First-Principles Theory of Anharmonicity and the Inverse Isotope Effect in Superconducting Palladium-Hydride Compounds

Palladium hydrides display the largest isotope effect anomaly known in the literature. Replacement of hydrogen with the heavier isotopes leads to higher superconducting temperatures, a behavior inconsistent with harmonic theory. Solving the self-consistent harmonic approximation by a stochastic approach, we obtain the anharmonic free energy, the thermal expansion, and the superconducting properties fully ab initio. We find that the phonon spectra are strongly renormalized by anharmonicity far beyond the perturbative regime. Superconductivity is phonon mediated, but the harmonic approximation largely overestimates the superconducting critical temperatures. We explain the inverse isotope effect, obtaining a $-0.38$ value for the isotope coefficient in good agreement with experiments, hydrogen anharmonicity being mainly responsible for the isotope anomaly.

Figure 1

Equilibrium lattice parameters as a function of temperature for PdH, PdD, and PdT compared to experimental results [12, 20]. In the inset the calculated thermal expansion coefficients are shown together with the measured values in Ref. 39.

Figure 2

Harmonic and SSCHA phonon spectra calculated for PdH (a), PdD (b), and PdT (c) at the equilibrium volume at 0 K. The experimental results obtained for nonstoichiometric ${\mathrm{PdD}}_{0.63}$ [14] and ${\mathrm{PdT}}_{0.70}$ [15] are shown. In the right panels the PDOS projected onto Pd and H atoms is plotted within the harmonic approximation (dotted lines) and the SSCHA (solid lines).

Figure 3

${\alpha }^{2}F(\omega )$ and $\lambda (\omega )=2{\int }_0^{\omega }d{\omega }^{\prime }{\alpha }^{2}F({\omega }^{\prime })/{\omega }^{\prime }$ for (a) PdH, (b) PdD, and (c) PdT within the harmonic approximation and the SSCHA.