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We report the electronic band structures and concomitant Fermi surfaces for afamily of exfoliable tetradymite compounds with the formula $T_2$$Ch_2$$Pn$,obtained as a modification to the well-known topological insulator binariesBi$_2$(Se,Te)$_3$ by replacing one chalcogen ($Ch$) with a pnictogen ($Pn$) andBi with the tetravalent transition metals $T$ $=$ Ti, Zr, or Hf. Thisimbalances the electron count and results in layered metals characterized byrelatively high carrier mobilities and bulk two-dimensional Fermi surfaceswhose topography is well-described by first principles calculations.Intriguingly, slab electronic structure calculations predict Dirac-like surfacestates. In contrast to Bi$_2$Se$_3$, where the surface Dirac bands are at the$\Gamma-$point, for (Zr,Hf)$_2$Te$_2$(P,As) there are Dirac cones of strongtopological character around both the $\bar {\Gamma}$- and $\bar {M}$-pointswhich are above and below the Fermi energy, respectively. For Ti$_2$Te$_2$P thesurface state is predicted to exist only around the $\bar {M}$-point. Inagreement with these predictions, the surface states that are located below theFermi energy are observed by angle resolved photoemission spectroscopymeasurements, revealing that they coexist with the bulk metallic state. Thus,this family of materials provides a foundation upon which to develop novelphenomena that exploit both the bulk and surface states (e.g., topologicalsuperconductivity).

Converting topological insulators into topological metals within the tetradymite family