Geometric Origin of Intrinsic Spin Hall Effect in an Inhomogeneous Electric Field

TL;DR

This paper explores how inhomogeneous electric fields influence the intrinsic spin Hall effect in a 2D system, revealing geometric origins and addressing inconsistencies in theoretical approaches.

Contribution

It derives an expression for the spin Hall conductivity under inhomogeneous fields using gauge-invariant geometric quantities and resolves discrepancies between wave packet and Kubo-Greenwood methods.

Findings

Intrinsic spin Hall conductivity expressed via geometric quantities.

Inhomogeneous electric fields significantly affect spin Hall effect.

Identified and addressed inconsistency in theoretical approaches.

Abstract

In recent years, the spin Hall effect has received great attention because of its potential application in spintronics and quantum information processing and storage. However, this effect is usually studied under the external homogeneous electric field. Understanding how the inhomogeneous electric field affects the spin Hall effect is still lacking. Here, we investigate a two-dimensional two-band time-reversal symmetric system and give an expression for the intrinsic spin Hall conductivity in the presence of the inhomogeneous electric field, which is shown to be expressed through gauge-invariant geometric quantities. On the other hand, when people get physical intuition on transport phenomena from the wave packet, one issue appears. It is shown that the conductivity obtained from the conventional wave packet approach cannot be fully consistent with the one predicted by the Kubo-Greenwood formula. Here, we attempt to solve this problem.