The intrinsic image and image spin-Hall conductivity of a periodically modulated two-dimensional electron gas in a perpendicular magnetic field

We evaluate the intrinsic dc spin-Hall conductivity of a periodically modulated two-dimensional electron gas in the presence of the Rashba spin–orbit interaction (SOI) and of a perpendicular magnetic field B. When the Zeeman term and the SOI are absent, the modulation potential broadens the Landau levels into spin-degenerate bands whose width oscillates with B. Combined with the Zeeman term, the SOI lifts this degeneracy, by creating two spin branches, and makes the bandwidth spin-dependent. The Weiss oscillations show beating patterns with a period determined by that of the modulation potential and the strength of the SOI. Importantly, we predict oscillations of the spin accumulation when the field B is varied. The ac SHC exhibits a structure that is explained with the help of inter- and intra-spin-branch transitions.