Linear Magnetization Dependence of the Intrinsic Anomalous Hall Effect

The anomalous Hall effect is investigated experimentally and theoretically for ferromagnetic thin films of ${\mathrm{Mn}}_5{\mathrm{Ge}}_3$. We have separated the intrinsic and extrinsic contributions to the experimental anomalous Hall effect and calculated the intrinsic anomalous Hall conductivity from the Berry curvature of the Bloch states using first-principles methods. The intrinsic anomalous Hall conductivity depends linearly on the magnetization, which can be understood from the long-wavelength fluctuations of the spin orientation at finite temperatures. The quantitative agreement between theory and experiment is remarkably good, not only near 0 K but also at finite temperatures, up to about $\sim 240\mathrm{K}$ ($0.8T_C$).

Figure 1

(a) Temperature-dependent spontaneous magnetization $M$. The upper inset shows the RHEED pattern of a ${\mathrm{Mn}}_5{\mathrm{Ge}}_3$ thin film grown on $\mathrm{Ge}/\mathrm{GaAs}(111)$. The lower inset shows the temperature dependence of the inverse magnetic susceptibility, measured at 0.7 T. The green/gray line is a linear fit and extrapolates to $T_C=298\mathrm{K}$. (b) $M$ versus $T^2$. The solid line is a linear fit, showing that $\Delta M(T)/M(0)\propto T^2$ up to $\sim 220\mathrm{K}$.

Figure 2

Field-dependent Hall resistivity ${\rho }_H$ at various temperatures

Figure 3

(a) ${\rho }_{\mathrm{AH}}$ as a function of temperature before (square-dotted line) and after (circle-dotted line) subtracting the skew scattering contribution. ${\rho }_{xx}$ is also shown. (b) ${\rho }_{\mathrm{AH}}/(M{\rho }_{xx})$ as a function of ${\rho }_{xx}$ before (square-dotted line) and after (circle-dotted line) subtracting the skew scattering term. The solid line across the circle dots is a fit crossing the origin.

Figure 4

(a) Calculated ${\sigma }_{\mathrm{IAH}}$ versus the effective spin-orbit coupling strength $\xi$ relative to the real value ${\xi }_0$ for ${\mathrm{Mn}}_5{\mathrm{Ge}}_3$. (b) Calculated ${\sigma }_{\mathrm{IAH}}$ versus the magnitude of magnetization for different relaxation time $\tau$. (c) Experimental AHC as a function of $M$ before and after subtracting the skew scattering (S.S.) contribution. The solid line across the solid dots is a linear fit. Calculated ${\sigma }_{\mathrm{IAH}}$ values versus the $z$ component of the magnetization $M_z$ are also shown.