# Repeater Insertion for Multi-Walled Carbon Nanotube Interconnects
## 2 ESC Model of the MWCNT Interconnect  
For easy reference and completion of the model, the ESC modeling procedure of an MWCNT interconnect is reproduced here. Figure 2 shows the schematic of an MWCNT placed above the ground  
Figure 1: Repeater insertion in a long multi-walled carbon nanotube (MWCNT) interconnect and the equivalent single-conductor (ESC) model for one segment of the MWCNT interconnect. plane. In the figure, \(r_{s}\) and \(r_{1}\) represent the outmost and innermost diameters, respectively, \(s\) is the shell number, and \(d\) is the distance between the MWCNT and ground plane. Here, the innermost shell diameter is assumed to be half of its outermost diameter, i.e., \(r_{1}=r_{s}/2\)[25]. Supposing the spacing between adjacent shells in an MWCNT is \(\delta=0.34\) nm, the number of the shells can be given as  
\[s=1+\text{Inter}\bigg{[}\frac{r_{s}-r_{1}}{\delta}\bigg{]} \tag{1}\]  
where \"Inter[\(\cdot\)]\" indicates that only the integer part is considered. The radius of the \(j\)th shell is \(r_{j}=r_{1}+(j-1)\delta\). As aforementioned, the MWCNT can be modeled as an ESC model [14] as shown in Figure 1. \(R_{c}\) is the total terminal contact resistance, and it can be given as  
\[R_{c}=2\left[\sum_{j=1}^{s}\left(\frac{R_{Q}}{2n_{j}}+R_{\text{mc},j}\right)^{ -1}\right]^{-1} \tag{2}\]  
where \(R_{Q}=h/\left(2e^{2}\right)=12.9\) k\(\Omega\), \(h\) is the Planck's constant, \(e\) is the electron charge, and \(R_{\text{mc},j}\) is the imperfect contact resistance of the \(j\)th shell. \(n_{j}\) is the number of conducting channels of the \(j\)th shell of an MWCNT, and it can be expressed as [12; 26]  
\[n_{j}=\left\{ \begin{array}{c}k_{1}r_{j}+k_{2},\ r_{j}>1.5\ \text{nm}\\ 2/3,\ r_{j}<3\ \text{nm}\end{array} \right. \tag{3}\]  
where \(k_{1}=0.122\) nm\({}^{-1}\) and \(k_{2}=0.425\). \(r\) is the equivalent quantum resistance per-unit-length (p.u.l. ), and it is in parallel connection of the p.u.l. series resistances of each shell, i.e.,  
\[r=\left(\sum_{j=1}^{s}\frac{n_{j}\lambda_{j}}{R_{Q}}\right)^{-1}=R_{Q}\left( \sum_{j=1}^{s}n_{j}\lambda_{j}\right)^{-1} \tag{4}\]  
where \(\lambda_{j}\approx 2000r_{j}\) is the effective mean free path of the \(j\)th shell in the MWCNT. \(l\) is the equivalent p.u.l. inductance, and it can be obtained by  
\[l=l_{K}+l_{E} \tag{5}\]  
where \(l_{K}\) is the p.u.l. equivalent kinetic inductance, and \(l_{E}\) is the p.u.l. magnetic inductance of the external shell. According to [14], as the p.u.l. mutual inductance between adjacent shells is more than  
Figure 2: Geometrical configuration of the cross-section of an MWCNT. 3 orders of magnitude smaller than the p.u.l. kinetic inductance, it can be neglected. Therefore, the equivalent kinetic inductance is given as  
\[l_{K}=\frac{L_{K/channel}}{n_{total}} \tag{6}\]  
where \(L_{K/channel}\approx 8\) nH/\(\upmu\)m is the kinetic inductance per channel, and \(n_{total}=\sum_{j=1}^{s}n_{j}\) is the total number of conducting channels of the MWCNT. \(c\) is the equivalent p.u.l. capacitance, and it can be given as [14]  
\[c=\frac{c_{Q}c_{E}}{c_{Q}+c_{E}}. \tag{7}\]  
The equivalent quantum capacitance is given as [14]  
\[c_{Q}=\alpha s+\beta r_{1}+\gamma \tag{8}\]  
where \(c_{Q}\) is expressed in nH/m, \(r_{1}\) is in nanometer, \(\alpha=2.56\times 10^{-2}\)nF/m, \(\beta=7.525\times 10^{-2}\)F/m\({}^{2}\), and \(\gamma=9.887\times 10^{-2}\)nF/m. \(c_{E}\) is the p.u.l. electrostatic capacitance of the external shell. The relationship between \(l_{E}\) and \(c_{E}\) can be expressed as  
\[l_{E}=\frac{\mu_{0}\varepsilon_{0}\varepsilon_{r}}{c_{E}} \tag{9}\]  
where \(\mu_{0}\) and \(\varepsilon_{0}\) are the vacuum permeability and permittivity, respectively, and \(\varepsilon_{r}\) is the relative permittivity of the surrounding dielectric. Based on the typical interconnect configurations shown in Figure 3, the p.u.l. external capacitance \(c_{E}\) is extracted by an electromagnetic simulator (e.g., ANSYS Maxwell). In this study, two technology nodes (i.e., the 14 nm and 7 nm technology nodes) are selected.