Tunable Intrinsic Spin Hall Conductivities in Bi$_2$(Se,Te)$_3$ Topological Insulators*

It has been recently shown by spin-transfer torque measurements that Bi$_2$Se$_3$ exhibits a very large spin Hall conductivity (SHC)[1]. It is expected that Bi$_2$Te$_3$, a topological insulator with similar crystal and band structures as well as large spin-orbit coupling, would also exhibit a giant SHC. In this study we have calculated intrinsic spin Hall conductivities of Bi$_2$Se$_3$ and Bi$_2$Te$_3$ topological insulators from a tight-binding Hamiltonian including two nearest-neighbor interactions. We have calculated the Berry curvature, used the Kubo formula in the static, clean limit and shown that both materials exhibit giant spin Hall conductivities, consistent with the results of Ref. 1 and larger than previously reported Bi$_{1-x}$Sb$_x$ alloys[2]. The density of Berry curvature has also been computed from the full Brillouin zone in order to compute the dependence of the SHC in these materials on the Fermi energy. Finally we report the intrinsic SHC for Bi$_2$(Se,Te)$_3$ topological insulators, which changes dramatically with doping or gate voltage. [1] A.R. Mellnik et al., Nature 511, 449 - 451 (2014) [2] C.\c{S}ahin and M.E.Flatt\'e, arXiv:1410.7319