dc properties of series-parallel arrays of Josephson junctions in an external magnetic field

A detailed dc theory of superconducting multijunction interferometers has previously been developed by several authors for the case of parallel junction arrays. The theory is now extended to cover the case of a loop containing several junctions connected in series. The problem is closely associated with high-${\mathit{T}}_{\mathit{c}}$ superconductors and their clusters of intrinsic Josephson junctions. These materials exhibit spontaneous interferometric effects, and there is no reason to assume that the intrinsic junctions form only parallel arrays. A simple formalism of phase states is developed in order to express the superconducting phase differences across the junctions forming a series array as functions of the phase difference across the weakest junction of the system, and to relate the differences in critical currents of the junctions to gaps in the allowed ranges of their phase functions. This formalism is used to investigate the energy states of the array, which in the case of different junctions are split and separated by energy barriers of height depending on the phase gaps. Modifications of the washboard model of a single junction are shown. Next a superconducting inductive loop containing a series array of two junctions is considered, and this model is used to demonstrate the transitions between phase states and the associated instabilities. Finally, the critical current of a parallel connection of two series arrays is analyzed and shown to be a multivalued function of the externally applied magnetic flux. The instabilities caused by the presence of intrinsic serial junctions in granular high-${\mathit{T}}_{\mathit{c}}$ materials are pointed out as a potential source of additional noise.