Theory of dielectric breakdown in metal-loaded dielectrics

We analyze a simple model of dielectric breakdown in metal-loaded dielectrics. We show that the breakdown electric field in a large, but finite, sample tends to zero in a universal manner as the fraction of metal in the material approaches the percolation threshold. In addition, the average breakdown electric field decreases logarithmically with the linear dimension of the system when the volume fraction of metal, p, is below the percolation threshold, $p_c$. The average initial breakdown field, $E_1$, behaves as $E_1$≊1/[A(p)+B(p)ln(L)], where L is the linear dimension of the system. The coefficient of the logarithmic term diverges as p approaches the percolation threshold, B(p)∼($p_c$-p${)}^{\mathrm{-}\nu }$. The exponent ν is the power-law exponent for the divergence of the percolation correlation length as p→$p_c$. The breakdown field distribution function $F_L$(E) has the form $F_L$(E)≊1-exp[-${\mathrm{cL}}^d$exp(-k/E)], where d is the dimensionality of the system. These predictions are verified by computer simulations of random mixtures of conducting and nonconducting elements. The breakdown process is modeled by allowing the insulating elements to only be able to withstand a fixed voltage difference before they fail and become conducting elements. This process continues until a macroscopic dielectric failure occurs by the formation of a conducting path across the system. We find that the electric field necessary to cause this complete breakdown is the same as the field necessary to cause a single microscopic failure.