Intrinsic spin-Hall conductivity of a periodically modulated two-dimensional electron gas in a perpendicular magnetic field: Interplay between the Zeeman term, spin-orbit interaction, and modulation potential

Abstract

We evaluate the intrinsic spin-Hall conductivity of a periodically modulated two-dimensional electron gas in the presence of the Rashba term of the spin-orbit interaction (SOI) and of a perpendicular magnetic field. The interplay between the Zeeman term, the SOI, and the strength of the modulation potential is studied. At high magnetic fields, the spin-Hall conductivity shows Shubnikov--de Haas oscillations whose phase, relative to that without SOI, may be shifted by $\ensuremath{\pi}$ when the magnetic field is varied. At low magnetic fields, the modulation-induced Weiss oscillations corresponding to the two spin branches are split by the SOI and show beating patterns with period determined by the period of the modulation potential and the strength of the SOI. The Zeeman term can enhance the amplitude of the Weiss oscillations. Some limited results of the literature are readily recovered and various limits are discussed. We also evaluate the spin-Hall conductivity for finite frequencies and observe a structure that can be explained with the help of the spin-dependent spectrum and the allowed transitions.