# Mini-Review: Modeling and Performance Analysis of Nanocarbon Interconnects
## 3 CNT Interconnects  
Similar with GNR, CNT possesses long MFP, high ampacity, and large thermal conductivity [14; 16; 50; 51; 52]. Based on the Luttinger liquid theory, [PERSON] firstly developed the transmission line model of a metallic SWCNT interconnect [53]. The equivalent circuit model of an isolated SWCNT interconnect is the same as the circuit model shown in Figure 2a, with the CNT diameter denoted as \(D_{\text{cnt}}\). The number of conducting channels of a metallic SWCNT is 2, and the MFP is usually \(1000D_{\text{cnt}}\). Although an SWCNT possesses many unit properties and some efforts have been devoted to reducing the SWCNT resistance by doping, an isolated SWCNT is still too resistive for interconnect applications in high-performance ICs [54; 55]. It can only be used in some specific applications such as subthreshold circuits and sub-10 nm circuits [56; 57; 58]. To reduce the CNT resistance, three kinds of CNT interconnects, i.e., monolayer SWCNT, bundled SWCNT, and MWCNT interconnects, have been widely studied, as shown in Figure 9.  
As discussed in [59], for short local interconnects, the interconnect resistances are not important, while their capacitances play a key role. So the electrical performance can be improved by using low aspect ratio interconnects such as monolayer metallic SWCNT. It can be seen in Figure 10a that monolayer SWCNT interconnect possesses smaller parasitic capacitance than Cu wire. Therefore, it can be expected that monolayer SWCNT interconnects can reduce power dissipation and crosstalk noise [60]. Figure 10b depicts the energy-delay product (EDP) ratio between Cu wire and monolayer SWCNT interconnect. Here, the imperfect contact resistance is neglected as it is highly dependent on the process. As the length exceeds 20 \(\upmu\)m, the EDP ratio becomes larger than 1, indicating that the monolayer SWCNT interconnect can provide better performance than the Cu counterpart. Moreover, the advantage of monolayer SWCNT interconnect becomes more evident for smaller inverter size. Figure 8: (**a**) Unit cell for simulation. (**b**) Maximum temperature rise on interconnects versus input power. (**c**) Temperature profiles of the unit cell with a power of 5 \(\upmu\)W input at the topmost global interconnect [44]. To further reduce the interconnect resistance, as shown in Figure 9b, the bundled SWCNTs have been proposed and manufactured [61; 62; 63]. It is worth noting that for the pristine SWCNTs with random distribution of chirality, one-third are metallic and two-thirds are semiconducting. Researchers have been continually tasked to improve the metallic fraction. For instance, [PERSON], et al. [64] successfully fabricated a bundled SWCNTs with very high metallic fraction of 0.91. Based on physical models, [PERSON], et al. [65] developed the circuit model for bundled SWCNT interconnect and demonstrated its potential for solving several major challenges facing gigascale integrated systems. Besides the applications as horizontal interconnects, bundled SWCNTs were also proposed as on-chip vias to reduce temperature rise and increase the electromigration resistance [66; 67; 68; 69]. Unlike SWCNTs, MWCNTs are always metallic. By assuming one-third of shells are metallic, the number of conducting channels of a shell in an MWCNT can be calculated by [70]  
\[N_{ch}=\left\{ \begin{array}{ll}2.04\times 10^{5}TD+0.425,&D>D_{T}/T\\ 2/3,&D<D_{T}/T\end{array} \right. \tag{8}\]  
where \(D_{T}=1300\leavevmode\
obreak\ \mathrm{nm}\cdot\mathrm{K}\). For an MWCNT interconnect, as shown in Figure 9c, it is usually assumed that the innermost shell diameter is half of the outermost shell diameter, and the aspect ratio is 2, i.e., two parallel MWCNTs are stacked. In general, an MWCNT possesses a longer MFP but smaller number of conducting channels than the bundled SWCNTs with the same cross-sectional dimensions. For MWCNT interconnects, [PERSON], et al. [71] developed a multiconductor transmission line model, which was further simplified as an ESC transmission line model [40].