## 4 Enhanced Series Resistance Model  
As mentioned earlier, hole reflection at the hetero-interface has been suggested to be the origin of the forward bias established at the pn junction under illumination and was taken as a basis for a model that interprets the _FF_ degradation due to the thermionic emission barrier at the a-Si:H/c-Si hetero-interface [35]. So far, however, such a model lacks answers to the several questions raised in the introduction. An attempt is made in the following analysis to develop a higher level model that gives acceptable and justified answers to these questions. Enhanced a-Si:H Resistance under Illumination and Its Relationship with Hole Reflection at the Thermionic Barrier  
By inspecting the band diagrams of Figure 5a,b more closely, following the behavior of the hole quasi-Fermi energy and the quasi-Fermi energy split at the junction, and studying the cell short circuit closed loop equation, the forward bias V\({}_{\text{SC}}\) at the junction should be given by  
\[\text{V}_{\text{SC}}=\text{V}_{\text{a-Si}}+\delta \tag{4a}\] \[\text{V}_{\text{SC}}=\Delta\text{V}+\text{V}_{\text{a-Si(p+)}}+\delta \tag{4b}\]  
where V\({}_{\text{a-Si}}\) is the total voltage drop on the double a-Si:H layer, \(\Delta\text{V}\) is the sharp voltage drop on the a-Si:H(i) layer, well identified in the band diagram of Figure 5b, and V\({}_{\text{a-Si(p+)}}\) is the voltage drop on the doped a-Si:H(p\({}^{+}\)) layer. As concluded in Section 3.2.1, the presence of a naturally high dark a-Si:H resistance that justifies the voltage drop V\({}_{\text{a-Si}}\) is ruled out, since it does not show when drift-diffusion transport is assumed. Consequently, the voltage drop on the double a-Si:H layer could \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Cell Na-Si [cm\({}^{-3}\)]** & **Vb,i-Si [V]** & **Vb,i-Si [V]** & **Vb,i-total = Vb,i-Si + Vb,i-Si + 8 [V]** \\ \hline SHJ\({}_{1}\): \(5\times 10^{18}\) & 0.609 & 0.480 & 1.089 + 0.08 = 1.169 \\ SHJ\({}_{4}\): \(1\times 10^{20}\) & 0.413 & 0.842 & 1.255 + 0.08 = 1.335 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Built-in potential in cells SHJ\({}_{1}\) and SHJ\({}_{4}\) under illumination-SC. Figure 5: Band diagrams obtained from AFORS-HET simulations of cell SHJ\({}_{1}\) under illumination-SC with (**a**) drift-diffusion and (**b**) thermionic emission transport [35]. only be explained by the presence of an enhanced a-Si:H resistance under illumination. The origin of such enhanced resistance and its exclusive occurrence under thermionic emission transport needs to be clarified. The hole concentration profile in cell SHJ\({}_{1}\) displayed in Figure 6a shows that, when thermionic emission transport is assumed, the peak hole concentration p\({}_{\text{peak}}\) at the hetero-interface increases from typically 8.55 \(\times\) 10\({}^{18}\) cm\({}^{-3}\) at equilibrium to 3.74 \(\times\) 10\({}^{19}\) cm\({}^{-3}\) under illumination-SC, which represents a substantial boost of 400%. On the other hand, the value of p\({}_{\text{peak}}\) practically does not change from its equilibrium value when drift-diffusion hole transport is assumed, and is maintained around 8.7 \(\times\) 10\({}^{18}\) cm\({}^{-3}\). Therefore, such a boost in p\({}_{\text{peak}}\) under thermionic emission transport can surely be attributed to incomplete hole collection, or equivalently to holes reflected at the thermionic barrier and piling-up as excess holes at the hetero-interface. These excess holes create a dipole with the negative acceptor ions in the doped a-Si:H(p\({}^{+}\)) that enhances the electric field crossing the a-Si:H(i) layer well above its value at equilibrium, as depicted in Figure 6b. Such a high field repels the free holes spilling-over in a-Si:H(i) back towards the a-Si:H(p\({}^{+}\)) layer, causing full depletion of the a-Si:H(i) layer, which explains the very low and flat hole concentration as well as the very high and flat field distribution in the layer depicted in Figure 6a,b, respectively.