# Spatial Monitoring of Wafer Map Defect Data Based on 2D Wavelet Spectrum Analysis
## 4 Application and Discussion  
In this section, we applied the suggested algorithm to wafer bin map (WBM) data provided by a semiconductor company in Korea. This data contained the defective information of WBMs during the manufacturing processes, which provided the X-Y coordinate of defects in each WBM. As shown in Table 1, the identification code of the lot, timestamp, and relative position of defects from the center (\(0,0\)) could be obtained from the data. In order to monitor the continuous status in a single lot, the data generated from a specified lot were considered. Since the suggested procedure aims to control and detect the spatial defects of time series processing based on image analysis, we selected four main types of defective WBM images, as shown in Figure 2. These images consisted of a faulty condition caused by multiple patterns of local defects including scratch, circle, and semicircle (half). Each of these images was separately applied to each step in the suggested algorithm to illustrate the effectiveness of the proposed procedure clearly. \begin{table}
\begin{tabular}{c c c c c} \hline
**Lot ID** & **Timestamp** & **Defect ID** & **X** & **Y** \\ \hline
15 & 10:00:50 & 1 & 7.293e+03 & 1.267e+01 \\
15 & 10:00:50 & 2 & 2.695e+02 & \(-2.502\)e+04 \\
15 & 10:00:50 & 3 & \(-5.369\)e+00 & 0.012e+02 \\ \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) & \(\cdots\) \\ \hline \end{tabular}
\end{table} 
Table 1: Format of the original data for a single wafer bin map. Proceeding with the proposed method, the original data were transformed into image-type format, where the locations with faults were represented as colored points. Every image had a size of \(2^{10}\times\,2^{10}\) for 2D DWT, and the size of windows was fixed to \(2^{5}\times\,2^{5}\). With respect to the 2D DWT function, \"la8\" wavelet, the Daubechies least asymmetry length eight, was used. Since random defects in the overall area can be produced during semiconductor manufacturing, global defects were eliminated for effective quantification of faults using universal thresholding with the maximum decomposition level. In addition to noise elimination, the image should be made gray-scale to integrate WBM in one layer. The result of image construction and pre-processing is given in Figure 3, where the scattered points indicating global defects are minimized. After the pre-processing step, the image was divided into square-sized sub-images as windows. As mentioned in Step 3 in Section 3, the hierarchy of wavelet coefficients was obtained by 2D DWT per each window. In Step 4, the energy of the detailed coefficients for resolution levels was calculated so that the linear regression based on wavelet spectrum analysis could be conducted. Figure 4 presents the energy profiles of all windows derived from WBMs, where a single line describes the trend of single windows and the color of lines (red, blue, and black) represents the direction of wavelet transforms: diagonal (HH), horizontal (HL), and vertical (LH) coefficients, respectively. From the comparison of WBMs between the normal condition and abnormal condition with a circle defect, we can observe that the WBM with more faults had greater variation of energies, where the Hurst exponents (slopes of the linear equation derived from each window) were more distributed than the normal status. Figure 3: (**a**) Construction of wafer bin map image; (**b**) preprocessed image from (a). Figure 2: Four types of defective wafer bin maps (WBMs): (**a**) scratch, (**b**) circle, (**c**) half, and (**d**) circle with vertical defects. Using the Hurst exponents from every window, a distance matrix \(D\) was calculated from Step 4 in Section 3, where the derivation of distances from the background enabled us to quantify the strength of faults. Based on the matrix, the density of the distance elements, \(dist_{l}\) for \(l=1,2,\ldots,n^{2}\), was estimated.