Optical sum rules and effective-medium theories for a polycrystalline material: Application to a model for polypyrrole.

We derive sum rules for the effective dielectric function ee(v) of a polycrystalline material, under the assumption of macroscopic isotropy. If the material comprising the polycrystal is a quasi-one-dimensional or quasiplanar Drude metal, we predict that part of the oscillator strength of the polycrystal is pushed up in frequency to form an ‘‘impurity’’ band of confined plasmonlike excitations. Under an additional condition of ‘‘strong isotropy,’’ we calculate the center of gravity of this band, in terms of the zero-frequency conductivity of the polycrystal. Analogous predictions are given for the energy-loss function, 2Ime e (v). The effectivemedium theory for a polycrystal composed of approximately spherical crystallites is shown to satisfy this condition of strong isotropy. A more general effective-medium theory for ellipsoidal crystallites does not satisfy strong isotropy. It does, however, obey the only sum rule which is valid for any microstructure, namely, the sum rule on the spectral density. As an application, we describe a simple effective-medium model which qualitatively accounts for the ac electromagnetic properties of polypyrrole, over a broad range of frequencies, based on the assumption of polycrystallinity. Many features of the observed optical constants are found consistent with the existence of a broad localized plasmon band arising from polycrystallinity.