Scaling characteristics of ac conductivity and dielectric function in fractal regime of the metal-insulator transition

Abstract

An analysis of the ac conductivity σ_ac(ω), and the ac dielectric constant, ɛ(ω), of the metal-insulator percolation systems is presented in the critical regime near the transition threshold. It is argued that the polarization and relaxation of the finite fractal metallic clusters play dominant roles in controlling the dynamic response of the system on both sides of the threshold. The relaxation time constant of a fractal cluster is shown to scale with its size as$\tau _s \sim r_s^{d_t }$ withd _t = 4 − 2d +d _c + μ/ν, whered is tge Euclidean dimension, andd _c , μ, and ν are the scaling indices for the charging, the dc conductivity, and the correlation length respectively. The average time dependent response of the system is shown to scale with a new time scale$\tau _0 \xi ^{d_t }$, where ζ is the correlation length and τ₀ is a microscopic time constant. It is shown that at frequencies$\begin{array}{*{20}c} {\xi ^{ - d_t } \ll \omega \tau _0 \ll 1,} & {\sigma _{ac} (\omega ) \sim \omega ^{{\mu \mathord{\left/ {\vphantom {\mu {\nu d_t }}} \right. \kern-\nulldelimiterspace} {\nu d_t }}} } \\ \end{array}$ and$\varepsilon (\omega ) \sim \omega ^{{\mu \mathord{\left/ {\vphantom {\mu {\nu d_t - 1}}} \right. \kern-\nulldelimiterspace} {\nu d_t - 1}}}$ with μ/νd_t ≃ 1, in close agreement with experiments. The effects of the anomalous transport along the infinite cluster and the medium polarizability are also discussed.