Magneto-spin Hall conductivity of a two-dimensional electron gas

It is shown that the interplay of long-range disorder and in-plane magnetic field gives rise to an out-of-plane spin polarization and a finite spin Hall conductivity of the two-dimensional electron gas in the presence of Rashba spin-orbit coupling. A key aspect is provided by the electric-field$\hbox{--} $induced in-plane spin polarization. Our results are obtained first in the clean limit where the spin-orbit splitting is much larger than the disorder broadening of the energy levels via the diagrammatic evaluation of the Kubo formula. Then the results are shown to hold in the full range of the disorder parameter α pF$\tau $ by means of the quasiclassical Green function technique