Single-loop-like energy oscillations and staircase vortex occupation in superconducting double networks

The magnetic-field dependence of the energy and vortex occupation is calculated for the recently realized superconducting double network consisting of two interlaced subnetworks of small and large loops. Two different approaches are employed, both based on the J2 model: Mean-field analysis that minimizes the network energy assuming random-vortex configurations and numerical simulations in which energy is minimized avoiding this assumption. In the mean-field analysis the vortex population in both subnetworks increases linearly with the applied field. In contrast the simulations show that while the population of the large loops increases linearly with field, the occupation of the small loops grows in steps, resembling the behavior of an ensemble of decoupled loops. This decoupling is also reflected in the waveform of the energy versus applied field. A modified mean-field analysis, which introduces decoupling between the small loops, yields results in excellent agreement with the simulations. These findings suggest that the behavior of a single loop is reflected in the double network and thus constitute it as a favorable system for the experimental study of quantization effects in superconducting loops.