Superconductivity, valence skipping, and topological crystalline metallic phase in AgSnSe 2

The recent suggestion of valence-skipping phenomenon driving a two-gap superconductivity in $Ag$-doped SnSe by Kataria et al. [Phys. Rev. B 107, 174517 (2023)] has brought to the fore a long-standing issue once again. The absence of crystallographically inequivalent Sn sites corroborated by electronic properties of ${\mathrm{AgSnSe}}₂$, calculated using first-principles density functional theory, however, does not appear to provide strong support in favor of valence skipping in ${\mathrm{AgSnSe}}₂$. Interestingly, the signature of avoided band crossings (with the inclusion of spin-orbit coupling) and nonzero mirror Chern number (${n}_{\mathcal{M}}$) suggest a nontrivial topology. The presence of mirror symmetry-protected surface states along the mirror planes indicates that ${\mathrm{AgSnSe}}₂$ could be a potential candidate for topological crystalline metals (TCMs), with a transition from a topological crystalline insulator phase to a TCM phase under Ag doping. Moreover, using the Migdal-Eliashberg theory of electron-phonon coupling and anisotropic superconductivity in ${\mathrm{AgSnSe}}₂$, we predict a single superconducting gap and a critical temperature ${T}_c\ensuremath{\approx}7\mathrm{K}$, consistent with the experimental value. The interplay of topology and superconductivity in this three-dimensional material appears quite intriguing and may provide insights into the nature of superconductivity in systems with nontrivial topology.