Abstract
We have developed a rapid computational algorithm that allows for a fully self-consistent solution of the three-dimensional Bogoliubov-de Gennes equations for a Josephson junction. This microscopic model is appropriate for short-coherence-length superconductors, Josephson junctions with strongly correlated proximity-coupled weak links and systems where the barrier thickness is the same order of magnitude as the coherence length. This is a regime that is usually not described by the highly successful analytic theories of Josephson junctions developed over the past 35 years. The formalism is applied to the simplest possible model as an example, but can easily incorporate correlation effects (via the dynamical mean field theory) with relatively little extra cost. We examine current-phase relations, effects of non-magnetic impurities, interfacial scattering and the local density of states within the barrier. This last `theoretical spectroscopy' shows the evolution of Andreev bound states in the presence of a Josephson current, illustrating the expected Doppler shift. We also calculate the figure of merit, IcRN, and find that our self-consistent solutions produce a variation of this product, which can be dramatically increased for coherent SNSNS junctions which have an additional thin superconducting layer within the normal-metal region.