Ab initio theory of plasmonic superconductivity within the Eliashberg and density-functional formalisms

We extend the two leading methods for the ab initio computational description of phonon-mediated superconductors, namely Eliashberg theory and density-functional theory for superconductors (SCDFT), to include plasmonic effects. Furthermore, we introduce a hybrid formalism in which the Eliashberg approximation for the electron-phonon coupling is combined with the SCDFT treatment of the dynamically screened Coulomb interaction. The methods have been tested on a set of well-known conventional superconductors by studying how the plasmon contribution affects the phononic mechanism in determining the critical temperature $({T}_{C})$. Our simulations show that plasmonic SCDFT leads to a good agreement between predicted and measured ${T}_{C}$'s, whereas Eliashberg theory considerably overestimates the plasmon-mediated pairing and, therefore, ${T}_{C}$. The hybrid approach, on the other hand, gives results close to SCDFT and overall in excellent agreement with experiments.