Anomalous frequency-dependent transport in composites

Abstract

We consider transport in metal-insulator composites where the metallic component has a range of conductivities described by the distribution function h(\ensuremath{\sigma})\ensuremath{\propto}${\ensuremath{\sigma}}^{\mathrm{\ensuremath{-}}\ensuremath{\alpha}}$ (0<\ensuremath{\alpha}<1). Near the percolation threshold ${p}_{c}$, the dc conductivity of such a composite is known to vary as (p-${p}_{c}$${)}^{t(\ensuremath{\alpha})}$, where t(\ensuremath{\alpha}) is a nonuniversal exponent and p is the volume fraction of metal. In this paper, we show that on the insulating side of the percolation threshold, the conductivity has an anomalous, nonanalytic frequency dependence; ${\ensuremath{\sigma}}_{\mathrm{eff}(\mathrm{\ensuremath{\omega}})\mathrm{\ensuremath{\propto}}{\ensuremath{\omega}}^{2\mathrm{\ensuremath{-}}\ensuremath{\alpha}}}$ at low frequencies. Analogous behavior is also shown to occur in composites for which the insulating component has a range of dielectric constants described by the distribution function g(\ensuremath{\epsilon})\ensuremath{\propto}${\ensuremath{\epsilon}}^{\mathrm{\ensuremath{-}}\ensuremath{\beta}}$ (1<\ensuremath{\beta}<2). In this case, on the conducting side of the percolation threshold, the real part of the composite dielectric constant has an anomalous frequency dependence, Re[${\ensuremath{\epsilon}}_{\mathrm{eff}(\mathrm{\ensuremath{\omega}})]\mathrm{\ensuremath{\propto}}{\ensuremath{\omega}}^{\mathrm{\ensuremath{-}}(2\mathrm{\ensuremath{-}}\ensuremath{\beta})}}$, while on the insulating side, as was previously known, the static dielectric constant diverges according to the law ${\ensuremath{\epsilon}}_{\mathrm{eff}(\mathrm{\ensuremath{\omega}}=0)\mathrm{\ensuremath{\propto}}({p}_{c}\mathrm{\ensuremath{-}}\mathrm{p}{)}^{\mathrm{\ensuremath{-}}s(\ensuremath{\beta})}}$ where s(\ensuremath{\beta}) is nonuniversal. We also suggest a scaling form which interpolates between the limiting behavior on both sides of the percolation threshold. For a composite of normal metal and superconductor, an anomalous frequency dependence is predicted when the superconducting component has a singular distribution of inverse kinetic inductances or the normal metal has a singular distribution of conductivities. Several possible mechanisms are suggested for producing singular distributions in real materials. These mechanisms imply that nonuniversal behavior may be prevalent in a variety of real composite media.