Electron-phonon vertex correction effect in superconducting H3S

TL;DR

This paper investigates the impact of electron-phonon vertex corrections on superconductivity in H3S and Pb, showing significant effects in H3S where non-adiabatic effects are important, leading to more accurate Tc predictions.

Contribution

It introduces a first-principles method incorporating vertex corrections into Eliashberg theory, improving superconductivity predictions in non-adiabatic regimes like H3S.

Findings

Vertex corrections significantly affect H3S superconductivity.

In Pb, vertex corrections are negligible, aligning with standard ME results.

Enhanced Tc prediction accuracy when including non-adiabatic effects.

Abstract

The Migdal-Eliashberg (ME) formalism provides a reliable framework for describing phonon-mediated superconductivity in the adiabatic regime, where the electronic Fermi energy exceeds the characteristic phonon energy. In this work, we go beyond this limit by incorporating first-order vertex corrections to the electron-phonon (e-ph) interaction within the Eliashberg formalism and assess their impact on the superconducting properties of H3S and Pb using first-principles calculations. For H3S, where the adiabatic assumption breaks down, we find that vertex corrections to the e-ph coupling are substantial. When combined with phonon anharmonicity and the energy dependence of the electronic density of states, the predicted critical temperature (Tc) is in very good agreement with experimental observations. In contrast, for elemental Pb, where the adiabatic approximation remains valid, vertex corrections have a negligible effect, and the calculated Tc and superconducting gap closely match the predictions of the standard ME formalism. These findings demonstrate the importance of non-adiabatic corrections in strongly coupled high-Tc hydrides and establish a robust first-principles framework for accurately predicting superconducting properties across different regimes.