Critical currents in superconductors with quasiperiodic pinning arrays: One-dimensional chains and two-dimensional Penrose lattices

Abstract

We study the critical depinning current ${J}_{c}$, as a function of the applied magnetic flux $\ensuremath{\Phi}$, for quasiperiodic (QP) pinning arrays, including one-dimensional (1D) chains and two-dimensional (2D) arrays of pinning centers placed on the nodes of a fivefold Penrose lattice. In 1D QP chains of pinning sites, the peaks in ${J}_{c}(\ensuremath{\Phi})$ are shown to be determined by a sequence of harmonics of long and short periods of the chain. This sequence includes as a subset the sequence of successive Fibonacci numbers. We also analyze the evolution of ${J}_{c}(\ensuremath{\Phi})$ while a continuous transition occurs from a periodic lattice of pinning centers to a QP one; the continuous transition is achieved by varying the ratio $\ensuremath{\gamma}={a}_{S}∕{a}_{L}$ of lengths of the short ${a}_{S}$ and the long ${a}_{L}$ segments, starting from $\ensuremath{\gamma}=1$ for a periodic sequence. We find that the peaks related to the Fibonacci sequence are most pronounced when $\ensuremath{\gamma}$ is equal to the ``golden mean.'' The critical current ${J}_{c}(\ensuremath{\Phi})$ in a QP lattice has a remarkable self-similarity. This effect is demonstrated both in real space and in reciprocal $k$ space. In 2D QP pinning arrays (e.g., Penrose lattices), the pinning of vortices is related to matching conditions between the vortex lattice and the QP lattice of pinning centers. Although more subtle to analyze than in 1D pinning chains, the structure in ${J}_{c}(\ensuremath{\Phi})$ is determined by the presence of two different kinds of elements forming the 2D QP lattice. Indeed, we predict analytically and numerically the main features of ${J}_{c}(\ensuremath{\Phi})$ for Penrose lattices. Comparing the ${J}_{c}$'s for QP (Penrose), periodic (triangular) and random arrays of pinning sites, we have found that the QP lattice provides an unusually broad critical current ${J}_{c}(\ensuremath{\Phi})$, that could be useful for practical applications demanding high ${J}_{c}$'s over a wide range of fields.