Dielectric resonances of lattice animals and other fractal structures

Abstract

Electrical and optical properties of binary inhomogeneous media are currently modelled by a random network of metallic bonds (conductance $σ_0$, concentration $p$) and dielectric bonds (conductance $σ_1$, concentration $1-p$). The macroscopic conductivity of this model is analytic in the complex plane of the dimensionless ratio $h=σ_1/σ_0$ of the conductances of both phases, cut along the negative real axis. This cut originates in the accumulation of the resonances of clusters with any size and shape. We demonstrate that the dielectric response of an isolated cluster, or a finite set of clusters, is characterised by a finite spectrum of resonances, occurring at well-defined negative real values of $h$, and we define the cross-section which gives a measure of the strength of each resonance. These resonances show up as narrow peaks with Lorentzian line shapes, e.g. in the weak-dissipation regime of the $RL-C$ model. The resonance frequencies and the corresponding cross-sections only depend on the underlying lattice, on the geometry of the clusters, and on their relative positions. Our approach allows an exact determination of these characteristics. It is applied to several examples of clusters drawn on the square lattice. Scaling laws are derived analytically, and checked numerically, for the resonance spectra of linear clusters, of lattice animals, and of several examples of self-similar fractals.