Physics-Informed Machine Learning for Structural Stability and Bandgap Engineering in Lead-Free Sn–Ge Perovskites for Photovoltaic Applications

Technical Summary: Physics-Informed Machine Learning for Structural Stability and Bandgap Engineering in Lead-Free Sn–Ge Perovskites

Problem Statement
The rapid advancement of halide perovskite photovoltaics (ABX₃) has been hindered by the toxicity of lead (Pb²⁺), which poses significant environmental and health risks, particularly given its high water solubility and strict regulatory constraints (e.g., REACH). While Tin (Sn²⁺) and Germanium (Ge²⁺) offer promising, less toxic alternatives with similar electronic structures, they suffer from critical instability issues. Specifically, Sn²⁺ is prone to rapid oxidation to Sn⁴⁺, leading to lattice collapse and material degradation. Furthermore, the vast chemical search space for optimizing stoichiometric ratios in Sn–Ge mixed-cation systems creates a bottleneck for traditional manual experimental screening, which is slow and resource-intensive.

Methodology
This study implements a Physics-Informed Machine Learning (PIML) framework to systematically map the trade-off between structural stability and bandgap optimization in lead-free Sn–Ge perovskites. Rather than relying on a "black-box" approach, the methodology integrates fundamental crystallographic principles directly into the machine learning architecture.

• Descriptors: The model utilizes physically grounded descriptors derived from coordination chemistry and lattice strain theory, specifically the Goldschmidt tolerance factor ($t$) and the octahedral factor ($\mu$). These are calculated using Shannon effective ionic radii to define geometric constraints.
• Dataset: A curated dataset of over 1,000 lead-free Sn–Ge mixed-cation compositions was generated, drawing from the Materials Project database. The dataset construction followed Vegard's Law to ensure realistic lattice parameter transitions from Sn-rich to Ge-rich phases.
• Algorithm: A Random Forest regression model (ensemble of 500 decision trees) was trained on this dataset. The framework balances chemical diversity against structural constraints to predict bandgaps and stability windows.
• Validation: The model's performance was validated against physics-based analytical formulas and benchmarked against reported experimental and theoretical literature values. Uncertainty quantification was performed by analyzing the variance among individual decision trees.

Key Results

• Predictive Accuracy: The PIML model achieved a high predictive fidelity with a coefficient of determination ($R^2$ = 0.9624) and a Mean Absolute Error (MAE) of 28.2 meV. Notably, this error margin is comparable to the thermal energy at room temperature ($k_BT \approx 25.7$ meV), suggesting the model captures intrinsic lattice dynamics without overfitting.
• Stability Window Identification: The study identified a specific "Novelty Zone" or stability window where compositions exhibit both structural integrity and optimal optoelectronic properties. This window corresponds to a Goldschmidt tolerance factor range of $0.88 \le t \le 0.94$.
• Bandgap Engineering: The model successfully mapped the non-linear bandgap tuning (bowing effect) resulting from Sn–Ge alloying. It identified a stable composition range achieving a bandgap of approximately 1.35 eV, which is suitable for photovoltaic applications and lies between the values of pure CsSnI₃ (1.3 eV) and CsGeI₃ (1.5–1.6 eV).
• Feature Importance: Analysis revealed that the B-site cation fraction (Sn vs. Ge) is the dominant descriptor, contributing 65% to the model's decision weight, directly influencing frontier orbitals and band edges. The A-site mixing (Cs vs. MA) acts primarily as a structural stabilizer through steric hindrance and chemical pressure.
• Stability Mechanisms: The research highlights that introducing Ge creates chemical pressure within the lattice, which suppresses octahedral tilting and reduces the oxidation of Sn²⁺. Additionally, small Ge fractions (10–20%) facilitate the formation of a protective GeO₂ passivation layer, enhancing moisture stability.

Significance and Claims
The paper claims to provide a robust, non-toxic roadmap for the next generation of perovskite photovoltaics. By anchoring AI in physical laws, the study demonstrates that data-driven materials acceleration can effectively bridge the gap between computational discovery and electronic device optimization.

• Environmental Impact: The study asserts that replacing lead with Sn–Ge alternatives significantly reduces the Toxicity Index and CO₂ footprint (estimated at an 80% reduction in total environmental impact score compared to lead-based systems).
• Experimental Guidance: The identified stability window offers a direct, reproducible guide for the experimental synthesis of high-performance, lead-free semiconductor thin films, moving beyond the limitations of trial-and-error experimentation.
• Sustainability: The findings align with global sustainability goals and 2027 energy directives, offering a viable path toward carbon-neutral photovoltaic technologies that are both efficient and environmentally safe.

The authors emphasize that this framework validates the efficacy of integrating crystallographic constraints with machine learning to classify stability-efficiency controls, providing a scientifically accurate and physically consistent approach to materials discovery.