Excess of topological defects induced by confinement in vortex nanocrystals

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 confinement effects in the first-order transition of extremely layered vortex nanocrystals.

Figure 1

Vortex nanocrystals with a vortex density of 16 G nucleated in micron-sized ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ disks with diameters of (a) 50, (b) 40, and (c) $30\mu \mathrm{m}$. Left panels: Vortices imaged in white by means of field-cooling magnetic decorations performed at 4.2 K. Central panels: Fourier transforms of the vortex positions. Right panels: Delaunay triangulations of the vortex structure depicting nonsixfold (sixfold) coordinated vortices in red (blue) with plastic deformations highlighted in gray. The scale bar indicates $10\mu \mathrm{m}$.

Figure 2

Top: Lanes considered to calculate the local vortex displacements $u\left(r\right)$ in the case of the 16 G vortex nanocrystal nucleated in the $50\mu \mathrm{m}$ disk of Fig. 1. The lanes running parallel to one of the three principal directions of the vortex structure are identified with the same color. Bottom: Schematics of the vortex displacements computed in order to calculate the displacement correlator $W^{*}\left(r\right)$ along $\left[u_{\mathrm{L}}\left(r\right)\right]$, and perpendicular to $\left[u_{\mathrm{T}}\left(r\right)\right]$, a given lane at a distance $r$ from its starting point. The real positions of the vortices are indicated with large dots, whereas the positions corresponding to a perfect triangular lattice are shown in small gray dots.

Figure 3

Evolution of the normalized displacement correlator $W^{*}\left(r\right)/a^2$ with $r/a$ for vortex nanocrystals nucleated in micron-sized ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ disks. (a) Data for 16 G vortex structures nucleated in macroscopic samples (full squares) and disks (open symbols). The inset shows the transversal-to-longitudinal displacement correlator ratio and dashed lines indicate the 1.44 value expected theoretically. Magnetic field evolution of the displacement correlator for disks of (b) 50 and (c) $40\mu \mathrm{m}$ diameters. Full lines are fits to the data with an algebraic decay with an exponent $\nu$ (with an error of $\pm 0.1)$ indicated for each case at the right.

Figure 4

Radial density of topological defects ${\rho }_{\mathrm{def}}\left(r\right)$ as a function of $r/a$ for nanocrystalline ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ vortex matter. (a) Vortex nanocrystals with a density of 16 G nucleated in 30, 40, and $50\mu \mathrm{m}$ diameter disks. Top panel: Circular shells considered to calculate this magnitude and location of topological defects (indicated with dots). (b) Vortex nanocrystals with densities of 16, 32, and 50 G nucleated in 50-$\mu \mathrm{m}$-diameter disks. Vertical dashed lines indicate the sample edge, $d/2$, and horizontal lines indicate the topological defect density for the parent macroscopic sample, ${\rho }_{\mathrm{def}}^{\mathrm{macro}}$. The full lines are fits to the data with a function ${\rho }_{\mathrm{def}}\left(r\right)=A_1+A_{2}exp\left[\right(r/a-d/2a)/\alpha ]$. (c) Evolution of the healing length of vortex nanocrystals, $\alpha ·a$, with the sample diameter $d$ for the three studied vortex densities.