We consider the behavior of ordered and disordered three-dimensional weakly coupled arrays of superconducting grains embedded in a nonsuperconducting host and placed in a magnetic field $B$. In ordered simple-cubic arrays, with $\overrightarrow{B}$ parallel to a crystal axis and nearest-neighbor interactions, both Monte Carlo and molecular-field calculations show that $T_c$ is periodic in $B$ with a period of one flux quantum per unit square perpendicular to the field, and with complex substructure, as found previously in two-dimensional ordered arrays. The dependence of $T_c$ upon $\overrightarrow{B}$ is also shown to be highly anisotropic. Positionally disordered arrays are shown to behave at sufficiently strong fields very much like a spin-glass: They are, in fact, a physical realization of the "gauge glass" discussed by several authors. Varying the magnetic field, at strong fields, is equivalent to jumping from one "spin-glass replica" to another. Monte Carlo calculations for a model of dilute Pb spheres in a Zn host show a continuous transformation from an "$\mathrm{xy}$ ferromagnet" to spin-glass behavior: $T_c\mathrm{}\left(B\right)\mathrm{}$ first drops with increasing field, then saturates at strong fields. For weaker disorder, $T_c\mathrm{}\left(B\right)\mathrm{}$ is predicted to be a damped oscillating function of $B$. Both ordered and disordered samples are predicted to be anisotropic superfluids in a magnetic field. The implications of these predictions for measurements of kinetic inductance and other transport properties are briefly discussed.

• Received 17 January 1984

DOI:https://doi.org/10.1103/PhysRevB.30.134