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 variation 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.
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