Lattice dynamical properties of superconducting ${\mathrm{SrPt}}_3\mathrm{P}$ studied via inelastic x-ray scattering and density functional perturbation theory

We present a study of the lattice dynamical properties of superconducting ${\mathrm{SrPt}}_3\mathrm{P}$ ($T_c=8.4$ K) via high-resolution inelastic x-ray scattering (IXS) and ab initio calculations. Density functional perturbation theory including spin-orbit coupling results in enhanced electron-phonon coupling (EPC) for the optic phonon modes originating from the Pt(I) atoms, with energies $\sim 5$ meV, resulting in a large EPC constant $\lambda \sim 2$. An overall softening of the IXS powder spectra occurs from room to low temperatures, consistent with the predicted strong EPC and with recent specific-heat experiments ($2{\mathrm{\Delta }}_0/k_{\mathrm{B}}{T}_c\sim 5$). The low-lying phonon modes observed in the experiments are approximately 1.5 meV harder than the corresponding calculated phonon branch. Moreover, we do not find any changes in the spectra upon entering the superconducting phase. We conclude that current theoretical calculations underestimate the energy of the lowest band of phonon modes indicating that the coupling of these modes to the electronic subsystem is overestimated.

Figure 1

(a) Crystal structure of ${\mathrm{SrPt}}_3\mathrm{P}$. Platinum atoms occupy two distinct positions, Pt(I) and Pt(II). (b) Calculated and experimental [5] lattice parameters. (c) Calculated phonon dispersions $\omega \left(k\right)$ and relative phonon linewidths $\gamma /\omega$ (red vertical bars, amplified for clarity). SOC was included in the calculations (see text). (d) Calculated PDOS (black lines) vs $\omega$. The colored regions emphasize the dominant atoms determining the dispersion at low [Pt(I), blue] and high (P, orange) energies. (e) Calculated isotropic Eliashberg function ${\alpha }^{2}F\left(\omega \right)$ (blue line) and EPC constant $\lambda \left(\omega \right)$ (red line) vs $\omega$. Percentages indicate the contribution of ${\alpha }^{2}F\left(\omega \right)$ to $\lambda \left(\omega \right)$ in different energy ranges.

Figure 2

(a) Inelastic data from combined energy scans at $Q=5.7$ Å${}^{-1}$ obtained at 300 K (circles) and 10 K (diamonds). Powder-average calculations of the dynamical structure factors are shown for $Q=5.7$ Å${}^{-1}$ as green (no SOC) and red (with SOC) lines. (b) A representative energy scan measured between $-10$ meV and 30 meV (empty symbols). The inelastic phonon component (full symbols) was obtained after subtracting the fit to the elastic peak contribution (blue line). (c) Scaling of the Eliashberg function ${\alpha }^{2}F\left(\omega \right)$ by a shift in energy ${\omega }_{sh}\to \omega +1.7$ meV.

Figure 3

(a) Inelastic data from combined energy scans at $Q=5.7$ Å${}^{-1}$ obtained above (diamonds) and below (squares) $T_c$. The difference between these spectra is displayed in (b). (c), (d) Example of calculations of phonon line shapes based on the theory of Allen et al. [24] for the first optic phonon mode at the $\mathrm{\Gamma }$ point for a single crystal. Red and blue curves correspond to temperatures above and below $T_c$, respectively. The broadening of the dashed curves is due only to the EPC effect presented in Fig. 1, while for the solid lines, an additional instrumental broadening is also included (see text).