Spontaneous magnetization in multi-junction superconducting π rings when the junctions’ critical currents are not equal

We theoretically investigate the spontaneous magnetization in double-junction and multi-Junction pi rings when the junctions' superconducting critical currents are not equal. The analytical solutions for the relations between the total magnetic flux Phi and the external magnetic flux Phi(x), the free energy f and the circulating current i, and the spontaneous magnetization conditions of the double-junction pi rings are deduced. For the multi-junction pi rings consisting of more than two junctions, mathematical computation is needed to analyse the Phi-Phi(x) and f-i relations. In the conditions where the absolute critical current value of each junction is equal and there are hystereses on the Phi-Phi(x) characteristic curve, the system has spontaneous magnetization solutions. However, the non-spontaneous magnetization state with zero circulating current is a metastable state with local minimum free energy; therefore, the multi-junction pi ring system probably would not spontaneously magnetize.