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We investigate the spin conductivity of topological insulators taking intoaccount both the surface and quasi-two-dimensional bulk states. We apply alow-energy expansion of the Hamiltonian up to the third order in momentum andtake into account the vertex corrections arising due to the short rangedisorder. Hexagonal warping gives rise to the additional anisotropic componentsin the spin conductivity tensor. Typically, isotropic part of the spinconductivity is larger than anisotropic one. The helical regime for the bulkstates, in which the electrons in the Fermi level have the same projection ofthe spin on the direction of momentum, have been studied in a more detail. Inthis regime, a substantial increase of the spin conductivity contribution fromthe bulk states at the Fermi level is observed. We find that the bulk spinconductivity is insensitive to disorder if Rashba spin-orbit coupling is largerthan disorder strength, otherwise, it is strongly suppressed. The contributionto the spin conductivity from the surface states is almost independent of thechemical potential, robust to disorder and its value is comparable to the spinconductivity contribution from the bulk states per layer. The obtained resultsare in agreement with experimental data.

Bulk and surface spin conductivity in topological insulators with hexagonal warping