  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.
