## 4 Modeling of Process Chamber Vacuum Leakage  
The presented in situ vacuum leak detection method follows ideal gas Equation (1) characteristics. \[PV=nRT \tag{1}\]  
Each variable in the equation can be related to the chamber parameters, where: \(P\) is the chamber pressure; \(V\) is the chamber volume; \(n\) is the amount of substance of gas; \(R\) is the ideal or universal gas constant; and \(T\) is the temperature of the gas. The volume (\(V\)) and the gas constant (\(R\)) are regarded as constants, and the pressure (\(P\)), gas amount (\(n\)), and temperature (\(T\)) are considered as variables. \[P\propto nT \tag{2}\]  
Note that \(P\) is proportional to \(n\) and \(T\) (2). In summary, an increase in gas input and chamber temperature causes a rise in pressure inside the process chamber. Based on this relationship, the following hypotheses were drawn:  
**Hypothesis 1 (H1). **: _When the chamber body temperature increases, the APC position value increases;_ \begin{table}
\begin{tabular}{c c} \hline \hline
**Condition** & **Value** \\ \hline O\({}_{2}\)-N\({}_{2}\) total gas flow rate (sccm) & 14,000 \\ Source power (W) & 4400 \\ Pressure (mTorr) & 1500 \\ Chuck Temperature (\({}^{\circ}\)C) & 250 \\ Main process time (s) & 20 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Process values. Figure 2: Schematic of the screw-type thermocouple sensor (PSK). **Hypothesis 2 (H2). **: _When a vacuum leak occurs, the APC position value increases;_  
**Hypothesis 3 (H3). **: _When a vacuum leak occurs, the chamber body temperature does not change._  
In detail, the APC changes its position value by reacting to the current pressure of the chamber to maintain a certain target value. When the chamber pressure rises, due to an increase in chamber body temperature or due to an increase in outside air caused by the chamber vacuum leak, the APC position increases (becomes more open). On the contrary, when the chamber pressure decreases, due to a decrease in chamber body temperature or eliminated chamber vacuum leaks, the APC position decreases (becomes more closed). In addition, the outside air caused by the vacuum leak is considerably smaller than the volume of the chamber and amount of supply gas, meaning that the temperature of the chamber body will be maintained without any changes. The assumption was confirmed through a relationship test between the temperature and APC position. **5. Test of Relationship between Temperature and APC Position**  
In this section, a simple test with two parameters (APC position and body temperature) empirically proves the proposed conception to be valid. In the equipment idle status, a total of 520 process runs were performed using same values. In those, 472 normal process runs were conducted and 48 faults (vacuum leak valve open) were reproduced. The test results are shown in Figure 3.  
The blue circles indicate the APC position as single point values of the stable state before plasma ignition begins, so the obtained values in that section were not affected by other factors of the plasma process. The orange circles indicate the APC position as single point values of the disturbed state where the vacuum leak valve was opened. The red circles belong to the chamber body temperature that was acquired in parallel to the APC position values. The chamber body temperature was keep maintained at 60 \({}^{\circ}\)C in the idle state of equipment where the heater chuck temp was 250 \({}^{\circ}\)C. As the process continued, the thermal energy generated from the plasma raised the chamber body temperature. The gas supplied to the chamber received thermal energy from the chamber body and the pressure increased. The test results below prove the assumption to be true. **Proof of Hypothesis 1 (H1). ** The APC position opens (APC position value increases) more to reduce the elevated pressure. For normal overall data, the correlation coefficient between  
Figure 3: Test results of relationship between chamber body temperature and APC position.