Electron-phonon vs. electron-impurity interactions with small electron bandwidths

It is a common practice to try to understand electron interactions in metals by defining a hierarchy of energy scales. Very often, the Fermi energy is considered the largest, so much so that frequently bandwidths are approximated as infinite. The reasoning is that attention should properly be focused on energy levels near the Fermi level, and details of the bands well away from the Fermi level are unimportant. However, a finite bandwidth can play an important role for low frequency properties: following a number of recent papers, we examine electron-impurity and electron-phonon interactions in bands with finite widths. In particular, we examine the behavior of the electron self-energy, spectral function, density of states, and dispersion, when the phonon spectral function is treated realistically as a broad Lorentzian function. With this phonon spectrum, impurity scattering has a significant nonlinear effect.