Performance modeling and analysis of carbon nanotube bundles for future VLSI circuit applications

Abstract

In this work, we have presented a comprehensive analysis of the performance of copper (Cu) and existing carbon nano tube (CNT) bundle structures (i.e. SWCNT, DWCNT and MWCNT) across nanometer technology nodes like 45, 32, 22 and 16 nm at local, intermediate and global level interconnects. Double walled carbon nano tubes (DWCNTs) and multi walled carbon nano tubes (MWCNTs) are modeled like simple single walled carbon nano tube (SWCNT) equivalent model with high accuracy. The analytical closed form delay expressions for SWCNT, DWCNT and MWCNT bundles have been found out. It has been observed that sparse SWCNT bundle interconnects show about 50 % performance improvement for 20 $\upmu$m long local level interconnects over Cu in 16 nm technology node, whereas the performance advantage numbers for MWCNT and sparse DWCNT bundles are 50 and 35 % respectively. For 200 $\upmu$m long intermediate level interconnects, the performance advantage numbers are 85, 80 and 75 % for dense SWCNT, MWCNT and dense DWCNT bundles respectively in 16 nm node. For 10 mm long global level interconnects, the performance advantage numbers are 85, 85 and 75 % for dense SWCNT, MWCNT and dense DWCNT bundles respectively in 16 nm node. It is also observed that the performance numbers improve with scaling for all levels of interconnects. It is also shown that the ratio of delay of CNT bundles and Cu for various levels of interconnects agree well with the existing work.

Appendix

1.1 Optimized repeater insertion methodology for copper wire

1.1.1 Local level interconnects

Electrical model of a Cu wire at the local level is shown in Fig. 22. Definitions for various parameters shown in the figure are tabulated in Table 15. Here, we approximate the whole system as a single dominant pole transfer function. The effective time constant ($\tau$) of the system is given as,

$$\begin{aligned} \tau = \frac{R_{d0}}{S}\Bigl (C_{d0}S+C_wL+C_{g0}S\Bigr )\,+\,R_wL\Bigl (\frac{C_wL}{2}\,+\,C_{g0}S\Bigr ) \end{aligned}$$

(39)

Differentiating (39) w.r.t S , the expression in (27) can be derived. By replacing S in (39), the expression in (26) can be found out.

Fig. 22

RC model of the Cu wire with repeaters at the local level

Table 15 Definition of parameters

1.1.2 Intermediate and global level interconnects

Electrical model of a Cu wire at the intermediate and global level is shown in Fig. 23. The effective time constant of a single stage (i.e. from one repeater to the immediately succeeding one) is denoted by $\tau _{stage}$ and it can be calculated as,

$$\begin{aligned} \tau _{stage}= \frac{R_{d0}}{S}\Bigl (C_{d0}S+\frac{C_wL}{N}+C_{g0}S\Bigr )+\frac{R_wL}{N}\Bigl (\frac{C_wL}{2N}+C_{g0}S\Bigr ),\nonumber \\ \end{aligned}$$

(40)

and, the effective time constant ($\tau$) of the whole system is given by,

$$\begin{aligned} \tau = N\tau _{stage} \end{aligned}$$

(41)

Differentiating (41) w.r.t S , the expression in (29) can be derived. Similarly, differentiating (41) w.r.t N , the expression in (30) can be derived. By replacing S and N in (41), the expression in (31) can be found out. Also one point to note is that as a single dominant pole transfer function is considered here, so the delay ratio of CNT bundle to Cu is equivalent to the ratio of their time constants.

Fig. 23

RC model of the Cu wire with repeaters at the intermediate and global level

1.2 Optimized Repeater insertion methodology for CNT bundles

1.2.1 Local level interconnects

Electrical model of a CNT bundle at the local level is shown in Fig. 24. The effective time constant ($\tau$) of the whole system is given by,

$$\begin{aligned} \tau&= \frac{R_{d0}}{S}\Bigl (C_{d0}S+C_wL+C_{g0}S\Bigr )+R_f\Bigl (C_wL+C_{g0}S\Bigr )\nonumber \\&+\,\,R_wL\Bigl (\frac{C_wL}{2}+C_{g0}S\Bigr )+R_fC_{g0}S. \end{aligned}$$

(42)

Differentiating (42) w.r.t S , the expressions in (34) can be derived. Finally, replacing S in (42), the expression in (33) can be found out.

Fig. 24

RC model of the CNT bundles with repeaters at the local level

1.2.2 Intermediate and global level interconnects

Electrical model of a CNT bundle at the intermediate and global level is shown in Fig. 25. Definitions for various parameters shown in the figure are tabulated in Table 15. The effective time constant ($\tau$) of the whole system is given by,

$$\begin{aligned} \tau&= 2R_{d0}\bigl (C_{d0}+C_{g0}\bigr )+2R_fSC_{g0}+N\Bigl (\frac{R_{d0}}{S}\Bigl (C_{d0}S\nonumber \\&+\,\,\frac{C_wL}{N}+C_{g0}S\Bigr )+\frac{R_wL}{N}\Bigl (\frac{C_wL}{2N}+C_{g0}S\Bigr )\Bigr ). \end{aligned}$$

(43)

Differentiating (43) w.r.t S and N, the expressions in (37) and (36) can be derived. Finally, replacing S and N in (43), the expression in (38) can be found out.

Fig. 25

RC model of the CNT bundles with repeaters at the intermediate and global level