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Georgi Diankov, Chi-Te Liang, François Amet, Patrick Gallagher, Menyoung Lee, Andrew J. Bestwick, Kevin Tharratt, William Coniglio, Jan Jaroszynski, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), David Goldhaber-Gordon

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[Robust fractional quantum Hall effect in the N=2 Landau level in bilayer graphene](https://mdr.nims.go.jp/datasets/5750e327-bb81-44f3-b99c-3e74b19805f3)

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Robust fractional quantum Hall effect in the N=2 Landau level in bilayer grapheneARTICLEReceived 2 Mar 2016 | Accepted 11 Nov 2016 | Published 21 Dec 2016Robust fractional quantum Hall effect in theN¼ 2 Landau level in bilayer grapheneGeorgi Diankov1,*, Chi-Te Liang1,2,*, François Amet3,4, Patrick Gallagher1, Menyoung Lee1,Andrew J. Bestwick1, Kevin Tharratt1, William Coniglio5, Jan Jaroszynski5, Kenji Watanabe6,Takashi Taniguchi6 & David Goldhaber-Gordon1The fractional quantum Hall effect is a canonical example of electron–electron interactionsproducing new ground states in many-body systems. Most fractional quantum Hall studieshave focussed on the lowest Landau level, whose fractional states are successfully explainedby the composite fermion model. In the widely studied GaAs-based system, the compositefermion picture is thought to become unstable for the NZ2 Landau level, where competingmany-body phases have been observed. Here we report magneto-resistance measurementsof fractional quantum Hall states in the N¼ 2 Landau level (filling factors 4o|n|o8) in bilayergraphene. In contrast with recent observations of particle–hole asymmetry in the N¼0/N¼ 1Landau levels of bilayer graphene, the fractional quantum Hall states we observe in the N¼ 2Landau level obey particle–hole symmetry within the fully symmetry-broken Landau level.Possible alternative ground states other than the composite fermions are discussed.DOI: 10.1038/ncomms13908 OPEN1 Department of Physics, Stanford University, Stanford, California 94305, USA. 2 Department of Physics, National Taiwan University, Taipei 106, Taiwan.3 Department of Physics, Duke University, Durham, North Carolina 27708, USA. 4 Department of Physics and Astronomy, Appalachian State University,Boone, North Carolina 28608, USA. 5 National High Magnetic Field Laboratory, Tallahassee, Florida 32310, USA. 6 Advanced Materials Laboratory, NationalInstitute for Materials Science, 1-1 Namiki, Tsukuba 305, Japan. * These authors contributed equally to this work. Correspondence and requests for materialsshould be addressed to C.-T.L. (email: ctliang@phys.ntu.edu.tw) or to D.G.-G. (email: goldhaber-gordon@stanford.edu).NATURE COMMUNICATIONS | 7:13908 | DOI: 10.1038/ncomms13908 | www.nature.com/naturecommunications 1mailto:ctliang@phys.ntu.edu.twmailto:goldhaber-gordon@stanford.eduhttp://www.nature.com/naturecommunicationsAfractional quantum Hall (FQH) state was first observed atLandau level (LL) filling factor n¼ 1/3 in a GaAs/AlGaAstwo-dimensional electron system1. This many-body statewas successfully explained by the Laughlin wave-function2.Within the lowest LL, the n¼ 2/3¼ 1� 1/3 state wasinterpreted as the particle–hole conjugate of the n¼ 1/3Laughlin state3. In the lowest LL, the observed fractional statesat filling factors n¼ p/(2mp±1), with m and p positive integers,can be successfully explained by the composite fermion (CF)picture4, in which an even number of magnetic flux quanta areattached to an electron. In addition to the CF model, ahierarchical scheme of parent and daughter states5,6 and amodel based on cyclotron braid subgroups7 offer competingexplanations for the pronounced FQH states at n¼ p/(2mp±1) inthe lowest LL.Graphene, whose band structure leads to the manifestationof relativistic quantum mechanical effects in the solid state8–10,has also revealed a rich FQH effect11,12, in which combined spinand valley degrees of freedom lead to new ground states,including multicomponent FQH states13 with an unconve-ntional sequence14. Numerous FQH states have been observedin the N¼ 0 and N¼ 1 LLs of monolayer graphene13,15 but not inthe N¼ 2 LL (6o|n|o10 for monolayers).With advances in sample preparation, the FQH effect was alsorecently seen in bilayer graphene, revealing surprising results suchas tunability of states with electric field normal to the plane16,indications of even-denominator FQH states17 at n¼ � 1/2 andat n¼ � 5/2 and, in scanning compressibility measurements,particle–hole asymmetry in the N¼ 0/N¼ 1 LLs and incipientFQH states in the N¼ 2 LL at n¼ 14/3, 17/3, 20/3 and 23/3,not forming a complete CF sequence18. Highlighting the roleof sample-to-sample variability, both symmetric and asymmetricFQH states were seen in the same measurement set in anotherstudy19.Here we report observations of particle–hole symmetricFQH states in the N¼ 2 LL in bilayer graphene. In contrast, inhigh (NZ2, n44) LLs of GaAs-based two-dimensional electronsystems, aside from possible evidence for n¼ 4þ 1/5 andn¼ 4þ 4/5 FQH states20, competing charge-ordered states suchas Wigner crystal bubbles and nematic stripes are thought tobe the many-body ground states21–23. Why might charge-orderedstates be expected to supplant FQH and specifically CF statesin high LLs? In high (N40) LLs, the more extended electronwave-functions may destabilize the FQH states21–25. In GaAs,such wave-functions have nodes at particular momentacorresponding to spatial separation between orbitals on theorder of the magnetic length, favouring charge ordering with thatspacing23. In bilayer graphene, the wave-functions in the N¼ 2LL have no complete nodes and hence might be expected tosupport FQH states over charge-ordered states.26 A numericalstudy that does not rely on the mean-field approximation orotherwise assume the CF picture predicts pronounced single-component FQH states at 1/3, 2/3 and 2/5 in the N¼ 2 LL26.ResultsSample characterization. Observation of the FQH effect requiresultra-clean systems with disorder energy scale smaller thanthe energy gaps of the elementary excitations from the fractionalground states. We achieve the desired cleanliness by fabricatingopen-face bilayer graphene/hexagonal boron nitride (h-BN)/graphite stacks sitting on SiO2 (Fig. 1a and SupplementaryFig. 1), specifically avoiding encapsulation of the bilayergraphene with a top h-BN layer in order to keep the dielectricconstant low and thus enhance Coulomb interactions(Supplementary Note 1 and Supplementary Fig. 2). Alldevices studied in this work were operated atdensities B1.5–5� 1012 cm� 2 with zero-field mobility of100,000–250,000 cm2 V� 1 s� 1. In zero-field measurements,we typically observe (Fig. 1b) strongly insulating behaviournear the charge neutrality point as previously seen19,27, with awidth of the charge neutrality (Dirac) peak suggestive oflow (B1010 cm� 2) disorder density15. The longitudinal (Rxx)and Hall (Rxy) magnetoresistances at a constant density onthe hole side of the N¼ 2 LL of device 1 are shown as a functionof magnetic field from 11.4 to 45 T at T¼ 0.4 K (Fig. 1c).Confirming the low disorder in the sample, the onset ofbroken symmetry occurs by 2 T (Fig. 1c, inset) and fullysymmetry-broken Hall plateaux are seen by 5 T (SupplementaryNote 2 and Supplementary Fig. 3).FQH effect. We observe the FQH effect in the N¼ 2 LL for4o|n|o6 in pronounced Rxx minima at fractional fillingn¼ � 13/3, � 14/3, � 16/3 and � 17/3, with accompanyingplateau-like structures in Rxy at n¼ � 13/3 and � 14/3 (Fig. 1c).High-field measurements as a function of carrier density revealmore details about the FQH states in the N¼ 2 LL. Whensweeping the back-gate voltage at B¼ 30 T, we observe the stateswith denominator 3, as well as the more weakly formed 22/5,23/5, 27/5 and 28/5 states (Fig. 2b). The � 13/3 and� 14/3 states, seen in device 1 at 30 T on the hole side (Fig. 2a),have analogues at þ 13/3 and þ 14/3 on the electron side indevice 2 at fields as low as 7 T (Fig. 2d). A Landau fan diagramfrom device 3 shows the persistence of the � 13/3 and� 14/3 states from 7 to 14 T (Fig. 2c). We confirmed theassignment of these features to FQH states with both Rxx and Rxydata. In the low-field Rxx data as a function of filling factor(normalized carrier density), these states appear as vertical lines,supporting their assignment as quantum Hall states (Fig. 2d).Quantization of the fractional Rxy plateaux with denominator 3,when they are clearly discerned, is within 1% of (1/n)(h/e2)(Fig. 2a).To gain insight into the nature of the ground states of theFQH states for 4o|n|o6, we performed tilted magnetic-fieldmeasurements, which allow discrimination between the effects ofCoulomb interaction (tuned by perpendicular field) and those ofthe Zeeman splitting (tuned by total field). We compared Rxx andRxy measured at a perpendicular (total) field of 25 T with ameasurement carried out at the same perpendicular field with anin-plane component of B37 T (Supplementary Note 3and Supplementary Fig. 4). The � 13/3, � 14/3, � 16/3 and� 17/3 states show little Rxx minima variation, suggesting thatthese states are spin-polarized at these fields.The sequence of filling factors of FQH states that we see in theN¼ 2 LL appears consistent with the CF model’s accounting. Ineach of the first three fully symmetry-broken LLs within theN¼ 2 orbital LL, we see 1/3 and 2/3 states, and in the first two, wesee 2/5 and 3/5 more weakly than the states with denominator 3,as expected in the CF framework and as seen in the N¼ 0 LL inGaAs. We do not see 1/5 or 4/5 in any LL. The 19/3 and20/3 states that we observe in device 2 (Fig. 2b) are the highestobserved, to the best of our knowledge, within-LL particle–holesymmetric pairs reported in any quantum Hall system; we do notobserve any FQH states for n47. These essentially particle–holesymmetric results are unexpected in light of recent experimentalfindings of particle–hole asymmetry in the lowest LL in bilayergraphene.Measurements of the FQH gaps. The magnitude of theenergy gaps of FQH states is a measure of their stability. Fromthe strongly temperature-dependent Rxx (Fig. 3a showing data forARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms139082 NATURE COMMUNICATIONS | 7:13908 | DOI: 10.1038/ncomms13908 | www.nature.com/naturecommunicationshttp://www.nature.com/naturecommunications4o|n|o5 from device 3 on the hole side and SupplementaryFig. 6 for 4o|n|o6 for device 2 on the electron side), weobserve that the fractional states are largely suppressed aboveB3 K (Supplementary Note 4). We extracted the activated energygaps of four fractional states with denominator 3 (� 13/3,� 14/3) for � 5ono� 4 (device 3) and 4ono6 (device 2)(Supplementary Fig. 7) at several magnetic fields. The tempera-ture-dependence of the Rxx minima for the � 13/3 and � 14/3states at 14 T fits the usual Arrhenius law Rxxpe�D/(2T), withD the FQH energy gap divided by the Boltzmann constant andT the temperature (Fig. 3b). Based on the fits for the data atB¼ 14 T, we calculate D� 13/3¼ (2.6±0.1) K (in units ofCoulomb energy, B0.01 e2/elB) and D� 14/3¼ (7.9±0.4) K(B0.03 e2/elB), and for the states at 16/3 and 17/3,D16/3¼ (7.5±0.2) K (B0.032e2/elB) and D17/3¼ (7.0±0.6) K(B0.025e2/elB) where lB¼ (:/eB)1/2 is the magnetic length.Measured FQH gaps are normally significantly reduced bydisorder broadening28 and LL mixing29. In monolayer graphene15and the N¼ 1 LL in GaAs/AlGaAs systems30 measured FQH gapsare at least one order of magnitude smaller than values predictedin the absence of disorder. For states that follow the expectedCF sequence, the activation gap of particle–hole conjugate statesis expected to be the same, as seen in the single activationenergy 3D measured for n¼ 1/3 and 2/3 and 5D for 2/5 and 3/5in GaAs31. Therefore, assuming disorder equally affects theparticle–hole conjugate states n* and its conjugate 1� n*,where n*¼ 1/3 or 2/5, the relative magnitude of the trueactivation gaps should track that of the experimentallymeasured gap sizes. Consistent with this expectation, D16/3is close in value to D17/3 across the magnetic field range used inour study of activated gaps. In contrast, D� 14/3 is larger thanD� 13/3 across the field range and is almost three times larger at14 T. This difference could be due to LL mixing, which may bemore significant for � 13/3 than � 14/3, decreasing D� 13/3.Another possible cause is proximity to a transition in quantumnumbers of the partially filled LL: in the Landau fan of Fig. 2c, then¼ � 5 gap is seen to weaken and then re-emerge as magneticfield is tuned through 9–10 T (VgB� 1.8 V).DiscussionIn the N¼ 2 LL in bilayer graphene, we observe a sequenceof FQH states that appears to be consistent with the accountingof the conventional CF model, including particle–hole symmetry.These results are significant and unexpected in light of recentexperimental findings of particle–hole asymmetry in thelowest LL in bilayer graphene, prompting us to compare bilayergraphene FQH states in the N¼ 0/N¼ 1 LLs with those in theN¼ 2 LL. Given that the FQH effect does not survive in highLLs in GaAs-based systems, we also assess the applicability of theacbiachBNGraphiteBilayerSiO2T = 0.4 KVg = –5 VDevice 1B (T)B (T)VxxVxy–6–4 –5–7 –8133–914316317315 20 25 30 35 40 45Vg(V)T = 0.4 K B = 0 TDevice 1Rxx (kΩ)Rxx (kΩ)Rxy (h/e2 )Rxx (kΩ)G (mS) 00100500.1 0.2–0.1–0.20.511.5202505000.140020002.52.01.51 0.20.3––––Figure 1 | Device schematic and transport characteristics. (a) Schematic of our bilayer graphene device design. (b) Zero-field resistance Rxx and itsinverse, conductance G, as a function of graphite back-gate voltage Vg for device 1 (optical image in Supplementary Fig. 1). (c) Magnetoresistance Rxx andHall resistance Rxy as a function of magnetic field B at Vg¼ � 5 V. Corresponding Landau level filling factors are labelled. Inset: low-field magnetoresistancewith Shubnikov-de Haas oscillations showing the onset of degeneracy breaking among the integer states.NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13908 ARTICLENATURE COMMUNICATIONS | 7:13908 | DOI: 10.1038/ncomms13908 | www.nature.com/naturecommunications 3http://www.nature.com/naturecommunicationsCF model to the states we have observed in the N¼ 2LL of bilayer graphene.In the same samples in which we measured states in theN¼ 2 LL, we also observed particle–hole asymmetric FQH statesin the N¼ 0/N¼ 1 LLs that do not form a complete CFsequence. Rxx and Rxy (Supplementary Fig. 5a andSupplementary Fig. 5b) of device 4 were measured at 30 Tas a function of back-gate voltage in the accessible density range58201aRxy (h/e2 )4002000–6.0 –5.5 –5.0 –4.5 –4.0–0.15 Filling factor νRxx (Ω)Rxx (Ω)Device 1T = 0.32 K B = 30 T3163133–– –––––17314b0123–3 –2 –1141210831334316 d6743.5 4.5 5.5 6.5314313316352352252752831932040608017Device 2B (T)B (T)5225235275284 5 633133163171412108Filling factor ν1400.20.40.60.81Rxx(kΩ)Rxx(kΩ)cVg (V)Vg (V)T = 1.7 KB = 14 T–0.20–0.25–8–6 –5 –4Figure 2 | Particle–hole symmetric fractional quantum Hall effect in the N¼2 Landau level. (a) Longitudinal magnetoresistance Rxx and Hall resistanceRxy of device 1 at 30 T showing pronounced fractional states. (b) Fractional states seen in Rxx on the electron side of device 2 at 14 T. (c) Landau fan diagramof Rxx as a function of magnetic field and carrier density on the hole side for device 3. (d) Rxx as a function of filling factor (carrier density rescaled bymagnetic field) for device 2 on the electron side. Vertical features mark FQH states.ac db0Filling factor ν 1/T (1/K)10100–4.7 –4.6 –4.5 –4.4 –4.31002003000.5 1.0 1.5 2.07 8 9 10 11 12 13 1402468024687 8 9 10 11 12 13 14Device 3B = 14 T T = 0.47KT = 1.26KT = 2.08KT = 2.50KT = 3.48KT = 3.73KRxx (Ω)Rxx (Ω)Device 3B = 14 Tν = –13/3ν = –14/3ν = –13/3ν = –14/3ν = –16/3ν = –17/3Device 3B (T)B (T)Device 2Δ (K) Δ (K) –14/3 –23/5 –24/5 –13/3Figure 3 | Fractional quantum Hall gaps in the N¼2 Landau level. (a) Temperature dependence of the magnetoresistance Rxx for device 3 for� 5ono�4 at 14 T, showing that the Rxx minima for the states with denominator 3 deepen with decreasing temperature down to B1 K.(b) Rxx at n¼ � 13/3 and � 14/3 (device 3) plotted on a semilogarithmic scale as a function of inverse temperature. The linear fits yield activation gaps,greater for n¼ � 14/3. The lowest temperature data points depart from activated behaviour, as is typically seen in QH systems at the onset of variable-range hopping and stronger localization. (c) Measured gaps as a function of magnetic field for n¼ � 13/3 and � 14/3 (device 3) and (d) for n¼ 16/3 and17/3 (electron side in device 2). The error bars are due to the statistical error in fitting the data to the Arrhenius law Rxxpe�D/(2T).ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms139084 NATURE COMMUNICATIONS | 7:13908 | DOI: 10.1038/ncomms13908 | www.nature.com/naturecommunicationshttp://www.nature.com/naturecommunicationson the hole side, showing the states � 7/3, � 8/3 and � 11/3 inthe N¼ 0/N¼ 1 LL. The states 2/3, 4/3, 5/3 and 8/3 are seenin device 2 on the electron side (Supplementary Fig. 5c),confirming the general observation from all measured samplesthat the states in the lowest LL do not form a complete CFsequence, unlike those in the N¼ 2 LL (Supplementary Note 5).Other than a possible intra-LL electron–hole conjugate state—abarely resolved 5/3 state symmetric to the 4/3 state—the rest ofthe states we observe display particle–hole asymmetry. Weobserve 2/3, 4/3 and 8/3 in the N¼ 0/N¼ 1 LLs on both theelectron and hole sides while the 7/3 and � 5/3 states are absent,suggesting that valley degeneracy has not been broken. Wealso measured the energy gaps for several FQH states in theN¼ 1/N¼ 0 LLs, such as 5/3 (DB2.5 K), 4/3 (DB4 K) and8/3 (DB1.2 K) (Supplementary Fig. 8), showing that FQH gaps inthe N¼ 0/N¼ 1 LLs are smaller than those in the N¼ 2LL (Supplementary Note 6).Although our results for the FQH states in the N¼ 2 LLin bilayer graphene are consistent with the CF sequence andobey particle–hole symmetry, our finding should also beconsidered in light of an alternative model7,32,33 motivated bythe breakdown in a high LL of the approximations on which theCF model is based. As the CF model might be destabilized in thiscircumstance, an alternative framework based on braid subgrouphierarchy has been advanced7,34. In its simplest, single-loop form,it predicts FQH states at all the fifths (states with denominator 5)in the N¼ 2 LL (e.g. 21/5, 22/5, 23/5 and 24/5 for 4o|n|o5) andno thirds (states with denominator 3) in this LL, whereas weobserve just those fifths expected from the CF model (22/5 and23/5), as well as all the thirds expected from the CF model. It ispossible that the fifths with numerators 1 and 4 that we do notobserve (21/5 and 24/5) but that are expected in this frameworkare subsumed into integer plateaux, while the observed thirdscould be explained at higher order. However, given that theobserved thirds are stronger than the fifths in the N¼ 2 LL,our results more simply lend themselves to a CF interpretation,which would naturally yield both the states we observe and theirrelative strengths.Theory specific to graphene also predicts robust fractionalstates in the N¼ 2 LL of bilayer graphene26, consistent with ourexperimental data, setting this system apart from conventionalsemiconductor systems in which states in the N¼ 2 LL arecharge-ordered. Fractional states with denominators 3 and 5 inbilayer graphene have been predicted to be as strong in the N¼ 2LL as in lower LLs owing to a shorter-range pseudopotential thanin the N¼ 2 LL of other systems35. A larger set of stableFQH states is expected to be accessed in bilayer graphenecompared with their monolayer counterparts by electricallytuning the layer asymmetry in bilayer graphene36 though oursample design did not allow this.The sharp contrast between our observations and priorstudies of bilayer graphene may be related to differences betweenthe heterostructures studied: notably, we designed our hetero-structures as open-face, bottom-gated bilayers in order toenhance Coulomb interactions15. This points to the opportunityto rationally optimize van der Waals heterostructures to host-desired FQH states. Adding a suspended top gate to our style ofbilayer device37 (Supplementary Fig. 9) should enable theapplication of a large electric field normal to the grapheneplane in order to probe FQH states in both low38 and high LLs inbilayer graphene, without weakening the Coulomb interactionsthat drive the FQH effect (Supplementary Note 7).MethodsDevice fabrication. We fabricated open-face bilayer graphene samples sittingon atomically smooth h-BN layers. Briefly, a thin (B5� 10 nm) graphite sheetexfoliated on a SiO2/nþ þ Si substrate was chosen to serve as a local bottomgate for each bilayer graphene sample. h-BN flakes tens of nanometre thick wereseparately exfoliated on thin (B60 nm) polyvinyl alcohol films spin-coated on bareSi. A suitable h-BN flake was chosen using optical and atomic force microscopicimaging and was subsequently transferred onto a part of the bottom-gate graphiteusing polymethyl methacrylate transfer and then annealed in 10% O2 in Ar at500 �C (ref. 39). Bilayer graphene was then transferred on the h-BN sheets and Hallbars were fabricated. We made no attempt to rotationally align our bilayers withthe underlying h-BN flakes and we saw no signs of secondary (superlattice)Dirac peaks and fractal quantum Hall states in our devices40–42. For B41.5 T,the magnetic length lB¼ 26/ffiffiffiBp(in nm) is always shorter than the h-BN thickness(27 or 47 nm for all devices studied) so that the graphite back-gate suppressespotential fluctuations without substantially screening the short-range Coulombinteractions responsible for the FQH states. Fabrication of such open-face sampleswas successfully accomplished with both polymethyl methacrylate (wet) transfer15and polypropylene carbonate (PPC) on top of polydimethylsiloxane (PDMS) (dry)transfer43.Measurements. The experiments were performed in a cryogen-free dilutionrefrigerator and in a 3He cryostat using standard ac lock-in techniques.Measurements at fields 414 T were performed at the National High MagneticField Laboratory in Tallahassee, FL, USA.Data analysis. Gap values and error bars are obtained by plotting lnRxx versus1/T as shown in Fig. 3c and fitting to a line. The reported gap D is half the slope ofthe linear fit (in units of Kelvin). The error bars are the standard error associatedwith the linear least squares fit.Data availability. The data that support the findings of this study are availablefrom the corresponding authors upon request.References1. Tsui, D. C., Stormer, H. L. & Gossard, A. C. Two-dimensionalmagnetotransport in the extreme quantum limit. Phys. Rev. Lett. 48, 1559–1562(1982).2. Laughlin, R. B. Anomalous quantum Hall effect: an incompressible quantumfluid with fractionally charged excitations. Phys. Rev. Lett. 50, 1395–1398(1983).3. Girvin, S. M. Particle-hole symmetry in the anomalous quantum Hall effect.Phys. Rev. B 29, 6012–6014 (1984).4. Jain, J. K. Composite-fermion approach for the fractional quantum Hall effect.Phys. Rev. Lett. 63, 199–202 (1989).5. Haldane, F. D. M. Fractional quantization of the Hall effect: a hierarchy ofincompressible quantum fluid states. Phys. Rev. Lett. 51, 605–608 (1983).6. Halperin, B. I. Statistics of quasiparticles and the hierarchy of fractionalquantized Hall states. Phys. Rev. Lett. 52, 1586–1589 (1983).7. Jacak, J. & Jacak, L. Commensurability condition and fractional quantum Halleffect hierarchy in higher Landau levels. JETP Lett. 102, 19–25 (2015).8. Novoselov, K. S. et al. Electric field effect in atomically thin carbon films.Science 306, 666–669 (2004).9. Novoselov, K. S. et al. Room-temperature quantum Hall effect in graphene.Science 315, 1379 (2007).10. Zhang, Y., Tan, Y.-W., Stormer, H. L. & Kim, P. Experimental observation ofthe quantum Hall effect and Berry’s phase in graphene. Nature 438, 201–204(2005).11. Bolotin, K. I., Ghahari, F., Shulman, M. D., Stormer, H. L. & Kim, P.Observation of the fractional quantum Hall effect in graphene. Nature 62, 96–199 (2009).12. Du, X., Skachko, I., Duerr, F., Luican, A. & Andrei, E. Y. Fractional quantumHall effect and insulating phase of Dirac electrons in graphene. Nature 462,192–195 (2009).13. Dean, C. R. et al. Multicomponent fractional quantum Hall effect in graphene.Nat. Phys. 7, 693–696 (2011).14. Feldman, B. E., Krauss, B., Smet, J. H. & Yacoby, A. Unconventional sequenceof fractional quantum Hall states in suspended graphene. Science 337,1196–1199 (2012).15. Amet, F. et al. Composite fermions and broken symmetries in graphene.Nat. Commun. 6, 5838 (2015).16. Maher, P. et al. Tunable fractional quantum Hall phases in bilayer graphene.Science 345, 61–64 (2014).17. Ki, D.-K., Fal’ko, V. I., Abanin, D. A. & Morpurgo, A. F. Observation of evendenominator fractional quantum Hall effect in suspended bilayer graphene.Nano Lett. 14, 2135–2139 (2014).18. Kou, A. et al. Electron-hole asymmetric integer and fractional quantum Halleffect in bilayer graphene. Science 345, 55–57 (2014).19. Kim, Y. et al. Fractional quantum Hall states in bilayer graphene probed bytransconductance fluctuations. Nano Lett. 15, 7445–7451 (2015).NATURE COMMUNICATIONS | DOI: 10.1038/ncomms13908 ARTICLENATURE COMMUNICATIONS | 7:13908 | DOI: 10.1038/ncomms13908 | www.nature.com/naturecommunications 5http://www.nature.com/naturecommunications20. Gervais, G. et al. Competition between a fractional quantum Hall liquid andBubble and Wigner crystal phases in the third Landau level. Phys. Rev. Lett. 93,266804 (2004).21. Fogler, M. M. & Koulakov, A. A. Laughlin liquid to charge-density-wavetransition at high Landau levels. Phys. Rev. B 55, 9326–9329 (1997).22. Moessner, R. & Chalker, J. T. Exact results for interacting electrons in highLandau levels. Phys. Rev. B 54, 5006–5015 (1996).23. Koulakov, A. A., Fogler, M. M. & Shklovskii, B. I. Charge density wave in two-dimensional electron liquid in weak magnetic field. Phys. Rev. Lett. 76, 499–502(1996).24. Töke, C., Peterson, M. R., Jeon, G. S. & Jain, J. K. Fractional quantum Halleffect in the second Landau level: the importance of inter-composite-fermioninteraction. Phys. Rev. B 72, 125315 (2005).25. Xia, J. S. et al. Electron correlation in the second Landau level: a competitionbetween many nearly degenerate quantum phases. Phys. Rev. Lett. 93, 176809(2004).26. Shibata, N. & Nomura, K. Fractional quantum Hall effects in graphene and itsbilayer. J. Phys. Soc. Jpn 78, 104708 (2009).27. Maher, P. et al. Evidence for a spin phase transition at charge neutrality inbilayer graphene. Nat. Phys. 9, 154–158 (2013).28. Pan, W. et al. Experimental studies of the fractional quantum Hall effect in thefirst excited Landau level. Phys. Rev. B 77, 075307 (2008).29. Willett, R. L., Stormer, H. L., Tsui, D. C., Gossard, A. C. & English, J. H.Quantitative experimental test for the theoretical gap energies in the fractionalquantum Hall effect. Phys. Rev. B 37, 8476–8479 (1988).30. Choi, H. C., Kang, W., Das Sarma, S., Pfeiffer, L. N. & West, K. W. Activationgaps of fractional quantum Hall effect in the second Landau level. Phys. Rev. B77, 081301 (2008).31. Boebinger, G. S. et al. Activation energies and localization in the fractionalquantum Hall effect. Phys. Rev. B 36, 7919–7929 (1987).32. Łydżba, P. et al. Hierarchy of fillings for the FQHE in monolayer graphene. Sci.Rep. 5, 14287 (2015).33. Jacak, J. & Jacak, L. Explanation of n¼ � 1/2 fractional quantum Hall state inbilayer graphene. Proc. R. Soc. A Math. Phys. Eng. Sci. 472, 20150330 (2016).34. Jacak, J. & Jacak, L. Difference in hierarchy of FQHE between monolayer andbilayer graphene. Phys. Lett. A 379, 2130–2134 (2015).35. Chakraborty, T. & Apalkov, V. M. Traits and characteristics of interactingDirac fermions in monolayer and bilayer graphene. Solid State Commun.175-176, 123–131 (2013).36. Apalkov, V. M. & Chakraborty, T. Controllable driven phase transitions infractional quantum Hall states in bilayer graphene. Phys. Rev. Lett. 105, 036801(2010).37. Amet, F., Williams, J. R., Watanabe, K., Taniguchi, T. & Goldhaber-Gordon, D.Insulating behavior at the a neutrality point in single-layer graphene. Phys. Rev.Lett. 110, 216601 (2013).38. Papic, Z. & Abanin, D. A. Topological phases in the zeroth Landau level ofbilayer graphene. Phys. Rev. Lett. 112, 046602 (2014).39. Garcia, A. G. F. et al. Effective cleaning of hexagonal boron nitride for graphenedevices. Nano Lett. 12, 4449–4454 (2012).40. Ponomarenko, L. A. et al. Cloning of Dirac fermions in graphene superlattices.Nature 497, 594–597 (2013).41. Dean, C. R. et al. Hofstadter’s butterfly and the fractal quantum Hall effect inmoiré superlattices. Nature 497, 598–602 (2013).42. Yankowitz, M. et al. Emergence of superlattice Dirac points in graphene onhexagonal boron nitride. Nat. Phys. 8, 382–386 (2012).43. Wang, L. et al. One-dimensional electrical contact to a two-dimensionalmaterial. Science 342, 614–617 (2013).AcknowledgementsWe thank Eli Fox for experimental help. We thank Allan MacDonald, Jainendra Jain andTapash Chakraborty for theoretical discussions about FQH in higher LLs and MichaelZaletel, Zlatko Papic, Michael Peterson and Kiryl Pakrouski for such discussions and alsosharing unpublished calculations. Jurgen Smet shared enlightening thoughts on FQHphysics in high LLs in various material systems. Luis Balicas graciously commented on anearlier version of the manuscript. Some of the measurements were performed at NationalHigh Magnetic Field Laboratory, which is supported by the US National ScienceFoundation cooperative agreement no. DMR-1157490. Experiments were funded in partby the Gordon and Betty Moore Foundation through Grant GBMF3429 to D.G.-G. G.D.was supported partly by a clean-energy seed grant from the Precourt Institute at StanfordUniversity and by a Stanford Graduate Fellowship. P.G. acknowledges a StanfordGraduate Fellowship. M.L. acknowledges support from Samsung and Stanford Uni-versity. A.J.B. was supported by a Benchmark Stanford Graduate Fellowship. C.-T.L. wasfunded by the MOST, Taiwan (grant numbers MOST 103-2918-I-002-028 and MOST103-2622-E-002 -031).Author contributionsG.D., F.A. and D.G.-G. conceived the experiment; G.D., P.G. and F.A. designed andfabricated the samples; T.T. and K.W. grew BN crystals used for the sample fabrication;G.D., C.-T.L., F.A., A.J.B., M.L., K.T., L.B. and D.G.-G. conducted the measurements;W.C. and J.J. assisted extensively with measurements at the NHMFL; the manuscript waswritten by G.D., C.-T.L. and D.G.-G. with input from all authors.Additional informationSupplementary Information accompanies this paper at http://www.nature.com/naturecommunicationsCompeting financial interests: The authors declare no competing financialinterests.Reprints and permission information is available online at http://npg.nature.com/reprintsandpermissions/How to cite this article: Diankov, G. et al. Robust fractional quantum Hall effect in theN¼ 2 Landau level in bilayer graphene. Nat. Commun. 7,13908 doi: 10.1038/ncomms13908 (2016).Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims inpublished maps and institutional affiliations.This work is licensed under a Creative Commons Attribution 4.0International License. The images or other third party material in thisarticle are included in the article’s Creative Commons license, unless indicated otherwisein the credit line; if the material is not included under the Creative Commons license,users will need to obtain permission from the license holder to reproduce the material.To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/r The Author(s) 2016ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms139086 NATURE COMMUNICATIONS | 7:13908 | DOI: 10.1038/ncomms13908 | www.nature.com/naturecommunicationshttp://www.nature.com/naturecommunicationshttp://www.nature.com/naturecommunicationshttp://npg.nature.com/reprintsandpermissions/http://npg.nature.com/reprintsandpermissions/http://creativecommons.org/licenses/by/4.0/http://www.nature.com/naturecommunications title_link Results Sample characterization FQH effect Measurements of the FQH gaps Discussion Figure™1Device schematic and transport characteristics.(a) Schematic of our bilayer graphene device design. (b) Zero-field resistance Rxx and its inverse, conductance G, as a function of graphite back-gate voltage Vg for device 1 (optical image in Supplem Figure™2Particle-hole symmetric fractional quantum Hall effect in the N=2 Landau level.(a) Longitudinal magnetoresistance Rxx and Hall resistance Rxy of device 1 at 30thinspT showing pronounced fractional states. (b) Fractional states seen in Rxx on the e Figure™3Fractional quantum Hall gaps in the N=2 Landau level.(a) Temperature dependence of the magnetoresistance Rxx for device 3 for -5ltngrlt-4 at 14thinspT, showing that the Rxx minima for the states with denominator 3 deepen with decreasing temperatur Methods Device fabrication Measurements Data analysis Data availability TsuiD. C.StormerH. L.GossardA. C.Two-dimensional magnetotransport in the extreme quantum limitPhys. Rev. Lett.48155915621982LaughlinR. B.Anomalous quantum Hall effect: an incompressible quantum fluid with fractionally charged excitationsPhys. Rev. Lett.50 We thank Eli Fox for experimental help. We thank Allan MacDonald, Jainendra Jain and Tapash Chakraborty for theoretical discussions about FQH in higher LLs and Michael Zaletel, Zlatko Papic, Michael Peterson and Kiryl Pakrouski for such discussions and al ACKNOWLEDGEMENTS Author contributions Additional information