Moiré Bands in Twisted Double-Layer Graphene

Moiré bands in twisted double-layer graphene

Rafi 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) θ −θ A moiré pattern is formed when two copies of a periodic pattern 0 eið k Þ hkðθÞ¼−vk − θ −θ ; are overlaid with a relative twist. We address the electronic struc- e ið k Þ 0 ture of a twisted two-layer graphene system, showing that in its continuum Dirac model the moiré pattern periodicity leads to where v is the Dirac velocity, k is momentum measured from the moiré Bloch bands. The two layers become more strongly coupled layer’s Dirac point, θk is the momentum orientation relative to and the Dirac velocity crosses zero several times as the twist angle the x axis, and the spinor Hamiltonian acts on the individual is reduced. For a discrete set of magic angles the velocity vanishes, layer’s A and B sublattice degrees-of-freedom. We choose the co- the lowest moiré band flattens, and the Dirac-point density-of- ordinate system depicted in Fig. 1 for which the decoupled bilayer states and the counterflow conductivity are strongly enhanced. Hamiltonian is j1ihðθ∕2Þh1jþj2ihð−θ∕2Þh2j, where jiihij projects onto layer i. ow-energy electronic properties of few layer graphene (FLG) We derive a continuum model for the tunneling term by assum- π Lsystems are known (1–8) to be strongly dependent on stacking ing that the interlayer tunneling amplitude between -orbitals arrangement. In bulk graphite 0° and 60° relative orientations of is a smooth function tðrÞ of spatial separation projected onto the the individual layer honeycomb lattices yield rhombohedral and graphene planes. The matrix element Bernal crystals, but other twist angles also appear in many sam- αβ Ψð1Þ Ψð2Þ ples (9). Small twist angles are particularly abundant in epitaxial Tkp0 ¼h kα jHTj p0βi [1] graphene layers grown on SiC (10, 11), but exfoliated bilayers can also appear with a twist, and arbitrary alignments between adja- of the tunneling Hamiltonian HT describes a process in which an cent layers can be obtained by folding a single graphene layer electron with momentum p0 ¼ Mp residing on sublattice β in one (12, 13). layer hops to a momentum state k and sublattice α in the other Recent advances in FLG preparation methods have attracted layer. In a π-band tight-binding model the projection of the wave theoretical attention (14–20) to the intriguing electronic proper- functions of the two layers to a given sublattice are ties of systems with arbitrary twist angles, usually focusing on the 1 1 ð Þ ikðRþταÞ two-layer case. The geometry of the bilayer system is character- jψ α i¼pffiffiffiffi e jR þ ταi [2] θ k ∑ ized by a twist angle and by a translation vector d. Commensur- N R θ ability is determined only by . Sliding one layer with respect to and the other in a commensurate structure modifies the unit cell but 2 1 0 τ0 ð Þ ipðR þ β Þ 0 0 jψ β i¼pffiffiffiffi e jR þ τβi: [3] leaves the bilayer crystalline. In this work we find it convenient to p N ∑ regard the AB stacking as the aligned configuration. The posi- R0 tions of the carbon atoms in the two misaligned layers labeled Here τ 0, τ τ, and R is summed over the triangular Bravais by R and R0 are then related by R0 ¼ MðθÞðR − τÞþd, where A ¼ B ¼ lattice. Substituting Eqs. 2 and 3 in Eq. 1 and invoking the two- M is a 2-D rotation matrix within the graphene plane, and τ is center approximation, a vector connecting the two atoms in the unit cell. The problem is mathematically interesting because a bilayer 0 0 0 0 hR þ ταjH jR þ τ i¼tðR þ τα − R − τ Þ; [4] forms a two-dimensional crystal only at a discrete set of commen- T β β surate rotation angles; for generic twist angles Bloch’s theorem for the interlayer hopping amplitude in which t depends on the dif- does not apply microscopically and direct electronic structure ference between the positions of the two carbon atoms we find that calculations are not feasible. For twist angles larger than a few t¯ degrees the two layers are electronically isolated to a remarkable αβ kþG1 τ − τ −τ − 0 i½G1 α G2ð β Þ G2dδ T 0 ¼ e k¯þG ;p¯0þG0 : [5] degree, except at a small set of angles which yield low-order com- kp ∑ Ω 1 2 mensurate structures (16, 19). As the twist angles become smal- G1G2 ler, interlayer coupling strengthens, and the quasiparticle velocity Here Ω is the unit cell area, tq is the Fourier transform of the tun- at the Dirac point begins to decrease. neling amplitude tðrÞ, the vectors G1 and G2 are summed over re- 0 Here we focus on the strongly coupled small twist angle regime. ciprocal lattice vectors, and G2 ¼ MG2. The bar notation over We derive a low-energy effective Hamiltonian valid for any value momenta in Eq. 5 indicates that momentum is measured relative of d and for θ ≲ 10° irrespective of whether or not the bilayer struc- to the center of the Brillouin zone and not relative to the Dirac ture is periodic. We show that it is meaningful to describe the elec- point. Note that crystal momentum is conserved by the tunneling PHYSICS tronic structure using Bloch bands even for incommensurate twist process because t depends only on the difference between lattice angles and study the dependence of these bands on θ. positions.* Model We construct a low-energy continuum model Hamiltonian that Author contributions: R.B. and A.H.M. designed research; R.B. performed research; and R.B. and A.H.M. wrote the paper. consists of three terms: two single-layer Dirac–Hamiltonian terms The authors declare no conflict of interest. that account for the isolated graphene sheets and a tunneling – *A closely related but slightly different expression appears in ref. 19 in which we chose the term that describes hopping between layers. The Dirac Hamilto- origin at a honeycomb lattice point. The present convention is more convenient for the nian (21) for a layer rotated by an angle θ with respect to a fixed discussion of small rotations relative to the Bernal arrangement. coordinate system is 1To whom correspondence should be addressed. E-mail: [email protected]. www.pnas.org/cgi/doi/10.1073/pnas.1108174108 PNAS ∣ July 26, 2011 ∣ vol. 108 ∣ no. 30 ∣ 12233–12237 500

400 ]

2 300 °

200 [meV A q t 100

2 4 6 8 10 12 qa Fig. 1. Momentum-space geometry of a twisted bilayer. (A) Dashed line Fig. 2. Hopping amplitude. The Fourier transform of the hopping amplitude marks the first Brillouin zone of an unrotated layer. The three equivalent is plotted vs. momentum (a is the carbon-carbon distance in single-layer gra- Dirac points are connected by Gð2Þ and Gð3Þ. The circles represent Dirac phene). The different curves correspond to different models described in k 2k θ∕2 points of the rotated graphene layers, separated by θ ¼ D sinð Þ, where k refs. 19 (dotted), 22 (solid), and 23 (dashed). The vertical line crosses the x D is the magnitude of the Brillouin-zone corner wave vector for a single k a 0 axis at D . layer. Conservation of crystal momentum implies that p ¼ k þ qb for a tunneling process in the vicinity of the plotted Dirac points. (B) The three ϕ 3 0 i 1 equivalent Dirac points in the first Brillouin zone result in three distinct −iGð Þ ·d e T3 ¼ e − ϕ ϕ ; [7] hopping processes. Interference between hopping processes with different e i ei wave vectors captures the spatial variation of interlayer coordination that k defines the moiré pattern. For all the three processespffiffiffi jqj j¼ θ; however, ϕ 2π∕3 ’ 0; − 1 j 1 3∕2;1∕2 j 2 and ¼ . The three qj s in Eq. 6 are Dirac model momen- thepffiffiffi hopping directions are ð Þ for ¼ , ð Þ for ¼ , and ð− 3∕2;1∕2Þ for j ¼ 3. We interchangeably use 1, 2, 3, b, tr, and tl as sub- tum transfers that correspond to the three interlayer hopping scripts for the three momentum transfers qj . Repeated hopping generates a processes. k-space honeycomb lattice. The green solid line marks the moiré band For d ¼ 0 and a vanishing twist angle the continuum tunneling – 3 δ δ Wigner Seitz cell. In a repeated zone scheme the red and black circles mark matrix is w αA βB, independent of position. By comparing with the Dirac points of the two layers. the experimentally known (24) electronic structure of an AB stacked bilayer we set w ≈ 110 meV for exfoliated samples, however experiments suggest (25) that w may be smaller in some The continuum model for HT is obtained by measuring wave epitaxial graphene samples. As we show below the spectrum is vectors in both layers relative to their Dirac points and assum- independent of d for θ ≠ 0. In the following we therefore ing that the deviations are small compared to Brillouin-zone set d ¼ 0. dimensions. The model’s utility rests centrally on the observa- Results tion that, although tq is not precisely known, it should nevertheless fall to zero very rapidly with q on the reciprocal lattice Moiré Bloch Bands. In the continuum model hopping is local and vector scale. This behavior follows from the property that the periodic, allowing Bloch’s theorem to be applied at any rotation graphene layer separation d⊥ exceeds the separation between angle irrespective of whether or not the bilayer is crystalline. We carbon atoms within the layers by more than a factor of 2. Be- solve numerically for the moiré bands using the plane wave ex- pansion illustrated in Fig. 1. Convergence is attained by truncat- cause the two-center integralpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffitðrÞ varies with the three-dimen- ing momentum space at lattice vectors of the order of w∕ℏv. The sional separation R ¼ r2 þ d2 the strong small r hopping ⊥ dimension of the matrix, which must be diagonalized numerically, processes vary with r on the scale of d⊥. For this reason tq begins is roughly ∼10 θ−2 for small θ (measured in degrees), compared 1 4 −2 to decline rapidly for qd⊥ > . Fig. 2 plots tq values obtained to the ∼10 θ matrix dimension of microscopic tight-binding numerically from the π-band tight-binding models proposed in models (14, 16). refs. 19, 22, and 23. The largest tq values that enter the tunneling Up to a scale factor the moiré bands depend on a single para- α ∕ near the Dirac point have q ¼ kD, the Brillouin-zone corner meter ¼ w vkθ. We have evaluated the moiré bands as a func- (Dirac) wave vector magnitude, and correspond to the three tion of their Brillouin-zone momentum k for many different twist 110 θ 5 reciprocal vectors 0, Gð2Þ, and Gð3Þ where the latter two vectors angles; results for w ¼ meV, and ¼ °, 1.05°, and 0.5° are connect a Dirac point with its equivalent first Brillouin-zone summarized in Fig. 3. For large twist angles the low-energy spec- counterparts (See Fig. 1). When only these terms are retained, trum is virtually identical to that of an isolated graphene sheet, except that the velocity is slightly renormalized. Large interlayer coupling effects appear only near the high energy van Hove sin- 3 gularities discussed by Andrei and coworkers (26). As the twist αβ − αβ angle is reduced, the number of bands in a given energy window T ðrÞ¼w expð iqj · rÞT ; [6] ∑ j increases and the band at the Dirac point narrows. This narrow- j¼1 ing has previously been expected to develop monotonically with decreasing θ. As illustrated in Fig. 3, we instead find that the where w ¼ t ∕Ω is the hopping energy, kD Dirac-point velocity vanishes already at θ ≈ 1.05°, and that the − ϕ vanishing velocity is accompanied by a very flat moiré band which 11 2 0 i 1 −iGð Þ ·d e contributes a sharp peak to the Dirac-point density-of-states T1 ¼ 11;T2 ¼ e iϕ −iϕ ; e e (DOS). At smaller twists the Dirac-point velocity has a nonmo-

12234 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1108174108 Bistritzer and MacDonald other layer. The dependence of hðθÞ on angle is parametrically small and can be neglected. We have numerically verified that this approximation reproduces the velocity with reasonable accuracy down to the first magic angle (Fig. 4, Inset). ⋆ ∂ ϵ⋆ The renormalized velocity v ¼ k k jk¼0 follows from the ϵ⋆ spectrum k of the twisted bilayer. The Hamiltonian is expressed 0 Hð0Þ Hð1Þ as a sum of the k ¼ term and the k-dependent term k and solved to leading order in k. Consider the k ¼ 0 term in the Hamiltonian. We assume that Hð0Þ has zero energy eigenstates and prove our assumption by explicitly finding these states. The zero energy eigenstates must satisfy ψ − −1 †ψ j ¼ hj Tj 0: [9] Because −1 † 0 Tjhj Tj ¼ [10]

the equation for the ψ0 spinor is h0ψ 0 ¼ 0, i.e., ψ 0 is one of Fig. 3. Moiré bands. (A) Energy dispersion for the 14 bands closest to the ψ ð1Þ ψ ð2Þ Dirac point plotted along the k-space trajectory A → B → C → D → A (see the two zero energy states 0 and 0 of the isolated layer. ð0Þ w 110 θ 5 The two zero energy eigenstates of H then follow from Eq. 9. Fig. 1) for ¼ meV, and ¼ °(Left,), 1.05° (Middle), and 0.5° (Right). ðjÞ (B) DOS. (C) Energy as a function of twist angle for the k ¼ 0 states. Band Given that jψ 0 j¼1, the wave functions should be normalized 2 2 separation decreases with θ as also evident from A.(D) Full dispersion of by jΨj ¼ 1 þ 6α . The effective Hamiltonian matrix to leading the flat band at θ ¼ 1.05°. order in k is therefore − θ ðiÞ ð1Þ ðjÞ v ðiÞ† 2 −1† notonic dependence on , vanishing repeatedly at the series of hΨ jH jΨ i¼ ψ 0 σ · k þ w T h σ k 1 þ 6α2 ∑ j j magic angles illustrated in Fig. 4. j Partial insight into the origin of these behaviors can be achieved by examining the simplest limit in which the momentum-space −1 † ψ ðjÞ − ⋆ψ ðiÞ†σ ψ ðjÞ · khj Tj 0 ¼ v 0 · k 0 : lattice is truncated at the first honeycomb shell. Including the sublattice degree of freedom, this truncation gives rise to the Aside from a renormalized velocity Hamiltonian ⋆ 2 2 3 v 1 − 3α θ∕2 ¼ ; [11] hkð Þ Tb Ttr Ttl 1 6α2 † v þ 6 T h ð−θ∕2Þ 007 H 6 b kb 7 k ¼ 4 † 0 −θ∕2 0 5; [8] the Hamiltonian is identical to the continuum model Hamilto- Ttr hktr ð Þ † T 00h ð−θ∕2Þ nian of single-layer graphene. The denominator in Eq. 11 cap- tl ktl Ψ ’ tures the contribution of the j s to the normalization of the wave function whereas the numerator captures their contribution where k is in the moiré Brillouin-zone and k ¼ k þ q . This j j to the velocity matrix elements. For small α, Eq. 11 reduces to Hamiltonian acts on four two-component spinors Ψ ¼ðψ 0;ψ 1; ⋆ 2 the expression v ∕v ¼ 1 − 9α , first obtained by Lopes dos Santos ψ 2;ψ 3Þ. The first (ψ 0) is at a momentum near the Dirac point of ψ et al. (15). The velocity vanishes at the first magic angle because it one layer and the other three j are at momenta near qj and in the is in the process of changing sign. The eigenstates at the Dirac point are a coherent combination of components in the two layers that have velocities of opposite sign.

Counterflow Conductivity. The distribution of the quasiparticle velocity between the two layers implies exotic transport charac- teristics for separately contacted layers. Consider a counterflow geometry in which currents in the two layers flow antiparallel to one another. We focus on twist angles θ ≳ 2° for which the eight- band model is valid and to the semiclassical regime in which ϵ τ 1 σ F > and find the counterflow conductivity CF. We assume that the Fermi momentum is much smaller than kθ and that 1∕τ0 < ℏvkθ, where τ0 is single particle lifetime. Using the Kubo formula we find that PHYSICS 4e2 σ ψ x ψ 2 r ϵ 2 [12] CF ¼ π ∑jh kjvCFj kij ½ImfGkμð FÞg ; kμ

where Fig. 4. Renormalized Dirac-point band velocity. The band velocity of the 0 1 v⋆ α2 α w∕vk σ 000 twisted bilayer at the Dirac point is plotted vs. , where ¼ θ x for 0.18°<θ < 1.2°. The velocity vanishes for θ ≈ 1.05°, 0.5°, 0.35°, 0.24°, B 0 −σ 00C x − B x C and 0.2°. (Inset) The renormalized velocity at larger twist angles. The solid vCF ¼ v@ 00−σ 0 A [13] x line corresponds to numerical results and dashed line corresponds to analytic 00 0−σ results based on the eight-band model. x

Bistritzer and MacDonald PNAS ∣ July 26, 2011 ∣ vol. 108 ∣ no. 30 ∣ 12235 is the x component of the counterflow velocity operator (we set 6 6 ⋆ −1 r ω ω − ϵ ∕2τ AB the electric fields along the x axis), Gkμð Þ¼ð kμ þ i 0Þ 4 4 is the retarded Green function with μ labeling the two Dirac ϵ⋆ μ ⋆ 2 2 bands, and kμ ¼ v k is the energy dispersion of the twisted y y θ bilayer at small momenta. For an electron-doped system the θ 0 AB BA 0 BA k valence band can be neglected and k −2 AA AB −2 AA Z θ 2 ⋆ d k x 2 −4 −4 σ ≈ τν ϵ ψ μ ψ μ [14] CF e g ð FÞ 2π jh k jvCFj k ij ; −6 −6 −5 0 5 −5 0 5 ⋆ where ν is the DOS of the twisted bilayer. The vertex function k x kθ x θ ψ x ψ θ h kjvCFj ki¼vCF cos k; [15] Fig. 5. Moiré period. (Right) Moiré pattern obtained from two graphene layers overlaid with a relative twist angle θ. Distances are measured in units 2 2 1 3α ∕ 1 6α k−1 k 2k θ∕2 k where vCF ¼ vð þ Þ ð þ Þ follows directly from the of θ , where θ ¼ D sinð Þ with D being the Dirac wave vector. Blue previous section if we notice the sign differences between the dots denote areas with local AB coordination. (Left) Smallest positive energy counterflow velocity operator and the normal velocity operator. of the interlayer Hamiltonian. The energy vanishes for local AB or BA coor- The counterflow conductivity is then dination and reached a maximum of 3w for local AA coordination. 2 mitant to momentum space commensuration of the Dirac points vCF σ ¼ σ0 ; [16] CF v⋆ in the extended-zone scheme (see figures 2 and 3 in ref. 19). The commensurate vector g can therefore be found using the formulapffiffiffi σ ∼ 2ϵ τ∕π for the moiré periodicity if the lattice vector of graphene 3a where 0 e F is the conductivity of an isolated single graphene layer. As θ is reduced from a large value toward 1°, v⋆ is (where a is the carbon-carbon distance in a single-layer graphene) is replaced by the reciprocal lattice vector G. It follows that reduced and the DOS is correspondingly increased. The counter- θ ≈ ∕θ 10 24∕ 2 120∕ flow conductivity is enhanced because of an increased density gð Þ G . For example gð °Þ¼ a and gð °Þ¼ a.As of carriers, which is not accompanied by a decrease in counter- Fig. 2 demonstrates, the hopping amplitudes for these large wave vectors are indeed negligible compared to the value of t .We flow velocity. For a conventional measurement in which the kD therefore expect the continuum model to be very accurate up current in the bilayer is unidirectional vCF in Eq. 16 is replaced by v⋆. The increase in the DOS is then exactly compensated by to energies of approximately 1 eV and up to angles of approxi- the reduction of the renormalized velocity and the single-layer mately 10°. value of the conductivity is regained. The Bloch band model has a simple and appealing physical interpretation. The hopping Hamiltonian is local in space. At Dependence of the Spectrum on d. We now show that the spectrum each position, its 4 × 4 matrix, describes sublattice-dependent of misaligned bilayers is independent of linear translations of one interlayer hopping, which depends on the local coordination layer with respect to the other using a unitary transformation that between the atoms in the two layers. In Fig. 5 we have plotted the makes the Hamiltonian independent of d. Consider HQ where Q moiré pattern of atomic positions and the smaller of the two po- is a momentum in the first moiré Brillouin zone. With each mo- sitive eigenvalues of the hopping Hamiltonian as a function of mentum on the k-space triangular Bravais lattice (see Fig. 1) position on the same length scale. At each position, the local interlayer tunneling Hamiltonian, is that of a system in which the k ¼ Q þ nq1 þ mq2; local coordination is maintained through all space. At AB and BA pffiffiffi pffiffiffi points, for example, the tunneling Hamiltonian is that of AB and where q1 ¼ kθð1∕2; 3∕2Þ and q2 ¼ kθð−1∕2; 3∕2Þ, we associate BA systems, for which tunneling does not produce a gap so that the phase the smallest positive eigenvalue vanishes. On the other hand the gap reaches its maxima (6w) at AA points in the moiré pattern. ϕ 0 0 k ¼ nG2 · d þ mG3 · d: In summary we have formulated a continuum model descrip- tion of the electronic structure of rotated graphene bilayers. The The phase associated with momentum k − kθy^ on the other sub- resulting moiré band structure can be evaluated at arbitrary twist lattice is ϕk as well. In terms of the new basis states expðiϕkÞjkαi angles, not only at commensurate values. We find that the velocity the Hamiltonian HQ is d-independent. at the Dirac point oscillates with twist angle, vanishing at a series Physically, the lack of dependence on d can be understood of magic angles which give rise to large DOS and to large counter- by noticing that varying d just shifts the moiré pattern in space. flow conductivities. Many properties of the moiré bands are still The bilayer spectrum does depend on d at θ ¼ 0, and at other not understood. For example, although we are able to explain the commensurate angles. We expect that dependence on d will be largest magic angle analytically, the pattern of magic angles at observable only at short period (large θ) commensurate angles. smaller values of θ has so far been revealed only numerically. Additionally the flattening of the entire lowest moiré band at Discussion θ ≈ 1 05 θ . ° remains a puzzle. Interesting new issues arise when Twisted double-layer graphene is, for most values of , a quasi- our theory is extended to graphene stacks containing three or periodic structure that has no unit cell. Nevertheless, we find that θ ≲ 10 more layers. Finally, we remark that electron-electron interac- for ° it is meaningful to describe the electronic structure of tions neglected in this work are certain to be important at magic the system in terms of Bloch bands. The hidden periodic structure twist angles in neutral systems and could give rise to counterflow is shown to be related to the moiré pattern of the overlaid superfluidity (28, 29), flat-band magnetism (30), or other types of layers (27). ordered states. The leading corrections to the periodic moiré band Hamilto- nian involve hopping amplitudes with the smallest momenta g ACKNOWLEDGMENTS. We acknowledge a helpful conversation with Gene that satisfy the crystal momentum conservation condition in Eq. 5 Mele. This work was supported by Welch Foundation Grant F1473 and by and are larger than kD. As we showed in ref. 19, real space com- the National Science Foundation-Nanoelectronics Research Initiative South mensuration between the two rotated hexagonal lattice is conco- West Academy of Nanoelectronics program.

12236 ∣ www.pnas.org/cgi/doi/10.1073/pnas.1108174108 Bistritzer and MacDonald 1. McCann E, Falko VI (2006) Landau-level degeneracy and quantum hall effect in a 16. Shallcross S, Sharma S, Kandelaki E, Pankratov OA (2010) Electronic structure of graphite bilayer. Phys Rev Lett 96:086805. turbostratic graphene. Phys Rev B 81:165105. 2. Koshino M, Ando T (2007) Orbital diamagnetism in multilayer graphenes: Systematic 17. Shallcross S, Sharma S, Pankratov OA (2008) Quantum interference at the twist study with the effective mass approximation. Phys Rev B 76:085425. boundary in graphene. Phys Rev Lett 101:056803. 3. Nilsson J, Castro Neto AH, Guinea F, Peres NMR (2008) Electronic properties of bilayer 18. Mele EJ (2010) Commensuration and interlayer coherence in twisted bilayer graphene. and multilayer graphene. Phys Rev B 78:045405. Phys Rev B 81:161405(R). 4. Min H, MacDonald AH (2008) Chiral decomposition in the electronic structure of 19. Bistritzer R, MacDonald AH (2010) Transport between twisted graphene layers. Phys graphene multilayers. Phys Rev B 77:155416. Rev B 81:245412. 5. Koshino M, McCann E (2009) Trigonal warping and Berrys phase Nπ in ABC-stacked 20. Morell ES, Correa JD, Vargas P, Pacheco M, Barticevic Z (2010) Flat bands in slightly multilayer graphene. Phys Rev B 80:165409. twisted bilayer graphene: Tight-binding calculations. Phys Rev B 82:121407(R). 6. Zhang F, Sahu B, Min H, MacDonald AH (2010) Band structure of ABC-stacked 21. Castro Neto AH, Guinea F, Peres NMR, Novoselov KS, Geim AK (2009) The electronic graphene trilayers. Phys Rev B 82:035409. properties of graphene. Rev Mod Phys 81:109–162. 7. Wang Y, et al. (2010) Stacking-dependent optical conductivity of bilayer graphene. 22. Pereira VM, Castro Neto AH, Peres NMR (2009) Tight-binding approach to uniaxial ACS Nano 4:4074–4080. strain in graphene. Phys Rev B 80:045401. 8. Ghosh S, et al. (2010) Dimensional crossover of thermal transport in few-layer 23. Tang MS, Wang CZ, Chan CT, Ho KM (1996) Environment-dependent tight-binding graphene. Nat Mater 9:555–558. potential model. Phys Rev B 53:979–982. 9. Rong ZY, Kuiper P (1993) Electronic effects in scanning tunneling microscopy: Moiré 24. Kuzmenko AB, Crassee I, van der Marel D, Blake P, Novoselov KS (2009) Determination pattern on a graphite surface. Phys Rev B 48:17427–17431. of the gate-tunable band gap and tight-binding parameters in bilayer graphene ¯ 10. Hass J, et al. (2008) Why multilayer graphene on 4H-SiCð0001Þ behaves like a single using infrared spectroscopy. Phys Rev B 80:165406. sheet of graphene. Phys Rev Lett 100:125504. 25. Sprinkle M, et al. (2009) First direct observation of a nearly ideal graphene band 11. First PN, et al. (2010) Epitaxial graphenes on silicon carbide. MRS Bull 35:296–305. structure. Phys Rev Lett 103:226803. 12. Schmidt H, Luedtke T, Barthold P, Haug RJ (2010) Mobilities and scattering times in 26. Li G, et al. (2010) Observation of van Hove singularities in twisted graphene layers. decoupled graphene monolayers. Phys Rev B 81:121403(R). Nat Phys 6:109–113. 13. Dobrik G, Tapaszto L, Nemes-Incze P, Lambin Ph, Biro LP (2010) Crystallographically 27. Amidror I (2009) The Theory of the Moiré Phenomenon, (Springer, London), Vol I. oriented high resolution lithography of graphene nanoribbons by STM lithography. 28. Eisenstein JP, MacDonald AH (2004) Bose–Einstein condensation of excitons in bilayer Phys Status Solidi B 247:896–902. electron systems. Nature 432:691–694. 14. Trambly de Laissardière G, Mayou D, Magaud L (2010) Localization of Dirac electrons in 29. Min H, Bistritzer R, Su JJ, MacDonald AH (2008) Room-temperature superfluidity in rotated graphene bilayers. Nano Lett 10:804–808. graphene bilayers. Phys Rev B 78:121401(R). 15. Lopes dos Santos JMB, Peres NMR, Castro Neto AH (2007) Graphene bilayer with a 30. Arita R, Suwa Y, Kuroki K, Aoki H (2002) Gate-induced band ferromagnetism in an twist: Electronic structure. Phys Rev Lett 99:256802. organic polymer. Phys Rev Lett 88:127202. PHYSICS

Bistritzer and MacDonald PNAS ∣ July 26, 2011 ∣ vol. 108 ∣ no. 30 ∣ 12237