Large Intrinsic Spin Hall Conductivity in Bismuth, Antimony and Bi$_{1-x}$Sb$_{x}$ Alloys*

Bismuth and antimony, which are building blocks of 3 dimensional topological insulators, are expected to exhibit a large spin Hall conductivity due to their large spin-orbit couplings. Furthermore the semimetal characteristics of these materials that originate from slightly overlapping conduction and valence bands can be altered by opening a gap through alloying them up to certain concentration. This so called semi-metal semiconductor transition also allows Bi$_{1-x}$Sb$_x$ alloy to exhibit topologically protected states [1]. In this work we use a low-energy effective spin-orbit Hamiltonian within a tight-binding approach for Bi and Sb as well as Bi$_{1-x}$Sb$_x$ alloys. Beginning with this low-energy Hamiltonian and band structure we calculate the intrinsic spin Hall conductivity using a Berry's curvature technique in the clean static limit. We have also investigated the behavior of the Berry's curvature in a full zone picture and observed that several symmetry points contribute largely to the SHC due to extreme curvature. Robust spin-orbit couplings and Berry curvatures in bulk Bi, Sb and Bi$_{1-x}$Sb$_x$ alloys result in SHC which is comparable to platinum and considerably larger than conventional semiconductors and metals.\\[4pt] [1] Zhang et al., Nature Physics 5, 438, (2009)