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Role of surface termination in realizing well-isolated topological surface states within the bulk band gap in TlBiSe$_2$ and TlBiTe$_2$

Electronic structures associated with the flat (polar) Se/Te- or Tl-terminated surfaces of TlBiSe$_2$ and TlBiTe$_2$ are predicted to harbor not only Dirac cone states, but also trivial dangling bond states near the Fermi energy. However, the latter, trivial states have never been observed in photoemission measurements. In order to address this discrepancy, we have carried out {\it ab-initio} calculations for various surfaces of TlBiSe$_2$ and TlBiTe$_2$. A rough nonpolar surface with an equal number of Se/Te and Tl atoms in the surface atomic layer is found to destroy the trivial dangling bond states, leaving only the Dirac cone states in the bulk energy gap. The resulting energy dispersions of the Dirac states are in good accord with the corresponding experimental dispersions in TlBiSe$_2$ as well as TlBiTe$_2$. We also show that in the case of flat, Se terminated, high-index (221) and (112) surfaces of TlBiSe$_2$, the trivial surface states shift energetically below the Dirac node and become well-separated from the Dirac cone states.