## 4 Enhanced Series Resistance Model
The enhanced electric field causes further depletion of the a-Si:H(p\({}^{+}\)) layer and decays gradually as it penetrates deeper in the layer. Consequently, the resistance of the double a-Si:H layer is strongly enhanced, which justifies the voltage drop \(V_{a\text{-}Si}\) of Equation (4a), and may finally lead to a considerable increase in the cell series resistance that explains the distorted shape of the I-V characteristics displayed in Figure 4a. Furthermore, the integration of the enhanced and extended electric field in a-Si:H explains the increase in the built-in potential in a-Si:H, V\({}_{\text{bi,a-Si}}\), compared to its value at equilibrium, as discussed in Section 3.2.2. Such an increase is balanced by a decrease in the built-in potential in c-Si, V\({}_{\text{bi,c-Si}}\), in order to force the total built-in potential at short circuit to be equal to its value at equilibrium, taking into account the dark resistance voltage drop \(\delta\), as confirmed in Table 2.  
At short circuit condition and using Equation (4a), the enhanced a-Si:H resistance \(R_{a-Si}\) is determined from  
\[R_{a-Si}=\frac{\text{V}_{\text{SC}}-\delta}{I_{SC}}\] (5a) where V\({}_{\text{SC}}\) is to be extracted from the band diagram under illumination-SC, \[\delta\] to be accurately determined from the band diagram under drift diffusion transport or evaluated knowing the dark series resistance, and \[I_{SC}\] determined from the I-V characteristics under illumination. Typically, \[R_{a-Si}\] in cell SHJ\({}_{1}\) is estimated to 9 \[\Omega\] cm\({}^{2}\). In addition, the latter is to be divided into (1) a dominant a-Si:H(i) resistance given by \[R_{a-Si(i)}=\frac{\Delta\text{V}}{I_{SC}}\] (5b) and (2) a resistance of the depleted region in the a-Si:H(p\({}^{+}\)) layer to be determined from \[R_{a-Si(p+)}=R_{a-Si}-\ R_{a-Si(i)}. \tag{5c}\]  
Hence, \(R_{a-Si(i)}\) can be estimated with \(\Delta\)V in Equation (5b) extracted from the band diagram of Figure 5b, or can be calculated knowing the low flat free hole concentration in the a-Si:H(i) layer in Figure 6a and the layer thickness. Finally, \(R_{a-Si(p+)}\) follows from Equation (5c). Typically, \(R_{a-Si(i)}\) and \(R_{a-Si(p+)}\) in SHJ\({}_{1}\) at short circuit are estimated to be 6.25 and 2.75 \(\Omega\) cm\({}^{2}\), respectively. It is worth noting that the hole profile, the electric field, and the built-in potential close to the interface remain intact when drift-diffusion transport is assumed, which confirm the proposed relationship between hole reflection at the thermionic barrier and the enhanced a-Si:H series resistance under illumination. Finally, the proposed enhanced resistance model developed here explains clearly that the forward bias V\({}_{\text{SC}}\) revealed in [35] is compensated in the cell closed loop at short circuit or at any bias by the voltage drop on the enhanced a-Si:H resistance. In addition, it also becomes clear that, since the effect of hole reflection at the thermionic emission barrier is finally assimilated to the effect of an enhanced series resistance, the fact that only the cell _FF_ is degraded while the cell open circuit voltage is not affected is now clarified.