Unoccupied topological states on bismuth chalcogenides

The unoccupied part of the band structure of topological insulators Bi$_2$Te$_{x}$Se$_{3-x}$ ($x=0,2,3$) is studied by angle-resolved two-photon photoemission and density functional theory. For all surfaces linearly-dispersing surface states are found at the center of the surface Brillouin zone at energies around 1.3 eV above the Fermi level. Theoretical analysis shows that this feature appears in a spin-orbit-interaction induced and inverted local energy gap. This inversion is insensitive to variation of electronic and structural parameters in Bi$_2$Se$_3$ and Bi$_2$Te$_2$Se. In Bi$_2$Te$_3$ small structural variations can change the character of the local energy gap depending on which an unoccupied Dirac state does or does not exist. Circular dichroism measurements confirm the expected spin texture. From these findings we assign the observed state to an unoccupied topological surface state.