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Moiré bands in twisted double-layer grapheneRafi Bistritzer and Allan H. MacDonald1

Department of Physics, University of Texas at Austin, Austin, TX 78712

Contributed by Allan H. MacDonald, June 7, 2011 (sent for review December 8, 2010)

A moiré pattern is formed when two copies of a periodic patternare overlaid with a relative twist. We address the electronic struc-ture of a twisted two-layer graphene system, showing that in itscontinuum Dirac model the moiré pattern periodicity leads tomoiré Bloch bands. The two layers become more strongly coupledand the Dirac velocity crosses zero several times as the twist angleis reduced. For a discrete set of magic angles the velocity vanishes,the lowest moiré band flattens, and the Dirac-point density-of-states and the counterflow conductivity are strongly enhanced.

Low-energy electronic properties of few layer graphene (FLG)systems are known (1–8) to be strongly dependent on stacking

arrangement. In bulk graphite 0° and 60° relative orientations ofthe individual layer honeycomb lattices yield rhombohedral andBernal crystals, but other twist angles also appear in many sam-ples (9). Small twist angles are particularly abundant in epitaxialgraphene layers grown on SiC (10, 11), but exfoliated bilayers canalso appear with a twist, and arbitrary alignments between adja-cent layers can be obtained by folding a single graphene layer(12, 13).

Recent advances in FLG preparation methods have attractedtheoretical attention (14–20) to the intriguing electronic proper-ties of systems with arbitrary twist angles, usually focusing on thetwo-layer case. The geometry of the bilayer system is character-ized by a twist angle θ and by a translation vector d. Commensur-ability is determined only by θ. Sliding one layer with respect tothe other in a commensurate structure modifies the unit cell butleaves the bilayer crystalline. In this work we find it convenient toregard the AB stacking as the aligned configuration. The posi-tions of the carbon atoms in the two misaligned layers labeledby R and R0 are then related by R0 ¼ MðθÞðR − τÞ þ d, whereM is a 2-D rotation matrix within the graphene plane, and τ isa vector connecting the two atoms in the unit cell.

The problem is mathematically interesting because a bilayerforms a two-dimensional crystal only at a discrete set of commen-surate rotation angles; for generic twist angles Bloch’s theoremdoes not apply microscopically and direct electronic structurecalculations are not feasible. For twist angles larger than a fewdegrees the two layers are electronically isolated to a remarkabledegree, except at a small set of angles which yield low-order com-mensurate structures (16, 19). As the twist angles become smal-ler, interlayer coupling strengthens, and the quasiparticle velocityat the Dirac point begins to decrease.

Here we focus on the strongly coupled small twist angle regime.We derive a low-energy effective Hamiltonian valid for any valueof d and for θ ≲ 10° irrespective of whether or not the bilayer struc-ture is periodic. We show that it is meaningful to describe the elec-tronic structure using Bloch bands even for incommensurate twistangles and study the dependence of these bands on θ.

ModelWe construct a low-energy continuum model Hamiltonian thatconsists of three terms: two single-layer Dirac–Hamiltonian termsthat account for the isolated graphene sheets and a tunnelingterm that describes hopping between layers. The Dirac–Hamilto-nian (21) for a layer rotated by an angle θ with respect to a fixedcoordinate system is

hkðθÞ ¼ −vk 0 eiðθk−θÞ

e−iðθk−θÞ 0

� �;

where v is the Dirac velocity, k is momentum measured from thelayer’s Dirac point, θk is the momentum orientation relative tothe x axis, and the spinor Hamiltonian acts on the individuallayer’s A and B sublattice degrees-of-freedom. We choose the co-ordinate system depicted in Fig. 1 for which the decoupled bilayerHamiltonian is j1ihðθ∕2Þh1j þ j2ihð−θ∕2Þh2j, where jiihij projectsonto layer i.

We derive a continuummodel for the tunneling term by assum-ing that the interlayer tunneling amplitude between π-orbitalsis a smooth function tðrÞ of spatial separation projected onto thegraphene planes. The matrix element

Tαβkp0 ¼ hΨð1Þ

kα jHTjΨð2Þp0βi [1]

of the tunneling Hamiltonian HT describes a process in which anelectron with momentum p0 ¼ Mp residing on sublattice β in onelayer hops to a momentum state k and sublattice α in the otherlayer. In a π-band tight-binding model the projection of the wavefunctions of the two layers to a given sublattice are

jψ ð1Þkα i ¼

1ffiffiffiffiN

p ∑R

eikðRþταÞjR þ ταi [2]

and

jψ ð2Þpβ i ¼

1ffiffiffiffiN

p ∑R0eipðR

0þτ0βÞjR0 þ τ0βi: [3]

Here τA ¼ 0, τB ¼ τ, and R is summed over the triangular Bravaislattice. Substituting Eqs. 2 and 3 in Eq. 1 and invoking the two-center approximation,

hR þ ταjHTjR0 þ τ0βi ¼ tðR þ τα − R0 − τ0βÞ; [4]

for the interlayer hopping amplitude in which t depends on the dif-ference between the positions of the two carbon atoms we find that

Tαβkp0 ¼ ∑

G1G2

tkþG1

Ωei½G1τα−G2ðτβ−τÞ−G0

2d�δkþG1 ;p0þG02: [5]

Here Ω is the unit cell area, tq is the Fourier transform of the tun-neling amplitude tðrÞ, the vectors G1 and G2 are summed over re-ciprocal lattice vectors, and G0

2 ¼ MG2. The bar notation overmomenta in Eq. 5 indicates that momentum is measured relativeto the center of the Brillouin zone and not relative to the Diracpoint. Note that crystal momentum is conserved by the tunnelingprocess because t depends only on the difference between latticepositions.*

Author contributions: R.B. and A.H.M. designed research; R.B. performed research; andR.B. and A.H.M. wrote the paper.

The authors declare no conflict of interest.

*A closely related but slightly different expression appears in ref. 19 in which we chose theorigin at a honeycomb lattice point. The present convention is more convenient for thediscussion of small rotations relative to the Bernal arrangement.

1To whom correspondence should be addressed. E-mail: [email protected].

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The continuum model for HT is obtained by measuring wavevectors in both layers relative to their Dirac points and assum-ing that the deviations are small compared to Brillouin-zonedimensions. The model’s utility rests centrally on the observa-tion that, although tq is not precisely known, it should never-theless fall to zero very rapidly with q on the reciprocal latticevector scale. This behavior follows from the property that thegraphene layer separation d⊥ exceeds the separation betweencarbon atoms within the layers by more than a factor of 2. Be-cause the two-center integral tðrÞ varies with the three-dimen-sional separation R ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffir2 þ d2⊥

pthe strong small r hopping

processes vary with r on the scale of d⊥. For this reason tq beginsto decline rapidly for qd⊥ > 1. Fig. 2 plots tq values obtainednumerically from the π-band tight-binding models proposed inrefs. 19, 22, and 23. The largest tq values that enter the tunnelingnear the Dirac point have q ¼ kD, the Brillouin-zone corner(Dirac) wave vector magnitude, and correspond to the threereciprocal vectors 0, Gð2Þ, and Gð3Þ where the latter two vectorsconnect a Dirac point with its equivalent first Brillouin-zonecounterparts (See Fig. 1). When only these terms are retained,

TαβðrÞ ¼ w∑3

j¼1

expð−iqj · rÞTαβj ; [6]

where w ¼ tkD∕Ω is the hopping energy,

T1 ¼ 1 1

1 1

� �; T2 ¼ e−iG

ð2Þ0 ·d e−iϕ 1

eiϕ e−iϕ

� �;

T3 ¼ e−iGð3Þ0 ·d eiϕ 1

e−iϕ eiϕ

� �; [7]

and ϕ ¼ 2π∕3. The three qj’s in Eq. 6 are Dirac model momen-tum transfers that correspond to the three interlayer hoppingprocesses.

For d ¼ 0 and a vanishing twist angle the continuum tunnelingmatrix is 3wδαAδβB, independent of position. By comparing withthe experimentally known (24) electronic structure of an ABstacked bilayer we set w ≈ 110 meV for exfoliated samples,however experiments suggest (25) that w may be smaller in someepitaxial graphene samples. As we show below the spectrum isindependent of d for θ ≠ 0. In the following we thereforeset d ¼ 0.

ResultsMoiré Bloch Bands. In the continuum model hopping is local andperiodic, allowing Bloch’s theorem to be applied at any rotationangle irrespective of whether or not the bilayer is crystalline. Wesolve numerically for the moiré bands using the plane wave ex-pansion illustrated in Fig. 1. Convergence is attained by truncat-ing momentum space at lattice vectors of the order of w∕ℏv. Thedimension of the matrix, which must be diagonalized numerically,is roughly ∼10 θ−2 for small θ (measured in degrees), comparedto the ∼104 θ−2 matrix dimension of microscopic tight-bindingmodels (14, 16).

Up to a scale factor the moiré bands depend on a single para-meter α ¼ w∕vkθ. We have evaluated the moiré bands as a func-tion of their Brillouin-zone momentum k for many different twistangles; results for w ¼ 110 meV, and θ ¼ 5°, 1.05°, and 0.5° aresummarized in Fig. 3. For large twist angles the low-energy spec-trum is virtually identical to that of an isolated graphene sheet,except that the velocity is slightly renormalized. Large interlayercoupling effects appear only near the high energy van Hove sin-gularities discussed by Andrei and coworkers (26). As the twistangle is reduced, the number of bands in a given energy windowincreases and the band at the Dirac point narrows. This narrow-ing has previously been expected to develop monotonically withdecreasing θ. As illustrated in Fig. 3, we instead find that theDirac-point velocity vanishes already at θ ≈ 1.05°, and that thevanishing velocity is accompanied by a very flat moiré band whichcontributes a sharp peak to the Dirac-point density-of-states(DOS). At smaller twists the Dirac-point velocity has a nonmo-

Fig. 1. Momentum-space geometry of a twisted bilayer. (A) Dashed linemarks the first Brillouin zone of an unrotated layer. The three equivalentDirac points are connected by Gð2Þ and Gð3Þ. The circles represent Diracpoints of the rotated graphene layers, separated by kθ ¼ 2kD sinðθ∕2Þ, wherekD is the magnitude of the Brillouin-zone corner wave vector for a singlelayer. Conservation of crystal momentum implies that p0 ¼ kþ qb for atunneling process in the vicinity of the plotted Dirac points. (B) The threeequivalent Dirac points in the first Brillouin zone result in three distincthopping processes. Interference between hopping processes with differentwave vectors captures the spatial variation of interlayer coordination thatdefines the moiré pattern. For all the three processes jqj j ¼ kθ ; however,the hopping directions are ð0; − 1Þ for j ¼ 1, ð ffiffiffi

3p

∕2;1∕2Þ for j ¼ 2, andð− ffiffiffi

3p

∕2;1∕2Þ for j ¼ 3. We interchangeably use 1, 2, 3, b, tr, and tl as sub-scripts for the three momentum transfers qj . Repeated hopping generates ak-space honeycomb lattice. The green solid line marks the moiré bandWigner–Seitz cell. In a repeated zone scheme the red and black circles markthe Dirac points of the two layers.

2 4 6 8 10 12

0

100

200

300

400

500

qa

t q [meV

A°

2 ]

Fig. 2. Hopping amplitude. The Fourier transform of the hopping amplitudeis plotted vs. momentum (a is the carbon-carbon distance in single-layer gra-phene). The different curves correspond to different models described inrefs. 19 (dotted), 22 (solid), and 23 (dashed). The vertical line crosses the xaxis at kDa.

12234 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1108174108 Bistritzer and MacDonald

notonic dependence on θ, vanishing repeatedly at the series ofmagic angles illustrated in Fig. 4.

Partial insight into the origin of these behaviors can be achievedby examining the simplest limit in which the momentum-spacelattice is truncated at the first honeycomb shell. Including thesublattice degree of freedom, this truncation gives rise to theHamiltonian

Hk ¼hkðθ∕2Þ Tb Ttr Ttl

T†

b hkbð−θ∕2Þ 0 0

T†tr 0 hktrð−θ∕2Þ 0

T†

tl 0 0 hktlð−θ∕2Þ

2664

3775; [8]

where k is in the moiré Brillouin-zone and kj ¼ k þ qj. ThisHamiltonian acts on four two-component spinors Ψ ¼ ðψ0;ψ1;ψ2;ψ3Þ. The first (ψ0) is at a momentum near the Dirac point ofone layer and the other three ψ j are at momenta near qj and in the

other layer. The dependence of hðθÞ on angle is parametricallysmall and can be neglected. We have numerically verified that thisapproximation reproduces the velocity with reasonable accuracydown to the first magic angle (Fig. 4, Inset).

The renormalized velocity v⋆ ¼ ∂kϵ⋆k jk¼0 follows from thespectrum ϵ⋆k of the twisted bilayer. The Hamiltonian is expressedas a sum of the k ¼ 0 term Hð0Þ and the k-dependent term Hð1Þ

kand solved to leading order in k.

Consider the k ¼ 0 term in the Hamiltonian. We assume thatHð0Þ has zero energy eigenstates and prove our assumption byexplicitly finding these states. The zero energy eigenstates mustsatisfy

ψ j ¼ −h−1j T†j ψ0: [9]

Because

Tjh−1j T†j ¼ 0 [10]

the equation for the ψ0 spinor is h0ψ0 ¼ 0, i.e., ψ0 is one ofthe two zero energy states ψ ð1Þ

0 and ψ ð2Þ0 of the isolated layer.

The two zero energy eigenstates of Hð0Þ then follow from Eq. 9.Given that jψ ðjÞ

0 j ¼ 1, the wave functions should be normalizedby jΨj2 ¼ 1þ 6α2. The effective Hamiltonian matrix to leadingorder in k is therefore

hΨðiÞjHð1Þk jΨðjÞi ¼ −v

1þ 6α2ψ ðiÞ†0

�σ · k þ w2

∑j

Tjh−1†j σ

· kh−1j T†j

�ψ ðjÞ0 ¼ −v⋆ψ ðiÞ†

0 σ · kψ ðjÞ0 :

Aside from a renormalized velocity

v⋆

v¼ 1 − 3α2

1þ 6α2; [11]

the Hamiltonian is identical to the continuum model Hamilto-nian of single-layer graphene. The denominator in Eq. 11 cap-tures the contribution of the Ψj’s to the normalization of thewave function whereas the numerator captures their contributionto the velocity matrix elements. For small α, Eq. 11 reduces tothe expression v⋆∕v ¼ 1 − 9α2, first obtained by Lopes dos Santoset al. (15). The velocity vanishes at the first magic angle because itis in the process of changing sign. The eigenstates at the Diracpoint are a coherent combination of components in the two layersthat have velocities of opposite sign.

Counterflow Conductivity. The distribution of the quasiparticlevelocity between the two layers implies exotic transport charac-teristics for separately contacted layers. Consider a counterflowgeometry in which currents in the two layers flow antiparallel toone another. We focus on twist angles θ ≳ 2° for which the eight-band model is valid and to the semiclassical regime in whichϵFτ > 1 and find the counterflow conductivity σCF. We assumethat the Fermi momentum is much smaller than kθ and that1∕τ0 < ℏvkθ, where τ0 is single particle lifetime. Using the Kuboformula we find that

σCF ¼ 4e2

π ∑kμ

jhψkjvxCFjψkij2½ImfGrkμðϵFÞg�2; [12]

where

vxCF ¼ −v

σx 0 0 0

0 −σx 0 0

0 0 −σx 0

0 0 0 −σx

0BB@

1CCA [13]

Fig. 3. Moiré bands. (A) Energy dispersion for the 14 bands closest to theDirac point plotted along the k-space trajectory A → B → C → D → A (seeFig. 1) for w ¼ 110 meV, and θ ¼ 5° (Left,), 1.05° (Middle), and 0.5° (Right).(B) DOS. (C) Energy as a function of twist angle for the k ¼ 0 states. Bandseparation decreases with θ as also evident from A. (D) Full dispersion ofthe flat band at θ ¼ 1.05°.

Fig. 4. Renormalized Dirac-point band velocity. The band velocity of thetwisted bilayer at the Dirac point v⋆ is plotted vs. α2, where α ¼ w∕vkθ

for 0.18° < θ < 1.2°. The velocity vanishes for θ ≈ 1.05°, 0.5°, 0.35°, 0.24°,and 0.2°. (Inset) The renormalized velocity at larger twist angles. The solidline corresponds to numerical results and dashed line corresponds to analyticresults based on the eight-band model.

Bistritzer and MacDonald PNAS ∣ July 26, 2011 ∣ vol. 108 ∣ no. 30 ∣ 12235

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is the x component of the counterflow velocity operator (we setthe electric fields along the x axis), Gr

kμðωÞ ¼ ðω − ϵ⋆kμ þ i∕2τ0Þ−1is the retarded Green function with μ labeling the two Diracbands, and ϵ⋆kμ ¼ μv⋆k is the energy dispersion of the twistedbilayer at small momenta. For an electron-doped system thevalence band can be neglected and

σCF ≈ e2gτν⋆ðϵFÞZ

dθk2π

jhψkμjvxCFjψkμij2; [14]

where ν⋆ is the DOS of the twisted bilayer. The vertex function

hψkjvxCFjψki ¼ vCF cos θk; [15]

where vCF ¼ vð1þ 3α2Þ∕ð1þ 6α2Þ follows directly from theprevious section if we notice the sign differences between thecounterflow velocity operator and the normal velocity operator.The counterflow conductivity is then

σCF ¼ σ0

�vCFv⋆

�2

; [16]

where σ0 ∼ e2ϵFτ∕π is the conductivity of an isolated single gra-phene layer. As θ is reduced from a large value toward 1°, v⋆ isreduced and the DOS is correspondingly increased. The counter-flow conductivity is enhanced because of an increased densityof carriers, which is not accompanied by a decrease in counter-flow velocity. For a conventional measurement in which thecurrent in the bilayer is unidirectional vCF in Eq. 16 is replacedby v⋆. The increase in the DOS is then exactly compensated bythe reduction of the renormalized velocity and the single-layervalue of the conductivity is regained.

Dependence of the Spectrum on d.We now show that the spectrumof misaligned bilayers is independent of linear translations of onelayer with respect to the other using a unitary transformation thatmakes the Hamiltonian independent of d. Consider HQ where Qis a momentum in the first moiré Brillouin zone. With each mo-mentum on the k-space triangular Bravais lattice (see Fig. 1)

k ¼ Qþ nq1 þmq2;

where q1 ¼ kθð1∕2;ffiffiffi3

p∕2Þ and q2 ¼ kθð−1∕2;

ffiffiffi3

p∕2Þ, we associate

the phase

ϕk ¼ nG02 · dþmG0

3 · d:

The phase associated with momentum k − kθ y on the other sub-lattice is ϕk as well. In terms of the new basis states expðiϕkÞjkαithe Hamiltonian HQ is d-independent.

Physically, the lack of dependence on d can be understoodby noticing that varying d just shifts the moiré pattern in space.The bilayer spectrum does depend on d at θ ¼ 0, and at othercommensurate angles. We expect that dependence on d will beobservable only at short period (large θ) commensurate angles.

DiscussionTwisted double-layer graphene is, for most values of θ, a quasi-periodic structure that has no unit cell. Nevertheless, we find thatfor θ ≲ 10° it is meaningful to describe the electronic structure ofthe system in terms of Bloch bands. The hidden periodic structureis shown to be related to the moiré pattern of the overlaidlayers (27).

The leading corrections to the periodic moiré band Hamilto-nian involve hopping amplitudes with the smallest momenta gthat satisfy the crystal momentum conservation condition in Eq. 5and are larger than kD. As we showed in ref. 19, real space com-mensuration between the two rotated hexagonal lattice is conco-

mitant to momentum space commensuration of the Dirac pointsin the extended-zone scheme (see figures 2 and 3 in ref. 19). Thecommensurate vector g can therefore be found using the formulafor the moiré periodicity if the lattice vector of graphene

ffiffiffi3

pa

(where a is the carbon-carbon distance in a single-layer graphene)is replaced by the reciprocal lattice vector G. It follows thatgðθÞ ≈G∕θ. For example gð10°Þ ¼ 24∕a and gð2°Þ ¼ 120∕a. AsFig. 2 demonstrates, the hopping amplitudes for these large wavevectors are indeed negligible compared to the value of tkD . Wetherefore expect the continuum model to be very accurate upto energies of approximately 1 eV and up to angles of approxi-mately 10°.

The Bloch band model has a simple and appealing physicalinterpretation. The hopping Hamiltonian is local in space. Ateach position, its 4 × 4 matrix, describes sublattice-dependentinterlayer hopping, which depends on the local coordinationbetween the atoms in the two layers. In Fig. 5 we have plotted themoiré pattern of atomic positions and the smaller of the two po-sitive eigenvalues of the hopping Hamiltonian as a function ofposition on the same length scale. At each position, the local in-terlayer tunneling Hamiltonian, is that of a system in which thelocal coordination is maintained through all space. At AB and BApoints, for example, the tunneling Hamiltonian is that of AB andBA systems, for which tunneling does not produce a gap so thatthe smallest positive eigenvalue vanishes. On the other hand thegap reaches its maxima (6w) at AA points in the moiré pattern.

In summary we have formulated a continuum model descrip-tion of the electronic structure of rotated graphene bilayers. Theresulting moiré band structure can be evaluated at arbitrary twistangles, not only at commensurate values. We find that the velocityat the Dirac point oscillates with twist angle, vanishing at a seriesof magic angles which give rise to large DOS and to large counter-flow conductivities. Many properties of the moiré bands are stillnot understood. For example, although we are able to explain thelargest magic angle analytically, the pattern of magic angles atsmaller values of θ has so far been revealed only numerically.Additionally the flattening of the entire lowest moiré band atθ ≈ 1.05° remains a puzzle. Interesting new issues arise whenour theory is extended to graphene stacks containing three ormore layers. Finally, we remark that electron-electron interac-tions neglected in this work are certain to be important at magictwist angles in neutral systems and could give rise to counterflowsuperfluidity (28, 29), flat-band magnetism (30), or other types ofordered states.

ACKNOWLEDGMENTS. We acknowledge a helpful conversation with GeneMele. This work was supported by Welch Foundation Grant F1473 and bythe National Science Foundation-Nanoelectronics Research Initiative SouthWest Academy of Nanoelectronics program.

−5 0 5−6

−4

−2

0

2

4

6

kθ x

k θ y AB

AB

AB

BA

AA

−5 0 5−6

−4

−2

0

2

4

6

BA

AA

kθ x

k θ y

Fig. 5. Moiré period. (Right) Moiré pattern obtained from two graphenelayers overlaid with a relative twist angle θ. Distances are measured in unitsof k−1

θ , where kθ ¼ 2kD sinðθ∕2Þ with kD being the Dirac wave vector. Bluedots denote areas with local AB coordination. (Left) Smallest positive energyof the interlayer Hamiltonian. The energy vanishes for local AB or BA coor-dination and reached a maximum of 3w for local AA coordination.

12236 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1108174108 Bistritzer and MacDonald

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