  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.
