Microscopic theory of geometric Josephson junctions

In the present work, the dc Josephson effect has been investigated for geometric Josephson junctions. These investigations were based on the calculation of two-dimensional selfconsistent solutions for the propagators of microscopic Eilenberger theory of superconductivity. Using these calculations, geometric Josephson junctions in the form of microbridges consisting of superconductors with s- and d-wave symmetry of the pairing interaction have been considered. The typical dimensions l and w of the constriction which realizes the Josephson junction have been varied in the full range from l,w<<ξ0 to l,w>>ξ0, where ξ0 is the coherence length of the superconductor. Furthermore, screening effects have been investigated as well as the influence of an external magnetic field. All calculations were carried out for the full temperature range from T=0 up to the critical temperature Tc. In order to explore geometric Josephson junctions theoretically, an efficient method for the numerical calculation of selfconsistent solutions for the propagators of microscopic Eilenberger theory for extended two-dimensional geometries has been developed and implemented using parallelized algorithms. To investigate the static behaviour of the Josephson junctions under consideration, current-phase relations have been calculated and the respective critical currents have been extracted. A derivation of the according relation based on the Eilenberger functional has been given. In the first part the methods described above have been applied to a geometric Josephson junction in the form of a rectangular microbridge consisting of a conventional s-wave superconductor. In th second part, the rectangular microbridge geometry has been investigated for the case of the unconventional d-wave superconductor. The influence of d-wave symmetry on the Josephson effect emerging at geometric Josephson junctions has been considered for the first time. The contributions of higher harmonics to the resulting current phase relations have been analyzed. In the third part a special geometry for a constriction-type Josephson junction consisting of a d-wave superconductor has been proposed, which leads to the appearance of intrinsic phase differences at comparatively large extensions of the constriction in a particularly robust way