J Comput ElectronDOI 10.1007/s10825-014-0582-z

Delay uncertainty in MLGNR interconnects under processinduced variations of width, doping, dielectric thicknessand mean free path

K. Narasimha Reddy · Manoj Kumar Majumder ·Brajesh Kumar Kaushik

© Springer Science+Business Media New York 2014

Abstract This paper analyzes the delay performance ofmulti-layered graphene nanoribbon (MLGNR) interconnectunder process induced variations. An equivalent multi-conductor transmission line (MTL) model driven by CMOSgate is employed for the analysis. The propagation delay isanalyzed for different interconnect lengths and widths by tak-ing into account the variations in width, dielectric thickness,dielectric constant, interlayer distance and doping concentra-tion of MLGNR. Encouragingly, it is observed that the aver-age deviation in delay is below 3 % for all process inducedparameter variations except for the mean free path.

Keywords Multi-layered graphene nanoribbon(MLGNR) · Multi-conductor transmission line (MTL) ·Equivalent RLC model · Process variation ·VLSI interconnects

1 Introduction

The advancement in integrated circuit (IC) manufacturingover the past decade has ushered few undesirable effects,among which reliability issues due to process variation is oneof the leading concerns. Process variations not only impactsthe performance of devices but the interconnects as well. Dur-ing manufacturing process, the circuit performance becomes

K. N. Reddy · M. K. Majumder (B) · B. K. KaushikDepartment of Electronics and Communication Engineering,Microelectronics and VLSI Group, Indian Institute of TechnologyRoorkee, Roorkee 247667, Indiae-mail: manojbesu@gmail.com

K. N. Reddye-mail: narasimhareddy422@gmail.com

B. K. Kaushike-mail: bkk23fec@iitr.ac.in

less predictable due to poor control of physical parametersand characteristics. Uncertainties due to the variations in fab-rication process are reflected in variations of circuit parame-ters [1]. As feature size continues to scale down, the circuitperformance becomes more sensitive to process variations.Therefore, it is essential to accurately predict the variationsof circuit performance, such as propagation delay, due toprocess variations.

During recent past, multi-layer graphene nanoribbons(MLGNRs) have aroused lot of research interests for theirimpending applications in the area of field effect devicesand interconnect [2–5]. An MLGNR exhibits higher carriermobility and current densities of 2 × 105 cm2 V−1 s−1 and109 A/cm2, respectively, due to long mean free path (mfp)in the range of micrometers [6–8]. Higher electrical con-ductivity in MLGNR can be obtained either by enhancingthe carrier mobility or by increasing the number of carriers.The carrier mobility can be increased by intercalation dop-ing of arsenic pentafluoride (AsF5) vapor. Using the AsF5

doping, the conductivity of MLGNR can be increased upto3.2 × 105 S/cm which is almost 1.5 times higher than thecopper interconnects [9,10].

The performance of MLGNR interconnects is usually ana-lyzed by means of transmission line models. The circuitalparameters of the transmission line takes into account theresistive, capacitive and inductive parasitics of MLGNR. Theexpressions of these parameters are mainly governed by thephysical and geometrical quantities of graphene structures.The physical characteristic of graphene is mostly affected byuncertainties and variations during manufacturing process[11,12]. These variations in processes can produce remark-able alteration in parasitics, and thereby the delay perfor-mance. Therefore, a knowledge based on the nominal valueof the parameters is not sufficient for accurate analysis of thedelay performance. Their maximum and minimum range of

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variation is required for an effective and reliable design ofthe MLGNR interconnects. This work employs Monte Carloapproach to analyze the impact of process induced variationsin MLGNR interconnects. Although, this approach requireslarge number of simulation trials to analyze the delay perfor-mance, but is considered to effective means of evaluating theperformance under process variation.

This paper employs a multi-conductor transmission line(MTL) model where interconnect parasitics are obtainedusing the process induced variations of MLGNR parame-ters. A driver-interconnect-load (DIL) model [13] based onCMOS gate driven MLGNR interconnect is used to analyzethe maximum and minimum deviations in delay. This devia-tion is governed by the variability of process dependent para-meters such as MLGNR width, doping concentration, dielec-tric thickness, dielectric constant, mean free path, interlayerdistance, etc.

The paper is organized in four different sections includ-ing the current Sect. 1 that presents the importance of processvariation in MLGNR interconnects and briefs about the workcarried out. In Sect. 2, the MTL model of an MLGNR isemployed to demonstrate the dependency of interconnectparasitics on different process induced parameters. Section 3analyzes the delay uncertainty of a DIL system using MonteCarlo simulation approach. The propagation delay along withthe maximum and minimum deviations is also analyzed inthis section. Finally, Sect. 4 draws a brief summary of thepaper and the outcome achieved.

2 Multi-conductor transmission line model

MLGNR is a stack of graphene sheets with each sheet sepa-rated by a distance δ. Each graphene layer is a hexagonalarrangement of carbon atoms. Depending on the Vander-Waal’s force between neighboring carbon atoms, the inter-layer distance (δ) is approximately equal to 0.34 nm [2].Figure 1a shows the cross-section of MLGNR of width (w)

and thickness (t) placed above the ground plane at a dis-tance d. The total number of MLGNR layers (Nlayer ) can beobtained as

Nlayer = 1 + I nteger (t/δ) (1)

Based on the geometry of Fig. 1a, the equivalent electricalmodel of MLGNR is shown in Fig. 1b. The interconnect par-asitics are primarily dependent on the number of conductingchannels (Nch) of each layer in MLGNR. The Nch takes intoaccount the spin and sub-lattice degeneracy of carbon atomsand primarily depends on the width, Fermi energy (E f ), tem-perature (T ), etc and can be expressed as

Nch ={

a0 + a1w + a2w2 + a3 E f + a4wE f + a5 E2

f f or E f > 0b0 + b1w + b2w

2 f or E f = 0

(2)

where a0 to a5 and b0 to b5 are the fitting parameters obtainedfor armchair and zigzag GNRs at T = 300 K [14].

Depending on the fabrication process, the imperfectmetal-MLGNR contact resistance (Rmc) has a typical valueranging from 1 to 20 k� [15]. Additionally, each layer ofMLGNR exhibits quantum resistance (Rq,n) that is due tothe quantum confinement of carriers across the intercon-nect width. Thus, the equivalent lumped resistance can beexpressed as

Requ,n = Rmc,n + Rq,n where Rq,n = h/2e2

Nch .Nlayer(3)

However, for longer interconnects, the scattering resis-tance (r) primarily appears due to the static impurity scatter-ing, defects, line edge roughness scattering (LER), acousticphonon scattering, etc [16–20]. The per unit length (p.u.l.)scattering resistance (rn) primarily depends on the mean freepath (λm f p) and can be expressed as

rn = h/2e2

Nch .Nlayer .λm f p(4)

Each layer in MLGNR comprises of kinetic inductance(lk,n) and quantum capacitance (cq,n) that represents themobile charge carrier inertia and the density of electronicstates, respectively. The lk,n and cq,n in p.u.l. can beexpressed as

lk,n = lk0

2Nch .Nlayer; where lk0 = h

2e2vF(5)

cq,n = 2cq0.Nch .Nlayer ; where cq0 = 2e2

hvF(6)

where vF ≈ 8 × 105m/s represents the Fermi velocity ofcarriers in graphene [16,21]. The electrostatic capacitance(ce) is due to the electric field coupling between MLGNR andground plane, which is primarily dependent on the MLGNRwidth and the distance from the ground plane. Apart fromthis, the magnetic inductance (le,n) of MLGNR is due to thestored energy of the carriers in magnetic field. The le,n andce in p.u.l. can be expressed as

le,n = μ0d

w(7)

ce = ε0w

d(8)

The mutual inductance (lm,nm ) and capacitance (cm,n

m )

between m and n layers are mainly due to the electron tunneltransport phenomenon [15]. The lm,n

m and cm,nm in p.u.l. can

be expressed as

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w

dεr

δt

(a)

,12

mcR

Bottom layer

Top layer

1

2

n

,12

qR

1r

,1kl

,1el

,12

qR ,1

2mc

R,1qc

ec

1, 2m

c

, 22

qR , 2

2mc

R, 2qc

, 2el

, 2kl

2r

, 22

mcR

, 22

qR

1, 2m

l

,2

mc nR

,2

q nR n

r,k n

l

,e nl

1, nm

l

,q nc

,2

mc nR

,2

q nR

(b)

Fig. 1 a Geometry and b equivalent electrical model of MLGNR interconnect

lm,nm = μ0δ

w(9)

cm,nm = ε0w

δ(10)

3 Impact of process induced variation on delay

The performance of an MLGNR based interconnect isaffected by variability of several process induced parame-ters such as interconnect width, mean free path, dielectricthickness, dielectric constant, etc. Few other parameters suchas doping concentration, interlayer distance and imperfectmetal-MLGNR contact resistance can also affect the overalldelay performance.

For the process induced parameter variations, propaga-tion delay is analyzed using a driver-interconnect-load (DIL)system shown in Fig. 2. A CMOS gate is employed to drivethe interconnect line that primarily represents the equivalentMTL model of the MLGNR (Fig. 1b) with 20 distributed seg-ments. The interconnect line is terminated by a load capaci-tance CL = 1fF.

3.1 Modelling process variation

MLGNRs are synthesized using several growth techniquessuch as arc discharge, chemical vapor deposition (CVD),

VDD

Vin MLGNR Interconnect

CL

Fig. 2 A driver-interconnect-load system

laser ablation, etc [1]. The synthesization process signifi-cantly affects the physical dimensions of the MLGNR inter-connects. Depending on the width, MLGNR can exhibit longmfp up to 1−5 µm. Due to photo-lithographic errors and etch-ing process, the width of MLGNR varies by ±10 % [12],whereas the dielectric thickness between MLGNR and theground plane varies up to ±10 % due to the chemical mechan-ical polishing (CMP) during the fabrication process [11].However, the mfp varies due to the defect in chiral structure ofmetallic MLGNRs. The relative permittivity of the medium isvaried as per ITRS-2012 information [22] provided for low-kdielectric materials. The statistical uncertainty in the qualityof the overall metal-MLGNR contact can vary the contactresistance by ±50 % [12]. Table 1 presents the range of varia-tion of process induced geometrical and physical parametersof MLGNR. Depending on the process induced parameter

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Table 1 Variable parameters and their ranges

Name ofparameter

Nominal value Variation (%) Minimum value Maximum value Reason ofvariation

Reference

Width (w) 25 nm ±10 22.5 nm 27.5 nm Photo-lithographicand etchingprocess

[12]

Doping cncentration (E f ) 0.6 eV ±10 0.54 eV 0.66 eV Fermi energy of dopant atoms [12]

Dielectric thickness (d) 200 nm ±10 180 nm 220 nm Oxide deposition, CMP process [11]

Dielectric constant (εr ) 2.2 ±10 2.0 2.4 Permittivity ofdifferent low-kdielectricmaterials

[21]

Interlayer distance (δ) 0.34 nm ±10 0.31 nm 0.37 nm Vander Waal’s forcebetween neighbouringcarbon atoms

[11]

Contact resistance (Rmc) 3.2 k� ±50 1.6 k� 4.8 k� Statistical uncertainty in thequality of the overallmetal-MLGNR contact

[12]

Mean free path (λm f p) 1.04 µm ±50 0.52 µm 1.56 µm Defect in chiralstructure ofmetallicMLGNRs

[12]

Table 2 Nominal values and ranges of the transmission line (TL) parameters

Variation of TL parameters Process induced parameters

Width Doping concentration Dielectric thickness Dielectric constant Mean free path

Requ (ohm) Nominal 356.99 356.99 356.99 356.99 356.99

Range [324.57, 396.60] [322.66, 398.61] [356.99, 356.99] [356.99, 356.99] [356.99, 356.99]

r (�/µm) Nominal 686.54 686.54 686.54 686.54 686.54

Range [624.18, 762.69] [620.50, 766.55] [686.54, 686.54] [686.54, 686.54] [457.69, 1373.07]

lk (nH/µm) Nominal 0.44 0.44 0.44 0.44 0.44

Range [0.40, 0.49] [0.40, 0.49] [0.44, 0.44] [0.44, 0.44] [0.44, 0.44]

le (pH/µm) Nominal 2.51 2.51 2.51 2.51 2.51

Range [2.28, 2.79] [2.51, 2.51] [2.26, 2.76] [2.51,2.51] [2.51, 2.51]

cq (aF/µm) Nominal 3613.46 3613.46 3613.46 3613.46 3613.46

Range [3252.66, 3974.45] [3236.26, 3998.03] [3613.46, 3613.46] [3613.46, 3613.46] [3613.46, 3613.46]

ce (aF/µm) Nominal 9.74 9.74 9.74 9.74 9.74

Range [8.76, 10.71] [9.74, 9.74] [8.85, 10.82] [8.85, 10.82] [9.74, 9.74]

variation; Table 2 presents the nominal and the range of par-asitic values. The propagation delay of the DIL is analyzedunder nominal conditions and in the presence of parametervariations evaluated by parameters A+ and A−. The A+and A− represent the upper bound and lower bound percent-age variation with respect to the nominal delay and can beexpressed as (11) and (12), respectively.

A+ = Maximum delay − Nominal delay

Nominal delay× 100 (11)

A− = Nominal delay − Minimum delay

Nominal delay× 100 (12)

The behavior of the time delay for A+ and A− is stronglydependent on the values of the circuital parameters.

3.2 Process variation for different interconnect lengths

This sub-section presents the variation in delay for differ-ent interconnect lengths ranging from 5 to 2500 µm at fixedMLGNR width of 25 nm. For each of the above mentionedinterconnect length, the process induced width and inter-calative doping is varied by ±10 % [12] from its nominalvalue of 25 nm and 0.6 eV, respectively (Table 1). MLGNRwidth and doping has major impact on total number of

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Fig. 3 Propagation delay atdifferent interconnect lengthsfor process induced a width andb doping variations

0

30

60

90

120

150

180

0 500 1000 1500 2000 2500

Nom

inal

del

ay (

ps)

Interconnect length (µm)

NominalA+A-

0

30

60

90

120

150

180

0 500 1000 1500 2000 2500

Nom

inal

del

ay (

ps)

Interconnect length (µm)

NominalA+A-

(a) (b)

Fig. 4 Propagation delays atdifferent interconnect lengthsfor process induced a dielectricthickness and b dielectricconstant variations

0

30

60

90

120

150

180

0 500 1000 1500 2000 2500

Nom

inal

del

ay (p

s)

Interconnect length (µm)

NominalA+A-

0

30

60

90

120

150

180

0 500 1000 1500 2000 2500

Nom

inal

del

ay (p

s)

Interconnect length (µm)

NominalA+A-

(a) (b)

Fig. 5 Propagation delays atdifferent interconnect lengthsfor process induced a interlayerdistance and b contact resistancevariations

020406080

100120140160180

0 500 1000 1500 2000 2500

Nom

inal

del

ay (p

s)

Interconnect length (µm)

NominalA+A-

0

30

60

90

120

150

180

0 500 1000 1500 2000 2500

Nom

inal

del

ay (p

s)

Interconnect length (µm)

NominalA+A-

(b)(a)

conducting channels. The ce of MLGNR is also affected bythe process induced width variations (Eqn. 8). The conduct-ing channels primarily decide the resistance, inductance andquantum capacitance values as shown in Table 2. The differ-ence between average value of A+ and A− is 1.65 % forprocess induced width variations, whereas, the differenceis only 0.8 % for the variation in doping concentration asobserved from Fig. 3a and b, respectively.

The process induced dielectric thickness and dielectricconstant varies up to ±10 % from their nominal values of200 nm and 2.2, respectively. As the dielectric material liesbetween the MLGNR and the ground plane, it mainly influ-ences the ce (Eqn. 8). However, the dielectric thickness alsoaffects the le of each layer in MLGNR (Eqn. 7). Due to thecumulative effect of le and ce, the difference between the

average values of A+ and A− is just 0.9 % as shown inFig. 4a. However, under the variation of process induceddielectric constant, the difference between A+ and A− isalmost 0 % due to negligible variation in ce as shown inFig. 4b.

Monte Carlo simulations are performed for processinduced variations of interlayer distance, mean free path andcontact resistance. It is observed that the difference betweenaverage A+ and A− is almost 0 % for interlayer distanceand contact resistance variations (Fig. 5), whereas for meanfree path variations, the difference is alarmingly high toalmost 30.4 % as shown in Fig. 6a. The scattering resistanceof MLGNR is inversely proportional to the mean free path(Eqn. 4). It results in almost doubling of scattering resis-tance with the variation in mean free path. For all process

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Fig. 6 Propagation delays atdifferent interconnect lengthsfor process induced a mean freepath and b all geometricalparameter variations

0

20

40

60

80

100

120

140

160

180

0 500 1000 1500 2000 2500

Nom

inal

del

ay (

ps)

Interconnect length (µm)

NominalA+A-

0

30

60

90

120

150

180

0 500 1000 1500 2000 2500

Nom

inal

del

ay (

ps)

Interconnect length (µm)

NominalA+A-

(a) (b)

Fig. 7 Propagation delays atdifferent interconnect widths forprocess induced a width and bdoping variations

150

170

190

210

230

250

0 10 20 30

Nom

inal

del

ay (p

s)

Interconnect width (nm)

Nominal

A+

A-

6

7

8

9

10

11

12

13

150

170

190

210

230

250

0 10 20 30

A+

, A-(

%)

Nom

inal

del

ay (p

s)

Interconnect width(nm)

NominalA+A-

(a) (b)

Fig. 8 Propagation delays atdifferent interconnect widths forprocess induced a dielectricthickness and b dielectricconstant variations

150

170

190

210

230

250

0 10 20 30

Nom

inal

del

ay (

ps)

Interconnect width (nm)

NominalA+A-

150

170

190

210

230

250

0 10 20 30

Nom

inal

del

ay (

ps)

Interconnect width (nm)

NominalA+A-

(a) (b)

induced parameters variation taken together, the differencebetween average A+ and A− is 31.2 % as demonstrated inFig. 6b.

3.3 Process variation for different interconnect widths

For fixed interconnect length of 2500 µm and differentMLGNR widths ranging from 5 to 25 nm, Monte Carlo sim-ulations are run for different process induced parameter vari-ations. The range of variations for MLGNR width, dopingconcentration, dielectric thickness, dielectric constant, meanfree path, contact resistance and interlayer distance are pre-sented in Table 1. It is observed that the difference betweenaverage A+ and A− are 3.2 and 1.8 % for process inducedMLGNR width and doping variations as shown in Fig. 7a and

b, respectively. The width and doping concentration (Fermienergy) have major impact on the number of conductingchannels of each layer in MLGNR (Eqn. 2). The total num-ber of conducting channels primarily decides the resistance,inductance and quantum capacitance values. Apart from this,the ce is mainly influenced by the MLGNR width (Eqn. 8).Therefore, due to the cumulative effect of ce and other inter-connect parasitics, the deviation is more for process inducedwidth than the doping variations.

Under process induced dielectric thickness and dielectricconstant variations, the difference between average A+ andA− are observed as 1.5 and 0 %, shown in Fig. 8a and b,respectively. Here, the deviation is primarily due to the vari-ations in ce that appears between the MLGNR and the groundplane.

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Fig. 9 Propagation delays atdifferent interconnect widths forprocess induced a mean freepath and b contact resistancevariations

20

40

60

80

100

120

140

150

170

190

210

230

250

0 10 20 30

A+,

A-(

%)

Nom

inal

del

ay (

ps)

Interconnect width (nm)

NominalA+A-

0

2

4

6

8

10

150

170

190

210

230

250

0 10 20 30

A+,

A-(

%)

Nom

inal

del

ay (

ps)

Interconnect width (nm)

NominalA+A-

(b)(a)

Fig. 10 Propagation delays atdifferent interconnect widths forprocess induced a interlayerdistance and b all parametervariations

150

170

190

210

230

250

0 10 20 30

Nom

inal

del

ay (

ps)

Interconnect width (nm)

NominalA+A-

0

50

100

150

200

250

300

0 10 20 30

Nom

inal

del

ay (

ps)

Interconnect width (nm)

NominalA+A-

(a) (b)

For a fixed interconnect length of 2500 µm, the differencebetween average A+ and A− is 56.9 % for process inducedmean free path variations (Fig. 9a) whereas the difference isalmost 0 % for process induced contact resistance (Fig. 9b)and interlayer distance (Fig. 10a) variations. It is due to thedominating effect of scattering resistance that is inverselyproportional to the mean free path (Eqn. 4). Thus, the varia-tion of mean free path results in almost doubling of the valueof scattering resistance. The difference between average A+and A− is 86.5 % for overall variations as observed fromFig. 10b.

Table 3 presents the average deviation in A+ and A− (inpercentage) for different process induced parameter varia-

tions. For process induced width, doping, dielectric thick-ness, dielectric constant, interlayer distance, contact resis-tance and mean free path variations, IV and OV representsthe parameters individual and overall variations, respectively.For different interconnect lengths and widths, the differencebetween A+ and A− for overall variation is approximatelyequal to the difference between the summation of A+ andA− of parameters individual variation and can be expressedas

[∑Avg.A+

]I V

−[∑

Avg.A−]

I V

≈ [Avg.A+]OV − [Avg.A−]OV (13)

Table 3 Average deviation in A+ and A− (in percentage) for process induced parameter variations

Process induced parameter variations Measured at different interconnect lengths Measured at different interconnect widths

Avg. A+ Avg. A− Avg. A+ Avg. A−Width 10.46 8.65 17.31 14.05

Doping concentration 5.08 4.23 9.53 7.70

Dielectric thickness 5.67 4.71 8.17 6.67

Dielectric constant 5.12 5.10 7.35 7.31

Interlayer distance 0.03 0.03 0.03 0.02

Contact resistance 18.94 18.49 5.64 5.61

Mean free path 45.70 15.28 85.38 28.48

Parameters overall variation 72.09 40.81 125.95 63.85

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4 Conclusion

This paper investigated the impact of process induced varia-tions on propagation delay of MLGNR interconnects. A DILsystem is used to analyze the propagation delay by employ-ing an MTL model of MLGNR interconnect. Monte Carlosimulations are run to analyze the effect of process inducedMLGNR width, doping concentration, dielectric thickness,interlayer distance, mean free path and contact resistancevariations. It is observed that the process induced mean freepath has significant impact on the average deviation in delayat global interconnect lengths.

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