Hyperuniform vortex patterns at the surface of type-II superconductors

A many-particle system must possess 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 nonhyperuniform 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 ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ 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 $x\text{−}y$ plane for lengths $q^{-1}$ larger than the sample thickness due to finite-size effects in the $z$ direction. We argue these predictions may be observable and propose further imaging experiments to test them independently.

Figure 1

Simulation snapshot of an array of fluctuating directed elastic lines (green) modeling vortices in a type-II superconductor with the magnetic field applied in the $z$ direction. Two constant-$z$ cross sections are highlighted, the top and an inner layer of the sample. The hydrodynamics of the elastic lines predicts a class-II hyperuniform two-dimensional point pattern at each constant-$z$ cross section (full circles), in spite of repulsion between lines being short-ranged. The pattern frozen at the top layer can be accessed experimentally via bitter magnetic decoration experiments (see Sec. 3).

Figure 2

Typical structure factors for a liquid and a solid at equilibrium at finite $T$ for an ideal clean superconductor, as obtained from molecular dynamics simulations of an elastic-line array with short-ranged elasticity and repulsive short-ranged interactions. Axial symmetry in both phases is observed in the large-wavelength (low-$q)$ limit.

Figure 3

Normalized two-dimensional structure factor obtained from molecular dynamics simulations of a three-dimensional elastic-line array in the ideal clean case, for various lattice spacings $a_0$ and number of layers $N_z$. We use a finite-size crossover scale $q_{FS}\propto a_0/N_z$ and a normalization $S\left(q_{FS}\right)\propto T{a}_0^2/N_z$. (a) Liquid phase at equilibrium and (b) solid Abrikosov phase. Both phases display the same finite-size crossover at $q_{FS}$. For $q<q_{FS}$, the system effectively becomes a nonhyperuniform two-dimensional system, while for $q>q_{FS}$ it shows a fair $S\left(q\right)\sim q$ hyperuniform behavior at low $q$. Upward corrections at large wave vectors, $q{\lambda }_{ab}\gtrsim 1$, can be explained by the dispersivity of the compression modulus emerging from the particular interaction potential. Lines are a guide to the eye for $S\left(q\right)\sim q^{\alpha }$ with $\alpha =2$ (dashed line) and $\alpha =1$ (dash-dotted line). The insert shows non-normalized data.

Figure 4

Magnetic decoration images of the vortex structure (black dots) nucleated in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ samples with different types of disorder: (a) pristine sample with point pins; (b) electron-irradiated sample with extra point pins; (c) heavy-ion-irradiated sample with a low density of columnar defects $(B_{\mathrm{\Phi }}=30\mathrm{G}$, Xe-irradiated). The magnetic induction $B$ controlling vortex density is indicated in every panel. In all cases, the white bar corresponds to 5 $\mu \mathrm{m}$. Each decoration image has overimposed Delaunay triangulations joining near-neighbor vortices with blue lines. Non-sixfold-coordinated vortices are highlighted in red.

Figure 5

Two-dimensional structure factors $S(q_x,q_y)$ for the examples of vortex structures shown in Fig. 4. The intensity is shown in a logarithmic scale and the color level is the same for the three images; see color bar at the bottom. The gray crosses indicate the $q$ window affected by spurious effects arising from the field-of-view edges. Data in these crosses are not considered for the calculation of the angularly averaged structure factor $S\left(q\right)$.

Figure 6

Angular average of the structure factor for the magnetically decorated vortex structure at the surface of (a) pristine, (b) electron-irradiated, and (c) Xe-irradiated (CD correlated disorder) ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ samples. The vortex density in every case is indicated. Data are shown as a function of $q/q_0$ with $q_0=2\pi /a_0$ the Bragg wave vector. The red line is the best power-law fit $S\left(q\right)\sim q_{\mathrm{eff}}^{\alpha }$ for the low wave-vector range $a_0/6<q/q_0<a_0/2$. Number variance ${\sigma }_N^2$ as a function of $R/a_0$ corresponding to the the same samples of the top panel: (d) pristine, (e) electron-irradiated, and (f) Xe-irradiated samples. The red line shows functions $\sim R^{2-{\alpha }_{\mathrm{eff}}}$ with the ${\alpha }_{\mathrm{eff}}$ obtained from fitting the $S\left(q\right)$ data of the top panels. The dashed line with slope $\alpha =2$ and the dash-dotted line with slope $\alpha =1$ are guides to the eye.

Figure 7

Effective exponents obtained fitting the structure factor as $S\left(q\right)\sim q^{{\alpha }_{\mathrm{eff}}}$ for the magnetically decorated vortex structures in (a) pristine, (b) electron-irradiated, and (c) heavy-ion irradiated samples as a function of vortex densities $B$.

Figure 8

Constant-$z$ cross-section structure factor $S\left(q\right)$ for the three-dimensional line liquid without disorder. Top panel: Infinitely thick sample $L/{\lambda }_{ab}\gg 1$ for various $a_0/{\lambda }_{ab}$ corresponding to different applied magnetic fields. At high fields, corresponding to the smallest $a_0{\lambda }_{ab}$ values, a crossover from the type II hyperuniform scaling $S\left(q\right)\sim q$ to $S\left(q\right)\sim q^2$ is observed at $q{\lambda }_c\approx 1$ (dotted lines are a guide to the eye for the two regimes). At low fields, the crossover shows up at larger $q$. The largest value of $a_0/{\lambda }_{ab}=2$ (black curve) roughly corresponds to the typical field of 40 G applied in magnetic decorations for ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ with ${\lambda }_{ab}\left(T_{\mathrm{freez}}\right)=0.4\mu \mathrm{m}$. Bottom panel: Finite-thickness effect in the structure factor $S\left(q\right)$ for various $L/{\lambda }_{ab}$ ratios at a fixed field such that $a_0/{\lambda }_{ab}=2$. This effect produces the saturation of $S\left(q\right)$ in a region of small $q$ whose extension depends on $L/{\lambda }_{ab}$. The case of $L/{\lambda }_{ab}=1000$ (black points) corresponds to a ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ sample with a thickness of 400 $\mu \mathrm{m}$, a typical value in real samples. The gray shaded area corresponds to the $q{\lambda }_{ab}$ range accessed experimentally for our typical field of view in decorations (obtained considered again ${\lambda }_{ab}\left(T_{\mathrm{freez}}\right)=0.4\mu \mathrm{m})$. Finite-size effects in $S\left(q\right)$ are not dominant for the experimental window accessed in our experiments in ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$. However, dispersivity effects do play a role in our experimental window.