## 3 Results and Discussion
The detailed simulation parameters are summarized in Table 2, and the cell structure is shown in Figure 2a. Some values are taken from the literature [17]. The most important thing to assess the performance of PERC solar cells is the open-circuit voltage (\(V_{oc}\)), which can be calculated using the one-diode equation:  
\[V_{oc}=\frac{kT}{q}\ln\left(\frac{J_{L}}{J_{0b}+J_{0e}}+1\right) \tag{4}\]  
where \(k\) is the Boltzmann constant, \(T\) is the temperature, \(q\) is the electron charge, \(J_{L}\) is the generated light current, \(J_{0b}\) is the base saturation current density, and \(J_{0e}\) is the emitter saturation current. The rear surface passivation affects the value of \(J_{0b}\), which is given by  
\[J_{0b}=\frac{q{n_{i}}^{2}D_{p}}{LN_{D}}\frac{S_{ren,eff}\cosh\left(\frac{W}{L} \right)+\frac{D_{p}}{L}sinh\left(\frac{W}{L}\right)}{\frac{D_{p}}{L}\cosh \left(\frac{W}{L}\right)+S_{ren,eff}\sin\left(\frac{W}{L}\right)} \tag{5}\]  
where \(n_{i}\) is the intrinsic concentration of silicon, \(D_{p}\) is the hole diffusion coefficient, \(L\) is the diffusion length, and \(N_{D}\) is the n-type silicon doping concentration. It should be noted that, as a PERC solar cell has local openings on the rear side, the silicon wafer is in contact with either the HfO\({}_{2}\) passivation layer \begin{table}
\begin{tabular}{c c c c} \hline
**Temperature (\({}^{\circ}\)C)** & \(\tau_{off,low}\) (\(\mu\)s) & \(\tau_{off,high}\) (\(\mu\)s) & \(\tau_{off,low}\)/\(\tau_{off,high}\) \\ \hline As-deposited & 0.5 & - & - \\
400 & 14.3 & 15.2 & 0.94 \\
450 & 44.8 & 48.2 & 0.93 \\
500 & 67.2 & 74.7 & 0.9 \\
550 & 18.5 & 27.3 & 0.68 \\
600 & 14.3 & 22.1 & 0.65 \\
650 & 11.8 & 19.4 & 0.61 \\ \hline \end{tabular}
\end{table} 
Table 1: Low- and high-injection-level minority carrier lifetime and their ratios for the HfO\({}_{2}\)-passivated silicon wafers at different annealing temperatures. Figure 3: Fixed charge density (\(Q_{f}\)) and interface defect density (\(D_{it}\)) of the HfO\({}_{2}\)-passivated Si wafers with different annealing temperatures. Figure 5 shows simulated \(V_{oc}\) and \(\eta\) as a function of base saturation current density. In this work, \(J_{ob}\) values of 180-460 fA/cm\({}^{2}\) are obtained for the single HfO\({}_{2}\) rear passivation layer. For PERC solar cells with an SiN\({}_{x}\):H/HfO\({}_{2}\) layer, we obtain a \(S_{max,eff}\) of 22 cm/s with a \(J_{0b}\) of 64 fA/cm\({}^{2}\), a \(V_{oc}\) of 689 mV, and an \(\eta\) of 23.02%. This great improvement might be because the SiN\({}_{x}\):H layer can provide hydrogen atoms diffusing toward the substrate surface to further passivate the SiO\({}_{x}\)/Si interface, similar to the case of the SiN\({}_{x}\):H/Al\({}_{2}\)O\({}_{3}\) structure [19,20]. From Equation (5), the lower limit of \(J_{0b}\) is 33.9 fA/cm\({}^{2}\), corresponding to \(V_{oc}\) of 694 mV and an \(\eta\) of 23.28% for the ideal rear passivation (\(S_{max,pass}=0\)). The small gap between the cell with ideal passivation and the cell with an SiN\({}_{x}\):H/HfO\({}_{2}\) stacked layer indicates the high passivation quality of the HfO\({}_{2}\)-based technique for n-type PERC solar cells.