**5. Illustrative Example**

In the manufacturing industry, semiconductor lithography (photolithography) refers to the formation of three-dimensional images on the substrate for subsequent transfer of the pattern to the substrate. A keynote aspect of this process is the bake process, both the pre (soft)-bake and post (hard)-bake. In this section, we implement the Shewhart chart with the proposed outlier detection models on the flow width measurement of a hard bake process. In the subsequent subsections, we give a brief overview of the hard-bake process and then application of the Shewhart chart on the dataset extracted from such a process (the Basics of Microlithography n.d.).

### *5.1. The Post (Hard) Bake Process*

A typical photolithography process consist of the following sequence of operation: substrate preparation, photoresist spin coat, pre-bake, exposure, post-exposure bake, development and finally the post-bake. The hard-bake process, as the name implies, is used to harden the final resist image so that it will withstand the harsh environments of etching. This post-bake ensures complete removal of solvent, improving adhesion in wet etch processes and resistance to plasma etches. Practitioners use di fferent temperatures depending on the material under study. However, the temperature should be carefully chosen and not more than 200 ◦C. A major characteristic of this process is the wafer. Recall that the word lithography is a combination of two Greek words: lithos meaning stones and graphia, meaning to write. Our stones in this case are silicon wafers and the patterns are written with photoresist, which are sensitive polymers. Figures 6 and 7 depict a typical photolithography flowchart and the hard-bake process.

**Figure 6.** A flowchart of a photolithography process of semiconductor manufacturing industry.

**Figure 7.** Illustration of hard-bake process.

### *5.2. Application of Shewhart Control Charts with Outlier*

In this section, we implement the findings of this study on a set of data generated from a semiconductor manufacturing of a hard-bake process, which monitors the flow width measurement of wafers [1]. The variable of interest is the flow width measurement (in microns) for the hard-brake process. The data consist of 25 IC phase-I samples and 10 phase-II samples each of sample size 5. The process mean and standard deviation of the phase-I samples are 16.7163 and 3.5167, respectively. Therefore, we use these estimates to setup Shewhart chart control limits for monitoring phase-II samples. Figure 8 shows all phase-I sample points staying within the limits and 3 of the phase-II sample points stretching beyond the LCL making them OoC due to some assignable cause of variation.

Prior to setting the limits, we test the data for possible autocorrelation. The data is autocorrelation-free as the Durbin–Watson (DW) test result proves. The value of the DW test statistics is DW = 1.7564 and the critical values at 1% level of significance are *dL* = 1.19, *and dU* = 1.31. By the interpretation explained in Table 9, we fail to reject the null hypothesis and conclude that there is no evidence of autocorrelation in the data.

**Figure 8.** Scatter plot of phase-I sample and the Shewhart chart with estimated parameters.



Furthermore, we introduce a 5% of outliers to the phase-I samples, to illustrate the argumen<sup>t</sup> that the presence of outliers affects the performance of control charts. This subsequently increased the mean and standard deviation by 4% and 25% respectively resulting to an increased UCL and decreased LCL. The changes in the control limits implies a wider range of the boundaries. Therefore the resulting control charts is less efficient as compared to the previous one without outliers. Figure 9 depicts this.

**Figure 9.** Scatter plot of phase-I sample and the resulting Shewhart chart with estimated parameters and 5% of outliers with magnitude 3.

### *5.3. Application of Shewhart Outlier Detection Model*

Having established the deficiency of the Shewhart chart with outliers on the dataset; we employ our proposed outlier detection model with the Shewhart chart explained in Section 2.4 to rectify this shortcoming. Figure 10 shows the application of the Shewhart Tukey-based model. It is evident there in that the chart was not only able to restore the efficiency of the chart as there were no outliers, detecting 3 OoC sample points, but also to identify the outliers in the phase-I sample points. Similarly, Figure 11 portrays the scenario when the Shewhart MAD-based model is applied on the monitoring stage. Despite the presence of outlier in the dataset, the chart is able to detect the OC sample points as much as it does when there were no outliers.

**Figure 10.** Scatter plot of phase-I sample and the resulting Shewhart chart with Tukey outlier detection screening.

**Figure 11.** Scatter plot of phase-I sample and the resulting Shewhart chart with MAD-outlier detection screening.
