# Mini-Review: Modeling and Performance Analysis of Nanocarbon Interconnects
## 2 Graphene Nanoribbon (GNR) Interconnects
### Multilayer Graphene Nanoribbon (MLGNR) Interconnects
It can be seen in Figure 3, that as the length exceeds a certain value, the influences of the contact type and interlayer coupling can be ignored, and the top-contacted MLGNR can be treated as a side-contacted one [28]. Figure 4 shows the multi-conductor transmission line model and the simplified equivalent single-conductor (ESC) transmission line model of a side-contacted MLGNR interconnect. In the multi-conductor model, the mutual capacitance and inductance between the adjacent graphene layers have been considered. The MLGNR interconnects are assumed to be decoupled, because several decoupled graphene layers have been grown experimentally in References [33; 34], and demonstrated theoretically in Reference [39]. In this case, the resistance of the MLGNR interconnect is a parallel connection of the resistance of each layer, i.e., \(R_{S}=\sum_{i=1}^{N}R_{S}^{(i,i)}\). The per-unit-length (p.u.l.) magnetic inductance \(L_{M}\) and electrostatic capacitance \(C_{E}\) are determined by the MLGNR geometry and its surrounding dielectrics. By applying the boundary conditions, the equivalent kinetic inductance \(L_{K}\) and quantum capacitance \(C_{Q}\) in the ESC model can be obtained by recursive schemes [40; 41]. As the kinetic inductance is much larger than the magnetic inductance, the equivalent kinetic inductance in the ESC model can be expressed as [28]:  
\[L_{K}\approx\frac{20}{3 NWE_{F}}\mathrm{nH/\upmu m} \tag{4}\]  
The equivalent quantum capacitance is mainly determined by the mutual coupling one between the adjacent layers. As the number of layers is larger than eight, the equivalent quantum capacitance can be given by:  
Figure 4: (**a**) Multi-conductor transmission line model and (**b**) equivalent single-conductor (ESC) transmission line model of side-contacted MLGNR interconnect [38]. Figure 3: Effective resistances of top-contacted and side-contacted MLGNR interconnects with Fermi level \(E_{F}=0.6\) eV, specularity parameter \(p=1\), and \(c\)-axis resistivity \(\rho_{c}=0.3\,\Omega\cdot\mathrm{cm}\)[28]. \[C_{Q}\approx 120 WE_{F}\Bigg{(}1+\sqrt{1+\frac{1}{2.76E_{F}}}\Bigg{)}\mathrm{aF}/ \mathrm{\SIUnitSymbolMicro m} \tag{5}\]  
Based on the extracted parameters, the 50% propagation delay can be calculated by [42]:  
\[\tau_{500\%}=\left(1.48\xi+\mathrm{e}^{-2.9\xi^{1.35}}\right)\sqrt{L_{ESC}L(C_{ ESC}L+C_{L})} \tag{6}\]  
where \(R_{d}\) is the driver resistance, \(C_{L}\) is the load capacitance, \(L_{ESC}=L_{K}+L_{M}\), \(C_{ESC}=\left(1/C_{Q}+1/C_{E}\right)^{-1}\), and  
\[R_{c}=R_{mc}+R_{Q} \tag{1}\]0  
Due to large driver resistance, nanocarbon interconnects will not exhibit superior performance compared to conventional Cu wires at the local level. So our attention is focused on the intermediate level and global level MLGNR interconnects. Figure 5 shows the time delay ratios of MLGNR and Cu wires at the intermediate level and global level, respectively. The inverter size is assumed as 50 and 100 times larger than the minimum-sized gate, respectively, at these levels. The imperfect contact resistance is neglected, and the Fermi level is set as 0.6 eV. As shown in Figure 5, the MLGNR interconnects with fully specular edges show superior performance over Cu wires at both the intermediate and global levels. Such an advantage can be enhanced with the feature size scaling down. However, when the specularity parameter decreases to 0.8, the electrical performance of the MLGNR interconnects will degrade to the comparable level of Cu wires. Therefore, it can be concluded that the MLGNR interconnects have the potential to outperform conventional Cu wires at an advanced technology node, and more attention should be paid to the edge quality.