Magnetic correlations, geometrical frustration, and tunable disorder in arrays of superconducting rings

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A superconducting ring, biased in an external flux ${\mathrm{\ensuremath{\Phi}}}_{0}$/2, 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 2.4 \ifmmode\times\else\texttimes\fi{}${10}^{5}$ aluminum rings, each approximately 1.6 \ensuremath{\mu}m across. We have used a sensitive superconducting quantum interference device-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 ${\mathrm{T}}_{\mathrm{c}}$. This illustrates that the ring dynamics is dominated by an energy barrier between the two states which rises rapidly as the temperature is lowered below ${\mathrm{T}}_{\mathrm{c}}$. 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 the Hall microscope to directly image specific configurations of spins. We find significant antiferromagnetic nearest-neighbor correlations, but no evidence for any long-range ordering. 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 ${\mathrm{\ensuremath{\Phi}}}_{0}$. The observed short-range correlations drop rapidly at these higher fractions.