Theory of the anomalous Hall effect from the Kubo formula and the Dirac equation

Abstract

A model to treat the anomalous Hall effect is developed. Based on the Kubo formalism and on the Dirac equation, this model allows the simultaneous calculation of the skew-scattering and sidejump contributions to the anomalous Hall conductivity. The continuity and the consistency with the weak-relativistic limit described by the Pauli Hamiltonian is shown. For both approaches, Dirac and Pauli, the Feynman diagrams, which lead to the skew-scattering and the side-jump contributions, are underlined. In order to illustrate this method, we apply it to a particular case: a ferromagnetic bulk compound in the limit of weak-scattering and free-electrons approximation. Explicit expressions for the anomalous Hall conductivity for both skew-scattering and side-jump mechanisms are obtained. Within this model, the recently predicted "spin Hall effect" appears naturally.

Separation of the individual contributions to the spin Hall effect in dilute alloys within the first-principles Kubo-Středa approach

“…It is important to mention that our results clarify the situation concerning a long-standing question about the magnitude of the side-jump contribution [3,4,7,32,33]. While it is commonly believed that in the dilute limit the skew-scattering mechanism should be dominating [9][10][11][12], there was no clear understanding whether the side-jump contribution may ever be significant as well.…”

Non-collinear magnetoelectronics

“…A straightforward extension would be a treatment of spin-orbit interactions that could help to get to grips with spin-flips at interfaces that have been measured quite recently [265]. Spin-orbit interaction also causes the anomalous Hall effects [266] that is also beyond the applicability of the present approach, in spite of being semiclassical in nature. The coupling between mechanical and ferromagnetic degrees of freedom is another classic topic of magnetism that may turn out to produce surprises in very small structures [267,268].…”

Current-induced spin–orbit torques

“…SO coupling has well-known consequences on the transport properties of both magnetic and non-magnetic conductors, contributing to the anisotropic magnetoresistance [48], anomalous Hall effect (AHE) [49,50], and extrinsic [9,51] and intrinsic [52] spin Hall effect. These phenomena originate from asymmetric scattering from impurities as well as intrinsic band-structure properties [50].…”