Uniform and Piece-wise Uniform Fields in Memristor Models

The Strukov model was the phenomenological model that accompanied the announcement of the first recognised physical instantiation of the memristor and, as such, it has been widely used. This model described the motion of a boundary, $w$, between two types of inter-converting material, $R_{\mathrm{off}}$ and $R_{\mathrm{on}}$, seemingly under a uniform field across the entire device. In fact, what was intended was a field with a discontinuity at $w$, that was uniform between $0<x<w$. In this paper we show that the discontinuity is required for the Strukov model derivation to be completed, and thus the derivation as given does not describe a situation with a uniform field across the entire device. The discontinuity can be described as a Heaviside function, $H$, located on $w$, for which there are three common single-valued approximations for $H(w)$. The Strukov model as intended includes an approximation for the Heaviside function (the field is taken to be the same as that across the $R_{\mathrm{on}}$ part of the device). We compare approximations and give solutions. We then extend the description of the field to a more-realistic continuously varying sigmoidal transition between two uniform fields and demonstrate that the centro-symmetric approximation model (taking the field as being the average of the fields across $R_{\mathrm{on}}$ and $R_{\mathrm{off}}$) is a better single-point model of that situation: the other two approximations over or underestimate the field