# Fileset [s41467-025-56141-0.pdf](https://mdr.nims.go.jp/filesets/53de3aa0-e92c-43b3-b6e7-600e2672782f/download) ## Creator Jin Jiang, Qixuan Gao, Zekang Zhou, Cheng Shen, Mario Di Luca, Emily Hajigeorgiou, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Mitali Banerjee ## Rights [Creative Commons BY-NC-ND Attribution-NonCommercial-NoDerivs 4.0 International](https://creativecommons.org/licenses/by-nc-nd/4.0/) ## Other metadata [Direct probing of energy gaps and bandwidth in gate-tunable flat band graphene systems](https://mdr.nims.go.jp/datasets/fb61a423-a967-4a56-af55-ec0090a36e9e) ## Fulltext Direct probing of energy gaps and bandwidth in gate-tunable flat band graphene systemsArticle https://doi.org/10.1038/s41467-025-56141-0Direct probing of energy gaps andbandwidth in gate-tunable flat bandgraphene systemsJin Jiang1, Qixuan Gao1, Zekang Zhou1, Cheng Shen1, Mario Di Luca1,Emily Hajigeorgiou1, Kenji Watanabe 2, Takashi Taniguchi 3 &Mitali Banerjee 1,4Moiré systems featuring flat electronic bands exhibit a vast landscape ofemergent exotic quantum states, making them one of the resourceful plat-forms in condensed matter physics. Tuning these systems via twist angle andthe electric field greatly enhances our comprehension of their strongly cor-related ground states. Here, we report a technique to investigate the nuancedintricacies of band structures in dual-gated multilayer graphene systems. Weutilize the Landau levels of a decoupled monolayer graphene to extract theelectric field-dependent bilayer graphene charge neutrality point gap. Then,we extend this method to analyze the evolution of the band gap and the flatbandwidth in twisted mono-bilayer graphene. The band gap maximizes at thesame displacement field where the flat bandwidth minimizes, concomitantwith the emergence of a strongly correlated phase. Moreover, we extractinteger and fractional quantum Hall gaps to further demonstrate the strengthof thismethod. Our technique paves the way for improving the understandingof electronic band structures in versatile flat band systems.Understanding the band structure of a system is of fundamental sig-nificance in condensed matter physics. For instance, the linear conicalenergy spectrum of monolayer graphene (MG) gives rise to two in-equivalent K points, which results in the observation of the half-integerquantumHall effect1–3. A small twist angle between twographene sheetsinduces a long-range periodic pattern, resulting in angle-dependentmoiré Bloch bands4. Particularly, near a “magic angle” (~ 1.1°) of rotationbetween the layers, the coupling between two graphene sheets isstrongly reinforced, and the low-energy moiré bands become verynarrow, almost without any dispersion (flat)5. This gives rise to exoticquantum phases, including strongly correlated insulating states5–12,unconventional superconductivity9–15, ferromagnetism16–18, etc.Due to the flexible carrier density control by dual electrostaticgating, twistedmultilayer graphene flat band systems have become anappealing platform for studying strongly correlated quantum phases.For example, orbital Chern insulators at different integer filling factorswere observed in twisted monolayer-bilayer graphene (TMBG)19,20. Incontrast, spin-polarized insulating states were observed in twisteddouble-bilayer graphene (TDBG)21–23, etc.The dual gate tuning provides a fundamental degree of freedomto understand the rich phasesmentioned above. To study these exoticphases, an experimental technique that can accurately measure theresponse of the device under an applied electricfield is crucial. Severalsingle-gated local probe techniques have been developed to enrich theunderstanding of band structure in two-dimensional materials, likescanning tunneling microscopy/spectroscopy (STM/STS)4,8,23–27, singleelectron transistor (SET)28, planar tunneling junction29 and nano-SQUID30. In addition to local probes, global measurements, includingReceived: 30 June 2024Accepted: 9 January 2025Check for updates1Laboratory of Quantum Physics (LQP), Institute of Physics, École Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland. 2Research Centerfor Electronic and Optical Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 3Research Center for MaterialsNanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 4Center for Quantum Science and Engineering (QSECenter), École Polytechnique Fédérale de Lausanne (EPFL), 1015 Lausanne, Switzerland. e-mail: mitali.banerjee@epfl.chNature Communications | (2025) 16:1308 11234567890():,;1234567890():,;http://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/0009-0009-5690-5822http://orcid.org/0009-0009-5690-5822http://orcid.org/0009-0009-5690-5822http://orcid.org/0009-0009-5690-5822http://orcid.org/0009-0009-5690-5822http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-025-56141-0&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-025-56141-0&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-025-56141-0&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-025-56141-0&domain=pdfmailto:mitali.banerjee@epfl.chwww.nature.com/naturecommunicationsmagneto-transport5–15,19–23, nano angle-resolved photoemission spec-troscopy (Nano-ARPES)31,32, electronic compressibility33, nano-infraredimaging34, and Fourier transform infrared (FTIR) spectroscopy35,36 arewidely adopted. However, the limiting factor in most of these afore-mentioned techniques, like STM/STS and nano-ARPES, is that thesamples are designed only with one gate.Although electronic compressibility measurements, nano-infrared imaging, Nano-SQUID-on-tip microscopy, and FTIR spectro-scopy can investigate dual gate devices, these measurements eachhave their drawbacks30–32,34–36. For example, the large beam spot(~1mm) used in FTIR spectroscopy, compared with the size of typicaldevices, makes the experiments substantially challenging. While con-ventional transport measurements for extracting the energy gap, suchas resistance vs. temperature (R−T) measurements, have been widelyadopted, the thermally activated gap overshadows other details. Inaddition, this method cannot extract the bandwidth.Recently, the unique band structure in twisted trilayer graphenehas enabled the dissociation of intertwined bands and the quantifica-tion of energy gaps15. Similarly, using decoupled monolayer graphenehas facilitated the uncovering of spin ordering in twisted bilayergraphene37. These advancements inspire further investigation into theelectronic band structures of electric field-tunable systems, includingthe direct probing of energy gaps and bandwidths in gate-tunable flatband graphene systems.ResultsGate-tunable CNP gap in BGTo benchmark our technique, we perform electrical transport mea-surements on a dual-gated Hall bar fabricated on a monolayergraphene (MG) stacked on a bilayer graphene (BG) with a large twistangle (see Fig. 1a, b). The calculated band structures of TMBG indicatethat a large twist angle θ (≥ 10°) between MG and BG is necessary toeffectively decouple the two layers (Supplementary Figs. 1, 2)38.We trace the CNP of this decoupled MG (DMG), which is deno-ted by the red dashed line in phase diagrams of Fig. 1d, e, LLDN =0 andN is the Landau level index. In our device configuration, the BG iscloser to the bottom gate. Hence, the bottom gate cannot effec-tively tune the DMG due to the screening from the BG, which isevident from the trace of the CNP line of the DMG as it is almostparallel to the bottomgate direction.Moreover, we observemultipleLandau levels splitting under a finite magnetic field (SupplementaryFig. 3). These Dirac Landau levels of the DMG (LLDN) shuffle along theVtg direction when Vbg is fixed.Using Landau level spectroscopy, we estimated the ΔCNP value(see Methods). Since the DMG is decoupled from the BG, wecan directly estimate the chemical potential extracted from the LLDN .Figure 1f shows the band crossings between the LLDN and the CNP gapedges of the BG denoted by A and B. In order to estimate ΔCNP, we takeLLDN = 3 (Fig. 1g) as an example. μN,A, μN,B denote chemical potentials ofthe left and right CNP edges (indicated by two horizontal black dashedlines and inset schematics), following μN =μN =0 + vF �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2e_N sgnðNÞBp,where Fermi velocity vF = 1 × 106m/s15,39–41 and ℏ is the reduced Planckconstant. The chemical potential difference ΔN = ∣μN,A − μN,B∣ repre-sents the energy gap size of the bilayer graphene CNP.In Fig. 2a, at both points A and B, the chemical potential extractedthrough the different LLDN changes asffiffiffiBp, implying that A and B bothtrack the same DMG Landau levels. For different LLDN , theΔCNP valuesremain consistent, with small variation (Fig. 2c).Fig. 1 | Twisted decoupled monolayer graphene (DMG) with bilayergraphene (BG). a The schematic of the device and measurement configuration.b The schematic of DMG stacked on top of the BG. The blue arrow pointing to theDMG indicates thepositivedirectionof the applieddisplacementfieldD.cTheopticalimageof theDMG+BG (Device_A1) device.d Longitudinal resistanceRxx as a functionof the total carrier density (ntot) andD atB =0.5 T. The red (black) arrow indicates thedirection of the bottom gate (top gate). The Dirac cone of the DMG splits into manyDirac Landau levels along the top gate (Vtg) direction. e A colored schematic diagramof the main features of n−D mapping in Fig. 1d. Magenta anti-S-shaped regionrepresents theCNPof theBG (ΔCNP). The twoboundariesof theCNPgapof theBGareindicated by faint dashed lines intersecting at α, where D =0.13V/nm. A and B areedges of theCNPgap of the BG for the same displacement field (D =0V/nm). A seriesof Dirac LLs of DMG (red dashed line along the diagonal) shuffle along the top gatedirection. LLDN is theNthDiracLandau level ofDMG.The letters inside theparenthesesrepresent the carrier types for BG (first) and DMG (second). “e” stands for electrons,and “h” stands for holes. f Landau fan diagram of Rxx near the CNP of the BG atD =−0.5 V/nm. White dots A and B represent the band crossing between the thirdDirac LL of the DMG and the CNP of the BG. g A schematic diagram of extracting theCNPgap of BG. The insets schematize the third LL of DMG located at the edges of theCNP gap of the BG. The gap is extracted from the change in the chemical potential ofLL. The chemical potential is calculated from μN =μN =0 + vF �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2e_N sgnðNÞBp.Article https://doi.org/10.1038/s41467-025-56141-0Nature Communications | (2025) 16:1308 2www.nature.com/naturecommunicationsWe then extend our analysis to include various displacementfields (D), allowing us to observe the electric field dependence of ΔCNPin BG of two devices (Fig. 2d). The ΔCNP in BG increases significantlywith increasing ∣D∣ as a result of the difference in Coulomb potentialbetween the layers. The estimated ΔCNP at D = −0.5 V/nm is83.2 ± 2.3meV of Device_A1, is in agreement with previous findings42,43.Next, we take Device_A1 as the main device, and we observe afinite built-in gap (17.2 ± 1.3meV) at the CNP at D = 0V/nm,which results from layer polarization of the valence andconduction bands44,45. Furthermore, we observe the evolution ofthe CNP gap closing and reopening with increasing positive D(Fig. 2e–g and Supplementary Figs. 4–6). Particularly, atD3 = 0.13 V/nm, the gap is completely closed by a sign change ofCoulomb potential difference46, which is indicated by the α point inFig. 1e. This behavior results from the asymmetric response of thedisplacement field and layer polarization. The weaker response of theCNP gap at positive D (pointing to MG, D =0.25 V/nm, ΔCNP is14.3 ± 2.7meV) compared to negative D (pointing to BG, D = −0.25 V/nm, ΔCNP is 55.3 ± 4.0meV) supports this observation. In addition, weobserve that the CNP gap does not fully close within the moderatenegative D range (D2 to D1). Further investigation is needed into theband structure, particularly considering the overlap between the CNPof the DMG and the BG.Bandwidth and bandgap of TMBGTo establish the robustness of this technique, we study a more com-plicated electric field-tunable flat band system-TMBG. The DMG wasplaced on top of the monolayer graphene of the small twist angleTMBG (1.14° ± 0.02°) (Fig. 3a). The schematic compares the moirépatterns of TMBG and DMG+TMBG, the moiré length is equal to~13 nm, and does not change significantly with the addition of theDMG. A n−D map at B = 1 T is shown in Fig. 3d (Supplementary Fig. 7).We observe similar multiple Landau levels LLDN splitting as in Fig. 1d.Then the strongly correlated states are observed at all integer fillingfactors and are found to vary with the D.Figure 3f shows the extracted flat bandwidth and bandgap as afunction of D (see Methods and Supplementary Figs. 8–14). When thecorrelated state appears at the same D as point α (D = 0.33 V/nm) withfilling factor 3, the flat bandwidth is estimated to be about73.5 ± 1.8meV, which is in good agreementwith the experimentalworkin TMBG (70 ± 10meV)47. Furthermore, due to the narrow width of theLLDN (Fig. 2a), the experimental error is smaller ( 10°), and intermediate dis-placement field (D)15,39–41, where the Fermi velocity vF = 1 × 106m/s andℏ is reduced Planck constant. The steps involved in this observationareoutlined below:(i) Identify the gap edge of BG. Since in the gap of BLG, there’s noDOS, and the charging electrons will accumulate in LL of DMGshown in Fig. 1f, g. We obtain the exact CNP gap edge points Aand B for every LL index N by tracking the crossing state thatcorresponds to partial filling of Dirac LLs at the Fermi level,which is marked by an Rxx peak, indicating the edge of incom-pressible gaps of Dirac LLs. The Rxx peak represents a generalmark of extracting the energy scale without considering thelifting of LLs degeneracy. This process is illustrated by the reddashed line and thebluedashed line in the toppanel, Region I, ofSupplementary Fig. 8c and the right panel of Supplemen-tary Fig. 9c.(ii) Identify the Landau level index of DMG.(iii) To determine the magnetic field corresponding to the CNP gapedge points A and B at the gap boundary, we refer to Fig. 1f andg. The magnetic field dictates the chemical potential at eachpoint for a given Nth Landau level (LL). By fitting the high--12 -11 -10 -9 -8 -7 (cm -2 ) 1011010203040 Rxx ()1012141618 (h/(e2 *Rxy)-7 -6.5 -64.24.254.34.354.42 2.2 ( )Rxy k-7 -6.5 -64.24.254.34.354.4 B (T)0 200 400 ( )Rxx-7 -6.6 -6.2 (cm-2) 10110200400 Rxx ()2.02.22.42.6 Rxy () k-12 -10 -8 -6 -4 -2 (cm-2) 101123456 B (T)0 2 ( )Rxy k-12 -10 -8 -6 -4 -2 (cm-2) 101123456 B (T)0 5 10 ( )Rxx k (cm-2) 1011ABFQHE (cm-2) 1011D = 0.53 V/nmB = 4.35 TB = 3.4 TB = 3.4 TB = 3.4 Tν = -11+¹⁄¹⁄��(h/e2)¹⁄��(h/e2)¹⁄� (h/e2)ν = -11-11 -11-11+¹⁄ -11+¹⁄ntotntotntotntotabcdefntot ntotD = 0.53 V/nm LLDl = -6 LLT-12 -10 -8 -6 -4 -2 (cm-2) 101123456 B (T)-18-14-13-11-10 -9 -8-9-9-9-10-10-10-11-12-11-8-8-12 LLDl = -5 LLDl = -4 LLDl = -3ntotFig. 5 | Chern number cascade and fractional quantum Hall gaps.a, b Longitudinal resistance Rxx and Hall resistance Rxy as a function of ntot and B.c Schematic of quantum oscillation of Fig. 5a. LLT is TMBG flat band LLs composedwith DMG LLs Chern number offset. d Filling factor dependence of longitudinalresistance (blue dash line in Fig. 5a) and Hall resistance plateaus (red dash line inFig. 5b). e zoomed Landau fan taken from yellow rectangular in Fig. 5a, b. The solidgreen line indicates ν = −11 integer quantum Hall state. The green dashed linedenotes a fractional quantum Hall state of ν = −11 + 1/3. A, B are band edges of thefractional quantum Hall gap of νf = −6 + 1/3. The estimated gap△N = ∣μA − μB∣ =0.56± 0.14meV. f Evidence of a fractional quantum Hall state. Thedata of Rxx and Rxy are taken along the blue and red dash lines in Fig. 5e.Article https://doi.org/10.1038/s41467-025-56141-0Nature Communications | (2025) 16:1308 6www.nature.com/naturecommunicationsresistance states Rxx peak at the gap edge, we extracted thevalues of B1 and B2 along with their respective uncertainties (σ).Owing to the exceptional quality of our devices, thesemeasure-ments offer remarkably high energy resolution.(iv) Using formula μN =μN =0 + vF �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2e_N sgnðNÞBpto extract thechemical potential of the edges and then get the CNP gap value.Weobserved that thedegeneracy of Landau levels (LLs) is liftedbyapplying higher magnetic fields, altering the energy spectrum ofmassless Dirac fermions. Consequently, we introduce a generalizedenergy spectrum for graphene Landau levels that incorporates theeffects of degeneracy lifting, such as Zeeman splitting and valleypolarization.The modified equation is shown as below, EN, s,σ = vFffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2e_∣N∣Bp+12 gsμBBs +ΔσK3,41,59–61, where N is Landau level index, ℏ is ReducedPlanckconstant, gs is sping-factor (typically ~ 2 in graphene),μB is Bohrmagneton, s is the spin state (+1 for up, −1 for down), Δσ is Valleysplitting energy, representing energy differences between the K and K 0valleys, K is Valley index (±1).The degeneracy-lifted sub-LLs create incompressible Landau levelgaps in the partially filledDirac LLs, as observed in the originalRxxpeak(Supplementary Figs. 8, 9). In Supplementary Fig. 8c, from region I toIII, we track the crossing state at point A1, which corresponds to thecrossing between the lowest incompressible degeneracy-lifted Diracsub-LL (l = −6) and the flat band edge. This crossing is marked by aminimum in Rxx and corresponding kinks in Rxy (SupplementaryFig. 8d, e). A similar rule is applied to the isospin-polarized Dirac LLs(LLfN =�2,A2). Further details can be found in Supplementary Fig. 9.The errors in the energy extraction of gaps or bandwidths arisefrom three main factors:(i) The broadening of Landau levels.It plays a crucial role in determining the accuracy of energyextraction for gaps or bandwidths. This broadening can result fromseveral factors, including disorder, random strain fields, temperatureeffects, electron-phonon coupling, and many-body interactions. Toachieve a narrower bandwidth for Dirac LLs and reduce these errors, itis essential to use clean and uniform samples.(ii) Zeeman splitting, isospin-polarized sub-LLs, and fully-degeneracy lifting sub-LLs.As the magnetic field increases, the Zeeman splitting effect andthe lifting of LL degeneracy become significant and should not beneglected. To accurately extract the chemical potential from the LLspectrum of Dirac fermions, it is necessary to modify the spectrumequation, as shown above.For the Zeeman splitting term59,60, 12 gsμBBs, using two points atdifferent magnetic fields can introduce some errors. However, thisterm is relatively small compared to the gaps or bandwidths we mea-sure. For instance, for a change of 1 T magnetic field, the contributionfrom pure Zeeman splitting to Dirac LLs is approximately1 μBB ~ 0.058meV (~0.67 K)59, which is negligible in comparison to theCNP gap and bandwidth observed in the main text.For measuring the fractional quantum Hall gap, the magneticfields atpoints A andB in Fig. 5e are 4.310 ±0.005 and4.270 ± 0.005 T,respectively. The 0.04 T difference in magnetic field results in a pureZeeman splitting contribution of 0.04 × 0.058meV =0.002meV,which is much smaller than the gap we measured (0.56 ± 0.14meV).Therefore, we neglected the Zeeman splitting term in themain text forthe decoupled system when measuring the chemical potentials at thetwo edge points under different magnetic fields.At high magnetic fields, the Dirac LLs split into two isospin-polarized sub-LLs or four fully lifted degeneracy LLs, as described bythe LL filling factor l. To minimize the effects of this splitting, we trackthe crossings between the lowest incompressible, degeneracy-liftedDirac sub-LLs (l = −4) and the target state. These crossings are markedby a minimum in Rxx and corresponding kinks in Rxy, (SupplementaryFigs. 8d, 9, 10f)In Supplementary Fig. 9, we specifically discuss the influence onbandgap extraction from two different Dirac LLs: The left panel of thenormal case and the right panel for the isospin-polarized case. Wefound that the difference in the bandwidth extracted for both cases issmall, and we averaged the results to reduce errors.(iii) Valley splittingWenotice that the energy separation between theDirac LLs of twosub-LLs (l) is significantly different. In Supplementary Fig. 10d, weclearly observe the A1 and A2 sub-LLs of l = −6, −4, with a substantialmagnetic field separation (BA2 − BA1 = 0.5 T). In contrast, at point B, thisseparation is much smaller, making it difficult to clearly differentiatebetween the sub-LLs. This suggests that the observed effects mightarise from valley splitting. See more discussion in Supplemen-tary Fig. 10.Wemust acknowledge that the valley effect cannot be completelyignored, and further theoretical work is needed to incorporate thevalley splitting term into the modification of the Dirac fermion LLspectrum.We should also note that our technique is particularly suitable fordetecting systems whose band structures are not highly sensitive tomagnetic fields. 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K.W. and T.T. acknowledge support from the JSPS KAKENHIArticle https://doi.org/10.1038/s41467-025-56141-0Nature Communications | (2025) 16:1308 8www.nature.com/naturecommunications(Grant Numbers 20H00354 and 23H02052) and World Premier Inter-national Research Center Initiative (WPI), MEXT, Japan.Author contributionsJ.J. and M.B. conceived the project. J.J. made all stacks and fabricatedthe devices. Q.G. fabricated and analyzed the data of Device_A2. J.J.performed the measurements with the help of Z.Z. J.J. has analyzed thedata with inputs from C.S. and Z.Z. K.W. and T.T. provided the hBNcrystals. J.J. wrote the manuscript with inputs from M.B., M.D., E.H. aswell as all other authors.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-025-56141-0.Correspondence and requests for materials should be addressed toMitali Banerjee.Peer review information Nature Communications thanks Rui Wang,Cheng Zhang and the other anonymous reviewers for their contributionto the peer review of this work. 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To view a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.© The Author(s) 2025Article https://doi.org/10.1038/s41467-025-56141-0Nature Communications | (2025) 16:1308 9https://doi.org/10.1038/s41467-025-56141-0http://www.nature.com/reprintshttp://creativecommons.org/licenses/by-nc-nd/4.0/http://creativecommons.org/licenses/by-nc-nd/4.0/www.nature.com/naturecommunications Direct probing of energy gaps and bandwidth in gate-tunable flat band graphene systems Results Gate-tunable CNP gap in BG Bandwidth and bandgap of TMBG Integer and fractional quantum Hall gaps Methods Device fabrication Measurements Twisted angle determination Extracting the gaps and bandwidths Data availability Code availability References Acknowledgements Author contributions Competing interests Additional information