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The Memristor

The first new passive circuit element since the 1830s might transform computer hardware

Brian Hayes

Lost and Found

There is a long tradition of explaining electric circuits by hydraulic analogies. Thus a conductor is compared to a pipe; electric current is analogous to the flow of water through the pipe; and voltage is like the pressure difference that drives the flow. In this imaginary world of electrical plumbing, a resistor is a small orifice that restricts the flow of water through a pipe. Similarly, a diode (or rectifier) might be likened to a one-way check valve, with a flap that the water pushes open when flowing in the right direction; pressure in the opposite direction closes the flap, which prevents any flow.

What is the hydraulic equivalent of a memristor? The closest analogy I can think of is a sand filter, an item of apparatus used in water-purification plants. As contaminated water flows through a bed of sand and gravel, sediment gradually clogs the pores of the filter and thereby increases resistance. Reversing the flow flushes out the sediment and reduces resistance. Note that this behavior differs from that of a check valve. Although in both cases the direction of flow is what controls the state of the device, at any given instant the resistance of the sand filter is the same in both directions. The memristor, too, is symmetric in this sense.

Plumbing analogies offer intuition about how a component works, but engineers need more—they need a predictive mathematical theory. The memristor has such a theory. It was formulated by Leon O. Chua of the University of California, Berkeley, in the early 1970s—when he had no physical device to which the theory applied. Chua’s 1971 paper on the subject was titled “Memristor—the missing circuit element.” Williams and his colleagues titled their 2008 announcement “The missing memristor found.”

Chua’s theory has nothing to say about oxygen vacancies or other details of materials and structures. It is framed in terms of the basic equations of electric circuits. Those equations link four quantities: voltage (v), current (i), charge (q) and magnetic flux (φ). Each equation establishes a relation between two of these variables. For example, the best-known equation is Ohm’s Law, v=Ri, which says that voltage is proportional to current, with the constant of proportionality given by the resistance R. If a current of i amperes is flowing through a resistance of R ohms, then the voltage measured across the resistance will be v volts. A graph of current versus voltage for an ideal resistor is a straight line whose slope is R.

2011-03CompSciHayesFB.jpgClick to Enlarge ImageEquations of the same form but with different pairs of variables describe two more basic electrical properties, capacitance and inductance. And two more equations define current and voltage in terms of charge and flux. That makes a total of five equations, which bring together various pairings of the four variables v, i, q and φ. Chua observed that four things taken two at a time yield six possible combinations, and so a sixth equation could be formulated. The missing equation would connect charge q and magnetic flux φ and would describe a new circuit element, joining the resistor, the capacitor and the inductor. Those three devices had all been known since the 1830s, so the new element would be a very late and unexpected addition to the family. Chua named it the memristor.

No law of physics demanded that such a device exist, but no law forbade it either; the existing theory of circuits with resistance, capacitance and inductance could be augmented in a straightforward way to include memristance as well. Chua argued for the plausibility of the memristor on grounds of symmetry and completeness, suggesting an analogy with Dmitri Mendeleev’s construction of the periodic table. Nature is not required to fill every square of this table, but a blank spot is certainly a good place to look for a new chemical element—or a new circuit element.

What would a device linking charge and flux look like? Framing the question in this way may be part of the reason it took so long to identify a physical memristor. The variables q and φ invite visions of electric and magnetic fields interacting in some conspicuous way. But the memristor invented at Hewlett-Packard has no obvious connection with magnetic phenomena. Instead it works as a special kind of variable resistor. How can this device be described in terms of q and φ?

Chua’s answer is that q and φ are more important as mathematical variables than as physical quantities. The charge q is the time integral of an electric current: The current is a rate of flow—the number of electrons per second passing some point in the circuit—whereas the charge is the total number of electrons passing that point. A similar relation defines voltage in terms of magnetic flux. By making use of these definitions, we can describe the action of the memristor in terms of voltage and current instead of charge and flux.

The simplest form of the memristor equation is just a variant of Ohm’s Law: v=M(q)i. Where Ohm’s Law has a constant, R, representing resistance, the memristor equation has a function, M(q). M is not a constant; instead it varies as a function of the quantity of charge that has passed through the device. This functional dependence allows memristance to be controlled in ways that ordinary resistance cannot. (Nevertheless, memristance is expressed in the same unit of measure as resistance, namely the ohm.)

Long before Williams announced the TiO2 memristor, there were reports of “anomalous” resistance effects that can now be understood in terms of memristance. Chua has compiled a list of examples going back to 1976, and Williams himself had been exploring such phenomena since 1997. What changed in 2008 was the recognition that Chua’s memristor theory could be applied to these devices. The connection between theory and experiment is more than a formality; it allows memristors to be modeled in circuit-simulation software, an essential in the design of large-scale systems.

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