Main

The superconducting proximity effect relies on penetration of Cooper pairs from a superconductor (S) into a normal metal (N) and is most pronounced in systems with transparent SN interfaces and weak scattering, where superconducting correlations penetrate deep inside the normal metal. Despite being one atom thick and having a low density of states, which vanishes at the Dirac point, graphene (G) can exhibit low contact resistance and ballistic transport on a micrometre scale19,20 exceeding the distance between the superconducting leads by an order of magnitude. These properties combined with the possibility to electrostatically control the carrier density n offer tunable Josephson junctions in a regime that can be referred to as ballistic proximity superconductivity21. Despite intense interest in SGS devices3,4,5,6,7,8,9,10,11,12,13,14,15,16 that can show features qualitatively different from the conventional SNS behaviour2,3, ballistic graphene Josephson junctions15,16 remain little studied.

Our SGS devices are schematically shown in Fig. 1 and described in further detail in Supplementary Section 1. The essential technological difference from the previously studied SGS junctions4,5,6,7,8,9,10,11,12,13,14 is the use of graphene encapsulated between boron nitride crystals19,20 as well as a new nanostrip geometry of the contacts. This allows high carrier mobility, low charge inhomogeneity and low contact resistance. More than twenty SGS junctions with a width W between 3 and 8 μm and a length L between 0.15 and 2.5 μm were studied, all exhibiting a finite supercurrent at low temperatures (T), reproducible behaviour and consistent changes with L and W. First, we characterize the devices above the transition temperature TC ≈ 7 K of our superconducting contacts. Figure 1b shows examples of the normal-state resistance Rn as a function of back gate voltage Vg that changes n in graphene. The neutrality point (NP) was found to be shifted to negative Vg by a few volts, with the shift being consistently larger for shorter devices (Supplementary Information). This is due to electron doping induced by our Nb contacts. For ballistic graphene such doping is practically uniform away from the metal interface22. The observed smearing of Rn(Vg) curves near the NP allows an estimate for charge inhomogeneity in the graphene bulk of ≈2 × 1010 cm−2. For consistency, data for devices with different L are presented as a function of ΔVg, the gate voltage counted from the NP.

Figure 1: Ballistic SGS junctions.
figure 1

a, Top: Junctions schematic. Bottom: Electron micrograph of a set of four junctions with different L. The few nm-wide graphene ledge (top drawing) is referred to as a nanostrip contact. b, Typical behaviour for SGS junctions with different L but for the same set of junctions with W = 5 μm. To avoid an obscuring overlap between four oscillating curves, we plot Rn at negative ΔVg only for the two shortest junctions. For positive ΔVg > 5 V, the four curves overlap within the line width. The dashed curve shows calculated RQ(n). Inset: changes in the differential conductance dI/dV; L = 0.25 μm. Colour scale, −1 to 1 mS (red to blue).

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For positive ΔVg (electron doping) and n > 1011 cm−2, SGS junctions made from the same graphene crystal and having the same W exhibit the same Rn(ΔVg) dependence, independently of L (Fig. 1b). This shows that the mean free path is larger than the contact separation and yields carrier mobility >300,000 cm2 V−1 s−1, in agreement with the quality measured for similarly made Hall bar devices. The dashed curve in Fig. 1b indicates the behaviour expected in the quantum ballistic limit RQ = (h/e2)/4N where N = int(2W/λF) is the number of propagating electron modes, λF the Fermi wavelength that depends on n(ΔVg) and the factor 4 corresponds to graphene’s degeneracy. The difference between RQ and the experimental curves yields a record low contact resistivity, ≈35 Ω μm. This value corresponds to an angle-averaged transmission probability Tr ≈ 0.8 (Supplementary Section 2).

For hole doping, Rn becomes significantly higher indicating smaller Tr. This is because p–n junctions appear at the Nb contacts and lead to partial reflection of electron waves, which effectively creates a Fabry–Pérot (FP) cavity5,23. The standing waves lead to pronounced oscillations in Rn as a function of both Vg and applied bias Vb (Fig. 1b). The oscillatory behaviour indicates that charge carriers can cross the graphene strip several times, preserving their monochromaticity and coherence. Some FP oscillations could also be discerned for positive ΔVg but they were much weaker because of higher Tr. The observed FP behaviour in the normal state agrees with earlier reports5,23. Its details can be modelled accurately if we take into account that the position of p–n junctions varies with Vg so that the effective length of the FP interferometer becomes notably shorter than L at low hole doping (Supplementary Section 3).

After characterizing our SGS devices at T > TC, we turn to their superconducting behaviour. All of the junctions (including L = 2.5 μm) exhibited the fully developed proximity effect. Figure 2a, b show that the critical current Ic remained finite at the NP and rapidly increased with |ΔVg|, reaching densities >5 μA μm−1 for high electron doping and short L, notably larger than values of Ic previously reported4,5,6,7,8,9,10,11,12,13,14,15,16. Such high Ic values are due to ballistic transport and low contact resistance. Indeed, Ic can theoretically reach a value2,24

where Δ is the superconducting gap and α ≈ 2.1. Because in our devices Rn ≈ RQ, the equation implies that we approach the quantum limit Ic ≈ (eΔ/h)4N where the supercurrent is determined by the number of propagating electronic modes that transfer Cooper pairs between superconducting contacts, and each of the modes has the supercurrent carrying capacity3,17IQ ≈ eΔ/h.

Figure 2: Quantum oscillations in the supercurrent.
figure 2

a, Examples of I–V characteristics for ballistic SGS junctions in the superconducting state. The data are for the device in Fig. 1 with L = 0.25 μm. The arrows explain notions Ic and Ie. b, Absolute voltage drop |Vb| across the SGS junction in (a) for a wide range of doping. The black region corresponds to the zero-resistance state, and its edge exhibits clear FP oscillations. c, IcRn and IeRn for a device with L = 0.3 μm, W = 6.5 μm and Δ ≈ 1.0 meV estimated from its TC. Each data point is extracted from a trace such as in (a). Inset: Oscillatory part of IcRn is magnified. Similar behaviour was observed for other devices. d, Effect of the junction length on supercurrent for 12 devices with different W. Red symbols, W = 3 μm; blue, 5 μm; green, 6.5 μm. For each data set, Ic follows the same dependence as IcRn because Rn were practically independent of L for the same W. For the two longest devices in (d), the critical current falls below the plotted 1/L dependence, probably because of thermal fluctuations (Supplementary Section 4).

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Equation (1) suggests that IcRn should be a constant. This holds well in our SGS devices away from the NP (Fig. 2c) and indicates that at low T, external noise, fluctuations and other mechanisms4,5,6,7,8,9,10,11,12,13,14,15,16 that are dependent on n or Rn do not limit Ic. However, Fig. 2c yields α that is notably smaller than the expected constant in equation (1). For hole doping, this can be attributed to the presence of p–n junctions at the superconducting interfaces but even for electron doping and high Tr we find α ≈ 0.4 (Fig. 2c). Furthermore, we measured the excess current Ie > Ic as shown in Fig. 2a and found that, in the case of Ie, α also does not reach a value close to 2.1 (Fig. 2c). This corresponds to the fact that all our devices were in the limit of long L > Δ/hvF (vF is the Fermi velocity), as also follows from the 1/L dependence found for IcRn (Fig. 2d). In this long-junction regime, the critical current is given by Ic ≈ ETh/eRn being determined by the Thouless energy ETh rather than the superconducting gap2,17,24. For a ballistic system, ETh depends on the time that the charge carriers spent inside the FP cavity and can be estimated25 as ∼hvF/L. This yields Ic ∝ 1/L and IcRn ∝ 1/L because Rn is independent of the ballistic device’s length. Our detailed studies of Ic as a function of T and L show that all data for IcRn/ETh collapse on a universal curve f(T/ETh) with ETh ∼ hvF/L, which again agrees well with expectations for the long-junction limit (Supplementary Section 4). We estimate that to reach the transition regime ETh/Δ ∼ 1 for our SGS junctions would require L < 100 nm. Let us also mention that no definitive signs of specular Andreev reflection3,17 were found in our devices (Supplementary Section 4).

As a consequence of FP resonances in the normal state (Fig. 1), the supercurrent also exhibits quantum oscillations that are clearly seen in Fig. 2b for hole doping. Equation (1) implies that such oscillations in Ic should occur simply because Rn oscillates. Indeed, Rn and Ic are found to oscillate in antiphase, compensating each other in the final products Ic, eRn. However, we find that oscillations in the critical current are approximately 3 times stronger than those in Rn. This observation is consistent with the fact that Ic is not only inversely proportional to Rn but also depends on the Thouless energy as discussed above, whereas the latter is expected to oscillate because of the oscillating transparency of FP resonators (Supplementary Section 4). We note in passing that Rn and its FP oscillations exhibit little temperature dependence below 20 K, which justifies the use of Rn measured above TC in the above analysis of the superconducting behaviour.

In a magnetic field B, our ballistic junctions exhibit further striking departures from the conventional behaviour (Fig. 3). In small B such that a few flux quanta φ0 = h/2e enter an SGS junction, we observe the standard Fraunhofer dependence2

where ϕ = L × W × B is the flux through the junction area. Marked deviations from equation (2) occur in B > 5 mT (Fig. 3a). Figure 3b–e shows that, in this regime, the supercurrent no longer follows the oscillatory Fraunhofer pattern but pockets of proximity superconductivity can randomly appear as a function of n and B. At low doping, the pockets can be separated by extended regions of the normal state where no supercurrent could be detected with accuracy of a few nA ≪ IQ (Fig. 3c, e). Within each pocket, I–V characteristics exhibit a gapped behaviour (inset of Fig. 3d) with Ic ∼ IQ ≈ 40 nA, although the exact value depends on doping and Ic decreases to ≈10 nA close to the NP, possibly due to the rising contributions of electrical noise and thermal fluctuations that suppress apparent Ic (Fig. 3c). These proximity states persist up to B ≈ 1 T (ϕ/φ0 ∼ 103) and are highly reproducible, although occasional flux jumps in the Nb contacts can reset the proximity pattern (Supplementary Section 5). The correlation analysis presented in Supplementary Section 6 indicates that to suppress such superconducting states, changes in ϕ of ≈ φ0 and changes in the Fermi energy of ≈1 meV are required.

Figure 3: Fluctuating proximity superconductivity.
figure 3

a, Example of dV/dI as a function of applied current I and B. The purple regions correspond to the zero-resistance state and their edges mark Ic (see Supplementary Section 10). The map is symmetric in both I and B. The white curve is given by equation (2). The low-B periodicity is ≈0.4 mT, smaller than expected from the device’s area, which is attributed to the Meissner screening that focuses the field into the junction30. b, Continuation of the map from (a) above 0.1 T. Intervals with finite Ic continue to randomly appear, despite the Fraunhofer curve being indistinguishable from zero. c, Another high-B example but as a function of ΔVg in 0.5 T. d, Examples of low-current resistance (I = 2 nA) in different B. The dashed curve for 0.5 T shows that current I = 150 nA > IQ completely suppresses superconductivity. The arrows mark the expected onset of edge state transport. Inset: Typical I–V characteristics for high-B superconducting states. e, Local map of fluctuating superconductivity. T ≈ 10 mK; all colour scales are as in c. f–i, Electron–hole paths responsible for Andreev states in ballistic junctions in zero (f), intermediate (g,h) and high B (i). In (h), the cyclotron bending suppresses the transfer of Cooper pairs in the middle of the graphene strip but Andreev states can persist near the edges.

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The semiclassical description2,24,25,26,27 of the superconducting proximity relates the Cooper pair transfer between the leads to electrons and Andreev-reflected holes, which travel along same trajectories but in opposite directions (Fig. 3f). In low B, interference between many Andreev states traversing the graphene strip results in the Fraunhofer-type oscillatory suppression of Ic described by equation (2) (see Fig. 3a). Although not reported before, the Fraunhofer pattern in ballistic devices can be expected to break down in relatively small B because the cyclotron motion deflects electrons and holes in opposite directions so that they can no longer retrace each other (Fig. 3g). We have estimated the field required to suppress Andreev states in the bulk as B∗ ∼ Δ/eLvF (Supplementary Section 7). For the devices in Fig. 3, this yields B∗ ≈ 5 mT, in agreement with the field where strong deviations from the Fraunhofer curve are observed.

As for the random pockets of superconductivity at B ≫ B∗ that exhibit Ic much higher than that expected from equation (2), we invoke the previously noticed analogy18 between mesoscopic fluctuations in the normal-state conductance28, 〈δG2〉 and in the supercurrent17,18, 〈δIc2〉. Both types of fluctuations are due to interference of electron waves propagating along different paths but start and finish together. In contrast to the case of B = 0, for which semiclassical phases of counter-propagating electrons and holes near the Fermi level cancel each other because of time-reversal symmetry (Fig. 3f), electrons and holes propagating along non-retracing trajectories in a finite B acquire large and random phase differences (Fig. 3g, h). Averaging over all imaginable geometrical paths would lead to the complete suppression of the supercurrent18. However, for each given realization of either diffusive or chaotic ballistic SNS junctions, the characteristic values of fluctuations are set17,18,28,29,30 at 〈δG2〉1/2 ∼ e2/h and 〈δIc2〉1/2 ∼ eΔ/h. In the case of B ≫ B∗, non-retracing paths that can transfer Cooper pairs between superconducting contacts can occur only near graphene edges (see Fig. 3h and Supplementary Section 7). In a way, a combination of cyclotron motion and edge scattering provides a chaotic ballistic billiard near each graphene edge, and this leads to random pockets of superconductivity with Ic = 〈δIc2〉1/2 ∼ IQ. Moreover, the analogy with chaotic billiards allows us to estimate the change in the Fermi energy, which is needed to change the realization of the mesoscopic system and, therefore, suppress an existing pocket of superconductivity. The required change is again given by the Thouless energy ETh ∼ hvF/Λ where Λ is the typical length of Andreev paths in a strong magnetic field (Fig. 3h). At high B, we estimate Λ as ≈ (rcL)1/2 where rc is the cyclotron radius. This yields ETh ≤ 1 meV, in agreement with the observed changes in doping that are required to suppress the pockets of superconducting proximity (Supplementary Section 6). An interference pattern in mesoscopic systems is also known18,28,29,30 to change upon changing the flux ϕ through the system by ≈φ0. This scale agrees well with that observed experimentally (Supplementary Figure 8)

Finally, the discussed mesoscopic proximity effect can be expected to disappear if rc becomes shorter than L/2 (Fig. 3i). This condition is marked in Fig. 3d and seen more clearly in the data of Supplementary Section 8. It is also worth noting that the near-edge superconductivity was not observed for hole doping, which we attribute to the fact that Klein tunnelling in graphene collimates trajectories perpendicular to the p–n interface23, making it essentially impossible to form the closed-loop Andreev states shown in Fig. 3h (Supplementary Section 7). In principle, the effect of near-edge Andreev states could be further enhanced by the presence of extended electronic states at graphene edges16 but, based on our experimental data, no evidence for this or other spatial inhomogeneity was found in the studied samples (Supplementary Section 9).

Methods

The measurements were carried out in a helium-3 cryostat for T down to 0.3 K and in a dilution refrigerator for lower T. All electrical connections to the sample passed through cold RC filters (Aivon Therma) and additional a.c. filters were on the top of the cryostats. The differential resistance was measured in the quasi-four-terminal geometry (using 4 superconducting leads to an SGS junction) and in the current-driven configuration using an Aivon preamplifier and a lock-in amplifier. To probe the superconducting proximity, we used an excitation current of 2 nA.