Raman scattering from the CaC$_6$ superconductor

\def\prb{Phys. Rev. B} \def\prl{Phys. Rev. Lett.} \def\pla{Phys. Lett. A} \def\pr{Phys. Rev.} \def\MgB2{MgB$_{2}$} \def\cm-1{cm$^{-1}$\,} \def\cmT-1{cm$^{-1}$/T\,} \def\E2g{$E_{2g}$} \def\A1g{$A_{1g}$} \def\2DS{$2\Delta_{S}^{E}$} \def\DL{$2\Delta_{L}^{E}$} \def\2DA{$2\Delta^{A}$} \def\D0{$2\Delta_{0}$} \def\CaC6{CaC$_6$} A polarized Raman scattering study has been performed on bulk 1st stage intercalated graphite CaC$_6$ crystals at sub-$T_c$ temperatures. We identify all three Raman active E$_g$ bands expected for the Rm$\overline{6}$ space group of CaC$_6$ at 440, 1120 and 1508\,\cm-1 and find them to be in agreement with zone center modes predicted by first principles calculations of phonon dispersion.$^1$ In addition the equivalents of the graphite D and G bands are observed at respective frequencies. Inherent to the disorder induced double resonant scattering process$^2$ the D band shifts from 1308\,\cm-1 to 1332\,\cm-1 upon the change of the excitation laser wavelength from 647\,nm to 476 \,nm. Assuming linear dependence of the D band peak position as a function of excitation energy this translates to the frequency shift of 35\,\cm-1$/$\,eV. By comparing the integrated intensity of the G band at 1582\,\cm-1 in \CaC6 to the one in kish graphite the relative fraction of higher stage domains to the 1st stage intercalation is estimated to be less then 0.2\%. Finally upon the superconducting phase transition we observe a 2$\Delta$ peak with the frequency of 24\,\cm-1 at $5\,\rm{K}$. With temperature increase this peak persists shortly up to the SC phase transition at 11.6\,K and shows temperature dependence consistent with the strong coupling regime. $^1$M.~Calandra and F.~Mauri, PRL $\mathbf{95}$, 237002 (2005). $^2$C.~Thomson and S.~Reich, PRL $\mathbf{85}$, 5214 (2000).