  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
($\hbar$/e)($\Omega$cm)$^{-1}$ for bismuth to 96 ($\hbar$/e)($\Omega$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.
