Communications in Mathematical Sciences

Volume 13 (2015)

Number 6

Numerical study of a quantum-diffusive spin model for two-dimensional electron gases

Pages: 1347 – 1378

DOI: https://dx.doi.org/10.4310/CMS.2015.v13.n6.a1

Authors

Luigi Barletti (Dipartimento di Matematica, Università di Firenze (Florence), Italy)

Florian Méhats (INRIA IPSO Team, IRMAR, Université de Rennes 1, Campus de Beaulieu, Rennes, France)

Claudia Negulescu (IMT, Université Paul Sabatier, Toulouse, France)

Stefan Possanner (IMT, Université Paul Sabatier, Toulouse, France)

Abstract

We investigate the time evolution of spin densities in a two-dimensional electron gas subjected to Rashba spin-orbit coupling on the basis of the quantum drift-diffusive model derived in [L. Barletti and F. Méhats, J. Math. Phys., 51, 053304(20), 2010]. This model assumes that the electrons are in a quantum equilibrium state in the form of a Maxwellian operator. The resulting quantum drift-diffusion equations for spin-up and spin-down densities are coupled in a non-local manner via two spin chemical potentials (Lagrange multipliers) and via off-diagonal elements of the equilibrium spin density and spin current matrices, respectively. We present two space-time discretizations of the model, one semi-implicit and one explicit, which also comprise the Poisson equation in order to account for electron-electron interactions. In a first step, pure time discretization is applied in order to prove the well-posedness of the two schemes, both of which are based on a functional formalism to treat the non-local relations between spin densities. We then use the fully space-time discrete schemes to simulate the time evolution of a Rashba electron gas confined in a bounded domain and subjected to spin-dependent external potentials. Finite difference approximations are first order in time and second order in space. The discrete functionals introduced are minimized with the help of a conjugate gradient-based algorithm where the Newton method is applied in order to find the respective line minima. The numerical convergence in the long-time limit of a Gaussian initial condition towards the solution of the corresponding stationary Schrödinger–Poisson problem is demonstrated for different values of the parameters $\epsilon$ (semiclassical parameter), $\alpha$ (Rashba coupling parameter), $\Delta x$ (grid spacing), and $\Delta t$ (time step). Moreover, the performances of the semi-implicit and the explicit scheme are compared.

Keywords

Rashba spin-orbit coupling, spin relaxation, entropic quantum drift-diffusion, quantum Liouville, Schrödinger–Poisson drift-diffusion, spin diffusion

2010 Mathematics Subject Classification

65K10, 65M12, 65N25, 76Y05, 81R25, 82C10, 82D37

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Published 13 May 2015