Microscopic self-consistent theory of Josephson junctions including   dynamical electron correlations
  We formulate a fully self-consistent, microscopic model to study the retardation and correlation effects of the barrier within a Josephson junction. The junction is described by a series of planes, with electronic correlation included through a local self energy for each plane. We calculate current-phase relationships for various junctions, which include non-magnetic impurities in the barrier region, or an interfacial scattering potential. Our results indicate that the linear response of the supercurrent to phase across the barrier region is a good, but not exact indicator of the critical current. Our calculations of the local density of states show the current-carrying Andreev bound states and their energy evolution with the phase difference across the junction.   We calculate the figure of merit for a Josephson junction, which is the product of the critical current, Ic, and the normal state resistance, R(N), for junctions with different barrier materials. The normal state resistance is calculated using the Kubo formula, for a system with zero current flow and no superconducting order. Semiclassical calculations would predict that these two quantities are determined by the transmission probabilities of electrons in such a way that the product is constant for a given superconductor at fixed temperature. Our self-consistent solutions for different types of barrier indicate that this is not the case. We suggest some forms of barrier which could increase the Ic.R(N) product, and hence improve the frequency response of a Josephson device. 
P. Miller and J. K. Freericks