Strong electron-phonon coupling and phonon-induced superconductivity in tetragonal C$_3$N$_4$ with hole doping

C$_3$N$_4$ is a recently discovered phase of carbon-nitrides with the tetragonal crystal structure (arXiv:2209.01968) that is stable at ambient conditions. C$_3$N$_4$ is a semiconductor exhibiting flat-band anomalies in the valence band, suggesting the emergence of many-body instabilities upon hole doping. Here, using state-of-the-art first-principles calculations we show that hole-doped C$_3$N$_4$ reveals strong electron-phonon coupling, leading to the formation of a gapped superconducting state. The phase transition temperatures turns out to be strongly dependent on the hole concentration. We propose that holes could be injected into C$_3$N$_4$ via boron doping which induces, according to our results, a rigid shift of the Fermi energy without significant modification of the electronic structure. Based on the electron-phonon coupling and Coulomb pseudopotential calculated from first principles, we conclude that the boron concentration of 6 atoms per nm$^3$ would be required to reach the critical temperature of $\sim$55 K at ambient pressure.Comment: 10 pages incl. Supplemental Material, 8 figure