Spin Hall conductivity in 2DEG with spin-orbit coupling

Spin Hall conductivity
Spin Hall effect.

The spin Hall effect is named so in analogy to the (charge) Hall effect, that refers to a response of the system in the direction perpendicular to the excitation. The excitation is, in general an electric field in the field (say x-direction), and the response in the spin Hall case is a spin current in the perpendicular direction (here the y-direction). Spin current here means that there is no net charge current, but that electrons of opposite spin move in opposite y-directions. If the sample/system is finite, one can observe opposite spin polarisations on opposite borders of the sample. This can and has been measured by optical Kerr rotation measurements.
Two dimensional electron gas with Rashba spin-orbit coupling.

In a two dimensional electron gas inversion asymmetry can be induced by a static electric field. This can be realized by semiconductor doping or even by gating. This asymmetry results in Rashba (linear in k) spin-orbit coupling, that can be seen as an effective magnetic field, and that makes the energy band split into chiral bands. This is at the origin of the intrinsic spin Hall effect with a "universal" spin Hall conductivity value -e/8 pi [Sinova et al. PRL 94 (2004)].

What happens to this spin Hall conductivity in the presence of disorder?
It was shown soon after Sinova's article that this system was not robust to disorder. First order corrections to the spin Hall conductivity very exactly cancel out the zeroth order conductivity [Raimondi and Schwab, PRB 71 (2005)] and quantum corrections are zero [Chalaev and Loss, PRB 71 (2005)], meaning a total disappearance of the spin Hall effect. But what if the disordered impurities have a magnetic moment (or a spin)? Magnetic impurities break time reversal invariance, so the effect of this type of disorder is different.

We noticed through an analytical linear response calculation of the optical spin Hall conductivity that there is a singularity in the static limit. Numerically we find that the spin Hall conductivity keeps close to its clean value (-e/8pi), but large fluctuations were observed. To get the average value shown in the figure above we needed about 700 disorder configurations with 64 twisted boundary conditions for each configuration.

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