Kriging prediction from a circular grid: application to wafer diffusion

Abstract

In the production process of silicon wafers, which are crystalline slices used as substrate of electronic micro-circuits, the thickness of the SiO 2 deposition on their top is a main characteristic to be controlled during the process. The experimental design that is commonly used to monitor the thickness to the target value consists of a regular array of points lying on concentric circles, the silicon wafer itself being a disk. To speed up the control process, the engineers aim to use just only a limited subset of such points. To reconstruct the values on untried locations of the silicon wafer, the Kriging interpolation has been proposed because of its recognized ability in providing fairly good predictions. In this paper, we consider two methodological issues related to universal Kriging models. First, we discuss the modeling of the covariance structure among the measured points; in fact, spatial data usually show a strong correlation when they come from spatially near observed points. Second, we put forward an algebraic method to assess the identifiability of trend models, based both on the full experimental design and on special fractions of it. Our findings are illustrated by a data set from an industrial application.

Polar Gaussian Processes and Experimental Designs in Circular Domains

“…These two cases might correspond to the "two clusters of profiles over a circular grid" mentioned in [24] without any additional information about their origin. A knowledge of the process or historical data may help to choose which kernel is appropriate.…”

“…For such models, it is shown that optimal design of experiments are included in concentric circles [5]. More recently, a stochastic model consisting of a Gaussian process (GP), also called Kriging, has been proposed for microelectronics applications [24]. Among the existing interpolation and approximation methods, Kriging models are famous for their ability to provide both accurate prediction and uncertainty quantification, as pointed out in [14].…”

“…Here, Zernike polynomials [28] are good candidates since they constitute an orthogonal basis for the usual scalar product on D. Their shape including regular patterns are suited to describe symmetries or rotations. They were recently used for predicting on a disk [24]. The first Zernike polynomials, up to order 2, are shown in Fig.…”

Fast sensitivity analysis methods for computationally expensive models with multi-dimensional output

“…More recently, there has been considerable interest in using this kriging emulator approach for practical purposes such as GSA or inverse modelling (Marrel et al, 2009;Roustant et al, 2012). Examples of its application can be found in atmospheric modelling Lee et al, 2013), medicine (Degroote et al, 2012) and electrical engineering (Pistone and Vicario, 2013).…”

Response by author to interactive comment by RC1 on “Fast sensitivity analysis methods for computationally expensive models with multidimensional output” by Edmund Ryan et al.

“…More recently, there has been considerable interest in using this kriging emulator approach for practical purposes such as GSA or inverse modelling (Marrel et al, 2009;Roustant et al, 2012). Examples of its application can be found in atmospheric modelling Lee et al, 2013), medicine (Degroote et al, 2012) and electrical engineering (Pistone and Vicario, 2013).…”