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 [D.Laniel $\mathit{\text{et}}\mathit{\text{al.}}$, Adv. Mater. 2023, 2308030] 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 turn 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$36 K at ambient pressure.