  It has been hypothesized that the variation of the critical currents in
Nb/Al-AlO$_x$/Nb junctions is due to, among other effects, the presence of
grain boundaries in the system. Motivated by this, we examine the effect of
grain boundaries on the critical current of a Josephson junction. We assume
that the hopping amplitudes are dependent on the interatomic distance, and
derive a physically realistic model of distance-dependent hopping amplitudes.
We find that the presence of a grain boundary and associated disorder is
responsible for a very large drop in the critical current relative to a clean
system. We also find that when a tunnel barrier is present, grain boundaries
cause substantial variations in the critical currents due to the disordered
hoppings near the tunnel barrier. We discuss the applicability of these results
to Josephson junctions presently intended for use in superconducting
electronics applications.
