Analytic approach to transport in superconducting junctions with arbitrary carrier density

Particle transport across junctions between two superconductors is commonly described using a simplifying approximation (often called the Andreev approximation), which assumes that excitations are fixed at the Fermi momentum and only Andreev reflections, with no normal reflections, occur at interfaces. While this approximation is appropriate for superconductors with high carrier density (for which the chemical potential vastly exceeds the pairing gap), it breaks down for superconductors with low carrier density, such as topological superconductors, doped semiconductors, or superfluid quantum gases. Here, we present a general analytical framework for transport in superconducting junctions that does not rely on this limiting Andreev approximation. We apply our framework to describe transport in junctions between $s$-wave superconductors along the BCS-BEC crossover, which interpolates between the conventional high-carrier-density (BCS-)regime and moderate- as well as low-carrier-density regimes (unitary and BEC regimes), for which the high-carrier-density (Andreev) approximation is not valid. As the system is tuned from the BCS to the BEC regime, we find that the overall magnitude of a subgap current, which is attributed to multiple Andreev reflections, decreases. However, nonlinearities in the current-voltage characteristic become more pronounced near the intermediate unitary limit, giving rise to sharp peaks and dips in the differential conductance with even negative differential conductance at specific voltages. Microscopically, the negative differential conductance is related to the van Hove points in the band structures, at which enhanced normal reflection occurs and that become accessible only when the chemical potential is comparable to or smaller than the pairing gap. The subgap current due to multiple Andreev reflections vanishes at a critical interaction strength on the BEC side, which we identify as the splitting point where the particle dispersion changes curvature. Our work shows that a description of transport in low-density superconducting junctions necessarily requires a treatment beyond the standard Andreev approximation.