# Fileset [s41467-024-46672-3.pdf](https://mdr.nims.go.jp/filesets/a9ac6539-b9e3-414c-ae66-9cc6fa0ad1cd/download) ## Creator Mohit Kumar Jat, Priya Tiwari, Robin Bajaj, Ishita Shitut, Shinjan Mandal, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), H. R. Krishnamurthy, Manish Jain, Aveek Bid ## Rights [Creative Commons BY Attribution 4.0 International](https://creativecommons.org/licenses/by/4.0/) ## Other metadata [Higher order gaps in the renormalized band structure of doubly aligned hBN/bilayer graphene moiré superlattice](https://mdr.nims.go.jp/datasets/9b8fbf02-b910-4d01-9422-167397fb83ba) ## Fulltext Higher order gaps in the renormalized band structure of doubly aligned hBN/bilayer graphene moiré superlatticeArticle https://doi.org/10.1038/s41467-024-46672-3Higher order gaps in the renormalized bandstructure of doubly aligned hBN/bilayergraphene moiré superlatticeMohit Kumar Jat1,5, Priya Tiwari2,5, Robin Bajaj1,5, Ishita Shitut1, ShinjanMandal 1,Kenji Watanabe 3, Takashi Taniguchi 4, H. R. Krishnamurthy1,Manish Jain 1 & Aveek Bid 1This paper presents our findings on the recursive band gap engineering ofchiral fermions in bilayer graphene doubly aligned with hBN. Using twointerfering moiré potentials, we generate a supermoiré pattern that renor-malizes the electronic bands of the pristine bilayer graphene, resulting inhigher order fractal gaps even at very low energies. These Bragg gaps can bemappedusing aunique linear combination of periodic areaswithin the system.To validate our findings, we use electronic transportmeasurements to identifythe position of these gaps as a function of the carrier density. We establishtheir agreement with the predicted carrier densities and correspondingquantum numbers obtained using the continuum model. Our study providesstrong evidence of the quantization of the momentum-space area of quasi-Brillouin zones in aminimally incommensurate lattice. It fills important gaps inthe understanding of band structure engineering of Dirac fermions with adoubly periodic superlattice spinor potential.Heterostructures of graphene encapsulated between two thin, rota-tionally misaligned hBN flakes form a stimulating platform for probingtopological phases of matter1–6. The difference in the lattice constantsof hBN and graphene and the angularmisalignment between the layersgenerate two distinct long-wavelength moiré superlattices at the topand bottom interfaces of graphene with hBN7–11. The interferencebetween these patterns forms a supermoiré structure with multiplecomplex real-space periodicities, often with a spatial range larger thanthat of hBN/graphene moiré at each interface12–20. The supermoirépotential (caused by atomic scale modulation of the carbon-carbonhopping amplitudes by the spinor graphene-hBN interactionpotential)effectively folds the graphene band over a smaller Brillouin zone whileretaining the symmetries of the honeycomb lattice21. To first-order,this results in additional, finite-energy split moiré gaps (SMG) in thegraphene dispersion2,7,13,16,22–26. It was recently realized that thesuperlattice-induced Bragg reflection at the mini Brillouin zoneboundaries has additional subtler effects on the electronic dispersionof graphene to arbitrary low energies manifested in the formation of afamily of Bragg gaps, van Hove singularities, and even possibly flatbands13,15,27. Studying these high-order mini-bands and van Hove sin-gularities in graphene/hBNmoiré superlattice is essential for a detailedunderstanding of the emergent quantum properties ofquasicrystals15,28,29 and Dirac fermions in a periodic non-scalarpotential2,16.Recent momentum-space low-energy continuum model calcula-tions (valid in the low-energy regime of interest13,30,31) predict that thepositions of these Bragg gaps form a fractal pattern reminiscent of theHofstadter butterfly14. Consequently, the number density of chargecarriers at which Bragg scattering (with supermoiré harmonics) occurscan be described by a unique set of Bragg indices (quantumnumbers)14. These indices, which are integers, relate directly to thefilled bands below the gaps and are associated with the quasi-BrillouinReceived: 4 May 2023Accepted: 27 February 2024Check for updates1Department of Physics, Indian Institute of Science, Bangalore 560012, India. 2Braun Center for Submicron Research, Department of Condensed MatterPhysics, Weizmann Institute of Science, Rehovot, Israel. 3Research Center for Electronic and Optical Materials, National Institute for Materials Science, 1-1Namiki, Tsukuba305-0044, Japan. 4ResearchCenter forMaterialsNanoarchitectonics, National Institute forMaterials Science, 1-1Namiki, Tsukuba305-0044,Japan. 5These authors contributed equally: Mohit Kumar Jat, Priya Tiwari, Robin Bajaj. e-mail: mjain@iisc.ac.in; aveek@iisc.ac.inNature Communications | (2024) 15:2335 11234567890():,;1234567890():,;http://orcid.org/0000-0003-1551-6555http://orcid.org/0000-0003-1551-6555http://orcid.org/0000-0003-1551-6555http://orcid.org/0000-0003-1551-6555http://orcid.org/0000-0003-1551-6555http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0001-9329-6434http://orcid.org/0000-0001-9329-6434http://orcid.org/0000-0001-9329-6434http://orcid.org/0000-0001-9329-6434http://orcid.org/0000-0001-9329-6434http://orcid.org/0000-0002-2378-7980http://orcid.org/0000-0002-2378-7980http://orcid.org/0000-0002-2378-7980http://orcid.org/0000-0002-2378-7980http://orcid.org/0000-0002-2378-7980http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-46672-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-46672-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-46672-3&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-46672-3&domain=pdfmailto:mjain@iisc.ac.inmailto:aveek@iisc.ac.inZones (qBZ) formed by the multiple reciprocal lattice vectors of thesupermoiré lattice. These indices are topological invariants of thesystem intimately related to the second Chern numbers14,32. Addition-ally, these minimally incommensurate moiré lattices form an idealplatform to probe the topological properties of quasicrystals. Despiteconcrete theoretical predictions, this aspect of moiré superlatticeremains experimentally unexplored.Here, we experimentally probe these characteristics of a quasi-periodic lattice using high-mobility heterostructures of bilayer gra-phene (BLG) doubly aligned with hBN as a model system. Fromcombined measurements of quantum oscillations, longitudinal resis-tance Rxx and transverse resistance Rxy of Dirac fermions in thissupermoiré potential, we observe and identify a multitude of higherorder Bragg gaps and van Hove singularities of the supermoiré struc-ture; these had escaped detection in previous studies13–18,33. We mapthese gaps uniquely to the recently predicted topological Bragg indi-ces of the underlying supermoiré lattice28. Furthermore, our con-tinuum modeling of the system shows these zone quantum numbersto have an elegant physical interpretation based on the quantizedareas of the qBZ at these Bragg gaps. This model explains the Bragggaps corresponding to the linear combinations of moiré reciprocallattice vectors, pGb1 +qGb2 + rGt1 + sGt2. Additionally, our analysisexplains several unexplained experimental features in graphene/hBNsupermoiré systems reported in recent publications17 (SupplementaryNote 8), which were previously studied based on symmetry-basedapproach26.We demonstrate that the BLG supermoiré is different from itssingle-layer counterpart in several critical aspects – for example, in thesymmetry of the moiré Brillouin zone, which has direct consequencesfor the anomalous Hall effect34 and electron-electron scattering35,36, interms of the positions and magnitudes of the higher order Bragg gaps(SupplementaryNote 11). Additionally, the ability to electrically controlthe layer and valley degrees of freedom in BLG promises exotic phasesthat are absent in its single-layer counterpart, e.g. electric fieldswitchable Chern insulators37.ResultsDevice characteristicsHeterostructures of BLGdoubly alignedwith hBNwith twist angles lessthan 0. 5° were fabricated using the dry transfer technique38,39 (seeSupplementary Note 1). The device is in a dual-gated field-effect tran-sistor architecture, allowing independent control on the charge carrierdensity n and displacement field D via n = [(CtgVtg +CbgVbg)/e + n0] andD = [(CbgVbg −CtgVtg)/2 +D0] across the device. Here Cbg (Ctg) is theback-gate (top-gate) capacitance, and Vbg (Vtg) is the back-gate (top-gate) voltage. The values of Ctg and Cbg are determined from quantumHall measurements. n0 and D0 are the residual charge carrier densityand displacement field due to channel impurities, respectively. A plotof the longitudinal resistance Rxxmeasured atD =0 and zeromagneticfield is shown in Fig. 1c. The appearance of split moiré resistance peaksat nb = ± 2.36 × 1016m−2 and nt = ± 2.80× 1016m−2 indicates the alignmentof the BLG with both the bottom and top hBN layers. Their presence isalso apparent in the 2D map of Rxx in the Vbg −Vtg plane (Fig. 1d).Figure 1e shows the 2Dmap of Gxx(n, B) at D =0 V/nm in the n −Bplane – one finds Landau fans emerging from the charge neutralitypoint (CNP) and from the secondaryDiracpointsnb andntwith Landaufilling ν = ± 4m (m∈ integer) (Supplementary Note 9). The faint hor-izontal streaks in the plot are the Brown-Zak oscillations originatingfrom the recurring Bloch states in the superlattice40,41. These featuresget accentuated at high temperatures, where thermal smearingaDnV bg (V)6040200-20-40-60Vtg (V)c103102Rxx (�)VbgRxxRxyIb d0.03 Rxx(k��8.5VbgVtghBNBLGDhBNB = 0 TB = 0 T0-4 -2 42 6-6 -20 -10 0 10 20�54321-3 -2 -1 0 1 2 3 n (1016 m-2 )n (1016 m-2 )B (T) e 0.5 100 Gxx (e2/h)-nt ntnCNP-nb nb-nb nbnCNP-nt nt13.97 nm12.84 nm5 �mFig. 1 | Characteristics of the doublemoiré device. a Schematic of doubly alignedBLGwith top and bottomhBN. The black and thewhite hexagonsmark the primarymoiré and supermoiré plaquettes, respectively. b An optical image of the device(before adding the top gate) labeled with the measurement configuration (scalebar: 5 μm). Top inset: Schematic of the layer-stacking, with the direction ofincreasing displacement field D marked. c Plot of the longitudinal resistanceRxx(B =0) as a function of n. The black dashed line marks the charge neutralitypoint. Magenta and dark green lines indicate the secondary Dirac points emergingfrom top and bottom moiré respectively, with carrier density (moiré wavelength)nt = ± 2.80× 1016m−2 (λt = 12.84 nm) and nb = ± 2.36 × 1016m−2 (λb = 13.97 nm),respectively. Yellow and blue arrows mark the higher order Bragg gaps at carrierdensity n = − 3.3 × 1016m−2 and n = − 4.8 × 1016m−2, respectively. d Map of Rxx as afunction of the back gate voltage, Vbg and top gate voltage, Vtg. The black andwhitearrows indicate the directions of increasing D and n, respectively. The color of thesmall arrows has the same interpretation as in c. e Landau-fan diagram Gxx(n, B)showing the emergence of Landau levels from the primary Dirac point and the twosecondary Dirac points. The cyan horizontal arrows on the right of the plot markthe weak Brown-Zak features. The measurements were done at T = 2 K.Article https://doi.org/10.1038/s41467-024-46672-3Nature Communications | (2024) 15:2335 2diminishes the effect of Landau quantization on the magnetotran-sport. This is seen clearly from Fig. 2a, which presents the magneto-conductance ΔGxx(B) plotted in the n − 1/B plane; the data weremeasured at 100K. The Fourier transform of a representative datameasured at n = 3.3 × 1016 m−2 (Fig. 2b) yields multiple frequenciesf = 24.5T, 29T, and 4T (Fig. 2c). f is related to the real-space area Sof thesuperlattice by f =ϕ0/S, where ϕ0 = h/e is the flux quantum42–45. Thecarrier densities (considering two-fold spin and two-fold valleydegeneracies) that fill the two first-order moiré Bloch bands are cal-culated from f = 24.5 T and 29T to be 2.36 × 1016m−2 and 2.80× 1016m−2.These number densities match nb and nt exactly, identifying these twooscillation frequencies to be associated with the moiré supercellformed at the bottom and top interfaces of BLG, respectively (Sup-plementary Note 2). The corresponding moiré wavelengths areλb = 13.97 nmand λt = 12.84nm, respectively. TheBrown-Zak frequencyfs = 4 T yields ns =0.39 × 1016m−2 – this number density corresponds toa real-space wavelength of λs = 34.6 nm which is the size of the super-moiré unit cell in our heterostructure (SupplementaryNote 2).We thusidentify fs to be the supermoiré Brown-Zak frequency.To verify that the split peaks at nb and nt are not artifacts due tolarge angle-inhomogeneity in the device, we repeated the measure-ments on a control device (labeled Dsingle) where only the top-hBNforms amoiré with the BLG (Supplementary Note 2). To achieve this, asingle-layer WSe2 was interposed between half of the BLG and thelower hBN. The n −Rxx plot of this single-moiré device had a singlesecondary peak at n = nt (See Supplementary Fig. 3). This helps ascer-tain that both the top- and bottom-hBN crystals have the same relativerotation direction with the intervening graphene layer for the double-moiré device, with twist angles θb = 0.03° ± 0.03° betweenbottomhBNand graphene and θt =0.44° ± 0.03° between top hBN and graphene(SupplementaryNote 2). The very small values of the twist angles placeour device in the commensurate limit46.Continuum HamiltonianHaving established the presenceof the supermoiré structure, wemoveon to discuss its effect on the bilayer graphene band structure usingthe Bistritzer-MacDonald continuum model47. The 4 × 4 effectiveHamiltonian (eliminating the sub lattice basis of hBN using second-order perturbation theory) is written as:Hef f =HG +VbhBN UyBLGUBLG HG +VthBN" #ð1Þwhere, in the low-energy limit,V ‘hBN =U‘yð�HhBNÞ�1U‘ = v0 + v1eiξG‘1 :r + v2eiξG‘2 :r + v3eiξG‘3 :r ð2ÞHere ℓ = t, b and ξ = ± 1 is the valley index. G‘1 and G‘2 are the reciprocallattice vectors of the ℓmoiré andG‘3 = � G‘1 � G‘2.UBLG is the inter-layerpotential between the layers of the BLG.Figure 3a shows the theoretically constructed density of states(DOS) versus carrier density plot; the zeros in the DOS correspond tothe gaps in the energy spectrum. To gain a physical understanding ofthe origin of these gaps, we follow the procedure laid out in ref. 14.Recall that a nearly commensurate system with dual periodicity isdefined by a set of four distinct reciprocal wave vectors:Gt1,2 being thetwo primitive reciprocal lattice vectors of the moiré lattice at the tophBN-graphene interface and Gb1,2 those for the second moiré lattice atthe bottom graphene-hBN interface. One can form quasi-Brillouinzones bounded by multiple Bragg planes defined by a linear combi-nation of these four primary reciprocal vectors. The m1,m2:m3,m4� �th– order Bragg-gap appears in the electronic spectrum when the totalcharge carrier density equals14,33:nðm1,m2,m3,m4Þ=4X4i = 1miAi=ð2πÞ2: ð3ÞHere A1 = jGb1 ×Gb2j, A2 = jGt1 ×Gt2j, A3 = jGb1 ×Gt2j, and A4 = jGt1 ×Gb2j arethe areas of the projections of the parallelograms formed by the fourreciprocal lattice vectors Gi. The quantityP4i= 1 miAi is the area (inreciprocal space) of the multifaceted quasi-Brillouin zone, and thefactor of four on the right-hand side of Eq. (3) arises from the spin andvalley degeneracies. The areas Ai for the experimentally obtained twistangle θb = 0.03° and θt = 0.44° are 0.277, 0.233, 0.181 and 0.290 nm−2.The integers mi are Bragg indices (quantum numbers) of the gap andare topological invariants of the system14,28. Note that this formalism ismathematically identical to that utilized by previous workers based ondifferences between the multiples of the aligned and rotatedreciprocal vectors13,15,17,33 with the added advantage of being intuitivelytransparent.Note that in our theoretical calculations, we used the latticeconstant of graphene and hBN to be 0.246nm and 0.2504nm,respectively (corresponding to a strain, ϵ =0.018). We also consideredother values of the strain parameter in the commonly used range0.0165 ≤ ϵ ≤0.0185. The theoretical values generated with ϵ = 0.018match best with our measured experimental results.Experimental observation of the Bragg gapsUsing the above formalism, the band gaps corresponding to the den-sities nb, nt, and ns are identified to be Bragg gaps with Bragg indices(0, 1, 0, 0), (1, 0, 0, 0), and ð1,1,�1,�1Þ respectively (see SupplementaryNote 5). We obtain the positions of additional Bragg gaps by com-paring calculated DOS (Fig. 3a) with the experimentally determinedtransverse resistance Rxy(B) and the extracted Hall carrier densitynH =B/(eRxy) measured in the presence of a small, non-quantizingmagnetic field B = 0.7T (Fig. 3b, c).The zeroes (and several prominent non-zero dips) in the calcu-lated DOS are reflected in the experimental data as a discontinuity inthe nH − n plot. Recall that in a multi-carrier type system and for smallB, a change in sign of Rxy (or a corresponding divergence in nH) caneither indicate a bandgap or a van Hove singularity48,49. The sign of nHon either side of a band gap reflects the local band curvature (andhence, the carrier type). Thus, for instance, with EF > 0, one can haveboth positive and negative nH; a positive (negative) value of nH impliesan electron-like (a hole-like) band (we take the electronic chargeto be e). A band gap can be said to exist at a certain number density ifthe following three conditions are simultaneously met: (1) the DOS inFig. 3a goes to zero, (2)nH in Fig. 3c changes sign, and (3) there is a localmaximum in the d2Gxx/dn2 (minima in Gxx) data in Fig. 3d. Using thiscriterion, we identify the principal gaps at nb = − 2.36 × 1016m−2,nt = − 2.80× 1016m−2, and nCNP =0 m−2 as Bragg gaps with quantumnumbers ð0,�1,0,0Þ, ð�1,0,0,0Þ, (0, 0, 0, 0), and respectively. We alsoidentify several higher order Bragg gaps, for example, atn = − 3.3 × 1016m−2 ð2,2,�1,�4Þ and − 4.8 × 1016m−2 ð�4,�1,0,3Þ. Wemark all theBragg gaps with solid gray lines in Fig. 3a–f.There are certain number densities for example, at−0.39 × 1016m−2 ð�1,�1,1,1Þ and − 6.6 × 1016m−2 ð�1,2,�3,�1Þ, (marked by dottedblue lines in Fig. 3a–d) where the DOS goes to zero and the Gxx has aminima, however nH does not reach zero. We tentatively identify themas narrow Bragg gaps that are masked by thermal/impurity broad-ening. Note also that there are no gaps at positive energies with theexception of the supermoiré gap at ns = 0.39 × 1016m−2 ð1,1,�1,�1Þ. Thereason why the supermoiré gap survives the band overlaps (thatquenches all other gaps at positive energies) is at present unclear.Additionally, there are features at which nH changes sign accom-panied byminima in d2Gxx/dn2 (maxima in Gxx) and a peak in DOS –weidentify these to be due to van Hove singularities. Two of theseArticle https://doi.org/10.1038/s41467-024-46672-3Nature Communications | (2024) 15:2335 3(at n = − 2.05 × 1016m−2 and n = − 2.6 × 1016m−2) have been marked withpurple dotted lines in Fig. 3e, f. Note that, in addition to the onesmarked, the calculated DOS plotted in Fig. 3a shows several dips atwhich the measured longitudinal and Hall resistances are featureless.We find that at these points, either the DOS is finite with no band gap,or the calculated band gaps Δ are substantially smaller than 1meV (e.g.at n = − 2.18 × 1016m−2, Δ = 0.74meV) and hence not resolvable in ourelectrical transport measurements.The fact that the data from measurements of three independentphysical quantities (quantum oscillations, Hall resistance, and0 0.2 0.4-6-4-2024-2 -1 0 1 -8 -4 0 4 8 -40 -20 0 20 40n (1016 m-2)Rxy (k�) nH (1016 6 m-2)DOS D/� (10 V/m)-1.5-2.0-2.5-3.00 0.2 -8 -4 0 4 8Bragg gapsDOS nH (1016 m-2) n (1016 m-2)-5 -4 -3 -2 -1 0 n (1016 m-2)( 0 0 0 0 )( 0 0 0 0 )( 2 2 1 4 )( 2 2 1 4 )- -( 4 1 0 3 )-- -------- -( 4 1 0 3 )- -( 1 2 3 1 )( 1 0 0 0 ) -nt-nt( 1 0 0 0 ) -( 0 1 0 0 )( 0 1 0 0 )-(VHS)(VHS)( 1 1 1 1 )ae fb c dgnCNPnCNP-nb-nb-ns( 1 0 0 0 )--nt( 0 1 0 0 )--nb-0.2 V/nm-0.1 V/nm 0.0 V/nm 0.1 V/nm 0.2 V/nm-0.5 0.5 1.5 2 0 0 d2Gxx /dn2 (arb. units)oFig. 3 | Experimentally obtained and theoretically calculated Bragg gaps. a Plotof the calculated density of states (DOS) for θb =0.026° and θt =0.44°. b Plot oftransverse resistance Rxy versus n measured at B =0.7 T and T = 2K. c Plot of Hallcarrier density nH versus n. d Map of the normalized d2Gxx(B =0)/dn2 in the n −Dplane; the d2Gxx(B =0)/dn2 have been plotted on a logarithmic scale. The indices ofthe Bragg gaps aremarked on the right. In a–d, the solid gray lines mark the valuesof n at which the Bragg gaps open with DOS =0, Rxy =0 and d2Gxx(B =0)/dn2 havinga maxima; the dotted blue lines mark the values of n at which the Bragg gaps openwith DOS =0 and d2Gxx(B =0)/dn2 having a maxima, but Rxy does not reach zero.e, f Zoomed-in plots of DOS and nH versus n in a narrow range on the hole-side. Thesolid gray lines indicate the Bragg gaps, while the dotted purple lines indicate thelocations of the vanHove singularities. g Plot of the positionof a few representativeBragg gaps versus n over a range of electric fields. The corresponding quantumnumbers of the gaps are marked on the right.n = 3.3 x 1016 m-21/B (1/T)�Gxx (mS)0.20.10-0.10.1 0.3 0.5 bNorm. amplitude34.6 nm13.97 nm 12.84 nm f (T)1.20.80.40.00 20 40 60 80 c3.0 2.5 2.0 1.5 0.1 0.3 0.5n (1016 m-2)1/B (1/T) a-0.1�Gxx(mS)0.2Fig. 2 | Brown-Zak oscillation indoublemoiré device. aBrown-Zak oscillations ofmagnetoconductance ΔGxx plotted in the n − 1/B plane; the data were measured atT = 100K. b Plot of ΔGxx as a function of 1/B for carrier density n = 3.3 × 1016m−2.c The Fourier spectrum of the data in b, shown as a function of f(T); the peaks aremarked with the corresponding moiré super-lattice wavelengths.Article https://doi.org/10.1038/s41467-024-46672-3Nature Communications | (2024) 15:2335 4zero-magnetic-field longitudinal resistance) and from continuum-model-based calculations match emphasizes the validity of our ana-lysis. We note in passing that the positions of the primary gaps innumber density are independent of applied small displacementfields (Fig. 3g).From the activated temperature-dependent resistance data, weextract the band-gap at CNP to be 6meV at zero displacement field.This value is in the same range as our theoretically calculated band gap4meV and is in agreement with the recent theoretical work in super-moiré system31 and experimental studies in transport19. The energygaps at the primary moiré gaps on the hole-side are extracted to beΔb = 1.46meV and Δt = 3.39meV respectively.Quasi Brillouin ZonesThe electronic carrier densities at which we observe the gaps in ourdoubly-periodic 2D system are related to the areas of the underlyingqBZ. In order to identify these zone boundaries, wefind the k-points atwhich the gaps open. One can observe the gap opening points byunfolding the supermoiréband structure to theunit cell of theBLG.Wemodulate the strength of top and bottom moiré potential in thereduced Hamiltonian (Eq. (1)) with strength parameter η ranging from0 to 1 (See Supplementary Note 6). The unfolded band structure(Fig. 4a) can be seen along a given k-path using unfolded spectralweights as:Aðq,ϵÞ=XnkXXj q,X jψnk� �j2δðϵ� ϵnkÞ ð4Þwhere X =A1, B1,A2, B2 denote the atoms of bilayer graphene, ∣ψnk�andϵnk denote the eigenstates and eigenvalues, respectively, q is thecrystalmomentum in the bilayer graphene unit cell BZ. Theq is relatedto k in the supermoiré BZwith amoiré reciprocal lattice vectorGSM viathe relation q =k +GSM21.Figure 4b–e shows the calculated qBZ for a few Bragg indicesusing the above procedure. These shapes and the correspondingBragg indices have simple geometrical interpretations. Consider, forexample, the qBZ of the supermoiré cell plotted in Fig. 4c; it is formedby the reciprocal lattice vectors Gb1 � Gt1 and Gb2 � Gt2. The area of thisqBZ can be expressed as:ðGb1 � Gt1Þ× ðGb2 � Gt2Þ= ðGb1 ×Gb2Þ+ ðGt1 ×Gt2Þ � ðGb1 ×Gt2Þ � ðGt1 ×Gb2Þ= jA1j+ jA2j � jA3j � jA4jð5ÞThis gives the Bragg indices of the qBZ of the supermoiré to be ð1,1,�1,�1Þ(see Eq. (3)) with the number density required to fill the bandns =0.39 × 1016m−2.We thus find the area of the supermoiré qBZ arrivedat using two very different theoretical routes (continuum model cal-culations and band geometric considerations) to be in excellentagreement with that extracted frommeasured Brown-Zak oscillations.A closer inspection reveals that several of the qBZ are three-foldsymmetric; two examples are provided in Fig. 4c, d. The source of thisC3 symmetry can be traced back to the triangular symmetry of theconstant energy contours of bilayer graphene energy dispersion (SeeSupplementary Note 7). Figure 4e shows an example of the fractal orflower-like qBZ for higher order gap Bragg predicted for doublyaligned graphene14.DiscussionWe note in passing that throughout the above discussion, we haveavoided any mention of the strength of the interlayer coupling. Asnoted in previous studies, the interlayer coupling strength affects onlythe magnitude of the Bragg gaps, leaving their positions unaffected13.To summarize, we have shown that the low-energy dispersion ofbilayer graphene can be significantly altered by the supermoirépotential. Our study provides an elegant physical picture of the Bragggaps opening in themoiré spectrum (basedon areaquantization of theqBZ) and helps identify the relevant topological quantum numbers.Our experimental results match very well with the predictions of thesubtle effects of nearly commensurate supermoiré structures on gra-phene bands. Importantly, our calculations establish that the qBZ ofthe supermoiré lattice in bilayer graphene are C3 symmetric (in con-trast to single-layer supermoiré), making it an ideal system to hostintrinsic Berry curvature dipoles. The scope of topology has beenlimited to strictly periodic systems, but our study represents a crucialstep toward expanding it to encompass quasicrystals and their topo-logical properties. To fully comprehend the physics of these intriguingmaterials and unlock their complete potential, additional experimentsΓKMAB150100500-50-100-150Energy (meV)A K BG2tG1tG2bG1bG2tG1tG2bG1bG2tG1tG2bG1bG2tG1tG2bG1bK(0 1 0 0) (1 1 1 1)(1 0 0 0) (1 1 0 3)kxkxk yk yn = -3.64 x 1016m-2n = -2.36 x 1016 m-2 n = -0.38 x 1016m-2n = -2.78 x 1016 m-2a bd ecFig. 4 | Unfolded band structure and qBZs. a Unfolded band structure along thepath AKB (shown in the subplot). Primary gaps from the top and bottommoiré areshown in red and blue. Gaps arising from both layers, i.e. supermoiré gaps, areshown in green. b–e Plots of the calculated qBZ for the Bragg gaps at numberdensities −2.36 × 1016m−2, −0.38 × 1016m−2, −2.78 × 1016m−2, and −3.64 × 1016m−2,respectively. The Bragg indices are indicated in the insets of each panel. Thek-pointswhere the gapopens are shown as dots in red,blue andgreen. The kx and kyrange between [−0.8,0.8] nm−1.Article https://doi.org/10.1038/s41467-024-46672-3Nature Communications | (2024) 15:2335 5and theoretical calculations that incorporate interaction effects arenecessary.MethodsDevice fabricationDevices of bilayer graphene (BLG) heterostructures doubly alignedwith single crystalline hBN were fabricated using a dry transfer tech-nique (for details, see SupplementaryNote 1). Flakes of hBNandbilayergraphene were exfoliated on the Si/SiO2 substrate with a thickness of280nm. Raman spectroscopy and AFM were used to determine thenumber of layers and thickness uniformity, respectively. A poly-dimethylsiloxane (PDMS) dome coatedwith a sacrificial polycarbonate(PC) layer was used to pick up the flakes sequentially. The rotationstage was coupled with the 3D stage manipulator to control the dis-tance and angle between the flakes independently. The hetero-structure was aligned under the microscope to form a moirésuperstructure with less than 1° misalignment. The final constructeddevice was vacuum annealed at 280 °C for 10 h. The devices werepatterned using standard electron beam lithography, followed byreactive ion etching (using mixture of CHF3 (40sccm) and O2 (4sccm))and thermal deposition of Cr/Au (5 nm/55 nm) contacts. The dual-gated device architecture allows for independent tuning of chargecarrier density and displacement field. The capacitance values of thetop gate and back gate were extracted from quantum hallmeasurements.Transport measurementsElectrical transport measurements were performed in a cryogen-freerefrigerator (with a base temperature of 2 K and magnetic field up to14 T). Thesemeasurements were performed at low frequency (18.8Hz)using standard low-frequency measurement techniques at a bias cur-rent of 10 nA.Uncertainity in twist angle estimationThe twist angle is estimated using the relationn=8½ϵ2 + 2ð1 + ϵÞð1� cosðθÞÞ�ffiffiffi3pa2ð1 + ϵÞ2ð6ÞHere, n is carrier density corresponding to the fully filledsuperlattice unit cell, a = 0.246 nm is the lattice constant of gra-phene, ϵ = 0.018 is the lattice mismatch between the hBNand graphene, and θ is the relative rotational angle betweenthe two lattices. Uncertainty in the twist angle (δθ) is estimatedfrom the uncertainty in the carrier density (δn) by the followingrelation:δθ=ffiffiffi3pa2ð1 + ϵÞ16 sinðθÞ δn ð7ÞThe impurity carrier concentration from the transport measure-ment is extracted to be δn = 7 × 1014m−2. At the twist angle of θ ≈0. 5°,uncertainty in the twist angle is extracted to be δθ =0.03°.Data availabilityThe authors declare that the data supporting the findings of this studyare available within the main text and its Supplementary Informationand at https://doi.org/10.6084/m9.figshare.25195817. Other relevantdata are available from the corresponding author upon request.Code availabilityThe code that support the findings of this study are available from thecorresponding author upon request.References1. González, D. A. G., Chittari, B. L., Park, Y., Sun, J.-H. & Jung, J.Topological phases in n-layer abc graphene/boron nitride moirésuperlattices. Phys. Rev. B 103, 165112 (2021).2. Song, J. C.W., Samutpraphoot, P. & Levitov, L. S. Topological blochbands in graphene superlattices. Proc. Natl Acad. Sci. 112,10879 (2015).3. Ponomarenko, L. A. et al. Cloning of dirac fermions in graphenesuperlattices. Nature 497, 594 (2013).4. Wang, P. et al. Topological winding number change and brokeninversion symmetry in a hofstadter’s butterfly. Nano Lett. 15,6395 (2015).5. Endo, K. et al. Topological valley currents in bilayer graphene/hexagonal boron nitride superlattices. Appl. Phys. Lett. 114,243105 (2019).6. Chen, G. et al. Tunable correlated chern insulator and ferro-magnetism in a moiré superlattice. Nature 579, 56 (2020).7. Yankowitz, M. et al. Emergence of superlattice dirac points in gra-phene on hexagonal boron nitride. Nat. Phys. 8, 382 (2012).8. Yankowitz, M., Watanabe, K., Taniguchi, T., San-Jose, P. & LeRoy, B.J. Pressure-induced commensurate stacking of graphene on boronnitride. Nat. Commun. 7, 13168 (2016).9. Yang, Y. et al. In situmanipulation of van derwaals heterostructuresfor twistronics. Sci. Adv. 6, eabd3655 (2020).10. Finney, N. R. et al. Tunable crystal symmetry in graphene-boronnitride heterostructures with coexisting moiré superlattices. Nat.Nanotechnol. 14, 1029 (2019).11. Kim, H. et al. Accurate gap determination in monolayer and bilayergraphene/h-bn moiré superlattices. Nano Lett. 18, 7732 (2018).12. Shi, J., Zhu, J. & MacDonald, A. H. Moiré commensurability and thequantum anomalous hall effect in twisted bilayer graphene onhexagonal boron nitride. Phys. Rev. B 103, 075122 (2021).13. Anðelković, M.,Milovanović, S. P., Covaci, L. & Peeters, F.M. Doublemoiré with a twist: Supermoiré in encapsulated graphene. NanoLett. 20, 979 (2020).14. Oka, H. & Koshino, M. Fractal energy gaps and topological invar-iants in hbn/graphene/hbn doublemoiré systems. Phys. Rev. B 104,035306 (2021).15. Leconte, N. & Jung, J. Commensurate and incommensurate doublemoire interference in graphene encapsulated by hexagonal boronnitride. 2D Mater. 7, 031005 (2020).16. Lu, X. et al. High-order minibands and interband landau levelreconstruction in graphene moiré superlattices. Phys. Rev. B 102,045409 (2020).17. Wang, Z. et al. Composite super-moiré lattices in double-alignedgraphene heterostructures. Sci. Adv. 5, eaay8897 (2019).18. Sun, X. et al. Correlated states in doubly-aligned hbn/graphene/hbn heterostructures. Nat. Commun. 12, 7196 (2021).19. Kuiri, M. et al. Enhanced electron-phonon coupling in doublyaligned hexagonal boron nitride bilayer graphene heterostructure.Phys. Rev. B 103, 115419 (2021).20. Yankowitz, M.,Ma, Q., Jarillo-Herrero, P. & LeRoy, B. J. van der waalsheterostructures combining graphene and hexagonal boronnitride. Nat. Rev. Phys. 1, 112 (2019).21. Mayo, S. G., Yndurain, F. & Soler, J. M. Band unfoldingmade simple.J. Phys.: Condens. Matter 32, 205902 (2020).22. Dean, C. R. et al. Hofstadter’s butterfly and the fractal quantum halleffect in moiré superlattices. Nature 497, 598 (2013).23. Hunt, B. et al. Massive dirac fermions and hofstadter butterfly in avan der waals heterostructure. Science 340, 1427 (2013).24. Moon, P. & Koshino, M. Electronic properties of graphene/hex-agonal-boron-nitride moiré superlattice. Phys. Rev. B 90,155406 (2014).Article https://doi.org/10.1038/s41467-024-46672-3Nature Communications | (2024) 15:2335 6https://doi.org/10.6084/m9.figshare.2519581725. Jung, J., Raoux, A., Qiao, Z. & MacDonald, A. H. Ab initio theory ofmoiré superlattice bands in layered two-dimensional materials.Phys. Rev. B 89, 205414 (2014).26. Wallbank, J. R., Patel, A. A., Mucha-Kruczyński, M., Geim, A. K. &Fal’ko, V. I. Genericminiband structure of graphene on a hexagonalsubstrate. Phys. Rev. B 87, 245408 (2013).27. Moriya, R. et al. Emergence of orbital angular moment at van hovesingularity in graphene/h-bn moiré superlattice. Nat. Commun. 11,5380 (2020).28. Koshino, M. & Oka, H. Topological invariants in two-dimensionalquasicrystals. Phys. Rev. Res. 4, 013028 (2022).29. Mao, D. & Senthil, T. Quasiperiodicity, band topology, and moirégraphene. Phys. Rev. B 103, 115110 (2021).30. Zhu, Z., Carr, S., Massatt, D., Luskin, M. & Kaxiras, E. Twisted trilayergraphene: A precisely tunable platform for correlated electrons.Phys. Rev. Lett. 125, 116404 (2020).31. Zhu, Z., Carr, S., Ma, Q. & Kaxiras, E. Electric field tunable layerpolarization in graphene/boron-nitride twisted quadrilayer super-lattices. Phys. Rev. B 106, 205134 (2022).32. Fujimoto, M., Koschke, H. & Koshino, M. Topological chargepumping by a slidingmoiré pattern. Phys. Rev. B 101, 041112 (2020).33. Wang, L. et al. New generation of moiré superlattices in doublyaligned hbn/graphene/hbn heterostructures. Nano Lett. 19,2371 (2019).34. Khalifa, A., Murthy, G. & Kaul, R. K. Lattice model for the quantumanomalous hall effect in moiré graphene. Phys. Rev. B 107,235137 (2023).35. Wallbank, J. R. et al. Excess resistivity in graphene superlatticescaused by umklapp electron-electron scattering. Nat. Phys. 15,32 (2019).36. Moulsdale, C. & Fal’ko, V. Umklapp electron-electron scattering inbilayer graphenemoiré superlattice. Phys. Rev. B 107, 144111 (2023).37. Ezzi, M.M. A., Ariando, J. Hu, Guinea, F. & Adam, S., Topological Flatbands in graphene super-moiré lattices. Preprint at https://arxiv.org/abs/2306.10116 (2023).38. Pizzocchero, F. et al. The hot pick-up technique for batch assemblyof van der waals heterostructures. Nat. Commun. 7, 1 (2016).39. Wang, L. et al. One-dimensional electrical contact to a two-dimensional material. Science 342, 614 (2013).40. Zak, J. Magnetic translation group. Phys. Rev. 134, A1602 (1964).41. Brown, E. Bloch electrons in a uniform magnetic field. Phys. Rev.133, A1038 (1964).42. Hofstadter, D. R. Energy levels and wave functions of bloch elec-trons in rational and irrational magnetic fields. Phys. Rev. B 14,2239 (1976).43. Kumar, R. K. et al. High-temperature quantum oscillations causedby recurring bloch states in graphene superlattices. Science 357,181 (2017).44. Barrier, J. et al. Long-rangeballistic transport of brown-zak fermionsin graphene superlattices. Nat. Commun. 11, 5756 (2020).45. Huber, R. et al. Band conductivity oscillations in a gate-tunablegraphene superlattice. Nat. Commun. 13, 2856 (2022).46. Woods, C. et al. Commensurate–incommensurate transition ingraphene on hexagonal boron nitride. Nat. Phys. 10, 451 (2014).47. Bistritzer, R.&MacDonald, A.H.Moirébands in twisteddouble-layergraphene. Proc. Natl Acad. Sci. 108, 12233 (2011).48. He, P. et al. Graphene moiré superlattices with giant quantumnonlinearity of chiral bloch electrons. Nat. Nanotechnol. 17,378 (2022).49. Wu, S. et al. Multiple hot-carrier collection in photo-excited gra-phene moiré superlattices. Sci. Adv. 2, e1600002 (2016).AcknowledgementsThe authors acknowledge Simrandeep Kaur for help with device fabri-cation. A.B. acknowledges funding from U.S. Army DEVCOM Indo-Pacific (Project number: FA5209 22P0166) and Department of ScienceandTechnology,Govt of India (DST/SJF/PSA-01/2016-17).M.J. andH.R.K.acknowledge the National Supercomputing Mission of the Departmentof Science and Technology, India, and the Science and EngineeringResearch Board of the Department of Science and Technology, India, forfinancial support under Grants No. DST/NSM/R&D_HPC Applications/2021/23 and No. SB/DF/005/2017, respectively. M.K.J. and R.B.acknowledge the funding from the Prime Minister’s research fellowship(PMRF), MHRD.Author contributionsM.K.J., P.T., and A.B. conceived the idea of the study, conducted themeasurements, and analyzed the results. T.T. andK.W. provided the hBNcrystals. M.J., R.B., S.M., I.S., and H.R.K. developed the theoreticalmodel. All the authors contributed to preparing the manuscript.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-024-46672-3.Correspondence and requests for materials should be addressed toManish Jain or Aveek Bid.Peer review information Nature Communications thanks NicolasLeconte, Xirui Wang and the anonymous reviewer(s) for their contribu-tion to the peer review of this work. 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To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2024Article https://doi.org/10.1038/s41467-024-46672-3Nature Communications | (2024) 15:2335 7https://arxiv.org/abs/2306.10116https://arxiv.org/abs/2306.10116https://doi.org/10.1038/s41467-024-46672-3http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Higher order gaps in the renormalized band structure of doubly aligned hBN/bilayer graphene moiré superlattice Results Device characteristics Continuum Hamiltonian Experimental observation of the Bragg�gaps Quasi Brillouin�Zones Discussion Methods Device fabrication Transport measurements Uncertainity in twist angle estimation Data availability Code availability References Acknowledgements Author contributions Competing interests Additional information