The electronic properties of graphene are described by a Dirac Hamiltonian with a four-fold symmetry of spin and valley. This symmetry may yield novel fractional quantum Hall (FQH) states at high magnetic field depending on the relative strength of symmetry-breaking interactions. However, observing such states in transport remains challenging in graphene, as they are easily destroyed by disorder. In this work, we observe in the first two Landau levels the two-flux composite-fermion sequences of FQH states between each integer filling factor. In particular, the odd-numerator fractions appear between filling factors 1 and 2, suggesting a broken-valley symmetry, consistent with our observation of a gap at charge neutrality and zero field. Contrary to our expectations, the evolution of gaps in a parallel magnetic field suggests that states in the first Landau level are not spin-polarized even up to very large out-of-plane fields.
In a large magnetic field B, the band structure of two-dimensional electrons becomes a discrete set of highly degenerate Landau levels (LLs)1,2,3,4. With kinetic energy quenched, the electron interactions determine the ground state of a partially filled LL. This yields new incompressible phases known as fractional quantum Hall (FQH) states5,6, where the longitudinal resistance ρxx vanishes exponentially at low temperatures and the transverse resistance ρxy is quantized as h/νe2, where e is the electron charge, h is the Planck’s constant and ν=nh/eB, the number of filled LLs for an electron density n.
In GaAs quantum wells, the most robust FQH states are observed when ν=p/(2kp±1)=1/3, 2/3, 2/5, 3/5...(k and p integers). This may be understood within the composite fermion (CF) theory, where each electron is imagined to bind an even number 2k of magnetic flux quanta φ0=h/e (refs 7, 8, 9, 10). CFs experience an effective residual magnetic field B*=B−2knφ0, and incompressible phases occur when the number of filled CF LLs p=nφ0/B* is an integer.
In graphene, the ground state in each of these FQH phases is characterized by the two internal degrees of freedom of the Hamiltonian, the spin and the valley isospin, the latter originating from the hexagonal crystal structure of graphene. These grant LLs an approximate SU(4) symmetry1,2,3,4 broken at high magnetic fields11 by Zeeman splitting EZ=gμBB and valley symmetry breaking on order (refs 12, 13, 14), where a is the graphene lattice constant, lB is the the magnetic length, Ec is the the typical energy of Coulomb interactions at a length scale lB and ε is the the dielectric constant. While the nature of the broken-symmetry phases at integer filling factors has been under intense scrutiny11,12,13,14,15,16,17, little is known about the impact of symmetry-breaking interactions on FQH states18,19,20,21,22,23,24,25,26.
In this work, we present magneto-transport measurements on high-quality graphene devices at magnetic fields up to 45 T and temperatures down to 30 mK. Between each integer filling factor and up to ν=6, we observe sequences of FQH states with unprecedented detail in transport. These follow the two-flux CF sequence δν=p/(2p±1), with p≤5 integer. Several devices are gapped at zero field, as a result of their substrate-induced broken sublattice symmetry. These host FQH states at odd filling factors 5/3 and 7/5, which were absent in devices with a preserved valley symmetry. In addition, we report the first study of the in-plane field dependence of ρxx at fixed out-of-plane field in the FQH regime of graphene. We observe a pronounced weakening of FQH states in the first LL as the in-plane field is increased, suggesting that they are not yet spin-polarized up to very large field.
Characterization at low magnetic field
Observation of FQH states requires that disorder-induced Fermi level fluctuations δEF be smaller than FQH energy gaps. To minimize δEF, we place monolayer graphene on an atomically flat flake of hexagonal boron nitride27,28, which is typically 15–25 nm thick in the seven heterostructures we studied. The carrier density is tuned with a voltage VBG applied to a graphite back gate (Fig. 1a), which also acts as a screening layer, recently shown to make the potential landscape in graphene devices cleaner29,30,31. Potential fluctuations should be suppressed on length scales larger than the distance to the back gate, while electron interactions remain on the scale of the magnetic length , less than the distance to the back gate for the relevant magnetic field range. The devices included in this study have field-effect mobilities ranging from 4 × 105 to 106 cm2 V−1 s−1, as extracted from a linear fit to the conductivity σ(n) at low density (typically n<2 × 1011 cm−2). This high quality is seen from the longitudinal resistivity ρxx, plotted in logarithmic scale as a function of the carrier density for Device A, at temperature T=4 K and B=0 T (Fig. 1b). ρxx drops to 20 Ω at n=5 × 1012 cm−2, although in this regime the mean free path is limited by boundary scattering32 and the sheet resistivity is not well defined. The quantum mobility is also estimated from the onset of the ν=2 gaps in ρxx(VBG, B) (ref. 33), which for example occurs at 25 mT for Device B (Fig. 1b, inset), indicating a mobility of at least 4 × 105 cm2 V−1 s−1.
Five of our seven devices are strongly insulating at the charge neutrality point, with the peak resistivities ρNP exceeding 100 kΩ at 4 K, well above the theoretical limit of πh/4e2~20 kΩ expected for gapless pristine graphene34,35,36 (the last two devices are comparable in quality but with ρNP <20 kΩ). This phenomenon has only been seen in extremely clean graphene on boron nitride29,30 and has been attributed to sublattice symmetry breaking29,30,37,38,39. We made no attempt to rotationally align our graphene and BN flakes, suggesting that very close rotational alignment is not required to produce this insulating behaviour (in agreement with ref. 30, where a graphene flake with misalignment up to 4° on BN still showed activated transport). In only two of our five insulating devices do we see superlattice peaks at attainable densities (5.3 and 6.0 × 1012 cm−2, respectively), indicating a rotational misalignment under 1.5° (see Supplementary Note and Fig. 1). In our most insulating device (Device A), ρxx(VBG) spans six orders of magnitude up to 20 MΩ at the neutrality point (Fig. 1b, in log scale). The temperature dependence of its peak resistivity is activated above 20 K (Fig. 1b): an Arrhenius fit ρxx~exp(−Δ/2kBT) yields a gap Δ=350±60 K, similar to that reported recently30.
As B is increased, vanishing ρxx indicates quantum Hall phases at integer filling factor (dark blue regions on Fig. 1c). Broken-symmetry states ν=1 and 3 are visible at B~1.5 T, and the zero field insulating state continuously undergoes a transition to the ν=0 phase below 1 T (refs 29, 30. Remarkably, FQH states at ν=−8/3, −10/3, −11/3 are seen for B as low as 5 T, significantly lower than in refs 40, 41, 42. From now on we focus on FQH states in the hole-doped regime. Contrary to the case of bilayer graphene43, the sequence of FQH states in monolayer graphene is electron–hole symmetric, but our data are significantly cleaner on the hole side. We suppress the negative sign of ν when we refer to values of filling factor.
FQH effect in the zeroth LL
The zeroth LL in monolayer graphene is populated at filling factors |ν|≤2. Between ν=0 and 1, we observe a detailed series of FQH states following the two-flux composite-fermion sequence44 at ν (or 1−ν)=1/3, 2/5, 3/7, 4/9 (Fig. 2a). Corresponding plateaus appear in the transverse conductivity , despite modest mixing between the longitudinal and transverse signals.
The sequence of FQH states between ν=1 and 2 is more intriguing: the transport and local compressibility measurements40,44,45 in graphene both suspended and on boron nitride have lacked incompressible states at odd-numerator fractional filling factors. The associated ground states are believed to be spin-polarized, but susceptible to valley-textured excitations46. Unlike more familiar spin skyrmions47, spin-polarized valley skyrmions can be spatially extended even at high magnetic fields46 because they do not involve spin flips. They provide low-lying delocalized excitations and explain the absence of incompressible states at these odd-numerator fractions in refs 40, 44. Yet in Fig. 2b, we see minima of ρxx not only at even-numerator filling factors 4/3 and 8/5 but also at 5/3 and 7/5. These were absent in refs 40, 44 and suggest that the valley skyrmions are suppressed in our samples.
We extract activation gaps from Arrhenius fits to the temperature dependence of ρxx at 25 T (Fig. 2c). Surprisingly, we find Δ5/3 to be the most robust at 10.4±1.4 K, followed by Δ4/3=8.9±0.6 K, Δ2/3=4.6±0.3 K and Δ1/3=3.5±0.2 K. Theoretical estimates of these gaps are one order of magnitude larger. Δ4/3 and Δ2/3 were predicted to be the largest gaps, ranging from 0.08 to 0.11 e2/εlB (or ), for ε=(1+εBN)/2≈2.5) (refs 21, 24). Theoretical estimates for Δ1/3 and Δ5/3 are slightly smaller: 0.03 to 0.1 e2/εlB (or ) (refs 24, 48). The observed gaps are most likely reduced relative to predictions due to disorder, the density at which FQH states occur fluctuating spatially due to remaining charged impurities44. This sensitivity to disorder causes FQH gaps to sometimes decrease after successive cool downs and to vary slightly between samples. It is also possible that gaps are reduced as filling factors approach ν=0 due to the competing insulating phase at charge neutrality40,44.
The boron nitride substrate can dramatically affect the band structure of graphene due to the very similar crystal lattices of these two materials49. Satellite Dirac peaks emerge at high densities30,50,51 and the quantum Hall spectrum changes dramatically, exhibiting a fractal pattern known as the Hofstadter butterfly when lB is comparable to the superlattice moiré cell size30,50,51. We observed superlattice effects in some of our devices at very high densities (Supplementary Fig. 1), but restricted our study to samples and ranges of (n, B) where such effects do not impact the quantum Hall spectrum. Even for twist angles up to a few degrees, boron nitride may break the sublattice symmetry30,38, resulting in a zero-field insulating behaviour at the neutrality point as observed in Fig. 1b29,30. In the zeroth LL, wavefunctions corresponding to the two valleys are each localized on a different sublattice. Sublattice symmetry breaking should therefore yield valley-polarized ground states, making extended valley skyrmions energetically costly. This is consistent with the observation of FQH states at ν=5/3 (ref. 30) and 7/5 in samples that are also gapped at zero field.
FQH effect in the first LL
FQH states in the first LL of graphene were predicted to be much more robust than in non-relativistic two-dimensional electron gases (2DEGs)48,52; yet, only states with denominator 1/3 have been reported in transport and scanning compressibility measurements have not reached sufficient density to probe beyond the zeroth LL. In contrast, we observe a detailed sequence of FQH states in transport, with sharp local minima of ρxx visible at factors ν+δν, where ν=2, 3, 4 and δν=1/3, 2/3, 2/5, 3/5, 3/7, 4/7... (Fig. 3a, Device A, T=300 mK). Minima of ρxx occur at constant filling factors (vertical blue lines) indicating that they are indeed FQH states, some of them emerging at particularly low fields. Between ν=2 and 3, additional FQH states are noticeable at 45 T, such as 17/7, 22/9 and 27/11 (Fig. 3f). We do not observe fractions beyond ν=5 and even the integer QH state ν=5 is not always clearly defined, a feature we have observed in several devices and cannot presently explain.
As expected, ρxx minima have a strong temperature dependence, captured at 14 T for a different device (B) in Fig. 3b–d for temperatures between 500 mK and 4 K. For each filling factor, ρxx(T) is fitted to an Arrhenius law ρxx(v)~exp(−Δν/2kBT). At that field, we find Δ8/3=5.5±1.5 K and Δ10/3=6.3±2.1 K, while Δ7/3=5.4±1.4 K, Δ13/3=1.2±0.2 K and Δ14/3=2.3±0.5 K. The temperature dependence of ρxx(11/3) is not strong enough to extract a gap value. As expected, FQH states with higher denominator are weaker than denominator-3 states at nearby filling factors, with Δ12/5=1.4±0.3 K, Δ13/5=2.6±0.8 K and Δ17/5=1.8±0.4 K.
Probing the nature of the FQH states
The extracted gaps are at least one order of magnitude smaller than theoretically predicted, and are likely reduced by disorder. In addition, their perpendicular field (B⊥) dependence does not follow that of the Coulomb energy at the magnetic length scale . Rather, the field dependence we observe is not monotonic in the first LL, with a noticeable decrease at high fields we did not observe in the zeroth LL (Device A, Fig. 3g). This could be due to emerging superlattice effects at high field and density, or to approaching FQH phase transitions between different ground states.
To probe the spin-polarization of the ground states at each filling factor, we follow the evolution of the gap when the sample is tilted with respect to B, which allows us to change the relative strength of the Zeeman coupling, proportional to the total field B, and Coulomb interactions, controlled by the perpendicular field B⊥ (Fig. 4, inset). In what follows, we call the in-plane field B||, so . Previous graphene transport measurements in the FQH regime required very large B⊥, making it challenging to tilt samples with respect to the magnetic field while preserving FQH states. Here the fractions are observed at fields as low as 6 T, so we can measure ρxx at a fixed B⊥=17 T while substantially changing the total field (Fig. 4). In the zeroth LL, the minima of ρxx at ν=−5/3, −4/3, −2/3, −3/5, −2/5 and −1/3 do not change strength when increasing B|| with B⊥ fixed. This observation is consistent with all ground and low-lying excited states being spin-polarized, in agreement with the theoretical expectations between ν=−2 and −1, where less than a quarter of the zeroth LL is filled and electrons in a FQH state can occupy a single hyperspin state with spins aligned with B (ref. 46). Between ν=−1 and 0, however, full spin-polarization would be surprising: electrons should occupy at least two hyperspin states, and based on the zero-field insulating behaviour at the charge neutrality point sublattice symmetry breaking strongly favours valley polarization. This would mean that spins cannot all be aligned with B. The absence of in-plane field dependence for these fractions may mean that the ground state and low-lying excited states have the same (not fully polarized) spin structure, or simply that the Zeeman coupling remains small compared with a particularly large gap associated with the broken-valley symmetry of the zeroth LL, especially in samples such as these, where the sublattice symmetry is broken by the BN substrate.
This behaviour contrasts with our observations in the first LL, where most FQH gaps are seen to decrease with B|| (Fig. 4b and Supplementary Fig. 2). Minima at 17/5 and 18/5 disappear, while 10/3 and 11/3 are visibly weakened. Only 8/3 remains robust. We observed the same behaviour in two different samples at perpendicular field up to 25 T (Supplementary Note 2). We would naively expect that such a high field should be strong enough to completely polarize the spin, but gaps weakened by increasing B|| suggest instead that the first LL FQH ground states are not yet spin-polarized even at 25 T. Integer quantum Hall gaps follow a similar evolution than the one observed in ref. 12: at half-filling, the region of vanishing ρxx widens with B|| suggesting a strengthening of Δ4, while gaps at quarter-filling Δ3 and Δ5 remain the same. This was attributed in ref. 12 to a spin-polarized ground-state at ν=4 with low-lying excitations involving spin flips, while ground states at quarter-filling have valley-textured excitations.
The spin-polarization of the ground state at a given filling factor depends on the competition between electron interactions and the Zeeman splitting. Neglecting spin, FQH gaps between two successive integer filling factors are expected to scale like Coulomb interactions, with Δν∝|1−2δν|e2/εlB, where δν=p/(2p+1). The prefactor |1−2δν| stems from the effective field B*=B⊥|1−2δν| experienced by CFs, which vanishes when approaching half-filling. This explains the decreasing gaps for FQH states with high index p, a well-documented fact in GaAs-based 2DEGs (refs 53, 54).
For CFs with a spin, the spin-polarization of the ground state at a given filling factor depends on the competition between electron interactions and the Zeeman splitting55,56: Zeeman splitting of the two consecutive CF LLs should produce a LL crossing when gμB equals the Coulomb contribution to the gap stated above, which only scales like . Below the transition, activation gaps should decrease linearly with an in-plane field, which could explain our observations in the first LL. Such crossings and associated transitions of ground states have been observed in transport in GaAs-based 2DEGs (ref. 53), and more recently in the zeroth LL of graphene using compressibility measurements45.
How should field scales for these transitions differ between GaAs and graphene? The dielectric constant is five times lower in graphene on BN than in GaAs, so at a given field electron interactions are stronger in graphene. Yet, Zeeman splitting is also much larger in graphene, with g≥2 compared with g≈0.4 in GaAs, and it is therefore unclear why the field required to observe transitions to spin-polarized states should be larger in graphene. However, for a given ratio e2/εlB, FQH gaps in the first LL of graphene were predicted to be significantly stronger than gaps in both non-relativistic 2DEGs48,52 and graphene’s zeroth LL, a result of the peculiar form factor of relativistic electron in n≠0 LLs. We could not quantitatively compare gaps between zeroth and first LLs at the same field in our devices (Supplementary Note 3). However, at low field (B≤14 T), several devices showed better developed FQH states in the first LL (Supplementary Fig. 3), in qualitative agreement with theoretical expectations48,52. This could explain the need for larger Zeeman splitting to overcome interaction-induced gaps in the first LL and induce phase transitions to spin-polarized states, but would not explain what is special about ν=8/3, the only fractional state in the first LL which is not weakened by an in-plane field at B⊥=17 T.
The graphene/h-BN/graphite structure is fabricated using the transfer method described infit29,57. Few-layer graphite is exfoliated on SiO2. Graphene and h-BN flakes are separately exfoliated on a polymer stack of polymethyl-methacrylate (400 nm) atop polyvinyl alcohol (60 nm). We transfer h-BN first, then graphene, on top of the graphite flake. For each transfer, the polyvinyl alcohol is dissolved in deionized water, the PMMA membrane lifts-off, is transferred on a glass slide, heated at 110 °C and aligned on top of the target with a micromanipulator arm. After each transfer, samples are annealed in flowing argon and oxygen (500 and 50 s.c.c.m. respectively) in a 1" tube furnace at 500 °C to remove organic contamination58. E-beam lithography is combined with oxygen plasma etching to define the graphene flake geometry, and with thermal evaporation to deposit Cr/Au contacts (1 nm/100 nm).
The devices were measured in a 3He cryostat and a dilution fridge in a current-biased lock-in setup, with an ac excitation current of 20 nA at 13 Hz. Measurements at fields higher than 14 T were performed at the National High Magnetic Field Laboratory in Tallahassee.
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We thank Mark Goerbig, Dmitri Abanin, Steve Kivelson, Chi-Te Liang and Inti Sodemann for theoretical and conceptual insights. We are grateful for Jonathan Billings’ and Alexey Suslov’s technical help. This work was supported by the Center for Probing the Nanoscale, an NSF NSEC, under grant No. PHY-0830228. Part of the measurements were done at the National High Magnetic Field Laboratory, which is supported by the US National Science Foundation cooperative agreement no.DMR-0654118, the State of Florida and the US Department of Energy. A.J.B. was supported by a Benchmark Stanford Graduate Fellowship.
The authors declare no competing financial interests.
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Amet, F., Bestwick, A., Williams, J. et al. Composite fermions and broken symmetries in graphene. Nat Commun 6, 5838 (2015). https://doi.org/10.1038/ncomms6838
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