## 3 Results of Simulations  
### Cell Characterization at Equilibrium: Energy Band Diagrams  
The simulated band diagram at equilibrium reproduced from [35] for cell SHJ\({}_{1}\) and obtained in the present work for cell SHJ\({}_{4}\) are displayed in Figure 2a,b, respectively. Conduction and valence band offsets (discontinuities) \(\Delta\)E\({}_{\text{C}}\) and \(\Delta\)E\({}_{\text{V}}\) occur at the a-Si:H(i)/c-Si hetero-interface due to different values of electron affinity and energy band-gap in a-Si:H and in c-Si. As predicted by band line-up and the electron affinity rule [42; 43], these offsets are independent of the doping level. Hence, for both cells  
\[\Delta\text{E}_{\text{c}}=\chi_{\text{c-Si}}-\chi_{\text{a-Si}}=4.05-3.9=0.15\ \text{eV} \tag{1a}\] \[\Delta\text{E}_{\text{G}}=\text{E}_{\text{G,a-Si}}-\text{E}_{\text{G,c-Si}}=1.72- 1.124=0.596\ \text{eVZ}\] (1b) \[\Delta\text{E}_{\text{V}}=\Delta\text{E}_{\text{G}}-\Delta\text{E}_{\text{c}}=0.4 46\ \text{eV} \tag{1c}\]  
where \(\chi_{\text{c-Si}}\) and E\({}_{\text{G,c-Si}}\) represent the electron affinity and the energy band-gap of the c-Si substrate. At the hetero-interface, the valence band bends and nears the Fermi energy, which induces p-type inversion in the c-Si side of the interface. Such an inversion layer, confirmed experimentally [44; 45; 46], plays a major role in boosting \(V_{OC}\) of the cell and in controlling the overall SHJ cell performance, as previously detailed [8; 9; 12]. The total band bending or built-in potential \(\rm V_{bi,total}\) in the structure is given by the difference between the work functions of the doped a-Si:H(p\({}^{+}\)) layer and of the n-type c-Si substrate. Such a potential is distributed between the two regions: (1) \(\rm V_{bi,a\text{-}Si}\) in a-Si:H, and (2) \(\rm V_{bi,c\text{-}Si}\) in the c-Si substrate, such that  
\[\rm V_{bi,total}=\rm V_{bi,a\text{-}Si}+\rm V_{bi,c\text{-}Si} \tag{2}\]  
The values of \(\rm V_{bi,total}\), \(\rm V_{bi,a\text{-}Si}\), and \(\rm V_{bi,c\text{-}Si}\) at equilibrium extracted from the band diagrams of Figure 2a,b are listed for both cells in Table 1. Due to the larger band bending in c-Si in cell SHJ\({}_{4}\) than in cell SHJ\({}_{1}\), the induced inversion layer in the former is much stronger, and its peak hole concentration at the hetero-interface is much higher and amounts to 4.8 \(\times\) 10\({}^{19}\) cm\({}^{-3}\) compared to 8.55 \(\times\) 10\({}^{18}\) cm\({}^{-3}\) in cell SHJ\({}_{1}\)[35], as depicted in Figure 3a. Additionally, the free hole concentration in the p-type a-Si:H(p\({}^{+}\)) of cell SHJ\({}_{4}\) is much larger and amounts to 2.1 \(\times\) 10\({}^{19}\) cm\({}^{-3}\), compared to 3.5 \(\times\) 10\({}^{16}\) cm\({}^{-3}\) in cell SHJ\({}_{1}\)[35]. The electric field distributions depicted in Figure 3b show that, due to the larger band bending and larger inversion hole population in cell SHJ\({}_{4}\), the electric field at the hetero-interface is higher and mainly confined in the depleted a-Si:H(i) layer due to the availability of a large concentration of negative dopant ions in the a-Si:H(p\({}^{+}\)) layer. On the other hand, the electric field at the hetero-interface in SHJ\({}_{1}\) is much smaller, and extends deep in the a-Si:H(p\({}^{+}\)) layer with fewer less negative ions capable of terminating the field. In addition, the much larger concentration of free holes spilling-over in the intrinsic a-Si:H(i) layer of cell SHJ\({}_{4}\) compared to cell SHJ\({}_{1}\) reduces the layer effective thickness in the former, as depicted in Figure 3a. This explains the gradual increase in the electric field in the a-Si:H(i) layer of SHJ\({}_{4}\) and the peak shown close to the edge of the layer. The flat electric field distribution over the whole a-Si:H(i) layer of cell SHJ\({}_{1}\) is due to much less hole spill-over, as explained above. Figure 3: (**a**) Hole concentration profile, and (**b**) electric field distribution, simulated for cells SHJ\({}_{1}\) and SHJ\({}_{4}\) at equilibrium.