A Finite-Volume Scheme for a Spinorial Matrix Drift-Diffusion Model for Semiconductors

TL;DR

This paper develops and analyzes an implicit finite-volume numerical scheme for a complex spinorial matrix drift-diffusion model in semiconductors, ensuring stability, positivity, and energy dissipation, with applications demonstrated through simulations.

Contribution

It introduces a novel finite-volume scheme that preserves key physical properties and proves its stability and boundedness for semiconductor spin transport models.

Findings

Scheme preserves positivity and energy dissipation.

Proved existence of bounded discrete solutions.

Numerical simulations demonstrate applicability to ferromagnetic devices.

Abstract

An implicit Euler finite-volume scheme for a spinorial matrix drift-diffusion model for semiconductors is analyzed. The model consists of strongly coupled parabolic equations for the electron density matrix or, alternatively, of weakly coupled equations for the charge and spin-vector densities, coupled to the Poisson equation for the elec-tric potential. The equations are solved in a bounded domain with mixed Dirichlet-Neumann boundary conditions. The charge and spin-vector fluxes are approximated by a Scharfetter-Gummel discretization. The main features of the numerical scheme are the preservation of positivity and L $\infty$ bounds and the dissipation of the discrete free energy. The existence of a bounded discrete solution and the monotonicity of the discrete free energy are proved. For undoped semiconductor materials, the numerical scheme is uncon-ditionally stable. The fundamental ideas are reformulations using spin-up and spin-down densities and certain projections of the spin-vector density, free energy estimates, and a discrete Moser iteration. Furthermore, numerical simulations of a simple ferromagnetic-layer field-effect transistor in two space dimensions are presented.