Unified description of intrinsic spin-Hall effect mechanisms

Abstract

The intrinsic spin-Hall effects (SHEs) in p-doped semiconductors (Murakami et al Science 301 1348) and two-dimensional electron gases with Rashba spin-orbit coupling (Sinova et al 2004 Phys. Rev. Lett. 92 126603) have been the subject of many theoretical studies, but their driving mechanisms have yet to be described in a unified manner. The former effect arises from the adiabatic topological curvature of momentum space, from which holes acquire a spin-dependent anomalous velocity. The SHE in Rashba systems, on the other hand, results from momentum-dependent spin dynamics in the presence of an external electric field. Our motivation in this paper is to address the disparity between the two mechanisms and, in particular, to clarify whether there is any underlying link between the two effects. In this endeavor, we consider the explicit time dependence of SHE systems starting with a general spin-orbit model in the presence of an electric field. We find that by performing a gauge transformation of the general model with respect to time, a well-defined gauge field appears in time space which has the physical significance of an effective magnetic field. This magnetic field is shown to precisely account for the SHE in the Rashba system in the adiabatic limit. Remarkably, by applying the same limit to the equations of motion of the general model, this magnetic field is also found to be the underlying origin of the anomalous velocity due to the momentum-space curvature. Thus, our study unifies the two seemingly disparate intrinsic SHEs under a common adiabatic framework. © IOP Publishing Ltd and Deutsche Physikalische Gesellschaft.