Memristors - how fundamental, and how useful?

You may have heard about an electronic device called a memristor, a term originally coined by Leon Chua back in 1971, and billed as the "missing fourth fundamental circuit element".  It's worth taking a look at what that means, and whether memristors are fundamental in the physics sense that resistors, capacitors, and inductors are.  Note that this is an entirely separate question from whether such devices and their relatives are technologically useful! 

In a resistor, electronic current flows in phase with the voltage drop across the resistor (assuming the voltage is cycled in an ac fashion).  In the dc limit, current flows in steady state proportional to the voltage, and power is dissipated.  In a capacitor, in contrast, the flow of current builds up charge (in the usual parallel plate concept, charge on the plates) that leads to the formation of an electric field between conducting parts, and hence a voltage difference.  The current leads the voltage (current is proportional to the rate of change of the voltage); when a constant voltage is specified, the current decreases to zero once that voltage is achieved, and energy is stored in the electric field of the capacitor.  In an inductor, the voltage leads the current - the voltage across an inductor, through Faraday's law, is proportional to the rate at which the current is changing.  Note that in a standard inductor (usually drawn as a coil of wire), the magnetic flux through the inductor is proportional to the current (flux = L I, where L is the inductance).  That means that if a certain current is specified through the inductor, the voltage drops to zero (in the ideal, zero-resistance case), and there is energy stored in the magnetic field of the inductor.  Notice that there is a duality between the inductor and capacitor cases (current and voltage swapping roles; energy stored in either electric or magnetic field).

Prof. Chua said that one could think of things a bit differently, and consider a circuit element where the magnetic flux (remember, in an inductor this would be proportional to the time integral of the voltage) is proportional to the charge that has passed through the device (the time integral of the current (rather than the current itself in an inductor)).  No one has actually made such a device, in terms of magnetic flux.  However, what people have made are any number of devices where the relationship between current and voltage depends on the past history of the current flow through the device.  One special case of this is the gadget marketed by HP as a memristor, consisting of two metal electrodes separated by a titanium oxide film.  In that particular example, at sufficiently high bias voltage, the flow of current through the device performs electrochemistry on the titanium oxide, either reducing it to titanium metal, or oxidizing it further, depending on the polarity of the flow.  The result is that the resistance (the proportionality between voltage and current; in the memristor language, the proportionality between the time integral of the voltage and the time integral of the current) depends on how much charge has flowed through the device.  Voila, a memristor.

I would maintain that this is conceptually very different and less fundamental than the resistor, capacitor, or inductor elements.  The resistor is the simplest possible relationship between current and voltage; the capacitor and inductor have a dual relationship and each involve energy storage in electromagnetic fields.  The memristor does not have a deep connection to electromagnetism - it is one particular example of the general "mem"device, which has a complex electrical impedance that depends on the current/voltage history of the device.  Indeed, my friend Max di Ventra has, with a colleague, written a review of the general case, which can be said to include "memcapacitors" and "meminductors".  The various memgizmos are certainly fun to think about, and in their simplest implementation have great potential for certain applications, such as nonvolatile memory.