Bounded weak solutions to matrix drift-diffusion model for spin-coherent electron transport in semiconductors

TL;DR

This paper proves the existence and uniqueness of bounded weak solutions for a matrix drift-diffusion model describing spin-coherent electron transport in semiconductors, using advanced mathematical techniques and numerical validation.

Contribution

It introduces a novel approach to analyze a spinorial matrix drift-diffusion system by transforming variables to diagonalize the diffusion matrix, establishing global existence and entropy decay.

Findings

Solutions are globally unique and bounded.

Entropy decays exponentially to equilibrium.

Numerical experiments confirm theoretical results.

Abstract

The global-in-time existence and uniqueness of bounded weak solutions to a spinorial matrix drift-diffusion model for semiconductors is proved. Developing the electron density matrix in the Pauli basis, the coefficients (charge density and spin-vector density) satisfy a parabolic $4\times 4$ cross-diffusion system. The key idea of the existence proof is to work with different variables: the spin-up and spin-down densities as well as the parallel and perpendicular components of the spin-vector density with respect to the magnetization. In these variables, the diffusion matrix becomes diagonal. The proofs of the $L^\infty$ estimates are based on Stampacchia truncation as well as Moser- and Alikakos-type iteration arguments. The monotonicity of the entropy (or free energy) is also proved. Numerical experiments in one space dimension using a finite-volume discretization indicate that the entropy decays exponentially fast to the equilibrium state.