Model for dielectric breakdown in metal-insulator composites

We consider continuum and lattice models for dielectric breakdown in random metal-insulator composites. The continuum model consists of a random composite below the metallic percolation threshold $p_c$. The dielectric is assumed to break down and become metallic when the local electric field exceeds a certain critical value. For fixed external boundary conditions and sample size, if the initial breakdown field $E_b$ varies as $E_b$≊($p_c$-p${)}^y$, we prove the inequality y>s/2, where s is the exponent governing the divergence of the dielectric constant in a metal-insulator composite, in both two and three dimensions. The lattice model we consider involves a random mixture of insulating and metallic bonds. The model has two main features: (i) an insulating bond breaks down and becomes metallic once the local electric field across it exceeds a critical value, $E_c$; and (ii) once broken down, the bond stays conducting. For fixed size, we find that $E_b$ varies roughly as ($p_c$-p${)}^y$, with y≊1.1±0.2 in d=2, y≊0.7±0.2 in d=3, for both site and bond percolation. The initial breakdown field appears to vary with sample size in a manner consistent with the logarithmic dependence proposed by Duxbury et al. Once breakdown starts, it generally cascades to completion, i.e., a completed conducting path, without further increase in the external potential difference.