Hopping conductivity of a nearly 1Dfractal: A model for conducting polymers

We suggest treating a conducting network of oriented polymer chains as an anisotropic fractal whose dimensionality D=1+ε is close to 1. 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 ε as →ε0, e.g., the critical exponent of conductivity is ε^-2exp(-1-1/ε). 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 ε 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-one-dimensional Mott law. The present results are consistent with electrical properties of poorly conducting polymers.