Accepted Wednesday Sep 16, 2009
The anomalous Hall effect (AHE) occurs in solids with broken time reversal symmetry, typically in a ferromagnetic phase, as a consequence of spin-orbit coupling. We present a review of experimental and theoretical studies of the 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. 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 crystal. 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 which incorporates an anomalous contribution to wavepacket group velocity due to momentum-space Berry curvatures and correctly combines the roles of intrinsic and extrinsic (skew scattering and side-jump) scattering-related mechanisms. In addition, we review 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. Building on results from recent experiment and theory, we propose a tentative global view of the AHE which summarizes the roles played by intrinsic and extrinsic contributions in the disorder-strength vs. temperature plane. Finally we discuss outstanding issues and avenues for future investigation.