Frequency response and bandwidth analysis of multi-layer graphene nanoribbon andmulti-walled carbon nanotube interconnects

Manoj Kumar Majumder, Narasimha Reddy Kukkam, Brajesh Kumar Kaushik

Department of Electronics and Communication Engineering, Indian Institute of Technology Roorkee, Roorkee 247667, IndiaE-mail: manojbesu@gmail.com

Published in Micro & Nano Letters; Received on 2nd January 2014; Revised on 19th June 2014; Accepted on 20th June 2014

This reported research analyses and compares the bandwidth and absolute frequency response of a multi-layer graphene nanoribbon (MLGNR)and a multi-walled carbon nanotube (MWCNT) at local, semi-global and global interconnect lengths. The transfer function of the driver-interconnect-load system is obtained by representing the interconnect line with an equivalent single conductor model of either a MLGNRor a MWCNT. Using absolute frequency response, it is observed that the bandwidth of the MLGNR is higher by almost ten times andfour times in comparison to the MWCNT for local and global interconnect lengths, respectively.

Figure 1 Cross-section of MLGNR and MWCNT interconnecta MLGNRb MWCNT

1. Introduction: During the recent past, multi-walled carbonnanotubes (MWCNTs) and multi-layer graphene nanoribbons(MLGNRs) have attracted much attention in scientific researchbecause of their unique physical properties such as higher thermalconductivity, current carrying capability, mechanical strength andso on [1–3] in comparison to Cu interconnects. Both the MLGNRand MWCNT support large current densities up to 109 A/cm2 andhave long mean free paths ranging from 1 to 5 µm. On the basis ofchirality (rolled up direction of graphene sheets), MWCNTs andMLGNRs exhibit metallic as well as semiconducting properties.However, only the metallic nature is important for VLSIinterconnect applications. The fabrication of a MLGNR is morecontrollable than for a MWCNT because of its planar nature [1, 2].Previously, Cui et al. [3] derived an analytical expression of the

output voltage waveform of a MLGNR interconnect. Nasiri et al.[4] also approximated the input–output transfer function of aMLGNR using the fourth-order Padé’s approximation by assuminga driver resistance equal to zero (Rdr = 0) and perfect metal-MLGNR contacts (Rmc = 0). The authors plotted Nyquist diagramsfor MLGNR lengths up to only the semi-global range. However,Fathi et al. [5] plotted the Nyquist diagrams of carbon nanotube(CNT) bundle interconnects by correctly assuming the driver parasi-tics and contact resistance only for local interconnects. In recentyears, Nishad and Sharma [6] and Kumar and Naeemi [7] proposedanalytical time and frequency domain models of side contact (SC)and top contact (TC) MLGNR interconnects. However, the authorsneglected the driver resistance (Rdr) and capacitance (Cdr) while de-riving the transfer function of SC andTCMLGNRs.While these pre-vious works [3–7] demonstrate the potential advantages of either aMLGNR or a MWCNT interconnect separately, they do notprovide a comprehensive comparison between the MLGNR andMWCNT interconnects for bandwidth and frequency response.This Letter presents a comparative analysis of the frequency re-

sponse and the bandwidth of MLGNR and MWCNT interconnects.To reduce the computational effort required for analysing the RLCcircuit, a simplified but accurate equivalent single conductor (ESC)model is employed by considering an imperfect metal-MLGNR/MWCNT contact resistance (Rmc). Using a driver-interconnect-load(DIL) system, a transfer function of the distributed transmissionline is derived to obtain the bandwidth at local, semi-global andglobal interconnect lengths. The transfer function accurately takesinto account the driver resistance (Rdr) and capacitance (Cdr).

2. Interconnect model: MLGNRs are considered as unrolledMWCNTs formed by hexagonal rings of carbon atoms. Owing to

Micro & Nano Letters, 2014, Vol. 9, Iss. 9, pp. 557–560doi: 10.1049/mnl.2013.0742

the Vander-Waal’s force between the neighbouring carbon atoms,the inter layer (or inter shell) distance of the MLGNR (orMWCNT) is δ≃ 0.34 nm [2]. Fig. 1 shows the cross-section of theside-contact MLGNR and MWCNT with an equivalent width andthickness (or diameter) of w and t (or d ), respectively. The totalnumber of layers (Nlayer) and shells (Nshell) in the MLGNR andMWCNT primarily depends on the w, t and outer shell diameter(d ), respectively. Therefore, the areas of the MLGNR andMWCNT are obtained as w·t and π(d/2)2, respectively. Dependingon the geometry, the ESC model of either the MLGNR orMWCNT is used to represent the interconnect line of the DIL inFig. 2. The driver resistance, driver capacitance and the loadcapacitance are represented as Rdr, Cdr and CL, respectively.

The interconnect parasitics (i.e. resistance, inductance and capa-citance) are modelled using the number of conducting channels (Nch)of each layer (or shell) in the MLGNR (or MWCNT). Nch takesinto account the effect of spin and sub-lattice degeneracy of carbonatoms and depends on the number of sub-bands, Fermi energy, tem-perature and the dimensions of the MLGNR andMWCNT intercon-nects [8, 9]. Nch of the MLGNR and MWCNT can be expressed as

Nch(MLGNR) = c0 + c1w+ c2w2

+ c3EF + c4wEF + c5E2F (1)

and

Nch(MWCNT) ≃ k1TDi + k2, Di . dT/T

≃ 2/3, Di ≤ dT/T(2)

557& The Institution of Engineering and Technology 2014

Figure 3 Cascaded connection of the DIL of Fig. 2

Table 1 Equivalent parasitics of MLGNR and MWCNT interconnects

Interconnect parasitic Parasitic values at t = d = 13.92 nm

MLGNR MWCNT

R′ESC, Ω/μm 150 21L′ESC, nH/μm 0.097 0.056C′ESC, aF/μm 6.74 89.78R1, kΩ 1.614 1.612Rdr, Ω 292.36Cdr, aF 45.51

Figure 2 DIL system representing the ESC model of MLGNR/MWCNTinterconnects terminated by load capacitance CL = 250 aF

where c0 to c5 are the constant parameters for metallic GNRs at roomtemperature with the Fermi energy EF > 0 [8]. On the other hand, Di

represents the diameter of the ith shell in the MWCNT, k1 and k2are equal to 3.87 × 10−4 nm−1 K−1 and 0.2, respectively [9]. Thethermal energy of electrons and the gap between the sub-bands deter-mine the quantitative value of dT equivalent to 1300 nm.K at roomtemperature (T = 300 K). Thus, the total number of conducting chan-nels (Ntotal) in a MLGNR (or MWCNT) is obtained by the summa-tion of Nch of each layer (or shell).

The lumped resistance (R1) in Fig. 2 is the series combination ofimperfect metal-MLGNR/MWCNT contact resistance (Rmc) andquantum resistance (Rq). Depending on the fabrication process,Rmc exhibits a typical value of 3.2 kΩ. Resistance, Rq appearsmainly because of the quantum confinement of electrons anddepends on the number of conducting channels of each layer (orshell) in the MLGNR (or MWCNT) interconnects. In addition, thescattering resistance (R′ESC) is because of the static impurity scatter-ing, defects, line edge roughness scattering and primarily dependson the effective mean free path (λmfp) of electrons [3]. Thelumped R1 and per unit length (p.u.l.) R′ESC can be expressed as

R1 = Rmc +Rq

Ntotal

( )/2; where Rq =

h

2e2(3)

R′ESC = h/2e2

2Ntotallmfp; where

lmfp,i =103Di

T/T0( )− 2

, T0 = 100K

(4)

The total inductance (L′ESC) of the MLGNR and MWCNT in Fig. 1is the summation of kinetic (L′kESC) and magnetic (L′eESC) induc-tances that primarily represent the stored energy in the magneticfield and the inertial mass of mobile charge carriers, respectively[3, 9]. The effective p.u.l. L′ESC can be expressed as

L′ESC = L′kESC + L′eESC (5)

where

L′kESC = L′k02Ntotal

; L′k0 =h

2e2vF(6)

and

L′eESC = m010C′eESC0

(7)

where vF represents the Fermi velocity of the CNT and graphene≃8 × 105 and C ′eESC0 is the effective electrostatic capacitance of theESC embedded in free space.

The outermost layer (or shell) of the MLGNR (or MWCNT)experiences an electrostatic capacitance (C ′eESC) with respect tothe ground that appears in series with the quantum capacitance(C ′qESC) shown in Fig. 2. Thus, the effective p.u.l. equivalent

558& The Institution of Engineering and Technology 2014

capacitance (C ′ESC) can be expressed as

C′ESC = 1/C′

qESC + 1/C′eESC

( )−1(8)

C′eESC,MLGNR = 101rw

H; C′

eESC,MWCNT = 2p101rcosh−1 H/d

( ) (9)

C′qESC = 2NtotalC

′q0; where C′

q0 =2e2

hvF(10)

where εr≃ 2.2 and H represent the dielectric permittivity and thedistance between the outermost layer (or shell) of the MLGNR(or MWCNT) and the ground plane, respectively. For a fixed t =d = 13.92 nm, the quantitative values of the driver and interconnectparasitics of the MLGNR and the MWCNT are summarised inTable 1. The interconnect parasitics are primarily dependent onNtotal and obtained using the parasitics expressions (1)–(10).

3. Distributed transfer function of DIL system: Transfer functionof the DIL (Fig. 2) is obtained using the driver parasitics and theMLGNR and MWCNT interconnect parasitics (Table 1). Toobtain the overall gain (Vout/Vin), the DIL of Fig. 2 is representedas a cascaded connection of several two-port networks (Fig. 3). Thebest choice of the two-port parameter is the ABCD parameter.The ABCD matrix parameters of each two-port network arerepresented as g1, g2, g3, g4 and g5. The Telegrapher’s equationof distributed transmission line [5, 10] is used to obtain theABCD matrix parameter (gain) g4. Using this method, the voltageand current at any point x of the DIL can be expressed as

∂V

∂x= − R′

ESC + sL′ESC( )

I(x) and∂I

∂x= −sC′

ESC · V (x) (11)

Solving expression (11)

V = V0 cosh (gnx)+ I0Z0 sinh (gnx) (12a)

and

I = I0 cosh (gnx)+ V0/Z0 sinh (gnx) (12b)

Micro & Nano Letters, 2014, Vol. 9, Iss. 9, pp. 557–560doi: 10.1049/mnl.2013.0742

Table 2 Bandwidth of MLGNR and MWCNT interconnects

l,μm

t = d,nm

MLGNR bandwidth, GHz MWCNT bandwidth, GHz

Absolute Analytical,second order

Absolute Analytical,second order

5 7.12 36.4 36.26 19 18.8810.52 44.4 44.59 20.6 20.5313.92 63.3 63.36 45.1 44.67

500 7.12 2.25 2.20 0.04 0.0410.52 3.09 3.08 0.68 0.6813.92 3.5 3.48 1.5 1.49

2500 7.12 0.11 0.11 0.001 0.00110.52 0.14 0.14 0.02 0.0213.92 3.5 3.48 0.06 0.06

Thus

VI

[ ]= cosh (gnx) Z0 sinh (gnx)

1/Z0 sinh (gnx) cosh (gnx)

[ ]V0

I0

[ ](13)

where n, x and g =sC

′ESC R

′ESC + sL′ESC

( )√are the number of

distributed segments, length of each segment and the propagationconstant, respectively.Using the transmission matrix parameter for a uniform RLC

transmission line of length l [5, 10], the resultant transmissionmatrix parameter of the DIL configuration (Fig. 3) can be expressedas

Tresult = g1.g2.g3.g4.g5

= 1 Rdr

0 1

[ ]1 0

sCdr 1

[ ]1 R1

0 1

[ ]

× cosh (gnx) Z0 sinh (gnx)

1/Z0 sinh (gnx) cosh (gnx)

[ ]1 R1

0 1

[ ]= A B

C D

[ ](14)

Now, using expression (14), the distributed transfer function of theDIL (Fig. 2) can be expressed as

Vout

Vin= TF = 1

A+ sCLB(15)

where

A = 1+ sR′ESCC

′ESC(nx)

2

2+ RdrCdr + C′

ESC(nx)(R1 + Rdr)

[ ]

+ s2L′ESCC

′ESC(nx)

2

2

[+ R′2

ESCC′2ESC(nx)

4

4!+ R

′ESCRdrC

′ESCCdr(nx)

2

2

+R′ESCC

′2ESC(nx)

3(R1 + Rdr)

3!+ R1RdrCdrC

′ESC(nx)

](15a)

(see (15b))

4. Cutoff frequency and bandwidth analysis: The transferfunction of (15) is used to analyse the cutoff frequency ( fc) thatprimarily depends on the MLGNR and MWCNT interconnectparasitics presented in Table 1. The parasitic values are obtainedfor similar diameters of the MWCNT and thicknesses of theMLGNR interconnects. To obtain the analytical expression of thecutoff frequency of the DIL, the transfer function of (15) can be

B = 2R1 + Rdr + R′ESC(nx)

)( + sR′ESCC

′ESC(n

2

[

+R′2ESCC

′ESC(nx)

3

3!+ L′ESC(nx)+ R2

1C′ESC(n

+ s2 2R1 + Rdr

( ) L′ESCC′ESC(nx)

2

2+ R′2

ES

([

+ 2R′ESCL

′ESCC

′ESC(nx)

3

3!+ R′3

ESCC′2ESC(nx)

5

5!

+ R′2ESCRdrCdrC

′ESC(nx)

3

3!+RdrCdr L

′ESC +(

Micro & Nano Letters, 2014, Vol. 9, Iss. 9, pp. 557–560doi: 10.1049/mnl.2013.0742

expressed to a simplified second-order expression as

TF = 1

a0 + a1s+ a2s2(16a)

where a0 = 1.

a1 =R

′ESCC

′ESC(nx)

2

2+ RdrCdr + C′

ESC(nx) R1 + Rdr

( )( )

+ CL 2R1 + Rdr + R′ESC(nx)

( ) (16b)

a2 =L′ESCC

′ESC(nx)

2

2+ R′2

ESCC′2ESC(nx)

4

4!+ R′

ESCRdrC′ESCCdr(nx)

2

2

[

+R′ESCC

′2ESC(nx)

3(R1 + Rdr)

3!+ R1RdrCdrC

′ESC(nx)

]

+ CLR′ESCC

′ESC(nx)

2

22R1 + Rdr

( ) + 2R1RdrCdr +R′2ESCC

′ESC(nx)

3

3!

[

+ L′ESC(nx)+ R21C

′ESC(nx)+ R′

ESCRdrCdr(nx)+ R1RdrC′ESC(nx)

](16c)

The roots of (16) can be expressed as

s1 =−a1 +

a21 − 4a2

√2a2

and

s2 =−a1 −

a21 − 4a2

√2a2

x)22R1 + Rdr

( )+ 2R1RdrCdr

x)+ R′ESCRdrCdr(nx)+ R1RdrC

′ESC(nx)

]

CC′2ESC(nx)

4

4!

)+ R1RdrR

′ESCC

′ESCCdr(nx)

2

+ R21R

′ESCC

′2ESC(nx)

3

3!

R21C

′ESC

)(nx)

]

(15b)

559& The Institution of Engineering and Technology 2014

Figure 4 Frequency response of MLGNR and MWCNTa For different interconnect lengths at t = d = 13.92 nmb For different thickness and diameter at l = 2500 µm

The cutoff frequency can be expressed as

fc =1

2p

(2a2 − a21)+

(2a2 − a21)

2 + 4a22

√2a22

√√√√(17)

Table 2 compares the bandwidth of the MLGNR and MWCNTobtained from the absolute frequency response of higher orderand the analytical expression of reduced order (17). It is observedthat bandwidths using both the methodologies are in goodagreement with each other for the MLGNR and MWCNTinterconnects. Apart from this, the bandwidth of both theMLGNR and MWCNT substantially reduces for longerinterconnects. The reduction in bandwidth follows anexponentially decaying function and can be expressed throughcurve fitting as

Bandwidth = 1+ 2× exp (−length) (18)

Using the analytical expression of (17), the frequency response ofthe DIL is compared with the HSPICE-RF simulated results for

560& The Institution of Engineering and Technology 2014

different interconnect lengths (l ), MLGNR thickness (t) andMWCNT diameter (d ) as shown in Figs. 4a and b. The obtainedresults are in good agreement with each other. However, irrespect-ive of interconnect lengths, it is observed that the MWCNT exhibitsa lower fc in comparison to the MLGNR. This fact can be explainedin terms of dominating interconnect parasitics R′ESC and C ′ESC. Itleads to form an RC lowpass filter that has a bandwidth close tothe cutoff frequency, fc = 1/2π·R′ESC·C ′ESC. For similar interconnectdimension, the time constant (R′ESC·C′ESC) of the MWCNT is highercompared with the MLGNR (Table 1) which results in a lower fc forMWCNT interconnects. Therefore, irrespective of interconnectdimensions, the bandwidth of the MWCNT is alarmingly smallerby almost ten times and four times for local and global intercon-nects, respectively, in comparison to the MLGNR.

5. Conclusion: This Letter presents a comparative analysis of thefrequency response and bandwidth of MLGNR and MWCNTinterconnects. Using the DIL system, represented by the ESCmodel of MLGNR/MWCNT interconnects the transfer function isderived that is further used to obtain the cutoff frequency andbandwidth. It is observed that the MLGNR exhibits higherbandwidth compared with the MWCNT for various interconnectdimensions.

6 References

[1] Li H., Xu C., Srivastava N., Banerjee K.: ‘Carbon nanomaterials fornext-generation interconnects and passives: physics, status, and pro-spects’, IEEE Trans. Electron Devices, 2009, 56, (9), pp. 1799–1820

[2] Xu C., Li H., Banerjee K.: ‘Modeling, analysis, and design ofgraphene nanoribbon interconnects’, IEEE Trans. Electron Devices,2009, 56, (8), pp. 1567–1578

[3] Cui J.-P., Zhao W.-S., Yin W.-Y.: ‘Signal transmission analysis ofmultilayer graphene nano-ribbon (MLGNR) interconnects’, IEEETrans. Electromag. Compat., 2012, 54, (1), pp. 126–132

[4] Nasiri S.H., Moravvej-Farshi Md.K., Faez R.: ‘Stability analysis ingraphene nanoribbon interconnects’, IEEE Electron Device Lett.,2010, 31, (12), pp. 1458–1460

[5] Fathi D., Forouzandeh B., Mohajerzadeh S., Sarvari R.: ‘Accurateanalysis of carbon nanotube interconnects using transmission linemodel’, Micro Nano Lett., 2009, 4, (2), pp. 116–121

[6] Nishad A.K., Sharma R.: ‘Analytical time-domain models for per-formance optimization of multilayer GNR interconnects’, IEEEJ. Sel. Top. Quantum Electron., 2014, 20, (1), p. 3700108

[7] Kumar V., Naeemi A.: ‘Analytical models for the frequency responseof multi-layer graphene nanoribbon interconnects’. Proc. IEEE Int.Symp. Electromagnetic Compatibility (EMC’12), 2012, pp. 440–445

[8] Nasiri S.H., Faez R.: ‘Compact formulae for number of conductionchannels in various types of grapheme nanoribbons at various tem-peratures’,Mod. Phys. Lett. B, 2012, 26, (1), pp. 11500041–1150045

[9] Naeemi A., Meindl J.D.: ‘Performance modeling for single- andmultiwall carbon nanotubes as signal and power interconnects ingigascale systems’, IEEE Trans. Electron Devices, 2008, 55, (10),pp. 2574–2582

[10] Sadiku M.N.O.: ‘Elements of electromagnetics’ (Oxford universityPress, USA, 2009, 5th edn)

Micro & Nano Letters, 2014, Vol. 9, Iss. 9, pp. 557–560doi: 10.1049/mnl.2013.0742