  Planar arrays of magnetoinductively coupled rf SQUIDs belong to the emergent
class of superconducting metamaterials that encompass the Josephson effect.
SQUID metamaterials acquire their electromagnetic properties from the resonant
characteristics of their constitutive elements, i.e., the individual rf SQUIDs,
which consist of a superconducting ring interrupted by a Josephson junction. We
investigate the response of a two-dimensional SQUID metamaterial to frequency
variation of an applied alternating magnetic field in the presence of disorder,
arising from critical current fluctuations of the Josephson elements; in
effect, the resonance frequencies of individual SQUIDs are distributed randomly
around a mean value. Bistability is observed in the total current-frequency
curves both in ordered and disordered SQUID metamaterials; moreover,
bistability is favoured by disorder through the improvement of synchronization
between SQUID oscillators. Relatively weak disorder widens significantly the
bistability region by helping the system to self-organize itself and leads to
nearly homogeneous states that change smoothly with varying frequency.
Moreover, the total current of the metamaterial is enhanced compared with that
of uncoupled SQUIDs, through the synergetic action of coupling and
synchronization. Multistability of nearly homogeneous states allows the
metamaterial to exhibit different magnetic responses corresponding to different
values of the magnetic permeability. At low power of the incident field,
high-current states exhibit extreme diamagnetic properties corresponding to
negative magnetic permeability in a narrow frequency region.
