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

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 ${\sigma }_{yx}^z$ and evaluate it in a systematic way using standard diagrammatic techniques for disordered systems. We find that in the diffusive regime both Boltzmann and the weak localization contributions to ${\sigma }_{yx}^z$ 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.

Figure 1

Diagrams for the spin-Hall conductivity in the zero-loop approximation: The right diagram represents the vertex correction. Dashed lines denote averaging over the impurities.

Figure 2

Weak localization diagrams. The wavy line represents the Cooperon (23). Each diagram contains one spin current vertex (left) and one charge current vertex (right).

Figure 3

The weak localization diagrams from Fig. 2 with the renormalized spin current vertex $[\propto L\left(\mathbf{p}\right)]$; see Eq. (36). The analogous renormalization of the current vertex $[\propto R\left(\mathbf{p}\right)]$ is of higher order in ${\left(p_{F}l\right)}^{-1}$ and can be neglected.

Figure 4

The time evolution of $⟨{\dot{\widehat{s}}}_y\left(t\right)⟩$ for $x=4$. The period of oscillation is $1∕\Delta$, and the exponential decay time is $\tau$. Note that for any $x$, $⟨{\dot{\widehat{s}}}_y\left(t\right)⟩=0$ at $t=0$.