Superconductivity, valence-skipping and topological crystalline metal in AgSnSe${}_2$

The recent suggestion of valence-skipping phenomenon driving a two-gap superconductivity in $Ag$-doped SnSe, by Kataria, $\mathit{\text{et}}\mathit{\text{al.}}$ [Phys. Rev. B 107, 174517 (2023)], has brought to the fore a long-standing issue once again. The absence of crystallographically inequivalent Sn cites corroborated by electronic properties of AgSnSe${}_2$, calculated using first-principles density functional theory, however, does not appear to provide a strong support in favor of valence-skipping in this system. Interestingly, the signature of avoided band-crossings (with the inclusion of SOC) and non-zero $\mathit{\text{mirror}}$ Chern number ($n_{\mathcal{M}}$) confirm a non-trivial topology. The presence of mirror symmetry-protected surface states along the mirror planes indicates that AgSnSe${}_2$ could be a potential candidate for topological crystalline metals (TCMs). Moreover, our calculation of electron-phonon coupling and anisotropic superconducting properties of AgSnSe${}_2$, using Migdal-Eliashberg theory, gives a single-gap superconductivity with critical temperature $T_c\approx 7$K, consistent with the experimental value of $5$K. The interplay of topology and superconductivity in this three-dimensional material appears quite intriguing and it may provide new insights into the exploration of superconductivity and topology.