# Mini-Review: Modeling and Performance Analysis of Nanocarbon Interconnects
## 2 Graphene Nanoribbon (GNR) Interconnects
### Multilayer Graphene Nanoribbon (MLGNR) Interconnects  
In comparison to the CNT, the GNR can be easily controlled horizontally and is more compatible with conventional lithography [24]. By using the tight-binding approximation, the conductance of a monolayer GNR has been calculated as a function of width \(W\), Fermi level \(E_{F}\), and specularity parameter \(p\) for edge diffuse scattering in Reference [25]. It was demonstrated that the atomically thick monolayer GNR could outperform Cu wire for widths below 8 nm. Figure 2a shows the schematic  
Figure 1: Schematics of (**a**) monolayer graphene nanoribbon (GNR), (**b**) single-walled carbon nanotube (SWCNT), and (**c**) multi-walled carbon nanotube (MWCNT). diagram of a monolayer GNR interconnect with its equivalent circuit model. In the model, \(R_{c}\) is the contact resistance, and it can be calculated by:  
\[R_{c}=R_{mc}+R_{Q} \tag{1}\]  
where \(R_{mc}\) is the imperfect contact resistance, and it highly depends on the fabrication technology \(R_{Q}\) is the quantum contact resistance, and it is calculated as \(R_{Q}=h/\left(2e^{2}N_{ch}\right)\), where \(h\) is the Planck's constant and \(e\) is the electron charge. The number of conducting channels in a monolayer GNR can be calculated by [26; 27]:  
\[N_{ch}=\sum_{n}\frac{1}{1+\mathrm{e}^{(E_{n}-E_{F})/(k_{B}T)}}+\sum_{n}\frac{1 }{1+\mathrm{e}^{(E_{n}+E_{F})/(k_{B}T)}} \tag{2}\]  
where \(k_{B}\) is the [PERSON]'s constant and \(T\) is the temperature. For a given width of the metallic armchair GNR with \(W>10\) nm and \(E_{F}>0.1\) eV, \(N_{ch}\) can be approximated as \(1.2 WE_{F}\), where \(W\) and \(E_{F}\) are in units of nanometers and electron volts, respectively [28]. The total resistance of a monolayer GNR is given by:  
\[R_{total}=R_{c}+R_{S}L=\frac{h}{2e^{2}}\left[\sum_{i}\left(1+\frac{L}{\lambda_ {\text{eff},i}}\right)^{-1}\right]^{-1} \tag{3}\]  
where \(L\) is the interconnect length, and \(\lambda_{\text{eff},i}\) is the effective MFP (mean free path) for the \(i\)th subband. The MFP is determined by various scattering mechanisms, and it is a function of the specularity parameter \(p\)[29]. Herein, the fully diffusive GNR can be represented by \(p=0\), meanwhile, \(p=1\) represents that the GNR is fully specular. However, it is worth noting that the physical properties of monolayer GNR is susceptible to the substrate influence, which can be avoided for the case of multilayer GNR (MLGNR). Moreover, MLGNR has superior characteristics for signal propagation as the multilayer structure can effectively reduce the interconnect resistance [29; 30; 31; 32]. Figure 2b,c show the schematic diagrams of MLGNR interconnects with top contacts and side contacts, respectively. In the figure, \(\delta\) denotes the van der Waal's gap between the adjacent graphene layers, \(H\) is the thickness of the MLGNR interconnect, and the number of graphene layers \(N=1+\text{Inter}(H/\delta)\), where \"\(\text{Inter}(\cdot)\)\" represents that only the integer part is considered. To date, most of the experiments on multilayer graphene have employed top contacts, as shown in Figure 2b [33; 34; 35]. For the top-contacted MLGNR interconnect, [PERSON], et al. [36] developed a resistor circuit model with the consideration of in-plane resistance \(R_{layer}\) and perpendicular resistance between adjacent layers \(R_{perp}\). It was found that the perpendicular resistance makes the conductance a nonlinear function of the number of graphene layers. Further, [PERSON], et al. [37] conducted a benchmark study, which indicated such top-contacted MLGNR can exhibit advantages under certain circumstances. With  
Figure 2: Schematics of (**a**) monolayer GNR interconnect, (**b**) top-contacted MLGNR, and (**c**) side-contacted MLGNR interconnect. the increasing interconnect length, however, the in-plane resistance increases, while the perpendicular resistance decreases continually.