Metal halide perovskites (MHPs), with their formula ABX3, show promise for optoelectronic applications due to their excellent properties. However, challenges remain, including the toxicity of lead (Pb) and instability, particularly in organic cation-based MHPs. Alloying, or substituting elements at different sites within the MHP structure, offers a potential solution to tune stability and optoelectronic properties while reducing Pb content. High-entropy alloys, where multiple elements occupy a single site, have shown promise in enhancing stability through configurational entropy. While experimental approaches are limited in exploring the vast chemical space of MHPs, computational methods offer an attractive alternative. Previous computational studies have utilized DFT calculations or combinations of DFT and ML models, but many lack exhaustive exploration of all possible atomic configurations for each composition. This study aims to address this gap by developing a DFT/ML combined framework to design B-site-alloyed MHPs with improved stability and optoelectronic properties by considering all possible atomic configurations, thereby identifying the most stable configurations for each composition and expanding the explored domain to quaternary alloying with consideration of mixing entropy.

Several studies have investigated the use of DFT calculations and DFT/ML combinations to screen the compositional space of MHPs. Wang et al. investigated the effect of entropy on Cs2B+B+Cl6 double-perovskite alloys using DFT and the special quasi-random structure (SQS) approach. Yang et al. performed DFT calculations on 495 ABX3 perovskite alloys, employing the SQS method to model mixed perovskites. Choubisa et al. developed a crystal site feature embedding (CSFE) representation for predicting DFT energies and bandgaps of mixed MHPs, discovering the impact of Cd doping on bandgap. Mannodi Kanakkithodi et al. proposed an ML-driven high-throughput screening framework based on stability, bandgap, and defect tolerance. While these studies offer valuable insights, most do not explore the full range of atomic configurations within each composition. Yamamoto et al. used the cluster expansion approach to identify the ground state of B-site mixed iodide perovskites, focusing on thermodynamic stability. This current research distinguishes itself by aiming for exhaustive exploration of all possible atomic configurations to achieve a more complete understanding of the properties of B-site-alloyed MHPs.

This study employed a DFT/ML combined framework (Figure 1d). A crystal graph convolutional neural network (CGCNN) was trained to predict three target properties: decomposition enthalpy (ΔHdecomp), bandgap (Eg), and band type (indirect vs. non-indirect). The training dataset consisted of 3159 DFT (PBEsol) calculated structures of B-site-alloyed MHPs modeled with a 20-atom unit cell and a compositional step of 1/4. The A-site comprised Cs, K, or Rb, and the X-site comprised Br, Cl, or I. The six B-site elements were Cd, Ge, Hg, Pb, Sn, and Zn, chosen for their electronic similarity to Pb²⁺ to minimize structural perturbation. The trained CGCNN model was then used to explore a much larger chemical space (41,400 compositions) with a four-fold enlarged unit cell (80 atoms) and a finer compositional step of 1/16. For each composition, all possible atomic configurations were evaluated to identify the most stable configuration using the predicted ΔHdecomp. The mixing entropy term (-TASmix) was added to ΔHdecomp at 298 K to account for entropy-driven stabilization. Bartel's tolerance factor (τ) was used to classify perovskite formability. Compounds with τ < 4.18 and ΔHdecomp - TASmix < 0 were considered thermodynamically stable. The CGCNN predicted bandgaps (ECGCNNgap) were used for initial screening, selecting compounds with ECGCNNgap < 0.5 eV (predicted to have PBE0-calculated bandgaps of approximately 1.2–1.4 eV). Selected compounds underwent further validation using DFT calculations with the hybrid PBE0 functional including spin-orbit coupling (SOC) for more accurate bandgap estimation, along with calculations for carrier effective mass, optical absorption spectra, and spectroscopic limited maximum efficiency (SLME).

Analysis of the training dataset revealed that the inclusion of the mixing entropy term shifted the ΔHdecomp distribution towards lower energies, with greater shifts observed for compounds with more mixed elements. The presence of Ge resulted in a lower ΔHdecomp distribution compared to compounds without Ge, while Zn had the opposite effect. Bartel's tolerance factor (τ) showed a positive correlation with ΔHdecomp - TASmix, but inconsistencies were observed at τ > 4.18. The CGCNN model demonstrated excellent prediction accuracy for ΔHdecomp (MAE of 0.45 meV/atom) and bandgap (MAE of 0.037 eV). Band type classification also yielded high accuracy (0.96) and recall (0.90). The chemical space exploration identified 110 compounds meeting initial screening criteria (non-indirect bandgap, ECGCNNgap < 0.5 eV, τ < 4.18, ΔHdecomp - TASmix < 0). Subsequent DFT (PBE0) calculations narrowed this down to 31 compounds with bandgaps close to the Shockley-Queisser limit (1.2-1.4 eV) or the ideal bandgap for a tandem solar cell top cell (1.73 eV). Ten of these compounds were selected based on optimal bandgaps (Table 1). Comparison with experimental bandgaps showed an RMSE of approximately 0.30 eV (Supplementary Table 4). The selected compounds exhibited carrier effective masses generally below 1 me (except for two), suggesting good electron mobility. CsGe0.3125Sn0.6875I3 and CsGe0.0625Pb0.3125Sn0.625Br3 were identified as promising candidates for single-junction and tandem perovskite solar cells, respectively. (Figures 2, 3, 4, 5).

The DFT/ML framework successfully explored the vast chemical and configurational space of B-site-alloyed all-inorganic perovskites, identifying promising candidates for solar cell applications. The use of CGCNN significantly accelerated the screening process, allowing for the evaluation of a large number of compositions and configurations. The results highlight the importance of considering both the thermodynamic stability and the electronic properties when designing MHPs. The accuracy of the CGCNN model in predicting thermodynamic stability was notable, but the accuracy in predicting band type and bandgap decreased for the larger 80-atom unit cells. This indicates the limitations of the current model to extrapolate to unseen compositional spaces, highlighting the need for further model development. While the SLME values provide a useful comparative metric, they don't represent real-world power conversion efficiency. Furthermore, the study did not account for the oxidation of Ge and Sn, an aspect that requires further investigation. The discrepancies between predicted and experimental bandgaps highlight the need for continuous refinement of DFT methods and machine learning models for accurate bandgap prediction in this material class.

This study presents a powerful DFT/ML framework for designing B-site-alloyed all-inorganic perovskites. The framework successfully identified promising candidates for single-junction and tandem solar cells. Future work should focus on addressing the limitations of the ML model's predictive capabilities, particularly for bandgap, and incorporating factors like oxidation and defect formation energy for a more comprehensive understanding. Expanding the framework to include A- and X-site alloying will further enhance its predictive power.

The study's limitations include the underestimation of bandgaps by PBE0 compared to experimental values, the decreased accuracy of the CGCNN model for the larger 80-atom unit cell, and the omission of other critical parameters such as defect formation energy and surface/interface stability. The oxidation of Ge and Sn during fabrication, which can negatively affect performance, was also not explicitly considered in the stability calculations.