Off-Fermi surface cancellation effects in spin-Hall conductivity of a two-dimensional Rashba electron gas

We calculate the spin-Hall conductivity of a disordered two-dimensional Rashba electron gas within the self-consistent Born approximation and for arbitrary values of the electron density, parametrized by the ratio $E_F /E_0$, where $E_F$ is the Fermi level and $E_0$ is the spin-orbit energy. We confirm earlier results indicating that in the limit $E_F/E_0 \gg 1$ the vertex corrections suppress the spin-Hall conductivity. However, for sufficiently low electron density such that $E_F\lesssim E_0$, we find that the vertex corrections no longer cancel contributions from the Fermi surface, and they cannot therefore suppress the spin current. This is instead achieved by contributions away from the Fermi surface, disregarded in earlier studies, which become large when $E_F\lesssim E_0$.