Antiferromagnetic ordering in arrays of superconducting $\pi$-rings

We report experiments in which continuous $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }\text{−}\mathrm{Nb}$ junctions with internal sign changes in the Josephson coupling, as well as one-dimensional (1D) and two-dimensional (2D) arrays of $\mathrm{Y}{\mathrm{Ba}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }\text{−}\mathrm{Nb}$ $\pi$-rings, are cooled through the superconducting transition temperature of the Nb in various magnetic fields. These systems have degenerate ground states with either clockwise or counter-clockwise spontaneous circulating supercurrents. The final flux state of each facet corner in the junctions and each ring in the arrays was determined using scanning superconducting quantum interference device (SQUID) microscopy. In the continuous junctions, fabricated with facets alternating between alignment parallel to a [100] axis of the YBCO and rotated 90° to that axis, half-fluxon Josephson vortices order strongly into an arrangement with alternating signs of their magnetic flux. We demonstrate that this ordering is driven by phase coupling and model the cooling process with a numerical solution of the Sine-Gordon equation. The 2D ring arrays couple to each other through the magnetic flux generated by the spontaneous supercurrents. Using $\pi$-rings for the 2D flux coupling experiments eliminates one source of static disorder seen in similar experiments using conventional superconducting rings [Davidovic et al., Phys. Rev. Lett. 76, 815 (1996)], since $\pi$-rings have doubly degenerate ground states in the absence of an applied field. Although anti-ferromagnetic ordering occurs, with larger negative bond orders than previously reported for arrays of conventional rings, ordering over more than a few lattice spacings is never observed in the 2D arrays, even in geometries without geometric frustration. Monte Carlo simulations of the 2D array cooling process are presented and compared with the experiment.

Figure 1

Schematics of the 3 types of facetted junctions studied in this work. (a) A continuous, single facetted junction. (b) Junction with the individual half-fluxons electrically isolated from one another. (c) Double row of continuous facetted junctions.

Figure 2

Schematic diagram of two types of $\pi$-rings used for the 2D arrays reported in this paper. Type (a) was used in the square arrays, and type (b) was used in the triangular, honeycomb, and kagomé arrays.

Figure 3

Scanning electron microscopy images of 4 arrays of $\pi$-rings, with $2.7\mu \mathrm{m}$ junctions and $11.5\mu \mathrm{m}$ ring to ring spacings.

Figure 4

Scanning SQUID microscope images of continuous facetted YBCO-Au-Nb $0-\pi$ junctions, cooled in nominally zero field, and imaged with a $4\mu \mathrm{m}$ diameter pickup loop. (a) Continuous junction with $40\mu \mathrm{m}$ facet lengths. (b) Electrically disconnected junction with $40\mu \mathrm{m}$ between facet corners. (c) Electrically disconnected junction with $20\mu \mathrm{m}$ between facet corners. Two parallel facetted $0-\pi$ junctions with $40\mu \mathrm{m}$ (d), $20\mu \mathrm{m}$ (e), and $10\mu \mathrm{m}$ (f) between facet corners. The apparent curvature of the junctions in these images is an artifact of the scanning mechanism.

Figure 5

Scanning SQUID microscope images of a facetted YBCO-Nb $0-\pi$ junction with 10 facets each $40\mu \mathrm{m}$ long, cooled in fields of $0\mathrm{nT}$ (a), $32\mathrm{nT}$ (b), $74\mathrm{nT}$ (c), and $110\mathrm{nT}$ (d), and imaged at $4.8\mathrm{K}$ with an $8\mu \mathrm{m}$ square pickup loop.

Figure 6

SQUID microscopy images of four electrically disconnected arrays of $\pi$-rings, in the geometries illustrated in Fig. 3, with values for the ring parameters given in Table . These images were taken at $4.2\mathrm{K}$ with a $4\mu \mathrm{m}$ diameter pickup loop after cooling in nominally zero field at $1–10\mathrm{mK}∕\mathrm{s}$. The spin up fractions $x_{+}$ and full scale variations in the scanned SQUID sensor flux $\left(\Delta {\Phi }_s\right)$ were $x_{+}=0.56$, $\Delta {\Phi }_s=0.12{\Phi }_0$ for the square lattice; $x_{+}=0.51$, $\Delta {\Phi }_s=0.11{\Phi }_0$ for the honeycomb lattice; $x_{+}=0.50$, $\Delta {\Phi }_s=0.10{\Phi }_0$ for the triangular lattice; and $x_{+}=0.54$, $\Delta {\Phi }_s=0.12{\Phi }_0$ for the Kagome lattice, respectively.

Figure 7

SQUID microscopy images of a honeycomb array in the geometry illustrated in Fig. 3, cooled in nominally zero field through the Nb superconducting transition temperature at different cooling rates. Each panel is labeled with the cooling rate and experimentally determined bond order $\sigma$. The white circles superimposed on the images label 6-ring loops in the honeycomb lattice in which the rings are perfectly anti-ferromagnetically ordered.

Figure 8

Difference image obtained from subtracting the image of Fig. 7c from Fig. 7b, to determine which rings flipped sign after successive cooldowns.

Figure 9

Modeling of the cooldown of the facetted junction of Fig. 5. The junction has 10 faces, each 40 microns long, with alternating 0- and $\pi$-intrinsic phase shifts.

Figure 10

Results from cooldown of the facetted junction of Fig. 5 in several externally applied magnetic fields. Half-fluxons with magnetic fields in one direction are represented by open symbols, those with the opposite sense with closed symbols. The left two panels are experimental results from two successive runs; the right panel represents modelling as described in the text.

Figure 11

Calculated energies for the honeycomb ring arrays of Fig. 3 as a function of the reduced temperature. $E_b$ is the barrier to thermally activated flipping of the sign of the spontaneous magnetization, $E_c$ is the strength of the spin-spin coupling energy, and $k_{B}T$ is the thermal energy.

Figure 12

Bond order as a function of $J∕k_{B}T$ for a honeycomb lattice. In this simulation the barrier to supercurrent reversal was taken to have the form $E_{bi}=2J{\sum }_j{s}_j{(a∕r_j)}^3$. The sum $j$ was over the three 1st nearest neighbors (circles), also over the six 2nd nearest neighbors (diamonds), also over the three third nearest neighbors (triangles), or over all the rings in the lattice (squares). The dashed line represents the mean-field prediction for the Neel temperature for a 2D honeycomb lattice Ising model, including only nearest neighbor interactions. In this simulation ${10}^{10}$ Monte Carlo cycles were taken for each step $\Delta J∕k_B{T}_c=0.01$ for the 1st, $1\text{st}+2\text{nd}$, and $1\text{st}+2\text{nd}+3\text{rd}$ nearest neighbor shell calculations, and ${10}^8$ steps were taken for each step $\Delta J∕k_B{T}_c=0.02$ for the all neighbors calculation.

Figure 13

Bond order as a function of reduced temperature $T∕T_c$ calculated using Monte Carlo techniques as described in the text, for various cooling rates $\eta$. Scales for the reduced barrier energy $E_b∕k_{b}T$ and ring-ring coupling energies $E_c∕k_{B}T$ are included in the top of the figure.

Figure 14

Low temperature limit of the bond order, as determined experimentally with SQUID microscopy imaging (solid symbols) for the honeycomb lattice of Fig. 6, and as calculated for the same lattices using Monte Carlo techniques as described in the text (open symbols), for two values of the mutual inductance between nearest neighbor rings. The calculations labeled 1st include only nearest neighbor spin-spin coupling, those labeled $1\text{st}+2\text{nd}$ also include second nearest neighbors, and those labeled $1\text{st}+2\text{nd}+3\text{rd}$ also include third nearest neighbors.

Figure 15

(a) Measured dependence of the bond order $\sigma$ upon fraction of spin-up rings $x_{+}$, for the four different types of arrays shown in Fig. 6, obtained by cooling in various fields. The solid line is the theoretical maximum negative bond order for a geometrically unfrustrated array; the dashed curve is that for a frustrated array. Some of the experimental points at very low and high $x_{+}$ fall below the theoretical maximum negative bond order because of edge effects due to the finite array sizes imaged. (b) Results of a Monte Carlo simulation of the cooling process for these arrays, as described in the text.