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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 usingmaximally localized Wannier function (MLWF). From the calculated results, weanalyze the topological nature of Bi$_{1-x}$Sb$_x$, and found the followings:(1) pure Bi crystal is topologically trivial; (2) topologically non-trivialphase can be realized by reducing the strength of SOC via Sb doping; (3) theindirect bulk band gap, which is crucial to realize the true bulk insulatingphase, can be enhanced by uniaxial pressure along c axis. (4) The calculatedsurface states can be compared with experimental results, which confirms thetopological nature; (5) We predict the spin-resolved Fermi surfaces and showedthe vortex structures, which should be examined by future experiments.

Electronic Structures and Surface States of Topological Insulator Bi$_{1-x}$Sb$_{x}$