Josephson effect in a Fibonacci quasicrystal

Quasiperiodicity has recently been proposed to enhance superconductivity and its proximity effect. Simultaneously, there has been significant experimental progress in the fabrication of quasiperiodic structures, including in reduced dimensions. Motivated by these developments, we use microscopic tight-binding theory to investigate the DC Josephson effect through a ballistic Fibonacci chain attached to two superconducting leads. The Fibonacci chain is one of the most-studied examples of quasicrystals, hosting a rich multifractal spectrum, containing topological gaps with different winding numbers. We study how the Andreev-bound states (ABS), current-phase relation, and the critical current depend on the quasiperiodic degrees of freedom, from short to long junctions. While the current-phase relation shows a traditional $2\pi$ sinusoidal or sawtooth profile, we find that the ABS develop quasiperiodic oscillations and that the Andreev reflection is qualitatively altered, leading to quasiperiodic oscillations in the critical current as a function of junction length. Surprisingly, despite earlier proposals of quasiperiodicity enhancing superconductivity compared to crystalline junctions, we do not, in general, find that it enhances the critical current. However, we find significant current enhancement for reduced interface transparency because of the modified Andreev reflection. Furthermore, by varying the chemical potential, e.g., by an applied gate voltage, we find a fractal oscillation between superconductor-normal metal-superconductor (SNS) and superconductor-insulator-superconductor (SIS) behavior. Finally, we show that the winding of the subgap states leads to an equivalent winding in the critical current, such that the winding numbers, and thus the topological invariant, can be determined.