In this work, the completed investigation of a possible superconducting phase in monolayer indium selenide and blue phosphorene is determined using first-principles calculations for both the hole and electron doping systems. Monolayer InSe: the hole-doped dependence of the Fermi surface is exclusively fundamental for monolayer InSe. It leads to the extensive modification of the Fermi surface from six separated pockets to two pockets by increasing the hole densities. For low hole doping levels of the system, below the Lifshitz transition point, superconductive critical temperatures T c ~55-75 K are obtained within anisotropic Eliashberg theory depending on varying amounts of the Coulomb potential from 0.2 to 0.1. However, for some hole doping above the Lifshitz transition point, the combination of the temperature dependence of the bare susceptibility and the strong electron-phonon interaction gives rise to a charge density wave that emerged at a temperature far above the corresponding T c . Having included nonadiabatic effects, we could carefully analyze conditions for which either a superconductive or charge density wave phase occurs in the system.
Monolayer blue phosphorene: we have found that the superconductive phase could be formed only for very low doping levels for the hole-doped systems. For deeper hole doping levels very strong phonon softening finally leads to charge density wave formation at temperatures above the superconductive transition temperature. For the electron-doped cases, optical phonons have more contribution to ?. While phonon softening is present in the electron-doped system, it is less severe in comparison with hole-doped cases. In contrast with the hole-doped regime, although Kohn anomaly does not lead to CDW formation even in very low temperatures, yet the Kohn anomaly persists even for non-adiabatic phonons. We carefully consider the effect of non-adiabaticity in the self-consistent method for the estimation of ? and a maximum of ? = 2.11 was achieved for electron-doped regime which results in superconductive critical temperature T c = 27 K.