Intrinsic Versus Extrinsic Anomalous Hall Effect in Ferromagnets

A unified theory of the anomalous Hall effect (AHE) is presented for multiband ferromagnetic metals with dilute impurities. In the clean limit, the AHE is mostly due to extrinsic skew scattering. When the Fermi level is located around anticrossing of band dispersions split by spin-orbit interaction, the intrinsic AHE to be calculated ab initio is resonantly enhanced by its nonperturbative nature, revealing the extrinsic-to-intrinsic crossover which occurs when the relaxation rate is comparable to the spin-orbit coupling.

Figure 1

Energy dispersions for ${\widehat{H}}_0$ in Eq. (2).

Figure 2

(a) The total anomalous Hall conductivity ${\sigma }_{xy}^{\mathrm{tot}}$ against $E_F$ and $\hslash /\tau$ in an energy unit of $E_{\mathrm{res}}=1.0$. (b) The intrinsic contribution ${\sigma }_{xy}^{\mathrm{int}}$ for the same parameter values. Note the difference of the scales for ${\sigma }_{xy}$ in (a) and (b).

Figure 3

${\sigma }_{xy}^{\mathrm{tot}}$ and ${\sigma }_{xy}^{\mathrm{int}}$ as a function of $\hslash /\tau$ for the same parameter values as Fig. 2 with $E_F=0.5$, 1.0, and 1.5 for (a), (b), and (c), respectively.

Figure 4

Scaling plot of ${\sigma }_{xy}$ versus ${\sigma }_{xx}$ for the same sets of parameter values as in Fig. 3b except $\nu =2m{v}_{\mathrm{imp}}$.