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 repeated 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 parameter f 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 cN = 1 / (2 × 4 N ), where N is the gasket order. In the fractal regime ( f > f cN ) 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 cN , as predicted by theory. The fine structure of L ( f ) radically changes in the euclidian regime ( f < f cN ), where it is determined by the commensurability of the vortex lattice with the effective 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.