Hopping Conductivity of a Nearly-1d Fractal: a Model for Conducting Polymers

TL;DR

This paper models the hopping conductivity of nearly one-dimensional fractal polymer networks, revealing nonanalytic critical behavior and universal scaling in conductivity distribution, with implications for understanding poorly conducting polymers.

Contribution

It introduces a fractal model for conducting polymers near one dimension, analyzing critical exponents and conductivity distribution using real space renormalization group methods.

Findings

Critical exponents are strongly nonanalytic functions of psilon.

The conductivity distribution has a universal scaling form.

Both DC and AC hopping conductivities follow a quasi-1D Mott law.

Abstract

We suggest treating a conducting network of oriented polymer chains as an anisotropic fractal whose dimensionality D=1+\epsilon is close to one. Percolation on such a fractal is studied within the real space renormalization group of Migdal and Kadanoff. We find that the threshold value and all the critical exponents are strongly nonanalytic functions of \epsilon as \epsilon tends to zero, e.g., the critical exponent of conductivity is \epsilon^{-2}\exp (-1-1/\epsilon). The distribution function for conductivity of finite samples at the percolation threshold is established. It is shown that the central body of the distribution is given by a universal scaling function and only the low-conductivity tail of distribution remains $\epsilon $-dependent. Variable range hopping conductivity in the polymer network is studied: both DC conductivity and AC conductivity in the multiple hopping regime are found to obey a quasi-1d Mott law. The present results are consistent with electrical properties of poorly conducting polymers.