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Type  Passive 

Working principle  Memristance 
Invented  Leon Chua (1971) 
Electronic symbol  
Computer memory types 

Volatile 
RAM 
In development 
Historical 

Nonvolatile 
ROM 
NVRAM 
Early stage NVRAM 
SAM 
In development 
Historical 

A memristor (/ˈmɛmrɪstər/; a portmanteau of memory resistor) is a hypothetical nonlinear passive twoterminal 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 socalled nonvolatility 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]}
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 RRAM devices to the memristor concept; the HP result was published in Nature.^{[3]} Following this claim, Leon Chua has argued that the memristor definition could be generalized to cover all forms of twoterminal nonvolatile memory devices based on resistance switching effects.^{[2]} There are, however, some serious doubts as to whether the memristor can actually exist in physical reality.^{[5]}^{[6]}^{[7]} Additionally, some experimental evidence contradicts Chua's generalization since a nonpassive nanobattery effect is observable in resistance switching memory.^{[8]} Chua also argued that the memristor is the oldest known circuit element, with its effects predating the resistor, capacitor and inductor.^{[9]}
These devices are intended for applications in nanoelectronic memories, computer logic and neuromorphic/neuromemristive computer architectures.^{[10]} Commercial availability of memristor memory has been estimated as 2018.^{[11]} 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.^{[12]}
In his 1971 paper, Chua extrapolated a conceptual symmetry between the nonlinear resistor (voltage vs. current), nonlinear capacitor (voltage vs. charge) and nonlinear inductor (magnetic flux linkage vs. current). He then inferred the possibility of a memristor as another fundamental nonlinear circuit element linking magnetic flux and charge. In contrast to a linear (or nonlinear) 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.
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 twoterminal 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 memristance M and is similar to variable resistance.
The memristor definition is based solely on the fundamental circuit variables of current and voltage and their timeintegrals, just like the resistor, capacitor and inductor. Unlike those three elements however, which are allowed in linear timeinvariant 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 timeinvariant 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.
According to the original 1971 definition, the memristor was the fourth fundamental circuit element, forming a nonlinear relationship between electric charge and magnetic flux linkage. In 2011 Chua argued for a broader definition that included all 2terminal nonvolatile memory devices based on resistance switching.^{[2]} Williams argued that MRAM, phase change memory and RRAM were memristor technologies.^{[15]} Some researchers argued that biological structures such as blood^{[16]} and skin^{[17]}^{[18]} fit the definition. Others argued that the memory device under development by HP Labs and other forms of RRAM were not memristors but rather part of a broader class of variable resistance systems^{[19]} and that a broader definition of memristor is a scientifically unjustifiable land grab that favored HP's memristor patents.^{[20]}
In 2011, Meuffels and Schroeder noted that one of the early memristor papers included a mistaken assumption regarding ionic conduction.^{[21]} In 2012, Meuffels and Soni discussed some fundamental issues and problems in the realization of memristors.^{[5]} They indicated inadequacies in the electrochemical modelling presented in the Nature paper "The missing memristor found"^{[3]} because the impact of concentration polarization effects on the behavior of metal−TiO_{2−x}−metal structures under voltage or current stress was not considered. This critique was referred to by Valov et al.^{[8]} in 2013.
In a kind of thought experiment, Meuffels and Soni^{[5]} furthermore revealed a severe inconsistency: If a currentcontrolled memristor with the socalled nonvolatility 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^{[6]} in 2013.
Within this context, Meuffels and Soni^{[5]} pointed to a fundamental thermodynamic principle: Nonvolatile 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.^{[6]} The free energy barriers must therefore be high enough to ensure a low biterror probability of bit operation.^{[22]} Consequently, there is always a lower limit of energy requirement – depending on the required biterror probability – for intentionally changing a bit value in any memory device.^{[22]}^{[23]}
In the general concept of memristive system the defining equations are (see: #Theory):
Here, u(t) is an input signal and y(t) an output signal. The vector x represents a set of n state variables describing the different internal memory states of the device. ẋ is the timedependent 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 nonvolatile memory elements, e.g., resistive randomaccess 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:
Here, ξ(t) is, e.g., white Gaussian current or voltage noise. On base of an analytical or numerical analysis of the timedependent 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 poweroff mode?
Such an analysis was performed by Di Ventra and Pershin^{[6]} with regard to the genuine currentcontrolled memristor. As the proposed dynamic state equation provides no physical mechanism enabling such a memristor to cope with inevitable thermal fluctuations, a currentcontrolled memristor would erratically change its state in course of time just under the influence of current noise.^{[6]}^{[24]} Di Ventra and Pershin^{[6]} 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 socalled "stochastic catastrophe". A currentcontrolled memristor can thus not exist as a solid state device in physical reality.
The abovementioned thermodynamic principle furthermore implies that the operation of 2terminal, nonvolatile memory devices (e.g. "resistance switching" memory devices (RRAM)) cannot be associated with the memristor concept, i.e., such devices cannot per se 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 (RRAM) processes. If the free energy barriers are not high enough, the memory device can even switch without having to do anything.
When a 2terminal, nonvolatile memory device is found to be in a distinct resistance state {j}, there exists therefore no physical onetoone relationship between its present state and its foregoing voltage history. The switching behavior of individual nonvolatile memory devices can thus per se not 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, viz. no extra energy remains in the system after it has been brought from one resistance state x_{i} to another one x_{j}. Thus, the internal energy of the memristor device in state x_{i}, U(V, T, x_{i}), would be the same as in state x_{j}, U(V, T, x_{j}), 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 (hightolow resistance switching) and reset time (lowtohigh resistance switching) and do not provide ionic mobility values consistent with experimental data. Nonlinear ionic drift models have been proposed to compensate for this deficiency.^{[25]}
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 (physicsbased) models such as Pickett's model or Menzel's ECM model (Menzel is a coauthor of that paper) 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.^{[26]}
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 it was not a memristor.^{[27]}
In the article "The Missing Memristor has Not been Found", published in Scientific Reports in 2015 by Vongehr and Meng,^{[7]} 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 RRAMs, and the conceptual memristors may have no equivalence at all.^{[7]}^{[28]}
Chua suggested experimental tests to determine if a device may properly be categorized as a memristor:^{[29]}
According to Chua^{[30]}^{[31]} 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 redoxbased 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.^{[8]}
The memristor was originally defined in terms of a nonlinear functional relationship between magnetic flux linkage Φ_{m}(t) and the amount of electric charge that has flowed, q(t):^{[1]}
The magnetic flux linkage, Φ_{m}, 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 Φ_{m} may be regarded as the integral of voltage over time.^{[32]}
In the relationship between Φ_{m} 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 chargedependent rate of change of flux with charge.
Substituting the flux as the time integral of the voltage, and charge as the time integral of current, the more convenient form is
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.
Device  Characteristic 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 = dΦ_{m} / dI 
Memristor (M)  Memristance (Wb / C, or ohm)  M = dΦ_{m} / dq 
The above table covers all meaningful ratios of differentials of I, Q, Φ_{m}, and V. No device can relate dI to dq, or dΦ_{m} to dV, because I is the derivative of Q and Φ_{m} is the integral of V.
It can be inferred from this that memristance is chargedependent 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
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, I^{2}R.
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 R_{ON} ≪ R_{OFF} the memristance function was determined to be
where R_{OFF} represents the high resistance state, R_{ON} represents the low resistance state, μ_{v} 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 nonlinear ionic drift and boundary effects.
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 R_{on} to R_{off} in time T_{on} to T_{off}, the charge must change by ΔQ = Q_{on}−Q_{off}.
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 capacitorbased. 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 "hardswitching regime".^{[3]} Another kind of switch would have a cyclic M(q) so that each offon event would be followed by an onoff event under constant bias. Such a device would act as a memristor under all conditions, but would be less practical.
The memristor was generalized to memristive systems in Chua's 1976 paper.^{[29]} 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 empiricallyobserved 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 randomaccess memory relating the theory to active areas of research.
In the more general concept of an nth order memristive system the defining equations are
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 currentcontrolled 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 voltagecontrolled 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) .
One of the resulting properties of memristors and memristive systems is the existence of a pinched hysteresis effect.^{[33]} For a currentcontrolled 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 twoterminal 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 noncrossing pinched hysteresis curves (denoted TypeII) cannot be described by memristors.^{[34]}
Some researchers have raised the question of the scientific legitimacy of HP's memristor models in explaining the behavior of ReRAM.^{[19]}^{[20]} and have suggested extended memristive models to remedy perceived deficiencies.^{[8]}
One example^{[35]} attempts to extend the memristive systems framework by including dynamic systems incorporating higherorder derivatives of the input signal u(t) as a series expansion
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 zerocrossing 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 zerocrossing pinched hysteresis effect.^{[8]}
In 2017, Campbell formally introduced the SelfDirected Channel (SDC) memristor.^{[36]} The SDC device is the first memristive device available commercially to researchers, students and electronics enthusiast worldwide.^{[37]} The SDC device is operational immediately after fabrication. Switching occurs in the Ge2Se3 active layer, where a key feature of the device, GeGe homopolar bonds, are found. The three layers consisting of Ge2Se3/Ag/Ge2Se3, directly below the top tungsten electrode, mix together during deposition and jointly form the silversource layer. This silversource layer is not in direct contact with the active layer. This allows the device to have significantly higher processing and operating temperatures (above 250 °C and at least 150 °C, respectively) since silver does not migrate into the active layer at high temperatures, and the active layer maintains a high glass transition temperature (~350 °C). These processing and operating temperatures are higher than most ionconducting chalcogenide device types, including the Sbased glasses (e.g. GeS) that need to be photodoped or thermally annealed. It is a combination of these factors that allow the SDC device to operate over a wide range of temperatures, including longterm continuous operation at 150 °C.
Interest in the memristor revived when an experimental solid state version was reported by R. Stanley Williams of Hewlett Packard in 2007.^{[38]}^{[39]}^{[40]} The article was the first to demonstrate that a solidstate 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 TiO_{2} memristor, the resistance switching characteristics of titanium dioxide were originally described in the 1960s.^{[41]}
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 nondepleted layer. When an electric field is applied, the oxygen vacancies drift (see Fast ion conductor), changing the boundary between the highresistance and lowresistance 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.^{[42]}
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=∫Idt, but rather keeps q at an upper bound and M fixed, thus acting as a constant resistor until current is reversed.
Memory applications of thinfilm 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.^{[43]} Samsung has a U.S. patent for oxidevacancy based switches similar to that described by Williams.^{[44]} Williams also has a U.S. patent application related to the memristor construction.^{[45]}
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,^{[46]} which bodes well for the future of the technology.^{[47]} At these densities it could easily rival the current sub25 nm flash memory technology.
In 2004, Krieger and Spitzer described dynamic doping of polymer and inorganic dielectriclike materials that improved the switching characteristics and retention required to create functioning nonvolatile memory cells.^{[48]} 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.^{[49]}
In 2010, Alibart, Gamrat, Vuillaume et al.^{[50]} introduced a new hybrid organic/nanoparticle device (the NOMFET : Nanoparticle Organic Memory Field Effect Transistor), which behaves as a memristor ^{[51]} and which exhibits the main behavior of a biological spiking synapse.This device, also called synapstor (synapse transistor), was used to demonstrate a neuroinspired circuit (associative memory showing a pavlovian learning) ^{[52]}
In 2012, Crupi, Pradhan and Tozer described a proof of concept design to create neural synaptic memory circuits using organic ionbased memristors.^{[53]} The synapse circuit demonstrated longterm 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.
In 2014, Bessonov et al. reported a flexible memristive device comprising a MoO_{x}/MoS_{2} heterostructure sandwiched between silver electrodes on a plastic foil.^{[54]} The fabrication method is entirely based on printing and solutionprocessing technologies using twodimensional 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.
The ferroelectric memristor^{[55]} 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: R_{OFF} ≫ R_{ON} (an effect called Tunnel ElectroResistance). 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 R_{ON} or R_{OFF}, 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.
In 2013, Ageev, Blinov et al.^{[56]} reported observing memristor effect in structure based on vertically aligned carbon nanotubes studying bundles of CNT by scanning tunneling microscope.
Later it was found^{[57]} that CNT memristive switching is observed when a nanotube has a nonuniform elastic strain ΔL0. It was shown that the memristive switching mechanism of strained СNT is based on the formation and subsequent redistribution of nonuniform elastic strain and piezoelectric field Edef in the nanotube under the influence of an external electric field E(x,t).
Chen and Wang, researchers at diskdrive manufacturer Seagate Technology described three examples of possible magnetic memristors.^{[58]} 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.^{[59]} 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.^{[60]}
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 (spintransfer torque).
Based on research performed between 1999 and 2003, Bowen et al. published experiments in 2006 on a magnetic tunnel junction (MTJ) endowed with bistable spindependent states^{[61]}(resistive switching). The MTJ consists in a SrTiO3 (STO) tunnel barrier that separates halfmetallic 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 spinpolarized current flows from the halfmetallic 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 MgObased memristive switching within MgObased MTJs appeared starting in 2008^{[62]} and 2009.^{[63]} While the drift of oxygen vacancies within the insulating MgO layer has been proposed to describe the observed memristive effects,^{[63]} another explanation could be charge trapping/detrapping on the localized states of oxygen vacancies^{[64]} and its impact^{[65]} 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^{[66]} or multiferroicity.^{[67]}
The magnetization state of a MTJ can be controlled by Spintransfer 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.^{[68]}
The combination of intrinsic (spintransfer torque) and extrinsic (resistive switching) mechanisms naturally leads to a secondorder memristive system described by the state vector x = (x_{1},x_{2}), where x_{1} describes the magnetic state of the electrodes and x_{2} denotes the resistive state of the MgO barrier. In this case the change of x_{1} is currentcontrolled (spin torque is due to a high current density) whereas the change of x_{2} is voltagecontrolled (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 synapseneuron system.^{[69]}
A fundamentally different mechanism for memristive behavior has been proposed by Pershin^{[70]} and Di Ventra.^{[71]}^{[72]} The authors show that certain types of semiconductor spintronic structures belong to a broad class of memristive systems as defined by Chua and Kang.^{[29]} 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,^{[73]} 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.
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Williams' solidstate 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 nonvolatile solidstate memory, which would allow greater data density than hard drives with access times similar to DRAM, replacing both components.^{[14]} HP prototyped a crossbar latch memory that can fit 100 gigabits in a square centimeter,^{[74]} and proposed a scalable 3D design (consisting of up to 1000 layers or 1 petabit per cm^{3}).^{[75]} In May 2008 HP reported that its device reaches currently about onetenth the speed of DRAM.^{[76]} The devices' resistance would be read with alternating current so that the stored value would not be affected.^{[77]} In May 2012 it was reported that access time had been improved to 90 nanoseconds if not faster, approximately one hundred times faster than contemporaneous flash memory, while using one percent as much energy.^{[78]}
Memristor patents include applications in programmable logic,^{[79]} signal processing,^{[80]} neural networks,^{[81]} control systems,^{[82]} reconfigurable computing,^{[83]} braincomputer interfaces^{[84]} and RFID.^{[85]} Memristive devices are potentially used for stateful logic implication, allowing a replacement for CMOSbased logic computation. Several early works in this direction are reported.^{[86]} ^{[87]}
In 2009, a simple electronic circuit^{[88]} consisting of an LC network and a memristor was used to model experiments on adaptive behavior of unicellular organisms.^{[89]} 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.^{[89]} 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.^{[90]} MoNETA is the first largescale neural network model to implement wholebrain circuits to power a virtual and robotic agent using memristive hardware.^{[91]} Application of the memristor crossbar structure in the construction of an analog soft computing system was demonstrated by MerrikhBayat and Shouraki.^{[92]} In 2011 they showed^{[93]} how memristor crossbars can be combined with fuzzy logic to create an analog memristive neurofuzzy 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 nonvolatile analog memories and can mimic classic habituation and learning phenomena.^{[94]}
According to Allied Market Research memristor market was worth $3.2 million in 2015 and will be worth $79.0 million by 2022.^{[95]}
In 2009, Di Ventra, Pershin and Chua extended^{[96]} 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.^{[6]}
In September 2014, MohamedSalah Abdelouahab, Rene Lozi and Leon Chua, published a general theory of 1st, 2nd, 3rd order and nth order memristive element using fractional derivatives.^{[97]}
Leon Chua postulated a new twoterminal circuit element characterized by a relationship between charge and flux linkage as a fourth fundamental circuit element.^{[1]}
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.^{[29]}
On May 1 Strukov, Snider, Stewart and Williams published an article in Nature identifying a link between the 2terminal resistance switching behavior found in nanoscale systems and memristors.^{[3]}
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.^{[96]}
On July 7, 2015 Knowm Inc announced Self Directed Channel (SDC) memristors commercially.^{[98]}
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