New topological surface state in layered topological insulators: unoccupied Dirac cone

The unoccupied states in topological insulators Bi$_2$Se$_3$, PbSb$_2$Te$_4$, and Pb$_2$Bi$_2$Te$_2$S$_3$ are studied by the density functional theory methods. It is shown that a surface state with linear dispersion emerges in the inverted conduction band energy gap at the center of the surface Brillouin zone on the (0001) surface of these insulators. The alternative expression of $\mathbb{Z}_2$ invariant allowed us to show that a necessary condition for the existence of the second $\bar\Gamma$ Dirac cone is the presence of local gaps at the time reversal invariant momentum points of the bulk spectrum and change of parity in one of these points.