Localization of Dirac Electrons in RotatedGraphene BilayersG. Trambly de Laissardiere,*,† D. Mayou,*,‡ and L. Magaud*,‡

†Laboratoire de Physique Theorique et Modelisation, Universite de Cergy-Pontoise-CNRS, F-95302 Cergy-PontoiseCedex, France, and ‡Institut Neel, CNRS - Universite Joseph Fourier, F-38042 Grenoble, France

ABSTRACT For Dirac electrons the Klein paradox implies that the confinement is difficult to achieve with an electrostatic potentialalthough it can be of great importance for graphene-based devices. Here, ab initio and tight-binding approaches are combined andshow that the wave function of Dirac electrons can be localized in rotated graphene bilayers due to the Moire pattern. This localizationof wave function is maximum in the limit of the small rotation angle between the two layers.

KEYWORDS Graphene, graphite, Moire pattern, electron localization, ab initio, tight binding

Graphene is a two-dimensional carbon material thattakes the form of a planar lattice of sp2 bondedatoms. Its honeycomb lattice consists in two sublat-

tices which gives a specific property to the wave function,the so-called chirality. The linear dispersion relation closeto the charge neutrality point implies that, in this energyrange, the electrons obey an effective massless Diracequation.1-3 The properties of electrons in graphene, deriv-ing from the Dirac equation, are fundamentally differentfrom those deriving from the Schrodinger equation. Inparticular the quantum Hall effect is quantized with integerplus half values4,5 and can even be observed at roomtemperature.6 Another example of the unique behavior ofDirac electrons is the so-called Klein paradox which isintimately related to the chirality of their wave function. TheKlein paradox is the fact that, in a one-dimensional config-uration, a potential barrier is perfectly transparent forelectrons. As a consequence it is difficult to localize Diracelectrons with an electrostatic potential, although it can beof great interest to realize this confinement, in particular forthe production of elementary devices.7,8

Therefore it is necessary to improve our knowledge of thebehavior of Dirac electrons in various types of potentials. Fordisordered potentials specific behavior related to the chiralityof Dirac electrons, such as weak antilocalization effects, havebeen predicted theoretically9 and observed in epitaxialgraphene.10-13 Recent STM measurements, performed onepitaxial graphene, have also confirmed the absence ofbackscattering related to the Klein paradox.14 For periodicsystems several studies have been published concerningeither a simple graphene sheet in an applied periodicpotential or periodic bilayers.15,16 The bilayer system17-21

has shown an unexpected and rich behavior. The well-

known Bernal AB stacking leads to massive quasi-particuleswith quadratic dispersion close to the Dirac point. Yet it hasbeen recognized recently in the scientific community (seefor instance the recent review in ref 22) that anotherimportant case is that of rotated graphene bilayers, whichare observed for example in epitaxial graphene on theC-terminated face of SiC.17

In a rotated bilayer the superposition of the two honey-comb lattices generates a Moire pattern with a longer period.The two Dirac electron gases are then coupled by a periodicinteraction, with a large supercell, which can restore a Dirac-like linear dispersion. Graphene-like Dirac cones were re-cently observed by angle-resolved photoemission spectro-copy (ARPES) on a sample that consists in 11 C layers onthe C face of SiC.23 However fundamental issues are con-troversial. In particular Lopez dos Santos et al.19 conclude,from a tight-binding perturbative treatment of the interplanecoupling, that the velocity can be renormalized and tend todecrease in a rotated bilayer with respect to its value in puregraphene. On the contrary Shallcross et al.20 conclude fromab initio calculations and from some general arguments thatthe velocity is unchanged in rotated graphene bilayers.

In this Letter we combine ab initio and tight-binding (TB)approaches. In agreement with ref 19, we conclude thatthere is a renormalization of the velocity for bilayers ascompared to pure graphene. Our main result is that thevelocity tends to zero in the limit of small twist angles whichmeans that this type of coupling is able to confine the twoDirac electron gases. We also show that the electronic wavefunction tends to localize in regions of the bilayer which arelocally similar to the AA arrangement.This new localizationregime should be observable since angles close to a degree,for which we predict strong localization effects, are foundin some rotated multilayers. Indeed recent ARPES resultsshow the existence of many possible rotation anglesslargeand small.23 Furthermore, large Moire patterns sand then

* To whom correspondence should be addressed: guy.trambly@u-cergy.fr;didier.mayou@grenoble.cnrs.fr; laurence.magaud@grenoble.cnrs.fr.Received for review: 09/7/2009Published on Web: 02/01/2010

pubs.acs.org/NanoLett

© 2010 American Chemical Society 804 DOI: 10.1021/nl902948m | Nano Lett. 2010, 10, 804–808

small rotation angless have been observed on STM imageof graphene multilayers on top of SiC C-face24 and also ongraphite.25

Moire Patterns and Geometry of Rotated Bilayer. Moirepatterns can be obtained in two cases: when two latticeswith slightly different parameters are superimposed or whentwo identical lattices are rotated by an angle θ.26 The presentsituation corresponds to the later case. A commensuratestructure can be defined if the rotation changes a latticevector V (m, n) to V′(n, m) with n, m the coordinates withrespect to the basis vectors ab1 (�3/2, -1/2) and ab2 (�3/2, 1/2). The rotation angle is then defined as

and the commensurate cell vectors correspond to

The commensurate unit cell contains N ) 4(n2 + nm + m2)atoms. It is important to notice that θ ∼ 0 results in a largecell slarge n and ms and small |m - n|. Large commen-surate cells can be also obtained for large angles θ ∼ 30°,then |m - n| is large.26 θ ) 60° is the perfect AB stacking,and θ close to 60° is obtained for n ) 1 (m ) 1) and large m(n).

For small angles, if the rotation axis, perpendicular to theplanes, passes through atomic positions in both layers,atoms in the four corners of the supercell are directlysuperimposed (Figure 1). This corresponds to the so-calledAA stacking. For cells (n, m) such that m ) n + 1, zone withBernal AB stacking are located at 1/3 and 2/3 of the longdiagonal (Figure 1).

Quite generally the structures that we study are fullycharacterized by the two index (n, m) of the rotation and also

by the value of the translation vector between the two layers.However for sufficiently small values of the rotation angleθ, the Moire pattern depends essentially on the rotationangle and the translation of one layer only results in a rigidshift of the overall Moire structure. Therefore we expect andindeed observe from our results that the rotation angle θ isthe main parameter that dictates the electronic structure.

Also in the small angle limit, all dimensions of the Moirepattern scale like 1/θ and in particular the size of locally AAor AB stacked regions and the distance between theseszones are proportional to 1/θ. Let us emphasize that the limitof small angle is discontinuous: when the rotation angledecreases the dimensions of the Moire structure scale like1/θ and therefore the structure does not tend to that forexactly zero angle. This singular, discontinuous, geometricevolution is associated to singular electronic properties aswe show in this work.

Ab Initio and Tight-Binding Calculations ofElectronic Structure. Our approach combines ab initiocalculations that can be used for unit cells containing up to600 atoms and tight-binding (TB) calculations that can beused for unit cells containing up to 15000 atoms or evenmore. Ab initio calculations are performed with the codeVASP27 and the generalized gradient approximation (GGA-PW 91).28 The C ultrasoft pseudopotential29 has been ex-tensively tested previously.17,30 The plane wave basis cutoffis equal to 211 eV. The empty space width is equal to 21 Å,the interlayer distance is fixed to its experimental value(=3.35 Å) and no atoms are allowed to relax for directcomparison with TB calculations. For multilayers, van derWaals forces should be important, but a precise treatmentof the electron correlations is unknown for such largesystems. We acknowledge this fact, yet following previouswork on multilayers,17-20,30 we expect that a sound physicalrepresentation of electronic band structure is given by TBand ab initio calculations. This is for example the basisof the famous band model of graphite (Slonczewski-Weiss-McClure 3D band structure) which is used to inter-pret many electronic and optical properties of graphite andmultilayer graphene (ref 31 and references therein).

In the tight-binding scheme only pz orbitals are taken intoaccount since we are interested in electronic states close tothe Fermi level. Since the planes are rotated, neighbors arenot on top of each other (as is the case in the Bernal ABstacking). Interlayer interactions are then not restricted toppπ terms but some ppσ terms have also to be introduced.For both terms, a parameter and a characteristic length havebeen fitted on AA, AB, (1,3), and (1,4) cells32 to reproducethe ab initio dispersion curves according to

FIGURE 1. Commensurate bilayer cell (n, m) ) (6, 7) for a rotationof θ ) 5.08°. Full (dashed) line circle AB (AA) region.

cos θ ) n2 + 4nm + m2

2(n2 + nm + m2)(1)

tb ) V′ ) nab1 + mab2; t′f ) -mab1 + (n + m)ab2(2)

Vppπ ) -γ0 exp(qπ(1 - da)) (3)

© 2010 American Chemical Society 805 DOI: 10.1021/nl902948m | Nano Lett. 2010, 10, 804-–808

d is the distance bewteen two orbitals, a is the nearestneighbor distance within a layer, a ) 1.42 Å, and a1 is theinterlayer distance, a1 ) 3.35 Å. The first neighbor interac-tion in a plane is taken equal to γ0 ) 2.7 eV and secondneighbor interaction in a plane equal to 0.1γ0

33 which fixesa value of the ratio q/a in eqs 3 and 4. γ1 ) 0.48 eV is theinterlayer coupling between two orbitals in AA stacking. Allpz orbitals have the same on-site energy in both planes. Wehave chosen this on-site energy such that the energy of theDirac point equals 0.

Localization by Moire Patterns. We have performedband structure calculations on a large number of structures.For structures that could be studied by both ab initio andTB methods, the agreement is always excellent.32 In all (n,m) bilayers, excepted the Bernal AB stacking, band disper-sions E(kb) arround Dirac points K are linear with k for energyE > 0.5 meV. Note that the small energy range, E < 0.5 meV,is not experimentally relevant in most circumstances. Forexample Figure 2 presents the band structure calculated withVASP and with the TB method for (n, m) ) (6, 7) andtherefore θ ) 5.08° and N ) 508. From calculated banddispersions along Γ-K, we compute the velocity V

V ) 1p

∂E∂k

of Dirac states around the K point. VASP and TB agree verywell and show a decrease of the velocity by about 15% inthe (6, 7) bilayer with respect to monolayer graphene veloc-ity. We have checked that the velocity remains isotropic inthe linear region up to a few percent. We note also that forangles within a few degrees of 0° or 60° the bands becomeflat and the energy range in which the bands are lineardecreases (Figure 3).

As stated above the geometry of the Moire dependsessentially on the angle θ, in the limit of small θ. This reflectsin the band dispersion which depends essentially on therotation angle θ between the two layers for all cases studiedhere and in particular in the small angle limit.

We performed a systematic study of the renormalizationof the velocity close to the Dirac point, compared to its value

in a monolayer graphene, for rotation angles θ varyingbetween 0° and 60° (Figure 4). The renormalization of thevelocity varies symmetrically around θ ) 30°. Indeed, thetwo limit cases θ ) 0° (AA stacking) and θ ) 60° (ABstacking) are different, but Moire patterns when θf 0° andwhen θf 60° are similar because a simple translation by avector transforms an AA zone to an AB zone.

Focusing on angles smaller than 30°, we define threeregimes as a function of the rotation angle θ (Figure 4). Forlarge θ (15° e θ e 30°) the Fermi velocity is very close tothat of graphene. For intermediate values of θ (3° e θ e15°) the perturbative theory of Lopez dos Santos et al.19

predicts correctly the velocity renormalization. But for thesmall rotation angles (θ < 3°) a new regime occurs wherebands are very flat (Figure 3) and the velocity tends to zero.This remarkable localization regime cannot be described bythe perturbative theory of Lopez dos Santos et al.19

This new localization mechanism by the potential of theMoire with very small angle leads to a strong peak in the local

Vppσ ) γ1 exp(qσ(1 - da1

)) withqσ

a1)

qπ

a(4)

FIGURE 2. Calculated band structure E(kb) in commensurate bilayercell (n, m) ) (6, 7), θ ) 5.08°, N ) 508.

FIGURE 3. TB band structure E(kb) in commensurate bilayer cell (a)(n, m) ) (19, 20), θ ) 1.70°, N ) 4564, and (b) (n, m) ) (33, 34), θ) 0.99°, N ) 13468. The dashed line is the monolayer graphene.

FIGURE 4. Velocity ratio Vbi /Vmono for a commensurate (n, m) bilayercell versus rotation angle θ: circle, VASP; cross, TB calculations. Theline is the model of Lopez dos Santos et al.:19 Vbi/Vmono ) 1 - 9 [t/(VmonoK sin(θ/2))]2 , with t ) 0.11 eV and VmonoK ) 2γ0π(31/2) ) 9.8eV.

© 2010 American Chemical Society 806 DOI: 10.1021/nl902948m | Nano Lett. 2010, 10, 804-–808

density of states (local DOS) on atoms in the AA zone (Figure5b) whereas for a larger angle no strong localization occurs(Figure 5a). To analyze quantitatively this localization, wecompute the participation ratio of each eigenstate |ψ⟩defined by p(ψ) ) (N∑i

N|⟨i|ψ⟩|4)-1. |i⟩ is the pz orbital on atomi and N is the number of atoms in a unit cell. For completelydelocalized eigenstate p ) 1, whereas state localized on 1atom have a small p value: p ) 1/N. The p value for states atK point with Dirac energy E = 0 is 0.80, 0.66, 0.44, 0.14,and 0.12 for (6, 7), (12, 13), (19, 20), (30, 31), and (33, 34)bilayers, respectively. For very small θ angles these statesare strongly localized. Their spatial repartition shows thatthey are localized on the AA zones of the Moire. This isillustrated in Figure 5b showing 80% of the weight of aneigenstate with energy very close to 0 in the (30, 31) bilayercell (θ ) 1.08°).

The localization regime that we predict here derives froma peculiar complex crystallographic structure in the large unitcell due to a large Moire pattern (small rotation angle). Inthe literature, it is known that complex structures canlocalize electron states, like it is the case in approximantsof quasi-crystals (ref 34 and references within). It would beinteresting to check whether these localized states in rotatedgraphene bilayers can be observed from STS experiments.Indeed, rotated multilayer graphene with small θ is observed

for θ= 1° or even less. In these systems, the strong peak inlocal DOS in AA zone at Dirac energy should increasesignificantly the contrast between AA zone and AB zone inMoire pattern. This new localization should also reduceconsiderably the mobility of the electrons and thus modifytransport properties. A study of quantum transport by thesestates would be of great interest.

Finally, these Moire patterns induce strong changes in thelocal density of states that depend on the local stacking. Itshould then induce strong differences in the chemicalreactivity of the different superstructure regions. Rotatedgraphene layers could then provide new templates for theselective adsorption of atoms or clusters. This is alreadythe case for graphene on transition metal surfaces where thelattice parameter discrepancy gives rise to a Moire patternthat is used to self-organize arrays of metallic or magneticclusters.35

Conclusions. In summary, by combining ab initio andtight-binding calculations, we have demonstrated that thedispersion relation stays linear in rotated graphene bilayersclose to the Dirac point. Yet the velocity of electrons isrenormalized. This renormalization depends essentially ona single parameter which is the rotation angle θ. For large θ(15° e θ e 30°), the Fermi velocity is very close to that ofgraphene. For intermediate values of θ (3° e θ e 15°), theperturbative theory of Lopez dos Santos et al.19 predictscorrectly the velocity renormalization. But for the smallrotation angles (θ < 3°), a new regime occurs where thevelocity tends to zero. In this regime electrons with energyclose to the Dirac point are localized in AA stacked regions.We believe that this localization regime should be observablesince angles as small as a fraction of a degree occur in somerotated multilayers.

Acknowledgment. We thank C. Berger, E. Conrad, W. deHeer, P. Mallet, F. Varchon, J. Y. Veuillen, and P. First forfruitful discussions. Computer time has been granted byCiment/Phynum and S.I.R (Universite de Cergy-Pontoise).

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© 2010 American Chemical Society 808 DOI: 10.1021/nl902948m | Nano Lett. 2010, 10, 804-–808