Spectral representation of the electrical properties of layered materials

We present a spectral representation for the effective conductivity of two homogeneous layers joined at a rough interface. This spectral representation is closely related to the Bergman–Milton spectral representation for bulk composites, and is easily extended to multilayered materials. By comparing the layered system to a reference layered system that has a flat interface, we form a surface spectral density that captures all the effects of surface structure on the effective conductivity of the layered sample, and is independent of the conductivities of the two layers. Because of the anisotropy of the layered system there are two surface spectral densities, one for the case where the applied field is parallel to the interface, and one for the case where the applied field is perpendicular to the interface. We discuss the relationship between these two spectral representations and present sum rules that are directly related to the degree of surface roughness. We present numerical calculations of the surface spectral density for Gaussian random surfaces which have been extensively used to study light scattering from rough surfaces. r 2003 Elsevier B.V. All rights reserved. PACS: 68.35.Ct; 72.80.Tm; 77.84.Lf; 78.20.Ci