Andreev reflection in spin-orbit nanowires proximized by superconductors

Superconductivity is one of the most important macroscopic quantum coherent phenomena in physics, where at sufficiently low temperature electrons form Cooper pairs sharing the same phase. When an interface is realized between a superconductor (S) and a normal (N) material, a phenomenon called Andreev reflection occurs: an electron impinging from the normal side towards the interface cannot penetrate the superconductor as a single quasiparticle and is back-reflected as a hole. Recently, one of the most interesting and studied setups involving a normal and a superconducting material is the case of semiconductor nanowires with spin-orbit coupling (SOC), where a nanowire portion, covered by a superconducting film, acquires a superconducting pairing by proximity effect. Nanowires characterized by strong SOC, such as InAs and InSb, exhibit ballistic transport and a wide tunability of the SOC, and have been predicted to host Majorana quasiparticles that are robust to decoherence, with potential applications in quantum technology. While experimental evidence for Majorana quasiparticles is still debated, a key open problem is to understand how Andreev Reflection is affected by SOC inhomogeneities. Indeed, while most theoretical models assume a homogeneous SOC, in actual experiments inhomogeneities are present mainly for two reasons: firstly, when a nanowire portion is covered by a superconducting film, its structural inversion asymmetry underlying the Rashba SOC is modified as compared to the bare nanowire portion. Secondly, the SOC in the normal portion can be controlled in magnitude and direction by wrap metallic gates, with the purpose of tuning the conduction properties of the system. So far, these aspects have been mainly overlooked in the literature. This MS Thesis is an attempt to bridge this gap, by investigating how Andreev reflection is affected by the inhomogeneities of the SOC. In order to describe a homogeneous single channel semiconductor nanowire proximized by a superconducting pairing, we first consider the generalization of the Bardeen-Cooper-Schrieffer model including the effects of SOC and of an external magnetic field. Then, by adopting the Bogoliubov-de Gennes formalism, we further extend this model to describe inhomogeneous systems, specifically a Normal-Superconducting (N/S) nanowire junction where the SOC profile is non-uniform across the interface. By combining analytical and numerical methods, we analyze the effects of two specific types of spin-orbit inhomogeneities on Andreev reflection. The first one is an inhomogeneity in the direction of the SOC field, defined by a misalignment angle between the N and S sides. The second type is the inhomogeneity in the magnitude of the Rashba SOC. Our results show that, while Andreev Reflection is independent of the misalignment angle, it can be significantly affected by the difference in the spin-orbit magnitude. We analyze this effect in two distinct regimes, where the spin-orbit energy is much larger than the superconducting gap, and in the opposite case of a large superconducting gap. Furthermore, we also discuss how these effects are modified by the presence of a magnetic field. In conclusion, this work provides a rigorous framework for modeling quantum transport in N/S nanowire junctions. Our findings clarify the distinct roles of directional and magnitude-based SOC inhomogeneities in modulating Andreev reflection, providing a solid basis for the design of future quantum devices.