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We report on direct imaging of vortex matter nucleated in micron-sized Bi2Sr2CaCu2O8+y superconducting samples that incidentally present a bridge structure. We find that when nucleating vortices in a field-cooling condition the deck of the bridge acts as a converging lens for vortices. By means of Bitter decoration images allowing us to quantify the enhancement of vortex-vortex interaction energy per unit length in the deck of the bridge, we are able to estimate that the deck is thinner than 0.6\,$\mu$m. We show that the structural properties of vortex matter nucleated in micron-sized thin samples are not significantly affected by sample-thickness variations of the order of half a micron, an important information for type-II superconductors-based mesoscopic technological devices., Comment: 7 pages, 4 figures

Inferring the nature of disorder in the media where elastic objects are nucleated is of crucial importance for many applications but remains a challenging basic-science problem. Here we propose a method to discern whether weak-point or strong-correlated disorder dominates based on characterizing the distribution of the interaction forces between objects mapped in large fields-of-view. We illustrate our proposal with the case-study system of vortex structures nucleated in type-II superconductors with different pinning landscapes. Interaction force distributions are computed from individual vortex positions imaged in thousands-vortices fields-of-view in a two-orders-of-magnitude-wide vortex-density range. Vortex structures nucleated in point-disordered media present Gaussian distributions of the interaction force components. In contrast, if the media have dilute and randomly-distributed correlated disorder, these distributions present non-Gaussian algebraically-decaying tails for large force magnitudes. We propose that detecting this deviation from the Gaussian behavior is a fingerprint of strong disorder, in our case originated from a dilute distribution of correlated pinning centers., Comment: 12 pages, 6 figures

A many-particle system must posses long-range interactions in order to be hyperuniform at thermal equilibrium. Hydrodynamic arguments and numerical simulations show, nevertheless, that a three-dimensional elastic-line array with short-ranged repulsive interactions, such as vortex matter in a type-II superconductor, forms at equilibrium a class-II hyperuniform two-dimensional point pattern for any constant-$z$ cross section. In this case, density fluctuations vanish isotropically as $\sim q^{\alpha}$ at small wave-vectors $q$, with $\alpha=1$. This prediction includes the solid and liquid vortex phases in the ideal clean case, and the liquid in presence of weak uncorrelated disorder. We also show that the three-dimensional Bragg glass phase is marginally hyperuniform, while the Bose glass and the liquid phase with correlated disorder are expected to be non-hyperuniform at equilibrium. Furthermore, we compare these predictions with experimental results on the large-wavelength vortex density fluctuations of magnetically decorated vortex structures nucleated in pristine, electron-irradiated and heavy-ion irradiated superconducting BiSCCO samples in the mixed state. For most cases we find hyperuniform two-dimensional point patterns at the superconductor surface with an effective exponent $\alpha_{\text{eff}} \approx 1$. We interpret these results in terms of a large-scale memory of the high-temperature line-liquid phase retained in the glassy dynamics when field-cooling the vortex structures into the solid phase. We also discuss the crossovers expected from the dispersivity of the elastic constants at intermediate length-scales, and the lack of hyperuniformity in the $x\,-y$ plane for lengths $q^{-1}$ larger than the sample thickness due to finite-size effects in the $z$-direction., Comment: 17 pages, 8 figures

We study the penetration field $H_{\rm P}$ for vortex nanocrystals nucleated in micron-sized samples with edges aligned along the nodal and anti-nodal directions of the d-wave superconducting parameter of Bi$_2$Sr$_2$CaCu$_2$O$_{8 - \delta}$. Here we present evidence that the $H_{\rm P}$ for nanocrystals nucleated in samples with edges parallel to the nodal direction is larger than for the antinodal case, $\sim 72$\,\% at low temperatures. This finding supports the theoretical proposal that surface Andreev bound states appearing in a sample with edges parallel to the nodal direction would produce an anomalous Meissner current that increases the Bean-Livingston barrier for vortex penetration.This has been detected thanks to the nucleation of vortex nanocrystals with a significant surface-to-volume ratio., Comment: 4 pages, 4 figures

Order-disorder transitions between glassy phases are quite common in nature and yet a comprehensive survey of the microscopic structural changes remains elusive since the scale of the constituents is tiny and in most cases few of them take part in the transformation. Vortex matter in type-II superconductors is a model system where some of the experimental challenges inherent to this general question can be tackled by adequately choosing the host superconducting sample. For instance, Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8 + \delta}$ is a type-II superconductor with weak point disorder that presents a transition between two glassy phases on increasing the constituents' (vortices) density. At low vortex densities, the impact of disorder produces the nucleation of a glassy yet quasi-crystalline phase, the Bragg glass. For high vortex densities the stable phase, coined as $\textit{vortex glass}$, was proposed to be disordered, but its structural properties have remained elusive up to now. Here we answer this question by combining surface and bulk vortex imaging techniques, and show that the vortex glass is neither a messy nor a hexatic phase: in the plane of vortices it presents large crystallites with positional correlations growing algebraically and short-ranged orientational order. However, no dramatic change in the correlation length along the direction of vortices is observed on traversing the order-disorder transformation.

We study the local vortex-vortex interaction force $\textbf{f}_i$ of the structure frozen during a field-cooling process in an electron-irradiated Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ sample. We compute this magnitude from snapshots of the vortex structure obtained via magnetic decoration experiments at various fields H in the same sample. Since the observed structures correspond to the equilibrium ones frozen at T$\sim$T$_{irr}$(H), at this temperature the local modulus of $\textbf{f}_i$ roughly equals the local pinning force at the decorated surface of the sample. We estimate the most probable local pinning force from the mode value of the f$_i$(r) distribution, f $_{p}^{m}$. We found that f $_{p}^{m}$ grows algebraically with H and in electron-irradiated samples is 50-20% smaller than for samples with columnar defects.

We directly image individual vortex positions in nanocrystals in order to unveil the structural property that contributes to the depletion of the entropy-jump entailed at the first-order transition. On reducing the nanocrystal size the density of topological defects increases near the edges over a characteristic length. Within this "healing-length" distance from the sample edge vortex rows tend to bend while towards the center of the sample the positional order of the vortex structure is what is expected for the Bragg-glass phase. This suggests that the healing-length may be a key quantity to model the entropy-jump depletion in the first-order transition of extremely-layered vortex nanocrystals.

We show that the dynamical freezing of vortex structures nucleated at diluted densities in Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8}$ samples with a dense distribution of columnar defects, $B \sim 10^{-2} B_{\Phi}$ with $B_{\Phi}=5$\,kG, results in configurations with liquid-like correlations. We propose a freezing model considering a relaxation dynamics dominated by double-kink excitations driven by the local stresses obtained directly from experimental images. With this model we estimate the relaxation barrier and the freezing temperature. We argue that the low-field frozen vortex structures nucleated in a dense distribution of columnar defects thus correspond to an out-of-equilibrium non-entangled liquid with strongly reduced mobility rather than to a snapshot of a metastable state with divergent activation barriers as for instance expected for the Bose-glass phase at equilibrium., Comment: 12 pages, 7 figures

We detect the persistence of the solidification and order-disorder first-order transition lines in the phase diagram of nanocrystalline Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8}$ vortex matter down to a system size of less than hundred vortices. The temperature-location of the vortex solidification transition line is not altered by decreasing the sample size although there is a depletion of the entropy-jump at the transition with respect to macroscopic vortex matter. The solid order-disorder phase transition field moves upward on decreasing the system size due to the increase of the surface-to-volume ratio of vortices entailing a decrease on the average vortex binding energy., Comment: 4 pages, 4 figures

We investigated configurational changes in mesoscopic vortex matter with less than thousand vortices during flux penetration in freestanding 50 $\mu$m diameter disks of Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8 +\delta}$. High-resolution AC and DC local magnetometry data reveal oscillations in the transmittivity echoed in peaks in the third-harmonics magnetic signal fainting on increasing vortex density. By means of extra experimental evidence and a simple geometrical analysis we show that these features fingerprint the discretized entrance of single-shells of vortices having a shape that mimics the sample edge.

We study geometrical confinement effects in Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8 +\delta}$ mesoscopic vortex-matter with edge-to-surface ratio of $7-12$%. Samples have in-plane square and circular edges, 30\,$\mu$m widths, and $\sim 2\,\mu$m thickness. Direct vortex imaging reveals the compact planes of the structure align with the sample edge by introducing topological defects. The defects density is larger for circular than for square edges. Molecular dynamics simulations suggest this density is not an out-of-equilibrium property but rather determined by the geometrical confinement., Comment: accepted in Journal of Low Temperature Physics

The pinning of quantized flux lines, or vortices, in the mixed state is used to quantify the effect of impurities in iron-based superconductors (IBS). Disorder at two length scales is relevant in these materials. Strong flux pinning resulting from nm-scale heterogeneity of the superconducting properties leads to the very disordered vortex ensembles observed in the IBS, and to the pronounced maximum in the critical current density jc at low magnetic fields. Disorder at the atomic scale, most likely induced by the dopant atoms, leads to "weak collective pinning" and a magnetic field-independent contribution jcoll. The latter allows one to estimate quasi-particle scattering rates., Comment: 4 pages; ECRYS 2011 - International School and Workshop on Electronic Crystals, Carg\`ese : France (2011)

The interrelation between heterogeneity and flux pinning is studied in Ba(Fe1-xCox)2As2 single crystals with widely varying Co-content x. Magnetic Bitter decoration of the superconducting vortex ensemble in crystals with x = 0:075 and x = 0:1 reveal highly disordered vortex structures. The width of the Meissner belt observed at the edges of the crystals, and above the surface steps formed by cleaving, as well as the width of the intervortex distance distribution, indicate that the observed vortex ensemble is established at a temperature just below the critical temperature Tc. The vortex interaction energy and pinning force distributions extracted from the images strongly suggest that the vortex lattice disorder is attributable to strong pinning due to spatial fluctuations of Tc and of the superfluid density. Correlating the results with the critical current density yields a typical length scale of the relevant disorder of 40 - 60 nm., Comment: 8 pages, Submitted o Phys. Rev. B