  We investigate the electronic structures of the alloyed Bi$_{1-x}$Sb$_x$
compounds based on first-principle calculations including spin-orbit coupling
(SOC), and calculate the surface states of semi-infinite systems using
maximally localized Wannier function (MLWF). From the calculated results, we
analyze the topological nature of Bi$_{1-x}$Sb$_x$, and found the followings:
(1) pure Bi crystal is topologically trivial; (2) topologically non-trivial
phase can be realized by reducing the strength of SOC via Sb doping; (3) the
indirect bulk band gap, which is crucial to realize the true bulk insulating
phase, can be enhanced by uniaxial pressure along c axis. (4) The calculated
surface states can be compared with experimental results, which confirms the
topological nature; (5) We predict the spin-resolved Fermi surfaces and showed
the vortex structures, which should be examined by future experiments.
