# Repeater Insertion for Multi-Walled Carbon Nanotube Interconnects
## 1 Introduction  
As the feature size of complementary metal-oxide semiconductor (CMOS)-integrated circuits continues to shrink, the interconnect dimensions are scaled down, which increases the electron scatterings at the surface and grain boundaries. Therefore, the interconnect delay increases dramatically and outweighs the gate delay, becoming the main bottleneck of improving the performance of modern integrated circuits [1]. More importantly, as predicted by the International Technology Roadmap for Semiconductors (ITRS), the current density will exceed the maximum allowable value of the conventional Cu interconnects within the next 5 years [2]. In order to solve such an issue, some innovative architectural concepts and materials are needed. As alternative conductor materials, carbon nanomaterials, including carbon nanotubes (CNTs) and graphene, have aroused lots of interest due to their outstanding physical properties [3; 4; 5; 6; 7; 8; 9]. According to the number of carbon atom layers in a CNT wall, CNTs can be classified into single-walled CNTs (SWCNTs) and multi-walled CNTs (MWCNTs). SWCNTs can be either semiconducting or metallic, while MWCNTs are always metallic. Moreover, MWCNTs have a simpler fabrication process than SWCNTs due to easier control of the growth process, and under certain circumstances, MWCNTs possess a negative temperature coefficient on electrical resistance, which cannot be observed in SWCNTs [10; 11]. Several research efforts have been made on the modeling and fabrication of MWCNT interconnects. Based on a physical model, it was found that the MWCNT conductivity increases with length and tends to be stable as the influence of contact resistance can be reduced by increasing the length [12]. As the length exceeds tens of micrometers, MWCNTs exhibit superior performance to their Cu and SWCNT counterparts, which makes MWCNTs suitable for the intermediate and global interconnect levels [13]. Although the complete multi-conductor transmission line proposed in [13] is accurate, it is time-consuming in dealing with complex MWCNT interconnects. Therefore, a simplified equivalent single-conductor (ESC) transmission line model was provided in [14] to reduce the computational complexity, with an expression for equivalent capacitance in the ESC model derived analytically. Furthermore, the impact of intershell tunneling conductance was investigated in [15; 16], respectively, and it was found that the ESC model can provide an accurate prediction of time delay. Based on the ESC model, the robust design of the propagation characteristics and the crosstalk of MWCNT interconnects was carried out [17; 18; 19]. On the other hand, a repeater insertion methodology is essential for the implementation of high-performance VLSI designs. It is intuitive to insert some repeaters at intermediate- and global-level MWCNT interconnects, and a related investigation was carried out in [20]. As is well-known, however, contact resistance is inevitable in real-world applications of nanotube-like structures [21; 22]. As shown in Figure 1, each inserted repeater in an MWCNT interconnect would, apparently, introduce new contact resistance, which would surely influence the propagation delay. The previous study [20] neglected the contact resistances induced by the inserted repeaters, thereby leading to overestimated electrical performance. The impact of contact resistance on optimal repeater insertion was studied for a monolayer SWCNT, an SWCNT bundle, and MWCNT interconnects in [23]. It was demonstrated that for MWCNT interconnects, the contact resistance must be taken into account when it exceeds a certain value. However, the study in [23] is just based on the resistance-capacitance (RC) model, and the optimal repeater size and number were obtained numerically. As indicated in [24], the measured CNT kinetic inductance may be 15 times larger than its theoretical value, and the inductive effect should be considered.