Scaling of the anomalous Hall effect in perpendicularly magnetized epitaxial films of the ferrimagnet Ni Co 2 O 4

We investigate the anomalous Hall effect in perpendicularly magnetized ferrimagnet $\mathrm{Ni}{\mathrm{Co}}_{2}{\mathrm{O}}_{4}$ epitaxial films 3 to 30 nm thick. We find that the anomalous Hall conductivity ${\ensuremath{\sigma}}_{\mathrm{AHE}}$ has complex relations with the longitudinal electrical conductivity ${\ensuremath{\sigma}}_{\mathrm{xx}}$. When the ${\ensuremath{\sigma}}_{\mathrm{xx}}$ is relatively large (larger than ${10}^{3}\phantom{\rule{0.16em}{0ex}}{\mathrm{\ensuremath{\Omega}}}^{--1}\phantom{\rule{0.16em}{0ex}}\mathrm{c}{\mathrm{m}}^{--1}$), the ${\ensuremath{\sigma}}_{\mathrm{AHE}}$ is almost independent of the ${\ensuremath{\sigma}}_{\mathrm{xx}}$, but scales with the magnetization. This unique scaling behavior of ${\ensuremath{\sigma}}_{\mathrm{AHE}}$ indicates that the anomalous Hall effect is of intrinsic origin dominated by the Berry curvature of occupied electronic states. On the other hand, when the ${\ensuremath{\sigma}}_{\mathrm{xx}}$ is lowered, the scaling relation is changed to ${\ensuremath{\sigma}}_{\mathrm{AHE}}\ensuremath{\sim}{\ensuremath{\sigma}}_{\mathrm{xx}}^{1.6}$, indicating a crossover of the dominant contribution to ${\ensuremath{\sigma}}_{\mathrm{AHE}}$, from the intrinsic Berry phase to the impurity scatterings. Our results show the intrinsic contribution from Berry curvature plays a crucial role in the anomalous Hall effect in $\mathrm{Ni}{\mathrm{Co}}_{2}{\mathrm{O}}_{4}$ films.