A Finite Element Framework for Assessing Diffusion Length–Thickness (Ld–d) Limitations in MA–FA Mixed Perovskite Semiconductor Absorber Layers for Solar Cells

The cation composition of metal halide perovskites plays a critical role in determining internal charge carrier dynamics and volumetric recombination within the absorber layer. While methylammonium (MA)-based perovskites serve as a foundational baseline, the incorporation of formamidinium (FA) cations has been shown to enhance optoelectronic properties under various environmental conditions [1, 4]. Despite these advancements, the quantitative relationship between specific MA:FA ratios and the resulting carrier survival within the perovskite domain remains a subject of intense research. In this study, the internal generation–recombination balance and its dependence on absorber thickness (d) are investigated using a one-dimensional (1D) finite element method (FEM) framework, strictly validated against champion experimental data obtained from sol–gel-fabricated solar cells [5].

Three representative compositions were analyzed: pure MA (MA100), MA-rich (MA70:FA30), and equimolar MA-FA (MA50:FA50). Experimental characterization reveals distinct performance trends across the compositions. The pure MA baseline achieved a power conversion efficiency (PCE) of 14.4% with an open-circuit voltage (Voc) of 1.040 V. In contrast, the MA-rich (MA70:FA30) composition emerged as the champion, reaching a PCE of 18.6% and a Voc of 1.029 V. The equimolar MA-FA composition resulted in a PCE of 14.1% and a Voc of 0.948 V. These results provide the empirical foundation for a numerical assessment of internal loss mechanisms.

The simulation follows a structured two-stage approach. First, specific volumetric recombination rates are extracted by calibrating the model to match the experimental results at the baseline thickness. Second, a systematic thickness sweep is performed to evaluate the carrier collection efficiency. The spatial distribution of SRH recombination obtained from the FEM simulations enables the extraction of a volume-averaged effective carrier lifetime (tau_eff). Based on this, the carrier diffusion length is calculated as Ld = sqrt(D * tau_eff). These calculated diffusion lengths are directly compared with the absorber thickness (d) to explicitly demonstrate diffusion length-limited (Ld < d) and transport-balanced (Ld >= d) regimes [3,7]. This Ld-d comparison establishes a direct quantitative link between observed performance trends and recombination-dominated transport, providing a robust framework for thickness optimization in mixed-cation perovskite photovoltaics [5, 7].

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