Topological insulators in the quaternary chalcogenide compounds and ternary famatinite compounds

We present first-principles calculations to predict several three dimensional (3D) topological insulators in quaternary chalcogenide compounds which are made of I$_2$-II-IV-VI$_4$ compositions and in ternary compositions of I$_3$-V-VI$_4$ famatinite compounds. Among the large members of these two families, we give examples of naturally occurring compounds which are mainly Cu-based chalcogenides. We show that these materials are candidates of 3D topological insulators or can be tuned to obtain topological phase transition by manipulating the atomic number of the other cation and anion elements. A band inversion can occur at a single point $\Gamma$ with considerably large inversion strength, in addition to the opening of a bulk band gap throughout the Brillouin zone. We also demonstrate that both of these families are related to each other by cross-substitutions of cations in the underlying tetragonal structure and that one can suitably tune their topological properties in a desired manner.