# Equivalent Electrical Circuit Modeling of CNT-Based Transparent Electrodes
## 1 Introduction  
The remarkable electrical properties of carbon nanotubes (CNTs) have led to intense research in various fields of electronics for the development of materials and devices for a wide range of applications. Among the most promising is certainly the use of CNTs in transport conductive films (TCFs) in order to replace more traditional materials, such as ITO (Indium Tin Oxide) and other metal oxides as transparent electrodes in several devices, such as touch panels, LCD screens, OLED displays, photovoltaic cells, and many others. ITO is the best-performing doped metal oxide in terms of low resistance and high transparency. Moreover, it has good durability, as it is quite resistant to humidity, and can be processed achieving a very precise patterning. Nevertheless, its superior electrical and optical properties are accompanied by some drawbacks, the main one being the scarcity of its main element, Indium, which is rare and expensive. Furthermore, its ceramic nature entails unsatisfactory elastic properties, which become significant when the application requires bending capabilities. In particular, repeated bending stress is reported as a cause of degradation of ITO optoelectronic properties, thus making the material unsuitable for the development of flexible electronics [1]. Thus, the search for new materials for the development of transparent electrodes is mainly driven by the necessity of replacing ITO with solutions less expensive, more available, and suitable for the development of flexible devices in next-generation electronics. In the recent past, different possibilities have been explored. For instance, conductive polymers have proved capable of dealing with the issues of flexibility and cost-effectiveness though did not prove equally performing in terms of optoelectronic properties. For example, PEDOT:PSS films exhibit a sheet resistance of about 200 \(\Omega/\square\) and a transparency above 80%, coming with poor thermal stability and a slight tendency to blue color [2,3]. Rather than depositing a conductive layer, one possibility for achieving a good flexibilityof conductive films is to pattern a conductive grid onto a substrate. Random or ordered networks of nanowires of Silver or Copper have already proven performance comparable to those of ITO. Carbon nanotubes thin films, at the moment, lag behind, with slightly worse performance due to the high resistance at the nanotube-nanotube interface. Thanks to their cheapness and versatility, they seem the most promising ITO competitor. To realize a CNT-based transparent electrode, CNTs are either treated by a dry or wet process to form a conductive network and deposited onto a transparent substrate [2], or dispersed in a transparent polymer creating a conductive compound [4; 5]. Several techniques have been reported to obtain large area depositions of CNT networks as well as precise patterning of conductive areas with little efforts. Unfortunately, a better electrical performance is usually obtained at the expense of optical transparency [6]. Nevertheless, some attempts at improving the electrical conductivity without losing transparency to visible light have been reported [7]. Lately, graphene also arose as a possible candidate to replace ITO in TCFs, though the issues related to the quality of exfoliated graphene have not been overcome yet. Solution-processed graphene has demonstrated optoelectronic properties still two orders of magnitude far from the state-of-art, mainly due to the grain boundaries and defects of graphene flakes [8]. Up to now, CNTs seem to be the best trade-off between costs, versatility, and performance. The raw material is abundant and easy to find and costs for the processing are affordable, as many solutions for the fabrication of TCFs based on CNTs involve operations at room temperature. CNTs are also very versatile, and in the form of thin film within the 1-100 nm range of thickness they exhibit good transparency in the visible range, neutral color, good flexibility, and appealing electrical properties.