Enhanced superconductivity in palladium hydrides by non-perturbative electron-phonon effects

Abstract

Palladium hydrides exhibit the largest isotope-effect anomaly in superconductivity: replacing hydrogen with heavier isotopes increases the superconducting critical temperature. Although this behavior is commonly attributed to strong anharmonic hydrogen vibrations, \textit{ab initio} treatments have so far incorporated anharmonic effects only through phonon renormalization, neglecting non-linear contributions to the electron-phonon interaction vertices. While such approaches reproduce the anomalous isotope trend, they severely underestimate the critical temperatures. Here, we show that non-linear electron-phonon coupling is essential in palladium hydrides. A straightforward inclusion of higher-order perturbative terms leads to a qualitative breakdown: the critical temperature is overestimated and the isotope anomaly is lost. We therefore adopt a non-perturbative framework based on an explicit evaluation of the ion-mediated electron-electron interaction, enabling anharmonic effects to be treated consistently in both the phonon spectra and the interaction vertices. Applied to PdH and PdD, it restores the anomalous isotope effect and brings calculated critical temperatures into significantly improved agreement with experiments.

Figures (7)

• Figure 1: First-order electron-phonon coupling constant $\lambda$ of PdH calculated by finite differences, both with forward (Eq. \ref{['eq:fwd_derivative']}) and central differences (Eq. \ref{['eq:central_derivative']}), for different $d^a$ steps, making use of SSCHA phonon frequencies and polarization vectors. On the horizontal axis the ratio between $d^a$ and equilibrium root-mean-square displacement $\sigma^a=\sqrt{\langle{(u^a)^2}\rangle_{{{\textup{eq}}}}}$ along the same direction is indicated (the same ratio $d^a/\sigma^a$ is considered for all the $a$). The horizontal dashed line represents the value of $\lambda$ obtained with DFPT.
• Figure 2: (Left) Electron self-energy diagrams constructed with first-order bare $g$ vertices [Eq. \ref{['eq:def_g_normal']}] and averaged first-order $\bigl\langle\mkern-1mu g \mkern-1mu \bigr\rangle{}$ vertices [Eq. \ref{['eq:def_gavgn_normal']}]. $G$ and $D$ denote electron and phonon propagators, respectively. The corresponding self-energies are labeled $gGDg$ and $\bigl\langle\mkern-1mu g \mkern-1mu \bigr\rangle{} GD \bigl\langle\mkern-1mu g \mkern-1mu \bigr\rangle{}$. Below, the first-order $\overset{{{(1)}}}{\alpha^2\!F}(\omega)$ and $\overset{{{(1)}}}{\lambda}(\omega)$ calculated with $gGDg$ (blue) and $\bigl\langle\mkern-1mu g \mkern-1mu \bigr\rangle{} GD \bigl\langle\mkern-1mu g \mkern-1mu \bigr\rangle{}$ (red), for both PdH and PdD using SSCHA anharmonic phonon propagators. The final value of $\overset{{{(1)}}}{\lambda}$ is indicated. (Middle) Same analysis for the second-order contribution: $\overset{{{(2)}}}{\alpha^2\!F}(\omega)$ and $\overset{{{(2)}}}{\lambda}(\omega)$ computed using bare second-order vertices $\overset{{{(2)}}}{g}$ and averaged second-order vertices $\bigl\langle\mkern-1mu \overset{{{(2)}}}{g} \mkern-1mu \bigr\rangle{}$. The corresponding self-energies are labeled $\overset{{{(2)}}}{g}GDD\overset{{{(2)}}}{g}$ and $\bigl\langle\mkern-1mu \overset{{{(2)}}}{g} \mkern-1mu \bigr\rangle{}GDD\bigl\langle\mkern-1mu \overset{{{(2)}}}{g} \mkern-1mu \bigr\rangle{}$. The resulting $\overset{{{(2)}}}{\lambda}$ is indicated. (Right) Total spectral functions and coupling strengths obtained from the sum of first- and second-order contributions. The final value $\lambda=\overset{{{(1)}}}{\lambda}+\overset{{{(2)}}}{\lambda}$ is reported.
• Figure S1: Self-consistent harmonic phonon dispersions of PdH and PdD.
• Figure S2: Left panels: values of $\overset{{{(1)}}}{\lambda}$ and $\overset{{{(2)}}}{\lambda}$ for PdH, computed from the $gGDg$ and $\overset{{{(2)}}}{g}GDD\overset{{{(2)}}}{g}{}$ self-energies, respectively, as a function of the $k$-point grid used for the Brillouin-zone integration and of the Gaussian smearing employed to approximate the delta functions. Right panels: values of $\overset{{{(1)}}}{\lambda}$ and $\overset{{{(2)}}}{\lambda}$ for PdH, computed from the $\bigl\langle\mkern-1mu g \mkern-1mu \bigr\rangle{} GD\bigl\langle\mkern-1mu g \mkern-1mu \bigr\rangle{}$ and $\bigl\langle\mkern-1mu \overset{{{(2)}}}{g} \mkern-1mu \bigr\rangle{}GDD\bigl\langle\mkern-1mu \overset{{{(2)}}}{g} \mkern-1mu \bigr\rangle{}$ self-energies, respectively, using the selected $k$-point grid and Gaussian smearing, as a function of the stochastic population size employed to evaluate the averaged vertices.
• Figure S3: Left panels: values of $\overset{{{(1)}}}{\lambda}$ and $\overset{{{(2)}}}{\lambda}$ for PdD, computed from the $gGDg$ and $\overset{{{(2)}}}{g}GDD\overset{{{(2)}}}{g}{}$ self-energies, respectively, as a function of the $k$-point grid used for the Brillouin-zone integration and of the Gaussian smearing employed to approximate the delta functions. Right panels: values of $\overset{{{(1)}}}{\lambda}$ and $\overset{{{(2)}}}{\lambda}$ for PdD, computed from the $\bigl\langle\mkern-1mu g \mkern-1mu \bigr\rangle{} GD\bigl\langle\mkern-1mu g \mkern-1mu \bigr\rangle{}$ and $\bigl\langle\mkern-1mu \overset{{{(2)}}}{g} \mkern-1mu \bigr\rangle{}GDD\bigl\langle\mkern-1mu \overset{{{(2)}}}{g} \mkern-1mu \bigr\rangle{}$ self-energies, respectively, using the selected $k$-point grid and Gaussian smearing, as a function of the stochastic population size employed to evaluate the averaged vertices.