# Mini-Review: Modeling and Performance Analysis of Nanocarbon Interconnects
## 5 Cu-Nanocarbon Interconnect  
### Cu-Graphene Interconnect  
During the past decades, tremendous progress has been made in the fabrication/integration of nanocarbon interconnects. However, the gap between theoretical studies and practical applications still exists. For example, the assumption of closed packed CNTs is invalid as the density of CNTs still cannot satisfy the requirements [96]. Also, the application of MLGNR interconnects encounters a serious challenge, i.e., graphene tends to behave more like graphite as the number of graphene layers increases [97]. In this perspective, Cu/low-k interconnect may be still a good choice for next-generation ICs [98]. In the application of Cu/low-k interconnect, a highly resistive diffusion barrier layer is required to prevent the diffusion of copper atoms into substrate. This barrier layer would occupy a certain area, thereby decreasing the conduction area of interconnects and increasing the effective resistivity. More importantly, the barrier layer thickness cannot scale as rapidly as the interconnect dimensions [4]. This problem becomes more and more serious with the technology advanced. To resolve this problem, graphene, the thinnest 2-D material in nature, has been proposed as the ultimate barrier layer [99; 100; 101]. It has been demonstrated that graphene barrier layer can help improving the breakdown current density, enhancing the electromigration lifetime, increasing the Cu grain size, and reducing the scattering probability at the surface [102; 103; 104; 105; 106]. Moreover, low-temperature deposition techniques for producing graphene on Cu and dielectric implies that the fabrication of graphene barrier layer can be compatible with traditional CMOS technology [101; 104; 107; 108]. Figure 16 shows the schematic view of a Cu-graphene interconnect, i.e., Cu wire encapsulated with graphene barrier layers. To gain an in-depth understanding of such a Cu-graphene interconnect, it is necessary to develop its circuit model and evaluate its electrical and thermal performances. In [109],  
Figure 15: **(a)** Schematic of a simulation model. **(b–d)** Temperature profiles of Cu, SWCNT vias/Cu wire, and SWCNT/MLG interconnects [95]. a distributed resistor model for Cu-graphene interconnect has been established, and the Cu-graphene interface resistance and graphene-graphene interface resistance have been considered and treated appropriately. The effective resistance of Cu-graphene interconnect can be derived as  
\[R_{\text{eff}}=\sum\limits_{m=1}^{M}\sum\limits_{m=1}^{M}\left(\frac{M}{R_{\text{ Cu}}}\Big{[}I^{1}\Big{]}_{M\times M}+\sum\limits_{i=t,b,l,r}[A_{21}]_{i}[A_{21}]_{i}^{-1 }\right)^{-1} \tag{9}\]  
where \(M\) represents the number of segments meshed along the interconnect length, \(R_{\text{Cu}}\) is the resistance of the central Cu wire, \(\big{[}I^{1}\big{]}_{M\times M}\) is the \(M\times M\) unit diagonal matrix, and \([A_{21}]\) can be obtained using the Kirchhoff's voltage law. The subscripts \(t\), \(b\), \(l\), and \(r\) represent the respective corresponding quantities as graphene barrier layers are placed at the top, bottom, left, and right surfaces of the Cu wire. It was demonstrated that the graphene barrier layers can share part of the current, thereby reducing the current passing through the central Cu wire and improving the reliability. Moreover, as the interconnect length exceeds several tens of micrometers, the central Cu wire, and graphene barrier layers can be treated as being parallel connected, and the effective resistance can be simplified as \(R_{\text{eff}}=\left[1/R_{\text{Cu}}+N\sum\limits_{i=t,b,l,r}\left(1/R_{\text{ gr}}^{i}\right)\right]^{-1}\), where \(R_{\text{gr}}\) denotes the resistance of graphene barrier layer and \(N\) is the layer number [110]. Figure 17a depicts the effective resistivity of Cu-graphene interconnects versus temperature. According to ITRS prediction, the barrier layer thickness would scale down to 1.3 nm at the 22 nm node [2]. However, such prediction is too optimistic and too challenging.