Three-species drift-diffusion models for memristors and brain-inspired neuromorphic computing

Abstract

More than 50 years ago, Moore predicted that the number of transistors on a microchip doubles every two years. This exponential growth is ending today because of physical limits, and new technologies are needed. Neuromorphic computing seems to be a promising avenue. An encouraging device as technology enabler of neuromorphic computing is the memristor. Memristors can be seen as nonlinear resistors with memory, mimicking the conductance response of biological synapses. The dynamics of the electron, hole, and oxide vacancy densities in memristors is typically modeled by drift-diffusion equations. While the analysis of two-species drift-diffusion equations is well understood in the literature, the multi-species equations are challenging, and previous results regard (simpler) Nernst-Planck models, two species, or two space dimensions. In this talk, the existence of global-in-time weak solutions in any space dimension is shown. Moreover, we present the uniform-in-time boundedness of the solutions and the fast-relaxation limit in two dimensions. The key ideas of the proof are the use of the entropy method and a combination of local and global compactness results. Numerical experiments in one dimension reproduce hysteresis effects in the current-voltage characteristics, which are a fingerprint for memristive devices.