Experimental design methods for nano-fabrication processes

Most design of experiments assumes predetermined design regions. Design regions with uncertainty are of interest in the first chapter. When a design region is restricted by inequality engineering constraints like solubility constraints (Chrastil, 1982), there might exist uncertainty in a design region. This uncertainty comes from the fact that engineering models are based on simplified assumptions. If a design region has uncertainty on its boundary, any data collection plan (e.g., optimal designs) that depends on the location of the design region boundary is not appropriate. Optimal designs tend to place many design points at the extreme limits of boundary regions. However, the boundary of the region is not precisely known in many engineering experiments (e.g., mechanical and chemical experiments). This chapter proposes optimal designs under a two-part model to handle the uncertainty in the design region. In particular, the logit model in the two-part model is used to assess the uncertainty on the boundary of the design region. This chapter derives the information matrix of the two-part model and constructed optimal designs. Through several examples, we show how the two-part models explain uncertainty in design regions and can be used for inequality engineering constraint estimation.;The second chapter proposes an efficient and effective multi-layer data collection scheme (Layers of Experiments) for building accurate statistical models to meet tight tolerance requirement commonly encountered in nano-fabrication. In nanofabrication processes, due to high material costs and processing time for physical experiments, number of experimental runs is very limited. However, the limited resources make it difficult to estimate statistical models that are required to be accurate enough to meet a tight tolerance requirement. To overcome these difficulties, “Layers-of-Experiments” (LOE) obtain sub-regions of interest (layer) where the process optimum is expected to lie and collect more data in the sub-regions with concentrated focus. An evaluation metric is developed to measure the performance of statistical models for nano-fabrication quality prediction and the metric is used to decide whether further layers are needed. This chapter also discusses appropriate types of designs for each layer, e.g. space-filling designs or optimal designs.;The third chapter contributes a new design criterion combining model-based optimal design and model-free space-filling design in a constraint and compound manner. Optimal design criterion is for precise statistical inference, while the space-filling design criterion is for exploration over the design space. The weights between the two criteria in the combined design is controlled by an adaptive parameter (κ) depending on the available information provided for a specific application (see chapter 4 for more examples). The proposed design is useful when the fitted statistical model is required to have both characteristics: accuracy in statistical inference and design space exploration. We showed that combined designs have properties between optimal designs and space-filling designs and they are robust against model misspecification. Moreover, combined designs perform better than space-filling designs or optimal designs where partial information about underlying model is available.;The fourth chapter proposes a method to determine the adaptive parameter (κ) sequentially in the layers of experiments. The parameter reflects the uncertainty of each layer, e.g., less uncertainty on the design space, more weights on model-based optimal criteria. This chapter also develops methods to improve model quality by combining information from various layers and from engineering models. Combined designs are modified to improve its efficiency by incorporate collected field data from several layers of experiments. Updated engineering models are used to build more accurate statistical models.