Magnetic-field-dependent surface resistance and two-level critical-state model for granular superconductors

We develop a two-level critical-state model for calculating the intergranular fluxon density of a granular superconductor as a function of the external magnetic field, and compare it to our measurements of magnetic-field hysteresis of the microwave surface resistance of ${\mathrm{YBa}}_2$ ${\mathrm{Cu}}_3$ ${\mathrm{O}}_7$ bulk samples and of Nb-Cu-Nb proximity-effect Josephson-junction arrays. Our model deals with the magnetic-flux profiles on both macroscopic and local levels, as the intergranular critical current density is significantly smaller than the intragranular critical current density. We first solve the simplified special case of an infinitely large flat-slab geometry with an ideal ordered superconducting lattice microstructure, where the resulting intergranular fluxon density leads to anomalous magnetic hysteresis as observed experimentally. Then we generalize to disordered samples by introducing a type of clusters of grains. In this case, fluxons move freely in and out of the sample through percolative paths between the clusters, forming a structure topologically identical to our simplified ordered structure. The sizes of clusters depend on the microstructure and the stiffness of flux-line lattices, and on whether the sample is cooled in zero magnetic field or in a magnetic field. These dependences are studied both theoretically and experimentally. Finally, we use the model to explain features observed in transport critical-current and flux-creep measurements.