Low-temperature conductivity of thin metallic slabs at arbitrary fuchs boundary conditions

Abstract

The conductivity of three- and two-dimensional metallic slabs, thin with respect to the mean free path (m.f.p.), is calculated in the isotropic metal model under arbitrary disordered impurity scattering and arbitrary Fuchs boundary conditions. The general expression for the conductivity for an angle-dependent specularity coefficient ϵ is also found. The detailed calculations are performed for three variants: (i) the same, angle-dependent ϵ on both borders, (ii) one diffuse and one arbitrary border, (iii) specular borders for suitably small electron incidence angles. In cases (i) and (ii) the O(1) term of the conductivity disappears unless we deal with the specular (i) case, in agreement with Fuchs' result. This term appears in case (iii). It is shown that the two first asymptotic terms of the conductivity averaged over the slab thickness coincide with those at the relaxation time approximation in cases (i) and (ii). The results for case (i) in their integral form pass into those by Fuchs at d = 3, but not in their asymptotic form, when his results become unphysical, giving, e.g., the averaged conductivity being a diminishing function of ϵ vanishing at ϵ = 1. In fact, our small parameter in case (i) is − q ln q(1−ϵ) with q being the slab thickness in units of the m.f.p.