Current induced magneto-optical Kerr effect as a probe of Dirac carriers in Bi$_{1-x}$Sb$_x$ alloy
We study the current-induced magneto-optical Kerr effect (MOKE) in Bi$_{1-x}$Sb$_x$ semi-metalic alloys. The MOKE signal is found to be the largest in pure Bi ($x=0$), exceeding that of transition metals by nearly four orders of magnitude, and decreases monotonically with increasing Sb concentration. We find the MOKE signal scales with the resistivity ($\rho$) as $\rho^{1.7 \pm 0.6}$ and with the mobility ($\mu_\mathrm{c}$) as $\mu_\mathrm{c}^{2.0 \pm 0.2}$. Model calculations show that such exponent can be accounted for if the Dirac electrons are responsible for the generation of spin current. This is in contrast to the $\rho^{2}$ and $\mu_\mathrm{c}^{-2}$ scaling of the MOKE signal induced by the free electrons in parabolic band. The scaling of the MOKE amplitude with the resistivity also partly accounts for the order of magnitude differences of the signal observed between metals, semimetals, and semiconductors. These results demonstrate that current induced MOKE serves as an effective means to characterize the nature of spin current in materials with diverse electronic structures.