# Equivalent Electrical Circuit Modeling of CNT-Based Transparent Electrodes
## 3 Electrical Modeling of CNT Networks  
The assessment of the electrical properties of carbon nanotube networks have led to the development of different models to simulate the current conduction in a CNT layer. The chirality of the CNTs composing the network play an important role in the film electric properties. It is reported in [2] that, in general, in a CNT network, one third of the nanotubes are metallic CNTs, whereas the remaining two thirds are semiconducting CNTs with different chirality, diameter, length, and band gap. While the metallic-to-metallic CNT conduction is good (and for this reason these CNTs are better suited for application in transparent conductive films [26]), the charge transport between metallic and semiconducting nanotubes is impaired by the highly resistive Schottky barrier at the junction, which contributes to the overall film resistance to a greater extent with respect to the nanotube segment resistance [27; 28; 29; 30]. The closer the nanotubes are, the smaller the tunneling resistance is. Eventually, a conductive path exists over two different points of the network, i.e., a small-resistance route across nanotubes connecting two different spots in the network. The establishment of conductive paths can be described by the percolation theory [31; 32]; according to this theory, a percolation threshold for the nanoparticles' density exists, over which conduction across two points is made possible and the network conductivity can be described by an exponential law. This kind of approach is particularly useful for the representation of the electrical behavior of nanocomposites, where the insulating behavior of the hosting material switches to a conductive behavior [33] when conductive CNTs form a percolative path. The dependence of the obtained conductivity on the filler density, aspect ratio, chirality, band gap, density of states (DOS), doping, etc., has been the object of many studies [34; 35; 36; 37]. The estimation of the number and morphology of percolation paths usually relies on statistical approaches such as Monte Carlo simulations of the nanotubes' distribution. Recently, machine learning approaches to the prediction of a nanomaterial's electrical properties have also been considered [38].  
The CNT network also has an intrinsic capacitive effect that can be considered to model the frequency-dependent variation of impedance. The degree of complexity needed in a model depends on the application under study.  
Whether the CNT network is deposited on a transparent, insulating, sometimes flexible substrate, or mixed in a polymeric matrix generating a conductive compound, its conductivity is usually computed by means of an equivalent electric circuit. If the CNTs are deposited in the form of a free-standing network, or the CNT-based layer is very thin, a planar equivalent circuit can be used, in which the equivalent electrical network develops on a surface and conduction only happens in-plane. If the CNTs are dispersed inside a hosting material, conduction paths also involve the vertical direction, thus the distance between the nanotubes that determines the tunneling resistance can also spread out of plane. In this case, a 3D equivalent network must be considered.  
In the following, we describe some examples found in the literature of circuital models that have been proposed to assess the electrical performance of CNT networks with different levels of complexity.