Impact of shrinking measurement error budgets on qualification metrology sampling and cost

When designing an experiment to assess the accuracy of a tool as compared to a reference tool, semiconductor metrologists are often confronted with the situation that they must decide on the sampling strategy before the measurements begin. This decision is usually based largely on the previous experience of the metrologist and the available resources, and not on the statistics that are needed to achieve acceptable confidence limits on the final result. This paper shows a solution to this problem, called inverse TMU analysis, by presenting statistically-based equations that allow the user to estimate the needed sampling after providing appropriate inputs, allowing him to make important “risk vs. reward” sampling, cost, and equipment decisions. Application examples using experimental data from scatterometry and critical dimension scanning electron microscope (CD-SEM) tools are used first to demonstrate how the inverse TMU analysis methodology can be used to make intelligent sampling decisions before the start of the experiment, and then to reveal why low sampling can lead to unstable and misleading results. A model is developed that can help an experimenter minimize the costs associated both with increased sampling and with making wrong decisions caused by insufficient sampling. A second cost model is described that reveals the inadequacy of current TEM (Transmission Electron Microscopy) sampling practices and the enormous costs associated with TEM sampling that is needed to provide reasonable levels of certainty in the result. These high costs reach into the tens of millions of dollars for TEM reference metrology as the measurement error budgets reach angstrom levels. The paper concludes with strategies on how to manage and mitigate these costs.