Non-adiabatic phonons within the doped graphene layers of XC$_6$ compounds

We report the first Raman scattering measurements of BaC$_6$, SrC$_6$, and YbC$_6$, which along with new data on CaC$_6$, permits a systematic study of the phonons and the electron-phonon interaction within the doped graphene layers of these compounds. The out-of-plane carbon phonon softens as the spacing of the graphene layers is reduced in the series BaC$_6$, SrC$_6$, YbC$_6$, and CaC$_6$. This is due to increasing charge in the $\pi^*$ electronic band. Electron-phonon interaction effects between the in-plane carbon modes at $\approx1500$\invcm and the $\pi^*$ electrons cause a strong non-adiabatic renormalization. As charge is transferred into the $\pi^*$ band these non-adiabatic effects are found to increase concurrent with a reduction in the phonon lifetime.