## 3 Results and Discussions
The actual residual stress (\(\sigma\)) of the AZO thin films can be calculated via Equations (5) and (6) as below [43]:  
\[\sigma=\frac{2c_{13}^{2}-c_{33}(c_{11}+c_{13})}{2c_{13}}\frac{c-c_{0}}{c_{0}} \tag{5}\]  
\[2d\sin\theta=\lambda \tag{6}\]  
where, the elastic constants of ZnO single crystal are \(c_{11}\) = 208.8 GPa, \(c_{33}\) = 213.8 GPa, \(c_{12}\) = 119.7 GPa, and \(c_{13}\) = 104.2 GPa; \(c_{12}\) = 0.5201 nm is the stress-free ZnO lattice constant. \(d\) is the space between (002) planes of the AZO thin films; \(c=2d\) is the lattice constant for AZO thin films. \(\lambda\) is the X-ray wavelength (1.5418 A), and \(\theta\) is the Bragg diffraction angle. As shown in Figure 4b, the calculated tensile stress (\(\sigma\)) values decreased from \(6.74\times 10^{8}\) Pa to \(4.0\times 10^{8}\) Pa and \(0.08\times 10^{8}\) Pa for the AZO thin films with Al doping of 4.65 wt.%, 5.22 wt.%, and 6.34 w.t.%, respectively. Figure 4: (a) X-ray diffraction patterns of AZO (002) with different Al doping concentrations and (b) AZO thin film grain size, residual strain, and \(qV_{bi}\) of AZO/n-Si junctions with different Al doping concentrations. Additionally, the average crystal grain sizes (\(D\)) for the AZO thin films were also extracted by using Equation (7) as follows:  
\[D=\frac{K\lambda}{\beta\cos\theta} \tag{7}\]  
where \(\lambda\) is the X-ray wavelength, \(\beta\) is the full width at half-maximum (FWHM) of the (002) diffraction peak, \(\theta\) is the Bragg diffraction angle, and \(K\) is a constant of 0.94. The average grain size for the AZO with Al dopings of 4.65 w.t.%, 5.22 w.t.% and 6.34 w.t.% was 12.4 nm, 17.3 nm, and 34.2 nm, respectively. As shown in Figure 4b, the casted AZO thin film structural parameters (i.e., residual stress and grain size) were compared with the AZO/Si heterojunction \(qV_{bi}\) values for difference Al doping in AZO. One can see that by moderately increasing Al doping in AZO films from 4.65 w.t.% to 6.34 w.t.%, the thin film residual stress reduced dramatically from 6.74 \(\times\) 10\({}^{8}\) Pa to 0.08 \(\times\) 10\({}^{8}\) Pa. As a result, the crystal grain size increased from 12.4 nm to 34.2 nm, due to the low surface free energy and enhanced ZnO thin film crystallinity provided by the residual stress release [44]. The built-in potential (\(qV_{bi}\)) of AZO/Si junction was also improved from 0.21 eV to 0.74 eV, indicating the heterojunction and interface quality enhancement. As previously reported, due to relatively large lattice and thermal expansion mismatch between AZO and Si, AZO thin films deposited on Si were expected to have high residual stress and heterojunction interfacial defects [45; 46; 47]. In this work, junction engineering was performed by adding moderate Al doping into AZO thin films via the dual-beam co-sputtering technique. The AZO thin film residual stress was significantly released with enhanced AZO crystallinity and junction built-in potential. This approach benefits our proposed ITO/AZO/Si heterojunction photovoltaic structures. Finally, the fabricated ITO/AZO/n-Si solar cell device was characterized to determine the photovoltaic performance. The ITO/n-Si devices without AZO inter-layers were also fabricated for comparison. As shown in Figure 5a, the ITO/n-Si device provided a J\({}_{\text{sc}}\) value of 16 mA/cm\({}^{2}\) with a relatively low V\({}_{\text{oc}}\) and FF performance of only 0.14 V and 0.28, respectively. By introducing AZO inter-layers with moderate Al doping for junction engineering, a significant improvement could be achieved, as shown in Figure 5b. The V\({}_{\text{oc}}\) and FF showed improvement from 0.14 V to 0.42 V and from 0.24 to 0.46, respectively. This improved device performance confirms that the AZO film with a delicately controlled Al doping can improve the AZO crystallinity and the thin film stress release, resulting in better AZO/Si heterojunction characteristics. Moreover, the photocurrent (J\({}_{\text{sc}}\)) enhancement was observed with increasing Al doping in AZO.