# Fileset

[s41467-024-47342-0.pdf](https://mdr.nims.go.jp/filesets/39544240-72da-4e67-b453-5ca414cd3e12/download)

## Creator

Anna M. Seiler, Nils Jacobsen, Martin Statz, Noelia Fernandez, Francesca Falorsi, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Zhiyu Dong, Leonid S. Levitov, R. Thomas Weitz

## Rights

[Creative Commons BY Attribution 4.0 International](https://creativecommons.org/licenses/by/4.0/)

## Other metadata

[Probing the tunable multi-cone band structure in Bernal bilayer graphene](https://mdr.nims.go.jp/datasets/da51d08a-6b7a-413b-b96a-7f18eb90ad1e)

## Fulltext

Probing the tunable multi-cone band structure in Bernal bilayer grapheneArticle https://doi.org/10.1038/s41467-024-47342-0Probing the tunable multi-cone bandstructure in Bernal bilayer grapheneAnna M. Seiler 1, Nils Jacobsen1, Martin Statz 1, Noelia Fernandez1,Francesca Falorsi1, Kenji Watanabe 2, Takashi Taniguchi 3, Zhiyu Dong4,Leonid S. Levitov 4 & R. Thomas Weitz 1,5Bernal bilayer graphene (BLG) offers a highly flexible platform for tuning theband structure, featuring two distinct regimes. One is a tunable band gapinduced by large displacement fields. Another is a gapless metallic bandoccurring at low fields, featuring rich fine structure consisting of four linearlydispersing Dirac cones and van Hove singularities. Even though BLG has beenextensively studied experimentally, the evidence of this band structure is stillelusive, likely due to insufficient energy resolution. Here, we use Landau levelsasmarkers of the energy dispersion and analyze the Landau level spectrum in aregime where the cyclotron orbits of electrons or holes in momentum spaceare small enough to resolve the distinct mini Dirac cones. We identify thepresence of four Dirac cones and map out topological transitions induced bydisplacementfield. By clarifying the low-energy properties of BLGbands, thesefindings provide a valuable addition to the toolkit for graphene electronics.Graphene, a single layer of carbon atoms arranged in a hexagonallattice, exhibits intriguing electronic properties due to its linearlydispersing bands forming Dirac cones at the K and K’ points. Yet, oneof the key limitations of monolayer graphene is its zero bandgap,which renders it nonideal for digital electronic applications and con-trolling electronic interactions1. Several attempts have been made toartificially open up a bandgap in monolayer graphene, including che-mical doping2–4, strain engineering5–8 and the creation of moirépatterns9,10. However,while thesemethodsmay allow to create a gap inan otherwise gapless dispersion, they also create disorder in the sys-tem or necessitate a complex experimental setup10,11. Opening a tun-able band gap in pristine monolayer graphene by electrostatic gatingpresently appears to be out of reach since itwould require electricfieldcontrol with atomic precision to induce a potential difference betweenthe two sublattices.The simple Bernal-stacked bilayer graphene (BLG), to the contrary,does allow electrostatic tunability of a band gap and the high-energyparabolic dispersion – as shown by experimental spectroscopy andtransport measurements12–16 as well as theoretical calculations17,18.However, perhaps surprisingly, there is no consensus between experi-ment and theory regarding the low-energy band structure of BLG. Forexample, quantum Hall measurements identified an eightfold degen-eracy of the lowest Landau level (LLL), facilitated by a two-fold spin,valley and orbital degeneracy, consistent with a low-energy parabolicdispersion15,19–22.While such quantumHallmeasurements provide someinformation about band symmetries, they leave several key questionsunanswered. First, upon including higher-order hopping terms, at zeromagnetic field and low carrier density one expects a metallic band thatremains gapless at finite and not-too-strong displacement fields. Thismetallic fine-structure band features four Dirac cones with differentchiralities at each valley and prominent van Hove singularities19,23–26,which appear inconsistent with the picture inferred from quantumoscillations. Furthermore, there are outstanding questions aboutstrong exchange-driven phases in suspended bilayer graphene15,20,22,27,28which are hard to reconcile with linear dispersion at low energies.Lastly, signatures of changes in Fermi surface topology due to trigonalwarping have been identified in the case of strong displacement fieldswhere sizablebandgaps areopened. Specifically, anunusual orderingofReceived: 26 February 2024Accepted: 27 March 2024Check for updates11st Physical Institute, Faculty of Physics, University of Göttingen, Friedrich-Hund-Platz 1, Göttingen, Germany. 2Research Center for Electronic and OpticalMaterials, National Institute for Materials Science, 1-1 Namiki, Tsukuba, Japan. 3Research Center for Materials Nanoarchitectonics, National Institute forMaterials Science, 1-1 Namiki, Tsukuba, Japan. 4Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts, USA. 5Interna-tionalCenter for AdvancedStudies of EnergyConversion (ICASEC), University ofGöttingen,Göttingen,Germany. e-mail: thomas.weitz@uni-goettingen.deNature Communications |         (2024) 15:3133 11234567890():,;1234567890():,;http://orcid.org/0000-0002-9883-9220http://orcid.org/0000-0002-9883-9220http://orcid.org/0000-0002-9883-9220http://orcid.org/0000-0002-9883-9220http://orcid.org/0000-0002-9883-9220http://orcid.org/0000-0001-7791-3981http://orcid.org/0000-0001-7791-3981http://orcid.org/0000-0001-7791-3981http://orcid.org/0000-0001-7791-3981http://orcid.org/0000-0001-7791-3981http://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/0000-0002-4268-731Xhttp://orcid.org/0000-0002-4268-731Xhttp://orcid.org/0000-0002-4268-731Xhttp://orcid.org/0000-0002-4268-731Xhttp://orcid.org/0000-0002-4268-731Xhttp://orcid.org/0000-0001-5404-7355http://orcid.org/0000-0001-5404-7355http://orcid.org/0000-0001-5404-7355http://orcid.org/0000-0001-5404-7355http://orcid.org/0000-0001-5404-7355http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-47342-0&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-47342-0&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-47342-0&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-47342-0&domain=pdfmailto:thomas.weitz@uni-goettingen.dequantum Hall states is observed in the presence of a strong electricdisplacement field in strongly biased BLG29–32, Bernal trilayergraphene33–35, rhombohedral trilayer graphene36,37 and Bernal tetralayergraphene38, which is consistent with the theoretical band structurepredicting a trigonally-warped low-energy Fermi surface topology.The present work aims to resolve the puzzle of why the linearlydispersing bands and electric-field-induced changes of topology havenot beenobserved experimentally in BLG. Signsof topological changesin the band structure involving a van Hove singularity should bedetectable, e.g., in a quantizingmagnetic field inwhich the presenceoffour Dirac cones results in an exotic sequence of Landau levels distinctfrom the previously studied instances of Landau levels. Accessing thiscomplex field-tunable band structure with multiple Dirac cones andfield-tunable van Hove singularities is of clear interest for the physicsof strongly correlated systems as well as graphene band engineering.ResultsTo reveal the detailed low-energy band structure and detect electric-field controllable linear dispersing bands, herewehave chosen toworkwith the hexagonal boron nitride (hBN) encapsulated Bernal-stackedBLG sample system over suspended BLG—even though both systemsyield samples of similar quality. In the suspendedBLG, however, due tothe low-dielectric constant of the dielectric (vacuum), the exchangeenergy scale seems to dominate the low-energy physics leading to avariety of nontrivial groundstates15,20,22,27,28. Our encapsulated BLG isequipped with graphite top and bottom gates and two-terminal gra-phite contacts (see “Methods” and Supplementary Fig. 1 and 2). Allmeasurementswere performed in a cryostat at a temperature of 10mKemploying standard lock-in techniques at 78Hz and an ac bias currentof 1 nA. By varying both gate voltages, we were able to tune the chargecarrier density (n) and the electric displacement field (D) indepen-dently. Since we, on the one hand, use two-terminal measurements(i.e., contact resistance cannot be easily determined) and, on the otherhand, explore the behavior at relatively low B and down to B =0, therespective LL are not fully developed (quantized), we have found itconvenient to use dG/dn values as the markers of the band structure.We complement our measurements with tight binding band structurecalculations of the quantum Hall states in BLG expected at zero andlow D-fields. We use a standard BLG bandstructure model in which weinclude also the weaker inter-layer coupling parameters γ3 and γ4as well as an energy difference between dimer and non-dimer atomicsites Δ17,39 (see “Methods” for technical details of the calculations).Tight binding calculations of Landau LevelsWe first discuss the tight binding calculations of LLs that are usedbelow to experimentally identify the transition from a metallic bandwith four Dirac cones to a gapped parabolic dispersion. In the absenceof an interlayer potential difference U and in case trigonal warp-ing effects are ignored or are made irrelevant by disorder, the low-energy band structure of BLG exhibits a nearly parabolic dispersion(Fig. 1b (left))15,17,23. Consistentwith previous experiments conducted inquantizing magnetic fields B >0.5 T15,19,27, this leads to an eight-folddegeneracy of the lowest Landau level (LLL) due to spin, valley andorbital degrees of freedom, and to a four-fold degeneracy of all higherLandau levels due to spin and valley degrees of freedom (see “Meth-ods”). In case that the low-energy band structure at the charge neu-trality point can be resolved below a Fermi energy E ~ ±1meV, the bandstructure dramatically changes when including trigonal warping, andthree off-center and one center cones emerge in each valley (alsoreferred to as mini Dirac cones), resulting from the weaker skewinterlayer hopping term γ3. The four cones with a Dirac-like spectrumresemble a four-fold copy of the spectrum ofmonolayer graphene, forwhich the LLL is shared equally by electrons and holes, overall leadingto a 16-fold degeneracy (2 spins, 2 valleys, 4 mini Dirac cones). Thiswould result in the appearance of quantum Hall states with fillingfactors ν = hneB = ±819,23–25. In addition, the skew interlayer hopping termγ4 and the on-site parameter Δ’, describing the energy differencebetween atoms A and A’ or B and B’, create an energetic asymmetrybetween these cones17. While the center cone formed by the conduc-tion and valence bands occurs at zero energy, the off-center conesoccur at higher energies (Fig. 1b, Fig. 2a, b; more information on theimpact of γ3, γ4, and Δ’ is given in Supplementary Fig. 3). In quantumHall measurements, these changes in the band structure can be dis-cerned only at B < 0.2T since here, the inverse of the magnetic lengthlB =ffiffiffiffiffiffiffiffiffiffiffiffiffiffi_=ðeBÞpwith _=h=2π and Planck’s constant h is smaller than thedistance in momentum space between two adjacent mini Dirac cones(i.e., below the fields at which magnetic breakdown occurs)31,32.Figure 2c shows the calculated inverse compressibility (∂μ/∂nwithchemical potential μ, n charge carrier density) as function of n and B atD = 0 (Landau fan diagram). Here, larger energy gaps in the Landaulevel spectrum (Fig. 2d)manifest as prominent peaks corresponding toquantum Hall states that are labeled by numerals (the calculationsinclude γ0, γ1, γ3, γ4 and Δ’, spin splitting is includedmanually in the LLfor the figures, both valleys are included in the calculation and are fullydegenerate; see “Methods” for further details). While quantum Hallstates with ν = ±8 indeed exhibit the largest compressibility and godown to the lowest B in the valence/ conduction band also the quan-tum Hall state with ν = −4 (but not ν = +4) is very robust, i.e., it can beresolved until very low B (Fig. 2c), which is a manifestation of theelectron-hole asymmetry. Neglecting the spin and valley degrees offreedom, the three off-center cones exhibit a three-fold degenerateLLL and are shifted to higher energies. Thus, the center cone LLL isnon-degenerate with the LLL that belong to the off-center cones(Fig. 2d). Since the LLL is shared between electrons and holes, the non-degenerate center cone as well as one of three LL originating from thethree off-center cones contribute to hole transport and give rise toquantum Hall states with ν = −8 and ν = −4 respectively. The other twoLL stemming from the three off-center cones contribute to electrontransport and give rise to a quantum Hall state with ν = +8. Thequantum Hall state with ν = +4 only emerges at larger B where thedegenerate LL diverge. With increasing n, the conventional sequencequantum Hall states with filling factors ν = ±12, ±16, ±20 is recoveredand the Fermi level lies above the Lifshitz transition where the Fermisurface is fully connected.Layer 1  A BLayer 2  A‘ B‘(a)0134 =    = 03 4Δ´ = 0Ekxky(b)  = 0,   = 03 4  = 0,   = 03 4E = 0Δ´ = 0 Δ´ = 0ΔE ≈ 1 meVFig. 1 | Lattice andband structure of Bernal bilayer graphene. a Lattice structureof Bernal bilayer graphene. The interlayer hopping processes described bythe parameters γ0,γ1,γ3 and γ4 are indicated. b Band structure of bilayer graphenein an energy (E) range of −2 mV to +2mV at zero electric displacement field (D =0)calculated with a tight binding model including various subsets of hopping para-meters and on-site parameter Δ’, featuring four Dirac cones of differentchiralities and three van Hove singularities in each valley. The center conesare shaded darker.Article https://doi.org/10.1038/s41467-024-47342-0Nature Communications |         (2024) 15:3133 2Quantum Hall measurements at zero displacement fieldThe theory thus shows in detail how the presence of four Dirac conescan unambiguously be identified in experiment. Figure 2e shows thenormalized derivative of themeasured two-terminal conductance |dG/dn| as a function of n and B at D =0V/nm. Quantum Hall states appearas plateaus in the conductance and thus as dips in |dG/dn| and can beassigned by their corresponding slopes in the Landau fan diagram (see“Methods”). Consistent with our theoretical simulations, quantumHallstates with ν = ±8 are the most robust and can be observed at thesmallest B, down to B ≈0.05 T which reveals the presence of four miniDirac cones. Additionally, due to electron-hole asymmetry, the quan-tum Hall state with ν = −4 appears at slightly larger magnetic fields(B ≈0.15 T), while the ν = +4 quantum Hall state only appears above0.2 T when themagnetic breakdown occurs (indicated by dashed linesin Fig. 2c, e). Atmagnetic fields above0.3T, a sequence of even integerquantum Hall states appears which is consistent with previous mea-surements in freestanding BLG15,20,27,40 and which reveals the highquality of our sample. Here, the spin degeneracy is likely lifted due toCoulomb interactions resulting in a two-fold degeneracy (valley)instead of the predicted four-fold degeneracy (spin and valley)23,41,42.Notably, some of the non-four-fold degenerate quantum Hall statesincluding the quantum Hall states with ν = ±6 also go down to below0.3 T and then merge with the quantum Hall states with ν = ±4 anddemand further investigation. Since spin and valley splitting are bothneglected in our theoretical simulations, this two-fold degeneracy isonly observed inour experimental data but not visible in the calculatedinverse compressibility.Landau level spectrum at finite displacement fieldWhile the measurements at zero displacement fields show the exis-tence of four Dirac cones near charge neutrality, we now show thetunability of the Dirac cones followed by bandgap opening controlledbyD.Wenote in passing that for very smallD-fieldswhile the fourDiraccones will be individually gapped, there will be no overall gap in thespectrum due to the inherent energetic offset between the center andthe three off-center Dirac cones. While theoretical tight binding cal-culations of this regime are discussed in theMethods and are shown inSupplementary Fig. 4, this regime is not within our experimentalresolution.We first discuss our calculations and corresponding chargetransport measurements at constant small D, where already an overallbandgaphasopenedup in thepreviously linearDirac spectrumof BLG.This goes along with drastic changes of the band structure: the centercone diminishes whereas the three off-center cones (Fig. 3a, b) withhowever a parabolic dispersion remain, which we consequently referto as pockets.In the valence band, where the three off-center cones alreadydominate at D =0 due to electron-hole asymmetry, the number ofpockets changes from four to three at finite D (Fig. 3b), resulting in a8 -1k (10  m )x  0.50-0.5  1-1 0.8 -2E (meV) 8-1k (10 m)y0-0.5-1 0.5 1(b)10 -2n (10  cm )0-60.0 dG/dn (arb. units)0 1B ( T)0.50.40.30.20.1-0.1-0.2-0.3-0.4-0.5-4 -2 2 4 6-12 -10-8-6-4-22 4128610-2-4-6 2 4 6-8-10-12 81012-4 48-8-12 1210 -2n (10  cm )0-6 -4 -2 2 4 60.030.50.40.30.20.1B (T)-15 2 ∂μ/∂n (10  eVm )(a)  2E (meV) 8-1k (10 m)y0-0.5-1 0.5 18 -1k (10  m )x  0.50-0.5  1-1(c) (e)(d)B (T)E (meV)0 0.1 0.2 0.3 5 -5 00.1 0 = 0,3  = 0,4-1D = 0 mV nm , U = 0 meVΔ´ = 0 = 0,3  = 0,4Δ´ = 0-4-881240.20-12  0holeselectronsFig. 2 | Fermi surface contours and quantum Hall states of Bernal bilayer gra-phene atD =0. a, b Fermi surface contour of the conduction band (a) and valenceband (b) of bilayer graphene at different Fermi energy levels. All hopping para-meters (γ0,γ1,γ3 and γ4) and the on-site parameter Δ’ are included. c Calculatedinverse compressibility (∂μ/∂n) as a functionof charge carrier density andmagneticfield at zero electric displacement field (D =0) and temperature T =0.1 K. Thecorresponding quantum Hall states are labeled by numerals. The regions in whichquantum Hall states with filling factors ν = ±4 terminate are highlighted by dashedcircles.d Evolutionof Landau levels as a functionofmagneticfield atD =0.The fourlowest Landau levels are colored, whereas Landau levels contributing to holetransport are colored in red and Landau levels contributing to electron transportare colored in blue. The lowest red colored Landau level originates from the centermini Dirac cone, the other three lowest Landau levels originate from the three off-center mini Dirac cones. Filling factors are indicated by numerals. A larger versionof this plot is shown in Supplementary Fig. 3. e Derivative of the normalized con-ductance measured as a function of the charge carrier density and the magneticfield at D =0. Blue regions correspond to vanishing differential conductance (i.e., aconductance plateau). The slopes of the lowest quantum Hall states are traced bylines in the mirror image. Solid lines correspond to zero-interaction (free-particle)quantumHall states that allow for comparisonwith theory (Fig. 2c). Their colors areadapted from Fig. 2d. Dashed lines correspond to interaction-induced quantumHall states in which the spin or valley degrees of freedom is broken. The regions inwhich quantum Hall states with filling factors ν = ±4 end are highlighted by dashedcircles.Article https://doi.org/10.1038/s41467-024-47342-0Nature Communications |         (2024) 15:3133 3reordering of expected quantum Hall states (Fig. 3c–e)39,43. We expectthe LLL to be six-fold degenerate at low B due to the remaining threeleg pockets (3 pockets, 2 spins, the 2 valleys are degenerate,Fig. 3d, e)31,32 and the ν = −6 quantum Hall state is expected to be themost robust for hole doping (Fig. 3c). These theoretical considerationsare confirmed by our measurements. As shown in Fig. 3e, forD = 50mV/nm, the ν = −6 quantum Hall state can be resolved down tovery lowmagnetic fields of B = 100mT. Surprisingly, this also holds forthe ν = −3 quantum Hall state which could result from spin or valleypolarization at low B and low n due to Stoner ferromagnetism that canoccur in the vicinity of the Lifshitz transition29,30,44. At B = 600 mT, asudden change in the degeneracy of Landau level takes place for n <0,which can be attributed to the magnetic breakdown. Here the effectsof the trigonal warping are no longer relevant, and we can observe allinteger quantum Hall states. It is worth noting that at larger densities,quantumHall states with ν = −7 and ν = −9 appear below B = 500mT. Inthis regime, the Fermi energy level lies above the Lifshitz transition andquantum Hall states start to become valley and spin polarized withincreasing magnetic field.The effects of band flattening and disappearing of the low-energyDirac spectrumcanbe also seen in the conduction band (Fig. 3a)wherethe center pocket also becomes less prominent with increasing D.However, due to electron-hole asymmetry the center cone is stilldominating atU = 17meV and the band becomesflatterwith increasingU until U ≈ 60 meV45. At U = 17meV, the degeneracy of quantum Hallstates is not as much affected by trigonal warping as in the valenceband and quantum Hall states with even ν appear first in the magneticfield in our conductance measurements (Fig. 3e). Remarkably, thequantumHall stateswith ν = +3 and ν = +4disappear at amagnetic fieldof 0.5 T and then reemerge at about 0.6T (Fig. 3c, e) resulting from acrossing of two bands that correspond to different valleys (Fig. 3d).Controlling the D-field induced Lifshitz transitionsThe active control and lifting of the four-fold Dirac spectrum can bealso traced by controlling the displacement field at constant B(Fig. 4a, b). For example, atB =0.25 T (Fig. 4a) themagnetic breakdownhas already occurred for lowD resulting in the appearance of eight-folddegenerate quantum Hall states at D =0mV/nm (2 valleys, 2 spins, 2orbits). At 3mV/nm<|D | < 25mV/nm thequantumHall statewith ν = ±2appears due to valley polarization27 while at |D | > 20mV/nm the threepockets can be resolved individually and a crossover from a two-fold(two spins) to a three-fold degenerate Landau level spectrum (threepockets) appears at hole doping (yellow circle in Fig. 4a). Higher LL arefour-fold degenerate (2 valleys, 2 spins) above a Lifshitz transition sincetheir corresponding Fermi surface is fully connected. For larger B, e.g.,at B =0.4 T (Fig. 4b), the crossover to the parabolic shifts to larger Dwhere the three pockets aremore pronounced. Here also the crossingsof LL stemming from different valleys in the conduction band can bediscerned in the n vs. B plot.DiscussionIn conclusion, we present measurements demonstrating that bilayergraphene exhibits a highly tunable band structure at low energieswhere four distinctDirac cones, under tuning transversefield, undergoFig. 3 | Band structure and quantumHall states of bilayer graphene at finiteD.a, b Fermi surface contour of the conduction band (a) and valence band (b) ofbilayer graphene at different Fermi energy levels at U = 17meV. c Calculatedinverse compressibility (∂μ/∂n) as a function of charge carrier density and mag-netic field at U = 17meV and temperature T =0.1 K. The corresponding quantumHall states are labeled by numerals. Regions corresponding to Landau levelcrossings are marked by dotted circles. d Evolution of Landau levels as a functionof the magnetic field. Landau levels contributing to transport in valley K = 1 arecolored in purple, Landau levels contributing to transport in valley K = −1 arecolored in grey. e Derivative of the normalized conductance measured as a func-tion of the charge carrier density and the magnetic field at D = 50mV/nm. Solidlines correspond to the non-interaction induced quantumHall states that allow forcomparison with theory (Fig. 3c). Dashed lines correspond to interaction-inducedquantum Hall states in which the spin degree of freedom is broken. The corre-sponding quantumHall states are labeled by numerals. The regions correspondingto Landau level crossings are marked by dotted circles.Article https://doi.org/10.1038/s41467-024-47342-0Nature Communications |         (2024) 15:3133 4topological transitions and merge into a parabolic band or threepockets with a gapped parabolic dispersion. The topological transi-tions result in a complex series of Landau levels that we extract byvirtue of numerical diagonalization methods based on a realistic tightbinding model and measurements in high-quality hBN-encapsulatedsamples. These results show that the simple and seemingly well-understood Bernal bilayer graphene is a true example of a long-soughttunable Dirac material with linear dispersion at low energies and anaccessible band gap of up to 250meV14. This, coupled with the pre-sence of tunable van Hove singularities and a cascade of correlatedstates emerging at large D (that have been extensively discussedin earlier works29,30,44,45), makes BLG a promising material forexploring ordered states of interacting electrons. Furthermore, theseattributes make BLG an attractive material for developing low-energyfast electronics, including graphene-based digital logic devices withtrue off-states1,46 and improved types of graphene transistors in whichthe mobility characteristics of monolayer graphene remainuncompromised1. The role of electron interactions and trigonalwarping effects, in particular their impact on the renormalized many-body band structure, is an interesting topic for futurework that can beaddressed in high-quality freestanding BLG samples where interactioneffects dominate even at low electric fields15,27.MethodsTight-binding calculationsThe Landau level calculations were based on a realistic tight bindingmodel for the π-electrons as described in ref. 17.The model includes different hopping processes which aredescribed by different parameters: γ0 describes the tunneling forneighboring sites within a single graphene sheet; γ1 accounts forFig. 4 | Changing the degeneracy of quantum Hall states due to magneticbreakdown induced by an electric displacement field. a, b Derivative of thenormalized conductance measured as a function of the charge carrier density andelectric displacement field at B =0.25T (a) and B =0.4 T (b). Quantum Hall statesare labeled by numerals and traced by lines. Note, that the quantum Hall states aresymmetric for positive and negative values of D but the labelling was restricted toD <0 for better visibility. Transitions between quantum Hall states due to trigonalwarping and electron-hole asymmetry are highlighted by yellow dotted lines,crossings between Landau levels ofdifferent K valleys are highlighted by reddottedcircles. QuantumHall states are three-fold degenerate at largeDwhere the pocketsaremorepronounced, and themagnetic breakdownhasnot yet accord. Schematicsof Fermi contours corresponding to regions which different Fermi surface topol-ogies are shown in the insets. Apart from the number of Fermi surfaces, thedegeneracyof quantumHall states is also affected by spin and valley polarization atlarge B and D.Article https://doi.org/10.1038/s41467-024-47342-0Nature Communications |         (2024) 15:3133 5hopping processes between the aligned lattice sites of the two sheets,the so-called dimer sites. Finally, γ3 and γ4 are the hopping parametersbetween non-dimer sites; γ3 causes the trigonal warping of the Fermisea at low carrier densities. Further parameters are Δ0, an energy dif-ference between dimer and non-dimer sites, and the interlayerpotential U accounting for an out-of-plane displacement field. Themagnitudes of these parameters were taken from ref. 39.We used a two-band model which is reduced to the non-dimersites and includes direct hoppings via γ3 and γ4, as well as hoppings viadimer sites. It is expected to capture the correct low-energy physics ifγ0 and γ1 are the relevant energy scales of the problems.Writing π = ξpx + ipy and π + = ξpx � ipy, where ξ = +1,−1 denotesthe valley index and px and py are the x and y components of themomentum vector, one may decompose the Hamiltonian ash=h0 +hw +has +hU withh0 = � 12m0 π +ð Þ2π2 0 !ð1Þhw = v30 ππ + 0� �� v3a4ffiffiffi3p_0 π +ð Þ2π2 0 !ð2Þhas =2vv4γ1+Δ0v2γ21 !π +π 00 ππ +� �ð3ÞhU = � U21 00 �1� �� 2v2γ21π +π 00 �ππ +� �" #ð4Þwhere we have introduced the velocities v=ffiffi3paγ02_ , v3 =ffiffi3paγ32_ , v4 =ffiffi3paγ42_with the lattice constant a and the band mass parameter m= γ12v2. Adiagonalization yields momentum space representations of two com-ponent wave functions Ψ = 1ffiffi2p ψ1,ψ2� �, with the components ψ1 2ð Þspecifying the wave functions projections on the non-dimer sites.The first term (h0) describes a simple parabolic two-band modelwith the dominant hopping processes between non-dimer sites, thesecond (hw) accounts for trigonal warping and adds a small correctionto the parabolic term, the third one (has) introduces an intrinsicelectron-hole asymmetry and the fourth one (hU) describes couplingto external fields.The effect of the latter may be understood by realizing that adisplacement field results in a potential difference U between the twographene sheets. A rigorous estimation of the magnitude of thepotential requires a self-consistent computation that includes thescreening effects due to the redistribution of charge carriers betweenthe two sheets in the presence of a displacement field17.Without aiming for an exact quantitative description, we estimatethe rough magnitude of U via a simple plate-capacitor calculation asU= ecDwhere c = 3.35 Å is the interlayer spacing. This estimate assumesminor importance of screening effects at small displacement fields.The out-of-plane magnetic field of magnitude B is introduced byadding a contribution from the vector potential to canonicalmomenta. In the Landau gauge this modifies the momentum operatoraccording to π = � iξ_∂x + _∂y � ieBx and π + = � iξ_∂x � _∂y + ieBx.The operators satisfy the same algebraic relation as the ladderoperators of the harmonic oscillator, namely [π,π+] = −2ℏBeξ. Onemayuse this fact to define the operators a= � iffiffiffiffiffiffiffiffi2_eBp π and a+ = iffiffiffiffiffiffiffiffi2_eBp π + forthe K + -valley or a+ = � iffiffiffiffiffiffiffiffi2_eBp π and a= � iffiffiffiffiffiffiffiffi2_eBp π + for the K�-valley.These operators act like raising and lowering operators for an oscil-lator with the cyclotron frequency ωc =eBm . Their action on the oscilla-tor wave functions reads aϕn =ffiffiffinpϕn�1 and a +ϕn =ffiffiffiffiffiffiffiffiffiffin+ 1pϕn+ 1The Hamiltonian in the K + -valley may then be rewritten asĥ=� U2 +C + a+ a Aða+ Þ2 � iRaAa2 + iRa+ U2 +C�aa+" #ð5ÞwithA= _ωc 1 +γ1γ36γ20 !ð6ÞR =γ3γ0ffiffiffiffiffiffiffiffiffiffiffiffiγ1_ωcpð7ÞandC ± = _ωc2γ4γ0+Δ0 ±Uγ1� �ð8ÞIn the opposite valley, the roles of creators and annihilators areinterchanged.In the presence of trigonal warping interaction, the eigenvalueproblem has no analytic solution. In order to get a numerical solution,we used a matrix representation of the Hamiltonian in a truncatedbasis. This method is expected to give good results for the low-energyspectrum, as the discarded high-energy states do not hybridize withthose at low energy.In the K+-valley, using0�=ϕ00�  ,1�=ϕ10�  , n,σ�=1ffiffiffi2p ϕnσϕn�2�  ðn≥ 2Þ ,σ = ±� �ð9Þthe matrix elements evaluate to0 h 0
 �=�U23,σ h 0
 �=σiRffiffiffi2p ð10Þ1 h 1
 �=�U2+C + 4,σ h 1
 �= σiR ð11Þn,σ h n,σ0
 �= σδσσ 0Affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin n� 1ð Þp� U21� δσσ 0� �+12C + n+ σσ0C� n� 1ð Þ� � ð12Þn +3,σ h n,σ0
 �=σiRffiffiffiffiffiffiffiffiffiffin + 1p2ð13ÞAll remaining matrix elements follow the requirement of thematrix being Hermitian.Analogously, for the K�-valley we used0�=0ϕ0�  ,1i= 0ϕ1�  ,n,σi= 1ffiffiffi2p ϕn�2σϕn�  ðn≥ 2Þ,σ = ±� �ð14Þto obtain0 h 0
 �=U23,σ h 0
 �=�iRffiffiffi2p ð15Þ1 h 1
 �=U2+C� 4,σ h 1
 �= � iR ð16ÞArticle https://doi.org/10.1038/s41467-024-47342-0Nature Communications |         (2024) 15:3133 6n,σ h n,σ0
 �= σδσσ0Affiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffin n� 1ð Þp� U21� δσσ 0� �+12C + n� 1ð Þ+ σσ0C�n� � ð17Þandn+3,σ h n,σ0
 �=�σ0iRffiffiffiffiffiffiffiffiffiffin+ 1p2ð18ÞFor the calculation, an upper cutoff for the Landau level indexwasset at nmax = 300 by observing the convergence behavior of the low-lying energy levels.To illustrate the impact of γ3, γ4, and Δ0 on the low-energy bandand Landau levels, the band structure is shown in Supplementary Figs.3 and 5 with γ3, γ4, and Δ0 not included in (a), with only γ3 included in(b) and γ3, γ4, and Δ0 all included in (c). Also, Supplementary Figs. 3and 5 show the evolution of Landau levels as a function of themagneticfield, aswell as the inverse compressibility in the quantizedHall regimethat was calculated as a function of charge carrier density andmagnetic field.The electron-hole asymmetry in the band structure in combina-tion with trigonal warping gives rise to a semi-metallic behavior at alow interlayer bias U. The interlayer bias gaps out the cones individu-ally, however if sufficiently small, the gaps do not exceed the energeticoffset of the leg pocket with respect to the center pocket, such that aglobal gap only emerges at a certain threshold value of U ≈ 1meV(Supplementary Fig. 4). Below this interlayer potential, a finite densityof states remains in the overlap region of the electron and the holeband. Owing to the small energy window for this phenomenon toappear, this regime is not directly accessible by spectroscopy meth-ods. The magnetotransport measurements presented in the main textsupported by the Landau level calculations serve as an indirect prooffor the existence of this regime.Calculation of the inverse compressibilityThe inverse compressibility is defined by ∂μ∂n. In contrast to transportcoefficients such as the conductance G, this quantity can be extracteddirectly from the Landau level spectrum, which makes it a suitablequantity for theoretical considerations. While distinct from con-ductance, its behavior is related to thatof conductance: Divergences inthe inverse compressibility indicate positions of filled Landau levelswhere the conductance exhibits a plateau.By fixing the temperature T , it is straightforward to calculate thecharge carrier density n as a function of μ by populating the energylevels ϵi� �according to the Fermi function. Each level comes with adegeneracy of g = ABe2π_, where A is system area, leading to a chargecarrier density of nðμÞ=2PiBe2π_11 + expðϵi�μkBTÞ with the prefactor of 2accounts for spin degeneracy. The chemical potential μ nð Þ and theinverse compressibility were obtained by a numerical inversion of thisfunction. To this end, we defined a reservoir M of 105 equally spacedvalues of μ in a range that was roughly adjusted to the lowest andhighest Landau level energies accessible to the considered chargecarrier densities. For these values of μ, the carrier densities werecomputed. The contribution of the lower half of the spectrum (thehole Landau levels) had to be subtracted as an offset.Fixing the carrier density to n*, the corresponding chemicalpotential could then be determined as μ n*� �= min μ 2 Mjn μð Þ>n*� �.The inverse function defined in this way may attain all values from thereservoir from the bottom to the top when n is increased. Thenumerical error of this procedure is controlled by the spacing withinthe reservoir. For the practical implementation the temperatureentering in the Fermi distributionwas chosen to be 0.1 K. This is higherthan the usual cryogenic temperatures of the actual experimentalrealization. However, the resulting broadening may also mimic thefinite width of Landau levels due to disorder in the sample.Device fabricationWe performed quantum Hall measurements in two different devices.The results from the first device, denoted as Stack 99 in ref. 47, arepresented in the main text. The following section has been adoptedfrom ref. 47To observe low-energy band structure effects in Bernal bilayergraphene, it is crucial to have high-quality devices. Mechanical exfolia-tion was used to obtain bilayer graphene, hBN and graphite flakes thatwere then combined to a delicate 2D heterostructure consisting of abilayer grapheneflake that is encapsulated inhBN, serving as adielectricmaterial and graphite, serving as electrical gates. Two graphite flakes lieon top of the bilayer graphene and serve as electrical contacts. Afterexfoliation, thecomponents for bilayer grapheneheterostructureswereidentified and selected using optical microscopy, Raman spectroscopyand atomic force microscopy. Suitable flakes were chosen regarding totheir size, cleanliness, and homogeneity. It was ensured that all flakeswere free of dirt andwrinkles and that they hadnot been in contactwithchemicals to ensure that they were not contaminated. Thickness mea-surements via atomic forcemicroscopy revealed the lower hBN flake tobe 42-nm thick and the upper hBN flake to be 34-nm thick. A stampingtechnique48 was employed to successively pick up the selected flakesusing a home-made “stamping setup47”within an argon-filled glove box.A stamp consisting of a block of polydimethylsiloxane (PDMS) used as acushion layer, and a thin film of polycarbonate (PC) used as a transfermedium, was employed to pick up the individual flakes. A schematic ofthe stamping process is shown in Supplementary Fig. 1. A detailed step-by-step description of the stamping process is given in the caption ofSupplementary Fig. 2 and is based on optical images taken during theprocess of assembling the device.Before measuring the electronic properties of the 2D hetero-structures, electrical contacts were applied to the conducting layers.Contact lines and pads were added to the graphite contacts and gatesusing electron-beam lithography, metal evaporation, and wire bond-ing, enabling electrical connections from the sample to a chip carrier.The second device (refer to Section E) features four graphitecontacts instead of two that go across the bilayer graphene flake asshown in Supplementary Fig. 9a. Since stamping a device with fourgraphite contacts is more complex and requires additional stampingsteps compared to stamping a device with two graphite contacts, thebottom part, consisting of the lower hBN flake and the bottom gra-phite flake, was stamped first and was then melted onto a clean wafer.Using a new stamp, the upper hBN flake, four graphite flakes serving ascontacts and the bilayer graphene flake were then successively pickedup. Two of the graphite flakes serving as contacts were located next toeach other on the same wafer so that the graphite contacts could bepicked up within three steps. The stamp was then melted onto thepreviously cleaned bottom part. The graphite top gate was stamped ina last step. The thicknesses of the hBNflakesweredetermined as 58nm(bottom hBN flake) and 15 nm (top hBN flake) via atomic forcemicroscopy.Device characterizationIn our dually gatedbilayer graphene samples the charge carrier densityn as well as the electric displacement field D can be tuned individuallyvia the use of graphite top and bottom gates. They are defined asn= ε0εhBNðV t=dt +Vb=dbÞ=e ð19ÞandD= εhBNðVt=dt � Vb=dbÞ=2 ð20ÞArticle https://doi.org/10.1038/s41467-024-47342-0Nature Communications |         (2024) 15:3133 7where Vt (Vb) is the gate voltage applied to the top (bottom) gate, dt(db) the thickness of the upper (lower) hBN flake serving as a dielectric,e the charge of an electron, εhBN the dielectric constant of hBN and ε0the vacuum permittivity.In order to determine εhBN and to thereby assign n and D, integerquantum Hall plateaus at finite magnetic fields were aligned with theircorresponding slopes in the fan diagram. All observed LL crossingsshow excellent agreement with those observed previously (see ref. 29.where data from the same device is shown). For example, at B =0.4 T(Fig. 4b) one can see the known LL crossings of the ν = ± 1 and ν = 0quantum Hall states at D ≈ 15mV/nm as well as crossings at ν = ± 2 andD ≈0mV/nm.Having aligned the sample by using the slopes of thequantumHallstates would in principle allow to determine the contact resistance bycomparing the measured resistance with the expected quantum Hallresistance. However, due to the use of graphite contacts in a two-terminal device configuration the contact resistance increases linearlywith increasingmagneticfield (see, for example, Supplementary Fig. 7,more details are given in ref. 29). Furthermore, there is a line ofdecreased conductance across zero displacement field (Supplemen-taryFig. 6). This region is only bottom- but not top-gatedependent andstems from the region of the BLG that is located below the graphitecontacts where the top graphite contacts screen the field of the topgate but not of the bottom gate. Thus, the contact resistance is addi-tionally dependent on the top gate voltage (therefore also on n andD).To not confuse the reader with the line of decreased conductance weonly show the derivative of the conductance in the main text. Anotheradvantage of showing the derivative of the conductance is that itallows to track quantum Hall states at lower magnetic fields where theconductance is not fully quantized yet as traceable fluctuations nearincompressible quantum Hall states can appear27,49,50. Exemplary, theconductance including a subtracted contact resistance is shown inSupplementary Fig. 7 for D =0mV/nm and in Supplementary Fig. 8 forD = 50mV/nm. Here a contact resistance was subtracted that linearlyincreases with B. However, we did not account for the dependence onthe charge carrier density. Therefore, the resistance values are onlyvalid in a small density regime (negative densities close to the bandedge). In Supplementary Fig. 7, we included data taken at largermagnetic fields up to B = 1.5 T which we did not show in the main text.At B >0.6 T and D =0V/nm the quantumHall states are fully polarizeddue to additional valley imbalances implying a small residual dis-placement field. In agreement with previous studies15,27,41,42, the eveninteger quantum Hall states still show wider plateaus compared to theodd integer quantum Hall states.Measurements conducted in a second deviceElectrical measurements conducted in a second device are shown inSupplementary Fig. 9. The measurements show agreement with thetheoretical simulations and the electrical measurements discussed inthe main text.Data availabilityRelevant data supporting the key findings of this study are availablewithin the article and the Supplementary Information file. All raw datagenerated during the current study are available from the corre-sponding authors upon request.References1. Weiss, N. O. et al. Graphene: an emerging electronic material. Adv.Mater. 24, 5782–5825 (2012).2. Shinde, P. P. & Kumar, V. Direct band gap opening in graphene byBN doping: ab initio calculations. Phys. Rev. B 84, 125401 (2011).3. Wang, X. et al. N-doping of graphene through electrothermalreactions with ammonia. Sci. (N. Y., N. Y.) 324, 768–771 (2009).4. Rani, P. & Jindal, V. K. Designing band gap of graphene by B and Ndopant atoms. RSC Adv. 3, 802–812 (2013).5. Guinea, F., Katsnelson, M. I. & Geim, A. K. Energy gaps and a zero-field quantum Hall effect in graphene by strain engineering. Nat.Phys. 6, 30–33 (2010).6. Peng, Z., Chen, X., Fan, Y., Srolovitz, D. J. & Lei, D. Strain engineeringof 2D semiconductors and graphene: from strain fields to band-structure tuning and photonic applications. Light Sci. Appl. 9,190 (2020).7. Ni, Z. H. et al. Uniaxial strain on graphene: Raman spectroscopystudy and band-gap opening. ACS Nano 2, 2301–2305 (2008).8. Gui, G., Li, J. & Zhong, J. Band structure engineering of graphene bystrain: first-principles calculations. Phys. Rev. B 78, 75435 (2008).9. Hunt, B. et al. Massive dirac fermions and Hofstadter butterfly in aVan der Waals heterostructure. Science 340, 1427–1430 (2013).10. Ribeiro-Palau, R. et al. Twistable electronics with dynamicallyrotatable heterostructures. Science 361, 690–693 (2018).11. Inbar, A. et al. The quantum twisting microscope. Nature 614,682–687 (2023).12. Novoselov, K. S. et al. Electric field effect in atomically thin carbonfilms. Science 306, 666–669 (2004).13. Ohta, T., Bostwick, A., Seyller, T., Horn, K. & Rotenberg, E. Con-trolling the electronic structure of bilayer graphene. Science 313,951–954 (2006).14. Zhang, Y. et al. Direct observation of a widely tunable bandgap inbilayer graphene. Nature 459, 820–823 (2009).15. Weitz, R. T., Allen, M. T., Feldman, B. E., Martin, J. & Yacoby, A.Broken-symmetry states in doubly gated suspended bilayer gra-phene. Science 330, 812–816 (2010).16. Icking, E. et al. Transport spectroscopy of ultraclean tunableband gaps in bilayer graphene. Adv. Elect. Mater. 8, 2200510(2022).17. McCann, E. & Koshino, M. The electronic properties of bilayer gra-phene. Rep. Prog. Phys. 76, 56503 (2013).18. McCann, E., Abergel, D. S. & Fal’ko, V. I. The low energy electronicband structure of bilayer graphene. Eur. Phys. J. Spec. Top. 148,91–103 (2007).19. Novoselov, K. S. et al. Unconventional quantum Hall effect andBerry’s phase of 2π in bilayer graphene. Nat. Phys. 2,177–180 (2006).20. Velasco, J. et al. Transport spectroscopy of symmetry-brokeninsulating states in bilayer graphene. Nat. Nanotechnol. 7,156–160 (2012).21. Xiang, F. et al. Intra-zero-energy Landau level crossings in bilayergraphene at high electric fields. Nano Lett. 23, 9683–9689 (2023).22. Mayorov, A. S. et al. Interaction-driven spectrum reconstruction inbilayer graphene. Sci. (N. Y., N. Y.) 333, 860–863 (2011).23. McCann, E. & Fal’ko, V. I. Landau-level degeneracy and quantumHall effect in a graphite bilayer. Phys. Rev. Lett. 96, 86805 (2006).24. Novoselov, K. S. et al. Two-dimensional gas of massless Dirac fer-mions in graphene. Nature 438, 197–200 (2005).25. Gail, R. D., Goerbig, M. O. &Montambaux, G. Magnetic spectrum oftrigonally warped bilayer graphene: Semiclassical analysis, zeromodes, and topological winding numbers. Phys. Rev. https://doi.org/10.1103/PhysRevB.86.045407 (2012).26. Partoens, B. & Peeters, F. M. From graphene to graphite: electronicstructure around the K point. Phys. Rev. B 74, 75404 (2006).27. Geisenhof, F. R. et al. Quantum anomalous Hall octet driven byorbital magnetism in bilayer graphene. Nature 598, 53–58 (2021).28. Geisenhof, F. R. et al. Impact of electric field disorder on broken-symmetry states in ultraclean bilayer graphene. Nano Lett. 22,7378–7385 (2022).29. Seiler, A. M. et al. Quantum cascade of correlated phases in tri-gonally warped bilayer graphene. Nature 608, 298–302 (2022).Article https://doi.org/10.1038/s41467-024-47342-0Nature Communications |         (2024) 15:3133 8https://doi.org/10.1103/PhysRevB.86.045407https://doi.org/10.1103/PhysRevB.86.04540730. Zhou, H. et al. Isospin magnetism and spin-polarized super-conductivity inBernal bilayergraphene.Science375, 774–778 (2022).31. Varlet, A. et al. Tunable fermi surface topology and Lifshitz transi-tion in bilayer graphene. Synth. Met. 210, 19–31 (2015).32. Varlet, A. et al. Anomalous sequence of quantum Hall liquidsrevealing a tunable lifshitz transition in bilayer graphene. Phys. Rev.Lett. 113, 116602 (2014).33. Zibrov, A. A. et al. Emergent Dirac gullies and gully-symmetry-breaking quantum Hall states in ABA trilayer graphene. Phys. Rev.Lett. 121, 167601 (2018).34. Winterer, F. et al. SpontaneousGully-polarized quantumHall Statesin ABA Trilayer graphene. Nano Lett. 22, 3317–3322 (2022).35. Campos, L. C. et al.Quantumandclassical confinement of resonantstates in a trilayer graphene Fabry-Pérot interferometer. Nat. Com-mun. 3, 1239 (2012).36. Zhou, H. et al. Half and quarter metals in rhombohedral trilayergraphene. Nature 598, 429–433 (2021).37. Zhou, H., Xie, T., Taniguchi, T., Watanabe, K. & Young, A. F. Super-conductivity in rhombohedral trilayer graphene. Nature 598,434–438 (2021).38. Shi, Y. et al. Tunable Lifshitz transitions and multiband transportin tetralayer graphene. Phys. Rev. Lett. 120, 96802 (2018).39. Jung, J. & MacDonald, A. H. Accurate tight-binding models forthe π bands of bilayer graphene. Phys. Rev. B 89, 35405 (2014).40. Feldman, B. E., Martin, J. & Yacoby, A. Broken-symmetry states anddivergent resistance in suspended bilayer graphene. Nat. Phys. 5,889–893 (2009).41. Lee, K. et al. Chemical potential and quantum Hall ferromagnetismin bilayer graphene. Science 345, 58–61 (2014).42. Li, J., Tupikov, Y., Watanabe, K., Taniguchi, T. & Zhu, J. EffectiveLandau level diagram of bilayer graphene. Phys. Rev. Lett. 120,47701 (2018).43. Nilsson, J., Castro Neto, A. H., Guinea, F. & Peres, N. M. R. Electronicproperties of bilayer and multilayer graphene. Phys. Rev. B 78,45405 (2008).44. La Barrera et al. Cascade of isospin phase transitions in Bernal-stacked bilayer graphene at zero magnetic field. Nat. Phys. 18,771–775 (2022).45. Seiler, A. M. et al. Interaction-driven (quasi-) insulating groundstates of gapped electron-doped bilayer graphene, Preprint athttp://arxiv.org/pdf/2308.00827v1 (2023).46. Kim, K., Choi, J.-Y., Kim, T., Cho, S.-H. & Chung, H.-J. A role forgraphene in silicon-based semiconductor devices. Nature 479,338–344 (2011).47. Seiler, A. M. Correlated phases in the vicinity of tunable van Hovesingularities in Bernal bilayer graphene. (Georg-August-UniversitätGöttingen, 2023).48. Purdie, D. G. et al. Cleaning interfaces in layered materials hetero-structures. Nat. Commun. 9, 5387 (2018).49. Lee, D. S., Skákalová, V., Weitz, R. T., Klitzing, Kvon & Smet, J. H.Transconductance fluctuations as a probe for interaction-induced quantum Hall states in graphene. Phys. Rev. Lett. 109,56602 (2012).50. Kumar, M., Laitinen, A. & Hakonen, P. Unconventional fractionalquantum Hall states and Wigner crystallization in suspended Cor-bino graphene. Nat. Commun. 9, 2776 (2018).AcknowledgementsR.T.W. and A.M.S. acknowledge funding from the Deutsche For-schungsgemeinschaft (DFG, German Research Foundation) under theSFB 1073 project B10. N.J. acknowledges funding from the InternationalCenter for Advanced Studies of Energy Conversion (ICASEC). R.T.W.acknowledges funding from the DFG SPP 2244. K.W. and T.T. acknowl-edge support from the JSPS KAKENHI (Grant Numbers 21H05233 and23H02052) and World Premier International Research Center Initiative(WPI), MEXT, Japan. The work at MIT was supported by the Science andTechnology Center for Integrated Quantum Materials, National ScienceFoundation grant No. DMR1231319.Author contributionsA.M.S. and M.S. fabricated the devices and conducted the measure-ments with the help of N.F. and F.F. A.M.S. conducted the data analysis.N.J. performed the theoretical simulations supervised by L.S.L., N.J.,Z.D., and L.S.L. contributed to the theory part. K.W. and T.T. grew thehexagonal nitride crystals. All authors discussed and interpreted thedata. R.T.W. supervised the experiments and the analysis. The manu-script was prepared by A.M.S., N.J., L.S.L. and R.T.W. with input from allauthors.FundingOpen Access funding enabled and organized by Projekt DEAL.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-024-47342-0.Correspondence and requests for materials should be addressed toR. Thomas Weitz.Peer review information Nature Communications thanks the anon-ymous reviewers for their contribution to the peer review of this work. Apeer review file is available.Reprints and permissions information is available athttp://www.nature.com/reprintsPublisher’s note Springer Nature remains neutral with regard to jur-isdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing,adaptation, distribution and reproduction in any medium or format, aslong as you give appropriate credit to the original author(s) and thesource, provide a link to the Creative Commons licence, and indicate ifchanges were made. The images or other third party material in thisarticle are included in the article’s Creative Commons licence, unlessindicated otherwise in a credit line to the material. If material is notincluded in the article’s Creative Commons licence and your intendeduse is not permitted by statutory regulation or exceeds the permitteduse, you will need to obtain permission directly from the copyrightholder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2024Article https://doi.org/10.1038/s41467-024-47342-0Nature Communications |         (2024) 15:3133 9http://arxiv.org/pdf/2308.00827v1https://doi.org/10.1038/s41467-024-47342-0http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Probing the tunable multi-cone band structure in Bernal bilayer graphene Results Tight binding calculations of Landau�Levels Quantum Hall measurements at zero displacement�field Landau level spectrum at finite displacement�field Controlling the D-field induced Lifshitz transitions Discussion Methods Tight-binding calculations Calculation of the inverse compressibility Device fabrication Device characterization Measurements conducted in a second�device Data availability References Acknowledgements Author contributions Funding Competing interests Additional information