Band-filling effects on electron-phonon properties of normal and superconducting state

We address the effect of band filling on the effective electron mass $m^*$ and the superconducting critical temperature $T_c$ in a electron-phonon system. We compare the vertex corrected theory with the non-crossing approximation of the Holstein model within a local approximation. We identify two regions of the electron density where $m^*$ and $T_c$ are enhanced or decreased by the inclusion of the vertex diagrams. We show that the crossover between the enhancement at low density and the decrease towards half filling is almost independent of the microscopic electron-phonon parameters. These different behaviors are explained in terms of the net sign of the vertex diagrams which is positive at low densities and negative close to half filling. Predictions of the present theory for doped MgB$_2$, which is argued to be in the low density regime, are discussed.