The Art and Science of Constructing a Memristor Model

Abstract

Constructing an accurate and predictive compact mathematical model for an electronic circuit element that displays memristor behavior is extremely challenging, but it is also essential for designing and modeling complex integrated circuits that contain the element. Although the fundamental equations that specify the device physics may be known, they usually comprise a set of coupled nonlinear integro-differential equations that are extremely challenging to solve in three dimensions. A numerical solution of the equations can require supercomputers and long times, and thus this approach is useless for interactive simulation of large circuits that contain many such elements. Thus, the equations must be simplified dramatically, and it is not always clear which terms are the most important for the behavior of the device. On the other hand, a purely black box approach of fitting a set of experimental measurements to a convenient functional form runs the risk of poorly representing the behavior of the device in operating regimes outside the range in which the data were collected. Thus, a hybrid approach is necessary, in which the mathematical formalism for a memristor provides the framework for the model and knowledge of the device physics defines the state variable(s), operating limits, and asymptotic behavior necessary to make the model useful. After describing the challenge, the art and science of constructing a memristor model are illustrated by two examples: a locally active and volatile device based on a thin film of niobium dioxide that undergoes an insulator-to-metal transition because of Joule heating and a nonvolatile memory device based on titanium dioxide in which the effective width of an electron tunnel barrier is determined by oxygen vacancy drift caused by an applied electric field.