Memristive Systems are Generalisations of Memristors, which are Resistors with Memory.
Resistors, capacitors and inductors are the three fundamental circuit elements that are familiar from high school level electronics. In 1971 Leon Chua postulated a fourth two-terminal circuit element characterized by a relationship between charge and flux linkage. Predicting that conductance would depend on the history of current flow through it, he dubbed it memristor for memory and resistor. This is described in L. O. Chua, Memristor—The Missing Circuit Element, EEE Trans. Circuit Theory 18 507 (1971).
More recently, Strukov, Snider, Stewart and Williams published a theoretical article (Strukov et al., The missing memristor found, Nature 453, 80 (2008)) showing that memristive behaviour existed in previously observed resistance switching behaviour when “electronic and ionic transport are coupled”, especially in nanoscale films. It remains controversial whether indeed this class of system implements the memristor as envisaged by Chua but the term has come to be used by many authors as a generic reference to 2-terminal non-volatile memory devices based on resistance switching, blurring the line with resistive random-access memory (RRAM). This meaning of the term is adopted herein, i.e. a “memristor” according to the invention is a 2-terminal non-volatile memory device based on resistance switching.
The resistance of a memristor depends on its history, i.e. on what current has passed through the memristive device previously. This results in a pinched hysteresis, or Lissajous loop. This time dependent current-voltage characteristic can then be incorporated into a circuit to serve as a memory, as described in, for example, Nonvolatile memristor memory: device characteristics and design implications by Ho et al., Proceedings of the 2009 International Conference on Computer-Aided Design, pages 485-490 (ISBN 978-1-60558-800-1, doi 10.1145/1687399.1687491), the contents of which are incorporated herein by reference.
Subsequent research has realised memristive behaviour in a wide variety of systems, and has identified applications where the memory effect may lead to dramatic advantages in simplicity and power versus conventional electronics. These range from ultra dense, low power memory envisaged as a successor to today's RAM and flash, through novel stateful logic processes, and even neuromorphic computing where the memristor is seen as a synthetic analog of a neuron.
As with many other electronic devices, the behaviour of memristive systems is increasingly affected by quantum phenomena as the device size shrinks to ever smaller dimensions. Memristors based on ionic motion can be scaled down to less than 10 nm, and other candidates of memristive systems include even smaller objects, e.g. molecules. At the nanoscale, electron transport through devices is strongly influenced by the discrete energy spectrum and interference effect, which may even emerge at room temperature in certain molecules. It is therefore crucial to understand the non-classical behaviour that emerges in nanoscale memristive systems.
For the purposes of the present invention, the terms “memristive device” and “memristive system” are employed interchangeably and are intended to refer to quantum memristive devices: that is, memristive devices whose characteristics and behaviour can be explained only at a quantum-mechanical level.
In quantum mechanics, a memristive device can be considered as a quantum open system: the internal state of the device interacts with the environment, typically the electrodes, such that the internal state affects the electron transport (the resistance), and the evolution of the internal state depends on external signals (current and voltage).
This effect (i.e. the fact that the device characteristics alter dramatically when the environment changes, e.g. via a change in the charge state in the immediate environment of the device) is frequently a frustration in experimental work. However, the present inventors have unexpectedly found that it may be usefully harnessed as the basis of memristive function.
An example of such an open quantum system is a pair of capacitively-coupled quantum dots (a “double quantum dot device” or “double dot device”), where only one dot participates in the electron transport while the other (called the control dot) controls the transport via the capacitive coupling. The environment (electrodes) is configured such that the evolution of the control dot depends on the applied voltage.
The term “capacitive coupling” is a well known term in the field of electronic circuit design and the conventional meaning is adopted herein. In brief, it means that energy is transferred by means of displacement currents induced by existing electric fields between nodes of a circuit or circuits. Further details can be found in standard electrical engineering or circuit design handbooks.
Quantum dots based on nanoparticles are well-known for optical applications owing to their bright, pure colours along with their ability to emit a wide variety of colours of light coupled with their high efficiencies, long lifetimes and high extinction coefficients. However, no quantum-dot-based memristive devices have been described up until now. To date, no purely quantum-mechanical memristors have been demonstrated; the present invention therefore represents the first implementation of a quantum-mechanical memristive device, and in particular the first implementation of a quantum-dot-based memristor.
A double quantum dot device exhibits current-voltage hysteresis curves, which are a characteristic property of memristive systems. However, unlike classical memristive systems, the current in quantum memristive systems shows stochastic behaviour and may not converge to a periodic curve under periodic driving due to quantum jumps between different values of the quantised conductance.
Room temperature operation of the devices of the invention could potentially be achieved in ultra-scale semiconductor devices or by using single-molecule transistors. While device geometries employing capacitive coupling between two quantum dots are known, the use of a double quantum dot system as a memristive device has not been explored to date and thus memristors based on capacitively-coupled double quantum dot systems have also been unknown before now.
The present invention provides a simple and practical scheme for realising memristive systems with quantum dots. The present invention harnesses a phenomenon that is commonly seen as a bane of nanoelectronics, i.e. switching of a trapped charge in the vicinity of the device, and demonstrates that this can be put to good use in the realisation of a quantum-dot-based memristive system. Quantum-dot memristive systems as described herein have hysteresis current-voltage characteristics and quantum jump induced stochastic behaviour.
Quantum dots (QDs) display unique electronic properties which are intermediate between those of bulk semiconductors and discrete molecules. Electrons in quantum dots are confined in a small space which leads to the quantum dot behaving as a “quantum box” similar to the well-known “particle in a box” model as described in, for example, Molecular Quantum Mechanics by P. W. Atkins and R. S. Friedman (Oxford University Press, 5th ed. 2010). When the radius of the quantum dot is smaller than the exciton Bohr radius (the exciton Bohr radius is the average distance between the electron in the conduction band of the semiconductor and the hole it leaves behind in the valence band of the semiconductor), there is quantization of the energy levels according to Pauli's exclusion principle. The discrete, quantized energy levels of quantum dots relate their behaviour more closely to atoms than bulk materials. Generally, as the size of the quantum dot decreases, the difference in energy between the highest valence band and the lowest conduction band increases.
In many cases, quantum dots are tiny particles (nanocrystals) of metal or semiconducting material. Such particles typically have a diameter of about 2 to about 10 nm, and/or an overall content of about 10 to about 50 atoms or about 10 to about 50 molecules. The unique electronic properties of such particles are therefore partly the result of the unusually high surface-to-volume ratios of such particles. Quantum dots of the particulate (nanocrystalline) type may be used in the memristive devices of the present invention.
Particulate quantum dots can be classified into different types based on their composition and structure, as described below. All such quantum dots may suitably be applied in the devices of the present invention.
1. Core-Type Quantum Dots. These may be single component materials with uniform internal compositions, such as chalcogenides (selenides or sulfides) of metals like cadmium or zinc, for example CdSe or CdS. The electronic properties of core-type quantum dots can be fine-tuned by simply changing the particle size.
2. Core-Shell Quantum Dots. These comprise a core of a first semiconducting material surrounded by at least one shell of another, higher-band-gap semiconducting material such that the quantum dots comprise small regions of one material embedded in another with a wider band gap. These are known as core-shell quantum dots (CSQDs) or core-shell semiconducting nanocrystals (CSSNCs). For example, quantum dots with CdSe in the core and ZnS in the shell exhibit greater than 80% quantum yield in radiative electron-hole recombinations. Coating quantum dots with shells improves quantum yield by passivizing nonradiative recombination sites and also makes them more robust to processing conditions for various applications. This method has been widely explored as a way to adjust the photophysical properties of quantum dots.
3. Alloyed (multicomponent) Quantum Dots. Multicomponent quantum dots offer an alternative method to tune properties without changing crystallite size. Alloyed semiconductor quantum dots with both homogeneous and gradient internal structures allow tuning of the optical and electronic properties by merely changing the composition and internal structure without changing the crystallite size. For example, alloyed quantum dots of the compositions CdSxSe1-x/ZnS of 6 nm diameter emit light of different wavelengths by just changing the composition (i.e. by varying x). Alloyed semiconductor quantum dots formed by alloying together two semiconductors with different band gap energies exhibit interesting properties distinct not only from the properties of their bulk counterparts but also from those of their parent semiconductors.
Within the scope of the present invention, quantum dots are not restricted to being particulate (atomic or molecular nanocrystalline) QDs such as those described above. In the memristive devices of the invention, a QD may be any structure which causes electron confinement in a region which behaves as a “quantum box”, i.e. any structure which leads to quantisation of energy levels.
Thus, in certain embodiments of the invention, the quantum dots employed in the memristive devices of the invention may be single atoms or single molecules. Alternatively, the quantum dots may be nanocrystalline. Where atomic, molecular or nanocrystalline quantum dots are employed, these may be anchored to their respective electrode or electrodes using linker molecules which permit electron tunneling between the QD and the anchored electrode(s).
In certain embodiments of the invention the quantum dots may be metallic or semiconductor “islands” which may be lithographically or electrostatically defined. Alternatively the quantum dots may be formed by individual atoms, molecules or nanoparticles.
Thus, in certain embodiments of the invention, the quantum dots employed in the memristive devices of the invention may be regions of material which are formed from the same piece of material as at least one electrode (e.g. a single piece of material comprising a source electrode, drain electrode and a quantum dot, or a single piece of material comprising a reservoir electrode and a quantum dot). In such embodiments, although there is a continuum of material between at least one electrode and a quantum dot, a quantum dot may be defined as a region of that material which causes electron confinement locally such that the region behaves as a quantum box. For example, the quantum dot may be formed by creating constrictions in the material (e.g. regions in which the material is narrower) which alter the local electron transport properties of the material to create a tunneling barrier. Such barriers may be created, for example, by using notches in the material. Such constrictions or notches may be formed lithographically, for example by creating constrictions in a lithographic channel. These notches or constrictions create a tunneling barrier which causes the region of material on one side of that barrier to behave as a quantum dot and the region of material on another side of that barrier to behave as an electrode, such that electrons can travel between the quantum dot and the electrode only via quantum tunneling. Any suitable lithographic technique may be employed, such as electron beam or optical lithography, wet chemical etching, reactive ion etching or plasma etching. Suitable materials for use in such embodiments include metals, semiconductors such as Si or gallium arsenide, or two dimensional materials such as graphene.
In certain embodiments of the invention, the quantum dots may be regions of a continuous material which are caused to behave as quantum dots by the creation of electrostatically-induced tunnel barriers. These barriers act to separate the QD region from the remainder of the material such that the QD region acts as a quantum box and such that electrons can travel in and out of the QD region only via quantum tunneling. Such barriers can be employed in lithographically defined channels such as those described above, for example in place of constrictions or notches such that the QD region is defined by a gate electrode which alters the electrostatic properties of the material to induce QD behaviour instead of using a physical constriction to induce this. Electrostatically-induced tunnel barriers may also be employed to create one or more QD regions in a two-dimensional electron/hole gas such as a patterned semiconductor material or a semiconductor heterostructure.
Quantum Description of Memristive Systems
The following provides a generic description of a quantum mechanical memristive system.
Any voltage-controlled memristive system is defined by equations
                              I          =                                    G              ⁡                              (                                  x                  ,                  V                  ,                  t                                )                                      ⁢            V                          ,                            (        1        )                                                      dx            dt                    =                      f            ⁡                          (                              x                ,                V                ,                t                            )                                      ,                            (        2        )            where the conductance G depends on the voltage V and the state parameter x. In such a memristive system, the evolution of the state is controlled by the voltage, and the state equation [Equation (2)] is assumed to have a unique solution for any initial condition.
For a quantum system, the state is described by a reduced density matrix ρ. If ρ is the state of the total system including both the system (device) and the environment (e.g. electrodes coupled to the device), the current going through the device readsI=Tr(Îφ,  (3)where the current operator Î is an operator of the total system. When the coupling between the system (device) and the environment is weak, the influence of the system on the environment is small, and the state of the total system is approximately a tensor product ρ≈ρS ⊗ρE, where ρS is the state of the system and ρE is the state of the environment. Under the weak-coupling condition, the current is approximatelyI=Tr(ÎSρS),  (4)where ÎS=TrE(ÎS ⊗ρE) is an operator of the device depending on the state of the environment. S is the identity matrix for the state of the system. Because the state of the environment depends on the voltage, i.e. electrochemical potentials in electrodes, the operator ÎS is voltage-dependent.
For a memristive system, the current is zero whenever the voltage is zero, regardless of the state of the device, which distinguishes a memristive system from an arbitrary dynamical system. Therefore, if I|V-0=0 is satisfied, equation (4) is a quantum-mechanical version of equation (1), in which the conductance G=I/V depends on both the voltage and the state of the device.
The evolution of a Markov open quantum system is given by the master equation
                                                        d              ⁢                                                          ⁢                              ρ                S                                      dt                    =                                    ℒ              ⁡                              (                V                )                                      ⁢                          ρ              S                                      ,                            (        5        )            which has a unique solution for any initial condition. Here, the generator  of the semigroup is determined by the state of the environment, i.e. the voltage V. Therefore, when the condition I|V-0=0 is satisfied, equations (4) and (5) describe a memristive system in quantum mechanics (a quantum mechanical memristive system).