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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.
##
(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.
##
(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 |
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