RF and Crosstalk Analysis of Copper and MLGNR Interconnects Using Different Repeaters in Sub-10 nm Regime

This paper presents RF and crosstalk analysis of Copper (Cu) and multi-layer Graphene nanoribbon (MLGNR) based interconnects using multi-gate (FinFET) and virtual-source carbon nanotube field effect transistor (CNFET) based repeater insertions in sub-10 nm regime. The SPICE based analysis utilizes an accurate π-type equivalent single conductor (ESC) model for mutually coupled interconnects at 7 nm technology node. The transfer function and 3-dB bandwidth results of lithium-doped MLGNRs offer many fold improved RF performance than Cu. The out-of-phase crosstalk induced (OPXT) delay results with FinFET repeaters demonstrate 27.54 and 67.6 % reductions for pristine and lithium-doped MLGNRs as compared to Cu, whereas CNFET repeaters demonstrate 20.48 and 81.88 % reductions at interconnect length of 1000 µm. The peak far-end crosstalk (FEXT) noise voltage results demonstrate 86.03 and 62.5 % using FinFET repeaters and 88.14 and 69.9 % reductions using CNFET repeaters for pristine and Li-doped MLGNRs than Cu at 1000 µm length. Further, the energy-delay-product (EDP) results demonstrate 59.7 and 97 % reductions using FinFET repeaters for pristine and Li-doped MLGNRs than Cu at length of 1000 µm. The EDP results using CNFET repeaters exhibit 34 % degradations for pristine-MLGNRs than Cu while Li-MLGNR exhibit 98.61% reductions than Cu at length of 1000 µm. (FEXT). In-Phase Crosstalk (IPXT). M ultilayer graphene nanoribbon (MLGNR). Near-End Crosstalk (NEXT). Out-of-Phase Crosstalk (OPXT)


Introduction
The continual feature size shrinking in modern integrated circuits (ICs) has paved the way for performance enhancements. However, at nanoscale technology nodes there starts a greater disparity in scaling factors of transistors and interconnects. This resulted in making interconnects a limiting factors in overall system performance. Moreover, the performance of CMOS transistors have started degrading significantly at nanoscale nodes. So, research community is coming up with new interconnect materials and alternatives of CMOS transistors for future ICs. The international technology roadmaps for semiconductor (ITRS) has forecasts that existing widely used interconnect material Cu is going to cross its critical current density value very soon [1]. Emerging interconnect materials having novel properties to overcome limitations of Cu at nanoscale. Carbon based GNR nanomaterials are becoming top choice of researchers to explore them for interconnect applications due to their scalability to nanoscale and novel properties such as two-dimensional current flow, ballistic transport, very high current handling capacity, larger mean free path, greater thermal stability, and improved mechanical strength [1][2][3]. Further, greater momentum relaxation time and the planar structure facilitates smooth control of their chiral vectors during fabrication process. This results in easy fabrication either as semiconducting or metallic as per requirement. Hence, GNRs are evolving as potential replacement of Cu for chip interconnect applications. Basically, a GNR comprise of a strip of Graphene sheet with width generally less than 50 nm. They are further categorized into armchair (ac) and zig-zag (zz) depending on profile of their edges. Normally, many Graphene layers are stacked to enhance conductivity and it is known as multi-layer GNRs or MLGNRs.
The signal integrity degradation due to higher packing density, dielectric imperfections, EMI/EMC issues, and skin effect etc. worsen at higher frequencies and could not be omitted in modern integrated circuits (ICs) [1][2][3]. So, the necessity of radio frequency and crosstalk analysis in performance characterization of chip interconnects has arisen in modern ICs. With increasing switching rate of signalling, inductance has started playing dominant role in nanoscale ICs. The global level interconnects are generally wider wires having higher conductivity as Manjit Kaur 1 , Sanjeev Kumar 2 , Balwinder Raj 3 , Neena Gupta 1 , and Arun K. Singh 1* compared to local/intermediate level. So, they demonstrate larger inductive related activities. This necessitates inclusion of inductance as important parameter in modelling of emerging as well Cu interconnects at nanoscale technology nodes. It has been demonstrated that exclusion of inductance in interconnect models may result in errors up to 35% for propagation delay calculations [4].
The scaling down of feature size below 22 nm raised many issues at process and device level which has led to search for new device structures [1,3]. Multi-gate transistors generally called Fin-FETs due to their Fin type shape are just structural improvement of traditional CMOS. They demonstrate better gate control over channel current with minimal presence of short channel effects (SCEs) as reported in [5][6][7]. CNFET has attracted attention of many researchers due to use of CNTs as channel material and their novel properties [8]. A CNFET compact model supporting multiple chiralties, diameters, and semiconducting/metallic CNT based channels has been introduced by Deng et al. [9]. This model includes optical/ acoustical phonon scattering, charge screening, and quantum confinement etc. Deng and Wong then demonstrated standard HSPICE CNFET model considering experimental device non-linearities [10]. Lee et al. introduced a virtual-source based semi-empirical compact CNFET model which appropriately considers effects of geometrical scaling, tunnelling currents, and parasitic resistance/capacitance etc. [11][12]. The virtual-source CNFET model (VSCNFET) extends CNFET technology to sub-10-nm technology nodes [13].
The RF analysis of MLGNRs is discussed by only few researchers [14][15][16][17]. The skin-depth and surface impedance based high frequency performance of pristine and intercalated MLGNRs is studied in [14][15]. However, the validation of RF behaviour at circuit level is missing. The Bode plots based relative stability of Cu, side-contact (SC) and top-contact (TC) MLGNRs is investigated in [16] for only local/intermediate level pristine MLGNR omitting the global level interconnects. Further, A research article compared MLGNRs and MWCNTs using driver-interconnect-load (DIL) system in terms absolute transfer function and bandwidth [17]. To improve the performance of nowadays ICs, repeaters are invariably inserted into longer resistive interconnect lines. Ismail and Friedman have derived analytical expressions for calculating the optimum number and size of repeaters [4]. M. Kaur et. al. presented crosstalk analysis using different types of repeaters for coupled MLGNR interconnects at 21 nm technology node [18]. They demonstrated that CNFET repeaters have better OPXT and IPXT delay reduction than FinFET counterparts. Different repeater insertion techniques for bundled SWCNT/MWCNT interconnects have been investigated in [19][20]. The impact of contact resistances on performance of interconnects among the CNTs and metallic electrodes for repeater insertions have been analysed in [21]. They authors of [21] and [22] introduced repeater insertion techniques for optimizing power for RC interconnect models. Zhao et al. analysed optimum repeater insertions for CNT/Cu based nanointerconnects to minimize power consumption and delay considering impact of contact resistance. They have used particle swarm optimization (PSO) method to estimate optimum number of repeaters and their size [23].
The rest of this paper is organized as follows. Section 2 introduces RLC models of Cu/MLGNR interconnects. The repeater insertions are enumerated in Section 3 along with calculation of optimum number and sizing of repeaters. Section 4 discusses about RF analysis of Cu and MLGNR interconnects. Section 5 illustrates crosstalk induced delay and noise results. The comparative results for Cu/MLGNR interconnects using FinFET/CNFET repeaters are covered in Section 6. The conclusions are presented in Section 7.

Interconnect Model
The Cu resistivity at nanoscale started increasing sharply due to electron scattering caused by surface and grain boundary effects [1]. The per unit length (p.u.l.) resistance of Cu [20][21] is estimated using (1) as where, W and H are width and thickness of Cu interconnects. The resistivity (ρ) at nanoscale is a function of surface and grain boundary scattering and is calculated using (2) as gb ss where, where,  is λcu R / g(1 -R). g and R represent mean grain size (≈360nm) and reflection coefficient (≈0.4) at grain boundaries [22]. The p.u.l. inductance (Lcu) and capacitance (Ccu) of copper interconnects are calculated using analytical models [20] (4) and (5) as . .
where, ε0 and εr are free space and relative permittivities. The free space permeability is denoted by μ0. The parameters L, T, and S represent interconnect length, height above ground plane, and spacing between adjacent interconnect lines.
In this paper, a simplified π-type ESC model [38] depicted in Fig. 1

A. Resistance Model
where, Rj,q and Rj,mc are j th layer quantum resistance and imperfect metal contact resistance, respectively. The growth process of GNRs decides the value of Rj,mc resistance and its value may range from few kilo-ohms to hundreds of kilo-ohms [24][25].

Fig. 1 Equivalent π-type ESC model used in driver-interconnect load (DIL) configuration [15]
The single layer quantum resistance (Rq) for each conducting channel [24][25] is estimated as in (7) j ch j ch where, h, e, and Nch denote Planck's constant, electronic charge, and conducting channel count, respectively.
The conducting channel count for each layer [26][27] is estimated as in (8) where, EF denote Fermi level, Ei is the sub-band energy below/ above the Fermi level, k denote Boltzmann's constant, and T temperature, respectively. The terms nv and nc represent valence and conduction sub-bands, respectively.
The p.u.l magnetic inductance of MLGNR [28][29] is estimated using (12) as where, μr and μ0 are relative and free space permeabilities, respectively. Due to much smaller value of magnetic inductance, it is ignored for MLGNR nanointerconnects in this work.

C. Capacitance Model
The equivalent quantum (Cq) and electrostatic (Ce) capacitances are present in MLGNRs. The Ce is usually due to electric field coupling between bottommost layer and the ground plane. The Cq and Ce capacitances for MLGNR interconnects [28][29] is estimated using (13) and (14) as where, ε0 and εr are permittivities of free space and material, respectively.

Repeater Insertions
To improve the performance of nowadays ICs, repeaters are invariably inserted into longer resistive interconnect lines. The RC time constant of an interconnect line is RTCT = RCl 2 with square dependence on length. So, dividing the interconnect line into smaller segments and inserting repeaters to reduce delay is most commonly used strategies in modern ICs [4]. At global level large number of repeaters are required and they consume significant proportion of area and total circuit power.

Fig. 2 Repeater insertion to improve delay performance of interconnects
To mitigate larger propagation delay and to improve signal levels on global lines, sufficient numbers of repeaters of uniform size are generally inserted. To insert repeaters an RLC interconnect line is divided into n number of uniform segments as shown in Fig. 2. All repeaters are of uniform size and h times larger than minimum sized buffer in that particular technology node. Accordingly, the repeater output resistance (Rdr), capacitance (Cdr), and load capacitance (Cl) are scaled to Rdr /h, hCdr , and hCl, respectively. The total propagation delay of a global interconnect line with inserted repeaters is summation of delay of n line segments and delay of of inserted repeaters (including driver).
A very important role is played by sizing of interconnect sections in estimating optimal number of repeaters required and the repeater sizing [4,18]. The analytical expressions (20)(21)(22) have been derived in [4] for determining optimal number of repeaters (nopt) insertions required and optimal repeater sizing (Sopt) for chip interconnects.
In this paper, we have calculated the optimal number of required FinFET and CNFET inverting repeaters and their sizing using expressions (16)(17) for uniform segmented nanoscale Cu, pristine and Li-doped MLGNRs interconnects. The Table 2 lists the optimal number of CNFET and FinFET repeaters required for Cu, pristine and Li doped MLGNRs interconnects at different lengths. The number of repeaters required for pristine and Li-doped interconnects in Table 2 clearly indicate that emerging nanomaterials need far less repeaters than conventional Cu material. As evident from Table 2, the pristine and lithium-doped MLGNRs need 56 and 94 % less repeaters in comparison to Cu at length of 1000 µm.

RF Analysis
The RF behaviour of Cu, pristine and Li-doped MLGNRs have been calculated and characterized in this section considering skin-depth, transfer gain, and 3-dB bandwidth RF centric parameters.

Skin-Depth Analysis
The current conduction in a conductor restricts to its outer periphery at radio frequencies mainly attributed to skin-effect. The skin-depth for GNRs [14][15] is calculated using (19)(20) as where, δ' is skin depth at lower frequencies, ω, τ, and σ are angular frequency, momentum relaxation time, and conductivity, respectively. The µ0 is free space permeability.
The skin-depth variations with frequency sweeping from 1 GHz to 1 THz are plotted for Cu, pristine and Lithium doped MLGNR materials in Fig. 3. The plots reveal skin-depth degradations for all materials. This happens due to reduced effective conductor area for current flow at RF frequencies due to skin-effect. For all calculations in this work, we have taken material dependent parameters such as τ, σ, and λeff from experimental results reported in [1, 14, 32-34].

Fig. 3 Skin-depth degradations for Cu and GNR materials
The skin-depth degradations observed for Cu, pristine and Lithium-doped MLGNRs are 9.56, 2.29, and 2.72 μm, respectively as frequency is varied from 1 GHz to 1 THz. As can be seen from Fig. 3 the pristine MLGNR undergoes highest skin-depth degradation as its conductivity is about 17 and 12 times lesser than Cu and Lithium doped MLGNR materials [14,34]. It is observed that skin-depth performance is dependent on effective mean free path, conductivity and momentum relaxation time parameters of a material. So, interconnect material choice for optimal RF performance can be made based on these parameters.

Transfer Gain Analysis
The transfer gain is one of very important parameter used to describe RF performance. The transfer gain of Cu, pristine and Lithium-doped MLGNRs at 7 nm technology node is calculated and analysed. The π-type ESC model of Fig. 1 is used in all simulations. The minimum-size inverter parameters for FinFET and CNFET transistors are taken from [1, 37]. We have considered driver and repeater parameters 50 times the minimum-size inverter size. The transfer gain graphs as frequency is varied from 1 GHz to 1 THz for different interconnect materials considered for evaluation are illustrated in Figs. 4a-d. As evident from graphs in Fig. 4a for Cu, CNFET repeater demonstrate 5 folds better frequency response than FinFET counterparts at global length of 1000 µm. For pristine MLGNRs shown in Fig. 4b, CNFET repeaters exhibit 3 folds better response. The frequency response of Lithium doped MLGNR calculated at transfer gain value of 0.5 is 110 and 115 folds better than Cu and pristine-MLGNRs at length of 1000 µm using CNFET repeaters as is evident from Fig. 4d. Similarly, the frequency response of Lithium doped MLGNR is 223 and 140 folds better than Cu and pristine-MLGNRs at length of 1000 µm with FinFET repeaters. The improved frequency response of Lithium doped MLGNR is mainly due to its much lower distributed resistance (≈ 5.9 kΩ) than pristine MLGNR (≈ 410 kΩ), and Cu (≈ 270 kΩ) at 1000 µm length.

3-dB Bandwidth Gain Analysis
The 3-dB bandwidth (f3dB) values for Cu, pristine and Lithium-doped MLGNR interconnect materials are extracted using FinFET and CNFET repeaters at 7 nm technology node and analysed with length varying from 100 to 1000 µm. The 3-dB bandwidth graphs are illustrated in Fig 5.

Crosstalk Induced Delay and Noise Results
Crosstalk is redundant electromagnetic coupling/interference between the signal carrying nearby interconnect lines in ICs. The major source of crosstalk is unwanted capacitive, inductive, and conductive couplings. With ever-increasing signal switching rates, crosstalk is becoming a major design issue to be handled carefully in modern ICs. In this paper, a standard three-line bus model is utilized to predict the crosstalk behaviour of Cu and MLGNR based interconnects in sub-10 nm regime. The crosstalk induced delay is caused predominantly by capacitive coupling in global level chip interconnects. The capacitive coupling of Cu [24,35] and MLGNR [36] are estimated using analytical expressions (21)(22). In case of Cu, the coupling capacitance (CCCU) is approximated using semi-empirical approach in which the CCCU contemplates variations in electric flux with reference to geometrical parameters illustrated in Fig. 6.
Where, H, T, and W describe geometrical parameters of the interconnect structure shown in Fig. 6. The parameter S specify the adjacent interconnect line spacing. The conformal mapping method introduced in [36] has been used to estimate the coupling capacitance (CGNR) of GNR interconnects. The whole geometry is segregated into elementary vertical and horizontal interconnect capacitances. Then, these capacitances are added to estimate the overall coupling capacitance (CGNR) of MLGNR interconnects as  The standard three-line bus model adopted to estimate crosstalk induced delay is enumerated in Fig. 7. The crosstalk performance is characterized in terms of out-of-phase/in-phase crosstalk induced delay and nearend/far-end crosstalk noise in this paper. The multi-gate (FinFET) transistor model parameters are adopted from predictive technology model (PTM) [24] while the Stanford virtual-source CNFET model is utilized for CNFET based transistors [11][12][13].  Table 3 lists device parameters of FinFET and virtual-gate CNFET based drivers and repeaters used in this work. The crosstalk induced delay is primarily of two types-out-of-phase and in-phase depending upon phase of victimline signal with respect to aggressor-lines. A step input of 0.7V is applied to aggressors as well as victim-line at 1GHz frequency. A load capacitance of 3fF is utilized for all simulations. The OPXT delay is worst-type as both aggressor-lines influence the victim-line signal in opposite phase direction. This causes severe deterioration of signal quality on victim-line resulting in increased delay. The Fig. 8a illustrates victim-line out-of-phase waveforms achieved at near/far end of the line driven by FinFET driver and repeaters for Cu, pristine-MLGNR, and Li-MLGNR interconnect materials at 1000 µm length. As the waveforms reveal near-end nodes experience minimum crosstalk-induced delay while the far-end node undergoes maximum delay. The Fig. 8b describes victim-line out-of-phase waveforms achieved at near/far end of the line driven by CNFET driver and repeaters.
In case of IPXT delay, since victim-line signal is in phase with aggressor-line signals, it almost does not experience any signal deterioration. The results in Fig. 10a-b illustrate peak near-end and far-end crosstalk noise voltages calculated using three-line bus model with interconnect length varying from 100 to 1000 µm for Cu, pristine-MLGNR, and Li-MLGNR materials driven by FinFET and CNFET based drivers and repeaters, respectively. As can be seen from Fig. 10a

Transient Analysis
The transient analysis of equivalent π-type ESC model of bundled MLGNR has been presented in this work and compared against the conventional Cu nanointerconnects. The SPICE simulations were performed to calculate the delay, power, and energy-delay-product results. The Fig. 11a illustrates comparative delay results of Cu, pristine-MLGNR, and Li-doped MLGNR for interconnect length varying from 100 to 1000 µm. The delay results of Cu using FinFET repeaters demonstrate reductions of 13 to 23.6 % as compared to CNFET repeaters at lengths of 100 and 1000 µm. In case of pristine-MLGNR, FinFET repeaters exhibit 26.1 and 46.3% reductions as compared to CNFET repeaters. For Li-MLGNR using CNFET repeaters, reductions of 34.7 and 29.8 % are observed in delay over FinFET repeaters.  The Fig. 11b illustrates comparative average power results of Cu, pristine-MLGNR, and Li-doped MLGNR for interconnect length varying from 100 to 1000 µm. The average power results of Cu using CNFET repeaters demonstrate reductions of 56.4 and 57.2 % for lengths of 100 and 1000 µm as compared to FinFET repeaters. In case of pristine MLGNR, CNFET repeaters exhibit 24.1 and 22.6 % reductions as compared to FinFET repeaters. For Li-MLGNR using CNFET repeaters, average power reductions of 92.8 and 88.7 % are observed over FinFET repeaters. The Fig. 12 illustrates comparative energy-delay-product results of Cu, pristine-MLGNR, and Li-doped MLGNR for length varying from 100 to 1000 µm. The EDP results of Cu using CNFET repeaters demonstrate reductions of 42.5 and 26.7 % as compared to FinFET repeaters at lengths of 100 and 1000 µm, respectively. In case of pristine MLGNR, FINFET repeaters exhibit 28.3 and 62.7 % EDP reductions as compared to CNFET repeaters. For Li-MLGNR using CNFET repeaters, EDP reductions of 97 and 94.5 % are observed over FinFET repeaters.

Conclusion
This paper presents RF and crosstalk analysis of Cu, pristine-MLGNR, and Lithium-doped MLGNR nanointerconnects at 7 nm technology node. The results reveal that pristine and lithium-doped MLGNRs need 56 and 94 % less repeaters as compared to Cu at length of 1000 µm. The Lithium-doped MLGNRs demonstrates 219 and 110 folds enhancement in f3dB bandwidth with FinFET and CNFET repeaters in comparison to Cu at 1000 μm. The OPXT induced delay results for Cu and Pristine MLGNR demonstrate 9.8 and 18.5% reductions using FinFET repeaters for global level length of 1000 µm as compared to CNFET repeaters. However, for Li-MLGNR, CNFET repeaters exhibit 37.6 % reduction in OPXT delay than FinFET repeaters. The peak NEXT noise voltage calculations for Cu and pristine-MLGNR reveal 22.6 and 40.8% reductions using CNFET repeaters at length of 1000 µm while for Li-MLGNR FinFET repeaters exhibit 24.4 reduction in comparison to CNFET repeaters. Similarly, peak FEXT noise calculations for Cu and Li-MLGNR demonstrate 13.75 and 41.22 % reductions using FinFET repeaters while for pristine-MLGNRs CNFET repeaters exhibit 14.9 % reductions. CNFET repeaters demonstrate 26.7 and 94.5 % EDP reductions for Cu and Li-MLGNR materials whereas in case of pristine-MLGNR FinFET repeaters provide 62.8 % EDP reductions.