Large planar arrays of Josephson junctions form two-dimensional systems with
many interesting properties, especially arrays containing superconducting
π-junctions with unconventional superconducting d-wave symmetry. Here we present an
analytical investigation of spontaneous magnetization in two-dimensional superconducting
π-ring arrays, starting from an ansatz that each of the identical
π-rings carries a magnetic flux with the same frustration index. Our results show that the
flux structure with full antiparallel spontaneous magnetization is the most favourable of all
possible arrangements in both square and triangular ring arrays. In a hexagonal
π-ring (triangular flux) array, although there is no full antiparallel flux pattern, the
antiparallel bond of nearest-neighbour spontaneous magnetization fluxes is still
energetically favourable. These results are in agreement with those recently observed
experimentally by Hilgenkamp et al (2003 Nature 422 50–3) in a triangular flux array.
We present an analytical investigation on the spontaneous magnetization
of infinitely large two-dimensional identical hexagonal superconducting
π-ring arrays based on an ansatz that each of the
π-rings carries a magnetic flux with the same frustration index. The free energies per ring,
Un, in various states characterized by the number of zero-current (‘quiet’) junctions
n
in the ring are calculated. A ladder relation of the energies has been demonstrated as
, which basically shows that the fewer the quiet junctions of a state, the more
favourable the energy for the state formation. This result is in qualitative
agreement with the recent experimental observations in the large-scale hexagonal
π-loop (triangular half flux quantum) array (Hilgenkamp et al 2003 Nature 422 50–3).
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