# Equivalent Electrical Circuit Modeling of CNT-Based Transparent Electrodes
## 2 Modeling of a Carbon Nanotube
### Equivalent Transmission Line (TL) Modeling  
One technique for the modeling of the CNT electrical properties is based on the equivalent transmission line (TL) electrical circuit. First proposed for the electrical modeling of metallic single-walled nanotubes (SWCNTs) [17], and later improved to represent the electrical properties both in the frequency domain and in the time domain of SWCNT bundles [22; 23], it successfully addresses the needs for a reliable method for the investigation of the RF signal propagation and the current transport capabilities of CNT-based interconnects. Due to the modular nature of TL models, they can be used to represent larger areas of conductive material, such as electrodes and interconnects, also above the micrometric scale. The components of the TL model representing a single conducting channel of a metallic SWCNT presented in [17] are three per-unit-length (p.u.l.) parameters: the kinetic inductance, \(L_{K}\), the quantum capacitance, \(C_{Q}\), and the electrostatic capacitance, \(C_{E}\), respectively defined as  
\[L_{k}=\frac{h}{2e^{2}v_{f}} \tag{1}\]  
\[C_{Q}=\frac{2e^{2}}{h\ v_{f}} \tag{2}\]  
\[C_{E}=\frac{2\pi\varepsilon_{0}\varepsilon_{\tau}}{\cosh^{-1}\left(\frac{d}{a }\right)} \tag{3}\]  
where \(h\) is [PERSON]'s constant, \(e\) the electron charge, \(v_{f}\) the Fermi velocity, \(\varepsilon_{0}\) the vacuum dielectric permittivity, \(\varepsilon_{\tau}\) the surrounding dielectric material relative permittivity, \(d\) the distance between the CNT and the underlying metal plane, \(a\) the CNT radius. This model was improved by [PERSON] et al. in [24] by proposing an analogy with the electron waveguide. The upgraded model of a SWCNT accounting for \(n\) conductive channels includes an external inductance, \(L_{E}\), and a diffusion resistance \(R\), respectively defined according to  
\[L_{E}=\frac{\mu_{0}}{2\pi}\cosh^{-1}\left(\frac{d}{a}\right) \tag{4}\]  
\[R=\frac{h}{2e^{2}v_{f}\tau} \tag{5}\]  
in which \(\mu_{0}\) is the vacuum magnetic permittivity and \(\tau\) is the relaxation time. Furthermore, the model also accounts for the intrinsic resistance \(R_{0}\) of each channel  
\[R_{0}=\frac{h}{2e^{2}} \tag{6}\]  
and for the additional nanotube-metal contact resistance \(R_{m}\). The resulting equivalent circuit for an \(n\)-channel SWCNT is represented in Figure 2.  
The described representation was also expanded to represent bundles of SWCNTs and multi-wall nanotubes (MWCNTs) [25]. Figure 2: The equivalent transmission line (TL) circuit model of an \(n\)-channel SWCNT. The intrinsic resistance \(R_{0}\) is split in two resistors placed at the two sides of the conductive path, close to the CNT-metal resistances \(R_{m}\). According to the simulations reported in [24], the signal propagation in a bundle of SWCNTs is mainly dependent on the shunt capacitive branch value of capacitance of every CNT of the bundle. At lower frequencies (in the proximity of \(f=1\) GHz for the simulated CNT bundle), the input and output currents are small and almost equal, due to the high impedance of the capacitive branch that behaves like an open circuit. Furthermore, the current is equally distributed along the tubes. By increasing the frequency, the difference between the input and output current increases due to the decrease of the shunt impedance that drains current to the ground and to the increased value of the impedance along the conduction path. With respect to MWCNTs, it was demonstrated in [25] that the kinetic inductance and the quantum capacitance strongly depend on the number of shells. In particular, \(L_{k}\) is inversely proportional and \(C_{Q}\) is linearly increasing with the number of shells. Referring to the external parameters, the magnetic inductance \(L_{E}\) and the electrostatic capacitance \(C_{E}\) only depend on the ratio between the distance of the CNTs from the ground plane and their radius.