Maximum intrinsic spin-Hall conductivity in two-dimensional systems with k-linear spin-orbit interaction

TL;DR

This paper analytically investigates the intrinsic spin-Hall conductivity in two-dimensional systems with k-linear spin-orbit interaction, revealing its dependence on the spin-orbit matrix, asymmetry, and Berry curvature effects.

Contribution

It provides a comprehensive analytical framework connecting the spin-orbit matrix properties to the magnitude and symmetry of the intrinsic spin-Hall conductivity.

Findings

ISHC can exceed e/8π but is bounded by e/4π.

The spin-Hall conductivity is generally asymmetric and influenced by the Berry curvature.

The spin-orbit matrix determinant relates to the SU(2) gauge field strength.

Abstract

We analytically calculate the intrinsic spin-Hall conductivity (ISHC) ($\sigma^z_{xy}$ and $\sigma^z_{yx}$) in a clean, two-dimensional system with generic k-linear spin-orbit interaction. The coefficients of the product of the momentum and spin components form a spin-orbit matrix $\widetilde{\beta}$. We find that the determinant of the spin-orbit matrix $\detbeta$ describes the effective coupling of the spin $s_z$ and orbital motion $L_z$. The decoupling of spin and orbital motion results in a sign change of the ISHC and the band-overlapping phenomenon. Furthermore, we show that the ISHC is in general unsymmetrical ($\sigma^z_{xy}\neq-\sigma^z_{yx}$), and it is governed by the asymmetric response function $\Deltabeta$, which is the difference in band-splitting along two directions: those of the applied electric field and the spin-Hall current. The obtained non-vanishing asymmetric response function also implies that the ISHC can be larger than $e/8\pi$, but has an upper bound value of $e/4\pi$. We will that the unsymmetrical properties of the ISHC can also be deduced from the manifestation of the Berry curvature at the nearly degenerate area. On the other hand, by investigating the equilibrium spin current, we find that $\detbeta$ determines the field strength of the SU(2) non-Abelian gauge field.