Memristor

Memristor

Leon Chua (1971)
Electronic symbol

A memristor (

Chua and Kang later generalized the concept to memristive systems.

ReRAM

.

The identification of memristive properties in electronic devices has attracted controversy. Experimentally, the ideal memristor has yet to be demonstrated.[3]^[4]

As a fundamental electrical component

Chua in his 1971 paper identified a theoretical symmetry between the non-linear resistor (voltage vs. current), non-linear capacitor (voltage vs. charge), and non-linear inductor (magnetic flux linkage vs. current). From this symmetry he inferred the characteristics of a fourth fundamental non-linear circuit element, linking magnetic flux and charge, which he called the memristor. 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 introduced a mathematical generality.

Derivation and characteristics

The memristor was originally defined in terms of a non-linear functional relationship between magnetic flux linkage Φ_m(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, 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.^[5]

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 charge-dependent rate of change of flux with charge.

${\displaystyle M(q)={\frac {\mathrm {d} \Phi _{\rm {m}}}{\mathrm {d} q}}}$

Substituting the flux as the time integral of the voltage, and charge as the time integral of current, the more convenient forms are;

${\displaystyle M(q(t))={\cfrac {\mathrm {d} \Phi _{\rm {}}/\mathrm {d} t}{\mathrm {d} q/\mathrm {d} t}}={\frac {V(t)}{I(t)}}}$

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.

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 = dΦm / dI
Memristor (M)	Memristance (Wb / C, or ohm)	M = dΦm / 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 charge-dependent

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.

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.

Analogously, we can define a ${\displaystyle W(\phi (t))}$ as memductance.[1]

${\displaystyle i(t)=W(\phi (t))v(t)}$

The

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

at some q). A negative value would mean that it would perpetually supply energy when operated with alternating current.

Modelling and validation

In order to understand the nature of memristor function, some knowledge of fundamental circuit theoretic concepts is useful, starting with the concept of device modeling.^[6]

Engineers and scientists seldom analyze a physical system in its original form. Instead, they construct a model which approximates the behaviour of the system. By analyzing the behaviour of the model, they hope to predict the behaviour of the actual system. The primary reason for constructing models is that physical systems are usually too complex to be amenable to a practical analysis.

In the 20th century, work was done on devices where researchers did not recognize the memristive characteristics. This has raised the suggestion that such devices should be recognised as memristors.^[6] Pershin and Di Ventra^[3] have proposed a test that can help to resolve some of the long-standing controversies about whether an ideal memristor does actually exist or is a purely mathematical concept.

The rest of this article primarily addresses memristors as related to

devices, since the majority of work since 2008 has been concentrated in this area.

Superconducting memristor component

Dr. Paul Penfield, in a 1974 MIT technical report

. This was an early use of the word "memristor" in the context of a circuit device.

One of the terms in the current through a Josephson junction is of the form:

${\displaystyle {\begin{aligned}i_{M}(v)&=\epsilon \cos(\phi _{0})v\\&=W(\phi _{0})v\end{aligned}}}$

where ${\displaystyle \epsilon }$ is a constant based on the physical superconducting materials, ${\displaystyle v}$ is the voltage across the junction and ${\displaystyle i_{M}}$ is the current through the junction.

Through the late 20th century, research regarding this phase-dependent conductance in Josephson junctions was carried out.^{[8][9][10][11]} A more comprehensive approach to extracting this phase-dependent conductance appeared with Peotta and DiVentra's seminal paper in 2014.^[12]

Memristor circuits

Due to the practical difficulty of studying the ideal memristor, we will discuss other electrical devices which can be modelled using memristors. For a mathematical description of a memristive device (systems), see Theory.

A discharge tube can be modelled as a memristive device, with resistance being a function of the number of conduction electrons ${\displaystyle n_{e}}$.^[2]

${\displaystyle {\begin{aligned}v_{M}&=R(n_{e})i_{M}\\{\frac {dn_{e}}{dt}}&=\beta n+\alpha R(n_{e})i_{M}^{2}\end{aligned}}}$

${\displaystyle v_{M}}$ is the voltage across the discharge tube, ${\displaystyle i_{M}}$ is the current flowing through it and ${\displaystyle n_{e}}$ is the number of conduction electrons. A simple memristance function is ${\displaystyle R(n_{e})={\frac {F}{n_{e}}}}$. ${\displaystyle \alpha ,\beta }$ and ${\displaystyle F}$ are parameters depending on the dimensions of the tube and the gas fillings. An experimental identification of memristive behaviour is the "pinched hysteresis loop" in the ${\displaystyle v-i}$ plane. For an experiment that shows such a characteristic for a common discharge tube, see "A physical memristor Lissajous figure" (YouTube). The video also illustrates how to understand deviations in the pinched hysteresis characteristics of physical memristors.^[13][14]

Thermistors can be modelled as memristive devices.^[14]

${\displaystyle {\begin{aligned}v&=R_{0}(T_{0})\exp \left[\beta \left({\frac {1}{T}}-{\frac {1}{T_{0}}}\right)\right]i\\&\equiv R(T)i\\{\frac {dT}{dt}}&={\frac {1}{C}}\left[-\delta \cdot (T-T_{0})+R(T)i^{2}\right]\end{aligned}}}$

${\displaystyle \beta }$ is a material constant, ${\displaystyle T}$ is the absolute body temperature of the thermistor, ${\displaystyle T_{0}}$ is the ambient temperature (both temperatures in Kelvin), ${\displaystyle R_{0}(T_{0})}$ denotes the cold temperature resistance at ${\displaystyle T=T_{0}}$, ${\displaystyle C}$ is the heat capacitance and ${\displaystyle \delta }$ is the dissipation constant for the thermistor.

A fundamental phenomenon that has hardly been studied is memristive behaviour in pn-junctions.^[15] The memristor plays a crucial role in mimicking the charge storage effect in the diode base, and is also responsible for the conductivity modulation phenomenon (that is so important during forward transients).

Criticisms

In 2008, a team at

The HP Labs result was published in the scientific journal Nature.^[17][19] 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.

These devices are intended for applications in

electrical resistance.^[31]

According to the original 1971 definition, the memristor is the fourth fundamental circuit element, forming a non-linear relationship between electric charge and magnetic flux linkage. In 2011,

In 2011, Meuffels and Schroeder noted that one of the early memristor papers included a mistaken assumption regarding ionic conduction.

in 2013.

In a kind of

in 2013.

Within this context, Meuffels and Soni[21] 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.^[22] The free-energy barriers must therefore be high enough to ensure a low bit-error probability of bit operation.^[39] 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.^[39][40]

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. ẋ 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^[22] 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.^[22][41] Di Ventra and Pershin^[22] 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 two-terminal non-volatile memory devices (e.g. "resistance-switching" memory devices (

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 two-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 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

A 2014 article from researchers of

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.^[44]

Experimental tests

• 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

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 (

Theory

In 2008, researchers from HP Labs introduced a model for a memristance function based on thin films of titanium dioxide.^[17] For R_ON ≪ R_OFF 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 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 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 R_on to R_off in time T_on to T_off, the charge must change by ΔQ = Q_on−Q_off.

${\displaystyle E_{\mathrm {switch} }=\ V^{2}\int _{T_{\mathrm {off} }}^{T_{\mathrm {on} }}{\frac {\mathrm {d} t}{M(q(t))}}=\ V^{2}\int _{Q_{\mathrm {off} }}^{Q_{\mathrm {on} }}{\frac {\mathrm {d} q}{I(q)M(q)}}=\ V^{2}\int _{Q_{\mathrm {off} }}^{Q_{\mathrm {on} }}{\frac {\mathrm {d} q}{V(q)}}=\ V\Delta Q}$

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

, 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".^[17] 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

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

. 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

Memristive networks and mathematical models of circuit interactions

The concept of memristive networks was first introduced by Leon Chua in his 1965 paper "Memristive Devices and Systems." Chua proposed the use of memristive devices as a means of building artificial neural networks that could simulate the behavior of the human brain. In fact, memristive devices in circuits have complex interactions due to Kirchhoff's laws. A memristive network is a type of artificial neural network that is based on memristive devices, which are electronic components that exhibit the property of memristance. In a memristive network, the memristive devices are used to simulate the behavior of neurons and synapses in the human brain. The network consists of layers of memristive devices, each of which is connected to other layers through a set of weights. These weights are adjusted during the training process, allowing the network to learn and adapt to new input data. One advantage of memristive networks is that they can be implemented using relatively simple and inexpensive hardware, making them an attractive option for developing low-cost artificial intelligence systems. They also have the potential to be more energy efficient than traditional artificial neural networks, as they can store and process information using less power. However, the field of memristive networks is still in the early stages of development, and more research is needed to fully understand their capabilities and limitations. For the simplest model with only memristive devices with voltage generators in series, there is an exact and in closed form equation (Caravelli–Traversa–Di Ventra equation, CTDV)^[49] which describes the evolution of the internal memory of the network for each device. For a simple memristor model (but not realistic) of a switch between two resistance values, given by the Williams-Strukov model ${\displaystyle R(x)=R_{off}(1-x)+R_{on}x}$, with ${\displaystyle dx/dt=I/\beta -\alpha x}$, there is a set of nonlinearly coupled differential equations that takes the form:

${\displaystyle {\frac {d{\vec {x}}}{dt}}=-\alpha {\vec {x}}+{\frac {1}{\beta }}(I-\chi \Omega X)^{-1}\Omega {\vec {S}}}$

where ${\displaystyle X}$ is the diagonal matrix with elements ${\displaystyle x_{i}}$ on the diagonal, ${\displaystyle \alpha ,\beta ,\chi }$ are based on the memristors physical parameters. The vector ${\displaystyle {\vec {S}}}$ is the vector of voltage generators in series to the memristors. The circuit topology enters only in the projector operator ${\displaystyle \Omega ^{2}=\Omega }$, defined in terms of the cycle matrix of the graph. The equation provides a concise mathematical description of the interactions due to Kirchhoff 's laws. Interestingly, the equation shares many properties in common with a Hopfield network, such as the existence of Lyapunov functions and classical tunnelling phenomena.^[50] In the context of memristive networks, the CTD equation may be used to predict the behavior of memristive devices under different operating conditions, or to design and optimize memristive circuits for specific applications.

Extended systems

Some researchers have raised the question of the scientific legitimacy of HP's memristor models in explaining the behavior of

One example[51] 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 ${\displaystyle a}$ to account for an observed nanobattery effect which violates the predicted zero-crossing pinched hysteresis effect.^[25]

${\displaystyle {\begin{aligned}y(t)&=g_{0}({\textbf {x}},u)(u(t)-a),\\{\dot {\textbf {x}}}&=f({\textbf {x}},u)\end{aligned}}}$

Implementation of hysteretic current-voltage memristors

There exist implementations of memristors with a hysteretic current-voltage curve or with both hysteretic current-voltage curve and hysteretic flux-charge curve [arXiv:2403.20051]. Memristors with hysteretic current-voltage curve use a resistance dependent on the history of the current and voltage and bode well for the future of memory technology due to their simple structure, high energy efficiency, and high integration [DOI: 10.1002/aisy.202200053].

Titanium dioxide memristor

Interest in the memristor revived when an experimental solid-state version was reported by

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

The HP device is composed of a thin (50

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.^[57] Samsung has a U.S. patent for oxide-vacancy based switches similar to that described by Williams.^[58]

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,^[59] which bodes well for the future of the technology.^[60] At these densities it could easily rival the current sub-25 nm flash memory technology.

Silicon dioxide memristor

It seems that memristance has been reported in

thin films of silicon dioxide as early as the 1960s .^[61]

However, hysteretic conductance in silicon was associated to memristive effects only in 2009. ^[62] More recently, beginning in 2012, Tony Kenyon, Adnan Mehonic and their group clearly demonstrated that the resistive switching in silicon oxide thin films is due to the formation of oxygen vacancy filaments in defect-engineered silicon dioxide, having probed directly the movement of oxygen under electrical bias, and imaged the resultant conductive filaments using conductive atomic force microscopy. ^[63]

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.

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.^[65]

In 2010, Alibart, Gamrat, Vuillaume et al.^[66] introduced a new hybrid organic/nanoparticle device (the NOMFET : Nanoparticle Organic Memory Field Effect Transistor), which behaves as a memristor^[67] and which exhibits the main behavior of a biological spiking synapse. This device, also called a synapstor (synapse transistor), was used to demonstrate a neuro-inspired circuit (associative memory showing a pavlovian learning).^[68]

In 2012, Crupi, Pradhan and Tozer described a proof of concept design to create neural synaptic memory circuits using organic ion-based memristors.

that act as spatiotemporal filters that process visual signals such as edges and moving lines.

In 2012, Erokhin and co-authors have demonstrated a stochastic three-dimensional matrix with capabilities for learning and adapting based on polymeric memristor.^[70]

Layered memristor

In 2014, Bessonov et al. reported a flexible memristive device comprising a

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.

Ferroelectric memristor

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 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 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.

Carbon nanotube memristor

In 2013, Ageev, Blinov et al.[78] reported observing memristor effect in structure based on vertically aligned carbon nanotubes studying bundles of CNT by scanning tunneling microscope.

Later it was found^[79] 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).

Biomolecular memristor

Biomaterials have been evaluated for use in artificial synapses and have shown potential for application in neuromorphic systems.^[80] In particular, the feasibility of using a collagen‐based biomemristor as an artificial synaptic device has been investigated,^[81] whereas a synaptic device based on lignin demonstrated rising or lowering current with consecutive voltage sweeps depending on the sign of the voltage^[82] furthermore a natural silk fibroin demonstrated memristive properties;^[83] spin-memristive systems based on biomolecules are also being studied.^[84]

In 2012,

Spin memristive systems

Spintronic memristor

Chen and Wang, researchers at disk-drive manufacturer

Memristance in a magnetic tunnel junction

The

).

Extrinsic mechanism

Based on research performed between 1999 and 2003, Bowen et al. published experiments in 2006 on a

). 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^[91] and 2009.^[92] While the drift of oxygen vacancies within the insulating MgO layer has been proposed to describe the observed memristive effects,^[92] another explanation could be charge trapping/detrapping on the localized states of oxygen vacancies^[93] and its impact^[94] 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^[95] or multiferroicity.^[96]

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.^[97]

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.^[98]

Spin memristive system

A fundamentally different mechanism for memristive behavior has been proposed by Pershin and

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, Kris Campbell formally introduced the self-directed channel (SDC) memristor.[102] The SDC device is the first memristive device available commercially to researchers, students and electronics enthusiast worldwide.^[103] 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).

Implementation of hysteretic flux-charge memristors

There exist implementations of memristors with both hysteretic current-voltage curve and hysteretic flux-charge curve [arXiv:2403.20051]. Memristors with both hysteretic current-voltage curve and hysteretic flux-charge curve use a memristance dependent on the history of the flux and charge. Those memristors can merge the functionality of the arithmetic logic unit and of the memory unit without data transfer [DOI: 10.1002/adfm.201303365]. 

Time-integrated Formingfree memristor

Time-integrated Formingfree (TiF) memristors reveal a hysteretic flux-charge curve with two distinguishable branches in the positive bias range and with two distinguishable branches in the negative bias range. And TiF memristors also reveal a hysteretic current-voltage curve with two distinguishable branches in the positive bias range and with two distinguishable branches in the negative bias range. The memristance state of a TiF memristor can be controlled by both the flux and the charge [DOI: 10.1063/1.4775718]. A TiF memristor was first demonstrated by Heidemarie Schmidt and her team in 2011 [DOI: 10.1063/1.3601113]. This TiF memristor is composed of a BiFeO₃ thin film between metallically conducting electrodes, one gold, the other platinum. The hysteretic flux-charge curve of the TiF memristor changes its slope continuously in one branch in the positive and in one branch in the negative bias range (write branches) and has a constant slope in one branch in the positive and in one branch in the negative bias range (read branches) [arXiv:2403.20051]. According to Leon O. Chua [Reference 1: 10.1.1.189.3614] the slope of the flux-charge curve corresponds to the memristance of a memristor or to its internal state variables. The TiF memristors can be considered as memristors with a constant memristance in the two read branches and with a reconfigurable memristance in the two write branches. The physical memristor model which describes the hysteretic current-voltage curves of the TiF memristor implements static and dynamic internal state variables in the two read branches and in the two write branches [arXiv:2402.10358].

The static and dynamic internal state variables of a non-linear memristors can be used to implement operations on non-linear memristors representing linear, non-linear, and even transcendental, e.g. exponential or logarithmic, input-output functions.

The transport characteristics of the TiF memristor in the small current – small voltage range are non-linear. This non-linearity well compares to the non-linear characteristics in the small current – small voltage range of the basic former and present building blocks in the arithmetic logic unit of von-Neumann computers, i.e. of vacuum tubes and of transistors. In contrast to vacuum tubes and transistors, the signal output of hysteretic flux-charge memristors, i.e. of TiF memristors, is not lost when the operation power is switched off before storing the signal output to the memory. Therefore, hysteretic flux-charge memristors are said to merge the functionality of the arithmetic logic unit and of the memory unit without data transfer [DOI: 10.1002/adfm.201303365]. The transport characteristics in the small current – small voltage range of hysteretic current-voltage memristors are linear. This explains why hysteretic current-voltage memristors are well established memory units and why they can not merge the functionality of the arithmetic logic unit and of the memory unit without data transfer [arXiv:2403.20051].

Potential applications

Memristors remain a laboratory curiosity, as yet made in insufficient numbers to gain any commercial applications. Despite this lack of mass availability, according to Allied Market Research the memristor market was worth $3.2 million in 2015 and was at the time projected to be worth $79.0 million by 2022.^[104] In fact, it was worth $190.0 million in 2022.^[105]

A potential application of memristors is in analog memories for superconducting quantum computers.^[12]

Memristors can potentially be fashioned into

Memristors have applications in

In 2009, a simple electronic circuit[123] consisting of an LC network and a memristor was used to model experiments on adaptive behavior of unicellular organisms.^[124] 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.^[124] 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.^[125] MoNETA is the first large-scale neural network model to implement whole-brain circuits to power a virtual and robotic agent using memristive hardware.^[126] Application of the memristor crossbar structure in the construction of an analog soft computing system was demonstrated by Merrikh-Bayat and Shouraki.^[127] In 2011, they showed^[128] 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.^[129]

Derivative devices

Memistor and memtransistor

The memistor and memtransistor are transistor-based devices which include memristor function.

Memcapacitors and meminductors

In 2009, Di Ventra, Pershin, and Chua extended^[130] 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.^[22]

Memfractance and memfractor, 2nd- and 3rd-order memristor, memcapacitor and meminductor

In September 2014,

History

Precursors

]

Theoretical description

Twenty-first century

On May 1, 2008, Strukov, Snider, Stewart, and Williams published an article in Nature identifying a link between the two-terminal resistance switching behavior found in nanoscale systems and memristors.[17]

On 23 January 2009,

In July 2014, the MeMOSat/

]

On 7 July 2015, Knowm Inc announced Self Directed Channel (SDC) memristors commercially.[138] These devices remain available in small numbers.

On 13 July 2018, MemSat (Memristor Satellite) was launched to fly a memristor evaluation payload.^[139]

In 2021,

In May 2023, TECHiFAB GmbH [https://techifab.com/] announced TiF memristors commercially. [arXiv: 2403.20051, arXiv: 2402.10358] These TiF memristors remain available in small and medium numbers.

In the September 2023 issue of Science Magazine, Chinese scientists Wenbin Zhang et al. described the development and testing of a memristor-based integrated circuit, designed to dramatically increase the speed and efficiency of Machine Learning and Artificial Intelligence tasks, optimized for Edge Computing applications.^[142]

See also

• 3D XPoint
• Electrical element
• Hybrid Memory Cube
• List of emerging technologies
• Neuromorphic engineering
• Trancitor

References

Further reading

• Chen, Dongmin; Chua, Leon O.; Hwang, Cheol Seong; Wang, Shih-Yuan; Waser, Rainer; Williams, R. Stanley; Yang, Jianhua, eds. (March 2011). "Special Issue: Memristive and Resistive Devices and Systems". Applied Physics A. 102 (4).
• Mazumder, P.; Kang, S. M.; Waser, R., eds. (June 2012). "Special Issue: MEMRISTORS: DEVICES, MODELS, AND APPLICATIONS".
  doi:10.1109/JPROC.2012.2197452
  .
• Tetzlaff, Ronald, ed. (2013). Memristors and Memristive Systems.
  ISBN 978-1-4614-9068-5
  .
• Adamatzky, Andrew; Chua, Leon, eds. (2013). Memristor Networks.
  S2CID 39739718
  .
• Atkin, Keith (May 2013). "An introduction to the memristor". Physics Education. 48 (3): 317–321.
  S2CID 121268844
  .
• Gale, Ella (2014-10-01). "TiO₂-based memristors and ReRAM: materials, mechanisms and models (a review)". Semiconductor Science and Technology. 29 (10): 104004.
  S2CID 5686212
  .
• Traversa, Fabio Lorenzo; Di Ventra, Massimiliano (November 2015). "Universal Memcomputing Machines". IEEE Transactions on Neural Networks and Learning Systems. 26 (11): 2702–2715.
  S2CID 1406042
  .
• Caravelli, Francesco; Carbajal, Juan Pablo (January 2019). "Memristors for the curious outsiders". Technologies. 6 (4): 118.
  S2CID 54464654
  .
• Maan, Akshay Kumar; Jayadevi, Deepthi Anirudhan; James, Alex Pappachen (August 2017). "A Survey of Memristive Threshold Logic Circuits". IEEE Transactions on Neural Networks and Learning Systems. 28 (8): 1734–1746.
  S2CID 1798273
  .
• Ghosh, M., Singh, A., Borah, S. S., Vista, J., Ranjan, A., Kumar, S. (2022). "MOSFET-based memristor for high-frequency signal processing". IEEE Transactions on Electron Devices. 69 (5): 2248–2255.
  S2CID 247889089
  .
• Singh, A., Borah, S. S., Ghosh, M. (2021), Simple grounded meminductor emulator using transconductance amplifier, IEEE

External links

Wikimedia Commons has media related to Memristors.

• Finding the missing memristor on
  YouTube
• Interactive database of memristor papers (2013)
• Simonite, Tom (2015-04-21). "Machine Dreams". Technology Review. Retrieved 2017-12-05.
• "Leon Chua: A bulb versus Google go player" - (in Polish) an interview with Leon Chua, the creator of memristor
• "Leon Chua: A bulb versus Google go player" - (in English) an interview with Leon Chua, the creator of memristor