Unconventional fractional quantum Hall states and Wigner crystallization in suspended Corbino graphene

Competition between liquid and solid states in two-dimensional electron systems is an intriguing problem in condensed matter physics. We have investigated competing Wigner crystal and fractional quantum Hall (FQH) liquid phases in atomically thin suspended graphene devices in Corbino geometry. Low-temperature magnetoconductance and transconductance measurements along with IV characteristics all indicate strong charge density dependent modulation of electron transport. Our results show unconventional FQH phases which do not fit the standard Jain’s series for conventional FQH states, instead they appear to originate from residual interactions of composite fermions in partially filled Landau levels. Also at very low charge density with filling factors \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nu \,\lesssim\, 1/5$$\end{document}ν≲1∕5, electrons crystallize into an ordered Wigner solid which eventually transforms into an incompressible Hall liquid at filling factors around ν ≤ 1/7. Building on the unique Corbino sample structure, our experiments pave the way for enhanced understanding of the ordered phases of interacting electrons.

In this supplement for the manuscript "Unconventional fractional quantum Hall states and Wigner crystallization in suspended Corbino graphene" we provide additional information on sample fabrication, current annealing of graphene to enhance mobility, sample characterization, the identification of fractional states using transconductance, transport mechanisms in our samples, rf-measurements of Wigner crystal, and the nature of its pinning.

Supplementary Note 1: Sample Fabrication
Our sample fabrication is based on well-chosen combination of resists with differential selectivity, which allowed the fabrication steps for the deposition of a suspended top contact. In this work, we employed heat assisted mechanical exfoliation technique [1]. The application of heat assisted exfoliation on top of LOR resist produced large graphene flakes, which was the basic requirement for a good Corbino device. A 250 + 250 nm thick base layer of LOR-3A [2] was spun in two steps on SiO 2 /Si++ and baked at 200 • C for 5 and 12 min, respectively. A graphite covered tape was pressed against the LOR-covered wafer for 1 min and subsequently baked on a hot plate at 100 • C for 2 min. The tape was peeled from the chip after it had cooled down to room temperature. The exfoliated graphene flake was first characterized optically and then using Raman spectroscopy at 633 nm.
In the first lithography step, the lead for the outer ring and its contact pad were fabricated. A double layer PMMA (950K/A3: top layer and 50K/A11: bottom layer) spun and patterned using e-beam. A Cr/Au sandwich of 5/70 nm thickness was evaporated at pressure p < 6 × 10 −8 mBar. After metal deposition, special care was taken during the metal lift-off process: the lift off was performed in 80 • C xylene and the unwanted residual metal from the circular hole of outer lead was removed using micromanipulation. The outer ring electrode is of rectangular geometry extending over the graphene flake with a circular hole of diameter 2 − 10 µm, see Supplementary Fig. 1c. On the processed sample, another 500 nm thick LOR was spun. A via in the center of the circular hole of outer lead was opened in the top LOR layer using e-beam lithography. A dose of 1000 µC/cm 2 was used for this step. The exposed pattern was developed in ethyl lactate for 60 s and rinsed in hexane. The diameter of the via varied between 0.5 − 5 µm, see Supplementary  Fig. 1d. This via acts as foundation for the inner electrode contact in the concentric Corbino geometry. For fabrication of the inner lead, a double layer PMMA of 200 nm was spun and patterned using e-beam lithography. On the patterned device, Cr (5 nm)/Au (100 nm) was deposited, and the lift off was done in hot xylene with a rinse in hexane, see Supplementary Fig. 1e.
In final steps of the lithography, the graphene between inner and outer ring leads was suspended by exposing LOR to a dose of 1000 µC/cm 2 . In order to expose the outer lead bonding pad, LOR on the larger structures was exposed to an e-beam dose of 700 µC/cm 2 . The device was developed and cleaned in ethyl lactate for 60s and rinsed in hexane for 60s. A schematic cross section of the final device is depicted in Supplementary Fig. 1f and an SEM image of a complete sample structure is shown in Fig. 1 of the main text. For bonding, part of the pad was pressed against the SiO 2 , as shown in Supplementary Fig. 1g. These pressed pads were used for bonding. The finished devices were electrically characterized at room temperature, and devices with low resistance were selected for low temperature measurements.  Figure 1: Fabrication process steps: a) LOR layer with thickness of ∼500 nm is spun on a cleaned Si/SiO 2 chip. b) Mechanical exfoliation of graphene on top of the LOR layer. c) Fabrication of outer lead and its bonding pad. d) Formation of a via for the inner lead using a second LOR layer. e) Air bridge connecting inner lead and pad is patterned using an e-beam lithography step with PMMA. f) Suspension of graphene between the inner and outer leads and uncovering of the outer lead-pad by e-beam exposure. g) Status after wire bonding: gold pads are deformed and pushed partly down to silicon. h) A schematic top view of a sample with suspended top contact: black color shows suspended graphene, blue color shows the exposed window in LOR, light yellow color shows the covered outer lead and bright yellow color indicates clean gold of the inner lead and the outer leads.

Supplementary Note 2: Sample characterization
Our measurements down to 20 mK were performed in a BlueFors LD-400 dilution refrigerator. The measurement lines were twisted pair phosphor-bronze wires supplemented by three stage RC filters with a nominal cut-off given by R = 100 Ohms and C = 5 nF. However, due high impedance of the quantum Hall samples the actual cutoff is determined by the sample resistance. For magnetoconductance measurement, we used an AC peak-to-peak current excitation of 0.1 nA at f = 3.333 Hz.
For transconductance g m = dI/dV g we measured both magnitude Mag{g m } and phase Arg{g m } using low frequency lock-in detection. The best results for g m correlation analysis were obtained by recording Arg{g m } at a bias voltage V that corresponded to the onset of the V α regime. Consistent information was obtained by analyzing the simultaneously acquired Mag{g m }. The gate frequency in AC transconductance measurements was set at f = 17.777 Hz, while the peak-to-peak AC excitation amplitude was adjusted to correspond to charge of one electron over the sample. The DC bias between source and drain was varied in the range V = 0.1 -0.5 mV.

Annealing
Our suspended Corbino graphene samples were predominantly of p-type. During typical annealing process, the doping shifts to n-type before reaching the desired charge-neutral state. For achieving charge neutrality, a high current annealing was performed. We employed voltage bias for our current annealing process: In contrast to current-biased devices, voltage-biased samples are protected against possible rise in resistance due to shifting of the Dirac point. A typical bias voltage value used in annealing of our samples was 1.6 ± 0.1 V; in spite of strong heating of the refrigerator, the sample was kept at temperatures T < 1 K during annealing. Slight asymmetry in the concentricity of the electrodes in our Corbino geometry, inverse radial decay of current, and local variation of the contact resistance result in non-uniformity of current annealing. Owing to this non-uniformity, our suspended Corbino disk becomes split into differently annealed graphene domains, each having their own charge neutrality point. This can be seen in Supplementary Fig. 2a. Eventually, during annealing process several of these peaks merge to give a single Dirac peak or closely degenerate Dirac peaks. See Supplementary Fig. 2b for the final R d vs V g curve of the sample EV * after annealing.

Residual Doping
The level of residual charge doping was extracted from G vs n -plots on loglogscale as the point where conductance G levels off when approaching the Dirac point. We employed straight line fits (power laws with exponent around 1/2) to the conductance data in order to better define the position of this cross-over point, as shown in Supplementary Fig. 2c. The residual doping for other samples was extracted in the same way: n 0,XD = 2 × 10 10 cm −2 and n 0,EV = 6 × 10 9 cm −2 . The magenta curve at the top is the first differential resistance R d trace measured after annealing current I ≈ 1.5 mA (V ≈ 1.3 V) while the black curve, obtained after I ≈ 2 mA (V ≈ 1.7 V), denotes the final annealing result (sample EV in Table I of the main paper). b) R d vs V g curve for sample EV * after annealing. c) G vs n for sample EV * on loglogscale: red and blue circles represent data on the hole and electron sides, respectively. The black lines represent power law trends in the data. The cross-over point of the curves from power law to constant conductance is taken as n 0 , which yields here n 0 = 6 × 10 9 cm −2 . R c was not subtracted in this analysis. d) Mobility as a function of electron density n (n < 0 corresponds to holes) determined for samples EV * , EV, and EV2 at zero magnetic field. Contact resistance R c (see text) has been subtracted off from the measured resistance, i.e., G = 1 R d −R c in this analysis. e) DC conductance measured for sample EV2 assuming an external series resistance, a contact resistance of R c = 380 Ω (black circles), R c = 420 Ω (blue circles), and R c = 450 Ω (red circles). The solid red curve illustrates a theoretically calculated curve which agrees with the data when the contact resistance is set to R c = 420 Ω, i.e., G = 1 R d −R c for the data points.

Supplementary Note 3: Mobility and Contact resistance
Field effect mobility of our samples was calculated using the formula where σ 0 is the measured minimum conductivity, and n denotes the charge carrier density. The conductivity in Corbino geometry can be calculated from measured conductance G using where r o and r i are the radii of outer and inner contacts, respectively.
When calculating the mobility using Supplementary Eqs. 1 and 2, we take into account the contact resistance R c . We estimate R c from the comparison of measured data to the theory based on evanescent mode transport [3]. Supplementary  Fig. 2e displays such a fitting for sample EV2, which yields R c = 420 Ω for the contact resistance. Similar comparison was done for EV and EV * , and we obtained R c = 410 Ω for EV * and R c = 550 Ω for EV.
After taking the contact resistance into account, we obtain for the maximum mobility of our samples: µ f,XD ≈ 5 × 10 4 cm 2 /Vs, µ f,EV * ≈ 1.4 × 10 5 cm 2 /Vs, µ f,EV ≈ 2.0 × 10 5 cm 2 /Vs, and µ f,EV2 ≈ 2.5 × 10 5 cm 2 /Vs. These values are listed in Table I in the main paper. According to mobility, the quality of our samples follows the order EV2 > EV > EV * > XD. Supplementary Fig. 2d illustrates the dependence of µ on the change density. The mobility on the hole side appears to be by factor of 3 − 5 worse than on the electron side. We assign this difference to the presence of pn interfaces near the chromium-gold contacts which are known to have appreciable negative doping [4]. The pn interfaces reduce the conductance and result in a lowered apparent mobility for holes.

Contact Resistance
The full calculation of conductance through graphene with contact transparency is quite a tedious task [4]. Here we will describe in simple terms, how the contact resistance can be viewed as an additive quantity on top of the graphene conductance which arises for graphene Corbino disk from the Dirac equation [3].

Conductance through a double barrier graphene system
To derive the total transmission through the sample with contact resistance, we follow the procedure of Ref. [4]. First we note that usually phase coherence is more sensitive to disorder than reflection and transmission, and we expect that the former is destroyed but the latter is not affected by disorder (ballistic regime). This assumption is supported by the experimental results on the Fabry-Pérot resonances which are found to be weak. Furthermore, when destroying the Fabry-Pérot resonances fully by an applied bias, the overall conductance does not change appreciably. Hence, we conclude that incoherent treatment of transmission probabilities is well justified in our sample.
In general, for the case of incoherent tunneling through a symmetric graphene barrier with equal transmission T for the left and right slopes, the total probability of transmission through the barrier is given by [4] If a finite contact resistance exists, then a finite transmission probability 0 ≤ T c ≤ 1 should be included in to the value of T in Eq. (3). By applying the incoherent addition model, we may write where |t| 2 denotes the transmission probability through the graphene sample and T c gives the transmission at the contacts. This equation can be written in a more transparent form using resistances scaled by the quantum resistance R Q = h 2e 2 : |t| 2 = R ideal /R Q , T c = R c /R Q , and T = R/R Q . Then we obtain By identifying R ideal R Q − 1 as the scaled sample resistance (zero in the ballistic limit), we find direct additivity of the contact resistance with the ideal, calculable resistance of the sample.

Extracting Contact Resistance
Transport properties of a graphene Corbino disk were calculated in Ref. [3]. The conductance is determined by where s = 8 and T j is the transmission coefficient of the jth channel. Transmission coefficient is given by with where H is the Kth Hankel function and k = π|n| denotes the wave vector.
By using Supplementary Eq. 6 one can generate theoretical curves corresponding to our devices. In Supplementary Fig. 2e, the measured conductance is compared with three theoretical curves and the data are found to agree well when a contact resistance of 420 Ω is taken into account. Note that if we would linearize the dependence of G vs. n using R c , a procedure which is expected be valid for long-range Coulomb scattering, then the contact resistance would grow from the above value. In our understanding, the estimated R c is mostly coming from the central leads where the multiple e-beam steps may create the lattice defects. Additionally, in our fabrication process, the development process for central leads has limitation to wash out the residual resists. These resists may forms the dominant charge density under the central metallic leads [5]. Either or both phenomena could explain the origin of this contact resistance of 1 kΩ per micrometer, which is quite close to the experimental findings of Russo et al. [6].

Shubnikov-de Haas Oscillations and Quantum Mobility
Samples were characterized at low magnetic fields around a few hundred mT using Shubnikov-de Haas oscillations (the well-defined quantum Hall states in our samples first develop around 0.5 T). These oscillations yield information about the characteristic properties of charge carriers like mobility, quantum life time of carriers, and their effective mass. The diagonal resistance R xx oscillates periodically with inverse magnetic field, while temperature suppresses these oscillation due to enhanced scattering of carriers. The variation of resistance is given by equation [7]: where R 0 is the resistance at zero field, X(T ) = 2π 2 k B T /hω c with cyclotron frequency ω c = eB/m * , E F specifies the Fermi level, µ q denotes the quantum mobility of charge carriers, and θ is a phase factor. Looking at resistance oscillations alone, Supplementary Eq. 9 can be written as: with A as a constant defined by X(T ). Supplementary Fig. 3a displays the SdH oscillations seen in sample EV. The quantum mobility obtained from the fit amounts to µ q 6 × 10 4 cm 2 /Vs, which is by a factor of two smaller than the value of the field effect mobility in Table I of the main paper. This value for µ q corresponds to the quantum scattering time τ d 0.3 ps which yields for the Landau level width δ LL =h/2τ q 10 K. In order to achieve a better analysis of the oscillations themselves, a base line marked by the red line in Supplementary Fig. 3a was subtracted from R. These pure oscillations are displayed in Supplementary Fig. 3b. Quantum mobility was extracted from the envelope curve of these oscillations, as indicated in Supplementary Fig. 3b. In the case of sample EV, the quantum mobility is µ q = 6 × 10 4 cm 2 /Vs (red lines in Supplementary Fig. 3b ).  Supplementary Fig. 3a). The solid red curves represent envelope of the oscillations and they correspond to quantum mobility µ q = 6 × 10 4 cm 2 /Vs.

Supplementary Note 4: Analysis of Transconductance Lines
The transconductance g m measurements were performed to identify fragile fractional fractional quantum Hall states in a well-controlled, indisputable manner. For g m (n, B) measurement, an ac excitation of the 2.7 mV rms is applied to the back gate on top of the dc voltage, and the sample is voltage biased with V ∼ 0.1 − 3 mV. Both amplitude and phase of the current modulation were recorded using a lock-in amplifier. The gate modulation frequency was selected to correspond to the roll-off frequency of RC time constant, governed by the sample resistance of a few MΩ. Consequently, the change in g m was observed both in the phase angle and amplitude of the recorded signal. It turned out that, when using long integration times in our lock-in detection, the phase signal was easier to analyze for identification of FQH states than the amplitude signal. Hence, we found our analysis mostly on the phase of the transconductance signal.
Transconductance-based Landau fan type of diagram for sample XD is displayed in Supplementary Fig. 4b. Compared with Fig. 2b in the main paper, this scan has more overlapping fringes, which tends to broaden the observed features and to smear the results of the correlation analysis. The number of visible features indicates a stronger disorder potential present in this device compared with sample EV, which is in agreement with the worse field-effect mobility listed in Table I for sample XD.
The FQH state signatures in g m are identified using correlation analysis, which is based on the work of Lee et al. [8]. A line with a slope corresponding to a filling factor ν, is drawn on the pixel map of g m (n, B) and the nearest pixels (n i , B j ) are selected. The values at selected points (n i , B j ) are transferred to vectors A w k where the index k covers the full line and the index w keeps track of the vertical (B) coordinate of the drawn lines (see Supplementary Fig. 4a). The cross-correlation h(ν) w of pixels within a single vector A w k is calculated using where the Kronecker delta δ kl removes the auto-correlation part from the function and S is the total number of pixel products in the summation over k and l. The total unscaled correlation function is obtained by summing over all values of w: h(ν) = ∑ w h(ν) w . In order to find dominant fractional slopes in the data, the cross-correlation was calculated for all ν ∈ {0...1.5} and histograms of h(ν) were plotted. A peak in the correlation histogram indicates presence of a dominating feature corresponding to the filling factor ν. For a better inter-comparison of different analyzed regions, we scale our correlation histogram by the maximum correlation: The correlation map for sample XD is shown in Supplementary Fig. 4c. As already mentioned above, the correlation peaks in this picture are broadened in comparison with Fig. 2c of the main paper. This is because the disorder potential in sample XD is larger than in EV, and disorder will in general suppress the observation of FQH states via life time broadening. Nevertheless, the transconductance measurement is able to bring about many of the fractional states in spite of the disorder.
The data in Fig. 2b in the main text display also clear fringes with slopes ν = 1 and 2 overlaid in the middle of the picture on the bulk fringes. This coexistence of fringes representing different filling factors ν requires the local charge density n loc > eB/h. Furthermore, these slopes are by far off from the possible slopes facilitated by the width of the residual charge distribution based on log G(V g ) analysis. Hence, we conclude that ν = 1 and 2 fringes are located at the boundary regions near the metallic contacts. Due to absence of any polarizing medium close to suspended graphene, the screening of charge near metallic contacts is rather weak. Consequently, strong doping from a metallic contact can extend appreciably in to the suspended graphene near the contact [9]. The amount of edge doping depends on the details of the metallic contacts, which may be in the range of E F ∼ −100 meV [4]. Such a doping corresponds to a charge density of n c = 7.2 × 10 11 cm −2 > eB/h under the contact metal, which is sufficient for accommodating QH states with ν = 2.  Figure 4: Transconductance analysis: a) Vector A w k with slope ν is formed by selecting a line y = νx + w on the measured n − B plane and taking the pixels closest to the line on each row (i.e. a particular field B) as illustrated in the figure. Blue and red lines correspond to filling factor slopes ν 1 and ν 2 , respectively, and the black dots denote data pixels: the selected data points for correlation analysis are indicated by blue and red circles for these two filling factors. b) Measured transconductance phase data at high field 7 − 9 T for sample XD (the blank upper right corner was not scanned in the experiment). The dashed yellow lines mark the boundaries of the sections in which the correlation function was evaluated. The color on top of each stripe corresponds to the analyzed histogram in Supplementary Fig. 4c. c) Scaled correlation function c(ν) obtained from the data in Supplementary Fig. 4b. The identified fractional states are marked by arrows; states marked in red could not be resolved in conductance measurements directly.

Supplementary Note 5: Magnetoconductance
The Landau fan diagram measured on sample EV (see Table I in the main paper) is illustrated in Supplementary Fig. 5a. In addition to several basic integer Landau levels (2,6,10,14) and broken symmetry states (1, 3, 4), several fractional states such as 1/3, 2/5, 3/7, 4/7, 3/5, and 2/3 are visible in the conductance. However, the levels are visible on the electron side but not clearly on the hole side 1 . Consequently, our magnetoconductance work concentrates on the FQH states on the electron side. Supplementary Fig. 5b illustrates the data of Supplementary Fig. 5a after differentiation with respect to charge density. By comparing these two figures for G d and d log(G d )/dn, it becomes immediately clear that the differentiated data provides a much better resolution for identifying weak fractional states, including those of interacting composite fermions (CF). This is well evidenced by the data displayed in Fig. 2a in the main paper.
The interacting composite fermion states are visible also in our data on the differential inverse magnetoconductance G −1 d = dV /dI| V =0 , which is depicted in Fig. 3a in the main paper for sample EV2. The data display two interacting CF states at ν = 4/11 and ν = 4/13, located on opposite sides of 1/3 state. The G −1 d data on sample EV2 is replotted in Supplementary Fig. 5c as a surface plot on ν vs. B plane. We note that, in this plot, all peaks assigned to FQH states are well vertical and they do not shift with field over the range B = 6 − 8.5 T.
The data in Supplementary Fig. 5c clearly indicates the independence of the peaks at ν = 1/3 and at ν = 4/11: the peak at ν = 4/11 appears narrower and it emerges at clearly different field value as the peak at ν = 1/3. The same conclusions apply for the comparison of ν = 1/3 with the peak at ν = 4/13. The signature at ν = 4/11 was also confirmed in mixing current experiments probing non-linear conductance at higher frequencies around 20 MHz where the hopping conductance becomes stronger.
In addition to sample EV2, the interacting CF states 4/11 and 4/13 are well visible in the G d = dI/dV vs. ν traces of sample EV measured at magnetic fields 6 − 7.5 T. These data are displayed in Supplementary Fig. 5d. In these data, the state 4/11 is better visible than 4/13, which is to be expected since the former fraction is related to 2Φ 0 composite fermion series while the latter belongs to interacting 4Φ 0 CF particles.  Supplementary Fig. 5a differentiated with respect to charge density. The high field region, where the scale is by a factor of 100 smaller than the low-B region, is separated using a horizontal black stripe. c) Inverse magnetoconductance G −1 d on the plane spanned by the filling factor ν and magnetic field B measured for sample EV2. Interacting CF states 4/13 = 0.308 and 4/11 = 0.363 are observed next to ν = 1/3 state above critical field values of 6.8 and 6.3 T, respectively. d) The differential conductivity σ xx = G d ln(r o /r i ) 2π vs. the filling factor ν for sample EV, measured at magnetic fields B = 6 − 7.5 T.

Supplementary Note 6: Basic transport processes
The IV characteristics of our Corbino samples contain contributions from many different transport processes: 1) quantum tunneling, 2) thermally assisted quantum tunneling, 3) thermal activation, 4) phonon assisted hopping, 5) electron crystal sliding a la charge density wave. All these processes leave their characteristics in to the IV curves. Furthermore, we have measured IV characteristics under rfirradiation (see Supplementary Note 9) at which the transport is influenced by photon assisted hopping, 6). Here we will discuss only those topics which are most relevant for the explanation of our data, namely items 3 and 6 above. However, one should keep in mind that processes in items 2 and 3 are intimately connected, and parameters for thermal activation may become adjusted by quantum tunneling near the top of the energy barrier.

Thermal activation
The conductance in the bulk of a quantum Hall conductor depends intimately on the potential landscape of the state of the system. The potential landscape leads to localized states as illustrated in Supplementary Fig. 6a. Theoretical models in the case of a long-range random disorder potential predict thermally activated conductivity, where the activation energy U s C∆ ν is proportional to the gap ∆ ν in FQH states with C 1, and the prefactor A (e * ) 2 /h is governed by effective charge e * of charge carriers [10,11]; in thermal activation, the effective charge is related to the filling factor e * = νe [10,11], whereas in quantum tunneling regime it may be either νe or e [12]. The suppression factor C of ∆ ν in the activation formula depends on the significance of quantum tunneling near the top of the barrier. In a similar approximative approach, the conductivity in the quantum tunneling regime can be expressed as σ = A exp(−U s /2k B T cr ), where T cr denotes the cross-over temperature between thermal activation and quantum tunneling. The cross-over temperature is difficult to determine theoretically but it can be estimated from the saturation temperature of σ (T ) scans, which in our experiments varies in the range T = 1 − 0.5 K.
Our conductance at high T agrees with the thermal activation process [10,11] and, in this regime, the data can be employed to obtain estimates of the excitation energy gaps of the FQH states by assuming ∆ ν = U s . By determining the activation energy across ν = 1/2 and assuming linear dependence for ∆ ν vs ν, we obtain approximatively 0.2 meV for the composite fermion LL width. This can be regarded as a systematic uncertainty (offset) for the energy gaps quoted in our paper for FQH states. Supplementary Fig. 6b displays conductance data for the state ν = 4/3 for the sample EV2. The data indicate thermally activated conduction at temperatures above 1 K. By using Supplementary Eq. 12, we obtain for the energy gap ∆ 4/3 = 0.7 K. At low temperatures, especially at low filling factors, the role of thermal activation becomes weaker, and the conductance and IV characteristics display a more complex dependence on charge density as will be discussed in Supplementary Note 7.

Photon assisted hopping
Hopping conduction in a disordered conductor having localized states is enhanced by microwave irradiation. This problem has been reviewed e.g. by Efros and Shklovskii [13]. We will deal with this issue only qualitatively, and our discussion is inspired by the review of Galperin and Gurevich [14].
It was first shown in Ref. [15] that, under microwave irradiation, the hopping conduction across localized states is determined by so called close pairs. Each such pair consists of one localized state which is occupied by an electron and one localized state which is empty, the distance between them being much smaller than the average distance between localized states given by r = ( 4π 2 3 N A ) −1/2 where N A is the density of localized states. Due to absorption, electron occupancies in the pairs are changed, which is equivalent to an enhancement of the average energy of bound electrons. Thus, microwave absorption leads effectively to heating of the system of localized states. Below we determine the conductivity σ (ω) that is relevant for the absorption of irradiation in the presence of localized states in a three dimensional system. As usual in hopping conductance, we don't expect any qualitative change between different dimensionalities, only a change in the exponent α of the conductivity σ ∝ exp( E 0 E α ) in terms of characteristic energy.
This difference in the exponent α is irrelevant for the following discussion.
We are interested in the regime in which e 2 /(4πε r ) >>hω, k B T , i.e. the charging energy of the localized state is the largest energy scale. In the resonant case (hω matches the energy difference between the levels in question), the Ohmic microwave conductivity is given by Here g denotes the single particle density of states, a corresponds to the radius of the localized state, and r ω = a log Λ 0 hω (14) specifies the minimum distance between pairs with energy separationhω; Λ 0 denotes the off-diagonal elements in the tunnel coupling matrix. As r ω does not depend on temperature, the resonant conductance σ res 0 (ω) decreases with increasing T . Similarly, a strong drive will result in more equalized populations, which leads to decrease of absorption. In both cases of reduced absorption, relaxation of the close pairs becomes of major importance.
In the non-resonant (relaxational) case with fast relaxation ωτ 1 << 1, where 1/τ 1 describes the relaxation rate due to the environment, the conductance becomes In our experiments based on mixing current detection of the FQH states, the relevant frequency is around 20 MHz (see Note 10). We expect that the localized states will remain incoherent in this regime (i.e. relaxation time τ 1 < 50 ns). In this case, the relevant pairs have an energy separation E ∼ k B T and a life time τ min ∼ ω, and the characteristic minimum distance between pairs becomes By using electron-phonon supercollision scattering rate near the Dirac point [16], we observe that r e increases with T , which leads to growth of relaxational conductance σ rel 0 (ω) with temperature. To sum up this section, hopping conductance grows with ω and, in magnetic fields where B ∼ r , we may have σ (ω) >> σ (0). In addition, σ res 0 (ω) due to resonant conductance is expected to dominated by non-resonant relaxation, which will enhance both as a function of T and microwave power in the limit of ωτ 1 << 1. This microwave absorption will result in heating of graphene that, in turn, will govern the change in low frequency (or DC) conductance in the measurements on our sample. Supplementary Figure 6: Transport process: a) Schematic picture of localized states with variation in charge density marked in green (enhanced density) and red (reduced density). The localized states are embedded within the incompressible FQH liquid with constant electron density (brown plane). Landau levels (LLs) are bending upwards when approaching the red regions while they bend down around the green bumps. Consequently, skipping orbits of edge state electrons reside outside and inside the borders of the low and high density regions, respectively. White-blue arrows mark the directions of the currents of these localized edge states. Electric transport at low temperatures takes place via quantum tunneling between the localized states through a barrier due to the energy gap ∆ ν of the incompressible FQH region. In case of considerable variation of electron density (i.e. the charge dips/hillocks reaching the next FQH charge density plane), there will be additional localized fractional states within such regions. Since these states are localized within a bounded region with an edge state, their effect on charge transport can be neglected. b) Logarithm of the zero-bias conductance G vs. inverse temperature 1/T for sample EV2 at magnetic field B = 4 T. The dashed line corresponds to an energy gap of ∆ 4/3 = 0.7 K

Supplementary Note 7: IV-characteristics
Our IV characteristics at low filling factor (ν = 0.16...0.33) are illustrated in Supplementary Fig. 7a for sample EV. At ν = 0.33, the IV curve displays first a linear part that extends nearly up to 0.5 − 0.7 mV, above which there is a strong increase in current in a power-law-like fashion as V α . Such an IV curve can be understood by transport via quantum tunneling along a percolating path of several localized states where the non-linear regime would be a sign of competing order of correlated tunneling of chiral edge states [17,18]. At higher temperatures, this conductance at ν = 0.33 becomes weakly temperature dependent, which is an indication of thermally assisted quantum tunneling. The combination of thermal activation and quantum tunneling leads to a lowered effective gap as determined from the temperature dependence of conductance [11]. At V > V T 1 meV, the IV characteristics becomes a power law V α with an exponent of α = 3 − 4. We assign this to edge-to-edge tunneling of fractional charge between localized states along the percolating path 2 .
At filling factor ν = 0.20, even the low voltage part appears as non-linear and it is difficult to resolve any Ohmic part in the IV curve. This behavior is identified as a cross-over regime between FQH and Wigner crystal order: Here pinning of the crystallites in the Wigner phase is not fully developed and linear quantum tunneling and non-linear sliding processes of charge are competing, resulting in an IV curve that is difficult to classify.
The behavior of our non-linear IV characteristics for sample XD is illustrated in Fig. 4b in the main paper: the data at ν = 0.16 are seen to display exponential increase in current between V ∼ 0.1 − 2.0 mV. Here we display the IV characteristics of sample EV2 in Supplementary Fig. 7b at filling factor ν = 0.15. The data indicate thermal depinning of Wigner crystal which is non-linear to the lowest detectable currents. Eq. 1 of the main paper has been used to fit the data using parameters e * = e, T = 0.2 K, ∆ = 170 µV, and N = 5. These values are consistent with results on the XD sample as well as with the results of the same sample under rf irradiation (see below). Supplementary Fig. 7c displays the logarithm of total conductance G tot = I/V on the ν − V plane for sample XD. The linear low-bias structure of the IV curves at ν > 0.2 is seen as constant conductance regions below the power-law regime. Above the power-law regime, G tot gradually approaches the high-bias slope dI/dV ∼ 100 kΩ −1 . Supplementary Fig. 7c indicates that conductance is strongly suppressed between ν = 0.12 − 0.20, but good conductance re-emerges again at ν = 0.10 − 0.12. For sample EV2, the Wigner crystal state found over a slightly smaller regime covering ν = 0.15 − 0.18.
The IV characteristics of EV2 in the re-entrant good conductance regime at filling factor ν = 0.14 is illustrated in Supplementary Fig. 7d. The behavior at ν = 0.14 resembles that of IV curves at ν = 1/3, where the initial linear behavior changes to a power law with an exponent α = 3 − 4, but now with an exponent α 7. The shape of the IV curve hints towards re-entrance of FQH states, but we are not able to identify any peak in the gate sweeps, which would make this conjecture more plausible. Nevertheless, as displayed in Fig. 3B in the main paper, we have determined the energy gap using G(T ) measurements and we can state that this intermediate conductance state is gapped. At smaller charge density, for example at ν = 0.12, the IV curve displays a larger gap than in the Wigner crystal regime, and the behavior gradually develops towards Zener tunneling IV curves.

Supplementary Note 8: Resonance modes in pinned Wigner crystal
The nature of pinning in disorder Wigner crystal depends strongly on the characteristics of the disorder. In suspended graphene membrane there are two quite different disorder elements present. First, there are membrane ripples (dimples), which can be either static or alternatively dynamic due to thermal fluctuations. Second, residual charge density is related to charged impurity disorder, most likely caused by residues of the lithography process, which are present in spite of the current cleaning. Static ripples can be modeled by the interfacial disorder model of Ref. [20], which relates the pinning potential of the Wigner crystal to the depth and width of the interfacial dimples. The pinning potential caused by charged impurities in clean graphene has a rather long-scale variation so that the particle size (magnetic length l B ) is smaller than the correlation length of the disorder ξ . Such a pinning case has been treated for example in Ref. [21]. We expect that both pinning mechanisms will contribute to the pinning potential and contribute to the pinning frequency. By assuming that the disorder contributions are independent, we obtain the pinning frequency ω p as a combination of these two processes: where ω q and ω r refer to charged impurities and interfacial disorder, respectively. Below we will outline the basic pinning energy considerations in both disorder cases.

Pinning by charged impurities
A Wigner solid is pinned in the presence of disorder, which results in breaking of the long-range order of the crystal. Consequently, the crystal splits into coherent, independent crystallites, and a gap opens up in the transverse phonon modes of the Wigner lattice. In our graphene case, the correlation length ξ > l B , and the particle locally sees a smooth potential, for which the particle size l B is not directly relevant. Under these conditions, the crystallite size L c over which collective pinning takes place is independent of the magnetic field. The crystallite size specifies the collective-pinning-mode frequency of the Wigner solid. One obtains for the pinning frequency in the simplest approximation where c is the shear modulus of the crystal, a is the lattice constant of the Wigner crystal, and R a = ca 2 /n 0 ∆ is the length scale at which the fluctuation of the electrons in the Wigner crystal starts to be on the order of a; n 0 denotes the average charge density. Consequently, these kinds of pinning sites yield a contribution to pinning frequency which decreases as B −1 . Using realistic values for ∆ and ξ , we obtain pinning frequencies in the GHz range. The relation of the pinning frequency f p to the crystallite size is given by L c = (2πc/n i eB f p ) 1/2 . By assuming the classical form for the shear modulus in a 2D solid, c = 0.245e 2 n 3/2 i /4πε 0 ε g , and using ε g ∼ 3, we obtain for the domain size of the Wigner crystal L c = 0.70 µm at B = 9 T. This is approximately half of our sample disk width r o − r i = 1.15 µm.
Commonly, charged impurities are viewed as a paradigm for a strongly pinned system [22], where certain locations of the lattice are basically fixed and, as such, eliminated as degrees of freedom. However, it is argued in Ref. [20] that one may consider a strongly pinned Wigner crystal as a weakly pinned crystal with point defects. If so, then the strongly pinned Wigner crystal for the purposes of AC conductance is equivalent to a weakly pinned, defective WC. Such arguments provide further support for the general applicability of the weak pinning formulas used in the main paper.

Pinning by ripples of membrane
Here we assume that the dimple lattice of suspended graphene can be approximated by the model of interfacial pits of depth ∆z and width w developed in Ref. [20]. The scanning tunneling microscope results of Ref. [23] indicate, for a typical currentcleaned graphene sample such as ours, an amplitude of corrugation of Z = 0.4 nm with a radius of curvature R = 1.5 nm (i.e. wave length ∼ 5 nm). Hence, we may assume ∆z = 0.4 nm and w = 3 nm. In our sample, the ratio of ∆z/ε is the nearly the same as that used in the evaluations in Ref. [20] (both ε g = 3 and ∆z ∼ 0.4 nm are by a factor of ∼ 3 smaller in graphene when compared with GaAs heterostructures). The potential energy gained by placing an electron into a dimple that is much larger than the magnetic length is given by ∆E = 2πn 0 e 2 ∆z/(4πε 0 ε g ) ≈ 7 K in temperature units. Since a typical dimple is generally be much smaller than the magnetic length, the pinning energy of an electron trapped in a single dimple will be on order of ∆Ew 2 /l 2 B

∆E.
As in Ref. [20], we will assume the average distance between dimples 1/ √ n i is smaller than the lattice spacing of electrons a but larger than the magnetic length l B . This means that each unit cell of the Wigner crystal contains several dimples on to which an electron could be trapped. We will ignore the spread in the size distribution of dimples and assume that their distribution can be approximated by single size and uniform density (> 10 11 cm −2 ), the latter of which will form a fitting parameter, to be determined self-consistently in the model.
The average number of dimples occupied by an electron in a single domain of size L c if placed at a random location is given by N pin = πn 0 L 2 c l 2 B n i . Consequently, the energy per electron gained from the disorder potential u p is By assuming purely transverse distortions in the lattice, we obtain an estimate for the distortion energy per electron where c is the shear modulus of the lattice. Minimizing the total energy u d − u p with respect to L c , we obtain for the optimum domain size The resulting pinning energy per particle is found by substituting this value of L c into our expression for u p in Supplementary Eq. 18. Using standard lattice parameters for the shear modulus, the resulting pinning energy per particle can be written as In the simplest approximation, the pinning frequency is given by the average binding energy per site, i.e.,hω pin ≈ u p . Note that within this approximation, ω pin ∝ l −2 B ∝ B, which leads to a pinning frequency that increases with magnetic field. Using n i ≈ 2.5 × 10 11 cm −2 , l B ≈ 9 nm, w = 3 nm, and ∆E/k B = 7 K, the pinning frequency becomes ∼ 1.5 GHz. This value is strongly dependent on ∆E or the dimple size w, which could even be larger than 3 nm [23].
To sum up this section, we have two independent pinning potentials for the electrons in the Wigner phase. Charged impurities lead to pinning frequency ω q ∝ B −1 while dimples yield ω r ∝ B. These pinning frequencies are added as independent contributions so that they yield the total pinning frequency ω 2 p = ω 2 q + ω 2 r . Since the pinning frequency does not display much magnetic field dependence, |d f p /dB| < 100 MHz/T, we conclude that both contributions are approximately of equal magnitude. Consequently, the pinning frequency used in deducing the crystallite size, is only approximative and it yields an upper limit for the crystallite size.

Supplementary Note 9: Wigner crystal under rf irradiation
Commonly, microwave conductance measurements have been conducted on GaAs 2DEG by capacitively coupling RF power to the 2DEG and measuring its transmission through the sample (at high frequency). FQHSs, Wigner phases, and CDWphases have been studied with this technique [24,25,26,27,28]. Similar to earlier works, we have also probed the Wigner crystal by microwave spectroscopy. Our measurement method has two distinct features; first, the rf power is directly coupled to the Wigner crystal via the inner contact of the Corbino geometry and, second, the transport across the Wigner crystal is measured at DC. Irrespective of these differences compared with earlier studies in 2DEG, the essence of our incompressibility measurements of the Wigner crystal under rf radiation will be the same.
A schematic of the measurement circuit is depicted in Supplementary Fig. 8. We employed bias-T circuits (Minicircuits) at the mixing chamber plate in order to apply simultaneously DC and AC bias across the sample. DC (or low-frequency) conductance was measured in voltage bias configuration through inductors while microwaves were coupled via capacitors to the sample. In these conductance measurements, a DC bias of V = 50 µV was applied to the inner contact while the resulting current at the outer contact was tracked using a Stanford SR570 current (transimpedance) preamplifier operating at a gain of 10 10 V/A followed by a HP 34410A multimeter. The DC bias was provided using HP 33120A generator connected through a 1:2000 voltage divider.
Anritsu MG3692B rf-generator was used to generate the applied rf irradiation at frequencies f = 10 MHz -10 GHz. The rf-input line had an attenuation of 50 dB at f = 1 GHz, with additional ∼10 dB per decade in frequency (i.e. attenuation was roughly 40 dB at 100 MHz, and 60 dB at 10 GHz). The rf-signal was fed to the inner contact through the bias-T, while the rf-line was terminated using a 50 Ω resistor at the outer contact.
There should always be hopping conductance of Efros-Shklovskii type, even in the case of a disordered Wigner crystal [29]. As discussed in some detail in Supplemetary Note 6, this conduction is mediated by resonant pairs of localized states, the absorption of which grows with enhanced frequency. Since there is a large spread of energy differences among the resonant pairs in our sample, we expect to have an increase of absorption over a large band of irradiation frequencies. This is apparent in our experiments at large irradiation power as displayed in Supplementary Fig. 9 for the strongly insulating regime at the filling factor ν = 0.12. The rf absorption leads to increase of conductance via heating, and this enhancement in G is seen to increase monotonically with frequency in Supplementary Fig. 9.
At high irradiation power, Joule heating P = σ (ω)v 2 ac becomes relevant, and it leads to an increase of temperature and conductance in the sample. At low irradiation powers (around −70 . . . − 65 dBm), however, only one strong peak in DC conductance around 3 GHz is observed (see Fig. 4a in the main paper). The resonant frequency coincides well with estimates for a resonance in a pinned, disordered Wigner crystal. The amplitude of this conductance resonance with respect to a broad background current enhancement grows with temperature. We assign this behaviour to a combined action of activated conductance and enhanced Joule heating by the microwave irradiation in the Wigner crystal. The Joule heating P = σ (ω)v 2 ac of the electron system is strongly frequency dependent due to the resonant behavior of σ (ω) at pinning frequency ω p . At the resonance, increase in temperature leads to an increase in observed conductance, and the change in conductance per unit Kelvin (∆T ) will grow with increasing temperature due to increased microwave absorption. The conductance enhancement will continue till the Wigner crystal melts and the resonance absorption at σ (ω p ) is lost. This scenario supports the identification of the maximum in Fig. 4c of the main paper as the melting temperature of the Wigner crystal. Similar identification of the Wigner crystal melting has been employed in Ref. [30]. For reference, we also show G(T ) without rf irradiation in Supplementary Fig. 10. This plot on linear scale indicates beginning of melting at T mb = 1.4 K, which is lower but consistent with the melting temperature T m = 1.7 K obtained from rf resonance measurements.

Supplementary Note 10: FQH states under rf irradiation
In addition of studies of the FQH states under GHz irradiation, we have studied them at MHz frequencies using mixing currents facilitated by mechanical motion [31]. The mechanical motion with amplitude z yields a mixing current via differential conductance G d (ω) as when the sample is biased using a frequency modulated signal V FM (t) = V AC (cos(2π f t + ( f A / f L ) sin(2π f L t)). Here V AC is the amplitude of the carrier frequency signal, f L is the frequency modulation, and f A defines the modulation depth. In the Wigner crystal and FQH transport regimes, the hopping conduction is enhanced by operation frequency so that the mixing current method may be more sensitive than the low frequency or DC conductance measurement. We have carefully measured the region around ν = 1/3 and our experiments confirm the finding of the fraction state of ν = 4/11 observed in the low frequency measurements.