Main

The evolution of magnetic properties of an infinite superconductor when crossing κc is shown in Fig. 1. Type I superconductors with κ < κc expel magnetic field H until it reaches a critical field Hc beyond which superconductivity is destroyed (Fig. 1b, e). In type II superconductors with κ > κc, superconductivity extends into a wider region, Hc1 < H < Hc2, where magnetic field penetrates the sample in the form of Abrikosov vortices, tiny filaments of the normal phase surrounded by encircling supercurrents (Fig. 1a, d), each carrying a quantum magnetic flux Φ0 = πℏ/ce.

Figure 1: Superconducting phase diagrams and corresponding magnetization curves.
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a–c, Phase diagrams for type II superconductor (a), type I superconductor (b) and type I finite-size superconductor (c). d–f, The corresponding magnetization curves M(H) for type II superconductor (d), type I superconductor (e) and type I finite-size superconductor (f). Note that Hc3 > Hc holds only as long as κ > 0.42.

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Finite-size systems acquire new features enriching their phase diagram. Most importantly, type I superconductors fall into an intermediate state, comprising alternating domains of normal and superconducting phases with the period for H ≃ 0.5Hc (ref. 10), where ξ is the coherence length and D is the sample thickness. The intermediate state forms in the interval (1 − n)Hc < H < Hc (n < 1 is the shape-dependent demagnetization factor) triggered by the local magnetic field near the edges of the sample exceeding the critical value Hc and locally destroying superconductivity (Fig. 1c, f). In type II superconductors, nucleation of superconductivity occurs first near the sample boundary at a specific surface critical field Hc3 > Hc2. In type I superconductors Hc3 can exceed Hc if κ ≲ κc, as shown in Fig. 1b, e.

Near κ-induced criticality, with domains containing only a few flux quanta, the intermediate state is unstable towards breaking into an Abrikosov lattice and transient effects become important. To analyse transient behaviour, we consider a sample with κ ≲ κc containing a single domain or droplet of the normal phase, that is, a sample with the lateral size L comparable to the period d of the domain structure. This droplet is nothing but a giant vortex with a normal core comprising several flux quanta11. Its critical fission occurs by splitting an N-quanta droplet (Nq-droplet) into a (N − 1)q-droplet and a single 1q vortex moving away (see Fig. 2). To calculate the energy difference between the Nq-droplet and the configuration consisting of the residual (N − 1)q-droplet and the separated vortex, we construct a perturbation theory in the vicinity of the Bogomolny point over the small parameter γ = κ2 − κc2 (Supplementary Information) and identify three contributions to the interaction energy (see Fig. 2c):

where l is the distance between the vortex and the droplet. The intrinsic interaction energy of the (N − 1)q-droplet with the separated vortex calculated in ref. 12 (see also Supplementary Information and ref. 13) is

where λ is the London electromagnetic screening length. This term yields attraction at γ < 0 as expected. Magnetostatic repulsion energy due to stray fields generated by vortices near the sample surface is

Finally, the confinement energy due to interaction of the external field H with the vortex, holding the droplet together is

As follows from equation (4), decreasing the field reduces the confinement strengths. At some threshold field, the repulsive forces begin to dominate and a single vortex splits from the droplet. On further decreasing the field, individual vortices sequentially detach from the droplet and escape from the sample. This disintegration mechanism is analogous to the instability introduced by Lord Rayleigh5 in 1882 leading to fragmentation of charged liquid droplets due to the competition between long-range Coulomb repulsion forces and a short-range molecular attraction.

Figure 2: Droplet fission.
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a–c, Distribution of the magnetic flux in the (a) Meissner state, (b) vortex droplet state showing the calculated stray fields and (c) sketch of the vortex interaction forces (Fe, Fc, and Fi) corresponding to the interaction energies (Ue, Uc and Ui) between the droplet and a separated single vortex. d, Vortex droplet before start of the fission. e, Initial stage of separation of a single vortex from the droplet. f, Complete separation of the vortex from the droplet.

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The threshold field Hinst(N) at which the Nq-droplet becomes unstable is determined from the instability point when the energy U(l) changes its curvature and transforms from a convex function to a concave one, and equations (1)–(4) yield

Direct disintegration of an Nq-droplet into N single vortices requires surmounting a higher confinement energy barrier than one-by-one vortex decoupling.

The Rayleigh instability can be observed if the field Hinst(N) falls into the region of the existence of the vortex droplet. On the descending field branch, the vortex droplet appears as a residual of the normal state in the finite sample below the surface critical field Hc3 = 2.39κHc. Alternatively, on the ascending branch, the droplet can form as a result of the field penetration in a Meissner state. The threshold is defined by the condition that the external field at the sample edges, H/(1 − n), exceeds the field of first penetration into an infinite sample Hp ≃ (1/21/4κ1/2)(1 + 5.44κ/1 + 4.78κHc) (ref. 14), which gives the superheating field Hsh = (1 − n)Hp for the lower bound of field penetration into a finite superconducting sample.

Criticality can be tuned by temperature variation of κ(T) = λ(T)/ξ(T). In a Pb superconductor κ(T) changes from κ(0) ≍ 0.68 at T = 0, which is slightly less than κc, to κ ≃ 0.38 at T = Tc(7.2 K) and is well described by the phenomenological formula κ(T) ≃ κ(0)/(1 + T2/Tc2) (see ref. 15 and Fig. 3b). Therefore, micrometre-sized samples of Pb, an exemplary type I superconductor, offer a natural laboratory to study vortex droplet fission. We selected a triangular-shaped Pb meso-crystal with lateral side dimensions of ~2.2 μm, thickness of ~0.7 μm, and critical temperature Tc = 7.18 K shown in the lower inset of Fig. 3a. The measurements on the crystal were done using a two-dimensional (2D) electron gas ballistic Hall micro-probe array magnetometer16,17 (see Fig. 3a lower inset). The temperature variation of κ gives rise to the phase diagram of Pb shown in Fig. 3c. The temperature dependence for Hc is standard, Hc(T) = Hc(0)[1 − (T/Tc)2] with μ0Hc(0) ≍ 78 mT (ref. 18). The critical fields Hsh and Hc3 are expressed through Hc as given above with the best-fit value n = 0.37. The curves for Hc, Hsh and Hc3 cross pair-wise near approximately Tx ≃ 6.3 K. The dotted lines show the instability field Hinst(N) for various N calculated from equation (5). We further focus on the temperature region Tx < T < Tc, which is the most favourable for the experimental observation of Rayleigh instability of the vortex droplet. At T < Tx the lines Hinst(N) for large N fall out from the range of existence of superconductivity, implying that there the droplet may become unstable with respect to splitting into single vortices. Our 3D numerical simulations, done using the phenomenologically adapted Ginzburg–Landau theory to account for the correct temperature dependence of κ and Hc (ref. 19), show the intermediate regime with a mixture of droplet and one-quanta vortices (see Fig. 4). Note that in the temperature range Tx < T < Tc where Hc3 < Hc < Hsh, the droplet can form only in the descending field regime, because in the ascending field the sample remains in the Meissner state until the field reaches Hsh at which superconductivity vanishes.

Figure 3: Experimental phase diagrams.
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a, Experimental phase diagram of the meso-Pb crystal. The top inset shows the magnetization curve M(H) at T = 0.36 K and demonstrates how Hsh, Hc3 and Hc were obtained experimentally. The black lines are guides to the eye. Top inset: M(H) at T = 0.36 K demonstrating how Hsh, Hc3 and Hc were obtained experimentally. Bottom inset: experimental set-up of the crystal and the Hall probe array. b, The κ(T) dependence of Pb (adopted from ref. 15). The inset shows M(H), as in the top inset in a, but at T = 2.0 K. c, Theoretical temperature dependencies of Hc, Hc3 and Hsh. Numbered dotted lines show the calculated locus of instabilities with respect to N → N − 1 vortex droplet fission. Red dots show the corresponding instabilities at T = 6.7 K and at T = 7.0 K, associated with the data in Fig. 5. d, Expanded interval Tx < T < Tc. The insets in b and d show the experimental M(H) dependencies in the diluted vortex gas regime at T = 2 K and in the metastable vortex droplet fission regime at T = 6.7 K, respectively.

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Figure 4: Exemplary vortex configurations in the critical region.
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Evolution of the ground-state vortex structures on decreasing magnetic field in a mesoscale triangular superconducting Pb prism at κ ~ κc in the extended critical region where the vortex droplet can coexist with separate vortices. This particular simulation is performed for T = 5.5 K, and parameters of the Pb sample Tc = 7.2 K, ξ(0) = 66 nm and λ(0) = 45 nm, with two-fluid model temperature dependencies19.

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The temperature dependencies of Hc, Hsh and Hc3 shown in Fig. 3c are in a good agreement with those of Fig. 3a obtained experimentally. The data were extracted from field-dependent magnetization curves as shown in the upper inset of Fig. 3a. From Hsh and Hc one obtains the temperature dependence of the Ginzburg–Landau parameter κ(T) through Hp. The temperature dependence of the Ginzburg–Landau parameter κ(T) = λ(T)/ξ(T) corresponds to the bulk behaviour (shown in Fig. 3b). Together with Hc(T) this gives a penetration λ(T) ≍ λ(0)/[1 − (T/Tc)2] with λ(0) ≍ 41 nm and a zero temperature coherence length ξ(0) ≍ 66 nm. To see the droplet fission, we use the individual vortex observation technique, analogous to that used in ref. 16 for observation of entrance and exit of individual vortices in small type II superconductors. The M(H) dependencies at T = 2 K and at T = 6.7 K shown in the insets of Fig. 3b and Fig. 3d have a different character, the difference stemming from the temperature dependence of κ(T).

At T = 2 K, where κ ≃ 0.6–0.7 and is slightly less than κc, the lower inset of Fig. 3b delineates the mixed-state-like behaviour of M(H) in which the individual vortices are stabilized by the repulsion due to the stray field. First, on increasing the applied field to Hsh ≃ 51 mT from the zero-field-cooled state, the absolute value of the magnetization grows proportionally to H owing to Meissner screening. Beyond Hsh, the magnetic flux starts to penetrate the sample and the magnetization decreases smoothly. An extrapolation of the linear drop of the absolute value of M(H) to zero agrees with the bulk value Hc ≃ 71 mT, but the diamagnetic signature of superconductivity disappears only at H = Hc3 ≃ 96 mT (Fig. 3a, b). At the reversing branch, the onset of the transition is observed at H ≍ Hc3 but the magnetization remains close to zero, as long as the magnetic flux can freely leave the sample. On further change of H, the magnetization becomes modulated by a saw-like structure, which reflects the effect of pinning that traps vortices within the sample. The drops in M(H) dependence correspond to the one-by-one escape of vortices from the sample, similar to what is observed in refs 16, 20. On switching the sign of the field, vortices leave the sample, which finally falls into the Meissner state and the process repeats itself cyclically.

The full M(H) curve at T = 6.7 K, shown in the inset of Fig. 3d, is exemplary for the single droplet regime at Tx < T < Tc where Hc3, Hc < Hsh are close to each other and where by tuning the field we can control the vortex droplet fission. An expanded view of one quadrant of the data is shown in Fig. 5 for T = 6.7 K and 7.0 K. On the ascending field at T = 6.7 K, the Meissner state is maintained up to Hsh. At H = Hsh the magnetization abruptly drops to zero. Moving from high field along the descending branch, one sees that superconductivity emerges at Hc3(T), but the system falls into a vortex droplet state.

Figure 5: Magnetization curves showing vortex droplet fission.
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a, b, Experimental M(H) curves at 6.7 K (a) and 7.0 K (b). The applied magnetic field is normalized by Hc(T) = Hc(0)[1 − (T/Tc)2], with Hc(0) = 78 mT and Tc = 7.18 K. a.u., arbitrary units. On increasing field (red curves) the Meissner state (N = 0) survives up to Hc ≃ Hsh. On decreasing field (blue curves), the Nq-vortex droplet undergoes sequential Rayleigh decay (black arrows). The green curves show the full stability ranges of the droplet states, obtained by reversing the field sweep direction.

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After formation of superconductivity, M(H) at the descending branch follows the envelope shape 4πM = (n−1 − 1)(H–Hc) modulated by the single quantum jumps due to one-by-one escape of vortices from the sample. Deviation from this dependence starts at H = 0.85Hc marking the transition of the intermediate state to the metastable regime of the vortex droplet containing N = 5 bounded vortices. On further field reduction, the disintegration of the droplet follows the scenario of instability, governed by equation (5). We marked experimentally observed values of Hinst(N) for N = 5,4,3 and 2 on the theoretical phase diagram in Fig. 3d by filled red dots. The data show a perfect agreement with theoretical predictions. The final two-quanta jump corresponds to disappearance of the last 2-quanta droplet: the last 2q vortex droplet splits symmetrically so that both vortices leave the sample simultaneously. A similar behaviour is observed in the T = 7.0 K data (Fig. 5b), where the maximum quantum number is N = 3. Besides, near Tc the coherence and screening lengths become comparable to the size of the sample, and the proposed theory applies only marginally. Thus, although, in general, the observations are consistent with the theoretical phase diagram of Fig. 3c, d, the experimental points appear slightly off the theoretical instability curves.

Methods

Micrometre-sized lead superconducting crystals were grown on a highly oriented pyrolytic graphite substrate synthesized using an electrochemical process, which we developed earlier21. By carefully selecting the electrodeposition parameters, we can grow a plethora of 3D-shaped mesoscopic Pb superconductors with various geometries such as pyramids, pentagons, needles and brushes.