Quasiclassical approach to the spin-Hall effect in the two-dimensional electron gas

TL;DR

This paper develops a quasiclassical Green's function approach to study spin-charge transport in a 2D electron gas, connecting spin-Hall effects with Berry phase, and provides numerical solutions for Rashba systems.

Contribution

It introduces a quasiclassical method to analyze spin and charge transport across regimes and links the spin-Hall conductivity to Berry phase, with numerical solutions for Rashba systems.

Findings

Established connection between spin-Hall conductivity and Berry phase.

Numerically solved Eilenberger equation for Rashba coupling.

Predicted universal spin currents and edge spin polarization.

Abstract

We study the spin-charge coupled transport in a two-dimensional electron system using the method of quasiclassical ($\xi$-integrated) Green's functions. In particular we derive the Eilenberger equation in the presence of a generic spin-orbit field. The method allows us to study spin and charge transport from ballistic to diffusive regimes and continuity equations for spin and charge are automatically incorporated. In the clean limit we establish the connection between the spin-Hall conductivity and the Berry phase in momentum space. For finite systems we solve the Eilenberger equation numerically for the special case of the Rashba spin-orbit coupling and a two-terminal geometry. In particular, we calculate explicitly the spin-Hall induced spin polarization in the corners, predicted by Mishchenko et al. [13]. Furthermore we find universal spin currents in the short-time dynamics after switching on the voltage across the sample, and calculate the corresponding spin-Hall polarization at the edges. Where available, we find perfect agreement with analytical results.