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Shuichi Iwakiri, Alexandra Mestre-Torà, Elías Portolés, Marieke Visscher, Marta Perego, Giulia Zheng, [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), Manfred Sigrist, Thomas Ihn, Klaus Ensslin

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[Tunable quantum interferometer for correlated moiré electrons](https://mdr.nims.go.jp/datasets/f9c37fc5-5691-4a6b-8ec6-6a8e347b1834)

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Tunable quantum interferometer for correlated moiré electronsArticle https://doi.org/10.1038/s41467-023-44671-4Tunable quantum interferometer forcorrelated moiré electronsShuichi Iwakiri 1 , Alexandra Mestre-Torà 1 , Elías Portolés 1,Marieke Visscher1, Marta Perego1, Giulia Zheng 1, Takashi Taniguchi 2,Kenji Watanabe 3, Manfred Sigrist 4, Thomas Ihn 1,5 & Klaus Ensslin 1,5Magic-angle twisted bilayer graphene can host a variety of gate-tunablecorrelated states – including superconducting and correlated insulatorstates. Recently, junction-based superconducting moiré devices have beenintroduced, enabling the study of the charge, spin and orbital nature ofsuperconductivity, as well as the coherence of moiré electrons in magic-angletwisted bilayer graphene. Complementary fundamental coherence effects—inparticular, the Little–Parks effect in a superconducting ring and theAharonov–Bohm effect in a normally conducting ring – have not yet beenreported in moiré devices. Here, we observe both phenomena in a singlegate-defined ring device, where we can embed a superconducting or normallyconducting ring in a correlated or band insulator. The Little–Parks effect isseen in the superconducting phase diagram as a function of density andmagnetic field, confirming the effective charge of 2e. We also find that thecoherence length of conductingmoiré electrons exceeds severalmicrons at 50mK. In addition, we identify a regime characterized by h/e-periodic oscillationsbut with superconductor-like nonlinear transport.Magic-angle twisted bilayer graphene (MATBG) with its moiré flatband1,2 constitutes a condensed-matter system to realize a wide varietyof correlated states, such as superconducting and correlated insulatorstates, that are tunable by gating3–9. A class of gate-defined nanode-vices, including Josephson junctions10,11 and SQUIDs12, have beenrecently realized in MATBG. These structures have provided excellentplatforms for controlling mesoscopic superconductivity and char-acterizing MATBG. Extending this approach to a doubly-connectedgeometry without any junction, namely a ring, promises uniquemicroscopic information about the material and the device.The physical properties of a ring threaded by a magnetic field arein general periodic in flux quanta13,14Φ0 =he*, with e* being the charge ofthe carrier. In a superconducting ring (e* = 2e), h/2e-periodic oscilla-tions of critical temperature and critical current appear. These oscil-lations are known as the Little–Parks effect and were the firstexperimental evidence for the 2e charge pairing in conventionalsuperconductors15,16. In fact, the Little–Parks effect can be used todetermine the charge of the superconducting carriers17,18, com-plementing the Josephson junction and SQUID experiments10,12.Moreover, properties of unconventional superconductors can berevealed by anomalies of the Little–Parks effect, such as a phaseshift19–24 or a change in periodicity24–29, and thereby help to understandthe underlying superconducting symmetry.By contrast, a normally conducting ring (e* = e) shows h/e-peri-odic oscillations of resistance, the Aharonov–Bohmeffect, andworksas a direct probe to quantify the phase coherence of electrons. Giventhe low Fermi velocity and the large effective mass in MATBG, apossible non-Fermi liquid nature of its flat band electrons4,30, andintrinsic disorders introduced by twist-angle inhomogeneity31,quantifying the phase coherence length is key to understanding theReceived: 19 October 2023Accepted: 20 December 2023Check for updates1Laboratory for Solid State Physics, ETH Zurich, CH-8093 Zurich, Switzerland. 2Research Center for Materials Nanoarchitectonics, National Institute forMaterials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. 3Research Center for Electronic and Optical Materials, National Institute for Materials Science, 1-1Namiki, Tsukuba 305-0044, Japan. 4Institute for Theoretical Physics, ETH Zurich, CH-8093 Zurich, Switzerland. 5Quantum Center, ETH Zurich, CH-8093Zurich, Switzerland. e-mail: siwakiri@phys.ethz.ch; amestre@phys.ethz.chNature Communications |          (2024) 15:390 11234567890():,;1234567890():,;http://orcid.org/0000-0003-2668-8328http://orcid.org/0000-0003-2668-8328http://orcid.org/0000-0003-2668-8328http://orcid.org/0000-0003-2668-8328http://orcid.org/0000-0003-2668-8328http://orcid.org/0009-0000-1010-2922http://orcid.org/0009-0000-1010-2922http://orcid.org/0009-0000-1010-2922http://orcid.org/0009-0000-1010-2922http://orcid.org/0009-0000-1010-2922http://orcid.org/0000-0001-7202-777Xhttp://orcid.org/0000-0001-7202-777Xhttp://orcid.org/0000-0001-7202-777Xhttp://orcid.org/0000-0001-7202-777Xhttp://orcid.org/0000-0001-7202-777Xhttp://orcid.org/0000-0003-3503-4940http://orcid.org/0000-0003-3503-4940http://orcid.org/0000-0003-3503-4940http://orcid.org/0000-0003-3503-4940http://orcid.org/0000-0003-3503-4940http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0002-1467-3105http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0003-3701-8119http://orcid.org/0000-0002-8627-5093http://orcid.org/0000-0002-8627-5093http://orcid.org/0000-0002-8627-5093http://orcid.org/0000-0002-8627-5093http://orcid.org/0000-0002-8627-5093http://orcid.org/0000-0002-5587-6953http://orcid.org/0000-0002-5587-6953http://orcid.org/0000-0002-5587-6953http://orcid.org/0000-0002-5587-6953http://orcid.org/0000-0002-5587-6953http://orcid.org/0000-0001-7007-6949http://orcid.org/0000-0001-7007-6949http://orcid.org/0000-0001-7007-6949http://orcid.org/0000-0001-7007-6949http://orcid.org/0000-0001-7007-6949http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-44671-4&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-44671-4&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-44671-4&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-023-44671-4&domain=pdfmailto:siwakiri@phys.ethz.chmailto:amestre@phys.ethz.chdynamics ofmoiré electrons. In addition, the phase coherence lengthenables the estimation of the penetration depth of a super-conducting wave function into the normally conducting state(proximity effect)32,33, which plays an important role in gate-definedsuperconducting devices. However, the exploration of these funda-mental quantum-interference effects has been hampered by thelack of a suitable device architecture and the sensitivity of themoiré superlattice to disorder31, which poses a challenge to theconventional approach of fabricating a ring by physical/chemicaletching.Here, we report the observation of Little–Parks and theAharonov–Bohm effects in MATBG. The device architecture allows usto define a ring consisting of a loop that can be tuned to be super-conducting or normally conducting, surrounded by a correlated orband insulator. We confirm 2e pairing via the Little–Parks effect, andshow that the phase coherence length of moiré electrons surpassesseveral microns at 50 mK, evidenced by h/e-periodic Aharonov–Bohmoscillations. We also discover an intriguing regime in which h/e-peri-odic oscillations appear alongside superconductor-like transport.These results highlight the promise of the quantum interferometer inMATBG for studying interference phenomena of exotic quantumstates of 2D materials.ResultsHighly tunable quantum interferenceWe develop the gate-defined ring architecture shown in Fig. 1a. Webase the design on the proof-of-principle device reported in ref. 34using Bernal bilayer graphene. The MATBG is encapsulated in hex-agonal boron nitride (hBN) and is contacted by four electrodes. Thesample is dual-gated with a graphite back gate and a metallic ring-shaped top gate (ring gate). We operate the ring by first tuning theback gate voltage Vbg, which affects the entire MATBG area andinduces a global density ng. Then we tune the density under the ringgate, nr, via the voltage Vrg. The ring has a lithographic inner radius ofrin = 600 nm and outer radius of rout = 1000nm. Through the electro-des, the sample is biased with a current I, and the voltage drop V ismeasured in a four-terminal configuration. Unless stated otherwise,the measurements are performed in a 3He–4He dilution refrigerator ata temperature of 50mK.The resistance Rof theMATBG (Fig. 1b) as a function of ng withVrgset to zero shows pronounced peaks at charge neutrality, and in thecorrelated insulator (CI) and band insulator (BI) regimes on the holeside (ng < 0). From the density at the BI peak, we estimate an averagetwist angle of 1.1∘. When tuning the density beyond the correlatedinsulator, we observe superconductivity (SC) through a resistanceFig. 1 | Highly tunable quantum interference. a Device and measurement sche-matics. The graphite–hBN–MATBG–hBN heterostructure is contacted by fourelectrodes (yellow). The ring gate (purple) is formed on top of an aluminium oxidelayer (green). The lithographic inner (rin = 600nm) and outer (rout = 1000nm)radius of the ring gate are indicated. DC-voltage sources are connected to the ringgate and back gate. A four-terminal measurement is performed by applying aconstant current and measuring the voltage drop across the ring. b Resistance ofthe MATBG as a function of carrier density ng at ring (top) gate voltage Vrg = 0 V at50 mK. The regions with colour background correspond to the density ranges ofthe states introduced in (c). The inset shows the resistance for both the electron(grey background) and hole side (white background). A background resistance of130Ω is subtracted from the resistance trace. c Band structure schematics ofMATBG. The colour-coded labels indicate the correlated insulator (CI), super-conducting (SC), normally conducting in the flat band (Nflat), normally conductingin the dispersive band (Ndisp) and band insulator (BI) states, respectively.d Overview of quantum-interference effects. A ring-shaped conducting path isdefinedby surrounding the ringwith aCI (left panel, blue-shaded)orBI (right panel,green-shaded) state. For both insulating states outside the ring, the magneto-resistance oscillations and their FFT spectrum are shown for the three conductingstates: SC (top row), Nflat (middle row) and Ndisp (bottom row). The triangle in thespectrummarks the peak frequency. The insets show the effective radius of the ringfor both h/2e and h/e oscillations (solid and dashed circles, respectively), on top ofthe lithographic area of the ring gate (purple).Article https://doi.org/10.1038/s41467-023-44671-4Nature Communications |          (2024) 15:390 2drop. We also access two normally conducting regimes: one inside theflat band (Nflat) and the other in the dispersive band (Ndisp). Figure 1csummarizes the relevant quantum states that form in the device. Fur-ther details of the experimental setup are given in theMethods section.In order to define a conducting path, we tune the ring-shapedregion into the SC, Nflat, or Ndisp regimes. This conducting path is thensurrounded by either CI or BI states to confine electrons. As wewill seebelow, the insulating state does not influence the observed inter-ference pattern but has an effect on the quantum state distributionacross the structure (see Supplementary Fig. 8). Figure 1d shows theresistance (R) oscillations in perpendicular magnetic field (B) for thesix regimes at zero bias current and a temperature of 150mK, togetherwith their fast Fourier transform (FFT) spectra. When calculating theFFT spectrum, we subtract a smooth background extracted with theSavitzky–Golay filter (smoothing window of 10 mT and polynomialorder 2). In this process, long-period signal in magneto-resistance,such as the universal conductance fluctuations, is filtered out. Weconvert the peakof the spectrum into an area assuming either h/e orh/2e-periodicity as the relevant flux quantum and then compare theresult with the lithographic radius of the ring (see inset circlesin Fig. 1d).In the case of a superconducting ring (top row in Fig. 1d,nr = − 1.89 × 1012 cm−2), the frequency peak appears at 0.92/mT for (SC,CI) and at 1.20/mT for (SC, BI), respectively. Hereafter, we denote thestate inside and outside the ring as (inside, outside). Assuming h/2e-periodicity, the observed frequency peaks correspond to an effectiveradius reff of 767 nm and 855nm, respectively. These values arecomparable to the center-line radius of the ring gatermid =rin + rout2 = 800nm. In contrast, the reff when assumingh/e-periodicity does not match the lithographic dimension of the ringgate (reff > 1000 nm= rout). In these regimes, we further observe criticalcurrent and critical density oscillations, as we discuss in Fig. 2. Basedon our findings, we attribute these oscillations to the h/2e-periodicLittle–Parks effect, confirming that the charge of the superconductingcarrier is 2e.When the ring is tuned into the dispersive band (bottom row inFig. 1d, nr = − 3.58 × 1012 cm−2), h/e-periodic oscillations appear with aspectrumcovering a significant range in 1/B. The peak frequency of theoscillations is 0.621/mT for both (Ndisp, CI) and (Ndisp, BI). This fre-quency is approximately half of those observed in the super-conducting case, suggesting anh/e-periodicity. The effective radii bothassuming h/e- (reff = 873 nm) and h/2e-periodicity (reff = 617 nm) fitwithin the ring dimensions. However, the oscillation amplitude decaysexponentially in temperature and strives at highermagneticfields thanthe Little–Parks oscillations (see Supplementary Figure 2). The ampli-tude decay in temperature is characteristic of the Aharonov–Bohmoscillations due to the smearing of the Fermi function and the reduc-tion of the phase coherence length35. We therefore attribute theoscillations to the h/e-periodic Aharonov–Bohm effect. The measure-ment of the temperature dependence also allows us to estimate thephase coherence length Lφ of electrons in the dispersive band (seeSupplementary Fig. 2). At 50 mK, the coherence length is ~ 12.3 ± 0.3μm in (Ndisp, CI) and ~18.7 ± 1.0 μm in (Ndisp, BI). These values exceedthe perimeter of the ring (2πreff≃ 4.80μm). Now, we compare thephase coherence length of bilayer graphene in the literature. Thecoherence length Lϕ of the ring of exfoliated Bernal (non-twisted)bilayer graphene encapsulated with hBN is 1.5μm at 36 mK for theetching-defined ring36 and4.4μmat 300mK for the gate-defined ring34Fig. 2 | TunableLittle–Parks oscillations. a Phase diagramof the device in the (SC,BI) configuration. The SC, Nflat and BI states are represented. The square indicatesthe (T, n) domain for which we sketch the schematics in (b). b Schematics of thesuperconducting dome in the phase diagram of MATBG when the flux threadingthe ring isΦ = nΦ0 (denotedasΦ0) andΦ = n2Φ0 (denoted asΦ0/2),whereΦ0 is thesuperconducting flux quantum. Tc(Φ0) and Tc(Φ0/2) mark the boundary betweenthe Nflat (orange) and the SC dome (in pink for Φ0 and purple for Φ0/2). Uponapplying a magnetic field, the phase diagram oscillates between Tc(Φ0) and Tc(Φ0/2). c Voltage drop V across the ring as a function of nr and B, at I = 5.0 nA. The whitedashed curve (a guide to the eye) marks the phase boundary nc(B) between the SCand Nflat states. d dVdI as function of ring density nr and DC current IDCwhen definingthe ring in the BI state with ng = − 2.94 × 1012 cm−2. eMagneto-resistance oscillationof the critical current taken at the densities indicated indwith a blackcircle, square,and star, respectively. f Magneto-resistance oscillations taken at IDC = 0 nA and atthe same densities as in (e).Article https://doi.org/10.1038/s41467-023-44671-4Nature Communications |          (2024) 15:390 3(maximum values reported in each paper). In our experiment, Lϕ ofmoiré electrons in MATBG exceeds 10μm at 50 mK. Considering thatthe amplitude of the Aharonov–Bohm oscillations of these threereports is all proportional to 1/T, one can estimate (or normalize) thecoherence length to that of 50 mK. Then, we obtain Lϕ = 2.1, 26.4, and10 μm, for the etched (Bernal bilayer), gate-defined (Bernal bilayer),and gate-defined (MATBG) ring, respectively.This result demonstrates that the phase coherence of moiréelectrons in the dispersive band is well preserved despite severalsources of disorder such as twist-angle inhomogeneity and straindistribution.Furthermore, there is a striking contrast between the oscillation inthe normally conducting flat band regimes depending on the sur-rounding insulators (middle row in Fig. 1d, nr = − 3.10 × 1012 cm−2). Inthe (Nflat, CI) regime, we observe an oscillation with a frequency of0.630/mT, even lower than in the SC and Ndisp cases. The effectiveradius assuming h/2e periodicity is reff = 432 nm, even smaller than rin,while reff = 605 nm for h/e periodicity matches rin. Though the geo-metric argument points towards h/e periodicity, this regime exhibitssuperconductor-like transport as well. We discuss this point in moredetail later in Fig. 3. On the other hand, in the (Nflat, BI) regime, weobserve magneto-resistance oscillations with very small amplitude. InFig. 3 | Spectroscopy of the quantum oscillations. a FFT spectrum of themagneto-resistance oscillations. The white lines in the spectrum indicate the rangeof frequencies expected for h/e and h/2e oscillations, taking the inner radius (rin)and mean radius (rmid) of the ring. b Resistance as a function of nr, the differentquantum states (CI, SC, Nflat, BI andNdisp) are indicated. c,d dVdI as a functionofB andIDC c: in (Nflat, CI) regime andd: in (Ndisp, CI) regime. In the side panel, linecut of dVdI atB =0. e, f Electrostatic simulation of the carrier density and quantum statedistribution e: in the (SC, CI) regimeand f: in (Nflat, CI) regime (f). The illustrations atthe top show the superconducting regions (red) on the lithographic structure ofthe ring (black). At the bottom, plot of the spatial distribution of the carrier densityalong the radial axis of the ring (arrow in the illustration). The black vertical linesindicate the lithographic dimensions of the ring which is 400 nm. The horizontalgrey dashed lines indicate the density range for superconductivity. Differentquantum states (CI, SC, and Nflat) are attributed depending on the density.Article https://doi.org/10.1038/s41467-023-44671-4Nature Communications |          (2024) 15:390 4this regime, the frequency peak appears close to the one in the Ndispregime. The effective radius is reff = 813 nm assuming h/e-periodicityand reff = 585 nm assuming h/2e-periodicity. This regime shows neithernonlinear transport nor a drop in resistance with temperature.Therefore, we attribute the oscillations in the (Nflat, BI) regime to h/e-periodic Aharonov–Bohm oscillations. The phase coherence length,estimated from the temperature dependence, is Lφ ~ 6.51 ± 1.32 μm,which is by a factor of 2–3 smaller than that for Ndisp. This relativelyshort Lφ in the flat band can be attributed to a large electron effectivemass. In fact, the phase coherence length is proportional to the Fermivelocity vFermi, which depends inversely on the effective mass m*(Lφ∝ vFermi∝ 1/m*), and the measured m* in the flat and dispersivebands are different by a factor of 1–104.These results demonstrate that one can switch between theLittle–Parks and the Aharonov–Bohm effects of MATBG by gate tun-ing. They also reveal the 2e charge pairing and long coherence lengthsof the moiré electrons. In the following sections, we discuss eachregime in more detail.Tunable Little–Parks oscillationsThe Little–Parks effect is essentially themagneto-oscillation of the freeenergy of the superconducting state37, which results in the oscillationof the critical temperature Tc and the critical current Ic. While thecritical current is readily measurable, the current injection inevitablydrives the system out of equilibrium, possibly giving rise to unwantedeffects such as local breakdown of superconductivity38–40. Therefore, ameasurement at equilibrium is preferable. However, measurement ofthe Tc oscillation is experimentally challenging as the expectedamplitude is in the sub-mK range37. Here, taking advantage of the in-situ tunability of the carrier density in MATBG, we demonstrate analternative route to detect Little–Parks oscillations. We probe theoscillation of Tc by translating it into the oscillation of the criticaldensity nc at which the superconducting transition occurs, enablingthe detection of the Little–Parks effect near equilibrium. Figure 2ashows the phase diagram of the device in the (SC, BI) regime, mea-suring the resistance as a function of temperature T and nr at zeromagnetic field. Increasing the temperature from 50mK to 600mK, thedensity range of the SC state shrinks, forming a superconductingdome. Due to the Little–Parks effect, the Tc of the superconductingring oscillateswith themagnetic flux. As depicted in Fig. 2b, this resultsin a compression and expansion of the superconducting phaseboundary. For a fixed temperature Tmeas, such breathing can betranslated into an oscillation of the critical density nc, betweennc(Φ =Φ0) and ncðΦ= 12Φ0Þ. It is therefore possible to probe themagneto-oscillation of the phase boundary by fixing the temperatureand sweeping the carrier density.Figure 2c shows the measured voltage drop V across the ring as afunction of nr and B, at a DC bias current of 5 nA and in a density rangeclose to the high-density edge of the superconducting dome (seeSupplementary Figure 4 for the temperature and current dependenceof the map). At a fixed B, a jump from zero to a finite voltage Vmarksthe transition from the superconducting to the normally conductingregime. Themagnetic field dependence of the density nc(B) representsthe phase boundary. The shape of this boundary oscillates with aperiod of 0.87 mT, agreeing with an h/2e-periodicity (reff ≈ rmid). Thesmooth shift of the phase boundary with increasing B to higher elec-tron densities reflects the shrinking of the superconducting dome inB.We observe the same result with a reversed current (−5 nA). Thedemonstration of the Little–Parks effect near equilibrium by con-structing the phase diagram is one of the distinct advantages of thegate-defined architecture.We can also track the development of the oscillations as the statedeparts from the vicinity of the SC-to-Nflat transition into deep insidethe superconducting dome by tuning the density nr. Figure 2d showsdV/dI as a function of DC current IDC and nr at zeromagnetic field, andFig. 2e the B- and IDC-dependence of dV/dI for nr = − 1.85, − 1.87 and− 1.93 × 1012 cm−2 (see solid horizontal lines in Fig. 2d). The periodicoscillations of the critical current Ic emerge on top of a decreasingbackground inmagnetic field. The periodicity of the Ic oscillations ish/2e (0.87 mT), in agreement with the (SC, BI) data in Fig. 1d. As nr istuned to a value that shows larger Ic (deep inside the superconductingdome), the amplitude of the Ic oscillations increases, and they arealso observed up to higher magnetic fields. Moreover, the magneticfield-origin of the oscillations shifts from zero and also depends onthe direction of the applied current (see Supplementary Fig. 9), whichcan be due to the inductance of the ring. At the lowest density(Fig. 2e, top panel), we observe that the critical current vanishes at ~5mT and re-emerges at a higher magnetic field. This pattern resemblesthe Fraunhofer pattern of Josephson junctions. However, aswe discussin the Supplementary Information section II.C, the width of thishypothetical Josephson junction does not fit any of our ringgeometries. We therefore attribute the observed pattern rather to theexistence of multiple interference paths within the ring, generating abeating pattern. Similar critical current oscillations but withouta beating pattern appear in the (SC,CI) regime and are shown inSupplementary Fig. 5.The differential resistance traces at zero bias current (IDC = 0 nA)for the three densities are shown in Fig. 2f. Close to the SC-to-Nflattransition (bottom panel, star symbol), the magneto-resistance oscil-lations are barely visible as the differenceof resistancebetween SC andthe normal states is vanishing. When the density is increased (middlepanel, square symbol), we observe pronounced oscillations where achain of parabolas appears on top of a smooth parabolic background.When the density is further reduced (top panel, circle symbol), theoscillation amplitude drops again. This is because the state is deep inthe superconducting dome and zero current is too remote from thetransition to fully capture the Little–Parks effect. Deeper in thesuperconducting dome, one can only observe the critical currentoscillations.Spectroscopy of the quantum oscillationsFigure 3a shows the magneto-oscillation spectrogram as a function ofthe density nr when the state outside the ring is CI (see Fig. 1d, leftcolumn). The spectrogram is constructed by first measuring the dif-ferential resistance at zero bias current sweeping the magnetic fieldbetween ± 20 mT. Then the FFT spectrum is calculated after sub-tracting the smooth background from the raw data. In the spectro-gram,weobserve several regionswith prominent peaks in the FFT. Theresistance R at zero magnetic field and corresponding quantum statesare shown in Fig. 3b.Anh/2epeak is observed (reff ≈ rmid) at the edge of the SC region inthe vicinity of the CI regime. This can be understood by the fact thatthe Little–Parks oscillation only appears at the onset of the super-conducting transition. When the density is tuned further down to the(Nflat, CI) regime, a prominent peak at 0.313 /mT appears between−3.40 × 1012 cm−2 < nr < −2.80 × 1012 cm−2. This corresponds to an effec-tive radius of reff = 605 nm assuming h/e-periodicity, and reff = 432nmassuming h/2e. As the latter reff is considerably smaller than rin, weattribute an h/e-periodicity to the oscillations. The density windowacross which this peak extends is unexpectedly wide and includes theBI regime, meaning that the oscillations persist even when the ring ismostly insulating. When entering the (Ndisp, CI) regime, the spectrumbecomes broader and features a peak within the h/e-periodic range(rin < reff < rmid). This broad spectrum is not unexpected for conven-tional Aharonov–Bohm oscillations as the aspect ratio of the gate-defined ring is small (rin/rout≃0.6), making possiblemany interferencepaths with different effective enclosed areas. The oscillations in (Nflat,CI) and in (Ndisp, CI) not only differ in peak frequency and frequencyextent but also show different responses to IDC. Figure 3c, d show theIDC andBmapping in (Nflat, CI), with nr = 3.07 × 1012 cm−2, and (Ndisp, CI),Article https://doi.org/10.1038/s41467-023-44671-4Nature Communications |          (2024) 15:390 5with nr = 3.51 × 1012 cm−2 respectively. Interestingly, a superconductor-like behaviour of the differential resistance (dip at IDC =0 nA) isobserved in the (Nflat, CI) regime, despite the high resistance reachingup to a few kΩ. In addition, a periodic chain of low-resistance statesappears by sweeping IDC and B. By contrast, the (Ndisp, CI) regimeexhibits a cusp in differential resistance at IDC =0 nA, and no char-acteristic pattern is observed in the IDC and B mapping. This againsupports the interpretation that Aharonov–Bohm oscillations areobserved for (Ndisp, CI).We attribute the superconducting behaviour of the (Nflat, CI)regime to the presence of a small residual superconducting region,emerging from the smooth evolution of the carrier density from insideto outside of the ring. To support this idea, weperform an electrostaticsimulation of the carrier density distribution (see Supplementary Fig-ure 8). As shown in Fig. 3e, in the (SC, CI) regime, SC is the only stateinside the ring. However, when the ring is detuned from the (SC, CI) tothe (Ndisp, CI) regime (Fig. 3f), the spatial distribution of the quantumstates becomes complex, and a superconducting region can beembedded around the ring. This spurious superconducting regionconsists of a small ringwith radius ~400 nmandapath surrounding thegated region, as Fig. 3f depicts.The origin of the high visibility of the oscillations and its peri-odicity in the (Nflat, CI) regime remains elusive, while careful inspectionof the electrostatic environment might give an insight into it. Forexample, the spurious superconducting ring shown in Fig. 3f has aradius of ~400 nm, which is much smaller than the lithographic radius.However, it matches the effective radius of the oscillations in thisregime, assuming h/2e periodicity (see Supplementary Fig. 8). Thismeans that the spurious superconducting ring that is formed due tothe electrostatic requirement might be giving the Little–Parks oscilla-tions in the (Nflat, CI) regime. Further improvement of the simulation,taking the proximity effect at the interface of different quantum statesinto account, will help the understanding of this regime.DiscussionIn conclusion, we presented a gate-defined quantum interferometer inMATBG that provides a versatile platform for investigating the quan-tum coherence and the charge of correlated electrons from super-conducting to normally conducting regimes. We observe theLittle–Parks effect by constructing the superconducting phase dia-gram as well as bymeasuring the oscillation of themagneto-resistanceand the critical current, confirming the charge-2e pairing. We alsoobserve the Aharonov–Bohm effect for the dispersive and flat bandelectrons in the same device. From it, we find that the phase coherencelength exceeds a few microns, highlighting its robustness. We find aregime that exhibits magneto-resistance and critical current oscilla-tions even detuned from the superconducting regime, whichmight bedue to the electrostatic constriction of the device.These experiments demonstrate that exploring the Little–Parkseffect has the potential to provide insight into both the charge and spinnature of the Cooper pair in MATBG. Notably, the measurement underan in-plane magnetic field along with the perpendicular field couldenable the observation of phase shifts in Little–Parks oscillations. Suchshifts could serve as indicators of unconventional pairings such as spin-triplet superconductivity41. Our observation of the Aharonov–Bohmoscillations and the estimation of the phase coherence length of themoiré electrons provide the experimental foundation for the under-standing of Andreev reflection physics in exotic quantum interfacessuch as superconducting v.s. Chern or Mott insulating interface. More-over, the gate-defined architecture and the measurement scheme pre-sented here can generally be implemented in other 2D superconductors(e.g., twisted multilayer graphene42, Bernal bilayer graphene43, andbilayer graphene/transition metal dichalcogenide44), opening up thepath towards the direct quantification of charge, spin, and coherence ofcorrelated electrons in a plethora of exotic quantum states.MethodsTwist angle estimationWe extract the twist angle of the sample using the relationθ= 2 arcsin a2L� �3. In this expression,a is the lattice constant of grapheneand L is the moiré periodicity, which represents the distance betweentwo adjacent AA-stacked regions. In turn, L is related to the area A ofthe moiré unit cell via L= 2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2A=ffiffiffi3pq. Within a moiré unit cell, fourelectrons can be accommodated due to spin and valley degeneracy.Then, the band insulator peak due to the twist appears at the electrondensity nBI which corresponds to the occupation of 4 electrons permoiré unit cell A= 4nBI. We obtain nBI from the Landau fan and densitymapping (see Supplementary Fig. 3). Our analysis yields an approx-imate twist angle of 1.11°.Device fabricationThe device stack is assembled using the dry pick-up method45. Weexfoliate graphene and hexagonal boron nitride (hBN) flakes on a285 nm p:Si/SiO2 wafer. We start by scratching a graphene flake in twousing a tungsten needle with a tip diameter of 2μm controlled by amicromanipulator. We pick up all the flakes using a poly-dimethylsiloxane/polycarbonate stamp. We first pick up the top hBNflake, with a thickness of 18 nm, at 90 °C. Then we proceed to pick upthe graphene and assemble the twisted structure. For this, we first pickup half of the pre-cut graphene, rotate the microscope stage by 1.1°and then pick up the other half of the graphene, all at 40 °C. Weencapsulate the graphene by picking up a bottom hBN flake, with athickness of 55 nm. For the encapsulation, the stack isfirst contacted tothe bottom hBN at 40 °C and the temperature of the stage is raised to80 °C. Finally, wepickup a graphiteflake of 29 nmat 100 °C that servesas a backgate. The stack is thendeposited at 160 °Conap:Si/SiO2 chip.After deposition, we clean the polycarbonate stamp usingdichloromethane.We contact the MATBG with edge contacts made by electronbeam lithography followedby reactive ion etching, usingCHF3/O2 (40/4 sccm, 60W). The contacts are then evaporated using Cr/Au (10/80nm). We define the electrode lines in two steps using electron beamlithography and depositing Cr/Au (10/60 nm for the first step and 5/50nm for the second). Then we etch the stack to define the mesa anddeposit a 20 nm thick layer of aluminium oxide by atomic layerdeposition. For the ring-shaped top gate, we again use electron beamlithography and evaporate Cr/Au (5/35 nm). The electrode line for thetop gate is also done by electron beam lithography and evaporationusing Cr/Au (10/70 nm). In Supplementary Figure 1 we show opticalpictures of thedifferent fabrication steps andanSEM imageof a similardevice.Measurement setupWe carry out all themeasurements in a dilution refrigerator that uses amixture of 3He and 4He with a base temperature of 55mK. We apply aconstant current bias between a pair of contacts across the ring andmeasure the voltage drop between another pair, also across the ring.To generate the bias current, we use an in-house-built d.c. source inseries with a 100MΩ resistor. We use a d.c. amplifier, also built in-house, and measure its output with a Hewlett Packard 3441A digitalmultimeter. Each gate is connected to a different voltage source of thesame type as the one used for generating a d.c. current.We convert thevoltages we apply to the gates to electron densities by a parallel platecapacitormodel.We estimate the capacitance per unit area of the backand ring (top) gate to be Cbg = ε0εhBN/dbot and Crg = ε0εhBNεAlOx/(εhBNdtop + εAlOxdAlOx), where ε0 is the vacuum permittivity, εhBN = 3.3and εAlOx = 9.5 are the relative permittivities of the hBN and the alu-minium oxide, dtop and dbot are the thicknesses of the top and bottomhBN and dAlOx is the thickness of the aluminium oxide layer. WeArticle https://doi.org/10.1038/s41467-023-44671-4Nature Communications |          (2024) 15:390 6calculate the electron density of the bulk as ng =CbgVbg/e and of theregion below the top gate as nr = ðCbgVbg +CrgV rgÞ=e, where e is theelementary charge.Error/uncertainty analysisElectronic noise that comes from the noise in the measurementequipment (e.g. amplifiers) was minimized by carefully removing theground loops. The measured voltage was typically integrated by 200ms to reduce the uncertainty of the data points. The error in the esti-mation of the coherence length was calculated from the standarddeviation of the least-square fitting to the data and the error-propagation rule.Data availabilityThe data that support the findings of this study will be made availableonline through the ETH Research Collection at hdl.handle.net/20.500.11850/644891.Code availabilityThe code used for plotting the figures will be made available onlinethrough the ETH Research Collection at hdl.handle.net/20.500.11850/644891.References1. Suárez Morell, E., Correa, J. D., Vargas, P., Pacheco, M. & Barticevic,Z. Flat bands in slightly twisted bilayer graphene: tight-bindingcalculations. Phys. Rev. B 82, 121407 (2010).2. Zhang, Y.-H., Mao, D., Cao, Y., Jarillo-Herrero, P. & Senthil, T. Nearlyflat Chern bands in moiré superlattices. Phys. Rev. B 99, 075127(2019).3. Cao, Y. et al. Correlated insulator behaviour at half-filling in magic-angle graphene superlattices. Nature 556, 80–84 (2018).4. Cao, Y. et al. 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Kim, K. et al. van der waals heterostructures with high accuracyrotational alignment. Nano Lett. 16, 1989–1995 (2016).AcknowledgementsWethank PeterMärki, Thomas Bähler, and the staff of the ETHcleanroomfacility FIRST for technical support. We thank LevGinzburg and RebekkaGarrais for their help with the experiments. We thank Andreas Trabe-singer for his valuable input during the writing process of this manu-script. We acknowledge financial support by the European GrapheneFlagship Core3 Project, H2020 European Research Council (ERC)Synergy Grant under Grant Agreement 951541, the European Union’sHorizon 2020 research and innovation programme under grant agree-ment number 862660/QUANTUM E LEAPS, the European InnovationCouncil under grant agreement number 101046231/FantastiCOF, NCCRQSIT (Swiss National Science Foundation, grant number 51NF40-185902). K.W. and T.T. acknowledge support from the JSPS KAKENHI(Grant Numbers 21H05233 and 23H02052) and World Premier Interna-tional Research Center Initiative (WPI), MEXT, Japan. E.P. acknowledgessupport of a fellowship from “la Caixa” Foundation (ID 100010434)under fellowship code LCF/BQ/EU19/11710062.Author contributionsS.I. and A.M.T. equally contributed to the project. A.M.T. fabricated thedevicewith the advice and assistance of S.I. S.I. andA.M.T. performed themeasurements and analyzed the data. S.I., A.M.T., E.P., M.P., G. Z., andM.S. discussed thedatawith comments fromall authors.M. V. performedthe electrostatics simulationswith the help of S.I. andA.M.T. T.T. and K.W.supplied the hBN crystals. T.I. and K.E. supervised the project. S.I. andA.M.T. wrote the manuscript with comments from all authors.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-023-44671-4.Correspondence and requests for materials should be addressed toShuichi Iwakiri or Alexandra Mestre-Torà.Peer review information Nature Communications thanks the anon-ymous reviewers for their contribution to the peer review of this work. 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To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2024Article https://doi.org/10.1038/s41467-023-44671-4Nature Communications |          (2024) 15:390 8https://doi.org/10.1038/s41467-023-44671-4http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/http://creativecommons.org/licenses/by/4.0/ Tunable quantum interferometer for correlated moiré electrons Results Highly tunable quantum interference Tunable Little–Parks oscillations Spectroscopy of the quantum oscillations Discussion Methods Twist angle estimation Device fabrication Measurement�setup Error/uncertainty analysis Data availability Code availability References Acknowledgements Author contributions Competing interests Additional information