Prediction of Nontrivial Band Topology and Superconductivity in Mg$_2$Pb

The interplay of BCS superconductivity and nontrivial band topology is expected to give rise to opportunities for creating topological superconductors, achieved through pairing spin-filtered boundary modes via superconducting proximity effects. The thus-engineered topological superconductivity can, for example, facilitate the search for Majorana fermion quasiparticles in condensed matter systems. Here we report a first-principles study of Mg$_2$Pb and predict that it should be a superconducting topological material. The band topology of Mg$_2$Pb is identical to that of the archetypal quantum spin Hall insulator HgTe, while isostructural and isoelectronic Mg$_2$Sn is topologically trivial; a trivial to topological transition is predicted for Mg$_2$Sn$_{1-x}$Pb$_x$ for x~0.77. We propose that Mg$_2$Pb-Mg$_2$Sn quantum wells should generate robust spin-filtered edge currents in analogy to HgTe/CdTe quantum wells. In addition, our calculations predict that Mg$_2$Pb should become superconducting upon electron doping. Therefore, Mg$_2$Pb is expected to provide a practical material platform for studying emergent phenomena arising from the interplay of superconductivity and band topology.