Perturbation-Moment Method: Application to Band Structure of Impure Semiconductors

Abstract

A contour integral method of calculating spectral moments from the perturbation expansion of a characteristic function is described. The moments are used to provide a sort of "analytic continuation" of the perturbation expansion for the density of states into a Legendre polynomial expansion valid for low energies. The method is tested on the one-dimensional impurity band structure problem. The results are less accurate than Klauder's best diagram-summing approximation. The inaccuracy of the results is attributed to poor convergence of the perturbation expansion for the moments. We apply the method to the three-dimensional impurity problem where the impurities are represented by randomly located screened Coulomb potentials. At intermediate densities the perturbation expansion for the moments converges much faster than in one dimension, hence, the results should be more accurate. Although the Thomas-Fermi method should not be accurate at intermediate densities, it agrees remarkably well with the perturbation-moment results.