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A. Díez-Carlón, J. Díez-Mérida, P. Rout, D. Sedov, P. Virtanen, S. Banerjee, R. P. S. Penttilä, P. Altpeter, [K. Watanabe](https://orcid.org/0000-0003-3701-8119), [T. Taniguchi](https://orcid.org/0000-0002-1467-3105), S.-Y. Yang, K. T. Law, T. T. Heikkilä, P. Törmä, M. S. Scheurer, D. K. Efetov

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[Probing the Flat-Band Limit of the Superconducting Proximity Effect in Twisted Bilayer Graphene Josephson Junctions](https://mdr.nims.go.jp/datasets/2825efeb-993d-470e-a408-cd829ec7ad11)

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Probing the Flat-Band Limit of the Superconducting Proximity Effect in Twisted Bilayer Graphene Josephson JunctionsProbing the Flat-Band Limit of the Superconducting Proximity Effect in Twisted BilayerGraphene Josephson JunctionsA. Díez-Carlón ,1,2 J. Díez-Mérida ,1,2 P. Rout ,1,2 D. Sedov ,3 P. Virtanen ,4 S. Banerjee ,3R. P. S. Penttilä ,5 P. Altpeter,1 K. Watanabe ,6 T. Taniguchi ,7 S.-Y. Yang ,8 K. T. Law,9 T. T. Heikkilä ,4 P. Törmä ,5M. S. Scheurer ,3 and D. K. Efetov 1,2,*1Fakultät für Physik, Ludwig-Maximilians-Universität, Schellingstrasse 4, 80799 München, Germany2Munich Center for Quantum Science and Technology (MCQST), München, Germany3Institute for Theoretical Physics III, University of Stuttgart, 70550 Stuttgart, Germany4Department of Physics and Nanoscience Center, University of Jyväskylä,P.O. Box 35 (YFL), FI-40014, Finland5Department of Applied Physics, Aalto University School of Science, FI-00076 Aalto, Finland6Research Center for Functional Materials, National Institute for Materials Science,1-1 Namiki, Tsukuba 305-0044, Japan7International Center for Materials Nanoarchitectonics, National Institute for Materials Science,1-1 Namiki, Tsukuba 305-0044, Japan8Southern University of Science and Technology, Shenzhen 518055, People’s Republic of China9Department of Physics, Hong Kong University of Science and Technology, Hong Kong, China(Received 5 February 2025; revised 15 May 2025; accepted 1 October 2025; published 20 November 2025)While extensively studied in normal metals, semimetals, and semiconductors, the superconducting (SC)proximity effect remains elusive in the emerging field of flat-band systems. In this study, we probeproximity-induced superconductivity in Josephson junctions (JJs) formed between superconductingNbTiN electrodes and twisted bilayer graphene (TBG) weak links. Here, the TBG acts as a highlytunable topological flat-band system, which, due to its twist-angle-dependent bandwidth, allows us toprobe the SC proximity effect at the crossover from the dispersive to the flat-band limit. Contrary to ouroriginal expectations, we find that the induced superconductivity remains strong even in the flat-band limitand gives rise to broad, dome-shaped SC regions, in the filling-dependent phase diagram. In addition, wefind that, unlike in conventional JJs, the critical current Ic strongly deviates from a scaling with the normalstate conductance GN . We attribute these findings to the onset of strong electron interactions, which cangive rise to an excess critical current. By also studying the dependence of Ic on the filling and twist angleacross multiple samples, we further uncover the importance of quantum geometric terms as well asmultiband pairing mechanisms in describing the induced superconductivity in the TBG flat bands as theirbandwidth decreases. To the best of our knowledge, our results present the first detailed study of the SCproximity effect in the flat-band limit and shed new light on the mechanisms that drive the formation of SCdomes in flat-band systems.DOI: 10.1103/ccb4-tqxq Subject Areas: Condensed Matter PhysicsI. INTRODUCTIONThe superconducting proximity effect at the interface ofa superconductor (S) and a normal metal (N) is understoodby the leaking of superconducting correlations into thenormal region, through the creation of phase coherentAndreev pairs [1,2]. This process is explained by deGennes’ theory [1–5], which considers wavefunctionmatching across the N/S interface and typically workswell for normal metals with dispersive bands, and a largeFermi velocity and Fermi surface. However, these con-ditions are dramatically altered in materials with a vanish-ing bandwidth, as found in various flat-band systems, likekagome metals [6,7], moiré materials [8], and Lieb lattices[9–11]. Recent theoretical studies have derived new for-malisms to describe the superconducting (SC) proximityeffect in such systems [12,13]. They show that, in atomicflat bands, the proximity effect can be strongly quenched,as their Fermi velocity vanishes and electrons localize.However, strong electronic interactions and quantum geo-metric terms can give rise to an enhanced conduction and*Contact author: dmitri.efetov@lmu.dePublished by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI.PHYSICAL REVIEW X 15, 041033 (2025)2160-3308=25=15(4)=041033(11) 041033-1 Published by the American Physical Societyhttps://orcid.org/0000-0001-8124-4549https://orcid.org/0000-0002-9811-4318https://orcid.org/0000-0001-6362-939Xhttps://orcid.org/0000-0002-7241-7572https://orcid.org/0000-0001-9957-1257https://orcid.org/0000-0002-4213-8599https://orcid.org/0009-0005-7170-0121https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0002-1467-3105https://orcid.org/0000-0002-4446-526Xhttps://orcid.org/0000-0002-7732-691Xhttps://orcid.org/0000-0003-0979-9894https://orcid.org/0000-0002-9439-5159https://orcid.org/0000-0001-5862-0462https://ror.org/05591te55https://ror.org/04xrcta15https://ror.org/04vnq7t77https://ror.org/05n3dz165https://ror.org/020hwjq30https://ror.org/026v1ze26https://ror.org/026v1ze26https://ror.org/049tv2d57https://ror.org/00q4vv597https://crossmark.crossref.org/dialog/?doi=10.1103/ccb4-tqxq&domain=pdf&date_stamp=2025-11-20https://doi.org/10.1103/ccb4-tqxqhttps://doi.org/10.1103/ccb4-tqxqhttps://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/superfluidity [14–16] and, as a result, allow for a sizable SCproximity effect [12,13].Twisted bilayer graphene (TBG) presents a highlysuitable platform to test these predictions. It containstwo adjacent flat bands with a bandwidth that can be tunedbelow w < 10 meV in devices with a magic twist angle ofθm ∼ 1.1°, which is 3 orders of magnitude lower than innormal metals. They were previously shown to host a broadnumber of strongly correlated phases—such as supercon-ductivity [17,18], correlated insulators [19,20], and non-trivial topological states [21–23]—and were argued tocontain quantum geometric terms that can enhance thesuperconducting state [16]. In addition, the close proximityof the two flat bands in energy could, in principle, allow forthe formation of more complex Andreev pairs, which couldreside in both bands [24–26].In this work, we probe the flat-band limit of the SCproximity effect by performing a detailed study of threetwist-angle controlled S/TBG/S Josephson junctions (JJs)and their evolution as the bandwidth is tuned from thedispersive limit w ∼ 66 meV (θ ∼ 1.24°), w ∼ 18 meV(θ∼0.94°) to the flat-band limit of w<10meV (θ∼1.00°).Surprisingly, we find that, even in devices with the flattestbands, the SCproximity effect can be comparable in strengthto the dispersive bands. When the Fermi energy is tunedthrough the flat bands, the critical current does not scalewiththe normal state conductance; it forms dome-shaped regionsclose to half filling of the bands and shows unconventionalinterference patterns. We discuss our findings in the contextof possible contributions to the SC proximity effect throughstrong electron interactions, quantum geometric terms, andmultiband pairing processes and provide constraints on itsunderlying symmetries.II. RESULTSA. Proximity-induced superconductivity in a TBG JJAs depicted in Fig. 1(a), our devices consist of a van derWaals heterostructure of TBG that is encapsulated withhexagonal boron nitride dielectrics (hBN) and patterned(b)(d)(e)(c) (f)(a)VIVSiO2Si++TBGhBNhBNNbTiNFIG. 1. Superconducting proximity effect in a TBG Josephson junction. (a) Device schematic of a TBG sheet acting as the weak linkof a JJ. The voltage V across the junction is recorded as a current bias I is applied through the superconducting electrodes in a two-probemeasurement. The carrier density is tuned by a gate voltage Vg to the doped Si. (b) Resistance R in red (right axis) at zero current bias, asa function of Vg (bottom) and of the corresponding moiré filling factor ν (top). Shaded vertical lines indicate the presence of the charge-neutrality point (gray), the correlated insulators at half filling of the flat bands (red), and the band insulators between the flat anddispersive bands (yellow). Regions with low resistance have a finite critical current Ic (blue, left axis), extracted from the nonlinearcharacteristics measured in panel (f). (c) Band structure of TBG for θ ¼ 1.00°. (d) The I − V curve measured at Vg ¼ 50 V. The solidand dashed lines have opposite sweep directions as indicated by the arrows. (e) Interference pattern recorded at the dispersive bands, forVg ¼ 60 V, which agrees well with a uniform single-slit junction (white dashed line). (f) Differential resistance dV=dI map, where darkblue regions represent superconducting states. Their contour along positive values of I was used for extracting Ic in panel (b). All datawere obtained in device D2.A. DÍEZ-CARLÓN et al. PHYS. REV. X 15, 041033 (2025)041033-2into a rectangular mesa of width W ∼ 1.5 μm. The devicesare capacitively coupled to a SiO2=Si back gate that allowsus to control the carrier concentration in the TBG, and theyare contacted with sputtered s-wave superconductingNbTiN leads that form one-dimensional edge contacts,resulting in junctions of length L ∼ 200 nm. It is worthnoting that while several previous TBG JJ experimentsused the intrinsic SC state of TBG and a gate-defined weaklink [27–29], the device design presented here stronglysimplifies the modeling of such JJs, as it uses a SC statewith a known pairing mechanism and defines a simpler andsharper junction interface.We first focus on device D2 with a twist angleθ ∼ 1.00� 0.01°, slightly smaller than θm. All data wereobtained at a temperature of 35 mK. Figure 1(b) shows two-terminal resistance R measurements as a function of gatevoltage Vg as we tune through the dispersive and flat-bandregions [see Fig. 1(c)], which show peaks at integer fillingsof the moiré unit cell ν ¼ 0, �2, and �4. This behavior ischaracteristic of strongly correlated TBG devices in therange of twist angles of θ ∼ 1.0°–1.2°, where ν ¼ �2marksthe occurrence of correlated insulator states [8,17–20].Additionally, zero-resistance states at various fillings arerecorded due to the SC proximity effect, as shown inFig. 1(c), where we measure the differential resistancedV=dI as a function of dc I for the same range of Vg asFig. 1(b). We clearly see superconducting regions with zeroresistance (dark blue) that are limited by the critical currentIc [displayed in Fig. 1(b)], which exist across almost theentire density range. We observe broad dome-shaped SCregions in the dispersive bands at jνj > 4 and also in the flatbands near the charge-neutrality point (CNP) ν ¼ 0, andbetween fillings ν ¼ �2 and ν ¼ �4.The formation of the Josephson effect is confirmed bythe observation of nonlinear current-voltage characteristics,such as the one shown in Fig. 1(d) in the dispersive bands(Vg ¼ 50 V, ν ¼ 7.3). The switching from the zero-resis-tance state to the normal state is detected as a sharptransition in voltage and presents a hysteretic behaviorbetween the retrapping (Ir) and critical (Ic) currents, as iscommon for underdamped junctions or due to self-heatingeffects [4]. The phase coherence of the JJ is furtherdemonstrated by the observation of Fraunhofer interferencepatterns when a perpendicular magnetic field B is applied tothe junction, as seen in Fig. 1(e). The period of themeasured oscillations ΔB ∼ 2.5� 0.2 mT matches wellwith the expected periodicity ΔBphys ∼ 2.3� 0.1 mTdefined by the physical area of the junction around 0.33�0.07 μm2 when flux-focusing effects are included (seeSupplemental Material [30] for more details).A natural question is whether some of the zero-resistancestates observed could arise from an intrinsic SC state ofTBG. However, several experimental signatures argueagainst such an assumption. First, the observed Ic in thesuperconducting domes at ν < −2 reaches its maximum atν ∼ −2.9 and spans down to ν ∼ −3.5 [Fig. 1(b)], exceed-ing the range of filling where TBG typically shows anintrinsic SC state [8,17,18]. Moreover, while sweeping νacross this range, we see no additional switching tran-sitions in the dV=dI curves and no abrupt changes in Ic orin the interference patterns (see Fig. S20 in theSupplemental Material [30]). In the latter, the period ofthe IcðBÞ oscillations coinciding with the junction areaalso suggests a lack of intrinsic superconductivity since Icwould otherwise not oscillate and instead decay mono-tonically with the field [28,31,32]. We then conclude thatthe superconductivity observed in our TBG JJs arises fromthe proximity effect.B. Excess of supercurrent due to strong electroninteractions in the flat bandsThe maximal critical currents that we observe in the flatbands of Ic ∼ 65 nA are only a factor of 5 lower ascompared to the dispersive bands Ic ∼ 350 nA. Thisfinding is surprising, as we would have expected a muchstronger suppression of the SC proximity effect in the flatbands as compared to the dispersive bands, given thelarge reduction in the bandwidth and Fermi velocity (seeSupplemental Material [30]) [12,13,16,33]. Since the Icvalues themselves do not allow for a direct estimate of thestrength of the SC proximity effect, it is instead typicallyapproximated with the IcRN product [2,4,5]. Here, RN isthe normal state resistance of the JJ, which we extractthrough resistance measurements at a current I > Ic (seeSupplemental Material [30] for more details). Figure 2(a)shows IcRN vs ν, where we find especially large values inthe dome-shaped proximity-induced regions betweenfillings ν ¼ �2 and ν ¼ �4, which are comparable tothe ones in the dispersive bands. This finding furtherconfirms that, unlike the initial expectations, the SCproximity effect is surprisingly large in the flat bandsof TBG [2,5,33].The unexpected strength of the SC proximity effect inthe flat bands is not the only peculiarity, as the fillingdependence of the Ic inside the flat bands is also highlyunusual. In typical JJs, Andreev pairs undergo dephasingprocesses, the strength of which scales with the normalstate resistance RN. It is therefore often found that Iccorrelates with GN ¼ R−1N , the normal state conductance[2,4,5]. We demonstrate that this is the case in JJs withsingle-layer graphene or small twist-angle TBG as the weaklinks (see Supplemental Material [30] for comparison) [34].The same trend is also found when looking at the dispersivebands of our device D2, as can be seen in Fig. 2(b), whichshows the filling dependence of the critical current Ic vs νand overlays it with the filling dependence of the normalstate conductance GN vs ν. In this case, the Ic generallyfollows the density of states by becoming stronger withincreasing filling and peaking between ν ∼ 7 and ν ∼ 8 (seeSupplemental Material [30], Fig. S14).PROBING THE FLAT-BAND LIMIT OF THE … PHYS. REV. X 15, 041033 (2025)041033-3In strong contrast, the flat bands of our devices show theexact opposite trend, as pointed out by vertical dashedarrows in Fig. 2(b) [see also Fig. 3(c)]. Here, both Ic andGN increase when doping away from the CNP, but while Icpeaks at jνj ∼ 0.3 and decreases beyond these points, finallyvanishing at jνj ∼ 1, GN continues to increase until jνj ∼ 1.At higher doping, with the Ic domes at jνj > 2, the sameprocess appears: Because GN shows smaller values thannear the CNP, we expect a vanishing Ic, and yet we observesimilar (ν > 2) or even bigger values (ν < −2). Analogoustrends are also found for other devices in their flat bands[see Figs. 3(a) and 3(b)].The deviation of the observed scaling of Ic andGN in theflat bands could be explained by the existence of excessvalues of Ic in certain fillings, which could come from anextra contribution that is independent of GN and thus ofband dispersion. Such an often-neglected term, Iintc , indeedexists theoretically and scales with an attractive interactioncoupling between electrons, contributing to the total criticalcurrent because it boosts the Andreev pair transport throughthe JJ [13]. It is then an important contribution in (quasi-)flat bands, where the range of pair correlations withoutinteractions can become short due to localization in non-interacting transport. Similarly, as in the superfluid weight,part of this increase is related to the quantum geometry andis independent of band dispersion [12,13,35,36]. As hasbeen shown in Ref. [10] and Supplemental Material [30],Iintc is independent ofGN, so the total critical current Ic doesnot necessarily correlate with GN in regions where Iintcdominates, as could be the case for our devices in the rangesmentioned above. We note that, for a quantitative explan-ation of the observed unconventional relation between Icand GN, future theoretical models should also derive thefull contribution from the band dispersion to the Ic andcompare it with Iintc .Previous experiments using gate-defined JJs inTBG [27–29] demonstrated that a finite supercurrentcan flow through the flat bands and could, in principle,have accessed similar proximity-induced phenomena.(b)(c)(a)FIG. 3 Proximitized flat bands with varying bandwidth bytuning the twist angle. (a)–(c) Colormaps of measured differentialresistance dV=dI as a function of dc I (left axis) and filling ν ofthe flat band, for devices D3, D1, and D2, respectively. The solidred line corresponds to the normal state conductance GN (rightaxis). The critical current Ic of each device is extracted byfollowing the contour of the dark blue regions in the colormaps,marked by white-dashed lines. Following the same color code,the dashed vertical arrows indicate the regions where Ic and GNfollow opposite trends, pointing to whether the correspondingquantity is an increasing or decreasing function of jνj. In D1 andD3, the observed Josephson effect holds throughout almost theentirety of the flat bands, given the absence of correlated states atinteger fillings. In the case of D2 (the closest to θm), thesuperconducting phases are interrupted by insulating states athalf filling.(b)(a)FIG. 2. Strength of the proximity effect and relation betweenthe critical current and normal conductance. (a) Product of thecritical current and normal state resistance, IcRN, as a function ofthe moiré filling factor ν. (b) Critical current Ic in blue (left axis)and normal state conductance GN in red (right axis), both as afunction of ν. Following the same color code, the dashed verticalarrows indicate whether the corresponding quantity has reached amaximum or minimum. This highlights the unusual relationbetween Ic and GN found at the flat bands. All data correspond toD2. Shaded vertical lines indicate the presence of the charge-neutrality point (gray), the correlated insulators (red), and theband insulators (yellow).A. DÍEZ-CARLÓN et al. PHYS. REV. X 15, 041033 (2025)041033-4However, device limitations such as varying junctionlengths with filling and the fact that the weak link cannotbe studied independently from adjacent gate-tunableregions may obscure subtle deviations from the conven-tional Josephson effect. In addition, because the super-conducting state of TBG itself may already have anunconventional pairing [17,37] as well as quantum geom-etry contributions to its superfluidity and coherence length[16,36,38], it becomes difficult to disentangle whether theobserved features originate from the weak link or from theintrinsic properties of the flat-band superconductor. Byusing external s-wave superconducting leads, our experi-ment cleanly isolates the TBG weak link and enablessystematic comparisons between flat and dispersive bandsas well as across different twist angles, which we willexplore next, offering a clearer study on how bandstructure, interactions, and quantum geometry affect theproximity-induced superconductivity.C. Examining potential effects due to quantumgeometry and multiband pairingWhile we now have postulated the existence of enhancedIc regions in the flat bands and a potential explanationrelated to strong electron interactions, the strong variationof Ic and the formation of dome-shaped regions, especiallyclose to the band insulators between fillings ν ¼ �2 andν ¼ �4, remain unclear. In order to obtain a better under-standing of the driving forces behind these domes, we usethe powerful tuning knob of TBG, the bandwidth w, whichcan be controlled directly by the twist angle. As we will seenext, by tuning w, we can effectively tune the banddispersion and interactions.Figure 3 compares three samples with different twistangles: D1 (θ ∼ 0.94� 0.01°), D2 (θ ∼ 1.00� 0.01°), andD3 (θ ∼ 1.24� 0.01°). All three devices show a finite Icacross almost the entire flat band, with some dome-shapedregions in between half filling ν ¼ �2 and the band edgesν ¼ �4. It appears that as w is lowered [Figs. 4(d)–4(f)],the dome-shaped SC regions move closer to the band edges[see Figs. 3(a)–3(c)]. While Ic is suppressed in D1 and D3close to the CNP, in D2, which has the lowest bandwidth,the region around the CNP shows enhanced Ic values.We try to qualitatively understand the observed variationof Ic with filling and twist angle. We use the Eliashbergformalism and perturbation theory to compute the linearresponse function to the contact-induced pairing from an s-wave superconductor into TBG at the Fermi level EF.Under the slowly varying field approximation, the responseis captured by the superconducting pair correlationfunction ϕR¼Ru:c: dahc↓;−ðRþ aÞ ·c↑;þðRþ aÞi. To modelthe TBG, we use the Bistritzer-MacDonald (BM) con-tinuum model [39] and calculate the band structure, thebandwidth [see Figs. 4(d)–4(f)], and the pair correlator ϕRat a distance R in moiré unit cells coinciding with the centerof the junctions D1-3 (see Supplemental Material [30] formore details). Here, besides the dominant-induced Andreevpair components with opposite spin ð↑;↓Þ and valleysð−;þÞ, a possible, small, admixed intravalley componentin TBG is expected to be quickly suppressed with R. Sincethe correlator ϕR can be understood as a response functionof the TBG bands to an external superconducting field, aqualitative comparison can be made with Ic.The strength of this formalism lies in the fact that we canexplicitly account for the different contributions to thecorrelator ϕR, which include the band dispersion andthe quantum geometric terms. Importantly, the inducedAndreev pairs do not need to reside only in a single band atthe Fermi level EF, but it can also have weight acrossmultiple bands. The contribution from multiband pairing toϕR, however, diminishes more the further these bands arefrom EF and the larger their dispersion is; thus, it isexpected to be significant only in the flattest bands (seeSupplemental Material [30] for more details).In order to highlight the effect of each contribution andwork out the effects of the quantum geometry and multi-band pairing, we separate these terms out and plot themindependently in Fig. 4. Here, we consider three cases.First, we consider ϕS:B:R , which contains only dispersion-driven effects, by taking the atomic limit of the flat bands,where the Bloch states are completely momentum inde-pendent and the quantum geometric contribution is sup-pressed, and Andreev pairs that are only formed inside asingle band at EF [Fig. 4(a)]. We next consider ϕQ:G:R , whichcontains dispersion-driven effects but now departing fromthe atomic limit where we incorporate the overlap betweenBloch states at different momenta, which is determined bythe quantum metric of the bands, and Andreev pairs that areonly formed inside one band at EF [Fig. 4(b)]. Lastly, weconsider ϕM:B:R , which contains dispersion-driven effectswith a suppressed quantum geometric contribution, like inthe first case, but now additional Andreev pairs formedinside other bands can interfere with the ones from the bandat EF [Fig. 4(c)].Our calculations are shown in Figs. 4(g)–4(i) for thethree different bandwidths w corresponding to the threedevices shown in Fig. 3, with D3 (w ∼ 66 meV), D1(w ∼ 18 meV), and D2 w ∼ 4 meV) [BM band structuresshown in Figs. 4(d)–4(f)]. Here, we find that, when thefilling is varied, ϕS:B:R just follows the density of states forall w (Supplemental Material [30], Fig. S13 shows thisstatement explicitly). It shows peaks close to half fillingν ¼ �2, which rapidly decrease close to the band edgesν ¼ �4 and the CNP. Although this scenario very wellmatches the broadest bandwidth w ∼ 66 meV [Fig. 4(g)]with our Ic measurements [Fig. 3(a)], it dramatically fails todescribe the observed features of devices D1 and D2 withlower w [see Figs. 3(b)–3(c) and 4(h)–4(i)]. For the flattestbands, w ∼ 4 meV, the peaks in ϕS:B:R strongly broaden andshow an almost constant value across the entire range of ν[Fig. 4(i)], failing to produce dome-shaped regions. By nowPROBING THE FLAT-BAND LIMIT OF THE … PHYS. REV. X 15, 041033 (2025)041033-5plotting ϕQ:G:R including quantum geometric and ϕM:B:Rincluding multiband terms, we find that these are stronglyaltered from ϕS:B:R as the bandwidth is lowered, and bothgive rise to dome-shaped features close to the band edges,just as observed for the devices D1 with w ∼ 18 meV andD2 with w ∼ 4 meV. We find that, overall, the ϕQ:G:R termbetter matches the findings of the w ∼ 4 meV case, as itallows for a finite-induced superconducting phase in thecenter of the band and close to the CNP, as observed indevice D2. We also find that D1 with w ∼ 18 meV is bettermatched with ϕM:B:R . The moderate mismatch in D2compared to D1 and D3 could come from the fact thatonly the former shows correlated states due to interactions,which are not considered in our model.We want to stress that, to fully capture and explain theproximity effect in this correlated system, future theoreticalmodels should further account for interactions, as ourcurrent approach does not explain the vanishing Ic due(g)(h)(i)(a)(b) (c)(d)(e)(f)MultibandFIG. 4. Quantum geometric and multiband contributions to the induced superconductivity in the flat bands. (a)–(c) Sketchesillustrating the different contributions to the induced superconductivity. The color of the spin of the electrons forming theAndreev pair indicates which band they are coming from. In panel (a), the contribution comes only from the dispersion of thesame band where the Fermi level EF lies, whereas in panel (c), the interference with more bands is also accounted for, i.e.,multiband pairing. In panel (b), the quantum geometric contribution is considered along with the dispersion of the single band inpanel (a). (d)–(f) Band structure of the TBG continuum model along high-symmetry points for the three different twist anglescorresponding to samples D3, D1, and D2, respectively. The bandwidth w of the flat bands is also shown. (g)–(i) Computedsuperconducting correlator ϕR vs filling factor ν for the three processes illustrated in panels (a)–(c), following the same color code asthe sketches of the Andreev pairs. Shaded vertical lines indicate the presence of the charge-neutrality point (gray), the correlatedinsulators (red), and the band insulators (yellow).A. DÍEZ-CARLÓN et al. PHYS. REV. X 15, 041033 (2025)041033-6to the correlated states at integer fillings. Overall anddespite the simplicity of the continuum model in describingthe flat bands of TBG, the qualitative agreement with thedata highlights that, when the bandwidth reaches the flat-band limit, quantum geometric and multiband processescould become important in understanding the susceptibilityof TBG to develop superconducting phases [25]. It mayexplain the formation of dome-shaped SC regions betweenhalf filling and the band edges, which roughly coincidewith the regions that typically also show intrinsic super-conductivity in TBG devices at the magic angle. It istherefore interesting to consider whether similar effects asthe ones that were worked out here can also explain theposition of the SC domes in TBG.D. Symmetry breaking and the Josephson diode effectThe presence of interactions in the TBG JJs at halffillings of the bands is further suggested by a consistentobservation of a symmetry-broken Josephson effect, whichwill be the focus of the remainder of this work. We start bystudying an interference pattern in D2 at ν ¼ −2.5, shownin Fig. 5(a). When recording Ic vs B in opposite directionsof the dc, Iþc ðBÞ and I−c ðBÞ, we find that Iþc ðBÞ ≠ jI−c ðBÞj,as clearly seen in the middle panel of Fig. 5(a). Such anobservation indicates that inversion symmetry is broken inour JJs and is unlike the conventional symmetric patternsfound near the CNP or in the dispersive bands [see Fig. 1(e)and Supplemental Material [30] ]. This nonreciprocity is ahallmark of the Josephson diode effect (JDE), which(a)(d) (e)(b)(c)× Inv.  TRS  FIG. 5. Josephson diode effect and inversion symmetry breaking at the jνj > 2 domes. (a) Top panel: measured differentialresistance dV=dI as a function of dc bias I and magnetic field B, at ν ¼ −2.5 for D2. The critical current of the dark-coloredoscillations is extracted for positive (Iþc ) and negative (I−c ) directions of dc. These results are represented for the same andopposite values of the magnetic field in the middle and bottom panels, respectively. (b) The dV=dI traces measured at oppositemagnetic fields. Shaded regions mark the I values at which the Josephson diode is operational. The measurements are performedat ν ¼ −2.9 for D2. (c) Demonstration of the reversible JDE in panel (b), performed by switching between the superconductingand normal states when opposite currents are applied, in this case, þ34 nA and −34 nA. Reversibility of the direction of thediode is achieved by applying an exactly opposite B. (d) The dV=dI colormap vs I (left axis) and ν. Represented alongwith the error bars is the extracted diode efficiency η (right axis) vs ν, computed at a magnetic flux Φ ¼ Φ0=2. All datacorrespond to D2. (e) Maximum value of ηðBÞ between −2Φ0 and 2Φ0, represented with error bars and as a function of ν for alljunctions near the magic angle D1-3. The line plots correspond to the Ic (in arbitrary units) from Figs. 3(d)–3(f). The shadedregions correspond to fillings where a finite asymmetry was recorded. The interference patterns where η was extracted can befound in Supplemental Material [30].PROBING THE FLAT-BAND LIMIT OF THE … PHYS. REV. X 15, 041033 (2025)041033-7requires both inversion C2z and time-reversal symmetry(TRS) breaking [40–42]. Importantly, in our devices, TRSis broken by applying an external perpendicular magneticfield to the graphene layers, which rules out Rashba spin-orbit coupling as a possible mechanism, where an in-planemagnetic field that is perpendicular to the direction of thecurrent is needed to produce the nonreciprocity [42].Furthermore, the TBG weak link itself does not intrinsi-cally break TRS, as no asymmetry is recorded at zero field;i.e., we find Iþc ð0Þ ¼ jI−c ð0Þj. This finding is furtherconfirmed by having Iþc ðBÞ ¼ jI−c ð−BÞj [see bottom panelof Fig. 5(a)]; an expression that conserves this symmetryand makes the diode programmable by applying exactlyopposite fields. Such programmability is illustrated inFigs. 5(b) and 5(c), where the dV=dI curves validate theaforementioned symmetry relations and the operation of thediode is demonstrated in the rectification measurements.Notably, we find that the key features reported inFigs. 5(a)–5(c) extend across the entire filling of the dome,spanning from ν ∼ −2 to ν ∼ −3.5. This case is shown inFig. 5(d), where the diode efficiency parameter ηðBÞ ¼ðIþc ðBÞ − jI−c ðBÞjÞ=ðIþc ðBÞ þ jI−c ðBÞjÞ, calculated at halfthe superconducting magnetic flux quantumΦ0=2, is foundto correlate with the Ic of the dome. Thus, we see that theasymmetry is most pronounced at the center of the dome atν ¼ −2.9, but it is no longer detectable at the edges of it atν ¼ −2.3 and ν ¼ −3.4. Furthermore, our observation ofinversion symmetry breaking consistently appears in thedomes at the hole side of ν ∼ −2 for all our TBG JJs closeto θm (D1-3) and, in some cases (D1-2), on the electron sideof ν ∼ 2. Figure 5(e) shows that, for all devices, η correlateswith the Ic of their respective domes. In this case, tocompare the JDE between different devices, we havecalculated the maximum value of η between −2Φ0 and2Φ0 (see Supplemental Material [30] for more details).Such an extent of the asymmetry with filling suggests adistinct phase is responsible for the JDE.Several interacting ground states that spontaneouslybreak the C2z and spinless time-reversal symmetries ofTBG have been proposed. Valley polarization was sug-gested to explain the abundance of orbital magnetism andbroken inversion found at these bands [29,43–45], althoughoriginal nematicity measurements [46,47] showed C3zsymmetry breaking instead. In addition, recent experimentsin scanning tunneling microscopy have pointed towards themost likely candidates at such fillings having intervalleycoherent or incommensurate Kekulé spiral orders [48],which do not break C2z. Therefore, none of the abovecandidates is consistent with our findings, and a sublattice-polarized phase emerges as the only candidate that fulfillsour observed symmetry relations [25,49,50], where eachvalley carries opposite Chern numbers C ¼ 1 and C ¼ −1.Since we cannot quantify the amount of strain in oursamples, unlike in STM [48,51], we cannot rule out thatthis tuning parameter is favoring such a ground state withbroken C2z symmetry at these fillings. We note that thiscase is different from a strain-induced structural breaking ofC2z, which would lead to a JDE at all carrier densities.Given the range of twist angles and fillings where theJDE is found, a natural question is whether the presence ofthe intrinsic superconducting phase of TBG could play asignificant role. However, the observation of oscillations inIc, with their period matching the total area of the junction[Fig. 5(a)], argues against the TBG being fully intrinsicallysuperconducting in these devices [28,31,32]. The inde-pendence in temperature of the inductance associated withthe junction is also consistent (see Supplemental Material[30]), in contrast with previous observations of asymmetricoscillations in SQUIDs with high kinetic inductance [52].Finally, the inversion symmetry breaking resulting fromgeometrical factors, such as a nonuniform junction withdifferent widths of contacts, can be ruled out in our case,given that this effect would be independent of the electrondensity, and yet, the JDE is only observed in the flat bandsand in devices close to θm (see Supplemental Material[30]). Self-field effects caused by inhomogeneous currentbias and screening currents [2,53] cannot be the cause ofthe JDE either since our small Ic results in a largerJosephson penetration length λJ ¼Φ0tW=4πμ0Icλ2L∼7 μmcompared to the dimensions of the junction [2,54,55].Here, t ∼ 0.6 nm is the thickness of TBG, and λL ∼400 nm is the London penetration length of NbTiN[56]. In Fig. 5(a), it can be observed that the nodes ofthe Ic are lifted, which is a signature of an asymmetry inthe supercurrent density profile [57,58]. Nevertheless, thisnode-lifting effect is, in general, not enough to provide aJDE though; in addition, higher-order terms that bring thecurrent-phase relation into a nonsinusoidal form areneeded [42,59]. Such higher harmonics or anomalousphases could be a result of a symmetry-broken state asdiscussed above [60,61], or they can be a result oftopological edge states [62,63]. The latter could also berelated to our samples, given that the supercurrent ismostly carried by the edges where the JDE is observed(see Supplemental Material [30]). Testing these possiblemechanisms would require phase-sensitive measure-ments, which we leave for future works.III. CONCLUSIONSTo summarize, we have explored the versatility of TBGto support an extrinsic s-wave-mediated Josephson effect inits entire electronic band structure. The dispersive bands, aswell as other devices away from the magic angle, served asa reference to compare the proximity effect with the flat-band limit and test its predictions. Our observation of anunconventional scaling between the critical current and thenormal state conductance in the flat bands suggests an extraterm in the Ic, potentially due to an attractive interactioncoupling between electrons. Furthermore, our study of theformation of dome-shaped SC regions as a function ofA. DÍEZ-CARLÓN et al. PHYS. REV. X 15, 041033 (2025)041033-8filling and twist angle hints at quantum geometric andmultiband pairing contributions being increasingly impor-tant in their enhancement as the bandwidth of the flat bandsnarrows. Overall, to the best of our knowledge, our workconstitutes the first experimental effort to characterize indetail the superconducting proximity effect in a flat-bandsystem. Future works should focus on fully incorporatingstrong interactions in the mechanism of induced super-conductivity in TBG, in order to explain symmetry-brokeneffects such as the Josephson diode effect.ACKNOWLEDGMENTSWe thank Srijit Goswami for help in sample fabrication.D. K. E. acknowledges funding from the EuropeanResearch Council (ERC) under the European Union’sHorizon 2020 research and innovation program (GrantAgreement No. 852927), the German Research Foundation(DFG) under the priority program SPP2244 (ProjectNo. 535146365), the EU EIC Pathfinder Grant “FLATS”(Grant Agreement No. 101099139), and the Keele, Kavli,Tschira, and Wells Foundations as part of the SuperCCollaboration. K.W. and T. T. acknowledge support fromthe Elemental Strategy Initiative conducted by the MEXT,Japan (Grant No. JPMXP0112101001) and JSPSKAKENHI (Grants No. 19H05790, No. 20H00354, andNo. 21H05233). R. P. S. P. acknowledges financial supportfrom the Fortum and Neste Foundation. This work wassupported by the Research Council of Finland underProjects No. 339313 and No. 354735, by EuropeanUnion’s HORIZON-RIA programme 331 (GrantAgreement No. 101135240 JOGATE), by the Jane andAatos Erkko Foundation, the Keele Foundation, and theMagnus Ehrnrooth Foundation as part of the SuperCCollaboration, and by a grant from the SimonsFoundation (SFI-MPS-NFS-00006741-12, P. T.) in theSimons Collaboration on New Frontiers inSuperconductivity. D. S., S. B., and M. S. S. acknowledgefunding by the European Union (ERC-2021-STG, ProjectNo. 101040651—SuperCorr).A. D. C., S. Y. Y., and D. K. E. conceived and designedthe experiments; A. D. C. and P.R. fabricated the devices;A.D.C. performed the measurements and analyzed the data;D. S., P. V., S. B., R. P. S. P., T. T. H., P. T., and M. S.S. performed the theoretical analysis; T. T. and K.W.provided materials; D. K. E. supported the experiments;A. D. C. and D. K. E. wrote the paper with input from J. D.M., D. S., P. V., T. T. H., P. 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X 15, 041033 (2025)041033-11https://doi.org/10.1103/PhysRevLett.127.126405https://doi.org/10.1038/s41565-022-01222-0https://doi.org/10.1038/s41565-022-01222-0https://doi.org/10.1038/s41467-022-31256-whttps://doi.org/10.1038/s41467-022-31256-whttps://doi.org/10.1103/PhysRevB.81.144515https://doi.org/10.1103/PhysRevB.81.144515https://doi.org/10.1103/PhysRevApplied.11.064053https://doi.org/10.1103/PhysRevApplied.11.064053https://doi.org/10.1103/PhysRevB.3.3015https://doi.org/10.1038/nphys3036https://doi.org/10.1038/s41467-024-52271-zhttps://doi.org/10.1103/PhysRevResearch.5.L032033https://doi.org/10.1103/PhysRevLett.131.016003https://doi.org/10.1103/PhysRevB.98.075430https://doi.org/10.1103/PhysRevB.98.075430https://doi.org/10.1021/acs.nanolett.3c01416https://doi.org/10.1021/acs.nanolett.3c01416 Probing the Flat-Band Limit of the Superconducting Proximity Effect in Twisted Bilayer Graphene Josephson Junctions I. INTRODUCTION II. RESULTS A. Proximity-induced superconductivity in a TBG JJ B. Excess of supercurrent due to strong electron interactions in the flat bands C. Examining potential effects due to quantum geometry and multiband pairing D. Symmetry breaking and the Josephson diode effect III. CONCLUSIONS ACKNOWLEDGMENTS DATA AVAILABILITY References