Theory of the Band Structure of Very Degenerate Semiconductors

In very degenerate semiconductors it is possible to achieve a situation in which the parameter $r_s$ that characterizes the strength of electron-electron or electron-impurity interactions is small compared to unity. In this limit, many-body perturbation theory may be used to study certain properties of the system. We here consider the density of states for such a semiconductor, writing this quantity in terms of the electron propagator and self-energy. It is shown that the effect of electron-electron interactions is to screen impurity fields, and produce a nearly rigid downward motion of the energy band. For electrons near the Fermi surface the influence of impurity potentials is small; in particular, the diagonal matrix element of the impurity potential is exactly canceled by the corresponding electron-electron interaction term. As one proceeds lower into the band the electron-impurity interaction becomes progressively more important, and the straightforward perturbation theory appears to diverge for energies less than ${r_s}^{\frac{1}{2}}$ times the Fermi energy. A propagator modification technique is suggested for extending the range of perturbation theory.