A finite‐volume scheme for a spinorial matrix drift‐diffusion model for semiconductors

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 electric 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 nonnegativity and bounds of the densities 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 unconditionally 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. © 2015 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 819–846, 2016