Anomalous thermal Hall conductivity of ferromagnetic transition metals

In ferromagnetic metals where the anomalous Hall effect (AHE) occurs, one expects a corresponding thermal Hall transport due to electrons that may be dubbed the anomalous thermal Hall effect (ATHE). The same mechanisms that are responsible for AHE, such as the intrinsic Berry curvature effect (usually the most important) or the side-jump mechanism, should play significant roles in ATHE as well. Despite the obvious correlations between AHE and ATHE expected on generalized Wiedemann-Franz law, however, the actual thermal Hall conductivities from each of these contributions have not been evaluated quantitatively. Here, we investigate the intrinsic and the impurity-scattering-induced side-jump contributions to the AHE and the ATHE in $3d$ ferromagnetic metals, bcc Fe, hcp Co, and fcc Ni. We find that the sum of the intrinsic and the side-jump contributions to anomalous thermal Hall conductivity (ATHC) agrees with the experimental values in Fe and Co rather well. In Ni, the calculated ATHC is much larger than the experimental value. We attribute this difference to the importance of electron-phonon scattering in Ni.