Existence of dual topological phases in Sn-based ternary chalcogenides

It is quite intriguing to investigate the transition from a topological insulator (TI) phase to a topological crystalline insulator (TCI) phase in a material, as the latter has an advantage over the former in controlled device applications. This work investigates the existence of this dual topological behavior in the family of Sn-based ternary chalcogenides $\mathrm{PbSn}{X}_{2}$ ($X$ = S, Se, Te) under hydrostatic pressure using a first-principles approach. These materials are dynamically stable at ambient and elevated pressure conditions, up to which the topological phase transitions (TPTs) are studied. This family has a topologically trivial ground state with direct band gap values 0.338, 0.183, and 0.217 eV for ${\mathrm{PbSnS}}_{2}, {\mathrm{PbSnSe}}_{2}$, and ${\mathrm{PbSnTe}}_{2}$, respectively. The first TPT, i.e., the TI phase, for these materials is observed under the effect of external pressures of 5, 2.5, and 3.5 GPa, with a single band inversion at the $F$ point in the bulk band structure and an odd number of Dirac cones along the (111) surface. A further increase in pressures to 5.5, 3, and 4 GPa results in another band inversion at the $\mathrm{\ensuremath{\Gamma}}$ point and an even number of Dirac cones along the (111) plane. The even number of band inversions suggest that the ($\overline{1}2\overline{1}$) surface has mirror symmetry around the ($10\overline{1}$) plane, which confirms the existence of the TCI phase. This TCI phase is further corroborated by the even value of the mirror Chern number calculated using the winding of Wannier charge centers.