Electronic structure of bilayer graphene: A real-space Green’s function study

Z. F. Wang,1 Qunxiang Li,1,* Haibin Su,2 Xiaoping Wang,1 Q. W. Shi,1,† Jie Chen,3 Jinlong Yang,1 and J. G. Hou1

1Hefei National Laboratory for Physical Sciences at Microscale, University of Science and Technology of China, Hefei,Anhui 230026, People’s Republic of China

2School of Materials Science and Engineering, Nanyang Technological University, 50 Nanyang Avenue, 639798, Singapore3Electrical and Computer Engineering, University of Alberta, Alberta, Canada T6G 2V4

�Received 26 October 2006; revised manuscript received 18 December 2006; published 14 February 2007�

In this paper, a real-space analytical expression for the free Green’s function �propagator� of bilayergraphene is derived based on the effective-mass approximation. Green’s function displays highly spatial an-isotropy with threefold rotational symmetry. The calculated local density of states �LDOS� of a perfect bilayergraphene produces the main features of the observed scanning tunneling microscopy �STM� images of graphiteat low bias voltage. Some predicted features of the LDOS can be verified by STM measurements. In addition,we also calculate the LDOS of bilayer graphene with vacancies by using the multiple-scattering theory �scat-terings are localized around the vacancy of bilayer graphene�. We observe that the interference patterns aredetermined mainly by the intrinsic properties of the propagator and the symmetry of the vacancies.

DOI: 10.1103/PhysRevB.75.085424 PACS number�s�: 73.61.Wp, 61.72.Ji, 68.37.Ef

I. INTRODUCTION

Since Novoselov et al. fabricated ultrathin monolayergraphite devices,1 the electronic properties of a graphitemonolayer �graphene� have attracted a great deal of researchinterest due to the Dirac-type spectrum of charge carriers inits gapless semiconductor material. Many interesting proper-ties of single-layer graphene, such as the Landau quantiza-tion, the defect-induced localization, and spin current states,have been studied experimentally and theoretically by sev-eral research groups.2–7 These nonorthodox properties, in-cluding massless Dirac fermions around the Dirac points inthe first Brillouin zone, result from graphene’s particularband structure.

Recent research attention has focused on the electronicstructure of bilayer and multilayer graphene.8–14 Studies haveshown that a bilayer graphene has some unexpectedproperties.11 For example, a bilayer graphene shows anoma-lies in its integer quantum Hall effect and in its minimalconductivity on the order of e2 /h. The common and distinc-tive electronic features of single-layer and bilayer grapheneare highlighted in Ref. 15. Charge carriers in a bilayergraphene are mainly quasiparticles with a finite density ofstates at zero energy and they behave similar to conventionalnonrelativistic electrons. Like the relativistic particles orquasiparticles in single-layer graphene, we can describe thesequasiparticles by using spinor wave functions. Althoughthese “massive chiral fermions” do not exist in the fieldtheory, their existence in condensed-matter physics offers aunique opportunity to investigate the importance of chiralityand to solve the relativistic tunneling problem.

The unusual physical properties of bilayer graphene areattributed to two key factors. �i� The relatively weak inter-layer coupling. Bilayer graphene inherits some propertiesfrom single-layer graphene material, such as the existence ofDirac points in the first Brillouin zone and the degeneracy ofelectron and hole bands. �ii� The special geometry of bilayergraphene with the Bernal stacking �A-B stacking� betweenadjacent graphene layers. There are two kinds of nonequiva-

lent sites �A and B� in each layer as shown in Figs. 1�a� and1�b�. Experimental scanning tunneling microscopy �STM�graphite images at low bias voltage have verified that onlysite B is visible and exhibits a triangular structure. In addi-tion, the orbital overlap coupling between two adjacent lay-ers is contributed mainly by the carbon orbitals at site A.16–18

To date, few analytical studies that attempt to uncover theunique electronic properties of the bilayer graphene havebeen done.12

In this paper, we first develop an analytical formula ofelectronic structure in a bilayer graphene with the Bernal

FIG. 1. �Color online� The crystal structure of a bilayergraphene. The unit cell consists of two layers with two nonequiva-lent sites: A �yellow� and B �gray�. �a� Top view with surface unitvectors a1 and a2; �b� side view. �c� The first Brillouin zone of abilayer graphene, where K1–K6 are the Dirac points. These Diracpoints can be further divided into two nonequivalent sets: K1

��K1 ,K3 ,K5� and K4��K2 ,K4 ,K6�.

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stacking based on real-space free Green’s function �propaga-tor� and the effective-mass approximation. We observe thatthe physical properties of the bilayer graphene are closelyrelated to its propagator and the bilayer graphene behavessimilar to a massive chiral fermions system. Since the localdensity of states �LDOS� can be measured by STM, we thencompute the LDOS of graphene in various forms based onGreen’s function explicitly. Finally, we highlight the impactof vacancies’ patterns �in terms of their geometry and sym-metry of the computed LDOS� on interferences in STM pic-tures.

II. REAL-SPACE GREEN’S FUNCTION AND ELECTRONICSTRUCTURE OF PERFECT BILAYER GRAPHENE

In what follows, we derive the analytical expression of thefree Green’s function for bilayer graphene in real spacebased on the effective-mass approximation. Bilayer grapheneconsists of two hexagonal lattice layers coupled by the Ber-nal stacking as shown in Figs. 1�a� and 1�b�. In each layer,there are two nonequivalent sites, A and B. Type-A atomshave direct neighbors in their adjacent layer, but type-B at-oms do not and locate at hollow sites. We assume that thereis one valence electron per carbon atom in a bilayergraphene. Provided that the difference between wave func-tions orthogonal to the graphene plane can be neglected, thebilayer graphene can be modeled as an effective two-dimensional system. Since the physical properties ofgraphene are determined by � bands near the Dirac point,only the contribution from the � band is considered in thispaper. We further limit our analysis by considering only the

nearest interactions of the graphene’s pz orbitals. The tight-binding Hamiltonian of the bilayer graphene is

H = �i

�2p��i�11�i� + �i�22�i�� + V1��ij�

��i�11�j� + �i�22�j� + H.c.�

+ V2��ij�

��i�12�j� + H.c.� , �1�

where �r � i�l, l= �1,2� is a wave function at site i at layer l.�2p is the on-site energy, V1 is the nearest hopping parameterwithin the layer, and V2 is the nearest hopping parameterbetween two layers. Our presented calculations are con-ducted with �2p=0 eV, V1=−3.0 eV, and V2=0.4 eV.

The Bloch orbits for two nonequivalent sites, A and B,are written as �kA�l= �1/�N�� jA

eik·rjA � jA�l and �kB�l

= �1/�N�� jBeik·rjB � jB�l. The summation covers all sites A and

B on layer l. rA and rB denote the real-space coordinates ofsites A and B, respectively. Here, N is the number of unitcells in the crystal. The Hamiltonian can be rewritten as

H =�2p V1�* V2 0

V1� �2p 0 0

V2 0 �2p V1�

0 0 V1�* �2p

, �2�

where �=eikya+ei�−�3akx/2−aky/2�+ei��3akx/2−aky/2� and a=1.42 Å is the c-c bond length. By defining the retardedGreen’s function as, G0

ret=1/ ��−H+ i��, where � is an in-finitesimal quantity, we obtain the reciprocal space Green’sfunction for bilayer graphene

G0ret�k,�� =

1

�t�t2 − V1

2��*� V1�*�t2 − V12��*� t2V2 tV1V2�

V1��t2 − V12��*� t�t2 − V1

2��* − V22� tV1V2� V1

2V2�2

t2V2 tV1V2�* t�t2 − V12��*� V1��t2 − V1

2��*�tV1V2�* V1

2V2�*2 V1�*�t2 − V12��*� t�t2 − V1

2��* − V22� , �3�

where �= �t2−V12��*− tV2��t2−V1

2��*+ tV2� and t=�−�2p

+ i�. From �=0, the dispersion relation is expressed as�= ±V2 /2±���k�2+ �V2 /2�2 with �=3aV1 /2, which is con-sistent with the previous theoretical result.12

The real-space Green’s function of the bilayer graphenecan be derived by taking the Fourier transform of G0

ret�k ,��.For simplicity, these calculations are carried out around sixcorners in the first Brillouin zone within the low-energy re-gion based on the effective-mass approximation. Six Diracpoints can be divided into two equivalent sets of points,K1 ,K3 ,K5 and K2 ,K4 ,K6, which are shown in Fig. 1�c�. Theyform two 360° integrals around K1 and K4. By using themathematical relation

eik·�r�−r��� = J0�k�r� − r���� + 2�n=1

inJn�k�r� − r����cos�nr�,r��� ,

where �� ,��= �A ,B�, r�,r��

is the angle between k and r�

−r��, and Jn is the type-I n-order Bessel function. We cansimplify the real-space Green’s function of the top layer �l=1� to

G0ret�rA,rA� ,�� = cos�K1 · �rA − rA���F1��rA − rA� �,�� ,

G0ret�rB,rB� ,�� = cos�K1 · �rB − rB���F2��rB − rB� �,�� ,

G0ret�rA,rB� ,�� = sin�K1 · �rA − rB�� + �rA,rB�

�F3��rA − rB� �,�� ,

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G0ret�rB,rA� ,�� = sin�K1 · �rB − rA�� − �rB,rA�

�F3��rB − rA� �,�� .

�4�

Here K1= �4� /3�3a ,0�, cos r�,r��=cos��−�r�,r

��� and � is

the angle between the k and x axis. �r�,r��

is the angle be-tween r�−r�� and x axis. The definitions of F1, F2, and F3 are

F1��r� − r���,�� =S

�

0

kc

dk kJ0�k�r� − r����

� t

t2 − ��k�2 − tV2+

t

t2 − ��k�2 + tV2� ,

F2��r� − r���,�� =S

�

0

kc

dk kJ0�k�r� − r����

� t − V2

t2 − ��k�2 − tV2+

t + V2

t2 − ��k�2 + tV2� ,

F3��r� − r���,�� =S

�

0

kc

dk k2J1�k�r� − r����

� �

t2 − ��k�2 − tV2+

�

t2 − ��k�2 + tV2� ,

�5�

where S=3�3a2 /2 is the area of the unit cell in real spaceand kc is the cutoff wave vector. The corresponding cutoffwavelength 2� /kc is comparable to the lattice constant a.With the same approach in our previous work to solve elec-tronic structure in the single-layer graphene, kc is set to be1.71/a.19 From Eq. �4�, it is clear that the real-space Green’sfunction of the bilayer graphene is constructed by multiply-ing two terms. The first term is spatially anisotropic, whichcan be represented by sine and cosine functions determinedfrom two nonequivalent sets of the Dirac points as shown inFig. 1�c�. The second term including F1, F2, or F3 is spatiallyisotropic in real space and depends only on the distance be-tween two sites.

Many physical properties of bilayer graphene can be de-duced from this explicit expression. Interestingly, when kc→, the functions, F1, F2, and F3, have simple analyticalforms

F1��r� − r���,�� =− S

��2�tK0� it1

��r� − r����

+ tK0� it2

��r� − r����� ,

F2��r� − r���,�� =− S

��2��t − V2�K0� it1

��r� − r����

+ �t + V2�K0� it2

��r� − r����� ,

F3��r� − r���,�� =− S

��2�it1K1� it1

��r� − r����

+ it2K1� it2

��r� − r����� , �6�

where t1=�t2− tV2 and t2=�t2+ tV2. K0 and K1 are the zerothorder and the first order terms of the type-I modified Besselfunction, respectively. Compared with our previous study,19

the expressions of the Green’s function derived here are thesame as those of the single-layer graphene once the nearesthopping parameter between two layers �V2� is set to zero.That is to say, it is straightforward to compare theoreticalresults of bilayer graphene with previous theoretical and ex-perimental data of the monolayer graphene by settingV2=0.19

Figure 2 shows the LDOS of bilayer graphene with �=−0.1 eV. Here, the LDOS of site r� is calculated by setting�0�r� ,��=−�1/��Im G0

ret�r� ,r� ,��. We can clearly observethe contrast between the LDOS at site A and that at site B.Site B is highlighted and forms a triangular lattice with three-fold symmetry. According to the Tersoff-Hamann model20

which has been successfully used to explain experimentalresults,21,22 the STM image can be simulated using theLDOS of the sample surface. It is reasonable for us to com-pare the LDOS of bilayer graphene with previous experi-mental STM observations of graphite surfaces at low biasvoltage �i.e., 0.3 V�. Our calculated LDOS can clearly pro-duce the main features as shown in several observed STMimages reported in Refs. 16–18.

One direct approach to find the difference between bilayerand monolayer graphene is to study the LDOS at sites A andB for both monolayer and bilayer graphenes. Figures 3�a�and 3�b� present the LDOS of one carbon site �A or B� in themonolayer graphene and two sites �A and B� in the bilayergraphene, respectively. For a monolayer graphene, sites Aand B are equivalent and thus only one curve of LDOS is

FIG. 2. �Color online� LDOS of bilayer graphene with �=−0.1 eV. The LDOS at B site is larger than that at A site, repre-sented by the brighter contour.

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shown in Fig. 2�a�. However, these two sites are nonequiva-lent for a bilayer graphene. Figure 2�b�, therefore, clearlyshows that the LDOS of bilayer graphene at site B is greaterthan that at site A, when ��� is less than approximately0.4 eV. This LDOS difference leads to the brighter spots atsite B as shown in Fig. 2. When �� � �0.4 eV, the differencebetween the LDOS at sites A and B in the bilayer graphenegradually diminishes. This feature is easy to verify throughSTM dI /dV mapping at a relatively large bias voltage. Forexample, sites A and B can be both observed in STM dI /dVmapping with the applied bias voltage of 0.6 V. This phe-nomenon can also be elaborated by our analytic expressionof Green’s function.

The difference between the LDOS of site A and that ofsite B can be defined as

��0��� = −1

�Im�G0

ret�rB,rB,�� − G0ret�rA,rA,���

= −1

�Im�F2�0,�� − F1�0,��� . �7�

After some simple math operations, shown in the Appendix,Eqs. �A5� and �A6�, ��0��� can be simplified to

��0��� = �0 ��� � V2,

SV2

2��2 ��� � V2.�8�

From this equation, the LDOS of the bilayer graphene canbe divided into two regions based on the interlayer hoppingparameter V2. One region is �� � �V2, where the LDOS of site

B is larger than that of site A by a constant, SV2 /2��2. Theother region is �� � �V2, where the LDOS of site A equalsthat of site B. Our calculations agree well with our numericalresults when V2=0.4 eV. The calculations show the disper-sion relation is approximately quadratic dependent in thelow-energy region and the results match the previous theo-retical results.12 We find that the low-energy LDOS of bothsites A and B are linearly proportional to the energy ��� �referto the Appendix�. Note that the LDOS of one unit cell at thevery low-energy level, however, is a constant. The LDOScan be expressed as ����= ��0�rA ,rA ,��+�0�rB ,rB ,����SV2 /2��2, which is the same as those in the conventionaltwo-dimensional system. This remarkable feature of the bi-layer graphene offers a direct method to measure the inter-layer hopping parameter V2, which is also the thresholdneeded to distinguish LDOS contrast in STM images. Theintralayer hopping parameter V1 can be deduced based on theLDOS difference measured according to STM images ���0=SV2 /2��2 with �=3aV1 /2�.

III. LOCAL DENSITY OF STATES OF BILAYERGRAPHENE WITH LATTICE VACANCY

To validate the accuracy of real-space Green’s functionderived in this paper, we use the promoted formula to solvethe electronic structure of a bilayer graphene with a singlevacancy. Based on the tight-binding scheme, a vacancy canbe simulated by introducing a large on-site energy at thevacancy site on a bilayer graphene.23 Assuming that thesingle vacancy locates at site B with on-site energy U, thetransport matrix or the T matrix24,25 can be written as T=U�1−UG0

ret�−1. If U is large, the T matrix becomes

T�0,0,�� � − G0ret−1

�0,0,�� = − F2−1�0,�� . �9�

Here the position of the vacancy is set as the coordinateorigin. That is to say, RB=0 in the x-y plane. Using theDyson equation, we have

Gret�r�,r�,�� = G0ret�r�,r�,��

+ G0ret�r�,0,��T�0,0,��G0

ret�0,r�,�� ,

�10�

where Gret�r� ,r� ,�� is the Green’s function of bilayergraphene with a single vacancy in real space. The LDOS onsite r� can be determined by

��r�,�� = −1

�Im Gret�r�,r�,�� ,

which can be simply rewritten as

��rA,�� = �0�rA,�� + sin2�K1 · rA + �rA�H1�rA,�� ,

��rB,�� = �0�rB,�� + cos2�K1 · rB�H2�rB,�� . �11�

In the above equation,

FIG. 3. �a� LDOS of monolayer graphene on one site �sites Aand B are equivalent�. �b� LDOS of bilayer graphene on two sites�sites A and B are nonequivalent�.

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�0�rA,�� = −1

�Im�F1�0,��� ,

�0�rB,�� = −1

�Im�F2�0,��� ,

H1�r�,�� =1

�Im�F3

2��r��,��F2�0,�� � ,

H2�r�,�� =1

�Im�F2

2��r��,��F2�0,�� � . �12�

Figure 4�a� shows the calculated LDOS with a singleB-site vacancy on the top sheet of the bilayer graphene. Aninteresting feature is clearly observed: Bright spots localizednear the vacancy around site A have nice threefold rotationalsymmetry. This phenomenon can be explained based on Eq.�11�. Since the function H2 in Eq. �11� has a very small value�close to zero�, its magnitude changes little as the distancefrom the vacancy increases and thus the magnitude of��rB ,�� remains very small. Although the cosine term in Eq.�11� reaches its maximum when rB= �3�3a /2�n, n=0, ±1, ±2, . . ., the sine function in ��rA ,�� reaches itsmaximum along directions with angles, �rA

=30°, 90°, 150°,210°, 270°, and 330°. Compared to H2, the value of thefunction H1 has a relatively larger value, i.e., 0.4 eV−1 at1.0 Å away from the vacancy for �=−0.1 eV. The functionH1, however, decays rapidly with increasing distance fromthe vacancy. Note that sites A along the directions �rA

=90°,210°, and 330° are located closer to the vacancy than thosealong the directions �rA

=30°, 150°, and 270°. The calculatedsite-A LDOS value is between that of the nearest and that ofthe next nearest to the B-site vacancy, that is 0.56 and0.36 eV−1, respectively. Therefore, the sites along the �rA=90°, 210°, and 330° directions are brighter than those alongthe �rA

=30°, 150°, and 270° directions. These bright spotsshow the localized character of the region surrounding thevacancy. By comparing to the dI /dV image of a vacancy onthe single-layer graphene,19 the enhanced localization ofLDOS in the bilayer graphene is caused by the additionalinterlayer channel propagated from the vacancy point.Green’s function can oscillate in a significantly longer dis-tance starting from the vacancy in the monolayer graphenethan in the bilayer graphene. The asymptotic behavior of H1can help us understand this phenomenon. When the distanceis longer than 4.0 Å, H1 is less than 0.1 eV−1 and thus thesebright spots show the localized character of the region withinnearest lattice surrounding the vacancy.

The LDOS with vacancy near site A is also simulated asshown in Fig. 4�b�. We observe the similar features that thevacancy around site B has bright spots with threefold sym-metry. The calculated site-B LDOS value is between that ofthe nearest and that of the next nearest to the site-A vacancy,which is between 0.35 and 0.23 eV−1.

The extension of our calculation to consider several va-cancies on bilayer graphene is straightforward. The scatter-ing T matrix in Eq. �10� includes all contributions from va-

cancies. Similar to a single-vacancy case, a large on-siteenergy U for all vacancy sites is used in our simulations.Figure 4�c� shows the LDOS near an AB-type vacancy �acouple of the neighboring sites A and B� on a bilayergraphene surface. The LDOS is still localized around thevacancies. However, the clear threefold rotational symmetry

FIG. 4. �Color online� LDOS of vacancies in bilayer graphene.�a� Single B-site vacancy; �b� A-site vacancy; �c� a pair of AB va-cancy. Here, �=−0.1 eV.

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disappears in this case. The bright spots localized above thevacancy correspond to the B sites. Note that the LDOS at thebright spots is about 0.03 eV−1, which is less than those inFigs. 4�a� and 4�b� by an order of magnitude. This phenom-enon results in the destructive interference due to themultiple-scattering caused by two vacancies. The asymmet-ric pattern in Fig. 4�c� reflects the residual contribution fromthe destructive interference since two sites A and B are non-equivalent in the bilayer graphene. Our result suggests thatthe interference pattern is determined mainly by vacancies,provided that the spatial anisotropy of the Green’s function isfixed. The symmetry will be lost in LDOS whenever vacan-cies break the threefold rotational symmetry in bilayergraphene.

IV. CONCLUSION

In this paper, an analytical form of the real-space Green’sfunction �propagator� of bilayer graphene is constructedbased on the effective-mass approximation. Green’s functiondemonstrates an elegant spatial anisotropy with a threefoldsymmetry for defect-free bilayer graphene. The LDOS of thebilayer graphene determines the main features of experimen-tal STM images on graphite surfaces with low-bias voltage.The predicted features according to our simulated results canbe verified by STM measurements. For example, two non-equivalent atomic sites can be observed in STM dI /dV im-ages with different bias voltages. The information of inter-layer and intralayer hopping strength can be deduced basedon the contrast of STM images. Moreover, we also calculatethe LDOS of bilayer graphene with vacancies by using themultiple-scattering theory. The interference patterns are de-termined mainly by the properties of Green’s function andthe symmetry of the vacancies. Once the vacancies break theintrinsic symmetry of the graphene, the threefold rotationalsymmetry of the LDOS vanishes. Our model provides exactresults near the Dirac points. We have discovered some in-teresting STM patterns of the second layer, but these resultswill be published elsewhere. In this paper, the bilayergraphene is described by using the simple noninteractivetight-binding scheme. We are currently investigating howCoulomb interaction impacts the electronic structure of thebilayer graphene.

ACKNOWLEDGMENTS

This work is partially supported by the National NaturalScience Foundation of China under Grants No. 10574119,10674121, 20533030, 20303015, and 50121202 by NationalKey Basic Research Program under Grant No.2006CB0L1200, by the USTC-HP HPC project, and by theSCCAS and Shanghai Supercomputer Center. Work at NTUis supported in part by COE-SUG Grant No. M58070001.J.C. would like to acknowledge the funding support from theDiscovery program of Natural Sciences and Engineering Re-search Council of Canada �No. 245680�.

APPENDIX

From Eq. �3�, Green’s functions at the top layer in recip-rocal space are expressed as

G0AAret �k,�� =

1

2� t

t2 − V12��* − tV2

+t

t2 − V12��* + tV2

� ,

G0BBret �k,�� =

1

2� t − V2

t2 − V12��* − tV2

+t + V2

t2 − V12��* + tV2

� ,

G0ABret �k,�� =

1

2� V1�*

t2 − V12��* − tV2

+V1�*

t2 − V12��* + tV2

� ,

G0BAret �k,�� =

1

2� V1�

t2 − V12��* − tV2

+V1�

t2 − V12��* + tV2

� .

�A1�

By taking the Fourier transform of G0��ret �k ,�� in the first

Brillouin zone �1BZ�, we can obtain the exact expression ofGreen’s function in real space for bilayer graphene

G0ret�r� ,r�� ,��=�1BZdk G0��

ret �k ,��eik·�r�−r���, where � and �are for site-A or site-B atoms, respectively. Based on theeffective-mass approximation, we can sum k points near thesix corners �labeled by i=1–6� in the first Brillouin zone.The real-space bilayer graphene Green’s function can bewritten as

G0ret�rA,rA� ,�� =

S

�2��2�i=1

6 dkxi dky

i� t

t2 − V12�i�i

* − tV2

+t

t2 − V12�i�i

* + tV2�ei�Ki+ki�·�rA−rA��,

G0ret�rB,rB� ,�� =

S

�2��2�i=1

6 dkxi dky

i� t − V2

t2 − V12�i�i

* − tV2

+t + V2

t2 − V12�i�i

* + tV2�ei�Ki+ki�·�rB−rB��,

G0ret�rA,rB� ,�� =

S

�2��2�i=1

6 dkxi dky

i� V1�i*

t2 − V12�i�i

* − tV2

+V1�i

*

t2 − V12�i�i

* + tV2�ei�Ki+ki�·�rA−rB��,

G0ret�rB,rA� ,�� =

S

�2��2�i=1

6 dkxi dky

i� V1�i

t2 − V12�i�i

* − tV2

+V1�i

t2 − V12�i�i

* + tV2�ei�Ki+ki�·�rB−rA��. �A2�

In Eq. �A2�, the integral around K1 ,K3 ,K5 and K2 ,K4 ,K6

can be summed together separately to form two 360° inte-grals around K1 and K4, that is

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G0ret�rA,rA� ,�� =

S

�2��2 �eiK1�rA−rA�� + eiK4�rA−rA��� 0

2�

d� 0

kc

dk k� t

t2 − ��k�2 − tV2+

t

t2 − ��k�2 + tV2�eik·�rA−rA��,

G0ret�rB,rB� ,�� =

S

�2��2 �eiK1�rB−rB�� + eiK4�rB−rB��� 0

2�

d� 0

kc

dk k� t − V2

t2 − ��k�2 − tV2+

t + V2

t2 − ��k�2 + tV2�eik·�rB−rB��,

G0ret�rA,rB� ,�� =

S

�2��2�eiK1�rA−rB�� 0

2�

d� 0

kc

dk k2��− i sin � − cos ��� 1

t2 − ��k�2 − tV2+

1

t2 − ��k�2 + tV2�eik·�rA−rB�� + eiK4�rA−rB��

0

2�

d� 0

kc

dk k2��− i sin � + cos ��� 1

t2 − ��k�2 − tV2+

1

t2 − ��k�2 + tV2�eik·�rA−rB��� ,

G0ret�rB,rA� ,�� =

S

�2��2�eiK1�rB−rA�� 0

2�

d� 0

kc

dk k2��i sin � − cos ��� 1

t2 − ��k�2 − tV2+

1

t2 − ��k�2 + tV2�eik·�rB−rA�� + eiK4�rB−rA��

0

2�

d� 0

kc

dk k2��i sin � + cos ��� 1

t2 − ��k�2 − tV2+

1

t2 − ��k�2 + tV2�eik·�rB−rA��� . �A3�

By using relation

eik·�r�−r��� = J0�k�r� − r���� + 2�n=1

inJn�k�r� − r����cos�nr�,r��� ,

and integrating the angle part of Eq. �A3�, we can easily get Eqs. �4� and �5�. If we let kc→, Eq. �5� is simplified to Eq. �6�.Next, we briefly derive Eq. �8�. At the same site A or B, the corresponding Green’s functions are

G0ret�rA,rA,�� =

S

�

0

kc

dk k� t

t2 − ��k�2 − tV2+

t

t2 − ��k�2 + tV2� ,

G0ret�rB,rB,�� =

S

�

0

kc

dk k� t − V2

t2 − ��k�2 − tV2+

t + V2

t2 − ��k�2 + tV2� . �A4�

The LDOS at sites A and B are

�0�rA,rA,�� = −1

�Im�G0

ret�rA,rA,��� = −S

2����2 Im 0

kc

d����k�2 + �V2

2�2�

� t

� + i� −V2

2−���k�2 + �V2

2�2

−t

� + i� −V2

2+���k�2 + �V2

2�2

+t

� + i� +V2

2−���k�2 + �V2

2�2

−t

� + i� +V2

2+���k�2 + �V2

2�2�

= �S�����2 ��� � V2,

S���2��2 ��� � V2,

�A5�

ELECTRONIC STRUCTURE OF BILAYER GRAPHENE: A… PHYSICAL REVIEW B 75, 085424 �2007�

085424-7

�0�rB,rB,�� = −1

�Im�G0

ret�rB,rB,��� = −S

2����2 Im 0

kc

d����k�2 + �V2

2�2�

� t − V2

� + i� −V2

2−���k�2 + �V2

2�2

−t − V2

� + i� −V2

2+���k�2 + �V2

2�2

+t + V2

� + i� +V2

2−���k�2 + �V2

2�2

−t + V2

� + i� +V2

2+���k�2 + �V2

2�2�

= �S�����2 ��� � V2,

S���� + V2�2��2 ��� � V2.

�A6�

Then the difference between LDOS at A and B sites is

��0��� = �0�rB,rB,�� − �0�rA,rA,�� = �0 ��� � V2,

SV2

2��2 ��� � V2.�A7�

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