Electron-phonon interaction and phonon renormalization in the lamellar cobaltate Na _x CoO ₂

We study theoretically the electron-phonon interaction in Na _x CoO ₂. For the A _1g and E _1g phonon modes found in Raman experiments, we calculate the matrix elements of the electron-phonon interaction. Analyzing the feedback effect of the conduction electrons on the phonon frequency ω, we investigate the doping dependence of these two phonon modes. Due to the momentum dependence of the electron-phonon interaction, we find the strongest renormalization of the E _1g mode around the Brillouin zone boundary which should be observed in the neutron scattering. At the same time, the A _1g mode shows the strongest coupling to the conducting electrons around the Γ point and reveals its doping dependence in the Raman experiments. Our results shed light on the possible importance of the electron-phonon interaction in the lamellar sodium cobaltates.