Superconductivity near a 3-Dimensional Dirac Semimetal: Topological Crystalline Insulator (Pb$_{1-x}$Sn$_x$)$_{1-y}$In$_y$Te near the Inversion Transition*

Superconductivity in topological insulators is expected to show very unconventional features such as $p+ip$ order parameter, Majorana fermions etc... However, so far, the intrinsic superconductivity has been observed only in Cu-intercalated Bi$_2$Se$_3$, where due to strong inhomogeneities, the pairing symmetry is still a matter of debate. Here, we show that in the topological crystalline insulator (TCI) (Pb$_{1-x}$Sn$_x$)$_{1-y}$In$_y$Te, superconductivity occurs near the gap inversion transition, when the system is nearly a 3D Dirac semimetal (DSM). The existence of superconductivity near the 3D DSM is highly unusual. We suggest that it is related to an intrinsic instability of a 3D DSM in (Pb$_{1-x}$Sn$_x$)$_{1-y}$In$_y$Te and ``flattening'' of the bulk valence (conduction) band on the inverted, TCI side of the phase diagram that favors the pairing instability if the chemical potential is pinned to these flat regions.