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Giulia Zheng, Elías Portolés, Alexandra Mestre-Torà, Marta Perego, [Takashi Taniguchi](https://orcid.org/0000-0002-1467-3105), [Kenji Watanabe](https://orcid.org/0000-0003-3701-8119), Peter Rickhaus, Folkert K. de Vries, Thomas Ihn, Klaus Ensslin, Shuichi Iwakiri

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[Gate-defined superconducting channel in magic-angle twisted bilayer graphene](https://mdr.nims.go.jp/datasets/1b7aa67e-059a-4d13-abb0-8488ce9ca92e)

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Gate-defined superconducting channel in magic-angle twisted bilayer graphenePHYSICAL REVIEW RESEARCH 6, L012051 (2024)LetterGate-defined superconducting channel in magic-angle twisted bilayer grapheneGiulia Zheng ,1 Elías Portolés ,1 Alexandra Mestre-Torà ,1 Marta Perego ,1 Takashi Taniguchi ,2 Kenji Watanabe ,3Peter Rickhaus ,1 Folkert K. de Vries ,1 Thomas Ihn ,1,4 Klaus Ensslin ,1,4 and Shuichi Iwakiri 1,*1Laboratory for Solid State Physics, ETH Zurich, CH-8093 Zurich, Switzerland2Research Center for Materials Nanoarchitectonics, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan3Research Center for Electronic and Optical Materials, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan4Quantum Center, ETH Zurich, CH-8093 Zurich, Switzerland(Received 18 December 2023; revised 17 January 2024; accepted 29 January 2024; published 6 March 2024)Magic-angle twisted bilayer graphene (MATBG) combines in one single material different phases such asinsulating, metallic, and superconducting. These phases and their in situ tunability make MATBG an importantplatform for the fabrication of superconducting devices. We realize a split-gate-defined geometry which enablesus to tune the width of a superconducting channel formed in MATBG. We observe a smooth transition fromsuperconductivity to highly resistive transport by progressively reducing the channel width using the splitgates or by reducing the density in the channel. Using the gate-defined constriction, we control the flow ofthe supercurrent, either guiding it through the constriction or throughout the whole device or even blockingits passage completely. This serves as a foundation for developing quantum constriction devices such assuperconducting quantum point contacts, quantum dots, and Cooper-pair boxes in MATBG.DOI: 10.1103/PhysRevResearch.6.L012051Introduction. Magic-angle twisted bilayer graphene(MATBG) offers highly tunable quantum states [1,2], whichhave enabled a novel class of gate-defined superconductingnanodevices such as Josephson junctions [3–5], supercon-ducting quantum interference devices (SQUIDs) [6], and ringgeometries showing Little-Parks oscillations [7]. All thesedevices rely on defining interfaces between resistive andsuperconducting phases. In a gate-defined narrow MATBGconstriction, the interface between the superconducting andresistive phases can be continuously controlled and studied.In Bernal bilayer graphene, a constriction (e.g., a quantumpoint contact [8,9]) can be formed due to its gate-tunableband gap of a few 100 meV [10,11]. However, in the case ofMATBG, the size of the band gap between the flat band andthe dispersive band is one order of magnitude smaller [4].Therefore it is challenging to form a similar constriction andthe question arises, whether it is possible at all to realizea narrow superconducting channel by tuning the width ofconfining resistive states.In this Letter, we demonstrate a gate-defined supercon-ducting channel in MATBG. We observe a smooth transitionfrom supercurrent to highly resistive transport (pinch-off) byprogressively reducing the channel width using split gates orby reducing the density in the channel using the channel gate.We find that the supercurrent in the constriction can be turnedon and off. These results will serve as the foundation for de-*siwakiri@phys.ethz.chPublished by the American Physical Society under the terms of theCreative Commons Attribution 4.0 International license. Furtherdistribution of this work must maintain attribution to the author(s)and the published article’s title, journal citation, and DOI.veloping constriction-based devices such as superconductingquantum point contacts, quantum dots, or Cooper-pair boxesin MATBG.Device and bulk characterization. The fabrication starts byencapsulating MATBG (light green) between two hexagonalboron nitride (hBN) layers of thickness 59 and 61 nm (darkgreen). Additionally, a graphite layer (dark gray) is picked up,serving as the back gate. The lateral device layout is shownin Fig. 1(a). To form electric contacts (orange in the figure),we etch the top hBN using reactive ion etching (RIE) andevaporate Cr(5 nm)/Au(65 nm). On top of the top hBN, weevaporate Cr(5 nm)/Au(15 nm) split gates (blue) with a gapof 150 nm. After depositing a 40-nm layer of aluminum oxide,an additional channel gate (yellow) with a width of 400 nm (inthe direction of current flow) is deposited, covering the regionin between the split gates. Our gate configuration defines threedifferent areas in the MATBG mesa: The leads shown in lightgreen and tuned only by the back gate, the split-gated region inblue, tuned by split gates and the back gate combined, and thechannel in between the split gates and below the channel gate.Since the thickness of the top hBN (59 nm) is comparable tothe lateral dimension of the split gate (150 nm), the channelregion is tuned not only by the channel gate and the back gate,but also by the split gates due to fringe fields. We perform two-terminal transport measurements at a temperature of 24 mK.To characterize the twist angle between the graphene lay-ers of the sample, we apply a dc current (I = 10 nA) andmeasure the resulting voltage drop V . We sweep the back-gate voltage Vbg while keeping the split-gate voltage Vsg andthe channel-gate voltage Vcg at zero, thus probing the bulk(leads, split-gated region, and channel together). The resultingresistance R = V/I is shown in Fig. 1(b), exhibiting sev-eral peaks originating from different filling factors ν of themoiré unit cell. The peak at around Vbg = 0 indicates charge2643-1564/2024/6(1)/L012051(6) L012051-1 Published by the American Physical Societyhttps://orcid.org/0000-0003-3503-4940https://orcid.org/0000-0001-7202-777Xhttps://orcid.org/0009-0000-1010-2922https://orcid.org/0009-0007-4830-4934https://orcid.org/0000-0002-1467-3105https://orcid.org/0000-0003-3701-8119https://orcid.org/0000-0003-3828-8153https://orcid.org/0000-0001-6732-6513https://orcid.org/0000-0002-5587-6953https://orcid.org/0000-0001-7007-6949https://orcid.org/0000-0003-2668-8328https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevResearch.6.L012051&domain=pdf&date_stamp=2024-03-06https://doi.org/10.1103/PhysRevResearch.6.L012051https://creativecommons.org/licenses/by/4.0/GIULIA ZHENG et al. PHYSICAL REVIEW RESEARCH 6, L012051 (2024)FIG. 1. (a) False-colored scanning electron microscopy (SEM)picture of the sample. The white scale bar at the bottom left indicates1 µm. MATBG (light green), contacts (orange), split gates (blue),and a channel gate (yellow) are shown. Inset: Schematic of thestack (thickness not to scale). (b) Characterization of the device withVcg = 0. Line cut at Vsg = 0 with annotations of filling factor ν = −1,−1/2, 0, 1/2, and 1 states. (c) Mapping of the resistance as a functionof Vsg and Vbg. Black dashed lines labeled ν ′ = −1, −1/2, 0, 1/2indicate filling factors in the split-gated area. The pink dashed linehighlights the transition from low to high resistance in the hole side.The inset shows the superconducting current-voltage characteristictaken at Vsg = 0 and Vbg = −7.44 V (indicated by a black star). Thered double-sided arrow is the gate voltage range in which Fig. 2(a) istaken.neutrality (ν = 0). At negative Vbg, we observe full-filling(ν = −1, where the Fermi level moves into the dispersiveband overcoming a small band gap) and half-filling (ν =−1/2) peaks for holes. At positive voltages we observe halffilling (ν = 1/2) for electrons, where the Fermi level resideswithin the flat bands. Superconductivity appears at fillingfactors slightly larger than 1/2. Filling ν = 1 for electronscannot be reached due to the voltage range of the back gatebeing limited by leakage currents. We extract the twist anglebetween the graphene layers of the device from the density ofthe ν = −1 peak, which leads to 1.1◦ (see Appendix A).To demonstrate the area-selective tunability of moiré fillingfactors, Fig. 1(c) shows R as a function of Vsg and Vbg. Fillingfactors ν ′ = −1, −1/2, 0, 1/2, and 1 below the split gates areindicated with black dashed lines. The inset in Fig. 1(c) showsa typical current-voltage (I-V ) characteristic in the supercon-ducting regime at Vbg = −7.44 V, Vsg = 0, and Vcg = 0. Here,a constant contact resistance of 10 k�, observed as the offsetresistance at zero-bias current in the superconducting regime,is subtracted from the raw data. In the remainder of this Letter,we consider transport to be superconducting when such an I-Vcharacteristic is observed.A question arising from the map shown in Fig. 1(c) isthe origin of the highly resistive area that spans out in theν ′ < −1 region (see the blue region delimited by the dashedpink line). In a standard back-gated MATBG without anytop gates [1,2], only a resistance peak can be observed asa function of Vbg and not a resistive area as a functionof Vbg and Vsg. In addition, at first glance, one would ex-pect the edge of this high resistive area (dashed pink line)to coincide with the predicted ν ′ = −1 (the labeled dashedblack line). The explanation for why this is not the case liesbelow.Tunable superconducting channel: Control via split gates.We now study the effect of the split gates on the supercurrent.We tune the leads to the superconducting regime by settingVbg = −7.44 V [cf. red arrow in Fig. 1(c)] and keep Vcg =0. Figure 2(a) shows the dV/dI characteristics numericallycalculated from the measured dc V -I data as a function ofthe current I and the split-gate voltage Vsg. We assign thestate of the carrier gas below the split gates as a functionof Vsg according to Fig. 1(c) and a capacitance model (seeAppendix B) as indicated in Fig. 2(a) by the colored bar abovethe color plot. When −6 V < Vsg < −2.8 V, the filling factorunder the split gate is smaller than −1, and thus the carrier gasin the region below the split gates is either a band insulator(BI) or the Fermi energy is in the dispersive band. When thevoltage is increased to Vsg > −2.8 V, the Fermi energy in theregion below the split gates is in the flat bands.In the range −6 V < Vsg < −5.5 V [before the pink dotin Fig. 2(a)], the device shows a large zero-bias resistanceof 0.48 M�. This can also be seen in the large slope of theI-V characteristic in Fig. 2(b) (black dashed line). Such ahigh resistance despite superconducting (SC) leads means thatnot only is the region below the split gates resistive, but theresistive region extends into the channel region between themdue to fringe fields.When the split-gate voltage is increased to −5.5 V < Vsg <−2.8 V [beyond pink dot in Fig. 2(a)], the device showslow resistance at low bias and a nonlinear I-V characteris-tic reminiscent of a superconductor as shown by the yellowcurve in Fig. 2(b). In this regime, the transport characteristicacross the sample remains superconducting even though thearea below the split gates is resistive. This indicates thata supercurrent flows through the channel while the split-gated region confines the flow. When Vsg is increased beyond−2.8 V, the nonlinear transport and the critical currents be-come more prominent [see green curve in Fig. 2(b)]. In thisregime, the supercurrent extends from the channel into thesplit-gated regions, eventually rendering the entire devicesuperconducting.L012051-2GATE-DEFINED SUPERCONDUCTING CHANNEL IN … PHYSICAL REVIEW RESEARCH 6, L012051 (2024)FIG. 2. Differential resistance (dV/dI) as a function of current I and gate voltages. (a) Split gate Vsg is swept while keeping Vcg = 0 V.(b) Voltage drop as a function of dc current and zoom-in picture of dV/dI at Vsg = −5.8 (pink), −4 (yellow), −1 (green) V. (c) Channel gateVcg is swept while keeping Vsg = −4.5 V. (d) I-V trace and zoom-in picture of dV/dI at Vcg = −2.8, −1, and 2 V. The states under the splitgate (dispersive band/BI or flat band/SC) are indicated by the bars above the top axes. In all cases, Vbg is kept at zero. Pink, yellow, and greendashed vertical lines show the gate voltage at which the line cuts are taken. Illustrations of the device’s state (pinch-off, SC channel, or SCbulk) are also shown. Red circle and star symbols show the onset of pinch-off.Tunable superconducting channel: Control via channelgate. In Figs. 2(c) and 2(d) we show that the conductanceof the superconducting channel can also be tuned with thechannel-gate voltage. It is only the channel that is tuned byVcg because the metallic split gate screens the electric field ofthe channel gate. We keep the leads superconducting (Vbg =−7.44 V) and tune the Fermi energy in the split-gated regionsinto the dispersive bands (Vsg = −4.5 V). This corresponds tothe condition in Fig. 2(a) indicated by the black triangle on theupper axis.Under these conditions, we see in Fig. 2(c) at Vcg = 0nonlinear transport indicating superconducting behavior con-sistent with Fig. 2(a). By reducing Vcg below −1.5 V, therebytuning the Fermi energy in the channel towards the BI orthe dispersive band, the device exhibits a high resistanceof 1.1 M� and a nonlinear I-V characteristic with a largeslope around zero bias as shown in Fig. 2(d) (black dashedline). This means that the superconducting channel is pinchedoff. Note that the light green line in Fig. 2(c) is an artifact.When Vcg is increased above 0 V, the superconductivity inthe channel is enhanced as illustrated by the green curve inFig. 2(d). The confinement of the superconducting current tothe channel and the local gate tunability of the channel fromthe superconducting to the highly resistive (pinch-off) state ofthe constriction are the central experimental findings of thisLetter.Vcg vs Vsg mapping of resistance. To further highlight thatthe pinch-off in Fig. 2(c) originates from the channel, we showthe resistance of the device as a function of Vsg and Vcg inFig. 3. Pink symbols indicate the gate voltages at which thechannel is pinched off as observed in the data of Figs. 2 and 5(see Appendix C). These values lie on a line with a negativeslope (see the black line in the figure). This means that thereexists a regime in which the channel conductance is controlledby both Vsg and Vcg. This is due to the fringe fields of thesplit gates, such that the channel conductance is controlledby all three gates. This additionally explains the origin of thediscrepancy between the expected ν ′ = 1 (the labeled dashedblack line) and the high resistance edge (dashed pink line)in Fig. 1(c). In between the two dashed lines, the resistanceis low because even if the split-gated region is resistive, asupercurrent can still flow through the channel. Only when theL012051-3GIULIA ZHENG et al. PHYSICAL REVIEW RESEARCH 6, L012051 (2024)FIG. 3. Mapping of resistance as a function of Vsg and Vcg ata constant back-gate voltage of Vbg = −7.44 V. Colored symbolsand black line interpolating between them indicate the thresholdgate voltages of the pinch-off. The gate-voltage regime where highresistances (R > 20 k�) are observed is annotated as “Pinch off.”channel is pinched off due to the effect of the fringing fieldsdo we measure a high resistance.Discussion. In narrow superconducting channels, it hasbeen predicted that the critical current becomes quantized asthe channel size is reduced when the BCS coherence lengthof the superconductor is longer than the channel length [12].A stepwise change of supercurrent, if not the predicted exactquantization, has been observed in various materials such asInAs [13–16], Ge/Si [17,18], and SrTiO3 [19]. In our device,we did not observe such quantization of the critical current. InMATBG, the BCS coherence length is estimated to be from afew tens of nm up to 100 nm [3,7]. The length of our channelis 400 nm, which is much longer than the estimated coherencelength. In order to observe the quantization of the criticalcurrent, it is necessary to reduce the channel length downto less than 100 nm, which is possible by advanced electronbeam lithography.Furthermore, we discuss the conditions at which the pinch-off occurs. In Figs. 2(a) and 2(c), the points at which thechannel is pinched off are (Vbg,Vsg,Vcg) = (−7.44,−5.3, 0)and (−7.44,−4.5,−1.5) V, respectively. Using the capaci-tance model (see Appendix B), we estimate the correspondingFIG. 4. Resistance as a function of Vsg and Vcg. at a constant back-gate voltage of Vbg = −7.44 V with an extended plot area.FIG. 5. dV/dI as a function of Vsg with fixed Vbg = −7.44 V andVcg = −4 V. Here, the pinch-off of the channel is observed at Vsg =−2.7 V (see the pink rotated triangular symbol). (b) shows dV/dI asa function of Vcg with fixed Vbg = −7.44 V and Vsg = −3.5 V.carrier densities nch in the channel to be −2.23 × 1012 cm−2and −2.56 × 1012 cm−2. For simplicity, we do not take fringefield effects into account in the model. By assuming anisotropic Fermi surface and fourfold degeneracy, the corre-sponding Fermi wavelengths λF = 2√π/|nch| are 23.7 and22.1 nm, respectively. Therefore, one can estimate the min-imal width of the channel that supports normal conductingtransport to be around this value (∼20 nm), which is signifi-cantly smaller than its lithographic value (150 nm). Moreover,in this narrow-channel limit, one can expect conductancequantization in the normal conducting regime. However, wedid not observe any signature of normal conductance quan-tization, presumably due to disorder and the relatively lowresistance of the insulating state in MATBG (∼1 M� inthis device) compared to gapped Bernal bilayer graphene(∼100 M� [20]). Nevertheless, the superconducting channelcan be formed as long as its resistance, which is ideally zero,is low enough compared to the resistance of the split-gatedarea.Conclusion. We have realized a gate-defined supercon-ducting channel in MATBG by implementing a device witha back gate and two layers of top gates (split gates and achannel gate). We observe a transition from superconduct-ing to highly resistive (up to ∼1 M�) transport through thechannel by tuning the split-gate and channel-gate voltages.L012051-4GATE-DEFINED SUPERCONDUCTING CHANNEL IN … PHYSICAL REVIEW RESEARCH 6, L012051 (2024)This shows that it is possible to define a narrow super-conducting channel by tuning the width of the confiningresistive areas. The threshold at which the pinch-off occursdepends not only on the channel gate but also on the split-gate voltage, indicating the essential role of the fringe fieldeffects. Our findings serve as a foundation for developingquantum constriction devices such as superconducting quan-tum point contacts, quantum dots, and Cooper-pair boxes inMATBG.Acknowledgments. We are grateful for fruitful discus-sions and technical support from P. Maerki, T. Baehler,R. Garreis, C. Tong, B. Kratochwil, W. Huang, and theETH FIRST cleanroom facility staff. We acknowledge fi-nancial support by the European Graphene Flagship Core3Project, H2020 European Research Council (ERC) SynergyGrant under Grant Agreement No. 951541, the EuropeanUnion’s Horizon 2020 research and innovation program underGrant Agreement No. 862660/QUANTUM E LEAPS, theEuropean Innovation Council under Grant Agreement No.101046231/FantastiCOF, and NCCR QSIT (Swiss NationalScience Foundation, Grant No. 51NF40-185902). K.W. andT.T. acknowledge support from the JSPS KAKENHI (GrantsNo. 21H05233 and No. 23H02052) and World Premier In-ternational Research Center Initiative (WPI), Ministry ofEducation, Culture, Sports, Science and Technology (MEXT),Japan. E.P. acknowledges support of a fellowship from “laCaixa” Foundation (ID 100010434) under fellowship codeLCF/BQ/EU19/11710062.Appendix A: Extraction of twist angle. We extract the twistangle of the sample using the relation θ = 2 arcsin( a2L ). Here,a is the lattice constant of graphene, and L is the moiré pe-riodicity, representing the distance between two AA-stackedregions. L is related to the area A of the moiré unit cell viaL = 2√2A/√3. A moiré unit cell can host four electronsdue to spin and valley degeneracy. Using this, the full fillingresistance peak appears at the density that corresponds to theoccupation of four electrons per moiré unit cell A = 4nν=1.Appendix B: Capacitance Model. To estimate the carrierdensity in the device, we estimate the capacitance per unitarea of the back gate, split gate, and channel gate to be Cbg =ε0εhBN/dbot, Csg = ε0εhBN/dtop, and Ccg = ε0εhBNεAlOx/(εhBNdtop + εAlOxdAlOx). Here, ε0 = 8.854 × 10−12 is thevacuum permittivity, εhBN = 3.3 and εAlOx = 9.5 are therelative permittivities of the hBN and the aluminumoxide, dtop and dbot are the thicknesses of the top andbottom hBN, and dAlOx is the thickness of the aluminumoxide layer. We then calculate the carrier density of thelead as nbg = CbgVbg/e, the region below the split gate asnsg = (CbgVbg + CsgVsg)/e, and the region below the channelgate as ncg = (CbgVbg + CcgVcg)/e, where e is the elementarycharge.Appendix C: Extended map of R(Vcg,Vsg). The same mapas Fig. 3 but with an extended range is shown in Fig. 4. Thethreshold for pinch-off extracted from the I-V characteristicmeasurements are plotted as symbols. Even though the datacannot be measured due to the leakage of the gate that hap-pened at the last stage of the entire experiment, one can clearlysee the linear relation between the threshold value of Vcg andVsg.Appendix D: Extended map for dV/dI (I,Vcg) anddV/dI (I,Vsg). We measure the behavior of the superconduct-ing channel in the same manner as in Fig. 2 but in two moreconfigurations as shown in Fig. 5. In Fig. 5(a), Vsg is sweptwith fixed Vbg = −7.44 V and Vcg = −4 V. Here, the pinch-off of the channel is observed at Vsg = −2.7 V (see the pinkrotated triangular symbol). In Fig. 5(b), Vcg is swept with fixedVbg = −7.44 V and Vsg = −3.5 V. Here, the pinch-off of thechannel is observed at Vcg = −2.88 V (see the pink rotatedtriangular symbol). These points are plotted in Figs. 3 and 4.[1] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E.Kaxiras, and P. Jarillo-Herrero, Unconventional superconduc-tivity in magic-angle graphene superlattices, Nature (London)556, 43 (2018).[2] M. Yankowitz, S. Chen, H. Polshyn, Y. Zhang, K. Watanabe,T. Taniguchi, D. Graf, A. F. Young, and C. R. Dean, Tuningsuperconductivity in twisted bilayer graphene, Science 363,1059 (2019).[3] F. K. de Vries, E. Portolés, G. Zheng, T. Taniguchi, K.Watanabe, T. Ihn, K. Ensslin, and P. Rickhaus, Gate-definedJosephson junctions in magic-angle twisted bilayer graphene,Nat. Nanotechnol. 16, 760 (2021).[4] D. Rodan-Legrain, Y. Cao, J. M. Park, S. C. de la Barrera, M. T.Randeria, K. Watanabe, T. Taniguchi, and P. Jarillo-Herrero,Highly tunable junctions and non-local Josephson effect inmagic-angle graphene tunneling devices, Nat. Nanotechnol. 16,769 (2021).[5] J. Díez-Mérida, A. Díez-Carlón, S. Y. Yang, Y.-M. Xie, X.-J. Gao, J. Senior, K. Watanabe, T. Taniguchi, X. Lu, A. P.Higginbotham, K. T. Law, and Dmitri K. Efetov, Symmetry-broken Josephson junctions and superconducting diodes inmagic-angle twisted bilayer graphene, Nat. Commun. 14, 2396(2023).[6] E. Portolés, S. Iwakiri, G. Zheng, P. Rickhaus, T. Taniguchi,K. Watanabe, T. Ihn, K. Ensslin, and F. K. de Vries, A tunablemonolithic squid in twisted bilayer graphene, Nat. Nanotechnol.17, 1159 (2022).[7] S. Iwakiri, A. Mestre-Torà, E. Portolés, M. Visscher, M. Perego,G. Zheng, T. Taniguchi, K. Watanabe, M. Sigrist, T. Ihn, and K.Ensslin, Tunable quantum interferometer for correlated moiréelectrons, Nat. Commun. 15, 390 (2024).[8] H. Overweg, H. Eggimann, X. Chen, S. Slizovskiy, M. Eich,R. Pisoni, Y. Lee, P. Rickhaus, K. Watanabe, T. Taniguchi,V. Fal’ko, T. Ihn, and K. Ensslin, Electrostatically inducedquantum point contacts in bilayer graphene, Nano Lett. 18, 553(2018).[9] S. Nakaharai, J. R. Williams, and C. M. Marcus, Gate-definedgraphene quantum point contact in the quantum Hall regime,Phys. Rev. Lett. 107, 036602 (2011).[10] T. Ohta, A. Bostwick, T. Seyller, K. Horn, and E. Rotenberg,Controlling the electronic structure of bilayer graphene, Science313, 951 (2006).L012051-5https://doi.org/10.1038/nature26160https://doi.org/10.1126/science.aav1910https://doi.org/10.1038/s41565-021-00896-2https://doi.org/10.1038/s41565-021-00894-4https://doi.org/10.1038/s41467-023-38005-7https://doi.org/10.1038/s41565-022-01222-0https://doi.org/10.1038/s41467-023-44671-4https://doi.org/10.1021/acs.nanolett.7b04666https://doi.org/10.1103/PhysRevLett.107.036602https://doi.org/10.1126/science.1130681GIULIA ZHENG et al. PHYSICAL REVIEW RESEARCH 6, L012051 (2024)[11] J. B. Oostinga, H. B. Heersche, X. Liu, A. F. Morpurgo, andL. M. K. Vandersypen, Gate-induced insulating state in bilayergraphene devices, Nat. Mater. 7, 151 (2008).[12] C. W. Beenakker and H. van Houten, Josephson current througha superconducting quantum point contact shorter than the coher-ence length, Phys. Rev. Lett. 66, 3056 (1991).[13] T. Bauch, E. Hürfeld, V. M. Krasnov, P. Delsing, H. Takayanagi,and T. Akazaki, Correlated quantization of supercurrent andconductance in a superconducting quantum point contact, Phys.Rev. B 71, 174502 (2005).[14] S. Abay, D. Persson, H. Nilsson, H Q Xu, M. Fogelström,V. Shumeiko, and P. Delsing, Quantized conductance and itscorrelation to the supercurrent in a nanowire connected to su-perconductors, Nano Lett. 13, 3614 (2013).[15] H. Irie, Y. Harada, H. Sugiyama, and T. Akazaki, Joseph-son coupling through one-dimensional ballistic channel insemiconductor-superconductor hybrid quantum point contacts,Phys. Rev. B 89, 165415 (2014).[16] H. Takayanagi, T. Akazaki, and J. Nitta, Observationof maximum supercurrent quantization in a superconduct-ing quantum point contact, Phys. Rev. Lett. 75, 3533(1995).[17] J. Xiang, A Vidan, M. Tinkham, R. M. Westervelt, and C. M.Lieber, Ge/Si nanowire mesoscopic Josephson junctions, Nat.Nanotechnol. 1, 208 (2006).[18] N. W. Hendrickx, M. L. V. Tagliaferri, M. Kouwenhoven, R. Li,D. P. Franke, A. Sammak, A. Brinkman, G. Scappucci, and M.Veldhorst, Ballistic supercurrent discretization and micrometer-long Josephson coupling in germanium, Phys. Rev. B 99,075435 (2019).[19] E. Mikheev, I. T. Rosen, and D. Goldhaber-Gordon, Quantizedcritical supercurrent in SrTiO3-based quantum point contacts,Sci. Adv. 7, eabi6520 (2021).[20] H. Overweg, Electrostatically induced nanostructures in bilayergraphene, Ph.D. thesis, ETH Zurich, 2018, https://doi.org/10.3929/ethz-b-000278503.L012051-6https://doi.org/10.1038/nmat2082https://doi.org/10.1103/PhysRevLett.66.3056https://doi.org/10.1103/PhysRevB.71.174502https://doi.org/10.1021/nl4014265https://doi.org/10.1103/PhysRevB.89.165415https://doi.org/10.1103/PhysRevLett.75.3533https://doi.org/10.1038/nnano.2006.140https://doi.org/10.1103/PhysRevB.99.075435https://doi.org/10.1126/sciadv.abi6520https://doi.org/10.3929/ethz-b-000278503