Memristor - Theory

Theory

The memristor was originally defined in terms of a non-linear functional relationship between magnetic flux linkage Φ_m(t) and the amount of electric charge that has flowed, q(t):

The variable Φ_m ("magnetic flux linkage") is generalized from the circuit characteristic of an inductor. It does not represent a magnetic field here, and its physical meaning is discussed below. The symbol Φ_m may simply be regarded as the integral of voltage over time.

In the relationship between Φ_m 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.

Substituting that the flux is simply the time integral of the voltage, and charge is the time integral of current, we may write the more convenient form

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.

Note that the above table covers all meaningful ratios of differentials of I, Q, Φ_m, and V. No device can relate dI to dq, or dΦ_m to dV, because I is the derivative of Q and Φ_m is the integral of V.

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 M does not vary with charge. Of course, nonzero 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 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.

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 would mean that it would perpetually supply energy when operated with alternating current.

In 2008 researchers from HP Labs introduced a model for a memristance function based on thin films of titanium dioxide. For R_ON<OFF the memristance function was determined to be

where R_OFF represents the high resistance state, R_ON represents the low resistance state, μ_v represents the mobility of dopants in the thin film, and D represents the film thickness. The paper from the HP Labs group noted that "window functions" were necessary to compensate for differences between experimental measurements and their memristor model due to nonlinear ionic drift and boundary effects.