A memristor (; a

portmanteau A portmanteau word, or portmanteau (, ) is a blend of wordstwo-terminal

electrical component An electronic component is any basic discrete device or physical entity in an electronic system used to affect electrons or their associated fields. Electronic components are mostly industrial products, available in a singular form and are n ...

relating

electric charge Electric charge is the physical property of matter that causes charged matter to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative'' (commonly carried by protons and electrons res ...

and magnetic flux linkage. It was described and named in 1971 by

Leon Chua Leon Ong Chua (; ; born June 28, 1936) is an American electrical engineer and computer scientist. He is a professor in the electrical engineering and computer sciences department at the University of California, Berkeley, which he joined in 1971. ...

, completing a theoretical quartet of fundamental electrical components which comprises also the

resistor 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 e ...

capacitor A capacitor is a device that stores electrical energy in an electric field by virtue of accumulating electric charges on two close surfaces insulated from each other. It is a passive electronic component with two terminals. The effect of ...

and

inductor 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 c ...

. Chua and Kang later generalized the concept to memristive systems. Such a system comprises a circuit, of multiple conventional components, which mimics key properties of the ideal memristor component and is also commonly referred to as a memristor. Several such memristor system technologies have been developed, notably

ReRAM Resistive random-access memory (ReRAM or RRAM) 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. ReRAM bears s ...

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

As a fundamental electrical component

Two-terminal non-linear circuit elements

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 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. Hi ...

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''): :

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. 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. :

M(q)=\frac

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

M(q(t)) = \cfrac=\frac

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. 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 resistance. If ''M''(''q''(''t'')) is a constant, then we obtain

Ohm's Law Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equa ...

''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 :

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 Alternating current (AC) is an electric current which periodically reverses direction and changes its magnitude continuously with time in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in whic ...

, 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. Analogously, we can define a

W(\phi(t))

as menductance. :

i(t)=W(\phi(t))v(t)

The power consumption characteristic recalls that of a resistor, ''I''²''R''. :

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.

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. 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. Pershin and Di Ventra 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 mentions the memristor in connection with Josephson junctions. 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: :

\begin
             i_M(v) &= \epsilon\cos(\phi_0)v \\
&=W(\phi_0)v
\end

where

\epsilon

is a constant based on the physical superconducting materials,

v

is the voltage across the junction and

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. A more comprehensive approach to extracting this phase-dependent conductance appeared with Peotta and DiVentra's seminal paper in 2014.

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 theory is a rational type of abstract thinking about a phenomenon, or the results of such thinking. The process of contemplative and rational thinking is often associated with such processes as observational study or research. Theories may ...

. A discharge tube can be modelled as a memristive device, with resistance being a function of the number of conduction electrons

n_e

. :

\begin
v_M&=R(n_e)i_M\\
\frac&=\beta n+\alpha R(n_e)i^2_M
\end

v_M

is the voltage across the discharge tube,

i_M

is the current flowing through it and

n_e

is the number of conduction electrons. A simple memristance function is

R(n_e)=\frac

\alpha,\beta

and

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

v-i

plane. For an experiment that shows such a characteristic for a common discharge tube, se
"A physical memristor Lissajous figure" (YouTube)
The video also illustrates how to understand deviations in the pinched hysteresis characteristics of physical memristors. Thermistors can be modelled as memristive devices. :

\end

\beta

is a material constant,

T

is the absolute body temperature of the thermistor,

T_0

is the ambient temperature (both temperatures in Kelvin),

R_0(T_0)

denotes the cold temperature resistance at

T=T_0

C

is the heat capacitance and

\delta

is the dissipation constant for the thermistor. A fundamental phenomenon that has hardly been studied is memristive behaviour in pn-junctions. 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

HP Labs 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, in ...

found experimental evidence for the Chua's memristor based on an analysis of a

thin film 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 (a process referred to as deposition) is a fundamental step in many ...

titanium dioxide Titanium dioxide, also known as titanium(IV) oxide or titania , is the inorganic compound with the chemical formula . When used as a pigment, it is called titanium white, Pigment White 6 (PW6), or CI 77891. It is a white solid that is insolu ...

, thus connecting the operation of

devices to the memristor concept. According to HP Labs, the memristor would operate in the following way: the memristor's

electrical resistance The electrical resistance of an object is a measure of its opposition to the flow of electric current. Its reciprocal quantity is , measuring the ease with which an electric current passes. Electrical resistance shares some conceptual parallel ...

is not constant but depends on the current that had previously flowed through the device, i.e., its present resistance depends on how much electric charge has previously flowed through it and in what direction; the device remembers its history—the so-called ''non-volatility property''. When the electric power supply is turned off, the memristor remembers its most recent resistance until it is turned on again. The HP Labs result was published in the scientific journal ''

Nature Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans are ...

''. 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. Chua also argued that the memristor is the oldest known

circuit element 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 electri ...

, with its effects predating the

, and

. There are, however, some serious doubts as to whether a genuine memristor can actually exist in physical reality. Additionally, some experimental evidence contradicts Chua's generalization since a non-passive nanobattery effect is observable in resistance switching memory. A simple test has been proposed by Pershin and Di Ventra to analyze 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 (

) which can pass the test. These devices are intended for applications in nanoelectronic memory devices, computer logic, and neuromorphic/neuromemristive computer architectures. In 2013, Hewlett-Packard CTO Martin Fink suggested that memristor memory may become commercially available as early as 2018. In March 2012, a team of researchers from

HRL Laboratories HRL Laboratories (formerly Hughes Research Laboratories) is a research center in Malibu, California, established in 1960. Formerly the research arm of Hughes Aircraft, HRL is currently owned by General Motors Corporation and Boeing. The resea ...

and the

University of Michigan , mottoeng = "Arts, Knowledge, Truth" , former_names = Catholepistemiad, or University of Michigania (1817–1821) , budget = $10.3 billion (2021) , endowment = $17 billion (2021)As o ...

announced the first functioning memristor array built on a

CMOS Complementary metal–oxide–semiconductor (CMOS, pronounced "sea-moss", ) is a type of metal–oxide–semiconductor field-effect transistor (MOSFET) fabrication process that uses complementary and symmetrical pairs of p-type and n-type MOSF ...

chip. 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, Chua argued for a broader definition that includes all two-terminal non-volatile memory devices based on resistance switching. Williams argued that MRAM, phase-change memory and

are memristor technologies. Some researchers argued that biological structures such as blood and skin fit the definition. Others argued that the memory device under development by

and other forms of

are not memristors, but rather part of a broader class of variable-resistance systems, and that a broader definition of memristor is a scientifically unjustifiable land grab that favored HP's memristor patents. In 2011, Meuffels and Schroeder noted that one of the early memristor papers included a mistaken assumption regarding ionic conduction. In 2012, Meuffels and Soni discussed some fundamental issues and problems in the realization of memristors. They indicated inadequacies in the electrochemical modeling presented in the ''

'' article "The missing memristor found" 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.'' in 2013. In a kind of

thought experiment A thought experiment is a hypothetical situation in which a hypothesis, theory, or principle is laid out for the purpose of thinking through its consequences. History The ancient Greek ''deiknymi'' (), or thought experiment, "was the most anc ...

, Meuffels and Soni furthermore revealed a severe inconsistency: If a current-controlled memristor with the so-called ''non-volatility property'' exists in physical reality, its behavior would violate

Landauer's principle Landauer's principle is a physical principle pertaining to the lower theoretical limit of energy consumption of computation. It holds that "any logically irreversible manipulation of information, such as the erasure of a bit or the merging of t ...

, which places a limit on the minimum amount of energy required to change "information" states of a system. This critique was finally adopted by Di Ventra and Pershin in 2013. Within this context, Meuffels and Soni 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 In statistical mechanics, thermal fluctuations are random deviations of a system from its average state, that occur in a system at equilibrium.In statistical mechanics they are often simply referred to as fluctuations. All thermal fluctuations b ...

. When unprotected against

, the internal memory states exhibit some diffusive dynamics, which causes state degradation. The free-energy barriers must therefore be high enough to ensure a low bit-error probability of bit operation. 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. In the general concept of memristive system the defining equations are (see

): :

\begin
              y(t) &= g(\mathbf,u,t) u(t), \\
  \dot &= f(\mathbf,u,t),
\end

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 Resistive random-access memory (ReRAM or RRAM) 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. ReRAM bears s ...

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

. The dynamic state equation in its general form then finally reads: :

\dot = f(\mathbf, u(t) + \xi(t), t),

where ''ξ''(''t'') is, e.g., white

Gaussian Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponym ...

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 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. Di Ventra and Pershin 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 Johnson–Nyquist noise (thermal noise, Johnson noise, or Nyquist noise) is the electronic noise generated by the thermal agitation of the charge carriers (usually the electrons) inside an electrical conductor at equilibrium, which happens reg ...

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 (

)) 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 Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...

nature. The probability for a transition from state to state 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 → 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 → 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 (

) 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 , 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 The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...

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 (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. A 2014 article from researchers of

concluded that Strukov's (HP's) initial/basic memristor modeling 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. Martin Reynolds, an electrical engineering analyst with research outfit

Gartner Gartner, Inc is a technological research and consulting firm based in Stamford, Connecticut that conducts research on technology and shares this research both through private consulting as well as executive programs and conferences. Its client ...

, 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.

Experimental tests

Chua suggested experimental tests to determine if a device may properly be categorized as a memristor: * The

Lissajous curve A Lissajous curve , also known as Lissajous figure or Bowditch curve , is the graph of a system of parametric equations : x=A\sin(at+\delta),\quad y=B\sin(bt), which describe the superposition of two perpendicular oscillations in x and y dire ...

in the voltage-current plane is a pinched

hysteresis Hysteresis is the dependence of the state of a system on its history. For example, a magnet may have more than one possible magnetic moment in a given magnetic field, depending on how the field changed in the past. Plots of a single component of ...

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 all resistive switching memories including

, 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 (

) 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.

Theory

In 2008, researchers from

introduced a model for a memristance function based on thin films of

. For R_ON ≪ R_OFF the memristance function was determined to be :

M(q(t)) = R_\mathrm \cdot \left(1-\frac 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. :

E_
=\ V^2\int_^ \frac
=\ V^2\int_^\frac
=\ V^2\int_^\frac =\ V\Delta Q

Substituting ''V'' = ''I''(''q'')''M''(''q''), and then ∫d''q''/''V'' = ∆''Q''/''V'' for constant ''V''To produces the final expression. This power characteristic differs fundamentally from that of a

metal oxide semiconductor The metal–oxide–semiconductor field-effect transistor (MOSFET, MOS-FET, or MOS FET) is a type of field-effect transistor (FET), most commonly fabricated by the controlled oxidation of silicon. It has an insulated gate, the voltage of which ...

transistor upright=1.4, gate (G), body (B), source (S) and drain (D) terminals. The gate is separated from the body by an insulating layer (pink). A transistor is a semiconductor device used to Electronic amplifier, amplify or electronic switch, switch ...

, 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

, also called the "hard-switching regime". 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 :

\begin
              y(t) &= g(\textbf,u,t)u(t), \\
  \dot &= f(\textbf,u,t)
\end

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 d''q''/d''t'' = ''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

effect. 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

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.

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, CTD) 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

R(x)=R_ (1-x)+R_ x

, with

dx/dt=I/\beta-\alpha x

, there is a set of nonlinearly coupled differential equations that takes the form: :

\frac  = -\alpha \vec+\frac (I-\chi \Omega X)^ \Omega \vec S

where

X

is the diagonal matrix with elements

x_i

on the diagonal,

\alpha,\beta,\chi

are based on the memristors physical parameters. The vector

\vec S

is the vector of voltage generators in series to the memristors. The circuit topology enters only in the projector operator

\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 A Hopfield network (or Ising model of a neural network or Ising–Lenz–Little model) is a form of recurrent artificial neural network and a type of spin glass system popularised by John Hopfield in 1982 as described earlier by Little in 1974 b ...

, such as the existence of Lyapunov functions and classical tunnelling phenomena. 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

. and have suggested extended memristive models to remedy perceived deficiencies. One example 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 :

\begin
              y(t) &= g_0(\textbf, u)u(t) + g_1(\textbf, u) +
                        g_2(\textbf, u) + \ldots +
                        g_m(\textbf, u), \\
  \dot &= f(\textbf, u)
\end

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 function In mathematics, a continuous function is a function such that a continuous variation (that is a change without jump) of the argument induces a continuous variation of the value of the function. This means that there are no abrupt changes in val ...

s. This equation produces the same zero-crossing hysteresis curves as memristive systems but with a different

frequency response In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of s ...

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. :

\begin
              y(t) &= g_0(\textbf,u)(u(t)-a), \\
  \dot &= f(\textbf,u)
\end

Implementations

Titanium dioxide memristor

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

R. Stanley Williams Richard Stanley Williams (born 1951) is a research scientist in the field of nanotechnology and a Senior Fellow and the founding director of the Quantum Science Research Laboratory at Hewlett-Packard. He has over 57 patents, with 40 more patents ...

Hewlett Packard The Hewlett-Packard Company, commonly shortened to Hewlett-Packard ( ) or HP, was an American multinational information technology company headquartered in Palo Alto, California. HP developed and provided a wide variety of hardware components ...

in 2007. 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 The nanoscopic scale (or nanoscale) usually refers to structures with a length scale applicable to nanotechnology, usually cited as 1–100 nanometers (nm). A nanometer is a billionth of a meter. The nanoscopic scale is (roughly speaking) a lo ...

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. The HP device is composed of a thin (50 nm)

film between two 5 nm thick

electrode An electrode is an electrical conductor used to make contact with a nonmetallic part of a circuit (e.g. a semiconductor, an electrolyte, a vacuum or air). Electrodes are essential parts of batteries that can consist of a variety of materials ...

s, one

titanium Titanium is a chemical element with the symbol Ti and atomic number 22. Found in nature only as an oxide, it can be reduced to produce a lustrous transition metal with a silver color, low density, and high strength, resistant to corrosion i ...

, the other

platinum Platinum is a chemical element with the symbol Pt and atomic number 78. It is a dense, malleable, ductile, highly unreactive, precious, silverish-white transition metal. Its name originates from Spanish , a diminutive of "silver". Pla ...

. Initially, there are two layers to the titanium dioxide film, one of which has a slight depletion of

oxygen Oxygen is the chemical element with the symbol O and atomic number 8. It is a member of the chalcogen group in the periodic table, a highly reactive nonmetal, and an oxidizing agent that readily forms oxides with most elements ...

atoms. The oxygen vacancies act as

charge carrier In physics, a charge carrier is a particle or quasiparticle that is free to move, carrying an electric charge, especially the particles that carry electric charges in electrical conductors. Examples are electrons, ions and holes. The term i ...

s, 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. Since the HP device displays fast ion conduction at nanoscale, it is considered a

nanoionic device Nanoionics is the study and application of phenomena, properties, effects, methods and mechanisms of processes connected with fast ion transport (FIT) in all-solid-state nanoscale systems. The topics of interest include fundamental properties of ...

. 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

. It ceases to integrate ''q''=∫''I'' d''t'', 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.

Samsung The Samsung Group (or simply Samsung) ( ko, 삼성 ) is a South Korean multinational manufacturing conglomerate headquartered in Samsung Town, Seoul, South Korea. It comprises numerous affiliated businesses, most of them united under the ...

has a U.S. patent for oxide-vacancy based switches similar to that described by Williams. 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, which bodes well for the future of the technology. At these densities it could easily rival the current sub-25 nm

flash memory Flash memory is an electronic non-volatile computer memory storage medium that can be electrically erased and reprogrammed. The two main types of flash memory, NOR flash and NAND flash, are named for the NOR and NAND logic gates. Both use ...

technology.

Silicon dioxide memristor

It seems that memristance has been registered in

thin films realized with silicon dioxide since 60s . However, hysteretic conductance in silicon has been associated to memristive effect in 2009 only , while Tony Kenyon and his group has clearly demonstrated that the resistive switching in silicon oxide thin films is due to silicon nanoinclusions in highly nonstoichiometric suboxide phases .

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. 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. In 2010, Alibart, Gamrat, Vuillaume et al. introduced a new hybrid organic/nanoparticle device (the NOMFET : Nanoparticle Organic Memory Field Effect Transistor), which behaves as a memristor 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). In 2012, Crupi, Pradhan and Tozer described a proof of concept design to create neural synaptic memory circuits using organic ion-based memristors. The synapse circuit demonstrated

long-term potentiation In neuroscience, long-term potentiation (LTP) is a persistent strengthening of synapses based on recent patterns of activity. These are patterns of synaptic activity that produce a long-lasting increase in signal transmission between two neurons ...

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

Layered memristor

In 2014, Bessonov et al. reported a flexible memristive device comprising a MoO_x/ MoS₂ heterostructure sandwiched between silver electrodes on a plastic foil. The fabrication method is entirely based on printing and solution-processing technologies using two-dimensional layered

transition metal dichalcogenides : 220px, Cadmium sulfide, a prototypical metal chalcogenide, is used as a yellow pigment. A chalcogenide is a chemical compound consisting of at least one chalcogen anion and at least one more electropositive element. Although all group 16 elements ...

(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. in the Akinwande group at the University of Texas, 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. The work was later extended to monolayer

hexagonal boron nitride Boron nitride is a thermally and chemically resistant refractory compound of boron and nitrogen with the chemical formula BN. It exists in various crystalline forms that are isoelectronic to a similarly structured carbon lattice. The hexagona ...

, which is the thinnest memory material of around 0.33 nm. 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 conducting 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 or h-BN atomristors, indicating a new application of memristors for 5G, 6G and THz communication and connectivity systems. In 2020, atomistic understanding of the conductive virtual point mechanism was elucidated in an article in nature nanotechnology.

Ferroelectric memristor

The

ferroelectric Ferroelectricity is a characteristic of certain materials that have a spontaneous electric polarization that can be reversed by the application of an external electric field. All ferroelectrics are also piezoelectric and pyroelectric, with the ad ...

memristor is based on a thin ferroelectric barrier sandwiched between two metallic electrodes. Switching the polarization of the

material by applying a positive or negative voltage across the junction can lead to a two order of magnitude resistance variation: (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. reported observing memristor effect in structure based on vertically aligned carbon nanotubes studying bundles of CNT by scanning tunneling microscope. Later it was found that CNT memristive switching is observed when a nanotube has a non-uniform elastic strain Δ''L''0. 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. In particular, the feasibility of using a collagen‐based biomemristor as an artificial synaptic device has been investigated, whereas a synaptic device based on lignin demonstrated rising or lowering current with consecutive voltage sweeps depending on the sign of the voltage furthermore a natural silk fibroin demonstrated memristive properties; spin-memristive systems based on biomolecules are also being studied. In 2012, Sandro Carrara and co-authors have proposed the first biomolecular memristor with aims to realize highly sensitive biosensors. Since then, several memristive

sensors A sensor is a device that produces an output signal for the purpose of sensing a physical phenomenon. In the broadest definition, a sensor is a device, module, machine, or subsystem that detects events or changes in its environment and sends ...

have been demonstrated.

Spin memristive systems

Spintronic memristor

Chen and Wang, researchers at disk-drive manufacturer

Seagate Technology Seagate Technology Holdings plc is an American data storage company. It was incorporated in 1978 as Shugart Technology and commenced business in 1979. Since 2010, the company has been incorporated in Dublin, Ireland, with operational headquarte ...

described three examples of possible magnetic memristors. 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 ''IEEE Spectrum'' is a magazine edited by the Institute of Electrical and Electronics Engineers. The first issue of ''IEEE Spectrum'' was published in January 1964 as a successor to ''Electrical Engineering''. The magazine contains peer-reviewe ...

. 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.

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 Spin-transfer torque (STT) is an effect in which the orientation of a magnetic layer in a magnetic tunnel junction or spin valve can be modified using a spin-polarized current. Charge carriers (such as electrons) have a property known as spin ...

=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 ( 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 and 2009. While the drift of oxygen vacancies within the insulating MgO layer has been proposed to describe the observed memristive effects, another explanation could be charge trapping/detrapping on the localized states of oxygen vacancies and its impact 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 or multiferroicity.

=Intrinsic mechanism

= The magnetization state of a MTJ can be controlled by

Spin-transfer torque Spin-transfer torque (STT) is an effect in which the orientation of a magnetic layer in a magnetic tunnel junction or spin valve can be modified using a spin-polarized current. Charge carriers (such as electrons) have a property known as spin ...

, 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. 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 = (''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 current-controlled (spin torque is due to a high current density) whereas the change of ''x''₂ 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.

Spin memristive system

A fundamentally different mechanism for memristive behavior has been proposed by Pershin and Di Ventra. The authors show that certain types of semiconductor spintronic structures belong to a broad class of memristive systems as defined by Chua and Kang. 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, but was not described in terms of memristive behavior. On a short time scale, these structures behave almost as an ideal memristor. 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. The SDC device is the first memristive device available commercially to researchers, students and electronics enthusiast worldwide. 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 , the device has significantly higher processing and operating temperatures at and at least , 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 .

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 will be worth $79.0 million by 2022. A potential application of memristors is in analog memories for superconducting quantum computers. Memristors can potentially be fashioned into non-volatile solid-state memory, which could allow greater data density than hard drives with access times similar to

DRAM Dynamic random-access memory (dynamic RAM or DRAM) is a type of random-access semiconductor memory that stores each bit of data in a memory cell, usually consisting of a tiny capacitor and a transistor, both typically based on metal-oxid ...

, replacing both components. HP prototyped a crossbar latch memory that can fit 100

gigabit The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represented ...

s in a square centimeter, and proposed a scalable 3D design (consisting of up to 1000 layers or 1 petabit per cm³). In May 2008 HP reported that its device reaches currently about one-tenth the speed of DRAM. The devices' resistance would be read with

alternating current Alternating current (AC) is an electric current which periodically reverses direction and changes its magnitude continuously with time in contrast to direct current (DC) which flows only in one direction. Alternating current is the form in whic ...

so that the stored value would not be affected. 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. Memristor have applications in programmable logic

signal processing Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing '' signals'', such as sound, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, ...

Super-resolution imaging Super-resolution imaging (SR) is a class of techniques that enhance (increase) the resolution of an imaging system. In optical SR the diffraction limit of systems is transcended, while in geometrical SR the resolution of digital imaging sensors ...

physical neural network A physical neural network is a type of artificial neural network in which an electrically adjustable material is used to emulate the function of a neural synapse or a higher-order (dendritic) neuron model. "Physical" neural network is used to emp ...

control systems A control system manages, commands, directs, or regulates the behavior of other devices or systems using control loops. It can range from a single home heating controller using a thermostat controlling a domestic boiler to large industrial c ...

reconfigurable computing Reconfigurable computing is a computer architecture combining some of the flexibility of software with the high performance of hardware by processing with very flexible high speed computing fabrics like field-programmable gate arrays (FPGAs). Th ...

, in-memory computing, brain–computer interfaces and

RFID Radio-frequency identification (RFID) uses electromagnetic fields to automatically identify and track tags attached to objects. An RFID system consists of a tiny radio transponder, a radio receiver and transmitter. When triggered by an electromag ...

. 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. In 2009, a simple electronic circuit consisting of an LC network and a memristor was used to model experiments on adaptive behavior of unicellular organisms. 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. Applications of such circuits may include, e.g.,

pattern recognition Pattern recognition is the automated recognition of patterns and regularities in data. It has applications in statistical data analysis, signal processing, image analysis, information retrieval, bioinformatics, data compression, computer graphics ...

. The

DARPA The Defense Advanced Research Projects Agency (DARPA) is a research and development agency of the United States Department of Defense responsible for the development of emerging technologies for use by the military. Originally known as the A ...

SyNAPSE In the nervous system, a synapse is a structure that permits a neuron (or nerve cell) to pass an electrical or chemical signal to another neuron or to the target effector cell. Synapses are essential to the transmission of nervous impulses from ...

project funded HP Labs, in collaboration with the

Boston University Boston University (BU) is a private research university in Boston, Massachusetts. The university is nonsectarian, but has a historical affiliation with the United Methodist Church. It was founded in 1839 by Methodists with its original cam ...

Neuromorphics Lab, has been developing neuromorphic architectures which may be based on memristive systems. In 2010,

Versace Gianni Versace S.r.l. (), usually referred to as Versace ( ), is an Italian luxury fashion company founded by Gianni Versace in 1978 known for flashy prints and bright colors. The company produces Italian-made ready-to-wear and accessories, as ...

and Chandler described the MoNETA (Modular Neural Exploring Traveling Agent) model. MoNETA is the first large-scale neural network model to implement whole-brain circuits to power a virtual and robotic agent using memristive hardware. Application of the memristor crossbar structure in the construction of an analog soft computing system was demonstrated by Merrikh-Bayat and Shouraki. In 2011, they showed how memristor crossbars can be combined with

fuzzy logic Fuzzy logic is a form of many-valued logic in which the truth value of variables may be any real number between 0 and 1. It is employed to handle the concept of partial truth, where the truth value may range between completely true and completel ...

to create an analog memristive

neuro-fuzzy In the field of artificial intelligence, neuro-fuzzy refers to combinations of artificial neural networks and fuzzy logic. Overview Neuro-fuzzy hybridization results in a hybrid intelligent system that these two techniques by combining the human ...

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.

Derivative devices

Memistor and memtransistor

The

and memtransistor are transistor based devices which include memristor function.

Memcapacitors and meminductors

In 2009, Di Ventra, Pershin, and Chua extended 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.

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

In September 2014, Mohamed-Salah Abdelouahab, Rene Lozi, and

published a general theory of 1st-, 2nd-, 3rd-, and nth-order memristive elements using fractional derivatives.

History

Precursors

Sir Humphry Davy is said by some to have performed the first experiments which can be explained by memristor effects as long ago as 1808. However the first device of a related nature to be constructed was the

(i.e. memory resistor), a term coined in 1960 by

Bernard Widrow Bernard Widrow (born December 24, 1929) is a U.S. professor of electrical engineering at Stanford University. He is the co-inventor of the Widrow–Hoff least mean squares filter (LMS) adaptive algorithm with his then doctoral student Ted Hoff. ...

to describe a circuit element of an early artificial neural network called ADALINE. A few years later, in 1968, Argall published an article showing the resistance switching effects of TiO₂ which was later claimed by researchers from Hewlett Packard to be evidence of a memristor.

Theoretical description

postulated his new two-terminal circuit element in 1971. It was characterized by a relationship between charge and flux linkage as a fourth fundamental circuit element. Five years later he and his student Sung Mo Kang generalized the theory of memristors and memristive systems including a property of zero crossing in the

characterizing current vs. voltage behavior.

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. On 23 January 2009, Di Ventra, Pershin, and Chua extended the notion of memristive systems to capacitive and inductive elements, namely

s and

s, whose properties depend on the state and history of the system. In July 2014, the MeMOSat/ LabOSat group (composed of researchers from Universidad Nacional de General San Martín (Argentina), INTI,

CNEA The National Atomic Energy Commission ( es, Comisión Nacional de Energía Atómica, CNEA) is the Argentine government agency in charge of nuclear energy research and development. The agency was created on May 31, 1950, with the mission of dev ...

, and CONICET) put memory devices into a

Low Earth orbit A low Earth orbit (LEO) is an orbit around Earth with a period of 128 minutes or less (making at least 11.25 orbits per day) and an eccentricity less than 0.25. Most of the artificial objects in outer space are in LEO, with an altitude never m ...

. Since then, seven missions with different devices are performing experiments in low orbits, onboard Satellogic's Ñu-Sat satellites. On 7 July 2015, Knowm Inc announced Self Directed Channel (SDC) memristors commercially. These devices remain available in small numbers. On 13 July 2018, MemSat (Memristor Satellite) was launched to fly a memristor evaluation payload. In 2021,

Jennifer Rupp Jennifer L. M. Rupp FRSC (born January 27, 1980) is a material scientist and professor at the Technical University of Munich, visiting professor at the Massachusetts Institute of Technology and the CTO for battery research at TUM International ...

and Martin Bazant of MIT started a "Lithionics" research programme to investigate applications of

lithium Lithium (from el, λίθος, lithos, lit=stone) is a chemical element with the symbol Li and atomic number 3. It is a soft, silvery-white alkali metal. Under standard conditions, it is the least dense metal and the least dense soli ...

beyond their use in battery electrodes, including lithium oxide-based memristors in neuromorphic computing.

References

External links

*
Interactive database of memristor papers (2013)
*
"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 {{Authority control Electrical components Emerging technologies American inventions Electronic circuits in computer storage Experimental electrical components