Memristors

Memristors /memˈrɪstɚ/ ("memory resistors") are a class of passive two-terminal circuit elements that maintain a functional relationship between the time integrals of current and voltage. This results in resistance varying according to the device's memristance function. Specifically engineered memristors provide controllable resistance useful for switching current. The memristor is a special case in so-called "memristive systems", a class of mathematical models useful for certain empirically observed phenomena, such as the firing of neurons^[3]. The definition of the memristor is based solely on fundamental circuit variables, similar to the resistor, capacitor, and inductor. Unlike those more familiar elements, the necessarily nonlinear memristors may be described by any of a variety of time-varying functions. As a result, memristors do not belong to linear time-invariant (LTI) circuit models. A linear time-invariant memristor is simply a conventional resistor.^[4]

Memristor theory was formulated and named by Leon Chua in a 1971 paper. Chua strongly believed that a fourth device existed to provide conceptual symmetry with the resistor, inductor, and capacitor. This symmetry follows from the description of basic passive circuit elements as defined by a relation between two of the four fundamental circuit variables, namely voltage, current, charge and flux^[5]. A device linking charge and flux (themselves defined as time integrals of current and voltage), which would be the memristor, was still hypothetical at the time. He did acknowledge that other scientists had already used fixed nonlinear flux-charge relationships.^[6] However, it would not be until thirty-seven years later, on April 30, 2008, that a team at HP Labs led by the scientist R. Stanley Williams would announce the discovery of a switching memristor. Based on a thin film of titanium dioxide, it has been presented as an approximately ideal device.^[7]^[8]^[9] Being much simpler than currently popular MOSFET switches and also able to implement one bit of non-volatile memory in a single device, memristors may enable nanoscale computer technology.^[10] Chua also speculates that they may be useful in the construction of artificial neural networks.^[11]

Memristor theory

The memristor is formally defined^[6] as a two-terminal element in which the magnetic flux Φ_m between the terminals is a function of the amount of electric charge q that has passed through the device. Each memristor is characterized by its memristance function describing the charge-dependent rate of change of flux with charge.

Noting from Faraday's law of induction that magnetic flux is simply the time integral of voltage ^[12], and charge is the time integral of current, we may write the more convenient form

It can be inferred from this that memristance is simply charge-dependent resistance. If M(q(t)) is a constant, then we obtain Ohm's Law R(t) = V(t)/I(t). If M(q(t)) is nontrivial, however, the equation is not equivalent because q(t) and M(q(t)) will vary with time. Solving for voltage as a function of time we obtain

This equation reveals that memristance defines a linear relationship between current and voltage, as long as charge does not vary. Of course, nonzero current implies instantaneously varying charge. Alternating current, however, may reveal the linear dependence in circuit operation by inducing a measurable voltage without net charge movement—as long as the maximum change in q does not cause much change in M.

Furthermore, the memristor is static if no current is applied. If I(t) = 0, we find V(t) = 0 and M(t) is constant. This is the essence of the memory effect.

As long as M(q(t)) varies little, such as under alternating current, the memristor will appear as a resistor. If M(q(t)) increases rapidly, however, current and power consumption will quickly stop.

Magnetic flux in a passive device

In circuit theory, magnetic flux Φ_m typically relates to Faraday's law of induction, which states that the voltage in terms of energy gained around a loop (electromotive force) equals the negative derivative of the flux through the loop:

This notion may be extended by analogy to a single passive device. If the circuit is composed of passive devices, then the total flux is equal to the sum of the flux components due to each device. For example, a simple wire loop with low resistance will have high flux linkage to an applied field as little flux is "induced" in the opposite direction. Voltage for passive devices is evaluated in terms of energy lost by a unit of charge:

Observing that Φ_m is simply equal to the integral of the potential drop between two points, we find that it may readily be calculated, for example by an operational amplifier configured as an integrator.

The upshot is that a passive element may relate some variable to flux without storing a magnetic field. Indeed, a memristor always appears instantaneously as a resistor. As shown above, assuming non-negative resistance, at any instant it is dissipating power from an applied EMF and thus has no outlet to dissipate a stored field into the circuit. This contrasts with an inductor, for which a magnetic field stores all energy originating in the potential across its terminals, later releasing it as an electromotive force within the circuit.

Physical restrictions on M(q)

An applied constant voltage potential results in uniformly increasing Φ_m. Numerically, infinite memory resources, or an infinitely strong field, would be required to store a number which grows arbitrarily large. Three alternatives avoid this physical impossibility:

Operation as a switch

For some memristors, applied current or voltage will cause a great change in resistance. Such devices may be characterized as switches by investigating the time and energy that must be spent in order to achieve a desired change in resistance. Here we will assume that the applied voltage remains constant and solve for the energy dissipation during a single switching event. 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.

The third expression results from changing the variable of integration. To arrive at the final expression, substitute V=I(q)M(q), and then ∫1/Vdq = ∆Q/V for constant V. This power characteristic differs fundamentally from that of a metal oxide semiconductor transistor, which is a capacitor-based device. 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 and enters hysteresis, also called the "hard-switching regime."^[13] 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.

Titanium dioxide memristor

Interest in the memristor revived in 2008 when an experimental solid state version was reported by R. Stanley Williams of Hewlett Packard.^[14]^[15]^[16] A solid-state device could not be constructed until the unusual behavior of nanoscale materials was better understood. 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 using a chemical mechanism.

The HP device is composed of a thin (5 nm) titanium dioxide film between two electrodes. Initially, there are two layers to the film, one of which has a slight depletion of oxygen atoms. The oxygen vacancies act as charge carriers, meaning that the depleted layer has a much lower resistance than the non-depleted layer. When an electric field is applied, the oxygen vacancies drift (see Fast ion conductor), changing the boundary between the high-resistance and low-resistance layers. Thus the resistance of the film as a whole is dependent on how much charge has been passed through it in a particular direction, which is reversible by changing the direction of current.^[8] Since the HP device displays fast ion conduction at nanoscale, it is considered a nanoionic device.^[17]

Memristance is only displayed when the doped layer and depleted layer both contribute to resistance. When enough charge has passed through the memristor that the ions can no longer move, the device enters hysteresis. It ceases to integrate q=∫Idt but rather keeps q at an upper bound and M fixed, thus acting as a 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.^[18] Samsung has a pending U.S. patent application for several oxide-layer based switches similar to that described by Williams.^[19] Williams also has a pending U.S. patent application related to the memristor construction.^[20]

Although the HP memristor is a major discovery for electrical engineering theory, it has yet to be demonstrated in operation at practical speeds and densities. Graphs in Williams' original report show switching operation at only ~1 Hz. Although the small dimension of the device seem to imply fast operation, the charge carriers move very slowly, with an ion mobility of 10^-10 cm²/(V·s). In comparison, the highest known drift ionic mobilities occur in advanced superionic conductors, such as rubidium silver iodide with about 2x10^-4 cm²/(V·s) conducting silver ions at room temperature. Electrons and holes in silicon have a mobility ~1000 cm²/(V·s), a figure which is essential to the performance of transistors. However, a relatively low bias of 1 volt was used, and the plots appear to be generated by a mathematical model rather than a laboratory experiment.^[8]

Polymeric memristor

In July 2008 Victor Erokhin and Marco P. Fontana in Electrochemically controlled polymeric device: a memristor (and more) found two years ago^[21] claim to have developed a polymeric memristor before the titanium dioxide memristor more recently announced.

Spin memristive systems

A fundamentally different mechanism for memristive behavior has been proposed by Yuriy V. Pershin and Massimiliano Di Ventra in their paper “Spin memristive systems”^[22]. The authors show that certain types of semiconductor spintronic structures belong to a broad class of memristive systems as defined by Chua and Kang^[3]. 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 a hysteresis-type behavior. This result was anticipated in the study of spin extraction at semiconductor/ferromagnet interfaces ^[23], but was not described in terms of memristive behavior. On a short time scale, these structures behave almost as an ideal memristor^[6]. This result broadens the possible range of applications of semiconductor spintronics and makes a step forward in future practical applications of the concept of memristive systems.

Potential applications

Williams' solid-state memristors can be combined into devices called crossbar latches, which could replace transistors in future computers, taking up a much smaller area. They can also be fashioned into non-volatile solid-state memory, which would allow greater data density than hard drives with access times potentially similar to DRAM, replacing both components.^[24] HP prototyped a crossbar latch memory using the devices that can fit 100 gigabits in a square centimeter.^[11] For comparison, as of 2008 the highest-density flash memories hold 32 gigabits. HP has reported that its version of the memristor is about one-tenth the speed of DRAM.^[25]

The devices' resistance would be read with alternating current so that they do not affect the stored value.^[26]

Some patents related to memristors appear to include applications in programmable logic,^[27] signal processing,^[28] neural networks,^[29] and control systems.^[30]

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