Introduction

To satisfy modern technology requirements, device dimensions are being reduced. However, merely shrinking device sizes is insufficient to significantly reduce overall chip delay, as interconnect delay remains a major contributing factor1. Therefore, it is equally, if not more, crucial to scale down interconnect dimensions. Scaled-down copper interconnects exhibit significantly higher effective resistivity compared to bulk copper. Copper interconnects are increasingly unable to meet current technological demands due to issues like grain-boundary scattering and electro-migration degradation, which push them to their operational limits2. The IRDS 2021 roadmap3 indicates that with the rise in both operating frequency and transistor count, sustaining both simultaneously has become virtually unfeasible. To ensure ICs can function within practical thermal limits, it has become necessary to stabilize either the number of transistors or the operating frequency. This issue is targeted here by analyzing Cu–Carbon hybrid interconnects at GHz frequency range. Nowadays high performance mixed-signal systems tap into all the domains i.e. digital, analog, RF etc. For designs with lower supply voltages, higher clock frequencies, dynamic logic usage, analog and microwave requirements, noise effects become prominent since these applications have lower noise margin4,5. Specially, these noise effects needs to be studied for interconnects as they run in parallel and are prone to crosstalk and interference6. As a result, finding an alternative to the conventional interconnect system has become imperative. Carbon nanomaterials, such as carbon nanotubes (CNTs) and graphene nanoribbons, have been extensively studied and identified as promising candidates for next-generation interconnect applications7. Fundamental research on carbon-based nano interconnects, as reported in8,9,10,11, indicates that these materials offer higher ampacity, longer mean free path (MFP), and better thermal conductivity compared to copper interconnects.

Figure 1
figure 1

Coupled dual metal line structure of Cu–Carbon hybrid interconnect.

Graphene nanoribbons have already been considered as a suitable candidate for RF interconnects12. Chen et. al. in13 experimentally demonstrated fully integrated carbon nanotube and graphene interconnects for high-speed CMOS electronics at gigahertz frequency range which is a crucial requirement in present on-chip technology. Copper carbon nanotube (Cu-CNT) composites, composed of CNT bundles and copper, are being considered as potential replacements for conventional interconnects in the near future14,15. The CNT bundles enhance performance due to their exceptional nanoscale properties. Additionally, their rectangular orientation allows for better alignment with contact pads compared to cylindrical CNTs, improving compatibility with CMOS fabrication processes. Sundaram et al. demonstrated in16 that the electro-deposition method can achieve a CNT volume fraction in the composite of more than 0.6.

Zhao et. al. conducted a high-frequency analysis of Cu-CNT through-silicon via (TSV) and Cu-graphene heterogeneous interconnects in17,18 and19, respectively. They found that co-depositing CNT with copper in the composite led to improvement in the electrical conductivity of the TSVs when compared to pure CNT TSVs and also that the resistive loss can be suppressed with multilayer graphene introduced in copper.

Cu–Carbon hybrid introduced as a novel structure in20, depicted in Fig. 1 is build upon the Cu-CNT composite interconnects explored in previous works, such as14,15. Our work introduces a graphene liner to this configuration, aiming to further reduce resistive losses and improve overall performance in high-frequency applications. This addition marks a significant advancement by addressing the limitations observed in earlier designs, such as increased impedance at high frequencies. We presented an electrical model for this Cu–Carbon hybrid and evaluated its electrical performance in terms of delay, noise, and power metrics, demonstrating its superior performance as an interconnect20. Recently, Park et al. fabricated this structure21 and confirmed that it exhibited excellent electrical, mechanical, and thermal properties, leading to overall performance improvements. However, to further validate its potential as an alternative to copper, other aspects of signal transmission need exploration. This paper introduces, for the first time, an ABCD parameter-based high-frequency model for Cu–Carbon hybrid interconnects, analyzing and comparing its transmission parameters with Cu-CNT composite and copper interconnects. The ABCD parameter-based22 analytical AC model is utilized to accurately estimating noise parameters in nanoscale systems. Figure 2 illustrates the configurational differences between the innovative Cu–Carbon hybrid and existing structures, including copper and Cu-CNT composites.

Figure 2
figure 2

Structural representation of copper, Cu-CNT and Cu–Carbon hybrid interconnect structure.

The remainder of the paper is organized as follows: “Results and discussion” section presents the simulation results. “Formulation and methodology” section proposes the high frequency model of Cu–Carbon hybrid interconnect structure. Finally, “Conclusion” section concludes this paper.

Results and discussion

The modeling and simulations were conducted using MATLAB software, version R2021b, on a standard desktop environment. The proposed model was verified with ADS software. This study assumes armchair CNT and zigzag graphene nanoribbon (GNR) configurations due to their metallic properties. The diameter of a single-walled carbon nanotube (SWCNT) in a bundle is set at 1 nm, with a CNT filling ratio (\(F_{cnt}\)) of 0.6 in the hybrid structure. For global-level interconnects, the line width and length are set at 15 nm and 1 mm, respectively. The thickness of the multi-layer graphene nanoribbon (MLGNR) barrier layer is considered to be 2.5 nm, equivalent to 6 layers of graphene. The parameters for the calculations in this paper were sourced from the IRDS 2017 roadmap1 for the 10 nm technology node.

Figure 3
figure 3

AC magnitude vs frequency plot of Cu–Carbon hybrid, Cu-CNT composite and copper interconnects vs frequency.

Figure 4
figure 4

(a) Per unit length AC resistance and (b) per unit length AC inductance of Cu–Carbon hybrid, copper and Cu-CNT composite interconnects.

Figure 3 shows the AC magnitude vs frequency plot of Cu–Carbon hybrid, copper and Cu-CNT composite interconnects. Cu–Carbon hybrid interconnect has the highest Bandwidth (BW) due to its least resistance among other alternatives.

Figure 4 shows the frequency-dependent resistances and inductances of Cu–Carbon hybrid, copper and Cu-CNT composite metal lines, respectively. The resistances of the metals increases while inductances decreases with increasing frequency. Cu–Carbon hybrid interconnects have the lowest impedance among all. By incorporating CNT bundles in copper metal line, the impedance can be suppressed, resulting in improved performance in the composite as compared to copper interconnect. As expected, the resistance of Cu–Carbon hybrid increases with the decrease of \(F_{cnt}\).

Figure 5
figure 5

Return loss coefficient \(S_{11}\) of single Cu–Carbon hybrid, copper and Cu-CNT composite interconnect lines. Verification with ADS software is reflected through dotted points.

Figure 6
figure 6

Return loss coefficient \(S_{11}\) of 2-line coupled Cu–Carbon hybrid, copper and Cu-CNT composite interconnect lines. Verification with ADS software is reflected through dotted points.

The magnitudes (in dB) of return loss coefficient \(S_{11}\) for single and 2-line coupled interconnects made of Cu–Carbon hybrid, Cu-CNT composite and copper are plotted in Figs. 5 and 6, respectively. \(S_{11}\) value increases with increase in frequency. Cu–Carbon hybrid encounters lowest return loss when compared to other alternatives due to its lower impedance. Cu–Carbon hybrid interconnect has \(\sim\)68% lower \(S_{11}\) value than copper at lower frequencies but even at 100 GHz frequency, the improvement is \(\sim\)43% for single line and \(\sim\)48% for 2-line coupled interconnects. The observed lower return loss coefficient for Cu-CNT compared to Cu–Carbon hybrid interconnects in the low-frequency range can be attributed to the distinct impedance characteristics and material properties of CNTs. The Cu-CNT composite interconnect benefits from the unique quantum conductance and reduced scattering rates of CNTs, which contribute to a lower impedance profile at lower frequencies. In contrast, the Cu–Carbon hybrid, while possessing superior high-frequency characteristics due to the graphene liner, exhibits slightly higher impedance at lower frequencies, resulting in a higher return loss coefficient. This behavior reflects the trade-offs inherent in optimizing interconnect materials for different frequency regimes. Data points estimated from ADS simulator matches well with the plots obtained from our model as shown in the figures.

Figure 7
figure 7

Forward transmission coefficient (\(S_{21}\)) of single Cu–Carbon hybrid, copper and Cu-CNT composite interconnect lines.

Figure 8
figure 8

Forward transmission coefficient (\(S_{21}\)) of 2-line coupled Cu–Carbon hybrid, copper and Cu-CNT composite interconnect lines.

The magnitudes of forward transmission coefficient \(S_{21}\) of single line and 2-line coupled interconnects made of Cu–Carbon hybrid, copper and Cu-CNT composite are demonstrated in Figs. 7 and x 8, respectively. \(S_{21}\) value decreases with increase in frequency. Single line Cu–Carbon hybrid interconnect structure has higher bandwidth for gain below \(-3\) dB. Cu–Carbon hybrid interconnect has highest transmission coefficient because it exhibits least impedance. At 100 GHz, Cu–Carbon hybrid interconnect has \(\sim\)30% and \(\sim\)38% higher \(S_{21}\) values than copper for single line and 2-line coupled interconnects, respectively.

Figure 9
figure 9

Noise figure corresponding to input at port 1 and output from port 4 of 2-line coupled Cu–Carbon hybrid interconnect for different \(F_{cnt}\).

The effect of \(F_{cnt}\) on the noise parameter of Cu–Carbon hybrid interconnects is depicted in Fig. 9. This study tells us that a CNT fraction of 0.8 and 0.6 is a good choice for high frequency noise constrained applications. \(F_{cnt}\) = 0.6 is selected for further evaluations because it performs better at certain range of frequencies and also it has been experimentally demonstrated by Sundaram et. al. in16. The subsequent noise analysis is done for 2-line coupled interconnect systems to study the effects of crosstalk and interference in high frequency applications by applying input at port 1 of line 1 and observing output at port 4 of line 2.

Figure 10
figure 10

Noise figure corresponding to input at port 1 and output from port 4 for 2-line coupled Cu–Carbon hybrid, copper and Cu-CNT composite interconnect lines.

Figure 10 gives an understanding of the noise figure in the 2-line coupled interconnect system. At lower frequencies, all interconnects have comparable noise profiles as observed in Fig. 10. Noise in copper remains almost constant with increasing frequency. Cu-CNT and Cu–Carbon hybrid interconnect experiences a steep dip in noise figure followed with a gradual rise. The percentage improvement (in dB) in the noise figure of Cu–Carbon hybrid interconnect as compared to copper is \(\sim\)48% at 100 GHz as a result of its lower mutual inductance. The noise factor corresponding to input at port 1 and output from port 4 for 2-line coupled interconnect lines is demonstrated in Fig. 11. As seen from the figure, the degradation in signal to noise ratio in Cu–Carbon hybrid interconnect is far better as compared to copper due to its lower coupling capacitance and mutual inductance, with an improvement of \(\sim\)98% at 100 GHz, which is zoomed and shown in the inset of the graph. The improvement is \(\sim\)44% when compared to Cu-CNT composite interconnect. Thus, both noise figure and noise factor parameters indicate that Cu–Carbon hybrid interconnect induces less noise as compared to copper especially at high frequencies.

Figure 11
figure 11

Noise factor corresponding to input at port 1 and output from port 4 for 2-line coupled Cu–Carbon hybrid, copper and Cu-CNT composite interconnect lines. Zoomed section for higher frequencies is shown in the inset.

Formulation and methodology

Figure 12
figure 12

Equivalent electrical single metal line model for Cu–Carbon hybrid interconnect structure.

Figure 12 describes the electrical circuit model of a single-line Cu–Carbon hybrid interconnect structure. Here, \(R_{icon}\) denotes the imperfect contact resistance, here assumed to be 10 K\(\Omega\)23. The equivalent distributed per unit length (p.u.l) resistance of Cu–Carbon hybrid which is defined as,

$$\begin{aligned} R_{Cu-Carbon} = Re\left( \frac{1}{{\sigma _{\mathrm{{eff}}}}}\right) \frac{1}{W T} \end{aligned}$$
(1)

where \(\sigma _\mathrm{{eff}}\) represents the conductivity of Cu–Carbon hybrid which can be defined as,

$$\begin{aligned} {\sigma _{\mathrm{{eff}}}} = \left( {1 - {F_{\mathrm{{cnt}}}}} \right) {\sigma _{{{\textrm{Cu}}}}} + {F_{\mathrm{{cnt}}}} {\sigma _{\mathrm{{cnt}}}}+\sigma _{gnr} \end{aligned}$$
(2)

where is the CNT filling ratio in the hybrid structure which is defined as15,

$$\begin{aligned} {F_{\mathrm{{cnt}}}} = \frac{\pi {N_{cnt} {{\left( {D_{cnt} + 0.31\ {\text {nm}}} \right) }^2}}}{{4 W T}} \end{aligned}$$
(3)

\(\sigma _{{{\textrm{Cu}}}}\) is the conductivity of copper as described in24. The conductivity of single walled carbon nanotube (SWCNT) is expressed as25,

$$\begin{aligned} {\sigma _{\mathrm{{gnr}}}} = \frac{L}{W T \sum \nolimits _{j = t,b,l,r}Z_\mathrm{{gnr}}}, \quad \quad {\sigma _{\mathrm{{cnt}}}} = \frac{4L}{\pi {{\left( {D_{cnt} + 0.31\ \mathrm{{nm}}} \right) }^2}Z_{\mathrm{{cnt}}}} \end{aligned}$$
(4)

where, \(j = t,b,l,r\) denote the top, bottom, left and right sides of GNR barrier, respectively. \(Z_{\mathrm{{cnt}}}\) and \(Z_{\mathrm{{gnr}}}\) are intrinsic self-impedance of an isolated SWCNT and an isolated MLGNR, which are expressed as26,

$$\begin{aligned} {Z_{\mathrm{{cnt}}}} = \frac{L}{{N_{ch}}^{c}}\left( {\frac{R_{quant}}{\lambda _{eff}^{c}}} + j\omega L_K \right) \end{aligned}$$
(5)
$$\begin{aligned} {Z_{\mathrm{{gnr}}}} = \frac{L}{N_{ch}^{g}N_L}\left( \frac{R_{quant}}{\lambda _{eff}^{g}} + j\omega L_K \right) \end{aligned}$$
(6)

where \(R_{quant}\) denotes the quantum resistance and \(L_K\) is the kinetic inductance calculated by utilizing the recursive method explained in27,28. \(N_{ch}^{c}\) and \(N_{ch}^{g}\) are the number of conducting channels in SWCNT and MLGNR, respectively. \(\lambda _{eff}^{c}\) and \(\lambda _{eff}^{g}\) are the effective electron mean free paths for the ith shell of SWCNT and jth layer of MLGNR which can be evaluated by the Matthiessen’s equation27,29.

The p.u.l equivalent capacitance of Cu–Carbon hybrid interconnect can be expressed as,

$$\begin{aligned} {C_{Cu-Carbon}} = {\left[ {{{\left( {\sum \nolimits _{j = t,b,l,r} {C_{gnr}^j}} \right) }^{-1}} +\frac{1}{{{C_{cnt}}}} + \frac{1}{{{C_{e}}}}} \right] ^{ - 1}} \end{aligned}$$
(7)

where the p.u.l equivalent capacitance contributed graphene barrier layers bordering Cu-CNT composite metal line is denoted by (\(C_{gnr}^t\), \(C_{gnr}^b\), \(C_{gnr}^l\), \(C_{gnr}^r\)) which can be evaluated using the recursive method provided in28. \({C_{cnt}}\) represents the p.u.l quantum capacitance of SWCNT bundle which can be calculated by utilizing the recursive method explained in27. The p.u.l electrostatic ground capacitance (\(C_{e}\)) is accessed from28.

The p.u.l equivalent inductance of Cu–Carbon hybrid interconnect consists of internal and external inductances which are described as,

$$\begin{aligned} {L_{Cu-Carbon}} = L_{in}+L_{ex} = {{\textrm{Im}}}\left( \frac{1}{\sigma _{eff}}\right) + \frac{{{\mu _0}{\varepsilon _0}{\varepsilon _{{r}}}}}{{{C_{e}}}} \end{aligned}$$
(8)
Figure 13
figure 13

Equivalent electrical 2-line coupled system model for Cu–Carbon hybrid interconnect system.

Figure 13 demonstrates the electrical equivalent circuit of a 2-line coupled Cu–Carbon hybrid interconnect system. Here, \(C^c_{Cu-Carbon}\) and \(L^m_{Cu-Carbon}\) represent the p.u.l coupling capacitance and mutual inductance developed between the Cu–Carbon hybrid metal lines, respectively which are taken from30,31.

The effective complex conductivity of Cu–Carbon hybrid interconnect is calculated by substituting \(Z_{cnt} = R_{cnt} + j\omega L_{cnt}\), \(Z_{gnr} = R_{gnr} + j\omega L_{gnr}\) and (4) into (2), one can obtain,

$$\begin{aligned} \begin{aligned} \sigma _{eff}&= \left( {1 - {F_{\mathrm{{cnt}}}}} \right) {\sigma _{{{\textrm{Cu}}}}} + \left( \frac{{k_1F_{\mathrm{{cnt}}}{R_{\mathrm{{cnt}}}}}}{{R_{\mathrm{{cnt}}}^2 + {\omega ^2}L_{\mathrm{{cnt}}}^2}} + \frac{{k_2{R_{\mathrm{{gnr}}}}}}{{R_{\mathrm{{gnr}}}^2 + {\omega ^2}L_{\mathrm{{gnr}}}^2}}\right) \\&- j \left( \frac{{k_1F_{\mathrm{{cnt}}}\omega {L_{\mathrm{{cnt}}}}}}{{R_{\mathrm{{cnt}}}^2 + {\omega ^2}L_{\mathrm{{cnt}}}^2}} + \frac{{k_2\omega {L_{\mathrm{{gnr}}}}}}{{R_{\mathrm{{gnr}}}^2 + {\omega ^2}L_{\mathrm{{gnr}}}^2}}\right) \\ \end{aligned} \end{aligned}$$
(9)

and the ratio between Re(\(\sigma _{eff}\)) and Im(\(\sigma _{eff}\)) is described by (10).

$$\begin{aligned} \begin{aligned} \left| {\frac{{{\textrm{Re}} \left( {{\sigma _{\mathrm{{eff}}}}} \right) }}{{{\textrm{Im}} \left( {{\sigma _{\mathrm{{eff}}}}} \right) }}} \right|&= \frac{{\left( {1 - {F_{\mathrm{{cnt}}}}} \right) {\sigma _{{{\textrm{Cu}}}}}\left( {R_{\mathrm{{cnt}}}^2 + {\omega ^2}L_{{\textrm{cnt}}}^2} \right) \left( {R_{\mathrm{{gnr}}}^2 + {\omega ^2}L_{{\textrm{gnr}}}^2} \right) + k_1F_{cnt}{R_{cnt}}}\left( {R_{{\textrm{gnr}}}^2 + {\omega ^2}L_{{\textrm{gnr}}}^2} \right) }{\omega \left( {k_1F_{cnt}L_{cnt}} \left( R_{\mathrm{{gnr}}}^2 + {\omega ^2}L_{{\textrm{gnr}}}^2\right) + k_2L_{gnr} \left( R_{{\textrm{cnt}}}^2 + {\omega ^2}L_{{\textrm{cnt}}}^2\right) \right) } + \\&\frac{ k_2{R_{gnr}}\left( {R_{cnt}^2 + {\omega ^2}L_{cnt}^2} \right) }{\omega \left( {k_1F_{cnt}L_{cnt}} \left( R_{{\textrm{gnr}}}^2 + {\omega ^2}L_{{\textrm{gnr}}}^2\right) + k_2L_{gnr} \left( R_{{\textrm{cnt}}}^2 + {\omega ^2}L_{{\textrm{cnt}}}^2\right) \right) } \\ \end{aligned} \end{aligned}$$
(10)
Figure 14
figure 14

Absolute magnitude of effective complex conductivity as a function of frequency as defined in (9).

Figure 15
figure 15

Modulus of ratio of real and imaginary parts of effective complex conductivity as a function of frequency as defined in (10).

The results depicted in Figs. 14 and 15 provide critical insights into the frequency-dependent behavior of the effective complex conductivity of Cu–Carbon hybrid interconnects compared to Cu-CNT composite interconnects. In Fig. 14, the absolute magnitude of the effective complex conductivity \(|\sigma _{eff}|\) is shown to initially decrease with increasing frequency before reaching a saturation point at higher frequencies. This behavior is particularly pronounced in the Cu–Carbon hybrid interconnects with a CNT filling ratio (\(F_{cnt}\)) of 0.6, which exhibits the highest conductivity among the compared structures. The superior performance of the Cu–Carbon hybrid interconnect is attributable to the synergistic effects of the copper matrix and the embedded CNTs, which reduce scattering and enhance electron transport at nanoscale dimensions. Moreover, the reduced impedance in the Cu–Carbon hybrid interconnects due to the inclusion of graphene nanoribbons (GNRs) further contributes to the higher conductivity observed.

Figure 15 further elucidates the complex conductivity by presenting the modulus of the ratio between the real and imaginary components of the effective complex conductivity \(|Re(\sigma _{eff})/Im(\sigma _{eff})|\). This ratio highlights the dominance of the reactive component (imaginary part) in Cu-CNT composite interconnects at higher frequencies, contrasting with the Cu–Carbon hybrid interconnects where both real and imaginary parts contribute more evenly. The balanced contribution in the Cu–Carbon hybrid interconnects can be linked to the enhanced material properties of the graphene liner, which stabilizes the conductivity and reduces the reactive losses, making these interconnects more suitable for high-frequency applications. These findings are significant because they suggest that Cu–Carbon hybrid interconnects not only outperform traditional Cu and Cu-CNT interconnects in terms of real part conductivity but also maintain a lower and more stable reactive component at high frequencies. This characteristic is crucial for minimizing signal distortion and loss in high-speed, high-frequency nanoscale systems, thus reinforcing the potential of Cu–Carbon hybrids as a superior alternative for next-generation interconnect technology.

ABCD parameter based model

Single line interconnect model

The equivalent electrical model shown in Fig. 12 can be simplified as a transmission line whose transmission characteristics can be expressed by utilizing an ABCD matrix22. The total ABCD transmission parameter matrix of the configuration shown in Fig. 12 is defined by (11).

$$\begin{aligned} \begin{bmatrix} A & B \\ C & D \\ \end{bmatrix} = \begin{bmatrix} 1 & R_{sd} \\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 \\ sC_{sd} & 1 \end{bmatrix} \begin{bmatrix} 1 & R_{icon} \\ 0 & 1 \end{bmatrix} \begin{bmatrix} cosh(\varphi h) & Z_{0}\sinh (\varphi h) \\ {\frac{1}{Z_{0}}} \sinh (\varphi h) & \cosh (\varphi h) \end{bmatrix} \begin{bmatrix} 1 & R_{icon}\\ 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0\\ sC_{gl} & 1 \end{bmatrix} \end{aligned}$$
(11)

where \(Z_0\) and \(\varphi\) are the characteristic impedance and the propagation constant of single line interconnect. These parameters can be expressed as,

$$\begin{aligned} \varphi = \sqrt{s(\alpha C_{Cu-Carbon})(R_{Cu-Carbon}+s(L_{Cu-Carbon}))} \end{aligned}$$
(12)
$$\begin{aligned} Z_0 = \sqrt{\frac{(R_{Cu-Carbon}+s(L_{Cu-Carbon}))}{s(\alpha C_{Cu-Carbon})}} \end{aligned}$$
(13)

Coupled line interconnect model

Similarly, for a 2-line coupled interconnect system configuration as shown in Fig. 13, the total ABCD transmission parameter matrix can be expressed by (14).

$$\begin{aligned} \begin{aligned} \begin{bmatrix} A_{11} & A_{12} & B_{11} & B_{12} \\ A_{21} & A_{22} & B_{21} & B_{22} \\ C_{11} & C_{12} & D_{11} & D_{12} \\ C_{21} & C_{22} & D_{21} & D_{22} \\ \end{bmatrix}&= \begin{bmatrix} 1 & 0 & R_{sd} & 0 \\ 0 & 1 & 0 & R_{sd} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ sC_{sd} & 0 & 1 & 0 \\ 0 & sC_{sd} & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & R_{icon} & 0 \\ 0 & 1 & 0 & R_{icon} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \\&\begin{bmatrix} CC^n \end{bmatrix} \begin{bmatrix} 1 & 0 & R_{icon} & 0 \\ 0 & 1 & 0 & R_{icon} \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ sC_{gl} & 0 & 1 & 0 \\ 0 & sC_{gl} & 0 & 1 \end{bmatrix} \\ \end{aligned} \end{aligned}$$
(14)

where \(CC^n\) is the ABCD matrix representing a 2-line coupled system constructed using n number of cascaded infinitesimal sections and is given by,

$$\begin{aligned} {CC^{n}=\begin{bmatrix} \frac{1}{2}\left( cosh(\varphi _1 L) + cosh(\varphi _2 L)\right) & \frac{1}{2}\left( cosh(\varphi _1 L) - cosh(\varphi _2 L)\right) & \frac{1}{2}\left( Z_1sinh(\varphi _1 L) + Z_2sinh(\varphi _2 L)\right) & \frac{1}{2}\left( Z_1sinh(\varphi _1 L) - Z_2sinh(\varphi _2 L)\right) \\ \frac{1}{2}\left( cosh(\varphi _1 L) - cosh(\varphi _2 L)\right) & \frac{1}{2}\left( cosh(\varphi _1 L) + cosh(\varphi _2 L)\right) & \frac{1}{2}\left( Z_1sinh(\varphi _1 L) - Z_2sinh(\varphi _2 L)\right) & \frac{1}{2}\left( Z_1sinh(\varphi _1 L) + Z_2sinh(\varphi _2 L)\right) \\ \frac{1}{2}\left( Z_1sinh(\varphi _1 L) + Z_2sinh(\varphi _2 L)\right) & \frac{1}{2}\left( Z_1sinh(\varphi _1 L) - Z_2sinh(\varphi _2 L)\right) & \frac{1}{2}\left( cosh(\varphi _1 L) + cosh(\varphi _2 L)\right) & \frac{1}{2}\left( cosh(\varphi _1 L) - cosh(\varphi _2 L)\right) \\ \frac{1}{2}\left( Z_1sinh(\varphi _1 L) - Z_2sinh(\varphi _2 L)\right) & \frac{1}{2}\left( Z_1sinh(\varphi _1 L) + Z_2sinh(\varphi _2 L)\right) & \frac{1}{2}\left( cosh(\varphi _1 L) - cosh(\varphi _2 L)\right) & \frac{1}{2}\left( cosh(\varphi _1 L) + cosh(\varphi _2 L)\right) \end{bmatrix}} \end{aligned}$$
(15)

Here, \(Z_1\) and \(Z_2\) are the characteristic impedances of the 2-line coupled interconnects. \(\varphi _1\) and \(\varphi _2\) are the propagation constants of the 2-line coupled interconnects. These parameters are defined as,

$$\begin{aligned} & \varphi _1 = \sqrt{s(\alpha C_{Cu-Carbon})(R_{Cu-Carbon}+s(L_{Cu-Carbon}+L^m_{Cu-Carbon}))} \end{aligned}$$
(16)
$$\begin{aligned} & \varphi _2 = \sqrt{s(\alpha C_{Cu-Carbon})(R_{Cu-Carbon}+s(L_{Cu-Carbon}-L^m_{Cu-Carbon}))} \end{aligned}$$
(17)
$$\begin{aligned} & Z_1 = \sqrt{\frac{(R_{Cu-Carbon}+s(L_{Cu-Carbon}+L^m_{Cu-Carbon}))}{s(\alpha C_{Cu-Carbon})}} \end{aligned}$$
(18)
$$\begin{aligned} & Z_2 = \sqrt{\frac{(R_{Cu-Carbon}+s(L_{Cu-Carbon}-L^m_{Cu-Carbon}))}{s(\alpha C_{Cu-Carbon})}} \end{aligned}$$
(19)

Here \(\alpha\)=(1-\(\eta _1\)-\(\eta _2\)), \(\beta\)=(1+\(\eta _1\)-\(\eta _2\)). Where \(\eta _1\)=\(\frac{b}{a^2-b^2}\), \(\eta _2\)=\(\frac{a}{a^2-b^2}\) with \(a=\frac{C_{Cu-Carbon}+C^c_{Cu-Carbon}+}{C_{Cu-Carbon}}\) and \(b=\frac{C^c_{Cu-Carbon}}{C_{Cu-Carbon}}\)

Conclusion

In this paper, for the first time, a high frequency electrical model of single and 2-line coupled Cu–Carbon hybrid metal lines is proposed and various transmission parameters are analyzed at 10 nm technology node. The high-frequency performance of Cu–Carbon hybrid is studied and compared with existing copper and Cu-CNT composite interconnects. The ABCD parameter model for single and 2-line coupled interconnects are developed which are utilized to obtain the scattering parameters. This model is also validated with ADS software. Here, Cu–Carbon hybrid interconnects have the lowest impedance among all which tends to increase with the decrease of \(F_{cnt}\). Compared with copper, Cu–Carbon hybrid interconnect (with \(F_{cnt}\)=0.6) possesses \(\sim\)80% lower impedance at 100 GHz frequency. The reactive part Cu-CNT composite interconnects is dominant at higher frequencies while that is not the case with Cu–Carbon hybrid interconnects where both real and imaginary parts contribute equally in the effective complex conductivity. Cu–Carbon hybrid interconnect experiences lowest return loss and highest forward transmission gain among all others. Compared to copper, Cu–Carbon hybrid interconnect has \(\sim\)43% and \(\sim\)48% lower \(S_{11}\) values and has \(\sim\)30% and \(\sim\)38% higher \(S_{21}\) values at 100 GHz for single and 2-line coupled interconnects, respectively. At lower frequencies, all interconnects have comparable noise profiles. Noise figure (in dB) and noise factor of Cu–Carbon hybrid interconnect when compared to copper shows \(\sim\)48% and \(\sim\)98% improvement at 100 GHz, respectively. The improvement in noise factor is \(\sim\)44% when compared to Cu-CNT composite interconnect. Although Cu–Carbon and Cu-CNT have comparable noise figures, still Cu–Carbon is recommended due to its lower AC impedances and better signal transmission. Cu–Carbon hybrid is an emerging interconnect structure with very promising performance metric and phenomenal advantages over copper interconnects. Therefore, it needs to be studied and analyzed extensively in theory and experiment for various aspects of VLSI applications.