Three-species drift-diffusion models for memristors

TL;DR

This paper analyzes a coupled drift-diffusion and Poisson system modeling charge dynamics in memristors, proving global solutions, boundedness, and exploring the fast-relaxation limit with numerical illustrations of hysteresis effects.

Contribution

It provides the first rigorous analysis of a three-species drift-diffusion model for memristors, including existence, boundedness, and limit behavior results.

Findings

Global existence of weak solutions in any dimension

Uniform boundedness of solutions in two dimensions

Numerical simulations showing hysteresis in current-voltage curves

Abstract

A system of drift-diffusion equations for the electron, hole, and oxygene vacancy densities in a semiconductor, coupled to the Poisson equation for the electric potential, is analyzed in a bounded domain with mixed Dirichlet-Neumann boundary conditions. This system describes the dynamics of charge carriers in a memristor device. Memristors can be seen as nonlinear resistors with memory, mimicking the conductance response of biological synapses. In the fast-relaxation limit, the system reduces to a drift-diffusion system for the oxygene vacancy density and electric potential, which is often used in neuromorphic applications. The following results are proved: the global existence of weak solutions to the full system in any space dimension; the uniform-in-time boundedness of the solutions to the full system and the fast-relaxation limit in two space dimensions; the global existence and weak-strong uniqueness analysis of the reduced system. Numerical experiments in one space dimension illustrate the behavior of the solutions and reproduce hysteresis effects in the current-voltage characteristics.