A memristor (; a portmanteau of memory resistor) is a non-linear two-terminal electrical component relating electric charge and magnetic flux linkage. It was described and named in 1971 by Leon Chua, completing a theoretical quartet of fundamental electrical components which also comprises the resistor, capacitor and inductor. 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. The identification of memristive properties in electronic devices has attracted controversy. Experimentally, the ideal memristor has yet to be demonstrated.
Electric 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 Φm(t) and the amount of electric charge that has flowed, q(t):
f ( Φ m ( t ) , q ( t ) ) = 0. {\displaystyle f(\mathrm {\Phi } _{\mathrm {m} }(t),q(t))=0.}
The magnetic flux linkage, Φm, is generalized from the circuit characteristic of an inductor. It does not represent a magnetic field here. Its physical meaning is discussed below. The symbol Φm may be regarded as the integral of voltage over time. In the relationship between Φm and q, the derivative of one with respect to the other depends on the value of one or the other, and so each memristor is characterized by its memristance function describing the charge-dependent rate of change of flux with charge:
M ( q ) = d Φ m d q . {\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:
M ( q ( t ) ) = d Φ
/ d t d q / d t = V ( t ) I ( t ) . {\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.
The above table covers all meaningful ratios of differentials of I, q, Φm, and V. No device can relate dI to dq, or dV to dΦm, because I is the time derivative of q and V is the time derivative of Φm. It can be inferred from this that memristance is charge-dependent resistance. If M(x) is a constant function (i.e. has the same value for all x), then we obtain Ohm's law: R(t) = V(t)/I(t). If M(x) is nontrivial, however, the equation is not equivalent because q(t) and thus M(q(t)) vary with time. Solving for voltage as a function of time produces
V ( t ) = M ( q ( t ) ) ⋅ I ( t ) . {\displaystyle V(t)=\ M(q(t))\cdot I(t)\,.}
This equation reveals that memristance defines a linear relationship between current and voltage, as long as M does not vary with charge. Non-zero current implies time-varying charge. Alternating current, however, may reveal the linear dependence in circuit operation by inducing a measurable voltage without net charge movement—as long as the maximum value of q does not cause much change in M compared to the initial value M(0). Furthermore, the memristor has a constant memristance if no current is applied. If I(t) = 0, M(q(t)) is constant due to q(t) being constant. This is the essence of the memory effect. Analogously, we can define a W(ϕ(t)) as memductance (portmanteau of memory and conductance):
I ( t ) = W ( ϕ ( t ) ) ⋅ V ( t ) . {\displaystyle I(t)=W(\phi (t))\cdot V(t)\,.}
Memductance with respect to flux is the inverse of memristance with respect to charge, i.e.
W ( ϕ ( t ) ) = 1 M ( q ( t ) ) , {\displaystyle W(\phi (t))={\frac {1}{M(q(t))}},}
and therefore the unit of memductance is the same as the unit of conductance ‒ Siemens. The power consumption characteristic recalls that of a resistor, I2R:
P ( t ) = I ( t ) ⋅ V ( t ) = I 2 ( t ) ⋅ M ( q ( t ) ) . {\displaystyle P(t)=\ I(t)\cdot V(t)=\ I^{2}(t)\cdot 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 for some charge q implies it acts as a power source at this charge level. A negative value for all charges 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 that 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 ReRAM 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:
i M ( v ) = ϵ cos ( ϕ 0 ) v = W ( ϕ 0 ) v {\displaystyle {\begin{aligned}i_{M}(v)&=\epsilon \cos(\phi _{0})v\\&=W(\phi _{0})v\end{aligned}}}
where ϵ is a constant based on the physical superconducting materials, v is the voltage across the junction and iM 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 Di Ventra's seminal paper in 2014.
Memristor circuits Study of the ideal memristor poses practical difficulties, so other electrical devices are commonly used as models for 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 ne.
v M = R ( n e ) i M d n e d t = β n + α R ( n e ) i M 2 {\displaystyle {\begin{aligned}v_{\mathrm {M} }&=R(n_{\mathrm {e} })i_{\mathrm {M} }\\{\frac {\mathrm {d} n_{\mathrm {e} }}{\mathrm {d} t}}&=\beta n+\alpha R(n_{\mathrm {e} })i_{\mathrm {M} }^{2}\end{aligned}}}
vM is the voltage across the discharge tube, iM is the current flowing through it, andne is the number of conduction electrons. A simple memristance function is R(ne) = F/ne. The parameters α, β, and F depend 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. Thermistors can be modeled as memristive devices:
v = R 0 ( T 0 ) exp [ β ( 1 T − 1 T 0 ) ] i ≡ R ( T ) i d T d t = 1 C [ − δ ⋅ ( T − T 0 ) + R ( T ) i 2 ] {\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 {\mathrm {d} T}{\mathrm {d} t}}&={\frac {1}{C}}\left[-\delta \cdot (T-T_{0})+R(T)i^{2}\right]\end{aligned}}}
β is a material constant, T is the absolute body temperature of the thermistor, T0 is the ambient temperature (both temperatures in Kelvin), R0(T0) denotes the cold temperature resistance at T = T0, C is the heat capacitance and δ is the dissipation constant for the thermistor. A fundamental phenomenon that has hardly been studied is memristive behaviour in p-n 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 (which is so important during forward transients).
Criticisms In 2008, a team at HP Labs found experimental evidence for Chua's memristor based on an analysis of a thin film of titanium dioxide, thus connecting the operation of ReRAM devices to the memristor concept. According to HP Labs, the memristor would operate in the following way: the memristor's electrical resistance 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. Chua 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, with its effects predating the resistor, capacitor, and inductor. However, there are doubts as to whether a memristor can actually exist. 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 (ReRAM) 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 and the University of Michigan announced the first functioning memristor array built on a CMOS 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 ReRAM 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 HP Labs and other forms of ReRAM 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 Nature article "The missing memristor found" because the impact of concentration polarization effects on the behavior of metal−TiO2−x−metal structures under voltage or current stress was not considered. In a kind of thought experiment, 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, 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. When unprotected against thermal fluctuations, 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 a memristive system, the defining equations are (see § Theory):
y ( t ) = g ( x , u , t ) u ( t ) , x ˙ = f ( x , u , t ) , {\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 {\displaystyle \mathbf {x} } represents a set of n state variables describing the different internal memory states of the device. x ˙ {\displaystyle {\dot {\mathbf {x} }}} is the time-dependent rate of change of the state vector x {\displaystyle \mathbf {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:
x ˙ = f ( x , u ( t ) + ξ ( t ) , t ) , {\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 the basis 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., whether the system would 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 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 (ReRAM)) cannot be associated with the memristor concept, i.e., such devices cannot by themselves remember their current or voltage history. Transitions between distinct internal memory or resistance states are of a probabilistic nature. The probability for a transition from state {i} to state {j} depends on the height of the free-energy barrier between both states. The transition probability can thus be influenced by suitably driving the memory device, i.e., by lowering the free-energy barrier for the transition {i}→{j} by means of, for example, an externally applied bias. A resistance switching event can simply be enforced by setting the external bias to a value above a certain threshold value. This is the trivial case, i.e., the free-energy barrier for the transition {i}→{j} is reduced to zero. In case one applies biases below the threshold value, there is still a finite probability that the device will switch in course of time (triggered by a random thermal fluctuation), but – as one is dealing with probabilistic processes – it is impossible to predict when the switching event will occur. That is the basic reason for the stochastic nature of all observed resistance-switching (ReRAM) processes. If the free-energy barriers are not high enough, the memory device can even switch without having to do anything. When a 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 xi to another one xj. Thus, the internal energy of the memristor device in state xi, U(V, T, xi), would be the same as in state xj, U(V, T, xj), even though these different states would give rise to different device's resistances, which itself must be caused by physical alterations of the device's material. Other researchers noted that memristor models based on the assumption of linear ionic drift do not account for asymmetry between set time (high-to-low resistance switching) and reset time (low-to-high resistance switching) and do not provide ionic mobility values consistent with experimental data. Non-linear ionic-drift models have been proposed to compensate for this deficiency. A 2014 article from researchers of ReRAM 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, 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 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 all resistive switching memories including ReRAM, MRAM and phase-change memory meet these criteria and are memristors. However, the lack of data for the Lissajous curves over a range of initial conditions or over a range of frequencies complicates assessments of this claim. Experimental evidence shows that redox-based resistance memory (ReRAM) includes a nanobattery effect that is contrary to Chua's memristor model. This indicates that the memristor theory needs to be extended or corrected to enable accurate ReRAM modeling.
Theory In 2008, researchers from HP Labs introduced a model for a memristance function based on thin films of titanium dioxide. For Ron ≪ Roff the memristance function was determined to be
M ( q ( t ) ) = R o f f ⋅ ( 1 − μ v R o n D 2 q ( t ) ) {\displaystyle M(q(t))=R_{\mathrm {off} }\cdot \left(1-{\frac {\mu _{v}R_{\mathrm {on} }}{D^{2}}}q(t)\right)}
where Roff represents the high resistance state, Ron represents the low resistance state, μv represents the mobility of dopants in the thin film, and D represents the film thickness. The HP Labs group noted that window functions were necessary to compensate for differences between experimental measurements and their memristor model due to non-linear ionic drift and boundary effects.
Operation as a switch For some memristors, applied current or voltage causes a substantial change in resistance. Such devices may be characterized as switches by investigating the time and energy that must be spent to achieve a desired change in resistance. This assumes that the applied voltage remains constant. Solving for energy dissipation during a single switching event reveals that for a memristor to switch from Ron to Roff in time Ton to Toff, the charge must change by ΔQ = Qon − Qoff.
E s w i t c h = V 2 ∫ T o f f T o n d t M ( q ( t ) ) = V 2 ∫ Q o f f Q o n d q I ( q ) M ( q ) = V 2 ∫ Q o f f Q o n d q V ( q ) = V Δ Q {\displaystyle {\begin{aligned}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\end{aligned}}}
Substituting V = I(q)M(q), and then ∫dq/V = ∆Q/V for constant V to produces the final expression. This power characteristic differs fundamentally from that of a metal oxide semiconductor transistor, which is capacitor-based. Unlike the transistor, the final state of the memristor in terms of charge does not depend on bias voltage. The type of memristor described by Williams ceases to be ideal after switching over its entire resistance range, creating hysteresis, also called the hard-switching regime. 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
y ( t ) = g ( x , u , t ) u ( t ) , x ˙ = f ( x , u , t ) {\displaystyle {\begin{aligned}y(t)&=g({\textbf {x}},u,t)u(t),\\{\dot {\textbf {x}}}&=f({\textbf {x}},u,t)\end{aligned}}}
where u(t) is an input signal, y(t) is an output signal, the vector x represents a set of n state variables describing the device, and g and f are continuous functions. For a current-controlled memristive system the signal u(t) represent