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 $L_{xy}={\kappa }_{xy}/{\sigma }_{xy}T$, the ratio of thermal $\left({\kappa }_{xy}\right)$ to electrical $\left({\sigma }_{xy}\right)$ Hall conductivities divided by temperature $\left(T\right)$. We show that Lorenz ratio for the intrinsic anomalous Hall current (AHC) $\left(L_{xy}{}^{\text{A}}\right)$ is almost constant and coincides with $L_0=2.44\times {10}^{-8}\text{W}\Omega /{\text{K}}^2$ (Wiedemann-Franz law) in the clean region (e.g., ${\rho }_{xx}=1–3\times {10}^{-6}\Omega \text{cm}$), 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 $3–6\times {10}^{-6}\Omega \text{cm}$, which indicates the crossover from the scattering-free to scattering-dependent nature of AHE. The scattering rate $\left(\hslash /\tau \right)$ corresponding to the crossover resistivity is comparable with the gap magnitude formed at the band anticrossing point caused by the spin-orbit interaction.

Figure 1

(Color online) Temperature dependences of electrical resistivity $\left({\rho }_{xx}\right)$, thermal conductivity $\left({\kappa }_{xx}\right)$, and the anomalous parts of electrical Hall conductivities $\left(\sigma _{xy}{}^{\text{A}}\right)$ for ${\text{Ni}}_{1-x}{\text{Cu}}_x$, ${\text{Fe}}_{1-y}{\text{Co}}_y\left({\text{Si}}_y\right)$, and ${\text{Co}}_{1-z}{\text{Si}}_z$.

Figure 2

(Color online) Magnetic field dependences of the thermal Hall conductivities divided by $T$ (${\kappa }_{xy}/T$, circles) and electrical Hall conductivities (${\sigma }_{xy}$, solid lines) at various temperatures for (a) ${\text{Ni}}_{1-x}{\text{Cu}}_x$, (b) ${\text{Fe}}_{1-y}{\text{Co}}_y\left({\text{Si}}_y\right)$, and (c) ${\text{Co}}_{1-z}{\text{Si}}_z$.

Figure 3

(Color online) Temperature dependences of the Lorenz ratios for longitudinal conduction ($L_{xx}$, dashed lines), the normal Hall current ($L_{xy}{}^{\text{N}}$, triangles), and the anomalous Hall current ($L_{xy}{}^{\text{A}}$, circles). For ${\text{Fe}}_{0.97}{\text{Co}}_{0.03}$, $L_{xy}{}^{\text{N}}$ is not shown above 120 K because the normal part of Hall resistivity is too small to accurately estimate.

Figure 4

(Color online) Comparison between ${\left(L_{xx}/L_0\right)}^2$ and $L_{xy}{}^{\text{N}}/L_0$ for pure Ni and Co.

Figure 5

(Color online) The ${\rho }_{xx}$ dependence of $L_{xy}{}^{\text{A}}/L_0$ for (a) ${\text{Ni}}_{1-x}{\text{Cu}}_x$, (b) ${\text{Fe}}_{1-y}{\text{Co}}_y\left({\text{Si}}_y\right)$, and (c) ${\text{Co}}_{1-z}{\text{Si}}_z$. The values of $L_{xy}{}^{\text{A}}\left(T=0\right)$ are estimated by extrapolation from the finite-$T$ data in Fig. 3. We omit the low-$T$ data governed by the skew scattering for ${\text{Fe}}_{0.97}{\text{Co}}_{0.03}$ and ${\text{Fe}}_{0.997}{\text{Si}}_{0.003}$. The upward arrows indicate the critical ${\rho }_{xx}$ value, above which $L_{xy}{}^{\text{A}}$ tends to deviate from its $T=0$ value $\left(\sim L_0\right)$. (For open triangles, see text.)