## 4 Enhanced Series Resistance Model
### Model Verification and Predictions
#### 4.2.3 Bias Dependence of the Cell Series Resistance  
Under illumination, the total inversion layer peak hole concentration \(\mathrm{p_{peak}}\) starts to decrease when the forward bias applied to the cell exceeds 0.2 V, as depicted in Figure 10a and listed in Table 5, which leads to a weaker excess dipole and reduced enhancement in the electric field at the hetero-interface, as depicted in Figure 10b and also shown in Table 5. Accordingly, the proposed model predicts less depletion of the double a-Si:H layer and smaller enhanced a-Si:H resistance with increasing the cell forward bias, which are both confirmed in Figure 10a and Table 5, respectively. The enhanced double a-Si:H layer resistance \(R_{a-Si}(V)\) is estimated as a function of the cell forward bias from  
\[R_{a-Si}(V)=\frac{\mathrm{V_{j}(V)-I(V)\times R_{S}-V}}{I(V)} \tag{6}\] \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline
**a-Si:H thick.** & \(\mathbf{V_{SC}}\)– & **Peak** & **Max. E-Field** & **Peak** & **Max. E-Field** & **I\({}_{SC}\)** & **A** & **B** & **C = A-B** \\
**[nm]** & **[mV]** & **Equilibri.** & **Equilibri.** & **Illumin.SC** & **Illumin.SC** & **Illumin.SC** & **[mA.cm\({}^{-2}\)]** & **\(R_{a-Si}\)** & **\(R_{a-Si}\)** & **\(R_{a-Si}\)** \\  & & & **[cm\({}^{-3}\)]** & **[KV/cm]** & **[cm\({}^{-3}\)]** & **[KV/cm]** & **[cm\({}^{-3}\)]** & **[KV/cm]** & **[mÅ.cm\({}^{2}\)]** & **[mÅ\({}^{2}\)]** & **[mÅ\({}^{2}\)]** & **[mÅ\({}^{2}\)]** \\ \hline
5 & 382 & \(8.55\times 10^{18}\) & 268 & \(3.74\times 10^{19}\) & 549 & 42.34 & 9.00 & 6.25 & 2.75 \\
4 & 332 & \(1.01\times 10^{19}\) & 290 & \(3.76\times 10^{19}\) & 551 & 42.34 & 7.84 & 5.00 & 2.84 \\
3 & 278 & \(1.19\times 10^{19}\) & 314 & \(3.77\times 10^{19}\) & 552 & 42.34 & 6.56 & 3.75 & 2.81 \\
2 & 221 & \(1.43\times 10^{19}\) & 342 & \(3.69\times 10^{19}\) & 554 & 42.34 & 5.21 & 2.50 & 2.71 \\
0 & 108 & \(2.09\times 10^{19}\) & 407 & \(3.99\times 10^{19}\) & 560 & 43.85 & 2.55 & N/A & 2.55 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Enhanced cell resistance in cell SHJ\({}_{1}\)versus a-Si:H(i) thickness. Figure 8: Simulated I-V characteristics under illumination for cell SHJ\({}_{1}\) with thermionic emission transport for different values of the Richardson constant A\({}^{*}\). The I-V characteristics assuming drift-diffusion (Curve 5) is also plotted for comparison. Figure 9: Simulated I-V characteristics under illumination for cell SHJ\({}_{1}\) with thermionic emission hole transport for different values of a-Si:H(i) layer thickness. where V is the cell terminal voltage, V\({}_{\rm j}\)(V) is the total bias at the junction to be extracted from the band diagram, and \(I\)(V) is the cell current to be extracted from the I-V characteristics. For instance, at short circuit condition V = 0, V\({}_{\rm j}\)(0) = V\({}_{\rm SC}\), \(I\)(0) = \(I_{\rm SC}\), \(I\)(0) \(\times\)\(R_{\rm s}\) = \(\delta\) such that Equation (6) reduces to Equation (5a). The resulting values of \(R_{\rm a\text{-}Si}\)(V) listed in Table 5 for SHJ\({}_{1}\) confirm that the enhanced a-Si:H resistance is bias-dependent, decreasing with increasing the forward bias, which explains the bias dependence of V\({}_{\rm SC}\). The accuracy of the enhanced resistance model is checked by comparing the I-V characteristics of cell SHJ\({}_{1}\) regularly simulated assuming thermionic emission transport, with the I-V characteristics of the same cell simulated assuming drift-diffusion transport with a bias dependent resistance, having the values listed in Table 5, connected in series with the cell. A perfect matching is obtained, as depicted in Figure 11, which confirms the validity of the approach and the accuracy of the model. The reference I-V characteristics of cell SHJ\({}_{1}\) under drift-diffusion transport are also displayed in Figure 11 for comparison. Figure 11: Simulated I-V characteristics under illumination for cell SHJ\({}_{1}\) with 1. thermionic emission transport, 2. drift-diffusion transport, and 3.