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Fractional Quantum Hall States in Bilayer Graphene Probed by Transconductance Fluctuations Youngwook Kim, Dong Su Lee, Suyong Jung, Viera Skakalova, Takashi Taniguchi, Kenji Watanabe, Jun Sung Kim, and Jurgen H. Smet Nano Lett., Just Accepted Manuscript • DOI: 10.1021/acs.nanolett.5b02876 • Publication Date (Web): 19 Oct 2015 Downloaded from http://pubs.acs.org on October 19, 2015

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Fractional Quantum

Hall States in Bilayer Graphene Probed by

Transconductance Fluctuations Youngwook Kim1,2#, Dong Su Lee3#, Suyong Jung4, Viera Skákalová5,6, T. Taniguchi7, K. Watanabe7, Jun Sung Kim1*, and Jurgen H. Smet2*

1

Department of Physics, Pohang University of Science and Technology, Pohang 790–784,

Korea. 2

Max-Planck-Institut für Festköperforschung, 70569 Stuttgart, Germany

3

KIST Jeonbuk Institute of Advanced Composite Materials, Jeonbuk 565–905, Korea

4

Center for Quantum Measurement Science, Korea Research Institute of Standards and

Science, Daejeon, 305-340, Korea 5

University of Vienna, Faculty of Physics, Boltzmanngasse 5, 1090 Vienna, Austria

6

STU Center for Nanodiagnostics, Vazovova 5, 812 43 Bratislava, Slovakia

7

Advanced Materials Laboratory, National Institute for Materials Science, 1-1 Namiki,

Tsukuba, 305-0044, Japan

*e-mail: [email protected] and [email protected]

#

These authors contributed equally.

1 ACS Paragon Plus Environment

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Abstract We have investigated fractional quantum Hall (QH) states in Bernal-stacked bilayer graphene using transconductance fluctuation measurements. A variety of odd-denominator fractional QH states with νQH → νQH + 2 symmetry, as previously reported, are observed. However, surprising is that also particle-hole symmetric states are clearly resolved in the same measurement set. We attribute their emergence to the reversal of orbital states in the octet level scheme induced by a strong local charge imbalance, which can be captured by the transconductance fluctuations. Also the even-denominator fractional QH state at filling −1/2 is observed. However, contrary to a previous study on a suspended graphene layer [Ki et al., Nano Lett. 2014, 14, 2135], the particle-hole symmetric state at filling 1/2 is detected as well. These observations suggest that the stability of both odd and even denominator fractional QH states is very sensitive to local transverse electric fields in bilayer graphene.

Keyword: bilayer graphene; transconductance fluctuations; fractional quantum Hall states; electron-hole symmetry


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In two-dimensional electron systems (2DESs) subject to a strong perpendicular magnetic field, the kinetic energy of electrons gets quenched as Landau levels (LLs) develop and are gradually depopulated. In this regime, Coulomb interactions become dominant for low energy excitations and induce strongly-correlated incompressible ground states when the uppermost Landau level at the chemical potential is filled such that the overall filling of the Landau levels, ν = nh/eB, takes on a rational value of the form p/q, where p and q are mutual primes and q is odd. The hallmarks of this fractional quantum Hall effect are a plateau in the Hall resistance with a value of (h/e2)/(p/q) and a vanishing of the longitudinal resistance1-4. Here n is the carrier density, e the elementary charge and h is the Planck constant. When the 2D electrons have multiple internal degrees of freedom, inherent symmetries have a strong impact on electron correlations, resulting in an unusual series of fractional quantum Hall (FQH) states5-9. For instance in monolayer graphene, with a four-fold SU(4) symmetry due to spin and valley degrees of freedom10, indeed an unconventional sequence of FQH states has been observed. In state-of-the-art suspended samples11,12,13 as well as in graphene on hexagonal BN14, the FQH states with odd-numerator fractional filling factors are missing between ν = 1 and 2. This has been attributed to a preservation of the valley-symmetry15. It illustrates that graphene serves as an intriguing platform for the study of multi-component FQH states whose stability is strongly affected by symmetry-breaking fields.

Bernal stacked bilayer graphene exhibits even richer symmetry-breaking physics due to the additional degeneracy of the zeroth (N = 0) and first (N = 1) orbital Landau levels on top of the spin and valley related SU(4) symmetry already present in monolayer graphene.16-20 Depending on the order in which the eight-fold degeneracy is lifted and particularly on the orbital mixing of the zeroth and first LLs, the fractional quantum Hall effect (FQHE) alters significantly in bilayer graphene.21-24 The eight-fold degeneracy can also be readily lifted by introducing a charge imbalance between the two layers.25,26 It represents a powerful additional knob to introduce symmetry break and alter the FQH behaviour. Theory has recently suggested that bilayer graphene may also host even denominator fractional quantum Hall states of the Moore Read type with excitations that possess exotic topological properties.27,28 Indeed, an experimental study on suspended bilayer graphene29 reported, in addition to several odd-denominator states, the even-denominator state at ν = −1/2, while for electrons at ν = 1/2 no fractional quantum Hall physics was observed. In subsequent studies25,30 using bilayer graphene on a BN substrate, however, only odd-denominator states


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were observed. More generally, the observed FQH sequences are different for different studies indicative of how fragile the symmetry breaking physics is.25,29,30 Clearly, further experimental investigations are called for.

Here, we report the observation of multiple FQH-states in Bernal stacked bilayer graphene supported by hexagonal boron nitride (h-BN) using transconductance fluctuation measurements. It has been demonstrated that such measurements can capture the phenomenon of charge localization in compressible quantum dots (QDs) that form in the QH regime.13 Charge localization produces a bundle of parallel lines in the transconductance map with a slope identical to the slope of constant filling factor (νQH) of the underlying quantum Hall state. This enables to identify in a macroscopic measurement quantum Hall ground states, including very fragile fractional quantum Hall states that form only locally, even though these states are washed away due to disorder on a macroscopic scale and do not appear in conventional magnetotransport measurements. With this technique, broken symmetry states at all integer filling factors from −14 to 14 have been observed here. Also a variety of FQH states can be identified in three samples including the following sequence of odd-denominator states νQH = −11/3, −10/3, −4/3, −2/3, −1/3, 1/3, 2/5, 2/3, 4/3, 10/3 as well as the even-denominator states at νQH = ±1/2. This provides experimental confirmation of most of the previously observed FQH-states as well as first evidence of an even-denominator state in non-suspended bilayer graphene. In contrast with the previous conventional transport study of even-denominator fractional quantum Hall physics in suspended graphene29, such physics is here not just observed for holes, but also for electrons.

The bilayer graphene devices were fabricated by transferring mechanically exfoliated bilayer graphene on top of a h-BN flake, which itself had been previously transferred to a highly-doped silicon substrate with a 270 nm-thick thermally grown oxide layer. The thickness of the h-BN flakes used for further processing ranges from 10 to 20 nm. An optical microscope image of a typical device is shown in the inset to Fig. 1 (a). Bernal stacking of the bilayer graphene layers was confirmed by Raman spectroscopy. The Raman data are deferred to the supporting information. In total, three bilayer devices (D1-D3) were studied, and all devices showed qualitatively similar transport behavior at zero magnetic field. Here, we present the results obtained from sample D1 unless otherwise noted.


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Figure 1 (a) displays the two-terminal resistivity ρ as a function of back gate voltage Vg at 4.2 K and in the absence of a magnetic field. The sharp resistance peak with a full-width-athalf-maximum of ~ 0.5 V occurs at a low backgate voltage of Vg ~ −4.5 V. The charge inhomogeneity can be estimated from a logarithmic plot of the carrier-density dependence of the conductance G(n) as shown in Fig. 1(b). This yields ~ 9 × 109 cm−2 for electrons and ~ 2 × 1010 cm−2 for holes. These values are comparable to those previously reported on high-quality bilayer graphene devices on h-BN substrates.25, 26, 29-33 The carrier mobility µ = (1/C)(dσ/dVg) extracted at these densities for electrons and holes is larger than 50,000 cm2/Vs and 30,000 cm2/Vs, respectively. Here, σ is the conductivity and C is the capacitance of the gate dielectric layer consisting of h-BN and the SiO2 layer.

Two-terminal conductance traces G(n) are illustrated in Fig. 1(c) for magnetic fields from 1 to 12 T in 1 T steps. The integer quantum Hall states expected for bilayer graphene at νQH = ±4, ±8, ±12 are fully resolved at 2 T. The QH states associated with symmetry breaking in the lowest LL emerge starting from 3 T. At higher magnetic fields, QH plateaus corresponding to

νQH = 0, −1 and −2 become well-developed and a weak feature for νQH = −3 state is also visible at 12 T. The broken-symmetry states for electrons are less clear, presumably due to the strong conductance fluctuations. The apparent complete lifting of the initial eight-fold degeneracy indicates the good quality of our bilayer device. Nevertheless, FQH states remain hidden in this macroscopic transport data.

In order to probe more fragile quantum Hall states, the transconductance gm = δIds/δVg has been measured as a function of carrier density and magnetic field as this method has been applied successfully previously for the identification of fractional quantum Hall states in monolayer graphene even though such states were not apparent in conventional transport.13 A cartoon of the circuit diagram is shown in Fig. 2(a). The measurements are performed in a two-terminal configuration. A dc bias voltage Vds, typically of 0.5 mV, is applied across the source and the drain. By sweeping a dc backgate voltage Vg, the density of the sample is tuned. A small ac voltage corresponding to a density variation of 7.7 × 108 cm−2 is superimposed on top of the dc backgate voltage and the induced oscillating source drain current, δIds, is measured at fixed Vds with the help of a lock-in amplifier. The underlying physics, which helps us to identify an incompressible ground state at filling νQH, is the formation of anti-dots or dots surrounded by an incompressible sea and their Coulomb


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blockade behaviour when localization sets in near filling νQH. Electrons attempt to screen the bare disorder potential by setting up a spatially dependent density profile n(x) with average overall density n as illustrated in Fig. 2(b). This linear screening is preserved in a magnetic field as long as the local density n(x) does not exceed the available number of Landau level states: νeB/h. When increasing the average density n, n(x) shifts up and eventually the required density locally exceeds the level degeneracy. Full screening of the bare disorder potential fails at these locations as further charge carrier accumulation is prevented by the energy gap separating the next available Landau level. These regions merge into an incompressible sea enclosing compressible anti-dots. 13, 34-37 The latter can accommodate further charges when increasing the backgate voltage however only when the Coulomb blockade is overcome and the dot becomes compressible. These local variations in the compressibility with backgate voltage due to Coulomb blockade physics also forces changes in the microscopic current path. As a result fluctuations in the transconductance are generated. An identical landscape of dots reemerges at other values of the magnetic field when the same absolute number of electrons (or holes) is missing in the partially filled Landau level. Accordingly, the same pattern of fluctuations is expected. In the (n, B)-plane these fluctuations therefore coalesce into bundles of parallel lines with a slope corresponding to filling νQH. Hence, by identifying the slope of these sets of transconductance features running parallel, incompressible quantum Hall ground states, which are available locally, can be identified. As has been demonstrated in GaAs-based 2DESs, the localization process for fractionally-charged quasi-particles is identical to that of electrons in the integer quantum Hall regime.35, 38-40 Hence, also FQH states will give rise to sets of parallel lines but with a slope that takes on a rational value. Using this method on monolayer graphene several fractional quantum Hall states were revealed at relatively low magnetic fields and the observed states were consistent with those reported based on conventional transport measurements on samples with state-of-the-art quality and at much higher magnetic fields of ~ 30 T as well as on more sophisticated local compressibility measurements using a scanning single electron transistor.11,12,14

Figure 3 displays a colour rendition of the derivative of the transconductance gm(n, B) with respect to the average density n recorded on the bilayer graphene device D1. The derivative amplifies the transconductance fluctuations and improves their visibility. For the sake of completeness, the raw transconductance data have been included in the supporting content.


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Parallel lines associated with single particle states with SU(4) symmetry at filling νQH = ±4, ±8, ±12, ±16 and ±20 as well as broken symmetry states at filling νQH = ±1, ±2, ±3 can be discerned in panels a and b of Fig. 3. The top row of graphs in Fig. 3b displays enlargements for a number of these fillings. They have been labelled with letters a through d. The corresponding regions in the main colour rendition are highlighted by boxes carrying the same label. The most prominent bundles of parallel lines were used to accurately determine the carrier density. Lines with slope νQH along the boundary of the color rendition in Fig. 3a and inside the enlargements in Fig. 3b aid in identifying the transconductance features of weaker quantum Hall states such as νQH = ±3. Note that the νQH = ±3 states, which were not seen in the conventional transport measurement shown in Fig. 1(c), are well-resolved in the transconductance (window labelled c in Fig. 3b). The same holds for a variety of fractional quantum Hall states. Enlargements of regions in the (n, B)-plane where transconductance fluctuations, running parallel to the filling factor slope of fractional quantum Hall states, can be visually distinguished are plotted at the bottom of Fig. 3b. These are νQH = −10/3, −4/3, 1/2 and 10/3.

In some areas of the (Vg, B)-plane, the charging lines are very dense and several sets with different slope overlap in the same region. This is particularly true for the hole side (see for instance upper left corner of Fig. 3a) where a larger inhomogeneity has been deduced from the conductance data in Fig. 1(c) recorded in the absence of a magnetic field. Because of stronger local density variations multiple pockets may contribute transconctance features of different slope in the same area of the(Vg, B)-plane. Under these circumstances, it is more difficult to visually identify the underlying incompressible ground states responsible for the observed transconductance features from the raw data. Here a quantitative approach may help. It relies on the calculation of a correlation function, C(ν) = Σ (DklDpqδν), that yields a high value if within a selected analysis window of the entire data set several charging lines run parallel to the selected filling factor slopeν. The sum in this correlation function runs over all pairs of data points where Dij corresponds to a data point at carrier density ni and magnetic field Bj located within the analysis window. A pair of data points contributes to the sum, i.e. the delta function δν equals 1, if the line connecting the pair of selected data points Dkl and Dpq falls within a narrow band centred around a line with slopeν. Otherwise, δν equals 0. A plot of C(ν) then highlights those filling factors for which a significant number of charging events occur within the analysis window as these produce a bundle of lines with slope ν. A similar


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procedure has been successfully applied previously for data recorded on monolayer graphene.13 Fig. 4(a) plots examples of this correlation function for different analysis windows covering the filling factor interval [−5, 5]. For instance, FQH states are observed between ν = 3 and 4; ν = ±10/3 and −11/3. Panel b displays the correlation function for data analysis windows in the filling factor range from −3/2 to +3/2. Here, apart from the list of FQH states detected by visual inspection of the raw data, also the FQH states at filling ν = −4/3, −2/3, −1/2, −1/3, 1/3, 2/5, 1/3, 2/3 and 4/3 are present. In addition to the 3-denominator states, we also observe lines associated with the higher order fractional state at ν = +2/5. These states have been only found in the cleanest graphene, using local compressibility and transconductance studies11-13. In the hole counterparts, ν = −2/5 lines are not clearly resolved presumably because the region is very crowded with many parallel lines of different slope originating from several different pockets. As shown in the supporting information, we also found the states with ν = −11/3, −10/3 and 10/3 for device D2 and ν = −10/3 and −1/2 for device D3. We note that the transconductance method does not offer any redundancy as available in transport measurements when FQH states are fully developed. There, such states can be identified not only from the location on the filling factor axis but also from the plateau value. The broadness of some of the features in the correlation function may raise the question whether this method offers sufficient resolution to distinguish sufficiently accurately nearby FQH states and the underlying filling factor. A critical assessment contained in the supporting content of for instance the influence of the width and position of the analysis window on the correlation function indicates however that this scepticism is not justified for the distinction of the FQH states identified in this work. This procedure of systematically shifting the analysis window is also helpful to discard trajectories in the raw data that do not follow a fixed filling fraction. They are occasionally observed and such behaviour is attributed to nearby compressible dots that incidentally merge. They are not suitable for the determination of incompressible ground states.

We conclude that the following sequence of fractional quantum Hall states is observed in bilayer graphene samples: ν = −11/3, −10/3, −4/3, −2/3, −1/2, −1/3, 1/3, 1/2, 2/5, 2/3, 4/3 and 10/3. As expected, FQH states with denominator 3 are prevalent and the ν = −10/3 state is the most robust as has been reported in the literature25,30. Previous studies were consistent with the following hierarchy for the lifting of the 8-fold Landau level degeneracy of the zeroth LL of bilayer graphene: ∆Z > ∆v > ∆LL. Here ∆Z, ∆v and ∆LL are the spin, valley and orbital


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energy splitting26,30,41 Hence, the polarization of the spin and valley pseudospin is maximized first, followed by orbital polarization. The level arrangement is illustrated in Fig. 3(c). The spin (↓ or ↑), valley (K+ or K-) and Landau level index (N=0 or 1) of the levels have been marked. The level arrangement explains the robustness of the ν = −10/3 state. This state corresponds to a 2/3 filling of the partially occupied LL. The valley pseudospin and real spin are polarized, but no orbital splitting is required. Both orbital states (N = 0 and N = 1) can just be equally occupied by a fraction f=1/3 for FQH behavior to develop. The lowest energy excitation involves a valley pseudospin flip with a large energy cost, leading to robust FQH behavior. In contrast, for the ν = −11/3 (−4 + 1/3) state orbital splitting is crucial. Only one orbital state must be occupied up to filling 1/3 for FQH behavior to emerge and the lowest energy excitation only requires to bridge the small gap ∆LL. Accordingly, this state is much weaker and only faintly, if at all, observed28. This is consistent with our data. Similar behaviour has also been observed for the valley degeneracy in monolayer graphene with SU(2) symmetry. The ν = −5/3 (f = 1/3) was found to be more fragile than the ν = −4/3 (f = 2/3) state14,15.

Both theory42 as well as past activation energy studies on GaAs 2DES have demonstrated that the strength of FQH states associated with the partial filling of a level with orbital index 1 is substantially reduced with respect to those of a N = 0 level. Recent theoretical calculations specific for bilayer graphene28 predict the same. Following this rule for the level diagram in inset of Fig. 3.(c), FQH states with filling between 2k and 2k + 1 (k = −2, −1, 0 and 1) are more likely to be observed than FQH-states for which 2k + 1 < ν < 2k + 2 holds and an N = 1 Landau level is partially filled. This ν → ν + 2 symmetry as well as the resulting absence of particle-hole symmetry has indeed been suggested in recent transport29,30 and local compressibility25 studies of bilayer graphene. It explains the observation of FQH states with

ν = −10/3 (= −4 + 2/3), ν = −4/3 (= −2 + 2/3) and ν = 2/3 (= 0 + 2/3) as well as the ν = 1/3 (= 0 + 1/3) and ν = −11/3 (= −4 + 1/3) states in our device D1 as in all these cases a N = 0 level is partially occupied. However, contrary to the previous reports on bilayer graphene, also the particle hole symmetric states at ν = −2/3 and ν = 10/3 are observed in the transconductance even though for these states the chemical potential would, according to the level diagram in Fig. 3 (c), be located within an N = 1 level. We attribute the unexpected particle hole symmetric states to a reversed level scheme with opposite orbital splitting between the states of the same valley spin orientation K+ [See the Fig. 3(c)]. This is possible when a large transverse electric field is applied between the layers. Indeed in Ref 23 it has been shown


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that the orbital splitting for K− valley states is always positive irrelevant of the transverse electric field, however for the K+ valley states the orbital splitting strongly depends on the out-of-plane electric field and the sequence of the orbital states can be reversed. This would place the ν = −2/3 and ν = 10/3 states also in an N = 0 level. Our transconductance measurement probes charge localization in selected regions of the device only. Different transverse electric fields may apply in each of these regions. Hence, multiple compressible dots with a very different local transverse electric field may simultaneously contribute bundles of parallel transconductance lines. This can explain why the FQH sequence observed here is not consistent with that of previous observations25,29,30 but rather appears to be a collection of previously observed FQH states and equivalent states expected when the orbital splitting of the K+ valley state is reversed due to spatial variations of the local transverse electric field. In a conventional macroscopic transport measurement only those states corresponding to the level arrangement in the majority of the sample are bound to survive disorder averaging. This is distinct from the case of monolayer graphene where FQH sequences found in different experiments with different methods, conventional transport, local compressibility and transconductance measurements,11-14 are all similar as this mechanism of orbital level reversal is absent. The study of double gated samples may be particularly instrumental for further confirmation provided these gates allow the application of an overall transverse electric field that exceeds the amplitude of the local variation of the transverse field available due to the sample disorder.

We now turn our attention to the observed even denominator FQH states at ν = −1/2 and +1/2. So far the FQHE at even denominator has not been reported on supported monolayer graphene or bilayer graphene. In all 2DES even-denominator states are rarely observed. For a long time they were the exclusive privilege of the GaAs community3 until they were detected recently in suspended bilayer graphene (ν = −1/2) as well as ZnO based 2DES. 29,44 In both cases they were suggested to be of the Moore Read type. Note that the precise origin of the incompressible ground states is irrelevant for the observation of parallel transconductance features. The above charge localization picture holds and generates parallel spikes in the compressibility and transconductance provided the ground state is accompanied by a gap. The compressible composite fermion Fermi sea that normally forms at half filling would not give rise to this charge localization physics. An example of a local


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compressibility study for the even denominator FQH state at filling 5/2 in GaAs using a single electron transistor has been reported in Ref. 45.

Theory has argued that partial occupation of an N = 1 level is an important prerequisite for the observation of the Moore Read type states43,44 and this assertion was tested explicitly in ZnO experiments44 offering control over the orbital index of the level at the chemical potential by tilting the sample. Theoretical considerations28 for bilayer graphene argued that for the level diagram in Fig. 3(c) the even denominator state is stable when the LL filling fulfils 2k + 1 < ν < 2k + 2 (k = −2, −1, 0 and 1), due to the strong N = 1 orbital character in these regimes. Hence FQH behavior can be expected forν = −1/2 (= −2 + 3/2) state. However for ν = 1/2 (= 0 + 1/2) the partially filled level possesses primarily N = 0 character excluding FQH behavior. This accounts for the experimental observations on suspended bilayer graphene.29 Some uncertainty however remained since the density inhomogeneity for the electron side was substantially larger in the investigated device and it cannot be entirely excluded that this inferior quality was responsible for the lack of the 1/2 state. Our measurements probe both the ν = −1/2 and the ν = +1/2 state. The simultaneous observation of both states is surprising. Whereas the emergence of unexpected particle hole symmetric FQH states with denominator 3 can be reconciled with theory by invoking a reversal of the orbital splitting due to local transverse electric fields, this is not possible for this pair of even denominator states, since the ν = +1/2 state is associated with the partial filling of a K−-state, whose orbital splitting does not reverse sign as the transverse field varies. There is clearly a need for additional experiments on even denominator FQH physics in bilayer graphene in particular in geometries that enable the independent tuning of the density and the transverse electric field.

Method Sample preparation method: The h-BN/bilayer graphene stacks were fabricated using a so-called dry transfer method. PMMA was used as a transfer membrane. h-BN flakes were first transferred to a thermally oxidized Si substrate. The thickness of the h-BN flakes ranges from 10 to 20 nm. Subsequently, a bilayer graphene flake was transferred on top of the h-BN


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flake. It was made sure that the substrate of the sample is clean and flat by measuring atomic force microscopy after every transfer step. After bilayer graphene is transferred to the h-BN flake, the substrate was heated up to 200 ℃ for 3 minutes in order to improve the adhesion between bilayer graphene and h-BN. The PMMA membrane was then dissolved in acetone. The sample was annealed in forming gas (Ar/H2) at 350 ℃. This step was intended to remove PMMA residuals and decrease the number of bubbles and wrinkles formed between bilayer graphene and the h-BN flakes. A bubble- and wrinkle-free area is chosen for further electron beam lithography processing to fabricate Cr/Au (5 nm/50 nm) electrodes. An optical microscope image of a typical device is shown in Fig. 1 (a).

Transconductance measurements: The circuit diagram for the transconductance measurement is illustrated in Fig. 2 (a). A dc current was imposed through the sample by applying a bias across the source and drain terminals Vds = 0.5 mV. The charge carrier density was tuned with backgate voltage Vg. In order to record the transconductance a small ac voltage with a frequency of 433 Hz was added to the backgate voltage. Its root mean square amplitude was equal to 10 mV corresponding to a density variation of 7.7x108 cm−2. The dc conductance was recorded by a multimeter and a current-voltage amplifier and the ac current induced by the backgate modulation was acquired using a lock-in amplifier.

Acknowledgment We thank D.K Ki, D. Zhang and F. Paolucci for fruitful discussions and M. Hagel for assistance with sample preparation. This work was supported by the National Research Foundation

(NRF)

through

SRC

(Grant

No.

2011-0030785),

the

Max

Planck

POSTECH/KOREA Research Initiative (Grant No. 2011-0031558) programs, and also by IBS (No. IBSR014- D1-2014-a02). The work at KIST was supported by the Korea Institute of Science and Technology (KIST) Institutional Program. The work at KRISS was supported by NRF

through

the

Fusion

Research

Program

for

Green

Technologies

(NRF-2012M3C1a1048861) Program. VS acknowledges support of the Project UVP, OPVaV-2011/4.2/01-PN. JHS acknowledges financial support from the graphene flagship and the DFG Priority Program SPP 1459.


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Supporting Information Raman spectrum of a bilayer graphene device, a comparison of the gm and the dgm/dn renditions, transconductance fluctuation maps on samples D2 and D3, a study of the vertical line features and their robustness at low and zero magnetic field, the influence of the broadening of the fluctuation lines on the autocorrelation spectrum


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References 1.

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Tsukazaki, A. et al. Observation of the fractional quantum Hall effect in an oxide. Nature materials 9, 889–893 (2010).

10. Zhang, Y., Tan, Y.-W., Stormer, H. L. & Kim, P. Experimental observation of the quantum Hall effect and Berry’s phase in graphene. Nature 438, 201–204 (2005). 11. Feldman, B. E., Krauss, B., Smet, J. H. & Yacoby, A. Unconventional Sequence of Fractional Quantum Hall States in Suspended Graphene. Science 337, 1196–1199 (2012). 12. Feldman, B. E. et al. Fractional Quantum Hall Phase Transitions and Four-Flux States in Graphene. Phys. Rev. Lett. 111, 076802 (2013). 13. Lee, D. S., Skákalová, V., Weitz, R. T., von Klitzing, K. & Smet, J. H. Transconductance Fluctuations as a Probe for Interaction-Induced Quantum Hall States in Graphene. Physical Review Letters 109, 056602 (2012).


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14. Dean, C. R. et al. Multicomponent fractional quantum Hall effect in graphene. Nature Physics 7, 693–696 (2011). 15. Abanin, D. A., Feldman, B. E., Yacoby, A. & Halperin, B. I. Fractional and integer quantum Hall effects in the zeroth Landau level in graphene. Phys. Rev. B 115407, 1–17 (2013). 16. Novoselov, K. S. et al. Unconventional quantum Hall effect and Berry’s phase of 2π in bilayer graphene. Nat. Phys. 2, 177–180 (2006). 17. Zhao, Y., Cadden-Zimansky, P., Jiang, Z. & Kim, P. Symmetry Breaking in the Zero-Energy Landau Level in Bilayer Graphene. Physical Review Letters 104, 066801 (2010). 18. Feldman, B. E., Martin, J. & Yacoby, A. Broken-symmetry states and divergent resistance in suspended bilayer graphene. Nature Physics 5, 889–893 (2009). 19. Weitz, R. T., Allen, M. T., Feldman, B. E., Martin, J. & Yacoby, A. Broken-symmetry states in doubly gated suspended bilayer graphene. Science 330, 812–816 (2010). 20. Kim, S., Lee, K. & Tutuc, E. Spin-Polarized to Valley-Polarized Transition in Graphene Bilayers at ν=0 in High Magnetic Fields. Phys. Rev. Lett. 107, 016803 (2011). 21. Shibata, N. & Nomura, K. Fractional Quantum Hall Effects in Graphene and Its Bilayer. Journal of the Physical Society of Japan 78, 104708 (2009). 22. Apalkov, V. M. & Chakraborty, T. Controllable Driven Phase Transitions in Fractional Quantum Hall States in Bilayer Graphene. Physical Review Letters 105, 036801 (2010). 23. Côté, R., Luo, W., Petrov, B., Barlas, Y., & MacDonald. A. H., Orbital and interlayer skyrmion crystals in bilayer graphene, Physical Review B 82, 245307 (2010). 24. Snizhko, K., Cheianov, V. & Simon, S. H. Importance of interband transitions for the fractional quantum Hall effect in bilayer graphene. Physical Review B 85, 201415 (2012). 25. Kou, A., Feldman, B. E., Levin, A. J., Halperin, B. I., Watanabe, K., Taniguchi, T., Yacoby, A., Electron-Hole Asymmetric Integer and Fractional Quantum Hall Effect in Bilayer Graphene, Science 345, 55–57 (2014). 26. Lee, K. et al. Chemical potential and quantum Hall ferromagnetism in bilayer graphene. Science 345, 58–61 (2014).


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27. Apalkov, V. M. & Chakraborty, T. Stable Pfaffian State in Bilayer Graphene. Physical Review Letters 107, 186803 (2011). 28. Papić, Z. & Abanin, D. A. Topological Phases in the Zeroth Landau Level of Bilayer Graphene. Physical Review Letters 112, 046602 (2014). 29. Ki, D., Fal’ko, V. I., Abanin, D. A. & Morpurgo, A. F. Observation of even denominator fractional quantum Hall effect in suspended bilayer graphene. Nano Lett. 14, 2135 (2014). 30. Maher, P., Wang, L., Gao, Y., Forsythe, C., Taniguchi, T., Watanabe, K., Abanin, D., Papić, Z., Cadden-Zimansky, P., Hone, J., Kim, P., & Dean., C. R., Tunable Fractional Quantum Hall Phases in Bilayer Graphene, Science 345, 61–64 (2014). 31. Dean, C. R. et al. Boron nitride substrates for high-quality graphene electronics. Nature nanotechnology 5, 722–726 (2010). 32. Zomer, P. J., Dash, S. P., Tombros, N. & van Wees, B. J. A transfer technique for high mobility graphene devices on commercially available hexagonal boron nitride. Applied Physics Letters 99, 232104 (2011). 33. Maher, P. et al. Evidence for a spin phase transition at charge neutrality in bilayer graphene. Nature Physics 9, 154–158 (2013). 34. Ilani, S. et al. The microscopic nature of localization in the quantum Hall effect. Nature 427, 1299–1303 (2004). 35. Martin, J. et al. Localization of fractionally charged quasi-particles. Science 305, 980–983 (2004). 36. Martin, J. et al. The nature of localization in graphene under quantum Hall conditions. Nature Physics 5, 669–674 (2009). 37. Jung, S. et al. Evolution of microscopic localization in graphene in a magnetic field from scattering resonances to quantum dots. Nat. Phys. 7, 245–251 (2011). 38. Saminadayar, L., Glattli, D. C., Jin, Y. & Etienne, B. Observation of the e/3 Fractionally Charged Laughlin Quasiparticle. Physical Review Letters 79, 2526-2529 (1997). 39. de Picciotto, R. et al. Direct observation of a fractional charge. Nature 389, 162-164, (1997).


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Figure captions

Figure 1 (a) Resistivity as a function of backgate voltage (Vg). The inset shows the optical microscope image of the bilayer graphene device on h-BN (D1). The scale bar is 5 µm. (b) Conductance as a function of carrier density on logarithmic scales. The arrows indicate the density inhomogeneity for the electron and hole regimes. (c) Conductance as a function of carrier density at magnetic fields from 1 T to 12 T in 1 T step.

Figure 2 (a) Schematic circuit diagram for the transconductance measurements. (b) Schematic illustration of the formation of compressible quantum dots and the evolution of compressibility spikes in the (n, B)-plane.

Figure 3 (a) Color rendition of dgm/dn in the (n, B)-plane. (b) Enlarged windows from the dgm/dn rendition. Letters correspond to the regions marked in (a). (c) The octet level scheme of the zero energy Landau level with the spin (up, down), valley (K+, K−), and orbital index (0,1) degrees of freedom. For the K+ valley polarized state, the orbital order can be reversed due to the transverse electric field, indicated by the index in the parenthesis.


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Figure 4 Auto correlation spectra (a) across the large filling factor interval [−5, 5] and (b) in the small filling factor interval [−3/2, 3/2]. Letters on the spectra correspond to the data windows marked in Fig. 3(a).


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