Percolation and conductivity: A computer study. II

Abstract

In this paper we present a large number of computer solutions of various types of resistor networks. Some of these are analogous to physical problems such as impurity conduction in lightly compensated semiconductors and variable-range hopping in amorphous semiconductors. A significant extension of the standard relaxation techniques was required to implement these solutions. The results of these calculations are compared to percolation-model predictions based on concepts developed in the first paper of this series. A simple criterion is found for the applicability of the critical-percolation-path analysis to problems of this type and this is used to formulate an accurate prediction for the impurity-conduction case. Arguments based on percolation models are also given to show that the ${T}^{\ensuremath{-}\frac{1}{4}}$ and ${T}^{\ensuremath{-}\frac{1}{3}}$ dependence of ${log}_{10}\ensuremath{\sigma}$ often predicted for three-dimensional and two-dimensional variable-range hopping are indeed expected to be observed, and results on resistivity networks analogous to these problems are shown to be consistent with these arguments. Accurate empirical formulas are deduced from these computer calculations and we use them to analyze some recent data on films of $a$-Ge. Employing the results of the preceding paper, several experimental studies, and our computer models we have also examined the utility of the critical-volume-fraction rule of Sher and Zallen in solving various types of mixture conduction problems. We find that application of this rule is appropriate only in rather limited circumstances, and that in general a knowledge of the topological properties of these problems must be employed in finding the percolation threshold.