Role of spin-phonon and electron-phonon interactions in phonon renormalization of (Eu$_{1-x}$Bi$_x$)$_2$Ir$_2$O$_7$ across the metal-insulator phase transition: Temperature-dependent Raman and X-ray studies

We report temperature-dependent Raman scattering and X-ray diffraction studies of pyrochlore iridates, (Eu$_{1-x}$Bi$_x$)$_2$Ir$_2$O$_7$, for x=0, 0.02, 0.035, 0.05 and 0.1. The temperature variation in Raman experiments spans from 4 K to 300 K, covering the metal-insulator phase transition accompanied by paramagnetic to all-in/all-out (AIAO) spin ordering (T$_N$). These systems also show a Weyl semi-metal (WSM) phase at low temperatures (below ~50 K). We show that the Ir-O-Ir bond bending mode, A$_{1g}$ (510 cm$^{-1}$), shows anomalous softening in the magnetically ordered AIAO state, arising primarily from the spin-phonon interaction due to the phonon-modulation of the Dzyaloshinskii-Moriya (DM) spin-exchange interaction. The two stretching modes, T$_{2g}^1$ (307 cm$^{-1}$) and T$_{2g}^2$ (382 cm$^{-1}$) harden significantly in the magnetic insulating phase. The T$_{2g}$ phonons also show anomalous temperature dependence of their mode frequencies, hitherto unreported, due to strong electron-phonon coupling. The signatures of the WSM state are observed in phonon renormalization below 50 K due to strong electron-phonon interaction. Our experimental results establish strong magneto-elastic coupling below T$_N$ and significant electron-phonon interactions in the metallic phase above T$_N$ as well as in the low-temperature WSM state.