Dimensional crossover and hidden incommensurability in Josephson junction arrays of periodically repeated Sierpinski gaskets

We report a study of overdamped Josephson junction arrays with the geometry of periodically repented Sierpinski gaskets. These model superconductors share essential geometrical features with truly random (percolative) systems. When exposed to a perpendicular magnetic field B, their Euclidian or fractal behavior depends on the relation between the intervortex distance (imposed by B) and the size of a constituent gasket, and was explored with high-resolution measurements of the sample magnetoinductance L(B). In terms of the frustration parameterf expressing (in units of the superconducting flux quantum) the magnetic flux threading an elementary triangular cell of a gasket, the crossover between the two regimes occurs at f c N =1/(2×4 N ), where N is the gasket order. In the fractal regime (f>f c N ) a sequence of equally spaced structures corresponding to the set of states with unit cells not larger than a single gasket is observed at multiples of f c N , as predicted by theory. The fine structure of L(f) radically changes in the Euclidian regime (feffective potential created by the array. Anomalies observed in both the periodicity and the symmetry of L(f) are attributed to the effect of a hidden incommensurability, which arises from the deformation of the magnetic field distribution caused by the asymmetric diamagnetic response of the superconducting islands forming the arrays.