IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 61, NO. 4, APRIL 2014 1193

Performance Analysis of Graphene NanoribbonField Effect Transistors in the Presence

of Surface RoughnessMajid Sanaeepur, Arash Yazdanpanah Goharrizi, and Mohammad Javad Sharifi

Abstract— Device performance of armchair graphene nanorib-bon field effect transistors in the presence of surface roughnessscattering is studied. A 2-D Gaussian autocorrelation functionis employed to model the surface roughness. Tight-bindingHamiltonian and nonequilibrium Green’s function formalism areused to perform atomic scale electronic transport simulation.The effect of geometrical and surface roughness parameterson the ON-current, the OFF-current, the transconductance, andthe subthreshold swing is investigated. Surface roughness canstrongly affect the device performance depending on how largeis the roughness amplitude or how small is the roughnesscorrelation length.

Index Terms— Device performance, graphene field effecttransistors, NEGF, quantum transport, subthreshold swing,surface roughness, transconductance.

I. INTRODUCTION

THE CONTINUED scaling of CMOS technology is reach-ing its limits due to short channel effects [1], [2]. Many

new materials have been explored as successive candidatesfor silicon [3]–[5]. Graphene a honey comb-shaped 2-D car-bon material has attracted most interest in both theoreticaland experimental scopes due to its novel electronic char-acteristics, such as high mobility and good compatibilityto the common planar semiconductor technology [6], [7].Large graphene sheets have no bandgap, therefore, are notsuitable for logic applications. An energy bandgap can beprepared by tailoring the graphene sheets to nanoribbonswhich are called graphene nanoribbon (GNR) [8], [9]. GNRfield effect transistors (GNRFETs) suffer from the line-edge roughness (LER) as investigated in the previous works[10]–[12]. Furthermore, experimental observations show thatall suspended and supported graphene sheets have some cor-rugation [13], [14]. For electronic applications, GNR shouldbe placed on a substrate (almost standard SiO2). Any sub-strate modulates the transport properties of GNRs through theeffects, such as charge inhomogenity, surface roughness andcharged impurities. The charge inhomogenity effect arises in

Manuscript received August 13, 2013; revised November 2, 2013; acceptedNovember 4, 2013. Date of publication November 14, 2013; date of currentversion March 20, 2014. The review of this paper was arranged by EditorA. C. Seabaugh.

The authors are with the Electrical and Computer Engineering Department,Shahid Beheshti University, Tehran 19834, Iran (e-mail: m_sanaee@sbu.ac.ir;ar_yazdanpanah@sbu.ac.ir; m_j_sharifi@sbu.ac.ir).

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

Digital Object Identifier 10.1109/TED.2013.2290049

graphene sheets wider than 1 μm and affects the transportproperties near the Dirac point so it should be neglected inthe case of a few nanometers wide GNRs studied in thispaper [15] and [16]. The effect of charged impurities on thetransport properties of bulk GNRs and the device performanceof GNRFETs has already been studied [17], [18]. Many effortshave been made to study the surface roughness of grapheneon various substrates, such as SiO2, BN, and Mica [19]–[23].The measured amplitudes of height variations due to surfaceroughness are 168–360, 75, and 24 pm for graphene on SiO2,BN, and Mica, respectively [21], [23]. However, the role ofthe surface roughness on the performance of GNRFETs hasnot been studied so far. It seems that the surface roughnessscattering can significantly affect the device performance ofGNRFETs. Therefore, the study of surface roughness effect isof great importance from the device engineering point of view.Although the edge profile of GNRs may be armchair, zigzag,or somewhat between the two, the armchair one is best suitedfor transistor applications [24], [25]. In this paper, the deviceperformance of armchair GNRFETs is investigated in thepresence of surface roughness and various device parameters,such as the ON-current, the OFF-current, the transconductance,and the subthreshold swing are compared with those of idealdevices (with a flat GNR channel). The surface roughnessis considered as a statistical random process and an ensem-ble average is performed over many devices to obtain anaveraged electronic behavior. Tight-binding Hamiltonian andNEGF formalism are used for atomistic transport simulation[26]–[28]. This paper is organized as follows. In Section II, thesimulation approach is presented. The effects of geometricaland roughness parameters on the performance of GNRFETsare investigated in Section III. Section IV summarizes theconcluding remarks.

II. NUMERICAL MODELS REVIEW

Numerical models for the atomistic simulation of GNRFETsin the presence of surface roughness scattering are presented.

A. Surface Roughness

A rough surface can be modeled by a 2-D autocorrelationfunction [29], [30]. Gaussian and exponential autocorrelationfunctions are almost used to create rough surfaces. We havechosen a Gaussian autocorrelation function for more compat-ibility with experimental data [13], [20], and [21]. A 2-D

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1194 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 61, NO. 4, APRIL 2014

autocorrelation function can be represented as

R(x, y)=SR2 exp(−x2/L2x −y2/L2

y) x =ndx and y =ndy

(1)

where SR is the root mean square of height fluctuations andLx (L y) represents the correlation length along x(y) axis. Thesampling intervals along the x and y directions (dx and dy) arechosen to be equal to acc/2 and

√3acc/2, respectively with acc

being the nearest neighbor C–C bond length in flat graphenewhich is equal to 1.42 Å. Because of stochastic nature ofsurface roughness one can treat it as a statistical randomprocess. To create random rough surfaces with the samesurface roughness parameters, we first evaluate the Fouriertransform of the autocorrelation function which is called powerspectrum. Adding a random phase to the power spectrumfollowed by an inverse Fourier transform, the rough surface inreal space is achieved [29], [30].

B. Electronic Transport

It is well established that for electronic applications the pZ

orbital of carbon atom is enough to construct tight-bindingHamiltonian [31].

Surface roughness modulates the position of carbon atomsaccording to the roughness profile. This out-of-plain defor-mation affects the strength of C–C bonds and alignment oftheir pZ orbital. These effects can be modeled by appropri-ate modification of corresponding hopping parameters in thedevice Hamiltonian. In this paper, we have considered one pZ

orbital per carbon atom. A first nearest neighbor tight-bindingparameter of t0 = 2.7 eV is used [32]. In the tight-bindingapproximation, the dependence of hopping integral, on thenearest neighbor C–C bond length can be written as [33]–[35]

tC−C(l) = t0 exp (−3.37(l/aCC − 1)) (2)

where t0 represents the pZ – pZ hopping integral of uncor-rugated graphene. NEGF formalism is used to simulate thedevice performance because of its abilityto capture all quantummechanical aspects and accuracy in numerical atomic scalesimulation. In the general context of nano-transistor modeling,the retarded green function of the channel in between the rightand left contacts can be written as [36]

G D = [(E + iη)I − HD − �L − �R

]−1 (3)

where η is an infinitesimally small quantity and HD is thedevice Hamiltonian, which can be evaluated using a tight-binding model as explained before.

∑L and

∑R are the

contact self-energy functions The transmission probability ofcarriers through the device can be evaluated as

T (E) = Trace[�L G D�R G†

D

](4)

where � is the contact broadening function obtained as

�L ,R = i(�L ,R − �†

L ,R

). (5)

Because the surface roughness is a source of inelastic scat-tering and the coherent length of GNR is much larger than

Fig. 1. (a) Representative schematic of simulated device structure. The Al2O3gate insulator is 1.5-nm thick with relative dielectric constant εr = 9.8.(b) 7 × 25 nm roughed surface GNR. (c) Average transmission probabilityof a 2 × 20 nm GNR as a function of energy for various surface roughnessamplitudes.

the channel length, the current can be evaluated via Landauerformula [37]

I = 2e

h

∫T (E) [ fR(E) − fL (E)] DE. (6)

Here, fL and fR are the right and left contact’s Fermifunctions, respectively.

III. NUMERICAL RESULTS

In this section, simulation results are presented along withthe physical expressions. Fig. 1(a) shows a representativeschematic of the structure of simulated device. A 7 × 25 nm,surface rough GNR is shown in Fig. 1(b). Top-gate geometryis used through 1.5 nm Al2O3 gate oxide with εr = 9.8. Thesource and the drain contacts are supposed to be heavily dopedextensions of GNR channel having no surface roughness.Power supply voltage is VD = 0.5 V and room temperature(T = 300 K) operation is assumed. In the rest of this paperunless otherwise mentioned, the amount of root mean squareof surface roughness and the correlation length are assumedto be 200 pm and 25 nm, respectively, which are typicalfor graphene on SiO2 substrate [13], [20]. Up to 200 devicesamples are generated with the same geometrical and surfaceroughness parameters to perform the ensemble average. Auseful parameter that helps us better understand and describethe device performance is the effective transport gap defined,where the transmission probability in bulk GNR drops below

SANAEEPUR et al.: ANALYSIS OF GRAPHENE NANORIBBON FIELD EFFECT TRANSISTORS 1195

Fig. 2. Average transfer characteristics of GNRFETs in (a) logarithmic and(b) linear scale as a function of surface roughness amplitude (L = 25 nm,W = 1.6 nm, and Lx = L y = 25 nm). The insets show the ON- andOFF-currents of rough GNRFETs as functions of surface roughness amplitude.(c) Average transmission probability for different values of surface roughnessamplitude (the solid black curve belongs to the ideal device and the dashedbrown represents the fS– fD).

10−3 (reference transmission) under which it has no experi-mentally significance [32], [38], and [39]. The transmissionprobability of a bulk GNR with respect to the energy forvarious surface roughness amplitudes is shown in Fig. 1(c).The inset shows intersect of the reference transmission line(dashed red) and the transmission probability curves in moredetail. As shown in figure, the transport gap is uninfluencedby increasing the roughness amplitude. The results, at least forGNR dimensions used in this paper, are incontrast to the LERwhere the transport gap increases by increasing the roughnessamplitude [40].

A. Role of the Roughness Amplitude

Fig. 2(a) shows the effect of surface roughness amplitudeonthe average transfer characteristics of GNRFETs in linear scalefor a 1.6 × 25 nm device. The inset shows the roughnessamplitude dependence of the ON-current. As shown in fig-ure by increasing the roughness amplitude, the ON-currentdecreases. The observed decrease in the ON-current is dueto the decreased transmission probability in the presence ofsurface roughness. Fig. 2(b) illustrates the average transfercharacteristics of the device in logarithmic scale for variousroughness amplitudes. The inset shows the OFF-current ofthe device with respect to the surface roughness amplitude.Any disorder which breaks the periodicity of a solid canchange the infinitely extended Bloch states into localizedstates with exponential decaying amplitude around the disorder[41]–[43]. By increasing the roughness amplitude deviationfrom the perfect periodic GNR increases and more localizedstates form in the bandgap. Therefore, the observed increase inthe OFF-current with respect to the increased roughness should

Fig. 3. Comparison between the average (a) transconductance and(b) subthreshold swing of rough GNRFETs (symbols) and those of a GNRFETwith perfect channel (dashed green line) as functions of the surface roughnessamplitude. Symbols show the exact values, and dashed red lines are fitted tothe data points (L = 25 nm, W = 1.6 nm, and Lx = L y = 25 nm).

Fig. 4. (a) Average transfer characteristics of rough GNRFETs for variousroughness correlation lengths in (a) linear and (b) logarithmic scales, respec-tively (L = 25 nm, W = 1.6 nm, and SR = 200 pm). (c) Average transmissionprobability for various correlation lengths (the dashed-dot curve belongs tothe ideal device and the dashed brown represents the fS– fD).

be attributed to the more localized states in the bandgap.Fig. 2(c) shows the origin of tunneling current through thecomparison of average transmission probabilities for varioussurface roughness amplitudes. Apparently, by increasing theroughness amplitude, the tunneling transmission probabilityincreases.

Fig. 3(a) compares the transconductance of roughGNRFETs as a function of surface roughness amplitude withthat of the ideal device. As shown in figure by increasing theroughness amplitude from 25 pm (mica) to 350 pm (about themaximum surface roughness for SiO2) the transconductancedecreases due to the decreased transmission probability inrougher surfaces. The subthreshold swing of the rough deviceincreases by increasing the roughness amplitude which is dueto the roughness enhanced tunneling current [see Fig. 3(b)].

B. Role of the Roughness Correlation Length

Correlation length is a measure of how smooth is thesurface. The larger the correlation length the smoother the

1196 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 61, NO. 4, APRIL 2014

Fig. 5. Averages of (a) ON-current, (b) OFF-current, and (c) ON-/OFF-currentratio as functions of the correlation length. Symbols show the exact values,and the dashed lines are fitted to the data points (L = 25 nm, W = 1.6 nm,and SR = 200 pm).

Fig. 6. (a) Transconductance and (b) subthreshold swing as functions ofcorrelation length in the presence of surface roughness (L = 25 nm, W =1.6 nm, and SR = 200 pm). The red curve is fitted to the exact data and thegreen dashed line represents the corresponding parameter of the ideal device.

surface will be. Fig. 4 shows the average transfer char-acteristics of a 1.6 × 25 nm rough device for variouscorrelation lengths in (a) linear and (b) logarithmic scales,respectively. As shown, larger correlation lengths lead inthe more ON-current, but less OFF-current. The observedincrease in the ON-current is due to the increased trans-mission probability in the smoother surfaces. By increasingthe correlation length, a smoother surface is created. As aresult by increasing the correlation length, the OFF-currentdecreases [Fig. 4(b)]. Effect of various correlation lengthson the average transmission probability of rough devices arecompared in Fig. 4(c). As shown in the inset by increasingthe correlation length from 5 to 50 nm, the tunneling trans-mission decreases. The dashed-dot curve shows that the min-imum tunneling current corresponds to the ideal device. TheON-, the OFF-, and the ON-/OFF-current are shown as functionsof correlation length in Fig. 5. The ON-/OFF-current increasesby increasing the correlation length due to the increasedON- and decreased OFF-currents.

The transconductance and subthreshold swing as func-tions of correlation length are depicted in Fig. 6.

Fig. 7. Averages of (a) ON-current, (b) OFF-current, and (c) ON-/OFF-currentratio as functions of the device length. Symbols show the exact values, andthe dashed lines are fitted to the data points (W = 1.6 nm, SR = 200 pm,and Lx = L y = 25 nm).

Fig. 8. Comparison between the averages of (a) transconductance and(b) subthreshold swing of rough GNRFETs and those of a GNRFET withperfect channel as functions of the channel length. Symbols show theexact values and dashed lines are fitted to the data points (W = 1.6 nm,SR = 200 pm, and Lx = L y = 25 nm).

Apparently, by increasing the roughness correlation length,both the transconductance and the subthreshold swingimprove. The observed increase (decrease) in the transconduc-tance (subthreshold swing) for increased correlation lengthsis due to the increased (decreased) transmission probability(tunneling current) in smoother surfaces.

C. Role of the Device Length

Fig. 7(a) and (b) shows that by increasing the lengthfrom 10 to 70 nm, the ON- and OFF-currents for roughdevice decrease by factors of about 2 and 1000, respectively.Consequently the ratio of the ON- to OFF-current increases bya factor of about 50. The observed decrease in the ON-currentis due to the reduced transmission probability, while that of theOFF-current is attributed to the decreased tunneling current inthe long channel lengths. Fig. 8(a) compares the transconduc-tance of the surface roughed GNRFETs as a function of devicelength with that of ideal devices. In the case of ideal devices,the transconductance of GNRFETs almost remains constant

SANAEEPUR et al.: ANALYSIS OF GRAPHENE NANORIBBON FIELD EFFECT TRANSISTORS 1197

Fig. 9. Averages of (a) ON-current, (b) OFF-current, and (c) ON-/OFF-currentratio as functions of the device width. Symbols show the exact values, andthe dashed lines are fitted to the data points (L = 25 nm, SR = 200 pm, andLx = L y = 25 nm).

with the channel length due to ballistic transport of carriers.However, the transconductance of surface roughed GNRFETsdecreases due to the reduced transmission probability as aresult of carrier scattering in the presence of surface roughness.The subthreshold swing of surface roughed device and idealone are compared for various channel lengths in Fig. 8(b). Asapparent in larger lengths the subthreshold swing decreases forboth rough and ideal device due to the stronger electrostaticcontrol of the gate in the larger lengths. In short channeldevices a larger subthreshold swing is obtained for surfaceroughed devices due to the enhanced tunneling current.

D. Role of the Device Width

Fig. 9 shows that by increasing the device width from 0.7 to3 nm the ON-current of rough device increases linearly bya factor of 36 and the OFF-current increases by a factor of1.3 × 108; therefore, the ratio of the ON- to OFF-currentfor rough device decreases by a factor of 2.7 × 10−7.The increased ON-current is due to the increased number ofconducting channels which in turn causes an increase in thetransmission probability. The OFF-current increases due to thedecreased bandgap in larger device widths. In rough devicesby increasing the width the OFF-current increases even moredue to the enhanced tunneling current through the localizedstates in the bandgap. Both the ON- and OFF-currents howeverdegrade in the presence of surface roughness. Fig. 10 compares(a) transconductance and (b) subthreshold swing of roughGNRFETs with those of an ideal one at various device widths.As apparent by increasing the width, the transconductanceand the subthreshold swing increase for both rough and idealdevices. The observed increase in the transconductance isdue to the increased number of conducting channels, whilesubthreshold swing increases due to the decreased bandgap.Both the transconductance and the subthreshold swing are

Fig. 10. Comparison between the averages of (a) transconductance and(b) subthreshold swing of rough GNRFETs and those of a GNRFET with per-fect channel as functions of the channel width. Symbols show the exact values,and dashed lines are fitted to the data points (L = 25 nm, SR = 200 pm, andLx = L y = 25 nm).

degraded in the presence of surface roughness due to thedecreased transmission probability and the more tunnelingcurrent, respectively.

IV. CONCLUSION

In this paper, the effect of geometrical and surface roughnessparameters on GNRFETs performance metrics are compre-hensively studied by means a 2-D Gaussian autocorrelationfunction model for surface roughness and NEGF formalism.Simulation results show that surface roughness can stronglyaffect device performance metrics. By increasing the lengthin the presence of surface roughness device performanceimproves in terms of the OFF-current and subthreshold swing,while degrades in terms of the ON-current and transconduc-tance. By increasing the width device performance improves interms of the ON-current and transconductance, while degradesin terms of the OFF-current and subthreshold swing. Largerroughness amplitudes or smaller correlation lengths degradeall device performance parameters and vice versa.

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Majid Sanaeepur received the M.Sc. degree in elec-tronics engineering from Shahid Beheshti University,Tehran, Iran, in 2009, where he has been pursuingthe Ph.D. degree since 2011.

Arash Yazdanpanah Goharrizi received the Ph.D.degree in electronics engineering from the Univer-sity of Tehran, Tehran, Iran, in 2012.

He is currently with the Department of Electricaland Computer Engineering, Shahid Beheshti Univer-sity, Tehran.

Mohammad Javad Sharifi received the Ph.D.degree in electronics engineering from the AmirkabirUniversity of Technology, Tehran, Iran, in 1998.

He is currently with the Department of Electricaland Computer Engineering, Shahid Beheshti Univer-sity, Tehran.