Dielectric breakdown in media with defects

We investigate behavior of electrical circuit models for dielectric breakdown in media with defects of arbitrary residual resistivity. We first investigate the quasi-static model, where the lattice consists of a two-dimensional network of resistors that break down when the local electric field exceeds a critical value. We investigate breakdown in an infinite or semi-infinite homogeneous media with a seed defect and breakdown in a lattice network with random defects.; We use the results of the infinite lattice and percolation theory to understand the breakdown properties of the random lattice. We find that for defects with non-zero residual resistivity, the breakdown field reaches art asymptotic value as the defect lengthens causing the random lattice to reach the same value. We also find that depending on the initial length of the seed defect and the residual resistivity, the breakdown grows one-dimensionally, or spreads in more than one dimension. This behavior also appears in the random lattice.; Using the insight gained in the quasi-static model, we investigate the dynamics and geometry of dielectric breakdown paths of needle defects of arbitrary residual resistivity in an otherwise homogeneous medium using a time-dependent electrical-circuit model. This circuit model consists of a semi-infinite lattice of capacitors in parallel with resistors that break down to lower (residual) resistance. The breakdown occurs if the local field across a resistor exceeds a critical value for a breakdown delay time. We consider cases where the initial resistance is infinite or finite and where the residual resistance is finite or zero. We consider the model for the case where the applied field reaches the critical value adiabatically. We find, that as in the quasi-static case, the breakdown grows either one dimensionally or spreads with a fractal dimension (bifurcates). Also, we find that the propagation velocity of the needle oscillates spontaneously. We give the phase diagram for bifurcation and oscillations. We derive a simplified recursive map approximation to explain this behavior.