Anomalous Hall effect

TLDR

Synergy between experimental and theoretical work, especially the adoption of Berry‑phase concepts linking the AHE to topological Hall currents, has clarified that the intrinsic Berry‑phase mechanism dominates in moderately conductive metallic ferromagnets while extrinsic skew scattering prevails in highly conductive ones. The review aims to synthesize recent experimental and theoretical advances that have clarified the anomalous Hall effect and to outline remaining questions and future research directions. The authors examine modern semiclassical, Kubo, and Keldysh formalisms—including multiband effects—and review experimental studies across transition metals, oxides, spinels, pyrochlores, and dilute magnetic semiconductors to establish systematic trends. The combined theoretical and experimental evidence strongly supports an intrinsic Berry‑phase–related mechanism as the dominant contributor to the anomalous Hall effect in moderately conductive metallic ferromagnets.

Abstract

We present a review of experimental and theoretical studies of the anomalous Hall effect (AHE), focusing on recent developments that have provided a more complete framework for understanding this subtle phenomenon and have, in many instances, replaced controversy by clarity. Synergy between experimental and theoretical work, both playing a crucial role, has been at the heart of these advances. On the theoretical front, the adoption of Berry-phase concepts has established a link between the AHE and the topological nature of the Hall currents which originate from spin-orbit coupling. On the experimental front, new experimental studies of the AHE in transition metals, transition-metal oxides, spinels, pyrochlores, and metallic dilute magnetic semiconductors, have more clearly established systematic trends. These two developments in concert with first-principles electronic structure calculations, strongly favor the dominance of an intrinsic Berry-phase-related AHE mechanism in metallic ferromagnets with moderate conductivity. The intrinsic AHE can be expressed in terms of Berry-phase curvatures and it is therefore an intrinsic quantum mechanical property of a perfect cyrstal. An extrinsic mechanism, skew scattering from disorder, tends to dominate the AHE in highly conductive ferromagnets. We review the full modern semiclassical treatment of the AHE together with the more rigorous quantum-mechanical treatments based on the Kubo and Keldysh formalisms, taking into account multiband effects, and demonstrate the equivalence of all three linear response theories in the metallic regime. Finally we discuss outstanding issues and avenues for future investigation.