An electronic Fabry–Pérot interferometer constructed of two narrow constrictions and a central confined region can be operated in the quantum Hall regime to manipulate the edge states and thus is a good tool to study quantum interference.^[1–5] Especially, it is predicted to be able to demonstrate non-Abelian statistics of fractionally charged quasiparticle excitations,^[6,7] which may be useful in fault-tolerant topological quantum computation.^[8–10]

With a perpendicular magnetic field applied to such an interferometer, edge states will form and travel along the circumference of the device while the quasiparticle states inside the central region are localized owing to the quantum Hall effect. Such edge states are ballistic one-dimensional channels with suppressed backscattering of disorders.^[11] The resistance oscillations emerge due to weak tunneling of the edge states at the constrictions and reveal two distinct types of behaviors which have been investigated both experimentally^{[1,2,4,12–15]} and theoretically.^[16,17] In large interferometers up to tens μm² in size, the constant phase lines in the magnetic field and gate voltage plane have a negative slope which is attributed to the Aharonov–Bohm (AB) effect in the framework of noninteracting electrons.^[1,2] Under this circumstance, an edge state which is partially transmitted when penetrating through the constrictions is scattered by the constrictions and changes direction to the opposite edge. The backscattered edge states from the two constrictions interfere. The periodicity of resistance oscillations arises from the phase difference between the two interfering paths which is given by the AB phase, and the magnetic field oscillation period is determined by the flux quantum ϕ₀ = h/e and the area of the interference loop A as ϕ₀/A. While in small interferometers up to a few μm² in size,^[1,4,15,18] the constant phase lines in the two-dimensional plot of magnetic field and gate voltage have a positive slope in contrast. This is because the Coulomb interaction starts to play an important role and thus the interferences display Coulomb-dominated (CD) features of single electron charging.^[16,17] Suppressing the Coulomb interaction by inducing effective screening^[5,13,14] or making the constriction more open,^[3] the AB dominated interferences could also be observed in small interferometers. To our knowledge, previous works on electronic Fabry–Pérot interferometer realized by etching of mesa^[4,18] or gating^[1,2,14] exclusively focused on the conventional two-dimensional electron gas (2DES) systems but few interferometer works based on graphene were reported. It is attractive to study edge state interferences in graphene through such interferometers since high-Laudau-level fractional quantum Hall effect^[19–22] and even denominator fractional quantum Hall states have been found in graphene.^[23,24]

In this paper, we report on the realization of a quantum Hall Fabry–Pérot interferometer fabricated on a relativistic 2D material — monolayer graphene. The edge state interferences were carefully investigated by the periodic resistance oscillations in the integer quantum Hall regime. We observed CD oscillations near integer quantum Hall plateaus in the graphene interferometer both on the electron side and hole side. Our work takes the first step to study the interference in a graphene quantum Hall interferometer and paves the way to measure the fractionally charged quasiparticle excitations in graphene.

The device was fabricated from chemical vapor deposition (CVD)-grown high quality monolayer graphene.^[25] The starting procedure of fabrication was to transfer CVD-grown graphene onto the heavily doped silicon substrate with a 300 nm silicon oxide coating layer. The interferometer geometry was then defined by electron beam lithography (EBL) and subsequent reactive ion etching (RIE) with oxygen plasma. Finally, Ti/Au bilayer (10 nm/90 nm) contact electrodes were fabricated by an additional EBL procedure and electron beam evaporation (EBE). Figure 1(a) is the atomic force microscope (AFM) image of a device before depositing the electrodes. The bright yellow region is graphene, while the dark region is the silicon oxide substrate where graphene was etched out. The device consists of a interferometer and a standard Hall bar structure. Figure 1(b) is the zoom in of the interferometer in Fig. 1(a). It is constructed with two adjacent narrow constrictions, each of which can scatter the propagating edge state to the opposite edge, and a confined cavity between the constrictions. The interferometer is tuned by the back gate and in-plane side gates L1 and L2. Figure 1(c) is the schematic of edge state paths in the CD regime assuming that the outer edge state (close to the boundary) is fully transmitted at the constriction. The inner edge state is partially transmitted when electrons (holes) enter the central region by weak forward tunneling which is denoted by the horizontal dashed lines.

Fig. 1.

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Fig. 1. (a) AFM image of a typical device used in this study. Bright yellow region is graphene and dark region is silicon oxide substrate. It consists of a interferometer and a standard Hall bar structure. Diagonal resistance RD (between contacts 4 and 7) of the interferometer and Hall resistance RXY (between contacts 5 and 9) of the bulk were measured simultaneously while driving current from 1 to 2. (b) A zoom in of the interferometer in (a). It is constructed with two adjacent narrow constrictions, introducing inter-edge scattering, and a central confined cavity. The designed area of the cavity is about 0.8 μm2. The side gates labeled by L1 and L2 define the interference path. (c) Schematic of the Coulomb-dominated interference. Propagating edge states are marked by the lines with arrows along the device edge. For the sake of simplicity, this picture (not exact the same in real measurement) shows that only one outer edge (close to the device boundary) is fully transmitted at the constriction. The inner edge state is partially transmitted to the central region by weak forward tunneling which is denoted by the horizontal dashed lines.

The measurement of a graphene interferometer with lithographic area of 0.8 μm² was conducted in a ³He/⁴He dilution refrigerator at temperature T = 60 mK, with a magnetic field B applied perpendicular to the graphene plane. The diagonal resistance R_D = V₄₇/I of the interferometer and Hall resistance R_XY = V₅₉/I of the bulk were measured simultaneously by driving current I from contact 1 to 2 and measuring the voltage drop V₄₇ (V₅₉) between contacts 4 and 7 (5 and 9) using a lock-in technique with a current bias I = 10–100 nA at 13 Hz.

Figure 2(a) shows R_D and R_XY as a function of back gate voltage V_BG with side gates L1 and L2 set to zero and magnetic field B = 6 T. R_D and R_XY display quantized integer Hall plateaus at filling factor ν = ±4 (n + 1/2), n = 0, 1, 2, where ± represents electrons and holes, respectively, which is typical for monolayer graphene in the quantum Hall regime. R_D exhibits obvious resistance oscillations in contrast to R_XY which shows no such oscillatory feature. Figure 2(b) is a zoom in of the typical resistance oscillations near ν = −6 in Fig. 2(a). ΔR_D is the measured R_D with the smooth background subtracted. The inset of Fig. 2(b) displays the result of fast Fourier transform (FFT) performed on the oscillatory ΔR_D. The spectrum of FFT shows a sharp peak at a single frequency, demonstrating that the oscillations are periodic in back gate voltage. ΔR_D is then measured as a function of the magnetic field, with V_BG = −12.5 V (near ν = −10) and side gates L1 and L2 set to zero. The results are shown in Fig. 2(c). Again, the FFT of the oscillations in the inset shows a single sharp peak, which indicates the good periodicity in magnetic field. The origin of these oscillations is investigated below.

Fig. 2.

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Fig. 2. (a) Diagonal resistance RD (black curve) and Hall resistance RXY (red curve) as a function of back gate voltage under perpendicular magnetic field B = 6 T with side gates L1 and L2 set to zero. Numbered horizontal lines indicate well developed quantum Hall plateaus. Orange (blue) shaded area corresponds to electron (hole) carriers. (b) Zoom in of the resistance oscillations in (a) (highlighted by the red arrow). ΔRD is the measured RD with the gross background subtracted, showing oscillations at near filling factor ν = −6. Inset: the FFT of the ΔRD data which shows a sharp peak at a single frequency, demonstrating the oscillations are periodic in back gate voltage. (c) ΔRD as a function of perpendicular magnetic field B around ν = −10 at VBG = −12.5 V (denoted by the black arrow in (a)) with side gates L1 and L2 set to zero. Inset: the FFT results of the oscillations which shows a sharp peak at a single frequency, indicating the good periodicity in magnetic field.

Resistance oscillations in the quantum Hall interferometer observed in previous experimental works focusing on conventional 2DES are either interpreted to base on the CD charing effect^[1,4,15,18] or attributed to the AB interference.^{[1–3,5,13,14]} One of the distinct features to distinguish these two regimes is that the constant phase lines in the magnetic field and gate voltage plane have a positive (negative) slope for the CD (AB) oscillations. Thus, in Fig. 3(a), we show R_D as a function of magnetic field B and side gate voltage V_L1&L2 at V_BG = 17.4 V near ν = 2, V_L1&L2 is the gate voltage applied on side gates L1 and L2 which are jointed together. The reason we choose filling factor ν = 2 is that at higher filling factor, larger numbers of edge states with finite width^[26] result in counter-propagating edge states in close proximity inside the narrow constrictions^[27] which causes the edge states to be mixed by tunneling through the localized states arising from disorders and thus decreases the visibility of the interference. Note that here the type of carrier is electron. The constant phase lines exhibit a positive slope which is consistent with the small sized interferometers being in CD regime, indicating their CD interference origin.

Fig. 3.

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Fig. 3. RD as a function of B and VL1&L2 with back gate voltage VBG = 17.4 V near ν = 2. VL1&L2 is the gate voltage applied on side gates L1 and L2 which are jointed together. Note that here the carrier type is electron. The positive slope of the constant phase lines indicates that the interference is Coulomb-dominated. Magnetic period ΔB ≈ 45 mT corresponds to an area A ≈ 0.046 μm2. It gives a mean interfering area with radius ∼ 120 nm, which is smaller than the lithographic geometry. (b) Schematic profile of spatial carrier density distribution when starting to fill an empty Landau level. Red dashed lines represent carrier density in compressible region while blue dashed lines represent incompressible region. (c) The corresponding screened disorder potential of (b). Red solid line denotes the compressible region where the disorder potential is screened while blue solid lines denote incompressible region where the potential is unscreened. Thus, the compressible region acts as a Coulomb island, the size and position of which are determined by the details of the disorder potential. For clarity, only one non-degenerate Landau level is considered.

Having identified the CD regime, we introduce a qualitative description of this regime in the next to analyze the data. Assume both bulk and central confined region of the interferometer include an incompressible region with f₀ fully occupied Landau levels. Inside the central region, filling factor higher than f₀ with partially occupied Landau level will form a compressible region which is surrounded by the incompressible region. Electrons residing in the compressible region behave as a quasi-isolated Coulomb island and vary discretely.

The magnetic field varies the flux of the central region in the form δϕ = AδB + BδA′, where A is the area of the interference loop, and δA′ is the interaction induced area response caused by the coupling between the Coulomb island and edge states. When a whole flux quantum ϕ₀ is added to the central region, the occupation of electrons within Landau levels will redistribute by adding one electron into every fully filled Landau level and thus totally f₀ electrons into all fully occupied Landau levels. At the meantime, f₀ electrons must be moved out from the compressible region due to the Coulomb interaction. Thus, increasing magnetic field plays an role like applying more negative voltage of a gate which electrostatically couples to the Coulomb island. The magnetic field also has impact on the interference area in an observable way. In the process that the magnetic field increases, the area of interference loop A first shrinks by δA′ to keep constant flux and abruptly recovers to the original area A with δA′ = 0 when δϕ reaches ϕ₀.^[14]

For an ideal back gate, it influences the resistance oscillations through its effect on the carrier density. For example, the oscillation period in back gate voltage in Fig. 2(b) corresponds to a single electron added to the Coulomb island through capacitive coupling. For an ideal side gate, it varies the enclosed flux by slowly affecting the interference area A. Therefore, considering all the contributions above to the Coulomb blockade effect, the charging energy of the Coulomb island is determined by

In detail, carrier density n does not uniformly distribute in space but fluctuates in an inhomogeneous profile to screen the bare disorder potential. Figures 3(b) and 3(c) illustrate this scenario schematically. The dashed line in Fig. 3(b) indicates the profile of the carrier density. Assuming electrons start to fill an empty Landau level in Fig. 3(b), near the bottoms of the profile of carrier density n, the Landau level is completely empty (n = 0) and forms local incompressible regions (blue dashed line) within which no electron can be used to redistribute to screen the bare disorder potential. Where n > 0, the Landau level is partially filled and there are available states for electrons to rearrange to screen the disorder potential where a compressible region forms (red dashed line). Accordingly, in Fig. 3(c), the bare disorder potential is screened in the compressible region (red solid line), while in the incompressible region (blue solid line) the potential sticks out and forms a potential valley. When the compressible region is surrounded by an incompressible region, it acts as a localized Coulomb island and electron transport is dominated by Coulomb blockade physics. Therefore, the reason that the area of Coulomb island in Fig. 3(a) is smaller than the designed area may be attributed to the formation of the localized compressible region which is determined by the details of the disorder potential. This interpretation is further corroborated by the experiment of a quantum Hall interferometer studied by scanning gate microscopy (SGM)^[32] which demonstrates that the Coulomb island could exist anywhere inside the central region and an experiment in graphene nanoribbons in the quantum Hall regime which verifies that the single quantum dot forms.^[31]

Up to now, we have discussed the CD oscillations on the electron side. Next, oscillations for holes are investigated. Figure 4 shows the plots of resistance oscillations in the B–V_L1&L2 plane, with back gate voltage V_BG = −1 V on the hole side near ν = −2. To clarify, we point out here that the effect of gating for holes is contrary to that for electrons. Hence, the constant phase lines which have a negative slope in Fig. 4 are consistent with the CD mechanism rather than the AB mechanism. The oscillation period in magnetic field ΔB ≈ 7.5 mT and the corresponding area of the Coulomb island is about 0.267 μm², which gives a mean radius R ≈ 300 nm. Again, it is smaller compared to the lithographic area of the interferometer which can be explained in the same framework discussed above. But it is worth noting that the area of the Coulomb island here is different from the area of electron carrier. The inherent electron–hole asymmetry in graphene^[33–38] may be possibly responsible for this scenario. To verify this, we compare the width of ν = 2 plateau and that of ν = −2 plateau in Fig. 2(a), deducing that the dominated disorder potential is negatively charged because repulsive Coulomb potential leads to smaller scattering rate for the Dirac fermions in graphene^[33,36–38] which corresponds to the lager width of plateau for electrons. For a negative disorder potential, the localized density of states below (above) the Dirac point is enhanced (reduced).^[34,35] Accordingly, the screening effect of the disorder potential in the process of localization under quantum Hall conditions will be much stronger on the hole side than that on the electron side. Thus, the Coulomb island seen by holes will be larger than that by electrons.

Fig. 4.

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Fig. 4. Plots of RD in the B–VL1&L2 plane, with back gate voltage VBG = −1 V near ν = −2. Note here the carrier type is hole. The negative slope of the constant phase lines indicates the device being in the CD regime (for electron carriers, negative slope represents Aharonov–Bohm interference). The magnetic field oscillation period ΔB ≈ 7.5 mT corresponds to an interference loop with area A ≈ 0.276 μm2 which has a mean radius of ∼ 300 nm.

Our work takes the first step to study the edge state interferences in the graphene quantum Hall interferometer. The principle finding of this work is that the resistance oscillations on both the electron and hole sides are in the CD regime. We propose that the next-step experiment could encapsulate graphene in hexagonal boron nitride together with a graphite back gate to reduce the influence of external disorder potential.^[39,40] Since high-Laudau-level^[19–22] and even denominator^[23,24] fractional quantum Hall effect have been found in graphene, we expect that the graphene interferometer will offer an intriguing platform for future studies of non-Abelian statistics.