Selecting subgrids from a spatial monitoring network: Proposal and application in semiconductor manufactoring process

Abstract

The monitoring of spatial production processes typically involves sampling network to gather information about the status of the process. Sampling costs are often not marginal, and once the process has been accurately calibrated, it might be appropriate to reduce the dimension of the sampling grid. This aim is often achieved through the allocation of a brand new network of less dimension. In some cases that is not possible and it might be necessary the selection of a subgrid extracted from the original network. Motivated by a real semiconductor problem, we propose a method to extract a monitoring subgrid from a given one, based upon grid representativeness, accuracy, and spatial coverage of the subgrid and, if available, by expert knowledge of the weights to be assigned to those areas where production may need greater precision. Discussion is mainly focused on circular spatial domain, since, in microelectronics, the basic production support, called wafer, is a circle. Straightforward generalizations to different spatial domains are possible. Furthermore, conditionally upon the availability of experimental data, we check the loss of accuracy by fitting a dual mean‐variance response surface on the reduced grid. Joining the latter information and the criteria used to select the subgrid, we provide additional guidelines on how to fine‐tune the subgrid selection. Real case studies are used to show the effectiveness of the proposal.

Model‐based process capability indices: The dry‐etching semiconductor case study

“…Because the proportional effect is clearly shown by the sample average and standard deviation (see Figure 2), we propose to model them jointly using a Generalized Additive Models for Location Scale and Shape (GamLSS 23 ).…”

“…This extensive information is also used to make inference on the response surface over the whole wafer. The aim is to check whether quality standards are also matched where data have not been collected or are not available 212223 …”

Model selection based on validation criteria for Gaussian process regression: An application with highlights on the predictive variance

“…For example, in Geostatistics, this step is done most of the times, in the stationary case, via variogram/covariance manual fitting using expert knowledge on the underlying phenomenon (to capture realistic correlation lengths, anisotropy, nugget effect, variogram/covariance slope at the origin, …). However, when the design of experiments is atypical (sparse data, asymmetric, large empty zones, …), this approach becomes critical 10 . Numerically, different techniques are proposed, such as the maximum likelihood method 11 or a cross‐validation strategy, 8 where the validation procedure and the associated criteria need careful attention.…”

“…), this approach becomes critical. 10 Numerically, different techniques are proposed, such as the maximum likelihood method 11 or a cross-validation strategy, 8 where the validation procedure and the associated criteria need careful attention.…”

The ENBIS‐16 quality and reliability engineering international special issue

“…Motivated by a real semiconductor problem, Zappa and Borgoni 5 propose a method to extract a subgrid from an initial monitored network to effectively maintain the spatial representativeness.…”