# Equivalent Electrical Circuit Modeling of CNT-Based Transparent Electrodes
## 3 Electrical Modeling of CNT Networks
### 3D Models for CNT Networks Simulations
#### 3.4.1 Resistive 3D Networks  
The 3D model proposed in [48] for the modeling of SWCNT networks is based on the computation of the equivalent resistance of a network of resistances. The model, originally presented in [52] for the modeling of the transport mechanisms in CNT-polymer compounds, is based on a MatLab(r) (The Matworks, Inc.) procedure that generates a random distribution of CNTs, represented by impenetrable solid cylinders of fixed length and radius, inside a fixed volume. Starting from the position of each CNT, randomly generated according to a uniform probability distribution, an electrical equivalent network of parallel resistive branches is generated following the conduction paths created across the CNTs distribution. The conduction across two points of the network achieves a convergence in correspondence of a given minimum density of CNTs; such information is crucial for the design of efficient CNT-based conductive materials. Each resistive branch composing the parallel network represents a conductive path over the two terminals across which the equivalent resistance must be computed. On each branch, there is the resistance of the CNT located across the \(l_{mn}\) distant points \(m\) and \(n\), \(R_{mn}\), computed assuming the CNTs of the conductive path have the resistivity of semiconducting CNTs \(\rho_{\text{CNT}}\) and cross-sectional area \(S_{\text{CNT}}\), and the tunneling resistance \(R_{tunned}\), introduced in correspondence with each nanotube-nanotube junction to account for the electron tunneling effect occurring between close CNTs. These resistances are computed according to the equations  
\[R_{mn}=\rho_{CNT}\frac{l_{mn}}{S_{CNT}} \tag{11}\]  
and  
\[R_{tunned}=\frac{h^{2}d}{S_{CNT}e^{2}\sqrt{2m_{e}\lambda}}\times\exp\left( \frac{4\pi d\sqrt{2m_{e}\lambda}}{h}\right) \tag{12}\]  
in which \(h\) is [PERSON]'s constant, \(d\) is the distance between the two CNTs, \(e\) is the electron charge, \(m_{e}\) is the electron effective mass, \(\lambda\) is the energy barrier height. Although also including the terms accounting for the CNT/metal contact resistance, which is crucial in devices where the electrode length is comparable with the nanotube length, suggestions for its quantification are not provided. The result is the computation of the equivalent resistance of the volume representing the conductive layer, as if the material were a homogeneous medium. The simulations of the on-resistance of CNT-based thin film transistors (TFT) with different channel widths and lengths were validated against DC measurements of TFT with channel made of high purity semiconducting CNTs. Recently, a simulation tool accounting for the DOS in semiconducting nanotubes arranged in a 3D random network was proposed in [49]. As the first step of the multiscale modeling of a random CNT network, a distribution of non-intersecting CNTs is generated with random orientation, chirality and length (the latter picked according to a lognormal distribution, whose mean and variance are inferred by experiments). The resistance of each nanotube is computed according to  
\[R(V)=R_{0}(V)\left(1+\frac{L}{\lambda_{eff}}\right) \tag{13}\]  
where \(L\) is the nanotube length, \(\lambda_{eff}\)is the electron effective mean free path, and \(R_{0}\) is the quantum resistance computed by dividing the applied voltage by the nanotube current  
\(I(V)\). The current is given by  
\[I(V)=\frac{e}{\hbar}\int\limits_{-\infty}^{+\infty}D(E)\frac{\gamma_{1}\gamma_ {2}}{\gamma_{1}+\gamma_{2}}[f_{1}(E,V)-f_{2}(E,V)]\;dE \tag{14}\]  
in which \(e\) is the electron charge, \(\hbar\) the reduced Plank constant, \(D(E)\) the DOS, \(\gamma_{i}\) the broadening functions, \(f_{1}(E,V)\) the Fermi distributions, and the subscripts 1 and 2 localize thecoefficients and the functions at the two contacts. Since the current depends on the potentials at the two contacts, the solution is obtained by an iterative procedure. The peculiarity of this model is that including the DOS allows simulating different materials.