## 4 Enhanced Series Resistance Model
### Model Verification and Predictions
#### 4.2.3 Bias Dependence of the Cell Series Resistance
_(Symb)_ drift-diffusion transport with \(R_{\rm a\text{-}Si}\)(\(V\)) of Table 5 connected in series. Figure 10: Simulated (**a**) hole concentration profile and (**b**) electric field distribution for cell SHJ\({}_{1}\) under illumination for different cell forward bias. .2.4 Dependence of the Enhanced a-Si:H Resistance on the Doping Concentration in a-Si:H(p\({}^{+}\))  
As discussed in Section 3.1 and depicted in Figure 3a,b, the equilibrium value of the peak inversion layer hole concentration \(\mathrm{p_{peak}}\) at the hetero-interface and of the electric field are much larger in cell SHJ\({}_{4}\) than in cell SHJ\({}_{1}\). Since hole reflection is not expected to be very different in either cell at identical values of A\({}^{*}\), of \(\Delta\mathrm{E_{V}}\), and of T, the relative increase in the hole concentration at the hetero-interface and the associated enhancement in the electric field upon hole reflection under illumination are expected to be much smaller in cell SHJ\({}_{4}\) than in cell SHJ\({}_{1}\). With such a smaller enhancement in the electric field, and with an initially much larger concentration of free holes spilling-over in a-Si:H(i), the depletion of the latter and of the a-Si:H(p\({}^{+}\)) layer is expected to be much less significant in SHJ\({}_{4}\); consequently, the enhanced a-Si:H resistance is expected to be much smaller than in SHJ\({}_{1}\). Moreover, at similar current, the voltage drop on \(R_{a-Si}\) would be much smaller in SHJ\({}_{4}\), which will be reflected on the value of \(\mathrm{V_{SC}}\). By interpolation, it can be concluded that the enhanced series resistance under illumination decreases continuously as the doping level in a-Si:H(p\({}^{+}\)) increases, which perfectly explains the doping dependence of \(\mathrm{V_{SC}}\) recorded and not interpreted in [35]. By inspecting the results of the simulations carried out for cell SHJ\({}_{4}\) under illumination-SC, it appears that the value of \(\mathrm{p_{peak}}\) is enhanced by 17% reaching 5.6 \(\times\) 10\({}^{19}\) cm\({}^{-3}\) compared to 4.8 \(\times\) 10\({}^{19}\) cm\({}^{-3}\) at equilibrium and that the electric field in a-Si:H(i) is enhanced by only 4% on the average, as depicted in Figure 12a,b, respectively. The value of \(\mathrm{V_{SC}}\) extracted from the simulated band diagram for cell SHJ\({}_{4}\) under illumination-SC amounts to 104 mV including the 80 mV dark cell series resistance voltage drop \(\delta\), which results in a total enhanced a-Si:H resistance of 0.64 \(\Omega\) cm\({}^{2}\), mainly due to the partial depletion of the hole spill-over region in a-Si:H(i) layer, compared to 9 \(\Omega\) cm\({}^{2}\) in cell SHJ\({}_{1}\). Finally, apart from the effect of \(\delta\), the expanded plot of the valence band diagram displayed in Figure 13 shows only a slight change in the built-in potential in a-Si:H and in c-Si under illumination-SC compared to the situation at equilibrium, which is consistent with the relatively small degradation of the \(FF\) in cell SHJ\({}_{4}\) from 78.4% with drift-diffusion transport to 75.5% with thermionic emission transport. Such findings are consistent, agree perfectly with the model predictions and give a clear explanation to the doping dependence of the I-V characteristics depicted in Figure 4a.