# Kernel-Density-Based Particle Defect Management
## 4 Experimental Results  
This section describes an experiment for verifying the proposed method and its results. The particle maps used for this experiment were retrieved from the etching process. As several \"killer-particles\" originate from the deposition of a layer of by-product on the inside of the plasma-etching chamber, the etching process is sensitive to particles [12]. Therefore, we used particle data from a real etching process in a Korean semiconductor manufacturing company, obtained over a period of five months. The entire dataset consists of 600 wafers, generating 600 particle maps. Among these particle maps, we defined the particle defect types using field knowledge. A dense particle indicates that the facility is abnormal. Therefore, the dataset with dense and even particles was selected. The four defect types are shown in Figure 10. Our dataset consists of 300 normal wafers, 100 wafers of Type 1, 100 wafers of Type 2, 50 wafers of Type 3, and 50 wafers of Type 4. These types were determined by experienced engineers. As shown in Figure 10, Type 1 shows an illustrative particle map of a small defect. This map includes only one particle cluster, shown on the left side of the map. Type 2 map has two or more  
Figure 9: Comparative particle defect (PD) and kernel-density-based particle defect (KDPD). small defects. The Type 3 map has only one large-sized defect and the Type 4 map shows the particle distribution on the entire surface of the wafer. According to the specified defect types, we generated four datasets for the experiment, as shown in Table 1.  
Dataset 1 includes non-defect wafer images and wafer images with Defect Types 1, 2, 3 and 4. On the other hand, Dataset 4 consists of non-defect wafers and Type 4 defect wafers only. In order to validate the monitoring performance for these datasets, we evaluated the use of \(KDPD\), compared to the use of the conventional monitoring statistics, to identify a defect. We used the monitoring method illustrated in Figure 7 to classify non-defect and defect particle maps. The threshold in the monitoring chart corresponds to the upper control limit (UCL) in the statistical process control chart. Therefore, we compared the classification performance of the particle defect map by adjusting the decision threshold of the defect/non-defect. Further, we used the receiver operating characteristic (ROC) curve, a graphical plot that illustrates the performance of a binary classification. Moreover, in order to compare these curves, the area under the ROC (AUROC) curve was considered [13]. A broader AUROC can detect a particle defect more accurately. As defect classification is more important than non-defect classification, the ROC curve uses a false negative rate for the \(X\)-axis and a true negative rate for the \(Y\)-axis. Figure 11 depicts the ROC curves for both PD and KDPD of the four datasets defined in Table 1, and the AUROC examines the monitoring performances. The ROC curves of Datasets 1, 2 and 3 show that the AUROCs of \(KDPD\) are broader than those of PD. Therefore, KDPD outperforms PD. Moreover, the results indicate an increase in the performance gap between PD and KDPD in proportion to the degree of the defects. On the other hand, Dataset 4 demonstrates that the conventional statistics (PD) outperforms KDPD. This dataset includes the Type \begin{table}
\begin{tabular}{c c} \hline \hline
**Dataset** & **Combination of Defect Types** \\ \hline Dataset 1 & Types 1, 2, 3 and 4 \\ Dataset 2 & Types 2, 3 and 4 \\ Dataset 3 & Types 3 and 4 \\ Dataset 4 & Type 4 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Defect datasets. Figure 10: Levels of the defects in the etching process. 4 defect as specified in Table 1 and is referred to as a fatal defect. This can be monitored using either PD or KDPD. Therefore, the proposed KDPD is appropriate for detecting Type 1, 2, and 3 defects. Since the Type 1, 2 and 3 defects indicate potential factors that can result in fatal defectives of Type 4, field engineers can detect critical defects in early stages.