Abstract
Superconducting correlations may propagate between two superconductors separated by a tiny insulating or metallic barrier, allowing a dissipationless electric current to flow^{1,2}. In the presence of a magnetic field, the maximum supercurrent oscillates^{3} and each oscillation corresponding to the entry of one Josephson vortex into the barrier^{4}. Josephson vortices are conceptual blocks of advanced quantum devices such as coherent terahertz generators^{5} or qubits for quantum computing^{6}, in which ondemand generation and control is crucial. Here, we map superconducting correlations inside proximity Josephson junctions^{7} using scanning tunnelling microscopy. Unexpectedly, we find that such Josephson vortices have real cores, in which the proximity gap is locally suppressed and the normal state recovered. By following the Josephson vortex formation and evolution we demonstrate that they originate from quantum interference of Andreev quasiparticles^{8}, and that the phase portraits of the two superconducting quantum condensates at edges of the junction decide their generation, shape, spatial extent and arrangement. Our observation opens a pathway towards the generation and control of Josephson vortices by applying supercurrents through the superconducting leads of the junctions, that is, by purely electrical means without any need for a magnetic field, which is a crucial step towards highdensity onchip integration of superconducting quantum devices.
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In 1963 Rowell^{3} observed the oscillations of the critical current in superconducting junctions subject to a magnetic field. Since his pioneering discovery, the effect has been reported in numerous superconductor–normal metal–superconductor (SNS) weak links^{4,9}, including many examples in which the links were made of recently discovered materials such as graphene^{10} and topological insulators^{11,12}. This macroscopic quantum interference effect is commonly interpreted as a sequence of Josephson vortices^{4} penetrating the junction. By analogy with the Abrikosov vortices in type II superconductors^{13}, Josephson vortices were defined as regions with zero net circulating current and enclosing a magnetic flux quantum, Φ_{0} = h/2e. Several theoretical works, motivated by superconducting vortex pinning at grain boundaries^{14,15,16,17,18}, studied the largescale evolution of Abrikosov^{18} and Pearl^{19,20} vortices in the Sparts of zerowidth tunnel junctions. Yet, the spectral fingerprint of Josephson vortices and their spatial organization in the Nparts of extended Josephson junctions have so far remained undecided, and the subject of controversy. The common wisdom is that Josephson vortices lack a specific spectral signature and therefore cannot be identified. By contrast, recent microscopic calculations predicted that in diffusive SNS junctions Josephson vortices should be manifested as a spatial modulation of the proximity minigap in the quasiparticle excitation spectrum of the normal region of the junction^{7}. Thus, Josephson vortices could presumably have normal cores—regions where the proximity gap vanishes, making possible their detection and imaging by, for example, scanning tunnelling microscopy (STM), as is readily done for Abrikosov vortices in superconductors^{21}.
With this idea in mind, we created (see Methods) a lateral SNS network of superconducting Pb nanocrystals (yellow areas in Fig. 1a) linked together by an atomically thin Pb wetting layer (brown regions in Fig. 1a). Depending on the growth conditions, the Pb wetting layer is superconducting when it is crystalline^{16,22}, whereas it is metallic when its structure is amorphous^{23}. The Pb islands become superconducting below a critical temperature T_{c} ≈ 6.5 K (refs 24, 25, 26, 27, 28). The local tunnelling conductance spectra measured on top of the superconducting islands at T = 0.3 K ≪ T_{c} (see Methods) exhibit a superconducting gap Δ = 1.2 meV (Fig. 1d, curve S). Here, the wetting layer is nonsuperconducting, and the conductance spectra measured far from the islands show no superconducting gap, but rather a tiny dip centred at the Fermi level (Fig. 1d, curve WL). This dip is a fingerprint of the Altshuler–Aronov zerobias anomaly^{29} due to electron–electron interaction in this twodimensional diffusive metal^{23}. Very close to the islands the superconducting correlations induce a small proximity gap in tunnelling spectra of the wetting layer^{23} (Fig. 1d, curve P). In the zerobias conductance map in Fig. 1b the proximity gap appears as a bluish halo extending over a few tens of nanometres away from islands. In locations where the edges of neighbouring islands get very close, this proximity halo becomes reinforced and appears as a deeper blue. The phenomenon reflects the overlap of superconducting correlations induced by both islands^{30,31,32}, which enhance the proximity gap (Fig. 1d, curve J). This results in the formation of several SNS Josephson junctions, such as those seen in locations J1–J4 (Fig. 1b, c).
We now follow the evolution of the proximity links in an applied magnetic field. The spectroscopic map acquired at 60 mT shows the usual penetration of Abrikosov vortices in the large islands, whereas the small islands remain in the vortexfree Meissner state^{28} (Fig. 1c). By contrast, unusual features are revealed in proximity links inside the SNS junctions, and identified as Josephson vortices. Precisely, whereas at zero field the proximity gap is observed in all three locations A, B, C of the junction J1 (see Fig. 1b and zoom of J1 in Fig. 1e), at 60 mT no proximity gap is observed in B (red spectrum in Fig. 1f), but instead a normal state with its Altshuler–Aronov zerobias dip is recovered. In the neighbouring positions A and C, however, the proximity gap persists (Fig. 1e, f). This behaviour is confirmed by the direct inspection of the conductance map in Fig. 1f, where two clear Josephson links remain at 60 mT in locations A and C (low conductance in blue), whereas in B the proximity gap vanishes (high conductance in yellowred). The normal region in B surrounded by gapped areas is thus indeed a Josephson vortex core. The spectroscopic maps in Fig. 2a, b focus on junctions J2, J3 and J4, located in the bottom framed part of Fig. 1c. They show how Josephson vortex configurations change with magnetic field. Both J2 and J3 junctions accept one Josephson vortex at 120 mT (Fig. 2a) and two at 180 mT (Fig. 2b). A striking effect is observed inside the junction J2. At 120 mT the proximity gap is destroyed in its centre, where a Josephson vortex core settles (Fig. 2a). At 180 mT, however, this region returns to the proximity gapped state, thus separating two neighbouring Josephson vortex cores (Fig. 2b). By contrast, the very short and narrow junction J4 does not accept any Josephson vortex in its centre up to 180 mT. In all studied SNS junctions, Josephson vortex cores are in the normal state, as predicted^{7}.
The interpretation of our observation in terms of Josephson vortex cores is fully corroborated by the numerical simulations in Fig. 2c, d; the calculation method is described in Fig. 3. We first calculated the Abrikosov vortex configurations in the islands using the Ginzburg–Landau formalism (Supplementary Information 1). As an example, in Fig. 3a the Cooperpair density map calculated at 60 mT reproduces the experimentally observed Abrikosov vortex configuration (Fig. 1c). In Fig. 3b the corresponding phase portrait of the superconducting order parameter shows that the interplay between Meissner and vortex currents results in a spatially evolving phase inside islands. At this point, the physical origin of Josephson vortices can be understood by considering the gaugeinvariant local phase difference across the junction, where φ(r_{i}) are local phases of the order parameter at two island edges at positions r_{i} (on opposite sides of a given junction) and A is the vector potential (Supplementary Information 2). At locations in which φ^{∗} is 0, ±2π, ±4π, ±6π, …, the superconducting correlations induced by both islands are in phase, their constructive interference resulting in a welldeveloped proximity gap. In contrast, in locations where φ^{∗} is ±π, ±3π, ±5π… the superconducting correlations interfere destructively and the proximity gap is suppressed^{33}. This latter situation corresponds to Josephson vortex cores. Using φ^{∗} we find a current circulating around Josephson vortices. As a rough approximation, the Josephson current^{2} can be considered locally as simply proportional to sin(φ^{∗}). In junction J1, for instance, in locations between A and B, φ^{∗} varies continuously from 0 to π, sin(φ^{∗}) > 0 and some net Josephson current flows locally from one island edge to the other island. In the opposite case, from B to C the phase difference φ^{∗} varies continuously from π to 2π, sin(φ^{∗}) < 0, and the current flows in the opposite direction. Therefore, Josephson current circulates around the point B, thus justifying the term ‘Josephson vortex’. The above phase considerations served as a basis for numerical simulations. First, the superconducting phase portrait φ_{GL}(r) was generated using Ginzburg–Landau formalism for independent islands. This phase portrait is not unique: it varies depending on the choice of the gauge. Figure 3b shows one of such phase portraits obtained using the symmetric gauge A(r) = (1/2)B × r, where B is the magnetic field, with the origin in the image centre. Because islands are decoupled, the phase portrait is defined up to arbitrary constant phases α_{i} for each ith island (that is, one can freely use without affecting the superconducting state in the islands). Once the islands are proximitycoupled, the difference α_{i} − α_{j} between two island phases of the same junction is no longer arbitrary but respects two constraints: first, in the absence of an external circuit the total Josephson current crossing a junction is null; second, the free energy of the Josephson junctions is minimized. Starting from the phase portrait in Fig. 3b, we calculated, for each location inside the junctions, the strength of the superconducting correlations as the interference between evanescent waves having a gaugeindependent phase difference φ^{∗}(r). In these calculations we fulfilled the above constraints and fixed the island phases α_{i} selfconsistently, with no need for adjustable parameters (for more details see Supplementary Information 2). The result of our calculations is shown in Fig. 3c, ideally reproducing the experimentally observed position and extent of the Josephson vortex in the junction J1 (Fig. 1c). In the same manner the correlation maps in Fig. 2c, d were calculated, again matching the experimental findings (Fig. 2a, b). The success of our simulation clearly highlights that the Josephson vortices in our system originate from quantum interference.
Next, we calculated the spatial evolution of the density of states inside the Josephson vortex core, which cannot be obtained from correlation maps, by using the microscopic Usadel approach^{7,34} (Supplementary Information 3). Figure 4a shows the computed map of the local density of states at the Fermi energy for a junction similar to J1. The map demonstrates the suppression of the proximity gap inside Josephson vortex cores^{7}. This suppression is further detailed in Fig. 4b, where we show the local density of states as a function of energy at various distances from the centre of a vortex. Using these results we generated the corresponding tunnelling conductance spectra (Fig. 4c). In this calculation the Altshuler–Aronov zerobias effect was taken into account^{23}. As one can see, the Usadel approach reproduces qualitatively the observed vanishing of the proximity gap in Fig. 1e, f.
In the present experimental study, the gaugeinvariant phase difference that generates Josephson vortices was created with the help of an applied magnetic field. Equivalently, the gaugeinvariant phase differences may be generated by edge supercurrents circulating in Selectrodes. Indeed, in zero magnetic field, taking the gauge A = 0, we get a simple expression relating the current and phase, J ≈ J_{c}(ξ/π)∇ φ, where ξ is the effective coherence length, and J_{c} is the critical current in the electrode, ∇ φ ≤ (π/ξ). Therefore, if the superconducting electrodes of the junction carry oppositely directed currents along the electrode edge, a gaugeinvariant phase difference appears across the junction. It evolves with the lateral position r inside the junction as φ^{∗}(r) ≈ 2π(J/J_{c})(r/ξ), leading to the formation of a Josephson vortex. To confirm this idea, we carried out calculations of the local density of states in SNS junctions where currents flow inside Sleads along the junction edges with no applied magnetic field (Supplementary Information 3). We show in Fig. 4d a typical result for a junction similar to J1, where we have assumed that opposite currents circulate along both interface edges. As one can see, there appear Josephson vortices with normal cores, very similar to those induced magnetically (see Fig. 4a). In this case, the density and size of generated Josephson vortices is simply proportional to the intensity of circulating edge currents. Moreover, vortex generation may also be achieved if only one superconducting lead carries a supercurrent, as we demonstrate in Fig. 4e. By tuning intensities of supercurrents in leads one may pin vortices at one or other edge. In a SNS device, such as the one sketched in Fig. 4f, it should be possible to create Josephson vortices by simply applying currents through superconducting leads. Such a method would open new pathways for generation and control of quantum objects by purely electrical means, without the need of any externally applied magnetic field.
Methods
The 7 × 7 reconstructed nSi(111) (n ≈ 10^{19} cm^{−3}) was prepared by direct current heating to 1,200 °C, followed by annealing procedure between 900 and 500 °C. Subsequently, a few atomic layers of Pb were evaporated on the Si(111)7 × 7 kept at room temperature, using an electron beam evaporator calibrated with a quartz microbalance. The resulting flattop (111)oriented single nanocrystals of Pb are interconnected via a disordered atomic wetting layer of Pb (refs 23, 24, 25, 26, 27, 28). During sample preparation the pressure did not exceed P = 3 × 10^{−10} mbar. The sample structure was controlled in both real and reciprocal space by scanning tunnelling microscopy and lowenergy electron diffraction.
The scanning tunnelling spectroscopy measurements were performed in situ with a homemade apparatus, at a base temperature of 320 mK and in a ultrahigh vacuum with P < 3 × 10^{−11} mbar; the electron temperature was estimated to be 390 mK. Mechanically sharpened Pt/Ir tips were used. The bias voltage was applied to the sample with respect to the tip. Typical setpoint parameters for spectroscopy are 150 pA at V = −5 mV. The tunnelling conductance curves dI(V)/dV were numerically derived from raw I(V) experimental data. Each conductance map is extracted from a set of data consisting of spectroscopic I(V) curves measured at each point of a 512 × 512 grid, acquired simultaneously with the topographic image. The magnetic field was applied perpendicular to the sample surface.
The frameworks for the theoretical simulations in this paper are the phenomenological Ginzburg–Landau (GL) theory^{4} and the microscopic Usadel approach^{7,34}. The GL simulations were implemented on a Cartesian grid for the exact geometry of the islands and expected electron meanfree path in the samples (Supplementary Information 1), using STM mapping from the experiment, with a grid spacing of 1 nm. The equations were solved selfconsistently in three dimensions and contain higherorder derivatives; for this demanding computation we have used GPU parallel computing. The description of the local density of states and tunnelling spectra of the SNS junctions was carried out using the Usadel approach, as explained in detail in the Supplementary Information 3.
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Acknowledgements
T.C., C.B., F.D., V.S. and D.R. acknowledge financial support from the French ANR project ELECTROVORTEX and the FrenchRussian program PICSCNRS/RAS. The authors also thank V. Cherkez for assistance during experiments and V. Vinokur (Argonne National Laboratory, Illinois USA) and A. Buzdin (University of Bordeaux 1, France) for stimulating discussions. J.C.C. acknowledges financial support from the Spanish MICINN (Contract No. FIS201128851C1). V.H.L.B. acknowledges support from CNPq Brazil and productive discussions with Prof. A. Chaves (UFC, Brazil). M.V.M. acknowledges support from Research Foundation Flanders (FWOVlaanderen) and CAPES Brazil (PVE project BEX1392/115).
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The experiments were conceived by D.R., T.C., C.B. and F.D. The experiments were performed by C.B., D.R., L.SG., T.C., V.S. and F.D. Theoretical support was provided by J.C.C., V.H.L.B. and M.V.M. Superconducting correlation maps were calculated by T.C. The manuscript was written by D.R., J.C.C. and T.C. with comments and input from all authors.
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Roditchev, D., Brun, C., SerrierGarcia, L. et al. Direct observation of Josephson vortex cores. Nature Phys 11, 332–337 (2015). https://doi.org/10.1038/nphys3240
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DOI: https://doi.org/10.1038/nphys3240
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