Proper-connected-diagram expansion of electrical conductivity for electron-impurity system

Abstract

The straightforward expansion of the electric conductivityσ for an electron-impurity system in powers of the impurity densityn _s is known to diverge term by term. It is shown that this difficulty which arises from the average potential field of impurities, can be removed by using the expansion in terms of proper connected diagrams. This expansion, however, leads to a complicated dependence ofσ onn _s . The analytical behavior ofσ(n _s ) in the neighborhood ofn _s =0 is investigated qualitatively with the assumption that the interaction potential between electron and impurity is arbitrarily strong but has a finite range. It is found that this investigation doesnot support the argument recently advanced by several authors thatσ(n _s ) should contain in its asymptotic expansion terms in the combination ofn _s and lnn _s The present theory starts with Kubo's current correlation function formula forσ and proceeds with the aid of connected diagrams. It is developed for a quantum system but also applies for a classical system with few modifications in definitions.