On confidence in characterization and application of variation models

In this paper we study statistics of statistics. Statistical modeling and analysis have become the mainstay of modern design-manufacturing flows. Most analysis techniques assume that the statistical variation models are reliable. However, due to limited number of samples (especially in the case of lot-to-lot variation), calibrated models have low degree of confidence. The problem is further exacerbated when production volumes are low (¿ 65 lots) causing additional loss of confidence in the statistical analysis (since production only sees a small snapshot of the entire distribution). The problem of confidence in statistical analysis is going to be further worsened with advent of 450mm wafers. We mathematically derive the confidence intervals for commonly used statistical measures (mean, variance, percentile corner) and analysis (SPICE corner extraction, statistical timing). Our estimates are within 2% of simulated confidence values. Our experiments (with variability assumptions derived from test silicon data from a 45nm industrial process) indicate that for moderate characterization volumes (10 lots) and low-to-medium production volumes (15 lots), a significant guardband (e.g., 34.7% of standard deviation for single parameter corner, 38.7% of standard deviation for SPICE corner, and 52% of standard deviation for 95%-tile point of circuit delay) is needed to ensure 95% confidence in the results. The guardbands are non-negligible for all cases when either production or characterization volume is not large. We also study the interesting one production lot case which may be common for prototyping as well as for academic designs. The proposed methods require are not runtime-intensive (always within 10s) as they require Monte-Carlo simulations on closed form expressions.