Critical Percolation and Transport in Nearly One Dimension

TL;DR

This paper investigates transport properties near the percolation threshold in nearly one-dimensional fractal networks, revealing nonanalytic critical indices and strong temperature dependence consistent with experimental data on conducting polymers.

Contribution

It introduces a nonanalytic analysis of critical behavior in nearly 1D fractal networks using real space renormalization, providing new insights into conductivity and dielectric properties.

Findings

Critical indices are nonanalytic functions of epsilon.

Distribution of conductivity is Gaussian with exponentially small relative width.

DC conductivity follows a Mott's law with strong temperature dependence.

Abstract

A random hopping on a fractal network with dimension slightly above one, $d = 1 + \epsilon$, is considered as a model of transport for conducting polymers with nonmetallic conductivity. Within the real space renormalization group method of Migdal and Kadanoff, the critical behavior near the percolation threshold is studied. In contrast to a conventional regular expansion in $\epsilon$, the critical indices of correlation length, $\nu =\epsilon ^{-1}+O(e^{-1/\epsilon })$, and of conductivity, $t\simeq \epsilon^{-2} exp (-1-1/\epsilon )$, are found to be nonanalytic functions of $\epsilon$ as $\epsilon \to 0$. Distribution for conductivity of the critical cluster is obtained to be gaussian with the relative width $\sim \exp (-1/\epsilon )$. In case of variable range hopping an ``1-d Mott's law'' $exp [ -( T_t/T)^{1/2}]$ dependence was found for the DC conductivity. It is shown, that the same type of strong temperature dependence is valid for the dielectric constant and the frequency-dependent conductivity, in agreement with experimental data for poorly conducting polymers.