## 3 Results and Discussions
The capacitance of the AZO/Si heterojunction can be described as below:  
\[\frac{1}{C^{2}}=\frac{2}{qA^{2}}\frac{\varepsilon_{AZO}N_{AZO}+\varepsilon_{Si} N_{Si}}{\varepsilon_{0}\varepsilon_{AZO}\varepsilon_{Si}N_{AZO}N_{Si}}(V_{bi}-V) \tag{3}\]  
where \(\varepsilon_{0}\): free space permittivity, \(\varepsilon_{AZO}\), and \(\varepsilon_{Si}\): relative dielectric constants of AZO and Si, \(N_{AZO}\) and \(N_{Si}\): doping concentrations of AZO and Si, \(V_{bi}\): build-in voltage, \(V\): bias voltage, and \(A\): device area. Since the doping level of AZO is much higher than that of Si (i.e., \(N_{AZO}\gg N_{Si}\)), Equation (3) can be rewritten as  
\[\frac{1}{C^{2}}=\frac{1}{A^{2}}\frac{2}{q\varepsilon_{0}\varepsilon_{Si}N_{Si}} (V_{bi}-V) \tag{4}\]  
Using Equation (4), the \(qV_{bi}\) values were extracted via the linear fitting of \(\frac{1}{C^{2}}\) vs. \(V\). As shown in Table 1, one can see that moderately increasing Al doping concentration in AZO thin films from 4.65 wt.% to 6.34 wt.% could significantly improve AZO/n-Si \(qV_{bi}\) values from 0.21 eV to 0.74 eV. The higher \(qV_{bi}\) value could benefit the photogenerated carrier separation and suppress recombination at the junction interface, providing better current transport efficiency with higher V\({}_{\text{oc}}\) and FF performance [41]. \begin{table}
\begin{tabular}{c c} \hline
**Al Doping in AZO (wt.\%)** & **Built-in Potential (\(qV_{bi}\)) of AZO/n-Si Junction** \\ \hline
4.65 wt.\% & 0.21 eV \\
5.78 wt.\% & 0.46 eV \\
6.34 wt.\% & 0.74 eV \\ \hline \end{tabular}
\end{table} 
Table 1: Built-in potential (\(qV_{bi}\)) of AZO/n-Si heterojunction with different Al dopings in AZO. Figure 3: Secondary ion mass spectrometry (SIMS) analysis of ITO/AZO/n-Si heterojunctions. (**a**)–(**c**) showed the depth profiles of indium (In), oxide (O), zinc (Zn), aluminum (Al), and silicon (Si) in ITO/AZO/n-Si heterostructures with 4.07 wt%, 5.22 wt %, and 6.34 wt% of Al doping in the AZO inter-layers, respectively. ITO (\(\sim\)250 nm) was grown by a DC magnetron sputtering. AZO (\(\sim\)40 nm) layer was grown by the dual-beam co-sputtering method. In order to understand the increased \(qV_{bi}\) values and the related junction interface characteristics, the AZO crystallinity and the thin film residual stress were systematically investigated. In this work, the AZO thickness was relatively thin (40 nm), thus the residual stress is a significant factor of AZO thin film properties as reported [42]. As shown in Figure 4a, the AZO thin films with various Al doping concentrations (i.e., 4.65 wt%, 5.22 wt %, and 6.34 wt%) all showed XRD diffraction peak positions of 20 around \(\sim\)34.4\({}^{\circ}\), indicating a preferential crystalline orientation of (002) with the c axis perpendicular to the substrates. For the AZO with Al doping of 4.65 wt.% (as shown in the upper panel of Figure 4a), the XRD peak was at 20 of 34.39\({}^{\circ}\), which is slightly lower than the pure (002) ZnO powder value (i.e., 20 = 34.47\({}^{\circ}\)). This slight deviation indicated a tensile residual stress parallel to the c axis caused by the imperfection of the AZO crystal structure and the atom impacting effects during the sputtering process. For AZO with slightly higher Al doping concentrations of 5.22 wt.% and 6.34 wt.% (as shown in the middle and bottom panels of Figure 4a), the peak position shifted to larger 20 angles of 34.43\({}^{\circ}\) and 34.49\({}^{\circ}\), respectively. This shift implied that the spacing between AZO (002) planes was shortened, and the tensile stress was reduced with the increase of Al doping. The ionic radius of Al\({}^{3+}\) (-53 pm) is smaller than that of Zn\({}^{2+}\) (-72 pm). When moderate Al doping atoms were introduced into AZO thin films substituting the Zn sites, the length between (002) planes was reduced with tensile stress released\({}^{2}\).