Spin-Hall conductivity due to Rashba spin-orbit interaction in disordered systems

TL;DR

This paper analyzes the spin-Hall conductivity in disordered two-dimensional electron gases with Rashba spin-orbit interaction, showing that the spin current vanishes in the stationary limit due to its relation to the magnetization's time derivative.

Contribution

It derives a generalized Kubo-Greenwood formula for spin-Hall conductivity and demonstrates that the spin current vanishes in the stationary limit for disordered systems with Rashba interaction.

Findings

Spin-Hall conductivity vanishes in the zero frequency limit.

Both Boltzmann and weak localization contributions cancel out.

The uniform spin current is related to the time derivative of magnetization.

Abstract

We consider the spin-Hall current in a disordered two-dimensional electron gas in the presence of Rashba spin-orbit interaction. We derive a generalized Kubo-Greenwood formula for the spin-Hall conductivity $\sigma$ and evaluate it in an systematic way using standard diagrammatic techniques for disordered systems. We find that in the diffusive regime both Boltzmann and the weak localization contributions to $\sigma$ are of the same order and vanish in the zero frequency limit. We show that the uniform spin current is given by the total time derivative of the magnetization from which we can conclude that the spin current vanishes exactly in the stationary limit. This conclusion is valid for arbitrary spin-independent disorder, external electric field strength, and also for interacting electrons.