Enhanced pinning for vortices in hyperuniform pinning arrays and emergent hyperuniform vortex configurations with quenched disorder

Disordered hyperuniformity is a state of matter exhibiting both isotropic liquid-like properties and crystalline-like properties such as minimal density fluctuations over long distances. Such states arise for jammed particle assemblies and in nonequilibrium systems. An open question is whether the properties of disordered hyperuniformity can be harnessed for technological applications. A major issue for applications of type-II superconductors is preventing the motion or depinning of magnetic vortices in order to achieve high critical currents, so there is great interest in identifying optimal pinning site geometries. Using large-scale simulations, we show that a disordered hyperuniform pinning arrangement produces enhanced vortex pinning compared to an equal number of purely randomly arranged pinning sites, and that the enhancement is robust over a wide parameter range for both short- and long-range vortex-vortex interactions. In disordered hyperuniform arrays, pinning density fluctuations are suppressed, permitting higher pin occupancy and preventing weak links that lead to easy-flow channeling. We also show that in amorphous vortex states on either random or disordered hyperuniform pinning arrays, the vortices themselves exhibit disordered hyperuniformity due to the repulsive nature of the vortex-vortex interactions.

Figure 1

The pinning site locations (open circles) for (a) a disordered hyperuniform array and (b) a random array. (c) The structure factor $S\left(\mathbf{k}\right)$ for the disordered hyperuniform pinning array, where the weight vanishes at small $\mathbf{k}$ and the system is isotropic. (d) $S\left(\mathbf{k}\right)$ for the random pinning array, where the system is isotropic but the weight approaches a finite value at small $\mathbf{k}$. (e) $S\left(k\right)$ vs $k=\left|\mathbf{k}\right|$ for the disordered hyperuniform array. The dashed line is a fit to $S\left(k\right)\propto k^2$. (f) $S\left(k\right)$ vs $k$ for the random array approaches a constant value at small $k$.

Figure 2

The number variance ${\sigma }^2$ vs region radius $R$ for the pinning site configurations in Fig. 1. Red squares: disordered hyperuniform array, with the orange dashed line indicating a fit to ${\sigma }^2\propto R^{1.0}$; blue circles: random array, with the green dashed line indicating a fit to ${\sigma }^2\propto R^{2.0}$.

Figure 3

Vortex velocity $\langle V\rangle$ vs driving force $F_D$ for a disordered hyperuniform pinning array (red, lower curves) and random pinning array (blue, upper curves) for systems with pinning density $n_p=0.7$ and pinning strength $F_p=2.55$. (a) $B/B_{\varphi }=0.3$, where $B_{\varphi }$ is the field at which there is one vortex per pinning site. The ratio of the depinning threshold for the disordered hyperuniform array to that of the random array is $R_e=1.125$. (b) At $B/B_{\varphi }=1.0,R_e=1.8$. (c) At $B/B_{\varphi }=2.7,R_e=1.2$.

Figure 4

(a) The depinning force $F_c$ vs $B/B_{\varphi }$ for disordered hyperuniform arrays with $F_p=2.55$ (dark blue circles), 1.05 (dark green squares), and 0.53 (dark red left triangles), and for random pinning arrays with $F_p=2.55$ (light blue diamonds), 1.05 (light green up triangles), and 0.53 (orange down triangles). The inset shows a blowup of the behavior at higher fields. (b) The depinning threshold ratio $R_e=F_c^{\mathrm{hyper}}/F_c^{\mathrm{random}}$ vs $B/B_{\varphi }$ for $F_p=2.55$ (dark blue circles), 1.05 (light blue triangles), and 0.53 (green squares), showing that the pinning is consistently enhanced for the disordered hyperuniform pinning arrays.

Figure 5

The depinning force $F_c$ vs $F_p$ for disordered hyperuniform arrays at $B/B_{\varphi }=0.6$ (dark blue circles) and $B/B_{\varphi }=1.9$ (dark green diamonds) and for random arrays at $B/B_{\varphi }=0.6$ (light blue squares) and $B/B_{\varphi }=1.9$ (light green triangles). Inset: the depinning current ratio $R_e$ vs $F_p$ for $B/B_{\varphi }=0.6$ (red squares) and $B/B_{\varphi }=1.9$ (pink circles).

Figure 6

(a) Fraction $P_v$ of vortices located at pinning sites vs $B/B_{\varphi }$ for disordered hyperuniform arrays at $F_p=2.55$ (dark blue circles), 1.05 (dark green squares), and 0.53 (red diamonds), and random arrays at $F_p=2.55$ (light blue up triangles), 1.05 (light green left triangles), and 0.53 (orange down triangles), showing that there is a consistently higher fraction of occupied pinning sites in the disordered hyperuniform arrays. (b) $P_v$ vs $F_p$ for disordered hyperuniform arrays at $B/B_{\varphi }=1.9$ (dark blue circles) and 0.6 (dark green squares) and random arrays at $B/B_{\varphi }=1.9$ (light blue diamonds) and 0.6 (light green triangles), showing a similar trend. (c) The vortex (blue filled circles) and pinning site (orange open circles) locations in a small portion of the sample for a disordered hyperuniform array at $F_p=2.55$ and $B/B_{\varphi }=1.5$. (d) Vortex (blue filled circles) and pinning site (orange open circles) locations in a small portion of the sample for the random array under the same conditions showing that a higher fraction of pinning sites are unoccupied.

Figure 7

(a)–(c) $S\left(k\right)$ of the vortex positions for $n_p=0.7$ at $F_p=0.53$ (green), 1.05 (orange), and 2.55 (purple) for a random pinning array at (a) $B/B_{\varphi }=0.6$, (b) $B/B_{\varphi }=1.9$, and (c) $B/B_{\varphi }=2.7$. (d)–(f) $S\left(k\right)$ of the vortex positions for $n_p=0.7$ at the same $F_p$ values as above for a disordered hyperuniform pinning array at (d) $B/B_{\varphi }=0.6$, (e) $B/B_{\varphi }=1.9$, and (f) $B/B_{\varphi }=2.7$. In each case $k$ goes to zero as a power law $S\left(k\right)\propto {\left|k\right|}^{\alpha }$, as indicated by the dashed lines which are all power law fits with exponent $\alpha =2.0$.

Figure 8

Vortex (blue filled circles) and pinning site (orange open circles) location in the entire sample at $B/B_{\varphi }=1.9$ at $F_p=2.55$ for (a) a disordered hyperuniform pinning array and (b) a random pinning array. (c) Structure factor $S\left(\mathbf{k}\right)$ for the vortex positions in panel (a). (d) $S\left(\mathbf{k}\right)$ for the vortex positions in panel (b).

Figure 9

(a)–(c) ${\sigma }^2$ of the vortex positions vs $R$ at $F_p=0.53$ (green), 1.05 (orange), and 2.55 (purple) for the system in Figs. 7–7 with a random pinning array with $n_p=0.7$ at (a) $B/B_{\varphi }=0.6$, (b) $B/B_{\varphi }=1.9$, and (c) $B/B_{\varphi }=2.7$. (d)–(f) ${\sigma }^2$ of the vortex positions vs $R$ at the same $F_p$ values as above for the system in Figs. 7–7 with a disordered hyperuniform pinning array at (d) $B/B_{\varphi }=0.6$, (e) $B/B_{\varphi }=1.9$, and (f) $B/B_{\varphi }=2.7$. The dashed brown lines in each panel are fits to ${\sigma }^2\propto R^{d-1}$.

Figure 10

Vortex velocity $\langle V\rangle$ vs $F_D$ for a model of vortices in a thin-film superconductor with $ln\left(r\right)$ vortex-vortex interaction potentials using the same parameters as in Fig. 3 with $n_p=0.7$ and $F_p=2.55$. The red lower curves are for a disordered hyperuniform pinning array and the blue upper curves are for a random pinning array. (a) $B/B_{\varphi }=0.44$. (b) $B/B_{\varphi }=0.67$. (c) $B/B_{\varphi }=0.89$. In all cases there is an enhancement of the pinning for the disordered hyperuniform pinning arrays.

Figure 11

(a) The depinning force $F_c$ vs $B/B_{\varphi }$ for random pinning arrays (squares) and disordered hyperuniform pinning arrays (circles) with $F_p=2.55$ for the system in Fig. 10 with long-range vortex-vortex interactions. (b) The depinning threshold ratio $R_e$ vs $B/B_{\varphi }$ shows a strong enhancement of $F_c$ in the disordered hyperuniform array. (c) $F_c$ vs $F_p$ for random (squares) and disordered hyperuniform (circles) pinning arrays at $B/B_{\varphi }=0.67$. (d) The corresponding $R_e$ vs $F_p$ shows pinning enhancement in the disordered hyperuniform array.

Figure 12

Pinning site locations (open circles), vortex positions (red dots), and vortex trajectories (blue lines) for the thin-film superconductor model from Fig. 10 at $B/B_{\varphi }=0.67,F_p=2.55$, and $F_D=0.7$. (a) For a random pinning array, numerous 1D filamentary flow channels form. (b) In the disordered hyperuniform pinning array, the filamentary channels are suppressed.

Figure 13

$S\left(k\right)$ of the vortex positions in the thin-film superconductor model from Fig. 10 with a random pinning array at $F_p=2.55$ (purple), 1.05 (orange), and 0.53 (green). (a) $B/B_{\varphi }=0.67$. (b) $B/B_{\varphi }=1.9$. As in Fig. 7, we find a power law decay $S\left(k\right)\propto {\left|k\right|}^{\alpha }$ indicative of disordered hyperuniformity, where the dashed lines are fits with exponent $\alpha =4.0$.

Figure 14

(a) Schematic proposed phase diagram of temperature $T$ vs disorder strength $F_p$ for a system of repulsively interacting particles in the presence of quenched disorder. Between the crystalline state and a random phase with Poisson characteristics, there could be a disordered hyperuniform state (DHyper). (b) Schematic proposed modified vortex phase diagram for a high-temperature superconductor as a function of magnetic field $H$ in arbitrary units vs reduced temperature $T/T_c$, where $T_c$ is the critical temperature of the material. As a function of increasing $H$, there is a transition from a dislocation-free Bragg glass into a disordered hyperuniform state, followed by a transition to a random glassy state with Poisson characteristics at higher fields.

Figure 15

$\langle V\rangle$ vs $F_D$ for samples with nonzero thermal fluctuations of magnitude $F^T=2.0$. Here $n_p=0.7,F_p=2.55$, and $B/B_{\varphi }=0.67$. The red lower curves are for a disordered hyperuniform pinning array and the blue upper curves are for a random pinning array. (a) Bulk vortices with short-range Bessel function interactions. (b) Thin-film vortices with long-range $ln\left(r\right)$ interactions. In both cases the enhancement of the pinning by the disordered hyperuniform array remains robust at finite temperatures.

Figure 16

$S\left(k\right)$ of the vortex positions in samples with random pinning arrays at $B/B_{\varphi }=1.9$ and $F_p=2.55$ for differing levels of thermal fluctuations $F^T=10$ (purple), 5 (orange), and 1 (green). (a) Bulk vortices with short-range Bessel function interactions. Dashed lines are fits to $S\left(k\right)\propto {\left|k\right|}^{\alpha }$; the upper dashed line has $\alpha =1.25$ and the lower dashed line has $\alpha =2.25$. (b) Thin-film vortices with long-range interactions. The upper dashed line is a power law fit with $\alpha =2.0$ and the lower dashed line is a fit with $\alpha =4.0$.