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

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 nearequilibrium 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.1.1. Model equations. We assume that the dynamics of the (Hermitian) density matrix N(x, t) ∈ C 2×2 , the current density matrix J(x, t) ∈ C 2×2 , and the electric potential V (x, t)

Quantum drift-diffusion equations for a two-dimensional electron gas with spin-orbit interaction

“…An existence result for a diffusion model for the spin accumulation with fixed electron current but nonconstant magnetization was proved in [12,21]. Matrix spin drift-diffusion models were analyzed in [15,17] with constant precession axis and in [23] with nonconstant precession vector. Numerical simulations for this model can be found in [7].…”