A Unified Delay, Power and Crosstalk Model for Current Mode Signaling Multiwall Carbon Nanotube Interconnects

Abstract

Multiwall carbon nanotube (MWCNT) has been investigated as a potential interconnect material for future advanced technology nodes. The present paper analyzes performance of MWCNT interconnects using current mode signaling (CMS) scheme. The novelty of the present work can be stated as: Firstly, a unified model is proposed for both copper and MWCNT interconnects using finite-difference time-domain (FDTD) technique. Secondly, this model is applicable for both the conventional voltage mode signaling and more versatile CMS schemes. Furthermore, the presented FDTD-based model is valid for single as well as M-line coupled interconnects in integrated circuits. The model also incorporates CMOS gate as driver for MWCNT interconnect. Thirdly, power model using FDTD technique is investigated for the first time. Accurate formulation and computation of power dissipation in CMS MWCNT interconnects are presented in the paper. Propagation delay, power dissipation and power_delay_product (PDP) are the performance metrics considered for single-line CMS MWCNT interconnect. Crosstalk is analyzed for 2-Line and 5-Line coupled interconnects. It is investigated that CMS scheme leads to about 4 times lesser propagation delay and 2.5 times reduced PDP in MWCNT interconnect than the conventional copper interconnect for interconnect length of 4500 $\upmu$m. The technology node considered is 32 nm. The response of the system is accurately computed using the proposed FDTD-based model. The maximum percentage error between results obtained by the proposed analytical model and SPICE simulation model is <3% for the various test cases.

Appendices

Appendix 1

The parasitic elements of MWCNT interconnect can be defined as:

The lumped resistance ($R_\mathrm{lump}$) comprises of quantum resistance ($R_\mathrm{q}$) which is due to quantum confinement of carriers along the interconnect dimensions. The contact resistance ($R_\mathrm{c}$) is due to imperfect contact between the interconnect and substrate. $R_\mathrm{c}$ depends on the fabrication process and varies from 1 to 20 K$\Omega$ [10]. $R_\mathrm{lump}$ is expressed as [19]

$$\begin{aligned} R_\mathrm{lump} =\left[ {\sum _{i=1}^N {\left( {\frac{h}{2e^{2}\cdot N_\mathrm{ch}^i}+R_\mathrm{c}^i} \right) } ^{-1}} \right] ^{-1} \end{aligned}$$

(34)

where h is Planck’s constant and e represents charge on electron.

$R_\mathrm{lump}$ is distributed equally along the two ends of the interconnect as $R_\mathrm{lump}^{\prime }$. It is presented as

$$\begin{aligned} R_\mathrm{lump}^{\prime } =\frac{R_\mathrm{lump}}{2} \end{aligned}$$

(35)

The distributed resistance ($R_\mathrm{dis}$) in the ESC model represents the scattering resistance per unit length (p.u.l.) [22]. It is primarily due to optical and acoustic phonon scattering. $R_\mathrm{dis}$ is predominant when interconnect length is greater than electron mean free path [28]. It is defined as [23]

$$\begin{aligned} R_\mathrm{dis} =\sum _{i=1}^N {\left( {\frac{h}{2e^{2}\cdot N_\mathrm{ch}^i \cdot \lambda _i}} \right) } \end{aligned}$$

(36)

where $\lambda _{i}$ corresponds to effective electron mean free path of ith shell and is obtained as [24]

$$\begin{aligned} \lambda _i =\frac{10^{3}d_i}{\left( {T/{T_0}} \right) -2} \end{aligned}$$

(37)

where $T_{0}$=100 K

The distributed inductance ($L_\mathrm{dis}$) comprises of kinetic inductance p.u.l. ($L_\mathrm{k}$) and magnetic inductance p.u.l. ($L_\mathrm{m}$).

The kinetic inductance p.u.l. per channel ($L_\mathrm{k/channel}$) is given as [19]

$$\begin{aligned} L_\mathrm{k/channel} =\left( {\frac{h}{2e^{2}v_\mathrm{f}}} \right) \left( {\frac{1}{2}} \right) \end{aligned}$$

(38)

where $v_\mathrm{f}$ is Fermi velocity.

Using $L_\mathrm{k/channel}$, kinetic inductance p.u.l. per shell ($L_\mathrm{k/shell}$) is computed.

$$\begin{aligned} L_\mathrm{k/shell}^i =\left( {\frac{L_\mathrm{k/channel}}{N_\mathrm{ch}^i}} \right) ;\quad 1\le i\le N \end{aligned}$$

(39)

The p.u.l. mutual shell to shell inductance ($L_\mathrm{m\_shell\_shell}$) is defined as [23]

$$\begin{aligned} L_\mathrm{m\_shell\_shell}^{i,i+1} =\frac{\mu }{2\pi }\ln \left( {\frac{d_{i+1}}{d_i}} \right) ;\quad 1\le i\le N-1 \end{aligned}$$

(40)

The equivalent kinetic inductance p.u.l. of ith shell ($L_\mathrm{equ}^i$) is obtained using recursive expression as [23]

$$\begin{aligned} L_\mathrm{equ}^i =\left[ {\frac{1}{L_\mathrm{equ}^{i-1} +L_\mathrm{m\_shell\_shell}^{i-1,i}}+\frac{1}{L_\mathrm{k/shell}^i}} \right] ^{-1};\quad 2\le i\le N \end{aligned}$$

(41)

where

$$\begin{aligned} L_\mathrm{equ}^1 =L_\mathrm{k/shell}^1 \end{aligned}$$

(42)

The equivalent kinetic inductance p.u.l. of MWCNT interconnects ($L_\mathrm{k}$) is given by:

$$\begin{aligned} L_\mathrm{k} =L_\mathrm{equ}^N \end{aligned}$$

(43)

The equivalent magnetic inductance p.u.l. of MWCNT interconnect ($L_\mathrm{m}$) is computed as [24].

$$\begin{aligned} L_\mathrm{m} =\frac{\mu }{2\pi }\cosh ^{-1}\left( {\frac{d_N +2h_\mathrm{g}}{d_N}} \right) \end{aligned}$$

(44)

where $d_{N}$ is the outermost shell diameter of MWCNT and $h_\mathrm{g}$ represents the distance between MWCNT and ground plane.

The equivalent inductance in the ESC model ($L_\mathrm{dis}$) is computed as

$$\begin{aligned} L_\mathrm{dis} =L_\mathrm{k} +L_\mathrm{m} \end{aligned}$$

(45)

Similarly, distributed capacitance ($C_\mathrm{dis}$) of MWCNT in the ESC model is obtained. It comprises of p.u.l. quantum capacitance ($C_\mathrm{q}$), and electrostatic capacitance ($C_\mathrm{e}$). $C_\mathrm{dis}$ is expressed as

$$\begin{aligned} C_\mathrm{dis} =\frac{C_\mathrm{q} \cdot C_\mathrm{e}}{C_\mathrm{q} +C_\mathrm{e}} \end{aligned}$$

(46)

$C_\mathrm{e}$ is obtained as [24]

$$\begin{aligned} C_\mathrm{e} =\frac{2\pi \varepsilon }{\cosh ^{-1}\left( {\frac{d_N +h_\mathrm{g}}{d_N}} \right) } \end{aligned}$$

(47)

$C_\mathrm{q}$ is obtained by solving recursive formulae given below [23]

$$\begin{aligned} C_\mathrm{equ}^i= & {} \left[ {\frac{1}{C_\mathrm{equ}^{i-1} }+\frac{1}{C_\mathrm{c\_shell\_shell}^{i-1,i}}} \right] ^{-1}+C_\mathrm{q/shell}^i ;\quad 2\le i\le N \end{aligned}$$

(48)

$$\begin{aligned} C_\mathrm{equ}^1= & {} C_\mathrm{q/shell}^1 \end{aligned}$$

(49)

$$\begin{aligned} C_\mathrm{q}= & {} C_\mathrm{equ}^N \end{aligned}$$

(50)

where $C_\mathrm{equ}^i$is the equivalent capacitance of ith shell. $C_\mathrm{q/shell}$ and $C_\mathrm{c\_shell\_shell}$ are p.u.l. quantum capacitance per shell and coupling capacitance between shells of MWNCT interconnect, respectively [11]. These are defined as [19]

$$\begin{aligned} C_\mathrm{q/shell}^i= & {} 2N_\mathrm{ch}^i \left( {\frac{2e^{2}}{hv_\mathrm{f}}} \right) ;\quad 1\le i\le N \end{aligned}$$

(51)

$$\begin{aligned} C_\mathrm{c\_shell\_shell}^{i,i+1}= & {} \frac{2\pi \varepsilon }{\ln \left( {{d_{i+1}}/{d_i}} \right) };\quad 1\le i\le N-1 \end{aligned}$$

(52)

The p.u.l. mutual inductance ($M_\mathrm{i}$) and coupling capacitance ($C_\mathrm{c}$) between two parallel MWCNT interconnects are given as [11, 21, 24, 31]

$$\begin{aligned} M_\mathrm{i}= & {} \frac{\mu }{2\pi }\left[ {\ln \left( {\frac{l}{h_\mathrm{c} }+\sqrt{1+\left( {\frac{l}{h_\mathrm{c}}} \right) ^{2}}} \right) -\sqrt{1+\left( {\frac{h_\mathrm{c}}{l}} \right) ^{2}}+\frac{h_\mathrm{c}}{l}} \right] \end{aligned}$$

(53)

$$\begin{aligned} C_\mathrm{c}= & {} \frac{\pi \varepsilon }{\ln \left( {\frac{s_p}{d_N}+\sqrt{\left( {\frac{s_p}{d_N}} \right) ^{2}+1}} \right) } \end{aligned}$$

(54)

where l is the length of interconnect and $s_{p}$ is the separation between two interconnects. $h_\mathrm{c}$ is the center-to-center distance between two interconnects and equals to ($s_{p}+d_{N}$).

Appendix 2

The parasitic elements for copper interconnects can be defined as [30, 39]:

$$\begin{aligned} R_\mathrm{dis}= & {} \frac{\rho }{w\cdot t} \end{aligned}$$

(55)

$$\begin{aligned} L_\mathrm{dis}= & {} \frac{\mu }{2\pi }\left[ {\ln \left( {\frac{2l}{w+t}} \right) +\frac{1}{2}+\frac{0.22\left( {w+t} \right) }{l}} \right] \end{aligned}$$

(56)

$$\begin{aligned} C_\mathrm{dis}= & {} \varepsilon \left[ {\begin{array}{l} \frac{w}{h_\mathrm{g}}+2.22\left( {\frac{s_p}{s_p +0.7h_\mathrm{g}}} \right) ^{3.19} \\ +1.17\left( {\frac{s_p}{s_p +1.51h_\mathrm{g}}} \right) ^{0.76}\cdot \left( {\frac{t}{t+4.53h_\mathrm{g}}} \right) ^{0.12} \\ \end{array}} \right] \end{aligned}$$

(57)

$$\begin{aligned} M_\mathrm{i}= & {} \frac{\mu }{2\pi }\left[ {\ln \left( {\frac{2l}{h_\mathrm{c}}} \right) -1+\frac{h_\mathrm{c}}{l}} \right] \end{aligned}$$

(58)

$$\begin{aligned} C_\mathrm{c}= & {} \varepsilon \left[ {\begin{array}{l} 1.14\left( {\frac{t}{s_p}} \right) \left( {\frac{h_\mathrm{g}}{h_\mathrm{g} +2.06s_p}} \right) ^{0.09}+0.74\left( {\frac{w}{w+1.59s_p}} \right) ^{1.14} \\ +1.16\left( {\frac{w}{w+1.87s_p}} \right) ^{0.16}\cdot \left( {\frac{h_\mathrm{g}}{h_\mathrm{g} +0.98s_p}} \right) ^{1.18} \\ \end{array}} \right] \end{aligned}$$

(59)

where $\rho$ is resistivity of copper material. w and t are width and thickness of the copper interconnect. All other parameters for copper interconnects in (55)–(59) have their usual meaning as that for MWCNT interconnect. $R_\mathrm{lump}^{\prime }$in the ESC model for copper interconnect is zero.

Appendix 3

The Telegraph’s equations are given as [29]:

$$\begin{aligned} \frac{\partial \left[ V \right] }{\partial z}+\left[ L \right] \frac{\partial \left[ I \right] }{\partial t}+\left[ R \right] \left[ I \right]= & {} 0 \end{aligned}$$

(60)

$$\begin{aligned} \frac{\partial \left[ I \right] }{\partial z}+\left[ C \right] \frac{\partial \left[ V \right] }{\partial t}= & {} 0 \end{aligned}$$

(61)

where [V] and [I] are $M \times 1$ voltage and current variables along interconnect and are function of position and time. [R], [L] and [C] are $M \times M$ dimensional p.u.l. interconnect parasitics. These are given as

$$\begin{aligned} \left[ R \right]= & {} \hbox {diag}\left( {R_\mathrm{dis}^1,R_\mathrm{dis}^2 ,\cdots ,R_\mathrm{dis}^M} \right) , \\ \left[ L \right]= & {} \left( {{\begin{array}{lllll} {L_\mathrm{dis}^1}&{} {M_\mathrm{i}^{1,2}}&{} {M_\mathrm{i}^{1,3}}&{} \ldots &{} {M_\mathrm{i}^{1,M}} \\ {M_\mathrm{i}^{2,1}}&{} {L_\mathrm{dis}^2}&{} {M_\mathrm{i}^{2,3}}&{} \cdots &{} {M_\mathrm{i}^{2,M}} \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ {M_\mathrm{i}^{M,1}}&{} {M_\mathrm{i}^{M,2}}&{} {M_\mathrm{i}^{M,3}}&{} \cdots &{} {L_\mathrm{dis}^M} \\ \end{array}}} \right) \hbox {and} \end{aligned}$$

$$\begin{aligned} \left[ C \right] =\left( {{\begin{array}{lllll} {C_\mathrm{dis}^1 +\sum \limits _{j=2}^M {C_\mathrm{c}^{1,j}}}&{} {-C_\mathrm{c}^{1,2}}&{} {-C_\mathrm{c}^{1,3}}&{} \ldots &{} {-C_\mathrm{c}^{1,M}} \\ {-C_\mathrm{c}^{2,1}}&{} {C_\mathrm{dis}^2 +\sum _{j=1, 3}^M {C_\mathrm{c}^{2,j}}}&{} {-C_\mathrm{c}^{2,3}}&{} \cdots &{} {-C_\mathrm{c}^{2,M}} \\ \vdots &{} \vdots &{} \ddots &{} \vdots &{} \vdots \\ {-C_\mathrm{c}^{M,1}}&{} {-C_\mathrm{c}^{M,2}}&{} {-C_\mathrm{c}^{M,3}}&{} \ldots &{} {C_\mathrm{dis}^M +\sum \limits _{j=1}^{M-1} {C_\mathrm{c}^{M,j}}} \\ \end{array}}} \right) \end{aligned}$$

(62)

where $R_\mathrm{dis}$, $L_\mathrm{dis}$ and $C_\mathrm{dis}$ are distributed resistance, inductance and capacitance, respectively, of MWCNT/copper interconnect in ESC model. $M_\mathrm{i}$ and $C_\mathrm{c}$ represent p.u.l. mutual inductance and coupling capacitance between two parallel MWCNT/copper interconnects.