Non-adiabatic phonon renormalization in metallic versus insulating rutile oxides

We present a comparative Raman scattering study of metallic rutile oxides (RuO$_2$ and IrO$_2$) and insulating rutiles (TiO$_2$ and SnO$_2$). Temperature-dependent Raman spectra reveal that the metallic compounds exhibit pronounced phonon frequency hardening, $\omega(11~\mathrm{K})-\omega(300~\mathrm{K})=\Delta\omega \approx 6$-$10~\mathrm{cm}^{-1}$, whereas the insulating rutiles show only modest hardening, $\Delta\omega \approx 1$-$3~\mathrm{cm}^{-1}$. In contrast, the linewidth changes, $\Delta\Gamma \approx 1$--$7~\mathrm{cm}^{-1}$, do not display a systematic metallic-insulating classification. Fits with the conventional Klemens anharmonic decay model reproduce the overall temperature trends but yield inconsistent anharmonic parameters for the metallic compounds when benchmarked against insulating rutile analogues. A modified Klemens framework, incorporating an additional $T^{2}$ correction to the phonon frequency arising from the electronic contribution to the phonon self-energy, quantitatively accounts for the enhanced renormalization observed in metallic systems. These results establish finite non-adiabatic electron-phonon coupling in metallic rutiles and demonstrate that phonon renormalization can be identified even in the absence of observable Fano asymmetry in the phonon line shapes.