Critical current of an inhomogeneous superconductor as a percolation-breakdown phenomenon

A percolation model for the critical current in inhomogeneous superconductors is introduced. The model is a network of randomly configured superconducting (concentration p) and normal (concentration 1-p) bonds on a lattice. Each superconducting bond has a critical current $i_c$ above which it becomes a normal Ohmic resistor. The current distribution in the superconducting regions is solved using the linearized Landau-Ginzburg equations for a network of wires as proposed by de Gennes. The current distribution in the normal regions is solved using Kirchoff’s laws. The critical current and the voltage-current relations are studied numerically in two dimensions on a square lattice, and comparisons are made with recent voltage-current experimental data on high-${\mathrm{T}}_{\mathrm{c}}$ superconductors. The scaling concepts and statistics of extremes introduced by Duxbury, Leath, and Beale (DLB) for general breakdown behavior, based on the most critical defect (normal region) in the network, are tested and found to be accurate for the scale-size dependence of the critical current and for the predicted critical-current distribution of random samples. In particular, it appears that the critical current goes to zero logarithmically in the thermodynamic limit, as proposed by DLB.