Hybrid Spatial Modeling of Non-Stationary Process Variations

Accurate characterization of spatial variation is essential for statistical performance analysis and modeling, post-silicon tuning, and yield analysis. Existing approaches for spatial modeling either assume that: (i) non-stationarities exist due to a smoothly varying trend component or that (ii) the process is stationary within regions associated with a predefined grid. While such assumptions may hold when profiling certain classes of variations, many studies now suggest that spatial variability is likely to be highly non-stationary. In order to provide spatial models for non-stationary process variations, we introduce a new hybrid spatial modeling framework that models the spatially varying random field as a union of non-overlapping rectangular regions where the process is assumed to be locallystationary. To estimate the parameters in our hybrid spatial model, we introduce a host of techniques for efficient detection of regions over which the process variations are locally-stationary. We verify our models and results on measurements collected from 65nm FPGAs.