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Dirac Fermion Quantum Hall Antidot in Graphene

The ability to localize and manipulate individual quasiparticles in mesoscopic structures is critical in experimental studies of quantum mechanics and thermodynamics, and in potential quantum information devices, e.g., for topological schemes of quantum computation. In strong magnetic field, the quantum Hall edge modes can be confined around the circumference of a small antidot, forming discrete energy levels that have a unique ability to localize fractionally charged quasiparticles. Here, we demonstrate a Dirac fermion quantum Hall antidot in graphene in the integer quantum Hall regime, where charge transport characteristics can be adjusted through the coupling strength between the contacts and the antidot, from Coulomb blockade dominated tunneling under weak coupling to the effectively non-interacting resonant tunneling under strong coupling. Both regimes are characterized by single -flux and -charge oscillations in conductance persisting up to temperatures over 2 orders of magnitude higher than previous reports in other material systems. Such graphene quantum Hall antidots may serve as a promising platform for building and studying novel quantum circuits for quantum simulation and computation.


I. Introduction
Localization and manipulation of individual quasiparticles play an important role in studies of quantum mechanics and quantum thermodynamics, and in the applications of quantum information devices. A wide variety of mesoscopic systems, such as quantum dots [1][2][3][4] , nitrogenvacancy (NV) centers 5,6 , superconducting Cooper pair boxes 7-10 , superconducting quantum interference devices 11 , etc., have been extensively studied for quantum manipulation of charges, spins, and magnetic fluxes. Quantum Hall (QH) antidots, on the other hand, offer a promising approach for localizing quantum Hall quasiparticles. Due to quantum confinement, the chiral one-dimensional (1D) edge mode has its energy quantized into discrete levels, mimicking a large, tunable "artificial atom" which hosts QH quasiparticles. Compared to the other approaches, QH antidots are capable of localizing even exotic quasiparticles with fractional charges and nontrivial exchange statistics. It therefore holds promise for topological schemes of quantum computation 12,13 . Experimentally, pioneering studies of QH antidots have been carried out using GaAs two-dimensional electron gas (2DEG), where localization of integer 14,15 and fractionally charged 16 quasiparticles have been demonstrated. The potential of applying such QH antidots for quantum information applications has also been discussed 17 . On the other hand, due to the small energy-scale associated with Landau level (LL) spacing and energy quantization, the signatures of the localized QH edge states, namely charge and magnetic flux oscillations in conductance across the antidot, are fragile and require very low electron temperature (typically sub-100 milliKelvin) 18 to observe. A 2DEG system which can provide more robust localization of QH quasiparticles and stronger coherence is therefore desirable for realizing more complex devices and functionalities.
The development of 2D crystal graphene in the recent decade raised a new opportunity for studying localized QH states in the antidot setup. The Dirac-nature of the 2DEG in graphene 19 differs fundamentally from that in GaAs, due to its linear energy dispersion, chirality and non-trivial Berry's phase. As a result, graphene can achieve high charge carrier mobilities (> 10 5 cm 2 Vs ⁄ ) which persist even with densities down to ~10 9 cm −2 without localization. It has a large and energy-independent Fermi velocity ( ≈ 10 6 m/s), which leads to large LL spacing ( ∆ = √2 ℏ 2 = 35√ [T]meV) 20 , as well as large quantization energy spacing under confinement. Both are critical factors for realizing robust localization of QH states.
Technically, being a single atomic layer, the size of graphene devices may be pushed down to nanometer-scale with sharp definition 21 . And Ohmic electrical contacts with low contact resistance have been routinely achieved both for top contacts 22 and side contacts 23 . Despite all these promising characteristics, experimental work on confining QH edge states in monolayer graphene has been limited, and mainly focused on gate-defined quantum dots 24,25 . Charge transport studies on well-structured graphene-based QH antidots, on the other hand, have not been reported to our knowledge. In this work, we study Dirac electron QH antidots in graphene, and demonstrate robust localization of QH edge states in the lowest -LL (LLL) which persists up to 10-100 times higher temperature compared to previous reports.

II. Results and Discussion
The samples used in this work, illustrated in Figure 1, are point contact-coupled antidots embedded in hexagonal boron nitride (hBN)-encapsulated graphene field effect transistors. The two-terminal conductance consists of a background from the bulk 26 and the tunneling conductance through the QH antidot. Under well-developed QH effect, depending on the coupling strength between the point contacts and the antidot, the QH plateau resistances can vary between ℎ 2 for decoupled antidot, and ℎ 2 2 in the situation when antidot is split by the contacts (see Supplementary Information). In our experiments, the coupling between the point contacts and the antidot is carefully adjusted to be in between these two limits. The overall geometry of our devices is designed so that: 1) the diameter of the antidot ≫ = √ ℏ , where is the magnetic length ~10nm for the few-Tesla magnetic field applied here; 2) is sufficiently small for energy quantization; 3) the point contacts probing the antidot are sufficiently sharp to minimize invasive effects and to optimize phase coherent charge transport. Satisfying these criteria, we designed the diameter to be ~ 200 − 310nm and the width of the point contacts to be ~ 15nm. The dimensions of the devices are directly confirmed by SEM imaging. Figure   1C shows QH resistance as a function of filling factor = ℎ in various magnetic fields (through ramping of the gate voltage). We note that in measuring these curves the gate voltage ramping speed is relatively large and the fine oscillatory features (as discussed below) are washed out. At anomalous integer fillings = 4 ( + 1 2 ), the resistance values are observed to be between ℎ 2 and ℎ 2 2 due to the conduction through the antidot.
In quantizing magnetic fields, periodic conductance oscillations both in gate voltage and in magnetic field become prevalent at filling factors between = 1 and = 2, with typical amplitude of ~0.1 0 ( 0 = 2 ℎ is the conductance quantum). Depending on the coupling distance between the point contacts and the antidot, two types of periodicity are observed which we separately discuss below. In the case of relatively weak coupling, as shown in Figure 1B for a 310nm-diameter antidot sample with a few tens of nanometers contact-antidot separation, the conductance oscillations under ramping magnetic field show an approximately constant period of 21 mT. In Figure 2B, we plot the oscillatory conductance as a function of magnetic flux through the antidot, normalized over single electron magnetic flux quantum: , where is the diameter of the QH edge current encircling the antidot, and Φ 0 = ℎ . With = 350nm which is slightly larger than the physical diameter of the antidot = 310 (by the amount consistent roughly with the notion that the edge states are removed from the physical edge by the distance of an order of the magnetic length ), the observed conductance oscillations match closely with a flux period of 0.5 ± 0.1Φ 0 . We note that the observed This suggests their origin to be intrinsic to the antidots, and independent of the unintentional defects.
Corresponding periodic conductance oscillations are also observed as the gate voltage is swept at fixed magnetic field. Figure 2D shows the conductance oscillations in the 310 nm antidot as a function of gate voltage and charge number change over the area of the antidot: . Here is the geometrical capacitance per area. On average, each conductance oscillation corresponds to addition/removal of one electron into the edge state on the antidot.
Compared to the flux oscillations, the charge oscillations appear to be somewhat less regular indicating the presence of random fluctuations in the coupling capacitance between the back gate and the antidot. Direct correspondence between the charge and flux oscillations is evidenced by Figure 2E, where the conductance oscillations form tilted stripes on the "magnetic field-gate voltage" plane. Along the charge oscillation axis, however, there are significant random fluctuations which manifest the charge noise.
Next, we discuss the antidots with stronger coupling to the point contacts, where a different type of magnetic oscillation periodicity is observed, as illustrated by Figure 3A  . The discrepancy may be attributed to the errors in the estimations of the antidot size and effective gate capacitance at the antidot.
The observation of Φ 0 period (which obviously also includes the Electron-electron interaction leads to Coulomb-blockade-type effects. In the weak coupling limit ("closed'' antidot), the electron interaction energy 2meV, which is much smaller than the LL spacing but significantly larger than the Zeeman energy within the range of magnetic fields applied. In the presence of metallic contacts, should be screened and somewhat reduced.
Following the discussion above, the antidot can be described by a many-body Hamiltonian (see Supplementary Information): is the total charge number associated with the edge states; , is the occupation number of the state with orbital index j, spin index = ±1, and the single-particle energy , = + /2, where μB is the Bohr magneton, B -the magnetic field, and the gfactor for electrons in graphene is close to its free-electron value 27  Byers-Yang theorem 28 . In addition, in Coulomb interaction-dominated regime, the Hamiltonian also gives rise to finer flux oscillations with a periodicity of Φ 0 , which corresponds to ∆ ϕ = 1.
For a relatively weakly coupled ("closed") antidot (e.g., the 310 nm-diameter antidot sample in Figure 1B) and with the Fermi energy above the LLL ( = 2), equally spaced magnetic oscillations are observed with Φ 0 /2 period (consistent with the larger period Φ 0 ) independent of magnetic field. This suggests that the antidot tunneling is dominated by electronelectron interaction which results in a constant energy gap for the addition of individual electrons to the antidot that is independent, e.g., of electron spin. To find this addition energy, we measured the bias voltage dependence of the device conductance. In Figure 2F, we plot the background-normalized differential conductance (dI/dV) as a function of gate voltage and bias voltage (charge stability diagram), which resembles the Coulomb blockade diamonds, revealing the charge addition energy to be ~1.5meV for the 310nm diameter antidot sample. We note that the quality of the charge stability diagram is limited by the charge noise in the sample and the electrical noise in our measurement setup. Well defined periodicity of the conductance oscillations in this antidot implies that the addition energy is given by the Coulomb repulsion energy , with the single-particle energy level spacing ~ either too small to be noticeable in comparison to , or washed out by electron-electron relaxation. Compared to a simple "closedantidot" estimate of ~2.3meV, the observed Coulomb repulsion energy is somewhat smaller, which can be qualitatively explained by finite screening effect from the contacts.
For a strongly coupled ("open") antidot, the electron-electron interaction is largely screened by the electrodes. Neglecting electron correlations, the single-particle picture of energy level quantization and Zeeman splitting predicts AB oscillation with a primary oscillation period of Φ 0 and a B-dependent Zeeman splitting of the conductance peaks in strong magnetic fields.
This is indeed consistent with our observation where single-flux oscillations in weak magnetic field split and resolve into "M"-shape pairs in strong magnetic field. In Figure 3C, the conductance oscillation period in magnetic field is plotted as a function of magnetic field, both for the "M"-shaped pairs (Δ ) and for the magnetic field splitting within each pair (Δ ). Δ is roughly magnetic field independent with its corresponding flux period ~Φ 0 . The magnetic field dependence of Δ can be fit to a straight line intersecting the origin. Extrapolating the linear dependence, Δ ( 0 ) 2 4 = Φ 0 , we get 0~1 8.5T at which Zeeman splitting becomes equal to the quantization energy spacing = 2ℏ / . Based on this comparison, we can estimate 1meV. This value is confirmed by measuring the bias and gate dependence of the differential conductance ( Figure 3D) where the level spacing energy is a sum of the heights of the two adjacent "diamonds" (from Zeeman splitting) in the plot, which gives ~ 2-2.5meV.
From the level spacing, we can estimate the QH edge state velocity ~3.4 × 10 5 m/s , about 1/3 of the Fermi velocity of free Dirac electrons in graphene.
In the Coulomb blockade-dominated regime, the conductance oscillations are suppressed at elevated temperatures through thermal excitation/smearing ( Figure 4A). The corresponding antidot conductance can be calculated numerically through the linear response of the Coulomb blockade rate equations 29 to the external bias (see Supplementary Information). Figure 4B shows the temperature dependence of the averaged conductance oscillation amplitude (defined as the difference in conductance at successive minima and maxima), with the best fit calculated numerically with addition energy and a normalization prefactor as fitting parameters. The best fit of the addition energy, = 1.3meV, is in good agreement with the value obtained experimentally from the height of the Coulomb diamonds for the 310 nm antidot sample ( Figure   2F). As is characteristic of single electron devices, the linear conductance oscillations are nearly completely suppressed as the temperature exceeds half of the addition energy.
We note that in the weakly couple QH antidot discussed above, the Coulomb oscillations persist up to ~4K, which is 1-2 orders of magnitude higher than the previous reports for GaAsbased devices with comparable antidot size. The robustness of single charging effects can be further enhanced by increasing the electron-electron interaction energy . To demonstrate this, we study a QH antidot sample with ~250nm effective diameter made on suspended graphene, where is enhanced through a reduction in dielectric screening. In the suspended graphene sample, carrier mobility can be sequentially improved through repeated current annealing (controlled Joule heating which evaporates the contaminants). Rather than coupling to the antidot using protruding point contacts (which induces unwanted high current density at the sharp tips, causing damage during the annealing), straight source and drain electrodes are used which are separated from the antidot by ~200nm ( Figure 4D inset). Coupling to the antidot is possible only in low magnetic fields, through extended states from the electrodes. With high mobility > 10 5 cm 2 /Vs, QH plateaus become well developed in magnetic field as low as 1 Tesla. Figure 4D shows the conductance oscillations in gate voltage at B=1T. The antidot is weakly coupled as is evident from the nearly un-perturbed QH plateau resistance background ( Figure 4D). A thorough current annealing narrows the conductance peaks, as shown by Figure 4E inset. In absence of the hBN encapsulation, the effective dielectric constant and screening becomes significantly reduced, resulting in a much larger Coulomb gap. In Figure 4F, the charge stability diagram measures a Coulomb gap of ~8meV. The large Coulomb gap allows single charge oscillations to persist for temperatures over 10K, as shown in Figure 4E. In Figure 4F, we also observe evidence of energy quantization in the excited states, indicated by the bright lines outside the Coulomb diamonds. The energy level spacing is found to be ~5meV, and is spin-degenerate due to the small magnetic field. Formation of such quantized energy levels suggests robust coherence in these high mobility samples. which provides an additional mechanism of suppression of the conductance oscillations that becomes noticeable once the thermal broadening is weak.

III. Conclusion
We demonstrate the first experimental study on Dirac electron quantum Hall antidots in graphene. Depending on the coupling strength to the antidot, both Coulomb blockade dominated tunneling and effectively non-interacting resonant tunneling are achieved. Both regimes are characterized by single-flux and single-charge oscillations in conductance which, due to the Dirac nature of the electron gas in graphene, persist up to temperatures over 2 orders of magnitude higher than that reported in previous reports on conventional 2D electron gas.
The main advantages of using graphene in QH antidots come from its large edge state velocity (as a result of the Dirac spectrum), which results in the large energy level spacing, and large Landau level separation which permits scaling up of the characteristics antidot energies.
Another technical convenience of graphene in QH antidot samples comes from the observation that the diameter of the QH edge state encircling the antidot is very close to the physical size. All this opens the possibility of precise design of the antidots and their coupling into multiple antidot structures for possible applications in quantum information. We note that while in this work we mainly focused on direct metallic point contacts, coupling to the antidot can also be achieved through the more conventional QH edge-to-edge tunneling (see Supplementary Information).
With further work to optimize the device structure and mobility, the graphene QH antidot system demonstrated here may serve as a promising platform for studying localized QH states, and for building antidot-based quantum circuits for quantum simulation and computation. Besides graphene, the chiral edge state-based antidot structure can also be applied in many other 2D systems (e.g., 2D superlattice, layered topological material, etc.), providing an effective technique for studying novel chiral quasiparticles.

S1. Device fabrication
The samples used in this work are hexagonal boron nitride (hBN)-encapsulated graphene field effect transistors with an antidot etched inside the channel. The electrical contacts follow a twoterminal geometry with sharp point contacts that protrude toward the circumference of the antidot. The antidot is well isolated from the bulk edges by several microns so that cross-talk from the bulk edges as well as the formation of hot-spots can be neglected. Graphene was encapsulated with hBN using the standard dry-transfer method. A Poly(methyl methacrylate) PMMA mask and E-beam lithography were used to define the anti-dot, which was then etched at 60W, 40/4 sccm CHF3/O2. The same procedure was used to define the side-contact electrodes and a Ta(5nm)/Nb(40nm) bilayer was then deposited using RF magnetron sputtering at a base pressure of 3 × 10 −7 Torr. The Ta/Nb bilayer was used for its low contact resistance and the superior step coverage of sputtering 22 . While Nb is a superconductor, in this work we focus on magnetic field above 5T which is significantly above its Hc2 and therefore the effects of superconductivity are of no consequence. Development was done in a low temperature (-15°C) solution of MIBK/IPA(1:3) in order to achieve a sub-30nm resolution. Measurements were carried out in an Oxford Instruments VTI with a 14T superconducting magnet and a He3 refrigerator insert, using the standard lock-in method. A bank of room-temperature pi-filters and low temperature 2-stage RC filters were used to insure the electron temperature closely followed the bath temperature. Excitations used varied from 2.5nA to 10nA at frequencies between 17Hz and 97Hz. Measurements were done at a variety of experimental parameters, including the sweeping rate of gate voltage and magnetic field, in order to confirm the reproducibility of the conductance oscillations.

S2 Zero Field Device Characteristics
The mobility for the samples measured can be estimated from the zero field gating curves ( Figure S1) using = ⁄ , where is the conductivity and is the carrier density calculated from the geometrical gate capacitance. For this work, the mobility ranged from 28,000 to 50,000 cm 2 /Vs. The mobility values achieved in these samples are larger than the typical graphene samples fabricated on SiO2 substrate, but can be further improved significantly when compared to the state-of-the-art hBN-encapsulated graphene samples. It was found that the main factors limiting mobility are the remote charge disorder from the SiO2 substrate and contamination on the top hBN surface. Strategies to improve mobility were discussed at length in [1].

S3. Antidot Coupling
Alignments between the antidot and the point contacts were carefully carried out. Due to the resolution and stability limitation of our SEM, there are inevitable variations in the coupling distance/strength between the contacts and the antidot, hence variations in the basic device characteristics. Figure S2 shows the evolution of the gating curve and tunneling amplitude as the distance between the antidot and the leads decreases. The two middle figures are described in detail in the main text, so here we will focus on the two extremes where the contacts are several magnetic lengths away (top) and "cut into" the antidot (bottom). With fully invasive contacts, that is where the leads physically pass through the antidot edge, the resulting QH plateaus have double the usual conductance as would be expected when measuring two, two-terminal devices in parallel. In this case, the phase coherence of electrons in the antidot is destroyed by the metallic leads. It is certain that this sample is monolayer because it is on the same flake as the 200nm device in the main text. In the opposite limit shown in Figures S2 (top) and S3, where the leads are several magnetic lengths away from the ant-dot edge, there appear randomly spaced oscillations in conductance on a background of the usual conductance value. This is possibly due to the influence of impurities on the tunneling probability. The histogram for gate voltage separation of neighboring conductance peaks is centered on 41 mV, which is the value one would expect given the antidot's 200 nm diameter. The magnetic field separation shows peaks at roughly integer multiples of 0 /2.

S4 Edge State Coupling
In GaAs antidots, the boundary of the antidot was defined either through chemical etching or through depletion using a top gate. The electrons encircling the antidot were then confined to within a depletion length of the physical boundary. Then, again using either a top gate or chemical etching, the edge states of a typical four terminal Hall measurement setup were extended towards the center near the antidot boundary. The distance to the antidot could then be fine-tuned using top gates or front gates. We attempted to mimic this setup by etching a hole in the center of a narrow constriction with a two terminal geometry, in order to demonstrate the feasibility of coupling multiple antidots together by proximity alone (see Figure S5). In graphene, there is no immediately obvious mechanism with which to change the distance between neighboring antidots that is viable at the required length scales (sub 20nm), so the device geometry will be fixed after lithography. With a separation of around 35 nm from the edge states of the constriction to the antidot, oscillations in conductance we observed in magnetic fields between 1.5 and 3 Tesla (see Figure S6). This is consistent with the expectation that the separation should be close to one magnetic length in order to have sufficiently large tunneling probability. This also demonstrates that in high magnetic fields, it will be necessary to have multiple antidots separated by less than 10 nm for there to be adequate coupling. Figure S5. The sample considered is a two-terminal constriction with a hole etched in the center. The black arrows show the path of the edge states, and the red lines indicate tunneling paths between the edge states and the antidot. The grey area is graphene, the gold area is the gold contacts, and the white area is graphene that has been etched. In zero field, the device shows quantized conductance plateaus at multiples of 8e 2 /h as one would expect for a constriction carrying two modes.  Figure S6. Edge tunneling into a 250 nm antidot at a bath temperature of around 700mK for the ν = 2 plateau. In this sample a graphite gate was used and the hBN separating the sample and the back-gate was 15nm.

S5 Model Hamiltonian and Aharonov-Bohm Periodicity
The antidot geometry implies that it is natural to write the Schrödinger equation for electrons in graphene in the presence of magnetic field perpendicular to the graphene plane in the polar coordinates , and employing a symmetric gauge for the vector potential ⃗ =̂. The Hamiltonian 0 for electrons in the valley is (see, e.g. [2]) where is the band-structure velocity near the Dirac point, and is the potential energy which we can approximate to be constant. For the wavefunction with momentum components close to the Dirac point, the Schrödinger equation in the polar coordinates is ) , (2) and , are the wavefunction amplitudes on the two graphene sublattices [3], while is the total electron energy. Assuming rotational symmetry, one can see immediately that the angular part of this equation is satisfied by the solution in the usual angular momentum-conserving form These functions are equivalent to the LLL wavefunctions of electrons with nonrelativistic dispersion relation (see, e.g., [4]). For the full graphene plane, the condition that the wavefunction is regular at → 0 imposes the condition ≥ 0 on the index of the wavefunctions.
In the presence of an antidot, which makes the point = 0 unaccessible, this restriction does not exist, but the relevant states have ≫ 1, and the states with < 0 would be localized very closely to the edge of the antidot and are shifted in energy beyond the relevant energy range (cf. the discussion below).
In the ′ valley, the LLL states have the same sublattice structure: they vanish on one sublattice and are given by Eq. (4) on the other, but the sublattices are switched in comparison to the valley discussed above. This difference makes the response of these states to the boundary conditions at the graphene edge very different.
For the antidots with the diameter on the order of 100 nanometers, e.g., ≃ 300 nm, as in the most of the measurements reported in this work, and field on the order of several tesla, three characteristic distance scales are very different: the size of the antidot, the magnetic length ∼ 10 nm ( ≃ 11 nm for = 5 T), and the nearest-neighbor distance = 0.14 nm in the graphene lattice. The continuous description based on Eqs. (2) to (4) is justified by ≫ . Also, condition ≫ makes it possible to simplify the wavefunctions (4) further. Expanding the logarithm of the right-hand-side of Eq. (4) near the maximum = √2 , and keeping only the second-order terms, one sees, that in the region where it has significant amplitude, the wavefunctions (4) is Gaussian: Equation (5) gives the radial structure of the LLL wavefunctions at distances larger than away from the antidot edge, which on the large scale of the antidot size can be thought of as a circle of radius 2 ≡ ⁄ . Precise structure of the wavefunctions closer to is determined by the boundary conditions at the edge. Circular geometry of the antidot implies that the nature of the boundary condition on the microscopic scale of the graphene lattice changes with the position on the edge. We adopt here the ``zigzag'' boundary conditions which are known to describe most of the generic edges of a graphene lattice [5]. This condition consists in vanishing wavefunction amplitude on one of the graphene sublattices, and is automatically satisfied for LLL states near one of the Dirac points. This means that these states are not affected at all by the edge, and remain at zero energy (see, e.g., [6]) i.e., well below the energy level relevant for electron transport through the antidot. The LLL states at the other Dirac point are non-zero on the sublattice wavefunctions on which should vanish at the edge, and therefore are pushed up in energy by the boundary condition forming one band of the spin-degenerate states that transport the current along the edge and are responsible for the proper Hall conductance 2 2 ℎ ⁄ of the LLL in graphene. In the case of the antidot, this means that we are imposing the boundary condition ( = − ) = 0 on the wavefunctions (5), which become distorted by this condition to various degree depending on how small is the distance to the edge − relative to the magnetic length . Under the realistic assumption that the antidot energies are small in comparison to the energy separation between the Landau levels, the LLL states forming the antidot edge states should not be too close to the graphene edge − ≥ , and the wavefunctions are not modified strongly in comparison to their bulk shape (4). Still, the energy is increased from zero, so that the dispersion relation of the edge state encircling the antidot acquires some finite slope ℏ that corresponds to the drift velocity of electrons ∼ (see, e.g., [6]) along the antidot edge.
This means that introducing the coordinate = along the antidot edge, one can write the coordinate part of the spin-degenerate wavefunctions of the antidot edge states as (6) and in the ideal case without disorder, the energies of these states would form an equidistant spectrum = ℏ ⁄ + .
For antidots in this work with ≃ 150 nm, the energy spacing = − −1 in this singleparticle spectrum can be estimates as ∼ 10 K taking into account that the velocity should be somewhat smaller than the band velocity . Since the electron -factor in graphene should be close to its free value, the Zeeman splitting of the spin states in the typical magnetic field of several tesla, 2 ∼ 5 K.
So far, the discussion did not include electron-electron interaction. The mean-field effect of the average self-consistent potential energy ( ) created by external electrodes and conducting electrons in graphene itself is simply to change the drift velocity of the edge states. Indeed, the ``size'' of the states (6) in the radial direction is much smaller than the characteristic scale of the variation of such a potential, which is given by the antidot size, ≪ . In this case, the only effect of the potential energy on the edge states is to add a constant value of this energy ( ) at the location of the state to its energy (7). Such an addition effectively changes the energy spacing and the velocity .
Besides the mean-field effect, electron-electron interaction creates correlations among electrons similar to ``Coulomb-blockade'' correlations in small tunnel structures [7]. Indeed, as follows from the form of edge-state wavefunctions given by Eqs. (5) and (6) the nearest-neighbor states in energy are also the nearest neighbors in the real space, and the real-space distance between them in the radial direction is very small, = − −1 ≃ √2 ⁄ ≃ 2 ⁄ ≪ in comparison to the magnetic length which determines their size. This means that on the scale of one can neglect , i.e., assume that the few edge states which participate in electron transport through the antidot, are located at essentially the same radius. As a result, the density-density interaction of electrons in these states can be correctly described with one interaction constant . This feature makes the antidot transport similar to the Coulomb blockade transport, despite the fact that the origin of this interaction constant in the case of the antidot is very different from both the metallic tunnel junctions and quantum dots. Explicitly, can be evaluated as where ̅ = { , }, and the condition ≪ , with = , made it possible to neglect the circular geometry of the antidot states. For the wavefunctions given by Eqs. (5) and (6), this integral can be evaluated with logarithmic accuracy and gives Here = Δ ⁄ is the gate voltage normalized to the period Δ = ⁄ with which the backgate voltage modulates the antidot conductance in the ``Coulomb blockade'' regime when the interaction is the dominant energy. Taking into account that the antidot radius is much smaller than the distance to the back gate, one can model the electrostatics of the antidot as hole of this radius in the conducting plane in the perpendicular electric field ⁄ inside the parallel-plate capacitor formed by graphene and the back gate. Electrostatics of this model can be solved exactly in the oblate spheroidal coordinates and shows that the charge induced around the hole by the gate voltage coincides with the charge induced in the regular parallel plate capacitor (without the hole) in the area equal to the antidot area, = 2 . This gives the estimate of the gate-voltage period Δ = 0 ⁄ which is in general agreement with the period observed experimentally.
Interaction of the antidot with another control field, the magnetic flux Φ through the antidot area is determined by the two effects both of which stem from from the dependence of the LLL states (4) on the magnetic field through the magnetic length . One effect is the increase of the energies (7) of the antidot edge states with increasing magnetic field which can be expressed in the quasiclassical terms as the interaction between the electron current in the state and the flux Φ: This interaction gives the proper account of the Aharonov-Bohm effect, which in the case of the antidot manifest itself as the periodicity of the antidot energy spectrum in Φ with the period equal to the magnetic flux quantum, Φ 0 : Another effect that couples the flux Φ to the antidot dynamics is the charge flow towards the antidot with increasing Φ through the lower boundary of the strip that defines the charge on the antidot. For the wavefunctions (4), the shift (11) of the spectrum in the energy space is associated with the similar shift of the wavefunctions in the real space, by one state per flux quantum. To see this explicitly, one can notice that the effective radius of the state ( ) (5) corresponds to the condition that the orbit area encloses flux quanta: 2 = 2 ℏ / = Φ 0 , and therefore, the states shift by one −1 ( ) → ( ) upon increase of the magnetic field in such a way that the flux Φ through these states increases by Φ 0 . Since the number of the flux quanta through the states encircling the antidot is very large, ≫ 1, this condition defines the same change of for all states relevant for transport. The shift of the filled LLL states by one means that the antidot charge , more precisely, the number of electrons in the strip of fixed width that determines the interaction energy of the antidot, is increased by one per filled LLL when the flux Φ changes by Φ 0 . Therefore, depends on the flux Φ: ( being the number of occupied LLs including spin degeneracy which gives the number of the propagating edge modes) and to account for this dependence, the antidot Hamiltonian (10) can be written as follows, up to an irrelevant constant: The dependence of the charge (12) on the flux Φ through the antidot and the single particle energy spectrum (11) mean that in the relevant regime of the chemical potential fixed by the external contacts to the antidot, the Hamiltonian (13) as a whole is periodic in flux with the period Φ 0 in agreement with the general ``Byers-Yang theorem'' [8].

S6 IV Curves and Thermal Broadening
Next, we outline the calculation of the tunnel conductance for tunneling through the antidot between the left and the right metallic electrodes of the structure, governed by the Hamiltonian (13). We do this in the most basic regime, when the spacing of the antidot energy spectrum is negligible for some reason, e.g., because of the temperature that is large on the scale of . In this case, the charge tunneling is described by the standard ``Coulomb blockade'' rate equation (see, e.g., [7]) for the probabilities of the different charge states of the antidot: where the charge tunneling rates Γ are the sums over tunneling rates Γ , in the left and right contacts. The currents in the structure can be expressed in terms of the tunneling rates: with a similar expression for the current in the right contact. We are interested in the stationary situation, when ̇= 0, i.e. = ≡ . In this situation, the currents in the two contacts should also be equal, = ≡ , the condition that can expressed as = 0. This condition makes it possible to write, explicitly, the stationary solution of the rate equation: and 0 is obtained then from the normalization condition. In equilibrium, when the bias voltage between the electrodes vanishes, the tunneling rates satisfy the detailed balance condition, and this procedure confirms the equilibrium charge distribution on the antidot: To find the expression for the linear conductance of the antidot, = , one needs to find the linear-in-V correction to the equilibrium solution of the rate equation, which can be done directly in Eq. (14). In this way, we get In the continuous approximation used above, the tunneling rates depend on the energy change in the tunneling process as We use Eq. (16) to calculate numerically the linear tunnel conductance of the antidot in various regimes.
In the opposite regime, when the interaction is effectively negligible, the antidot conductance can be calculated directly from the transmission and reflection probabilities , = 1 − of the two contacts, = 1,2, using the standard double-barrier resonant transmission model: Here ( ) is the net electron transmission probability through the antidot between the two contacts that follows from the double-barrier model:

S6 Electron-Electron Scattering and Particle Lifetime
One more element of the physics of electron transport through the antidot, relevant at temperatures large on the scale of the single-particle level spacing , is electron relaxation and dephasing due to electron-electron interaction. Qualitatively, this relaxation smears out the single-particle energy levels and makes electron propagation around the antidot incoherent at large temperatures. The relaxation/dephasing rate can be calculated from the wavefunctions (5) with energy spacing given by (7). Since in this work we use this dephasing only in qualitative arguments, we calculate it below in the most basic approximation, neglecting electron spin. For convenience, we let = 2 be the antidot circumference. The dephasing rate is calculated with the interaction ( ) as a perturbation using Fermi's Golden Rule: = 2 ℏ ∑ |⟨ 1 , 2 | | 3 , 4 ⟩| 2 ( 2 )(1 − ( 3 ))(1 − ( 4 )) ( 1 + 2 − 3 − 4 ) 2 , 3 , 4 .
Integration of the coordinates gives the modified interaction Since the interaction matrix elements depend on weakly, through the logarithm, we can neglect this dependence when evaluating the sums over momenta for the relaxation rate. For the tunneling conductance measurements we are discussing, the relevant relaxation rate is for electrons at the Fermi energy, i.e. one should take 1 = 0. In the large-temperature range discussed here, the sums in this equation can be replaced by integrals over energy and give