Electronic Band Structure of Graphene Based on the Rectangular 4-Atom Unit Cell

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1 Journl of Modern Physics, 07, 8, ISSN Online: 53-0X ISSN Print: Electronic Bnd Structure of Grphene Bsed on the Rectngulr 4-Atom Unit Cell Akir Suzuki, Msshi Tnbe, Shigei Fuit Deprtment of Physics, Fculty of Science, Tokyo University of Science, Tokyo, Jpn Deprtment of Physics, University t Bufflo, Stte University of New York, Bufflo, NY, USA How to cite this pper: Suzuki, A., Tnbe, M. nd Fuit, S. (07) Electronic Bnd Structure of Grphene Bsed on the Rectngulr 4-Atom Unit Cell. Journl of Modern Physics, 8, Received: Februry 3, 07 Accepted: Mrch 7, 07 Published: Mrch 30, 07 Copyright 07 by uthors nd Scientific Reserch Publishing Inc. This work is licensed under the Cretive Commons Attribution Interntionl License (CC BY 4.0). Open Access Abstrct The Wigner-Seitz unit cell (rhombus) for honeycomb lttice fils to estblish k -vector in the D spce, which is required for the Bloch electron dynmics. Phonon motion cnnot be discussed in the tringulr coordintes, either. In this pper, we propose rectngulr 4-tom unit cell model, which llows us to discuss the electron nd phonon (wve pckets) motion in the k -spce. The present pper discusses the bnd structure of grphene bsed on the rectngulr 4-tom unit cell model to estblish n pproprite k -vector k for the Bloch electron dynmics. To obtin the bnd energy of Bloch electron in grphene, we extend the tight-binding clcultions for the Wigner-Seitz (- tom unit cell) model of Reich et l. (Physicl Review B, 66, Article ID: 0354 (00)) to the rectngulr 4-tom unit cell model. It is shown tht the grphene bnd structure bsed on the rectngulr 4-tom unit cell model revels the sme bnd structure of the grphene bsed on the Wigner-Seitz -tom unit cell model; the π -bnd energy holds liner dispersion ( ε k ) reltions ner the Fermi energy (crossing points of the vlence nd the conduction bnds) in the first Brillouin zone of the rectngulr reciprocl lttice. We then confirm the suitbility of the proposed rectngulr (orthogonl) unit cell model for grphene in order to estblish D k -vector responsible for the Bloch electron (wve pcket) dynmics in grphene. Keywords Grphene, Rectngulr 4-Atom Unit Cell Model, Primitive Orthogonl Bsis Vector, Bloch Electron (Wve Pcket) Dynmics, k -Vector, Dirc Points, Liner Dispersion Reltion. Introduction The electronic bnd structure of grphene plys n importnt role for understnding its unique properties [] [] [3] [4]. Grphene is perfect two- DOI: 0.436/mp Mrch 30, 07

2 dimensionl crystl consisting of single lyer of crbon toms rrnged in honeycomb lttice. A crbon tom contins four vlence electrons, one s - electron, nd three p -electrons. They re sp -hybridized, tht is, one s - electron nd two p -electrons form strong σ -bonds between crbon toms leding to the honeycomb structure with the crbon-crbon distnce of 0.4 nm. The remining p -electron occurs s p z -orbitl, which is oriented perpendiculrly to the plnr structure, nd forms π -bond with the neighbouring crbon toms. The σ -bonds re completely filled nd form deep vlence bnd []. The smllest gp between the bonding nd the nti-bonding σ -bnds is pproximtely ev. Therefore, the mority of low-energy physicl effects is determined by the π -bnds. Since the overlp with other orbitls ( s, px,p y ) is strictly zero by symmetry, p z -electrons forming the π -bonds cn be treted independently from other vlence electrons [5]. Indeed, the bnd structure of grphene cn be seen s tringulr lttice with bsis of two toms per unit cell. This -tom unit cell (Wigner-Seitz (WS) cell) model hs customrily been used to obtin the grphene bnd structure for the p z -electrons. The bnd structure provides useful informtion bout the energy dispersion reltion of electrons t zero temperture. At finite tempertures those thermlly excited electrons nd holes ply n essentil role for crrier trnsport properties. Following Ashcroft nd Mermin (AM) [6], if we dopt the semiclssicl model of electron dynmics in solids, it is necessry to introduce k -vector: k = k eˆ + k eˆ + k e ˆ, () where ˆ ( =,, ) x x y y z z e x y z re the Crtesin orthogonl unit vectors since the k -vector, k, is involved in the semiclssicl eqution of (wve pcket) motion: d ħk ħ k = q ( E+ v B ), () dt where E nd B re the electric nd mgnetic fields, respectively, nd the vector v ε v k = (3) ħ k is the electron velocity where ε is the electron energy. In the semiclssicl theory electrons re treted s wve pcket. The D crystls such s grphene cn lso be treted similrly, only the z -component being dropped. Grphene forms D honeycomb lttice. The WS unit cell is rhombus (dotted lines) s shown in Figure (). The potentil energy V ( r ) is lttice-periodic: V ( r+ R ) = V ( r ), (4) where Rmn = m + n re Brvis vectors with the primitive non-orthogonl vectors (, ) nd integers ( mn, ). In the field theoreticl formultion, the field point r is given by r= r + R mn, where r is the point defined within the stndrd WS unit cell. Eqution (4) describes the D lttice periodicity but mn 608

3 () (b) Figure. () Lttice structure of grphene. Crbon toms t vertices. Ech honeycomb lttice consists of equivlent crbon (C + ) ions lbeled by A nd B. The Wigner-Seitz -tom unit cell (dotted lines) spnned by the bsis (lttice unit) vectors ã nd ã ; (b) Reciprocl lttice showing the first Brillouin zone (BZ) of grphene including the high-symmetry points Γ, K, nd M. The BZ is spnned by two reciprocl lttice vectors b nd b constructed from the bsis vectors ã nd ã. does not estblish the k -spce for the Bloch electrons in grphene, which will be explined in Section 4, nd this fct hs motivted us to study the bnd structure for Bloch electron bsed on the rectngulr 4-tom unit cell model for grphene (see Figure ()), which defines the k -vectors plying n importnt roll for Bloch electron dynmics. Our purpose of this pper is to explore the suitbility of the rectngulr (orthogonl) unit cell model for the Bloch electron bnd structure nd to discuss the k -vector defined in the D spce for Bloch electrons in grphene. Bsed on the rectngulr 4-tom unit cell model of grphene, we obtin the bnd structure for the p z -electrons of grphene by extending the prevlent tight-binding clcultions for the Wigner-Seitz (WS) -tom unit cell model to the rectngulr (orthogonl) 4-tom unit cell model introduced by the present uthors [7]. In Section, the rectngulr 4-tom unit cell of grphene is introduced. Section 3 presents our tight-binding clcultions of the energy bnd of Bloch electron in grphene, bsed on the rectngulr 4-tom unit cell model described in Section. Section 4 illustrtes why we hve to utilize the rectngulr (orthogonl) unit cell rther thn the WS unit cell when considering the electron dynmics of grphene. Results nd discussion re given in Section 5. Finlly conclusions nd some remrks re given in Section 6.. The Rectngulr 4-Atom Unit Cell Model for Grphene Grphene is mde out of honeycomb lttice (crbon toms rrnged in hexgonl structure) nd the D honeycomb lttice hs reflection symmetry reltive to the x - nd y -xis (cp. Figure ()). Then the bnd structure for Bloch electron in grphene cn be constructed from rectngulr lttice with bsis of four toms per unit cell, which we cll the rectngulr 4-tom unit cell. 609

4 () (b) Figure. () Lttice structure of grphene, mde out of the rectngulr 4-tom unit cell ( squre dotted line) spnned by the bsis vectors nd the nerest-neighbor vectors with the constnt crbon-crbon distnce of τ i i =,,3 re. The ( ) τi = 3 = 0.4 nm ; (b) The first Brillouin zone of the rectngulr 4-tom unit cell. The b nd b re the reciprocl lttice vectors corresponding to the lttice unit vectors. The high-symmetric points within the rectngulr unit cell re indicted by Γ, X, Y, nd W within the reciprocl lttice plne. The reson why we choose rectngulr (orthogonl) unit cell rther thn tringulr (WS) unit cell for grphene will be explined lter. The lttice bsis (unit) vectors ( ) defined s i i =, of grphene (cp. Figure ()) re ( ) ( ) =, 0, = 0, 3. (5) The lttice constnts for the rectngulr (orthogonl) unit cell in the Crtesin coordintes re = 3 0 nd = 3 = 30, respectively, where 0 is the nerest-neighbor distnce between two crbon toms. The rectngulr 4-tom unit cell of grphene contins four crbon toms, sy A, B, C, nd D, s shown in Figure (). The three nerest-neighbor vectors ( ) spce re defined (cp. Figure ()) by τ i i =,, 3 in rel ( ) ( ) ( ) τ = 0, 3, τ =, 3, τ =, 3. (6) 3 The reciprocl-lttice vectors for the rectngulr unit cell re given (from Eqution (5)) by π b =, 0, b = 0, 3. (7) π ( ) ( ) The first Brillouin zone is rectngle s shown in Figure (b) with sides of BZ BZ length, bx = b = π nd by = b = π 3, nd re equl to 4π 3. There re four key loctions of high symmetry in the first Brillouin zone (FBZ) of the rectngulr 4-tom unit cell. In Figure (b), these loctions re identified s the Γ -point, the X-point, the Y-point, nd the W-point. The Γ -point is t the center of the Brillouin zone. The loctions of these points in the D reciprocl spce re: 60

5 π π π ( ) ( ) ( ) ( ) Γ= 0, 0, X =, 0, Y = 0, 3, W =, 3. (8) 3. Tight-Binding Approch The single-prticle bnd structure of grphene cn be nlyticlly clculted within the tight-binding pproximtion ssuming tht electrons re tightly bound to their C + ion [8] [9] [0] []. In order to obtin the bnd structure of grphene bsed on the 4-tom rectngulr unit cell model [7] for grphene, we follow the tight-binding pproch employed by Reich et l. [8]. The Schrödinger eqution for single electron in the lttice potentil field V ( r ) is expressed by where the Hmiltonin is given by nd the lttice potentil V ( ) = V ( ) ( r) ε ( r), Ψ = Ψ (9) ħ = + V ( r ), (0) m r r R hs lttice periodicity chrcteristic for grphene. The totl wve function Ψ for the electron in the p z -orbitl (Bloch electron) my be obtined from the liner combintion of the Bloch wve Φ of the form: i Φ ( r) r Φ = e kr φ ( r R ), () N R where N is the number of unit lttices nd the index refers to the respective crbon toms. In the bove expressions, the phse exp( ikr ) re introduced in order to stisfy the Bloch theorem in ech conduction chnnel. Thus the wve function for Bloch electron in grphene is then expressed by the liner combintions of these Bloch wve functions: Ψ( r) r Ψ () = λ k Φ r = λ k r Φ ( ) ( ) ( ), where r Φ s re given by Eqution () nd the wve function r φ for the p z -orbitl is normlized. The energy eigenvlues of Eqution (9) for the p z -electron cn be obtined in usul mnner by using the grphene Hmiltonin (0) nd the grphene wve function () long with Eqution (): Inserting Eqution () into Eqution (9) nd multiplying Φ from the left to i Eqution (9), we obtin four seprte equtions, which re simply expressed in the mtrix form s λ k = εsλ k ; λ k = εs λ k, (3) ( ) ( ) ( ) ( ) i i where the mtrices,, S nd λ( k ), re respectively expressed by AA AB AC AD BA BB BC BD = (4) CA CB CC CD DA DB DC DD 6

6 Here i Φi Φ SAA SAB SAC SAD SBA SBB SBC SBD S = (5) SCA SCB SCC SCD SDA SDB SDC SDD ( ) λ k ( k ) ( k ) ( k ) ( k ) λa λb =. λc λ D (6) re the mtrix elements of the Hmiltonin, which we cll hopping (or trnsfer) integrl, nd re the units of energy. Si Φi Φ re the overlp mtrix elements, which re given by the overlp integrl between Bloch functions, nd re unitless. Eqution (3) expresses the simultneous equtions for λa ( k ), λb ( k ), λc ( k ) nd λd ( ) nontrivil solutions for λ ( k ) ( = A,B,C,D) nnt k. In order to obtin the, the seculr eqution determi ε S of their simultneous equtions must be zero: ε S = 0. (7) Since the tomic wve functions re well loclized round the crbon toms, only the nerest-neighbor hopping of Bloch electrons re tken into considertion of the following clcultions. In other words, s for the electron in tom A orbitl, it cn hop to nerest orbitl of tom B or tom D. Similrly, the electron in the orbitl of tom B cn hop to the nerest orbitl of tom A or tom C, the electron in the orbitl of tom C to the nerest orbitl of tom B or tom D, nd the electron in the orbitl of tom D to the nerest orbitl of tom A or tom C. Let us consider the nerest-neighbor hopping between the orbitl of tom A nd tom B. The mtrix elements AB is expressed by the hopping integrl: AB = ΦA ΦB i ( B A) (8) = e k R R φ( r RA) φ( r RB). N RA RB Here we only tke into ccount the nerest-neighbor hopping between crbon toms. From the symmetry of the lttice strcture of grphene, the dcent hoppings re ll the sme. We introduce the prmetric prmeter t for the dcent hopping between the orbitl of tom A nd B: φ ( r R ) φ( r R ). (9) A B t Since the nerest-neighbor vector R B R A is given by τ nd τ 3 (cp. Eqution (6)), Eqution (8) cn be expressed by i ( B A) AB = e k R R φ( r RA ) φ( r RB ) N RA RB (0) i ky ik τ ik τ3 3 t( e + e ) = te cos kx. Similrly, the nerest-neighbor hopping integrls re given by 6

7 i ky i ky 3 3 AC = AD = te BC = te i ky 3 BD 0, CD te cos kx. 0,,, = = We note tht the hopping prmeter t is the nerest-neighbor hopping energy (hopping between dcent crbon toms). The mtrix element AA (on-site hopping) is evluted s AA = ΦA ΦA ik ( RA RA) = e φ( r RA) φ( r RA ) N RA RA () φ( r RA) φ( r RA) N RA t, 0 where N is the number of the bsic unit cells, nd t 0 is the prmetric prmeter for φ( r RA) φ( r RA). Similr evlution leds to = = = t. BB CC DD 0 Next we evlute the mtrix elements Si ( i, = A,B,C,D), which re given by the overlp integrl Φi Φ. Thus the mtrix element S AB is evluted from S = Φ Φ = φ A N AB A B i ( B A) e k R R R R A B ( r R ) φ( r R ) Introducing the prmetric prmeter s for the overlp integrl: ( ) φ ( ) s, A B B. () (3) φ r R r R (4) the mtrix element Similrly we obtin S AB is given by i ky 3 AB cos x. S = se k (5) i ky i ky 3 3 AC = AD = BC = i ky 3 BD 0, CD cos x. S 0, S se, S se, S = S = se k We note tht the mtrix element S = Φ Φ AA A A S AA is evluted s ik ( R R ) ( r R ) φ( r R ) ( r R ) φ( r R ) A A = e φ A N RA RA φ N RA A =. A A (6) (7) In similr mnner, we obtin SBB = SCC = SDD =. (8) 63

8 We cn solve Eqution (7) for ε by using Equtions (8)-(8) for the mtrix elements, nd obtin the four energy eigenvlues of Eqution (9) for the grphene π -electron bnds: where tively. ε c nd 3 t0 t + 4cos kx± 4cos ky cos kx εc =, 3 s + 4cos kx± 4cos ky cos kx 3 t0 + t + 4cos kx± 4cos ky cos kx ε v =, 3 + s + 4cos kx± 4cos ky cos kx (9) (30) ε v re the conduction-nd the vlence-bnd energies, respec- 4. Electron Dynmics of Grphene In semiclssicl (wve pcket) theory for electron dynmics, it is necessry to introduce wve vector k (nmely, k -vector) [6] [] since the k -vectors re involved in the semiclssicl eqution of motion (see Eqution ()). Here, we explin why we employed the rectngulr (orthogonl) unit cell for grphene in order to clculte one electron energy bnd for grphene. Grphene forms D honeycomb lttice. Let us first consider the Wigner- Seitz (WS) unit cell (rhombus, dotted lines shown in Figure ()). The potentil energy V ( r ) is lttice-periodic: V ( + ) = V ( ), r R r (3) where the position vector R in the potentil field V cn be represented by, nd Brvis vectors with the primitive (non-orthogonl) bsis vectors ( ) integers ( mn, ) : R R. mn = m + n (3) In the field theoreticl formultion, the field point r is given by r = r + R (33) where r is the point defined within the stndrd WS unit cell. Eqution (3) describes the (D) lttice periodicity but does not estblish the D k -vector of Bloch electron in grphene if we choose the non-crtesin coordintes system. Thus the WS unit cell is not pproprite when one dels with grphene trnsport problem, which is explined below. We ssume tht the wve pcket is composed of superposble plne-wves chrcterized by the k -vectors. The superposbility is the bsic property of the Schrödinger wve eqution. The Schrödinger wve eqution for Bloch electron (wve pcket) is ħ iħ ψ ( r) = ψ * ( r) + V ( r) ψ ( r ), (34) t m * where m is the effective electron mss. The Brvis vectors for the rhombic mn, 64

9 lttice, R mn, re given by Eqution (3) nd the system is lttice-periodic, cp. Eqution (3). The Bloch theorem cn be expressed in the form: ikr ( ) =, = e ( ) ψ r rkψ ψ r R (35) k for ll R in the Brvis lttice. The theorem in D cn be proved elementrily [6]. If n oblique lttice is considered nd set of nonorthogonl bsis vectors (, ) re introduced for D Brvis vector, cp. Eqution (3), there re no D Fourier trnsform vilble since there is no D k -vector to stisfy the periodicity condition for the Bloch wve, which must be of the form (35). Thus, the WS unit cell model for grphene is not suitble when one considers the electron dynmics in grphene. The wve function must be Fourier-nlyzble. In the rhombic system, however, if we choose the x -xis, sy long ã, then the potentil energy field V ( r ) is periodic in the x -direction, but it is periodic in the y -direction (cp. Figure ()) since these bsis vectors (, ) re not orthogonl ech other. Thus, if we choose the rhombic unit cell for grphene, we cnnot express k - vectors stisfying the condition for the Bloch wve (35). Then, there is no D k spce spnned by D k -vectors of Bloch wves if we choose the rhombic unit cell. If we omit the kinetic energy term, then we cn still use Eqution (3) nd obtin the ground stte energy (except the zero-point energy). Ashcroft nd Mermin (AM) [6] introduced trnsltion opertor R such tht cting on ny field f ( r ) it shifts its rgument by R : ( ) = f ( + ). k R f r r R (36) They used = (37) R R to estblish Eqution (35). The trnsltion opertor terms of differentil opertor: X cn be expressed in X = exp X (38) x s seen from exp X f ( x) = f ( x+ X). If the Brvis vector R is given in x terms of the orthogonl bsis vectors (, ) in the Crtesin coordintes nd choose the rectngulr unit cell (cp. Figure ()) s in Eqution (34), then Eqution (37) is stisfied, mening tht the Bloch wve for rectngulr unit cell cn be expressed in terms of the Bloch wve (35). If R is written in terms of nonorthogonl (, ), then R is product of differentil opertors exp X nd exp Y, which does not x y commute with the Lplcin = + in the Hmiltonin opertor x y. Hence, we cn not estblish Fourier-nlyzble D k -vectors when we choose the rhombic unit cell for grphene. In ctul clcultions, Eqution (37) 65

10 is vlid for the Bloch wve (35), where the Brvis vectors re expressed by the orthogonl bsis vectors in the Crtesin coordintes. It should be noted tht for n infinite lttice the periodic boundry is the only cceptble boundry condition for the Fourier trnsformtion. AM s second proof [6] lso fils in D if the Brvis vector R is expressed in nonorthogonl bse vectors. This is so becuse Lplcin terms cnnot be hndled in non-crtesin (oblique) coordintes. AM's Eqution (8.36) does not hold. We must choose the rectngulr unit cell in order to estblish the Bloch plne wves [3] [4] for the electron in D. We ssume tht the electron ( hole ') (wve pcket) hs the chrge e ( + e) nd size of the rectngulr unit cell, generted bove (below) the Fermi energy ε F. It ws shown [7] [5] tht () the electron nd the hole hve different chrge distribution nd different effective msses, (b) tht the electron nd hole move in different esy chnnels, (c) tht the electrons nd holes re thermlly excited with different ctivtion energies, nd (d) tht the electron ctivtion energy ε is smller thn the hole ctivtion energy ε : ε < ε (39) Thus, electrons re the mority crriers in grphene. The thermlly ctivted electron densities re then given by n T = nexp ε kt, n= constnt, (40) ( ) ( ) ( ) i i i B i where i = nd represent the electron nd hole, respectively. The prefctor n i is the density t the high-temperture limit. Mgnetotrnsport experiments by Zhng et l. [6] indicte tht the electrons re the mority crriers in grphene. Thus, our theory bsed on the rectngulr unit cell model is greement with experiments. At finite temperture phonons re present in the system. The excittion of phonons cn be discussed bsed on the sme rectngulr unit cell introduced for the conduction electrons. We note tht phonons cn be discussed nturlly bsed on the orthogonl unit cells. It is difficult to describe phonons in the WS cell model. 5. Results nd Discussion Bsed on the rectngulr 4-tom unit cell model, we obtined the bnd dispersion of grphene by pplying the tight-binding theory of Reich et l. [8]. The obtined ( π -) bnd structure bsed on the rectngulr 4-tom unit cell model for grphene computed from Equtions (9) nd (30) is shown in Figure 3, where the nerest-neighbor hopping prmeters t nd s re tken from the vlues obtined by the b initio clcultion [8] [7]: t 0 = 0 ev, t = ev, s = 0.9. This 3D figure illustrtes the conduction nd the vlence bnd within the first Brillouin zone. The two crossing points P nd Q indicted in the figure re of prticulr importnce for the physics of grphene. Figure 4 shows the energy bnd profile long the high symmetric points (indicted by Γ, X, Y, nd W) of the rectngulr 4-tom unit cell in reciprocl 66

11 Figure 3. (Color online) Bnd structure of grphene bsed on the rectngulr 4-tom unit cell model. The energy is mximl t the Γ -point in the conduction bnd, while there is sddle-point t the Γ -point in the vlence bnd in the first BZ. The disply shows the liner dispersion structure round the crossing-points P nd Q, t which ε ± ( k ) = 0. Figure 4. (Color online) Bnd profile long the symmetric points of grphene for the rectngulr 4-tom unit cell model. One of the cross in points (so-clled Dirc points) is indicted by P. Inset is the st Brillouin zone of the rectngulr unit cell model. Conduction bnds (red nd blue) re computed from Eqution (9) while vlence bnds (blck nd green) re computed from Eqution (30). Note tht the bnd profile is plotted within the one qurter of the first Brillouin zone. 67

12 spce. There re two crossing points P nd Q in the first BZ (cp. Figure 3), t ± ± which the bnd energy is crossing (i.e., ε ( ) ε ( ) grphene is zero gp semiconductor. P = Q = 0), suggesting tht Let us tke close look t the behvior of the bnd energy close to the crossing points, t which the bnd energy equls to zero. The conduction nd vlence bnds re degenerte t this point. The dispersion reltion for smll moment k ner the crossing points Pπ ( 3,0) nd Q( π 3,0) is given by Tylor expnding the bnd energy round the points P nd Q, resulting in unique liner energy dispersion: ( ) x y ε ± k = ± ħv k = ± ħv k + k = ± ħvk (4) F F F, where k ( = k ) is now in sphericl coordintes (their origin is now t the points P nd Q), 6 v the Fermi velocity given by v ( 3 ) t = m s F with = = 0.46 nm, t = ev F. This prticulr bnd structure re- sembles the physics of mssless Dirc fermions with velocity pproximtely 300 times smller thn the speed of light. Ner the crossing point P in the first Brillouin zone, the liner dispersion (Eqution (4)) is clerly seen in the plot in Figure 5 for the rectngulr 4-tom unit cell model for grphene. The liner dispersion for smll k lso holds for the crossing point Q s well. The second BZ of the reciprocl lttice for the rectngulr 4-tom unit cell is shown by the shded re in Figure 6. The high symmetric points in the reciprocl lttice of the rectngulr 4-tom unit cell re indicted by Γ, X, Y, Figure 5. (Color online) The liner dispersion ner the point P in the first BZ of the rectngulr 4-tom unit cell, where the two bnds (vlence nd conduction) cross ech other. The energy is mximl t the Γ -point, while there is sddle point t the Γ - point of the first BZ. The liner dispersion reltion (red lines) holds for the smll k ner the so-clled Dirc point, while the ctul bnds (blck lines) devite from the linerity ner ε ± ev. 68

13 Figure 6. The first Brillouin zone (rectngulr) nd the second Brillouin (shded) zone of the rectngulr 4-tom unit cell. The region of the first nd second BZ s (i.e., the hexgonl region) of the rectngulr 4-tom unit cell is identicl to the first BZ of the Wigner-Seitz -tom unit cell. nd W, while those in the reciprocl lttice of the -tom unit cell re indicted by Γ, M nd K. The Dirc points of the Wigner-Seitz -tom unit cell re indicted by K nd K in Figure 6, while the crossing (so-clled Dirc) points of the first BZ of the 4-tom unit cell re locted t the points P, Q in the first BZ of the 4-tom unit cell. Note tht the first Brillouin zone of the Wigner-Seitz -tom unit cell is identicl to the region of the first nd second Brillouin zone of the rectngulr 4-tom unit cell (hexgon. cp. Figure 6). The crossing points, P nd Q, in the first Brillouin zone of the rectngulr 4-tom unit cell correspond to the Dirc points, K nd K of the Wigner-Seitz -tom unit cell since the energy bnd structure of the rectngulr 4-tom unit cell model is identicl to the folded bnd structure of the -tom unit cell model long the boundry of the first BZ of the 4-tom unit cell model. The pproximte results ( π -bnds) obtined in this pper re bsed on the rectngulr 4-tom unit cell model for grphene by using tight-binding clcultions, where only the nerest-neighbor hopping is tken into ccount for the trnsport of Bloch electrons. The horizontl ε ( k ) = 0 line shows the chemicl potentil. Below the chemicl potentil, the sttes re ll occupied nd the conduction sttes re bove it. The chemicl potentil is t the points lbeled P nd Q in the first BZ, where the vlence nd conduction bnds re crossing. The bnds with the lowest energy re from the σ -bnds in the plnes between the crbon toms. Electrons in the σ -bnds re tightly bound. The energy bnds ner the chemicl potentil re from the π -bnds. They re formed from the crbon orbitl p z, which proects bove nd below the plne of grphene. These π -bnds re well described by nerest-neighbor tight-binding clcultions bsed on the 4-tom unit cell model. 6. Conclusions nd Some Remrks The current uthors proposed rectngulr 4-tom unit cell model for grphene 69

14 [7]. Bsed on this model, we obtined the bnd energy by pplying the tight- binding pproch employed by Reich et l. [8]. We proved the bnd structure bsed on the rectngulr 4-tom unit cell model for grphene gives the sme bnd structure of grphene bsed on the prevlent grphene model bsed on the Wigner-Seitz -tom unit cell model. The rectngulr 4-tom unit cell for grphene hs the sides perpendiculr to ech other (see Figure ). The k -vector k is defined s in Eqution () but with the orthogonl unit vector, nd the k is obtined s the Fourier conugte of the position vector r. The trnsport electrons in grphene originting from the π -bnd cn be described by the D Bloch k -vector defined in the proposed rectngulr unit cell. We showed tht the energy dispersion reltion ner the Dirc points (smll k ) shows the liner dependence of k (see Eqution (4)). A mteril (density) wve such s phonon wve cn be presented by trveling wve function of the form: Cexp( iωt+ ik r ) with C = mteril density, ω = ngulr frequency, nd k = k -vector. The direction of k points to the direction of the trveling plne wve. In the currently previling theory [] [3] [4] [5] [8] [9] [0] [] [7] the solid stte theory deling with hexgonl crystl strts with primitive nonorthogonl unit cell, the -tom unit cell. This theoreticl model hs difficulties in prticulr for superconductor. The ground stte of superconductor must be condensed in single prticle-stte in ccordnce with Nernst s theorem (the third lw of thermodynmics). Mny fermions hve distribution in energy nd hence mny-fermion system cnnot be superconductor. Only mny-boson system cn be superconductor. In the previling theory [] [3] [4] [5] [8] [9] [0] [] [7] -tom unit cell model is used to set up k -vectors with non-orthogonl unit vectors ã nd ã. A Bloch k -vector is k = k + k constructed with the Born-von Krmn boundry condition [6]. But this vector is not useful in deling with superconducting intercltion grphite compound. In the present work we hve shown tht the non-orthogonl -tom nd orthogonl 4-tom models give the sme bnd energies in the men field pproximtion. In seprte publiction we report the superconductivity in grphite intercltion compounds C 8 K, C 6 C nd in compound MgB. References [] Geim, A.K. nd Noboselov, K.S. (007) Nture Mterils, 6, [] Neto, A.H.C., Guine, F., Peres, N.M.R., Novoselov, K.S. nd Geim, A.K. (009) Reviews of Modern Physics, 8, [3] Srm, S.D., Adm, S., Hwng, E. nd Rossi, E. (0) Reviews of Modern Physics, 83, [4] Ktsnelson, M.I. (0) Grphene: Crbon in Two Dimensions. Cmbridge University Press, Cmbridge. 60

15 [5] Reich, S., Thomsen, C. nd Multzsch, J. (004) Crbon Nnotubes: Bsic Concepts nd Physicl Properties. Wiley-VCH Verlg GmbH, Berlin. [6] Ashcroft, N.W. nd Mermin, N.D. (976) Solid Stte Physics. Sunders, Phildelphi, 75, 33-35, 37-39, 8-9. [7] Fuit, S. nd Suzuki, A. (00) Journl of Applied Physics, 07, Article ID: [8] Reich, S., Multzsch, J., Thomsen, C. nd Ordeön, P. (00) Physicl Review B, 66, Article ID: [9] Wllce, P.R. (947) Physicl Review, 7, 6. [0] McClure, J.W. (957) Physicl Review, 08, 6. [] Slonczewski, J.S. nd Weiss, P.R. (958) Physicl Review, 09, 7. [] Fuit, S. nd Ito, K. (007) Quntum Theory of Conducting Mtter: Newtonin Equtions of Motion for Bloch Electron. Springer, New York, 0-, [3] Bloch, F. (99) Zeitschrift für Physik, 5, [4] Fuit, S., Jovini, A., Godoy, S. nd Suzuki, A. (0) Physics Letters A, 376, [5] Fuit, S., Tkto, Y. nd Suzuki, A. (0) Modern Physics Letters B, 5, 3. [6] Zhng, Y., Tn, Y.W., Stormer, H.L. nd Kim, P. (005) Nture, 438, [7] Sito, R., Dresselhus, G. nd Dresselhus, M.S. (005) Physicl Properties of Crbon Nnotubes. Imperil College Press, London. Submit or recommend next mnuscript to SCIRP nd we will provide best service for you: Accepting pre-submission inquiries through Emil, Fcebook, LinkedIn, Twitter, etc. A wide selection of ournls (inclusive of 9 subects, more thn 00 ournls) Providing 4-hour high-qulity service User-friendly online submission system Fir nd swift peer-review system Efficient typesetting nd proofreding procedure Disply of the result of downlods nd visits, s well s the number of cited rticles Mximum dissemintion of your reserch work Submit your mnuscript t: Or contct mp@scirp.org 6