Impact of the variability of the process parameters on CNT-based nanointerconnects performances: A comparison between SWCNTs bundles and MWCNT

924 IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 11, NO. 5, SEPTEMBER 2012

Impact of the Variability of the Process Parameterson CNT-Based Nanointerconnects

Performances: A Comparison BetweenSWCNTs Bundles and MWCNTPatrizia Lamberti, Member, IEEE, and Vincenzo Tucci, Member, IEEE

Abstract—A reliable estimation of the performances of two pos-sible realizations of a CNT-based nano-interconnect, namely oneobtained by using a bundle of SWCNT and another one employingan MWCNT, taking into account the variations of some physicaland geometrical characteristics is carried out. The ranges of theper unit length parameters of a transmission line modeling the in-terconnect and those of the propagation time delay are analyzedfor three different technologies (15, 21, and 32 nm) by means ofinterval analysis. This approach provides, at once, a worst caseanalysis and a sound assessment of the robustness and accuracyof the considered performance as a function of the interconnectlength. A comparison of the two technological alternatives is car-ried out in a more consistent way than that usually achieved byconsidering the nominal behavior.

Index Terms—Carbon nanotubes (CNT), interconnects,reliability.

I. INTRODUCTION

CARBON nanotubes (CNTs) have been proposed as a pos-sible alternative to copper interconnects for future very

large scale integration (VLSI) systems [1], [2]. The CNTs,characterized by electron mean-free path of the order of themicrometers, high current carrying capability, and remark-able thermal and mechanical stability, may be single-walled(SWCNTs, i.e., only one shell) or multiwalled (MWCNTs, i.e.,nested tubes). The most promising interconnect solutions allow-ing the reduction of the considerable contact resistance of a CNTare based on bundles of SWCNTs and on MWCNTs due to thepossibility to contact also their interior conducting shells [3].A comprehensive state of the art concerning the different re-search efforts connected with the production of CNTs in futurenanointerconnects can be found in a very recent paper [4].

Manuscript received March 25, 2011; accepted June 27, 2012. Date of pub-lication July 5, 2012; date of current version September 1, 2012. This workwas supported by EC’s research project Carbon nAnotube Technology forHigh-speed nExt-geneRation nano-InterconNEcts (CATHERINE) under Grant216215. The review of this paper was arranged by Associate Editor L. Dong.

The authors are with the Department of Electronic and Computer Engineer-ing, University of Salerno, Via Ponte Don Melillo 1, I-84084 Fisciano (SA),Italy (e-mail: [email protected]; [email protected]).

Color versions of one or more of the figures in this paper are available onlineat http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TNANO.2012.2207124

The propagation performances of CNTs-based interconnectsare usually evaluated by means of Transmission Line (TL)models whose per unit length (p.u.l.) circuital parameters takeinto account the specific conduction, electrostatic, and magneticcharacteristics of CNTs structures [5]–[11]. The expressions ofthe p.u.l. parameters of the TL, indeed, depend on the physi-cal and geometric quantities of the CNT structures, which, dueto the very complex manufacturing processes, may be affectedby significant uncertainties and variations whose extent cannotbe easily quantified. These variations, in turn, may determine aremarkable alteration of the observed performance, such as the50% time delay of the whole interconnect.

Therefore, it seems evident that a knowledge not only of thenominal value of either the parameters and the observed perfor-mance, but also of their range is required for a more accuratecomparison among the possible CNT technological alternatives(bundles of SWCNT or MWCNT) and for an effective and re-liable design of the interconnect. Stochastic tools based on theMonte Carlo (MC) approach [12], [13] have been adopted inorder to study the impact of process variation on interconnectbased on SWCNT bundles. However, such a general and widelytrusted search scheme requires large numbers of simulation tri-als and leads to an underestimation of an a priori unknown levelof the observed performances.

The idea of using interval analysis (IA) for taking into ac-count statistical variations of the relevant parameters as a rangehas also been considered [14], [15]. In [14] a classical approachbased on intervals is employed to approximate each uncertaincircuit parameter as a range. Such a method is used to studythe time dynamics in a simple RC timing circuit. However, thisapproach could only be used for quite small number of cellssimulating the interconnect, in order to limit the growing prop-agation error associated with long calculation chains. In [15]a standard model order reduction technique is considered forstudying a traditional interconnect. Then, the impact of statis-tical manufacturing variations on the time delay is obtainedfrom a statistical sampling to compact interval-valued model.However, in [15] the focus is rather on the efficiency of thenumerical IA procedure than on the interconnect performance.Furthermore, the peculiarities of the conduction mechanisms ofCNT structures and associated circuital parameters are not takeninto consideration.

In this paper, the ranges of the 50% time delay, due to thevariability of some relevant process-dependent parameters, for

1536-125X/$31.00 © 2012 IEEE

LAMBERTI AND TUCCI: COMPARISON BETWEEN SWCNTs BUNDLES AND MWCNT 925

three CNTs-based interconnect technology nodes (15, 21, and32 nm) made with a bundle of SWCNT or with an MWCNT areevaluated. The two technological alternatives may be obtainedby a controlled chemical vapour deposition (CVD) growth ofCNTs inside the pores of suitable templates (see, for exam-ple, [16]–[18]). As an example, in [18] the synthesis of CNTstructures grown inside the pores of aluminum oxide templatesfor the fabrication of next-generation nanointerconnects hasbeen described.

The differences in the performances of the two technologicalalternatives, discussed in this paper, are analyzed as a functionof the variability of some geometrical and physical quantitieswhich can be assumed to affect the manufacturing process ofboth solutions. In particular, the variability of the wire diam-eter (MWCNT or SW bundle), the distance of the wire fromthe ground plane, the contact resistances, and the relative per-mittivity of the external medium are considered. The analysisis performed under less favorable conditions (as evidenced inprevious papers [4]) concerning the cross section of the line andso this is assumed equal to the gate length of local (Metal 1)interconnect with unitary aspect ratio [19].

In order to compare the reliability of the two technologicalalternatives, the characteristics of the interval arithmetic (IA)are exploited. In particular, the monotonic inclusion property ofIA ([20]–[22]) allows us to obtain, in an analytic way, the rangesassociated with the uncertainties assumed by the circuital pa-rameters representing the interconnect. Then, by consideringsuch interval values in the analytic expression of the time de-lay reported in [22], the ranges of the propagation delay areachieved. The comparison between the two technological so-lutions is, consequently, carried out by analyzing not only thenominal value of the time delay, but also its worst case figuresas a function of the interconnect length. In this way, the relativerobustness of the two technological alternatives, i.e., an indica-tion of which one is less sensitive to the parameter variations,is achieved by considering the range amplitude of the prop-agation delay with respect to its nominal value. Furthermore,valuable information concerning the accuracy associated withthe calculated performance is determined. The obtained resultsare validated by means of an MC analysis.

This paper is organized as follows. In Section II, the two in-terconnect structures and the associated TL circuital models arepresented. In Section III, the results, obtained by using IA, con-cerning the evaluation of the ranges of the circuital parametersand the time delay are described and in Section IV the mainconclusions are drawn.

II. NANOINTERCONNECT BASED ON CNTS AND ASSOCIATED

TL CIRCUITAL MODEL

The schematic set-up of the considered interconnect structureis shown in Fig. 1. In particular, it provides the transmission ofa step voltage from a CMOS (driver) modeled by means of aresistance Rd to a load described by the input capacitance CL

of the CMOS.Three line widths w = 15, 21, and 32 nm, according to

the technological developments described by the International

Fig. 1. Scheme of the CNT-based interconnect.

Fig. 2. Cross section of the two considered technologies: (a) bundle ofSWCNT; (b) MWCNT structure.

TABLE IRELEVANT GEOMETRIC AND PHYSICAL CHARACTERISTICS OF THE

CONSIDERED STRUCTURES

Technology Roadmap for Semiconductors (ITRS) [18], are con-sidered for the interconnect which may assume a variable lengthl ranging from those corresponding to the local (0.1–1 μm) tointermediate ones (few hundreds of micrometers).

The cross sections of the two structures whose performancesare the objective of the comparison performed in this study areshown in Fig. 2(a) and (b), respectively, whereas the relevantgeometric and physical parameters of the interconnect configu-rations based on them are reported in Table I.

Both the bundle and the MWCNT are located above a perfectconducting ground plane from which they are separated by a lin-ear dielectric material, having relative electrical permittivity εr .The interconnect may be effectively modeled by an equivalentlossy TL, as depicted in Fig. 3.

The electrical model adopted to study the two cases is theso-called equivalent single conductor (ESC) model ([9], [10])that relates, as described later, the physical/geometrical param-eters to the p.u.l. components of the elementary unit cell in theTL and the lumped resistive parameter Rm . Such a model hasbeen effectively employed for evaluating the electromagneticperformances of CNT-based interconnects [9], [10].

926 IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 11, NO. 5, SEPTEMBER 2012

Fig. 3. TL modeling the CNT-based interconnect.

A. TL Model of an SWCNT Bundle

The line is realized by means of a bundle of N equally spacedSWCNTs, fixed at the Van der Waals distance δ, having thesame radius rCNT and in the hypothesis of a perfect hexagonalpacked structure with a lattice constant Δ = 2rCNT + δ [9]. IfNs is the number of tubes constituting the external side of thehexagon [e.g., Ns = 2 in Fig. 2(a)], the bundle radius is givenby rb = rCNT + Δ(NS−1) and it results

N = 1 +NS∑

i=1

6(i − 1). (1)

The bundle is contained in a circular cross section with aradius w/2 and therefore rb ≤w/2. It is assumed that the numberof SWCNTs in the hexagonal structure is

Ns = int part(

w/2 − rCNT

Δ

)+ 1 (2)

where int_part represents the integer part of the number. There-fore, the total number of CNTs in the bundle is a function of theline width.

Since it is normally assumed that only 1/3 of the nanotubesare conductive, the N SWCNTs are supposed to have the samenumber of spinless conducting channels n [3]:

n =

⎧⎪⎨

⎪⎩

2aTrCNT + b rCNT >dT

2T23

rCNT <dT

2T

(3)

where a = 3.87 × 10−4 nm−1 ·K−1 , b = 0.2, dT = 1300 nm·K,and T is the absolute temperature.

The total number of conducting channels ntot is given by thesum of the conducting channels of each SWCNT in the bundle;therefore, ntot = n·N.

The lumped parameter Rm is given by the parallel of theresistances of the N SWCNTs and takes into account the contactresistance of the jth SWCNT, Rmj , and the intrinsic resistanceof each conducting channel in every SWCNT, Ri = h/2e2 ,where h = 6.6262 × 10−34 J·s is the Planck constant and e =

1.602 × 10−19 C is the electron charge. By assuming Rmj =Rmc ∀ j = 1, . . ., N, it results

Rm =

⎡

⎣N∑

j=1

(Ri

2n+ Rmj

)−1⎤

⎦−1

=1N

(Ri

2n+ Rmc

). (4)

The factor 2 multiplying n takes into account the presence ofthe two spin orientations per channel. The conduction mecha-nism taking place in the CNTs is assumed to be ballistic if itslength is much shorter than the effective mean free path λmfp ofthe SWCNT given by [3], [24]

λmfp =2 ∗ 103rCNT

(T/T0) − 2(5)

and T0 = 100 K. Unless the interconnect is very short(l � λmfp ), its TL model includes the p.u.l. resistances R′ asso-ciated with the scattering mechanisms which, under the hypoth-esis of low bias excitation, can be approximated with less than10% error by [3]

R′ =Ri

2nNλmfp. (6)

By adopting the approach proposed in [9], the p.u.l. capaci-tance C′ and inductance L′ are given by

C ′ ={

cosh−1 [(rb + d)/rb ]2πε0εr

+1

2ntotCq

}−1

(7)

L′ =Lk

2ntot+ Lm (8)

where Cq = 2e2 /(hνF ) and Lk = h/(2e2νF ) are the quantumcapacitance and the kinetic inductance of each spinless conduct-ing channel, respectively, defined by using the Fermi velocityfor the graphene νF = 8 × 105 m/s. Also for these two param-eters the factor 2 multiplying ntot is justified by the two spinorientations per channel. Furthermore, ε0 = 8.854 × 10−12 F/mis the vacuum electrical permittivity, εr is the relative permit-tivity of the medium, and Lm = μ0 /(2π)∗cosh−1[(rb + d)/rb ]is the magnetic inductance of the bundle in which μ0 = 4π ×10−12 H/m is the vacuum magnetic permeability.

B. TL Model of an MWCNT-Based Interconnect

The line is realized by means of a single MWCNT with anexternal radius rN = w/2. The number of shells (walls) N isdetermined by fixing the internal radius r1 of the MWCNTs tobe rN /2 and the distance between adjacent shells δ:

N = int part[(rN − r1)

δ

]+ 1. (9)

The lumped parameter Rm takes into account the contactresistance Rmj of each shell and the intrinsic resistance Ri ofeach conducting channel:

Rm =

⎡

⎣N∑

j=1

(Ri

2nj+ Rm j

)−1⎤

⎦−1

(10)

LAMBERTI AND TUCCI: COMPARISON BETWEEN SWCNTs BUNDLES AND MWCNT 927

where nj is the number of spinless conducting channels of thejth shell [5]. The factor 2 takes into account the presence of thetwo spin orientations per channel. The number of channels nj

is dependent on the operating temperature T and the shell radiusrj :

nj =

⎧⎪⎨

⎪⎩

2aTrj + b rj >dT

2T23

rj <dT

2T

(11)

with a = 3.87× 10−4 nm−1 ·K−1 , b = 0.2, and dT = 1300 nm·K.The total number of conducting channels ntot is given by the

sum of the conducting channel of each shell:

ntot =N∑

j=1

nj .

The conduction mechanism taking place in the CNTs may benominally ballistic (lossless) or dissipative. In fact, every shellis characterized by a different mean free path λmfp,j that can beexpressed as [3]

λmfp,j =2 ∗ 103rj

(T/T0) − 2(12)

with T0 = 100 K. The TL model includes the p.u.l. resistancesR′ associated with the conduction mechanism

R′ =

⎛

⎝N∑

j=1

2nj

Riλmfp,j

⎞

⎠−1

. (13)

The p.u.l. capacitance C′, as proposed in [10], includes theeffect of the electrostatic capacitance of the MWCNT, Ce , andthe quantum capacitance of each conducting channel, Cq =2e2 /(hνF ). In particular, the capacitance Ce is given by thep.u.l. external capacitance of a cylindrical conductor having thesame radius rN of the outer shell of the MWCNT. The effectof the quantum capacitance of each conducting channel leadsto a quantum capacitance for each shell, Cj,j

q = 2njCq , thatcombined with the mutual capacitance between the shells leadsto consider a series capacitance C ′

q = C ′p,N in which C ′

p,N isobtained by applying the following recursive expression:

C ′p,j = C ′j,j

q +(1/C ′

p,j−1 + 1/C ′j−1,jm

)−1, j ∈ [2, N ]

C ′p,1 = C ′1,1

q = 2n1Cq

C ′j,j+1m being the p.u.l. electrostatic mutual capacitance between

the jth and the (j + 1)th shells:

C ′j,j+1m =

2πε0

ln(rj+1/rj ).

Therefore

C ′ =(

1Ce

+1Cq

)−1

=(

cosh−1 [(rN + d)/rN ]2πε0εr

+1Cq

)−1

.

(14)The p.u.l. inductance L′, according to [10], takes into account

both the magnetic Lm MWCNT and the kinetic inductance of

each conducting channel, Lk = h/(2e2νF ):

L′ = LmM W C N T +Lk

2ntot=

μ0

2πcosh−1

(rN + d

rN

)+

Lk

2ntot.

(15)

III. IMPACT OF THE PROCESS PARAMETERS ON THE TWO

TECHNOLOGICAL ALTERNATIVES

Many sources of variability may affect the performances of ananointerconnect based on SWCNT bundle or MWCNT. Someof them as the wire thickness and width, or the thickness of thedielectric layer interposed between the wires and between themand the ground plane are common with the Cu technology. Othersources such as interbundle variation in the spacing betweenCNTs inside the bundle, CNT diameter variation in CNT (bothfor SW and MWCNT), and contact resistance are peculiar of themanufacturing technique adopted for the realization of the CNT-based interconnect. Extensive analysis concerning the effects ofmany causes of uncertainties on the performances of the futureCNT interconnects has been presented in [12] and [13].

With respect to such studies the analysis proposed in this pa-per is characterized by the distinctive features that the causesof uncertainties concern some significant parameters affectingboth alternative technologies. In addition, also the variability ofan operation parameter such as the temperature is taken into ac-count. Furthermore, the analysis is performed by taking advan-tage of the specific features of the monotonic inclusion propertyof IA ([20]–[22]) allowing us to obtain, in an analytic way, theranges associated with the uncertainties assumed by the circuitalparameters representing the interconnect and the correspondingpropagation time delay.

A. Range Evaluation by Using IA

In the presence of parameter variations, the IA permits astraight determination of an interval that certainly includes thetrue range of a function [21], [22]. In particular, by substitutingthe operations on real numbers x ∈ �n with those on intervalsX ∈ I�n , with Xi = [ximin , ximax ], the interval extension (IE)F (X) of a function f(x): x → X ⇒ f(x) → f(X) is attained.Due to the “inclusion property” [20], the IE certainly includesthe range of f(x) ∀x ∈ X, f(X), i.e., f(X) ⊆ F (X). There-fore, if x0 is the nominal solution and xmin ≤ x0 ≤ xmax , thenf(x0) is certainly in the IE and the range assumed by the func-tion in the presence of variability around this nominal set can beeasily estimated by computing the upper and lower bounds ofthe IE in the compact X . In particular, the upper bound certainlyovercomes the maximum value assumed by the function whena parameter is fixed to a given value, whereas the other onesvary in their range. It represents a reliable reference in order toestimate, without any influence of the adopted algorithm, theenvelope of the maxima of the function. This approach will beused for evaluating the ranges of the circuital parameters of theTL model and the range of the propagation delay. This last cal-culation will allow us to perform a worst case analysis and toobtain noteworthy information on the accuracy and robustnessof the performance of both technological alternatives.

928 IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 11, NO. 5, SEPTEMBER 2012

TABLE IIVARIABLE PARAMETERS AND THEIR RANGES

B. Comparison of the Circuital Parameters forthe Two Technologies

In order to compare the two possible technological alterna-tives, the behavior of the interconnect is analyzed both undernominal conditions and in the presence of parameter variation.In particular, it is assumed that the relevant geometric and phys-ical parameters assume the ranges indicated in Table II. The lastthree quantities in Table I, i.e., N, rb , and rN , are the functionsof the parameters in Table II and their ranges can be easily de-rived from these ones. It has to be explicitly evidenced that inthis study, owing to the use of IA, the parameters vary in theirrange with a uniform probability and that the variations are as-sumed to be uncorrelated, thus allowing for a search of the trueworst case conditions [21]. The considered ranges are selectedby referring to direct experience gained in the research on CNTgrowth inside the AAO templates [18] or to studies available inthe literature [12], [19], [25]. In particular, as a comparison, thevalues of the 3σ ranges of the Gaussian distributions assumedin [12] can be considered. In Table II information concerningthe bibliographic source employed for the range selection isalso presented. The relative permittivity of the medium is as-sumed variable for each technology according to the indicationspresented in the ITRS [19] for low-k material.

In addition to the variations of the process-related parameters,a variable, as the temperature, depending on the operating con-ditions of the interconnect, is considered in order to evaluate theeffects on conduction characteristics of the CNTs in a realisticsituation for an interconnect [25].

Therefore, the nominal solutions and the ranges of the cir-cuital parameters in Fig. 2 can be obtained by considering f(·)= {Rm , R′, C′, L′} and x = (w, d, T, rCNT , Rmj , εr , δ) for theSWCNT bundle according to (4), (6)–(8) or x = (w, d, T, r1 ,Rmj , εr , δ) for the MWCNT according to (10), (13)–(15). InTable III the computed nominal values and corresponding rangesof the p.u.l. parameters R′, C′, L′ are reported together with thelumped parameter Rm . In the same table the nominal valuesand ranges of N and λmfp , due to the effect of the uncertainparameters in Table II, are also reported.

The behavior of the time delay for the two technologies,discussed in the sequel, is strongly dependent on the values ofthe circuital parameters. Therefore, an analysis of the differenceson both nominal values and ranges for the two technologicalsolutions may be valuable. To perform a significant comparisonon the nominal values (for all parameters), the ratio betweenthe value relevant to the MW and that of the SW bundle can be

considered, whereas the amplitude (i.e., the difference betweenright and left extremes) can be assumed as a measure for theranges.

As it concerns the lumped resistance Rm it can be notedthat the ratio between the nominal values is close to unity(1.39) for the 15 nm technology (i.e., the two parameters arealmost comparable), whereas it is 3.30 for the 21 nm and 4.53for the 32 nm respectively. As far as the ranges are considered,the amplitude is comparable for the 15 nm, whereas that per-taining to the MW solution is larger and increases with theline width for the other two technologies. Such a behavior(either for the nominal value or for the range) can be corre-lated with that of the number N of CNT shells, also reported inTable III, and with the dependence of Rm on N described by(4) and (10) for the SW bundle and MW-based technology, re-spectively. The larger increase of the nominal value and rangeamplitude of N for the SW bundle determines a more noticeabledecrease for Rm (nominal value and range) associated with thissolution.

A different behavior is exhibited by the values of the dis-tributed resistance R′. In particular, both nominal values andrange amplitude for the SWCNT bundle are about one order ofmagnitude greater than those for the MWCNT. This behavior,which holds true for all the analyzed technology nodes, can becorrelated mainly with the differences concerning λmfp associ-ated with the two approaches, as can be noticed from the last tworows of Table III. This characteristic parameter, as expressed by(12), is indeed fixed by the largest CNT shell.

If the p.u.l. capacitances, described by (7) and (14), are con-sidered, it can be observed that, for each technology node, thenominal values corresponding to the bundle of SWCNT andthose of the MWCNT are very similar. This is due to the simi-lar dependence on the geometrical dimensions of the prevailingelectrostatic term (the external radii for the two structures arecomparable). The quantum term provides a significant contribu-tion only for the 15 nm technology leading to a higher value forthe MWCNT. If the ranges are compared, it can be easily ob-served that the range for the MWCNT is always narrower thanthat corresponding to the bundle. This result can be ascribed tothe fact that in the bundle the amplitude of the rb range is due tocombined effects of both the uncertainties on the width w andthe CNT radius, whereas for the MW only the variability of wplays a role.

As it concerns the p.u.l. inductance, the contribution of themagnetic term in the MW structure is negligible with respectto the kinetic term [3]. The nominal values corresponding tothe SWCNT bundle and those of the MWCNT are of the sameorder: the former is lower than the latter one for the 21-nm and32-nm technology nodes, due to the greater number of con-ducting channels obtained for the SWCNT in these two cases.If the ranges are compared, those of the MWCNT are alwaysincluded and hence narrower than those corresponding to thebundle, since for the bundle the number of channels is moresensibly dependent on the parameter uncertainties. Summingup, for all the p.u.l. parameters of the TL model, the MWCNTappears to be less prone than the SW bundle to the variabilityof the geometric and physical quantities.

LAMBERTI AND TUCCI: COMPARISON BETWEEN SWCNTs BUNDLES AND MWCNT 929

TABLE IIINOMINAL VALUES AND RANGES OF THE TL PARAMETERS NOT DEPENDING ON THE INTERCONNECT LENGTH

TABLE IVADOPTED DRIVER AND LOAD PARAMETERS

C. Evaluation of the Worst Case Time Delay

For the examined performance of the interconnect, i.e., the50% propagation time delay, the analytic expression reportedin [22] is considered:

τ50% = (1.48ζ + e−2.9ζ 1 . 3 5)√

L′l(C ′l + CL ) (16)

in which

ζ =2(Rt + Rm )(C ′l + CL ) + R′l(C ′l + 2CL )

4√

(L′l(C ′l + CL )(17)

where Rt and CL represent, respectively, the input resistanceand the load capacitance. Their values are fixed to their min-imum sized gate (MSG) (maximum resistance and minimumcapacitance) in each ITRS technology node (see Table IV) [26].

By adopting the parameter ranges evaluated in Table III asinterval variables, the bounds of the propagation delay can beobtained for each length, by considering f(·) = τ 50% and x =(Rm , R′, C′, L′) according to (16). The physical and geometricalparameters affecting the performance are regarded as implicitlyuncorrelated and distributed with uniform probability in theassigned range and therefore all their combinations are keptinto account.

In Fig. 4, the propagation delay versus the interconnect lengthof the two considered solutions for the three technology nodesare shown. The three dashed lines are associated with theMWCNT, while the solid ones describe the performances ofthe SWCNT bundle. Moreover, the thin lines are for the 15 nmwidth, the lines with medium thickness are for the 21 nm, whilethe thickest ones correspond to the 32-nm gate length. The plotsin Fig. 4(a) depict the nominal behavior over all the investigatedlengths (up to 250 μm), whereas in Fig. 4(b) a zoom of theprevious curves for shorter lengths is presented.

From such plots several observations can be achieved. In par-ticular, it can be noted that, for the largest considered lengths, the

Fig. 4. Comparison of the time delay versus interconnect length for the twosolutions. Dashed lines are associated with the MWCNT, and solid ones withthe SWCNT bundle: thin line = 15 nm, medium line = 21 nm, and thick line =32 nm node. (a) Overall behavior; (b) zoom putting in evidence that for a criticallength specific for each technology the MW outperforms the SW bundle.

MW performs better than the SW bundle. This behavior holdstrue also for all the technologies. For example, for the 15 nmtechnology at 250 μm, the delay for the MWCNTs is 336 ps,whereas the SW bundle exhibits a 70% higher value (560 ps).At the same length, for the 21 (32) nm the delay is 303 (256) psfor the MW and 358 (269) ps for the SW bundle. In addition,as visibly put in evidence by the curves in Fig. 4(b) (wherethe time delay is plotted on a linear scale), as the interconnectlength increases, there is a “critical” value (specific for eachtechnology) where the performance of the MW becomes better

930 IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 11, NO. 5, SEPTEMBER 2012

Fig. 5. Upper bounds (obtained by IA) of the propagation time delay versusinterconnect length for the two technologies. Description of the curves as inFig. 4.

than that of the SW bundle. Such lengths can be determinedby the intersection of continuous and dashed lines in Fig. 4(b).In particular, the transition occurs at a length equal to 194 μmfor the 32 nm technology (the corresponding time delay τ 50%is 199 ps), at 112.8 μm for 21 nm (τ 50% = 134.9 ps), and at39.03 μm for 15 nm (τ 50% = 49.5 ps). It is worth noting thatsuch results are in accordance with those reported in [26]. Thisbehavior can be justified by the different contributions of thecontact resistance Rm and that of the TL p.u.l. resistance R′ re-ported in Table III. The influence of this last term on the globalresistance, and hence on the time delay, obviously increases withthe length of the interconnect. Moreover, a higher value of thecritical length for the 21 and 32 nm nodes may be attributed tothe larger differences between the contact resistances of the MWand that of the bundle. These increasing ratios imply that longernanotubes are required to make it possible that the contributionof the distributed resistances prevails over that of the contactresistance. Under this condition, the performance exhibited bythe MW may outperform that of the SW bundle.

In order to analyze the effects of the parameter uncertaintiesand tolerances, the upper bounds of the time delay as a functionof the interconnect length are depicted in Fig. 5(a) and (b). Alsoin this case the plots in Fig. 5(a) portray the overall behavior,whereas in Fig. 5(b) a zoom of the previous curves, evidencingthe transition region, is presented.

It is worth remarking that the curves in Fig. 5 are those toconsider in order to perform a worst case design. It can be noted

Fig. 6. Range amplitude of the propagation delay. Dashed lines are associatedwith the MWCNT, and solid ones with the SWCNT bundle: thin line = 15 nm,medium line = 21 nm, and thick line = 32 nm width.

that, by taking into account the possible parameter variability,τ 50% obtained for the bundle of SW is sensibly larger than thatcorresponding to the MWCNT even at very short lengths. Thecurves for the SW bundle, for the most part of the exploredlengths range, are above those of the corresponding MW solu-tions, indicating a larger delay. In particular, for the SW τ 50%can be greater than 103 ps at lengths equal to 79, 120, and157 nm, respectively, for the 15, 21, and 32 nm technologies.Such very large delays could lead to an unacceptable behaviorof the configuration based on the SW bundle [27]. On the con-trary, the delay for all the configurations based on MW remainslower than 103 ps even for the longest (250 μm) consideredinterconnect length. Moreover, as clearly evidenced by the plotsin Fig. 5(b), the “critical” lengths, where the performance transi-tion occurs, are much smaller than those obtained when consid-ering the curves relevant to the nominal behavior. Specifically,the transition occurs at a length equal to 6.93 μm for the 32 nmtechnology (the corresponding time delay τ 50% is 22.44 ps), at4.15 μm for 21 nm (τ 50% = 16.76 ps), and at 0.13 μm for 15 nm(τ 50% = 4.47 ps).

D. Robustness and Accuracy

Information concerning the relative robustness of the twotechnological alternatives, i.e., an indication of which one isless sensitive to the parameter variations, is achieved by consid-ering the range amplitude of the propagation delay. The rangeamplitude is obtained for each length by considering the dif-ference between the upper and lower bounds of τ 50% evaluatedby means of the IA. In particular, the lower the amplitude, theless sensitive the solution and hence the better the robustness.The results concerning the two technological alternatives aresummarized in Fig. 6.

It can be noted that, due to the smaller amplitude of theassociated TL circuital parameter ranges, the solution based onMWCNTs is characterized by a better robustness with respectto the SW bundle.

Another relevant information, achieved at no additional costsby the study based on IA, concerns the accuracy, i.e., a measureof how close to the nominal value are the calculations of thetime delay in the presence of the parameter variations. Since the

LAMBERTI AND TUCCI: COMPARISON BETWEEN SWCNTs BUNDLES AND MWCNT 931

Fig. 7. Lower (A−) and upper bound (A+ ) percent variations with respect tothe nominal delay time. Thin line = 15 nm, medium line = 21 nm, and thickline = 32 nm node.

nominal value is typically not the central value of all possiblecomputations, the accuracy is determined by considering theratio of the maximum deviation with respect to the nominalvalue of the upper and lower bounds of the time delay either forthe configuration derived from MW or for that based on the SWbundle. In particular, the plus or minus percent variations aregiven by

A± = 100 × |Y± − Yn |Yn

where Y± can be the upper or lower bound of the propagationtime delay and Yn is the nominal value. The results are sum-marized in Fig. 7 where, for the sake of clarity, A− and A+ arereported separately.

It is worth remarking that the analytical formula adopted toevaluate the time delay provides better accuracy as the length ofthe interconnect increases [23]. Moreover, it is known that theuse of IA leads to an overestimation of the bounds concerningthe observed performance [21].

In order to ascertain how much these two factors may affectthe validity of the obtained bounding, a comparison with theresults of an MC procedure has been carried out. In particular,the time delay is evaluated by performing 100 MC trials onthe TL circuits associated with both the MW and SW bundleswhen a unit step input is applied to the circuit in Fig. 1. Thecommercial package Simulink has been used to perform, foreach interconnect length, time-domain simulations on the TLuniformly discretized by considering nz = 100 sections havinglength Δz = l/nz . The values of the variable parameters areassumed to be distributed with uniform probability in their rangereported in Table III; moreover, such distributions are assumedas uncorrelated (i.e., a parameter can assume a specific valuewhichever are the values of all other ones).

The MC time-domain simulations of the output voltage ob-tained in the case of the 15 nm technology for an interconnectof a critical length (i.e., 39.03 μm) are reported in Fig. 8. Inthe same figures also the curves corresponding to the nominalvalues of the parameters shown in Table III (black solid lines)are depicted.

Fig. 8. Time-domain MC simulations of the output voltages obtained with aunit step input for a 15-nm technology device of critical length l = 39.03 μm.Upper plot: SW-based device; lower plot: MWCNT-based device.

TABLE VPERCENT VARIATIONS ACHIEVED BY MEANS OF THE MC SIMULATIONS AND

ANALYTICALLY BY IA

It can be noted that the time instants where the output voltagereach 50% of its final value are more wide spread for the SWbundle than for the MW illustrating the greater robustness ofthe MW-based solution. Moreover, it can be noted that bothnominal waveforms are close to the best case (i.e., the curvesexhibiting the lowest time delay), a result which is not desirablefor a robust design of the device.

Furthermore, in Table V the A+ and A− values, obtained atdifferent interconnect lengths by means of the MC simulations,are compared, for the 15 nm (most severe) technology, withthose evaluated analytically by IA. A+ and A− are, as before,computed with respect to the nominal time delay (achieved inthe simulation where the nominal values of the parameters areadopted) which is also reported in Table V.

The obtained bounds confirm the superior performances ofthe MW-based configuration, i.e., the A− and A+ values of theSW bundle are always greater than the corresponding MWCNTfor each considered length. It can be noted that the MC approachleads to achieve bounds of a quality comparable with those ob-tained by IA and always included in those ones. The data inTable V can also be used in order to estimate the worst casecondition in terms of propagation delay. As an example, for a

932 IEEE TRANSACTIONS ON NANOTECHNOLOGY, VOL. 11, NO. 5, SEPTEMBER 2012

10 μm interconnect, a time delay of about 10 ps is achieved inthe nominal case for both technological solutions. The presenceof uncertainty in the parameter values leads to a maximum delayfor the SW bundle between 36.6 ps (according to the MC under-estimation) and 40.7 ps (according to the IA overestimation);for the MW the maximum delay is between 17.4 ps (MC) and21.0 ps (IA).

Results of similar tone can be obtained also for the other twotechnologies. In any case, when compared to MC simulations(requiring 100 time consuming simulations), the IA approach(involving only one computation) is much more efficient, andremarkably faster for the same quality results.

IV. CONCLUSION

The impact on the performances of two CNT-based intercon-nect configurations, one using a bundle of SWCNT and anotheran MWCNT, taking into account the variations of some physicaland geometrical characteristics has been analyzed. IA has beenemployed to evaluate either the ranges of the circuital parame-ters of a TL circuit modeling the interconnect or the propagationtime delay for three different technologies (15, 21, and 32 nm).This approach has allowed us to perform the comparison of thetwo technological alternatives in a more consistent way thanthat usually achieved by considering only the nominal behavior.

In fact, the variability on the process parameters gives rise(for each considered technology) to different ranges of the TLparameters which in turn determine a remarkable impact on thepropagation time delay of the interconnect.

The nominal values of the TL parameters corresponding tothe bundle of SWCNT and those pertaining to the MWCNT arequite similar and hence the time delays obtained in the two con-figurations are comparable in a wide range of the interconnectlength.

Conversely, the range analysis performed in the presence ofthe parameter variation highlights the different behaviors of thetwo solutions. The ranges of the p.u.l parameters correspondingto the MWCNT are narrower than those corresponding to thebundle and, as a consequence, the MWCNT appears to be lessprone than the SW bundle to the variability of the influencingparameters. In fact, for short interconnects the performed eval-uation has put in evidence the existence of technology-specificcritical lengths above which the solution based on MWCNToutperforms that founded on the bundle of SW. For the longestconsidered interconnects (250 μm), the time delay obtained inthe worst case for the MWCNT is less than 1 ns for the mostsevere configuration (0.86 ns at 250 μm for the 15 nm node),whereas very large values are estimated for the SW bundle(7.87 ns at 250 μm for the 15 nm node).

The study, performed by means of the IA, concerning theamplitude of the propagation delay ranges is more efficient andfaster than the MC approach for the same quality results andleads to the conclusion that the solution based on MWCNTsis generally characterized by a better robustness and greateraccuracy with respect to the SW bundle.

REFERENCES

[1] J. Davis and J. D. Meindl, Interconnect Technology and Design forGigascale Integration. London: Springer, 2003.

[2] F. Kreupl, A. P Graham, G. S. Duesberg, W. Steinhogl, M. Liebau, E.Unger, and W. Honlein, “Carbon nanotubes in interconnect applications,”Microelectron. Eng., vol. 64, pp. 399–408, 2002.

[3] A. Naeemi and J. D. Meindl, “Carbon nanotubes interconnects,” Annu.Rev. Mater. Res, vol. 39, pp. 3.1–3.21, 2009.

[4] H. Li, C. Xu, N. Srivastava, and K. Banerjee, “Carbon nanomateri-als for next-generation interconnects and passives: Physics, status, andprospects,” IEEE Trans. Electron Devices, vol. 56, no. 9, pp. 1799–1821,Sep. 2009.

[5] A. Naeemi and J. D. Meindl, “Compact physical models for multiwallcarbon nanotube interconnects,” IEEE Electron Device Lett., vol. 27,no. 5, pp. 338–340, May 2006.

[6] P. J. Burke, “An RF circuit model for carbon nanotubes,” IEEE Trans.Nanotechnol., vol. 1, pp. 393–396, Mar. 2003.

[7] A. Maffucci, G. Miano, and F. Villone, “A transmission line model formetallic carbon nanotube interconnects,” Int. J. Circuit Theory Appl.,vol. 36, pp. 31–51, 2008.

[8] Y. Xu and A. Srivastava, “A model for carbon nanotube interconnects,”Int. J. Circuit Theory Appl., vol. 38, no. 6, pp. 559–575, Aug. 2010.

[9] M. D’Amore, M. S. Sarto, and A. Tamburrano, “New electron-waveguide-based modeling for carbon nanotube interconnects,” IEEE Trans. Nan-otechnol., vol. 8, no. 2, pp. 214–225, Mar. 2009.

[10] M. S. Sarto and A. Tamburrano, “Single-conductor trasmission line modelof multiwall carbon nanotubes,” IEEE Trans. Nanotechnol., vol. 9, no. 1,pp. 82–92, Jan. 2010.

[11] N. Srivastava, H. Li, F. Kreupl, and K. Banerjee, “On the applicabilityof single-walled carbon nanotubes as VLSI interconnects,” IEEE Trans.Nanotechnol., vol. 8, no. 4, pp. 542–559, Jul. 2009.

[12] Y. Massoud and A. Nieuwoudt, “On the impact of process variations forcarbon nanotube bundles for VLSI interconnect,” IEEE Trans. ElectronDevices, vol. 54, no. 3, pp. 446–455, Mar. 2007.

[13] Y. Massoud and A. Nieuwoudt, “On the optimal design, performance, andreliability of future carbon nanotube-based interconnect solutions,” IEEETrans. Electron Devices, vol. 55, no. 8, pp. 2097–2110, Aug. 2008.

[14] C. L. Harkness and D. P. Lopresti, “Interval methods for modeling uncer-tainty in RC timing analysis,” IEEE Trans. Comput.-Aided Des. Integr.Circuits Syst., vol. 11, no. 11, pp. 1388–1401, Nov. 1992.

[15] J. D. Ma and R. A. Rutenbar, “Fast interval-valued statistical modelingof interconnect and effective capacitance,” IEEE Trans. Comput.-AidedDes. Integr. Circuits Syst., vol. 25, no. 4, pp. 710–724, Apr. 2006.

[16] W. Y. Jang, N. N. Kulkarni, C. K. Shih, and Z. Yao, “Electrical char-acterization of individual carbon nanotubes grown in nanoporous anodicalumina templates,” Appl. Phys. Lett., vol. 84, no. 7, pp. 1177–1179,2004.

[17] W. Hu, L. Yuan, Z. Chen, D. Gong, and K. Saito, “Fabrication and char-acterization of vertically aligned carbon nanotubes on silicon substratesusing porous alumina nanotemplates,” J. Nanosci. Nanotechnol., vol. 2,pp. 203–207, 2002.

[18] P. Ciambelli, L. Arurault, M. Sarno, S. Fontorbes, C. Leone, L. Datas,D. Sannino, P. Lenormand, and B. Du Plouy Sle, “Controlled growthof CNT in mesoporous AAO through optimized conditions for mem-brane preparation and CVD operation,” Nanotechnology, vol. 22, no. 12,pp. 265613-1–265613-12, 2011.

[19] International Technology Roadmap for Semiconductors. (2010). [Online].Available: http://public.itrs.net

[20] R. E. Moore, Interval Analysis. Englewood Cliffs, NJ: Prentice-Hall,1966.

[21] P. Lamberti and V. Tucci, “Interval approach to robust design,” Int. J.Comput. Math. Electr. Electron. Eng., vol. 26, pp. 285–297, 2007.

[22] B. De Vivo, L. Egiziano, P. Lamberti, and V. Tucci, “Range analysis on thewave propagation properties of a single wall carbon nano tube,” presentedat the 12th IEEE Workshop Signal Propagation on Interconnects, Avignon,France, May 12–15, 2008.

[23] Y. Ismail and E. G. Friedman, “Effects of inductance on the propagationdelay and repeater insertion in VLSI circuits,” IEEE Trans. VLSI Syst.,vol. 8, no. 2, pp. 195–206, Apr. 2000.

[24] A. Naeemi and J. D. Meindl, “Physical modeling of temperature coefficientof resistance for single- and multi-wall carbon nanotube interconnects,”IEEE Electron Device Lett., vol. 28, no. 2, pp. 135–138, Feb. 2007.

[25] S. Im, N. Srivastava, K. Banerjee, and K. E. Goodson, “Scaling analy-sis of multilevel interconnect temperatures for high-performance ICs,”

LAMBERTI AND TUCCI: COMPARISON BETWEEN SWCNTs BUNDLES AND MWCNT 933

IEEE Trans. Electron Devices, vol. 52, no. 12, pp. 2710–2719, Dec.2005.

[26] H. Li, W.-Y. Yin, K. Banerjee, and J.-F. Mao, “Circuit modeling andperformance analysis of multi-walled carbon nanotube interconnects,”IEEE Trans. Electron Devices, vol. 55, no. 6, pp. 1328–1337, Jun. 2008.

[27] J. A. Davis, R. Venkatesan, A. Kaloyeros, M. Beylansky, S. J. Souri,K. Banerjee, K. C. Saraswat, A. Rahman, R. Reif, and J. D. Meindl,“Interconnect limits on gigascale integration (GSI) in the 21st century,”Proc. IEEE, vol. 89, no. 3, pp. 305–324, Mar. 2001.

Patrizia Lamberti (M’08) was born in 1974. Shereceived the Laurea degree in electronic engineeringand the Ph.D. degree in information engineering fromthe University of Salerno, Salerno, Italy, in 2001 and2006, respectively.

Since January 2005, she has been an AssistantProfessor of Electrotechnics with the Department ofElectronic and Computer Engineering (DIEII), Uni-versity of Salerno. Since 2002, she has been involvedin experimental research on materials and innovativecomposites for electrical engineering applications at

the DIEII Lab for electromagnetic characterization of materials. The researchactivity has led to several scientific publications in international journals andin proceedings of national and international conferences. The main researchsubjects concern numerical methods for electromagnetic fields, optimizationalgorithms, tolerance analysis, and robust design of electromagnetic systems.

Vincenzo Tucci (M’96) received the Laurea de-gree in electronic engineering (with laude) from theUniversity of Naples “Federico II,” Napoli, Italy, inMarch 1981.

In 1983, he joined the Department of ElectricalEngineering, University of Naples “Federico II” asa Researcher, and in 1992, he was appointed as anAssociate Professor at the Faculty of Engineering ofthe Polytechnic of Milan. Since 1994, he has beenwith the Department of Electronic and Computer En-gineering (DIEII), Universita degli Studi di Salerno,

Fisciano, Italy, where he has been appointed as a Full Professor of ElectricalEngineering in 2000. Recently, he has been involved in research activity con-cerning carbon nanotube technology for high-speed nanointerconnects. He isthe author or coauthor of more than 130 scientific papers in journals and theproceedings of international conferences. His research interests include the elec-tromagnetic characterization and treatment of materials, the numerical methodsfor the design of electromagnetic components, and the techniques for robustdesign.

Prof. Tucci serves as a member of the evaluation board in national and inter-national research projects. He is a reviewer of scientific journals and technicalcommittees of several international conferences and workshops. He is a mem-ber of IEEE (TC11 Nano-technology), Associazione Elettrotecnica Italiana, andWG 36B of Comitato Elettrotecnico Italiano.