Band structure of deformed armchair nanoribbon with bondalternation

Nguyen Ngoc Hieu a,n, Le Cong Nhan b

a Institute of Research and Development, Duy Tan University, Da Nang, Vietnamb Department of Environmental Science, Saigon University, Ho Chi Minh city, Vietnam

H I G H L I G H T S

� In the presence of bond alternation all armchair GNRs are semiconducting.� Slope of dependency of the energy gap on its stress depends on tension direction.� We can control the band gap of nanoribbon by its strain.

a r t i c l e i n f o

Article history:Received 28 November 2013Accepted 14 February 2014Available online 22 February 2014

Keywords:Deformed nanoribbonBond alternationTight-binding approximation

a b s t r a c t

Electronic energy band structure of deformed armchair graphene nanoribbons with bond alternation isstudied by the tight-binding approximation. In the presence of bond alternation, all armchair graphenenanoribbons become semiconducting with small band gap opened at center of the Brillouin zone. Undertensional strain, armchair graphene nanoribbons can become metallic at the critical values of deforma-tion and we can control the band gap of nanoribbon by its strain.

& 2014 Elsevier B.V. All rights reserved.

1. Introduction

Graphene is a single layer of carbon atoms packed in a honey-comb lattice [1]. Besides carbon nanotubes (CNTs), another one-dimensional carbon nanostructures, graphene nanoribbons (GNRs)can be obtained by cutting a graphene sheet along a certain directionin the form of a quasi-one dimensional wire. Due to edge structures,GNRs have extraordinary electronic properties which open manyways for applications in nanoelectronic devices.

Generally, the electronic properties of carbon low-dimensionalsystems are particularly sensitive to structural perfection, size, andgeometry [2,3]. For example, the electronic properties of CNTs aredetermined by their diameter and chirality. Because of the Peierlstransition, armchair CNTs become semiconducting at low tem-perature and Peierls distortions lead to the Kekule structure withtwo essentially different C–C bond lengths and a triple transla-tional period [4]. The ground state of the (6, 0) CNT is found tohave a Kekule structure with four types of bonds and differencebetween lengths of long and short bonds of about 0.005 nm [5].Due to the presence of dimerization structure, the semiconductor–

metal transitions in CNTs are actual in connection with perspec-tives of their applications in nanoelectronic and nanoelectrome-chanical devices.

Studies of structure and elastic properties of GNRs are also offundamental interest. Many works have studied the electronicproperties of both pristine and deformed GNRs [6–10]. Althoughthe lattice distortion leading to appearance of dimerization struc-ture (bond alternation) GNRs has also been studied by the tight-binding model at very early [11], however, up to date, GNRs withbond alternation seem to remain unclear and less understood. Inthis paper, we study the electronic band structure of armchairGNRs with bond alternation under strain using the simple tight-binding model. The dependence of band gap on the magnitude oftension and the bond lengths is also considered.

2. Deformed graphene with bond alternation

Under stress ε, in the framework of the elasticity theory, theposition vector of carbon atoms is given by the following relation-ship [12]

Ri ¼ ð1þεÞR0i; ð1Þ

Contents lists available at ScienceDirect

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

Physica E

http://dx.doi.org/10.1016/j.physe.2014.02.0141386-9477 & 2014 Elsevier B.V. All rights reserved.

n Corresponding author. Tel.: þ84 511 3827111; fax: þ84 511 3650443.E-mail address: hieunn@duytan.edu.vn (N.N. Hieu).

Physica E 60 (2014) 91–94

where R0i and Ri are the position vectors of carbon atoms beforeand after stress, ε is the strain tensor in the graphene plane.

x, y coordinates are shown in Fig. 1 and the strain tensor isgiven by Ref. [12]

ε ¼ εsin 2θ�s cos 2θ ð1þsÞ cos θ sin θ

ð1þsÞ cos θ sinθ cos 2 θ�s sin 2 θ

!; ð2Þ

where s is Poisson's ratio and θ is defined as the angle betweenthe applied tension vector and axis Ox. The axis Ox is parallel to thearmchair direction of graphene lattice. This implies that tensionalong the armchair (Ox) and zigzag (Oy) directions corresponds toθ¼ 0 and θ¼ π=2, respectively.

The unit vectors of the deformed graphene with bond alterna-tion are defined by a1 ¼ r1�r2 and a2 ¼ r1�r3, where ri is thedeformed bond vectors defined by Eq. (1). Then, the unit vectorsbecome

a1 ¼ ða1x; a1yÞ;a2 ¼ ða2x; a2yÞ; ð3Þ

where

a1x ¼ aþb2

� �ð1þε11Þ�

bffiffiffi3

p

2ε21;

a1y ¼ �bffiffiffi3

p

21þε22�

1ffiffiffi3

p ε12� �

þaε12;

a2x ¼ aþb2

� �ð1þε11Þþ

bffiffiffi3

p

2ε21;

a2y ¼bffiffiffi3

p

21þε22þ

1ffiffiffi3

p ε12� �

þaε12:

Here a and b are the C–C bond length and εij are thecomponents of the strain tensor.

In the nearest-neighbor tight-binding model, the Hamiltoniancan be expressed by

H¼ �∑RB

∑3

i ¼ 1tia

†RB þ ri

bRB þH:c:; ð4Þ

where a†RAðb†RB

Þ and aRA ðbRB Þ are, respectively, the create andannihilate operators an electron at RAðRBÞ site in sublattice A(B),ti is the nearest-neighbor hopping parameter. The dependence ofthe hopping parameter ti on the deformed bond length ri can bedescribed by Harrison's formula [13]

ti ¼ t0a0ri

� �2

; ð5Þ

where a0 ¼ 1:42 Å and t0 ¼ 2:7 eV are the parameters for the C–Cbond length and hopping of pristine (undimerized) graphene [14].

In the tight-binding approximation, the energy dispersions ofthe deformed graphite are given by Refs. [15,16]

EðkÞ ¼ 7ft21þt22þt23þ2t1t2 cos ½kðr1�r2Þ�þ2t2t3 cos ½kðr2�r3Þ�þ2t1t3 cos ½kðr3�r1Þ�g1=2; ð6Þ

where k is the two-dimensional wave vector.In the special case, when the tension is along armchair and

zigzag directions corresponding to θ¼ 0 and θ¼ π=2, respectively,the graphene with bond alternation as shown in Fig. 1 hast1 ¼ t0ða0=r1Þ2 and t2 ¼ t3 ¼ t0ða0=r2Þ2. Substituting Eq. (1) intoEq. (6) we obtain the energy dispersions of deformed graphenewith bond alternation

EtAðkx; kyÞ ¼ 7 t21þ4t1t2 cos aþb2

� �ð1�εsÞkx

� �cos

bffiffiffi3

p

2ð1þεÞky

" #(

þ4t22 cos 2 bffiffiffi3

p

2ð1þεÞky

" #)1=2

; ð7Þ

EtZðkx; kyÞ ¼ 7 t21þ4t1t2 cos aþb2

� �ð1þεÞkx

� �cos

bffiffiffi3

p

2ð1�εsÞky

" #(

þ4t22 cos 2 bffiffiffi3

p

2ð1�εsÞky

" #)1=2

; ð8Þ

where tAðtZÞ stands for tension along the armchair (zigzag)direction. Sublattice A(B) connects with three sublattices B(A) byone large and two small bonds. In comparison with equilibriumbond length a0, the two different bonds of the alternation latticechange their lengths by �2δ and δ, where δ can be positive ornegative [17]. This assumption is agreement with the quantumchemistry calculations for CNTs [4,5].

The position of the Dirac points of deformed graphene changesdue to tension and difference of C–C bond lengths. The shifting ofDirac points can change the electronic properties of CNTs andGNRs because it connects to the parallel k-lines passing or notthrough Dirac points of the two-dimensional Brillouin zone.

3. Band structure of deformed armchair graphene nanoribbonwith bond alternation

An armchair GNRs is strip of graphene that can be obtained bycutting a graphene sheet along the zigzag direction in the form ofa quasi-one dimensional wire. Atomic structure of deformedarmchair nanoribbon with bond alternation is shown in Fig. 1.All dangling bonds at the edges of graphene nanoribbons will beassumed to be terminated by hydrogen atoms. The width ofgraphene nanoribbon is defined via a number of dimer N. Theunit cell of the armchair GNR contains 2N carbon atoms. In thispaper we study the applied tension only along armchair andzigzag directions.

The energy spectrum of an GNR can be obtained from that ofgraphene by applying periodic boundary conditions at both edgesof the ribbon. For an armchair GNR as shown in Fig. 1, the periodicboundary condition leads to the discrete transverse wavenumber

ab

xy

1234

N

ly

lx

r1

r2

r3

a2

a1

Fig. 1. Atomic structure of graphene with bond alternation. a and b are the C–Cbond lengths. The shaded rhombus denotes the primitive cell of the length lx andthe width ly which contains two carbon atoms (filled circles). The unit cell ofarmchair graphene nanoribbon is shown as the dashed rectangle. All danglingbonds at the edges are terminated by hydrogen atoms (small empty circles). Filledand empty arrows stand for the tension along zigzag and armchair directions,respectively.

N.N. Hieu, L.C. Nhan / Physica E 60 (2014) 91–9492

given by

ky ¼ qπlyðNþ1Þ; q¼ 1;2;3;…;N; ð9Þ

where ly ¼ a1yþa2y.Substituting Eq. (9) into Eqs. (7) and (8), then we obtain the

one-dimensional energy spectrum E(k) of deformed armchair GNRwith bond alternation in the case of tension along armchair andzigzag directions

EtAðkÞ ¼ 7 4t1t2 cos aþb2

� �ð1�εsÞk

� �cos

qπNþ1

� ��

þ4t22 cos 2 qπNþ1

� �þt21

�1=2

; ð10Þ

EtZðkÞ ¼ 7 4t1t2 cos aþb2

� �ð1þεÞk

� �cos

qπNþ1

� ��

þ4t22 cos 2 qπNþ1

� �þt21

�1=2

: ð11Þ

Figs. 2 and 3 show, for example, the electronic band structure ofarmchair GNRs under various stresses. The possibility of the mini-mum of energy E(k) occurs at k¼0 with certain band index q. Hence,we have

Eð0Þ ¼ t1þ2t2 cosnπ

Nþ1

� �: ð12Þ

The band index corresponding to lowest energy of the conduc-tion band (and highest energy of the valence band) at k¼0 will bechanged due to stress and bond alternation, i.e., depending onhopping parameters t1 and t2 (see Figs. 2 and 3). In the absence ofapplied tension, all armchair GNRs with bond alternation becomesemiconducting, including armchair GNRs of N¼ 3n�1 as shownin Fig. 3(a). This is similar to appearance of band gap in zigzagcarbon nanotubes with bond alternation [5]. An effect of bondalternation on the energy gap of armchair GNRs is illustrated inFig. 4(a). We see that the band gap of armchair GNRs dependslinearly on the difference between C–C bond lengths δ. In thepresence of strain, the change in slope in Fig. 4(b) is due to thechange of the hopping parameters in Eq. (12) at certain band indexq corresponding to minimum of energy. Fig. 4(b) shows that theband gap of armchair GNRs can be equaled to zero at the criticalstrain which depends on applied strain and the width of ribbons. Itmeans that we can control the band gap of armchair GNRs by itsstrain.

The width dependence of the band gap for deformed arm-chair GNRs in both cases of tension along armchair and zigzagdirections is shown in Fig. 5. Almost energy gaps of armchairGNRs under tension along the zigzag direction are larger thanone in the case of tension along the armchair direction. Theband gap is roughly inversely proportional to the width ofribbon.

E, e

V

6

4

2

00 0.5 1

k, π/lx k, π/lx k, π/lx k, π/lx0 0.5 1 0 0.5 1 0 0.5 1

N = 5δ = 0ε = 0

Fig. 2. Band structure of armchair GNRs with bond alternation in the case of tension along armchair (Ox) direction. Only conduction bands are shown. The valence bands aresymmetric, about EF¼0, to the conduction bands.

0 0.5 1

N = 5= 0= 0.2

N = 5= 0.02 A

o

= 0.2

N = 6= 0.02 A

o

= 0.20 0.5 1 0 0.5 1 0 0.5 1

N = 7= 0= 0.2

6

4

2

, eV

0

tensile-armchairtensile-zigzag

k, π/lx k, π/lx k, π/lx k, π/lx

Fig. 3. Band structure of armchair GNRs with bond alternation under strain. Solid and dashed lines correspond to tension along armchair and zigzag directions, respectively.

N.N. Hieu, L.C. Nhan / Physica E 60 (2014) 91–94 93

4. Conclusions

In conclusion, we studied the π-electrons band structure ofdeformed armchair GNRs with bond alternation by the tight-binding approximation. Under stress, armchair GNRs with bondalternation can become semiconducting and the slope of thedependency of the energy gap on its stress depends on thedirection of tension. The band gap of armchair GNRs is sensitiveto bond alternation and uniaxial stress, however, we can controlthe energy gap by its deformation. Hence, GNRs can become thecontrolled object in the nanometer-scale strain sensor based onthe stretched GNRs.

Acknowledgments

This research was funded by the Vietnam National Foundationfor Science and Technology Development (NAFOSTED) underGrant No. 103.02-2012.14.

References

[1] K.S. Novoselov, A.K. Geim, S.V. Mozorov, D. Jiang, Y. Zhang, S.V. Dubonos,I.V. Gregorieva, A.A. Firsov, Science 306 (2004) 666.

[2] R. Saito, G. Dresselhaus, M.S. Dresselhaus, Physical Properties of CarbonNanotubes, Imperial College Press, London, 1998.

[3] A.H. Castro Neto, F. Guinea, N.M.R. Peres, K.S. Novoselov, A.K. Geim, Rev. Mod.Phys. 81 (2009) 109.

[4] N.A. Poklonski, E.F. Kislyakov, N.N. Hieu, O.N. Bubel', S.A. Vyrko, A.M. Popov,Yu.E. Lozovik, Chem. Phys. Lett. 464 (2008) 187.

[5] N.A. Poklonski, S.V. Ratkevich, S.A. Vyrko, E.F. Kislyakov, O.N. Bubel',A.M. Popov, Yu.E. Lozovik, N.N. Hieu, N.A. Viet, Chem. Phys. Lett. 545 (2012) 71.

[6] L. Brey, H.A. Fertig, Phys. Rev. B 73 (2006) 235411.[7] A.V. Rozhkov, S. Savelev, F. Nori, Phys. Rev. B 79 (2009) 125420.[8] Y. Li, X. Jiang, Z. Liu, Nano Res. 3 (2010) 545.[9] Y. Lu, J. Guo, Nano Res. 3 (2010) 189.[10] N.A. Poklonski, E.F. Kislyakov, S.A. Vyrko, O.N. Bubel', S.V. Ratkevich,

J. Nanophotonic 6 (2012) 061712.[11] M. Fujita, M. Igami, K. Nakada, J. Phys. Soc. Jpn. 66 (1997) 1864.[12] V.M. Pereira, A.H. Castro Neto, N.M.R. Peres, Phys. Rev. B 80 (2009) 045401.[13] W.A. Harrison, Electronic Structure and the Properties of Solids: The Physics of

the Chemical Bond, Dover Publications, New York, 1989.[14] S. Reich, J. Maultzsch, C. Thomsen, P. Ordejón, Phys. Rev. B 66 (2002) 035412.[15] P.R. Wallace, Phys. Rev. 71 (1947) 622.[16] L. Yang, M.P. Anantram, J. Han, J.P. Lu, Phys. Rev. B 60 (1999) 13874.[17] N.A. Viet, H. Ajiki, T. Ando, J. Phys. Soc. Jpn. 63 (1994) 3036.

E g, e

V

0

0.5

1.0

1.5

2.0

N = 5N = 6N = 7

ε = 0N = 5N = 6N = 7

0 0.01 0.02 0.03

δ, A

0.10 0.2 0.3ε

E g, e

V

0

0.5

1.0

1.5

2.0

δ = 0.02 A

Fig. 4. Dependence of band gap of armchair GNRs on: (a) difference of bond lengths δ and (b) tension at δ¼ 0:02 Å. tA (filled symbols) and tZ (empty symbols) stand fortension along armchair and zigzag directions, respectively.

0

0.5

1.0

1.5

2.0

5 10 15

E g, e

V

N

tensile-armchairtensile-zigzag

ε = 0.2δ = 0.02 A

Fig. 5. Dependence of band gap of armchair GNRs on the wide ribbon at δ¼ 0:02 Åand ε¼ 0:2.

N.N. Hieu, L.C. Nhan / Physica E 60 (2014) 91–9494