Hall conductivity in the presence of spin-orbit interaction and disorder

TL;DR

This paper develops a comprehensive theoretical framework for analyzing the Hall conductivity, including anomalous and ordinary contributions, in systems with spin-orbit interaction and disorder, applicable across temperature regimes.

Contribution

It introduces a cumulant-based expansion of the Kubo formula that handles both high and low temperatures, incorporates spin-orbit effects, and generalizes to strong disorder and mobility edges.

Findings

Recover the Karplus-Luttinger result for anomalous Hall effect

Derive a side-jump type formula using linear response

Show anomalous Hall coefficient varies with resistance as a power law

Abstract

Starting from the Kubo formula, we expand the Hall conductivity using a cumulant approach which converges quickly at high temperatures (k_BT > energy differences of initial and final scattering states) and can be extended to low temperatures. The theory can deal with the sign, the ordinary and the anomalous contributions to the Hall effect. When applied to include the spin-orbit interaction to first order, we recover what is essentially the Karplus-Luttinger result for the anomalous Hall effect. Contact is made to the Chazalviel and Nozieres-Lewiner formulae. A side-jump type formula is obtained by using an exact application of linear response. We show that there exists an exact rigid Hall current which is not a Fermi level property. We introduce a relationship between mass and diffusivity which allows us to generalize the theory to strong disorder and even introduce a mobility edge. The formalism provides a systematic and practical way of analyzing both ordinary and anomalous contributions to the Hall conduction including the changes of sign, and in the presence of serious disorder. As a byproduct of the method, we show that the anomalous Hall coefficient can vary with resistance to the power n, with 1 <= n <= 2 depending on the degree of coherence.