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L. Barletti, P. Holzinger, A. Jüngel:
"Quantum drift-diffusion equations for a two-dimensional electron gas with spin-orbit interaction";
in: "ASC Report 23/2020", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2020, ISBN: 978-3-902627-13-1, 1 - 16.

Quantum drift-diffusion equations are derived for a two-dimensional electron gas with spin-orbit interaction of Rashba type. The (formal) derivation turns out to be a non-standard application of the usual mathematical tools, such as Wigner transform, Moyal product expansion and Chapman-Enskog expansion. The main peculiarity consists in the fact that a non-vanishing current is already carried by the leading-order term in the Chapman-Enskog expansion. To our knowledge, this is
the first example of quantum drift-diffusion equations involving the full spin vector. Indeed, previous models were either quantum bipolar (involving only the spin projection on a given axis) or full spin but semiclassical.

http://www.asc.tuwien.ac.at/preprint/2020/asc23x2020.pdf