# Sequential and Comprehensive Algorithm for Fault Detection in Semiconductor Sensors
## 3 Results and Discussion  
The individual inspection was performed for the stabilization of each sensor. We could clearly discern the process ready to be initiated when all sensors were in control normally. The software we developed in this study could inspect each sensor's status through the SPC method. The V001 sensor was a timer representing the step change, showing that the process consisted of six steps (Figure 3A). V009 is exemplified for the peak sensor group, indicating the peak of which appeared twice: steps 2 and 4 (Figure 3B). Such an output was repetitive during the whole process. We separately extracted the peak at each step and applied the SPC method to each preprocessed data (Figure 2B). The fraction of the outlier over the dataset was 33.3% in both steps, respectively, during 25 processes. It might be in a stabilization where the sensor output could be fluctuated. When there was none of the data beyond UCL and LCL, we could proclaim the stable state. This SPC method made the process wait for the sensor output to become stable against time. As the sensor output becomes stable against time, the process gets the full-fledged initiation. Additionally, it helped to reduce the delay time of a wafer. Too long wafer delay time might damage the wafer due to the high temperature or dense chemical gas concentration [33]. This could become a trigger of some faults generated in the afterward manufacturing process. During the process, we assumed that the effect of time on the sensor output could be negligible as compared to the effect of the other property changes. At the end of every process, the initialization step (remarked as step 0) followed to become stable again. Some of the sensor pairs showed a correlation linearly or exponentially (Figure 4A). Such relations could be found in the semiconductor manufacturing process. For example, the thermal conductivity and electric conductivity were linearly related [34]. This relationship is called the conductivity-limiting phenomenon that the electric carrier mobility could control the bipolar thermal transport in semiconductors. Additionally, the mean energy of the electric carrier and electron concentration followed the Einstein relation \(\left(D=\frac{\mu_{\mathrm{C}}k_{B}T}{q}\right)\). In this equation, \(D\) is the diffusion coefficient, \(q\) is the electrical charge of a particle, \(u_{q}\) is the mobility of the charged particle, \(k_{B}\) is Boltzmann's constant, and \(T\) is the absolute temperature. This [PERSON] equation could be generalized for the semiconductor industry, which was a form of an exponential equation, as previously described [35]. For such correlated pairs, the faults could be found by mathematical calculations such as SVR. For the exponential relationship, the additional data processing was required, because it showed the threshold value to activate the meaningful reaction in semiconductors. Before the threshold value, the sensor output linearly changed with a gentle slope. Upon reaching the threshold, an exponential increase appeared. To determine the threshold value, we investigated the slope. When the slope was suddenly increased, the data point was meant to be the threshold value. However, there were some data points out of control in the area before the threshold. The algorithm could misinterpret the defective data to be a threshold since a sudden increase appeared at that point. For this reason, we concentrated on the slope that showed continuously sudden increases at least twice. Therefore, we divided it into two sectors and separately conducted a regression analysis to each sector (sector1: before  
Figure 4: Regression analysis to show the correlation between two sensors. (**A**) Grouping two sensors to observe whether they were correlated or not. Meaningful relationships in the semiconductor manufacturing process were linear and exponential. (**B**) Support vector regression method for the fault detection in the exponential relationship. This relationship could be divided into two sectors based on its threshold.