# Kernel-Density-Based Particle Defect Management
## 2 The Conventional Industrial Method  
In an actual semiconductor manufacturing process, conventional particle defect management involves manual action by engineers if, based on the monitoring chart, there is a cause for alarm. The procedure consists of three phases for the detection of the cause of the defect. In the first phase, when an alarm occurs, the insides of the manufacturing equipment are repeatedly cleaned. In the second phase, if the problem persists, the field engineers replace suspicious parts. In the final phase, if the previous replacement does not solve the prevailing problem, predictive maintenance (PM) of the equipment is performed. The second phase, in particular, consumes a substantial amount of time in terms of both decision-making and detection of the root cause, owing to an experience-based decision of the replacement. Therefore, reducing false alarms in the first phase is critical. Among the aforementioned particle defect management phases, we focus on the first phase, wherein the particle defects are monitored. Particle defects in process equipment are primarily due to equipment aging. Other defects occur due to environmental factors. The counts control chart (c-chart) is commonly used to monitor particle defects. The c-chart is based on the assumption that the distribution of the number of nonconformities is sufficiently well fitted by a Poisson distribution [8]. However, in the actual semiconductor manufacturing process, the particle count does not follow a Poisson distribution [9]. Especially, in our case, random variables are not independent since the particles are cumulated with a time-varying property. Figure 5 shows the metrology equipment gauging the number of particles on the wafer. The number of particles, \(N\), at each coordinate on the wafer is stored. Figure 4: Change in the particle defect size with respect to the pitch. Figure 5: Particle measurements on a test wafer. Moreover, these particles indicate the status of the equipment, not of the wafer. Therefore, in conventional monitoring statistics, the particle defect (PD) that indicates the number of particles on a test wafer can be denoted as follows:  
\[PD=\sum_{i}N_{x_{i},y_{i}} \tag{1}\]  
where \(x_{i}\) and \(y_{i}\) are the \(i\)-th particle coordinates, and \(N_{x_{i},y_{i}}\) indicates the number of particles at the designated coordinate. The particle coordinates can be used to generate a wafer map to obtain the distribution and number of particles. Figure 6 depicts four illustrative particle defect maps for four processes. The particle defect maps indicate high-density particles at a particular location on the wafer and the location is correlated with the structure of the process equipment. As indicated by the dotted ellipse in each particle defect map, a high-density area indicates that an ongoing wafer has a potential defect. The particle defect map of the oxidation is also affected by the structure of the equipment. Field engineers have discovered a problem at the center position, where the gas is injected into the furnace, using the particle map. The lithography process shows a higher density at the edges of the wafer. The lithography process includes a photo resist (PR) coating using a high-speed rotation of the spinner equipment in the wafer process. Therefore, the remnants of the chamber affect the density of the wafer edges owing to the characteristics of the high-speed spinning process. Both the etching and strip & clean processes exhibit the highest densities where the dotted ellipse is drawn. Engineers have checked the equipment that analyzes the causes of this phenomenon and discovered a particle resulting from the improper fastening of the O-ring in the chamber. Further, as the strip & clean process includes multiple baths that form bias chemicals at the bottom wafer, the particle defect map indicates high densities of the particles in the bottom area. A real domain conducts monitoring based on the control chart in Equations (2)-(4) to monitor the particle counts [9].