  Quenched disorder in semiconductors induces localized electronic states at
the band edge, which manifest as an exponential tail in the density of states.
For large impurity densities, this tail takes a universal Lifshitz form that is
characterized by short-ranged potential fluctuations. We provide both
analytical expressions and numerical values for the Lifshitz tail of a
parabolic conduction band, including its exact fluctuation prefactor. Our
analysis is based on a replica field integral approach, where the leading
exponential scaling of the tail is determined by an instanton profile, and
fluctuations around the instanton determine the subleading pre-exponential
factor. This factor contains the determinant of a fluctuation operator, and we
avoid a full computation of its spectrum by using a Gel'fand-Yaglom formalism,
which provides a concise general derivation of fluctuation corrections in
disorder problems. We provide a revised result for the disorder band tail in
two dimensions.
