Correlation Analysis between Wafer Acceptance Test and In-line Data for Process Control in Semiconductor Manufacturing
Date Issued
2009
Date
2009
Author(s)
Lu, Chun-Yao
Abstract
Yield ramping and feature size shrinking continuously lead to tightening of process control. In semiconductor fabrication, process control is a key element for successful IC manufacturing. As IC technology advances towards the nanometer era, the concept of advanced process control (APC) has been proposed and implemented in semiconductor manufacturing. Feed-forward control is one of the APC strategies for manufacturing processes.orrelation analysis of Wafer Acceptance Test (WAT) data and In-line data is needed for yield control and fault diagnosis of the fabrication processes. Through the measurement data and correlation analysis, statistical methods are used to characterize the process status and establish a correlation model. Currently, regression analysis is one of the popular correlation analysis methods for fitting correlation models between WAT and In-line data with “single-model” assumption. However, “multiple models” due to model indicator which is an index to separate multiple models and could be the recipe names, equipment/chamber IDs, and results in the unsatisfied R-square performance. Unfortunately, the model indicator may not manifest itself on the data. Therefore, the model indicator always does not appear in the dataset and it is a “hidden variable.” To cope with the multi-model issue in correlation analysis, the main challenge is neither hidden variables nor regression parameters is pre-known.n this research, we extend the Expectation-Maximization (EM) algorithm for identifying the multiple regression models. The key ideas are: given the initial solution first, then iteratively estimate the hidden variables and regression parameters until convergence. However, direct application of the standard EM algorithm for regression has two challenges: local optimum and overfitting. Local optimum comes from a greedy search strategy such as EM algorithm whose performance depends on the initial solution and it is easy to converge with one of the numerous local minima. To solve the local optimal problems, we try many different initial solutions to find the effective local optimum which approximates global optimum. Overfitting is a common problem in statistics and data mining that gives data too much explanation including minor fluctuations or random error in the data. The outcome of the multi-model is definitely better than single-model, but now there is no measurement and metric capable of determining if the model is single-model or two-model. So we need to use statistical metric for evaluating the single-model against the multi-model. Finally, we develop an EM-Based-Regression (EMR) algorithm which contains above ideas to resolve these two problems.o validate global optimum approximation and overfitting avoidance of EMR algorithm, we conduct simulations to evaluate performance. Three types of simulation models are designed: single model, parallel two models, and cross two models. For global optimum approximation, we propose likelihood as the criteria of convergence estimation as the variation of likelihood is minor and local optimum selection when likelihood maximization. Then confident interval of the estimated regression parameters covers setting of model coefficients, so there is high probability to approach global optimum. For the overfitting problem, we compare the performance of several statistical metrics, which consists of F-statistic, Bayesian Information Criterion (BIC), and empirical F-statistic, to evaluate the statistic significance. In conclusion, the empirical F-test and the BIC-test achieve a better performance in distinguish model significance, and control the false identification rate (or Type I Error) and Miss Detection Rate (or Type II Error) less than 0.05.
Subjects
process control
semiconductor manufacturing
correlation analysis
WAT-In-line regression modeling
multi-model identification
EM algorithm
Type
thesis
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