Percolation models for gate oxide breakdown

Computer calculations of the formation of a percolation path across a finite lattice are used to model dielectric breakdown. The classical scaling relations for percolation are expected to be valid only for large (finite) systems near $p_c.$ We investigate the opposite limit of very small samples, comparable to the lattice spacing. It is shown that relatively simple numerical calculations can quantitatively describe the statistics and thickness dependence of oxide breakdown in thin samples. The critical defect density for breakdown shows a strong decrease with thickness below about 5 nm, then becomes constant below 3 nm. Both of these features can be quantitatively explained by percolation on a finite lattice. The effective defect “size” of about 3 nm is obtained from the thickness dependence of the breakdown distributions. The model predicts a singular behavior when the oxide thickness becomes less than the defect size, because in this limit a single defect near the center of the oxide is sufficient to create a continuous path across the sample. It is found that a given percolation path has a probability of about ${10}^{\text{−3}}$ for initiating destructive breakdown. We investigate both homogeneous percolation and percolation in a nonuniform density of sites.