Nonadiabatic and anharmonic effects in high-pressure H3S and D3S superconductors

TL;DR

This work addresses how anharmonic lattice dynamics and nonadiabatic electron-phonon vertex corrections affect superconductivity in hydrogen-rich H3S and D3S under high pressure. It combines anharmonic phonon renormalization with first-order vertex corrections within a full-bandwidth Eliashberg framework to predict Tc at 160 and 200 GPa. The main finding is that both anharmonicity and vertex corrections suppress the effective e-ph coupling and Tc, with FBW calculations yielding Tc in close agreement with experiments for D3S at both pressures and for H3S at 200 GPa. The results underscore the need for including anharmonicity, vertex effects, and energy-dependent DOS in predictive models of high-Tc hydride superconductors.

Abstract

Superconductivity in compressed H3S arises from the interplay between high-frequency phonons and a pronounced van Hove singularity near the Fermi level. Using first-principles calculations, we investigate the superconducting properties of H3S and D3S at 160 and 200 GPa, explicitly incorporating anharmonic lattice dynamics and first-order vertex corrections to electron-phonon (e-ph) interactions, thereby going beyond the Migdal approximation underlying conventional Migdal-Eliashberg theory. We find that both anharmonicity and nonadiabatic vertex corrections suppress the effective e-ph coupling and reduce the superconducting critical temperature (Tc). Calculations performed within the energy-dependent full-bandwidth Eliashberg formalism, including both anharmonic and vertex effects, yield Tc values in close agreement with experimental measurements for D3S at both pressures and for H3S at 200 GPa.

Figures (4)

• Figure 1: Electronic band structure and density of states (DOS) of H$_3$S at 160 GPa (blue line) and 200 GPa (black line).
• Figure 2: a) Phonon dispersion and phonon density of states (PhDOS) of H$_3$S at 160 GPa calculated within the harmonic (black) and anharmonic (red) approximations. The Eliashberg spectral function $\alpha^2F(\omega)$ (solid lines) and its cumulative e-ph coupling $\lambda(\omega)$ (dashed lines) are shown along with the spectral function from vertex corrections $\alpha^2F^{\rm V}(\omega, \omega^\prime)$ and its integrated coupling strength $\lambda^{\mathrm{V}}$. The isotropic superconducting gap $\Delta(T)$ is computed using four approaches with anharmonic phonons and an effective Coulomb pseudopotential $\mu_{\mathrm{E}}^* = 0.16$: FBW (orange), FSR (teal-green), vertex-corrected FBW (dashed orange), and vertex-corrected FSR (dashed teal-green). b) Corresponding results for D$_3$S at 160 GPa.
• Figure 3: a) Phonon dispersion and phonon density of states (PhDOS) of H$_3$S at 200 GPa calculated within the harmonic (black) and anharmonic (red) approximations. The Eliashberg spectral function $\alpha^2F(\omega)$ (solid lines) and its cumulative e-ph coupling $\lambda(\omega)$ (dashed lines) are shown along with the spectral function from vertex corrections $\alpha^2F^{\rm V}(\omega, \omega^\prime)$ and its integrated coupling strength $\lambda^{\mathrm{V}}$. The isotropic superconducting gap $\Delta(T)$ is computed using four approaches with anharmonic phonons and an effective Coulomb pseudopotential $\mu_{\mathrm{E}}^* = 0.16$: FBW (orange), FSR (teal-green), vertex-corrected FBW (dashed orange), and vertex-corrected FSR (dashed teal-green). b) Corresponding results for D$_3$S at 200 GPa.
• Figure 4: Superconducting critical temperatures ($T_{\rm c}$) calculated with anharmonic phonons for H$_3$S and D$_3$S at 160 and 200 GPa using four approaches: FSR (open star), FBW (open diamond), ver- FSR (filled star), and ver- FBW (filled diamond). Experimental data extracted from the literature are shown for comparison and labeled as Exp-1 Drozdov2015, Exp-2 Einaga2016Mozaffari2019Nakao2019Eremets2016, and Exp-3 Du2025Minkov2020. Black symbols represent H$_3$S and red symbols represent D$_3$S.