The Critical Current Density of an SNS Josephson-Junction
  The critical current density (Jc) through a superconductor in high magnetic fields is controlled by the inclusions and microstructure of the material that hold fluxons stationary to keep the resistance zero and is described using Ginzburg-Landau (G-L) theory. Although the functional form of Jc for superconducting-normal-superconducting (SNS) Josephson-Junctions (J-Js) is known in the low field limit, includes the local properties of the junctions and has been confirmed experimentally in many systems, there are no general solutions for Jc of J-Js in high fields. Scaling laws describe the functional form (magnetic field, temperature and strain dependence) of Jc for polycrystalline superconductors in high fields but do not include local grain boundary properties. They are derived by considering isolated pinning sites where at criticality fluxons either depin or free fluxons shear past pinned fluxons as part of the flux line lattice. However, visualisation of solutions to the Time Dependent Ginzburg-Landau (TDGL) equations for polycrystalline materials have shown that fluxons cross the superconductor by flowing along the grain boundaries. Here we derive clean- and dirty- limit analytic equations for of SNS J-Js in high fields and verify them using computational solutions to TDGL theory. We consider SNS J-Js to be the basic building blocks for grain boundaries in polycrystalline materials since they provide flux-flow channels. The J-Js description provides a mathematical framework that includes the utility of scaling laws together with our microscopic understanding of barriers to supercurrent flow and helps identify the grain boundary engineering that can improve in low temperature polycrystalline superconductors used in high magnetic field applications. 
George J. Carty and Damian P. Hampshire
cond-mat.supr-con