Dynamics of dielectric breakdown paths

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. The circuit model consists of a semi-infinite lattice of capacitors in parallel with resistors that break down to a 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 quasistatic case, the breakdown grows either one dimensionally or spreads with a fractal dimension (bifurcates) depending on the values of residual resistance and breakdown delay time. 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.