Maxwell Garnett theory for mixtures of anisotropic inclusions: Application to conducting polymers

Abstract

The effective dielectric function ${\ensuremath{\epsilon}}_{e}$ for a medium of anisotropic inclusions embedded in an isotropic host is calculated using the Maxwell Garnett approximation. For uniaxial inclusions, ${\ensuremath{\epsilon}}_{e}$ depends on how well the inclusions are aligned. We apply this approximation to study ${\ensuremath{\epsilon}}_{e}$ for a model of quasi-one-dimensional organic polymers. The polymer is assumed to be made up of small single crystals embedded in an isotropic host of randomly oriented polymer chains. The host dielectric function is calculated using the effective-medium approximation (EMA). The resulting frequency-dependent ${\ensuremath{\epsilon}}_{e}(\ensuremath{\omega})$ closely resembles experiment. Specifically, $\mathrm{Re}{\ensuremath{\epsilon}}_{e}(\ensuremath{\omega})$ is negative over a wide frequency range, while $\mathrm{Im}{\ensuremath{\epsilon}}_{e}(\ensuremath{\omega})$ exhibits a broad ``surface plasmon'' band at low frequencies, which results from localized electronic excitations within the crystallites. If the host is above the conductivity percolation threshold, $\mathrm{Im}{\ensuremath{\epsilon}}_{e}(\ensuremath{\omega})$ has a low-frequency Drude peak in addition to the surface plasmon band, and $\mathrm{Re}{\ensuremath{\epsilon}}_{e}(\ensuremath{\omega})$ is negative over an even wider frequency range. We also calculate the cubic nonlinear susceptibility ${\ensuremath{\chi}}_{e}(\ensuremath{\omega})$ of the polymer, using a nonlinear EMA. At certain frequencies, ${\ensuremath{\chi}}_{e}(\ensuremath{\omega})$ is found to be strongly enhanced above that of the corresponding single crystals. Our results suggest that the electromagnetic properties of conducting polymers can be understood by viewing the material as randomly inhomogeneous on a small scale such that the quasistatic limit is applicable.