Performance and reliability improvement in intercalated MLGNR interconnects using optimized aspect ratio

In this work, aspect ratio of various intercalation doped MLGNR interconnects are optimized using a numerical approach to achieve improved performance and reliability. A numerical optimization method is presented to estimate optimized aspect ratio considering combined effects of performance, noise and reliability metrics for any arbitrary nano interconnect system. This approach is cost effective and will be extremely useful to industry for selection of aspect ratio of interconnects as it is a non-SPICE method and reduces fabrication iterations for achieving desired performance and reliability. Our numerical method suggests that by minimizing the figure of merit (i.e. Noise Delay Power Product / Breakdown Power \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$P_{BD}$$\end{document}PBD ratio), aspect ratio of FeCl3 doped MLGNR interconnect is optimized at 0.987, 0.61 and 0.579 for local, intermediate and global level, respectively at 7 nm node. 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Increase in contact resistance leads to significant decrease in performance and increase in optimized aspect ratio of local Fecl3 doped MLGNR interconnect. Scaling down from 10 to 7 nm node results in increase of optimized aspect ratio in all levels of interconnects. Even though the performance of MLGNR degrades with scaling down but when compared to copper, the performance improves with technology scaling. Finally, this study provides circuit designers a detailed guideline for selecting an optimized aspect ratio for achieving better performance, power efficiency and reliability in doped MLGNR interconnects.

www.nature.com/scientificreports/ bottom-up synthesis. These fabricational advancements have strengthened the claim of graphene nanoribbons as an effective alternative to commercial metals. Wang et al. in 13 also advertised graphene nanoribbon as a promising candidate for quantum electronic applications praising its high mobility and current-carrying capability. Nishad et al. in 14 optimized thickness of Lithium and AsF 5 intercalated Top-Contact MLGNR (TC-MLGNR) interconnects and compared with copper and pristine interconnects. Both Jiang et al. and Nishad et al. have not shown any dimensional optimization for improved performance and reliability issues which is a concerning factor to consider for commercialization of MLGNR interconnects in near-future VLSI circuits.
In this work, a numerical model is developed for optimization of aspect ratio (AR) by minimizing delay and FOM ( NPDP/P BD ratio) for local, intermediate and global level MLGNR interconnects considering different intercalation dopants. This model is supported by the simulation results provided in "Results and discussion" section . Delay, Delay/P BD , NDP and PDP, PDP/P BD and NPDP/P BD are compared by considering AR prescribed by IRDS 2018 roadmap 1 and the optimized AR obtained from this work (for FeCl 3 doped MLGNR). Impact of scaling on optimization of AR is studied for two representative nodes, 7 nm and 10 nm. Effect of contact resistance on numerically optimized AR is shown which acts as an important factor in sub-10 nm technology nodes. Our study is in accordance with the trends observed in IRDS roadmap. A realistic model including the effects of crosstalk and vias is adopted which is not considered in 10 . Modeling of coupled three conductor line system shown in Fig. 3 is performed in Verilog-A. This proposed numerical methodology is applicable to all types of nano-interconnects making it a generalized model. We have validated this model with experimental data from 10 and simulation data from 14 .
The remainder of the paper is organized as follows: "Circuit modeling of MLGNR interconnects-an overview" section presents the equivalent electrical model of MLGNR interconnects. "Formulation and methodology" section proposes an numerical model for optimizing AR. "Results and discussion" section presents the simulation results. Finally, "Conclusion" section concludes this paper.

Circuit modeling of MLGNR interconnects-an overview
The structural representation of MLGNR interconnect is shown in Fig. 1. Here, thickness and width are denoted by t and w, respectively and ht denotes the height of interconnect above ground plane. The spacing between two layers of MLGNR is represented by δ . The advantage of doping is that each layer of intercalated MLGNR can be understood as stacked single layer GNRs because these layers do not have any interaction with each other. So, every layer of MLGNR can be modeled as Equivalent Single Conductor (ESC) model as shown in Fig. 2.
Electrical equivalent model of MLGNR interconnect system is shown in Fig. 2 where driver and load (at active device level) are connected to the interconnect metal line though vias. Copper is chosen as the via material in this study whose dimensions are taken from IRDS 2018 roadmap.
The electrical circuit of MLGNR interconnect model consists of lumped resistance, ( R fix = R con +R qtm 2 ) where R con is the imperfect contact resistance (MLGNR to Nickel contact) which is taken as 6 � − µm/W 10 . R qtm is the quantum resistance which is given by 8 ,   where G pul represents the p.u.l conductance of a single layer MLGNR as expressed below 6 , where q is the elementary charge, h is Planck's constant, ν f = 10 6 m/s is the Fermi velocity, E f is the Fermi level, K b is Boltzmann's constant, T is the temperature (here room temperature is considered), w is the width of the MLGNR and f (i, D , w) as expressed in 10 , is a function of specularity parameter where, i = (1 − P) , D is the mean free path determined by the Matthiessen's equation 10 . P represents specularity index which is a measure of specularity of GNR edges. P = 1 means completely specular edges whereas P = 0 implies completely diffusive edges 6 . The p.u.l capacitance ( C pul ) is a series combination of quantum capacitance ( C qtm ) and electrostatic capacitance ( C elc ) as described below 15 .
where C gnd , C inter and C intra are wire to ground capacitance, inter-layer capacitance and intralayer capacitance, respectively explained in detail in 15 .
The coupling capacitance between two interconnect lines as shown in Fig. 3 is given by 8 , where ǫ 0 and ǫ r are the dielectric constant and permittivity in the free space, respectively. t denotes the thickness of MLGNR, ht represents the height of the interconnect above ground plane and S is spacing between adjacent interconnects. The p.u.l inductance ( L pul ) of MLGNR is sum of p.u.l kinetic inductance ( L kn ) and self inductance ( L sf ) and is expressed as L pul ≈ (L kn + L sf ) 15 . L sf and the electrostatic capacitance ( C elc ) of MLGNR are considered same as of copper interconnect having equal dimensions 10 . Here, we have ignored the mutual inductance because the analysis is done for low/mid frequency range where effect of mutual inductance is negligible.
The capacitance model used for copper and cobalt interconnect is taken from 8 and the resistance model is described in 4 where 30% of the copper line is occupied by liner and cobalt has no liner.  interconnects are considered as 500 nm, 10 µm and 1 mm, respectively. Nearly specular (i.e. P = 0.8 ) MLGNR interconnects is considered for all the calculations. Properties of various intercalated MLGNR interconnect materials are described in Table 1.
Aspect ratio is varied from 0.4 to 3.2 for intermediate level and 0.4 to 3.4 for global level interconnects. Width of local level (Metal line 1) interconnects is considered as w min as specified in IRDS roadmap 1 for 7 and 10 nm nodes. It is taken as 1.5 times the w min for intermediate level and 5 times the w min for global level as specified in IRDS roadmap 1 to reduce delay and power consumption 24 . Optimizing AR by minimizing various metrics. Figure 4 shows the optimized AR in AsF 5 , Fecl 3 , Lithium doped MLGNR, neutral MLGNR, cobalt and copper interconnects. Intercalated MLGNR experiences very low delay when compared to neutral MLGNR and conventional metals irrespective of interconnect length for smaller AR. The AR for intermediate level, Optimized AR at minimum NDP is shown in Fig. 5. AsF 5 , Fecl 3 , Lithium doped MLGNR is optimized at 0.8, 0.8 and 0.6 for intermediate level and at 0.6 for global level interconnects. Intermediate level copper and cobalt cut all the doped MLGNR at AR value of 1.6 and 2, respectively. Hence, intermediate level MLGNR interconnects are more prone to noise for higher AR as compared with copper and cobalt as shown in Fig. 5a. But Fig. 5b shows that this is not the case for global level interconnect. Figure 6a,b show power consumption in cobalt, copper, neutral and various doped MLGNR interconnects for intermediate and global level respectively. Switching power is the dominating factor in total power consumed by the repeaters. P switch increases as AR increases. So, interconnects with large AR consume more power as   Figure 7b shows optimization at 0.8 which is also the crossing point after which copper exceeds all the doped MLGNR interconnect. Figure 8 shows the variation of breakdown power with respect to aspect ratio of interconnects. Here, we can see an increasing curve of breakdown power as the aspect ratio increases. Thermal healing length increases with an increase in AR. Breakdown power is a function of thermal healing length for intermediate level when L TH is comparable to GNR length. doped MLGNRs appear to be the most reliable candidates among all as they have large breakdown power.
The optimization of AR by minimizing the metric (Power * Delay/P BD ) is shown in Fig. 9. The optimization for AsF 5 , Fecl 3 , Lithium doped MLGNR interconnects are obtained at 1.0, 1.0 and 0.6. In case of cobalt and copper interconnects, the metric decreases with increasing AR.
We have considered the metric (Noise * Delay * Power/P BD ) as the Figure of Merit (FOM) which gives a measure of performance, noise, power consumption and reliability effects. The optimization of AR by minimizing this FOM for intermediate level interconnects is shown in Fig. 10. Here, we get an optimization AsF 5 , Fecl 3 , Lithium doped MLGNR interconnects at 0.6, 0.6 and 0.5, respectively. The FOM first decreases and then saturates with increasing AR for neutral MLGNR. It keeps decreasing with increasing AR for cobalt and copper interconnects. Doped MLGNRs are far better candidates considering an overall performance and reliability aspect specially at lower AR. Although Lithium dopant gives the highest advantage but Fecl 3 is explored more in      10 . The comparison is shown in Table 2. All the dimensional parameters are taken from 10 for comparison. Also to validate our numerical model, comparison is shown in Table 3. Optimized AR estimated using our numerical model is compared with that obtained from existing works 10,14 . All the dimensional parameters are considered to be same as in respective papers for a fair comparison. Table 4 shows a comparison between our results (considering FeCl 3 doped MLGNR interconnect) and results obtained using IRDS roadmap guidelines. Here, minimum delay, NDP and PDP are calculated and compared considering the optimized AR obtained in our work and the AR prescribed by IRDS 2018 roadmap 1 . It can be observed that there is an insignificant advantage in intermediate level interconnect performance but when it comes to global level, a substantial improvement is registered. This improvement in global metal line becomes more valuable as the effect of via is more dominant in global metal path. Our results show improved performance in Fecl 3 doped MLGNR interconnect for optimized value of AR as compared to IRDS roadmap 2015 prescribed AR. Also the improvement in FOM metric is significant indicating lower AR should be adopted considering overall performance and reliability. Impact of contact resistance. Table 5 gives us an understanding of effect of contact resistance on numerically optimized AR and FOM. Here we have varied the contact resistance from 5 K to 20 K 25 . As we can see, intermediate and global lines are not affected by it. But when it comes to local lines, increase in contact resistance leads to significant decrease in performance and increase in optimized AR. With a 75% increase in contact resistance, ∼66%, ∼5.5% and ∼0.4% degradation in FOM of local, intermediate and global level Fecl 3 doped MLGNR Table 3. Validation of our numerical model (section "Formulation and methodology") with 10 and 14 .  www.nature.com/scientificreports/ interconnect, respectively is witnessed. Optimized AR also experiences an increase of ∼26%, ∼2.5% and ∼1.8% in local, intermediate and global level, respectively. This infers that when the contact resistance increases, then in order to compensate the decrease in performance, AR can be increased (which will lead to increase in number of layers and thus decrease in contact resistance). So the performance of MLGNR interconnects will not improve beyond a certain limit.

Optimization parameters Optimized AR (Existing Works) Optimized AR (Numerical Model)
Impact of scaling. The Impact of scaling (from 10 to 7 nm node) on optimized AR value of 10 µ m long Fecl 3 doped MLGNR interconnect is shown in Fig. 12

Formulation and methodology
This proposed methodology provides a detailed numerical dimensional optimization procedure and is applicable to any generic nano-interconnect system. This numerical methodology can avoid costly simulators set-ups and expensive fabrication procedures for providing the dimensional design guidelines to achieve such improvement in performance. Aspect ratio (AR) optimization serves to be an effective dimensional optimization technique for achieving superior performance and reliability metrics. Here, dependence of the RLC parameters on AR is discussed in order to establish relation between AR of interconnect and its delay, noise induced effects, power consumption and breakdown power.
And N ch is a function of width of the interconnect. Therefore R qtm ∝ (1/AR) as N ch is constant and N oL is a function of AR.
So, R pul ∝ (1/AR) as shown in Fig. 14 because G pul is a constant here as it is a function of width and N oL is directly proportional to AR.
Similarly, C pul is an increasing function of AR. C gnd , C intra and C inter collectively adds to C elc , where C inter can be neglected 15 . They can be described as functions of AR as mentioned in Eqs. (7) and (8), The dependence of C cup on AR is described as, Noise and power consumption are a function of capacitance and hence they increase with increasing AR. However, delay is the dominating factor in NDP and PDP metrics. So, the expression of NDP and PDP leads to an optimized AR value.

Delay centric design. Propagation delay in an interconnect is basically a function of its RC product. With
increasing AR, resistance decreases as shown in Fig. 14. But the capacitance increases, therefore an optimized value of AR is obtained for minimum delay point. The transfer function for the crosstalk delay or noise evaluation in the victim net will be denoted as H(s). The second-order Pade ′ s expansion of the transfer function is given by 16 : The two poles of the transfer function are:  16 . They can be defined as a function of AR as follows, The coefficients a 1 , a 2 , c 1 , c 2 , c 3 are described as: The step response, which is the inverse Laplace transform of 1 H(s) , is given by: The 50% propagation delay ( τ ) is given by 16 , Delay in terms of AR can be defined as: Setting the derivative of delay with respect to AR to zero, we can obtain the optimized AR at which delay is minimum: where u, v, ∂u ∂AR and ∂v ∂AR are described as: FOM centric design. Similar approach is adopted to obtain the optimized AR at minimum FOM. We define where τ is given in equation (4), peak noise voltage ( N peak ) is given by 17,18 , The total power consumed in the interconnect is mainly because of the power consumed by driver and load buffers which is given by 19 , where P switch , P short and P leak are switching, short circuit and leakage power of a repeater, respectively. The definition of various parameters are specified in detail in 20 . Switching power dominates the equation thus is considered for further calculation for simplicity 19 .
where V DD is power supply voltage, f clk is the clock frequency, L rep is the inter repeater stage length, S r is the ratio of buffer size to minimum sized buffer and S f is the switching factor, which is the fraction of repeaters on a chip that are switched during an average clock cycle. It can be taken as 0.15 20 . P switch as a function of AR can be defined as: (25)  − (c 1 + c 2 .AR + c 3 .AR 2 ) q 2 + 2q 3 .AR + 3q 4 .AR 2 + 4q 5 .AR 3 2 q 1 + q 2 .AR + q 3 .AR 2 + q 4 .AR 3 + q 5 .AR 4

Conclusion
This work focuses on numerically determining optimum aspect ratio in order to improve performance, reliability and minimize noise effects and power consumption. This approach will be extremely useful to industry for selection of AR of interconnects as it is a non-SPICE method. Our approach provides a detailed guideline for the Aspect ratio optimization and reduces fabricational cost to achieve high performance and reliability MLGNR interconnects by reducing iterations during fabrication process for achieving desired performance. The optimized AR of AsF 5 , FeCl 3 , Lithium doped MLGNR interconnects by minimizing delay is obtained at 1.