Quasicrystalline 30◦ twisted bilayer graphene asan incommensurate superlattice with stronginterlayer couplingWei Yaoa,b, Eryin Wanga,b, Changhua Baoa,b, Yiou Zhangc, Kenan Zhanga,b, Kejie Baoc, Chun Kai Chanc, Chaoyu Chend,Jose Avilad, Maria C. Asensiod,e, Junyi Zhuc,1, and Shuyun Zhoua,b,f,1

aState Key Laboratory of Low Dimensional Quantum Physics, Tsinghua University, Beijing 100084, China; bDepartment of Physics, Tsinghua University,Beijing 100084, China; cDepartment of Physics, The Chinese University of Hong Kong, Hong Kong, China; dSynchrotron SOLEIL, L’Orme des Merisiers, SaintAubin-BP 48, 91192 Gif sur Yvette Cedex, France; eUniversite Paris-Saclay, L’Orme des Merisiers, Saint Aubin-BP 48, 91192 Gif sur Yvette Cedex, France;and fCollaborative Innovation Center of Quantum Matter, Beijing 100084, P. R. China

Edited by Hongjie Dai, Department of Chemistry, Stanford University, Stanford, CA, and approved May 25, 2018 (received for review December 1, 2017)

The interlayer coupling can be used to engineer the electronicstructure of van der Waals heterostructures (superlattices) toobtain properties that are not possible in a single material. Sofar research in heterostructures has been focused on commen-surate superlattices with a long-ranged Moire period. Incom-mensurate heterostructures with rotational symmetry but nottranslational symmetry (in analogy to quasicrystals) are not onlyrare in nature, but also the interlayer interaction has often beenassumed to be negligible due to the lack of phase coherence.Here we report the successful growth of quasicrystalline 30◦

twisted bilayer graphene (30◦-tBLG), which is stabilized by thePt(111) substrate, and reveal its electronic structure. The 30◦-tBLGis confirmed by low energy electron diffraction and the interval-ley double-resonance Raman mode at 1383 cm−1. Moreover, theemergence of mirrored Dirac cones inside the Brillouin zone ofeach graphene layer and a gap opening at the zone boundarysuggest that these two graphene layers are coupled via a gen-eralized Umklapp scattering mechanism—that is, scattering of aDirac cone in one graphene layer by the reciprocal lattice vectorof the other graphene layer. Our work highlights the importantrole of interlayer coupling in incommensurate quasicrystallinesuperlattices, thereby extending band structure engineering toincommensurate superstructures.

twisted bilayer graphene | NanoARPES | incommensurateheterostructure | band structure engineering | quasicrystalline 30◦-tBLG

In van der Waals heterostructures (1), a long-ranged commen-surate Moire superlattice (2–7) forms only when satisfying

matop1 + natop

2 = m ′abottom1 + n ′abottom

2 (8), where atop1,2 and

abottom1,2 are primitive lattice vectors for the top and bottom lay-

ers, respectively, and m , n , m ′, and n ′ are integers. Twistedbilayer graphene (tBLG) can be viewed as a simple version of“hetero”-structure, with atop

1,2 rotated by a twisting angle θt withrespect to abottom

1,2 . In tBLG, commensuration occurs when θtsatisfies (9, 10)

cos θt =3p2 + 3pq + q2/2

3p2 + 3pq + q2, [1]

where p and q are two coprime positive integers. Fig. 1A plotsthe possible twisting angles for forming commensurate tBLGfor q = 1. It is clear that commensuration easily forms around0◦, while for larger twist angles, especially around 30◦, incom-mensurate superlattice is more common (11). In contrast tocommensurate superlattice with a long-ranged period (see exam-ple in Fig. 1B), incommensurate superlattice lacks a long-rangedtranslational symmetry in real space while preserving the rota-tional symmetry; for example, 30◦-tBLG (Fig. 1C) is similar toquasicrystals with a classic dodecagonal pattern (12).

Although the electronic structures of commensurate het-erostructures have been investigated in recent years (13–17),research on incommensurate heterostructures remains limitedfor two reasons. On one hand, incommensurate heterostruc-tures are difficult to be stabilized, and thus, they are quiterare under natural growth conditions. On the other hand,it is usually assumed that the interlayer interaction is sup-pressed due to the lack of phase coherence. For example,incommensurate tBLG has often been taken as a trivial com-bination of two noninteracting graphene layers (18–20). Howthe electronic structure is modified in such incommensu-rate heterostructures, with symmetry analogous to quasicrys-talline order, is a fundamental question. Here by growingepitaxial 30◦-tBLG successfully on Pt(111) substrate and reveal-ing its electronic structure using angle-resolved photoemis-sion spectroscopy (ARPES) and NanoARPES, we show thatthe interlayer coupling can modify the electronic structuresignificantly, leading to mirrored Dirac cones. Additionally,to understand the intriguing stability that is against com-mon belief on bilayer graphene, we performed first-principlescalculations and found that such incommensurate 30◦-tBLG

Significance

Interlayer interaction in van der Waals heterostructurescould induce many exotic phenomena. Unlike commensu-rate heterostructures, incommensurate ones have often beenneglected, due to the rarity in nature and the assumption ofsuppressed coherent interlayer movement of electrons. Howthe interlayer interaction affects the electronic structure ofsuch a nonperiodic heterostructure is a fundamental ques-tion. In this work, by successfully growing 30◦-twisted bilayergraphene (tBLG) as an example for incommensurate “het-erostructure” with quasicrystalline order, we report the emer-gence of mirrored Dirac cones. We identify that these mirroredDirac cones are a consequence of the interlayer interaction,showing its importance in the incommensurate structure,which has been overlooked before. These results broaden theapplication range of van der Waals heterostructure for futureelectronic devices.

Author contributions: S.Z. designed research; W.Y., E.W., C.B., Y.Z., K.Z., K.B., C.K.C., C.C.,J.A., M.C.A., and J.Z. performed research; W.Y. and S.Z. analyzed data; and W.Y., J.Z., andS.Z. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

Published under the PNAS license.1 To whom correspondence may be addressed. Email: syzhou@mail.tsinghua.edu.cn orjyzhu@phy.cuhk.edu.hk.y

This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1720865115/-/DCSupplemental.

Published online June 18, 2018.

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0 60

incommensuratecommensurate

commensurate superlattice

incommensurate heterostructure (Quasicrystalline)

A

B

C

Fig. 1. Commensurate and incommensurate tBLG. (A) The distribution ofall possible twisting angles θt for forming commensurate tBLG when q = 1in Eq. 1. The Moire period at corresponding twisting angles is representedby the length of the solid line (with logarithmic scale). (B) CommensuratetBLG with 7.34◦ twisting angle (indicated by the blue arrow in A), show-ing Moire pattern with periodic translational symmetry. Green arrows arethe Moire reciprocal lattice vectors. (C) Incommensurate 30◦-tBLG withoutany long-ranged period (indicated by red arrow in A), showing patterns ofdodecagonal quasicrystal.

heterostructure is stabilized under appropriate substrateconditions.

ResultsGrowth of 30◦-tBLG. The 30◦-tBLG sample was grown on thePt(111) substrate by carbon segregation from the bulk sub-strate (21, 22). Fig. 2A shows the low energy electron diffraction(LEED) of bottom monolayer graphene at 30◦ azimuthal ori-entation (blue arrow) relative to the substrate. Our previous

work shows that distinguished from other graphene/Pt with com-mensurate (e.g., 2 × 2, 3 × 3) Moire superstructures (21), theinterface between the 30◦ monolayer graphene and the Pt(111)substrate is incommensurate without forming any Moire pattern,leading to nearly free-standing monolayer graphene (22). Fur-ther increasing the annealing temperature and time can lead toa thicker graphene sample with a new set of diffraction peaksemerging at 0◦ orientation (red arrow in Fig. 2B) coexisting withthose at 30◦, and such a graphene sample is the focus of the cur-rent work. The lattice constants extracted from LEED show thatthere is negligible strain (<0.2%; see SI Appendix, Fig. S1 fordetails) between the 30◦ and 0◦ graphene layers, and the absenceof additional reconstructed diffraction spots in the LEED pat-tern further supports that they do not form commensurate super-lattice. If these diffraction peaks come from the same graphenedomains, this would imply that 0◦ graphene is stacked on top ofthe bottom 30◦ graphene layer—namely, a 30◦-tBLG is formed.In the following, we will provide direct experimental evidencefor the conjectured incommensurate 30◦-tBLG from Ramanspectroscopy and reveal its electronic structure from ARPESmeasurements.

Intervalley Double-Resonance (DR) Raman Mode. Raman spec-troscopy is a powerful tool for characterizing the vibrationalmode in graphene (23) and can provide direct information aboutthe sample thickness and stacking. Fig. 2G shows a typical opticalimage of the as-grown sample. The optical image shows strongintensity contrast, with a darker region of ≈10 µm overlappingwith the brighter region. The Raman spectra in Fig. 2C showstronger intensity for the darker region (red curve) than thebrighter region (blue curve), suggesting that the darker regionis thicker. The spectra for both regions show characteristic fea-tures of monolayer graphene (23), in which the 2D mode shows asingle Lorentzian peak with stronger intensity than the G mode.This suggests that the top and bottom flakes are both monolayergraphene, and the thicker region is not a Bernal (AB stack-ing) bilayer graphene but is rather a bilayer graphene with a

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-QphononhνE

hνS

Ktop Mbottom MbottomKtop

Pt

A

B G H

C D

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'

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Gbottom

Qphonon

Fig. 2. Observation of DR Raman mode in 30◦-tBLG. (A) The LEED pattern for the graphene sample after annealing at 1,200 ◦C measured at electronbeam energy of 135 eV. (B) The LEED pattern for the graphene sample at the higher annealing temperature of 1,600 ◦C. (C) Measured Raman spectrumfor monolayer and bilayer region, respectively, covering the range of G peak and the 2D peak. (D) Zoom in of C for a range from 1,300 to 1,430 cm−1. (E)The geometry for electron momentum transferring in the DR Raman process of the observed R mode. (F) The schematic drawing for the DR process of Rmode in graphene band structure (not in scale). An electron at one Dirac cone from the top layer is photo-excited to the conduction band, scattered byone reciprocal lattice vector of the bottom layer q = Gbottom, and subsequently scattered back by phonon with vector Qphonon. (G) The optical image for themeasured area. The red and blue dots indicate the measuring positions of spectra in C and D. (H) The Raman map for the D peak, integrating intensity from1,343 to 1,363 cm−1. (Scale bar: 3 µm.) (I) The Raman map for the R peak, integrating intensity from 1,373 to 1,393 cm−1. (Scale bar: 3 µm.) (J) The phononspectrum of the TO mode along high symmetric direction, taken from ref. 26. The arrow indicates the momentum of transferring R mode phonon. Insetshows the high-symmetry path in k-space for the phonon spectrum. The left hexagon is the first Brillouin zone (BZ).

Yao et al. PNAS | July 3, 2018 | vol. 115 | no. 27 | 6929

large twisting angle instead (24). More importantly, when zoom-ing in the spectra between 1,300 and 1,430 cm−1, the bilayerregion shows two peaks centered at 1,353 cm−1 and 1,383 cm−1,respectively (Fig. 2D). The peak at 1,353 cm−1 is the D modeof graphene, which is caused by the limited size or defects(23). The Raman mapping for this peak (Fig. 2H) shows thatit only appears at the edges of the bilayer region or defectcenters, consistent with the nature of the D peak. In contrast,the peak at 1,383 cm−1 (“R” peak) is observed in the wholebilayer graphene region (Fig. 2I), indicating that it is intrinsic tothe tBLG.

Since the energy of the R mode is close to that of the D mode,the R mode likely has a similar origin as the D mode: intervalleyDR Raman process (25, 26). The key process for the interval-ley DR Raman process involves a special scattering process (bydefect, phonon, or Moire pattern) in which photoexcited elec-trons at one valley are scattered to another valley by a nonzeromomentum transfer q and subsequently decay back to the orig-inal valley by emitting a phonon with wave vector Qphonon =q (see Fig. 2F). While D mode is induced by defect scatteringbetween two neighboring Dirac cones, instead we claim that theR mode in our case is caused by scattering between two oppositeDirac cones that are connected by one reciprocal lattice vec-tor of the bottom graphene layer with q = Gbottom (see Fig.2 E and F). This is verified by observing a phonon mode withmatching momentum and energy in the transverse optical (TO)phonon spectrum along the Γ-K-M-K′-Γ direction (see Fig. 2J).The phonon momentum of the R mode is not a Moire recipro-cal vector of any commensurate superlattice, suggesting that thebilayer region is an incommensurate 30◦-tBLG. It also revealsan unreported scattering process involving the reciprocal latticevector of the bottom layer in this interesting 30◦-tBLG region.This novel scattering process for the observed R mode is closelyrelated to the interlayer interaction to be discussed below andis supported by the exotic electronic structure of the 30◦-tBLG,which we will investigate next.

Observation of Mirrored Dirac Cone. The electronic structure ofthe sample was first measured by conventional ARPES with abeam size of ∼100 µm. Fig. 3A shows the ARPES intensitymap at −0.8 eV. Characteristic conical contours appear at the Kpoints of the graphene Brillouin zone (BZ) for both 0◦ and 30◦

layers (labeled as K0◦ and K30◦), in agreement with the coex-istence of 0◦- and 30◦-graphene layers observed in the LEEDpattern. Additional contours with weaker intensity are detectedinside the first BZ near K0◦ (green arrows in Fig. 3B) whenenhancing the weak features using the logarithmic scale. Thecurvature images (Fig. 3C) at different energies with higher visi-bility show that these contours expand from low to high bindingenergy similar to the Dirac cones at K0◦ , and they occur at thereflected position of K0◦ (indicated by a green arrow and labeledas KR) with respect to the 30◦ BZ edges (blue lines). The dis-persion near the KR also shows linear behavior, which is thereflected image of the Dirac cone at K0◦ (Fig. 3D). The mir-rored Dirac cone shows similar intensity asymmetry caused bythe dipole matrix element (27), with stronger intensity betweenK0◦ (KR) and the M point of the 30◦-graphene layer M30◦ ,suggesting that it comes from the original Dirac cone at theopposite momentum valley and is strongly related to the trans-fer mechanism discussed above for the DR Raman mode (moredetails are shown in Discussion). Moreover, the suppression ofintensity at the crossing point M30◦ between the original andmirrored Dirac cones indicates a gap opening (pointed by blackarrows in Fig. 3D). Such mirrored Dirac cones and gap open-ing are not observed in the bottom monolayer 30◦-graphene(see SI Appendix for details). Furthermore, the gap happens toappear at the BZ boundary of the bottom layer and is awayfrom the substrate bands (see SI Appendix for details), suggesting

-2.0

-1.5

-1.0

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0.0

E-E

F (e

V)

1.81.61.41.2k// (Å

-1)

KR M30° K0°

-1.0 -0.5 0.0

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k y (Å

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kx (Å-1)

BZ(0°)

BZ(replica)BZ(30°) M30°

kykx

K0°

KRE

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1.0

0.5

k y (Å

-1)

-0.8 eV K0°

K30°High

Low

K0°M30°KR

A B

C

D E

Fig. 3. Observation of mirrored Dirac cone in 30◦-tBLG. (A) Intensity mapat −0.8 eV. The BZs of 0◦-graphene and 30◦-graphene are indicated by thedashed lines. (B) The same map as A but with the logarithmic scale. Greenarrows indicate the traces of replica Dirac cones. (C) Schematic drawing ofthe BZ (Left) and the 2D curvature image of constant energy maps for theregion marked by the rectangle at energies from EF to −0.8 eV (Right). TheBZ boundary of 0◦, 30◦, and replica bands are shown as red, blue, and greenlines. (D) The measured Dirac-type dispersion near the K point of the 0◦

orientation. The three vertical arrows indicate the K point of 0◦-graphene,the M point of 30◦-graphene, and the K point of the replica BZ. The gapis indicated by the horizontal arrows. (E) Schematic illustration showing themirrored relation between the original and the emerging replica Dirac cone.

that they are caused not by the graphene–substrate interac-tion but by intrinsic properties of the 30◦-tBLG. Therefore,our ARPES results confirm that the 30◦ graphene layer spa-tially overlaps with the 0◦ graphene layer and they interact witheach other.

The Band Structures with Spatial Resolution. The electronic struc-ture of 30◦-tBLG is further confirmed by NanoARPES (28),which is capable of resolving the electronic band structure ofcoexisting graphene structures with spatial resolution at the≈120 nm scale. A spatially resolved intensity map is shown inFig. 4A, where we can distinguish different regions and map outtheir distinct electronic structure. Region labeled by B shows dis-persion along the Γ-M direction of the 30◦ bottom graphenelayer (Fig. 4B), while region C shows dispersions from the 30◦-tBLG, with dispersion along the Γ-K direction of the 0◦ topgraphene layer coexisting with that along the Γ-M direction ofthe bottom 30◦ graphene layer (Fig. 4C). Therefore, dispersionsmeasured by NanoARPES with spatial resolution provide defini-tive experimental evidence for the 30◦-tBLG, and the mirroredDirac cones indeed originate from this newly discovered tBLG

6930 | www.pnas.org/cgi/doi/10.1073/pnas.1720865115 Yao et al.

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1.81.61.41.21.0k// (Å

-1)

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B C DM30° M30° K0°KR

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(a.u

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-4 -3 -2 -1 0E-EF (eV)

k1

k13

k1 k13

B

C

B

C

Fig. 4. Confirmation of 30◦-tBLG and the electronicstructure by NanoARPES measurements. (A) Spatialintensity map obtained by integrating the energyfrom −1.5 eV to EF . The structures for regions B andC are schematically shown on Right. (B and C) Dis-persion cuts taken at regions marked by B and C inA. (D) The energy distribution curves (EDCs) taken inthe momentum range marked by the arrows in C.The EDC through the cross point of the original andmirrored bands is shown by the red color. Blue andgreen circles mark the peak positions.

structure. Moreover, by focusing only in the 30◦-tBLG region,sharper dispersions can be obtained, and the gap at the M pointbecomes more obvious. Analysis from the EDCs for 30◦-tBLG(Fig. 4D) shows a gap at the M30◦ point, suggesting the Diraccone at K0◦ and the mirrored Dirac cone are hybridized.

Band Gap in k Space. To track the evolution of the gap at thecrossing points between the original Dirac cone at K0◦ and themirrored Dirac cone at KR, we show in Fig. 5A dispersions mea-sured from cuts 1 to 5 (labeled in Fig. 5B). The gap size isquantified by the peak separation of the EDCs at the crossing

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Experiment Fitting Curve

A

B C D E

large

smallKR

K0°

M30°

1 2 3 4 5

510= 3.10 eVF= 1.00×106 m/s

Fig. 5. Evolution of gap size and gap position in k space. (A) Five different cuts located at different positions in the k space shown in B. The EDCs crossingthe gap are shown on the right of the spectrum as red symbols. The fitting curves are plotted as blue lines, and green curves are the two Lorentzian peaksfrom fitting results. Black markers show the positions of the two peaks. (B) Schematic BZ showing the positions of the five cuts in A. (C) The gap size alongthe ky direction. (D) The gap position along the ky direction. The green line is the fitting curve by the tight-binding model. (E) Cartoon diagram for the twoDirac cones and their intersecting line with a gap. The color indicates the gap size.

Yao et al. PNAS | July 3, 2018 | vol. 115 | no. 27 | 6931

points, and the extracted value is plotted in Fig. 5C. It is clear thatthe gap size decreases from the maximum value of ∼280 meVat the M30◦ point to ∼200 meV when deviating from the M30◦

point. In addition, we can define the gap position as the aver-age of the two peak positions. The gap position curve in Fig.5D shows that the gap is located at the intersecting line of thetwo Dirac cones; hence, it is possible to fit the gap positionsby the energy dispersion obtained from a tight-binding model(see SI Appendix for details). The fitting curve (green curvein Fig. 5D) is in good agreement with the experimental data,giving a nearest neighbor hopping parameter γ0 of 3.10 eV,which is closed to the value derived from theoretical calcula-tions (29). The Fermi velocity νF =

√3aγ0/2 (a ∼ 2.46 A) is

extracted to be 1.003±0.002× 106 m/s. A schematic summary ofthe gap between the original and mirrored Dirac cones is shownin Fig. 5E.

Theoretical Calculation. Ab initio calculations are performed toreveal the electronic structure of 30◦-tBLG and investigate thestability. Since Bloch theorem does not apply to the nonperiodicstructure and an infinitely large cell is beyond the calcula-tion capability, a finite supercell of (5× 5)/(3

√3× 3

√3)R30◦

is constructed to mimic the nonperiodic 30◦-tBLG system. Thecalculated electronic structure in Fig. 6A shows the band gapopening at the M point of the counterpart layer, with the gapvalue close to the experimental measurements. An interest-ing observation regards the stability of 30◦-tBLG when the Ptsubstrate is included in the calculation. While the formationenergy of free-standing 30◦-tBLG is 1.6 meV/(C atom) higherthan the more common AB-stacking BLG, inclusion of the sub-strate leads to formation energy of 4.1 meV/(C atom) lowerthan AB-stacking one, which explains why 30◦-tBLG emergesin our sample and suggests an important role of the Pt sub-strate in stabilizing the 30◦-tBLG. By investigating real-spaceprojection of the Bloch states, a significant coupling betweenthe carbon p orbital and platinum d orbital (shown in Fig. 6B)is revealed near Fermi level. Since such p–d coupling occurs

A B

180 meV117 meV

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Fig. 6. Electronic structure of 30◦-tBLG from first-principles calculations.(A) Band structure of a (5× 5)/(3

√3× 3

√3)R30◦-tBLG in the extended zone

scheme. The energy gap of π-band at the M point of the other layer isindicated. (B) Real-space projection of the Bloch state at Γ point near theFermi level, showing a clear bonding state between pz orbital of C (brownballs) and d orbital of Pt (white balls). (C) Schematic illustration for the bandstructure of 30◦-tBLG, showing the emergence of mirrored Dirac cones.

at the Γ point, and phase matching between graphene and thePt substrate is less critical (due to zero k -vector). Therefore, itcould thus exist in systems with weak and incoherent sample–substrate interaction such as our sample, contributing positivelyto the stability of grown 30◦-tBLG on the Pt substrate. Extrap-olating to a larger supercell, like 5 × 5, 7 × 7, and 16 × 16, thestability of 30◦-tBLG is further enhanced (see SI Appendix fordetails), and the electronic structure remains similar, suggest-ing that the calculation conclusion mentioned above still holdsat the limit of an infinitely large cell or the incommensuratestructure. Our calculation, although not perfect, still providessome theoretical insights into the physics and stability of the30◦-tBLG.

DiscussionUnlike the satellite Dirac cones appearing in the commensu-rate graphene system with long-ranged Moire pattern (30–32),the emergence of mirrored Dirac cones in incommensurate30◦-tBLG indicates an unusual scattering mechanism in thisstructure. Based on a generalized Umklapp scattering process(33), we build up a tight-binding model with second-order pertur-bation further included (see SI Appendix for details) and reveala general coupling condition for two Bloch states in one layer ofthe twisted bilayer system:

ku1 = ku

2 −Gu + Gd [2]

where ku1 and ku

2 are wave vectors of the two Bloch states inone layer, and Gu and Gd are the reciprocal lattice vectorsof the same layer and its counterpart, respectively. Applyingthis condition, we propose a scattering mechanism for elec-trons in 30◦-tBLG: Dirac cone at the K′ point of one graphenelayer K′0◦ is scattered by one reciprocal lattice vector of theother layer G30◦ , forming a mirrored Dirac cone KR near theopposite momentum valley at K0◦ as schematically shown inFig. 6C. This corresponds to the simplest case in Eq. 2 forGu = 0 and Gd = G30◦ . Such a mechanism is confirmed by threeexperimental observations. First of all, the momentum of themirrored Dirac point KR is connected to the other graphenevalley K′0◦ by just one reciprocal lattice vector of the other30◦ layer G30◦ . Second, the asymmetry of the intensity con-tour for the mirrored Dirac cone is identical to the one at K′

(see Fig. 3D or Fig. 4C), suggesting that the mirrored Diraccone is scattered from the K′ point. Third, such a scatteringmechanism is fully consistent with the observed R mode inRaman spectrum. Not only for the incommensurate structure,this scattering mechanism is also able to account for the appear-ance of replica Dirac cones in those commensurate graphenesystems following the same analysis (33). Thus, this mecha-nism can be applied to other van der Waals heterostructuresbeyond tBLG.

ConclusionsIn summary, we have successfully grown 30◦-tBLG, a typicalexample for incommensurate superlattice with quasicrytallineorder. The realization of 30◦-tBLG provides opportunities forinvestigating the intriguing physics of quasicrystalline superlat-tice. Moreover, by revealing the mirrored Dirac cones in a 30◦-tBLG, we provide direct experimental evidence for the stronginterlayer coupling through a coherent scattering process in suchan incommensurate superlattice. Such a scattering mechanismcan be applied to engineer the band structure of both commen-surate and incommensurate tBLG as well as other van der Waalsheterostructures.

Materials and MethodsSample Growth. The graphene sample was obtained by annealing onePt(111) substrate to about 1,600 ◦C. At this high temperature, the carbon

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impurities would segregate from the bulk to the surface, forminggraphene film.

ARPES and NanoARPES. Conventional ARPES measurements were performedat the home laboratory with a Helium discharge lamp and beamline 10.0.1of Advanced Light Source (ALS) with synchrotron radiation source. TheNanoARPES measurements were performed at the analysis nano-spot angle-resolved photoemission spectroscopy (ANTARES) endstation of SynchrotronSOLEIL, in France, with a lateral spatial resolution of 120 nm. The sampletemperature was kept at 20 K for measurements at ALS and 40 K for thoseat SOLEIL.

Calculations. The calculations were based on VASP code (34, 35) withplane wave basis set (36, 37). We adopted opt86b-vdW functional (38,39) to include the van der Waals interactions. A periodic (5 × 5) on the(3√

3×3√

3)R30◦ structure was created to simulate the 30◦-tBLG becauseit is impossible to directly simulate the very large nonperiodic bilayer. Thelower layer is slightly compressed about 1.8% to fit the Pt lattice, andthe upper layer is slightly stretched about 2% to partially compensate for

the compression from the lower layer. Extensive convergence tests havebeen performed in terms of vacuum, film thickness, and k-point sampling.Although our calculations were based on one periodic structure, such effectswould be extrapolated to nonperiodic structures of 30◦-tBLG.

ACKNOWLEDGMENTS. This work is supported by Ministry of Science andTechnology of China Grants 2016YFA0303004 and 2015CB921001, NationalNatural Science Foundation of China Grants 11334006 and 11725418, Sci-ence Challenge Project 20164500122, and Beijing Advanced InnovationCenter for Future Chip (ICFC). The theoretical work is supported by theGeneral Research Fund (Grant 2130490) from Research Grants Council inHong Kong. The Synchrotron SOLEIL is supported by the Center Nationalde la Recherche Scientifique (CNRS) and the Commissariat a l′Energie Atom-ique et aux Energies Alternatives (CEA), France. This work is supported bya public grant overseen by the French National Research Agency (ANR) aspart of the “Investissements d′Avenir” program (Labex NanoSaclay, refer-ence ANR-10-LABX-0035) as well as the French Ministre des affaires trangreset europennes (MAEE), the Center National de la Recherche Scientifique(CNRS) through Information and Communication Technology (ICT)-ASIAProgram Grant 3226/DGM/ATT/RECH.

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