# Repeater Insertion for Multi-Walled Carbon Nanotube Interconnects
## 4 Results and Discussion  
In order to clarify the influences of the contact resistance and CNT kinetic inductance, three types of MWCNT interconnects are taken into account, as listed in Table 2. Here, MWCNT 1 denotes an MWCNT interconnect with the minimum contact resistance \(R_{c,\text{ min}}\), i.e., only the quantum contact resistance is considered. However, in real-world fabrication, imperfect contact resistance is inevitable. So, a contact resistance \(R_{c}\) of 10 k\(\Omega\) is considered in the case of MWCNT 2. As aforementioned, the CNT kinetic inductance may be 15 times larger than its theoretical value. Therefore, MWCNT 3 represents an MWCNT interconnect with a contact resistance \(R_{c}\) of 10 k\(\Omega\) and a kinetic inductance per channel \(L_{K/channel}\) of 60 nH/\(\upmu\)m.  
Figure 6 shows the propagation delay for 1000 \(\upmu\)m-long intermediate-level Cu and MWCNT interconnects with the insertion of different numbers of repeaters. The optimal repeater size \(h_{opt}\) and the size of 50 times larger than the minimum gate size are considered. It can be seen that the minimum propagation delay \(T_{\text{min}}\) is achieved when the repeaters of optimal size are inserted, and the estimation from the closed-form expressions is closed to the simulated optimal number of repeaters, as listed in Table 3. It is found that with the technology node advanced, the propagation delay increases significantly and more repeaters need to be inserted, which has an adverse effect on the chip area and power consumption. Therefore, a tapered buffer insertion should be employed to avoid large power dissipation [30,31], and further investigation should be conducted with constraints of both delay and power [32, 33, 34], which is the subject of our next study. It is found that MWCNT interconnects require a much lower number of repeaters to optimize timing than Cu wires, which results in low complexity and power consumption. With the minimum contact resistance, MWCNT interconnects still exhibit superior performance over their Cu counterpart. However, the propagation delay of the MWCNT interconnects increases with the increasing contact resistance. At the 14 nm technology node, as the contact resistance increases to 10 k\(\Omega\), the minimum propagation delay of the MWCNT interconnect is 1.538 ns, which is much larger than that of Cu wire. \begin{table}
\begin{tabular}{c c c c} \hline
**Type** & **MWCNT 1** & **MWCNT 2** & **MWCNT 3** \\ \hline \(R_{c}\) (k\(\Omega\)) & \(R_{c,\text{ min }}\)1  & 10 & 10 \\ \(L_{K/channel}\) (nH/\(\upmu\)m) & 8 & 8 & 60 \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters of MWCNT interconnects. Moreover, the increase in the CNT kinetic inductance has negligible influence on the optimal repeater size and number. So, the minimum propagation delay is kept almost unchanged with the variation of CNT kinetic inductance. Further, the optimal repeater insertions in the global-level Cu and MWCNT interconnects are studied, as shown in Figure 7. The minimum propagation delay and optimal repeater size and number are summarized in Table 4. As a global-level interconnect has smaller resistivity than an intermediate one, it requires less repeaters to optimizing the timing. Similarly, the estimated repeater number agrees well with the simulated one. The propagation delay of the MWCNT interconnect decreases significantly with increasing contact resistance, implying that the impact of contact resistance cannot be simply neglected in the interconnect's optimization and it is always essential to improve the fabrication