  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.
