Correlations in Arrays of Superconducting Rings

A superconducting ring, biased in an external flux can be in either of two energetically degenerate fluxoid states. In one state, the supercurrent flows in a clockwise direction with a resulting downward magnetic moment; the current in the other state flows in a counterclockwise direction and its moment points up. There is thus a strong analogy between such a ring and an Ising spin. Two nearby but electrically isolated rings can interact magnetically; this interaction favors an antiparallel alignment of moments and is thus analogous to an antiferromagnetic spin-spin interaction. Regular arrays of such rings may thus be expected to exhibit effects of lattice geometry and geometrical frustration.

To study these issues, we have fabricated arrays containing up to 240,000 aluminum rings, each approximately 1.6 µm across. We have used a sensitive SQUID-based magnetometer to probe the global magnetic properties of the arrays; local information about particular spin configurations was obtained using a high-resolution scanning Hall probe microscope. The magnetic measurements show that individual rings do indeed behave as Ising spins, showing a paramagnetic susceptibility which freezes out only a few milliKelvin below the critical temperature Tc. This illustrates that the ring dynamics is dominated by a energy barrier between the two states which rises rapidly as the temperature is lowered below Tc. The magnetic measurements also show a hysteretic field dependence of the susceptibility which can be quantitatively interpreted in terms of an antiferromagnetic interaction between the rings.

To explore possible ordering of the spins, we have used a scanning Hall probe microscope to directly image specific configurations of spins. We find a significant amount of antiferromagnetic nearest-neighbor correlations; no evidence for any long-range ordering was found, however. We attribute this to a significant degree of disorder in the system related to small fluctuations in the areas of the aluminum rings. The effective disorder may be increased by working at higher fractions of . The observed short-range correlations drop rapidly at these higher fractions.

Arrays of aluminum rings in different geometries. The white scale bar is 5 µm long. (Clockwise from top left) Square, triangular, Kagome, and honeycomb lattices. The triangular and Kagome lattices have triplets of rings which lead to geometrical frustration with antiferromagnetic spins. For a superconducting ring, the current-vs-flux relationship and the energy are periodic in the applied flux, with period . At half-integral values of , such as , the energies of the n = 0 (with magnetic moment or "spin" down) and n = 1 (spin "up") states are the same. Thus it is possible for the moment of the rings to flip between the up and down orientations, exactly analogous to an Ising spin.

Due to the dipolar nature of the fields generated by their circulating supercurrents, a down spin (Ring 1) will create a field at Ring 2 which points up. This stabilizes Ring 2 in the up configuration. Ring 2's field in turn stabilizes Ring 1 in the down configuration. This "antiferromagnetic" configuration of the two spins is thus especially stable. We have searched for antiferromagnetic correlations between moments in the rings arrays using a variety of techniques, including SQUID susceptometry and scanning Hall probe microscopy. Above we show images for three lattices at applied fluxes slightly below, at, and above . At low fields, only a few spins are up, and at high fields most are up with only a few down. Just at half are up and half down. Although the configurations appear random, we find that there are definite short-range antiferromagnetic correlations.


Arrays of aluminum rings in different geometries. The white scale bar is 5 µm long. (Clockwise from top left) Square, triangular, Kagome, and honeycomb lattices. The triangular and Kagome lattices have triplets of rings which lead to geometrical frustration with antiferromagnetic spins.	For a superconducting ring, the current-vs-flux relationship and the energy are periodic in the applied flux, with period . At half-integral values of , such as , the energies of the n = 0 (with magnetic moment or "spin" down) and n = 1 (spin "up") states are the same. Thus it is possible for the moment of the rings to flip between the up and down orientations, exactly analogous to an Ising spin.