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We have investigated several classes of strong spin-orbit chalcogenidesrelated to the (Pb,Sn)Te series studied in connection with the Dirac fermionphysics in the 1980s. Our first-principle theoretical calculations suggest thatternary chalcogenides TlBiX$_2$ and TlSbX$_2$ (X=Te, Se, S) series harbor smallbandgap topological insulators with single Dirac cone on some selectivesurfaces whereas the isostructural and isoelectronic silver-based AgBiX$_2$ andAgSbX$_2$ (X=Te, Se, S) series do not. We find that several Tl-compounds are inthe vicinity of a topological critical point. We identify the precise surfacetermination that realizes the single Dirac cone. The single-Dirac-cone surfaceis shown to be correlated with a termination that minimizes dangling bondingeffects favorable for ARPES experiments. Our results further suggest that thisclass of topological semi-metals may harbor odd-parity topologicalsuperconductors similar to the possibility proposed for CuxBi2Se3 (T_c ~ 4K).

Single-Dirac-Cone topological surface states on pseudo-IV-VI-semimetal/semiconductors: Thallium-based III-V-VI2 Ternary Chalcogenides