Phase Diagram of the Spin Hall Effect

We obtain analytic formulas for the frequency-dependent spin-Hall conductivity of a two-dimensional electron gas (2DEG) in the presence of impurities, linear spin-orbit Rashba interaction, and external magnetic field perpendicular to the 2DEG. We show how different mechanisms (skew scattering, side jump, and spin precession) can be brought in or out of focus by changing controllable parameters such as frequency, magnetic field, and temperature. We find, in particular, that the dc spin-Hall conductivity vanishes in the absence of a magnetic field, while a magnetic field restores the skew-scattering and side jump contributions proportionally to the ratio of magnetic and Rashba fields.