## 2 Methodology  
The p\({}^{+}\)/n SHJ cell under study is sketched in Figure 1. The substrate of the device consists of 200 \(\mu\)m high quality crystalline silicon (c-Si) substrate with a Shockley-Read-Hall (SRH) carrier lifetime of 5 ms and a default n-type doping concentration N\({}_{\rm D}\) = 1.5 \(\times\) 10\({}^{16}\) cm\({}^{-3}\). A double ultrathin intrinsic a-Si:H(i)/p-type doped a-Si:H(p\({}^{+}\)) layer (5 nm/10 nm) is deposited on the surface of the c-Si substrate. The AFORS-HET device simulator is capable of dealing with pyramidal surface texturing, which is therefore implemented in the structure under study to ensure maximum light coupling and maximum \(I_{\rm SC}\). A TCO layer is normally deposited on top of the a-Si:H(p\({}^{+}\)) for the front contact. However, as mentioned earlier, the study deals with FF degradation due to the thermionic emission barrier at the hetero-interface, and any other source of performance degradation should be turned off. Hence, the TCO/a-Si:H(p\({}^{+}\)) interface (the front contact) is set always to a flat band condition, unless stated differently, and assumed to be perfectly ohmic with a surface recombination velocity of 10\({}^{6}\) cm/s, as expected at a metal/Si contact. Furthermore, the TCO is assumed to be transparent with no light absorption, so its material and thickness are irrelevant to the present study. Consequently, the TCO layer is omitted from the structure to be simulated setting the top of the cell at the a-Si:H(p\({}^{+}\)) top surface. Figure 1: Silicon hetero-junction (SHJ) solar cell under study (simulations done with TCO removed). Default values of the a-Si:H energy band-gap (E\({}_{\text{G,a-Si}}\)) and electron affinity (\(\chi_{\text{a-Si}}\)) are set to 1.72 eV and 3.9 eV, respectively, as commonly used [36; 37; 38; 39]. The default physical models and properties of a-Si:H implemented in the AFORS-HET simulator are maintained in all simulations. The work function of a-Si:H(p\({}^{+}\)) depends on the free hole concentration, hence on the chemical doping concentration in the layer and on the dangling bond defect density in the bulk and at the interfaces. These dangling bond defects are represented in the AFORS-HET simulator by Gaussian distributions with a peak density N\({}_{\text{DB0}}\) in the bulk and D\({}_{\text{it0}}\) at the a-Si:H/c-Si interface. It is well known that the dangling bond density in the bulk of a-Si:H increases as the chemical doping the concentration is increased [40; 41]. Therefore, the peak value N\({}_{\text{DB0}}\) cm\({}^{-3}\) eV\({}^{-1}\) in the bulk is set to the value of the doping concentration in the a-Si:H(p\({}^{+}\)) layer, and to a minimum value equal to 1.385 \(\times\) 10\({}^{16}\) cm\({}^{-3}\) eV\({}^{-1}\) in the intrinsic a-Si:H(i) layer. The peak dangling bond defect density at the a-Si:H(i)/c-Si hetero-interface, D\({}_{\text{it0}}\), is set to 10\({}^{10}\) cm\({}^{-2}\) eV\({}^{-1}\), which ensures excellent passivation and negligible recombination at the interface. Shockley-Read-Hall recombination and Auger recombination are considered in the simulations, while band-gap narrowing (BGN) is ignored in the heavily doped a-Si:H due to lack of data, and in the heavily populated inversion layer where it is expected to be negligible, as mentioned earlier. All simulations are carried out at room temperature (T = 300 K). Since the performance of the SHJ cell is fundamentally determined by the difference between the work function of a-Si:H and that of c-Si, and the former depends on the hole activation energy (E\({}_{\text{F}}\)-E\({}_{\text{V}}\)) and the associated p-type doping level N\({}_{\text{a-Si}}\) in a-Si:H(p\({}^{+}\)), two extreme doping levels will be considered: (1) the lowest doping level N\({}_{\text{a-Si}}\) = 5 \(\times\) 10\({}^{18}\) cm\({}^{-3}\) in cell SHJ\({}_{1}\), and (2) the highest doping level N\({}_{\text{a-Si}}\) = 1 \(\times\) 10\({}^{20}\) cm\({}^{-3}\) in cell SHJ\({}_{4}\).