Dynamic spin-Hall effect and driven spin helix for linear spin-orbit interactions

We derive boundary conditions for the electrically induced spin accumulation in a finite, disordered two-dimensional semiconductor channel. While for dc electric fields these boundary conditions select spatially constant spin profiles equivalent to a vanishing spin-Hall effect, we show that an in-plane ac electric field results in a nonzero ac spin-Hall effect, i.e., it generates a spatially nonuniform out-of-plane polarization even for linear intrinsic spin-orbit interactions. Analyzing different geometries in [001]- and [110]-grown quantum wells, we find that although this out-of-plane polarization is typically confined to within a few spin-orbit lengths from the channel edges, it is also possible to generate spatially oscillating spin profiles which extend over the whole channel. The latter is due to the excitation of a driven spin-helix mode in the transverse direction of the channel. We show that while finite frequencies suppress this mode, it can be amplified by a magnetic field tuned to resonance with the frequency of the electric field. In this case, finite-size effects at equal strengths of Rashba and Dresselhaus SOI lead to an enhancement of the magnitude of this helix mode. We comment on the relation between spin currents and boundary conditions.