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 current-phase relations are considered and described in terms of interacting phase variables associated with the sites of the gasket. Relying on a mean-field approach, two central issues are addressed: the fine structure of L(f) reflecting flux-quantization phenomena in loops with a hierarchical distribution of sizes and the low-field (f→0) scaling behavior of L(f) resulting from the self-similar 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/(2×4N), 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 ground-state phase configuration is known. Considerable simplification is achieved by making use of the triangle-star 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 self-similar structure of the SG, the frustration-induced inductance correction δL(f) is predicted to scale as fν 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 high-resolution measurements of L(f) performed on triangular arrays of periodically repeated gaskets of proximity-effect coupled Pb/Cu/Pb Josephson junctions.

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