Formal derivation of quantum drift-diffusion equations with spin-orbit interaction

Luigi Barletti; Philipp Holzinger; Ansgar Jüngel

doi:10.3934/krm.2022007

Article Contents

2022, Volume 15, Issue 2: 257-282. Doi: 10.3934/krm.2022007

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Formal derivation of quantum drift-diffusion equations with spin-orbit interaction

1.
Dipartimento di Matematica e Informatica "Ulisse Dini", Università degli Studi di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italia
2.
Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstrasse 8–10, 1040 Wien, Austria

^*Corresponding author: Ansgar Jüngel
^*Corresponding author: Ansgar Jüngel

Received: September 2021

Revised: January 2022

Early access: March 2022

Published: April 2022

The first author acknowledges support by Italian National Group for Mathematical Physics (INdAM-GNFM). The last two authors have been partially supported by the Austrian Science Fund (FWF), grants P30000, P33010, F65, and W1245. This work received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme, ERC Advanced Grant NEUROMORPH, no. 101018153.

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Abstract

Quantum drift-diffusion equations for a two-dimensional electron gas with spin-orbit interactions of Rashba type are formally derived from a collisional Wigner equation. The collisions are modeled by a Bhatnagar–Gross–Krook-type operator describing the relaxation of the electron gas to a local equilibrium that is given by the quantum maximum entropy principle. Because of non-commutativity properties of the operators, the standard diffusion scaling cannot be used in this context, and a hydrodynamic time scaling is required. A Chapman–Enskog procedure leads, up to first order in the relaxation time, to a system of nonlocal quantum drift-diffusion equations for the charge density and spin vector densities. Local equations including the Bohm potential are obtained in the semiclassical expansion up to second order in the scaled Planck constant. The main novelty of this work is that all spin components are considered, while previous models only consider special spin directions.

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Mathematics Subject Classification: 35K55, 35Q40, 35Q81, 82B10.

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Figure 1. Left: A two-dimensional electron gas (2DEG) is confined between two different semiconductor materials A and B (for instance, InAlAs and InGaAs). Right: The electrons of the 2DEG experience an effective magnetic field $ \alpha_R(p\times \mathit{\boldsymbol{e}}_3) $ orthogonal to both the electron momentum $ p $ and the confinement direction $ \mathit{\boldsymbol{e}}_3 $, where $ \alpha_R>0 $ and $ \mathit{\boldsymbol{e}}_3 = (0, 0, 1)^T $

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