Tunable Giant Spin Hall Conductivities in a Strong Spin-Orbit Semimetal: Bi${}_{1-x}$Sb${}_x$

Intrinsic spin Hall conductivities are calculated for strong spin-orbit Bi${}_{1-x}$Sb${}_x$ semimetals, from the Kubo formula and using Berry curvatures evaluated throughout the Brillouin zone from a tight-binding Hamiltonian. Nearly-crossing bands with strong spin-orbit interaction generate giant spin Hall conductivities in these materials, ranging from 474 ($\mathrm{\hslash }$/e)($\mathrm{\Omega }$cm)${}^{-1}$ for bismuth to 96 ($\mathrm{\hslash }$/e)($\mathrm{\Omega }$cm)${}^{-1}$ for antimony; the value for bismuth is more than twice that of platinum. The large spin Hall conductivities persist for alloy compositions corresponding to a three-dimensional topological insulator state, such as Bi${}_{0.83}$Sb${}_{0.17}$. The spin Hall conductivity could be changed by a factor of five for doped Bi, or for Bi${}_{0.83}$Sb${}_{0.17}$, by changing the chemical potential by 0.5 eV, suggesting the potential for doping or voltage tuned spin Hall current.