Anomalous Hall effect in 2D Dirac band: link between Kubo-Streda formula and semiclassical Boltzmann equation approach

TL;DR

This paper establishes a formal connection between the Kubo-Streda formula and semiclassical Boltzmann approaches for the anomalous Hall effect in 2D Dirac systems, clarifying the microscopic origins of various contributions.

Contribution

It demonstrates the equivalence of modified semiclassical transport theory and microscopic Kubo-Streda perturbation theory for the AHE, with explicit calculations in a 2D Dirac model.

Findings

Confirmed the equivalence through explicit calculations in a 2D Dirac model.

Identified specific Feynman diagrams corresponding to semiclassical contributions.

Clarified the microscopic origin of anomalous velocity and side jump contributions.

Abstract

The anomalous Hall effect (AHE) is a consequence of spin-orbit coupling in a ferromagnetic metal and is related primarily to density-matrix response to an electric field that is off-diagonal in band index. For this reason disorder contributions to the AHE are difficult to treat systematically using a semi-classical Boltzmann equation approach, even when weak localization corrections are disregarded. In this article we explicitly demonstrate the equivalence of an appropriately modified semiclassical transport theory which includes anomalous velocity and side jump contributions and microscopic Kubo-Streda perturbation theory, with particular unconventional contributions in the semiclassical theory identified with particular Feynman diagrams when calculations are carried out in a band-eigenstate representation. The equivalence we establish is verified by explcit calculations for the case of the two-dimensional (2D) Dirac model Hamiltonian relevant to graphene.