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Alexis Coissard, Adolfo G. Grushin, Cécile Repellin, Louis Veyrat, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Frédéric Gay, Hervé Courtois, Hermann Sellier, Benjamin Sacépé

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[Absence of edge reconstruction for quantum Hall edge channels in graphene devices](https://mdr.nims.go.jp/datasets/b2fc21d6-d3bc-4c8d-b02b-5bd189250fd0)

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Absence of edge reconstruction for quantum Hall edge channels in graphene devicesPHYS ICSAbsence of edge reconstruction for quantum Hall edgechannels in graphene devicesAlexis Coissard1, Adolfo G. Grushin1, Cécile Repellin2, Louis Veyrat1, Kenji Watanabe3,Takashi Taniguchi4, Frédéric Gay1, Hervé Courtois1, Hermann Sellier1, Benjamin Sacépé1†*Quantum Hall (QH) edge channels propagating along the periphery of two-dimensional (2D) electron gasesunder perpendicular magnetic field are a major paradigm in physics. However, groundbreaking experimentsthat could use them in graphene are hampered by the conjecture that QH edge channels undergo a reconstruc-tion with additional nontopological upstream modes. By performing scanning tunneling spectroscopy up to theedge of a graphene flake on hexagonal boron nitride, we show that QH edge channels are confined to a fewmagnetic lengths at the crystal edges. This implies that they are ideal 1D chiral channels defined by boundaryconditions of vanishing electronic wave functions at the crystal edges, hence free of electrostatic reconstruction.We further evidence a uniform charge carrier density at the edges, incompatible with the existence of upstreammodes. This work has profound implications for electron and heat transport experiments in graphene-basedsystems and other 2D crystalline materials.Copyright © 2023 TheAuthors, somerights reserved;exclusive licenseeAmerican Associationfor the Advancementof Science. No claim tooriginal U.S. GovernmentWorks. Distributedunder a CreativeCommons AttributionLicense 4.0 (CC BY).INTRODUCTIONIn 1982, 2 years after the discovery of the quantum Hall (QH) effect(1), Halperin (2) predicted the existence of edge states carrying theelectron flow along sample periphery. These edge states, which formunidirectional (chiral) ballistic conduction channels, have beenpivotal in understanding most of the transport properties of theQH effect (3, 4). They have served as an extraordinarily versatileplatform for a multitude of quantum coherent experiments (5), cul-minating recently in the evidence of fractional statistics in the frac-tional QH effect (6) and the possibility of anyon braiding throughinterferometry (7).The existence of edge states was initially inferred as a conse-quence of the boundary conditions imposed by the physical edgeson the electron wave functions (2). The energy of the electron statesthat are condensed into Landau levels increases upon approachingthe edge due to the hard-wall boundary conditions, opening con-duction channels—the QH edge channels—spatially located attheir intersection with the Fermi level (see Fig. 1, B and C) (2). In-clusion of a smooth electrostatic confining potential, which is ex-perimentally used to define edges in two-dimensional (2D)electron gases buried in semiconductor heterostructures, enrichesthe picture with the concept of edge reconstruction (8). There, theCoulomb interaction energy dominates the confining potential,leading to a transformation of the edge states into a series of widecompressible channels separated by incompressible strips. In theopposite case of a sharp potential, the Coulomb interaction is notrelevant, and the single-particle picture is valid. Edge reconstructionmechanisms have further proven to be of paramount importance inthe fractional QH regime where additional co- and/orcounterpropagative or even neutral modes (9–11) can emerge andcomplexify charge and heat transport (12–14).Nonreconstructed edge states can substantially clarify QH edgetransport with virtually ideal 1D edge states (15) and new regimes ofintra- and interchannel interactions. Contrary to semiconductorheterostructures, 2D crystalline materials like graphene, for whichphysical edges are crystal edges, may be archetypical systemshosting such edge states. For graphene and its massless, linearband structure, QH edge states without confining electrostatic po-tential are expected to be the exact eigenstates of the Dirac equationsderived with vanishing boundary conditions at the armchair orzigzag edge (16–18). Akin to Halperin’s original prediction (2),these solutions for edges states are maximally confined to a fewmagnetic lengths lB ¼ffiffiffiffiffiffiffiffiffiffiffih� =eBp(ℏ is the reduced Planck constant,e is the electron charge, and B is the magnetic field) from thecrystal edge, leaving no room for edge reconstruction.Here, we unveil the real-space structure of the QH edge states ofgraphene lying on an insulating hexagonal boron nitride (hBN)flake and evidence the absence of edge reconstruction by perform-ing scanning tunneling spectroscopy up to the graphene crystaledge under strong perpendicular magnetic field. We achieved thisby overcoming the long-standing experimental challenge (19–29) ofapproaching a scanning tunneling tip to the edge without crashingit on the insulating substrate that borders the graphene flake bymeans of a prior localization of the graphene edge by atomicforce microscopy (AFM). We purposely used a homemade hybridscanningmicroscope (30) capable of operating alternatively in AFMand scanning tunneling microscopy (STM) mode, thanks to a PtIrSTM tip glued onto a piezoelectric tuning fork acting as a forcesensor (31, 32) for AFM (see Fig. 1A). Our sample schematized inFig. 1 (A and B) consists of a graphene monolayer deposited on anhBN flake sitting on a Si/SiO2 substrate that serves as a back-gateelectrode (see Methods). The graphene flake is contacted by a Cr/Pt/Au trilayer that allows to apply a voltage bias Vb and collect atunnel current It via the STM tip. All experiments presented hereare performed at a temperature of 4.2 K and a perpendicular mag-netic field of 14 T.1Université Grenoble Alpes, CNRS, Grenoble INP, Institut Néel, Grenoble 38000,France. 2Université Grenoble Alpes, CNRS, LPMMC, Grenoble 38000, France.3Research Center for Functional Materials, National Institute for MaterialsScience, 1-1 Namiki, Tsukuba 305-0044, Japan. 4International Center for MaterialsNanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba305-0044, Japan.†Present address: Google Research, Mountain View, CA, USA.*Corresponding author. Email: benjamin.sacepe@neel.cnrs.frS C I ENCE ADVANCES | R E S EARCH ART I C L ECoissard et al., Sci. Adv. 9, eadf7220 (2023) 12 May 2023 1 of 8Downloaded from https://www.science.org at National Institute for Materials Science on May 19, 2023http://crossmark.crossref.org/dialog/?doi=10.1126%2Fsciadv.adf7220&domain=pdf&date_stamp=2023-05-12RESULTSQH edge states spectroscopyFigure 1E displays an STM topographic image taken in constantcurrent mode of the graphene edge, initially coarsely located byAFM (see fig. S1). The height profile of this image (Fig. 1D)shows a large flat area and a slight bump on the left part of thescan. This bump results from the tip-graphene interaction liftingup the graphene edge when the tip is right above it (33). Thisbump allows us to locate the edge of the graphene crystal with anaccuracy of a few nanometers (see the SupplementaryMaterials). Tothe left of the bump, the tip dips toward the hBN substrate, on whicha tip crash is avoided by a height limit of the STM controller.Atomic-scale imaging of the honeycomb lattice shown in Fig. 1Fgives insight into the graphene lattice termination. The edge orien-tation in Fig. 1E, which is reported in Fig. 1F with the blue line, in-dicates an armchair termination.The central result of this work is shown in Fig. 2, which presentsthe evolution of the Landau levels upon approaching the immediateproximity of the graphene edge in the region shown in Fig. 1E,under a magnetic field of 14 T. We first study charge-neutralgraphene by tuning the density with the back-gate voltage set atVg = −5.4 V. Tunneling spectroscopy of Landau levels (34–37)results in a series of peaks in the tunneling conductance G(Vb) =dIt/dVb that is proportional to the local density of states. We showin Fig. 2A the tunneling conductance G(dedge, Vb) as a function oftip distance perpendicular to the graphene edge dedge and biasvoltage Vb. Far from the edge, Landau levels are readily identifiedas bright conductance peaks that we label LLN, where N is theLandau level index. These conductance peaks are conspicuouslystable upon approaching the edge on the left of the figure. Within40 nm from the edge, we observe a suppression of the Landau levelpeak heights (see individual spectra in Fig. 2D) starting at distancesthat depend on the Landau level (the higher the Landau index, thefurther from the edge). Figure 2C shows spatial maps of the tunnel-ing conductance at the voltage bias of the Landau level peaks. Foreach Landau level peak, darker areas corresponding to Landau levelpeak suppression appear further and further from the edge as theLandau level index increases.These findings contrast with the expectation for a smooth con-fining potential at the edges, for which the Landau level spectrumFig. 1. Tunneling spectroscopyof QH edge states. (A and B) Schematics of the experiment. A PtIr tip is glued at the extremity of one prong of a piezoelectric tuning forkto enable imaging both in scanning tunneling microscopy (STM) (by regulating the tunneling current It) and in atomic force microscopy (AFM) (by regulating the fre-quency shift of the tuning fork). Graphene lies atop an insulating hBN flake and is contacted by a Cr/Pt/Au electrode to apply the sample bias Vb. A back-gate voltage Vgapplied to the Si/SiO2 substrate enables to tune the Fermi level EF in graphene. Graphene edges are first located by AFM under perpendicular magnetic field, B. The tip isthen moved from the graphene bulk to the edge in STM to perform tunneling spectroscopy of QH edge channels. (C) Landau level spectrum (16–18) as a function ofenergy E (normalized to the first cyclotron gap ε0) and position. The Landau levels disperse at an armchair edge on the scale of themagnetic length lB. Their intersect withthe Fermi level defines the QH edge channels. (D and E) Topographic image (E) and its z profile averaged on the y direction (D) of the graphene edge obtained in STM. Weconsider that the tip apex is located above the graphene edge at themaximumof the z profile. (F) Atomic resolution of the graphene honeycomb latticemeasured in STMa few nanometers away from the edge. The vertical blue line indicates the crystal edge orientation deduced from (E). (G) Kekulé bond order imaged in charge-neutralgraphene (30) at Vg= −5 V at a distance of 20 nm from the edge.S C I ENCE ADVANCES | R E S EARCH ART I C L ECoissard et al., Sci. Adv. 9, eadf7220 (2023) 12 May 2023 2 of 8Downloaded from https://www.science.org at National Institute for Materials Science on May 19, 2023would have continuously shifted in energy, following the confiningpotential as the edge is approached. Because the tunneling conduc-tance probes states on the scale of the electron wave function, that is,the cyclotron radius RNc ¼ lBð2 jN jþ1Þ1=2 for Landau level indexN,the suppression of the Landau level peaks, here, reflects a spreadingof the spectral weight to higher energy due to an abrupt edge statedispersion at the physical edge, on a very short scale of the order ofthe magnetic length (see Fig. 2B). This suppression of the tunnelingdensity of states of the Landau levels, which has been observed ongraphene on a conductive graphite substrate (27), is therefore directevidence of QH edge states sharply confined at the edges. Ultimate-ly, on the last few nanometers from the edge, the Landau level peaksdisappear completely, and the redistribution of Landau level spec-tral weight yields a V shape–like tunneling density of states(see Fig. 2D).In this measurement, we have set the Fermi level at charge neu-trality, that is, at Landau level filling factor ν = 0, which leads to asplitting of the zeroth Landau level (see split peaks labeled LL0+ andLL0− in Fig. 2A) with the opening of an interaction-induced gap atVb = 0 V [see (30)]. This splitting signals the broken-symmetry state(38) at charge neutrality with the Kekulé bond order (30, 39, 40).Weidentified the Kekulé bond order at 20 nm of the edge in Fig. 1G,indicating that this broken-symmetry state, which develops in thebulk, is robust even in the very proximity of the edge (41).To substantiate our finding, we performed numerical simula-tions of the local density of states of a charge-neutral grapheneribbon with an armchair edge under perpendicular magnetic field(see Fig. 3A) (16–18). We computed the Landau levels of the latticeHamiltonian of nearest-neighbor hopping energy t. We assumed aKekulé bond order with a gap at half-filling of the zeroth Landaulevel of 50 meV, as measured experimentally (30). The eigenstatesFig. 2. Sharp QH edge states. (A) Evolution of the tunneling conductance dIt/dVb as a function of the distance from graphene edge measured at charge neutrality (Vg =−5.4 V). The half-filled zeroth Landau level is split into two sublevels LL0+ and LL0− due to QH ferromagnetism (30). (B) Schematics of the tunneling into QH edge states.Because of the spatial extent RNc ¼ lBð2 jN jþ1Þ1=2 of the LLN wave functions, the tunneling electrons probe at one point contributions from all states up to distances ofabout RNc (red and orange Gaussians in the middle for LLN and LLN+1, respectively). The resulting density of states features sharp Landau level peaks in the bulk, i.e., atdistance d1 from the edge, and a smooth profile close to the edge, at a distance d2∼ lB, due to the energy broadening of Landau levels (16). In addition, when approachingthe edge, the tip starts to probe the edge states of the lower Landau levels, pushed at higher energies by the presence of the physical edge, and overlapping with thehighly degenerate bulk states. The resulting peaks in the density of states thus exhibit a spectral weight redistribution toward higher energies, which leads to a sup-pression of the Landau level peak height in the tunneling conductance (bottom; in solid blue, each individual N and N + 1 Landau level peak; and in dashed blue, theoverall density of states). (C) Spatial maps of the tunneling conductance dIt/dVb at the energies of the Landau levels. (D) Individual spectra taken from (A) at differentdistances from the edge indicated by the color-coded arrows in (A). a.u., arbitrary units.S C I ENCE ADVANCES | R E S EARCH ART I C L ECoissard et al., Sci. Adv. 9, eadf7220 (2023) 12 May 2023 3 of 8Downloaded from https://www.science.org at National Institute for Materials Science on May 19, 2023for a ribbon with periodic boundary conditions along ŷ are shownin Fig. 3B as white dashed lines. The Landau level eigenstates dis-perse as their average x position, locked to their kymomentum, ap-proaches the physical edge of graphene (42), as schematized inFig. 1B. Figure 3B shows the clean local density of states, which in-tegrates the eigenstates weighted by the amplitude of the wave func-tions, as a function of the distance to the edge normalized by lB,dedge/lB, averaged over each unit cell (see Methods). The range ofdedge/lB coincides with the range of displacement in Fig. 2A, allow-ing direct comparison with the experimental data. The resultingLandau level peaks are suppressed at higher values of dedge thehigher their Landau level index, and on the same spatial scale as ob-served experimentally in Fig. 2A. This reduction of spectral weightis more visible in Fig. 3C where we plot spectra for different dedge(solid lines) including a single realization of on-site disorder(dashed lines). The latter breaks the particle-hole symmetry of thespectrum and thus may contribute to the asymmetries observed inthe zeroth Landau level peaks.On the charge accumulation on the edgesThe question of charge carrier homogeneity is critical for graphenetransport. A body of work has shown anomalous asymmetry insome transport properties supplemented by scanning probe inves-tigations (43–45), which points to a charge carrier accumulation atthe graphene edges. Its origin may be either electrostatic stray fieldof the back-gate electrode (46) or chemical doping due to edge treat-ments (etching) or dangling bonds. In the QH effect, such anaccumulation could open up additional counterpropagative edgechannels and produce dissipation (44–46).In tunneling experiments, a charge inhomogeneity on the edgewould result in an energy shift of the Landau level spectrum as awhole due to a local change of the Landau level filling factor. Ourmeasurements in Fig. 2 provides a first insight on this issue with aremarkable stability of the Landau level peaks in energy that indi-cates that a possible charge accumulation is not large enough todepin the chemical potential from the zeroth Landau level (30).In particular, it is lower than the value δn = 6.8 × 1011 cm−2 requiredto fill the zeroth Landau level and reach ν = 2 at 14 T, which wouldproduce a visible energy shift of the Landau level spectrum that wedo not observe.To enhance the sensitivity of the spectroscopy to possible chargeinhomogeneities, we performed similar measurements at fillingfactor ν = 2 (Vg = 4 V), when the Fermi level is pinned by localizedstates in the cyclotron gap separating LL0 from LL1. There, becauseof the little density of localized states as compared to the highly de-generate Landau levels, a small variation of charge density wouldresult in a substantial shift of the Landau levels in the tunnelingspectra. Figure 4 displays the spatial evolution of the tunneling con-ductance up to the edge at ν = 2 and 14 T. As in Fig. 2, the Landaulevel peaks (LL0, LL−1, and LL−2) stay at the same energy over thescan and vanish at about 20 nm from the edge, clearly indicating theabsence of charge accumulation. We further performed systematicgate-tuned tunneling spectroscopy maps at various locations, from500 to 5 nm from the edge (see the Supplementary Materials).Figure 4 (B to D) displays three of these maps taken close to theFig. 3. Theoretical tunneling density of states. (A) Schematic of the simulated edge geometry. We considered a Kekulé bond order (30, 39, 40) as broken-symmetry state(38), with lattice vectors that triple the unit cell compared to pristine graphene. (B) Corresponding local density of states (LDOS) as a function of the distance from thearmchair graphene edge, dedge, normalized by lB, for charge-neutral graphene. The Kekulé bond order splits the zeroth Landau level into two sublevels LL0+ and LL0− withan energy gap chosen to match the experimentally measured value of 50 meV (30). The white dashed lines are the numerically computed Landau levels of graphenenanoribbonwith armchair termination. Because of positionmomentum locking, the Landau Levels disperse as they approach the physical edge, as sketched in Fig. 1B. (C)Individual spectra taken from (B) at different distances from the edge. Solid lines show cuts at different dedge, while dashed lines show spectral asymmetry emerging froma single disorder realization of an on-site disorder potential with strength W/t = 0.3 (see Methods).S C I ENCE ADVANCES | R E S EARCH ART I C L ECoissard et al., Sci. Adv. 9, eadf7220 (2023) 12 May 2023 4 of 8Downloaded from https://www.science.org at National Institute for Materials Science on May 19, 2023edge.We observe in Fig. 4B the usual staircase pattern of the Landaulevel peaks due to the successive pinning of the Fermi energy in theLandau levels (30, 47, 48), which allows us to precisely identify theback-gate voltage of the charge neutrality point VCNPg . As shown inFig. 4E that displays VCNPg as a function of the distance from theedge, there is no charge accumulation from 500 to 20 nm to theedge, and only within 20 nm of the edge we measure a variationδn = (−1.5 ± 1.1) × 1011 cm−2.Such a charge density variation near the edge at 14 T yields alittle variation δν = 0.4 of local filling factor, which would haveno consequence on the QH edge transport properties. Extrapolatingat lower field, however, δν = 2 would be reached at a magnetic fieldof 3 T, thus potentially affecting edge transport with additionalmodes. However, the very small spatial scale of this charge accumu-lation cannot explain recent scanning probes experiments evidenc-ing indirect, sometimes out-of-equilibrium responses withinhundreds of nanometers from the edge (44, 45). We conjecturethat this charge accumulation in our particular case is related tothe tip-graphene interaction when the tip reaches and lift up the gra-phene edge (see section SIII).DISCUSSIONThe issue of charge accumulation on the edge and the ensuingemergence of upstreammodes (44, 45) were put forth as an alterna-tive interpretation (49) for the signature of helical edge transport incharge-neutral graphene (50, 51). Although we cannot exclude thatthe stray field of the back-gate electrode may accumulate charges athigh back-gate voltages, that is, away from charge neutrality point,and over a long distance (46), our results show that this accumula-tion is absent at low back-gate voltage, thus invalidating the doubtsraised (49) on the existence of the QH topological insulator phase incharge-neutral graphene (50, 51). Still, it may be interesting torevisit nonlocal transport in nonlinear regime (49) in view of theexact spatial structure of the QH edge states in graphene.Regarding edge reconstruction, a wealth of fractional and integerQH states exhibits complex sequences of reconstructed edge chan-nels, including additional integer and/or fractional as well as neutralFig. 4. Charge density inhomogeneity on pristine edges. (A) Spatial evolution of the tunneling conductance up to the edge at filling factor ν = 2 for Vg = 4 V. (B to D)Tunneling conductance gate maps as a function of sample bias Vb and gate voltage Vg. At a distance dedge = 36 nm = 5.3 lB from the edge, in (B), we observe the staircasepattern of Landau levels of the graphene bulk. At closer distances from the edge, the Landau level peaks in the staircase pattern start to blur (C) andmostly vanish in (D). Inthe three panels, the opening of the ν = 0 gap as a function of Vg is indicated by white dashed lines, and the back-gate voltage of the charge-neutrality point VCNPg isidentified by themaximumof the gap. The blue dashed lines indicate the spectra at Vg = 4 V, which coincidewith the back-gate voltage of (A). The faint diagonal lines thattranslate into faint horizontal lines above the LL 0 peak in (A) result from residual charging effects in the tunneling process. Their downward dispersion near the edge in (A)is consistent with the charge carrier variation in (E). (E) Evolution of VCNPg and charge carrier density n0 determined from tunneling conductance gate maps as a function ofthe distance from the edge dedge. The position of the charge carrier density shift coincides with the upward shift of the LL0 in (A). Error bars correspond to the range ofgate voltage where the ν = 0 gap opens in the gate maps.S C I ENCE ADVANCES | R E S EARCH ART I C L ECoissard et al., Sci. Adv. 9, eadf7220 (2023) 12 May 2023 5 of 8Downloaded from https://www.science.org at National Institute for Materials Science on May 19, 2023modes (9–11). Whereas the smooth electrostatic potential in GaAsand other semiconductors reconstructs edge states into wide com-pressible stripes of the order of ⁓100 nm [see (28)], the grapheneQH edge states confined on a very short length scale, at fewmagnet-ic lengths on the physical edge, pose new constraints and limits forsuch a reconstruction, opening the investigation of universal trans-port and thermal properties (15). Moreover, in such a strongly con-fined configuration, an enhancement of inter-edge statesinteractions can be expected, which makes the picture of indepen-dent chiral channels irrelevant in this case, thus affecting charge andheat equilibration (52–54). This should affect QH interferometry(55) in graphene systems (56, 57) and other coherent experiments(5), for which the independence, exact positions, and nature of edgemodes are crucial parameters to address anyon physics and otherinteraction-driven phenomena, such as charging effects (58),spin-charge separation (59), or electron pairing (60). Note: A veryrecent work (61) reports a complementary tunneling spectroscopystudy of electrostatically defined QH edge states at a pn junction.METHODSSample fabricationThe graphene/hBN heterostructure was assembled from exfoliatedflakes with the van der Waals pickup technique using a polypropyl-ene carbonate polymer (62). The stack with graphene on top of thehBN flake was deposited using the method described in (63) on ahighly p-doped Si substrate with a 285-nm-thick SiO2 layer. Elec-tron beam lithography using a poly(methyl methacrylate) resistwas used to pattern a guiding markerfield on the whole 5 mm–by–5 mm substrate to drive the STM tip toward the device and tolocate the graphene edge. Cr/Pt/Au electrodes contacting the gra-phene flake were also patterned by electron beam lithography andmetalized by e-gun evaporation. The sample was thermally an-nealed at 350°C in vacuum under a halogen lamp to remove resistresidues and clean graphene before being mounted into the STMwhere it was heated in situ during the cooling to 4.2 K.MeasurementsExperiments were performed with a homemade hybrid STM andAFM operating at a temperature of 4.2 K in magnetic fields up to14 T. The sensor consists of a hand-cut PtIr tip glued on the freeprong of a tuning fork, the other prong being glued on aMacor sub-strate. Oncemounted inside the STM, the tip is roughly aligned overthe sample at room temperature. The AFM mode was used first forcoarse navigation at 4.2 K on the sample surface to align the tip ontographene and then for locating coarsely the graphene edge; see theSupplementary Materials. The STM imaging in constant-heightmode of the edge, done subsequently, yields a fine identification.Scanning tunneling spectroscopy was performed using a lock-inamplifier technique with a modulation frequency of 263 Hz androot mean square modulation voltage between 1 and 5 mV depend-ing on the spectral range of interest. Current imaging tunnelingspectroscopy (CITS) measurements were acquired by starting farfrom the edge, with a grid whose slow x axis is perpendicular tothe edge direction (as imaged by STM) and the y axis is parallelto the edge with a size of a few tens of nanometers. A safety condi-tion is added to the tip vertical z-position controller to prevent thecrashing into the hBN flake beyond the graphene edge: if the z po-sition reaches a threshold (typically 3 nm below the z position of thetip estimated close to the edge), the tip is withdrawn and the CITSends. Imaging of the Kekulé bond order was carried out in STMconstant-height mode after tuning the graphene to charge neutral-ity with the back gate, at a bias voltage corresponding to the energyof the LL0+ peak [see (30) for details].Theoretical simulationsTo compute the local density of states shown in Fig. 3, we use thesimulation software Kwant (64). First, we create a honeycomb latticein a square system of size Lx × Ly = 130 × 130, in units of graphene’slattice constant a. The unit cell for the Kekulé order is tripled com-pared to pristine graphene and is defined by the reciprocal vectorsa1 ¼ að3ffiffiffi3p=2; 3=2Þ and a2 ¼ að� 3ffiffiffi3p=2; 3=2Þ; see Fig. 3A. To cal-culate the local density of states ρ(E, x) at a given energy E andKekulé unit cell x, we average over the six sites weighted by the cor-responding wave function, ρ(E, x) =∑α ∣ψα(x)∣2δ(E − Eα), where αruns over the six unit cell sites. We compute the local density ofstates spectra, shown in Fig. 3B as a color map, using the kernelpolynomial method (65) with a target energy resolution of ΔE/t =0.005 and a magnetic field of ϕ/ϕ0 = 0.005 in units of the magneticflux ϕ0 = h/e. The dashed line spectra of Fig. 3B maps are obtainedfor a finite nanoribbon of width Ly, with an armchair edge parallel tothe ŷ direction, as in (42). We allow the edge to be misaligned withthe Kekulé lattice vectors, as observed experimentally in Fig. 1 (Fand G). Last, the solid lines in Fig. 3C show cuts of the localdensity of states spectra shown in Fig. 3B. The dashed lines are cal-culated adding a single disorder realization obtained by adding arandom on-site potential Vdis at each site to the clean localdensity of states spectra described above. The disorder strength ateach site is drawn from a uniform distribution in the interval[−W, W ] with W/t = 0.3.Supplementary MaterialsThis PDF file includes:Supplementary TextFigs. S1 to S7ReferencesREFERENCES AND NOTES1. K. Von Klitzing, G. Dorda, M. Pepper, New method for high-accuracy determination of thefine-structure constant based on quantized Hall resistance. Phys. Rev. Lett. 45,494–497 (1980).2. B. I. Halperin, Quantized Hall conductance, current-carrying edge states, and the existenceof extended states in a two-dimensional disordered potential. Phys. Rev. B 25,2185–2190 (1982).3. M. Büttiker, Absence of backscattering in the quantum Hall effect in multiprobe con-ductors. Phys. Rev. B 38, 9375–9389 (1988).4. C. Beenakker, H. van Houten, Quantum transport in semiconductor nanostructures. SolidState Phys. 44, 1–228 (1991).5. C. Bäuerle, D. C. Glattli, T. Meunier, F. Portier, P. Roche, P. Roulleau, S. Takada, X. Waintal,Coherent control of single electrons: A review of current progress. Rep. Prog. Phys. 81,056503 (2018).6. H. Bartolomei, M. Kumar, R. Bisognin, A. Marguerite, J. M. Berroir, E. Bocquillon, B. Plaçais,A. Cavanna, Q. Dong, U. Gennser, Y. Jin, G. Fève, Fractional statistics in anyon collisions.Science 368, 173–177 (2020).7. J. Nakamura, S. Liang, G. C. Gardner, M. J. Manfra, Direct observation of anyonic braidingstatistics. Nat. Phys. 16, 931–936 (2020).8. D. B. Chklovskii, B. I. Shklovskii, L. I. Glazman, Electrostatics of edge channels. Phys. Rev. B46, 4026–4034 (1992).S C I ENCE ADVANCES | R E S EARCH ART I C L ECoissard et al., Sci. Adv. 9, eadf7220 (2023) 12 May 2023 6 of 8Downloaded from https://www.science.org at National Institute for Materials Science on May 19, 20239. C. D. C. Chamon, X. G. Wen, Sharp and smooth boundaries of quantum Hall liquids. Phys.Rev. B 49, 8227–8241 (1994).10. C. L. Kane, M. P. A. Fisher, J. Polchinski, Randomness at the edge: Theory of quantum Halltransport at filling ν=2/3. Phys. Rev. Lett. 72, 4129–4132 (1994).11. U. Khanna, M. Goldstein, Y. Gefen, Fractional edge reconstruction in integer quantum Hallphases. Phys. Rev. B 103, L121302 (2021).12. V. Venkatachalam, S. Hart, L. Pfeiffer, K. West, A. Yacoby, Local thermometry of neutralmodes on the quantum Hall edge. Nat. Phys. 8, 676–681 (2012).13. M. Goldstein, Y. Gefen, Suppression of Interference in quantum Hall Mach-Zehnder ge-ometry by upstream neutral modes. Phys. Rev. Lett. 117, 276804 (2016).14. R. Bhattacharyya, M. Banerjee, M. Heiblum, D. Mahalu, V. Umansky, Melting of interferencein the fractional quantum Hall effect: Appearance of neutral modes. Phys. Rev. Lett. 122,246801 (2019).15. Z.-X. Hu, R. N. Bhatt, X. Wan, K. Yang, Realizing universal edge properties in graphenefractional quantum Hall liquids. Phys. Rev. Lett. 107, 236806 (2011).16. D. A. Abanin, P. A. Lee, L. S. Levitov, Spin-filtered edge states and quantum Hall effect ingraphene. Phys. Rev. Lett. 96, 176803 (2006).17. L. Brey, H. A. Fertig, Edge states and the quantized Hall effect in graphene. Phys. Rev. B 73,195408 (2006).18. D. A. Abanin, P. A. Lee, L. S. Levitov, Charge and spin transport at the quantum Hall edge ofgraphene. Solid State Commun. 143, 77–85 (2007).19. K. L. McCormick, M. T. Woodside, M. Huang, M. Wu, P. L. McEuen, C. Duruoz, J. S. Harris,Scanned potential microscopy of edge and bulk currents in the quantumHall regime. Phys.Rev. B 59, 4654–4657 (1999).20. A. Yacoby, H. F. Hess, T. A. Fulton, L. N. Pfeiffer, K. W. West, Electrical imaging of thequantum Hall state. Solid State Commun. 111, 1–13 (1999).21. J. Weis, K. von Klitzing, Metrology and microscopic picture of the integer quantum Halleffect. Phil. Trans. R. Soc. A 369, 3954–3974 (2011).22. H. Ito, K. Furuya, Y. Shibata, S. Kashiwaya, M. Yamaguchi, T. Akazaki, H. Tamura, Y. Ootuka,S. Nomura, Near-field optical mapping of quantum Hall edge states. Phys. Rev. Lett. 107,256803 (2011).23. K. Lai, W. Kundhikanjana, M. A. Kelly, Z.-X. Shen, J. Shabani, M. Shayegan, Imaging ofcoulomb-driven quantum Hall edge states. Phys. Rev. Lett. 107, 176809 (2011).24. M. E. Suddards, A. Baumgartner, M. Henini, C. J. Mellor, Scanning capacitance imaging ofcompressible and incompressible quantum Hall effect edge strips. New J. Phys. 14,083015 (2012).25. P. Weitz, E. Ahlswede, J. Weis, K. von Klitzing, K. Eberl, Hall-potential investigations underquantum Hall conditions using scanning force microscopy. Phys. E 6, 247–250 (2000).26. G. Nazin, Y. Zhang, L. Zhang, E. Sutter, P. Sutter, Visualization of charge transport throughLandau levels in graphene. Nat. Phys. 6, 870–874 (2010).27. G. Li, A. Luican-Mayer, D. Abanin, L. S. Levitov, E. Y. Andrei, Evolution of Landau levels intoedge states in graphene. Nat. Commun. 4, 1–7 (2013).28. N. Pascher, C. Rössler, T. Ihn, K. Ensslin, C. Reichl, W. Wegscheider, Imaging the conduc-tance of integer and fractional quantum Hall edge states. Phys. Rev. X 4, 011014 (2014).29. S. Kim, J. Schwenk, D. Walkup, Y. Zeng, F. Ghahari, S. T. Le, M. R. Slot, J. Berwanger,S. R. Blankenship, K. Watanabe, T. Taniguchi, F. J. Giessibl, N. B. Zhitenev, C. R. Dean,J. A. Stroscio, Edge channels of broken-symmetry quantum Hall states in graphene visu-alized by atomic force microscopy. Nat. Commun. 12, 2852 (2021).30. A. Coissard, D. Wander, H. Vignaud, A. G. Grushin, C. Repellin, K. Watanabe, T. Taniguchi,F. Gay, C. B. Winkelmann, H. Courtois, H. Sellier, B. Sacépé, Imaging tunable quantum Hallbroken-symmetry orders in graphene. Nature 605, 51–56 (2022).31. F. J. Giessibl, S. Hembacher, M. Herz, C. Schiller, J. Mannhart, Stability considerations andimplementation of cantilevers allowing dynamic forcemicroscopy with optimal resolution:The qPlus sensor. Nanotechnology 15, S79–S86 (2004).32. J. Senzier, P. S. Luo, H. Courtois, Combined scanning force microscopy and scanningtunneling spectroscopy of an electronic nanocircuit at very low temperature. Appl. Phys.Lett. 90, 043114 (2007).33. A. Georgi, P. Nemes-Incze, R. Carrillo-Bastos, D. Faria, S. Viola Kusminskiy, D. Zhai,M. Schneider, D. Subramaniam, T. Mashoff, N. M. Freitag, A. Georgi, M. Liebmann,M. Pratzer, L. Wirtz, C. R. Woods, R. V. Gorbachev, Y. Cao, K. S. Novoselov, N. Sandler,M. Morgenstern, Tuning the pseudospin polarization of graphene by a pseudomagneticfield. Nano Lett. 17, 2240–2245 (2017).34. T. Matsui, H. Kambara, Y. Niimi, K. Tagami, M. Tsukada, H. Fukuyama, STS observations ofLandau levels at graphite surfaces. Phys. Rev. Lett. 94, 226403 (2005).35. K. Hashimoto, K. Hashimoto, C. Sohrmann, J. Wiebe, T. Inaoka, F. Meier, Y. Hirayama,R. A. Römer, R. Wiesendanger, M. Morgenstern, QuantumHall transition in real space: Fromlocalized to extended states. Phys. Rev. Lett. 101, 256802 (2008).36. Y. J. Song, A. F. Otte, Y. Kuk, Y. Hu, D. B. Torrance, P. N. First, W. A. de Heer, H. Min, S. Adam,M. D. Stiles, A. H. MacDonald, J. A. Stroscio, High-resolution tunnelling spectroscopy of agraphene quartet. Nature 467, 185–189 (2010).37. E. Y. Andrei, G. Li, X. Du, Electronic properties of graphene: A perspective from scanningtunneling microscopy and magnetotransport. Rep. Prog. Phys. 75, 056501 (2012).38. M. O. Goerbig, From the integer to the fractional quantum Hall effect in graphene. arXiv:2207.03322v1 (2022). https://doi.org/10.48550/arXiv.2207.03322.39. S.-Y. Li, Y. Zhang, L.-J. Yin, L. He, Scanning tunneling microscope study of quantum Hallisospin ferromagnetic states in the zero Landau level in a graphene monolayer. Phys. Rev. B100, 085437 (2019).40. X. Liu, G. Farahi, C.-L. Chiu, Z. Papic, K. Watanabe, T. Taniguchi, M. P. Zaletel, A. Yazdani,Visualizing broken symmetry and topological defects in a quantum Hall ferromagnet.Science 375, 321–326 (2022).41. A. Knothe, T. Jolicoeur, Edge structure of graphene monolayers in the ν = 0 quantum Hallstate. Phys. Rev. B 92, 165110 (2015).42. P. K. Pyatkovskiy, V. A. Miransky, Spectrum of edge states in the ν = 0 quantum Hall phasesin graphene. Phys. Rev. B 90, 195407 (2014).43. Y.-T. Cui, B. Wen, E. Y. Ma, G. Diankov, Z. Han, F. Amet, T. Taniguchi, K. Watanabe,D. Goldhaber-Gordon, C. R. Dean, Z.-X. Shen, Unconventional correlation betweenquantum Hall transport quantization and bulk state filling in gated graphene devices. Phys.Rev. Lett. 117, 186601 (2016).44. A. Marguerite, J. Birkbeck, A. Aharon-Steinberg, D. Halbertal, K. Bagani, I. Marcus,Y. Myasoedov, A. K. Geim, D. J. Perello, E. Zeldov, Imaging work and dissipation in thequantum Hall state in graphene. Nature 575, 628–633 (2019).45. N. Moreau, B. Brun, S. Somanchi, K. Watanabe, T. Taniguchi, C. Stampfer, B. Hackens, Up-stream modes and antidots poison graphene quantum Hall effect. Nat. Commun. 12,1–7 (2021).46. P. G. Silvestrov, K. B. Efetov, Charge accumulation at the boundaries of a graphene stripinduced by a gate voltage: Electrostatic approach. Phys. Rev. B 77, 155436 (2008).47. A. Luican, G. Li, E. Y. Andrei, Quantized Landau level spectrum and its density dependencein graphene. Phys. Rev. B 83, 041405(R) (2011).48. J. Chae, S. Jung, A. F. Young, C. R. Dean, L. Wang, Y. Gao, K. Watanabe, T. Taniguchi, J. Hone,K. L. Shepard, P. Kim, N. B. Zhitenev, J. A. Stroscio, Renormalization of the graphene dis-persion velocity determined from scanning tunneling spectroscopy. Phys. Rev. Lett. 109,116802 (2012).49. A. Aharon-Steinberg, A. Marguerite, D. J. Perello, K. Bagani, T. Holder, Y. Myasoedov,L. S. Levitov, A. K. Geim, E. Zeldov, Long-range nontopological edge currents in charge-neutral graphene. Nature 593, 528–534 (2021).50. A. F. Young, J. D. Sanchez-Yamagishi, B. Hunt, S. H. Choi, K. Watanabe, T. Taniguchi,R. C. Ashoori, P. Jarillo-Herrero, Tunable symmetry breaking and helical edge transport in agraphene quantum spin Hall state. Nature 505, 528–532 (2014).51. L. Veyrat, C. Déprez, A. Coissard, X. Li, F. Gay, K. Watanabe, T. Taniguchi, Z. Han, B. A. Piot,S. Sellier, B. Sacépé, Helical quantum Hall phase in graphene on SrTiO3. Science 367,781–786 (2020).52. S. K. Srivastav, R. Kumar, C. Spånslätt, K. Watanabe, T. Taniguchi, A. D. Mirlin, Y. Gefen, A. Das,Vanishing thermal equilibration for hole-conjugate fractional quantum Hall states in gra-phene. Phys. Rev. Lett. 126, 216803 (2021).53. R. Kumar, S. K. Srivastav, C. Spånslätt, K. Watanabe, T. Taniguchi, A. D. Mirlin, Y. Gefen, A. Das,Observation of ballistic upstream modes at fractional quantum Hall edges of graphene.Nat. Commun. 13, 213 (2022).54. G. Le Breton, R. Delagrange, Y. Hong, M. Garg, K. Watanabe, T. Taniguchi, R. Ribeiro-Palau,P. Roulleau, P. Roche, F. D. Parmentier, Heat equilibration of integer and fractional quantumHall edge modes in graphene. Phys. Rev. Lett. 129, 116803 (2022).55. D. E. Feldman, B. I. Halperin, Fractional charge and fractional statistics in the quantum Halleffects. Rep. Prog. Phys. 84, 076501 (2021).56. C. Déprez, L. Veyrat, H. Vignaud, G. Nayak, K. Watanabe, T. Taniguchi, F. Gay, H. Sellier,B. Sacépé, A tunable Fabry–Pérot quantum Hall interferometer in graphene. Nat. Nano-technol. 16, 555–562 (2021).57. Y. Ronen, T. Werkmeister, D. Haie Najafabadi, A. T. Pierce, L. E. Anderson, Y. J. Shin, Y. H. Lee,J. Bobae, K. Watanabe, T. Taniguchi, A. Yacoby, P. Kim, Aharonov–Bohm effect in graphene-based Fabry–Pérot quantum Hall interferometers. Nat. Nanotechnol. 16, 563–569 (2021).58. B. I. Halperin, A. Stern, I. Neder, B. Rosenow, Theory of the Fabry-Pérot quantum Hall in-terferometer. Phys. Rev. B 83, 155440 (2011).59. T. Fujisawa, Nonequilibrium charge dynamics of tomonaga-luttinger liquids in quantumhall edge channels. Ann. Phys. 534, 2100354 (2022).60. H. K. Choi, I. Sivan, A. Rosenblatt, M. Heiblum, V. Umansky, D. Mahalu, Robust electronpairing in the integer quantum Hall effect regime. Nat. Commun. 6, 1–7 (2015).S C I ENCE ADVANCES | R E S EARCH ART I C L ECoissard et al., Sci. Adv. 9, eadf7220 (2023) 12 May 2023 7 of 8Downloaded from https://www.science.org at National Institute for Materials Science on May 19, 2023https://doi.org/10.48550/arXiv.2207.0332261. T. Johnsen, C. Schattauer, S. Samaddar, A. Weston, M. Hamer, K. Watanabe, T. Taniguchi,R. Gorbachev, F. Libisch, L. Morgenstern, Mapping quantum Hall edge states in grapheneby scanning tunneling microscopy. Phys. Rev. B 107, 115426 (2023).62. L. Wang, I. Meric, P. Y. Huang, Q. Gao, Y. Gao, H. Tran, T. Taniguchi, K. Watanabe,L. M. Campos, D. A. Muller, J. Guo, P. KimHone, J. Hone, K. L. Shepard, C. D. Dean, One-dimensional electrical contact to a two-dimensional material. Science 342,614–617 (2013).63. Y. Choi, J. Kemmer, Y. Peng, A. Thomson, H. Arora, R. Polski, Y. Zhang, H. Ren, J. Alicea,G. Refael, F. von Oppen, K. Watanabe, T. Taniguchi, S. Nadj-Perge, Electronic correlations intwisted bilayer graphene near the magic angle. Nat. Phys. 15, 1174–1180 (2019).64. C. Groth, M. Wimmer, A. R. Akhmerov, X. Waintal, Kwant: A software package for quantumtransport. New J. Phys. 16, 063065 (2014).65. A. Weisse, G. Wellein, A. Alvermann, H. Fehske, The kernel polynomial method. Rev. Mod.Phys. 78, 275–306 (2006).66. S. Das Sarma, E. H. Hwang, W.-K. Tse, Many-body interaction effects in doped and undopedgraphene: Fermi liquid versus non-Fermi liquid. Phys. Rev. B 75, 121406(R) (2007).Acknowledgments: We thank C. Déprez, B. Halperin, M. Feigelman, M. Goerbig, M. Guerra,D. Perconte, H. Vignaud, and W. Yang for valuable discussions. We thank F. Blondelle, D. Dufeu,Ph. Gandit, D. Grand, G. Kapoujyan, D. Lepoittevin, J.-F. Motte, and P. Plaindoux for technicalsupport in setting up the experimental system. Samples were prepared at the Nanofab facility ofthe Néel Institute. Funding: This work has received funding from the European Union’s Horizon2020 research and innovation program ERC grants QUEST no. 637815 and SUPERGRAPH no.866365, and the Marie Sklodowska-Curie grant QUESTech no. 766025. A.G.G. acknowledgesfinancial support by the ANR under grant ANR-18-CE30-0001-01 (TOPODRIVE). Authorcontributions: A.C. fabricated the sample and performed the measurements. A.C., A.G.G., C.R.,H.S., and B.S. analyzed the data. A.G.G. and C.R. conducted the theoretical analysis. L.V.assembled the STMmicroscope. K.W. and T.T. supplied the hBN crystals. F.G. provided technicalsupport on the experiment. B.S. conceived and supervised the project, designed theexperimental setup, and wrote the paper with inputs from all coauthors. Competing interests:The authors declare that they have no competing interests. Data and materials availability:All data needed to evaluate the conclusions in the paper are present in the paper and/or theSupplementary Materials.Submitted 30 November 2022Accepted 10 April 2023Published 12 May 202310.1126/sciadv.adf7220S C I ENCE ADVANCES | R E S EARCH ART I C L ECoissard et al., Sci. Adv. 9, eadf7220 (2023) 12 May 2023 8 of 8Downloaded from https://www.science.org at National Institute for Materials Science on May 19, 2023Use of this article is subject to the Terms of serviceScience Advances (ISSN ) is published by the American Association for the Advancement of Science. 1200 New York Avenue NW,Washington, DC 20005. The title Science Advances is a registered trademark of AAAS.Copyright © 2023 The Authors, some rights reserved; exclusive licensee American Association for the Advancement of Science. No claimto original U.S. Government Works. Distributed under a Creative Commons Attribution License 4.0 (CC BY).Absence of edge reconstruction for quantum Hall edge channels in graphenedevicesAlexis Coissard, Adolfo G. Grushin, Ccile Repellin, Louis Veyrat, Kenji Watanabe, Takashi Taniguchi, Frdric Gay, HervCourtois, Hermann Sellier, and Benjamin SacpSci. Adv., 9 (19), eadf7220. DOI: 10.1126/sciadv.adf7220View the article onlinehttps://www.science.org/doi/10.1126/sciadv.adf7220Permissionshttps://www.science.org/help/reprints-and-permissionsDownloaded from https://www.science.org at National Institute for Materials Science on May 19, 2023https://www.science.org/content/page/terms-service INTRODUCTION RESULTS QH edge states spectroscopy On the charge accumulation on the edges DISCUSSION METHODS Sample fabrication Measurements Theoretical simulations Supplementary Materials This PDF file includes: REFERENCES AND NOTES Acknowledgments