Magnetoinductance of a superconducting Sierpinski gasket
Abstract
A study of the magnetoinductance L(B) of a planar superconducting fractal lattice, the Sierpinski gasket (SG), exposed to a perpendicular magnetic field B is reported. Being inversely proportional to the superfluid density in the gasket, L(B) provides a tool to appreciate how frustration effects created by B and characterized by a parameter f~B affect phase coherence in a superconductor sharing essential geometrical elements with a truly percolating system near threshold. Both Josephson junction arrays (JJA) and superconducting wire networks (SWN) differing in their currentphase relations are considered and described in terms of interacting phase variables associated with the sites of the gasket. Relying on a meanfield approach, two central issues are addressed: the fine structure of L(f) reflecting fluxquantization phenomena in loops with a hierarchical distribution of sizes and the lowfield (f>0) scaling behavior of L(f) resulting from the selfsimilar geometry of the gasket. It is shown that for a particlar set of f values consistent with the requirement of fluxoid quantization in the central loop of a gasket generated by repeated juxtapositions of gaskets of lower order (f=P/(2×4^{N}), where N is the gasket order and P an integer) the problem of computing L(f) reduces to a calculation on a finite gasket and can be solved exactly once its groundstate phase configuration is known. Considerable simplification is achieved by making use of the trianglestar transformation of electric networks. The amplitude of the fine structure is found to depend crucially on the degree of anharmonicity of the phase interaction function. It vanishes (thereby implying that L is independent of f) in weakly coupled SWN with a strictly harmonic interaction and reaches its maximum strength in JJA with a cosinusoidal interaction. Using a perturbative decimation procedure which takes advantage of the selfsimilar structure of the SG, the frustrationinduced inductance correction δL(f) is predicted to scale as f^{ν} with ν=ln(125/33)/ln4~=0.96 in the asymptotic limit (f>0). This exact result as well as other theoretical predictions emerging from the model are found to agree with highresolution measurements of L(f) performed on triangular arrays of periodically repeated gaskets of proximityeffect coupled Pb/Cu/Pb Josephson junctions.
 Publication:

Physical Review B
 Pub Date:
 March 1995
 DOI:
 10.1103/PhysRevB.51.5914
 Bibcode:
 1995PhRvB..51.5914K
 Keywords:

 74.50.+r;
 74.80.Bj;
 75.10.Hk;
 Tunneling phenomena;
 point contacts weak links Josephson effects;
 Classical spin models