susceptibility in magnetic materials. The LSSE was originally found in ferro(ferri)magnets and later measured also in antiferromagnets and paramagnets. However, the LSSE has not been studied in magnetic materials with two-dimensional (2D) crystal structures, even though 2D materials, such as transition-metal chalcogenides, have drawn extensive research attention due to their extraordinary magnetic properties.
The layered ferromagnetic insulators CrSiTe\({}_{3}\) and CrGeTe\({}_{3}\) have been studied recently due to their intriguing physical properties. First-principles calculations predicted that the ferromagnetism in CrSiTe\({}_{3}\) survives even down to a monolayer thickness and indeed ferromagnetism in bilayer flakes of CrGeTe\({}_{3}\) was experimentally confirmed. The Cr layers possess a graphenelike honeycomb structure and exotic properties are expected to arise in heterostructures fabricated with related 2D materials by van der Waals epitaxy. Furthermore, it has been reported that CrGeTe\({}_{3}\) acts as an ideal ferromagnetic substrate for the growth of the popular topological insulator Bi\({}_{2}\)Te\({}_{3}\).
In this Rapid Communication, we studied the LSSE in the ferromagnets CrSiTe\({}_{3}\) and CrGeTe\({}_{3}\) in contact with Pt films. The crystal structure of CrSiTe\({}_{3}\) and CrGeTe\({}_{3}\) is illustrated schematically in Fig. 1(a). The Cr\({}^{3+}\) (spin 3/2) ions form a honeycomb lattice in the \(ab\) plane with the Si or Ge atoms in the center of the hexagon and the Cr\({}^{3+}\) atoms are surrounded by octahedra of Te atoms. The honeycomb layers stack along the \(c\) axis, held together by van der Waals interactions, forming a quasi-2D structure with a highly anisotropic magnetic environment. Interestingly, it was reported that CrSiTe\({}_{3}\), with a Curie temperature of \(T_{\rm C}\approx 31\) K, has short-range, in-plane ferromagnetic correlations which survive up to at least 300 K, whereas out-of-plane correlations disappear above 50 K. This is because the in-plane exchange coupling is more than five times greater than the out-of-plane coupling.
Single crystals of CrSiTe\({}_{3}\) and CrGeTe\({}_{3}\) were grown by a self-flux method, following the procedure described in the literature. First, high-purity powders of Cr, Si, Ge, and Te were placed in alumina crucibles in a molar ratio of Cr:Si:Te = 1:2:6 and Cr:Ge:Te = 1:3:18; the excess Si, Ge, and Te work as a flux for the crystal growth. The alumina crucibles were placed inside quartz tubes and sealed under argon atmosphere (pressure of 0.3 bar). The ampoules were then heated up to 1150 \({}^{\circ}\)C (700 \({}^{\circ}\)C), and maintained at theses temperatures for 16 h (22 days) and then slowly cooled to 700 \({}^{\circ}\)C (500 \({}^{\circ}\)C) for CrSiTe\({}_{3}\) (CrGeTe\({}_{3}\)), followed by centrifugation to remove excess flux. Single crystals were obtained as platelike forms with the size of several millimeters.
The sample structures were characterized by x-ray diffraction with Cu \(K\alpha 1\) radiation at room temperature. Figure 1(c) shows x-ray diffraction patterns of the CrSiTe\({}_{3}\) and CrGeTe\({}_{3}\) samples. Both samples show only sharp (00\(n\)) peaks; no impurity peaks were observed. The widest planes of the single crystals were determined as crystallographic \(ab\) planes. The lattice parameter was estimated to be \(c=20.67\) A for CrSiTe\({}_{3}\) and \(c=20.56\) A for CrGeTe\({}_{3}\), consistent with previous reports.
The temperature (\(T\)) dependences of the in-plane sheet resistance of CrSiTe\({}_{3}\) and CrGeTe\({}_{3}\) crystals are shown in Fig. 1(d).
(a) Schematic illustration of the crystal structure of CrSiTe\({}_{3}\) and CrGeTe\({}_{3}\). (b) Schematic illustration of the LSSE measurements. \(H\) denotes the external magnetic field and \(\Delta T\) (\(\nabla T\)) the temperature difference (gradient). (c) X-ray diffraction patterns of CrSiTe\({}_{3}\) and CrGeTe\({}_{3}\) single crystals. The inset is an atomic force microscope image of the surface of CrGeTe\({}_{3}\). (d) \(T\) dependence of the sheet resistance for CrSiTe\({}_{3}\) and CrGeTe\({}_{3}\).
The sheet resistance of CrGeTe\({}_{3}\) shows a similar \(T\) dependence, but is three orders of magnitude smaller than that of CrSiTe\({}_{3}\). In CrGeTe\({}_{3}\)/Pt devices ferromagnetic itinerant transport, such as the anomalous Nernst effect, is expected to be negligible below \(\approx\) 100 K.
Magnetic properties were measured with a vibrating sample magnetometer. Figure 2(a) shows the magnetization \(M\) versus magnetic field \(H\) applied in the \(ab\) plane of CrSiTe\({}_{3}\) and CrGeTe\({}_{3}\) crystals at \(T=5\) K. A magnetization curve with a saturation magnetization of about 2.7\(\mu_{\rm B}\) was observed for both samples, in good agreement with the spin 3/2 of Cr\({}^{3+}\) ions. In Fig. 2(b), we show the \(T\) dependence of \(M\) for CrSiTe\({}_{3}\) and CrGeTe\({}_{3}\) at \(H=1\) kOe. A sharp paramagnetic to ferromagnetic phase transition was observed in both samples.
We evaluated the critical temperature \(T_{\rm C}\) for CrSiTe\({}_{3}\) and CrGeTe\({}_{3}\) using a modified Arrott plot analysis. This method determines the critical exponents \(\beta\) and \(\gamma\) by the fitting of \(H/M\) and \(M\) with the Arrott-Noaks equation:
\[\left(\frac{H}{M}\right)^{1/\gamma}=a\frac{T-T_{\rm C}}{T}+bM^{1/\beta}, \tag{1}\]
Fitting of Eq. with a rigorous iterative method gives \(T_{\rm C}=31.3\) K for CrSiTe\({}_{3}\) and \(T_{\rm C}=64.7\) K for CrGeTe\({}_{3}\) [see Figs. 2(c) and 2(d), where the estimated critical exponents \(\beta\) and \(\gamma\) are also shown]. The values of \(\beta\) are smaller than in three-dimensional (3D) spin models (\(\beta_{\rm 3D\,Heisenberg}=0.365\) and \(\beta_{\rm 3D\,Ising}=0.325\)) but larger than 2D spin models (\(\beta_{\rm 2D\,Ising}=0.125\)), signaling a 2D character of the ferromagnetic transition.
To measure the LSSE we used samples of CrSiTe\({}_{3}\) (CrGeTe\({}_{3}\)) with dimensions \(L_{x}=1.5\) mm (0.8 mm), \(L_{y}=3.5\) mm (4.4 mm), and \(L_{z}=0.2\) mm (75 \(\mu\)m) and deposited a 5-nm-thick Pt film on the surface [see Fig. 1(b)]. To ensure clean and flat interfaces, their (as-grown) top (\(ab\)-plane) surfaces were exfoliated using adhesive tape before the Pt deposition; the resultant surface roughnesses (\(R_{a}\)) of the CrSiTe\({}_{3}\) and CrGeTe\({}_{3}\) samples were \(5.4\times 10^{-2}\) nm and \(4.1\times 10^{-2}\) nm, respectively, confirming the samples are very flat and smooth [see the atomic force microscope image for the CrGeTe\({}_{3}\) surface shown in the inset to Fig. 1(c)]. To apply a temperature gradient, \(\nabla T\), along the \(c\) axis [\(z\) axis in Fig. 1(b)]of the samples, the CrSiTe\({}_{3}\)/Pt and CrGeTe\({}_{3}\)/Pt were sandwiched between two sapphire plates; a 100 \(\Omega\) chip resistor was fixed on one plate, while the other is connected to a heat bath [see Fig. 1(b)]. By applying a charge current to the resistor, a constant temperature difference, \(\Delta T\), of 2 K (1 K) was generated for the CrSiTe\({}_{3}\)/Pt (CrGeTe\({}_{3}\)/Pt) sample that was measured by using type-E thermocouples attached to the two sapphire plates [see Fig. 1(b)]. The external magnetic field \(H\) was applied in the \(ab\) plane (along the \(x\) axis) and the thermal voltage \(V\) between the ends of the Pt film (along the \(y\) axis; its distance \(L_{y}\)) was measured. We define the LSSE voltage \(V_{\rm LSSE}\) as the antisymmetric contribution of the thermoelectric voltage: \([V(+H)-V(-H)]/2\). Hereafter, we mainly use the transverse thermopower \(S=(V_{\rm LSSE}/\Delta T)(L_{z}/L_{y})\) as the normalized LSSE voltage.
Figures 3(a) and 3(c) show the \(H\) dependence of \(S\) in the CrSiTe\({}_{3}\)/Pt and CrGeTe\({}_{3}\)/Pt samples at selected temperatures. In the ferromagnetic phase below \(T_{\rm C}\) clear LSSE signals were observed for both the samples [see the dark- and bright-blue solid lines in Figs. 3(a) and 3(c)]. The \(S\)-\(H\) curves for the CrSiTe\({}_{3}\)/Pt and CrGeTe\({}_{3}\)/Pt qualitatively agree with the \(M\)-\(H\) curves for the bulk CrSiTe\({}_{3}\) and CrGeTe\({}_{3}\) crystals as shown in Figs. 3(b) and 3(d); by increasing \(H\) from zero, the \(S\) amplitude rapidly increases and almost saturates above the
(a) \(H\) dependence of \(M\) (\(M\)-\(H\) curve) for CrSiTe\({}_{3}\) and CrGeTe\({}_{3}\) at 5 K. (b) \(T\) dependence of \(M\) for CrSiTe\({}_{3}\) and CrGeTe\({}_{3}\) at 1 kOe (below saturation). (c),(d) Modified Arrott plot of isotherms for (c) CrSiTe\({}_{3}\) and (d) CrGeTe\({}_{3}\), from which the \(T_{\rm C}\) values are determined. The red straight lines in (c) and (d) are Eq. at \(T=T_{\rm C}\).
This is a characteristic feature of the LSSE.
Interestingly, clear LSSE signals are observed at the paramagnetic phases above \(T_{\rm C}\) where long-range ferromagnetic ordering is absent. The red solid lines in Figs. 3(a) and 3(c) represent the \(S\)-\(H\) curves for the CrSiTe\({}_{3}\)/Pt and CrGeTe\({}_{3}\)/Pt samples at 35 and 70 K, higher than their \(T_{\rm C}\) values of 31.3 and 64.7 K, respectively. We attribute the nonlinear \(H\) dependence of \(S\) to the LSSE as the observed \(S\) signals should originate purely from the LSSE in the paramagnetic phase since the Nernst effect in CrSiTe\({}_{3}\) and CrGeTe\({}_{3}\) will be vanishingly small for the high resistivities at these temperatures.
We systematically measured the \(T\) dependence of the LSSE. The dark-blue plots in Figs. 4(a) and 4(c) show the \(S\) versus \(T\) results for the CrSiTe\({}_{3}\)/Pt and CrGeTe\({}_{3}\)/Pt samples, at a low field of \(H=15\) and 6 kOe, respectively. By increasing \(T\) from low temperature, \(S\) increases and takes a maximum value at around 15 K (30 K) for the CrSiTe\({}_{3}\)/Pt (CrGeTe\({}_{3}\)/Pt) sample. Further increasing \(T\), \(S\) decreases. The behavior is similar to the LSSE in a 3D ferro(ferri)magnet Y\({}_{3}\)Fe\({}_{5}\)O\({}_{12}\) (YIG). In contrast to the YIG case, however, the \(S\) signal persists at and above \(T_{\rm C}\) [see the dark-blue plots around dashed lines in Figs. 4(a) and 4(c)]. Upon further increasing \(T\), the LSSE signal disappears around 50 K (90 K) for the CrSiTe\({}_{3}\)/Pt (CrGeTe\({}_{3}\)/Pt) sample, which is higher than \(T_{\rm C}\) by \(\sim 20\) K.
\(H\) dependence of the normalized LSSE signal \(S\) in (a) CrSiTe\({}_{3}\)/Pt and (c) CrGeTe\({}_{3}\)/Pt and \(H\) dependence of \(M\) for (b) CrSiTe\({}_{3}\) and (d) CrGeTe\({}_{3}\) at several temperatures, where \(H\) was applied in the \(ab\) plane and swept between \(\pm 90\) kOe.
The observation of the LSSE in the paramagnetic phase above \(T_{\rm C}\) gives an insight into the different roles of the anisotropic (in-plane and out-of-plane) spin correlations in LSSE. The quasi-2D ferromagnetism of CrSiTe\({}_{3}\) and CrGeTe\({}_{3}\) is due to the significant difference in strength of the in-plane and out-of-plane exchange interactions. In CrSiTe\({}_{3}\) the in-plane exchange coupling strength \(J_{ab}\sim 15\) K is more than five times larger than the out-of-plane \(J_{c}\). Short-range in-plane ferromagnetic correlations even persist to room temperature, while the out-of-plane correlations rapidly diminish above \(T_{\rm C}\). If the LSSE we measured was driven by spin pumping from the in-plane (\(\perp\nabla T\)) spin correlations adjacent to the Pt interface, the LSSE signal would appear until 300 K. That is not the case [Fig. 5(a)]. Our experimental results show the vital role of spin transport between the planes (along the temperature gradient, \(\parallel\nabla T\)) to create the nonequilibrium magnon population essential for the appearance of LSSE [Fig. 5(b)].
\(T\) dependence of the normalized LSSE signal \(S\) (a) at 15.0, 42.0, and 87.0 kOe in CrSiTe\({}_{3}\)/Pt and (c) at 6.0, 42.0, and 87.0 kOe in CrGeTe\({}_{3}\)/Pt. \(T\) dependence of the magnetization \(M\) (b) at 15.0, 42.0, and 87.0 kOe in CrSiTe\({}_{3}\) and (d) at 6.0, 42.0, and 87.0 kOe in CrGeTe\({}_{3}\). (e) Calculated spin pumping and (f) magnetization for CrSiTe\({}_{3}\) at 15.0, 42.0, and 87.0 kOe using the Hamiltonian in Ref..
5(a) and 5(b)]. Here, the interface spin pumping refers to the injection of spin currents by magnetization dynamics at the interface, and should thus be distinct from the bulk spin transport. This decoupling has been shown on picosecond timescales in extreme nonequilibrium using terahertz pulses. But we demonstrate this in a conventional LSSE experiment in a nonequilibrium but steady state.
In the ferromagnetic phase below \(T_{\rm C}\), the LSSE can be understood in the same manner as conventional 3D ferromagnets, such as YIG; in the long-range ordered state, the out-of-plane exchange coupling facilitates the spin transport along the \(c\) axis (\(\parallel\nabla T\)) [Fig. 5(b)]. In the paramagnetic phase above \(T_{\rm C}\), there is no magnetization (in zero magnetic field) and magnons cease to exist. Spin-current transport is therefore not usually observed. In the case of CrSiTe\({}_{3}\) and CrGeTe\({}_{3}\), however, strong short-range ferromagnetic correlations above \(T_{\rm C}\) form in the \(ab\) plane. These spin correlations can be conveyed along the \(c\) direction [Fig. 5(b)] via spin-exchange coupling (\(J_{c}\)) and dipole-dipole interactions, giving a finite inverse spin-Hall voltage when a moderate \(H\) is applied to align spins. Since the LSSE response is observed in a limited \(T\) range above \(T_{\rm C}\) [Figs. 4(a) and 4(c)], the exchange coupling is likely to be the primary interaction mediating the interlayer spin transport.
We also examined the \(T\) dependence of the LSSE response with high magnetic fields up
Schematic illustrations of CrSiTe\({}_{3}\)/Pt or CrGeTe\({}_{3}\)/Pt under \(\nabla T\). (a) For \(T\gg T_{\rm C}\), interfacial spin pumping from the in-plane spin correlations in CrSiTe\({}_{3}\) and CrGeTe\({}_{3}\) adjacent to the Pt interface is absent (\(J_{s}^{\rm int}=0\)), where bulk spin transport (\(\parallel\nabla T\)) in CrSiTe\({}_{3}\) and CrGeTe\({}_{3}\) is inert (\(J_{s}^{\rm bulk}=0\)) since thermal fluctuations mask weak out-of-plane spin-exchange interaction \(J_{\rm c}\) at such high temperatures. (b) For \(T\leqq T_{\rm C}\) and the paramagnetic phase just above \(T_{\rm C}\), bulk spin transport between the planes becomes active (\(J_{s}^{\rm bulk}\neq 0\)) and creates nonequilibrium magnon population at the interface, causing finite interfacial spin pumping \(J_{s}^{\rm int}\).
4(a) and 4(c) (see orange and red dots). As \(H\) increases \(S\) also increases in magnitude and survives to higher temperatures. For CrSiTe\({}_{3}\)/Pt [Fig. 4(a)] the \(S\) signal at 87.0 kOe appears even at \(\sim 80\) K, more than twice the critical temperature \(T_{\rm C}\sim 31.3\) K. Under strong magnetic fields, the spin polarization is further enhanced by the Zeeman interaction, which may be responsible for the \(S\) increase. This breaks the symmetry of the Hamiltonian so the system no longer exhibits true critical behavior. This is seen in the \(M\)-\(T\) curves shown in Figs. 4(b) and 4(d) where the ferroparamagnetic transition becomes less defined and there is a significant magnetization above the true critical temperature. \(S\) decreases more rapidly than \(M\) with increasing \(T\) [Figs. 4(a) and 4(c)]. The spin Seebeck signal has almost disappeared even though \(M>0.5\mu_{\rm B}\), which also points to the importance of the out-of-plane spin transport in the LSSE. The coupling between the layers (\(J_{c}\)) is insensitive to magnetic fields so the decay rate of \(S\) is not relational as with applied fields. Hence \(S\) does not show the dependence on \(M\) that would be expected for a "3D" magnetic system.
To confirm the above scenario and separate the spin pumping and spin transport of the LSSE we performed atomistic spin dynamics calculations of the (in-plane) magnetization and spin pumping (in the absence of spin transport). This provides the pure spin pumping contribution of the LSSE without concerns about the magnon distribution at the interface [its formulation is given by Eq. in Ref.]. The Hamiltonian is based on the magnetic parameters of CrSiTe\({}_{3}\) reported in Ref.. Our calculations show a good agreement with the \(M\)-\(T\) curves of the experiments [Figs. 4(b) and 4(d)], although with a slightly higher critical temperature. Figure 4(e) shows \(T\) dependence of the spin pumping amplitude at 15, 42, and 87 kOe. Large \(H\) increases the spin pumping significantly at temperatures far above \(T_{\rm C}\) due to the Zeeman effect; it shows a maximum at approximately 70 K and decreases gently. At 100 K there is still a large difference between the different field values, roughly proportional to \(M\). This is also consistent with analytic approaches in paramagnets. The reason this is not seen in our experimental measurements is because the out-of-plane spin transport between the layers has vanished and so the magnon distribution at the interface is close to equilibrium. Therefore the paramagnetic spin pumping would not be observed.
To summarize, we studied the LSSE in the ferromagnetic transition-metal trichalcogenides CrSiTe\({}_{3}\) and CrGeTe\({}_{3}\) with Pt contact. In contrast to typical LSSE magnet YIG, the LSSE signal in the CrSiTe\({}_{3}\)/Pt and CrGeTe\({}_{3}\)/Pt was found to persist even above \(T_{\rm C}\). The LSSE above \(T_{\rm C}\) is dominated by thermal spin transport via out-of-plane exchange coupling between the 2D layers. At low magnetic fields the strong short-range ferromagnetic correlations inherent in CrSiTe\({}_{3}\) and CrGeTe\({}_{3}\) are the main spin-current carriers, while the Zeeman energy, comparable in strength to the exchange coupling, is dominant for the spin alignment under high magnetic fields. Our numerical simulation corroborates the strong magnetic-field effects and points out the importance of the interlayer spin transport regardless of the quasi-2D structure. The measurable spin transport across the ferromagnetic Cr layers in CrSiTe\({}_{3}\) and CrGeTe\({}_{3}\) could be useful in studying spin transports in heterostructures fabricated with other 2D materials and topological insulators.
The authors thank K. Ohgushi, K. Emi, K. Tanigaki, T. Ogasawara, K. Nawa, and M. Takahashi for their valuable comments on crystal growth. The authors also thank Y. Nambu, S. Takahashi, and K. Oyanagi for their fruitful discussion. This work was supported by JST ERATO "Spin Quantum Rectification Project" (Grant No. JPMJER1402), JSPS KAKENHI (Grants No. JP26103005, No. JP19H02424, No. JP19K21039, No. JP18H04215, No. JP18H04311, No. JP19K21031, and No. JP17K14102), JSPS Core-to-Core program "the International Research Center for New-Concept Spintronics Devices", and GP-Spin, Tohoku University. J.B. acknowledges support from the Royal Society through a University Research Fellowship. D.H. was supported by the Yoshida Scholarship Foundation through the Doctor 21 program.
| 10.48550/arXiv.1908.05807 | Spin Seebeck effect in the layered ferromagnetic insulators CrSiTe$_3$ and CrGeTe$_3$ | Naohiro Ito, Takashi Kikkawa, Joseph Barker, Daichi Hirobe, Yuki Shiomi, Eiji Saitoh | 800 |
10.48550_arXiv.0708.4050 | ## 3.
Firstly, the dependence of \(\alpha_{E}\) on \(H\) was measured at 1 kHz for transverse (in-plane) and longitude (out-of-plane) magnetic fields, respectively. With the rise of \(H\), \(\alpha_{E,31}\) increases first, reaches a maximum at \(H_{\rm m}\)=0.16 kOe, then decreases rapidly. In contrast, with the rise of \(H\), \(\alpha_{E,33}\) first increases to a maximum at \(H_{\rm m}\)=4.5
\begin{table}
\begin{tabular}{l c} Nickel aminosulfonate(g/l) & 600 \\ Nickelous chloride(g/l) & 20 \\ Boric acid(g/l) & 20 \\ Sodium lauryl sulfate(g/l) & 0.1 \\ PH & 4 \\ Temperature(\({}^{\circ}\)C) & 60 \\ Cathodic current density (A/dm\({}^{2}\)) & 5 \\ \end{tabular}
\end{table}
Table 1: Components and process parameters of the nickel electro-depositionkOe, and then decreases slowly. The magnitude and the field dependence of \(\alpha_{\rm E}\) are related to variation of the demagnetic effect. The ME coefficients are directly proportional to \(q\)\(\sim\)\(\delta\)\(\lambda\)/\(\delta H\), where \(\delta\lambda\) is the magnetostriction, and the \(H\)-dependence tracks the slope of \(\lambda\) vs \(H\). Saturation of \(\lambda\) at high field leads to \(\alpha_{\rm E}\)=0.
Secondly, \(\alpha_{\rm E}\) was measured at the bias field of \(H_{\rm m}\) as frequency of AC magnetic field _(f)_ varied from 1 kHz to 120 kHz. Typical \(\alpha_{\rm E}\) vs \(f\) profile for transverse fields is shown in Fig. 2(a). Both for \(\alpha_{\rm E,31}\) and \(\alpha_{\rm E,33}\), there is a sharp peak at about 88 kHz. However, the maximum of \(\alpha_{\rm E,33}\), about 1.08 V/cm Oe, is an order of magnitude smaller than that of \(\alpha_{\rm E,31}\), 33 V/cm Oe.
There is a resonance peak at about 89.9 kHz which is associated with the electromechanical resonance (EMR). The consistency of the frequency of the peak of dielectric constant and \(\alpha_{\rm E}\) indicates that the high ME effect is associated with the EMR.
The frequency of EMR shifts towards high frequency with the rise of Ni thickness, because it is directly proportional to the thickness of piezoelectric and piezomagnetic layers. insert also shows frequency dependence of dielectric constant is consistent with that of EMR for the samples with different Ni thickness. The resonance frequency of magnetoelectric voltage coefficient in inset is corresponding well that of dielectric constant in inset for the samples with the same Ni thickness.
Both \(\alpha_{\rm E,31}\) and \(\alpha_{\rm E,33}\) increase with the ratio of \(t_{\rm Ni}\) /(\(t_{\rm Ni}+t_{\rm PZT}\)). When the total thickness of Ni is about 0.4mm, \(\alpha_{\rm E,31}\) is up to 33 V/cm Oe, and \(\alpha_{\rm E,33}\) is up to 1.08 V/cm Oe. These results can be well reproduced and have good agreement with theoretical expression for layered ME composites:
\[\begin{split}\alpha_{E,31}=&\frac{-k(q_{21}+q_{11})d_{31}t_ {\rm Ni}t_{\rm PZT}}{(s_{11}^{\rm Ni}+s_{12}^{\rm Ni})\varepsilon_{33}kt_{\rm PZT}+(s_{11}^{\rm PZT}+s_{12}^{\rm PZT})\varepsilon_{33}t_{\rm PZT}-2(d_{31})^{2}kt_{\rm Ni}}\\ \alpha_{E,33}=&\frac{-2kq_{31}d_{31}t_{\rm Ni}t_{\rm PZT}}{(s_{11}^{\rm Ni}+s_{12}^{\rm Ni})\varepsilon_{33}kt_{\rm PZT}+(s_{11}^{\rm PZT}+s_{12}^{\rm PZT})\varepsilon_{33}t_{\rm Ni}-2(d_{31})^{2}kt_{\rm Ni}}\end{split} \tag{1}\]
Since the thickness of PZT is constant in our experiment, the ME voltage coefficient is directly proportional to the thickness of Ni. Based on equation, \(\alpha_{E,31}\) should increases with increasing \(t_{\rm Ni}\) because of \(d(\alpha_{E,31})/d(t_{\rm Ni})>\)0. Nan et al theoretically reported that \(\alpha_{E}\) increases with \(t_{\rm Ni}\) monotonously. Our experimental result is in good agreement with this theoretical prediction.
For the magnetoelectric laminate structure, the interfacial binding between magnetostrictive layer and piezoelectric layer is important to the magnetoelectric coefficient. Liu et al theoretical reported the influence of interfacial binder layer's thickness and shear modulus on the magnetoelectric effect. While interfacial binder layer's thickness rises or its shear modulus reduces, the ME responses will decrease rapidly. It is worthwhile to note that although the magnetostriction of Ni is two orders of magnitude smaller than that of Terfenol-D, the coefficient \(\alpha_{E}\) of the laminated Ni-PZT-Ni composite synthesized by electro-deposition is comparative to that of Terfenol-D/PZT/PVDF bulk samples. It is because the plastic interfacial layer of PVDF in the Terfenol-D/PZT/PVDF system was replaced by Ag metal interfacial layer in the electro-deposited Ni-PZT-Ni system. The shear modulus of Ag is much higher than that of PVDF, and the Ag layer is thinner than PVDF bonder layer. Hence, better interfacial coupling between PZT and Ni layers will supply the gap of Ni's small magnetostriction. It is promising to enhance the ME coefficient more by improving electro-deposition technology, such as electro-depositing metal with higher magnetostriction coefficient.
In summary, this article presents first report on ME interaction of Ni-PZT-Ni trilayered composites synthesized by electro-deposition. At the frequency of EMR, the coefficient \(\alpha_{\rm E}\) have a peak maximum, \(\alpha_{\rm E,31}\) up to 33 V/cm Oe when the thickness of Ni is about 0.4 mm. Tight bonding between PZT and Ni layers makes these samples exhibit large ME voltage coefficient among bulk magnetoelectric composites. Electro-deposition provide an effective method to enhance ME coefficient of magnetoelectric composites remarkably. Moreover, the electro-deposition method makes the preparation of ME composites with complex shape easily, and can control the structural parameters effectively. It will promote a rapid development of magnetoelectric composites' applications, such as various magnetoelectric coupling devices.
# Figure Captions Page
FIG. 1. Magnetoelectric voltage coefficient \(\alpha_{\rm E,31}\) and \(\alpha_{\rm E,33}\) at room temperature for Ni-PZT-Ni trilayered composites with total thickness of Ni about 0.4 mm.
FIG 2. Frequency dependence of \(\alpha_{\rm E,31}\) (a) and \(\alpha_{\rm E,33}\) (b) for the Ni-PZT-Ni trilayered composites with total thickness of Ni about 0.4mm at \(H_{\rm m}\) corresponding to maximum ME coupling (see FIG. 1.). The inset shows ME voltage coefficient around EMR frequency for the samples with different Ni thickness.
FIG. 3. Frequency dependence of dielectric constant and dielectric loss for the Ni-PZT-Ni trilayered composites with total thickness of Ni about 0.4 mm. The inset shows dielectric constant around EMR frequency for the samples with different Ni thickness.
FIG. 4. \(t_{\rm Ni}\) /(\(t_{\rm Ni}+t_{\rm PZT}\)) dependence of ME voltage coefficient for Ni-PZT-Ni trilayered composites at the frequency of EMR.
Magnetoelectric voltage coefficient \(\alpha_{\rm E,31}\) and \(\alpha_{\rm E,33}\) at room temperature for Ni-PZT-Ni trilayers with total thickness of Ni about 0.4 mm.
Frequency dependence of \(\alpha_{\rm E,31}\) (a) and \(\alpha_{\rm E,33}\) (b) for the Ni-PZT-Ni with total thickness of Ni about 0.4mm at \(H_{\rm m}\) corresponding to maximum ME coupling (see FIG 1.). The inset shows ME voltage coefficient around EMR frequency for the samples with different Ni thickness.
Frequency dependence of dielectric constant and dielectric loss for the Ni-PZT-Ni composites with total thickness of Ni about 0.4 mm. The inset shows dielectric constant around EMR frequency for the samples with different Ni thickness.
\(t_{\rm Ni}/(t_{\rm Ni}+t_{\rm PZT})\) dependence of ME voltage coefficient for Ni-PZT-Ni trilayers at the frequency of EMR. | 10.48550/arXiv.0708.4050 | Ni-PZT-Ni Trilayered Magnetoelectric composites Synthesized by Electro-deposition | D. A. Pan, Y. Bai, W. Y. Chu, L. J. Qiao | 4,296 |
10.48550_arXiv.1101.3538 | ##
\(R\)-\(T\) measurements of (a) 80 nm LAO/SNO and (b) 120 nm Si/SNO normalized to \(R\)(25 \({}^{\circ}\)C) of 1\({}^{\rm st}\) cycles. Insets: Raw \(R\)-\(T\) data for respective 3\({}^{\rm rd}\) cycles. (c) Resistance after applying 10 V, 1 s square pulses to in-plane 80 nm LAO/SNO device.
| 10.48550/arXiv.1101.3538 | Metal-insulator transition and electrically-driven memristive characteristics of SmNiO3 thin films | Sieu D. Ha, Gulgun H. Aydogdu, Shriram Ramanathan | 401 |
10.48550_arXiv.2012.04037 | ### Phase retrieval on image pairs of opposite defocus
State-of-the-art TIE formalism uses the differentiation of two images of opposite defocus to approximate the longitudinal intensity derivatives \(\partial I/\partial z\). The phase is then retrieved by solving the partial differential equation below using the intensity of the in-focus image, \(I_{0}\),
\[\nabla\cdot[I_{0}\nabla\varphi]=-\frac{2\pi}{\lambda}\frac{\partial I}{\partial z} \tag{1}\]
The choice of this image pair has significant impact on the accuracy and spatial resolution of the TIE retrieved phase, as is illustrated in the upper part of shows the ground truth phase used for creating the simulated LTEM dataset (Fig. S1). The knowledge of the ground truth is essential for evaluating the accuracy and spatial resolution of the retrieved phase, which is described in the **Methods** section. TIE retrieved phase using moderately defocused image pair (\(\Delta f\approx\pm 0.1\) mm) preserves a reasonable spatial resolution, but is less accurate and extremely susceptible to noise. TIE retrieved phase using strongly defocused image pair (\(|\Delta f|>1\)mm) is tolerant to noise (and 2e), but the result is blurry due to the much reduced spatial resolution.
We then performed AD phase retrieval on the same datasets. In the case of moderately defocused image pair, AD is not as accurate as TIE under noise free conditions, but is able to retain the same level of accuracy in the presence of noise.
Flow chart of phase retrieval in LTEM through differentiable programming. The experimental data and instrument parameters are color coded in grey. The imaging process that carries the phase information is color coded in blue while the path for back-propagation is color coded in red.
In the presence of noise, however, AD overfits to the noise, resulting in grainy images with low accuracy. Supplemental Figure S2b shows the evolution of the accuracy. It can be seen that AD initially reached a higher level of accuracy (orange line) than TIE (orange rectangle), but did not converge to the truth despite a monotonically decreasing loss function (Supplemental Figure S2c).
### AD phase retrieval on images of hybrid defocus
The high accuracy and high spatial resolution achieved in exemplifies the huge potential of AD as a viable phase retrieval method for LTEM, but its practical application is hampered by its instability in the presence of noise (Supplemental Figure S2b). To circumvent this, we explore a strategy that leverages the flexibility of the AD method to work with multiple images at different defocus conditions. We choose to show the gradient of the phase to highlight the effectiveness of our strategy.
(a) Ground truth for the phase. TIE retrieved phase using image pair with (b) 0% and (c) 10% of Gaussian noise at \(\Delta f=\pm\)0.15 mm. TIE retrieved phase using image pair with (d) 0% and (e) 10% of Gaussian noise at \(\Delta f=\pm\)1.6 mm. AD retrieved phase using image pair with (f) 0% and (g) 10% of Gaussian noise at \(\Delta f=\pm\)0.15 mm. AD retrieved phase using image pair with (h) 0% and (i) 10% of Gaussian noise at \(\Delta f=\pm\)1.6 mm. The numbers in the brackets denote the accuracy of retrieved phases.
As a result, the phase gradient gives a better assessment of the spatial resolution and accuracy of the retrieved phase. shows the ground truth for the gradient of the phase, with its direction and its magnitude respectively indicated by false color and grayscale contour. With 10% of Gaussian noise, the magnitude of the TIE retrieved phase gradient at moderate defocus is severely distorted at the center of the nanostructures, which is reflective of the low accuracy of 56.78%. Under strongly defocused conditions, the TIE retrieved phase gradient is distorted near the edge of the nanostructures, as a result of the low spatial resolution.
Next, we demonstrate that high spatial resolution, high accuracy and tolerance to noise can be simultaneously achieved in the AD formalism. We recall that while AD seems to guarantee a high spatial resolution regardless of the focusing conditions, extremely high accuracy was only observed with strongly defocused image pairs while tolerance to noise was only observed with moderately defocused image pairs. One advantage of the AD phase retrieval process is that it does not require the images to be pairs of opposite defocus. Indeed, pairing one image of moderate defocus with one of strong defocus effectively stabilized the accuracy evolution in the presence of 10% Gaussian noise (Fig.S3b, red line). The best result was obtained by combining two images of moderate defocus with two images of strong defocus, where an accuracy of 98.56% was reached (Fig.S3b, blue line), about 10% higher than the best value achievable with the TIE method. To ensure a fair comparison, the noise level of the four images was raised to 15%. This is to account for the factor of \(\sqrt{2}\) shot noise variation when the exposure time of each image is halved and the total exposure time remains unchanged. The much improved accuracy in the final retrieved phase is thus understood as due entirely to a stronger numerical constraint. shows the line profiles of the retrieved phase gradient across multiple nanostructures. It can be seen that AD with hybrid defocus images is the only method capable of reproducing faithfully the sharp variation of the phase gradient at the edge of the nanostructures.
### Application on experimental data
Finally, we demonstrate the viability of the proposed phase retrieval strategy on experimental LTEM images. The imaging conditions and information about the sample can be found in the **Methods** section. We have specifically chosen, for the purpose of verification, nanostructures with known magnetic configuration. Three different scenarios were tested, with 2, 4 and 20 imagestaken at various defocus conditions. The total counting time was 4 s in all three cases, to ensure a fair comparison under the same electron dose conditions. Similar to what was concluded with the simulated dataset, the convergence of the AD method improves with increasing number of images, with the best result obtained for a series of 20 images with hybrid defocus. and 4b show respectively the AD retrieved phase and its gradient on the 20 defocus images spanned between \(\Delta f=\pm 1.44\) mm. The exposure time was 0.2 s per image. and 4d show in comparison the TIE retrieved phase and its gradient at \(\Delta f=\pm 1.44\) mm, with 2 s of exposure time per image. As expected, AD retrieved phase appears to be sharper than the TIE retrieved one thanks to its inherent high spatial resolution. Line profiles were extracted across the center of the image. The results of the simulation study are shown in The results of the simulation study are shown in The results of the simulation study are shown in The results of the simulation study are shown in The results of the simulation study are shown in The results of the simulation study are shown in Fig.
Once again, AD (Fig. 4e, red line) outperformed TIE (black line) in retrieving the sharp variations of the phase gradient expected at the edges of the nanostructures. Elsewhere on the extracted line profiles, the two methods agree extremely well with each other, indicating that the accuracy of the AD method on the experimental data is at least equal to, if not better than that of TIE, under the same electron dose conditions.
## III Discussion
In this work, we have demonstrated a new method based on automatic differentiation for the phase retrieval in Lorentz TEM. The strengths and weaknesses of our proposed method against the conventional TIE approach are summarized in Fig. 5a, based on results presented in and Fig. S4. TIE in theory requires a pair of images, moderately defocused in the opposite directions.
(a) AD retrieved phase and (b) its gradient on images at defocus spanned between \(\Delta f=\pm 1.44\) mm, with 4 s of total exposure time. (c) TIE retrieved phase and (d) its gradient on images at defocus \(\Delta f=\pm 1.44\) mm, with 4 s of total exposure time. (e) shows line profile of the phase gradient. The position at which the line profile was extracted is indicated by the dashed line in (b).
While that works well with ideal data, the accuracy of the retrieved phase drops quickly in the presence of noise (green symbols). The tolerance to noise can be greatly enhanced by increasing the defocus distance of the image pair (orange symbols). This can be easily understood by looking back at Eq. 1, as any statistical fluctuations in the intensity is attenuated by a factor proportional to the defocus distance. The use of strongly defocused images is not without its problems. TIE is strictly valid only in the small defocus limit where the microscope transfer function remains linear for the given spatial frequency resolution. Therefore, the strong defocusing reduces the achievable spatial resolution, resulting in blurriness in the retrieved phase.
We then demonstrated the use of AD phase retrieval on the same set of data. We note that good spatial resolution is always guaranteed in the AD formalism regardless of the defocus distance of the image pair. But a new dilemma emerges. The accuracy of the retrieved phase is limited to about 90% when using moderately defocused images (red symbols). This can be understood by the fact that the transfer function for moderate defocus values does not have much variations to carry enough phase information into image intensity.
(a) Accuracy versus resolution plot for the TIE and the AD methods. The values are calculated from phase retrieved on the simulated dataset with various noise levels. Moderate defocus refers to \(\Delta f=\pm 0.15\) mm while strong defocus refers to \(\Delta f=\pm 1.6\) mm. The hybrid defocus uses 4 defocused images at -0.1, -0.2, +1.5 and +1.6 mm. (b) Accuracy of the AD retrieved phase using 2 images of hybrid defocus, for data with 10% of Gaussian noise. The defocus values cover the entire parameter space from \(\Delta f=+1.6\) mm (under-focus) to \(\Delta f=-1.6\) mm (over-focus). The values shown are the final accuracy of the retrieved phase after 500 iterations. (c) Stability of the AD retrieved phase covering the same parameter space as (b). The more “unstable” an image pair is, the earlier in the iterations does the AD retrieved phase diverges from the ground truth. The maximum value corresponds to those image pairs that continue to converge to the ground truth after 500 iterations.
However, the achieved high accuracy was limited to ideal data only, as AD tends to overfit to the presence of noise, resulting in divergence from the ground truth in noisy images. We solve this dilemma by leveraging the flexibility of the AD method to work with multiple images of mixed defocus conditions. In the most simple scenario, this involves pairing two images taken at different defocus distances. We have mapped out the entire parameter space to determine the achievable accuracy and stability (defined as the ability to stay converged to the ground truth in the presence of noise) in this scenario. The result is shown respectively in and c. It can be seen that while the maximum accuracy is obtained by pairing one strongly under-focused image with one strongly over-focused one (equivalent to having a strongly defocused image pair), the stability of the AD approach is not very high under these conditions. Instead, we can identify a sweet spot for high accuracy and high stability where a strongly defocused image is paired with a moderately defocused one (referred to as hybrid defocus).
We further demonstrated that, under the same electron dose condition, the accuracy and stability of AD phase retrieval improves with the number of images. For simulated data, the accuracy of the retrieved phase is 72.53% for 2 images of 10% Gaussian noise, and 98.56% for 4 images of 15% noise (Fig. 5, blue symbols). Additional tests on simulated data shows that the noisier the data is, the larger the number of images is required to maintain the stability (Fig. S5). The same was observed in the application of AD phase retrieval on experimental LTEM images, but for a slightly different reason. The retrieved phase appears to be more accurate when 20 images of hybrid defocus were used, as compared to 2 or 4 images of equal total exposure time. We believe that the use of a large number of images averages out the variations introduced by uncertainties in experimental parameters such as sample misalignment, leading to a highly accurate retrieved phase in our case.
The most significant advantage of our proposed method, as compared to the conventional TIE approach, is its ability to retrieve phase with simultaneous high spatial resolution and high accuracy. This is best illustrated on the line profiles of the gradient of the phase in both the simulated and experimental dataset. Compared to off-axis electron holography, it offers high resolution phase information over a large field of view, without the additional experimental complications. This is particularly important for applications such as determining interfacial electrostatic potentials in materials where it is critical to sample a statistically significant number of interfaces. The main limitation of our method is its requirement of _a priori_ knowledge of the microscope transfer function. Some of the parameters in the TF can be accurately determined in an aberration-corrected Lorentz TEM, such as spherical aberration and defocus distance. Other parameters like beam coherence are experimentally harder to evaluate, in which case, we recommend extending the AD approach to also retrieve these parameters along with the sample phase. It is in theory possible to recover the true amplitude as well, provided that sufficient amount of data is taken to ensure a strong numerical constraint. This would enable the reconstruction of the complete electron wavefunction, which can be adapted for high-resolution TEM imaging at the atomic scale. Finally, thanks to the simplicity of its implementation, the method can be easily incorporated with other iterative reconstruction algorithms such as electron beam tomography to determine three-dimensional electromagnetic fields..
## IV Methods
### Gradient descent optimization with automatic differentiation
We use the Adam optimizer implemented in Google's Tensorflow package for the gradient descent optimization. The initial learning rate is set to 1. Cyclic learning rate has occasionally been used but does not show additional improvement on the maximum achievable accuracy. Before computing the amplitude, the in-focus image was denoised with a total variation (TV) filter. For simulated data, the weight of the TV filter is set to the level of the Gaussian noise. For experimental data, it is set to the standard deviation measured on the background area. No denoising process was performed on any of the defocus images. The starting guess for the phase is 0.5 everywhere. Mean Squared Error was used as the loss function. The phase retrieval process is run on a remote Nvidia Tesla K80 GPU hosted on Google's Colaboratory. The amount of time per iteration varies depending on the number of defocus images used, but is typically 2 min per 5000 iterations.
### Accuracy and spatial resolution of the retrieved phase
The use of the simulated datasets allowed us to evaluate the accuracy of the retrieved phase, calculated as the correlation between the ground truth phase \(\varphi_{\text{tru}}\) and the retrieved phase \(\varphi_{\rm ret}\). The sum \(\Sigma\) runs over each pixel of the 2D image.
\[\Sigma|\varphi_{\rm ret}\varphi_{\rm tru}|/\sqrt{\Sigma|\varphi_{\rm ret}\varphi_{ \rm ret}|\Sigma|\varphi_{\rm tru}\varphi_{\rm tru}|} \tag{2}\]
The spatial resolution of the retrieved phase is estimated by fitting a Gaussian function to the sharp variation of the absolute of the phase gradient at the edges of the nanostructures. The spatial resolution is then taken as the FWHM of the Gaussian distribution. Because an error function can be considered as twice the integral of a normalized Gaussian function. This is equivalent to fitting an error function to the sharp variation of the phase at the same edges (Fig.
Simulated LTEM images were computed for magnetic nanostructures using parameters for Permalloy (Fig. S1). For a given noise level, a total of 65 images were produced with defocus values spanned evenly between \(\Delta f=\pm 1.6\) mm.
\[\varphi_{t}(\mathbf{r}_{\perp})=\sigma\int V(\mathbf{r}_{\perp},z)\;\mathrm{d }z+\frac{e}{\hbar}\int\mathbf{A}(\mathbf{r}_{\perp},z)\;\mathrm{d}z; \tag{3}\]
The simulated pattern was a slightly modified version of the 1951 USAF resolution test standard. Each LTEM image is composed of 512\(\times\)512 pixels of 5\(\times\)5 nm. Gaussian noise was added to the images after they are generated. The level of the Gaussian noise refers to the standard deviation of the distribution.
Experimental images (512\(\times\)512 pixels) were taken using an aberration-corrected JEOL 2100F Lorentz TEM operating at 200 kV. The sample consists of 10 nm thick of Permalloy magnetic nanostructures, sputter-deposited on a TEM grid and patterned with e-beam lithography. The pixel size was 6.9 nm.
| 10.48550/arXiv.2012.04037 | High resolution functional imaging through Lorentz transmission electron microscopy and differentiable programming | Tao Zhou, Mathew Cherukara, Charudatta Phatak | 3,167 |
10.48550_arXiv.0910.3522 | ###### Abstract
We present a dynamic model to study ordering of particles on arbitrary curved surfaces. Thereby the particles are represented as maxima in a density field and a surface partial differential equation for the density field is solved to the minimal energy configuration. We study annihilation of dislocations within the ordered system and premelting along grain-boundary scars. The obtained minimal energy configurations on a sphere are compared with existing results and scaling laws are computed for the number of excess dislocations as a function of system size.
Problems related to optimal ordering of particles on curved surfaces date back to the classical Thomson problem to find the ground state of \(N\) particles on a sphere interacting with a Coulomb potential. A classic theorem of Euler shows for a triangulation of the surface in which nearest neighbors are connected, that \(\sum_{i}(6-i)v_{i}=6\chi\), with \(v_{i}\) as the number of vertices with \(i\) nearest neighbors and \(\chi\) as the Euler characteristic of the surface. Thus for surfaces with the topology of a sphere (\(\chi=2\)), besides the expected triangular lattice with six-fold coordination, which would give the optimal packing in a plane, there must be at least 12 five-fold disclinations present. With each disclination an extra energy is associated (relative to a perfect triangular lattice in flat space) which grows proportional to \(r^{2}\), with \(r\) as the radius of the sphere. For a fixed lattice constant \(a\) we have \(N\sim(r/a)^{2}\). Thus for large \(N\) mechanisms are expected which reduce this extra energy by changing the ground-state configuration. One mechanism is a buckling transition of the disclinations, which form sharp corners and turn the sphere into a polygon. The transition depends on Young's modulus \(Y\) and bending rigidity \(b\) via the Foppl-von Karman number \(Yr^{2}/b\) and is intensively studied for viral capsids where protein subunits play the role of the particles. In cases where large surface tension limits significant buckling the energy can be reduced by introducing grain-boundary scars. Realizations are for example water droplets in oil, which are coated with colloidal particles. Such coated droplets are potential drug delivery vehicles. Similar configurations occur if a jammed layer of colloidal particles separats two immiscible fluids forming a so-called bijel, which has potential applications as an efficient micro-reacting media. A large number of ordered particles on curved surfaces is also required for fabrication of nanostructures on pliable substrates, e.g. to make foldable electronic devices. For all these applications a detailed understanding of the grain boundary scars is of interest as they may be sources of leaks, influence mechanical properties or lead to failure in electronic devices. We introduce an efficient way to compute the dynamics of these grain boundary scars and dislocations associated with them and provide an approach to compute optimal ordering of many particles on arbitrarily curved surfaces. As the grain boundary scars belong to the thermal and mechanical equilibrium our approach is based on energy minimization with the geometric frustration resulting from the curved surface incorporated.
For \(2\leq N\leq 100\) there is agreement of all numerical and theoretical methods for the Thomson problem, suggesting that the global minimum configuration has been found. However, for large \(N\), owing to an exponential growth in local minima, finding global minima becomes extremely difficult. Grain boundary scars are expected for \(N>360\). Numerical approaches to solve such problems are typically based on genetic algorithms, steepest decent minimization or coarse grained approaches, in which the elasticity field between grain boundary scars is solved. All approaches are devoted to finding the ground state. Dynamic models have been considered which allow us to describe experimentally observed dislocation glide within the grain boundary scars. We will introduce an approach without any coarse graining by directly addressing the dynamic evolution and rearrangement of the particles on an arbitrary curved surface. Our approach is based on a free energy functional for a number density. In the plane such free energy functionals have been used to characterize patterns.
\[\mathcal{F}[\rho]=\int_{\Omega}-|\nabla\rho|^{2}+\frac{1}{2}|\Delta\rho|^{2}+ f(\rho)\;d\Omega \tag{1}\]
The equilibrium state for \(\Omega=\mathbb{R}^{2}\) has a perfect six-fold symmetry. Evolutional laws associated with this energy are the \(L^{2}\)-gradient flow \(\partial_{t}\rho=-\delta\mathcal{T}/\delta\rho\), the Swift-Hohenberg model, and the \(H^{-1}\) gradient flow \(\partial_{t}\rho=\Delta\delta\mathcal{T}/\delta\rho\), the phase field crystal (PFC) model. The evolutions naturally contain elastic energy, as an expansion of the free energy around the equilibrium period spacing results in the potential energy of a spring, ieflooke's law. As the energyis rotationally invariant arbitrary orientationtions of periodic structures can emerge. Furthermore the model allows the formation of dislocations, which occur when two periodic structures of different orientation collide or when it is energetically favorable for them to nucleate. The PFC model has been used to simulate various crystal growth phenomena including epitaxial growth, nucleation, commensurate-incommensurate transitions and plastic deformations. In is shown how the model can be derived from a microscopic Smoluchowski equation via dynamical density functional theory. Formulating the energy in Eq.
\[\mathcal{F}^{\Gamma}[\rho]=\int_{\Gamma}-|\nabla_{\Gamma}\rho|^{2}+\frac{1}{2} |\Delta_{\Gamma}\rho|^{2}+f(\rho)\;d\Gamma \tag{2}\]
We will use the \(H^{-1}\) gradient flow of this energy to solve the generalized Thomson problem and to analyze the dynamics of rearrangements of particles on a curved surface.
\[\partial_{t}\rho = \Delta_{\Gamma}u \tag{3}\] \[u = \Delta_{\Gamma}v+2\,v+f^{\prime}(\rho)\] \[v = \Delta_{\Gamma}\rho. \tag{5}\]
The stable finite element discretization for the PFC model in the plane introduced in can be adapted to solve Eqs.- on a surface triangulation using parametric finite elements. The key idea is to use the surface operators on the discrete surface which consists of triangles \(T\). To do the integration on these triangles a parametrization \(F_{T}:\hat{T}\to T\) is used, with \(\hat{T}\) as the standard element in \(\mathbb{R}^{2}\). These allow us to transform all integrations onto the standard element using the finite element basis defined also only in \(\mathbb{R}^{2}\). The parametrization \(F_{T}\) is given by the coordinates of the surface mesh elements and provides the only difference between solving equations on surfaces and on planar domains. For a surface we have to allow \(F_{T}:\mathbb{R}^{2}\rightarrow\mathbb{R}^{3}\), whereas for a planar domain \(F_{T}:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}\). With this tiny modification any code to solve partial differential equations on Cartesian grids can be used to solve the same problem on a surface, providing a surface triangulation is given. The computational cost is the same as solving the problem in a planar domain. Within an efficient implementation this does not even require to recompile a running two-dimensional (2D) code, but only a change in a parameter file, as e.g. done in AMDiS. With this approach all available tools to solve partial differential equations on planar domains, such as adaptive refinement, multigrid algorithms or parallelization approaches are available also to solve equations on surfaces.
The approach is used to evolve a randomly perturbed constant initial configuration \(\rho=\rho_{0}\) towards an energy minimum. With the possibilities to use adaptive time stepping and the efficiency of parallelization of the finite element method problem sizes of \(1\times 10^{6}\) particles can be addressed. The simulation for with 6.064 particles required 1 day computing time on a single processor.
In order to validate our approach we compute minimal energy configurations for various numbers of \(N\). We systematically compute the minimal energy configuration for all \(N\in\). In our configuration \(N=2790\) corresponds to \(r=100\). The numerical results indicate, that the obtained minimal energy configurations are only sensitive to the defined lattice spacing and insensitive to a large extent to the other parameters. This might explain why triangular tessellations on spherical surfaces occur in very distinct occasions, for which the interactions involved may differ a lot. In the following we use \(\rho_{0}=-0.3\) and \(\epsilon=0.4\), which together with the radius \(r\) of the sphere determines \(N\). For all \(N\) the type and number of disclinations, as well as the computed energy is in agreement with known analytical or other numerical results. For \(N\leq 112\) the configurations and energies coincide with the known equilibrium values. The maximal deviation from the minimal energy for larger \(N\) is less than 0.1 %.
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \multicolumn{2}{c}{**N v4 v5**} & \multicolumn{2}{c}{**v6 v7 v8**} & \multicolumn{2}{c}{**energy**} \\ \hline
63 & 0 & 12 & 51 & 0 & 0 & 1708.87968150 \\ \hline
99 & 0 & 12 & 87 & 0 & 0 & 4357.13916313 \\ \hline
130 & 0 & 13 & 116 & 1 & 0 & 7632.16737891 \\ \hline
185 & 0 & 12 & 173 & 0 & 0 & 15723.72346397 \\ \hline
222 & 0 & 13 & 208 & 1 & 0 & 22816.0753076 \\ \hline
363 & 0 & 14 & 347 & 2 & 0 & 62066.53633167 \\ \hline
684 & 0 & 33 & 630 & 21 & 0 & 224048.60512144 \\ \hline
846 & 0 & 38 & 782 & 26 & 0 & 344267.84965308 \\ \hline
1073 & 0 & 45 & 995 & 33 & 0 & 556250.19927822 \\ \hline
1403 & 0 & 54 & 1307 & 42 & 0 & 955173.65896550 \\ \hline
2726 & 0 & 78 & 2584 & 64 & 0 & 3636897.41372145 \\ \hline \end{tabular}
\end{table}
Table 1: Comparison with known results for small \(N\). In order to compute the energy we identify the position of the maxima in the denisty field and compute the Coulomb energy according to this positions.
(Color online) Local minimal energy configuration for 6.064 particles on a sphere. (left) density profile, (right) color coded number of neighbors, 5-black, 6-red (gray) and 7-yellow (white).
For \(N>360\) we obtain additional defects in the ground state, which are pairs of fivefold and sevenfold coordinated particles (dislocations) and chains of alternation fivefold and sevenfold coordinated particles (grain-boundary scars). Since dislocations have vanishing total disclination charge there can be an arbitrary number of them in any spherical lattice configuration without violating the topological constraint on the total disclination charge discussed above. The grain-boundary scars found are unlike any grain boundaries found in flat space as they terminate freely inside the crystal at both ends. Due to the importance of dislocations and grain boundaries in bulk materials in determining material properties similar roles can be expected on surfaces. In the motion of dislocations is observed experimentally. Dislocation motion can be separated into glide and climb, where glide is motion parallel to the dislocation's Burger vector and requires only local rearrangement of the lattice, and climb is motion perpendicular to the Burger's vector and requires the presence of vacancies and interstitials. In agreement with the observations in and the computational results based on elasticity in we also observe only glide motion which leads to significant shape changes of the scars. shows a local rearrangement of a dislocation.
In the question is asked which effect a raising of temperature has on the spherical crystal. For bulk polycrystalline material there is indirect experimental evidence for the occurrence of grain boundary premelting, which could directly be visualized for colloidal crystals in. As a liquid film at grain boundaries will alter macroscopic properties and especially will lead to a drastically reduced resistance to shear stresses which can lead to material failure it is not only of theoretical interest if grain-boundary premelting is also present in spherical crystals. As grain-boundary scars belong to the equilibrium state it would be even more severe as it would be a general property of crystalline materials on curved surfaces. PFC models have already been used to study grain-boundary premelting in flat space. For high-angle grain boundaries an uniformly wetting is observed below the melting temperature in these studies.
We use the premelting of the dislocations to improve the local minima our evolution settles in. Within an iterative procedure we evolve the system according to Eqs. - until we reach a minimal state, increase the temperature and run the system until liquid layers have replaced all dislocations, and start the evolution again with the original temperature. The algorithm is terminated if the total energy does not further decrease. Typically this is achieved after 2-3 iteration cycles. In we plot the excess dislocations in a scar as a function of the system size \(\sqrt{N}=r/a\).
As already pointed out the method is not restricted to spherical geometries. Indeed the algorithm works for arbitrary surfaces with the only requirement that an appropriate surface mesh is needed on which the computation can be done. As an example we use toroidal crystals, which can be found e.g. in self-assembled monolayers of micelles and vesicles or carbon nanotori.
Excess dislocations as a function of system size. The obtained slope is \(0.388\pm 0.020\), which is in excelent agreement with the experimental measured data in, which give \(0.404\pm 0.062\), and the theoretical value of \(0.41\) from
(Color online) Time sequence showing premelting at grain boundary scars. The simulations indicate an initiation of the melting at the ends of the scars.
(Color online) Annihilation of dislocation by local rearrangment of 5-7 defect. The gray lines indicate the formation of a new neighboure at an intermediate state.
shows a typical configuration for aspect ratio \(R_{1}/R_{2}=2.78\).
In computations on more complicated surfaces, with convex and concave regions, we observe isolated fivefold and sevenfold disclinations which arrange according to the local curvature of the surface. Thereby fivefold disclinations are preferably found in convex regions, whereas sevenfold disclinations are present in concave regions, which is in accordance with the theory discussed in. The ability of the approach to work on arbitrary surfaces will also allow us to consider ordering on evolving surfaces. As discussed above for low surface tensions a buckling transition of the disclinations can turn the sphere into a polygon inorder to reduce the energy. The question arises if such a transition can intervene with grain boundary scar formation. In it is speculated that grain boundary scars could be formed on capsids at an intermediate stage of their evolution and that the release of the bending energy present in these scars into stretching energy could allow for shape changes. To model such shape changes requires us to evolve the surface. Appropriate continuum models which account for bending and surface tension are discussed in the mathematical review papers. Different concepts have been developed to solve differential equations on evolving surfaces, in the context of parametric finite elements (as considered here) and within a phase-field context. The coupling of the surface evolution with the evolution of the number density on it and with it the question if grain boundary scars can initiate a buckling transition, however remains open and requires further developments.
| 10.48550/arXiv.0910.3522 | Particles on curved surfaces - a dynamic approach by a phase field crystal model | Rainer Backofen, Axel Voigt, Thomas Witkowski | 3,492 |
10.48550_arXiv.1401.0891 | ## Abstract
Phase-coexistence in the manganese-oxide compounds or manganites with colossal magneto-resistance (CMR) has been generally considered to be an inhomogeneous ground state. An alternative explanation of phase-coexistence as the manifestation of a disorder-broadened first order magnetic phase transition being interrupted by the glasslike arrest of kinetics is now gradually gaining ground. This kinetic arrest of a first order phase transitions, between two states with long-range magnetic order, has actually been observed in various other classes of magnetic materials in addition to the CMR manganites. The underlying common features of this kinetic arrest of a first order phase transition are discussed in terms of the phenomenology of glasses. The possible manifestations of such glass-like arrested states across disorder-influenced first order phase transitions in dielectric solids and in multiferroic materials, are also discussed.
## 1 Introduction
First order phase transitions are defined by a discontinuous change in entropy at the transition temperature T\({}_{\rm C}\) (resulting in a latent heat) and a discontinuous change in either volume or magnetization (depending on whether T\({}_{\rm C}\) changes with pressure or with magnetic field). Since liquids and solids have different densities, the crystallization of a liquid entails motion at a molecular level that requires non-zero time. The concept of rapidly freezing a liquid out of equilibrium has been exploited for producing splat-cooled metallic glasses. As was noted by Greer, the glass would have a density close to that of the liquid.
We have noted recently that the kinetics of a first order phase transition is dictated by the time required for the latent heat to be extracted and, in addition to the well-known quenched metallic glasses, the kinetics of any first order phase transition (including the one between two long-range-order phases) could thus be arrested. We have studied many first order magnetic transitions whose kinetics has been arrested, and shall point out similar behavior in some published reports on first order dielectric transitions.
While temperature is one thermodynamic variable common to all these classes of first order phase transitions (viz. structural, magnetic, dielectric), the second thermodynamic variable would be pressure, magnetic field, and electric field, respectively. There has been a resurgence of interest in first order phase transitions with magnetic field induced transitions providing the impetus, because of possible applications envisaged for magneto-caloric materials, for materials showing large magneto-resistance, for magnetic shape memory alloys, etc. Unlike pressure, magnetic field does not require a medium for its application and controlling H thus does not complicate controlling temperature. This has enabled some very interesting observations on glasslike metastable states. While electric field has not yet been used extensively as a thermodynamic variable, the scenario is bound to change as more multiferroic materials are discovered.
The role of the second thermodynamic variable in glass formation has been recognized even in structural glasses because the freezing point shifts with pressure, but was exploited only a few years back when elemental germanium was vitrified under pressure, and the glassy state was retained on release of pressure. The use of magnetic field as a second thermodynamic parameter for arresting a first order magnetic transition was conceived and exploited by us over ten years back. If the first-order transition occurs over a range of temperatures (as happens in the presence of disorder, or when the transition is accompanied by large strain) and the glass formation temperature (T\({}_{\rm K}\)) falls within this range, then there is an interesting possibility that the transition is arrested while it is still partial and incomplete. We observed first order magnetic transitions being arrested at low temperatures while still incomplete, resulting in the two magnetic phases coexisting down to the lowest temperature.
First reported by us in the case of a ferromagnetic to anti-ferromagnetic transition in CeFe\({}_{2}\)-pseudobinary alloys, such persistent phase coexistence during cooling has now been reported across first order magnetic transitions in various classes of magnetic materials including CMR manganites. We shall discuss our studies following various thermomagnetic history paths where the cooling field and the warming field are different, that allow study of reentrant transitions, analogous to devitrification of glass followed by melting of the crystalline state.
## 2 Kinetic arrest of disorder broadened first order magneto-structural transition: Magnetic-glass
A first order magnetic/magneto-structural phase transition across a phase transition temperature T\({}_{\rm N}\)(H) line in the two parameter magnetic field (H) - temperature (T) phase space (see Fig.1) of a solid magnetic material is identified by a discontinuous change in entropy (i.e. measureable latent heat) or a discontinuous change in magnetization (M) as this T\({}_{\rm N}\)(H) line is crossed by varying either of the control variables T or H. The high- and low-temperature phases coexist at the transition temperature T\({}_{\rm N}\), and in the absence of any energy fluctuations the high temperature phase continues to exist as a supercooled metastable state until the temperature T*, which represents the limit of supercooling. A limit of metastability is similarly possible while heating at T**(H) \(>\) T\({}_{\rm N}\)(H), and this is not shown in the schematic (H,T) phase diagram (Fig.1) for the sake of clarity. Such limits of metastability H* and H** can also be defined across an isothermal magnetic field induced first order phase transition. In many real magnetic materials instead of latent heat (which is often difficult to determine experimentally) such phase-coexistence and metastability can be used as characteristic observables to identify a first order phase transition.
The actual composition of multi-component magnetic alloys, intermetallic compounds and metal-oxide compounds varies around some average composition due to disorder that is frozen in as the solid crystallizes from the melt. In a pioneering work Imry and Wortis showed that such static, quenched-in, purely statistical compositional disorder could broaden a first order phase transition. The associated latent heat could be significantly diminished or completely wiped out by the disorder. This disorder induced broadening of first order phase transition can take place for sufficiently large range of disorder correlations, greater than the order-parameter correlation length and the length defined by the ratio of the inter-phase surface tension and the latent heat. This phenomenon of disorder influenced first order phase transition drew significant attention of the theorists, and its possible role in the interesting properties of CMR manganites was highlighted by detailed computational studies. On the experimental front, finding of an intrinsic disorder-induced landscape of vortex solid melting transition temperatures/fields in a high temperature superconductor BiSrCaCuO had shown the applicability of such a concept in real materials.
It is well known that some liquids, called 'glass-formers', experience a viscous retardation of crystallization during first order phase transition in their supercooled state. Such supercooled liquid ceases to be ergodic within the experimental timescale, and enters a non-equilibrium state called a 'glassy state'. Glass can be considered within a dynamical framework, as a liquid where the atomic or molecular motions are arrested, which leads to the general definition of conventional glasses that 'glass is a noncrystalline solid material which yields a broad nearly featureless diffraction pattern'. There is another widely acceptable picture of glass as a liquid where the atomic or molecular motions (or kinetics) are arrested. Within this latter dynamical framework, 'glass is time held still'. Various dynamical features typically associated with glass formation are not necessarily restricted to materials with positional disorder.
It is reasonable to assume that during glass formation the viscous retardation of crystallization on an experimental timescale would occur below some temperature T\({}_{\rm K}\). In the case of metallic glasses T\({}_{\rm K}\)\(<\)T\({}^{*}\) the limit of supercooling of the liquid, hence the state in which the glass can be formed is unstable. In such cases the rapid quenching of the liquid is essential so that the kinetics is arrested before it can effect a structural change. On the other hand in the case of T\({}_{\rm K}\)\(>\)T* the glass is formed in a state that is metastable (in terms of free energy), and the glass formation can be caused by slow cooling, as in the well known glass-former O-terphenyl. In this case the system is already trapped in a deep valley in the potential energy landscape corresponding to a glass structure, although it has reached the spinodal point in the free energy configuration. The potential energy minimum corresponding to a crystalline structure is well separated from this deep valley and a glassy state does not get transformed into a crystalline state with finite energy fluctuations within a finite experimental timescale.
As shown in the schematic in Fig. 1, T\({}_{\rm N}\) drops with increasing H, just like the transition temperature (T\({}_{\rm M}\) ) drops with increasing pressure whenever a liquid expands on freezing. Within the realm of experimentally achievable steady pressures T\({}_{\rm M}\) drops by 25% or less, whereas in many ferromagnetic (FM) to antiferromagnetic (AFM) transitions, T\({}_{\rm N}\) drops almost by 100% with experimentally achievable steady magnetic fields. As the drop in the transition temperature slows the kinetics of the phase transition, it is more likely to produce a glass-like arrested state. We thus see the possibility of going from the scenario of T\({}_{\rm N}\)\(>>\)T* to T\({}_{\rm N}\)\(<\)T* with experimentally achievable magnetic fields, or of going from a metglass like situation (which requires rapid quenching from liquid state to avoid crystallization) to an O-terphenyl ( a standard glass-former) like situation in the same material. This glass-like arrested state formed by arresting a first order magnetic phase transition is referred to as a magnetic-glass, and shows various attributes (including time dependent behavior) similar to structural glasses. Indeed it has recently been observed that in certain regions of the H-T phase diagrams of the different classes of magnetic materials (CeFe\({}_{2}\) based alloys, Gd\({}_{5}\)Ge\({}_{4}\), CMR mangnites, Ni-Mn-In and Ni-Mn-Sn Heusler alloys and Fe-Rh ) there is a viscous retardation of nucleation and growth of the low temperature phase across a magneto-structural first order phase transition leading to such a magnetic-glass state. In all these materials quenched disorder broadens the first order phase transition as envisaged within the Imry-Wortis framework, and there is a landscape or band of transition temperature T\({}_{\rm N}\) (H) or field H\({}_{\rm M}\) (T) and limit of supercooling (or superheating) temperature T*(H) ((T**(H)) or field H*(T) ((H**(T)) instead of a single transition temperature and limit of supercooling (superheating). The schematic in Fig.2 shows disorder broadening of T\({}_{\rm N}\), as well as of T* and T**, because they are now defined over regions of the length-scale of the correlation length and have values dictated by local composition.
The schematic in Fig.3 shows how the transition proceeds over a range of temperatures for a fixed value of H. For an ultrapure material the material can be taken to the limit of superheating (supercooling) T** (T*) without any nucleation of the high (low) temperature phase and the phase transition onset temperature T\({}_{1}\) (T\({}_{3}\)) coincides with the T** (T*) (see Fig. 3(a)). In the presence of small disorder, nucleation and growth of the high (low) temperature phase starts before reaching T** (T*), and we denote the new positions of T\({}_{1}\) (T\({}_{3}\)) by T\({}_{1a}\) (T\({}_{3a}\)). In many materials of practical interest, the quenched-disorder will thus give rise to T** (T*) bands, corresponding to the range between start and completion of phase transition while cooling T\({}_{3a}\) and T* (and start and completion of phase transition while heating T\({}_{1a}\) and T** respectively). Fig. 3(b) shows the case when the band becomes very broad, and T\({}_{3}\) is shifted to a position which is higher than T\({}_{1}\). This schematic in Fig. 3(b) corresponds to the extensively studied Magnetic Shape Memory alloys (MSMA) where glasslike kinetic arrest of the first order martensitic transition has been reported. We must point out that this schematic visually emphasizes that the underlying thermodynamic transition temperature cannot be estimated as Tm=[Ms +Af]/2; it is best approximated by [Ms+Mf+As+Af]/4. Further, we now have the very interesting possibility that T\({}_{K}\) intersects the T* band over some range of H. In this range of H the broadened transition is initiated on cooling and proceeds, but is interrupted (or arrested) midway as the kinetics gets arrested. As stated above, T* is decreasing with rising H. Assuming that T\({}_{K}\), where the arrest (or interruption) occurs remains same as H rises, it follows that a smaller fraction of the transition has taken place when the transition is interrupted. One manifestation of this is that thermomagnetic irreversibility associated with zero field cooled(ZFC)/field cooled(FC) magnetization measurements would increase with rising H. This is in striking contrast with thermomagnetic irreversibilities associated with pinning of domain walls, of easy axis and spin-flip process, or even with spin-glasses. A clear cut manifestation of this contrasting behavior has been reported by Sharma et al in the case of NiMnIn shape memory alloy. Figure 4, presents the temperature dependence of magnetization of this NiMnIn alloy obtained under zero field cooled (ZFC), field cooled cooling (FCC) and field cooled warming (FCW) protocol. The austenite to martensite phase transition is accompanied by a sharp drop in magnetization. The thermal hysteresis in the FCC/FCW magnetization across the austenite-martensite transition is associated with the first order nature of this phase transition. In addition there is a distinct thermomagnetic irreversibility between the ZFC/FC magnetization in the matertensitic phase at the low applied filed H = 100 Oe, which gets reduced with the increase in H and becomes completely reversible for H = 10 kOe (see Fig.4). This hysteresis is attributed with the hindrance of domain wall motion in the ferromagnetic martensitic phase. However, in applied H beyond 10 kOe, the ZFC/FC thermomagnetc hysteresis reappears and then rises with further increase in H. This hysteresis is due to the glass-like kinetic arrest of the first order austenite-martensite phase transition in the presence of high H. The non-equilibrium nature of the magnetic-glass state is evidenced by the fragility (devitrification) of this state by multiple temperature cycling (see the inset of the upper figure in 4(b)). It has been further observed that in a Fe-doped NiMnIn Heusler alloy the kinetic arrest of the austenite-martensite phase transition takes place even in the absence of an applied magnetic field.
T\({}_{K}\) is also influenced by disorder; it can also have different values (even for a specific cooling rate) for different regions having the length-scale of the correlation length. Schematic-4 presented in shows this interesting scenario; the glasslike arrested state can now undergo only partial devitrification when H is varied at some temperature like (see point F in Fig.5). This gives rise to an anomalous behavior in the form of the virgin magnetization or resistivity curve lying outside the envelope curves as H is varied isothermally. This anomalous behaviour was first reported in Al-doped CeFe\({}_{2}\) in both resistivity measurements and in magnetization measurements (see and 7). With increasing H the AFM state undergoes a broad first order phase transition to the FM state as it crosses the superheating band, with the AFM state not being fully recovered even when H is reduced to below the supercooling T** band because it has only partially crossed the T\({}_{K}\) band. The remaining AFM state can be recovered by heating, which also causes devitrification. We would like to assert that it was shown in these pioneering works that de-arrest was caused both by raising T and by lowering H. As we shall discuss below, in materials like Gd\({}_{5}\)Ge\({}_{4}\) having an AFM-to-FM transition with lowering temperature, isothermal increase of H at the lowest temperature ( say T = 2K) causes devitrification of the kinetically arrested first order magnetic transition. In the light of these observations, we believe any rechristening of "kinetic arrest" as "thermal transformation arrest" would belie the underlying physics of the phenomenon.
In the case of materials like Gd\({}_{5}\)Ge\({}_{4}\) that have an AFM-to-FM transition with lowering T, the transition temperature T\({}_{\rm N}\) is lowered by decreasing H. Cooling in higher fields inhibits the process of "kinetic arrest", and less of the transition has proceeded before it is arrested (or interrupted). In the isothermal variation of H at low T, the glasslike arrested state is de-arrested with increasing H. The virgin ZFC state is the arrested AFM state, while the remnant zero-field state after isothermal cycling of H is the equilibrium FM state. This kinetic arrest of the AFM-FM transition in Gd\({}_{5}\)Ge\({}_{4}\) has actually been visualized with micro-Hall probe scanning study. Further the devitrification process of this arrested state has also been studied in details. In we show the T*, T**, and T\({}_{\rm K}\) bands for both cases of AFM ground state and of the FM ground state (with the simplifying assumption that the slopes of these bands have no H-dependence). Even though the ground states are very different, the manifestation of kinetic arrest in isothermal M-H curves is very similar. This behavior is exemplified in Fig.9 with very similar anomalous isothermal virgin M-H curves obtained in Pr\({}_{0.5}\)Ca\({}_{0.5}\)Mn\({}_{0.975}\)Al\({}_{0.025}\)O\({}_{3}\) and La\({}_{0.5}\)Ca\({}_{0.5}\)MnO\({}_{3}\) with ferromagnetic and antiferromagnetic ground state, respectively. A magnetic-glass state arising out of a kinetic arrest of a first order AFM (charge ordered) to FM transition has been reported in La5/8\(-\)yPryCa3/8MnO3 (LPCMO). Some anomalous metastable features associated with the AFM (charge ordered)-FM transition of this compound had actually been noticed earlier. Commonality of the kinetic arrest observed in LPCMO and in other classes of magnetic materials has been discussed in details in Ref.5.
Before proceeding further, we wish to assert that the virgin curve will lie outside the envelope magnetization (M)- field (H) (or Resistance (R) - field (H)) curve if the measuring temperature falls below T\({}_{\rm K}\)(H=0), but also if it lies between T*(H=0) and T**(H=0)! In the latter case it matters whether the measuring temperature is reached by cooling or heating. It is important to note these in view of some recent confusion (see in the reference) and associated discussion). We now address the similarity mentioned in the preceding paragraph. For a cooling field lying between H\({}_{1}\) and H\({}_{2}\) (schematic-5), the equilibrium state and an arrested state coexist at the lowest temperature. Cooling in a higher field increases the FM fraction, irrespective of the equilibrium state; cooling in different fields and varying the field isothermally at the lowest T enables one to tune the fraction of the coexisting phases. If the state so obtained is warmed in a field different from the cooling field, then the equilibrium phase fraction can increase. For an FM (AFM) ground state, the equilibrium fraction increases if the warming field is larger (smaller) than the cooling field, and one sees a reentrant behavior during warming. This 'cooling and heating in unequal field' (CHUF) protocol not only allows the determination of the equilibrium state, but provides conclusive evidence of the occurrence of glass-like kinetic arrest of a first order phase transition. If various values of the cooling field are used in conjunction with a single warming (or measuring) field, then one observes two temperatures where nonergodicity sets in; one for H\({}_{\rm Cool}>\) H\({}_{\rm Warm}\) and another for H\({}_{\rm cool}<\) H\({}_{\rm warm}\). This is because there is a glasslike arrest, and also an underlying first order phase transition. The CHUF protocol has been very useful in identifying magnetic-glass behavior in many magnetic systems, and this is exemplified in Fig.7 with the results obtained in Co-doped NiMnSn magnetic shape memory alloys.
It is also important to highlight here a few experimentally determined characteristics of magnetic glass and the relevant set of experiments, which will enable one to distinguish a magnetic glass unequivocally from the well-known phenomena of spin glass (SG) and reentrant spin glass (RSG). First of all a magnetic-glass arises out of the kinetic arrest of a first-order FM to AFM phase transition, which is accompanied by a distinct thermal hysteresis between the FCC and FCW magnetizations, whereas no such thermal hysteresis is expected in the case of a SG/RSG transition since this is considered to be a second-order phase transition or a gradual phase transformation. Second, the thermomagnetic irreversibility associated with the magnetic-glass rises with the increase in the applied H, and this is just the opposite in SG/RSG. Third, the FC state in the magnetic-glass systems is the non-equilibrium state showing glass-like relaxation, whereas in the SG/RSG the ZFC state is the non-equilibrium state, which shows thermal relaxation. Lastly, the newly introduced experimental protocol CHUF reveals distinct features in the \(T\) dependence of magnetization in magnetic-glass, which depends on the sign of inequality between the fields applied during cooling and heating; no such features are expected for a RSG system.
Both jamming and structural glass formation are manifestations of the slowing down of translational kinetics, and this leads to certain similarities between these two distinct phenomena. In the context of magnetic-glass, recently Chaddah and Banerjee argued that the magnetic-glass formation arose due to kinetic arrest of the underlying first order phase transition, and not due to jamming. The phenomenon of jamming does not need any underlying first order phase transition or latent heat. The argument of Chaddah and Banerjee is based on the idea that glasses are formed when the heat removal process preferentially removes specific heat, without removing latent heat. Within this picture magnetic glasses are likely to form in such systems where magnetic latent heat is weakly coupled to the thermal conduction process.
## 3 Disorder influenced first order phase transition in ferroelectric materials
The ferroelectric systems and ferromagnetic systems are quite similar in many ways. Below a critical temperature known as Curie temperature, electrical polarization and magnetic moment go to an ordered state in ferroelectric and ferromagnetic systems, respectively. In both the systems the order parameter can be coupled to lattice, and this in turn can lead to the piezoelectric/electro-strictive and magneto-elastic effects.
A first order phase transition can be induced both by temperature and electric field in the ferroelectric (FE) materials, leading to a sharp change in polarization accompanied with distinct thermal or electric field hysteresis. In many materials including Ba-doped PbZrO\({}_{3}\), La-doped PbZrO\({}_{3}\), La and Hf doped PbZrO\({}_{3}\), Pb\({}_{0.99}\)[(Zr\({}_{0.8}\)Sn\({}_{0.2}\))\({}_{0.96}\)Ti\({}_{0.04}\)]\({}_{0.98}\)Nb\({}_{0.02}\)O\({}_{3}\) and PbHfO\({}_{3}\) there is an electric field induced first order phase transition from antiferroelectric (AFE) to ferroelectric (FE) state giving rise to a double hysteresis loop in the polarization (P) versus electric field (E) curve in the isothermal field excursion between \(\pm\) E\({}_{\text{Max}}\).
The role of disorder in the ferroelectric materials has also been investigated in some details. In one of the classic ferroelectric material PbZrO\({}_{3}\) the width of the temperature range in which the ferroelectric (FE) phase is stable, depends on the amount of defects in the PbZrO\({}_{3}\). It has also been observed that both FE and antiferroelectric (AFE) phases coexisted in epitaxial PbZrO\({}_{3}\) films with high crystalline quality. Results of dielectric and x-ray diffraction measurements in Pb\({}_{0.90}\)Ba\({}_{0.10}\)ZrO\({}_{3}\) ceramics indicated that while cooling the AFE-FE phase transition in this material was not quite reversible. It was observed that the equilibrium AFE phase could be recovered from the metastable FE matrix on ageing at room temperature, and the kinetics of recovery of this AFE phase was quite slow.
There is a class of ferroelectric materials called relaxor-ferroelectrics, which are distinguished from ordinary ferroelectrics by the presence of a diffused phase transition and strong metastable behaviour. They also show electric-field annealing and the aging effects, which indicate the evolution of micropolar clusters under an external electric field. The glass-like behavior in many relaxor-ferroelectrics Li- and Nb-doped KTaO\({}_{3}\), PbMg\({}_{1/3}\)Nb\({}_{2/3}\)O\({}_{3}\) and Lab-substituted PbZr\({}_{1-x}\)Ti\({}_{x}\)O\({}_{3}\) families are now quite well known. While a detailed discussion of these materials is beyond the scope of the present paper, we will briefly discuss some phenomena in KTaO\({}_{3}\) which are of the interest in the present context. KTaO\({}_{3}\) is a quantum paraelectric where the ferroelectric transition is suppressed by zero-point fluctuations. A small substitution of Li for K in KTaO\({}_{3}\) creates a local electric dipole due to its off-centre position with respect to the cubic site and a sharp ferroelectric phase transition has been observed in K\({}_{0.937}\)Li\({}_{0.063}\)TaO\({}_{3}\). A distinct thermal hysteresis and metastable behaviour associated with sharp peaks in the dielectric permittivity while cooling and heating in electric fields 75 kV m\({}^{-1}\)\(<\) E \(<\) 300 kV m\({}^{-1}\) clearly indicates the first order nature of this transition. The nature of the low temperature phase of K\({}_{1-x}\)Li\({}_{x}\)TaO\({}_{3}\) (x \(<<\)1) is quite interesting, where signatures of both glass like and long-range ordered ferroelectric behaviour have been observed. Similarities between the characteristic behavior of relaxor-ferroelectrics and two-phase coexistence, magnetic-field annealing and the aging effect, and diffuse x-ray scattering in various CMR-manganite systems have already been documented. In fact Kimura et al highlighted that the "freezing of the two-phase coexistence state under the first-order phase transition accompanied by lattice distortion are common characteristics of CMR manganites and relaxor ferroelectrics". Such magnetic field annealing and metastable behaviors observed in the CMR-manganites can actually be explained quite naturally within the premise of the kinetic arrest of a first order magneto-structural phase transition. This is brought out in some detail by Kumar et al where detailed structure within the bands for supercooling and for kinetic arrest was also discussed. In this context various E-T history effects, metastability and glass-like behaviour in relaxor-ferroelectrics may be worth revisiting.
Signatures of the kinetic arrest of a first order phase transition and the associated glass-like metastable response have been observed in the multiferroic material LuFe\({}_{2}\)O\({}_{4}\). This system undergoes a three-dimensional charge order transition below 320 K, which is followed by a ferrimagnetic transition below 240 K, and a first order magnetic phase transition coupled with monoclinic distortion takes at 175 K. There is indication that this last magnetic transition is kinetically arrested giving rise to glass-like non-equilibrium behaviour. Kinetic arrest (dearrest) of the first order multiferroic to weak-ferromagnetic transition has been reported in the multiferroic system Eu\({}_{0.75}\)Y\({}_{0.25}\)MnO\({}_{3}\), which results in frozen (melted) magneto-electric glass states with coexisting phases.
## 4 Conclusions
Whenever a first order phase transition occurs between two states with close-lying energies, it gives rise to a delicate balance of competing order parameters that are influenced by slight disorder, making the transition susceptible to interruption by glass-like kinetic arrest with consequent phase coexistence. Complex physical phenomena are also observed when there is a delicate balance of competing order parameters, as in multiferroic materials. Recent reports of kinetic arrest of the first-order multiferroic transition, resulting in magneto-electric glass states with two coexisting phases, point to the ubiquitous nature of the phenomenology of glasslike kinetic arrest. Questions related to the formation of structural glasses have persisted for over a century with the experimentalists having to study newer materials for testing newer ideas. The phenomenology of the kinetic arrest of first order phase transition discussed by us allows a second thermodynamic control variable, and its validity across magnetic and dielectric transitions may enable the resolution of outstanding issues in understanding glasses.
# Figures :Schematic showing the fraction of transformed phase as a function of temperature in a first-order phase transition. (a) Sharp transition (b) Disorder-broadened transition [Ref. 26].
Schematic representation of broadened bands of phase-transition (\(H_{M}\),\(T_{N}\)), supercooling (\(H\)*,\(T\)*), and superheating (\(H\)**,\(T\)**) lines [Ref. 25].
Figure 4(a). Magnetization (\(M\)) versus temperature (\(T\)) plots for Ni\({}_{50}\)Mn\({}_{34}\)In\({}_{16}\) alloy in external magnetic field \(H\)=0.1, 1, 10, and 50 kOe. Open square represents ZFC data and open (close) triangle represents FCC (FCW) data [Ref.7].
Figure 4(b). Magnetization (\(M\)) versus temperature (\(T\)) plots for Ni\({}_{50}\)Mn\({}_{34}\)In\({}_{16}\) alloy in external magnetic field \(H\)=70 and 80 kOe. The inset shows the effect of thermal cycling on \(M_{\rm FCC}\) (\(T\)), thus highlighting the non-equilibrium nature of the low T magnetic-glass state [Ref.7].
Schematic representation of the relative position of the band (\(H_{K}\),\(T_{K}\)) across which the kinetics of the phase transition is hindered with respect to (\(H\)*,\(T\)*) band [Ref.25].
\(M\) vs \(H\) plots of Ce(Fe\({}_{0.96}\)Al\({}_{0.04}\))\({}_{2}\) obtained after cooling in zero field at \(T\)=5 K. Note that the virgin \(M\)-\(H\) curve lies outside the envelope \(M\)-\(H\) curve. To confirm this anomalous nature of virgin curve, the same is also traced in the negative field direction after zero-field cooling the sample from the temperature above the magnetic phase transition [Ref.25].
Resistivity (\(\rho\)) vs \(H\) plots of Ce(Fe\({}_{0.96}\)Al\({}_{0.04}\))\({}_{2}\) at \(T\)= 20 K, 5 K, and 3 K. Filled squares (dashed lines) represent virgin curve drawn in the positive (negative) field direction after zero-field cooling the sample [Ref.25].
Comparison of schematic phase diagrams showing transformation from high-temperature AFM phase to low-temperature FM phase, and from high-temperature FM phase to low- temperature AFM phase [Ref.33].
_M–H_ curves of CMR manganite compounds PCMAO and LCMO at 5 K after cooling the samples from 320 K in zero field. Note that the the virgin M-H curve lies outside the envelope M-H curve in both the cases [Ref.10].
M vs T curves obtained using the CHUF (cooling and heating in unequal field) protocol in Ni\({}_{45}\)Co\({}_{5}\)Mn\({}_{38}\)Sn\({}_{12}\) alloy. In (a), (b) the sample is cooled in a constant field of 6 or 3 T and measurements are carried out in various different fields. In (c), (d) the sample is cooled under different magnetic fields whereas measurements during warming are carried out in 4 or 1 T respectively [Ref.39]. | 10.48550/arXiv.1401.0891 | Phase-coexistence and glass-like behavior in magnetic and dielectric solids with long range order | S. B. Roy, P. Chaddah | 3,693 |
10.48550_arXiv.1305.2997 | ###### Abstract
Combining the Lagrangian-Laplace mechanics and the known pressure dependence of the length-stiffness relaxation dynamics, we have determined the critical, yet often-overlooked, short-range interactions in the O:H-O hydrogen bond of compressed ice. This approach has enabled determination of the force constant, cohesive energy, potential energy of the O:H and the H-O segment at each quasi-equilibrium state as well as their pressure dependence. Evidencing the essentiality of the inter-electron-pair Coulomb repulsion and the segmental strength disparity in determining the asymmetric O:H-O relaxation dynamics and the anomalous properties of ice, results confirmed that compression shortens and stiffens the O:H bond and meanwhile lengthens and softens the H-O bond.
*SI is accompanied.
PACS numbers: 61.20.Ja, 61.30.Hn, 68.08.BcWater and ice has attracted much attention because of its anomalous performance relating to issues from galaxy to geology, climate, biology, and to our daily lives. As the building unit, the hydrogen bond (O:H-O) relaxes in different manners under the change of environment conditions, which determines the anomalous properties of water and ice. Contributions have been made experimentally, computationally, and theoretically to the understanding of water and ice based on the polarizable or non-polarizable models, including the TIP\(n\)P (\(n\) varies from 1 to 5) series. Using _ab initio_ density functional theory and molecular dynamics calculations, one is able to reproduce some of the anomalies demonstrated by compressed ice with limited knowledge about the nature of the inter- and intra-molecular interactions.
The objective of this work is to explore analytically the energy relaxation dynamics of the segmented O:H-O bond of ice under compression based on the Lagrangian-Laplace mechanics. With the known length-stiffness relaxation dynamics of the O:H-O bond under compression as input, we have been able to determine the force constants, the potential well depths, and the cohesive energies of each part of the O:H-O bond as well as their pressure dependence.
A linear hydrogen bond is assumed for simplicity because the O:H-O bond angle in ice is valued at \(170\pm 4^{\circ}\). By averaging the surrounding background interactions of H\({}_{2}\)O molecules and protons and the nuclear quantum effect on fluctuations, we focus on the short-range interactions in this O:H-O bond with H being the coordination origin. As illustrated in Figure 1, the van der Waals force is limited to the O:H bond (denoted L), the exchange interaction is within the H-O polar-covalent bond (denoted H), and the Coulomb repulsion (denoted C) applies between the electron pairs attached to the adjacent oxygen ions, see Supplementary information (SI).
Because of the short-range nature of the interactions, only the solid lines in the shaded area in Figure1 are effective for the basic O:H-O unit. These interactions will switch off immediately outside the O:H-O region. These interactions determine the physical properties irrespective of the phase structures of the hydrogen-bonded networks but only O--O interaction bridged by H. The presence of inter-electron-pair Coulomb repulsion dislocates both O ions slightly away from their respective equilibrium position. \(\Delta_{x}\) (\(x=\) L for the O:H and \(x=\) H for the H-O bond) denotes the dislocations. \(d_{x0}\) is the interionic distance at equilibrium without the Coulomb repulsion being involved. \(d_{x}=d_{x0}+\Delta_{x}\) is the quasi-equilibrium bond length with the Coulomb repulsion being involved. The Coulomb repulsion raises the cohesive energies of the O:H and the H-O from \(E_{x0}\) to \(E_{x}\) by the same amount.
Schematic of the segmented O:H–O bond with springs representing the short-range interactions with H atom being the coordination origin: intramolecular exchange interaction limited to the H–O bond (H), intermolecular van der Waals (vdW) force limited to the O:H bond (L), and the inter-electron-pair Coulomb repulsion (C-repulsion) force between adjacent O—O (C). The red and grey spheres denote the oxygen and the hydrogen atoms, respectively.
The Coulomb repulsion pushes both O atoms away from their equilibrium positions.
The O:H-O bond is taken as two oscillators coupled by Coulomb interaction. The reduced mass of the H\({}_{2}\)O:H\({}_{2}\)O oscillator is \(m_{\rm L}\)=18\(\times\)18/(18+18)\(m_{0}\) = 9\(m_{0}\) and that of the H-O oscillator is \(m_{\rm H}\) =1\(\times\)16/(1+16)\(m_{0}\) = 16/17 \(m_{0}\) with \(m_{0}\) being the unit mass of 1.66\(\times\)10\({}^{-27}\) kg. The O:H-O bond motion follows the Lagrangian motion equation:
\[\frac{\rm d}{\rm d\it t}\bigg{(}\frac{\partial L}{\partial(\rm d\it q_{i}/\rm d \it t)}\bigg{)}-\frac{\partial L}{\partial\it q_{i}}=Q_{i}\]
The Lagrangian \(L=T-V\) consists of the total kinetic energy \(T\) and the total potential energy \(V\). \(Q_{i}\) denotes the generalized non-conservative forces. Here, it is the pressure \(f_{\rm P}\). The time-dependent \(q_{i}(t)\) represents the generalized variables, denoting the deviating displacements from the equilibrium position of the springs L and H here, i.e. \(u_{\rm L}\) and \(u_{\rm H}\).
\[T=\frac{1}{2}\Bigg{[}m_{\rm L}\bigg{(}\frac{\rm d\it u_{\rm L}}{\rm d\it t} \bigg{)}^{2}+m_{\rm H}\bigg{(}\frac{\rm d\it u_{\rm H}}{\rm d\it t}\bigg{)}^{ 2}\Bigg{]}\]
The potential energy \(V\) is composed of three terms: the van der Waals interaction \(V_{\rm L}\big{(}r_{\rm L}\big{)}\!=\!V_{\rm L}\big{(}d_{\rm L0}-u_{\rm L}\big{)}\), the exchange interaction \(V_{\rm H}\big{(}r_{\rm H}\big{)}\!=\!V_{\rm H}\big{(}d_{\rm H0}+u_{\rm H}\big{)}\), and the Coulomb repulsion \(V_{\rm C}\big{(}r_{\rm C}\big{)}\!=\!V_{\rm C}\big{(}d_{\rm C0}-u_{\rm L}+u_{ \rm H}\big{)}\!=\!V_{\rm C}\big{(}d_{\rm C}-u_{\rm C}\big{)}\). Here, \(d_{\rm C0}\!=\!d_{\rm L0}\!+\!d_{\rm H0}\) is the distance between the adjacent oxygen ions at equilibrium. \(d_{\rm C}\!=\!d_{\rm L}\!+\!d_{\rm H}\) denotes that distance at quasi-equilibrium. \(u_{\rm C}\!\!=\!u_{\rm L}\!+\!\Delta_{\rm L}\)-\(u_{\rm H}\)\(+\Delta_{\rm H}\) shows the change of the distance between the two oxygen ions at quasi-equilibrium. The \(u_{\rm L}\) and \(u_{\rm H}\) are assumed to be of the opposite sign because of the O:H and H-O dislocate in the same direction.
\[\begin{split} V&=V_{\text{L}}\big{(}r_{\text{L}}\big{)}+V_{ \text{H}}\big{(}r_{\text{H}}\big{)}+V_{\text{C}}\big{(}r_{\text{C}}\big{)}\\ &=\sum_{n}\left\{\frac{\text{d}^{\,n}V_{\text{L}}}{n!\text{d}r_{ \text{L}}^{\,n}}\Big{|}_{d_{\text{L}n}}\big{(}-u_{\text{L}}\big{)}^{n}+\frac{ \text{d}^{\,n}V_{\text{H}}}{n!\text{d}r_{\text{H}}^{\,n}}\Big{|}_{d_{\text{L}n} }\big{(}u_{\text{H}}\big{)}^{n}+\frac{\text{d}^{\,n}V_{\text{C}}}{n!\text{d}r_ {\text{C}}^{\,n}}\Big{|}_{d_{\text{C}}}\big{(}-u_{\text{C}}\big{)}^{n}\right\}\\ &\approx\big{[}V_{\text{L}}\big{(}d_{\text{L}0}\big{)}+V_{\text{H }}\big{(}d_{\text{H}0}\big{)}+V_{\text{C}}\big{(}d_{\text{C}}\big{)}\big{]}-V_ {\text{C}}^{\prime}u_{\text{C}}+\frac{1}{2}\Big{[}k_{\text{L}}u_{\text{L}}^{\, 2}+k_{\text{H}}u_{\text{H}}^{\,2}+k_{\text{C}}u_{\text{C}}^{\,2}\Big{]}\end{split} \tag{3}\]
Noting that the Coulomb potential never has an equilibrium point where the repulsion force is 0, we can then expand this potential at quasi-equilibrium point. Therefore, the terms of \(n=1\) is the force equaling 0 for the L and H segments at equilibrium, while equaling \(\ -V_{\text{C}}^{\prime}u_{\text{C}}\) for the C spring at quasi-equilibrium. Here, \(\ V_{\text{C}}^{\,\prime}\) denotes the first order derivative at the quasi-equilibrium position, i.e. \(\big{(}\text{d}V_{\text{C}}\big{/}\text{d}r_{\text{C}}\big{)}_{d_{\text{C}}}\). The coefficients of the \(n=2\) terms, or the curvatures of the respective potentials, denote the force constants, i.e., \(k_{x}=\text{d}^{\,2}V_{x}\Big{/}\text{d}r_{x}^{\,2}\Big{|}_{d_{x0}}\) for harmonic oscillators. The \(n\geq 3\) terms are the high-order nonlinear contributions that are insignificant, as it will be shown.
Substituting Eqs and into leads to the coupled Lagrangian equation,
\[\begin{split}&\begin{cases}m_{\text{L}}\,\frac{\text{d}^{\,2}u_{ \text{L}}}{\text{d}t^{\,2}}+\big{(}k_{\text{L}}+k_{\text{C}}\big{)}u_{\text{L} }-k_{\text{C}}u_{\text{H}}+k_{\text{C}}\big{(}\Delta_{\text{L}}-\Delta_{\text{ H}}\big{)}-V_{\text{C}}^{\prime}-f_{\text{P}}=0\\ m_{\text{H}}\,\frac{\text{d}^{\,2}u_{\text{H}}}{\text{d}t^{\,2}}+\big{(}k_{ \text{H}}+k_{\text{C}}\big{)}u_{\text{H}}-k_{\text{C}}u_{\text{L}}-k_{\text{C }}\big{(}\Delta_{\text{L}}-\Delta_{\text{H}}\big{)}+V_{\text{C}}^{\prime}+f_{ \text{P}}=0\end{cases}\end{split}\end{split} \tag{4}\]
A Laplace transformation turns out solutions to Eq,
\[\begin{split}&\begin{cases}u_{\text{L}}=\frac{A_{\text{L}}}{ \gamma_{\text{L}}}\sin\gamma_{\text{L}}t+\frac{B_{\text{L}}}{\gamma_{\text{H} }}\sin\gamma_{\text{H}}t\\ u_{\text{H}}=\frac{A_{\text{H}}}{\gamma_{\text{L}}}\sin\gamma_{\text{L}}t+ \frac{B_{\text{H}}}{\gamma_{\text{H}}}\sin\gamma_{\text{H}}t\end{cases}.\end{split}\end{split} \tag{5}\]The coefficients denote the vibrational amplitudes. \(\gamma_{\rm L}\) and \(\gamma_{\rm H}\) are the vibration angular frequencies of the respective segment, which depend on the force constants and the reduced masses of the oscillators. This set of general solutions indicates that the O:H and the H-O segments share the same form of eigen values of stretching vibration.
\[k_{\rm H,\,L}=2\pi^{2}m_{\rm H,\,L}c^{\,2}\left(\omega_{\rm L}^{2}+\omega_{\rm H }^{2}\right)-k_{\rm C}\pm\sqrt{\left[2\pi^{2}m_{\rm H,\,L}c^{\,2}\left(\omega_ {\rm L}^{2}-\omega_{\rm H}^{2}\right)\right]^{2}-m_{\rm H,\,L}k_{\rm C}^{\,2} \left/m_{\rm L,\,H}\right.} \tag{6}\]
Omitting the Coulomb repulsion, the coupled oscillators will be degenerated into the independent H\({}_{2}\)O:H\({}_{2}\)O and H-O oscillators with respective vibration frequencies of \(\sqrt{k_{\rm L}\,/\,m_{\rm L}}\) and \(\sqrt{k_{\rm H}\,/\,m_{\rm H}}\). With the measured \(\omega_{\rm L}\) and \(\omega_{\rm H}\), and the known \(k_{\rm C}\), one can obtain the force constants \(k_{x}\), the potential well depths \(E_{x0}\), and the cohesive energy \(E_{x}\).
The force constant due to Coulomb repulsion is, \(k_{\rm C}=q_{z}q_{\cdot}/\left(2\pi\varepsilon_{\rm r}\varepsilon_{0}d_{\rm C} ^{\,3}\right)\) at equilibrium. Here, \(\varepsilon_{\rm r}\) is the relative dielectric constant of ice, equaling to 3.2. \(\varepsilon_{0}=8.85\times 10^{-12}\) F/m, is the vacuum dielectric constant. The \(q_{i}=2\)e for the electron lone pair, and \(q_{i}=0.2\)e or so, is the effective charge referring to our density functional theory optimizations. In this situation, the \(k_{\rm C}\) equals to 0.17 eV/A\({}^{2}\) at 0 GPa. The \(\omega_{\rm L}\) and \(\omega_{\rm H}\) dependence of the \(k_{\rm L}\) and the \(k_{\rm H}\), in Figure 2, shows that the \(k_{\rm L}\) increases from 1.44 to 5.70 eV/A\({}^{2}\) while the \(k_{\rm H}\) increases from 21.60 to 42.51 eV/A\({}^{2}\) with their respective frequency. The \(k_{\rm L}(\omega_{\rm H})\) and the \(k_{\rm H}\) (\(\omega_{\rm L}\)) remains, however, almost constant. Therefore, Eq.
\[k_{\rm H,\,L}=4\pi^{2}c^{2}m_{\rm H,\,L}\omega_{\rm H,\,L}^{2}-k_{\rm C}\]
With the measured \(\omega_{\rm L}=237.42\) cm-1 and \(\omega_{\rm H}=3326.14\) cm-1 for the ice-VIII phase under the atmospheric pressure, Eq derives \(k_{\rm L}=1.70\) eV/A\({}^{2}\) and \(k_{\rm H}=38.22\) eV/A\({}^{2}\). With the known \(d_{\rm L}=0.1768\) nm and \(d_{\rm H}=0.0975\) nm under Coulomb repulsion, we can obtain the free length \(d_{\rm L0}\) is 0.1628 nm, and the \(d_{\rm H0}\) is 0.0969 nm. Then, with the derived values of \(k_{\rm L}\) and \(k_{\rm H}\), as well as the \(E_{\rm H0}=3.97\) eV, we can determine the parameters in the van der Waals and the Morse potentials, as well as the force fields of the O:H-O bond at the ambient pressure,
\[\begin{cases}k_{\rm L}=72\,E_{\rm L0}\Big{/}d_{\rm L0}^{2}=1.70\,{\rm eV}\Big{/} \text{A}^{2}\\ k_{\rm H}=2\alpha^{2}E_{\rm H0}=38.22\,{\rm eV}\Big{/}\text{A}^{2}\end{cases}\]
or
\[\begin{cases}E_{\rm L0}=1.70\times 1.628^{2}\Big{/}72=0.062\,{\rm eV}\\ \alpha=\big{(}38.22/3.97/2\big{)}^{\!\!1/2}=2.19\,\text{A}^{\scalebox{0.75}{ -1}}\end{cases}\]
Using the measured Raman shifts \(\omega_{x}\) and the interionic distances \(d_{x}\) as input, we can readily calculate the evolution of the force constant and cohesive energy of the respective segment, from one quasi-equilibrium to another, under compression based on Eq.. Table 1 and display the results.
Results indicate that the compression shortens and stiffens the softer O:H bond, meanwhile,lengthens and softens the H-O bond slightly through the Coulomb repulsion, which results in contraction of the O--O distance towards O:H and H-O length symmetry. The \(k_{\rm C}\) (curvature of the Coulomb potential) in Figure 3(a), keeps almost constant under compression because of the low compressibility of O--O distance. The \(k_{\rm L}\) increases more rapidly than \(k_{\rm H}\) reduces because of the coupling of the compression, the repulsion, and the potential disparity of the two segments.
As the \(d_{\rm L}\) shortens by 4.3% from 0.1768 to 0.1692 nm and the \(d_{\rm H}\) lengthens by 2.8% from 0.0975 to 0.1003 nm with pressure increasing from from 0 to 20 GPa. Figure 3(b) indicates that the increase of pressure from 0 to 20 GPa stiffens the O:H bond from 0.046eV to 0.190 eV while soften the H-O bond from 3.97 eV to 3.04 eV. When the pressure goes up to 60 GPa, the O:H bond almost equals to the elongated H-O bond in length of about 0.110 nm, forming a symmetric O:H-O bond. At 60 GPa, the \(k_{\rm L}=10.03\) eV/A\({}^{2}\) and \(k_{\rm H}=11.16\) eV/A\({}^{2}\), the \(E_{\rm L}\) recovers slightly, see Table 1. Results indicate that the nature of the interaction within the segment remains though the length and force constant approaches to equality, which means that the sp\({}^{3}\)-hybridized oxygen could hardly be de-hybridized by compression.
Pressure dependence of (a) the force constant \(k_{x}(P)\) and (b) the cohesive energy \(E_{x}(P)\) of the respective segment of the hydrogen bond. Solid lines in (b) results from the potential functions, which matching well the scattered data of the harmonic approximation.
Table 1 Pressure dependence of the O:H-O segmental cohesive energy (\(E_{x}\)), force constant (\(k_{x}\)), and the stepped deviation (\(\Delta_{x}\)) from the equilibrium position. Subscript \(x\) denotes L and H. The measured \(d_{x}(P)\) and \(\omega_{x}(P)\) are used as input in calculations.
\begin{tabular}{c|c c c c c c} \hline \(P\) (GPa) & \(E_{\rm L}\) (eV) & \(E_{\rm H}\) (eV) & \(k_{\rm L}\) (eV/A\({}^{*}\)) & \(k_{\rm H}\) (eV/A\({}^{*}\)) & \(\Delta_{\rm L}\)(10\({}^{-2}\) nm) & \(\Delta_{\rm H}\) (10\({}^{4}\) nm) \\ \hline
0 & 0.046 & 3.97 & 1.70 & 38.22 & 1.41 & 6.25 \\
5 & 0.098 & 3.64 & 2.70 & 35.09 & 0.78 & 6.03 \\
10 & 0.141 & 3.39 & 3.66 & 32.60 & 0.51 & 5.70 \\
15 & 0.173 & 3.19 & 4.47 & 30.69 & 0.36 & 5.26 \\
20 & 0.190 & 3.04 & 5.04 & 29.32 & 0.27 & 4.72 \\
30 & 0.247 & 2.63 & 7.21 & 25.31 & 0.14 & 3.85 \\
40 & 0.250 & 2.13 & 8.61 & 20.49 & 0.08 & 3.16 \\
50 & 0.216 & 1.65 & 9.54 & 15.85 & 0.05 & 2.71 \\
60 & 0.160 & 1.16 & 10.03 & 11.16 & 0.04 & 3.35 \\ \hline \end{tabular}
The oxygen ion (solid spheres in the bottom of Figure 4) in the O:H bond moves towards while the other in the H-O bond away from the H origin. The intrinsic equilibrium position of the oxygen in H-O almost superposes on its quasi-equilibrium position, with the distance of only 6.25\(\times\)10\({}^{-4}\) nm. However, for O:H, the distance is 1.41\(\times\)10\({}^{-2}\) nm, evidencing a very soft vdW bond. The cohesive energies of both segments relax along the contours as a resultant of the Coulomb repulsion and the compression.
In summary, with the aid of Lagrangian-Laplace mechanics, we have been able to formulate, correlate, clarify, and quantify the short-range interactions in the flexile, polarizable hydrogen bond of compressed ice. This approach has enabled us to determine the cohesive energy, force constant, potential field of each segment and their pressure dependence based on the measurements.
\(E_{x}\)-\(d_{x}\) relaxation dynamics of the O:H–O bond of compressed ice (from left to right, \(P=0\), 5, 10, 15, 20, 30, 40, 50, 60 GPa). Small solid circles in blue represent the intrinsic equilibrium coordinates (length and energy) of the oxygen without the Coulomb repulsion, and small open circles denote the quasi-equilibrium coordinates caused by both the Coulomb repulsion and the pressure. The leftmost solid (0 GPa) and the broken curves show the potentials at quasi-equilibrium while the thick solid lines are the contours of the \(E_{x}\)-\(d_{x}\) that approach the respective vdW and the Morse potential at equilibrium. Note scale difference between the two segments.
Financial support from National Natural Science Foundation (No. 11172254) of China is acknowledged.
| 10.48550/arXiv.1305.2997 | Missing short-range interactions in the hydrogen bond of compressed ice | Chang Q Sun | 1,411 |
10.48550_arXiv.0909.5215 | ## Abstract
Accelerated molecular dynamics (MD) simulations are implemented to model the sliding process of AFM experiments at speeds close to those found in experiment. In this study the hyperdynamics method, originally devised to extend MD time scales for non-driven systems, is applied to the frictional sliding system. This technique is combined with a parallel algorithm that simultaneously simulates the system over a range of slider positions. The new methodologies are tested using 2-dimensional and 3-dimensional Lennard-Jones AFM models. Direct comparison with the results from conventional MD shows close agreement validating the methods.
## 1.
While friction has presented many intriguing challenges since the onset of civilization, only in recent years have researchers been able to consider the origin of friction by directly investigating atomic-scale interactions. This change has been triggered mainly by the invention of Atomic Force Microscope (AFM), which made it possible to measure the friction force acting on a single-asperity contact at the nanometer scale. Moreover, the development of new hardware and computer simulation methodologies have played an important role in interpreting the origin of frictional forces and energy dissipation during sliding at this scale.
In an AFM experiment, only a small number of atoms at the end of a tip are in contact with the atoms in the substrate. The common physical picture of the process of frictional sliding is as follows. The atoms at the tip are initially equilibrated at a local free energy minimum, but as the slider moves and thecantilever deforms the initial minimum configuration becomes meta-stable. Due to thermal fluctuations, the atoms rearrange into a new more stable minimum releasing the elastic energy stored in the cantilever. This process: equilibration at a local minimum, escape from the minimum and establishment of a new local minimum, is repeated as the slider advances.
Molecular dynamics (MD) simulations have been used to model this process, but it has not been possible to perform the simulations at sliding velocities close to typical experimental values (nm/s \(\sim\)\(\mu\)m/s) due to MD's short time scale limited to sub-microseconds and velocities in excess of meters per second. In recent years, several novel methods to extend the MD time scale have been devised, but most of these are not suitable for driven systems, whose boundary conditions change with time. One notable exception is the extension of the parallel replica method for driven systems: this method has been applied to straining nano-tubes and stick-slip friction. However, the parallel replica method has only allowed a decrease in sliding speed by a factor corresponding to the number of processors, which is insufficient to reach experimental sliding speeds.
In this paper we present a novel method to extend MD time scale for driven systems using hyperdynamics and an alternative parallel algorithm. Like the parallel replica method for driven systems, the fundamental assumption of our method is that the system boundary conditions change slowly so that the system remains at a near equilibrium state. We apply the method here to a sliding system with a slider moving at a constant velocity. Instead of simulating multiple independent replicas simultaneously, we parallelize slider positions and perform the simultaneous hyperdynamics simulations with the models at different slider positions.
## 2 Method
In this section we will briefly review transition state theory (TST) and the hyperdynamics method. The hyperdynamics method is based on the TST assumptions. Our methodology is described in Sec. 2.3.
### Infrequent events and transition state theory
Most of the kinetic phenomena macroscopically observed in solid materials such as diffusion and creep are related to thermally activated changes in the configuration of the atoms comprising the material. Atoms make transitions from one meta-stable potential energy basin to another when they gain enough energy to overcome the energy barrier due to thermal fluctuations. In these dynamical systems, the information about the waiting times at each state and the transition mechanisms leading to other states as well as their relative probabilities is essential to understand the underlying physics of the phenomena.
In many cases, before hopping to other states, the system stays in the neighborhood of a potential energy minimum for a very long time compared to the typical atomic vibration period and a transition itself occurs in a relatively short time. This long waiting time hinders the utility of conventional MD methodologies.
\[p(t)=R\,\exp(-R\,t)\,\,\,, \tag{1}\]
Transition state theory can provide an analytical expression for the transition rates in an infrequently hopping system so long as we can construct a proper dividing surface that the system crosses in transitions.
\[R_{A}^{TST}=\frac{1}{2}\frac{\int d\,\,\vec{r}\int d\,\,\vec{v}\mid\nu_{n}\mid \delta_{S}(\,\vec{r}\,)\,\exp[-\beta(\,V+K)]}{\int_{A}d\,\,\vec{r}\int d\,\,\vec {v}\,\,\exp[-\,\beta(\,V+K)]}\,\,\,\,, \tag{2}\]
In many cases the dividing surface is chosen as a hyper-plane passing through the saddle point between two given minima and normal to the eigenvector corresponding to the lowest eigenvalue of the Hessian at the saddle point.
### Hyperdynamics for non-driven system
In this section we briefly review the hyperdynamics method and more detailed description is found in. Based on the assumptions of TST, hyperdynamics uses a potential modified from a given potential to reduce the energy barriers. As long as the modified potential does not alter the original potential along the TST dividing surface, it can be proven that the relative probabilities to neighboring states are identical in both potentials.
\[\alpha=\frac{\left(R_{A}^{TST}\right)_{b}}{R_{A}^{TST}}=\frac{\,\vec{t}_{A}\,} {\left(\vec{t}_{A}\right)_{b}}=\frac{\int_{A}\,\,\exp[-\,\beta\,V]\,d\,\,\vec{r }}{\int_{A}\exp[-\,\beta\,V_{b}\,]\,d\,\,\vec{r}}=\left<e^{+\,\beta\,\Delta V} \right>_{b}\,\,\,\,, \tag{3}\]
The bias potential \(\,\Delta V\,\) must satisfy the following condition.
\[\Delta V(\,\vec{r}\,)=\left\{\begin{array}{l}>0\,\,,\quad in\,\,A\\ =0\,,\quad along\,\,S\end{array}\right.\,\,\,. \tag{4}\]The difficulty of performing a hyperdynamics simulation arises particularly from the subtlety of constructing a computationally efficient bias potential. Voter's original bias potentials used the lowest eigenvalue and the corresponding eigenvector of the Hessian matrix, but calculating the eigenvalue and its derivative require significant computational overhead.
For this study we have devised new bias potentials. These bias potentials use local variables to approximate a TST dividing surface. If a local variable is bounded along the dividing surface, the variable can be used to construct a volume located inside the dividing surface. For example, the lowest eigenvalue of the Hessian matrix is negative at the dividing surface so that the configuration volume with positive lowest eigenvalues can be located inside the dividing surface. We considered a number of local variables that could be used in place of or in addition to the lowest eigenvalue of the Hessian. Possible choices are the potential energy slope or curvature along the direction vector connecting a configuration and the minimum of the potential energy basin, and the distance from this minimum.
\[\sigma=\nabla V\cdot\vec{s}=\frac{dV}{d\,s}\;\;, \tag{5}\]
\[\kappa=\frac{d^{\,2}\,V}{d\,s^{\,2}}\;\;, \tag{6}\]
and
\[d=\sqrt{\sum_{k=1}^{3N}\left(r_{k}-r_{O,k}\right)^{2}}\;\;\;, \tag{7}\]
In some specific systems, the slope in equation and/or the curvature in equation become smaller and the distance in equation increases when the system approaches a dividing surface. In such cases it may be possible to determine critical values for them. A more detailed review regarding the methodology we use to construct bias potentials is in preparation.
### New acceleration method for driven system: Hyperdynamics with a parallel algorithm
The original hyperdynamics method was developed assuming a time-invariant potential energy. In this study we extend the method for driven systems. Moreover, we incorporate this method into a parallel algorithm.
As boundary conditions in a driven system change with time, the system is not, strictly speaking, in equilibrium. However, as we will detail below, if the change rate is so slow that the system remains at a near-equilibrium state, then we can still apply the rate theory based on equilibrium assumptions and the instantaneous transition rate can be calculated using equation with an instantaneous potential function.
In our AFM model the external parameter changing in time is the slider position \(x_{S}\) and the potential energy is a function of the slider position as well as the atomic positions. By assuming that the sliding velocity is low enough, the slider position is updated by \(\Delta x_{S}\) after a time period \(\Delta\tau\) has elapsed instead of changing continuously as shown in \(\Delta\tau\) is determined by the sliding rate (\(\Delta\tau=\Delta x_{S}/v_{S}\) ). For this process to be equivalent to the continuous sliding, \(\Delta x_{S}\) should be small compared to the length scale that characterizes the surface corrugation. Moreover, \(\Delta\tau\) must be longer than the thermal equilibration time-scale (\(\Delta\tau>>\tau_{eq}\) ).
We turn to the interpretation of the hyperdynamics simulations within the stepped sliding scheme. In the stepped sliding, if we use the original potential with the slider, frozen for \(\Delta\tau\), we have to run the simulation for the same period \(\Delta\tau\).
\[1-e^{-R\Delta\tau}\ \, \tag{8}\]
If we perform the same simulation with a biased potential, which has the boost factor of \(\alpha\), then the transition rate increases (\(R_{{}_{b}}=\alpha\times R\) ). Note that we have the same probability for the shorter time period (\(\Delta\tau_{{}_{b}}=\Delta\tau/\alpha\) ) because \(1-e^{-R\Delta\tau}=1-e^{-(R\alpha)(\Delta\tau/\alpha)}=1-e^{-R_{b}\Delta\tau_{ b}}\). Therefore, with the biased potential we can reduce the simulation time.
Assuming that the slider moves so slowly that the system is fully equilibrated at each slider position, the dynamics at different slider positions are uncorrelated. Thus, the transition probabilities at each slider position during stepped sliding are independent of each other, and we do not necessarily need to perform the simulations successively. Rather, we can perform the simulations with different slider positions in parallel as illustrated in For example, if we use the conventional serial algorithm for the system shown in figure 2, we have to first perform a simulation at \(x_{S}=1\) ( (a)), and after it finishes, we perform another simulation at \(x_{S}=2\) ((b)), etc. until we observe a transition. This corresponds to throwing dice and throwing again after knowing the first result. However, if these two events are independent of each other, we can throw both simultaneously. Thus, we can perform four simulations at \(x_{S}=1\), 2, 3, 4, simultaneously. If we have a transition at \(x_{S}=3\) and this is the latest slider position that experienced a transition in the time interval, then we ignore the result at \(x_{S}=4\) and redistribute the jobs starting from \(x_{S}=3\) and restart to perform the simulations. The speed-up obtained by this parallel distribution method is roughly proportional to the number of processors used for one system.
## 3 Application
### 2-Dimensional L-J system
#### 3.1.1 Model
We begin by investigating a simple 2-dimensional AFM model illustrated in The substrate consists of 80 atoms marked in blue, and the tip contains 33 atoms marked in red. The tip and the substrate have a 2-dimensional crystalline structure corresponding to an FCC crystal in 3-dimension. The lattice parameters of the tip and the substrate are identical.
The atoms on the bottom layer of the substrate are fixed to prevent a rigid-body translation in the vertical direction, and the system is subject to the periodic boundary condition in the horizontal direction. The relative motions of the atoms on the top layer of the tip are constrained, but they can move like a rigid body. These top atoms are pulled by a spring and pushed downward by the applied normal force as shown in All the quantities are expressed in length units of \(\sigma\), the energy units of \(\varepsilon\), and the mass units of \(m\). Time is measured in the time units of \(\tau=\sqrt{m\,\sigma^{2}\,/\,\varepsilon}\). Hereafter the units are omitted unless there is ambiguity. We used a spring stiffness of \(k=5\) and an applied normal force of \(F_{N}=5\).
The interactions of the atoms are modeled by the Lennard-Jones potential,
\[V(r)=4\,\varepsilon_{ab}\left[\left(\frac{\sigma_{ab}}{r}\right)^{12}\,-\left( \frac{\sigma_{ab}}{r}\right)^{6}\,\right]\,, \tag{9}\]
We used the following parameters.
\[\sigma_{ss}=\sigma_{n}=\sigma_{ts}=1.0\,,\,\varepsilon_{ss}=\varepsilon_{n}=1. 0\,,\,\,\varepsilon_{ts}=0.5\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\We varied the sliding velocity by 5 orders of magnitude ranging from \(v_{S}=10^{-4}\) to \(10^{-8}\) and three different temperatures (\(T=0.1\), \(0.01\), \(0.001\)\(\varepsilon\)/\(k_{B}\)) have been simulated using the Nose-Hoover chain method. The equations of motion are solved using a modified velocity-Verlet algorithm.
#### 3.1.2 Results
The graphs shown in through are obtained from a simulation with \(v_{S}=10^{-6}\) and \(T=0.01\) and illustrate typical frictional behaviors of the model.
As in the Tomlinson model, apparent stick-slip motion is observed. The tip position increases linearly during the stick-phase and jumps at several discrete points corresponding to slip events. The average distance of these points corresponds to the lattice parameter of the substrate. Note that the tip position shown in the figure is the averaged quantity over a time period (otherwise the curve is very noisy due to thermal fluctuations).
The lateral force \(F_{R}\) is measured by the deformation of the spring, as in an AFM experiment, expressed as
\[F_{R}=k\left(x_{S}-x_{T}\right). \tag{10}\]
(a) and 5 (b) show the lateral force at the first slip as a function of the slider position and the tip position respectively. As expected from figure 4, where the tip position is linearly proportional to the slider position (\(x_{T}\approx k_{C}\ x_{S}\)), the lateral force exhibits a linear dependence on both the slider position and the tip position. However, the straight line extended from the initially linear portion of the curves illustrates that the lateral force deviates from the linear dependence near the transition.
The potential energy, which is a function of the tip position as well as the atom positions, \(V(\vec{r_{1}},\cdots,\vec{r}_{N}\); \(\vec{r}_{T}\)), is shown in (a) shows the potential energy as a function of the slider position, and (b) shows the potential energy as a function of the tip position. The increase in the potential energy is due to the elastic deformation of the tip and can be fit to a quadratic function. We expect that the time averaged potential energy \(\overline{V}\) has the following relation with the slider position.
\[\overline{V}\sim\frac{1}{2}\,k_{1}\ x_{S}^{2}\ \ \, \tag{11}\]
The linear and quadratic increases in the tip position and in the potential energy respectively imply a linear decrease in the energy barrier as the slider advances. However, Catastrophe theory predicts that the energy barrier decreases as a function of \(\left(x_{S}^{*}-x_{S}\right)^{3/2}\) near the transition, where \(x_{S}^{*}\) is the slider position where the energy barrier completely vanishes. Thus, as the slider approaches near transition we could expect the deviations.
shows the dependence of the lateral force on the sliding velocity. At a temperature of 0.01 the results from five different sliding velocities (10\({}^{\text{-}4}\), 10\({}^{\text{-}5}\), 10\({}^{\text{-}6}\), 10\({}^{\text{-}7}\), 10\({}^{\text{-}8}\)) are shown. It is apparent that as the sliding velocity decreases the tip makes a transition at an earlier slider position, which is consistent with the prediction of the modified Tomlinson model. The temperature dependence is shown in In this figure we can observe that the transition occurs at much earlier slider position at higher temperature, and the effective stiffness _k_eff, the slope of the lateral force vs. slider position curve, slightly reduces as temperature increases due to softening of the tip and contact stiffness.
Finally, we compare the results from various methods. summarizes the simulation results at various sliding velocities and various temperatures obtained from the four different methods. At each velocity and temperature, we prepared 10 samples for serial simulations and 5 samples for parallel simulations. Each sample has different initial conditions. The lateral forces in this graph are measured at the transition points and averaged over eight different transition points and over different samples.
The continuous sliding and the stepped sliding (using the original potential on a single processor) are tested at the velocities of 10\({}^{\text{-}4}\), 10\({}^{\text{-}5}\), 10\({}^{\text{-}6}\) and at temperatures of 0.001, 0.01, 0.1. All the data overlap and agree within the range of the standard deviation shown as the error bar in Most data ranges within the standard error.
At the sliding velocity of 10\({}^{\text{-}6}\) and the temperature of 0.01, all four methods are tested and all the measured lateral forces agree. The simulations on a single processor could not have been performed at the sliding velocities lower than 10\({}^{\text{-}6}\) due to excessive running time on a standard workstation, but all the data from the parallel simulations at 10\({}^{\text{-}7}\) and 10\({}^{\text{-}8}\) using either the original potential or the biased potential fall close to the trend line extended from the data obtained from continuous sliding.
The lateral forces show the expected logarithmic dependence on the sliding velocity, and no plateaus are found in any velocity range. The slope of the lateral force vs. ln _vS_ curve increases as temperature increases.
#### 3.1.3 Discussion
With the conventional method, we were not able to perform simulations at velocities lower than 10\({}^{\text{-}6}\) because of the extended running time on a standard workstation. With the velocities above this limit, the simulation results from the stepped sliding agree with the results of the continuous sliding. Thus, the basis for the other methods (the parallel method and the hyperdynamics methods) is well verified. Using the parallel method makes it possible to lower the sliding velocity by one order of magnitude, and with the hyperdynamics methodology we can lower the sliding velocity further.
In this 2-dimensional sliding system, we have found that the relative population density of the unboosted region in the phase space is altered as the slider position changes. When the slider position is far from the transition point, the pre-simulation using the original potential to calculate the boost factor does not sample any points in the unboosted region. Thus, the maximum boost factor, which is the inverse of the relative population density of the unboosted region, will be very large. However, as the slider approaches the transition point, some unboosted points are sampled and the maximum achievable boost factor reduces. Since as the sliding velocity decreases the transition occurs at earlier slider positions where the maximum boost factor is larger, we expect that we can reduce the sliding velocity even below \(10^{-8}\).
Although the lateral force shows a logarithmic dependence on the sliding velocity, this is expected due to the simplicity of the current model. Since the tip maintains its crystalline structure after transitions and no defects arise inside the tip due to much weaker interaction between the tip and the substrate, the only possible transition mechanisms are backward and forward hopping, which have the same energy barriers. As the slider advances, the forward hopping (in the sliding direction) becomes more favorable than the backward hopping. Moreover, the relative configurations of the system before and after transition do not change. However, in more realistic situations, the tip may lose atoms during sliding and its interface configuration may be altered during transitions or different pathways may be traversed at high and low temperature.
### 3-Dimensional L-J system
#### 3.2.1 Model
We now proceed to a 3-dimenstional system modeling an AFM tip and a substrate illustrated in The tip has 183 atoms shown in red and the substrate consists of 1800 atoms shown in blue. The substrate has FCC crystalline structure, and the tip is created by carving an FCC crystal with the same lattice parameter as the substrate into a conical shape with flat ends. The tip and the substrate are joined in the direction, and as shown on the right side of figure 10, nine atoms on the bottom of the tip are in contact with the substrate. Because the tip and the substrate have the same lattice parameter and are aligned in the same orientation the tip atoms are in registry with the substrate.
The sliding simulation is realized in the same way as the 2-dimensional model. A spring (\(k=10\)) is linked to the top layer of the tip and the bottom layer of the substrate is fixed. A normal force (\(F_{N}=5\)) is applied to the top of the tip.
The interaction between substrate atoms and the interaction between a substrate atom and a tip atom are modeled by the Lennard-Jones (L-J) potential, and the following parameters are used.
\[\sigma_{ss}=1.0\quad,\quad\varepsilon_{ss}=1.0\] \[\sigma_{st}=1.0\quad,\quad\varepsilon_{st}=0.2\quad\quad(\text{$s$: substrate, $t$: tip})\]
For the interaction between tip atoms, we used a harmonic potential, which does not allow any bond breaking to maintain the shape of the tip and prevent wear during sliding.
\[V(r)=\frac{1}{2}k\left(r-r_{O}\right)^{2}\, \tag{12}\]
The stiffness and the equilibrium length are chosen to be identical to the values of the L-J potential with \(\sigma_{u}=1\), \(\varepsilon_{st}=1\) at the equilibrium position.
We tested the same four methods as in 2-D simulations, and for the hyperdynamics simulations we used a bias potential using a local slope \(\sigma\) defined in equation and the lowest eigenvalue \(\varepsilon\) of the Hessian Matrix.
\[\Delta V(\sigma,\varepsilon)=\Delta V_{1}(\sigma)+\Delta V_{2}(\varepsilon)- \frac{1}{\Delta V_{\max}}\Delta V_{1}(\sigma)\times\Delta V_{2}(\varepsilon)\, \tag{13}\]
where
\[\Delta V_{1}(\sigma)=\begin{cases}\Delta V_{\max}&\sigma\geq\sigma_{U}\\ \Delta V_{\max}\left[1-\left(\frac{\sigma-\sigma_{U}}{\sigma_{L}-\sigma_{U}} \right)^{m}\right]^{2}&\sigma_{L}<\sigma<\sigma_{U}\\ 0&\sigma\leq\sigma_{L}\end{cases} \tag{14}\]
and
\[\Delta V_{2}(\varepsilon)=\begin{cases}\Delta V_{\max}&\varepsilon\geq \varepsilon_{U}\\ \Delta V_{\max}\left[1-\left(\frac{\varepsilon-\varepsilon_{U}}{\varepsilon_{ L}-\varepsilon_{U}}\right)^{m}\right]^{2}&\varepsilon_{L}<\varepsilon<\varepsilon_{U}\\ 0&\varepsilon\leq\varepsilon_{L}\end{cases}\, \tag{15}\]
Note that \(\Delta V(\sigma,\varepsilon)\) ranges from 0 to \(\Delta V_{\max}\).
We performed simulations at four different sliding velocities (\(10^{\text{-4}}\), \(10^{\text{-5}}\), \(10^{\text{-6}}\), \(10^{\text{-7}}\)) and at temperatures of 0.001, 0.01, 0.1. As in the 2-dimensional model, we used the Nose-Hoover chain method to control temperature and a modified velocity-Verlet algorithm to numerically solve the equations of motion.
#### 3.2.2 Results
We plotted the lateral force and the potential energy as functions of both the slider position and the tip position in and figure 12, where the data are obtained from the simulations with \(v_{S}=10^{-5}\) and \(T=0.01\). As in the 2-dimensional case, the lateral forces show the linear dependence, but deviates from the straight lines near transition points. The potential energy changes like a quadratic function of the slider position at earlier slider positions, but shows deviation from the quadratic fits unlike the 2-D models and very steep changes near transitions.
The lateral forces shown in the figure are the averages of the peak values at each transition over the samples and the peaks. At sliding velocities of \(10^{-4}\) and \(10^{-5}\), the lateral forces measured from the continuous sliding show close agreement with the forces from the stepped sliding. Thus, the fundamental assumption of our methodologies is verified with this 3-dimensional model. However, although the number of atoms in this model is not large, the simulations on a single processor using the conventional method at lower sliding velocities (\(<10^{-5}\)) are prohibitive, requiring more than one month on a standard workstation.
By the parallel method using 50 processors, we were able to perform MD simulations at a sliding velocity of \(10^{-6}\). The running time was less than a week. However, without the aid of hyperdynamics, the simulations at lower sliding velocities (\(<\)\(10^{-6}\)) are not attainable because whenever we lower the sliding velocity by a factor of 10, we need to increase the running time by the same factor. Using a bias potential constructed using the eigenvalue and the local slope, the simulations at a sliding velocity of \(10^{-7}\) were attainable.
All the data measured from the various methods show close agreement with the trend line obtained from the continuous method on a single processor within the standard deviation shown as the error bars in Moreover, as expected from the modified Tomlinson model, the lateral force exhibits the logarithmic dependence on the sliding velocity within the range of the parameters used in this simulation study.
## 4 Conclusion
We have devised a novel scheme to accelerate the MD simulations for driven systems extending the original hyperdynamics method. Combined with a parallel algorithm simultaneously running systems at different slider positions on multiple processors, this extended hyperdynamics method has been applied to the frictional sliding of the 2-dimensional and 3-dimensional AFM models.
The validity of the methodologies was well verified by comparison with conventional MD simulations. First, we observed that the stepped sliding serves as a reasonable approximation forcontinuous sliding, and the simulation results using the parallel methodology and hyperdynamics showed close agreements with the simulation results of the conventional method. Moreover, both 2-D and 3-D simulations showed that the average of the lateral forces at the transitions have the logarithmic dependence on the sliding velocity as predicted from the modified Tomlinson model.
The sliding velocities used in experiment (nm/s \(\sim\) mm/s) and MD simulation (\(\sim\) m/s) are different by several orders of magnitude and this difference cannot be completely overcome purely by parallelization methods such as the parallel replica method. The method applied here gains acceleration both from the boost factor in the hyperdynamics and from the parallel algorithm. Therefore, with this combined method we anticipate that it will be possible to simulate real systems investigated in AFM experiments over a comparable range of sliding velocities.
| 10.48550/arXiv.0909.5215 | Accelerated molecular dynamics simulation of low-velocity frictional sliding | Woo Kyun Kim, Michael L. Falk | 2,087 |
10.48550_arXiv.0811.4396 | ###### Abstract
We analyze the effect of tensional strain in the electronic structure of graphene. In the absence of electron-electron interactions, within linear elasticity theory, and a tight-binding approach, we observe that strain can generate a bulk spectral gap. However this gap is critical, requiring threshold deformations in excess of 20%, and only along preferred directions with respect to the underlying lattice. The gapless Dirac spectrum is robust for small and moderate deformations, and the gap appears as a consequence of the merging of the two inequivalent Dirac points, only under considerable deformations of the lattice. We discuss how strain-induced anisotropy and local deformations can be used as a means to affect transport characteristics and pinch off current flow in graphene devices.
Investigations in the context of carbon nanotubes reveal intrinsic strengths that make these systems the strongest in nature, Recently, graphene -- the mother of all \(sp^{2}\) carbon structures -- has been confirmed as the strongest material ever to be measured, being able to sustain reversible elastic deformations in excess of 20%.
These mechanical measurements arise at a time where graphene draws considerable attention on account of its unusual and rich electronic properties. Besides the great crystalline quality, high mobility and resilience to high current densities, they include a strong field effect, absence of backscattering and a minimum metallic conductivity. While many such properties might prove instrumental if graphene is to be used in future technological applications in the ever pressing demand for miniaturization in electronics, the latter is actually a strong deterrent: it hinders the pinching off of the charge flow and the creation of quantum point contacts. In addition, graphene has a gapless spectrum with linearly dispersing, Dirac-like, excitations. Although a gap can be induced by means of quantum confinement in the form of nanoribbons and quantum dots, these "paper-cutting" techniques are prone to edge roughness, which has detrimental effects on the electronic properties. Hence, a route to induce a robust, _clean_, bulk spectral gap in graphene is still much in wanting.
In this paper we inquire whether the seemingly independent aspects of mechanical response and electronic properties can be brought together with profit in the context of a tunable electronic structure. Motivated by recent experiments showing that reversible and controlled strain can be produced in graphene with measurable effects, we theoretically explore the effect of strain in the electronic structure of graphene within a tight-binding approach. Our calculations show that, in the absence of electron-electron interactions, a gap can be opened in a pure tight binding model of graphene for deformations beyond 20%. This gap opening is not a consequence of a broken sublattice symmetry but due to level crossing. The magnitude of this effect depends on the direction of applied tension, so that strain along a zig-zag direction is most effective in overcoming the gap threshold, whereas deformations along an armchair direction do not induce a gap. Unfortunately, such large threshold deformations render strain an ineffective means to achieve a bulk gapped spectrum in graphene. We discuss alternate means to impact transport and electronic structure using local strain profiles.
## I Model
We consider that electron dynamics of electrons hopping in the honeycomb lattice is governed by the nearest
(Color online) (a) Tension geometry considered in the text. The zig-zag direction of the honeycomb lattice is always parallel to the axis \(Ox\).
neighbor tight-binding Hamiltonian
\[H=\sum_{\mathbf{R},\mathbf{\delta}}t(\mathbf{R},\mathbf{\delta})a^{\dagger}(\mathbf{R})b(\mathbf{R}+\mathbf{ \delta})+\text{H. c.}\,. \tag{1}\]
Here \(\mathbf{R}\) denotes a position on the Bravais lattice, and \(\mathbf{\delta}\) connects the site \(\mathbf{R}\) to its neighbors; \(a(\mathbf{R})\) and \(b(\mathbf{R})\) are the field operators in sublattices A and B. The first thing to emphasize is that, under general stress conditions, the hopping \(t(\mathbf{R},\mathbf{\delta})\) will be generally different among different neighbors. We are interested in the elastic response, for which deformations are affine. This means that even though the hoppings from a given atom to its neighbors can be all different, they will be the same for every atom. Therefore, as depicted in Fig. 2, we need only to consider three distinct hoppings: \(t_{1}=t(\mathbf{\delta}_{1})\), \(t_{2}=t(\mathbf{\delta}_{2})\), and \(t_{3}=t(\mathbf{\delta}_{3})\). The relaxed equilibrium value for \(t_{i}=t(\mathbf{\delta}_{i})\) is \(t_{0}\approx 2.7\) eV. Our goal is to investigate the changes that strain induces in these hoppings, and what impact they have in the resulting electronic structure.
Throughout this paper we shall use the C-C equilibrium distance, \(a_{0}=1.42\) A, as unit of length, and will frequently use \(t_{0}\) as unit of energy.
## II Analysis of strain
We are interested in uniform planar tension situations, like the one illustrated in the graphene sheet is uniformly stretched (or compressed) along a prescribed direction. The fixed Cartesian system is chosen in a way that \(Ox\) always coincides with the zig-zag direction of the lattice. In these coordinates the tension, \(\mathbf{T}\), reads \(\mathbf{T}=T\cos(\theta)\,\mathbf{e}_{x}+T\sin(\theta)\,\mathbf{e}_{y}\).
As for any solid, the generalized Hooke's law relating stress, \(\tau_{ij}\) and strain \(\varepsilon_{ij}\) has the form
\[\tau_{ij}=C_{ijkl}\,\varepsilon_{kl}\,,\quad\varepsilon_{ij}=S_{ijkl}\,\tau_{ kl} \tag{2}\]
Since we address only states of planar stress, we resort to the 2-dimensional reduction of the stress and strain tensors. In general the components \(C_{ijkl}\) depend on the particular choice of the Cartesian axes. Incidentally, for an hexagonal system under planar stress in the basal plane, the elastic components are independent of the coordinate system. This means that graphene is elastically isotropic.
The analysis of strain is straightforward in the principal system \(Ox^{\prime}y^{\prime}\) where we simply have \(\mathbf{T}=T\mathbf{e}_{x^{\prime}}\):
\[\varepsilon^{\prime}_{ij}=S_{ijkl}\tau^{\prime}_{kl}=T\,S_{ijkl}\delta_{kx} \delta_{lx}=T\,S_{ijxx} \tag{3}\]
Given that only five compliances are independent in graphite (viz., \(S_{xxyy}\), \(S_{xxyy}\), \(S_{xxzz}\), \(S_{zzzz}\), \(S_{yzyz}\)), it follows that the only non-zero deformations are
\[\varepsilon^{\prime}_{xx}=T\,S_{xxxx}\,,\,\varepsilon^{\prime}_{yy}=T\,S_{xxyy }\,, \tag{4}\]
If we designate the tensile strain by \(\varepsilon=T\,S_{xxxx}\), the strain tensor can be written in terms of Poisson's ratio, \(\sigma=-S_{xxxy}/S_{xxxx}\):
\[\mathbf{\varepsilon}^{\prime}=\varepsilon\begin{pmatrix}1&0\\ 0&-\sigma\end{pmatrix}\,. \tag{5}\]
This form shows that graphene responds as an isotropic elastic medium. For Poisson's ratio we use the value known for graphite: \(\sigma=0.165\). It should be mentioned that when stress is induced in graphene by mechanically acting on the substrate (i.e. when graphene is adhering to the top of a substrate and the latter is put under tension, as is done in Ref.), the relevant parameter is in fact the tensile strain, \(\varepsilon\), rather than the tension \(T\). For this reason, we treat \(\varepsilon\) as the tunable parameter. Since the lattice is oriented with respect to the axes \(Oxy\), the stress tensor needs to be rotated to extract information about bond deformations.
\[\mathbf{\varepsilon}=\varepsilon\begin{pmatrix}\cos^{2}\theta-\sigma\sin^{2} \theta&(1+\sigma)\cos\theta\sin\theta\\ (1+\sigma)\cos\theta\sin\theta&\sin^{2}\theta-\sigma\cos^{2}\theta\end{pmatrix}\,. \tag{6}\]
## III Bond deformations
If \(\mathbf{v}^{0}\) represents a general vector in the undeformed graphene plane, its deformed counterpart is given, to leading order, by the transformation
\[\mathbf{v}=(\mathbf{1}+\mathbf{\varepsilon})\cdot\mathbf{v}^{0}\,. \tag{7}\]
Especially important are the deformations of the nearest-neighbor bond distances. Knowing \(\varepsilon_{ij}\) one readily obtains the deformed bond vectors using.
\[|\mathbf{\delta}_{1}| =1+\tfrac{3}{4}\varepsilon_{11}-\tfrac{\sqrt{3}}{2}\varepsilon_{1 2}+\tfrac{1}{4}\varepsilon_{22} \tag{8a}\] \[|\mathbf{\delta}_{2}| =1+\varepsilon_{22}\] (8b) \[|\mathbf{\delta}_{3}| =1+\tfrac{3}{4}\varepsilon_{11}+\tfrac{\sqrt{3}}{2}\varepsilon_{ 12}+\tfrac{1}{4}\varepsilon_{22} \tag{8c}\]
(Color online) (a) Honeycomb lattice geometry. The vectors \(\mathbf{\delta}_{1}=a(\sqrt{\frac{3}{2}},-\frac{1}{2})\), \(\mathbf{\delta}_{2}=a\)\(\mathbf{\delta}_{3}=a(-\frac{\sqrt{3}}{2},-\frac{1}{2})\) connect A-sites (red/dark) to their B-site (blue/light) neighbors. (b) The first Brillouin zone of undeformed graphene, with its points of high symmetry.
Of particular interest are the cases \(\theta=0\) and \(\theta=\pi/2\) since they correspond to tension along the zig-zag (\(\mathcal{Z}\)) and armchair (\(\mathcal{A}\)) directions:
\[\mathcal{Z}: |\mathbf{\delta}_{1}|=|\mathbf{\delta}_{3}|=1+\tfrac{3}{4}\varepsilon- \tfrac{1}{4}\varepsilon\sigma\,,\;|\mathbf{\delta}_{2}|=1-\varepsilon\sigma \tag{9a}\] \[\mathcal{A}: |\mathbf{\delta}_{1}|=|\mathbf{\delta}_{3}|=1+\tfrac{1}{4}\varepsilon- \tfrac{3}{4}\varepsilon\sigma\,,\;|\mathbf{\delta}_{2}|=1+\varepsilon \tag{9b}\]
The modification of these distances distorts the reciprocal lattice as well, and the positions of the high-symmetry points shown in Fig. 2(b) are shifted.
\[\mathbf{b}_{1}\approx\frac{2\pi}{\sqrt{3}}\left(1-\varepsilon_{11}- \frac{\varepsilon_{12}}{\sqrt{3}};\,\frac{1}{\sqrt{3}}-\varepsilon_{12}- \frac{\varepsilon_{22}}{\sqrt{3}}\right), \tag{10a}\] \[\mathbf{b}_{2}\approx\frac{2\pi}{\sqrt{3}}\left(-1+\varepsilon_{11}- \frac{\varepsilon_{12}}{\sqrt{3}};\,\frac{1}{\sqrt{3}}+\varepsilon_{12}-\frac {\varepsilon_{22}}{\sqrt{3}}\right). \tag{10b}\]
Most importantly, the symmetry point \(\mathbf{K}=(\frac{4\pi}{3\sqrt{3}},0)\) (that coincides with the Fermi point in the undoped, equilibrium situation, and chosen here for definiteness) moves to the new position
\[\mathbf{K}\approx\frac{4\pi}{3\sqrt{3}}\left(1-\frac{\varepsilon_{11}}{2}-\frac{ \varepsilon_{22}}{2};\,-2\varepsilon_{12}\right) \tag{11}\]
For uniaxial tension this reduces to
\[\mathbf{K}\approx\frac{4\pi}{3\sqrt{3}}\left(1-\frac{\varepsilon(1-\sigma)}{2};\, -\varepsilon(1+\sigma)\,\sin[2\theta]\right)\,. \tag{12}\]
The factor of \(2\theta\) means that the shift is the same for the \(\mathcal{A}\) and \(\mathcal{Z}\) directions, in leading order. These general results will be important for our subsequent discussion.
## IV Hopping renormalization
The change in bond lengths leads to different hopping amplitudes among neighboring sites. In the Slater-Koster scheme Slater and Koster, the new hoppings can be obtained from the dependence of the integral \(V_{pp\pi}\) on the inter-orbital distance. Unfortunately determining such dependence with accuracy is not a trivial matter. Many authors resort to Harrison's flyleaf expression which suggests that \(V_{pp\pi}(l)\propto 1/l^{2}\)Harrison. However this is questionable, insofar as such dependence is meaningful only in matching the tight-binding and free electron dispersions of simple systems in equilibrium (beyond the equilibrium distance such dependence is unwarrantedHarrison). It is indeed known that such functional form fails away from the equilibrium distanceHarrison, and a more reasonable assumption is an exponential decayHarrison.
\[V_{pp\pi}(l)=t_{0}e^{-3.37(l/a_{0}-1)}\,, \tag{13}\]
As a consistency check, we point out that, according to Eq., the next-nearest neighbor hopping (\(t^{\prime}\)) would have the value \(V_{pp\pi}(\sqrt{3}a_{0})=0.23\) eV, which tallies with existing estimates of \(t^{\prime}\) in grapheneHarrison.
## V Gap threshold
The bandstructure of Eq.
\[E(k_{x},k_{y})=\pm\left|t_{2}+t_{3}\,e^{-i\mathbf{k}.\mathbf{a}_{1}}+t_{1}\,e^{-i\mathbf{ k}.\mathbf{a}_{2}}\right|\,. \tag{14}\]
Here both \(t_{\alpha}\) and the primitive vectors \(\mathbf{a}_{\alpha}\) (see Fig. 2(a) for the definition of the vectors \(\mathbf{a}_{\alpha}\)) change under strain: the hoppings change as per Eqs., and the primitive vectors as per Eq.. This generalized dispersion has been previously discussed in Refs., under the assumption that only the hopping elements change, without lattice deformation. It was found that the gapless spectrum is robust, and that a gap can only appear under anisotropy in excess of 100% in one of the hoppings.
\[\left|\frac{t_{1}}{\left|t_{2}\right|}-1\right|\leq\frac{\left|t_{3}\right|} {\left|t_{2}\right|}\leq\left|\frac{\left|t_{1}\right|}{\left|t_{2}\right|} +1\right| \tag{15}\]
This condition corresponds to the shaded area in Fig. 3(a). Using the results in eqs. we have mapped the evolution of the hoppings with \(\varepsilon\) and \(\theta\). This allows us to identify the range of parameters that violate, and to obtain the threshold for gap opening. For a given \(\theta\), we follow the trajectory of the point (\(t_{1}/t_{2}\), \(t_{3}/t_{2}\)) as strain grows, starting from the isotropic point at \(\varepsilon=0\). The result is one of the arrowed curves in Fig. 3(a).
(Color online) (a) Plot of \(t_{1}/t_{2}\) vs \(t_{3}/t_{2}\) as a function of strain, \(\varepsilon\), and \(\theta\). Closed lines are iso-strain curves, and arrowed lines correspond to the trajectory of the point (\(t_{1}/t_{2},t_{3}/t_{2}\)) as \(\varepsilon\) increases, calculated at constant angle. The graph is symmetric under reflection on both axes. In the shaded area the spectrum is gapless. The blue iso-strain line (\(\varepsilon\approx 0.23\)) corresponds to the gap threshold. In panel (b) we show the angular dependence of \(t_{1,2,3}\) for \(\varepsilon=0.05\) and \(\varepsilon=0.23\).
From such procedure, summarized in Fig. 3, we conclude that: (i) the gap threshold is at \(\varepsilon\approx 0.23\) (\(\sim 20\%\)); (ii) the behavior of the system is periodic in \(\theta\) with period \(\pi/3\), in accord with the symmetry of the lattice; (iii) tension along the zig-zag direction (\(\theta=0,\pi/3,\dots\)) is more effective in overcoming the gap threshold; (iv) tension along the armchair direction never generates a gap.
The two panels of contain plots of the individual \(t_{\alpha}\) for two particular values of strain. It is clear that, for deformations along the \(\mathcal{Z}\) direction, the highest relative change occurs along the zig-zag bonds (\(t_{1,3}\)), and conversely for deformations along the \(\mathcal{A}\) direction. This could have also been anticipated from Eqs. and the smallness of \(\sigma\).
## VI Critical gap
The fact that the isotropic point \(\) in is surrounded by an appreciable shaded area, means that the gapless situation is robust, and the emergence of the gap requires a critical strain. The physical effect behind such critical gap lies in the fact that, under strain, the Dirac cones drift away from the points \(\mathbf{K}\), \(\mathbf{K}^{\prime}\) in the Brillouin zone (BZ). Before we proceed, it is pertinent to advance a crucial detail: in a deformed lattice, the Dirac points (i.e. the positions in the BZ where conduction and valence bands touch conically) and the symmetry points \(\mathbf{K}_{i}\) do not coincide. In what follows we shall distinguish them explicitly.
To be more definite, we examine the position of the minimum energy for the bands obtained from Eq., which can be done exactly if we assume that the lattice remains undeformed. Due to the particle-hole symmetry, we minimize \(E(\mathbf{k})^{2}\). Let us assume that \(t_{1}=t_{3}\neq t_{2}\), which applies for tension along either zig-zag or armchair directions.
\[\mathbf{k}_{\text{min}}=\left(\pm\frac{2}{\sqrt{3}}\arccos\!\left[-\frac{t_{2}}{2t _{1}}\right]\!;\,0\right), \tag{16}\]
The \(\pm\) sign refers to one possible choice for the two inequivalent valleys.
\[\mathbf{k}=\left(\frac{\pi}{\sqrt{3}};\,\frac{\pi}{3}\right)\!,\,\mathbf{k}=\left(0; \,\frac{2\pi}{3}\right)\!,\,\mathbf{k}=\left(\frac{\pi}{\sqrt{3}};\,-\frac{\pi}{3}\right) \tag{17}\]
(and all symmetry related points). These are just the points \(\mathbf{M}_{1}\), \(\mathbf{M}_{2}\), \(\mathbf{M}_{3}\) shown in and their position is independent of \(t_{i}\). The values of energy at these points are \(E(\mathbf{M}_{1})=|t_{1}+t_{2}-t_{3}|\), \(E(\mathbf{M}_{2})=|-t_{1}+t_{2}+t_{3}|\) and \(E(\mathbf{M}_{3})=|t_{1}-t_{2}+t_{3}|\).
The result in Eq. shows that the Dirac points drift away from the \(\mathbf{K}\) point, and the direction of that drift is dictated by the relative variations in \(t_{i}\). For example, for uniaxial tension along the \(\mathcal{Z}\) we have \(t_{2}>t_{1}=t_{3}\), and therefore the minimum of energy moves to the right (left) of \(\mathbf{K}_{1}\) (\(\mathbf{K}^{\prime}_{1}\)) [cfr. Fig. 2]. This means that the inequivalent Dirac points move toward each other, and will clearly meet when \(2t_{1}=t_{2}\). They meet precisely at the position of the saddle point \(\mathbf{M}_{2}\). Throughout this process, the dispersion remains linear along the two orthogonal directions, albeit with different Fermi velocities. If the hoppings change further so that \(2t_{1}>t_{2}\), the solution is no longer valid, and the minimum lies always at \(\mathbf{M}_{2}\). Since the energy at this saddle point is given exactly by \(E(\mathbf{M}_{2})=|2t_{1}-t_{2}|\), the system becomes gapped, with a gap \(\Delta=2|2t_{1}-t_{2}|\). Moreover, the dispersion becomes peculiar in that it remains linear along one direction (the \(y\) direction in this example) and quadratic along the other. The topological structure is also modified since the two inequivalent Dirac cones have merged.
These considerations assume that the hoppings can change but the lattice remains undeformed. Under a real deformation both lattice and hoppings are affected. The lattice deformation will distort the BZ but will not affect the aspects discussed above.
(Color online) Top row shows density plots of the energy dispersion, \(E(k_{x},k_{y})\), for \(\{\varepsilon=0,\,\theta=0\}\) (a), \(\{\varepsilon=0.2,\,\theta=\pi/2\}\) (b), and \(\{\varepsilon=0.2,\,\theta=0\}\) (c). In these plots, the central white dashed lines represent the boundary of the first BZ of the undeformed lattice, while the solid white lines mark the boundaries of the BZ for the deformed lattice. In (d) we have a cut of (c) along \(k_{y}=0\), showing the merging of the Dirac cones as strain increases, and the ultimate appearance of the gap. In panel (e) we compare the gap given by Eq. (line) with the result obtained from direct minimization of the energy in the full BZ (dots).
Fig. 4(a-c). The plotted dispersions include the deformation of the BZ and the change in the hoppings, simultaneously. For strain along the \(\mathcal{A}\) direction the nonequivalent Dirac cones move in opposite directions and never meet [Fig. 4(b)]. However, if the deformation is along the \(\mathcal{Z}\) direction, the cones always approach each other [Fig. 4(c)], and will eventually merge. This merging is seen in detail in Fig. 4(d) where a cut along \(k_{y}=0\) is presented to show the emergence of the gap beyond the threshold deformation. For tension along an arbitrary direction (except armchair) the cones always merge, the \(\mathcal{Z}\)direction being the optimal orientation, requiring less strain [cfr. Fig. 3(a)]. Precisely at the critical point, the dispersion is linear along \(k_{x}\) and quadratic along \(k_{y}\), as shown in This modification in the dispersion along one of the Cartesian directions has peculiar implications for the DOS and Landau level quantization.
The gap is a result of this Dirac cone merging process, and the origin of the high critical strain is now clear: one needs to deform enough to bring the two Dirac points to coincidence. This agrees with the existing understanding that the gapless Dirac spectrum in graphene is robust (topologically protected) with respect to small perturbations.
For strain along \(\theta=0\), as discussed above, the gap is conveniently given by
\[E_{g}(\varepsilon)=2\left|2t_{1}(\varepsilon)-t_{2}(\varepsilon)\right|\theta (t_{2}-2t_{1})\,. \tag{18}\]
An example of the strain dependence of \(E_{g}\) can be seen in Fig. 4(e). In it we see the agreement between the gap given by Eq. and the value extracted from a direct minimization of \(E(k_{x},k_{y})\) in the full (deformed) BZ.
From Fig. 4(b) one can see that pulling along an armchair direction imparts 1D-like features to the system: the dispersion becomes highly anisotropic. This is explained on account of the results plotted in Fig. 3(b) which show that stress along \(\mathcal{A}\) tends to weaken one bond only. In extreme cases, the weak bond can be highly suppressed leaving only a set of 1D chains. This means that strain along certain directions can be used as a means to induce preferred anisotropy in electric transport. In contrast, pulling along a zig-zag direction tends to dimerize the system for large deformations, which ultimately explains the appearance of the gap in this case.
## VII Position of the Dirac point
The fact that there are two concurrent effects determining the changes in the bandstructure (viz. the lattice distortion itself, and the modification in the nearest-neighbor hoppings) means that the position of the minimum of energy _does not coincide with the symmetry points of the deformed BZ_.
(Color online) (a) A close-up of the energy dispersion close to its minimum, for tension along \(\theta=0\) and \(\varepsilon=0.05\). The solid white lines show the intersection of the Bragg planes that define the boundary of the first BZ in the deformed lattice, while the dashed white lines represent the same boundary es in equilibrium. It is clear that the Dirac point lies neither at \(\mathbf{K}\), nor at its deformed counterpart. The energy contours are labeled in eV. (b) A contour plot of the absolute value of the Fermi velocity, \(\hbar v_{F}=\nabla_{\mathbf{k}}E(\mathbf{k})\) for the same region shown in (a).
(Color online) Closeup of the energy dispersion \(E(\mathbf{k})\) in the vicinity of the Dirac points. Panel (a) shows a pre-critical situation (in which the two cones persist) for a strain of \(\varepsilon=0.2\) along the zig-zag (\(\theta=0\)) direction. Notice how the saddle point approaches zero energy with increasing strain. In (b) we show the bandstructure at precisely the critical strain for which the two Dirac cones meet. The dispersion is quadratic along \(k_{x}\) (the direction of strain) but remains linear along \(k_{y}\). This can be seen in (c) where we show the critical dispersion viewed from the \(k_{x}\) and \(k_{y}\) axes, respectively.
Fig. 6(a) where we provide a close-up of the energy dispersion close to the Dirac point. This should be clear from the foregoing discussion on the merging of the Dirac points. In any case we want to stress this effect and illustrate it by analyzing small perturbations with respect to the undeformed situation. For definiteness let us focus again in the case \(t_{1}=t_{3}\). The position of the new \(\mathbf{K}\) was given already in eqs.. The position of the Dirac point is given by eq. when only the hoppings change, but not the lattice.
\[\mathbf{k}_{\text{min}}\approx\pm\Bigg{(}\frac{4\pi}{3\sqrt{3}}+2\frac{t_{2}-t_{1 }}{3t_{1}};\,0\Bigg{)}\,. \tag{19}\]
One can calculate the correction to this result simultaneously accounting for the lattice deformation. But the lengthy expression that results is less important than the qualitative effect: the corrections to the expression depend on the specific details of the variation of \(t_{i}\) with distance. Consequently, the Dirac point and the \(\mathbf{K}\) point of the deformed lattice do not coincide in general. The equilibrium situation, in which they coincide, is a very particular case.
In fact, even assuming a simple lattice distortion that does not change the hoppings will move the Dirac point away from the symmetry point of the resulting lattice. This can be seen from a low energy expansion of putting \(t_{i}=t\).
\[\mathbf{k}_{\text{min}}\approx\Bigg{(}\frac{4\pi}{3\sqrt{3}}(1-\varepsilon_{11}); \,-\frac{4\pi}{3\sqrt{3}}\varepsilon_{12}\Bigg{)}\,, \tag{20}\]
This fact is of critical relevance when interpreting results of similar calculations obtained _ab-initio_, as will be discussed below.
The fact that the Dirac point drifts from the corner of the BZ means that there are no longer 3 equivalent pairs of points in the first BZ for the neutral system, but only one pair of non-equivalent points in general (in other words, in the undoped and undeformed lattice the Fermi surface is distributed among the 6 degenerate K points in the boundary of the BZ, whereas for a general deformation we have only two within the rst BZ). In the situation shown in Fig. 6(a), for example, the Dirac point shown in the figure lies outside the first Brillouin zone. The one inside the first BZ is actually the (equivalent) Dirac point that moved away from \(\mathbf{K}_{2}\) (or \(\mathbf{K}_{3}\)), in the notation of Fig. 2(b).
## VIII Discussion
We have seen that, within the tight-binding Hamiltonian written in Eq., uniform tension can induce a bulk spectral gap in graphene. However, at least within a non-interacting tight-binding approach, the gap threshold is very difficult to overcome, if at all possible. Since a tensional strain in excess of 20% is required to observe such feature, several comments are in order.
### On the approximations employed
We start by noticing that in our calculation we kept only the lowest order terms in \(\varepsilon\). In addition, although strain magnitudes of \(\sim 20\%\) are not unreasonable, graphene is expected to be in the non-linear elastic regime at those deformations. Therefore, non-linear corrections can be relevant at the quantitative level in the vicinity of the threshold.
Notwithstanding, our main result is robust: no gap can be opened under planar tension situations, except in highly strained situations. This conclusion does not depend on having taken a linear approximation insofar as it should be valid up to deformations in the range of 5-10%.
With respect to our tight-binding parametrization including only nearest neighbor hopping, we should mention that Kishigi _et al._have shown that inclusion of next-nearest neighbor terms (\(t^{\prime}\)) can, alone, generate a gap. But this requires a very specific deformation of the lattice, unlikely to occur under simple tension. The presence of \(t^{\prime}\) can also lead to other effects, like tilted Dirac cones as discussed in Ref..
It is expected that the planar arrangement of carbon atoms in freely hanging graphene should become unstable with respect to a buckled or rippled configuration, or even experience mechanical failure for moderate to high tension. The presence of a substrate should provide more stability for the planar distribution of the carbon atoms. In fact, a recent experiment published during revision of this manuscript shows that reversible strain of the order of 18% can be induced in graphene deposited on flexible plastic substrates.
### On related _ab-initio_ calculations
Secondly, some _ab-initio_ calculations seem to show that a gap is present in graphene for arbitrarily small tensions. But these reports have some conflicting details. For example there is an order of magnitude discrepancy between the gap predicted in these two references for 1% strain. In addition, Ref. claims their _ab-initio_ result agrees with the bandstructure after a suitable choice of hoppings. As we showed above this cannot be the case, since there is always a (large) threshold for the appearance of the gap. Consequently, further clarification regarding _ab-initio_ under strain is desired.
One issue that requires special attention in interpreting density functional theory (DFT) calculations of graphene's bandstructure under strain is the shift of the Dirac point. As we stressed earlier, when graphene is strained the Dirac point (position of the energy minimum) does not lie at any symmetry point of the lat tice. This is paramount because DFT calculations of the bandstructure rely on a preexisting mesh in reciprocal space, at whose points the bandstructure is sampled. These meshes normally include points along high symmetry lines of the BZ. But in the current problem, the use of such a traditional mesh will not be particularly useful to distinguish between a gapped and gapless situation. Since the Fermi surface of undoped graphene is a single point, unless the sampling mesh includes that precise point, one will always obtain a gap in the resulting bandstructure.
Most recently, we became aware of two new developments from the _ab-initio_ front, that shed the light needed to interpret the earlier calculations mentioned above. One of them is a revision of the DFT calculations presented in Ref.. In this latter work it is shown that, upon careful analysis, the DFT calculation shows no gap under uniaxial deformations up to \(\sim 20\%\). In fact the authors mention explicitly that the shift of the Dirac point from the high symmetry point mislead the authors in their initial interpretation of the bandstructure. In another preprint, independent authors show that their DFT calculations reveal, again, no gap for deformations of the order of \(10\%\) in either the \(\mathcal{Z}\) or \(\mathcal{A}\) directions. These subsequent developments confirm our prediction that only excessive planar deformations are able to engender a bulk spectral gap in graphene.
### Anisotropic Transport
Several effects of tensional strain are clear from our results. Tension leads to one-dimensionalization of transport in graphene by weakening preferential bonds: transport should certainly be anisotropic, even for small tensions. One example of that is seen in Fig. 6(a) where a strain of \(5\%\) visibly deforms the Fermi surface. Fig. 6(b), where the Fermi velocity is plotted for the same region in the BZ, further shows that, for the chosen tension direction, the Fermi surface is not quite elliptical but slightly oval, in a reminiscence of the trigonal warping effects.
The anisotropy in the Fermi velocity can become quite large, as shown in There we plot the ratio \(v_{F(\text{max})}/v_{F(\text{min})}\) for \(E_{F}=50\) meV. As can be seen in the cuts of Fig. 4(d), near the critical strain the Fermi level may touch the van Hove singularity (midpoint between the two cones) where \(v_{F}=0\) along one direction, which leads to formal divergence of the ratio between the maximum and minimum Fermi velocities.
The Fermi surface anisotropy has been captured in transport experiments that reveal a considerable anisotropy in the resistivities parallel (\(R_{xx}\)) and perpendicular (\(R_{yy}\)) to the tension direction. The authors of Reference report in their resistivity anisotropies of up to one order of magnitude at \(\sim 19\%\) strain. We can make a simple estimate of the anisotropies expected in the light of our results for the deformed bandstructure. For that we follow a Boltzmann approach to the relaxation time and DC conductivity. As is usually the case, the relevant electron states are the ones lying in a narrow vicinity of the Fermi surface. On account of the Fermi surface anisotropy, we can take the Fermi velocities in the direction of the electric field only. For example, in the situation shown in Fig. 6(a) for uniaxial strain along \(Ox\), the relevant \(v_{F}\)'s determining \(R_{xx}\) and \(R_{yy}\) will be the ones along the major and minor axes, respectively.
For the purposes of our estimate, we take the Boltzmann longitudinal conductivity for scattering out of unscreened charged impurities of valence \(Z\) and concentration \(n_{i}\), given by
\[\sigma=\frac{2e^{2}}{h}\frac{\pi\hbar^{2}v_{F}^{2}n}{Z^{2}e^{2}n_{i}}\,, \tag{21}\]
We immediately see that this estimate leads to a resistance anisotropy of
\[\frac{R_{xx}}{R_{yy}}\approx\left(\frac{v_{F(\text{max})}}{v_{F(\text{min})}} \right)^{2}. \tag{22}\]
For the maximum strain used in the experiment (\(\sim 19\%\)) our plot in Fig. 6(b) yields a ratio of 2.8, which, per, corresponds to a resistance anisotropy of roughly 8 fold, consistent with the measured anisotropy.
Our results apply to exfoliated and epitaxial graphene alike. As matters currently stand, it is perhaps more relevant in the context of the latter, since graphene grown epitaxially on SiC is almost always under strain. The strained configurations are imposed by the lattice mismatch with the substrate, and can be controlled by changing the growth and annealing conditions. For these systems, the relaxed starting configuration is already deformed.
Lastly, it is important to point out that, even though a spectral gap seems to require extreme strain, one can generate a _transport_ gap by means of local, small, deformations. It has been shown in Refs. that tunneling across a locally strained region is highly suppressed, and leads to a transport gap (i.e., a suppression of electrical conductivity) at small densities, even in the absence of a bulk spectral gap.
## IX Conclusions
Within a non-interacting, nearest-neighbor tight-binding approach, we have shown that opening a spectral gap in strained graphene requires deformations of the order of \(20\%\). This result is confirmed by the most recent _ab-initio_ calculations. Such an extreme strain is required on account of the stability of the Dirac points in graphene, that renders the spectrum gapless unless the two inequivalent Dirac points merge. The merging requires substantial anisotropy in the hopping integrals, only achieved under high strain. General features of strained graphene are an anisotropic Fermi surface, anisotropic Fermi velocities, and the drift of the Dirac points away from the high symmetry points of the lattice.
Uniform planar stain appears to be an unlikely candidate to induce a bulk gap in graphene. Nevertheless, strain (local or uniform) can be an effective means of tuning the electronic structure and transport characteristics of graphene devices. Even if the bulk gap turns out to be challenging in practice, local strain could be used as a way to mechanically pinch off current flow.
| 10.48550/arXiv.0811.4396 | A tight-binding approach to uniaxial strain in graphene | Vitor M. Pereira, A. H. Castro Neto, N. M. R. Peres | 3,856 |
10.48550_arXiv.1911.02006 | ###### Abstract
In this work we derive sum rules for orbital angular momentum(OAM) resolved electron magnetic chiral dichroism (EMCD) which enable the evaluation of the strength of spin and orbital components of the atomic magnetic moments in a crystalline sample. We also demonstrate through numerical simulations that these rules appear to be only slightly dependent from the dynamical diffraction of the electron beam in the sample, making possible their application without the need of additional dynamical diffraction calculations.
Electron microscopy; electronic OAM; electron magnetic chiral dichroism
## I Introduction
Since the work of Schattschneider and collaborators, electron magnetic circular dichroism (EMCD) has stimulated the attention of many researchers in the field of electron microscopy because of its potential of providing information about magnetic properties of materials with sub-nanometric resolution. As in the case of X ray circular dichroism (XMCD) sum rules for EMCD, independently derived by Calmels and Russ permit in principle to quantify the orbital and the spin components of the magnetic moment per atom in the sample, even if a practical application of these rules is made complicated by dynamical diffraction effects, which introduce thickness dependent factors to be evaluated by separate dynamical calculations.
In this work we derive sum rules for the orbital and spin components of the magnetic moments of the atoms in a crystalline sample for the specific case of zone axis orbital angular momentum (OAM) resolved STEM-EMCD, a technique recently proposed in Ref.: in such a proposed experiment, both the energy and the OAM spectra of the electrons, having experienced a core-loss process, are measured and differences among the \(\ell\)=\(\pm 1\) spectra are expected in the case of magnetic materials. The main result of this analysis is that, if the probe evolution along the column is dominated by channeling ( as in most zone-axis STEM experiments), the OAM resolved EMCD sum rules are strongly simplified: practically, they exhibit a weak dependency from diffraction effects, making their application more straightforward in real life experiments.
## II Theory
The experimental quantity which can be directly evaluated through the combined action of an OAM sorter and an energy spectrometer in TEM is the OAM-resolved loss function \(I_{\beta}(\ell,\Delta E)\) which can be formally defined as
\[I_{\beta}(\ell,\Delta E)=\int_{0}^{\frac{2\pi\beta}{\Delta}}I(\ell,k,\Delta E) dk=\int_{0}^{\frac{2\pi\beta}{\Delta}}Tr[\hat{\rho}_{\gamma}\hat{P}_{\epsilon E}(k)]\,dk \tag{1}\]
With these definitions, \(\left.\frac{k\lambda}{2\pi}\right.\) represents scattering angle (given below in mrad units).
Assuming valid dipolar approximation, taking the material magnetization saturated along \(z\) axis and working in paraxial conditions for the incoming electrons, it is possible to write \(I(\ell,k,\Delta E)\) appearing in Eq. as
\[I(\ell,k,\Delta E)=\sum_{\zeta,l}^{x,y,z}N_{ij}(\Delta E)\sum_{ \mathbf{a}}p_{\mathbf{a}}^{ij}(\ell,k)\] \[+\,\,M(\Delta E)\sum_{\mathbf{a}}S_{\mathbf{a}}^{z}(\ell,k) \tag{2}\]
The functions \(p_{\mathbf{a}}^{ij}(\ell,k)\) and \(S_{\mathbf{a}}^{z}(\ell,k)\) define the effects of the dynamical diffraction of the electron beam in the sample: the summations over the atomic positions \(\mathbf{a}\) are restricted over the magnetic atoms which can give rise to the energy losses \(\Delta E\) in which we are interested; we point out here that Eq. is rigorously valid only if the magnetic atom of interest in the whole sample are equivalent one with the others, otherwise a dependency on \(\mathbf{a}\) should also appear for \(N_{ij}(\Delta E)\) and \(M(\Delta E)\).
Integrating Eq. over the scattering angle \(\left.\frac{k\lambda}{2\pi}\right.\), we can write the OAM resolved loss function as
\[I_{\beta}(\ell,\Delta E)=\sum_{\mathbf{i},l}^{x,y,z}N_{ij}(\Delta E) \sum_{\mathbf{a}}F_{\mathbf{a}}^{ij}(\ell,\beta)\] \[+\,\,M(\Delta E)\sum_{\mathbf{a}}S_{\mathbf{a}}^{z}(\ell,\beta) \tag{3}\]being
\[F_{\mathbf{a}}^{i,i}(\ell,\beta)=\int_{0}^{\frac{2\pi\beta}{4}}P_{\mathbf{a}}^{i,l}(\ell,k )dk\]
and
\[S_{\mathbf{a}}^{x}(\ell,\beta)=\int_{0}^{\frac{2\pi\beta}{4}}S_{x}(\ell,k)dk\]
In conventional STEM experiments, once the beam is centered on an atomic column, the electrons tend to be laterally confined along it, in a phenomenon called channeling: practically, the probing electrons form a laterally bound state in the projected potential exerted by the atoms so only a small fraction of the beam can go away from it (i.e. can de-channel), while propagating in the sample.
As core loss scattering is a strongly localized process, the main contributions to the overall inelastic signal come from the atoms of the column on which the STEM probe is focused, while those of the neighboring columns only partially contribute to such a quantity, as only a small portion of the incoming electrons de-channel towards these atoms: therefore one should expect that all the dynamical coefficients \(P_{\mathbf{a}}^{i,i}\) and \(S_{\mathbf{a}}^{x}\) have an intense strength for \(\mathbf{a}\) on the column on which the probe is centered ( from now on this column will be taken at in the \(xy\) plane) and a rapidly decreasing value for the atoms on the neighboring and more distant ones.
We now take into account the role of the dispersion in OAM performed through appropriate devices introduced in the TEM column: practically, we focus on the features of the OAM dispersed coefficients \(P_{\mathbf{a}}^{i,i}(\ell,k)\) and their integrated counterparts \(F_{\mathbf{a}}^{i,j}(\ell,\beta)\) evaluated for \(\ell=\pm 1\), respectively as functions of \(k\) and \(\beta\). As we formally demonstrate in Appendix A, some of these quantities are zero by symmetry once evaluated for the atoms along the central column with coordinates (0,0,a,) and they are expected to be non-zero only for the neighboring columns where, in optimal channeling conditions, the probe intensity is low, making negligible their contribution to the function \(I_{\mathbf{g}}(\ell,\Delta\ell)\).
The main results derived in Appendix A can be summarized by the following formulas
\[P_{(0,0,a_{2})}^{i,i}(\ell=\pm 1,k)=0,\ \forall k,\ \text{with}\ i=x,y,z \tag{4.1}\] \[P_{(0,0,a_{2})}^{xy,i}(\ell=\pm 1,k)=0,\ \forall k\] (4.2) \[P_{(0,0,a_{2})}^{xx}(\ell=\pm 1,k)=P_{(0,0,a_{2})}^{yy}(\ell=\pm 1,k)\ \forall k \tag{4.3}\]
Such relations, together with the fact that the contributions coming from the neighboring columns are negligible, suggest that the functions \(P_{\mathbf{a}}^{xx}(\ell,k)\) and \(P_{\mathbf{a}}^{yy}(\ell,k)\) (and so \(F_{\mathbf{a}}^{xx}(\ell,\beta)\) and \(F_{\mathbf{a}}^{yy}(\ell,\beta)\)) dominate the non magnetic part of the OAM resolved loss function, computed for \(\ell=\pm 1\). These rather general predictions are here confirmed and detailed in a few specific material systems of interest. Practically, we will describe the behavior of the summed quantities \(F^{i,j}(\ell=+1,\beta)=\sum_{\mathbf{a}}F_{\mathbf{a}}^{i,j}(\ell,\beta)\) as functions of the semi-collection angle \(\beta\) for the specific case of a Cobalt sample of 30 nm, oriented along the zone axis, keeping into account the contributions of all the magnetic atoms in the sample.
Further, in the Supplementary Material (SM), we also show the results of similar calculations performed for bcc Iron and FePt samples, in order to demonstrate the quite general validity of the considerations exposed above.
The calculation of the functions \(P^{i,j}(\ell=+1,k)\) (and so of the integrated counterparts \(F_{\mathbf{a}}^{i,j}(\ell,\beta)\)) has been performed by a multislice approach through a modified version of the software MATSV2, according to the procedure outlined in Refs.: details about the convergence parameters and the unit cells adopted to perform the simulations of the electron beam propagation in the crystal are given in the SM.
In all calculations presented in this work, the incident electron beam is assumed with energy of 300 keV and different semi-convergence angles, in a range from 7 to 22 mrad, to explore different channeling conditions for the STEM probe.
Calculations for \(\ell=-1\) are presented in the SM: from those results it is simple to realize that the considerations exposed here for \(\ell=+1\) can be easily generalized to the opposite OAM.
The results of these calculations are summarized in Figures 1 and 2. Looking at Fig. 1, it is simple to observe that the behavior of the functions \(F^{xx}(\ell=+1,\beta)\) with \(\beta\) turns out to be practically unmodified by modifying the STEM probe convergence within reasonable limits: the increase in its absolute value obtained increasing the probe convergence from 7 mrad to 16 mrad can be justified by the fact that the latter beam channels in a stronger way along the Co atomic columns than the former, and so an overall increase in the inelastic signal with \(\ell=+1\) is expected. Further, looking at the inset a) of Fig.1 we point out that the ratio \(\frac{p^{xx}(\ell=+1,k)}{p^{yy}(\ell=+1,k)}\) turns out to be almost constant with the scattering angle \(k\), with very small fluctuations around one, as predicted by Eq. 4.3, considering only the atoms on the column.
The small differences we notice in the inset a) of are only due to the atoms of the neighboring columns, whose contributions to the overall inelastic signal are much smaller than the ones from the column in.
Function \(F^{xx}(\ell=1,\beta)\) computed for STEM probes with different semi-convergence angle (from 7 mrad to 16 mrad) in the case of a hpc Cobalt sample of 30 nm oriented along the direction: the behavior of this function with \(\mathbf{\beta}\) turns out to be practically unmodified by changing the STEM probe convergence; the increase in its absolute value obtained passing from the 7 mrad probe to the one with convergence of 16 mrad is justified by the fact that the latter beam channels in a stronger way along the Co atomic columns than the former. In the inset a) the ratio \(\frac{p^{xx}(\ell=+1,k)}{p^{yy}(\ell=+1,k)}\) is presented (evaluated for a probe semi-convergence of 10 mrad): it is simple to notice that this quantity only weakly oscillates around one as a function of the scattering angle; finally, in inset b) the function \(\mathbf{P}^{xx}(\ell=+1,\mathbf{k})\) is shown for different STEM convergence angles (see main legend).
In we show \(\frac{p^{\mu x}(\ell=+1,\beta)}{p^{\mu x}(\ell=+1,\beta)}\) (in log scale) as a function of the semi-collection angle \(\beta\), computed for different STEM probe convergences. This ratio points out that \(F^{xx}(\ell=+1,\beta)\) is at least an order of magnitude larger than \(F^{xx}(\ell=+1,\beta)\), and such a ratio decreases by increasing the probe convergences, i.e. the channeling capability of the incoming electron beam: this can be justified remembering that the contribution of the atoms (0,0,a\({}_{2}\)) to the function \(P^{xx}(\ell=+1,k)\) (and so to \(F^{xx}(\ell=+1,\beta)\)) is zero by symmetry as clarified by Eq. 4.1.
Finally, in the cross terms \(F^{ij}(\ell=+1,\beta)\) are shown, for a probe convergence of 10 mrad: it is evident that these functions are all at least two orders of magnitude smaller than \(F^{xx}(\ell=+1,k)\) and \(F^{yy}(\ell=+1,k)\); again this behavior is related to the fact that the atoms on the column on which the beam is focused do not give contribution to these functions, so that their value turns out to be effectively negligible with respect to the one of \(F^{il}\), with \(i\)=x,y.
These calculations permit to simplify Eq.3, through the following reasonable assumptions:
* we neglect the contributions to \(I_{\beta}(\ell=\pm 1,\Delta E)\) due both to the cross terms \(F^{ij}(\ell=\pm 1,\beta)\) (with \(i\neq j\)) and to \(F^{xx}(\ell=\pm 1,\beta)\), as they are at least two orders of magnitude smaller than \(F^{il}(\ell=\pm 1,\beta)\), with \(i\)=x,y;
* as the ratio \(\frac{p^{xx}(\ell=+1,k)}{p^{xx}(\ell=+1,k)}\) only slightly fluctuates around one, we take \(P^{yy}(\ell=+1,k)\approx P^{xx}(\ell=+1,k)\), and so \(F^{xx}(\ell=\pm 1,\beta)\approx F^{yy}(\ell=\pm 1,\beta)=F_{\beta}(\ell=\pm 1)\).
We underline that these conclusions have general validity, independently from the chosen material and its symmetry: the only requirements are orienting the crystal along an high symmetry direction and using an electron probe characterized by strong channeling properties along the chosen atomic column.
Under these approximations, we can write the inelastic signal experimentally observed at energy \(\Delta E\) and at OAM \(\ell h=\pm h\) as
\[I_{\beta}(\ell=\pm 1,\Delta E)\approx\left[N_{xx}(\Delta E)+N_{yy}( \Delta E)\right]F_{\beta}(\ell=\pm 1)\\ +M(\Delta E)S^{x}(\ell=\pm 1,\beta) \tag{5}\]
## III Derivation of sum rules for OAM-resolved EMCD
The aim of this section is to derive sum rules for OAM resolved EELS experiment, starting from the expression of \(I_{\beta}(\ell=\pm 1,\Delta E)\) obtained in Section II: practically we will derive expressions of the orbital and spin components of the atomic magnetic moment as functions of the measured OAM resolved EEL spectra, directly available experimentally.
To do this we will relate the functions \(N_{\rm{if}}(\Delta E)\) and \(M(\Delta E)\)to the mixed dynamic form factor (MDFF) exploiting the rules provided in Ref. and reported for clarity in Section 1 of the SM.
If the sample, in the selected projection, is mirror symmetric with respect to a plane perpendicular to \(xy\) plane, we have demonstrated in Ref.
\[P^{xx}(\ell,k) = P^{xx}(-\ell,k)\] \[S^{x}(-\ell,k) = -S^{x}(\ell,k)\]
Using these properties it is simple to write
\[I_{\beta}(+1,\Delta E)+I_{\beta}(-1,\Delta E)\\ =2\left[N_{xx}(\Delta E)+N_{yy}(\Delta E)\right]\,F_{\beta}(+1) \tag{6.1}\]
\[I_{\beta}(+1,\Delta E)-I_{\beta}(-1,\Delta E)\ =\ 2M(\Delta E)|S^{x}(+1,\beta)| \tag{6.2}\]
Assuming to work with \(3d\) transition metals, we provide the following definitions
\[A_{2} = \int\limits_{L_{2}}d\Delta E\left[I_{\beta}(+1,\Delta E)-I_{ \beta}(-1,\Delta E)\right] \tag{7.1}\] \[A_{3} = \int\limits_{L_{2}}d\Delta E\left[I_{\beta}(+1,\Delta E)-I_{ \beta}(-1,\Delta E)\right]\] (7.2) \[A_{23} = \int\limits_{L_{2}+L_{2}}d\Delta E\left[I_{\beta}(+1,\Delta E)+I_ {\beta}(-1,\Delta E)\right] \tag{7.3}\]
\(A_{23}\) corresponds to the integral of the sum of the \(\ell=\pm 1\) spectra over the atomic edges \(L_{2}\) and \(L_{3}\), while \(A_{2}\) (\(A_{3}\)) is the integral over the single edge \(L_{2}\) (\(L_{3}\)) of their difference. We now use these quantities to obtain sum rules for the orbital and the spin components of the atomic magnetic moment.
### Orbital sum rule
Using Eq. 6 and 7 we can write
\[\frac{A_{2}+A_{3}}{A_{23}}=\frac{\int_{L_{2}+L_{3}}M(\Delta E)d\Delta E}{\int_{L_{ 2}+L_{1}}[N_{xx}(\Delta E)+N_{yy}(\Delta E)]d\Delta E}\frac{|S^{x}(1,\beta)|}{F _{\beta}(+1)} \tag{8}\]
As demonstrated in Appendix B, we have
\[\int_{L_{2}+L_{3}}[N_{xx}(\Delta E)+N_{yy}(\Delta E)]d\Delta E\]
\[=\frac{9L^{2}G_{L}^{2}}{(2L-1)(2L+1)}[D_{xx}+D_{yy}]\]
and
\[\int_{L_{2}+L_{3}}M(\Delta E)d\Delta E=\frac{9G_{L}^{2}}{2\mu_{B}(2L+1)}m_{orb}\]
By simple substitution of these expressions in Eq.
\[m_{orb}=\frac{2\mu_{B}l^{2}[D_{xx}+D_{yy}]}{2L-1}\frac{F_{\beta}(+1)\ A_{2}+ A_{3}}{|S^{x}(1,\beta)|\ A_{23}} \tag{9}\]
### Spin sum rule
Using Eqs. 6.1 and 6.2, we have
\[\frac{1}{L-1}A_{2}=\frac{2M_{2}}{L-1}|S^{x}(+1,\beta)|\]
\[\frac{1}{L}A_{3}=\frac{2M_{3}}{L}|S^{x}(+1,\beta)|\]
Exploiting the analytical expressions of \(M_{2}\) and \(M_{3}\) we can immediately find
\[\frac{1}{L}A_{3}-\frac{1}{L-1}A_{2}=\frac{18|S^{x}(+1,\beta)|G_{L}^{2}}{(2L-1) (2L+1)}\frac{2L-1}{3\mu_{B}}m_{spin}\]
Dividing this difference for the integral over the two edges of the sum of \(I_{\beta}(+1,\Delta E)\) and \(I_{\beta}(-1,\Delta E)\) we find
\[m_{spin}=\frac{3\mu_{B}l^{2}[D_{xx}+D_{yy}]}{2L-1}\frac{F_{\beta}(+1)\frac{1}{ L}A_{3}-\frac{1}{L-1}A_{2}}{A_{23}} \tag{10}\]
By computing the ratio \(\frac{m_{orb}}{m_{spin}}\) using Eqs. 9 and 10 we find
\[\frac{m_{orb}}{m_{spin}}=\frac{2}{3}\Bigg{[}\frac{A_{2}+A_{3}}{\Bigg{[}\frac{ 1}{L}A_{3}-\frac{1}{L-1}A_{2}\Bigg{]}}\]
the ratio of the orbital and spin components of the atomic magnetic moment does not depend on dynamical effects and _ab initio_ pre-factors, and so can be evaluated directly from the experimental spectra: such relation was first reported in Ref..
## IV Dependency of sum rules from dynamical diffraction effects
Looking at Eq. 9 and Eq. 10 we notice both the presence of terms which need to be evaluated _ab initio_ and also of a quantity which only depends on the dynamical diffraction of the electron beam in the sample; in the following we define this factor as \(R_{\beta}\), given by
\[R_{\beta}=\frac{F_{\beta}(+1)}{|S^{x}(+1,\beta)|}\]
Our simulations point out that this ratio is in reality almost independent from the crystal under study, the sample thickness and, for sufficiently large \(\beta\), from the OAM spectrometer semi-collection angle. This can be understood from Figure 3, where such a quantity is evaluated as a function of \(\beta\) for different convergences of the STEM probe for a Cobalt sample of 30 nm.
Such a general behavior can be understood from the calculations we present in Appendix A: practically, we find that once \(\mathbf{a}=(0,0,\text{a}_{n})\), the following relation holds
\[|S_{\mathbf{a}}^{x}(\ell=\pm 1,k)|=2P_{\mathbf{a}}^{xx}(\ell=\pm 1,k)\]
Therefore using the definitions of \(F_{\beta}\) and \(S^{x}(+1,\beta)\) and neglecting the contributions of the atoms not on the column we have \(R_{\beta}(0,0,a_{n})\to 1/\frac{1}{2}\). This asymptotic behavior could be confirmed taking as a probe a Gaussian beam with transverse size matching the one of the Bloch 1s state of the Cobalt crystal: such a probe exhibits almost perfect channeling along the atomic column on which the beam is centered i.e. it is able to propagate in the crystal almost without any diffraction effect, similar to Bessel beams in vacuum. Because of this property, such a beam is expected not to excite the atoms of the other columns so that we should provide \(R_{\beta}\to 1/\frac{1}{2}\).
Function \(\mathbf{R}\) computed for different semi-collection angles \(\mathbf{\beta}\) and for different STEM convergences. Notice that for a small convergence (i.e. 7 mrad) this function converges to a value quite larger than 0.5, because of the strong de-channelling suffered by this electron beam (see SM for more details). The dashed line corresponds to the result we would obtain for a perfectly channeling beam (like a gaussian beam ), which should be able to propagate in the crystal without exciting atoms not on the column on which it is centered.
The fact that \(R_{\beta}\) never converges to this limiting value for conventional probes is only due to the presence of the neighboring atomic columns; to further confirm this point, we have evaluated \(R_{\beta}\) with different probe convergences and we have collected the values assumed by these ratio for \(\beta=10\) mrad, i.e. when \(R_{\beta}\) starts assuming a flat trend, as a function of the semi-collection angle: the results are shown in We notice that by passing from a semi-convergence angle of 7 mrad to one of 13 mrad, \(R_{\beta}\) gets closer to the limiting value of 0.5, while for larger angles it stays almost constant: this is due to the fact that for the latter probes the degree of de-channelling is strongly reduced, with respect to the former one; this last point is further clarified through separate multislice calculations presented in the SM.
Therefore, from our results, it emerges that, by appropriately choosing the STEM convergence and the collection angle \(\beta\) so to have \(R_{\beta}\) as much as possible close to 0.5, sum rules for OAM resolved EMCD turn out to be only weakly dependent from dynamical diffraction effects.
To quantify the error in the evaluation of \(m_{orb}\) (but also of \(m_{spin}\)) once \(R\) is approximated by 0.5, we call \(R=0.5+\delta R\), where \(\delta R\) is the deviation from the limiting value only due to the atoms on the column.
\[m_{orb}^{exp}=m_{orb}^{exact}+2m_{orb}^{exact}\delta R\]
Therefore, the experimental orbital component overestimates the real one by a fraction \(\Delta\) given by
\[\Delta=\frac{m_{orb}^{exp}-m_{orb}^{exact}}{m_{orb}^{exact}}=2\delta R\]
## V Conclusions
In this work we have derived sum rules for OAM resolved EMCD and we have demonstrated how these relations are characterized by a weak dependence from the dynamical diffraction effects, which can also be minimized by an appropriate choice of the STEM probe to be used in experiments. These results should make the application of sum rules easier from the experimental point of view and, combined with the atomic resolution provided by OAM resolved EMCD, they could give access to atomically resolved maps for the orbital and spin components of the atomic magnetic moments in crystalline samples.
| 10.48550/arXiv.1911.02006 | Sum rules in zone axis STEM-orbital angular momentum resolved electron magnetic chiral dichroism | Matteo Zanfrognini, Enzo Rotunno, Jan Rusz, Rafal E. Dunin Borkowski, Ebrahim Karimi, Stefano Frabboni, Vincenzo Grillo | 2,877 |
10.48550_arXiv.1307.7617 | ###### Abstract
The enthalpies of solution of H in Zr binary intermetallic compounds formed with Cu, Cr, Fe, Mo, Ni, Nb, Sn and V were calculated by means of density functional theory simulations and compared to that of H in \(\alpha\)-Zr. It is predicted that all Zr-rich phases (formed with Cu, Fe, Ni and Sn), and those phases formed with Nb and V, offer lower energy, more stable sites for H than \(\alpha\)-Zr. Conversely, Mo and Cr containing phases do not provide preferential solution sites for H. In all cases the most stable site for H are those that offer the highest coordination fraction of Zr atoms. Often these are four Zr tetrahedra but not always. Implications with respect to H-trapping properties of commonly observed ternary phases such as Zr(Cr,Fe)\({}_{2}\), Zr\({}_{2}\)(Fe,Ni) and Zr(Nb,Fe)\({}_{2}\) are also discussed.
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Footnote †: journal: Journal Of Nuclear Materials
## 1 Introduction
Zirconium alloys are widely used as fuel cladding and structural materials in water cooled nuclear reactors. The principal role of the cladding is to provide a physical barrier between the fuel and the coolant thereby ensuring radionuclides remain contained within the fuel pin, while protecting the nuclear fuel from a strong flow of hot, potentially corrosive water or water/steam mix.
In the past four decades a number of compositional and processing changes have been made to improve the mechanical and corrosion resistance properties of Zr alloys. Concurrently with the increased durability and reliability of the cladding (and structural elements of the reactor core) the fuel burnup has also increased, leading to a reduction in energy production costs.
One of the limiting factors for a further increase in fuel burnup is the hydrogen pick-up of Zr alloys. The presence of H in the cladding is of concern for a variety of reasons including dimensional changes, reduced ductility of the metal, and the formation of hard, brittle hydrides, which in turn may increase corrosion rates or cause failure by delayed hydride cracking. New nuclear fuel cycle regulations thattake into account hydrogen levels in the cladding are also being considered. It is, therefore, essential to develop a deeper understanding of the processes responsible for the H pick-up, in order to deliver improved reactor performance, from both economic and safety standpoints. Recent work has been carried out to investigate, from first principles, the structure, stability and mechanical properties of hydrides within the Zr metal. The current work, instead, focuses on interaction between H and the second phase particles (SPPs) formed within the Zr alloys.
Most alloying elements that are typically used in Zr alloys have very limited solubility in \(\alpha\)-Zr, and therefore tend to precipitate as intermetallic SPPs. The main exceptions are Sn, which exhibits solid solubility in \(\alpha\)-Zr at alloying concentrations of interest, and Nb, which precipitates out as \(\beta\)-(Zr,Nb) solid solution at operating temperature and below, as well as forming intermetallic SPPs in the presence of Fe. Further details are provided in Section 3. The interaction between the SPPs and H is not yet well understood. It remains unclear whether the SPPs act as traps for H, provide nucleation sites for hydrides, or whether they may act as a preferred transport path through the outer oxide layer for ingress of H into the metal.
In our previous work we employed atomic scale computer simulations, based on density functional theory (DFT), to investigate the solubility of H in Zr intermetallics of particular importance for Zircaloy-2 and Zircaloy-4 alloys. Here, the study is extended to other binary intermetallics of Zr (containing Cu, Fe, Mo, Ni, Nb, Sn and V), which are either directly relevant to fuel cladding in water cooled nuclear reactors, or have been used in model alloys to understand the role SPPs play in controlling the H pick-up.
## 2 Computational Methodology
All DFT simulations were carried out using CASTEP. The exchange and correlation functional employed was the generalized gradient approximation, as formulated by the Perdew Burke and Ernzerhof (PBE).
Ultra-soft pseudo potentials with a consistent cut-off energy of 450 eV were used throughout. A high density of \(\mathbf{k}\)-points was employed for the integration of the Brillouin zone, following the Monkhost-Pack sampling scheme: the distance between sampling points was maintained as close as possible to 0.30 nm\({}^{-1}\) and never above 0.35 nm\({}^{-1}\). In practice this means a sampling grid of \(3\times 3\times 3\) points for the largest intermetallic supercells. The fast Fourier transform grid was set to be twice as dense as that of the wavefunctions, with a finer grid for augmentation charges scaled by 2.3. Due to the metallic nature of the system, density mixing and Methfessel-Paxton cold smearing of bands were employed with a width of 0.1 eV. Testing was carried out to ensure a convergence of \(10^{-3}\) eV/atom was achieved with respect to all of the above parameters. No symmetry operations were enforced and all calculations were spin polarised, taking particular care in finding the lowest energy spin state of phases containing magnetic elements.
The energy convergence criterion for self-consistent calculations was set to \(1\times 10^{-6}\) eV. Similarly robust criteria were imposed for atomic relaxation: energy difference \(<1\times 10^{-5}\) eV, forces on individual atoms \(<0.01\) eV A\({}^{-1}\) and for constant pressure simulations stress component on cell \(<0.05\) GPa.
## 3 Crystallography of intermetallic phases
Many different types of SPPs have been reported in the literature relating to Zr alloys, often with little agreement on their chemical composition and crystal structure. In this section we report a summary of the possible SPPs that can form between Zr and each of the alloying elements under investigation (see Table 1).
Most alloying additions tend to form intermetallic phases with Zr. Exceptions are Sn, which shows large solubility in \(\alpha\)-Zr and Nb, which is a \(\beta\) phase stabiliser in Zr and as such, tends to form precipitates of \(\beta\)-(Nb,Zr) solid solutions within the \(\alpha\)-Zr matrix, as well as the intermetallic Zr(Nb,Fe)\({}_{2}\) in the presence of Fe. Limitations in the current methodology do not allow the modelling of solid solutions, therefore the HCP and BCC phases of the pure elements were modelled instead.
\begin{table}
\begin{tabular}{l l l l l l l} \hline Phase & Space & Pearson & Prototype & \(N\) & \(d\) & [H] \\ & Group & Symbol & & & (nm) & _wt_ ppm \\ \hline \(\alpha\)-Zr & \(P6_{3}/mmc\) & \(hP2\) & Mg & 150 & 1.56 & 74 \\ \(\beta\)-Zr,Nb & \(Im\bar{3}m\) & \(cI2\) & W & 128 & 1.43 & 85–86 \\ \hline ZrM\({}_{2}\)_C15_ & \(Fd\bar{3}m\) & \(cF24\) & Cu\({}_{2}\)Mg & 192 & 1.41 & 57–81 \\ ZrM\({}_{2}\)_C14_ & \(P6_{3}/mmc\) & \(hP12\) & MgZn\({}_{2}\) & 96 & 1.00 & 114–163 \\ ZrM\({}_{2}\)_C36_ & \(P6_{3}/mmc\) & \(hP24\) & MgNi\({}_{2}\) & 96 & 1.00 & 114–163 \\ Zr\({}_{3}\)Fe & \(Cmcm\) & \(oS16\) & Re\({}_{4}\)B & 96 & 1.14 & 127 \\ Zr\({}_{2}\)(Fe,Ni) & \(I4/mcm\) & \(tI12\) & Al\({}_{2}\)Cu & 96 & 0.98 & 131–132 \\ ZrNi & \(Cmcm\) & \(oS8\) & TlI & 96 & 1.00 & 140 \\ Zr\({}_{2}\)Cu & \(I4/mmm\) & \(tI6\) & MoSi\({}_{2}\) & 96 & 1.12 & 128 \\ ZrCu & \(Pm\bar{3}m\) & \(cP2\) & CsCl & 128 & 1.31 & 102 \\ Zr\({}_{3}\)Sn & \(Pm\bar{3}n\) & \(cP8\) & Cr\({}_{3}\)Si & 64 & 1.13 & 160 \\ Zr\({}_{5}\)Sn\({}_{3}\) & \(P6_{3}/mcm\) & \(hP16\) & Mn\({}_{5}\)Si\({}_{3}\) & 128 & 1.16 & 78 \\ Zr\({}_{5}\)Sn\({}_{3.5}\) & \(P6_{3}/mcm\) & \(hP17\) & & 136 & 1.19 & 74 \\ Zr\({}_{5}\)Sn\({}_{4}\) & \(P6_{3}/mcm\) & \(hP18\) & Ga\({}_{4}\)Ti\({}_{5}\) & 143 & 1.20 & 68 \\ ZrSn\({}_{2}\) & \(Fddd\) & \(oF24\) & TiSi\({}_{2}\) & 288 & 1.71 & 32 \\ \hline \end{tabular}
\end{table}
Table 1: Overview of all compounds investigated for the accommodation of H, together with the size of the largest supercell simulated in terms of the number of atoms (\(N\)) and smallest distance between the H defect and its replicas (\(d\)). For Laves phases, M = Cr, Fe, Mo, Nb, V.
The most common intermetallic phases observed are the ZrM\({}_{2}\) Laves phases, where M = Cr, Fe, Mo, Nb, V (see Table 1). Among the literature many reports exist of both the cubic C15 and the two hexagonal C14 and C36 Laves phases. Due to the nominal composition of commercial alloys, these SPPs have mostly been identified as Zr(Cr,Fe)\({}_{2}\) and Zr(Nb,Fe)\({}_{2}\), however, they are also known to form with Mo and V additions.
The Ni-Zr binary phase diagram exhibits numerous intermetallic compounds. However, Ni containing SPPs in common Zr alloys tend to be stable as Zr rich phases, typically the body-centred tetragonal Zr\({}_{2}\)Ni phase, which -- in the presence of Fe -- forms Zr\({}_{2}\)(Ni,Fe), as it is commonly found in Zircaloy-2. For completeness, here we also consider the orthorhombic ZrNi structure.
In addition to the ZrFe\({}_{2}\) and Zr\({}_{2}\)Fe phases described above, the Zr-Fe system exhibits an orthorhombic Zr\({}_{3}\)Fe phase, which has been observed only sporadically in high Fe, low Ni low Cr alloys, \(\beta\)-quenched alloys and irradiated Zircaloys. A metastable Zr\({}_{4}\)Fe phase has also been reported. Unfortunately there is insufficient crystallographic information to conduct a reliable DFT study of Zr\({}_{4}\)Fe, therefore it was not considered further. Finally the \(Fd\bar{3}m\) structure of Zr\({}_{2}\)Fe was also considered, as documented by Buschow, but the formation energy of this phase was found to be 0.20 eV greater than the body-centred Zr\({}_{2}\)Fe phase described above and is therefore discarded from further investigation.
Copper additions in Zr-base alloys, tend to be observed mainly as a tetragonal Zr\({}_{2}\)Cu phase, even though the complete phase diagram presents a large number of possible intermetallic compounds. For completeness, all reported phases containing \(\geq 50\%\) Zr, were investigated. Table 2 contains a list of the phases considered, with those that are widely regarded as stable being marked by an asterisk. Enthalpies of formation from standard state (\(E_{f}^{\circ}\)) and from solid solution (\(E_{f}^{sol}\)) are also reported in Table 2, following reaction 1 and 2 respectively (Enthalpies of formation for the other possible SPP phases were reported previously).
\[n\mathrm{Zr}+\mathrm{Cu} \rightarrow\mathrm{Zr}_{n}\mathrm{Cu} \tag{1}\] \[x\mathrm{Zr}_{149}\mathrm{Cu} \rightarrow x\mathrm{Zr}_{n}\mathrm{Cu}+(149-n)x\mathrm{Zr} \tag{2}\]
The Zr\({}_{3}\)Cu phase was found to be thermodynamically unstable (positive formation energy). The tetragonal \(I4/mmm\) structure of Zr\({}_{2}\)Cu was found to possess a substantially more stable formation energy compared to the cubic \(Fd\bar{3}m\) structure. On the other hand, there is little difference in the formation energies calculated for the ZrCu compounds, suggesting that all three phases are likely to form in the Cu-Zr alloy. However, ZrCu is a high temperature phase, which decomposes in a eutectoid reaction at temperatures below 715 \({}^{\circ}\)C. DFT does not incorporate the effect of vibrational energy associated with temperature, therefore DFT results alone are generally not indicative of the stability of high temperature phases. This is especially true when the differences in formation energy are very small, as in the ZrCu phases. Experimental investigation suggests that the CsCl structure is the most stable at high temperatures. Following the above results,the body-centred Zr\({}_{2}\)Cu and the CsCl structure of ZrCu were studied.
Finally, we examine Zr-Sn intermetallic compounds. Although Sn is highly soluble in \(\alpha\)-Zr, there have been reports of Zr-Sn intermetallic SPPs in irradiated Zircaloy samples, suggesting that they form due to radiation enhanced diffusion of the alloying elements. Recent work confirms this from a thermodynamic viewpoint. However, redeposition of Sn during TEM sample preparation has also been suggested as a possible cause for the formation of Zr-Sn intermetallics.
Even ignoring the effects of irradiation and ternary alloying elements on the stability of Sn-SPPs, the equilibrium binary Sn-Zr system is rather complex. At levels of Sn greater than the \(\alpha\)-Zr solid solution regime, the stable phases are Zr\({}_{4}\)Sn, Zr\({}_{5}\)Sn\({}_{3+x}\) and ZrSn\({}_{2}\). With the exception of the latter, the other phases exhibit a high degree of disorder. Extensive experimental work by Kwon and Corbett subsequently modelled by Baykov _et al._ shows that at equilibrium Zr\({}_{5}\)Sn\({}_{3+x}\) has a large concentration of self-interstitial Sn, up to a stoichiometry of Zr\({}_{5}\)Sn\({}_{4}\). These studies also showed that Zr\({}_{4}\)Sn is a Zr-substitutional structure, which is derived from Zr\({}_{3}\)Sn with one fifth of the Sn sites occupied by Zr atoms, i.e. Zr\({}_{3}\)(Sn\({}_{0.8}\)Zr\({}_{0.2}\)). Due to limitations in our computational methodology, such complex phases cannot be modelled reliably, instead, their parent phases were simulated: the cubic Zr\({}_{3}\)Sn structure, face centered orthorhombic ZrSn\({}_{2}\), and the hexagonal Zr\({}_{5}\)Sn\({}_{3}\), as well as two of its interstitial derivatives, an ordered form of Zr\({}_{5}\)Sn\({}_{3.5}\) in which only the \(2b\) Wyckoff sites were occupied, and Zr\({}_{5}\)Sn\({}_{4}\), in which all the interstitial sites are occupied.
\begin{table}
\begin{tabular}{l l l l l l l} \hline Formula & Space & Pearson & Prototype & \(E_{f}^{\circ}\) & \(E_{f}^{sol}\) & Ref. \\ unit & group & symbol & structure & (eV) & (eV) & \\ \hline Zr\({}_{3}\)Cu & \(P4/mmm\) & \(tP4\) & CuInPt\({}_{2}\) & 0.29 & 0.00 & \\ Zr\({}_{2}\)Cu\({}^{*}\) & \(I4/mmm\) & \(tI6\) & MoSi\({}_{2}\) & -0.41 & -0.70 & \\ Zr\({}_{2}\)Cu & \(Fd\bar{3}m\) & \(cF24\) & AuBe\({}_{5}\) & -0.20 & -0.48 & \\ ZrCu\({}^{*}\) & \(Pm\bar{3}m\) & \(cP2\) & CsCl & -0.21 & -0.50 & \\ ZrCu & \(P121/m1\) & \(mP4\) & NiTi & -0.25 & -0.54 & \\ ZrCu & \(Cm\) & \(mS16\) & & -0.26 & -0.55 & \\ \hline \end{tabular}
\end{table}
Table 2: List of Zr-Cu intermetallic phases modelled in the current work. The standard enthalpy of formation \(E_{f}^{\circ}\) was calculated from \(\alpha\)-Zr and FCC-Cu metals. Whilst \(E_{f}^{sol}\),was calculated from an isolated substitutional Cu atom in a 150 atoms cell of \(\alpha\)-Zr. The quoted enthalpies are per formula unit.
## 4 Results and Discussion
### Choice of Supercell size
When simulating point defects in solids using the supercell approach, it is possible to do so under constant pressure or constant volume conditions. In the former, which is more computationally expensive and replicates alloying conditions, both the crystal's lattice parameters and atomic positions within the supercells are subject to energy minimisation; as a result the size and shape of the supercell is allowed to react to any internal stress resulting from the incorporation of a defect. In the latter case, constraints are applied to the lattice so that the cell's shape and volume are fixed during internal energy minimisation. This method most closely replicates the dilute case, however, if the supercell size is insufficiently large, defect-defect interactions will cause a sufficient stress build-up on the cell, such that a significant non-physical contribution to the calculated energies is added, which we term the constrained cell energy contribution.
The difference in energy between the two methods is a good measure of how effectively the supercell describes isolated defects in the bulk material. A convergence analysis was carried out with respect to the supercell size for the H interstitial defects in \(\alpha\)-Zr, see It is clear that an accuracy of the order of \(10^{-2}\) eV/atom is achieved with a defect-defect separation of 0.8 nm, and \(10^{-3}\) eV/atom at 1.56 nm (corresponding to a \(5\times 5\times 3\) supercell containing 150 Zr atoms). Note that the tetrahedral interstitial has a larger defect volume, therefore it is affected by the size of the supercell to a greater extent.
Starting from a fully relaxed unit cell, two supercells were generated: first a smaller one, containing \(\sim 50\) atoms, which was used for simulations of H defects in each of the crystallographically unique interstitial sites; then a larger one in which the lowest energy configurations of the defects, as identified from the smaller cell, were replicated for better accuracy. The larger supercells were chosen to have no supercell axis smaller than 1 nm prior to relaxation. Before adding the defects to the supercells, these were relaxed again to avoid any aliasing errors (misalignment of atoms within the supercell where the crystal boundaries were situated in the original unit cell) and errors arising from the use of non-identical sampling grids. This process was repeated for each intermetallic phase investigated.
### Hydrogen Accommodation
As previously reported, H was found to preferentially occupy the tetrahedral site over the octahedral one in \(\alpha\)-Zr, and all other interstitial sites were found to be metastable. The enthalpy of solution of H in the tetrahedral site is -0.464 eV. These findings are in agreement with both experimental results of Khoda-Bakhsh and Ross and the DFT results of Domain _et al._
Interestingly it was found that the lowest energy H solution site for nearly all intermetallic phases had a tetrahedral configuration. The only exceptions were ZrCu, ZrNi and Zr\({}_{5}\)Sn\({}_{3+x}\), which are discussed in greater detail below. Furthermore, it was found systematically that sites with the largest fraction ofneighbouring Zr atoms offered the lowest energy for H accommodation. Thus, for most of the intermetallic phases, the lowest energy site is one consisting of 4 neighbouring Zr atoms. For those compounds where, due to stoichiometry, no such sites are present, for example the ZrM\({}_{2}\) phases, H was found to preferentially occupy tetrahedral sites with 2-3 Zr atoms and only 1-2 M atoms, irrespective of the concentration of the M-species. A recent study reports an analogous behaviour for Laves phases. In the case of ZrCu and ZrNi, the most stable interstices are octahedral with coordination of 4 Zr and 2 Cu/Ni atoms. These sites offer a greater Zr-neighbour fraction compared to the available tetrahedral sites, which have 2 Zr and 2 Cu/Ni neighbours.
A summary of solution enthalpies for H in the most favourable interstitial site, in each of the intermetallic phases, is presented in Table 3.
\begin{table}
\begin{tabular}{l c c c c c} \hline \hline Phase & \(E_{sol}\)(H) & \(\Delta E_{sol}^{\alpha\text{-Zr}}\)(H) & error & \multicolumn{2}{c}{H interstitial} \\ & & & Position & Type & Zr/X \\ \hline \(\alpha\)-Zr & -0.46 & & & 4\(f\) & tet & 4/– \\ \(\beta\)-Zr & -0.62 & **-0.16** & & 12\(d\) & tet & 4/– \\ \(\beta\)-Nb & -0.46 & **0.00** & & 12\(d\) & tet & –/4 \\ \hline ZrFe\({}_{2}\)† & -0.03 & 0.50 & \(\pm 0.00\) & 96\(g\)/6\(h\) & tet & 2/2 \\ ZrMo\({}_{2}\) & -0.24 & 0.22 & \(\pm 0.06\) & 96\(g\)/6\(h\) & tet & 2/2 \\ ZrCr\({}_{2}\)† & -0.31 & 0.15 & \(\pm 0.03\) & 96\(g\)/6\(h\) & tet & 2/2 \\ ZrV\({}_{2}\) & -0.73 & **-0.26** & \(\pm 0.01\) & 96\(g\)/6\(h\) & tet & 2/2 \\ ZrNb\({}_{2}\) & -0.81 & **-0.35** & \(\pm 0.02\) & 96\(g\)/6\(h\) & tet & 2/2 \\ \hline ZrSn\({}_{2}\) & 0.28 & 0.75 & & 32\(h\) & tet & 2/2 \\ ZrCu & -0.28 & 0.19 & & 3\(d\) & oct & 4/2 \\ ZrNi & -0.37 & 0.09 & & 4\(c\) & oct & 4/2 \\ Zr\({}_{5}\)Sn\({}_{3+x}^{**}\) & -0.38 & **0.08** & \(\pm 0.14\) & 2\(a\)/2\(b\) & tri/oct & 3–6/– \\ Zr\({}_{2}\)Fe† & -0.45 & **-0.01** & & 16\(l\) & tet & 4/– \\ Zr\({}_{2}\)Cu & -0.52 & **-0.06** & & 4\(d\) & tet & 4/– \\ Zr\({}_{2}\)Ni † & -0.67 & **-0.20** & & 16\(l\) & tet & 4/– \\ Zr\({}_{3}\)Sn & -0.64 & **-0.18** & & 6\(d\) & tet & 4/– \\ Zr\({}_{3}\)Fe & -0.74 & **-0.27** & & 8\(f\) & tet & 4/– \\ \hline \end{tabular}
\end{table}
Table 3: Enthalpy of solution for an interstitial H, in the most stable site, for each of the intermetallic phases investigated. \(\Delta E_{sol}^{\alpha\text{-Zr}}\)(H) is the difference in enthalpy of solution of H in the given intermetallic, compared to the tetrahedral site in \(\alpha\)-Zr. The preferred interstitial site for H is indicated in Wyckoff notation, and it is described in terms of its geometry and coordination number with Zr and non-Zr elements X. All values are expressed in units of eV. \({}^{**}x=0\)–1. †From previous study.
Similarly, the three models of the Zr\({}_{5}\)Sn\({}_{3+x}\) phase were reported as one. The exact values of \(E_{sol}\)(H) were \(-0.56\), \(-0.34\), \(0.23\) eV for \(x=0.0,0.5,1.0\) respectively. For the case of Zr\({}_{5}\)Sn\({}_{4}\), the lowest energy site was the only unoccupied \(2b\) Wyckoff site. It is reasonable to expect a large number of unoccupied Sn self-interstitial sites in this phase, therefore the defect energy was calculated as an interstitial defect into a phase with one unoccupied Sn self-interstitial site, rather than the substitutional defect H\({}_{\rm Sn}\) in the fully occupied Zr\({}_{5}\)Sn\({}_{4}\) structure. Zr\({}_{5}\)Sn\({}_{3+x}\), exhibits two preferred sites: the \(2a\) site between \(3\) Zr atoms, and, directly above and below it, the \(2b\) site between \(6\) Zr atoms. The relative preference for H of one site over the other changes as a function of Sn content. At a content of three Sn atoms per formula unit, the trigonal \(2a\) site is preferred (\(-0.56\) eV against \(-0.39\) eV of the octahedral site). As the Sn content increase to \(3.5\), half of the \(2b\) sites are occupied by the excess Sn. The presence of a Sn atom in the \(2b\) site, causes a reduction in space in the neighbouring \(2a\) sites, while only marginally affecting the configuration of other (unoccupied) \(2b\) sites in the cell. This is reflected in the solution of H in the two sites: the \(2a\) site becomes significantly less favourable (\(-0.27\) eV), while the \(2b\) provides a similar solution enthalpy as in the previous case (\(-0.34\) eV). In the case of Zr\({}_{5}\)Sn\({}_{4}\) all of the \(2b\) sites are occupied, consequently the \(2a\) sites are compressed by two Sn atoms, one above and one below, reducing the volume available for accommodation of H even further, and the enthalpy associated with accommodating an H atom in that site becomes positive and large (\(2.06\) eV).
The current work shows that \(\beta\)-Zr accommodates H more readily than \(\alpha\)-Zr (in agreement with experimental data), and that \(\beta\)-Nb exhibits the same value of \(E_{sol}\)(H) as \(\alpha\)-Zr. This suggests that, if the \(\beta\)-(Nb,Zr) solid solution found in binary Zr-Nb alloys behaves similarly to its two end members, those alloys do not contain any strong sinks for H. Nevertheless, the formation of metastable ZrNb\({}_{2}\) phases may affect this, as discussed below.
Regarding the ZrM\({}_{2}\) Laves phases, the solution enthalpy of H generally decreases with increasing number of \(d\) electrons in the transition metal M: from highest to lowest affinity Nb, V, Cr, Mo and Fe. The same trend has been observed with respect to H solution capacity. Whilst H does not prefer to dissolve in the Laves phases containing the latter three elements (i.e. Cr, Mo, and Fe) compared to \(\alpha\)-Zr, the intermetallics formed with either Nb or V offer favourable sites for the accommodation of H. This suggests that if these binary SPPs are present in the cladding, H will likely segregate to them, which may deplete the H content in the zirconium metal. The beneficial effect of the H sinks may, however, be limited to the initial stages of the fuel cycle. At higher burnups, the intermetallic particles are likely to dissolve, amorphise or oxidise, thereby releasing any stored H.
In addition to ZrNb\({}_{2}\) and ZrV\({}_{2}\), all Zr-rich phases provided lower \(E_{sol}\)(H) values compared to \(\alpha\)-Zr. Furthermore, for each element where more than one stoichiometric phase is present (Cu, Fe, Ni and Sn), those with the largest Zr/M ratio provided the lowest solution enthalpy for H (see Figure 2). Zr is known to exhibit higher affinity for H compared to Cu, Fe and Ni (see the extremes of Figure 2), therefore a decrease in enthalpy of solution with increasing Zr content is expected. However, from a volumetric standpoint, intermetallic phases have a lower packing fraction compared to the pure metals, offering a larger number of interstitial sites with varying amounts of space. For this reason intermetallic phases are expected to exhibit lower defect volumes (and associated strain fields) when accommodating an H interstice. As a result of the two competing processes -- chemical bonding and volumetric effects -- the lowest solution enthalpies are found, as mentioned above, for Zr-rich intermetallic phases that provide interstitial sites with 4-fold Zr-coordination.
Unlike other alloying additions, Fe forms a wide range of intermetallic compounds, and their relative stability is greatly affected by other alloying elements: Zr\({}_{2}\)(Fe,Ni) SPPs are commonly observed in the absence of Cr additions, while in Cr-containing alloys, Zr(Cr,Fe)\({}_{2}\) Laves phases become the dominant SPPs. In the presence of Nb, hexagonal Zr(Nb,Fe)\({}_{2}\) Laves phases and cubic (Zr,Nb)\({}_{2}\)Fe phases have been reported. Similarly, the few records of Sn-Zr SPPs also report some Fe solubility into these particles. This suggests that the addition of Fe does not influence which intermetallic phases form, rather it will go into solution in all or most of the SPPs present. Assuming that a ternary phase behaves similarly to its binary end members, solution of Fe in ZrM\({}_{2}\) Laves phases and Zr\({}_{2}\)X phases, is expected to reduce their affinity to H, since ZrFe\({}_{2}\) and Zr\({}_{2}\)Fe exhibit less favourable \(\Delta E_{sol}\)(H) compared to all other phases with the same stoichiometry and structure. The opposite can be said for the Zr\({}_{3}\)Fe-type SPPs.
Owing to the fact that both ZrCr\({}_{2}\) and ZrFe\({}_{2}\) have unfavourable \(\Delta E_{sol}\)(H) values, it is predicted that the ternary Zr(Cr,Fe)\({}_{2}\) phase, found predominantly in Zircaloy-4, does not getter H from the surrounding \(\alpha\)-Zr. As for the Zr\({}_{2}\)(Fe,Ni), which is the predominant SPP in Zircaloy-2, both of its binary end members offer favourable solution enthalpies for H. Nonetheless, the difference in affinity to H with respect to \(\alpha\)-Zr is rather small (on the order of 0.1 eV), and is expected to diminish with increasing Fe content. Such small differences in energy may easily be overcome by thermal and entropic effects, consequently Zr\({}_{2}\)(Fe,Ni) are not expected to be strong sinks for H in solution.
Whilst it is possible to speculate on the behaviour of ternary phases where both binary end members have either positive or negative \(\Delta E_{sol}\)(H) values, it is much harder to predict the H affinity of other ternary SPPs such as Zr(Nb,Fe)\({}_{2}\) found in ZIRLO alloys. It is possible, however, to expect that the behaviour of such SPPs is strongly correlated to their chemical composition, and more specifically the Nb/Fe ratio of the intermetallic particle. If \(\frac{\rm Nb}{\rm Fe}>>1\) then the SPPs may act as H sinks, whilst with a composition of \(\frac{\rm Nb}{\rm Fe}<<1\) they are not likely to accommodate any H.
Under irradiation, Fe has been reported to diffuse out of the SPPs faster than other elements. In the case of Laves phases, and especially Nb-containing Laves phases, the current work suggests that this would increase the affinity of the residual SPP for H. However, concomitantly to the dissolution of Fe, the SPPs are reported to undergo amorphisation and at present it is impossible to predict how this will affect the interaction between H and the SPP.
## 5 Conclusions
DFT simulations have been employed to calculate the enthalpy of solution of H in pure Zr, Nb and in intermetallic phases of Zr. Pure \(\beta\)-Zr exhibits more favourable solution enthalpy for H relative to \(\alpha\)-Zr, however the difference is predicted to diminish in the presence of Nb. Regarding the Laves SPPs, the presence of Nb or V is predicted to increase the affinity with H, whilst the presence of Cr, Mo and Fe will reduce it. Cu, Ni and Sn additions may form a number of binary intermetallic phases, but tend to stabilise as Zr-rich phases. All Zr-rich phases, namely Zr\({}_{3}\)Fe, Zr\({}_{2}\)Ni, Zr\({}_{2}\)Cu and Zr\({}_{3}\)Sn, provide lower energy sites for H accommodation, compared to \(\alpha\)-Zr, suggesting that their presence in the alloy could provide sinks for H.
With regards to the more commonly observed ternary SPPs, it is predicted that Zr\({}_{2}\)(Fe,Ni) (found mainly in Zircaloy-2) and \(\beta\)-(Zr,Nb) precipitates (present in all Nb containing alloys) exhibit an affinity to H similar to that of \(\alpha\)-Zr, and are therefore not expected to strongly getter H from their surroundings. Zr(Cr,Fe)\({}_{2}\) SPPs found in Zircaloy-4 are predicted to have very unfavourable solution enthalpies for H and therefore not to accommodate any H. The affinity to H of the Zr(Nb,Fe)\({}_{2}\) SPPs, present in ZIRLO alloys, is expected to vary strongly with Nb/Fe ratio: high Nb content SPPs are expected to trap H, whilst high Fe content SPPs will reject it.
| 10.48550/arXiv.1307.7617 | Hydrogen solubility in zirconium intermetallic second phase particles | P. A. Burr, S. T. Murphy, S. C. Lumley, M. R. Wenman, R. W. Grimes | 2,642 |
10.48550_arXiv.1111.4667 | ###### Abstract
We present a computational study for the equilibrium shape of gold nanoparticles. By linking extensive quantum-mechanical calculations, based on Density-Functional Theory (DFT) to Wulff construction, we predict equilibrium shapes that are in good agreement with experimental observations. We discuss the effect of the interactions between a nanoparticle and the encapsulating material on the equilibrium shape. As an example, we calculate adsorption of CO on several different Au(\(hkl\)) and use the results to explain the experimentally observed shape change of Au nanoparticles.
Gold; Density Functional calculations; Nanoparticles; Surface Science; Adsorption; Nanomaterials pacs: 81.10.Aj, 82.65.+r, 68.43.Fg Bulk gold is the noblest of all metals, as demonstrated by delicate gold jewels manufactured several millennia ago which are found intact in excavations. On the other hand, catalysts that include oxide-supported gold nanoparticles were found to efficiently oxidize CO at room temperature; Au is by far the best such catalyst. Among the key factors that determine the efficiency of Au catalysts is the shape of Au nanoparticles, in particular the 5- and 6-fold coordinated atoms at its corners.
The shape of Au nanoparticles has a key role in every aspect of their functionality, from sensing and biolabeling applications to plasmonics and photonics. In optoelectronics, quantum leaps between electronic states transform light into electricity and vice versa. The probability of such a transition depends on the density of states and the dipole matrix elements according to Fermi's golden rule. For a given size, both wavefunctions and energies depend critically on the nanoparticle shape. For example, the lowest excitation energy for a cubic nanoparticle is 10% higher than that of a spherical nanoparticle of the same volume.
Gold nanoparticles are often found in their equilibrium shape.
\[\sum_{hkl}A_{hkl}\gamma_{hkl}, \tag{1}\]
\(A_{hkl}\) is the total area of faces parallel to (\(hkl\)) plane of the crystal and \(\gamma_{hkl}\) is the surface tension, i.e. the energy required to create a surface of unit area that is parallel to the (\(hkl\)) plane of the crystal. In order to predict equilibrium shape, one needs calculations of surface tensions for many different (\(hkl\)). Several such calculations exist in the literature either based on empirical potentials or limited to Miller indexes of 0 and, or using quantum-mechanics for low-index faces and empirical models for higher indexes. An accurate and systematic calculation of all high-index Au surfaces is missing.
The equilibrium shape is often found to change upon exposure to some interacting environment. As Au nanoparticles are used in CO oxidation catalysis, CO gas is the ideal candidate to test this idea. Changes to shapes of higher sphericity upon exposure to CO gas have been observed both experimentally and theoretically. The interface tension of a metal in equilibrium with a gas is found to depend on the surface tension, the adsorption energy and the coverage of adsorbates Eq.. In order to predict the equilibrium shape in an interacting environment using the Wulff construction method, it is necessary to have a systematic calculation of adsorption energies for all relevant (\(hkl\)) surfaces.
The Wulff construction has been used to predict equilibrium shapes in a variety of systems. Wulff polyhedra are often employed in observations and models for nanomaterials including Cu catalysts, or semiconductors. In the past decade, Wulff shapes employing surface tensions from first-principles calculations were used for the successful prediction of the shape of nanoparticles, including interactions with their environment. In the context of ammonia-synthesis catalysis, an _ab initio_ determination of a nanoparticle shape was used as a first step in the creation of a virtual nano-catalyst. In that work, the Wulff polyhedron was filled with atoms in order to create a realistic nanoparticle. The advantage of this method was that it allowed for detailed analysis of the atomic positions, making it possible to calculate structural quantities such as the number of active sites. This virtual nano-catalyst was used in other similar reactions, such as ammonia decomposition. Here, we expand this methodology by including all possible (\(hkl\)) orientations. Moreover, wetake into account changes in shape that may be induced by interactions between the nanoparticle and its environment. We apply our method to supported gold nanoparticles, a system of high technological importance.
The paper is organised as follows: in Section I, we briefly review Wulff's theory regarding the equilibrium shape. In Section II, we present a systematic calculation of the surface tension for every Au(\(hkl\)) with Miller indexes up to 4. In section III, we use these surface tensions to create atomistic models for Au nanoparticles of sizes up to 70 nm, and analyse their structural properties, such as the concentration of active sites. In Section IV, we generalize our methodology for nanoparticles in interacting environment. We provide a simple formula that relates the interface tension to the surface tension and adsorption energy. We calculate the minimum adsorption energy of CO on every Au surface with Miller indexes up to 3, and use these results to calculate the change in equilibrium shape of Au particles upon exposure to CO gas. We summarize our results in Section V.
## I The Wulff construction
The concept of "equilibrium shape" was postulated by Gibbs in the late 19th century. Under thermodynamic equilibrium, a given quantity of matter will attain a shape that minimizes the total surface energy of the system. More than a century ago, mineralogist G. Wulff proposed that the shape that minimizes Eq. is such that the distance of each face from the center is proportional to the surface tension of the respective (\(hkl\)) surface:
\[d_{hkl}\sim\gamma_{hkl}. \tag{2}\]
One begins the Wulff construction by drawing up a plane (for example,) at a distance \(d_{111}\) from the origin followed by planes parallel to (\(hkl\)) at distances \(d_{hkl}=d_{111}\gamma_{hkl}/\gamma_{111}\). The equilibrium shape will be the polyhedron enclosed by these planes, having thus the following properties:
* The shape depends on ratios between surface tensions, and not their absolute values.
* (\(hkl\)) planes with high surface tension (usually high-indexed ones) will be drawn at greater distances and are therefore less likely to appear in the equilibrium shape.
* Being steeper, high-index faces are usually hidden behind low-index ones, and tend to occupy smaller areas in the equilibrium shape even if \(\gamma_{hkl}\) is low.
* The extra energy associated with the formation of edges between two surfaces is not taken into account.
* The Wulff polyhedron belongs to the same point group as the crystal structure of the material.
In addition to Wulff construction, there exist other methods for the study of nanoparticles. Advances in computers allow for the direct simulation of nanoparticles of large sizes using empirical potentials, as done for example by McKenna. In that work, a large number of different shapes are tested to find the lowest-energy one. The Wulff construction is complimentary to that method. Here, we use Wulff construction coupled to first-principles calculations of surface tensions. This method offers a systematic, easy-to-follow recipe for the construction of atomistic models of nanoparticles.
## II Surface tension of gold surfaces
As we are interested in relatively large Au nanoparticles, we limit our study to nanoparticles where Au atoms far from the surfaces are in the ideal fcc lattice. This is observed in simulations of large clusters, although small gold clusters may have structures very different from fcc. We begin by calculating the surface tension, \(\gamma_{hkl}\), of Au by simulations of periodic (\(hkl\)) slabs using Density-Functional Theory (DFT). We use the open-source Dacapo/ASE suite ([https://wiki.fysik.dtu.dk](https://wiki.fysik.dtu.dk)). We use a plane wave basis with 340 eV cut-off. The core electrons are treated with Vanderbilt non-local ultrasoft pseudopotentials. The Brillouin zone of the-(1\(\times\)1) surface is modelled by a (10\(\times\)10\(\times\)1) Monkhorst-Pack grid of \(\vec{k}\)-points; number of \(\vec{k}\)-points in other surfaces is calculated in proportionality to the cell. We use the generalized gradient approximation (GGA) Perdew-Wang exchange-correlation functional PW91 for the clean surfaces and the revised Perdew-Burke-Ernzerhof functional RPBE for the surfaces covered with CO since this one gives better adsorption energies. For each set of calculations, we use the theoretical lattice constant which is found to be 4.22 A for RPBE and 4.18 A for PW91, very close to the experimental value of 4.08 A.
(Color online) Typical calculated Au nanoparticles for various sizes, \(d\): (a) \(d=12.1\) nm (b) \(d=27.2\) nm (c) \(d\rightarrow\infty\). In (a) and (b), step and kink atoms are shown in darker color. In (c), different colors correspond to different kinds of surfaces. (c) was created using Wulffman.
We model all (\(hkl\)) surfaces of fcc Au with indexes up to 4. Even for CO-covered nanoparticles, no (\(4kl\)) surfaces are observed in the Wulff construction; for this reason we do not consider (\(5kl\)) surfaces in this work. Atoms in the top two layers from each side are allowed to relax, while subsequent slabs are separated by 12 A of vacuum. Slab thickness is chosen independently for each (\(hkl\)) slab until the surface tension converges within 0.01 J/m\({}^{2}\).
\[E_{slab}=NE_{bulk}+2A\gamma_{hkl}, \tag{3}\]
The results are summarized in Table 1. Interestingly, the ratio of surface energies of different cells is very close to the ratio of the areal density of cleaved bonds. This is another example of the unique chemistry of Au: Au atoms have a closed \(d\)-shell and have the least preference for directional bonds in the entire periodic table. The calculated absolute value for \(\gamma_{111}\) is 0.69 J/m\({}^{2}\), very close to 0.64 J/m\({}^{2}\) reported by Wen and Zhang and within the same order of magnitude as the values reported by state-of-the-art relativistic all-electron calculations. The nanoparticle shape depends only on ratios between surface energies. As shown in Table 1, our results for the ratios between surface tensions agree with more detailed calculations within 5% or less.
## III Au Nanoparticles in Non-Interacting Environment
The Wulff construction for Au is shown in Fig. 1(c). It contains 144 vertices and 86 faces of 5 different kinds:,,, and in order of total area.
To construct atomistic models for nanoparticles, we start from a large fcc crystal. As has the lowest surface tension, we begin by choosing the number of layers. This determines the distance of plane from the center of the nanoparticle, \(d_{111}\), and, consequently, the nanoparticle size. For other faces, we use Eq., with the calculated values of \(\gamma_{hkl}\) and cut the crystal at the correct distances \(d_{hkl}\). We calculate the equation of the plane defined by every set of three surface atoms, and make sure that only faces consistent with the Wulff construction appear on the nanoparticle. We consider about 30000 different nanoparticles with diameters ranging from 1.7 nm to more than 100 nm.
At small sizes, some faces might not be large enough to accommodate a single atom, let alone a unit cell of this. Very small nanoparticles expose only and faces; in particular, our simulated 459-atom nanoparticle is identical to the one found from simulations and X-ray experiments. In all cases, the shape resembles a truncated octahedron consisting mainly of and faces, with their edges decorated by several (\(hkl\)) faces with indexes up to 3. For diameters up to 16.3 nm, we find only and faces; as the nanoparticle grows in size, different (\(hkl\)) orientations start to appear. The thermodynamic limit, shown in 1(c) is reached at diameters of the order of 100 nm.
Geometrical features or typical nanoparticles are shown in Table 2. The area of nanoparticles is calculated analytically using the coordinates of vertices; their volume is obtained by numerical integration. By fitting over a hundred particles of different diameters, we provide scaling relations of various properties with total number of atoms, in accordance with atom-counting models for nanoparticles.
## IV Au Nanoparticles in Interacting Environment
The equilibrium shape of nanoparticles that interact with their environment can be found by means of a Wulff construction based on interfacial tensions, \(\gamma_{hkl}^{int}\), between Au and its environment instead of surface tensions, \(\gamma_{hkl}\).
\begin{table}
\begin{tabular}{c c c c c} \hline \hline & This work & calc\({}^{14}\) & calc\({}^{16}\) & calc\({}^{15}\) & \(E_{ads}\) (eV) \\ \hline \(\gamma_{100}/\gamma_{111}\) & 1.23 & 1.11 & 1.15 & 1.27 & -0.25 (b) \\ \(\gamma_{110}/\gamma_{111}\) & 1.29 & 1.24 & 1.22 & 1.33 & -0.36 (b) \\ \(\gamma_{7120}/\gamma_{111}\) & 1.33 & 1.31 & 1.29 & -0.49 (t) \\ \(\gamma_{711}/\gamma_{111}\) & 1.17 & 1.19 & 1.18 & -0.34 (t) \\ \(\gamma_{7221}/\gamma_{111}\) & 1.14 & 1.16 & 1.15 & -0.35 (t) \\ \(\gamma_{710}/\gamma_{111}\) & 1.31 & 1.28 & 1.28 & -0.51 (t) \\ \(\gamma_{7311}/\gamma_{111}\) & 1.26 & 1.24 & 1.22 & -0.32 (b) \\ \(\gamma_{7320}/\gamma_{111}\) & 1.36 & 1.30 & 1.28 & -0.48 (t) \\ \(\gamma_{7321}/\gamma_{111}\) & 1.25 & 1.26 & 1.23 & -0.50 (t) \\ \(\gamma_{7322}/\gamma_{111}\) & 1.11 & 1.13 & 1.12 & -0.34 (t) \\ \(\gamma_{7332}/\gamma_{111}\) & 1.18 & 1.21 & 1.19 & -0.34 (t) \\ \(\gamma_{7332}/\gamma_{111}\) & 1.07 & 1.11 & 1.11 & -0.35 (t) \\ \(\gamma_{7410}/\gamma_{111}\) & 1.32 & 1.25 & 1.26 & -0.49\({}^{f}\) \\ \(\gamma_{7411}/\gamma_{111}\) & 1.27 & 1.23 & 1.22 & -0.34\({}^{f}\) \\ \(\gamma_{7421}/\gamma_{111}\) & 1.32 & 1.29 & 1.26 & -0.49\({}^{f}\) \\ \(\gamma_{7430}/\gamma_{111}\) & 1.34 & 1.29 & 1.27 & -0.49\({}^{f}\) \\ \(\gamma_{7431}/\gamma_{111}\) & 1.27 & 1.27 & 1.25 & -0.49\({}^{f}\) \\ \(\gamma_{7432}/\gamma_{111}\) & 1.19 & 1.20 & 1.18 & -0.49\({}^{f}\) \\ \(\gamma_{7433}/\gamma_{111}\) & 1.09 & 1.09 & 1.09 & -0.34\({}^{f}\) \\ \(\gamma_{7441}/\gamma_{111}\) & 1.22 & 1.22 & 1.21 & -0.34\({}^{f}\) \\ \(\gamma_{7443}/\gamma_{111}\) & 1.06 & 1.09 & 1.08 & -0.34\({}^{f}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Ratios of surface tensions of Au in comparison to other calculations. The last column contains the adsorption energy of CO at low coverage on the same (\(hkl\)) surface, and the adsorption geometry. b=bridge adsorption, t=on-top adsorption. \({}^{f}\): energies calculated from the linear fit shown in
compared to isolated Au surface and isolated encapsulating material. Eq. includes implicitly the effects of adsorbate-adsorbate interactions, as both the adsorption energy and the equilibrium coverage depend on such interactions.
To prove Eq., we use the definitions of \(\gamma^{int}\) and \(E_{ads}\) for a slab of metal in equilibrium with some material X:
\[E_{slab+X}=NE_{bulk}+N_{ads}E_{X}+2A\gamma^{int}_{hkl}, \tag{5}\]
\[E_{slab+X}=E_{slab}+N_{ads}E_{X}+N_{ads}E_{ads}. \tag{6}\]
In the above equations, \(E_{slab+X}\) is the total energy of the slab+X system, \(E_{X}\) is the total energy per molecule of X, and \(N_{ads}\) is the number of bonds between slab and X. The latter is related to the coverage, \(\theta\) and area per surface atom, \(A_{at}\), by \(\theta=N_{ads}/N_{surf}\) and \(A=A_{at}N_{surf}\). Substituting into Eqs. and and using Eq. yields Eq..
For a typical system (\(\theta=0.1\), \(E_{ads}\)=0.5 eV), the second term in Eq. is about 0.1 J/m\({}^{2}\), or 10% of \(\gamma_{hkl}\). Change in ratios between various \(\gamma_{hkl}\) will be of the order of 1%, resulting in very small change in the equilibrium shape. This explains the similarity of nanoparticle shapes observed in a wide variety of environments: our simulations nicely match experimental observations, not only for Au clusters but also Au particles on C nanotubes, on TiO\({}_{2}\) and on CeO\({}_{2}\).
On the other hand, shape can change dramatically for very small nanoparticles where bonding on faces might be very different from bonding on a large surface or when small molecules with high adsorption energy are adsorbed. The ideal adsorbate to test this idea is CO.
We calculate the minimum adsorption energy of CO on every Au(\(hkl\)) with \(h,k,l\leq 3\). We consider several different adsorption sites to ensure that the global minimum is found; as we are interested in very low CO coverage, neighbouring CO molecules maintain a distance of more than 4.2 A at all cases. In almost every case, CO binds atop the lowest-coordinated Au atom with adsorption energy being a linear function of the coordination number of this Au atom, \(z\). We use this linear fit to obtain adsorption energies for the nine (\(4kl\)) surfaces. Adsorption energies and adsorption sites are shown in Table 1.
We use calculated adsorption energies together with Eq. and obtain the equilibrium shape of Au nanoparticles at low CO coverage shown in For rough surfaces, \(\gamma_{hkl}\) will be relatively high, but at the same time \(E_{ads}\) will be quite low; this results in a compensation effect for the two terms in Eq.. As the different \(\gamma^{int}_{hkl}\) are close to each other, the shape has a much higher sphericity (98%) than the shape in vacuum (93%), in excellent agreement with experiments.
Exposure of the nanoparticle to CO gas makes it much more reactive. This effect has been observed in first-principles simulations of small Au clusters. We find that the same happens at larger sizes, although it is more prominent for smaller nanoparticles.
Gold nanoparticles are usually supported on oxides, such as MgO or rutile TiO\({}_{2}\).
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline Shape & d (nm) & N\({}_{corner}\) & N\({}_{edge}\) & N\({}_{surf}\) & N\({}_{tot}\) & Area (nm\({}^{2}\)) & Volume (nm\({}^{3}\)) & Faces \\ \hline (a) & 12.12 & 24 & 444 & 5208 & 42925 & 438 & 730 &, \\ & & & & & & & (86\%), (14\%) \\ \hline (b) & 27.17 & 96 & 2832 & 25998 & 473550 & 2275 & 8439 &,,, \\ & & & & & & & (58\%), (16\%), (15\%), (11\%) \\ \hline (c) & 0.31N\({}^{0.34}\) & 144 & 6N\({}^{0.40}\) & 3N\({}^{0.68}\) & N & 0.3N\({}^{0.67}\) & 0.02N &,,,, \\ \hline \end{tabular}
\end{table}
Table 2: Characteristic data for typical nanoparticles shown in Shape (a) is typical for particles up to \(d=16.3\) nm in diameter, shape (b) is typical for larger particles and (c) presents fitted values from over a hundred different particles with \(d<16.3\) Å. \(N_{corner}\), \(N_{edge}\) and \(N_{surf}\) are the total number of atoms at vertices, edges and faces of the nanoparticle, respectively; \(N_{total}\) is the total number of atoms. \(A\) is the total surface area and \(V\) the volume of the nanoparticle. (\(hkl\)) are the appearing surfaces in the shape with the percentage of the total area they occupy.
(Color online) Adsorption energy of CO on Au(\(hkl\)) as a function of the Au coordination number, \(z\), and its linear fit. Some characteristic adsorption geometries are shown.
The epitaxial growth will introduce strain in the nanoparticle. More important, the the values of \(\gamma_{hkl}\) for the faces attached to the supporting material will be very different. A qualitative picture of this interaction has been presented by Lopez _et al._.
## V Summary
We have developed a method for constructing and characterizing equilibrium-shaped nanoparticles in thermodynamic equilibrium with their environment. Using an atomistic version of the Wulff construction, we generate Cartesian positions of nanoparticles, which can then be used to analyse structural properties. Our results provide insight into large nanoparticles that are of interest to catalysis, but are inaccessible by direct atomistic simulations. The calculated nanoparticles match experimental results, including the similarity of shapes in weakly interacting systems as well as the change towards more spherical shapes upon exposure to reactive gas. The method is easily generalized to other materials, and might be useful for the improved design of nanomaterials with tailored physical and chemical properties.
| 10.48550/arXiv.1111.4667 | First-principles atomistic Wulff constructions for gold nanoparticles | Georgios D. Barmparis, Ioannis N. Remediakis | 239 |
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