“The presence (or absence) of an anomalous Hall effect is currently important to test whether a candidate magnetic semiconductor is (or isn't) a ferromagnet”

We were motivated by the possibility that the large changes in electrical resistivity in response to magnetic fields in some manganese-based oxides might be due to the trapping of electrons by lattice distortions. It was known from early work on semiconductors that such trapped electrons (polarons) would have characteristic Hall signatures in the ordinary Hall effect. This was the case. In the course of that work, we found the anomalous Hall effect to have characteristics that were not seen before and that we could understand them in the context of quantum mechanical phase effects (Berry's phase). It now appears that this mechanism may underlie the anomalous Hall effect in many cases.

The ordinary Hall effect is caused by the direct effect (Lorentz force) of magnetic fields on current carriers (electrons or holes) and is proportional to the applied field. The anomalous Hall effect is an additional contribution that arises in ferromagnetic materials that is proportional to the sample magnetization and therefore becomes constant at large fields. Unlike the ordinary Hall effect in metals, this contribution is strongly temperature dependent. It is caused by spin-orbit coupline, a combination of quantum-mechanical and relativistic effects. The presence (or absence) of an anomalous Hall effect is currently important to test whether a candidate magnetic semiconductor is (or isn't) a ferromagnet.

The Hall resistivity is the quantity that is directly measured—it is the Hall voltage divided by the current through the sample. If there is no anomalous Hall effect, the Hall coefficient is the Hall resistivity divided by the sample's internal magnetic field. This is a classical measurement. The situation is more complicated in the presence of an anomalous contribution. The quantity that can be calculated theoretically is the Hall conductivity; it can be determined experimentally from the Hall resistivity and the longitudinal resistivity; that is, the voltage across the sample divided by the current through it.

The materials we study are under investigation for possible magnetic memory devices. Our work contributes to an understanding of the underlying physics of the sensitivity of these materials to magnetic fields.

Our work and related theoretical work have emphasized the importance of quantum mechanical phase in the anomalous Hall effect (AHE). Most previous explanations of the AHE have relied on extrinsic mechanisms (scattering) and have been controversial. The Berry-phase approach may prove to be a unifying concept, particularly near magnetic transitions where the extrinsic explanations are inapplicable.