  The substantial amount of recent research into spin torques has been
accompanied by a revival of interest in the spin-Hall effect. This effect
contributes to the spin torque in many materials, including topological
insulator/ferromagnet devices, Weyl semimetals, and van der Waals
heterostructures. In general the relative sizes of competing spin torque
mechanisms remain poorly understood. Whereas a consensus is beginning to emerge
on the evaluation of a conserved spin current, the role of extrinsic disorder
mechanisms in the spin-Hall effect has not been clarified. In this work we
present a comprehensive calculation of the extrinsic spin Hall effect while
focussing on the bulk states of topological insulators as a prototype system
and employing a fully quantum mechanical formalism to calculate the proper spin
current. Our calculation of the proper spin current employs a 4x4 k.p
Hamiltonian describing the bulk states of topological insulators. At the same
time, we provide a qualitative explanation of the proper spin currents
calculated based on an effective $2\times2$ Hamiltonian obtained via a
Schrieffer-Wolf transformation. We find that the extrinsic contribution to the
proper spin current, driven by side jump, skew scattering and related
mechanisms, is of a comparable magnitude to the intrinsic contribution, making
it vital to take such disorder effects into account when seeking to understand
experiments. Among the scattering effects considered, side jump scattering is
the primary contributor to the extrinsic spin Hall effect. The total spin
susceptibility calculated here is too small to explain experimentally measured
spin torques, hence we expect the spin Hall effect to make a negligible
contribution to the spin torque in topological insulator structures.
