# Equivalent Electrical Circuit Modeling of CNT-Based Transparent Electrodes
## 3 Electrical Modeling of CNT Networks
### 2D Models for CNT Networks Simulations  
The modeling of the conductive path of a percolated network can be performed by means of lumped-parameters equivalent electrical circuits, made of resistive or resistive-capacitive 2D or 3D electrical networks. 2D networks consider only percolation paths developing on a plane. An example of 2D network morphology is shown in Figure 3.  
A resistive model featuring zero-width intersecting rod-like particles was used in [42] to study the effect of the filler length dispersity on the thin film electrical conductivity. Despite the simplicity of neglecting the nanotubes width and impenetrability, the model includes the interesting possibility of choosing the anisotropy of the particles alignment  
Figure 3: 2D morphology of CNT networks, like the ones adopted in [11; 42; 43]. by considering the rotation angle on the film plane of every particle. This feature allows the simulation of CNT films with a tunable degree of alignment in any direction on the plane. The authors considered random values of the particles' angle and length, the first following a normal distribution and the latter following a log-normal distribution, and computed the conductivity by accounting for the presence of an insulating substrate. In particular, the thin film plane was partitioned in a meshed grid, with every cell considered conductive if covered by a rod, and otherwise considered insulating. The cells were then replaced by four segments arranged crosswise with a fixed value of conductivity (equal to \(2\times 10^{6}\) if conductive, equal to \(2\) if non-conductive), in order to transform the film in a random resistive network, whose equivalent conductivity was computed by the Frank-Lobb algorithm [44]. This approach makes it possible to study the correlation between the percolation and the density and orientation of fillers, but simplistically neglects the rods volume and impenetrability, as well as the particles positioning at different heights in the material. Moreover, the conductivity of each segment is considered equal, despite the different lengths of the rods, and the different location of the junctions across the rod's length. The results obtained by means of this model were not supported by a comparison with experiments. A procedure based on graph theory and the solution of Kirchhoff's laws by means of incidence matrices was proposed in [43] to solve the analysis of a linear network of ideal resistors representing connected CNTs. In this case, both the tube segment resistance and the tube junctions tunneling resistance are considered. The proposed block-matrix manipulation can be used to either compute the total conductivity of the CNT network by knowing the tunneling resistances value or compute the tunneling resistance by measuring the total conductivity. In [11], a 2D resistive network of mono-dimensional semiconducting CNTs is obtained by accounting for the carrier density dependent charge transport for the first time. This model was developed in the view of studying the impact of the CNT network electrical properties on field-effect transistors performance, an application in which the dependence of the junction resistance on the carriers' density and chirality acquires a great importance. In these devices, in fact, nanotubes are mostly semiconducting and with different band gaps due to the uneven geometry and chirality, and the density of states (DOS) of each nanotube has a considerable impact on the CNT network electrical properties. Due to the tube-tube junction resistance being much higher than the nanotube resistance, the authors propose the modeling of a SWCNT network in form of a planar resistive network of junction conductances, given by  
\[G_{ij}(E_{F})=\iint G^{\prime}_{ij}\big{(}E_{F},E_{i},E_{j}\big{)}\cdot D_{i}( E_{i})\cdot D_{j}\big{(}E_{j}\big{)}\ dE_{i}dE_{j} \tag{9}\]  
where the subscripts \(i\) and \(j\) indicate two nanotubes of the network over which the charge transport happens.