Extreme value budgeting: Modeling, Sampling and Design
by Remco van der Hofstad - joint work with Bram Slachter, Tom Castenmiller of ASML, TE-SE-Technologie-Nodes
With the vast numbers of features in Integrated Circuits of modern chips, the allowed failure probabilities per feature are extremely small, leading to extreme-value events. Since also features become smaller and smaller, the stochastic fluctuations and inhomogeneities in the printing procedures, due to various sources including variable photon counts and changes in light intensities and focus, need to be properly taken into account. Stochastic models are essential in the design and optimization of the photolithography process.
Opportunity 鈥� Photoresist simulation
The semiconductor industries aim to make chips or dies with a yield of 90 to 99%. Dies can consist of up to 10^8 to 10^{10} parts or features that together form the Integrated Circuit (IC) of the die. In most dies, all features need to work for the die to work. As such, the semiconductor industries are looking for ways to obtain insight into the photolithography printing process, so as to optimize designs and parameter settings. For this, we have developed a stochastic model for the processes driving photolithography printing. Simulation and analysis of these help to improve design and insight into the printing processes.
Roadblock - Errors arise through extreme-value events
With 10^8 to 10^{10} parts or features per die and yields of 90 to 99%, error probabilities for printing a single feature are in the range of 10^{-9} to 10^{-12}. Such small error probabilities are called extreme-value events, and make the most common simulation schemes, such as Markov Chain Monte Carlo, infeasible. This makes it difficult to improve the process design, and to go to smaller critical dimensions (CDs) in the feature printing, while keeping the yield at healthy levels. Further, even when one has significant insight into which processes contribute in what extent to the failures of features and thus chips or dies, it is unclear how efforts should be prioritised to obtain optimal return on the investment (extreme-value budgeting).
Research hypothesis
Stochastic modelling of the photolithography process, as well as the inhomogeneities in the die design due to differences in focus, location on the wafer, etc., allows us to obtain a physical model for the printing process of the various features that occur on a die. It also allows to quantify the improvements of the various processes that may give rise to yield loss. Therefore, this allows us to budget, on the basis of the model, what improvements need to be prioritised, and how much slack the design can tolerate.
Solution, research questions and timeline
In order to obtain accurate approximations for extreme-value failure probabilities for given parameter and design settings, one needs to extrapolate from error probabilities that one can simulate, to levels that are several orders of magnitude smaller. We have made first steps in proposing the required physical photolithography models, to include the stochastic effects of photon and PAG distributions, as well as electron and acid blurs, resulting in a flexible class of models for the lithography process in different settings. We aim to extend these models to include, for example, focus differences and quenching.
We propose advanced simulation techniques, based on importance sampling, to obtain insight into the most likely way how the printing process of a feature can fail. Roughly speaking, the importance sampling technique changes the system parameters to enhance the error probabilities (for example, by increasing, or rather decreasing, the light intensities), and accounting for the change in parameters by twisting the error estimates in such a way that the estimates remain unbiased. While possibly being highly effective to accurately estimate very small probabilities, importance sampling relies on insight in the precise ways that such errors are most likely to occur if they do occur. For the rather involved photolithography process, it is crucial to enhance this insight.
In the coming four years, we aim to investigate the following questions:
- How do the error probabilities depend on the parameters of the photolithography process (PAG concentration, light intensity, focus, blurs, printing various shape such as contact holes or lines, overlay errors, etc.)?
- How can these parameters be optimized? How can we cleverly budget the efforts on the various error sources, given that every improvement requires investments, so that we get the most of our money?
An example where this strategy was successfully carried out was the so-called black border effect, where light from neighbouring fields interferes with the lithography process. This creates extra high-intensity areas close to the corners of the printing fields (figure on the right).
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