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

In recent years, high-aspect-ratio materials, such as metallic nanowires andcarbon nanotubes, have become attractive alternatives for the next generation oftransparent conductive films. The functionality of the films is represented bytheir opto-electric performance, which is primarily affected by the nano- ormicro-structures inside the films. In this study, we focus on the analysis ofthe electrical conductivity of two-dimensional networks of conducting rods bytreating 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 andelectrical potential is formulated using a block matrix equation. After a seriesof block-matrix manipulations, the equation can be reduced to yield severalsimple equations expressed in terms of the incidence matrices and the weightedgraph Laplacians. Among these, the equation representing theOhm's-law-like relation between the total current and the bias voltagecan be used to derive the explicit expression for the normalized conductivity,which can quantify the effect of the network. During the analysis, we alsodeduce the normalized number of edges, that is, the combination of variablesused in the system. The normalized number of edges can be related to the reducednumber density of rods by using a proper statistical model. Moreover, we foundthe direct relation between the normalized number of edges and the backbonefraction, which is a representative quantity related with the electricalconductivity.