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 L-xy = kappa(xy)/sigma T-xy, the ratio of thermal (kappa(xy)) to electrical (sigma(xy)) 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 L-0=2.44 x 10(-8) W Omega/K-2 (Wiedemann-Franz law) in the clean region (e.g., rho(xx) = 1-3 x 10(-6) Omega cm), indicating the scattering-free nature of intrinsic AHE. On the other hand, the Lorenz ratio for the AHC begins to decrease from L-xy = L-0 with increasing rho(xx) above 3-6 x 10(-6) Omega cm, which indicates the crossover from the scattering-free to scattering-dependent nature of AHE. The scattering rate ((h) over bar/tau) corresponding to the crossover resistivity is comparable with the gap magnitude formed at the band anticrossing point caused by the spin-orbit interaction.