Geometry and the Anomalous Hall Effect in Ferromagnets

Geometry crops up in physics in unexpected ways. The example in this talk traces an idea proposed 50 years ago. Mired in controversy from the start, it simmered for a long time as an unresolved problem, but has now re-emerged as a topic with modern appeal. In 1954, Karplus and Luttinger (KL) [1] discovered the earliest instance of the Berry phase [3] in solids when they calculated the charge current of electrons moving in a Bravais lattice with broken time-reversal symmetry (a ferromagnet). They uncovered a quantity Ω(k) that acts like a “magnetic field” in reciprocal space to produce a transverse velocity eE×Ω. The velocity leads to a dissipationless Hall current which, according to KL, accounts for the anomalous Hall effect (AHE) seen in all ferromagnets [1]. Because Berry-phase physics lay far into the future, the KL theory encountered stiff resistance. From the 60’s to late 70’s, Smit’s rival skew-scattering theory [2] gained ascendancy, with apparent support from experiments on dilute Kondo systems, e.g. CuMn (in hindsight, these experiments are irrelevant because the host lattice retains time-reversal invariance). In the past 4 years, interest in topological currents derived from the quantum Hall effects has led to the re-examination of the KL theory from the modern viewpoint [4–9]. The new treatments show that KL’s early perturbative approach has a much broader generality [4, 5]. Experiments [10–13] show that the Berry phase plays a key role in the Hall effect of magnetic materials. The evidence that the KL theory provides the correct explanation of the AHE and Nernst effect in ferromagnets is reviewed here. In a Hall experiment on a ferromagnet, the observed Hall resistivity ρxy is the sum of the AHE resistivity ρ′xy and the usual Lorentz term R0H (R0 is the ordinary Hall coefficient). We have