Analytical solution of the Eliashberg equations for strong-coupling superconductivity in hydrides

Abstract

The recent discovery of room-temperature superconductivity in hydrogen-rich materials under extreme pressures has renewed interest in phonon-mediated pairing mechanisms described by the Eliashberg theory, which generalizes the BCS framework by incorporating phonon retardation effects. In this article, we present an analytical solution to the isotropic Eliashberg equations for the superconducting critical temperature, formulated within the Debye model. This solution exhibits good agreement with fully selfconsistent numerical calculations across both weak- and strong-coupling regimes, correctly reproducing the exponential and square-root dependences on the electron-phonon coupling parameter. We further apply the solution to YH6, benchmarking against experiment and the McMillan-Allen-Dynes formula. More broadly, across a diverse set of hydride superconductors, the predicted critical temperature shows scaling consistency with ab initio calculations and experimental data.