## 4 Application to RTCVD System  
### RTCVD System  
Figure 4 displays the schematic of the rapid thermal chemical vapor deposition (RTCVD) system. Figure 4: Sketch for the RTCVD system. Figure 3: Flowchart of SPSA learning-based three-dimensional fuzzy modeling (three-dimensional abbreviated as 3D). A 6-inch silicon wafer is positioned on a revolving support inside the system's chamber and subjected to heat from a heating system. The heating system consists of three Lamp banks. Lamp bank 1 uniformly warms the whole surface area of the wafer, Lamp bank 2 specifically heats the periphery of the wafer, and Lamp bank 3 equally heats the wafer as a whole. Figure 5 displays the incident radiation flux emitted by the heating lights. A 10% concentration of silane gas (SiH4) is fed into the reactor. SiH4 undergoes decomposition to produce silicon (Si) and hydrogen gas (H2). A layer of polysilicon, with a thickness of 0.5 \(\upmu\)m, is applied onto the wafer by depositing it at temperatures of about 800 K or more, which takes about 1 min. During wafer processing, the support is turned to ensure that the temperature is evenly distributed in the azimuthal direction. Due to the thinness of the silicon wafer, all changes in temperature along the azimuth direction are disregarded. Hence, it is necessary to maintain consistent temperature distribution over the wafer radius in order to achieve a homogeneous deposition of polysilicon on the wafer. This may be achieved by regulating the power supplied to the three zones of lighting groups. A one-dimensional space model of the thermal dynamics may be expressed as the following PDE [25; 26]:  
\[ \begin{split}\partial T_{f}^{\prime}/\partial k^{\prime}=& _{\kappa_{0}}\Big{[}(1/f^{\prime})\partial T_{f}^{\prime}/\partial f^{ \prime}+\partial^{2}T_{f}^{\prime}/\partial f^{\prime 2}\Big{]}\\ &+\sigma_{0}\Big{(}1-T_{f}^{\prime 4}\Big{)}+\omega_{f}\varrho_{1 }\big{(}f^{\prime}\big{)}q_{1}\\ &+\omega_{f}\varrho_{2}\big{(}f^{\prime}\big{)}q_{2}+\omega_{f} \varrho_{3}\big{(}f^{\prime}\big{)}q_{3}.\end{split}  \tag{17}\]  
This is constrained by the specified border conditions:  
\[\left\{ \begin{array}{ll}\partial T_{f}^{\prime}/\partial f^{\prime}=\sigma_ {ed}\Big{(}1-T_{f}^{\prime 4}\Big{)}+\varrho_{ed}q_{2},&\text{when }f^{\prime}=1,\\ \partial T_{f}^{\prime}/\partial f^{\prime}=0,&\text{when }f^{\prime}=0. \end{array} \right. \tag{18}\]  
where the nondimensional wafer temperature, denoted by \(T_{f}^{\prime}=T_{f}/T_{amb}\), represents the ratio of the actual wafer temperature \((T_{f})\) to the ambient temperature \((T_{amb})\), which is 300 K. The nondimensional time, denoted by \(k^{\prime}=k/\tau\), represents the ratio of the actual time \((k)\) to \(\tau\), which is 2.9 s. The nondimensional radius position, denoted by \(f^{\prime}=f/V_{f}\), represents the ratio of the actual radius position \((f)\) to the wafer radius \((V_{f})\), which is 7.6 cm. The variables \(q_{1}\), \(q_{2}\), and \(q_{3}\) represent the percentage of the light source power. The variables \(\varrho_{1}(f^{\prime})\), \(\varrho_{2}(f^{\prime})\), and \(\varrho_{3}(f^{\prime})\) represent the wafer incident radiation flux from the three-zone warming lights, respectively. The parameters in (17) through (18) are listed as follows:  
\[ \begin{split}&\kappa_{0}=0.0021,\sigma_{0}=0.0012,\sigma_{ed}=0.0037,\\ &\varrho_{ed}=4.022,\omega_{f}=0.0256.\end{split} \]  
Figure 5: Radiation flux distribution of Lamp banks 1, 2, and 3. The RTCVD system is a complex system that exhibits both spatial and temporal characteristics and has an unlimited number of dimensions. For practical purposes, a certain number of sensors must be used. Assume that sensors positioned at \(f_{1},f_{2},\cdots,f_{P}\) are employed to gauge the system's output. Consider \(\bar{F}\) to be the spatial domain, where \(\bar{F}\) is a vector containing the elements \(f_{1},f_{2},\cdots,f_{P}\). Let \(w(\bar{F},k)\) be a vector denoted by \(w(f_{1},k),w(f_{2},k),\cdots,w(f_{P},k)\), where \(w(f,k)=T_{f}(f,k)\).