Ab-initio study of structural and electronic properties of thin film and bulk forms of Bi$_{2}$Q$_{3}$ (Q$=$ Se, Te) as topological insulators

Bi$_{2}$Q$_{3}$ (Q$=$Se, Te) are the best-known bulk thermoelectric materials, which have been demonstrated to be topological insulators (TI). TI's are insulators with conductive surface states consisting of a single Dirac cones. These materials have layered structures consisting of stacked quintuple layers (QL), with relatively weak coupling between the QL's. Therefore, it might be easy to prepare the Bi$_{2}$Q$_{3}$ in the form of thin films with particular thicknesses using the available experimental techniques. In this study, the electronic and structural properties of bulk Bi$_{2}$Se$_{3}$ are investigated using density functional theory. Our results show that the Bi$_{2}$Se$_{3}$ is an indirect semiconductor with energy gap of $\approx $ 0.27 eV. Additionally, the electronic structure dependence of Bi$_{2}$Se$_{3\, }$to the thicknesses of thin films (n-QL's with n$=$1,2\textellipsis 9) is considered. It is observed that the electronic structure of this kind of thin films depends on the number of QL's. For n-QL's with n larger than three, the thin film has a bulk band gap and has protected conducting states on its surface. Moreover, the effect of number of layers (n) on band-gap energy is studied. Similar calculations and discussions are carried out for Bi$_{2}$Te$_{3}$ and the results are compared to the Bi$_{2}$Se$_{3}$ case and also the available theoretical and experimental results.