Anomalous Hall effect in a two-dimensional Dirac band: The link between the Kubo-Streda formula and the semiclassical Boltzmann equation approach

Abstract

The anomalous Hall effect ͑AHE͒ is a consequence of spin-orbit coupling in a ferromagnetic metal and 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 semiclassical 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 explicit calculations for the case of the two-dimensional Dirac model Hamiltonian relevant to graphene.

Anomalous Hall effect in a two-dimensional electron gas

“…[10][11][12][13][14][15][16][17] In the present paper, we present a calculation with conclusions that are in disagreement with some previous studies. On such a background, we believe that previous articles have to be discussed in some detail.…”

“…12 due to a subtlety in the calculation of the vertex at the Fermi surface, which was later corrected in the appendix of Ref. 10. Starting with the equation for the renormalized vertex T x = ak x + b x + c y and with the assumption that the density of impurities is low, they find correctly that b = 0 to leading order in n i , i.e., b / a ϰ n i .…”

“…This work 13 intentionally avoids a discussion of the skew-scattering contribution due to the asymmetry of the collision term kernel, which is also an important mechanism of the Hall current and can even be parametrically similar to all other contributions, 10 in the case of Gaussian correlations. Therefore, the work in Ref.…”

“…7 In the Kubo formula approach, neglecting this correction would be equivalent to the unjustified omission of an important subset of Feynman diagrams. 10 Within the calculation presented here, all these terms are present.…”

Transport theory for disordered multiple-band systems: Anomalous Hall effect and anisotropic magnetoresistance

“…We restrict ourselves here to 2DEG Rashba Hamiltonian with an exchange field h ͑breve accent here means that h is in dimensional units͒ in order to obtain simple analytical results that connect directly with other microscopic linearresponse calculations 19,22,46 …”