Electron-phonon vertex correction effect in superconducting H3S

TL;DR

The paper develops a first-principles implementation of the lowest-order electron-phonon vertex corrections within isotropic Eliashberg theory, enabling non-adiabatic corrections to be incorporated beyond Migdal's approximation. Using full-bandwidth (FBW) and Fermi-surface restricted (FSR) formulations, and including anharmonic phonons, the authors apply the method to H$_3$S and Pb. They find strong non-adiabatic vertex effects in H$_3$S at 200 GPa, which, when combined with anharmonic phonons and energy-dependent DOS, yield TcValues in excellent agreement with experiment; in Pb, vertex corrections are negligible and Tc and the gap align with standard ME predictions. The study provides a robust, predictive framework for electronic-structure based superconductivity across regimes where non-adiabatic effects are significant or negligible, guiding future material discoveries in high-Tc hydrides and related systems.

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.

Figures (6)

• Figure 1: First and second-order Feynman diagrams illustrating the (a) e-ph vertex and (b) the electron self-energy contributions arising from e-ph interactions. In these diagrams, straight lines represent bare electron Green’s functions, straight double lines represent fully dressed electron Green’s functions, and wavy lines denote bare phonon propagators. The second phonon-mediated e-ph vertex is indicated by red dots.
• Figure 2: (a) Electronic band structure and total density of states (DOS), highlighting the vHs peak near $\varepsilon_{\rm F}$, and (b) phonon dispersion and phonon density of states (PhDOS) computed within the harmonic (black solid lines) and anharmonic approximations (red solid lines) for H$_3$S at 200 GPa.
• Figure 3: (a) Eliashberg spectral function $\alpha^2F(\omega)$ and integrated e-ph coupling strength $\lambda(\omega)$ for both harmonic and anharmonic phonons within the adiabatic Migdal approximation, and (b) vertex-corrected Eliashberg spectral function $\alpha^2F^{\rm V} (\omega, \omega^\prime)$ and integrated vertex-corrected e-ph coupling strength $\lambda^{\rm V}$ with harmonic (left) and anharmonic (right) phonons for H$_3$S at 200 GPa.
• Figure 4: Isotropic superconducting gap $\Delta (T)$ for H$_3$S, calculated with anharmonic phonons using four different approaches: adiabatic FBW (solid orange), adiabatic FBW+$\mu$ (solid purple), vertex-corrected FBW (dashed orange), and vertex-corrected FBW+$\mu$ (dashed purple) with a Coulomb parameter $\mu_{\rm E}^*$= 0.16.
• Figure 5: (a) Electronic band structure and total DOS, and (b) phonon dispersion and PhDOS for Pb.