Competition between phonon mediated superconductivity and carriers localization in field-effect doped molybdenum dichalcogenides

Superconductivity occurs in electrochemically doped molybdenum dichalcogenides samples thicker than four layers. While the critical temperature (Tc) strongly depends on the field effect geometry (single or double gate) and on the sample (MoS2 or MoSe2), Tc always saturates at high doping. The pairing mechanism and the complicate dependence of Tc on doping, samples and field-effect geometry are currently not understood. Previous theoretical works assumed homogeneous doping of a single layer and attributed the Tc saturation to a charge density wave instability, however the calculated values of the electron-phonon coupling in the harmonic approximation were one order of magnitude larger than the experimental estimates based on transport data. Here, by performing fully relativistic first principles calculations accounting for the sample thickness, the field-effect geometry and anharmonicity, we rule out the occurrence of charge density waves in the experimental doping range and demonstrate a suppression of one order of magnitude in the electron-phonon coupling, now in excellent agreement with transport data. By solving the anisotropic Migdal-Eliashberg equations, we explain the behaviour of Tc in different systems and geometries. From an analysis of mobility data, we propose that the Tc saturation is due to carriers localization and disorder.