Superconducting properties of MoTe$_2$ from the $ab~initio$ anisotropic Migdal-Eliashberg theory

Molybdenum ditelluride (MoTe$_2$) is attracting considerable interest since it is the archetypal type-II Weyl semimetal and a candidate for topological superconductivity. We investigate the superconducting phase diagram of two MoTe$_2$ polymorphs using the $ab~initio$ anisotropic Migdal-Eliashberg theory, and we show that the superconducting dome originates from the synergistic contribution of the density of states at the Fermi level and the transverse acoustic Te modes in the 1T$^\prime$ phase. We find that the electron and hole pockets carry trivial $s$-wave order parameters of slightly different magnitude, reminiscent of a two-gap structure as suggested by recent experiments. We suggest that a possible route for enhancing the superconducting critical temperature, and realizing $s_{+-}$ pairing, in the T$_d$ phase is to exploit its non-trivial band topology via electron doping.