The investigation of spontaneous magnetization in large-scale two-dimensional hexagonal superconducting  -ring arrays

Abstract

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).

Magnetic field induced phase branches of the superconducting transition in one-dimensional π-loop arrays

“…They studied the ordering of half flux quanta in these π-loop arrays, and moreover the possibility of manipulating the polarizations of individual half flux quanta in hexagonal π-loop arrays. Their experimental results as well as the analysis given by other authors [6,14] show that there is a general tendency to antiferromagnetic ordering of spontaneous magnetization fluxes of π-loop arrays in the absence of external field. Based on their results, the ordering of the half flux quanta in largescale one-dimensional π-loop arrays in a uniform external magnetic field is analysed via the free energy approach in this paper.…”

Magnetic field induced phase branches of the superconducting transition in two-dimensional square π-loop arrays

“…[11−14] Recently, Hilgenkamp et al have reported the realization of large-scale coupled π-loop arrays based on YBa 2 Cu 3 O 7 -Au-Nb Josephson contacts. [15,16] Their experimental results as well as the analysis given by other authors [8,17,18] showed a robust antiferromagnetic ordering of spontaneous magnetization in both square and triangular π-loop arrays and predominant anti-ferromagnetic nearest-neighbour coupling of half-flux quanta in hexagonal π-loop arrays in the absence of an external field. Very recently, the free energy densities in possible flux states of twodimensional square π-loop arrays in an external magnetic field were given analytically and compared numerically by Li et al [19] To complete the theory, in this paper an analytical treatment for the comparison of the free energy densities is provided with the same model described in Refs.…”

“…[8] Although ferromagnetic states do exist in the neighbourhood of φ e = n + 1/2, the mathematical treatment to provide a flux structure in the vicinity of φ e = n is difficult. The predominant anti-ferromagnetic nearest-neighbour coupling can be described by the bond-order parameter σ, [15,17,21,22] which equals to -1/3 at zero temperature.…”