Traditional fault detection methods have been model-based or knowledge-based approaches, which require considerable effort to build analytical models or knowledge-based systems. In order to address the difficulties that lie in model-based or knowledge-based methods, model-free statistical process monitoring (SPM) methods have been developed, which only require a good historical data set of normal operations. In today's semiconductor industry, massive amount of trace or machine data are generated and recorded. Because of the high dimensionality of the data, the principal component analysis (PCA) and partial least squares (PLS) based multivariate statistical fault detection methods, which were originally developed to monitoring continuous processes, have drawn increasing interest in semiconductor manufacturing industry. However, the unique characteristics of the semiconductor processes, such as most processes are batch (or single wafer) processes and each process consists of multiple steps, have posed some difficulties to these multivariate statistical methods. For example, in order to take explicit account of the time sequence, Multiway PCA (MPCA) method unfolds the data into two-dimensional data array. The consequence of MPCA un-folding is that the number of variables is increased dramatically, which requires significant amount of samples (wafers) to build a reliable PCA model in order to capture the key characteristics of the process.
In this work, two fault detection methods are developed to address some of the shortfalls of MPCA. One developed method is based on the k-nearest-neighbor (kNN) rule. The k-nearest-neighbor rule is an intuitive concept widely used in pattern classification where unlabeled samples are classified based on their similarities with samples in the training set . The developed fault detection method consists of three steps: model building, threshold determination and fault detection. In the first step, a model is built based on k-nearest-neighbor rule using training data set where kNN distance of each sample is calculated. In the second step, a threshold is determined based on a chi-square distribution of kNN distances. In the third step, the unlabeled sample is projected onto the model and its kNN distance is compared against the threshold to determine whether it is a normal or fault sample. The developed method only requires an integer k, a set of labeled samples (training data) and a metric to measure distances. Due to its simplicity, the implementation is straightforward.
The other developed method is based on the Mahalanobis distance which is widely used in pattern classification as a metric of distance. Mahalanobis distance is the statistical distance between two N dimensional points scaled by the statistical variation in each component of the point . Mahalanobis distance is a very useful measure of the similarities among different samples. It is superior to Euclidean distance because it takes parameter variabilities into account. However, its direct application in fault detection has not been reported. For fault detection, the normality of a new sample point can be identified with respect to reference samples based on its Mahalanobis distance to the reference base.
This work presents the proposed methods with a simple illustrative example, provides general guidance on model parameters and discusses their advantages over MPCA –simpler data pre-processing, elimination of data un-folding, nonlinearity and multimodal distributions handling, etc. The superior performance of the developed algorithms to MPCA is demonstrated using a data set  consisting of 21 variables from a LAM 9600 metal etcher over the course of etching 129 wafers where 21 different faults were intentionally introduced during the experiments.
Key words: semiconductor manufacturing, fault detection, statistical process monitoring, pattern recognition, k-nearest-neighbor rule
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