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C. Chainais-Hillairet, A. Jüngel, P. Shpartko:
"A finite-volume scheme for a spinorial matrix drift-diffusion model for semiconductors";
in: "ASC Report 03/2015", herausgegeben von: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2015, ISBN: 978-3-902627-08-7, S. 1 - 27.

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 positivity and L1 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.

Spinor drift-diffusion equations, semiconductors, finite volumes, energy dissipation

http://www.asc.tuwien.ac.at/preprint/2015/asc03x2015.pdf