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 ${\sigma }_0$, concentration $p$) and dielectric bonds (conductance ${\sigma }_1$, concentration $1-p$). The macroscopic conductivity of this model is analytic in the complex plane of the dimensionless ratio $h={\sigma }_1/{\sigma }_0$ 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.

arXiv comment25 pages, plain TeX. Figures (hard copies) available upon request, to appear in the Journal of Physics A

Cited by in corpus (11)

• Dielectric mixtures -- electrical properties and modeling
• Theory of impedance networks: The two-point impedance and LC resonances
• Dielectric resonances of binary random networks
• Impedance responses and size-dependent resonances in topolectrical circuits via the method of images
• Spectral Properties of Random Reactance Networks and Random Matrix Pencils
• Local-field distribution in resonant composites: Green's-function formalism
• Statistics of level spacing of geometric resonances in random binary composites
• Dielectric resonances of ordered passive arrays
• Random matrix approach to plasmon resonances in the random impedance network model of disordered nanocomposites
• Two-dimensional plasmons in the random impedance network model of disordered thin-film nanocomposites
• Interplay Between Structural Randomness, Composite Disorder, and Electrical Response: Resonances and Transient Delays in Complex Impedance Networks