Abstract
The fractional quantum Hall effect is a canonical example of electron–electron interactions producing new ground states in manybody systems. Most fractional quantum Hall studies have focussed on the lowest Landau level, whose fractional states are successfully explained by the composite fermion model. In the widely studied GaAsbased system, the composite fermion picture is thought to become unstable for the N≥2 Landau level, where competing manybody phases have been observed. Here we report magnetoresistance measurements of fractional quantum Hall states in the N=2 Landau level (filling factors 4<ν<8) in bilayer graphene. In contrast with recent observations of particle–hole asymmetry in the N=0/N=1 Landau levels of bilayer graphene, the fractional quantum Hall states we observe in the N=2 Landau level obey particle–hole symmetry within the fully symmetrybroken Landau level. Possible alternative ground states other than the composite fermions are discussed.
Introduction
A fractional quantum Hall (FQH) state was first observed at Landau level (LL) filling factor ν=1/3 in a GaAs/AlGaAs twodimensional electron system^{1}. This manybody state was successfully explained by the Laughlin wavefunction^{2}. Within the lowest LL, the ν=2/3=1−1/3 state was interpreted as the particle–hole conjugate of the ν=1/3 Laughlin state^{3}. In the lowest LL, the observed fractional states at filling factors ν=p/(2mp±1), with m and p positive integers, can be successfully explained by the composite fermion (CF) picture^{4}, in which an even number of magnetic flux quanta are attached to an electron. In addition to the CF model, a hierarchical scheme of parent and daughter states^{5,6} and a model based on cyclotron braid subgroups^{7} offer competing explanations for the pronounced FQH states at ν=p/(2mp±1) in the lowest LL.
Graphene, whose band structure leads to the manifestation of relativistic quantum mechanical effects in the solid state^{8,9,10}, has also revealed a rich FQH effect^{11,12}, in which combined spin and valley degrees of freedom lead to new ground states, including multicomponent FQH states^{13} with an unconventional sequence^{14}. Numerous FQH states have been observed in the N=0 and N=1 LLs of monolayer graphene^{13,15} but not in the N=2 LL (6<ν<10 for monolayers).
With advances in sample preparation, the FQH effect was also recently seen in bilayer graphene, revealing surprising results such as tunability of states with electric field normal to the plane^{16}, indications of evendenominator FQH states^{17} at ν=−1/2 and at ν=−5/2 and, in scanning compressibility measurements, particle–hole asymmetry in the N=0/N=1 LLs and incipient FQH states in the N=2 LL at ν=14/3, 17/3, 20/3 and 23/3, not forming a complete CF sequence^{18}. Highlighting the role of sampletosample variability, both symmetric and asymmetric FQH states were seen in the same measurement set in another study^{19}.
Here we report observations of particle–hole symmetric FQH states in the N=2 LL in bilayer graphene. In contrast, in high (N≥2, ν>4) LLs of GaAsbased twodimensional electron systems, aside from possible evidence for ν=4+1/5 and ν=4+4/5 FQH states^{20}, competing chargeordered states such as Wigner crystal bubbles and nematic stripes are thought to be the manybody ground states^{21,22,23}. Why might chargeordered states be expected to supplant FQH and specifically CF states in high LLs? In high (N>0) LLs, the more extended electron wavefunctions may destabilize the FQH states^{21,22,23,24,25}. In GaAs, such wavefunctions have nodes at particular momenta corresponding to spatial separation between orbitals on the order of the magnetic length, favouring charge ordering with that spacing^{23}. In bilayer graphene, the wavefunctions in the N=2 LL have no complete nodes and hence might be expected to support FQH states over chargeordered states.^{26} A numerical study that does not rely on the meanfield approximation or otherwise assume the CF picture predicts pronounced singlecomponent FQH states at 1/3, 2/3 and 2/5 in the N=2 LL^{26}.
Results
Sample characterization
Observation of the FQH effect requires ultraclean systems with disorder energy scale smaller than the energy gaps of the elementary excitations from the fractional ground states. We achieve the desired cleanliness by fabricating openface bilayer graphene/hexagonal boron nitride (hBN)/graphite stacks sitting on SiO_{2} (Fig. 1a and Supplementary Fig. 1), specifically avoiding encapsulation of the bilayer graphene with a top hBN layer in order to keep the dielectric constant low and thus enhance Coulomb interactions (Supplementary Note 1 and Supplementary Fig. 2). All devices studied in this work were operated at densities ∼1.5–5 × 10^{12} cm^{−2} with zerofield mobility of 100,000–250,000 cm^{2} V^{−1} s^{−1}. In zerofield measurements, we typically observe (Fig. 1b) strongly insulating behaviour near the charge neutrality point as previously seen^{19,27}, with a width of the charge neutrality (Dirac) peak suggestive of low (∼10^{10} cm^{−2}) disorder density^{15}. The longitudinal (R_{xx}) and Hall (R_{xy}) magnetoresistances at a constant density on the hole side of the N=2 LL of device 1 are shown as a function of magnetic field from 11.4 to 45 T at T=0.4 K (Fig. 1c). Confirming the low disorder in the sample, the onset of broken symmetry occurs by 2 T (Fig. 1c, inset) and fully symmetrybroken Hall plateaux are seen by 5 T (Supplementary Note 2 and Supplementary Fig. 3).
FQH effect
We observe the FQH effect in the N=2 LL for 4<ν<6 in pronounced R_{xx} minima at fractional filling ν=−13/3, −14/3, −16/3 and −17/3, with accompanying plateaulike structures in R_{xy} at ν=−13/3 and −14/3 (Fig. 1c). Highfield measurements as a function of carrier density reveal more details about the FQH states in the N=2 LL. When sweeping the backgate voltage at B=30 T, we observe the states with 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 +13/3 and +14/3 on the electron side in device 2 at fields as low as 7 T (Fig. 2d). A Landau fan diagram from device 3 shows the persistence of the −13/3 and −14/3 states from 7 to 14 T (Fig. 2c). We confirmed the assignment of these features to FQH states with both R_{xx} and R_{xy} data. In the lowfield R_{xx} 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 R_{xy} plateaux with denominator 3, when they are clearly discerned, is within 1% of (1/ν)(h/e^{2}) (Fig. 2a).
To gain insight into the nature of the ground states of the FQH states for 4<ν<6, we performed tilted magneticfield measurements, which allow discrimination between the effects of Coulomb interaction (tuned by perpendicular field) and those of the Zeeman splitting (tuned by total field). We compared R_{xx} and R_{xy} measured at a perpendicular (total) field of 25 T with a measurement carried out at the same perpendicular field with an inplane component of ∼37 T (Supplementary Note 3 and Supplementary Fig. 4). The −13/3, −14/3, −16/3 and −17/3 states show little R_{xx} minima variation, suggesting that these states are spinpolarized at these fields.
The sequence of filling factors of FQH states that we see in the N=2 LL appears consistent with the CF model’s accounting. In each of the first three fully symmetrybroken LLs within the N=2 orbital LL, we see 1/3 and 2/3 states, and in the first two, we see 2/5 and 3/5 more weakly than the states with denominator 3, as expected in the CF framework and as seen in the N=0 LL in GaAs. We do not see 1/5 or 4/5 in any LL. The 19/3 and 20/3 states that we observe in device 2 (Fig. 2b) are the highest observed, to the best of our knowledge, withinLL particle–hole symmetric pairs reported in any quantum Hall system; we do not observe any FQH states for ν>7. These essentially particle–hole symmetric results are unexpected in light of recent experimental findings of particle–hole asymmetry in the lowest LL in bilayer graphene.
Measurements of the FQH gaps
The magnitude of the energy gaps of FQH states is a measure of their stability. From the strongly temperaturedependent R_{xx} (Fig. 3a showing data for 4<ν<5 from device 3 on the hole side and Supplementary Fig. 6 for 4<ν<6 for device 2 on the electron side), we observe that the fractional states are largely suppressed above ∼3 K (Supplementary Note 4). We extracted the activated energy gaps of four fractional states with denominator 3 (−13/3, −14/3) for −5<ν<−4 (device 3) and 4<ν<6 (device 2) (Supplementary Fig. 7) at several magnetic fields. The temperaturedependence of the R_{xx} minima for the −13/3 and −14/3 states at 14 T fits the usual Arrhenius law R_{xx}∝e^{−Δ/(2T)}, with Δ the FQH energy gap divided by the Boltzmann constant and T the temperature (Fig. 3b). Based on the fits for the data at B=14 T, we calculate Δ_{−13/3}=(2.6±0.1) K (in units of Coulomb energy, ∼0.01 e^{2}/ɛl_{B}) and Δ_{−14/3}=(7.9±0.4) K (∼0.03 e^{2}/ɛl_{B}), and for the states at 16/3 and 17/3, Δ_{16/3}=(7.5±0.2) K (∼0.032e^{2}/ɛl_{B}) and Δ_{17/3}=(7.0±0.6) K (∼0.025e^{2}/ɛl_{B}) where l_{B}=(ħ/eB)^{1/2} is the magnetic length.
Measured FQH gaps are normally significantly reduced by disorder broadening^{28} and LL mixing^{29}. In monolayer graphene^{15} and the N=1 LL in GaAs/AlGaAs systems^{30} measured FQH gaps are at least one order of magnitude smaller than values predicted in the absence of disorder. For states that follow the expected CF sequence, the activation gap of particle–hole conjugate states is expected to be the same, as seen in the single activation energy ^{3}Δ measured for ν=1/3 and 2/3 and ^{5}Δ for 2/5 and 3/5 in GaAs^{31}. Therefore, assuming disorder equally affects the particle–hole conjugate states ν* and its conjugate 1−ν*, where ν*=1/3 or 2/5, the relative magnitude of the true activation gaps should track that of the experimentally measured gap sizes. Consistent with this expectation, Δ_{16/3} is close in value to Δ_{17/3} across the magnetic field range used in our study of activated gaps. In contrast, Δ_{−14/3} is larger than Δ_{−13/3} across the field range and is almost three times larger at 14 T. This difference could be due to LL mixing, which may be more significant for −13/3 than −14/3, decreasing Δ_{−13/3}. Another possible cause is proximity to a transition in quantum numbers of the partially filled LL: in the Landau fan of Fig. 2c, the ν=−5 gap is seen to weaken and then reemerge as magnetic field is tuned through 9–10 T (V_{g}∼−1.8 V).
Discussion
In the N=2 LL in bilayer graphene, we observe a sequence of FQH states that appears to be consistent with the accounting of the conventional CF model, including particle–hole symmetry. These results are significant and unexpected in light of recent experimental findings of particle–hole asymmetry in the lowest LL in bilayer graphene, prompting us to compare bilayer graphene FQH states in the N=0/N=1 LLs with those in the N=2 LL. Given that the FQH effect does not survive in high LLs in GaAsbased systems, we also assess the applicability of the CF model to the states we have observed in the N=2 LL of bilayer graphene.
In the same samples in which we measured states in the N=2 LL, we also observed particle–hole asymmetric FQH states in the N=0/N=1 LLs that do not form a complete CF sequence. R_{xx} and R_{xy} (Supplementary Fig. 5a and Supplementary Fig. 5b) of device 4 were measured at 30 T as a function of backgate voltage in the accessible density range on the hole side, showing the states −7/3, −8/3 and −11/3 in the N=0/N=1 LL. The states 2/3, 4/3, 5/3 and 8/3 are seen in device 2 on the electron side (Supplementary Fig. 5c), confirming the general observation from all measured samples that the states in the lowest LL do not form a complete CF sequence, unlike those in the N=2 LL (Supplementary Note 5). Other than a possible intraLL electron–hole conjugate state—a barely resolved 5/3 state symmetric to the 4/3 state—the rest of the states we observe display particle–hole asymmetry. We observe 2/3, 4/3 and 8/3 in the N=0/N=1 LLs on both the electron and hole sides while the 7/3 and −5/3 states are absent, suggesting that valley degeneracy has not been broken. We also measured the energy gaps for several FQH states in the N=1/N=0 LLs, such as 5/3 (Δ∼2.5 K), 4/3 (Δ∼4 K) and 8/3 (Δ∼1.2 K) (Supplementary Fig. 8), showing that FQH gaps in the N=0/N=1 LLs are smaller than those in the N=2 LL (Supplementary Note 6).
Although our results for the FQH states in the N=2 LL in bilayer graphene are consistent with the CF sequence and obey particle–hole symmetry, our finding should also be considered in light of an alternative model^{7,32,33} motivated by the breakdown in a high LL of the approximations on which the CF model is based. As the CF model might be destabilized in this circumstance, an alternative framework based on braid subgroup hierarchy has been advanced^{7,34}. In its simplest, singleloop form, it predicts FQH states at all the fifths (states with denominator 5) in the N=2 LL (e.g. 21/5, 22/5, 23/5 and 24/5 for 4<ν<5) and no thirds (states with denominator 3) in this LL, whereas we observe just those fifths expected from the CF model (22/5 and 23/5), as well as all the thirds expected from the CF model. It is possible that the fifths with numerators 1 and 4 that we do not observe (21/5 and 24/5) but that are expected in this framework are subsumed into integer plateaux, while the observed thirds could be explained at higher order. However, given that the observed thirds are stronger than the fifths in the N=2 LL, our results more simply lend themselves to a CF interpretation, which would naturally yield both the states we observe and their relative strengths.
Theory specific to graphene also predicts robust fractional states in the N=2 LL of bilayer graphene^{26}, consistent with our experimental data, setting this system apart from conventional semiconductor systems in which states in the N=2 LL are chargeordered. Fractional states with denominators 3 and 5 in bilayer graphene have been predicted to be as strong in the N=2 LL as in lower LLs owing to a shorterrange pseudopotential than in the N=2 LL of other systems^{35}. A larger set of stable FQH states is expected to be accessed in bilayer graphene compared with their monolayer counterparts by electrically tuning the layer asymmetry in bilayer graphene^{36} though our sample design did not allow this.
The sharp contrast between our observations and prior studies of bilayer graphene may be related to differences between the heterostructures studied: notably, we designed our heterostructures as openface, bottomgated bilayers in order to enhance Coulomb interactions^{15}. This points to the opportunity to rationally optimize van der Waals heterostructures to hostdesired FQH states. Adding a suspended top gate to our style of bilayer device^{37} (Supplementary Fig. 9) should enable the application of a large electric field normal to the graphene plane in order to probe FQH states in both low^{38} and high LLs in bilayer graphene, without weakening the Coulomb interactions that drive the FQH effect (Supplementary Note 7).
Methods
Device fabrication
We fabricated openface bilayer graphene samples sitting on atomically smooth hBN layers. Briefly, a thin (∼5−10 nm) graphite sheet exfoliated on a SiO_{2}/n^{++} Si substrate was chosen to serve as a local bottom gate for each bilayer graphene sample. hBN flakes tens of nanometre thick were separately exfoliated on thin (∼60 nm) polyvinyl alcohol films spincoated on bare Si. A suitable hBN flake was chosen using optical and atomic force microscopic imaging and was subsequently transferred onto a part of the bottomgate graphite using polymethyl methacrylate transfer and then annealed in 10% O_{2} in Ar at 500 °C (ref. 39). Bilayer graphene was then transferred on the hBN sheets and Hall bars were fabricated. We made no attempt to rotationally align our bilayers with the underlying hBN flakes and we saw no signs of secondary (superlattice) Dirac peaks and fractal quantum Hall states in our devices^{40,41,42}. For B>1.5 T, the magnetic length l_{B}=26/ (in nm) is always shorter than the hBN thickness (27 or 47 nm for all devices studied) so that the graphite backgate suppresses potential fluctuations without substantially screening the shortrange Coulomb interactions responsible for the FQH states. Fabrication of such openface samples was successfully accomplished with both polymethyl methacrylate (wet) transfer^{15} and polypropylene carbonate (PPC) on top of polydimethylsiloxane (PDMS) (dry) transfer^{43}.
Measurements
The experiments were performed in a cryogenfree dilution refrigerator and in a ^{3}He cryostat using standard ac lockin techniques. Measurements at fields >14 T were performed at the National High Magnetic Field Laboratory in Tallahassee, FL, USA.
Data analysis
Gap values and error bars are obtained by plotting lnR_{xx} versus 1/T as shown in Fig. 3c and fitting to a line. The reported gap Δ is half the slope of the linear fit (in units of Kelvin). The error bars are the standard error associated with the linear least squares fit.
Data availability
The data that support the findings of this study are available from the corresponding authors upon request.
Additional information
How to cite this article: Diankov, G. et al. Robust fractional quantum Hall effect in the N=2 Landau level in bilayer graphene. Nat. Commun. 7,13908 doi: 10.1038/ncomms13908 (2016).
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Acknowledgements
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 also sharing unpublished calculations. Jurgen Smet shared enlightening thoughts on FQH physics in high LLs in various material systems. Luis Balicas graciously commented on an earlier version of the manuscript. Some of the measurements were performed at National High Magnetic Field Laboratory, which is supported by the US National Science Foundation cooperative agreement no. DMR1157490. Experiments were funded in part by the Gordon and Betty Moore Foundation through Grant GBMF3429 to D.G.G. G.D. was supported partly by a cleanenergy seed grant from the Precourt Institute at Stanford University and by a Stanford Graduate Fellowship. P.G. acknowledges a Stanford Graduate Fellowship. M.L. acknowledges support from Samsung and Stanford University. A.J.B. was supported by a Benchmark Stanford Graduate Fellowship. C.T.L. was funded by the MOST, Taiwan (grant numbers MOST 1032918I002028 and MOST 1032622E002 031).
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G.D., F.A. and D.G.G. conceived the experiment; G.D., P.G. and F.A. designed and fabricated 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 was written by G.D., C.T.L. and D.G.G. with input from all authors.
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Diankov, G., Liang, CT., Amet, F. et al. Robust fractional quantum Hall effect in the N=2 Landau level in bilayer graphene. Nat Commun 7, 13908 (2016). https://doi.org/10.1038/ncomms13908
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