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

We present a study of the lattice dynamical properties of superconducting SrPt$_3$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 (SOC) 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 ($2\Delta_0 / k_{\mathrm{B}}T_c \sim 5$). 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.