# Equivalent Electrical Circuit Modeling of CNT-Based Transparent Electrodes
## 1 Introduction
As in any other field, being able to predict the behavior of a component is one key advantage during the design process. Simulation enables shorter times for the design, less efforts for prototyping and less failures. However, the fabrication of CNT-based TCFs has often been engineered by trial-and-error, achieving high quality and high performance, nonetheless. The lack of a model capable of effectively representing the behavior of a CNT-based transparent electrode becomes more significant during the process of designing devices that include such layers; CNT thin films and carbon-based nanomaterials, in fact, cannot be included in full device models in the form of ideally behaving elements. Rather, a truthful representation of the material characteristics, such as its electric properties, must be considered in order to carry out an engineered design of the final device. Due to the complex nature of materials for TCFs, it is not possible to simulate all levels of abstraction with one single method. The different lengths involved in the physical phenomena ruling the behavior of CNT thin films, as well as the stochastic nature of the CNT deposition outcomes, make it impossible to perform the simulation in one step. The modeling of CNT-based devices must be performed at several layers of abstraction, each of them related to a different order of magnitude of the lengths involved, starting from the atomistic modeling of carbon molecules, and ending at the macro-dimensional modeling of the final device. Such an approach, known as multiscale modeling, takes advantage of several methodologies according to the nature of the physics involved and of the quantities of interest, and is a recurring topic in the modeling of 2D material-based devices. Expanding the models developed for random CNT networks to other 2D material networks is considered a fundamental topic in the 2D materials simulation community [9]. In the case of transparent electrodes, the main quantity of interest is the ratio between electrical and optical conductivity. In this review, we will focus on the modeling of the first. Developing a reliable model for the simulation of an arbitrarily shaped transparent electrode made of CNT-based conductive films requires a valid representation of the conductive behavior of a randomly aligned network of differently sized carbon nanotubes. This implies having a solid model of the transport phenomenon, as related to the shape, chirality, quality, and quantity of nanotubes involved. Modeling the properties of the CNT is the first step for the development of the CNT network model. The CNT network model, in turn, is functional for describing the properties of the transparent electrode, both in caseswhere it is made of a CNT thin film deposited on an insulating substrate, and in cases where it is made of a nanocomposite material. For the purpose of including the TCF model in already existing circuit models of complex electronic devices, usually designed by means of automated electronic design systems and circuit simulators, lumped-parameters equivalent electric circuits are extensively used. The complexity of a circuit, the numbers of properties that its parameters take into account, and the kind of physical phenomena involved in the current conduction, should all be tailored according to the kind of application to be studied. For instance, the computation of the dc value of the electrical conductivity of a transparent electrode is often performed by referring to a simple resistive network, implementing the dominant nanotube-to-nanotube junction resistance. Conversely, the modeling of the CNT-based electrode for applications where the frequency dependence of the conductive layer impedance cannot be neglected, and has to include the capacitive effects arising in the CNT network. The design of devices for critical applications, e.g., high-frequency field-effect transistors, may require including in the CNT network model the dependence on aspects like the nanotubes doping [10], the carriers density [11], the trap-state density [12].