# Memristor

Last updated
Type Passive Memristance Leon Chua (1971)

A memristor (; a portmanteau of memory resistor) is a hypothetical non-linear passive two-terminal electrical component relating electric charge and magnetic flux linkage. It was envisioned, and its name coined, in 1971 by circuit theorist Leon Chua. [1] According to the characterizing mathematical relations, the memristor would hypothetically operate in the following way: the memristor's electrical resistance is not constant but depends on the history of current that had previously flowed through the device, i.e., its present resistance depends on how much electric charge has flowed in what direction through it in the past; the device remembers its history—the so-called non-volatility property. [2] When the electric power supply is turned off, the memristor remembers its most recent resistance until it is turned on again. [3] [4]

A portmanteau or portmanteau word is a linguistic blend of words, in which parts of multiple words or their phones (sounds) are combined into a new word, as in smog, coined by blending smoke and fog,, motel, from motor and hotel, or Pegahorn, from Pegasus and Unicorn. In linguistics, a portmanteau is defined as a single morph that represents two or more morphemes.

A hypothesis is a proposed explanation for a phenomenon. For a hypothesis to be a scientific hypothesis, the scientific method requires that one can test it. Scientists generally base scientific hypotheses on previous observations that cannot satisfactorily be explained with the available scientific theories. Even though the words "hypothesis" and "theory" are often used synonymously, a scientific hypothesis is not the same as a scientific theory. A working hypothesis is a provisionally accepted hypothesis proposed for further research, in a process beginning with an educated guess or thought.

Passivity is a property of engineering systems, used in a variety of engineering disciplines, but most commonly found in analog electronics and control systems. A passive component, depending on field, may be either a component that consumes but does not produce energy or a component that is incapable of power gain.

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In 2008, a team at HP Labs claimed to have found Chua's missing memristor based on an analysis of a thin film of titanium dioxide thus connecting the operation of ReRAM devices to the memristor concept. The HP result was published in the scientific journal Nature . [3] [5] Following this claim, Leon Chua has argued that the memristor definition could be generalized to cover all forms of two-terminal non-volatile memory devices based on resistance switching effects. [2] Chua also argued that the memristor is the oldest known circuit element, with its effects predating the resistor, capacitor, and inductor. [6] There are, however, some serious doubts as to whether a genuine memristor can actually exist in physical reality. [7] [8] [9] [10] [11] Additionally, some experimental evidence contradicts Chua's generalization since a non-passive nanobattery effect is observable in resistance switching memory. [12] A simple test has been proposed by Pershin and Di Ventra [13] to analyse whether such an ideal or generic memristor does actually exist or is a purely mathematical concept. Up to now, there seems to be no experimental resistance switching device (ReRAM) which can pass the test. [13] [14]

HP Labs is the exploratory and advanced research group for HP Inc. HP Labs' headquarters is in Palo Alto, California and the group has research and development facilities in Bristol, UK. The development of programmable desktop calculators, inkjet printing, and 3D graphics are credited to HP Labs researchers.

A thin film is a layer of material ranging from fractions of a nanometer (monolayer) to several micrometers in thickness. The controlled synthesis of materials as thin films is a fundamental step in many applications. A familiar example is the household mirror, which typically has a thin metal coating on the back of a sheet of glass to form a reflective interface. The process of silvering was once commonly used to produce mirrors, while more recently the metal layer is deposited using techniques such as sputtering. Advances in thin film deposition techniques during the 20th century have enabled a wide range of technological breakthroughs in areas such as magnetic recording media, electronic semiconductor devices, LEDs, optical coatings, hard coatings on cutting tools, and for both energy generation and storage. It is also being applied to pharmaceuticals, via thin-film drug delivery. A stack of thin films is called a multilayer.

Titanium dioxide, also known as titanium(IV) oxide or titania, is the naturally occurring oxide of titanium, chemical formula TiO
2
. When used as a pigment, it is called titanium white, Pigment White 6 (PW6), or CI 77891. Generally, it is sourced from ilmenite, rutile and anatase. It has a wide range of applications, including paint, sunscreen and food coloring. When used as a food coloring, it has E number E171. World production in 2014 exceeded 9 million metric tons. It has been estimated that titanium dioxide is used in two-thirds of all pigments, and pigments based on the oxide have been valued at 13.2 billion. These devices are intended for applications in nanoelectronic memories, computer logic, and neuromorphic/neuromemristive computer architectures. [15] [16] [17] In 2013, Hewlett-Packard CTO Martin Fink suggested that memristor memory may become commercially available as early as 2018. [18] In March 2012, a team of researchers from HRL Laboratories and the University of Michigan announced the first functioning memristor array built on a CMOS chip. [19] HRL Laboratories, was the research arm of Hughes Aircraft. It is a dedicated research center, established in 1960, in Malibu, California. Currently owned by General Motors Corporation and Boeing, the research facility is housed in two large, white multi-story buildings overlooking the Pacific Ocean. helloy everybody Complementary metal–oxide–semiconductor (CMOS), also known as complementary-symmetry metal–oxide–semiconductor (COS-MOS), is a type of MOSFET fabrication process that uses complementary and symmetrical pairs of p-type and n-type MOSFETs for logic functions. CMOS technology is used for constructing integrated circuits (ICs), including microprocessors, microcontrollers, memory chips, and other digital logic circuits. CMOS technology is also used for analog circuits such as image sensors, data converters, RF circuits, and highly integrated transceivers for many types of communication. ## Background In his 1971 paper, Chua extrapolated a conceptual symmetry between the non-linear resistor (voltage vs. current), non-linear capacitor (voltage vs. charge), and non-linear inductor (magnetic flux linkage vs. current). He then inferred the possibility of a memristor as another fundamental non-linear circuit element linking magnetic flux and charge. In contrast to a linear (or non-linear) resistor the memristor has a dynamic relationship between current and voltage including a memory of past voltages or currents. Other scientists had proposed dynamic memory resistors such as the memistor of Bernard Widrow, but Chua attempted to introduce mathematical generality. A memistor is a nanoelectric circuitry element used in parallel computing memory technology. Essentially, a resistor with memory able to perform logic operations and store information, it is a three-terminal implementation of the memristor. It is a possible future technology replacing flash and DRAM. Memristor resistance depends on the integral of the input applied to the terminals (rather than on the instantaneous value of the input as in a varistor). [3] Since the element "remembers" the amount of current that last passed through, it was tagged by Chua with the name "memristor". Another way of describing a memristor is as any passive two-terminal circuit element that maintains a functional relationship between the time integral of current (called charge) and the time integral of voltage (often called flux, as it is related to magnetic flux). The slope of this function is called the memristanceM and is similar to variable resistance. A varistor is an electronic component with an electrical resistance that varies with the applied voltage. Also known as a voltage-dependent resistor (VDR), it has a nonlinear, non-ohmic current–voltage characteristic that is similar to that of a diode. In contrast to a diode however, it has the same characteristic for both directions of traversing current. Traditionally, varistors were indeed constructed by connecting two rectifiers, such as the copper-oxide or germanium-oxide rectifier in anti-parallel configuration. At low voltage the varistor has a high electrical resistance which decreases as the voltage is raised. Modern varistors are primarily based on sintered ceramic metal-oxide materials which exhibit directional behavior only on a microscopic scale. This type is commonly known as the metal-oxide varistor (MOV). In mathematics, a function is a relation between sets that associates to every element of a first set exactly one element of the second set. Typical examples are functions from integers to integers or from the real numbers to real numbers. An electric current is the rate of flow of electric charge past a point or region. An electric current is said to exist when there is a net flow of electric charge through a region. In electric circuits this charge is often carried by electrons moving through a wire. It can also be carried by ions in an electrolyte, or by both ions and electrons such as in an ionized gas (plasma). The memristor definition is based solely on the fundamental circuit variables of current and voltage and their time-integrals, just like the resistor, capacitor, and inductor. Unlike those three elements however, which are allowed in linear time-invariant or LTI system theory, memristors of interest have a dynamic function with memory and may be described as some function of net charge. There is no such thing as a standard memristor. Instead, each device implements a particular function, wherein the integral of voltage determines the integral of current, and vice versa. A linear time-invariant memristor, with a constant value for M, is simply a conventional resistor. [1] Manufactured devices are never purely memristors (ideal memristor), but also exhibit some capacitance and resistance. A resistor is a passive two-terminal electrical component that implements electrical resistance as a circuit element. In electronic circuits, resistors are used to reduce current flow, adjust signal levels, to divide voltages, bias active elements, and terminate transmission lines, among other uses. High-power resistors that can dissipate many watts of electrical power as heat, may be used as part of motor controls, in power distribution systems, or as test loads for generators. Fixed resistors have resistances that only change slightly with temperature, time or operating voltage. Variable resistors can be used to adjust circuit elements, or as sensing devices for heat, light, humidity, force, or chemical activity. A capacitor is a device that stores electrical energy in an electric field. It is a passive electronic component with two terminals. An inductor, also called a coil, choke, or reactor, is a passive two-terminal electrical component that stores energy in a magnetic field when electric current flows through it. An inductor typically consists of an insulated wire wound into a coil around a core. ## Memristor definition and criticism According to the original 1971 definition, the memristor was the fourth fundamental circuit element, forming a non-linear relationship between electric charge and magnetic flux linkage. In 2011, Chua argued for a broader definition that included all 2-terminal non-volatile memory devices based on resistance switching. [2] Williams argued that MRAM, phase-change memory and ReRAM were memristor technologies. [22] Some researchers argued that biological structures such as blood [23] and skin [24] [25] fit the definition. Others argued that the memory device under development by HP Labs and other forms of ReRAM were not memristors, but rather part of a broader class of variable-resistance systems, [26] and that a broader definition of memristor is a scientifically unjustifiable land grab that favored HP's memristor patents. [27] In 2011, Meuffels and Schroeder noted that one of the early memristor papers included a mistaken assumption regarding ionic conduction. [28] In 2012, Meuffels and Soni discussed some fundamental issues and problems in the realization of memristors. [7] They indicated inadequacies in the electrochemical modelling presented in the Nature article "The missing memristor found" [3] because the impact of concentration polarization effects on the behavior of metal−TiO2−x−metal structures under voltage or current stress was not considered. This critique was referred to by Valov et al. [12] in 2013. In a kind of thought experiment, Meuffels and Soni [7] furthermore revealed a severe inconsistency: If a current-controlled memristor with the so-called non-volatility property [2] exists in physical reality, its behavior would violate Landauer's principle of the minimum amount of energy required to change "information" states of a system. This critique was finally adopted by Di Ventra and Pershin [8] in 2013. Within this context, Meuffels and Soni [7] pointed to a fundamental thermodynamic principle: Non-volatile information storage requires the existence of free-energy barriers that separate the distinct internal memory states of a system from each other; otherwise, one would be faced with an "indifferent" situation, and the system would arbitrarily fluctuate from one memory state to another just under the influence of thermal fluctuations. When unprotected against thermal fluctuations, the internal memory states exhibit some diffusive dynamics, which causes state degradation. [8] The free-energy barriers must therefore be high enough to ensure a low bit-error probability of bit operation. [29] Consequently, there is always a lower limit of energy requirement – depending on the required bit-error probability – for intentionally changing a bit value in any memory device. [29] [30] In the general concept of memristive system the defining equations are (see Theory): {\displaystyle {\begin{aligned}y(t)&=g(\mathbf {x} ,u,t)u(t),\\{\dot {\mathbf {x} }}&=f(\mathbf {x} ,u,t),\end{aligned}}} where u(t) is an input signal, and y(t) is an output signal. The vector x represents a set of n state variables describing the different internal memory states of the device. is the time-dependent rate of change of the state vector x with time. When one wants to go beyond mere curve fitting and aims at a real physical modeling of non-volatile memory elements, e.g., resistive random-access memory devices, one has to keep an eye on the aforementioned physical correlations. To check the adequacy of the proposed model and its resulting state equations, the input signal u(t) can be superposed with a stochastic term ξ(t), which takes into account the existence of inevitable thermal fluctuations. The dynamic state equation in its general form then finally reads: ${\displaystyle {\dot {\mathbf {x} }}=f(\mathbf {x} ,u(t)+\xi (t),t),}$ where ξ(t) is, e.g., white Gaussian current or voltage noise. On base of an analytical or numerical analysis of the time-dependent response of the system towards noise, a decision on the physical validity of the modeling approach can be made, e.g., would the system be able to retain its memory states in power-off mode? Such an analysis was performed by Di Ventra and Pershin [8] with regard to the genuine current-controlled memristor. As the proposed dynamic state equation provides no physical mechanism enabling such a memristor to cope with inevitable thermal fluctuations, a current-controlled memristor would erratically change its state in course of time just under the influence of current noise. [8] [31] Di Ventra and Pershin [8] thus concluded that memristors whose resistance (memory) states depend solely on the current or voltage history would be unable to protect their memory states against unavoidable Johnson–Nyquist noise and permanently suffer from information loss, a so-called "stochastic catastrophe". A current-controlled memristor can thus not exist as a solid-state device in physical reality. The above-mentioned thermodynamic principle furthermore implies that the operation of 2-terminal non-volatile memory devices (e.g. "resistance-switching" memory devices (ReRAM)) cannot be associated with the memristor concept, i.e., such devices cannot by itself remember their current or voltage history. Transitions between distinct internal memory or resistance states are of probabilistic nature. The probability for a transition from state {i} to state {j} depends on the height of the free-energy barrier between both states. The transition probability can thus be influenced by suitably driving the memory device, i.e., by "lowering" the free-energy barrier for the transition {i} → {j} by means of, for example, an externally applied bias. A "resistance switching" event can simply be enforced by setting the external bias to a value above a certain threshold value. This is the trivial case, i.e., the free-energy barrier for the transition {i} → {j} is reduced to zero. In case one applies biases below the threshold value, there is still a finite probability that the device will switch in course of time (triggered by a random thermal fluctuation), but – as one is dealing with probabilistic processes – it is impossible to predict when the switching event will occur. That is the basic reason for the stochastic nature of all observed resistance-switching (ReRAM) processes. If the free-energy barriers are not high enough, the memory device can even switch without having to do anything. When a 2-terminal non-volatile memory device is found to be in a distinct resistance state {j}, there exists therefore no physical one-to-one relationship between its present state and its foregoing voltage history. The switching behavior of individual non-volatile memory devices thus cannot be described within the mathematical framework proposed for memristor/memristive systems. An extra thermodynamic curiosity arises from the definition that memristors/memristive devices should energetically act like resistors. The instantaneous electrical power entering such a device is completely dissipated as Joule heat to the surrounding, so no extra energy remains in the system after it has been brought from one resistance state xi to another one xj. Thus, the internal energy of the memristor device in state xi, U(V, T, xi), would be the same as in state xj, U(V, T, xj), even though these different states would give rise to different device's resistances, which itself must be caused by physical alterations of the device's material. Other researchers noted that memristor models based on the assumption of linear ionic drift do not account for asymmetry between set time (high-to-low resistance switching) and reset time (low-to-high resistance switching) and do not provide ionic mobility values consistent with experimental data. Non-linear ionic-drift models have been proposed to compensate for this deficiency. [32] A 2014 article from researchers of ReRAM concluded that Strukov's (HP's) initial/basic memristor modelling equations do not reflect the actual device physics well, whereas subsequent (physics-based) models such as Pickett's model or Menzel's ECM model (Menzel is a co-author of that article) have adequate predictability, but are computationally prohibitive. As of 2014, the search continues for a model that balances these issues; the article identifies Chang's and Yakopcic's models as potentially good compromises. [33] Martin Reynolds, an electrical engineering analyst with research outfit Gartner, commented that while HP was being sloppy in calling their device a memristor, critics were being pedantic in saying that it was not a memristor. [34] In the article "The Missing Memristor has Not been Found", published in Scientific Reports in 2015 by Vongehr and Meng, [10] it was shown that the real memristor defined in 1971 is not possible without using magnetic induction. This was illustrated by constructing a mechanical analog of the memristor and then analytically showing that the mechanical memristor cannot be constructed without using an inertial mass. As it is well known that the mechanical equivalent of an electrical inductor is mass, it proves that memristors are not possible without using magnetic induction. Thus, it can be argued that the variable-resistance devices, such as the ReRAMs, and the conceptual memristors may have no equivalence at all. [10] [35] ## Experimental tests for memristors Chua suggested experimental tests to determine if a device may properly be categorized as a memristor: [36] • The Lissajous curve in the voltage-current plane is a pinched hysteresis loop when driven by any bipolar periodic voltage or current without respect to initial conditions. • The area of each lobe of the pinched hysteresis loop shrinks as the frequency of the forcing signal increases. • As the frequency tends to infinity, the hysteresis loop degenerates to a straight line through the origin, whose slope depends on the amplitude and shape of the forcing signal. According to Chua [37] [38] all resistive switching memories including ReRAM, MRAM and phase-change memory meet these criteria and are memristors. However, the lack of data for the Lissajous curves over a range of initial conditions or over a range of frequencies complicates assessments of this claim. Experimental evidence shows that redox-based resistance memory (ReRAM) includes a nanobattery effect that is contrary to Chua's memristor model. This indicates that the memristor theory needs to be extended or corrected to enable accurate ReRAM modeling. [12] ## Theory The memristor was originally defined in terms of a non-linear functional relationship between magnetic flux linkage Φm(t) and the amount of electric charge that has flowed, q(t): [1] ${\displaystyle f(\mathrm {\Phi } _{\mathrm {m} }(t),q(t))=0}$ The magnetic flux linkage , Φm, is generalized from the circuit characteristic of an inductor. It does not represent a magnetic field here. Its physical meaning is discussed below. The symbol Φm may be regarded as the integral of voltage over time. [39] In the relationship between Φm and q, the derivative of one with respect to the other depends on the value of one or the other, and so each memristor is characterized by its memristance function describing the charge-dependent rate of change of flux with charge. ${\displaystyle M(q)={\frac {\mathrm {d} \Phi _{m}}{\mathrm {d} q}}}$ Substituting the flux as the time integral of the voltage, and charge as the time integral of current, the more convenient forms are; ${\displaystyle M(q(t))={\cfrac {\mathrm {d} \Phi _{m}/\mathrm {d} t}{\mathrm {d} q/\mathrm {d} t}}={\frac {V(t)}{I(t)}}}$ To relate the memristor to the resistor, capacitor, and inductor, it is helpful to isolate the term M(q), which characterizes the device, and write it as a differential equation. DeviceCharacteristic property (units)Differential equation Resistor (R)Resistance (V / A, or ohm, Ω)R = dV / dI Capacitor (C)Capacitance (C / V, or farad)C = dq / dV Inductor (L)Inductance (Wb / A, or henry)L = dΦm / dI Memristor (M)Memristance (Wb / C, or ohm)M = dΦm / dq The above table covers all meaningful ratios of differentials of I, Q, Φm, and V. No device can relate dI to dq, or m to dV, because I is the derivative of Q and Φm is the integral of V. It can be inferred from this that memristance is charge-dependent resistance. If M(q(t)) is a constant, then we obtain Ohm's Law R(t) = V(t)/I(t). If M(q(t)) is nontrivial, however, the equation is not equivalent because q(t) and M(q(t)) can vary with time. Solving for voltage as a function of time produces ${\displaystyle V(t)=\ M(q(t))I(t)}$ This equation reveals that memristance defines a linear relationship between current and voltage, as long as M does not vary with charge. Nonzero current implies time varying charge. Alternating current, however, may reveal the linear dependence in circuit operation by inducing a measurable voltage without net charge movement—as long as the maximum change in q does not cause much change in M. Furthermore, the memristor is static if no current is applied. If I(t) = 0, we find V(t) = 0 and M(t) is constant. This is the essence of the memory effect. The power consumption characteristic recalls that of a resistor, I2R. ${\displaystyle P(t)=\ I(t)V(t)=\ I^{2}(t)M(q(t))}$ As long as M(q(t)) varies little, such as under alternating current, the memristor will appear as a constant resistor. If M(q(t)) increases rapidly, however, current and power consumption will quickly stop. M(q) is physically restricted to be positive for all values of q (assuming the device is passive and does not become superconductive at some q). A negative value would mean that it would perpetually supply energy when operated with alternating current. In 2008 researchers from HP Labs introduced a model for a memristance function based on thin films of titanium dioxide. [3] For RON ≪ ROFF the memristance function was determined to be ${\displaystyle M(q(t))=R_{\mathrm {OFF} }\cdot \left(1-{\frac {\mu _{v}R_{\mathrm {ON} }}{D^{2}}}q(t)\right)}$ where ROFF represents the high resistance state, RON represents the low resistance state, μv represents the mobility of dopants in the thin film, and D represents the film thickness. The HP Labs group noted that "window functions" were necessary to compensate for differences between experimental measurements and their memristor model due to non-linear ionic drift and boundary effects. ### Operation as a switch For some memristors, applied current or voltage causes substantial change in resistance. Such devices may be characterized as switches by investigating the time and energy that must be spent to achieve a desired change in resistance. This assumes that the applied voltage remains constant. Solving for energy dissipation during a single switching event reveals that for a memristor to switch from Ron to Roff in time Ton to Toff, the charge must change by ΔQ = QonQoff. ${\displaystyle E_{\mathrm {switch} }=\ V^{2}\int _{T_{\mathrm {off} }}^{T_{\mathrm {on} }}{\frac {\mathrm {d} t}{M(q(t))}}=\ V^{2}\int _{Q_{\mathrm {off} }}^{Q_{\mathrm {on} }}{\frac {\mathrm {d} q}{I(q)M(q)}}=\ V^{2}\int _{Q_{\mathrm {off} }}^{Q_{\mathrm {on} }}{\frac {\mathrm {d} q}{V(q)}}=\ V\Delta Q}$ Substituting V = I(q)M(q), and then ∫dq/V = ∆Q/V for constant VTo produces the final expression. This power characteristic differs fundamentally from that of a metal oxide semiconductor transistor, which is capacitor-based. Unlike the transistor, the final state of the memristor in terms of charge does not depend on bias voltage. The type of memristor described by Williams ceases to be ideal after switching over its entire resistance range, creating hysteresis, also called the "hard-switching regime". [3] Another kind of switch would have a cyclic M(q) so that each off-on event would be followed by an on-off event under constant bias. Such a device would act as a memristor under all conditions, but would be less practical. ### Memristive systems The memristor was generalized to memristive systems in Chua's 1976 paper. [36] Whereas a memristor has mathematically scalar state, a system has vector state. The number of state variables is independent of the number of terminals. Chua applied this model to empirically-observed phenomena, including the Hodgkin–Huxley model of the axon and a thermistor at constant ambient temperature. He also described memristive systems in terms of energy storage and easily observed electrical characteristics. These characteristics might match resistive random-access memory relating the theory to active areas of research. In the more general concept of an n-th order memristive system the defining equations are {\displaystyle {\begin{aligned}y(t)&=g({\textbf {x}},u,t)u(t),\\{\dot {\textbf {x}}}&=f({\textbf {x}},u,t)\end{aligned}}} where u(t) is an input signal, y(t) is an output signal, the vector x represents a set of n state variables describing the device, and g and f are continuous functions. For a current-controlled memristive system the signal u(t) represents the current signal i(t) and the signal y(t) represents the voltage signal v(t). For a voltage-controlled memristive system the signal u(t) represents the voltage signal v(t) and the signal y(t) represents the current signal i(t). The pure memristor is a particular case of these equations, namely when x depends only on charge (x = q) and since the charge is related to the current via the time derivative dq/dt = i(t). Thus for pure memristors f (i.e. the rate of change of the state) must be equal or proportional to the current i(t) . ### Pinched hysteresis One of the resulting properties of memristors and memristive systems is the existence of a pinched hysteresis effect. [40] For a current-controlled memristive system, the input u(t) is the current i(t), the output y(t) is the voltage v(t), and the slope of the curve represents the electrical resistance. The change in slope of the pinched hysteresis curves demonstrates switching between different resistance states which is a phenomenon central to ReRAM and other forms of two-terminal resistance memory. At high frequencies, memristive theory predicts the pinched hysteresis effect will degenerate, resulting in a straight line representative of a linear resistor. It has been proven that some types of non-crossing pinched hysteresis curves (denoted Type-II) cannot be described by memristors. [41] ### Extended memristive systems Some researchers have raised the question of the scientific legitimacy of HP's memristor models in explaining the behavior of ReRAM. [26] [27] and have suggested extended memristive models to remedy perceived deficiencies. [12] One example [42] attempts to extend the memristive systems framework by including dynamic systems incorporating higher-order derivatives of the input signal u(t) as a series expansion {\displaystyle {\begin{aligned}y(t)&=g_{0}({\textbf {x}},u)u(t)+g_{1}({\textbf {x}},u){\operatorname {d} ^{2}u \over \operatorname {d} t^{2}}+g_{2}({\textbf {x}},u){\operatorname {d} ^{4}u \over \operatorname {d} t^{4}}+\ldots +g_{m}({\textbf {x}},u){\operatorname {d} ^{2m}u \over \operatorname {d} t^{2m}},\\{\dot {\textbf {x}}}&=f({\textbf {x}},u)\end{aligned}}} where m is a positive integer, u(t) is an input signal, y(t) is an output signal, the vector x represents a set of n state variables describing the device, and the functions g and f are continuous functions. This equation produces the same zero-crossing hysteresis curves as memristive systems but with a different frequency response than that predicted by memristive systems. Another example suggests including an offset value a to account for an observed nanobattery effect which violates the predicted zero-crossing pinched hysteresis effect. [12] {\displaystyle {\begin{aligned}y(t)&=g_{0}({\textbf {x}},u)(u(t)-a),\\{\dot {\textbf {x}}}&=f({\textbf {x}},u)\end{aligned}}} ## Implementations ### Titanium dioxide memristor Interest in the memristor revived when an experimental solid-state version was reported by R. Stanley Williams of Hewlett Packard in 2007. [43] [44] [45] The article was the first to demonstrate that a solid-state device could have the characteristics of a memristor based on the behavior of nanoscale thin films. The device neither uses magnetic flux as the theoretical memristor suggested, nor stores charge as a capacitor does, but instead achieves a resistance dependent on the history of current. Although not cited in HP's initial reports on their TiO2 memristor, the resistance switching characteristics of titanium dioxide were originally described in the 1960s. [46] The HP device is composed of a thin (50 nm) titanium dioxide film between two 5 nm thick electrodes, one titanium, the other platinum. Initially, there are two layers to the titanium dioxide film, one of which has a slight depletion of oxygen atoms. The oxygen vacancies act as charge carriers, meaning that the depleted layer has a much lower resistance than the non-depleted layer. When an electric field is applied, the oxygen vacancies drift (see Fast ion conductor ), changing the boundary between the high-resistance and low-resistance layers. Thus the resistance of the film as a whole is dependent on how much charge has been passed through it in a particular direction, which is reversible by changing the direction of current. [3] Since the HP device displays fast ion conduction at nanoscale, it is considered a nanoionic device. [47] Memristance is displayed only when both the doped layer and depleted layer contribute to resistance. When enough charge has passed through the memristor that the ions can no longer move, the device enters hysteresis. It ceases to integrate q=∫I dt, but rather keeps q at an upper bound and M fixed, thus acting as a constant resistor until current is reversed. Memory applications of thin-film oxides had been an area of active investigation for some time. IBM published an article in 2000 regarding structures similar to that described by Williams. [48] Samsung has a U.S. patent for oxide-vacancy based switches similar to that described by Williams. [49] Williams also has a U.S. patent application related to the memristor construction. [50] In April 2010, HP labs announced that they had practical memristors working at 1 ns (~1 GHz) switching times and 3 nm by 3 nm sizes, [51] which bodes well for the future of the technology. [52] At these densities it could easily rival the current sub-25 nm flash memory technology. ### Polymeric memristor In 2004, Krieger and Spitzer described dynamic doping of polymer and inorganic dielectric-like materials that improved the switching characteristics and retention required to create functioning nonvolatile memory cells. [53] They used a passive layer between electrode and active thin films, which enhanced the extraction of ions from the electrode. It is possible to use fast ion conductor as this passive layer, which allows a significant reduction of the ionic extraction field. In July 2008, Erokhin and Fontana claimed to have developed a polymeric memristor before the more recently announced titanium dioxide memristor. [54] In 2010, Alibart, Gamrat, Vuillaume et al. [55] introduced a new hybrid organic/nanoparticle device (the NOMFET : Nanoparticle Organic Memory Field Effect Transistor), which behaves as a memristor [56] and which exhibits the main behavior of a biological spiking synapse.This device, also called synapstor (synapse transistor), was used to demonstrate a neuro-inspired circuit (associative memory showing a pavlovian learning) [57] In 2012, Crupi, Pradhan and Tozer described a proof of concept design to create neural synaptic memory circuits using organic ion-based memristors. [58] The synapse circuit demonstrated long-term potentiation for learning as well as inactivity based forgetting. Using a grid of circuits, a pattern of light was stored and later recalled. This mimics the behavior of the V1 neurons in the primary visual cortex that act as spatiotemporal filters that process visual signals such as edges and moving lines. ### Layered memristor In 2014, Bessonov et al. reported a flexible memristive device comprising a MoOx/MoS2 heterostructure sandwiched between silver electrodes on a plastic foil. [59] The fabrication method is entirely based on printing and solution-processing technologies using two-dimensional layered transition metal dichalcogenides (TMDs). The memristors are mechanically flexible, optically transparent and produced at low cost. The memristive behaviour of switches was found to be accompanied by a prominent memcapacitive effect. High switching performance, demonstrated synaptic plasticity and sustainability to mechanical deformations promise to emulate the appealing characteristics of biological neural systems in novel computing technologies. ### Atomristor Atomristor is defined as the electrical devices showing memristive behavior in atomically thin nanomaterials or atomic sheets. In 2018, Ge and Wu et al. [60] first reported a universal memristive effect in single-layer TMD (MX2, M = Mo, W; and X = S, Se) atomic sheets based on vertical metal-insulator-metal (MIM) device structure. These atomristors offer forming-free switching and both unipolar and bipolar operation. The switching behavior is found in single-crystalline and poly-crystalline films, with various metallic electrodes (gold, silver and graphene). Atomically thin TMD sheets are prepared via CVD/MOCVD, enabling low-cost fabrication. Afterwards, taking advantage of the low "on" resistance and large on/off ratio, a high-performance zero-power RF switch is proved based on MoS2 atomristors, indicating a new application of memristors. [61] ### Ferroelectric memristor The ferroelectric memristor [62] is based on a thin ferroelectric barrier sandwiched between two metallic electrodes. Switching the polarization of the ferroelectric material by applying a positive or negative voltage across the junction can lead to a two order of magnitude resistance variation: ROFF ≫ RON (an effect called Tunnel Electro-Resistance). In general, the polarization does not switch abruptly. The reversal occurs gradually through the nucleation and growth of ferroelectric domains with opposite polarization. During this process, the resistance is neither RON or ROFF, but in between. When the voltage is cycled, the ferroelectric domain configuration evolves, allowing a fine tuning of the resistance value. The ferroelectric memristor's main advantages are that ferroelectric domain dynamics can be tuned, offering a way to engineer the memristor response, and that the resistance variations are due to purely electronic phenomena, aiding device reliability, as no deep change to the material structure is involved. ### Carbon nanotube memristor In 2013, Ageev, Blinov et al. [63] reported observing memristor effect in structure based on vertically aligned carbon nanotubes studying bundles of CNT by scanning tunneling microscope. Later it was found [64] that CNT memristive switching is observed when a nanotube has a non-uniform elastic strain ΔL0. It was shown that the memristive switching mechanism of strained СNT is based on the formation and subsequent redistribution of non-uniform elastic strain and piezoelectric field Edef in the nanotube under the influence of an external electric field E(x,t). ### Spin memristive systems #### Spintronic memristor Chen and Wang, researchers at disk-drive manufacturer Seagate Technology described three examples of possible magnetic memristors. [65] In one device resistance occurs when the spin of electrons in one section of the device points in a different direction from those in another section, creating a "domain wall", a boundary between the two sections. Electrons flowing into the device have a certain spin, which alters the device's magnetization state. Changing the magnetization, in turn, moves the domain wall and changes the resistance. The work's significance led to an interview by IEEE Spectrum. [66] A first experimental proof of the spintronic memristor based on domain wall motion by spin currents in a magnetic tunnel junction was given in 2011. [67] #### Memristance in a magnetic tunnel junction The magnetic tunnel junction has been proposed to act as a memristor through several potentially complementary mechanisms, both extrinsic (redox reactions, charge trapping/detrapping and electromigration within the barrier) and intrinsic (spin-transfer torque). ##### Extrinsic mechanism Based on research performed between 1999 and 2003, Bowen et al. published experiments in 2006 on a magnetic tunnel junction (MTJ) endowed with bi-stable spin-dependent states [68] (resistive switching). The MTJ consists in a SrTiO3 (STO) tunnel barrier that separates half-metallic oxide LSMO and ferromagnetic metal CoCr electrodes. The MTJ's usual two device resistance states, characterized by a parallel or antiparallel alignment of electrode magnetization, are altered by applying an electric field. When the electric field is applied from the CoCr to the LSMO electrode, the tunnel magnetoresistance (TMR) ratio is positive. When the direction of electric field is reversed, the TMR is negative. In both cases, large amplitudes of TMR on the order of 30% are found. Since a fully spin-polarized current flows from the half-metallic LSMO electrode, within the Julliere model, this sign change suggests a sign change in the effective spin polarization of the STO/CoCr interface. The origin to this multistate effect lies with the observed migration of Cr into the barrier and its state of oxidation. The sign change of TMR can originate from modifications to the STO/CoCr interface density of states, as well as from changes to the tunneling landscape at the STO/CoCr interface induced by CrOx redox reactions. Reports on MgO-based memristive switching within MgO-based MTJs appeared starting in 2008 [69] and 2009. [70] While the drift of oxygen vacancies within the insulating MgO layer has been proposed to describe the observed memristive effects, [70] another explanation could be charge trapping/detrapping on the localized states of oxygen vacancies [71] and its impact [72] on spintronics. This highlights the importance of understanding what role oxygen vacancies play in the memristive operation of devices that deploy complex oxides with an intrinsic property such as ferroelectricity [73] or multiferroicity. [74] ##### Intrinsic mechanism The magnetization state of a MTJ can be controlled by Spin-transfer torque, and can thus, through this intrinsic physical mechanism, exhibit memristive behavior. This spin torque is induced by current flowing through the junction, and leads to an efficient means of achieving a MRAM. However, the length of time the current flows through the junction determines the amount of current needed, i.e., charge is the key variable. [75] The combination of intrinsic (spin-transfer torque) and extrinsic (resistive switching) mechanisms naturally leads to a second-order memristive system described by the state vector x = (x1,x2), where x1 describes the magnetic state of the electrodes and x2 denotes the resistive state of the MgO barrier. In this case the change of x1 is current-controlled (spin torque is due to a high current density) whereas the change of x2 is voltage-controlled (the drift of oxygen vacancies is due to high electric fields). The presence of both effects in a memristive magnetic tunnel junction led to the idea of a nanoscopic synapse-neuron system. [76] #### Spin memristive system A fundamentally different mechanism for memristive behavior has been proposed by Pershin [77] and Di Ventra. [78] [79] The authors show that certain types of semiconductor spintronic structures belong to a broad class of memristive systems as defined by Chua and Kang. [36] The mechanism of memristive behavior in such structures is based entirely on the electron spin degree of freedom which allows for a more convenient control than the ionic transport in nanostructures. When an external control parameter (such as voltage) is changed, the adjustment of electron spin polarization is delayed because of the diffusion and relaxation processes causing hysteresis. This result was anticipated in the study of spin extraction at semiconductor/ferromagnet interfaces, [80] but was not described in terms of memristive behavior. On a short time scale, these structures behave almost as an ideal memristor. [1] This result broadens the possible range of applications of semiconductor spintronics and makes a step forward in future practical applications. ### Self-directed channel memristor In 2017, Dr Kris Campbell formally introduced the self-directed channel (SDC) memristor. [81] The SDC device is the first memristive device available commercially to researchers, students and electronics enthusiast worldwide. [82] The SDC device is operational immediately after fabrication. In the Ge2Se3 active layer, Ge-Ge homopolar bonds are found and switching occurs. The three layers consisting of Ge2Se3/Ag/Ge2Se3, directly below the top tungsten electrode, mix together during deposition and jointly form the silver-source layer. A layer of SnSe is between these two layers ensuring that the silver-source layer is not in direct contact with the active layer. Since silver does not migrate into the active layer at high temperatures, and the active layer maintains a high glass transition temperature of about 350 °C (662 °F), the device has significantly higher processing and operating temperatures at 250 °C (482 °F) and at least 150 °C (302 °F), respectively. These processing and operating temperatures are higher than most ion-conducting chalcogenide device types, including the S-based glasses (e.g. GeS) that need to be photodoped or thermally annealed. These factors allow the SDC device to operate over a wide range of temperatures, including long-term continuous operation at 150 °C (302 °F). ## Applications Williams' solid-state memristors can be combined into devices called crossbar latches, which could replace transistors in future computers[ citation needed ], given their much higher circuit density. They can potentially be fashioned into non-volatile solid-state memory, which would allow greater data density than hard drives with access times similar to DRAM, replacing both components. [21] HP prototyped a crossbar latch memory that can fit 100 gigabits in a square centimeter, [83] and proposed a scalable 3D design (consisting of up to 1000 layers or 1 petabit per cm3). [84] In May 2008 HP reported that its device reaches currently about one-tenth the speed of DRAM. [85] The devices' resistance would be read with alternating current so that the stored value would not be affected. [86] In May 2012, it was reported that the access time had been improved to 90 nanoseconds, which is nearly one hundred times faster than the contemporaneous Flash memory. At the same time, the energy consumption was just one percent of that consumed by Flash memory. [87] Memristor patents include applications in programmable logic, [88] signal processing, [89] neural networks, [90] control systems, [91] reconfigurable computing, [92] brain-computer interfaces, [93] and RFID. [94] Memristive devices are potentially used for stateful logic implication, allowing a replacement for CMOS-based logic computation. Several early works have been reported in this direction. [95] [96] In 2009, a simple electronic circuit [97] consisting of an LC network and a memristor was used to model experiments on adaptive behavior of unicellular organisms. [98] It was shown that subjected to a train of periodic pulses, the circuit learns and anticipates the next pulse similar to the behavior of slime molds Physarum polycephalum where the viscosity of channels in the cytoplasm responds to periodic environment changes. [98] Applications of such circuits may include, e.g., pattern recognition. The DARPA SyNAPSE project funded HP Labs, in collaboration with the Boston University Neuromorphics Lab, has been developing neuromorphic architectures which may be based on memristive systems. In 2010, Versace and Chandler described the MoNETA (Modular Neural Exploring Traveling Agent) model. [99] MoNETA is the first large-scale neural network model to implement whole-brain circuits to power a virtual and robotic agent using memristive hardware. [100] Application of the memristor crossbar structure in the construction of an analog soft computing system was demonstrated by Merrikh-Bayat and Shouraki. [101] In 2011, they showed [102] how memristor crossbars can be combined with fuzzy logic to create an analog memristive neuro-fuzzy computing system with fuzzy input and output terminals. Learning is based on the creation of fuzzy relations inspired from Hebbian learning rule. In 2013 Leon Chua published a tutorial underlining the broad span of complex phenomena and applications that memristors span and how they can be used as non-volatile analog memories and can mimic classic habituation and learning phenomena. [103] According to Allied Market Research, memristor market was worth3.2 million in 2015 and will be worth 79.0 million by 2022. [104] ## Memcapacitors and meminductors In 2009, Di Ventra, Pershin, and Chua extended [105] the notion of memristive systems to capacitive and inductive elements in the form of memcapacitors and meminductors, whose properties depend on the state and history of the system, further extended in 2013 by Di Ventra and Pershin. [8] ## Memfractance and memfractor, 2nd- and 3rd-order memristor, memcapacitor and meminductor In September 2014, Mohamed-Salah Abdelouahab, Rene Lozi, and Leon Chua published a general theory of 1st-, 2nd-, 3rd-, and nth-order memristive elements using fractional derivatives. [106] ## Timeline ### 1971 Leon Chua postulated a new two-terminal circuit element characterized by a relationship between charge and flux linkage as a fourth fundamental circuit element. [1] ### 1976 Chua and his student Sung Mo Kang generalized the theory of memristors and memristive systems including a property of zero crossing in the Lissajous curve characterizing current vs. voltage behavior. [36] ### 2008 On May 1, Strukov, Snider, Stewart, and Williams published an article in Nature identifying a link between the 2-terminal resistance switching behavior found in nanoscale systems and memristors. [3] ### 2009 On January 23, Di Ventra, Pershin, and Chua extended the notion of memristive systems to capacitive and inductive elements, namely capacitors and inductors, whose properties depend on the state and history of the system. [105] ### 2015 On July 7, 2015 Knowm Inc announced Self Directed Channel (SDC) memristors commercially. [107] ### 2018 On July 13, 2018 MemSat (Memristor Satellite) was launched to fly a memristor evaluation payload. [108] ## See also ## Related Research Articles The Hall effect is the production of a voltage difference across an electrical conductor, transverse to an electric current in the conductor and to an applied magnetic field perpendicular to the current. It was discovered by Edwin Hall in 1879. For clarity, the original effect is sometimes called the ordinary Hall effect to distinguish it from other "Hall effects" which have different physical mechanisms. Positronium (Ps) is a system consisting of an electron and its anti-particle, a positron, bound together into an exotic atom, specifically an onium. The system is unstable: the two particles annihilate each other to predominantly produce two or three gamma-rays, depending on the relative spin states. The orbit and energy levels of the two particles are similar to that of the hydrogen atom. However, because of the reduced mass, the frequencies of the spectral lines are less than half of the corresponding hydrogen lines. Capacitance is the ratio of the change in an electric charge in a system to the corresponding change in its electric potential. There are two closely related notions of capacitance: self capacitance and mutual capacitance. Any object that can be electrically charged exhibits self capacitance. A material with a large self capacitance holds more electric charge at a given voltage than one with low capacitance. The notion of mutual capacitance is particularly important for understanding the operations of the capacitor, one of the three elementary linear electronic components. Electrical elements are conceptual abstractions representing idealized electrical components, such as resistors, capacitors, and inductors, used in the analysis of electrical networks. All electrical networks can be analyzed as multiple electrical elements interconnected by wires. Where the elements roughly correspond to real components the representation can be in the form of a schematic diagram or circuit diagram. This is called a lumped-element circuit model. In other cases infinitesimal elements are used to model the network, in a distributed-element model. Neuromorphic engineering, also known as neuromorphic computing, is a concept developed by Carver Mead, in the late 1980s, describing the use of very-large-scale integration (VLSI) systems containing electronic analog circuits to mimic neuro-biological architectures present in the nervous system. In recent times, the term neuromorphic has been used to describe analog, digital, mixed-mode analog/digital VLSI, and software systems that implement models of neural systems. The implementation of neuromorphic computing on the hardware level can be realized by oxide-based memristors, spintronic memories, threshold switches, and transistors. The fluctuation–dissipation theorem (FDT) or fluctuation–dissipation relation (FDR) is a powerful tool in statistical physics for predicting the behavior of systems that obey detailed balance. Given that a system obeys detailed balance, the theorem is a general proof that thermodynamic fluctuations in a physical variable predict the response quantified by the admittance or impedance of the same physical variable, and vice versa. The fluctuation–dissipation theorem applies both to classical and quantum mechanical systems. Phase-change memory (also known as PCM, PCME, PRAM, PCRAM, OUM and C-RAM or CRAM is a type of non-volatile random-access memory. PRAMs exploit the unique behaviour of chalcogenide glass. In the older generation of PCM, heat produced by the passage of an electric current through a heating element generally made of TiN was used to either quickly heat and quench the glass, making it amorphous, or to hold it in its crystallization temperature range for some time, thereby switching it to a crystalline state. PCM also has the ability to achieve a number of distinct intermediary states, thereby having the ability to hold multiple bits in a single cell, but the difficulties in programming cells in this way has prevented these capabilities from being implemented in other technologies with the same capability. In information theory, the Rényi entropy generalizes the Hartley entropy, the Shannon entropy, the collision entropy and the min-entropy. Entropies quantify the diversity, uncertainty, or randomness of a system. The entropy is named after Alfréd Rényi. In the context of fractal dimension estimation, the Rényi entropy forms the basis of the concept of generalized dimensions. Power in an electric circuit is the rate of flow of energy past a given point of the circuit. In alternating current circuits, energy storage elements such as inductors and capacitors may result in periodic reversals of the direction of energy flow. In mesoscopic physics, a Coulomb blockade (CB), named after Charles-Augustin de Coulomb's electrical force, is the decrease in electrical conductance at small bias voltages of a small electronic device comprising at least one low-capacitance tunnel junction. Because of the CB, the conductance of a device may not be constant at low bias voltages, but disappear for biases under a certain threshold, i.e. no current flows. Saturable absorption is a property of materials where the absorption of light decreases with increasing light intensity. Most materials show some saturable absorption, but often only at very high optical intensities. At sufficiently high incident light intensity, atoms in the ground state of a saturable absorber material become excited into an upper energy state at such a rate that there is insufficient time for them to decay back to the ground state before the ground state becomes depleted, and the absorption subsequently saturates. Saturable absorbers are useful in laser cavities. The key parameters for a saturable absorber are its wavelength range, its dynamic response, and its saturation intensity and fluence. They are commonly used for passive Q-switching. Chua's circuit is a simple electronic circuit that exhibits classic chaotic behavior. This means roughly that it is a "nonperiodic oscillator"; it produces an oscillating waveform that, unlike an ordinary electronic oscillator, never "repeats". It was invented in 1983 by Leon O. Chua, who was a visitor at Waseda University in Japan at that time. The ease of construction of the circuit has made it a ubiquitous real-world example of a chaotic system, leading some to declare it "a paradigm for chaos". A quantum point contact (QPC) is a narrow constriction between two wide electrically conducting regions, of a width comparable to the electronic wavelength. Quantum point contacts were first reported in 1988 by a Dutch group and, independently, by a British group. They are based on earlier work by the British group which showed how split gates could be used to convert a two-dimensional electron gas into one-dimension, first in silicon and then in gallium arsenide Resistive random-access memory is a type of non-volatile (NV) random-access (RAM) computer memory that works by changing the resistance across a dielectric solid-state material, often referred to as a memristor. This technology bears some similarities to conductive-bridging RAM (CBRAM), and phase-change memory (PCM). A biological neuron model, also known as a spiking neuron model, is a mathematical description of the properties of certain cells in the nervous system that generate sharp electrical potentials across their cell membrane, roughly one millisecond in duration, as shown in Fig. 1. Spiking neurons are known to be a major signaling unit of the nervous system, and for this reason characterizing their operation is of great importance. It is worth noting that not all the cells of the nervous system produce the type of spike that define the scope of the spiking neuron models. For example, cochlear hair cells, retinal receptor cells, and retinal bipolar cells do not spike. Furthermore, many cells in the nervous system are not classified as neurons but instead are classified as glia. A physical neural network is a type of artificial neural network in which an electrically adjustable resistance material is used to emulate the function of a neural synapse. "Physical" neural network is used to emphasize the reliance on physical hardware used to emulate neurons as opposed to software-based approaches which simulate neural networks. More generally the term is applicable to other artificial neural networks in which a memristor or other electrically adjustable resistance material is used to emulate a neural synapse. A carbon nanotube field-effect transistor (CNTFET) refers to a field-effect transistor that utilizes a single carbon nanotube or an array of carbon nanotubes as the channel material instead of bulk silicon in the traditional MOSFET structure. First demonstrated in 1998, there have been major developments in CNTFETs since. In astrophysics the chirp mass of a compact binary system determines the leading-order orbital evolution of the system as a result of energy loss from emitting gravitational waves. Because the gravitational wave frequency is determined by orbital frequency, the chirp mass also determines the frequency evolution of the gravitational wave signal emitted during a binary's inspiral phase. 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