  Recent discovery of topological superconductors (TSCs) has sparked enormous
interest. Realization of TSC requires a delicate tuning of multiple microscopic
parameters, which remains a great challenge. Here, we develop a
first-principles approach to quantify realistic conditions of TSC by solving
self-consistently Bogoliubov-de Gennes equation based on Wannier function
construction of band structure, in presence of Rashba spin-orbit coupling,
Zeeman splitting and electron-phonon coupling. We further demonstrate the power
of this new method by predicting the Mn-doped GeTe (Ge$_{1-x}$Mn$_x$Te)
monolayer - a well-known dilute magnetic semiconductor showing
superconductivity under hole doping - to be a Class D TSC with Chern number of
-1 and chiral Majorana edge modes. By constructing a first-principles phase
diagram in the parameter space of temperature and Mn concentration, we propose
the TSC phase can be induced at a lower-limit transition temperature of ~40 mK
and the Mn concentration of $x$~0.015%. Our approach can be generally applied
to TSCs with a phonon-mediated pairing, providing useful guidance for future
experiments.
