Fractal aspects of the dielectric response of charge carriers in disordered materials

Abstract

We put forward a fractal model for the anomalous low-frequency dispersion in the dielectric properties of materials. This dispersion consists of two fractional power laws in the frequency domain. We review the theory of the dielectric response due to fractal time processes and conduction on fractal structures. A fractal time process can arise from multiple trapping and trap-controlled hopping. The most studied fractal structure is percolation clusters. The anomalous low-frequency dispersion can be modeled by considering a combination of a fractal time process and a fractal structure. The power law at low frequencies is due to the fractal time process, while the high-frequency power law depends on both the conduction on the fractal structure and the fractal time process. The crossover between the two power laws corresponds to the correlation length of the material. We have compared this model to available experimental data. Experiments show the qualitative features predicted by the fractal model, but quantitative agreement is lacking in many cases. Even for percolation systems at the percolation threshold, discrepancies are apparent. This points to the need for further refinements of the theory. Some possible reasons for the discrepancies are pointed out.