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# 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

#### 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

Published 13 May 2015