Ferromagnetic Hall Effect in an Electron-Phonon Gas

The contribution of the electron-phonon interaction to the anomalous Hall effect in ferromagnetic metals is calculated, utilizing Holstein's transport theory of the electron-phonon gas. Specifically, the transverse dc conductivity σxy is calculated using Kubo's formalism and the temperature diagrammatic technique. Results based on the (ferromagnetically) polarized itinerant-electron model are consistent with the Luttinger's theory of the anomalous Hall effect for impurity scattering. This leads to the non-Boltzmann (so-called "anomalous-velocity") contribution as well as the skew-scattering contribution of the type first obtained by Smit. However, it is in disagreement with the result of Leribaux based on the same model. A principal feature of the treatment is the explicit representation in terms of charge-carrier motion between different Wannier cells as first suggested by Smit. However, in contrast to the latter author, who took the velocity operator of this intercell motion to be the ordinary Bloch velocity, it is found that there is an additional velocity component of intercell motion arising from the electron scattering mechanism (either electron-phonon interaction, as in this case, or electron-impurity interaction as in the previous treatments of Smit or Luttinger). This additional component gives rise, in turn, to the above-mentioned anomalous-velocity correction to σxy, and hence, to the Hall coefficient.