Renormalization of the chemical potential due to multiphonon effects at the surface of metals

We study the relation between renormalization of the chemical potential due to multiphonon effects at the surface of Be（0001） and doping by solving the strong-coupling self-consistent equations of a two-dimensional（2D） electron-phonon interaction system.We present the quasiparticle dispersions and inverse lifetimes of a 2D electron system interacting with Einstein phonons under the different dopings（corresponding to chemical potentials）.We find that the effect of electron-phonon interaction on electron structure is strongest at the half filling,but it has no effect on the chemical potential.However,the chemical potential shows distinct renormalization effects away from half filling due to the electron-phonon interaction.