Abstract
Depending on the Ginzburg–Landau parameter κ, superconductors can either be fully diamagnetic if (type I superconductors) or allow magnetic flux to penetrate through Abrikosov vortices if (type II superconductors; refs 1, 2). At the Bogomolny critical point, , a state that is infinitely degenerate with respect to vortex spatial configurations arises^{3,4}. Despite indepth investigations of conventional type I and type II superconductors, a thorough understanding of the magnetic behaviour in the nearBogomolny critical regime at κ ~ κ_{c} remains lacking. Here we report that in confined systems the critical regime expands over a finite interval of κ forming a critical superconducting state. We show that in this state, in a sample with dimensions comparable to the vortex core size, vortices merge into a multiquanta droplet, which undergoes Rayleigh instability^{5} on increasing κ and decays by emitting single vortices. Superconducting vortices realize Nielsen–Olesen singular solutions of the Abelian Higgs model, which is pervasive in phenomena ranging from quantum electrodynamics to cosmology^{6,7,8,9}. Our study of the transient dynamics of Abrikosov–Nielsen–Olesen vortices in systems with boundaries promises access to nontrivial effects in quantum field theory by means of benchtop laboratory experiments.
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Main
The evolution of magnetic properties of an infinite superconductor when crossing κ_{c} is shown in Fig. 1. Type I superconductors with κ < κ_{c} expel magnetic field H until it reaches a critical field H_{c} beyond which superconductivity is destroyed (Fig. 1b, e). In type II superconductors with κ > κ_{c}, superconductivity extends into a wider region, H_{c1} < H < H_{c2}, 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 Φ_{0} = πℏ/ce.
Finitesize 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 ≃ 0.5H_{c} (ref. 10), where ξ is the coherence length and D is the sample thickness. The intermediate state forms in the interval (1 − n)H_{c} < H < H_{c} (n < 1 is the shapedependent demagnetization factor) triggered by the local magnetic field near the edges of the sample exceeding the critical value H_{c} 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 H_{c3} > H_{c2}. In type I superconductors H_{c3} can exceed H_{c} if κ ≲ κ_{c}, as shown in Fig. 1b, e.
Near κinduced 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 ≲ κ_{c} 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 quanta^{11}. Its critical fission occurs by splitting an Nquanta droplet (Nqdroplet) into a (N − 1)qdroplet and a single 1q vortex moving away (see Fig. 2). To calculate the energy difference between the Nqdroplet and the configuration consisting of the residual (N − 1)qdroplet and the separated vortex, we construct a perturbation theory in the vicinity of the Bogomolny point over the small parameter γ = κ^{2} − κ_{c}^{2} (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)qdroplet 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 Rayleigh^{5} in 1882 leading to fragmentation of charged liquid droplets due to the competition between longrange Coulomb repulsion forces and a shortrange molecular attraction.
The threshold field H_{inst}(N) at which the Nqdroplet 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 Nqdroplet into N single vortices requires surmounting a higher confinement energy barrier than onebyone vortex decoupling.
The Rayleigh instability can be observed if the field H_{inst}(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 H_{c3} = 2.39κH_{c}. 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 H_{p} ≃ (1/2^{1/4}κ^{1/2})(1 + 5.44κ/1 + 4.78κH_{c}) (ref. 14), which gives the superheating field H_{sh} = (1 − n)H_{p} 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 κ_{c}, to κ ≃ 0.38 at T = T_{c}(7.2 K) and is well described by the phenomenological formula κ(T) ≃ κ(0)/(1 + T^{2}/T_{c}^{2}) (see ref. 15 and Fig. 3b). Therefore, micrometresized samples of Pb, an exemplary type I superconductor, offer a natural laboratory to study vortex droplet fission. We selected a triangularshaped Pb mesocrystal with lateral side dimensions of ~2.2 μm, thickness of ~0.7 μm, and critical temperature T_{c} = 7.18 K shown in the lower inset of Fig. 3a. The measurements on the crystal were done using a twodimensional (2D) electron gas ballistic Hall microprobe array magnetometer^{16,17} (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 H_{c} is standard, H_{c}(T) = H_{c}(0)[1 − (T/T_{c})^{2}] with μ_{0}H_{c}(0) ≍ 78 mT (ref. 18). The critical fields H_{sh} and H_{c3} are expressed through H_{c} as given above with the bestfit value n = 0.37. The curves for H_{c}, H_{sh} and H_{c3} cross pairwise near approximately T_{x} ≃ 6.3 K. The dotted lines show the instability field H_{inst}(N) for various N calculated from equation (5). We further focus on the temperature region T_{x} < T < T_{c}, which is the most favourable for the experimental observation of Rayleigh instability of the vortex droplet. At T < T_{x} the lines H_{inst}(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 H_{c} (ref. 19), show the intermediate regime with a mixture of droplet and onequanta vortices (see Fig. 4). Note that in the temperature range T_{x} < T < T_{c} where H_{c3} < H_{c} < H_{sh}, 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 H_{sh} at which superconductivity vanishes.
The temperature dependencies of H_{c}, H_{sh} and H_{c3} shown in Fig. 3c are in a good agreement with those of Fig. 3a obtained experimentally. The data were extracted from fielddependent magnetization curves as shown in the upper inset of Fig. 3a. From H_{sh} and H_{c} one obtains the temperature dependence of the Ginzburg–Landau parameter κ(T) through H_{p}. 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 H_{c}(T) this gives a penetration λ(T) ≍ λ(0)/[1 − (T/T_{c})^{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 κ_{c}, the lower inset of Fig. 3b delineates the mixedstatelike 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 H_{sh} ≃ 51 mT from the zerofieldcooled state, the absolute value of the magnetization grows proportionally to H owing to Meissner screening. Beyond H_{sh}, 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 H_{c} ≃ 71 mT, but the diamagnetic signature of superconductivity disappears only at H = H_{c3} ≃ 96 mT (Fig. 3a, b). At the reversing branch, the onset of the transition is observed at H ≍ H_{c3} 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 sawlike structure, which reflects the effect of pinning that traps vortices within the sample. The drops in M(H) dependence correspond to the onebyone 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 T_{x} < T < T_{c} where H_{c3}, H_{c} < H_{sh} 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 H_{sh}. At H = H_{sh} the magnetization abruptly drops to zero. Moving from high field along the descending branch, one sees that superconductivity emerges at H_{c3}(T), but the system falls into a vortex droplet state.
After formation of superconductivity, M(H) at the descending branch follows the envelope shape 4πM = (n^{−1} − 1)(H–H_{c}) modulated by the single quantum jumps due to onebyone escape of vortices from the sample. Deviation from this dependence starts at H = 0.85H_{c} 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 H_{inst}(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 twoquanta jump corresponds to disappearance of the last 2quanta 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 T_{c} 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
Micrometresized lead superconducting crystals were grown on a highly oriented pyrolytic graphite substrate synthesized using an electrochemical process, which we developed earlier^{21}. By carefully selecting the electrodeposition parameters, we can grow a plethora of 3Dshaped mesoscopic Pb superconductors with various geometries such as pyramids, pentagons, needles and brushes.
Change history
17 November 2014
In the version of this Letter originally published, the names of three of the authors were missing middle initials and should have read M. V. Milošević, S. J. Bending and F. M. Peeters. This error has now been corrected in all versions of the Letter.
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Acknowledgements
We would like to thank N. Nekrasov for illuminating discussions. The work was supported by the US Department of Energy, Office of Science Materials Sciences and Engineering Division (V.M.V., W.K.K., U.W., R.X., M.Z., Z.L.X., G.W.C. and partially I.L. through the Materials Theory Institute), by FP7IRSESSIMTECH and ITNNOTEDEV programs (I.L.), and by the Flemish Science Foundation (FWOVlaanderen) (M.V.M. and F.M.P.).
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I.L., V.M.V., W.K.K. and U.W. conceived the work, I.L. and V.M.V. carried out calculations and analysed the data, M.Z. and Z.L.X. grew the samples, A.R., R.X., U.W. and W.K.K. carried out the experiment, S.J.B. provided the Hall sensor, M.V.M. and F.M.P. performed numerical simulations, I.L. and V.M.V. wrote the manuscript, A.R., U.W., M.V.M., G.W.C., S.J.B., F.M.P. and W.K.K. discussed the results and contributed to writing the manuscript.
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Lukyanchuk, I., Vinokur, V., Rydh, A. et al. Rayleigh instability of confined vortex droplets in critical superconductors. Nature Phys 11, 21–25 (2015). https://doi.org/10.1038/nphys3146
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DOI: https://doi.org/10.1038/nphys3146
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