Charge transport modelling of perovskite solar cells accounting for non-Boltzmann statistics in organic and highly-doped transport layers

We present a drift-diffusion model of a perovskite solar cell (PSC) in which carrier transport in the charge transport layers (TLs) is not based on the Boltzmann approximation to the Fermi-Dirac (FD) statistical distribution, in contrast to previously studied models. At sufficiently high carrier densities the Boltzmann approximation breaks down and the precise form of the density of states function (often assumed to be parabolic) has a significant influence on carrier transport. In particular, parabolic, Kane and Gaussian models of the density of states are discussed in depth and it is shown that the discrepancies between the Boltzmann approximation and the full FD statistical model are particularly marked for the Gaussian model, which is typically used to describe organic semiconducting TLs. Comparison is made between full device models, using parameter values taken from the literature, in which carrier motion in the TLs is described using (I) the full FD statistical model and (II) the Boltzmann approximation. For a representative TiO₂/MAPI/Spiro device the behaviour of the PSC predicted by the Boltzmann-based model shows significant differences compared to that predicted by the FD-based model. This holds both at steady-state, where the Boltzmann treatment overestimates the power conversion efficiency by a factor of 27%, compared to the FD treatment, and in dynamic simulations of current-voltage hysteresis and electrochemical impedance spectroscopy. This suggests that the standard approach, in which carrier transport in the TLs is modelled based on the Boltzmann approximation, is inadequate. Furthermore, we show that the full FD treatment gives a more accurate representation of the steady-state performance, compared to the standard Boltzmann treatment, as measured against experimental data reported in the literature for typical TiO₂/MAPI/Spiro devices.