Freezing vortex rivers
Introduction
Superconductivity is generally regarded as a diamagnetic state of matter where dc electrical current can flow without dissipation. In type-II superconductors, however, a magnetic field can penetrate in form of flux tubes or vortices each of them bearing a quantum unit of flux Φ0. The repulsive interaction between vortices make them to distribute in a periodic triangular array known as Abrikosov lattice. The ultimate mechanism leading to this repulsive interaction between vortices in type-II superconductors is the increase of magnetic energy as vortices approach each other at distances of the superconducting penetration depth λ.
Interestingly, under certain circumstances, an attractive vortex–vortex interaction can appear even in type-II superconductors at equilibrium conditions. This is the case for vortices in an anisotropic superconductor tilted away from the principal symmetry axes [1], [2]. Here, the net attractive interaction results from the change of sign of the component parallel to the vortex direction of the field generated by a vortex line. A similar field reversal in the magnetic field distribution of an isolated vortex results from the non-local relationship between supercurrents and vector potential in clean and low-κ materials. It has been shown that this effect also leads to an attractive vortex interaction [3], [4], [5], [6]. Another example can be found in the case of two components superconductors where two weakly coupled order parameters, each of which belonging to a different type of superconductivity, coexist in the same material [7], [8].
Although all this plethora of possible vortex arrangements correspond to equilibrium thermodynamical configurations, vortex–vortex attraction can also be found when the system is driven out of equilibrium. For instance, a fast moving vortex line creates an excess of quasiparticles behind its core thus generating a wake of depleted order parameter which attracts other vortices [10], [11], [12]. Time dependent Ginzburg–Landau calculations showed that the resulting direction dependent interaction between vortices may gives rise to the formation of vortex rivers in narrow transport bridges, somehow similar to the vortex stripes in the static case. These fast moving vortices, known as kinematic vortices, have a highly anisotropic vortex core along the direction of motion which eventually evolves into a phase slip line beyond certain critical velocity [12], [13]. A way to promote the proliferation of these kinematic vortices at relative small currents can be achieved by introducing a periodic array of holes in the superconducting film. The reason is two fold, on the one hand the constriction imposed by the period of the hole array favors the formation of phase slip lines [13], [14]. On the other hand the patterned sample leads to an inhomogeneous current distribution which magnifies the current density in between the holes. This picture indicates that the high current resistive state of samples with a periodic array of antidots should be dominated by kinematic vortices. It was anticipated by Reichhardt et al. [9] that under these circumstances the competition between a long range repulsive interaction and more local attractive force in presence of an applied dc drive can lead to the formation of conglomerates of vortices such as dimmers, labyrinths, or stripes.
In the present work we directly visualize by scanning Hall microscopy the formation of vortex stripes perpendicular to the current direction in a conventional superconductor with a periodic array of holes. Due to the large integration time needed to acquire a single frame (about 3 min) the images are recorded after freezing the dynamic phase by quickly cooling the sample in presence of a bias current and applied field. The relevance of the periodic array of holes in stabilizing parallel vortex stripes is clearly evidenced by the lack of such stripes in a plain film without holes. Time dependent Ginzburg–Landau calculations support our interpretation and give further insight on the birth, growth and evolution of these flux rivers.
Section snippets
Experimental details
The investigated sample consist of a 50 nm thick Pb film with a square array of square holes made by electron beam lithography and subsequent lift-off. The experimental procedure for sample preparation can be found in Ref. [16]. The period of the pattern is d = 1.5 μm and the size of the holes is a = 0.6 μm. Electrical transport measurements performed in a sister sample show clear commensurability effects at H/H1 = 1/2, 1, 1.5, and 2, where H1 = Φ0/d2 is the field at which the density of vortices equals the
Static distribution of vortices
A series of images obtained at 4.2 K after field cooling procedure with fields ranging from −4 mT to +4 mT in steps of 0.02 mT allowed us to determine the remanent field with high accuracy. In the patterned sample we clearly identify the different commensurate vortex states as described in previous reports [17], [18], [19]. Fig. 1 shows SHPM images obtained at 4.2 K with H = 0, H = H1 = 9.2 G, H = 2H1, and H = 3H1. Fig. 1a shows an isolated vortex with a periodic background signal produced by the modulated
Pulsed transport measurements
An attempt to simultaneously record images while moving vortices with an external dc current for T < 6.8 K showed that vortices remain still until reaching the critical current beyond which a severe heat dissipation associated with the vortex motion drives the entire sample to the normal state. This effect becomes apparent in Fig. 2 where a voltage–current characteristic at T = 7.1 K has been recorded within 5 s time and 5 μs time. In this case, despite the fact that initially the system is under
Conclusion
To summarize, in this manuscript we report on the first successful visualization of vortex stripe patterns formed by an unconventional vortex attraction when these entities are driven to very high velocities. These, so called, kinematic vortices attract each other forming stripe patterns that can be stabilized by literally freezing their motion via a fast thermal quench and a strong pinning potential produced by an array of antidots. Contrary to other systems exhibiting stripe phases, vortices
Acknowledgements
This work was supported by Methusalem funding by the Flemish government, the Flemish Science Foundation (FWO-Vl), the Belgian Science Policy, and the ESF NES network. A.V.S., G.R.B., and J.V.d.V. acknowledge support from FWO-Vl. R.F.L. acknowledge support from I3P CSIC program and MAT2008-01022.
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