On the quantum theory of electrical conductivity : The conductivity tensor to zeroth order

Abstract

The Kubo formula, which gives in closed form a very general expression for the electrical conductivity tensor, is simplified. A first simplification is of general nature and replaces the double integration in the original formula by a single one. A second reduction is performed in the one electron approximation. The formula there obtained is evaluated using and extending Van Hove's methods for deriving the master equation. For simplicity the evaluation is restricted to the case of elastic scattering by static impurities. The conductivity tensor is calculated up to zeroth order, correcting and extending an incomplete discussion by Chester and Thellung based on the same techniques. The lowest order term (of order -2) reproduces the result usually obtained by the Boltzmann equation method. The order of magnitude of the next order term is governed by the dimensionless parameter[h/τ(η)η]1/2 where τ(η) is the relaxation time taken at the Fermi surface and η the Fermi energy. This term may be important in explaining the electrical resistance of binary alloys with rather high resistivity.