Energy band of graphene ribbons under the tensile force

Transcription

1 Energy band of graphene ribbons under the tensile force Yong Wei Guo-Ping Tong * and Sheng Li Institute of Theoretical Physics Zhejiang Normal University Jinhua 004 Zhejiang China ccording to the tight-binding approximation we investigate the electronic structures of graphene ribbons with zigzag shaped edges (ZGRs) and armchair shaped edges (GRs) drawn by the tensile force and obtain the analytic relations between the energy bands of π-electrons in ZGR GR and the tensile force based on only considering the nearest-neighbor interaction and the hydrogen-like atomic wave function is considered as π-electron wave function. Importantly we find the tensile force can open an energy gap at the K point for ZGR and GR and the force perpendicular to the zigzag edges can open energy gap more easily besides the gap values of ZGR and GR at the K point both increase as the tensile force increases. PCS number(s): 7.5.-m 7.0.Tx 7.6.Wp Graphene a single atomic layer of graphite has attracted much attention recently - because it not only has been produced successfully in experiment 4 and observed by an ordinary optical microscope 56 but also has a number of unusual transport properties 7-9 and exists the Klein paradox 0 weak localization etc. Moreover the graphene has many potential applications. The electronic structure of graphene has been investigated by Wallace since 947. But the energy dispersion relations of the π -electrons of graphene were given very well by Saito et al. using the tight-binding approximation although only considering the nearest-neighbor interaction. Then S.Reich et al. investigated the energy band of graphene using the same method including up to third-nearest neighbors in Nowadays. H. Castro Neto et al. give a description of conclusion for the electronic properties of graphene. Because of some significant properties of graphene that can be future electronic devices graphene nanoribbon (GNR) a strip of graphene of nanometers in width with zigzag shaped edges or armchair shaped edges on both sides and graphene ribbon that the width is far more than nanometer have also investigated by many investigators. 5- Whether the graphene nanoribbon or the graphene ribbon it must be effected by the tensile force when it is been electronic devices but it is not clear that the relations between the electronic structure of graphene and the tensile force which is solved by this paper. In this paper we build a simple model: the graphene ribbon with zigzag shaped edges or armchair shaped edges on both sides is drawn by the tensile force perpendicular to two graphene ribbon edges and parallel to the graphene ribbon plane (see Figs.(a) and (b)) the widths of graphene ribbons are so wide that the periodic boundary condition is satisfied and the

2 approximation of an infinite plane is used in calculating electronic structures of graphene ribbons besides the interaction between carbon atoms satisfies the Huke law. Then we investigate the relations between the π-electron bands of the graphene ribbons with zigzag shaped edges (ZGR) armchair shaped edges (GR) and the tensile force based on the tight-binding approximation. We first investigate the ZGR drawn by the tensile force. Similar to the method of Ref. using the tight-binding approximation and only considering the nearest-neighbor interaction we obtain the matrix Hamiltonian where H as H r e r R H r R ϕ ( ) ik ( R R) ϕ N R N p p () R N is the unit cell number of ZGR R is the site vector of carbon atom k is the wave vector H is a one-electron Hamiltonian ϕ can be considered as p z -orbital wave function of ϕ r R H ϕ r R is different from the atomic energy carbon atoms. Here p value for the free carbon atom because the Hamiltonian H contains all ZGR potential and changes as the tensile force changes so H BB for the same order of approximation. p p is a variable as the tensile force. Similarly FIG.. (a).the graph of graphene ribbon with zigzag shaped edges (ZGR) drawn by the tensile force. (b).the graph of graphene ribbon with armchair shaped edges (GR) drawn by the tensile force. (c).the graph of showing R R R α. site vectors and half-bond angle We also obtain the matrix Hamiltonian H B for only considering the nearest-neighbor interaction as ik R B { ϕ B N R ϕ ( R ) H e r R H r R ϕb + ik R e ϕ r R H r R R

3 ϕb } + ik R e ϕ r R H r R R where R R and R ik R ik R ik R e e e γ + γ + γ are the positions of the nearest-neighbor atom B relative to atom (see () Fig.(c)) the transfer integralsγ γ andγ are denoted by γ ϕ ϕ ( r R) H B ( r R R ) γ ϕ ϕ ( r R) H B ( r R R ) γ ϕ ϕ ( r R) H B ( r R R ) () where γ γ due to R Raccording to the symmetry of ZGR. The matrix element H B can be obtained by the Hermitian conjugation relation H H. Using the similar method used for H or H we obtain the overlap matrix S S as assuming that the atomic wave B ik R ik R function is normalized and also derive S se + s e + s e B B B BB ik R integrals: s s and s between the nearest and B atoms are denoted by s ϕ r R ϕ r R R B s r R r R R ϕb ϕ s r R r R R ϕb ϕ we can get s because of the symmetry and obtain S s conjugation relation. We solve the secular equation [ H ] dispersion relations of the π -electrons of ZGR as E ± ( k) where the overlap (4) B S B though the Hermitian ES and obtain the energy det 0 p γs O k ± γ s + P k (5) O( k ) ( p ) s Q( k) where the P k and Q k are denoted by cos O k γs + ( k R k R) + ( γs+ γs) cos( k R k R) + cos ( k R k R) cos ( ) cos P k γ ps γ ps k R k R + ( k R k R) ( γ s ) cos( k R k + p + R) (6)

4 E + ( γ ) s γs sin k R k R + sin k R k R (7) Q( k) s + cos( k R k R ) ss cos( k R k R) + cos( k R k R) (8) k and E k are called bonding π and antibondingπ energy bands. Now we explore the relations between the parameters of Eq. (5) and the tensile force in order to obtain the analytic expressions of π -electron energy band with the tensile force. ccording to the geometric knowledge we can easy to obtain the lattice vectors a a and the reciprocal lattice vectors b of ZGR as a R cos α + R R sin α b a ( R α + R R α ) cos sin π π b Rcosα + R Rsin α (9) b π π Rcosα + R Rsin α (0) where the relative positions R R R and the half-bond angleα (shown in Fig.(c)) are given due to the mechanical and geometric knowledge by ac c R ac c +Δ R π sin + ( + fn) + 4fN + f 6 R R N R ac c+δ R ac c( + fn) π α f N fn f N () where 0.4 nm is the nearest-neighbor distance between two carbon atoms without the ac c tensile force fn is a dimensionless value of f that the tensile force shares to a little component force of each carbon atom and fn f ( k0ac c) k0 is the elastic constant between two carbon atoms. ccording to Eq. (0) we can also obtain the positions of the Dirac points K and K in momentum space as

5 π π πr sinα K R R R R R cosα + sinα ( cos α + ) π πr sinα π K R R R R R. cosα + ( cos α + ) sinα () Based on Eqs. (5) (6) (7) (8) (0) and () if we can also obtain the relations between the transfer integrals the overlap integrals and the tensile force we will obtain the analytic relations between the energy band of the π -electrons of ZGR and the tensile force fn in all Brillouin zone. For one-electron Hamiltonian H if we only consider the nearest-neighbor interaction we can write H approximately as H + V( r R) + VB ( r R R ) ( + VB r R R + VB r R R ). () 5 Substituting Eq.() into Eq.() considering wave function ϕ( r) λ / πrcosθe λr as π -electron wave function and using the conclusions of Refs. and - we can obtain the analytic expressions of transfer integrals as J γ sλ( λ Z) / + ( Z) J+ Jn n γ γ γ s λ λ Z /+ Z J + Z J (4a) λ + η 6 e η ηn ηn ηn λ λη λe λe ληne Jn ( n ) + S K T U ηn ηn ηη n ηn η 6η (4b) S + η + η + η 5 5 e η sinh cosh ( n n ) ( η ) K ηn ηn ηn ηnη η η η + ηη + 9ηη + η sinh η T F + F + F + F + F η n e U F + F+ + 4 E ηn ( η η ) η ( η η ) ( η ) e e η n (4c)

6 ( η η η ) η ( η η η ) e e η 4 ( η 4η 4η 70η 70) e 4 ( η + η + η + η+ ) 4 ( 6η 60η 70η 60η 60) e 4 ( η + η + η + η+ ) + + η e η e 4 (4d) η η F η n + 4 F ηn + ηn + 4 F ηn + ηn + ηn F4 ηn + ηn + ηn + ηn F5 ηn + ηn + ηn + ηn + ηn t E ( ηn ) e dt (4e) ηn t where J is equal to displaced by λr and η λr ηn λrn n (4f) J that the η is displaced by λ R J is equal to η is displaced by λ R. J that the η is also ccording to the conclusions of Refs. and we can also derive the overlap integrals and p as p λ( λ ) Z / Z I i i I i λ R λ R R λr λ R i i i i i e λr i λ Ri λ R i si + λri + + e λ 5 5 R i ( i ) (5) where R R R are given by Eq. () but the R R R of Eqs. (4) and (5) are all dimensionless that have been divided by the Bohr radius a0 λ and Z are the Slater orbital parameter and the effective nuclear charge number of carbon atoms. Based on above equations we have established the analytic relations between the energy band of π -electrons in ZGR and the tensile force. In fact above equations also give the analytic relations between the energy band of

7 π -electrons in GR and the tensile force if the Eq. () is modified by R R ac c π cos ( fn) ( fn) 8fN R R ac c π α ( + fn) + ( + fn) + 8 N. f (6) For showing clearly the changeable relations of ZGR and GR as the tensile force Figure. is given to show the numerical relations of half-bond angleα bond-lengths R ( R ) R and transfer integralsγ ( γ ) γ with the tensile force which acts on the zigzag edges of graphene ribbon and the armchair edges of graphene ribbon. Here the dimensionless value fn f ( k0a c c) is taken as 0~0.. α(rad). Zigzag edges (a) rmchair edges (b) Bond-length(0-0 m) Transfer integral(ev) R R γ (a) (a) γ f N R R (b) (b) γ γ f N FIG.. (Color online) The relation graphs of half-bond angleα bond-lengths R ( R ) R and transfer integralsγ ( γ ) γ with the tensile force which acts on the zigzag edges of graphene ribbon (three graphs in left) and the armchair edges of graphene ribbon (three graphs in right). The data values of Figs. (a) (a) (b) and (b) are obtained by Eqs. () and (6) the data values of Figs.(a) and (b) are derived by Eqs. () (4) (5) and (6) based on effective nuclear charge number Z.6 Slater orbital parameter λ.8. In Figs. (a) and (b) the increasing or decreasing of half-bond angleα is obvious according to Eqs.() and (6) because of the tensile force. In Fig. (a) the increasing of bond-lengths R ( R ) R is also obvious due to the force and the values of R and R are closed because f N is very small and the edge effect of graphene ribbon is neglected which are also the reason of the bond-length R for GR is not changeable in Fig.(b) these phenomena

8 are reasonable according to the mechanical knowledge. In Figs.(a) and (b) the data relations of transfer integralsγ ( γ ) γ for ZGR and GR with the tensile force are obtained by Eqs. () (4) (5) and (6) based on effective nuclear charge number Z.6 Slater orbital parameter λ.8. These values of Z and λ are taken for getting the transfer integral without tensile force γ eV and the overlap integral without tensile force is between -.5 and -ev and s because γ 0 4 s 0 is below 0.for fitting the experimental or first-principles data. In Fig. (a) the changing of γ is closed to γ came from the changing values of R and R are closed because the transfer integral is related to the bond-length. In Fig. (b) the decreasing of γ is small which can be explained that the Hamiltonian H is changeable (because the Hamiltonian H changes as the tensile force changes) although R is a constant. 0.5 Zigzag edges 0.8 rmchair edges (a) 0.6 (b) Gap value(ev) Gap value(ev) f N f N FIG.. (Color online) The relation graphs between the energy gap values and the tensile force at K point. The left graph shows the gap values of the graphene ribbon with zigzag edges drawn by the tensile force the right graph shows the gap values of the graphene ribbon with armchair edges drawn by the tensile force. These data values are derived by the Eq. () and above analytic relations between the energy bands of π -electrons in ZGR GR and the tensile force. In many literatures p is considered as a parameter which is often taken as zero or closed to r R H r R is much larger than zero whether the zero 4 in fact the p ϕ ϕ ZGR and GR are drawn by the force or not though the calculation of above formulas based on 5 π -electron wave function ϕ( r) λ / πrcosθe λr and Z.6 λ.8 so the energy E ± values of graphene ribbons are dependent mainly from p besides other parameters are related in Eq. (5) which produces that the E ± of graphene ribbons without tensile force are not

9 closed to zero at K point due to p is not closed to zero but this defect can be neglected in calculating energy difference ΔE between two energy bands in all Brillouin zone because Δ E is independent approximately from p according to Eq. (5). So we can obtain the accurate data values of Δ E (including the gap values at K or K ' point) based on Eq. (5) because p can be neglected and γ 0 s 0 are taken for fitting the experimental or first-principles data though Z.6 λ.8. In Fig. we show the data relations between the gap values of ZGR (shown in Fig. (a)) GR (shown in Fig. (b)) and the tensile force at the K point according to the Eq. () and above analytic relations between the energy bands of π -electrons in ZGR GR and the tensile force based on Z.6 λ.8. In this figure we find that the tensile force can open an energy gap of ZGR or GR at the K point and the gap values of ZGR and GR both increase as the force increases moreover the gap values of ZGR increase more far than the GR at the K point as the tensile force increases. For these important phenomenon we can explain that the opening of energy gap for graphene ribbons is due to the deformation of graphene ribbons produced from the force and the deformation that zigzag edges are drawn by the fore gives an energy gap more easily. In conclusion we have investigated the electronic structures of ZGR and GR drawn by the tensile force according to the tight-binding approximation and obtained the analytic relations between the energy bands of π -electrons in ZGR GR and the tensile force based on only 5 considering the nearest-neighbor interaction and wave function ϕ( r) λ / πrcosθe λr is regarded as π -electron wave function. Though the numerical calculation we give the numerical relations of half-bond angles bond-lengths transfer integrals and the gap values at K point with the tensile force for ZGR and GR and find that the tensile force can open an energy gap of ZGR or GR at the K point and the tensile force perpendicular to the zigzag edges can open energy gap more easily besides two gap values of ZGR and GR at the K point both increase as the tensile force increases. * tgp646@zjnu.cn. K. Geim and K. S. Novoselov Nat. Mater. 6 8 (007).. H. Castro Neto F. Guinea N. M. R. Peres K. S. Novoselov and. K. Geim arxiv: [Rev.Mod. Phys. (to be published)]. L. Yang C.-H. Park Y.-W. Son M. L. Cohen and S. G. Louie Phys. Rev. Lett (007). 4 K. S. Novoselov. K. Geim S. V. Morozov D. Jiang Y. Zhang S. V. Dubonos I. V. Grigorieva and.. Firsov Science (004). 5 P. Blake E. W. Hill. H. Castro Neto K. S. Novoselov D. Jiang R. Yang T. J. Booth and. K. Geim ppl. Phys. Lett (007). 6 D. S. L. bergel. Russell and V. I. Fal ko ppl. Phys. Lett (007).

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