Interactions and Interference in Mesoscopic Systems: Josephson Current through Quantum Dots and Josephson Junction Arrays

Doctoral thesis, 2003

The focus of the thesis is on the interplay between interactions and interference effects in mesoscopic systems. More specifically, the two main projects included deal with Coulomb interactions and Josephson effect in two superconductor-normal metal systems: (i) a Josephson junction array and (ii) a superconductor-normal metal-superconductor (SNS) system in which the normal part is a quantum dot. Theoretical treatments of Josephson junction arrays (JJA) are numerous and cover a range of regimes. Our analysis is based on the simplest model of a one-dimensional JJA, which is known to exhibit a superconductor-insulator transition at zero temperature as the Josephson coupling is varied, in the limit of an infinitely long array. We consider the experimentally more realistic case in which the temperature, as well as the array length, is finite. The key approach is the well-known mapping onto a classical two-dimensional XY-model, which we proceed to analyze with both analytic and numerical methods. SNS systems also exhibit Josephson effects. In short junctions, the supercurrent is supported exclusively by discrete so-called Andreev states, which are localized near the junction due to the presence of the superconducting gap in the leads. In a density functional approach, we investigate the effects of Coulomb interactions on the transport of charge through the normal part of the system, which we take to be a quantum dot-like structure. States confined by the dot potential interact with charge associated with the Andreev state, hence producing an extra contribution to the total energy of the system, which is manifested as a strongly modified supercurrent--phase relationship. Our first paper on this system treats a quasi-one-dimensional case. The second paper in the series is a generalization to two dimensions, with somewhat greater emphasis on computational issues. In the final paper we take into account the influence of states with continuous energies outside the superconducting gap.