Conductivity of metallic films and multilayers

Abstract

We examine the quantum and semiclassical approaches to the calculation of the electrical conductivity using a model of free electrons with a finite lifetime. We evaluate the Kubo formula exactly for the free-electron model for homogeneous conductors, thin films, and multilayers. We use these results to explicitly demonstrate the relationships among the exact quantum approach, semiclassical approaches, and an approximate quantum approach. One popular semiclassical approach uses a specularity parameter to describe the fraction of electrons that scatter coherently at boundaries. We show that this parameter should depend on the angle at which the electrons are incident on the boundary. Another semiclassical theory employs the concept of a local mean free path. We show that this approximation works surprisingly well; the error in the calculated conductivity being significant only for very thin layers. A third approach uses an approximate solution to the Kubo formula. We show that this approximation works well for thick layers and extremely thin layers but not for intermediate layer thicknesses. We discuss the implication of these results to the study of the giant magnetoresistance.