Andreev Crystals in Hybrid Josephson Junction Arrays

We explore hybrid Josephson junction arrays formed by superconductor-semiconductor systems in a periodic configuration. When superconducting gates are short, Andreev bound states hybridize, leading to the emergence of Andreev bands, which we refer to as Andreev Crystals. Using a one-dimensional theoretical framework based on scattering matrices, we investigate the transport properties of these devices. We provide analytical results for systems for clean superconductor-semiconductor interfaces—where Andreev reflection dominates—and complement these with numerical simulations of disordered systems. We compute both local and non-local differential conductance, highlighting the non-reciprocal behavior of Andreev Crystals and discuss their properties as directional filters.