Dielectric breakdown in media with defects

We investigate the breakdown held and geometry of breakdown paths of an electrical circuit model for dielectric breakdown in media with defects of arbitrary residual resistivity. The circuit model consists of a two-dimensional square lattice network of resistors that break down from a high resistance to a lower (residual) resistance when the local electric field exceeds a critical value. We consider infinite and semi-infinite samples with a single cluster (needle) of defects as well as samples with a finite concentration of defects from the dilute limit to the percolation threshold. We find that for needle defects with nonzero residual resistivity, the breakdown field reaches a finite value as the defects lengthen, causing the random lattice to reach the same breakdown field in the thermodynamic limit. Furthermore, we find that depending on the initial length of the seed defect and the residual resistivity, the breakdown either grows one dimensionally, or spreads with a fractal dimension. We give the phase diagram and relevant exponents for this crossover, and report similar behavior in random lattices at dilute defect concentrations.