Diamagnetic susceptibility of superconducting clusters: Spin-glass behavior

Abstract

We present a theory for the diamagnetic response of weakly linked superconducting clusters. In the model, superconducting grains, each small compared to a penetration depth, are weakly coupled into closed loops. These support screening supercurrents in response to an external magnetic field. In a magnetic field, a large cluster can support many supercurrent-carrying states of nearly equal energy, but energy barriers between these states tend to inhibit hops from one state to another at low temperatures. The picture is similar to that often proposed for spin glasses. An important consequence is predicted to be a large difference between the dc and ac susceptibilities at low temperatures. The former, an equilibrium property, will fall off much more rapidly with field than the latter, which is generally a property of the metastable states. In addition, the magnetization of a cluster varies discontinuously with field; for a sufficiently large cluster, the magnetization is everywhere discontinuous. To check these conjectures, two examples are studied. The first consists of single loops of random areas and orientation, which can be solved analytically at zero temperature. The second involves random two-dimensional clusters of many closed loops, and is studied via careful Monte Carlo simulation at various temperatures and fields. Both examples display the expected strong differences between ac and dc susceptibilities at low temperatures. Our predictions are found to be quite similar to the experimental results of Bastuscheck et al. [Phys. Rev. B 24, 6707 (1981)] for the fibrous superconductors ${\mathrm{NbSe}}_{3}$ and ${\mathrm{TaSe}}_{3}$.