Defect distribution model validation and effective process control

Abstract

Assumption of the underlying probability distribution is an essential part of effective process control. In this article, we demonstrate how to improve the effectiveness of equipment monitoring and process induced defect control through properly selecting, validating and using the hypothetical distribution models. The testing method is based on probability plotting, which is made possible through order statistics. Since each ordered sample data point has a cumulative probability associated with it, which is calculated as a function of sample size, the assumption validity is readily judged by the linearity of the ordered sample data versus the deviate predicted by the assumed statistical model from the cumulative probability. A comparison is made between normal and lognormal distributions to illustrate how dramatically the distribution model could affect the control limit setting. Examples presented include defect data collected on SP1 the dark field inspection tool on a variety of deposited and polished metallic and dielectric films. We find that the defect count distribution is in most cases approximately lognormal. We show that normal distribution is an inadequate assumption, as clearly indicated by the non-linearity of the probability plots. Misuse of normal distribution leads to a too optimistic process control limit, typically 50% tighter than suggested by the lognormal distribution. The inappropriate control limit setting consequently results in an excursion rate at a level too high to be manageable. Lognormal distribution is a valid assumption because it is positively skewed, which adequately takes into account the fact that defect count distribution is typically characteristic of a long tail. In essence, use of lognormal distribution is a suggestion that the long tail be treated as part of the process entitlement (capability) instead of process excursion. The adjustment of the expected process entitlement is reflected and quantified by the skewness of lognormal distribution, yielding a more realistic estimate (defect count control limit). It is of particular importance to use a validated probability distribution when the sample size is small. Statistical process control (SPC) chart is generally constructed on the assumption of normality of the underlying population. Although this assumption is not true, as we discussed in the previous paragraph, the sample average will follow a normal distribution regardless of the underlying distribution according to the central limit theorem. However, this practice requires a large sample, which is sometimes impractical, especially in the stage of process development and yield ramp-up, when the process control limit is and has to be a moving target, enabling a rapid and constant yield-learning with minimal amount of production interruption and/or resource reallocation. In this work, we demonstrate that a validated statistical model such as lognormal distribution allows us to monitor the progress in a quantifiable and measurable way, and to tighten the control limits smoothly and systematically. To do so, we use the verified model to make a deduction about the expected defect count at a predetermined deviate, say 3s. The estimate error or the range is a function of sample variation, sample size, and the confidence level at which the estimation is being made. If we choose a fixed sample size and confidence level, the defectivity performance is explicitly defined and gauged by the estimate and the estimate error.