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Prasanna Rout, Nikos Papadopoulos, Fernando Peñaranda, [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), Elsa Prada, Pablo San-Jose, Srijit Goswami

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[Supercurrent mediated by helical edge modes in bilayer graphene](https://mdr.nims.go.jp/datasets/6a9e2f44-9891-4d3e-80e2-5351da273587)

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Supercurrent mediated by helical edge modes in bilayer grapheneArticle https://doi.org/10.1038/s41467-024-44952-6Supercurrent mediated by helical edgemodes in bilayer graphenePrasanna Rout1, Nikos Papadopoulos1, Fernando Peñaranda 2,Kenji Watanabe 3, Takashi Taniguchi 4, Elsa Prada 2, Pablo San-Jose 2 &Srijit Goswami 1Bilayer graphene encapsulated in tungsten diselenide can host a weak topo-logical phase with pairs of helical edge states. The electrical tunability of thisphase makes it an ideal platform to investigate unique topological effects atzeromagneticfield, such as topological superconductivity. Herewe couple thehelical edges of such a heterostructure to a superconductor. The inversion ofthe bulk gap accompanied by helical states near zero displacement field leadsto the suppression of the critical current in a Josephson geometry. Usingsuperconducting quantum interferometry we observe an even-odd effect inthe Fraunhofer interference pattern within the inverted gap phase. We showtheoretically that this effect is a direct consequence of the emergence ofhelical modes that connect the two edges of the sample. The absence of suchan effect at high displacement field, as well as in bare bilayer graphene junc-tions, supports this interpretation and demonstrates the topological nature ofthe inverted gap.Helical edge modes in two-dimensional (2D) systems are an importantbuilding block for many quantum technologies, such as dissipationlessquantum spin transport1–3, topological spintronics4,5, and topologicalquantum computation6. Electrons traveling along these modes cannotinvert their propagation direction unless their spin is flipped. As a con-sequence, they cannot backscatter as long as time-reversal symmetry ispreserved (e.g., even in the presence of arbitrary non-magnetic defects).Helical states are expected to appear at the edges of 2D topologicalinsulators7,8 or in one-dimensional semiconductors with large spin-orbit coupling (SOC)6. Interestingly, single-layer graphene was the firsttheoretically predicted quantum spin Hall insulator9, whereby theintrinsic SOC gives rise to helical edge states. However, the strength ofthis Kane-Mele type SOC (λKM) is too small in graphene (≈40μeV) torealize a topological phase in practice.With the advent of graphene-based van der Waals hetero-structures with exceptional electronic properties, several alternativeapproaches have been explored to create helical edge modes. Thesemodes are shown to exist in the quantum Hall regime of single-layergraphene at filling factor ν = 0 under the application of a large in-planemagnetic field10, or when placed on a substrate with an exceptionallylarge dielectric constant11. Using a double-layer graphene hetero-structure, helical transport was also observed by tuning each of thelayers to ν = ±112. While these experiments did not involve super-conductivity, they have been complemented by theoretical proposalsshowing that coupling the helical modes to a superconductor shouldgive rise to topological superconductivity13,14. Unlike the experimentsinvolving topological insulators15–18, the main practical drawback ofthese proposals is the requirement of large magnetic fields, which isdetrimental to any system involving superconductors.Recently, it was shown that helical modes can appear at zeromagnetic field in bilayer graphene (BLG) encapsulated with WSe2, atransition metal dichalcogenide (TMD)19. Several experiments haveshown that graphene coupled to TMDs gives rise to a proximityinduced Ising-type SOC, denoted λI20–27. In the case of BLG symme-trically encapsulated inWSe2 (Fig. 1a), λI has opposite signs on the twographene layers, thereby effectively emulating a Kane-Mele type SOC,Received: 14 June 2023Accepted: 4 January 2024Check for updates1QuTech and Kavli Institute of Nanoscience, Delft University of Technology, 2600 GA Delft, The Netherlands. 2Instituto de Ciencia de Materiales de Madrid(ICMM), CSIC. Sor Juana Inés de la Cruz 3, 28049Madrid, Spain. 3Research Center for Functional Materials, National Institute for Materials Science, Tsukuba305-0044, Japan. 4International Center for Materials Nanoarchitectonics, National Institute for Materials Science, Tsukuba 305-0044, Japan.e-mail: pablo.sanjose@csic.es; s.goswami@tudelft.nlNature Communications |          (2024) 15:856 11234567890():,;1234567890():,;http://orcid.org/0000-0003-2180-3593http://orcid.org/0000-0003-2180-3593http://orcid.org/0000-0003-2180-3593http://orcid.org/0000-0003-2180-3593http://orcid.org/0000-0003-2180-3593http://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/0000-0002-1467-3105http://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/0000-0001-7522-4795http://orcid.org/0000-0001-7522-4795http://orcid.org/0000-0001-7522-4795http://orcid.org/0000-0001-7522-4795http://orcid.org/0000-0001-7522-4795http://orcid.org/0000-0002-7920-5273http://orcid.org/0000-0002-7920-5273http://orcid.org/0000-0002-7920-5273http://orcid.org/0000-0002-7920-5273http://orcid.org/0000-0002-7920-5273http://orcid.org/0000-0002-9095-4363http://orcid.org/0000-0002-9095-4363http://orcid.org/0000-0002-9095-4363http://orcid.org/0000-0002-9095-4363http://orcid.org/0000-0002-9095-4363http://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-44952-6&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-44952-6&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-44952-6&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41467-024-44952-6&domain=pdfmailto:pablo.sanjose@csic.esmailto:s.goswami@tudelft.nlbut with a significantly larger magnitude of few meVs19,25,28. Using anelectric displacement field D across the BLG, it was shown19 that theband structure could be tuned continuously from a band-insulatorphase (BIP) at largeD, through a topological phase transition, and intoan inverted-gap phase (IGP). In this work we combine such a WSe2/BLG/WSe2 heterostructure with a superconductor and study theJosephson effect across this phase transition. In the normal phase wereplicate the previously measured19 transition between a BIP and anIGP. In the superconducting phase we show that the IGP is character-ized by a local minimum of the critical current Ic. In a topological IGP,theory predicts that currents should be able to circulate around theBLG sample via topological helical edgemodes. We show theoreticallythat this should in turn cause a unique even-odd modulation with fluxin the Fraunhofer interference pattern of the BLG Josephson junction(JJ). We present quantum interferometry measurements that exhibitthis modulation. Importantly, the modulation is found to disappearupon entering the BIP of our encapsulated samples, and is altogetherabsent in trivial BLG control devices without WSe2 encapsulation.Collectively, these observations suggest a topological nature for thesuperconducting IGP, and the presence of supercurrent-carryingtopological helical modes.ResultsA schematic of the devices is shown in Fig. 1a (“Methods” section andSupplementary Section 1 for details). The WSe2/BLG/WSe2 hetero-structure has superconducting NbTiN contacts as well as a back gate,VBG, and a top gate, VTG. We present results on two devices (Dev A andDev B) fabricated on the same heterostructure. Dev A has a largercontact separation, L = 3.7μm, which suppresses the Josephson effectand allows us to study the normal-state properties of the hetero-structure. The supercurrent transport is explored in a ballistic JJ (DevB) with a contact separation of L≈300nm. Analogous measurementson a third device Dev D are found to be consistent with the results forDev B, see Supplementary Sections 5, 6 and 7.Figure 1b showshow theBLGband structure is predicted to evolveas a function of displacement field D in the presence of WSe2-inducedSOC. For D = 0 the valence and conduction bands of the BLG areinverted due to the Ising SOC. The band inversion opens up a gap λI inthe bulk band structure. Increasing ∣D∣ introduces an additional com-peting energy u = −edD/ϵBLG. Here, d =0.33 nm is the BLG interlayerseparation and ϵBLG = 4.3 is the out-of-plane dielectric constant of BLG.At large ∣D∣,ubecomes the dominant energy scale compared to λI and agap associated with layer-polarized bands opens up. Thus, by tuning∣D∣ one can transition from the inverted phase (IGP: ∣u∣ < λI) to band-insulator phase (BIP: ∣u∣ > λI) via an inversion point, where the bulk gapcloses. The IGP is a weak topological insulator, with two associatedhelicalmodes per edge. If we neglect for themoment the possibility ofadditional trivial BLG/vacuum edge modes, the normal resistance forsufficiently large, pristine samples is then expected to be R ≈ h/4e2 = 6.4kΩ in the IGP, owing to the emergence of helical edge trans-port, and the exponential suppression of currents across the gap-ped bulk.We now compare this theory expectation to experiment. Beforeproceeding to the superconducting regime of our devices, we char-acterize the various phases in the normal regime. As a first step wemeasure the dependence of normal resistance with bottom and topgates, which independently control the carrier density, n, and thedisplacement field,D (Fig. 2a). The resistanceR as a function of n andDreveals a local maximum in R close to D =0 along the n = 0 line. This isconsistent with the predicted opening of a SOC gap in the IGP (λI > ∣u∣).A similar behavior has been observed previously in the form of anincompressible phase in capacitance measurements19. As we increase∣D∣, the gap is reduced and R decreases, reaching a minimum at thegap-inversion point λI = ∣u∣. Increasing ∣D∣ further leads to an increase in|D| >> 0kVTGVBGI VD = 0Fig. 1 | Band inversion in a bilayer graphene junction. a Device schematic of aNbTiN-based Josephson junction. The BLG is symmetrically encapsulated in WSe2leading to a proximity induced SOC. The hexagonal boron nitride (hBN) and hBN/AlOx act as bottom and top-gate dielectrics, respectively. The measurements areperformed using a quasi-four terminal current-biased circuit, where I is the currentbias, V is the measure voltage, and VTG(BG) is the voltage applied to top (bottom)gate.b Band structure of the encapsulated BLG for different displacement fieldsD.At high ∣D∣ (∣u∣ > λI) the bulk is in a band-insulator phase (BIP), whereas the SOCcreates an Ising gap in the bulk bands around D =0, driving the system into aninverted-gap phase (IGP). In this weak topological phase, gapless edge states arepresent all around the BLG stack.Article https://doi.org/10.1038/s41467-024-44952-6Nature Communications |          (2024) 15:856 2R as the band-insulator gap grows. The inversion points correspondingto Rminima are at D = −32 and 35mV/nm (Fig. 2a), which yield λI = 2.5and 2.7meV. The estimated λI matches quite well with previouslyreported values of 2.2 − 2.6meV19,25,26. While this trend is qualitativelyconsistent with theory, one should note that the magnitude of the IGPresistance is smaller than expected. As we will argue below, we inter-pret this as the result of a finite bulk conduction, enabled by thecharge-puddles in theBLGat smallD, which is known to arise as a resultof intrinsic disorder or defects29–32 and of charge inhomogeneity fromchemical or electrostatic doping33–35. The SOC gap can be probed bythermal excitation, see Fig. 2b. With increasing temperature, Rdecreases as would be expected for a thermally activated gap, exceptthat at low temperatures, R is capped to the finite value from bulkconduction. In addition, the local R(D) maximum at D = 0 graduallybecomes less pronounced as the temperature is increased, and com-pletely vanishes at higher temperatures. To see this effectmore clearlywe determine the difference in the resistances at D = 0 and at theinversion point Dinv = −32mV/mm, i.e., ΔR =R(D =0) −R(D =Dinv),which acts as an indicator of band inversion. While ΔR is positive closetoD = 0 in the IGP, it is almost zeroatT≈26K (side panel of Fig. 2b). Thegap extracted from fitting theR(T) dependence reveals the presenceofa gap maximum around D =0 and a gradual increase of the gap withincreasing D in the BIP (Supplementary Section 2).While the temperature dependence provides information aboutthe bulk gap, a verification of the existence and the helical character ofedge states can be done by breaking time-reversal symmetry. An in-plane magnetic field By introduces a Zeeman energy EZ which opens agap in the helical edge states19, and removes their contribution fromconduction. To check this effect wemeasure R at n = 0 in the presenceof an in-plane magnetic field (By) as presented in Fig. 2c. At D =0 weobserved a dip in magnetoresistance MR= [R(By) −R(0.3 T)]/R(0.3 T)for By < 0.15 T indicating the extra conduction from edge states. Athigher fields the absence of these gapless states results in MR valuesclose to zero. Outside the IGP where no helical edge is present, MR≈0for all By. Additionally, we can rule out that this effect is related to thebulk bands as EZ < λI. Moreover, the positive ΔR for the complete fieldrange is consistent with a band inversion originating from the SOC gap(side panel of Fig. 2c). At lower fieldswe once again see the presence ofconducting edge channels resulting in a dip in ΔR. This behavior isconsistent with the helicity of at least some of the edge modes in theIGP, as trivial spinful modes are not expected to be suppressed by By.After checking the plausible helicity of IGP edge states, we turnour focus towards inducing superconductivity in these edge modesusing JJs. The JJ (Dev B) is in the ballistic limit, as indicated by Fabry-Perot oscillations in the normal-state transport36,37 (SupplementarySection 3). We first check the evolution of the critical current (Ic) in theJJ as a function of n for fixedD =0 (Fig. 3a) and vice versa (Fig. 3b). Thedependence of Ic on n is similar to several previous studies36–40.Figure 3a shows an Ic minimum at n =0, coinciding with the R max-imum, while Ic increases with higher electron/hole doping. The moreinteresting behavior is the D dependence of Ic at n =0, displayed inFig. 3b.Weobserve a localminimum in Ic atD =0, corresponding to thelocal Rmaximum, and twomaxima at higher ∣D∣ values, concurrent tothe R minima at the band inversion points. At higher ∣D∣, Ic decreasesmonotonically due to the gradual opening of a band-insulator gap40.The suppression of the supercurrent around D =0 is a con-sequence of the higher resistance in the IGP. This suppression, how-ever, does not reveal whether the supercurrent flows through the bulkor through edge channels. One could attribute the suppression of Ic inthe IGP to the opening of a bulk SOC gap entirely. Therefore,demonstrating the existence of proximitized edge modes requires adifferent probe that can differentiate edge from bulk supercurrents. Acommon method employed for this purpose is superconductingquantum interferometry (SQI)15,16,18,39–42, which involves the measure-ment of Ic as a functionof a perpendicularmagneticfield Bz. In the caseof a short-and-wide JJ with homogeneous current transport across itswidthW, Ic(Bz) should display a standard Fraunhofer pattern, followingthe functional form Ic ∼ sinðπΦ=Φ0Þ=ðπΦ=Φ0Þ, where Φ = Bz/LW andΦ0 = h/2e is the superconducting flux quantum. This regime isobserved in our samples at high densities n (see Supplementary Sec-tion 3, Figure S7). On the other hand, for a JJ with only two super-conducting and equivalent edges (i.e., a symmetric SQUID) oneexpects a SQI pattern of the form cosðπΦ=Φ0Þ. However, the SQIpattern becomesmore complicated if there is some coupling betweenthese edges. An efficient inter-edge transport along edge channelsflowing along the twoSN interfaces (without becoming fully gappedbyproximity) can lead to the electrons and holes flowing around theplanar JJ, therebypickingup aphase from themagneticflux. Thisphaseintroduces a 2Φ0-periodic component into the Φ0-periodic SQUIDpattern of the edge supercurrent43, which then becomes a clear sig-nature of the existence of proximitizedNS edgemodes connecting thevacuum edges.Given its importance for the interpretation of our experimentalSQI results presented below, we first provide a theoretical analysis ofhow 2Φ0 harmonics arise in Ic(Φ) as a result of electrons and holescirculating around the sample. We compute Ic in a simple model con-taining the minimal ingredients to capture the effect (see Supple-mentary Section 9 for further details). The JJ with a gapped centralregion is abstracted into just four sites; one per corner of the BLGregion. The parent superconductor induces on the j-th corner a pairingFig. 2 | Bulkandedge transport in the inverted-gapphase. aResistanceR at3.3Kfor Dev A measured as a function of carrier density n = (CTGVTG +CBGVBG)/e anddisplacement field D = (CTGVTG −CBGVBG)/2ϵ0, where CTG(BG) is the capacitance oftop (bottom) gate and VTG(BG) is the voltage applied to the top (bottom) gate. Sidepanel: line cut of the R(n,D)map at n =0.bTemperature dependenceof resistanceR(T)measured atn =0 forDevA. Sidepanel: the disappearanceof the IGP (betweenthe two dotted lines) at higher temperatures is seen from the difference ΔRbetween R at D =0 and at the inversion point, Dinv = − 32mV/mm. c The magne-toresistance MR= [R(By) −R(0.3 T)]/R(0.3 T) measured at 40mK as a function of Dand in-plane magnetic field By for Dev A. The dotted lines represent the inversionpoints. The observed dip close to D =0 and By =0 is the result of the conductiondue to helical edge channels. Side panel: field dependence of ΔR.Article https://doi.org/10.1038/s41467-024-44952-6Nature Communications |          (2024) 15:856 3potential Δeiϕj , see Fig. 4a. The pairing phases ϕj depend on themagnetic fluxΦ through the BLG region, which is described by a gaugefield Ay =Bzx. This makes the tight-binding Hamiltonian H of the foursites strictly Φ0-periodic in Φ. Next, we add two different hoppingamplitudes between the corners, representing the existence of edgechannels flowing around the central region. These hoppings acquire aPeierls phase induced by Ay. The horizontal hoppings along thevacuumedges are denoted by t. They have zero Peierls phase, and thuspreserve the Φ0 periodicity of the Hamiltonian. The vertical hoppingalong the left NS interface is denotedby τ, againwith zeroPeierls phasesince it is located at x =0. On the right NS interface at x = L the hoppinghas the same modulus τ, but its Peierls phase is now e± iπΦ=Φ0 ,depending on the direction. This term makes the Hamiltonian 2Φ0-periodic when τ ≠0.The critical current Ic may be computed by maximizing theJosephson current I = (2e/ℏ)∂ϕF versus ϕ, where F is the free energycomputed from H. The resulting Ic, shown in Fig. 4b, thereforeinherits the periodicity of H with Φ. For a small Δ/t and a small butfinite τ/t, the resulting critical current is accurately approximatedby43Ic ∼ j cosðπΦ=Φ0Þ+ f j for a positive 0≤f < 1 that grows with τ, seethe Supplementary Sections 10 and 13. A finite f results in an even-odd modulation of the τ = 0 SQUID-like pattern. As τ approaches t, Icdevelops higher harmonics that deviate from this simple expression.The transport processes enabled by the coupling τ along the NSinterfaces, i.e., by Josephson currents looping around the normalregion, lead to the even-odd effect in the Fraunhofer pattern.We may improve the minimal model by adopting a more micro-scopic description with several trivial transport channels flowing alongvacuum edges39,40, and two helical modes flowing all around the BLGjunction, as expected from the band-inverted phase19,28,44. A coupling τbetween vacuum and helical states enables an inter-edge scatteringmechanism, see Fig. 4c. This time the Peierls phase is incorporated intothe tight-binding discretization of the different modes. Like in theminimal model, the inter-mode coupling and the superconductingproximity effect are both assumed to take place at the corners of thenormal sample, whose bulk is again assumed to be completely gapped.Although these simplifying assumptions are not strictly satisfied in theexperimental samples, they are enough to confirm that the conclusionsdrawn from the minimal model still hold in a more generic situation,with propagating helical modes in place of a direct inter-edge hoppingand with multiple trivial vacuum edge modes. The results for Ic in thecase of one and two vacuum edge modes are shown in Fig. 4d, e. Weonce more find a Φ0-periodic SQUID-like pattern at τ =0 (in red),representative of the non-inverted phase. By switching on the couplingτ, an even-odd modulation arises. This holds true also for higher num-ber of vacuummodes. Themain difference with respect to theminimalmodel is the behavior when τ→ t, which is now less drastic.Keeping these theoretical results in mind, we measure the SQIpatterns as a function of D at n = 0. Fig. 5a shows interference patternsat three different values of D. In contrast with the simulations, weobserve a large central peak at Bz =0. This is indicative of a finite bulkconduction, in line with our inference from the measurements ofnormal resistance. However, the SQI pattern at D = 0 (inside the IGP)shows a clear even-odd effect, i. e. the odd lobes are less intensecompared to subsequent even lobes. In contrast, at high D (inside theBIP) this effect is lost, giving rise to a more Fraunhofer-like pattern.Fig. 5c shows the evolution of the SQI as a function of D, where itbecomes clear that the even-off effect is present only in the IGP and isstrongest at D =0. As ∣D∣ is increased further (thus going deeper intothe BIP), the intensity of Bz =0 peak decreases, and a SQUID-like pat-tern without any even-odd effect emerges (Supplementary Section 7),indicating the presence of trivial and decoupled vacuum edge modes.To further confirm that the even-odd effect is associated with super-conducting helical modes, we perform control experiments on a barehBN/BLG/hBN JJ (Dev C), where we expect no induced SOC and hencenohelicalmodes. Indeed, we donot observe any even-odd effect in thecontrol SQI patterns although the trivial edges are still present athigher D (Fig. 5e, f and Supplementary Section 7).DiscussionEven-odd modulated SQI patterns have been experimentally reportedin various 2D systems with and without bulk conduction16,18,41,42. Themodulation is qualitatively described by the addition of a flux-independent supercurrent offset in the standard interference pattern(Supplementary Section 13). In InAs- and InSb-based planar JJs41,42, it wasproposed that the even-odd effect (and the flux-independent offset)stems from crossed Andreev reflection connecting the trivial edges ofthe JJs via conducting NS interfaces. Unfortunately, its origin was notfully clarified in these experiments. In contrast, our symmetricallyencapsulated BLG junctions offer a very natural candidate for anFig. 3 | Supercurrent in the inverted-gap phase. a Current bias I dependence ofthe differential resistance dV/dI for Dev B as a function of carrier density n atD =0and T = 40mK. The jump in dV/dI marks the critical current Ic. The lower panelshows the normal-state resistanceR(n) forD =0.b dV/dI (upper panel) and normal-state resistance R (lower panel) as a function of D for n =0.Article https://doi.org/10.1038/s41467-024-44952-6Nature Communications |          (2024) 15:856 4efficient inter-edge coupling mechanism, in the form of helical modesflowing along the two NS boundaries in the IGP. The even-odd effectthen acquires a special significance in our experiment, as a probe of theemergence of edge modes in the IGP. Their disappearance after theband inversion and their response to in-plane Zeeman fields offercomplementary evidence of their helicity and the probable topologicalnature of the inverted gap, and serves as a demonstration of how theiremergence can be controlled with an electric field.Our work provides the first experimental evidence of super-current flow along helical edges in graphene. The clear 2Φ0-periodicsignature in our SQI experiment suggests an enticing prospect ofdetecting topological 4π-periodic current-phase relations in thissystem8, which requires the investigation of the a.c. Josephsoneffect17,18,45 and/or non-equilibrium effects at finite bias in a.c.supercurrent46. The observation of gate tunable helical edge-mediatedsupercurrents opens a promising new avenue44 towards topologicalsuperconductivity and Majorana physics in van der Waals materials.MethodsFabricationWe exfoliate flakes of bilayer graphene (BLG), graphite (5–15 nm),tungsten diselenide WSe2 (10–35 nm) and hexagonal boron nitrideFig. 5 | Even-odd effect in superconducting quantum interferometry.a Normalized critical current Ic/Ic0 as a function of perpendicular magnetic field Bzmeasured at three different displacement fieldD for Dev Bwhile keeping n = 0. TheSQI patterns are vertically shifted by 0.2 from each other. b Zero-field criticalcurrent Ic0 extracted from Fig. 3b (solid line) and from (c) (circles) as a function ofD. c Normalized critical current Ic/Ic0 as a function Bz and D. An even-odd effect inthe SQI oscillatory pattern can be observed only within the IGP (marked by twovertical dotted lines).d–f Same as (a–c) but for Dev C where a BLG is encapsulatedin hBN without WSe2. The solid line in (d) is the extracted supercurrent in theSupplementary Section 4.Fig. 4 | Theory of the Fraunhofer even-odd effect from helical edge states.a Sketch of the four-site minimal model, in terms of the induced superconductingpairings at the corners (with amplitude Δ, phase ϕ, magnetic flux Φ and fluxquantum Φ0), intra-edge hoppings t and inter-edge hoppings τ, with their Peierlsphases in the gauge Ay = Bx. The origin of coordinates is chosen at the bottom-leftcorner. b Critical current Ic(Φ), normalized to a fixed I0, of the minimal modelexhibiting an even-odd modulation for τ ≠0. c Sketch of a more elaborate model,including in each spin sector one or more trivial edge modes (yellow) and onehelical mode (green) flowing around a gapped bulk. d, e Corresponding Ic(Φ) forone and two trivial edgemodes, respectively, exhibiting the even-odd effect whentrivial and helical modes become coupled by a hopping τ at the corners.Article https://doi.org/10.1038/s41467-024-44952-6Nature Communications |          (2024) 15:856 5hBN (20–55 nm) on different SiO2/Si substrates using Scotch tape.Then the stacks of hBN/WSe2/BLG/WSe2/hBN/graphite are assembledusing the van derWaals dry-transfer using polycarbonate (PC) films onPolydimethylsiloxane hemispheres. The flakes are picked up at 110 °C.The stacks are checked with atomic force microscopy before furtherfabrication.To fabricate the Josephson junctions (JJs) we spin coat a bilayerof 495 A4 and 950 A3 PMMA resists at 4000 rpm and bake at 175 °Cfor 5min, after each spinning. After the e-beam lithography pat-terning, the resist development is carried out in a cold H2O:IPA (3:1)mixture. We perform a reactive ion etching step using CHF3/O2mixture (40/4 sccm) at 80 μbar with a power of 60W to etch pre-cisely through the top hBN and WSe2. Then we deposit NbTi (5 nm)/NbTiN (110 nm) by dc sputtering for superconducting edge contactsand lift-off in NMP at 80 °C. In order to isolate the top-gate metalfrom the ohmic, we deposit a 30 nm dielectric film of AlOx usingatomic layer deposition at 105 °C. Finally we define the top gate bye-beam lithography, deposit Ti (5 nm)/Au (120 nm), followed bylift-off.MeasurementsAll measurements are performed in a dilution refrigerator with a basetemperature of 40mK. Four-probe transport measurements are per-formed with a combination of DC and AC current-bias scheme. Todetermine critical current we have employed a voltage switchingdetection method with current source and critical current measure-ment modules47. The magnetic fields are applied by a 3D vector mag-net, which enables us to align the field within ± 5° accuracy.Four-site modelThe spinless BdG Hamiltonian used for the four-site model, written inthe Nambu basis ðcσ ,cy�σÞTwhere cσ = (c1σ, c2σ, c3σ, c4σ), readsH =H0 H +ΔHΔ �H*0 !, ð1ÞwhereH0 =0 t 0 τt 0 τe�iπΦ=Φ0 00 τeiπΦ=Φ0 0 tτ 0 t 00BBB@1CCCA , HΔ =Δ1 0 0 00 eiϕ 0 00 0 eiϕ+ i2πΦ=Φ0 00 0 0 10BBB@1CCCA: ð2ÞH0 andHΔ are 4 × 4matrices corresponding to the normalHamiltonianof the particle sector and the onsite superconducting electron-holepairing terms induced by the leads, respectively. Δ and ϕ are theinduced pairing amplitude and phase from the parent superconduc-tors, respectively.Φ is the flux into the normal region encompassed bythe four sites. t and τ are the hopping amplitudes between sites alongthe vacuum edge interfaces (1↔2 and 3↔4) and the NS interfaces (1↔4and 2↔3), respectively. Further details, including the multimodegeneralization of this model, are given in Supplementary Sections 9and 12.Data availabilityRaw data and analysis scripts for all presented figures are available inthe Zenodo database under accession code https://doi.org/10.5281/zenodo.10046824.Code availabilityAll computer codes employed in this work are available in the Zenododatabase under accession code https://doi.org/10.5281/zenodo.7941266. Julia scripts can be executed after instantiating the packageenvironment.Mathematica notebooks canbe rundirectly fromthe topusing version 13 and above.References1. Murakami, S., Nagaosa, N. & Zhang, S.-C. Dissipationless quantumspin current at room temperature. Science 301, 1348–1351 (2003).2. Roth, A. et al. Nonlocal transport in the quantum spin Hall state.Science 325, 294–297 (2009).3. Brüne, C. et al. Spin polarization of the quantum spin Hall edgestates. Nat. Phys. 8, 485–490 (2012).4. Tokura, Y., Yasuda, K. & Tsukazaki, A. Magnetic topological insula-tors. Nat. Rev. Phys. 1, 126–143 (2019).5. He, Q. L., Hughes, T. L., Armitage, N. P., Tokura, Y. & Wang, K. L.Topological spintronics and magnetoelectronics. Nat. Mater. 21,15–23 (2022).6. Prada, E. et al. From Andreev to Majorana bound states in hybridsuperconductor-semiconductor nanowires. Nat. Rev. Phys. 2,575–594 (2020).7. Fu, L. & Kane, C. L. Superconducting proximity effect andMajoranafermions at the surface of a topological insulator. Phys. Rev. Lett.100, 096407 (2008).8. Beenakker, C. W. J., Pikulin, D. I., Hyart, T., Schomerus, H. & Dahl-haus, J. P. Fermion-parity anomaly of the critical supercurrent in thequantum spin-Hall effect. Phys. Rev. Lett. 110, 017003 (2013).9. Kane, C. L. &Mele, E. J. Quantum spin Hall effect in graphene. Phys.Rev. Lett. 95, 226801 (2005).10. Young, A. et al. Tunable symmetry breaking and helical edgetransport in a graphene quantum spin Hall state. Nature 505,528–532 (2014).11. Veyrat, L. et al. Helical quantum Hall phase in graphene on SrTiO3.Science 367, 781–786 (2020).12. Sanchez-Yamagishi, J. D. et al. Helical edge states and fractionalquantum Hall effect in a graphene electron–hole bilayer. Nat.Nanotechnol. 12, 118–122 (2017).13. San-Jose, P., Lado, J. L., Aguado, R., Guinea, F. & Fernández-Rossier,J. Majorana zero modes in graphene. Phys. Rev. X 5, 041042 (2015).14. Finocchiaro, F., Guinea, F. &San-Jose, P.QuantumspinHall effect intwisted bilayer graphene. 2D Mater. 4, 025027 (2017).15. Hart, S. et al. Induced superconductivity in the quantum spin Halledge. Nat. Phys. 10, 638–643 (2014).16. Pribiag,V.S. et al. Edge-modesuperconductivity ina two-dimensionaltopological insulator. Nat. Nanotechnol. 10, 593–597 (2015).17. Wiedenmann, J. et al. 4π-periodic Josephson supercurrent in HgTe-based topological Josephson junctions.Nat. Commun. 7, 1–7 (2016).18. Bocquillon, E. et al. Gapless Andreev bound states in the quantumspin Hall insulator HgTe. Nat. Nanotechnol. 12, 137–143 (2017).19. Island, J. et al. Spin-orbit-driven band inversion in bilayer grapheneby the van der Waals proximity effect. Nature 571, 85–89 (2019).20. Wang, Z. et al. Strong interface-induced spin-orbit interaction ingraphene on WS2. Nat. Commun. 6, 8339 (2015).21. Wang, Z. et al. Origin and magnitude of ‘designer’ spin-orbit inter-action in graphene on semiconducting transition metal dichalco-genides. Phys. Rev. X 6, 041020 (2016).22. Wakamura, T. et al. Strong anisotropic spin-orbit interaction inducedin graphene by monolayer WS2. Phys. Rev. Lett. 120, 106802 (2018).23. Wakamura, T. et al. Spin-orbit interaction induced in graphene bytransition metal dichalcogenides. Phys. Rev. B 99, 245402 (2019).24. Zihlmann, S. et al. Large spin relaxation anisotropy and valley-Zeeman spin-orbit coupling in WSe2/graphene/h-BN hetero-structures. Phys. Rev. B 97, 075434 (2018).25. Wang, D. et al. Quantum Hall effect measurement of spin–orbitcoupling strengths in ultraclean bilayer graphene/WSe2 hetero-structures. Nano Lett. 19, 7028–7034 (2019).26. Kedves, M. et al. Stabilizing the inverted phase of aWSe2/BLG/WSe2heterostructure via hydrostatic pressure.Nano Lett. 23, 9508 (2023).27. Tiwari, P., Srivastav, S. K., Ray, S., Das, T. & Bid, A. Observation oftime-reversal invariant helical edge-modes in bilayer graphene/WSe2 heterostructure. ACS Nano 15, 916 (2020).Article https://doi.org/10.1038/s41467-024-44952-6Nature Communications |          (2024) 15:856 6https://doi.org/10.5281/zenodo.10046824https://doi.org/10.5281/zenodo.10046824https://doi.org/10.5281/zenodo.7941266https://doi.org/10.5281/zenodo.794126628. Zaletel, M. P. & Khoo, J. Y. The gate-tunable strong and fragiletopology of multilayer-graphene on a transition metal dichalco-genide. Preprint at https://arxiv.org/abs/1901.01294v2 (2019).29. Oostinga, J. B., Heersche, H. B., Liu, X., Morpurgo, A. F. & Vander-sypen, L. M. Gate-induced insulating state in bilayer graphenedevices. Nat. Mater. 7, 151 (2008).30. Yan, J. & Fuhrer, M. S. Charge transport in dual gated bilayer gra-phene with Corbino geometry. Nano Lett. 10, 4521 (2010).31. Taychatanapat, T. & Jarillo-Herrero, P. Electronic transport in dual-gated bilayer graphene at largedisplacement fields. Phys. Rev. Lett.105, 166601 (2010).32. Zou, K. & Zhu, J. Transport in gapped bilayer graphene: The role ofpotential fluctuations. Phys. Rev. B 82, 081407 (2010).33. Chae, J. et al. Enhanced carrier transport along edges of graphenedevices. Nano Lett. 12, 1839–1844 (2012).34. Allen, M. T. et al. Spatially resolved edge currents and guided-waveelectronic states in graphene. Nat. Phys. 12, 128–133 (2016).35. Woessner, A. et al. Near-field photocurrent nanoscopy on bare andencapsulated graphene. Nat. Commun. 7, 10783 (2016).36. Calado, V. E. et al. Ballistic Josephson junctions in edge-contactedgraphene. Nat. Nanotechnol. 10, 761–764 (2015).37. Ben Shalom, M. et al. Quantum oscillations of the critical currentandhigh-field superconductingproximity inballistic graphene.Nat.Phys. 12, 318 (2016).38. Heersche, H. B., Jarillo-Herrero, P., Oostinga, J. B., Vandersypen, L.M. K. & Morpurgo, A. F. Bipolar supercurrent in graphene. Nature446, 56–59 (2007).39. Allen, M. T. et al. Spatially resolved edge currents and guided-waveelectronic states in graphene. Nat. Phys. 12, 128 (2016).40. Zhu, M. et al. Edge currents shunt the insulating bulk in gappedgraphene. Nat. Commun. 8, 1 (2017).41. de Vries, F. K. et al. h/e superconducting quantum interferencethrough trivial edge states in InAs. Phys. Rev. Lett. 120,047702 (2018).42. de Vries, F. K. et al. Crossed Andreev reflection in InSb flakeJosephson junctions. Phys. Rev. Res. 1, 032031 (2019).43. Baxevanis, B., Ostroukh, V. P. & Beenakker, C. W. J. Even-odd fluxquanta effect in the Fraunhofer oscillations of an edge-channelJosephson junction. Phys. Rev. B 91, 041409 (2015).44. Peñaranda, F., Aguado, R., Prada, E. & San-Jose, P. Majorana boundstates in encapsulated bilayer graphene. SciPost 14, 075 (2023).45. Laroche, D. et al.Observationof the4π-periodic Josephsoneffect inindium arsenide nanowires. Nat. Commun. 10, 1–7 (2019).46. Haidekker Galambos, T., Hoffman, S., Recher, P., Klinovaja, J. &Loss, D. Superconducting quantum interference in edge stateJosephson junctions. Phys. Rev. Lett. 125, 157701 (2020).47. Bouman, D. et al. Triplet-blockaded Josephson supercurrent indouble quantum dots. Phys. Rev. B 102, 220505 (2020).AcknowledgementsWe thank Joshua Island for discussions regarding heterostructureassembly and Tom Dvir for comments on the manuscript. This researchwas supported by an NWO Flagera grant and TKI grant of the DutchTopsectoren Program [P.R., N.P., S.G.], by the Spanish Ministry ofEconomy and Competitiveness through Grant PCI2018-093026 (Fla-gERA Topograph) [F.P., E.P., P.S-J.], by MCIN/AEI/10.13039/501100011033 and “ERDF A way of making Europe" through grantPID2021-122769NB-I00 [F.P., E.P., P.S-J.], and by the Comunidad deMadrid through Grant S2018/NMT-4511 (NMAT2D-CM) [E.P.].Author contributionsP.R. and N.P. fabricated the devices, performed the measurements, andanalyzed the experimental data. S.G. conceived and coordinated theexperiment. T.T. and K.W. provided hBN crystals. F.P., E.P., and P.S-J.developed the theoretical interpretation and models, and F.P. per-formed the simulations. All authors participated in discussions and inwriting of the manuscript.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41467-024-44952-6.Correspondence and requests for materials should be addressed toPablo San-Jose or Srijit Goswami.Peer review information Nature Communications thanks BenjaminSacepe, Henry Legg and the other anonymous reviewers for their con-tribution to the peer review of this work. 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