## 1 Introduction
On the contrary to the superior \(V_{OC}\) and equally high \(I_{SC}\), the \(FF\) in SHJ cells rarely exceeds the range 75-79% compared to over 84% achieved in high efficiency c-Si cells. One of the main reasons for such a deficiency relates to inefficient hole transport across the energy barrier at the hetero-interface, resulting from the difference between the energy band-gap of a-Si:H and that of c-Si. In principle, thermionic emission is the fundamental mechanism that controls such a transport. However, several tunneling mechanisms through this barrier have been evoked for a relatively long time. These mechanisms remain in dispute with no general agreement. Direct tunneling, capture-emission tunneling, multistep tunneling, tunnel by hopping via tail states, as well as mixed bias-dependent tunneling/diffusion modes have been suggested but yet not confirmed with full certainty [24; 25; 26; 27; 28; 29; 30; 31; 32]. Some studies have shown that tunneling does not prevail in p\({}^{+}\)/n SHJ cells e.g., [28] or may not be needed to describe the performance of the cell with energy band-gap up to 1.85 eV [32], while others have concluded that tunneling is critical and needed [26]. It is believed that selecting the optimum a-Si:H band-gap and thickness in conjunction with effective tunneling is necessary to achieve a high FF close to 84%, as is reported for the record efficiency SHJ cell [7] but is not always possible to ensure. Thermionic emission transport over the energy barrier, however, remains the fundamental mechanism controlling and, in many cases, hindering hole transport across the a-Si:H/c-Si hetero-interface. Therefore, a good understanding of the constraints imposed by such a transport mechanism and its effect on the device performance is necessary in order to optimize the cell design and technology. Using AFORS-HET device simulations [33; 34], it has been recently shown that thermionic emission hole transport across the hetero-interface in SHJ cells, with flat band perfectly ohmic contacts, results in distortion in the I-V characteristics under illumination that becomes more serious as the active doping level in a-Si:H decreases [35]. It was suggested that incomplete hole collection (hole reflection) at the thermionic barrier results in a hole forward current through the pn junction, establishing a forward bias V\({}_{SC}\), even at zero terminal voltage. The study concentrated on V\({}_{SC}\), which was found to depend on the applied bias and on the doping level in a-Si:H(p\({}^{+}\)) [35]. The link between V\({}_{SC}\) and the thermonic emission transport mechanism has been confirmed by studying the dependence of V\({}_{\rm SC}\) on the thermionic emission parameters. Although all the results presented in [35] are consistent and seem to indicate the role of hole reflection at the thermionic barrier in the degradation, the work lacks answers to several questions. For instance, it is not shown how the forward voltage V\({}_{\rm SC}\) established at the pn junction is compensated in the cell closed loop (at short circuit as well as under bias). Without such a compensation, the loop voltage equation is offended and the presence of V\({}_{\rm SC}\) becomes questionable. In addition, the bias dependence of V\({}_{\rm SC}\) was not justified, and its dependence as well as that of the distortion of the I-V characteristics on the doping level in a-Si:H(p\({}^{+}\)) were not discussed nor interpreted in [35]. Therefore, in an attempt to find answers to these questions, the present analysis elaborates on the hole reflection model proposed in [35] by carrying out extensive AFORS-HET simulations of SHJ cells and investigating carefully all possible details offered by the results of these simulations. The study aims at deriving a higher level model that can give more measurable and quantitative indications that answer the questions raised above.