The ordering of spontaneous magnetization moments in large-scale two-dimensional superconducting  -ring arrays

Abstract

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.

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

“…Moreover, we indicated that a 1D π ring chain and 2D π ring arrays will spontaneously magnetize to an 'antiferromagnetic' configuration in the absence of an external magnetic field. Tian et al [7,8] gave the same result after computing a large dimensional model of 1D π ring chains and 2D π ring arrays. This is in accordance with the scanning SQUID observations of the spontaneous magnetization with YBCO-Nb edge junction 1D chains and 2D arrays by Hilgenkemp et al [9], in which they definitely exhibit the 'antiferromagnetic' configuration.…”

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.…”

“…However, in hexagonal π-loop arrays the anti-ferromagnetic state does not exist. [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.…”

“…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. [19] and [8]. Based on the comparison, a phase branch diagram is obtained; the branch formation shows a similar behaviour to that in one-dimensional arrays.…”

“…This has been described in Refs. [19] and [8]. We also restrict ourselves to the symmetrical case, in which the loops have the same inductance L and the junctions have the same critical current amplitude I c .…”