Large-time asymptotics for a matrix spin drift-diffusion model

TL;DR

This paper proves that under certain conditions, the density matrix in a spin-polarized electron transport model converges exponentially fast to equilibrium, using a reformulation that simplifies the analysis of the equations.

Contribution

It introduces a reformulation of the matrix-valued equations that removes cross-diffusion terms, enabling the proof of exponential convergence to equilibrium.

Findings

Density matrix converges exponentially fast to equilibrium under small relaxation time.

Reformulation using spin densities simplifies the analysis of the model.

Time-uniform bounds are established for spin densities.

Abstract

The large-time asymptotics of the density matrix solving a drift-diffusion-Poisson model for the spin-polarized electron transport in semiconductors is proved. The equations are analyzed in a bounded domain with initial and Dirichlet boundary conditions. If the relaxation time is sufficiently small and the boundary data is close to the equilibrium state, the density matrix converges exponentially fast to the spinless near-equilibrium steady state. The proof is based on a reformulation of the matrix-valued cross-diffusion equations using spin-up and spin-down densities as well as the perpendicular component of the spin-vector density, which removes the cross-diffusion terms. Key elements of the proof are time-uniform positive lower and upper bounds for the spin-up and spin-down densities, derived from the De Giorgi-Moser iteration method, and estimates of the relative free energy for the spin-up and spin-down densities.