| In this dissertation, a general description of spontaneous magnetization in multi-junction π-rings is given by considering the free energy of the system. The magnetization flux of π-rings in the ground states as a function of screen parameter β is discussed.Based on the description, a model with infinitely large one-dimensional π-ring array in the absence of external magnetic field is analyzed. The result shows that the ground state corresponds to the full antiparallel pattern of magnetization flux generated in the π-rings, which is in agreement with the experimental results obtained by Hilgenkamp et al [Nature 422, 50 (2003)] in large-scale YBCO corner junction arrays.In addition, an analytical investigation of spontaneous magnetization in two-dimensional superconducting π-ring arrays has presented. The results show that the structure with full antiparallel spontaneous magnetization is the most favorable of all possible arrangements in both square and triangular ring arrays.In hexagonal ring (triangular flux) array, however, there is no full antiparallel flux pattern although in principle the antiparallel bond of spontaneous magnetization fluxes in neighboring rings would be certainly favored in their free energy. We present an analytical investigation in the spontaneous magnetization of infinitely large two-dimensional hexagonal π-ring array. The free energies per ring, Un, in various states characterized by the number of zero-current ("quiet") junctions in the ring are calculated. A ladder relation of the energies has been analytically proved asU0123456, which basically shows that the less "quiet" junctionsof a state, the more favorable in energy to the state formation.Moreover, current density and local magnetic field of spontaneous magnetization states in one-dimensional superconducting connected corner-junction arrays have been analyzed by solving the phase equation of the arrays. The solutions can be expressed by the Jacobian elliptic functions, which have been calculated numerically.
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