Anomalous Hall Effect and Magnetoresistance of Ferromagnetic Metals

The anomalous Hall effect and the magnetoresistance of ferromagnetic metals are investigated on the basis of the localized d-electron (or f-electron) model. First the Hamiltonian of the interaction between condition electrons and localized electrons is given, which is valid when the orbital angular momenta of localized electrons are not quenched. Then we find that the scattering matrix of a conduction electron depends in a complicate way on the directions of the initial and final wave vectors and of the magnetization. The transition probability calculated to the first Born approximation by using the above matrix elements leads to the electrical resistivity of the form expressed by Eq. (1. 1) in the text. To the second Born approximation we find that the transition probability from a state k to another state k′ is not equal to that from k′ to k and leads to the anomalous Hall effect. The magnitude and the temperature dependence of both effects are reasonable for iron and nickel when compared with experiment.