## 3 Results and Discussion
### Analysis of Passivation Properties  
Note that the implied open-circuit voltage (iV\({}_{\rm{oc}}\)), derived from implicit current-voltage characteristic curves of the QSSPC, is an essential parameter to evaluate the overall performance of a silicon solar cell, as it reflects the quality of surface passivation and the lifetime bulk carrier. The iV\({}_{\rm{oc}}\) of the double-sides symmetrical passivation structure, nc-SiO\({}_{\rm{x}}\)/SiO\({}_{2}\)/n-type Si/SiO\({}_{2}\)/nc-SiO\({}_{\rm{x}}\) layers, was studied exhaustively to estimate the surface passivation of the TOPCon structure. The iV\({}_{\rm{oc}}\) can be described using Equation (3):  
\[\rm{iV}_{\rm{oc}}=\frac{kT}{q}Ln\left(\Delta n\left(\frac{\Delta n+N_{dop}}{n _{i}^{2}}\right)\right) \tag{3}\]  
where \(\Delta n\) is the excess carrier density at one sun, k is the Boltzmann constant, T is the temperature, q is the elementary charge, N\({}_{\rm{dop}}\) is the bulk doping density, and n\({}_{i}\) is the intrinsic carrier concentration. N\({}_{\rm{dop}}\) and t of the wafer used for the calculation are \(5\times 10^{15}\) cm\({}^{-3}\) and 200 \(\mu\)m, respectively. Note that the carrier lifetime (\(\tau_{\rm{eff}}\)), which can be measured experimentally in a given sample, is an effective parameter that encompasses several different recombination and transport mechanisms, including surface recombination. \(\tau_{\rm{eff}}\) derived from the implicit current-voltage characteristic curves of the QSSPC, is an essential parameter to evaluate the overall performance of a silicon solar cell. The \(\tau_{\rm{eff}}\) of the double-sides symmetrical passivation structure, nc-SiO\({}_{\rm{x}}\)/SiO\({}_{2}\)/n-type Si/SiO\({}_{2}\)/nc-SiO\({}_{\rm{x}}\) layers, was studied exhaustively to estimate the surface passivation of the TOPCon structure. The \(\tau_{\rm{eff}}\) can be described using Equation (4):  
\[\tau_{\rm{eff}}=\frac{\Delta n(t)}{G_{\rm{L}}(t)-\frac{\Delta n}{\partial t}} \tag{4}\]  
where G\({}_{\rm{L}}\)(t) is the generation rate per unit volume. The deposited 30 nm thick nc-SiO\({}_{\rm{x}}\) sample was annealed at different temperatures in the range of 850-950 \({}^{\circ}\)C. As a result of the PDA, a substantial improvement was observed in the passivation characteristics after the PDA. As the thickness approached 30 nm, substantial improvements were observed in both the carrier lifetime (\(\tau_{\rm{eff}}\)) and implied open-circuit voltage (iV\({}_{\rm{oc}}\)). Figure 3a depicts the \(\tau_{\rm{eff}}\) and iV\({}_{\rm{oc}}\) of the 30 nm deposited sample at different post-deposition annealing temperatures (T\({}_{\rm{PDA}}\)). We found more prominent results with the samples in terms of \(\tau_{\rm{eff}}\) and iV\({}_{\rm{oc}}\), as compared to the as-deposited sample, but in the case of the 950 \({}^{\circ}\)C temperature, a more notable lifetime, and iV\({}_{\rm{oc}}\) of \(\sim\)941 \(\upmu\)s and \(\sim\)737 mV, respectively, was observed. In the case of Figure 3b, we attempt to demonstrate the response of TPDA on the 30 nm thick layer of the nc-SiO\({}_{\rm{x}}\) in terms of recombination current density (J\({}_{\rm{o}}\)). The gradual decrease in Jo valued up to a J\({}_{\rm{o}}\) of 1.26 fA/cm\({}^{2}\) was observed as the annealing temperature reaches 950 \({}^{\circ}\)C. At the annealing temperature in the range of 950 \({}^{\circ}\)C, increases in the \(\tau_{\rm{eff}}\) and iV\({}_{\rm{oc}}\), as well as the diminution in J\({}_{\rm{o}}\), were observed compared to the as-deposited state of the nc-SiO\({}_{\rm{x}}\) layer. After FGA treatment for 2 h at 400 \({}^{\circ}\)C by using rapid thermal processing, the passivation characteristic of the symmetric structure increased further. After FGA treatment, the maximum lifetime was improved by up to 1 ms and the iV\({}_{\rm{oc}}\) reached up to 738 mV, for the 30 nm thick nc-SiO\({}_{\rm{x}}\) which represented in Figure 4a. Under this condition, a substantial reduction was observed in J\({}_{\rm{o}}\) up to 1.1 fA/cm\({}^{2}\).