Unified Description of the Intrinsic Spin-Hall Effects

TL;DR

This paper provides a unified theoretical framework for understanding the intrinsic spin-Hall effects in semiconductors and Rashba systems by linking their mechanisms through an adiabatic gauge field approach.

Contribution

It introduces a gauge transformation approach that unifies the topological and dynamical origins of the intrinsic SHE under an adiabatic framework.

Findings

The gauge field in time space acts as an effective magnetic field.

This field accounts for SHE in Rashba systems in the adiabatic limit.

The same field explains the anomalous velocity from momentum space curvature.

Abstract

The intrinsic spin-Hall effects (SHE) in $p$-doped semiconductors [S. Murakami et al., Science 301, 1348 (2003)] and two-dimensional electron gases with Rashba spin-orbit coupling [J. Sinova et al., Phys. Rev. Lett. 92, 126603 (2004)] 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 the Rashba system, on the other hand, results from the momentum-dependent spin dynamics in the presence of an external electric field. The two effects clearly appear to originate from distinct mechanisms. Our motivation for this article is to address this apparent disparity and, in particular, to seek a unifying description of the effects. In this endeavor, we consider the explicit time-dependence of SHE systems starting with a general spin-orbit model. 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 carefully analyzing the equation of motion of the general model, this field component is also found to be the origin of the anomalous velocity due to the momentum space curvature. Our study therefore unifies the two seemingly disparate intrinsic SHEs under a common adiabatic framework.