Memristor

An image of a circuit with 17 memristors captured by an atomic force microscope. The wires are 50 nm - about 150 atoms - wide. Each memristor is composed of two layers of titanium dioxide, of different resistivities, connected to wire electrodes. As electric current is passed through the device, the boundary between the layers moves, changing the net resistance of the device. This change may be used to record information.

Memristors /memˈrɪstɚ/ ("memory resistors") are a class of passive ideal 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. In other cases, memristance theory is used as a mathematical model for empirically observed phenomena. The definition of the memristor is based solely on fundamental circuit variables, similarly to the resistor, capacitor, and inductor. Unlike those more familiar elements, memristors may be described by any of a variety of time-varying functions. As a result, memristors do not belong to linear circuit models including time. A time-invariant memristor is simply a conventional resistor.

Memristor theory was formulated and named by Leon Chua in a 1971 paper. Chua believed that a fourth device might exist to provide conceptual symmetry with the resistor, inductor, and capacitor. He also acknowledged that other scientists had already used fixed nonlinear flux-charge relationships. However, it would be 37 years until April 30, 2008, when a team at HP Labs led by the scientist R. Stanley Williams would announce the discovery of a switching memristor. The Hewlett Packard memristor, based on a thin film of titanium dioxide, appears to be practical and ideal in its initial incarnation. However, as of yet, none have been reported outside HP. Being much simpler than currently popular MOSFET transistor switches and also able to implement one bit of memory in a single device, memristors may enable nanoscale computer technology. (As they are passive, thus incapable of amplification, however, it is impossible to construct digital logic entirely from memristors.) Chua also speculates that they may be useful in the construction of artificial neural networks.

Memristors can implement memory on the principle that direct current applied in the component can adjust its apparent resistance. This resistance may then be observed using alternating current.

Memristor theory

Memristor symbol

The memristor is formally defined 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.

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

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

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

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

$V(t)=\ M(q(t))I(t)$

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.

The power consumption characteristic recalls that of a resistor, I²R.

$P(t)=\ I(t)V(t)=\ I^{2}(t)M(q(t))$

As long as M(q(t)) varies little, such as under alternating current, the memristor will appear as a 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:

${\mathcal {E}}=-\mathrm {d} \Phi _{\mathrm {m} }/\mathrm {d} t$

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:

$V=\mathrm {d} \Phi _{\mathrm {m} }/\mathrm {d} t$

$\Phi _{\mathrm {m} }=\int V\mathrm {d} t$

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.

Two unintuitive concepts are at play:

• Magnetic flux is generated by a resistance in opposition to an applied field or electromotive force. In the absence of resistance, flux due to constant EMF increases indefinitely. The opposing flux induced in a resistor must also increase indefinitely so their sum remains finite.
• Any appropriate response to applied voltage may be called "magnetic flux."

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 and also constant current. 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:

• M(q) approaches zero, such that Φ_m = ∫M(q)dq = ∫M(q(t))I dt remains bounded but continues changing at an ever-decreasing rate. Eventually, this would encounter some kind of quantization and unideal behavior.
• M(q) is cyclic, so that M(q) = M(q − Δq) for all q and some Δq, e.g. sin²(q/Q).
• The device enters hysteresis once a certain amount of charge has passed through, or otherwise ceases to act as a memristor.

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.

$E_{\mathrm {switch} }\ =\ V^{2}\int _{T_{\mathrm {off} }}^{T_{\mathrm {on} }}{\frac {1}{M(q(t))}}\mathrm {d} t\ =\ V^{2}\int _{Q_{\mathrm {off} }}^{Q_{\mathrm {on} }}{\frac {1}{M(q)I(q)}}\mathrm {d} q\ =\ V\Delta Q$

The middle expression results from changing the variable of integration, and the final expression reflects W/I = 1/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." 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. 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 (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. Due to fast ion conduction at nanoscale, titanium dioxide memristor relates to nanoionic devices.

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. Samsung has a pending U.S. patent application for several oxide-layer based switches similar to that described by Williams.

Potential applications

Williams's solid-state memristors can be combined into devices called crossbar latches, which would 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. HP prototyped a crossbar latch memory using the devices that can fit 100 gigabits in a square centimeter. For comparison, the highest-density flash memories at this time (2008) hold 16 gigabits. HP has reported that its version of the memristor is about one tenth the speed of DRAM.

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

Some patents related to memristors appear to include applications in programmable logic, signal processing, neural networks, and control systems.

Controversies

Although the major public response is feeling excited about the finding of the fourth basic element, a few critical comments are posted after the news reports.

Dispute over the fourth element

One concern is that the fundamental elements should be linear devices, which represent different cases of general AC impedance. Memristor should be of great interest as a non-linear two terminal device, instead of a "basic element".

Another opinion is that, when people are talking about fundamental element, either R or L or C, it is characterized by a specific way of linking voltage, current and/or time. It doesn't matter if resistance or capacitance or inductance is a constant value or a special function. But memristor links I & V in the same way as R and its identity can only be expressed through a function of q. In this sense the definition of memristance is breaking the mathematical symmetry.

People also argue that following the same logic, a "memcitor" can be defined as the 5th element, which links I, V, t in the same way as capacitor, but the capacitance value is function of charging history.

ReRAM or Memristor?

Some researchers argue that the memristor HP reported is just one version of ReRAM (also called resistance change memory devices), which has been widely studied before HP announces its first discovery of memristor. Moreover, the actual mechanism for ReRAMs is still under debate.

Just according to the theoretical model and equations, the memristor should be different with a typical resistance change memory. The latter usually switches at certain threshold voltage, while for a memristor, as its memristance is determined as a function of q, which is the integral of idt, will gradually change its state no matter how low the applied current is, and finally approach the highest conductivity state as the current has been hold for sufficient long time. Although this is not demonstrated in the Nature paper which reports the first memristor.

Non electro-magnetic

The memristor is an ionic device - unlike the three basic components R, L, C, which are electro-magnetic. As an ionic device it depends upon the transfer of massive ions likely leading to very large response times. This will cause them to be impractical as components in circuits that require high speed behavior, especially computers.

See also

• Electronic component
• Integrated circuit

External links

• HP Reveals Memristor, The Fourth Passive Circuit Element April 30, 2008
• BBC News - Electronics' 'missing link' found May 01, 2008
• Nature News - Found: the missing circuit element Apr 30, 2008
• Wired.com - Scientists Create First Memristor: Missing Fourth Electronic Circuit Element Apr 30, 2008
• EE Times - 'Missing link' memristor created: Rewrite the textbooks? April 30, 2008
• IEEE Spectrum - The Mysterious Memristor, by Sally Adee May 2008
• Solid-state thin-film memistor for electronic neural networks - Journal of Applied Physics, vol. 67 March 1990