variability of bandgap and carrier mobility caused by edge defects...

Solid-State Electronics xxx (2015) xxx–xxx

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Solid-State Electronics

journal homepage: www.elsevier .com/locate /sse

Variability of bandgap and carrier mobility caused by edge defectsin ultra-narrow graphene nanoribbons

http://dx.doi.org/10.1016/j.sse.2014.12.0120038-1101/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author.E-mail address: [email protected] (M. Poljak).

Please cite this article in press as: Poljak M et al. Variability of bandgap and carrier mobility caused by edge defects in ultra-narrow graphene nanorSolid State Electron (2015), http://dx.doi.org/10.1016/j.sse.2014.12.012

M. Poljak a,⇑, K.L. Wang b, T. Suligoj a

a Faculty of Electrical Engineering and Computing, University of Zagreb, Zagreb HR 10000, Croatiab Department of Electrical Engineering, University of California at Los Angeles, Los Angeles, CA 90095, USA

a r t i c l e i n f o

Article history:Available online xxxx

The review of this paper was arranged byB. Gunnar Malm

Keywords:Graphene nanoribbonsNEGF simulationEdge defectsTransport gapMobility simulationVariability

a b s t r a c t

We report the results of multi-scale modeling of ultra-narrow graphene nanoribbons (GNRs) that com-bines atomistic non-equilibrium Green’s function (NEGF) approach with semiclassical mobility modeling.The variability of the transport gap and carrier mobility caused by random edge defects is analyzed. Wefind that the variability increases as the GNR width is downscaled and that even the minimum variationof the total mobility reaches more than 100% compared to average mobility in edge-defectednanoribbons. It is shown that scattering by optical phonons exhibits significantly more variability thanthe acoustic, line-edge roughness and Coulomb scattering mechanisms. The simulation results demon-strate that sub-5 nm-wide nanoribbons offer no improvement over conventional bulk semiconductors,however, GNRs are comparable with sub-7 nm-thick silicon-on-insulator devices in terms of mobility-bandgap trade-off characteristics.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

When considering graphene as channel material for futurenanoelectronic applications, its main advantage is the high carriermobility that reaches up to �200,000 cm2/Vs in suspendedsamples at low temperature [1] and up to �10,000 cm2/Vs at roomtemperature when graphene is on SiO2 [2]. However, in order toachieve switching for digital applications a bandgap must beinduced since the valence and conduction band touch at Diracpoints which results in a zero bandgap [3]. The OFF state perfor-mance of a transistor, i.e. the OFF state leakage, and ON–OFFcurrent ratio are not crucial for analog applications. Therefore, inthis case the high carrier mobility could still be a deciding factorfor using graphene as channel material since carrier transportproperties determine the ON-state performance. The moststraightforward method of achieving the bandgap is the fabricationof narrow graphene nanoribbons (GNRs) in which the bandgap isinduced by quantum confinement along the nanoribbon width[4–6]. In contrast to large-area graphene, the GNRs exhibit massivecarriers which decreases the mobility [5,7] and, hence, it is neces-sary to study the impact of GNR width downscaling on the mobilityand bandgap simultaneously.

In order to investigate the suitability of ultra-narrow GNRs forhigh-speed field-effect transistors (FETs), both experimental andcomputational studies on bandgap [6,8,9], ON-state conductionand ON–OFF current ratio [10–12], and mobility behavior[13–16] have been reported previously. However, the feasibilitystudies of GNR FETs for the future extremely-scaled CMOStechnology nodes must also address device variability becausethe random variation from device to device is one of the mainissues when the manufacturability of GNR FETs is considered.There are only a few reports on the variability of device propertiesof GNR FETs, mostly because the atomistic NEGF simulations thatare used to investigate GNR devices are computationally verydemanding. For example, Basu et al. [17] obtained the draincurrent variation for �4 nm wide and �17 nm long GNRs on anensemble of N = 10, Yoon et al. [18] reported a histogram of theON current for N = 100 �1.4 nm-wide and 15 nm-long GNRs, whileLeong et al. [19] performed statistical simulations of the ON–OFFcurrent ratio for 2 nm-wide GNR devices on a sample of N = 50.Finally, Choudhury et al. [20] reported a statistical analysis of thefrequency and power dissipation of a ring oscillator implementedwith 2 nm-wide GNR FETs. Therefore, a thorough statisticalanalysis of variability of the main electronic and transportproperties of GNRs is necessary in order to assess the feasibilityof these devices for future ultra-scaled FETs.

ibbons.

http://dx.doi.org/10.1016/j.sse.2014.12.012

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Fig. 1. Illustration of a randomly-generated GNR with edge defects that areindicated with arrows. L = 10.1 nm and W = 1.1 nm.

2 M. Poljak et al. / Solid-State Electronics xxx (2015) xxx–xxx

We believe that, in order to perform meaningful analysis of GNRvariability relevant for transistor applications at the nanoscale, thefollowing steps are necessary: (i) to perform statistical analysis fora wider range of GNR widths, (ii) to increase ensemble size sincethe number of defects is large which leads to a very large numberof possible defect arrangements, and (iii) to take into account theskewness and kurtosis of the obtained distributions together withthe mean and standard deviation values. In this paper, we provideimportant information regarding the feasibility of GNR-basednanoelectronics from the statistical variability point-of-view. Westudy the ultra-narrow GNRs with edge defects and quantify thevariability of the effective bandgap, i.e. transport gap, and carriermobility caused by random edge defects that are present inrealistic nanoribbons. Our multi-scale approach combines atomis-tic quantum transport simulations and semiclassical mobilitysimulations of large ensembles (N = 200) of randomly-generated GNRs.Statistical analysis is performed for several device widths in the1–5 nm range as those GNRs exhibit bandgaps larger than �0.2 eV.

2. Numerical modeling

The investigation of transport properties of GNRs in this work isbased on a multi-scale approach. We first employ the atomisticquantum transport based on the non-equilibrium Green’s function(NEGF) formalism in order to explore the consequences of edgedefects on the transmission and effective bandgap. Second, basedon the knowledge obtained in the previous step we develop amodel for the density of states (DOS) for GNRs that covers the sto-chastic effects caused by edge defects. Finally, the DOS model isemployed to calculate carrier mobility and its variability causedby edge defects within a semiclassical approach.

The quantum transport study is based on a tight-bindingHamiltonian with a single pz orbital per carbon atom in the GNR,and it accounts for up to the third-nearest neighbor interaction.The Hamiltonian is given by the following equation:

H ¼X

i

eicyi ci þ

X3

k¼1

tk

Xi;j

cyi cj þ H:c:; ð1Þ

where ei is the on-site energy, and tk are the hopping parameters[21]. A modified hopping parameter is used for edge carbon–carbonbonds in order to take the edge-bond relaxation effect into account[22]. The properties of GNRs relevant for transport studies areobtained by using the NEGF formalism [23], in which the deviceGreen’s function is obtained as

Gd ¼ ðEþ i0þÞI � H � R1 � R2� ��1

; ð2Þ

where H is the device Hamiltonian calculated using (1), and R1,2 arethe contact self-energy matrices that describe the coupling betweenthe device and contacts [8]. Surface Green’s functions that areneeded in order to calculate R1,2 are found using an iterativemethod from [24]. The transmission function between the contacts1 and 2, i.e. source and drain, is found from

T12ðEÞ ¼ Trace C1GdC2Gyd� �

; ð3Þ

where C1,2 are contact broadening functions. The NEGF simulationswere performed on ensembles of semiconducting armchair GNRswith the widths of 1.10, 1.84, 2.58, 3.32. 4.06 and 4.80 nm. Thelength of GNRs is kept at 10.1 nm and, therefore, the results providethe transport properties of extremely-scaled GNR FETs for the12 nm CMOS technology node [25]. The simulated GNRs have roughedges and edge defects are implemented by randomly removing50% of edge carbon atoms as illustrated in Fig. 1. For each of theremoved carbon atoms its orbital is removed from the Hamiltonianby setting the respective hopping parameter to zero. In order to

Please cite this article in press as: Poljak M et al. Variability of bandgap and carrSolid State Electron (2015), http://dx.doi.org/10.1016/j.sse.2014.12.012

study the variability of transport properties of GNRs using the NEGFapproach, we simulate N = 200 randomly generated devices for eachwidth, and report the features of transmission and the extractedtransport gap.

The scattering-dominated transport in ultra-narrow GNRs isinvestigated by calculating carrier mobility within the semiclassi-cal approach, i.e. using the Fermi golden rule and momentumrelaxation time (MRT) approximation. Derivation of MRTs foracoustic (AP) and optical (OP) phonons, line-edge roughness(LER), and Coulomb scattering (CO) is based on spinor-like wave-functions for graphene and the decomposition procedure from[13]. The mobility simulations are based on the model reportedin detail in [16] and here we list the main equations as they areneeded for the discussions in Section 3. The MRT for AP scatteringfor the n-th subband is given by the following expression:

1sAP;nðEÞ

¼ pNphD2APEAP

4�hqv2s W

gGNR;nðEÞ; ð4Þ

where Nph, DAP and EAP is the phonon number, deformation potentialand energy, respectively. The AP energy is found using

EAP ¼ 2vS=vF

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � E2

n

q: Scattering by OPs is described by the follow-

ing MRT

1sOP;nðEÞ

¼ p�hD2OP

4qEOPWNph

1� f 0 EþEOPð Þ1� f 0ðEÞ

gGNR;n EþEOPð Þ�

þ Nphþ1� 1� f 0ðE�EOPÞ

1� f 0ðEÞgGNR;n E�EOPð ÞH E�En�EOPð Þ

; ð5Þ

where DOP is the deformation potential strength and EOP is the OPenergy. The LER scattering for the n-th subband is calculated using

1sLER;nðEÞ

¼ pE2nH2

�hW2

K

1þ 4k2yK

2gGNR;nðEÞ; ð6Þ

where H is the amplitude and K is the correlation length of theroughness, En is the subband energy, and ky is the wave-vector com-ponent in the transport direction that is calculated as

ky ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � E2

n

q�hvF= . Finally, the MRT in the case of CO scattering is

found as

1sCO;nðEÞ

¼ p2�h

e2

4pe0eenv

� �2 Nint

W2

Z W

0

Z W

0K0ð2kyDÞdx

2

dx0

� gGNR;nðEÞ; ð7Þ

where Nint is the interface charged impurity density. Screening isincluded in the calculation of the LER and CO MRTs according to[16]. The total MRT is obtained by summing up the contributionsof all subbands and all scattering mechanisms described above.The mobilities limited by each of the scattering mechanisms, i.e.lAP, lOP, lLER and lCO, are calculated using the Kubo–Greenwoodformula [13] and parameters listed in Table 1.

As previously stated, the sensitivity of carrier mobility to edgedefects is quantified by taking into account the results from theatomistic NEGF modeling via the analytical GNR DOS model thatenters Eqs. (4)–(7). Therefore, a direct inclusion of phonon and

ier mobility caused by edge defects in ultra-narrow graphene nanoribbons.


Table 1List of parameters used in the simulation of carrier mobility in GNRs. Parameters forAP and OP scattering are taken from [13].

Scattering mechanism Parameters

Acoustic phonons DAP = 16 eVq = 7.6 � 10�8 g/cm2

vs = 2 � 104 cm/s

Optical phonons DOP = 1.4 � 109 eV/cmEOP = 160 meV

Line-edge roughness H = 0.5 nmK = 2 nm

Coulomb eenv = 2.45Nint = 8.5 � 1012 cm�2

M. Poljak et al. / Solid-State Electronics xxx (2015) xxx–xxx 3

Coulomb scattering in the NEGF solver is omitted, which makesthis approach more computationally efficient and, consequently,allows an acceptable duration of large-ensemble mobility simula-tions. We expect the results presented hereafter for defected arm-chair GNRs to be qualitatively valid for zig-zag nanoribbons aswell, since the dependence of GNR transport properties on nano-ribbon type is suppressed in devices with defected edges [8].

3. Results and discussion

3.1. Variability of the transport gap

Fig. 2 reports the impact of edge defects on the transmission ofGNRs with the widths of 1.84, 3.32 and 4.80 nm, obtained usingNEGF simulations. From the comparison of transmissions in theideal GNRs (dashed line) and the averaged transmission of edge-defected GNRs (full line), we find that the transmission is greatlyreduced by edge defects. The deterioration is stronger at lowerenergies, which leads to the effective increase of the bandgap, i.e.

Fig. 2. Transmission of GNRs with edge defects for nanoribbon widths of (a)1.84 nm, (b) 3.32 nm and (c) 4.80 nm. Transmission of ideal devices is shown forcomparison. N = 200 for each width.


it results in the formation of the transport gap. If we define thetransport gap as the energy range where the averaged transmissionis lower than 0.01, we can clearly observe that the energy gapincreases in defected GNRs. For example, half-gap (Ed) increasesfrom �250 meV to �350 meV for W = 1.84 nm [Fig. 2(b)], whileFig. 2(d) gives an increase from �65 meV to �110 meV for the4.80 nm-wide GNR. Fig. 2 also contains the transmissions of allrandomly generated nanoribbons, which demonstrates thevariability of quantum transmission caused by stochasticarrangement of edge defects. The variability of transmission and,hence, the transport gap, increases as the GNR width is scaleddown. For the 4.80 nm-wide nanoribbons, Ed ranges from�60 meV to �280 meV, whereas for W = 1.84 nm the half-gapextends from �250 meV up to �1090 meV. In addition to thevariability of the half-gap, the reported transmission variabilityintroduces the sensitivity of ON and OFF state conductance, andON–OFF ratio on edge defects in ultra-scaled GNRs [12]. Moreover,it is reasonable to assume that the variability of Ed also has conse-quences on scattering-dominated transport, even in long GNRs,since MRTs described in Section 2. depend directly on the DOS.

The randomness of Ed demands a statistical approach fordescribing the properties of the half-gap in defected GNRs. Hence,for each GNR width we analyze the statistical distribution of Ed andprovide the main parameters such as the mean, mode, 25th and75th percentiles, together with the skewness (S) and kurtosis (K)of the distribution. Fig. 3 shows histograms of the half-gap forGNRs that are 1.84, 3.32 and 4.80 nm wide, in order to illustratethe impact of GNR width downscaling that modifies the distribu-tions both qualitatively and quantitatively. For example, the1.84 nm-wide devices [Fig. 3(a)] exhibit a mean half-gap (hEdi) of0.609 eV and standard deviation (rEd) of 0.133 eV, with a some-what skewed distribution (S = 0.527). We qualitatively observe asimilar positive skewness for W = 3.32 nm in Fig. 3(b), as evidencedquantitatively by the obtained S = 0.653. In contrast, the distribu-tion for the 4.80 nm-wide GNRs exhibits a considerable positiveskewness, i.e. the maximum of the distribution exhibits a strongershift towards lower values than in narrower devices, as evident inthe histogram in Fig. 3(c). The calculated skewness in this caseequals S = 1.718, which is almost 3� higher than for the 1.84 nm-wide GNR. The variation range and the half-gap increase withGNR width decrease is more readily understood from Fig. 4, whichreports box-and-whisker plots for all the examined devices. Start-ing from the top, the characteristic points are the maximum, 75thpercentile, median and mean value, 25th percentile and the mini-mum value of Ed for a given nanoribbon width. As the widthdecreases, hEdi of defected GNRs increases from 0.103 eV(W = 4.80 nm) up to 1.711 eV (W = 1.10 nm). Similarly, the varia-tion of the half-gap increases in its absolute value when the widthis scaled down. The box plots in Fig. 4 demonstrate that thedistributions in narrower GNRs are more normal-like since the per-centile borders are close to the center of the min–max whiskerrange and, additionally, the mean and mode are almost identical.For wider GNRs, the 25th–75th percentile range is narrow withthe mode and mean value being closer to the minimum reportedhalf-gap for the specific width, which means that the distributionis positively skewed. The half-gap distribution parameters for allthe examined GNRs are listed in Table 2, and are later used forthe random generation of DOS curves for ensemble mobility simu-lations. Table 2 shows that the skewness generally increases inwider nanoribbons, whereas the kurtosis is similar in all devicesexcept for the 4.80 nm-wide GNR that exhibits a highly-peakedand long-tailed distribution (S = 1.718, K = 2.343). The overallincrease of S and K is understandable because the impact of edgedefects is diminished in wider nanoribbons and, hence, thebandgap values of edge-defected GNRs concentrate tightly aroundthe mean value. The behavior of skewness is closer to a monotonic



Fig. 3. Histograms showing normalized frequencies of half-gap values obtained by simulation of N = 200 GNRs for the width of (a) 1.84 nm, (b) 3.32 nm and (c) 4.80 nm.

Fig. 4. Impact of GNR width downscaling on the half-gap statistics. The box plotsshow the mean and median values together with the variation range for each GNRwidth.

Table 2Width dependence of the mean and standard deviation, and of the skewness andkurtosis of the half-gap distribution.

Width (nm) Half-gap distribution parameters

Mean (eV) Std. dev. (eV) Skewness Kurtosis

1.10 1.711 0.301 0.527 �0.0291.84 0.609 0.133 0.693 0.1652.58 0.350 0.084 0.610 �0.0503.32 0.196 0.048 0.653 0.1934.06 0.111 0.034 0.856 0.1434.80 0.103 0.022 1.718 2.343


dependence on GNR width, while the same cannot be said for kur-tosis that exhibits positive and negative values. Nevertheless, wenote that kurtosis values for 1.10–4.06 nm wide nanoribbons areclose to zero, especially in comparison to the case whenW = 4.80 nm, which indicates that the corresponding distributionsare not strongly peaked. The non-monotonic behavior of S and K isattributed to the ensemble size (N = 200), which is clearly not largeenough to obtain a smooth dependence of skewness and kurtosison GNR width. The results presented in Figs. 3 and 4, includingTable 2, illustrate the importance of examining the statistical dis-tribution for various widths since the skewness and kurtosis ofthe distribution, not only mean and standard deviation, dependon the width of the nanoribbon.


3.2. Analytical DOS model for GNRs with edge defects

The DOS of GNRs with edge defects is modeled as

gGNRðEÞ ¼nsplv2p�hvF

Xn

EffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � ðnEdÞ2

q ; n ¼ 1;2;3 . . . ; ð8Þ

where the valley multiplicity equals lv = 2, while nsp = 2 whencalculating the DOS and nsp = 1 when calculating the mobility in(4)–(7) because the spin does not change under the influenceof scattering processes considered in this work. The randomly-generated half-gap of a single edge-defected GNR, which is equalto the bottom subband level in Eq. (8) for n = 1, is determinedaccording to the MATLAB function

Ed ¼ pearsrndðhEdi; rEd; S; K þ 3; N; 1Þ; ð9Þ

where hEdi, rEd, S and K are the half-gap distribution parameterslisted in Table 2. for each device width, and N is the ensemblesize. The variation of Ed induces the variability of the DOS indefected GNRs, which results in the sensitivity of lAP, lOP, lLER

and lCO to edge defects. For each GNR width, we generateN = 200 random DOS curves defined by (8) and calculate themobilities N times according to Eqs. (4)–(7) and the Kubo–Green-wood formula. This approach allows the examination of the var-iability and average carrier mobility in GNRs with edge defectswithout the need to incorporate scattering in the NEGF solver.This, in turn, makes our approach much less computationallydemanding and much less time-consuming. The mobility ofgraphene nanoribbons with ideal edges is calculated using thetheory in Section 2, and the DOS model and bandgap values forideal GNRs reported in [9].

3.3. Variability of mobility

The sensitivity of lAP, lOP, lLER and lCO to edge defects is studiedwith respect to mobility dependence on inversion charge density(Ninv). Fig. 5 reports the results for the 2.58 nm-wide GNR and itcontains the averaged mobility (full line) for comparison withthe mobility variation range. The AP-limited mobility ranges from�3900 cm2/Vs up to �28,800 cm2/Vs with the averaged lAP being�12,600 cm2/Vs at Ninv = 1012 cm�2. An even stronger spread isobserved for lOP, where the mobility ranges almost four ordersof magnitude, from �2600 cm2/Vs up to �5.5 � 106 cm2/Vs, withthe averaged lOP at Ninv = 1012 cm�2 of�5.4 � 105 cm2/Vs. The sen-sitivity of the LER-limited mobility is comparable to lAP, rangingfrom �3200 cm2/Vs up to �16,500 cm2/Vs with the average being�7100 cm2/Vs. Finally, the average lCO equals �830 cm2/Vs withthe lower and upper values of �2100 cm2/Vs and �4400 cm2/Vs,



Fig. 5. Sensitivity of (a) lAP, (b) lOP, (c) lLER and (d) lCO to edge defects in 2.58 nm-wide GNRs. The full black line shows the averaged GNR mobility curve. N = 200.


respectively, at the inversion density of 1012 cm�2. Therefore, thehigh variability of the transport gap caused by edge defects trans-lates into high variability of the low-field mobility, irrespective ofthe scattering mechanism. The results for the 4.06 nm-wide GNR,shown in Fig. 6, demonstrate that the sensitivity of the AP, LERand CO-limited mobilities to edge defects are similar or somewhatlower than in the case of the 2.58 nm-wide nanoribbon. However,the variation of lOP is considerably reduced in comparison to thenarrower nanoribbon, e.g. at Ninv = 1012 cm�2 the OP-limitedmobility ranges from �5.8 � 105 cm2/Vs up to �2.8 � 107 cm2/Vs.This variation range is slightly less than two orders of magnitude,compared to almost four orders of magnitude for W = 2.58 nm. Theresults presented in Figs. 5 and 6 demonstrate that wider GNRs areless sensitive to variability introduced by edge defects than narrownanoribbons, but the variation remains quite high nevertheless.

Before continuing onto the analysis of variability, the qualita-tive dependence of lAP, lOP, lLER and lCO on Ninv is brieflyaddressed. The AP-limited mobility decreases with the downscal-ing of GNR width, as evident from the comparison of Figs. 5(a)and 6(a). Similarly, lAP slightly decreases in strong inversion due

to increased EAP �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � E2

n

qand higher gGNR(E) at higher energies

[cf. Eq. (4)]. In the case of the OP-limited mobility, scatteringincreases as the width becomes smaller, and it increases consider-ably in strong inversion [see Figs. 5(b) and 6(b)]. This is a conse-quence of the higher carrier energy that allows the emission ofOPs as seen in the emission term of Eq. (5). For W = 4.06 nm inwhich the half-gap equals �0.1 eV, the deterioration of lOP beginsat Ninv � 2 � 1012 cm�2 which corresponds to EF � 200 meV. This

Fig. 6. Sensitivity of (a) lAP, (b) lOP, (c) lLER and (d) lC


energy level approximately coincides with first subband increasedby the OP energy, which is the lowest energy level for which OPemission is possible. From the comparison of Figs. 5(c) and 6(c),we observe a stronger degradation of lLER when the width is scaleddown. The lLER exhibits a weak increase in strong inversion, whichis a consequence of the increased screening since eD � gGNR(E). Sim-ilarly, the CO-limited mobility also increases for higher Ninv due toscreening, as reported in Figs. 5(d) and 6(d). The lCO generallydecreases with the downscaling of GNR width [15], however, thenarrow nanoribbons reported in Figs. 5 and 6 exhibit a comparableCO-limited mobility. This behavior is caused by the interplay of dif-ferent mechanisms such as subband population and screening insub-5 nm-wide GNRs [16].

Fig. 7 presents the influence of GNR width downscaling on themobility in ideal devices, average mobility and variability of thelAP, lOP, lLER and lCO for the GNR width range investigated byNEGF simulations. The shaded area shows the variation of mobilityin realistic GNRs obtained by simulation of N = 200 devices for eachGNR width. A monotonic mobility degradation is observed for boththe ideal and averaged mobility curves when the GNR width isscaled down, for all scattering mechanisms except CO scattering.Namely, the average lCO is relatively constant in sub-5 nm-wideGNRs with an enhancement effect observed for W � 2 nm, whilethe CO-limited mobility in ideal devices behaves similarly but witha mobility increase effect at W � 1 nm. As the width is scaleddown, the average and ideal mobility curves diverge, which indi-cates a more pronounced impact of edge defects on the mobilityin narrower devices. The average lAP and lLER have a similar

O to edge defects in 4.06 nm-wide GNRs. N = 200.



Fig. 7. Dependence of (a) lAP, (b) lOP, (c) lLER and (d) lCO on nanoribbon width in ideal and defected GNRs. The mobilities are extracted at Ninv = 1012 cm�2. The shaded areaindicates the range of mobility variation (min and max values) caused by random edge defects. For each device width, ensemble size is N = 200 in semiclassical mobilitysimulations.


mobility-width characteristic, with both curves decreasingfrom �105 cm2/Vs for W = 4.80 nm down to �30 cm2/Vs forW = 1.10 nm. As the width is decreased, the degradation of averagelOP equals six orders of magnitude. Finally, the averagedCO-limited mobility is �2000 cm2/Vs with a local maximum atW = 1.84 nm that equals �3600 cm2/Vs. The lAP, lOP, and lLER

monotonically decrease mainly due to �W�1 dependence of 1/sAP

and 1/sOP, and �W�2 dependence of 1/sLER [see Eqs. (4)–(6)]. Inaddition, the LER MRT also depends on � E2

n; which increases theLER scattering since the subbands are shifted up in narrower GNRs.The Coulomb scattering also exhibits a �W�2 dependence but nei-ther ideal nor average lCO experiences a strong deterioration withthe downscaling of GNR width. In contrast, the CO scattering issuppressed in ultra-narrow devices due to contributions ofincreased screening and a strongly decreasing integral in Eq. (7)

because of the dependence ky �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiE2 � E2

n

q:

The variation of lAP in GNRs with edge defects [Fig. 7(a)] is quitehigh, i.e. the lAP ranges �4.4 � 104–1.2 � 105 cm2/Vs forW = 4.80 nm and �0.1–320 cm2/Vs for W = 1.10 nm. As the widthdecreases, the shaded area generally widens [except for lOP

reported in Fig. 7(c)] which indicates that narrower GNRs are lessimmune to variability. Qualitatively, the variability of lLER andlCO is comparable to that of lAP, but the sensitivity of lOP to edgedefects is much higher than for the other scattering mechanisms.In order to quantify the variability, we plot in Fig. 8 the variation

Fig. 8. Dependence of the mobility variation range on GNR width for lAP, lOP, lLER

and lCO. The variation percentage is calculated relatively to the average mobility fora given width.


percentage for each GNR width, calculated as 100% � (lmax–lmin)/hli,i.e. the variation range compared to the average mobility. Thevariation percentage increases when the width is downscaled.The curves obtained for lAP, lLER and lCO are almost identical, withvariation values from 81% to 214% for 2–5 nm-wide nanoribbonsand up to 1032% for W = 1.10 nm. In contrast, the width-depen-dence of the variability of OP-limited mobility is not monotonic.The variation percentage sharply increases for W < 3.32 nm andreaches a maximum of 982%. The OP scattering is the only inelasticscattering mechanism considered in our simulations and 1/sOP

depends on gGNR(E ± EOP) [cf. Eq. (5)]. In turn, this form of theDOS for OP scattering is a source of increased sensitivity sincethe subbands in the already random DOS are shifted additionallyby ± EOP for OP emission and absorption. Nevertheless, it is notexpected that the very high variability of lOP will translate into avery high variation of the total mobility, at least for W > 2 nm, sincelOP is significantly higher than lAP, lLER and lCO.

The behavior of the total averaged mobility (calculated usingthe Matthiessen’s rule) together with its variation range for eachGNR width is shown in Fig. 9. The plot also contains the simulatedmobility curve for ideal devices and experimental data from [5]with horizontal error bars indicating the uncertainty of GNR width.We note that the measured data falls within the variation range forW < 2.5 nm, whereas the simulated total mobility is unrealisticallyhigh compared to experiment for W > 2.5 nm. This disagreement isattributed to the absence of scattering induced by remote phonons

Fig. 9. Impact of GNR width downscaling on the total mobility in ideal and defectedGNRs. The shaded area indicates the range of mobility variation (min and maxvalues) caused by random edge defects. Experimental data from [5] is shown forcomparison.



Fig. 11. Interdependence between the lTOT obtained by semiclassical mobilitysimulations and Ed obtained by atomistic NEGF modeling. The error bars show thevariability of lTOT and Ed in defected devices for each GNR width. Data points forbulk semiconductors and UTB SOI are inserted for comparison (taken from [28] and[29]).


from the substrate [26,27] and vacancy-induced scattering [12,15]in our model. The addition of these mechanisms would decreasethe highest values of the lTOT curve well below �500 cm2/Vs. Nev-ertheless, the results reported in Fig. 9 allow a qualitative assess-ment of the impact of edge defects on the total mobility. We findthat lTOT exhibits similar features as the lAP and lLER, i.e. totalmobility monotonically decreases as the width is scaled down inthe case of ideal and defected GNRs. The lTOT in ideal GNRsdecreases from �2280 cm2/Vs to �360 cm2/Vs when W is down-scaled from 4.80 nm to 1.10 nm. Similarly, the average lTOT indefected nanoribbons changes from �1800 cm2/Vs to �9 cm2/Vs,which is �1.3� and �40� lower compared to ideal nanoribbonsof the same width. The results in Fig. 9 also demonstrate a highvariation of the total mobility, inherited from the variability thatis introduced by all scattering mechanisms, and they also showthat the variability increases in narrower devices. The extent ofvariation is more readily understood from Fig. 10, which reportsthe percentage of mobility variation range relative to the averagedlTOT for a given nanoribbon width. The variation of lTOT followsclosely the variation of lAP, lLER and lCO, reported in Fig. 7, bothqualitatively and quantitatively. Namely, the variability increasesin narrower GNRs, ranging from �121% for W = 4.80 nm up to�968% in the 1.10 nm-wide nanoribbon. The results reported inFigs. 5–10 indicate that the sensitivity of carrier mobility to edgeroughness (assuming 50% edge defect density) would be prohibi-tively high for the sub-5 nm-wide GNRs in terms of device variabil-ity and process reliability.

Fig. 11 compares the mobility–half-gap trade-off of the simu-lated ideal and edge-defected GNRs, several bulk semiconductors(data taken from [28]), and a few sub-7 nm thick ultra-thin bodysilicon-on-insulator (UTB SOI) MOSFETs (mobility data from[29]). The optimum material would be positioned in the upper-right corner of the plot because higher mobility corresponds tohigher ON-state currents and higher bandgap leads to lower OFF-state current and higher breakdown voltages. However, even theconventional semiconductors exhibit a trade-off, meaning thatmaterials with higher carrier mobility usually have smaller band-gap. Fig. 11 clearly shows that neither ideal nor defected ultra-narrow GNRs offer improvement over conventional bulksemiconductors when both carrier mobility and correspondingbandgap are considered at the same time. Namely, data pointsfor all the materials except silicon lie beyond the curve obtainedfor ideal GNRs and beyond the variability range obtained for GNRswith edge defects. When the bandgap value of a GNR is close tothat of silicon, the corresponding GNR exhibits �2.5� lower

Fig. 10. Variation of the total mobility versus GNR width. The variation percentageis calculated relatively to the average lTOT for each nanoribbon width.


average lTOT than bulk silicon. Similarly, GNRs could achieve band-gaps equal to that of InAs, however, with carrier mobility being�21� lower in the nanoribbon than in bulk InAs. Nanoribbons inthe sub-5 nm range could achieve mobilities comparable to GaN,but the bandgap of �0.4 eV that corresponds to those GNRs wouldbe much lower than 3.4 eV in bulk GaN. If additional scatteringmechanisms were included in the model, the total mobility inGNRs would be even lower. In that case, even the bulk silicon datapoint in Fig. 11 would lie beyond the mobility–half-gap variabilityrange calculated for graphene nanoribbons. Despite the discussionabove, it would be unreasonable to dismiss GNRs completely aspossible future channel material because the comparison inFig. 11 is made between one-dimensional structures and bulkmaterials. Hence, it is more plausible to compare graphenenanoribbons to UTB MOSFETs since in low-dimensional transistorarchitectures, which are needed to improve the immunity toshort-channel effects, the mobility is greatly reduced [29,30].Fig. 11 reveals that ideal GNRs cannot reach mobility–half-gapvalues of 3–7 nm-thick UTB SOI devices due to low bandgap, eventhough the ideal GNRs exhibit considerably higher mobilities thanthe considered UTB SOI MOSFETs. Surprisingly, the �2 nm-widedefected GNRs are comparable in terms of average carrier mobilityand average bandgap to UTB SOI devices, which is a consequence ofthe increased bandgap caused by edge defects while, at the sametime, the GNR mobility is not reduced below the values reportedfor SOI devices.

4. Conclusion

The results of multi-scale transport modeling of sub-5 nm-wideGNRs are reported, with an emphasis on the variability of thebandgap and carrier mobility caused by edge defects. Semiclassicalsimulation of mobility is coupled with atomistic NEGF simulationin order to quantify the variation of mobility in realistic nanorib-bons. A statistical analysis of the transport gap, i.e. effective band-gap, is performed on an ensemble of N = 200 randomly defectednanoribbons, and the obtained half-gap distributions are quanti-fied in terms of the mean, standard deviation, skewness and kurto-sis values. The NEGF results are then used to incorporate the effectsof bandgap increase and variability into the mobility calculation.For each nanoribbon width, ensembles of N = 200 random GNR




mobilities are obtained and statistically analyzed. We report highmobility variation irrespective of the scattering mechanisms,which spans approx. one order of magnitude for lAP, lLER andlCO, while the OP scattering exhibits the greatest variability of upto four orders of magnitude. In comparison to the averaged lTOT,the variation range of the total mobility increases from �120% upto �970% as the width is downscaled from 4.80 nm to 1.10 nm.We have shown that realistic ultra-narrow GNRs with the assumed50% edge defects offer no improvement over conventional bulksemiconductors when both the carrier mobility and correspondingbandgap are considered at the same time, i.e. mobility-bandgapdata points of all relevant bulk semiconductors lie beyond the var-iability range obtained for edge-defected GNRs. Nevertheless, dueto the low-dimensional nature of nanoribbons it seems more plau-sible to compare their electronic and transport properties to thoseof low-dimensional semiconductor structures. Hence, we havefound that some devices among sub-5 nm-wide defected GNRsare comparable to sub-7 nm-thick UTB SOI MOSFETs in terms ofmobility-bandgap values. And yet, the very high variation of carriermobility and bandgap in realistic nanoribbons remains as the bot-tleneck in terms of device variability and process repeatability, andstresses the importance of finding a fabrication method with agood control of nanoribbon edges.

References

[1] Bolotin KI, Sikes KJ, Jiang Z, Klima M, Fudenberg G, Hone J, et al. Ultrahighelectron mobility in suspended graphene. Solid State Commun 2008;146:351–5.

[2] Novoselov KS, Geim AK, Morozov SV, Jiang D, Zhang Y, Dubonos SV, et al.Electric field effect in atomically thin carbon films. Science 2004;306:666–9.

[3] Castro Neto AH, Guinea F, Peres NMR, Novoselov KS, Geim AK. The electronicproperties of graphene. Rev Mod Phys 2009;81:109–62.

[4] Han MY, Özyilmaz B, Zhang Y, Kim P. Energy band-gap engineering ofgraphene nanoribbons. Phys Rev Lett 2007;98:206805.

[5] Wang X, Ouyang Y, Li X, Wang H, Guo J, Dai H. Room-temperature all-semiconducting sub-10-nm graphene nanoribbon field-effect transistors. PhysRev Lett 2008;100:206803.

[6] Li X, Wang X, Zhang L, Lee S, Dai H. Chemically derived, ultrasmooth graphenenanoribbon semiconductors. Science 2008;319:1229–32.

[7] Yang Y, Murali R. Impact of size effect on graphene nanoribbon transport. IEEEElectron Dev Lett 2010;31:237–9.

[8] Querlioz D, Apertet Y, Valentin A, Huet K, Bournel A, Galdin-Retailleau S, et al.Suppression of the orientation effects on bandgap in graphene nanoribbons inthe presence of edge disorder. Appl Phys Lett 2008;92:042108–042108-3.

[9] Poljak M, Wang M, Song EB, Suligoj T, Wang KL. Disorder-induced variability oftransport properties of sub-5 nm-wide graphene nanoribbons. Solid-StateElectron 2013;84:103–11.


[10] Yoon Y, Guo J. Effect of edge roughness in graphene nanoribbon transistors.Appl Phys Lett 2007;91:073103–073103-3.

[11] Sako R, Hosokawa H, Tsuchiya H. Computational study of edge configurationand quantum confinement effects on graphene nanoribbon transport. IEEEElectron Dev Lett 2011;32:6–8.

[12] Poljak M, Song EB, Wang M, Suligoj T, Wang KL. Influence of edge defects,vacancies, and potential fluctuations on transport properties of extremelyscaled graphene nanoribbons. IEEE Trans Electron Devices 2012;59:3231–8.

[13] Fang T, Konar A, Xing H, Jena D. Mobility in semiconducting graphenenanoribbons: phonon, impurity, and edge roughness scattering. Phys Rev B2008;78:205403.

[14] Bresciani M, Palestri P, Esseni D, Selmi L. Simple and efficient modeling of theE–k relationship and low-field mobility in Graphene Nano-Ribbons. Solid-State Electron. 2010;54:1015–21.

[15] Betti A, Fiori G, Iannaccone G. Atomistic Investigation of low-field mobility ingraphene nanoribbons. IEEE Trans Electron Devices 2011;58:2824–30.

[16] Poljak M, Suligoj T, Wang KL. Influence of substrate type and quality on carriermobility in graphene nanoribbons. J Appl Phys 2013;114:053701.

[17] Basu D, Gilbert MJ, Register LF, Banerjee SK, MacDonald AH. Effect of edgeroughness on electronic transport in graphene nanoribbon channel metal-oxide-semiconductor field-effect transistors. Appl Phys Lett 2008;92:042114–042114-3.

[18] Yoon Y, Fiori G, Hong S, Iannaccone G, Guo J. Performance comparison ofgraphene nanoribbon FETs with schottky contacts and doped reservoirs. IEEETrans Electron Devices 2008;55:2314–23.

[19] Leong Z-Y, Lam K-T, Liang G. Device performance of graphene nanoribbon fieldeffect transistors with edge roughness effects: a computational study. In: 13thInternational Workshop on Computational Electronics (IWCE ’09); 2009.p. 1–4.

[20] Choudhury MR, Yoon Y, Guo J, Mohanram K. Graphene nanoribbon FETs:technology exploration for performance and reliability. IEEE TransNanotechnol 2011;10:727–36.

[21] Hancock Y, Uppstu A, Saloriutta K, Harju A, Puska MJ. Generalized tight-binding transport model for graphene nanoribbon-based systems. Phys Rev B2010;81:245402.

[22] Son Y-W, Cohen ML, Louie SG. Energy gaps in graphene nanoribbons. Phys RevLett 2006;97:216803.

[23] Datta S. Quantum transport: atom to transistor. 2nd ed. New York (NY,USA): Cambridge University Press; 2005.

[24] Golizadeh-Mojarad R, Datta S. Nonequilibrium Green’s function based modelsfor dephasing in quantum transport. Phys Rev B 2007;75:081301.

[25] International Technology Roadmap for Semiconductors (ITRS). 2013 Edition:Semiconductor Industry Association; www.itrs.net.

[26] Fratini S, Guinea F. Substrate-limited electron dynamics in graphene. Phys RevB 2008;77:195415.

[27] Meric I, Han MY, Young AF, Ozyilmaz B, Kim P, Shepard KL. Current saturationin zero-bandgap, top-gated graphene field-effect transistors. Nat Nano2008;3:654–9.

[28] Skotnicki T, Fenouillet-Beranger C, Gallon C, Buf F, Monfray S, Payet F, et al.Innovative materials, devices, and CMOS technologies for low-power mobilemultimedia. IEEE Trans Electron Devices 2008;55:96–130.

[29] Uchida K, Watanabe H, Kinoshita A, Koga J, Numata T, Takagi S. Experimentalstudy on carrier transport mechanism in ultrathin-body SOI n- and p-MOSFETswith SOI thickness less than 5 nm. Int Electron Dev Meet 2002:47–50.

[30] Esseni D, Palestri P, Selmi L. Nanoscale MOS transistors: semi-classicaltransport and applications. Cambridge, UK: Cambridge University Press; 2011.


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