Intrinsic and extrinsic anomalous Hall effect as investigated by Lorenz number

We have investigated thermal and electrical Hall conductivities ($\kappa _{xy} $, $\sigma _{xy} )$ for typical itinerant ferromagnets of Fe, Co, Ni, and their impurity-doped samples to examine the effect of scattering on the intrinsic (Berry-phase-induced) and the extrinsic (skew-scattering-induced) anomalous Hall current (AHC) in terms of the Lorenz number,$L_{xy} =\kappa _{xy} /\sigma _{xy} T$. The$L_{xy} $coincides with the free electron value ($L_0 )$ in the absence of inelastic scattering but it deviates from$L_0 $ if the Hall current is affected by the inelastic scattering. We showed that the Lorenz number for the intrinsic AHC is almost constant and coincides with$L_0 $ at finite temperature, which indicates the scattering-free nature of the intrinsic AHC. On the other hand, that for the extrinsic AHC steeply decreases from $L_0 $as temperature is increased from$T=0$. This clearly shows that the extrinsic AHC is certainly dependent on the scattering rate.