Extreme-Value Statistics and Poisson Area Scaling With a Fatal-Area Ratio for Low- $k$ Dielectric TDDB Modeling

During the study of time-dependent dielectric breakdown (TDDB) of back-end-of-line low- k dielectrics, accurate statistical and area-scaling models are important for the final reliability lifetime projection. The extrapolated product lifetime from high percentiles to low percentiles has strong dependence on the choice of a statistical model. Meanwhile, the lifetime of a product chip is obtained typically by extrapolating TDDB data from small test structures to large chip areas by an area-scaling law. In this paper, a thorough investigation of low-k TDDB statistical distribution with large sample size and various metal areas was conducted to validate a physically relevant statistical model for low-k TDDB modeling. In addition, we explored the various TDDB dependence on metal area and introduced a new fatal-area-ratio concept for low-k TDDB area-scaling model in the event that the conventional Poisson area-scaling law failed based on the as-designed area ratio. A graphic shift-and-compare method was then developed to determine, experimentally, the fatal-metal-area ratio. The determination of the fatal-metal-area ratio should be critical for an accurate low-k TDDB lifetime projection and process assessment.