Band Gap of Strained Graphene Nanoribbons

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1 Nano Res () 3: DOI.7/s Research Aricle Band Ga of Srained Grahene Nanoribbons Yang Lu and Jing Guo ( ) Dearmen of Elecrical and Comuer Engineering, Universiy of Florida, Gainesville, FL 36, USA Received: December 9 / Revised: 9 January / Acceed: January The Auhor(s). This aricle is ublished wih oen access a Sringerlink.com ABSTRACT The band srucures of srained grahene nanoribbons (GNRs) are examined using a igh-binding Hamilonian ha is direcly relaed o he ye and magniude of srain. Comared o a wo-dimensional grahene whose band ga remains close o zero even if a large srain is alied, he band ga of a grahene nanoribbon (GNR) is sensiive o boh uniaxial and shear srains. The effec of srain on he elecronic srucure of a GNR deends srongly on is edge shae and srucural indices. For an armchair GNR, a weak uniaxial srain changes he band ga in a linear fashion, whereas a large srain resuls in eriodic oscillaion of he band ga. On he oher hand, shear srain always ends o reduce he band ga. For a zigzag GNR, he effec of srain is o change he sin olarizaion a he edges of GNR, and hereby modulae he band ga. A simle analyical model, which agrees wih he numerical resuls, is roosed o inerre he resonse of he band ga o srain in armchair GNRs. KEYWORDS Grahene nanoribbons (GNRs), band ga, srain. Inroducion Srain has been exensively used in he silicon elecronics indusry o boos device erformance and has layed an imoran role for he 9-nm echnology node []. Grahene is an aomically hin wo-dimensional (-D) maerial and is herefore srucurally more amenable han silicon o exernal modificaions including srain. Grahene can also susain a much larger srain han silicon. The effec of srain on -D grahene has been sudied boh exerimenally and heoreically, including he effecs of uniform [ 5] and local srains [6] on he elecronic srucure, as well as he ossibiliy o achieve quanum Hall saes in he absence of an exernal magneic field [7]. Two-dimensional grahene does no have a band ga, and he band ga remains close o zero even if a srain as large as % is alied. A band ga can be creaed by aerning he -D grahene ino a nanomeer-wide grahene nanoribbon (GNR); his has been rediced heoreically [8 ] and realized exerimenally [ 3]. GNRs resen ineresing ransor roeries where, for examle, disorder such as imerfec edges, can lay an imoran role [4]. Moreover, srain could be a useful way o furher ailor he elecronic roeries of GNRs. Based on densiy funcional heory (DFT), he effec of uniaxial srain on he elecronic roeries of GNRs has been sudied [5, 6]. These sudies revealed he oenial of uniaxial srain as a way of uning he elecronic roeries of GNRs. The underlying hysics of srain effecs on he band ga of GNRs, however, is buried in DFT simulaions and is no fully undersood. In his work, a sysemaic sudy of he effec of boh Address corresondence o guoj@ufl.edu

2 9 Nano Res () 3: uniaxial and shear srain on he band ga of GNRs is erformed using a igh-binding Hamilonian ha is direcly relaed o he magniude and ye of srain. An analyical model is develoed o describe he deendence of bandga on srain in GNRs. The work rovides exlici relaionshis beween he bandga and srain in GNRs, which enables a simle and deailed hysical undersanding. I is observed ha he band ga of a GNR is much more sensiive o srain han -D grahene and srongly deends on is edge shae and srucural indices. For zigzag GNRs (ZGNRs), uniaxial srain and shear srain modulae he sin densiy a he GNR edges hereby alering he band ga. For armchair GNRs (AGNRs), uniaxial srain and shear srain resul in qualiaively differen deendences of he band ga on srain. The effec of srain on he band ga is qualiaively differen for AGNRs wih differen srucural indices. The effecs of edge bond relaxaion [9] and hird neares neighbor couling [7] modify he quaniaive deendence of he band ga on srain.. Aroach The band srucures of he modeled GNRs are calculaed by using a igh-binding model, whose binding arameers have been arameerized by ab iniio calculaions in revious sudies of GNR band srucures in he absence of srain [9, 7]. For AGNRs, modeling he edge bond relaxaion and he hird neares neighbor couling are necessary o rea he edge effecs and describe all semiconducing band srucures [3], as rediced by he ab iniio calculaions [9,, 7]. For ZGNRs, inclusion of a Hubbard erm in he Hamilonian is needed o describe he edge sin olarizaion and oening of he band ga [8]. The binding arameers in he resence of srain are modified according o he Harrison binding arameer relaion. This aroach has been used and validaed before in he sudy of srain effecs on carbon nanoubes [9]. As shown in Fig. (a), he unsrained bond vecors for an AGNR are given by r = axˆ 3 r = axˆ ayˆ () 3 r3 = axˆ ayˆ where we se ˆx as he ransor direcion of he GNR. The alicaion of a uniaxial or shear srain causes he following changes: rix ( σ) rix riy ( ν σ) riy (Uniaxial srain) rix rix γ r iy (Shear srain) where i =,, 3 and r ix, r iy are he x and y comonens () Figure (a) The uni cell of AGNR (b) uni cell of ZGNR. In each figure, r, r, r 3 are he bond vecors, and he ransor direcion of he ribbon is se as he x direcion

3 Nano Res () 3: of r i. σ reresens he uniaxial srain along he x direcion, ν.65 is he Poisson raio [], and γ is he shear srain. Here we focus on he simle case of uniform srain, while in racice he deformaion of grahene may include long range and regular rile aerns. A igh-binding Hamilonian as arameerized by Gunlycke and Whie [], which includes he reamen of he edge bond relaxaion and he hird neares neighbor couling, is used o comue he band srucure of he AGNR. In he resence of srain, each binding arameer is scaled by a dimensionless r facor ξ =, where r he unsrained bond lengh, r and r is he bond lengh in he resence of srain. The bond lenghs of he zigzag GNR, as shown in Fig. (b), are modified in a similar manner o hose of he AGNR in he resence of srain. Due o he exisence of localized edge saes in ZGNRs, he sin olarized ineracion should be included in he Hamilonian of he sysem, which can be generally described as [] where H = c c U n n (3) c, iσ,, ij iσ jσ i σ i σ i, j, σ i, σ c jσ, and n iσ are creaion, annihilaion, and number oeraors, resecively, for an elecron of sin σ in he π-orbial cenered on he i -h C aom in he ribbon. ij, denoes he se of all neares neighbors, ij is he corresonding neares neighbor hoing arameer and U describes he srengh of he sindeenden field. n i, σ is he average elecron densiy wih sin σ a he locaion of i -h C aom, and can be calculaed self-consisenly from equilibrium carrier saisics. The Hamilonian described by Eq. (6) is in fac equivalen o he Harree Fock aroximaion alied o he Hubbard model [3]. This was firs sudied by Fujia e al. in heir aer on edge saes in ZGNRs [4]. 3. Resuls and discussion 3. Armchair GNRs urose of comarison, we firs neglec he effec of edge bond relaxaion and hird neares neighbor couling. In his simle case, he band srucure of a srained AGNR wih an index of n is similar o ha of a srained zigzag single-wall nanoube (SWNT) wih an index of n [7], exce for he lack of valley degeneracy. As shown in Fig. (a), he band ga scales linearly wih he magniude of he srain over a cerain range, and reeas iself eriodically as he srain is furher increased. The effec of srain on band ga is significan and qualiaively differen for AGNRs wih n = 3q, 3q, and 3q. For he n = 3 AGNR (3q), small ensile srains increase he band ga, wih only 5% uniaxial srain leading o he oening of a band ga of abou.4 ev. Small ensile srains also increase he band ga of he n = 4 AGNR (3q ), bu decrease he band ga of he n = 5 AGNR (3q ). In he resence of edge bond relaxaion and he hird neares neighbor couling, he band ga is non-zero in he absence of srain for any AGNR. Figure (b) shows los of he band ga of an AGNR under uniaxial srain when he effecs of edge bond relaxaion and hird neares neighbor couling are included. The qualiaive feaures of he relaionshi beween band ga and srain do no change, bu he quaniaive value of he band ga is erurbed. Furhermore, he maximum achievable band ga in he resence of comressive srain is smaller han ha in he resence of ensile srain. In order o obain a simle relaion beween srain and he band ga of an AGNR, we can calculae he lowes order conribuion of srain o band srucures. The eigenenergies of an AGNR a k = can be wrien as (see Aendix) π E ( ) = α ( β)cos n π α ( β)cos n 4( ( β) Δe) π sin n n (4) We firs consider he case of uniaxial srain. The band srucure En( k ) is calculaed and he band ga is obained by finding he minimum of En( k ) for all he band indices n and wave vecors k. For he where α = σ 3 σ ( 3 ν ) σ 3 β = γ ( 3 ν ) σ 4 4 (5)

4 9 Nano Res () 3: and β are he correcions due o srain. Equaion (4) gives n of he n eigenenergies of he sysem. The oher n eigenenergies, due o symmery, are jus he oosies of Eq. (4). Thus, he band ga can be calculaed as E = min E( ) (6) g P=, n From Eq. (4), i is observed ha here is no firs order conribuion o he band ga from shear srain. Also, because and Δ e are relaively small comared o, we hen reserve only he firs erm of Eq. (4), which is he dominaing facor in he qualiaive deendence of he band ga on srain, E π min α ( β)cos P=, n n g (7) To he firs order of uniaxial srain, Eq. (7) can be furher aroximaed as E min ( 3 π ) =, n n g (8) where 3 = ( ν ) σ 3 π Figure Plo of band ga versus uniaxial srain for AGNR; n = 3, 4, and 5 corresond o AGNRs wih n = 3q, 3q, and 3q, resecively. (a) is he simle case neglecing edge bond relaxaion and hird neares neighbor couling effecs and (b) includes hese wo effecs. In boh figures, he los of band ga versus srain show similar eriodic aerns. Locally, he band ga changes linearly when increasing (decreasing) he magniude of he srain Here, is he band index running from o n, is he neares neighbor hoing inegral in he absence of srain, σ and γ are he magniudes of uniaxial srain and shear srain, resecively, ν is he Poisson raio, is he hird neares neighbor couling srengh and Δ e is he correcion due o edge bond relaxaion. In Eq. (4), he firs erm corresonds o he band energy neglecing edge bond relaxaion and hird neares neighbor couling, whils he second and hird erms accoun for hese wo effecs. The quaniies α Equaion (8) imlies ha he band ga is roorional o he shores disance beween and n, he quanizaion grids in he widh direcion of he GNR. This is deiced in Fig. 3. I is observed ha when n = 3q, he shores disance of o he grids is zero, and for n = 3q or 3q, he shores disance of o he grids is. This exlains he feaures 3( n ) of Fig., as for all hree cases, small ensile srains shif he osiion of o he negaive direcion, which resuls in he minimum disance of o he grids being increased for n = 3q or 3q, and decreased for n = 3q. The behavior in comressive srain can be exlained similarly. Figure 3 also rovides a qualiaive exlanaion of he eriodic oscillaion of he band ga as he srain increases. Furher shifing he osiion of resuls in a eriodic reeiion of ha minimum disance, which gives he eriodic relaionshi beween band ga and srain. Furhermore, he maximum achievable band ga is roorional o

5 Nano Res () 3: Figure 4 Plos of band ga versus uniaxial srain for AGNRs wih differen widhs, wih he effec of edge bond relaxaion and hird neares neighbor couling included. The widh of he AGNR varies from o nm. Generally, he band gas sill show a eriodic deendence on he magniude of he srain, and are roughly inversely roorional o he widh of he ribbons Figure 3 Visualizaion of he osiion of in Eq. (8) relaive o he quanizaion grids, /(n ), where = o n. The disance of o he neares grid oins is differen for AGNRs wih differen indices. For n = 3q, i is ; for n = 3q or 3q, i is one hird of he grid sace, /[3(n )]. As indicaed by he red arrows, ensile srain shifs o he negaive direcion half he grid sace, /[(n )]. For an AGNR wih n = 3q or 3q, his corresonds o an increase of abou 5% in he band ga comared o he unsrained case. The exlanaion of band ga oscillaion under srain is similar o revious sudies of srain effecs on nanoubes, which aribued he change in band ga o he shifing of he Fermi oin under srain [9, 5]. Plos of he deendence of band ga on boh srain and ribbon widh are shown in Fig. 4. The simulaed range of uniaxial srain is from 5% o 5%, and he widh from o nm. The eriodic oscillaions of he band ga as a funcion of he magniude of uniaxial srain and he qualiaive difference beween 3q, 3q, and 3q grous are observed for he whole range of simulaion arameers. In general, increasing he widh of he ribbon reduces he maximum achievable band ga, due o he weaker confinemen in he widh direcion. The effec of shear srain on he band ga of an AGNR is qualiaively differen from ha of uniaxial srain, as shown in Fig. 5. As Eqs. (4) and (5) indicae, here is no firs order conribuion from shear srain o he band ga, so he deendence of he band ga on shear srain is due o he second and higher order Figure 5 Plos of band ga versus shear srain for AGNRs wih edge bond relaxaion and hird neares neighbor effec included. In his case, shear srain always ends o reduce he band ga

6 94 Nano Res () 3: erurbaion effecs. In he resence of edge bond relaxaion and hird neares neighbor couling, shear srain always reduces he band ga regardless of he srucural indices of he AGNR. 3. Zigzag GNRs The band ga of a ZGNR originaes from a oally differen mechanism from ha for an AGNR, as indicaed by Eq. (3). When he sin ineracion is included, he band searaion a he zone boundary can be aroximaed as [8] Δ E= U( n n ) (9),, The acual band ga is roorional o, bu smaller han, ΔE. In Fig. 6, we lo he band srucure of a ZGNR wih n = 6. The solid blue lines reresen he unsrained band srucure while he dashed red lines corresond o he band srucure under 5% uniaxial srain. Obviously, his ensile srain oens u he band ga. The deendence of he band ga of he ZGNR on uniaxial srain is shown in Fig. 7(a). In conras o AGNRs, he band ga of a ZGNR increases as ensile srain is alied and decreases as comressive srain is alied, regardless of is srucural index. A lo of he normalized band ga Eg/ E g versus srain is shown Figure 6 Band srucure of a ZGNR wih n =. The solid blue lines are he case wihou srain, and he dashed red lines are he case wih 5% uniaxial srain. In each case, he u sin and down sin band srucures are degenerae. I is obvious ha he ensile srain increases he band ga in he inse of Fig. 7(a), and is aroximaely he same for ZGNRs wih differen widhs. Fiing he curve gives he emirical relaion, E E g g = σ.6σ () where E g is he unsrained band ga. To exlain he effec of srain on band ga, we calculaed he edge sin olarizaion for various ZGNRs under uniaxial srain. As shown in Fig. 7(b), osiive (negaive) uniaxial srain always end o increase (decrease) he edge sin olarizaion. As indicaed by Eq. (9), sronger sin olarizaion will induce a larger band searaion, which is roughly roorional o he band ga. This jusifies he monoonic feaure of he lo of band ga versus srain in Fig. 7. The reason why ensile srain will increase edge sin olarizaion can be exlained by he analyical model roosed by Fujia [4]. For edge saes, he corresonding charge densiy is roorional o [cos( k / )] m a each non-nodal sie of he mh zigzag chain from he edge. So cos( k / ) reresen he daming lengh of he edge saes. If a srain is alied, due o he disorions of bond vecor and bond arameer, his daming facor should be modified as cos( k /), where, are bond arameers relaed o he bond vecors r, r in Fig. (b). For ensile srain ( σ > ), / <, he daming of edge saes becomes much quicker, which resuls in more localized edge saes. Due o elecron elecron ineracion, his will increase he sin olarizaion a edge sies, hereby increasing he band ga of he sysem. A similar argumen alies in he case of comressive srain ( σ < ). We also calculaed he effec of shear srain on he band ga of a ZGNR, as shown in Fig. 8 (a). Comared o he case of uniaxial srain, he change in band ga is relaively small, and shear srain always ends o reduce he band ga. These feaures can also be exlained by Fujia s model [4]. We find ha, under shear srain, he daming facor should k 3 k be modified as cos γ an, where γ 4

7 Nano Res () 3: Figure 7 (a) Band ga of a uniaxially srained ZGNR wih differen widhs (indicaed by he number of zigzag chains in he ransverse direcion); inse is a lo of he normalized band ga versus srain, in which ZGNRs wih differen widhs show a similar linear deendence. (b) The sin olarizaion (u sin densiy (N u ) minus down sin densiy (N dn )) a he edges of he ZGNR. Tensile (comressive) srain increases (decreases) sin olarizaion is he magniude of he shear srain. Because 3 k γ an, under shear srain he daming 4 of edge saes is slower han wihou srain. So he edge saes are less localized, hus decreasing he sin olarizaion a edge sies, and herefore reducing he band ga. This analysis is confirmed by Fig. 8(b), in which shear srain always reduce he sin olarizaion a he edges. Figure 8 (a) The band ga of shear srained ZGNR wih differen widhs (indicaed by he number of zigzag chains in he ransverse direcion). (b) The sin olarizaion (u sin densiy (N u ) minus down sin densiy (N dn )) a he edges of he ZGNR. Shear srain always ends o decrease sin olarizaion a he edges 4. Conclusions We have exlored he effec of srain on he band ga of GNRs. Two yes of srain (uniaxial srain and shear srain) and wo yes of GNRs (AGNRs and ZGNRs) have been sudied. The effec of srain is modeled as a modificaion o he igh-binding neares neighbor hoing inegral. I is found ha for an AGNR, uniaxial srain linearly shifs he band ga in a way which eriodically reeas iself as he magniude of he srain is increased. Shear srain makes no obvious

8 96 Nano Res () 3: conribuion o he oening u of he band ga and, in all cases, i ends o reduce he band ga. We exlained hese observaions by roosing a erurbaion model and i reroduced well he resuls of he numerical calculaions. For a ZGNR, we found ha srain changes he sin olarizaion a edge sies of he nanoribbon, hus furher affecing he band ga. Tensile srain increases he band ga while comressive and shear srain reduce he band ga. These resuls indicae ha, he band ga of a GNR is sensiive o magniude of he srain o which i is subjeced. By alying moderae srains, he elecronic roeries of GNRs can be readily engineered. Acknowledgemens This work was suored by Office of Naval Research (ONR) and he Naional Science Foundaion (NSF). Oen Access: This aricle is disribued under he erms of he Creaive Commons Aribuion Noncommercial License which ermis any noncommercial use, disribuion, and reroducion in any medium, rovided he original auhor(s) and source are credied. References [] Inernaional echnology road ma for semiconducors. h:// [] Pereira, V. M.; Casro Neo, A. H.; Peres, N. M. R. Tighbinding aroach o uniaxial srain in grahene. Phys. Rev. B 9, 8, 454. [3] Gui, G.; Li, J.; Zhong, J. X. Band srucure engineering of grahene by srain: Firs-rinciles calculaions. Phys. Rev. B 8, 78, [4] Farjam, M.; Rafii-Tabar, H. Commen on Band srucure engineering of grahene by srain: Firs-rinciles calculaions. Phys. Rev. B 9, 8, 674. [5] Gui, G.; Li, J.; Zhong, J. X. Rely o Commen on Band srucure engineering of grahene by srain: Firs-rinciles calculaions. Phys. Rev. B 9, 8, 674. [6] Pereira, V. M.; Casro Neo, A. H. Srain engineering of grahene s elecronic srucure. Phys. Rev. Le. 9, 3, 468. [7] Guinea, F.; Kasnelson, M. I.; Geim, A. K. Energy gas and a zero-field quanum Hall effec in grahene by srain engineering. Na. Phys. 9, 6, [8] Eazwa, M. Peculiar widh deendence of he elecronic roeries of carbon nanoribbons. Phys. Rev. B 6, 73, [9] Son, Y. W.; Cohen, M. L.; Louie, S. G. Energy gas in grahene nanoribbons. Phys. Rev. Le. 6, 97, 683. [] Barone, V.; Hod, O.; Scuseria, G. E. Elecronic srucure and sabiliy of semiconducing grahene nanoribbons. Nano Le. 6, 6, [] Han, M. Y.; Ozyilmaz, B.; Zhang, Y. B.; Kim, P. Energy band-ga engineering of grahene nanoribbons. Phys. Rev. Le. 7, 98, 685. [] Chen, Z. H.; Lin, Y. M.; Rooks, M. J.; Avouris, P. Grahene nano-ribbon elecronics. Physica E 7, 4, 8 3. [3] Li, X. L.; Wang, X. R.; Zhang, L.; Lee, S. W.; Dai, H. J. Chemically derived, ulrasmooh grahene nanoribbon semiconducors. Science 8, 39, 9 3. [4] Cresi, A.; Nemec, N.; Biel, B.; Niebler, G.; Triozon, F.; Cuniberi, G.; Roche, S. Charge ransor in disordered grahene-based low dimensional maerials. Nano Res. 8,, [5] Sun, L.; Li, Q. X.; Ren, H.; Su, H. B.; Shi, Q. W.; Yang, J. L. Srain effec on elecronic srucures of grahene nanoribbons: A firs-rinciles sudy. J. Chem. Phys. 8, 9, [6] Hod, O.; Scuseria, G. E. Elecromechanical roeries of susended grahene nanoribbons. Nano Le. 9, 9, [7] Whie, C. T.; Li, J.; Gunlycke, D.; Minmire, J. W. Hidden one-elecron ineracions in carbon nanoubes revealed in grahene nanosris. Nano Le. 7, 7, [8] Gunlycke, D.; Areshkin, D. A.; Li, J. W.; Minmire, J. W.; Whie, C. T. Grahene nanosri digial memory device. Nano Le. 7, 7, [9] Yang, L.; Han, J. Elecronic srucure of deformed carbon nanoubes. Phys. Rev. Le., 85, [] Blakslee, O. L.; Procor, D. G.; Seldin, E. J.; Sence, G. B.; Weng, T. Elasic consans of comression annealed yrolyic grahie. J. Al. Phys. 97, 4, [] Gunlycke, D.; Whie, C. T. Tigh-binding energy disersions of armchair-edge grahene nanosris. Phys. Rev. B 8, 77, 56. [] Guo, J.; Gunlycke, D.; Whie, C. T. Field effec on sinolarized ransor in grahene nanoribbons. Al. Phys. Le. 8, 9, 639. [3] Hubbard, J. Elecron correlaions in narrow energy bands. Proc. R. Soc. Lond. A 963, 76,

9 Nano Res () 3: [4] Fujia, M.; Wakabayashi, K.; Nakada, K. Peculiar localized sae a zigzag grahie edge. J. Phys. Soc. Jn. 996, 65, [5] Nagariya, K. S.; Berber, S.; Cohen-Karni, T.; Segev, L.; Srur-Lavi, O.; Tománek, D.; Joselevich, E. Origin of orsioninduced conducance oscillaions in carbon nanoubes. Phys. Rev. B 8, 78, Aendix Derivaion of Eq. (4) For an AGNR, in he igh-binding model he band ga always occurs a k = due o he symmery of he Hamilonian. A k =, he igh-binding Hamilonian reduces o a wo leg ladder laice sysem [9], as shown in Fig. A.. corresonds o he lef (righ) leg of he ladder. The eigensaes and eigenenergies of H are π H E = E E, E = cos, =,... n n (A.) where π sin n π ψ sin E = ψ = ψ, n... nπ sin n Equaion (A.) describes n of he n eigensaes we would like o discuss. The oher n eigensaes, due o symmery, jus have he oosie eigenenergies, E, =,... n. The erurbed Hamilonian and energy due o srain are d3 TL di = = d3 d H', TL, di TR d... d3 d3 d = d d3 T R d 3... d d Figure A. The k = Hamilonian used in he Aendix: (a) he unsrained Hamilonian; (b) he srain-induced erurbaion Hamilonian where H T I = I T = T... (A.) ΔE ψ TL di ( ψ ψ ) E ψ H' E di TR = = E E ψψ ψ ( TL TR) ψ d ψ ψ ψ ( TL TR) ψ = = d ψψ ψψ N (d d 3) kπ ( k ) π = d sin sin ( N ) n n k= π = d (d d 3)cos n (A.3) Using Eqs. (A.) and (A.3) we can obain he exressions for he erurbed bonding arameer,

10 98 Nano Res () 3: d = ( σ 3 σ ) 3 ( 3 ν ) σ 3 d = γ γ ( 3 ν ) σ ( 3 ν ) γσ 3 ( 3 ν ) σ 3 d 3 = γ γ ( 3 ν ) σ ( 3 ν ) γσ (A.4) = ( σ 3 σ ) () 3 ( 3 ν ) σ = n γ () γ ( 3 ν) σ ( 3 ν) γσ ( 3 ν ) σ 3 = n γ γ 4 (3) 3 3 ( 3 ν) σ ( 3 ν) γσ 4 (A.6) Subsiuing Eq. (A.4) ino Eq. (A.3), we have ΔE σ σ = ( 3 ) (3ν ) σ ( 3 ν) σ 3 π γ cos n (A.5) To include he effec of edge disorion and hird neares neighbor couling, we use a similar mehod o he above, namely he erurbed Hamilonian due o he hird neares neighbor couling and edge bond relaxaion, H T3 n H =, T3 n () () () (3) (3) () () () () T3 n = (3) (3) () () () edge Δe T e =, T = e T e Δ e where Δ e is he edge bond correcion and () (),, and are he hird neares neighbor couling ara- (3) meers under srain, which are given by exressions similar o Eq. (A.4) where is he unsrained hird neares neighbor couling arameer. ΔE The erurbed energy is given by ΔE E H3 E ψ ( T 3 T3 ) ψ = = E E n n n n () (3) () () (3) π ( ) π = ( )cos sin n n N ν σ = σ σ ( 3 ) 3 ( 3 ) γ 4 π ν σ ν σ 4 ( 3 ) ( 3 ) cos 4 n n 3 γ ( 3 ν ) σ π sin (A.7) 4 4 n e E Hedge E ψte ψ 4Δe π = = = sin E E n n n Adding Eqs. (A.), (A.5), (A.6), and (A.7) we ge where π E ( ) = α ( β)cos n π α ( β)cos n 4( ( β) Δe) π sin n n α = σ 3σ ( 3 ν ) σ 3 β = γ ( 3 ν ) σ 4 4 (A.8) (A.9) (A.)

11 Nano Res () 3: To see he effeciveness of his analyical aroxi- maion, we comared he resuls calculaed numerically by he igh-binding model in Secion wih hose using Eqs. (9) o (), as shown in Fig. A.. Figure A. Band ga of a srained AGNR wih n = 4, showing a comarison beween numerical calculaion (solid blue lines) and he analyical model (dashed red lines) develoed in his secion. In (a) edge disorion and hird neares neighbor couling are ignored, whils in (b) hese wo effecs are included. The analyical model agrees well wih he numerical simulaion resuls. The deviaions beween he wo reamens in he high srain region may be due o higher order effecs