Effect of scattering on intrinsic anomalous Hall effect investigated by Lorenz ratio

We have investigated the effect of scattering on intrinsic anomalous Hall effect (AHE) for typical itinerant ferromagnets of Fe, Co, Ni, and their impurity-doped specimens in terms of the Hall Lorenz ratio Lxy = κxy / σxy T, the ratio of thermal (κxy) to electrical (σxy) Hall conductivities divided by temperature (T). We show that Lorenz ratio for the intrinsic anomalous Hall current (AHC) (L xy A) is almost constant and coincides with L0 =2.44× 10^-8WΩ/ K2 (Wiedemann-Franz law) in the clean region (e.g., ρxx =1-3× 10^-6Ωcm), indicating the scattering-free nature of intrinsic AHE. On the other hand, the Lorenz ratio for the AHC begins to decrease from Lxy = L0 with increasing ρxx above 3-6× 10^-6Ωcm, which indicates the crossover from the scattering-free to scattering-dependent nature of AHE. The scattering rate (?/τ) corresponding to the crossover resistivity is comparable with the gap magnitude formed at the band anticrossing point caused by the spin-orbit interaction.