Systematic analysis for electrical conductivity of network of conducting rods by Kirchhoff's laws and block matrices

Abstract

In recent years, high-aspect-ratio materials, such as metallic nanowires and carbon nanotubes, have become attractive alternatives for the next generation of transparent conductive films. The functionality of the films is represented by their opto-electric performance, which is primarily affected by the nano- or micro-structures inside the films. In this study, we focus on the analysis of the electrical conductivity of two-dimensional networks of conducting rods by treating parts of the networks as a linear circuit system. For the analysis, multi-nodal representation is used to assign the nodes and edges of the circuit. Based on Kirchhoff's laws, the relation between the current and electrical potential is formulated using a block matrix equation. After a series of block-matrix manipulations, the equation can be reduced to yield several simple equations expressed in terms of the incidence matrices and the weighted graph Laplacians. Among these, the equation representing the Ohm's-law-like relation between the total current and the bias voltage can be used to derive the explicit expression for the normalized conductivity, which can quantify the effect of the network. During the analysis, we also deduce the normalized number of edges, that is, the combination of variables used in the system. The normalized number of edges can be related to the reduced number density of rods by using a proper statistical model. Moreover, we found the direct relation between the normalized number of edges and the backbone fraction, which is a representative quantity related with the electrical conductivity.