Nonlocal screening, electron-phonon coupling, and phonon renormalization in metals*

A new method for calculating the phonon self-energy in metals arising from the coupling between phonons and electrons near the Fermi surface is developed. The essence of this scheme is the separation of the inter-and intra-band parts of the electron polarizability. The intra-band contribution provides extra screening and is closely related to the electron-phonon coupling and phonon softening in metals. Applications of this scheme to phonons in MgB$_2$ give excellent results when compared with experiments and previous theoretical work. In addition, both electron and hole dopings are found to reduce the renormalization effect of the $E_{2g}$ phonon mode. This indicates weakened electron-phonon couplings in the doped systems. This is consistent with the experimental observations that the superconducting transition temperature of MgB$_2$ drops upon substituting Mg with either Al or Li.