Unexpected dependence of the anomalous Hall angle on the Hall conductivity in amorphous transition metal thin films

Abstract

The anomalous Hall effect (AHE), and magnetic and electronic transport properties were investigated in a series of amorphous transition metal thin films---${\mathrm{Fe}}_{x}{\mathrm{Si}}_{1--x}, {\mathrm{Fe}}_{x}{\mathrm{Ge}}_{1--x}, {\mathrm{Co}}_{x}{\mathrm{Ge}}_{1--x}, {\mathrm{Co}}_{x}{\mathrm{Si}}_{1--x}$, and ${\mathrm{Fe}}_{1--y}{\mathrm{Co}}_{y}\mathrm{Si}$. The experimental results are compared with density functional theory calculations of the density of Berry curvature and intrinsic anomalous Hall conductivity. In all samples, the longitudinal conductivity (${\ensuremath{\sigma}}_{xx}$), magnetization ($M$), and Hall resistivity (${\ensuremath{\rho}}_{xy}$) increase with increasing transition metal concentration; due to the structural disorder ${\ensuremath{\sigma}}_{xx}$ is lower in all samples than a typical crystalline metal. In the systems with Fe as the transition metal (including ${\mathrm{Fe}}_{1--y}{\mathrm{Co}}_{y}\mathrm{Si}$), the magnetization and AHE are large and in some cases greater than the crystalline analog. In all samples, the AHE is dominated by the intrinsic mechanism, arising from a nonzero, locally derived Berry curvature. The anomalous Hall angle (AHA) $(={\ensuremath{\sigma}}_{xy}/{\ensuremath{\sigma}}_{xx})$ is as large as 5% at low temperature. These results are compared with the AHAs reported in a broad range of crystalline and amorphous materials. Previous work has shown that in a typical crystalline ferromagnet the Hall conductivity (${\ensuremath{\sigma}}_{xy}$) and ${\ensuremath{\sigma}}_{xx}$ are correlated and are usually either both large or both small, resulting in an AHA that decreases with increasing ${\ensuremath{\sigma}}_{xy}$. By contrast, the AHA increases linearly with increasing ${\ensuremath{\sigma}}_{xy}$ in the amorphous systems. This trend is attributed to a generally low ${\ensuremath{\sigma}}_{xx}$, while ${\ensuremath{\sigma}}_{xy}$ varies and can be large. In the amorphous systems, ${\ensuremath{\sigma}}_{xx}$ and ${\ensuremath{\sigma}}_{xy}$ are not coupled, and there may thus exist the potential to further increase the AHA by increasing ${\ensuremath{\sigma}}_{xy}$.