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. 2026 Oct 22. Online ahead of print. doi: 10.1039/d6ra07191k

Discovery of germanium-based hybrid perovskites with high photovoltaic performance with machine-learning

Carrie Song a, R X Lian b, H Zhang b, Q Y Jin c,✉
PMCID: PMC13690253  PMID: 42923869

Abstract

The discovery of stable, high-performance lead-free hybrid perovskite materials remains a major challenge for next-generation photovoltaic technologies. Here, a high-throughput computational framework integrating machine learning, deep learning, and density functional theory (DFT) calculations was adopted to accelerate the discovery and validation of lead-free hybrid perovskites. An optimized XGBoost model was firstly employed to screen approximately 180k charge-neutral perovskite candidates according to formation-energy and bandgap criteria. The resulting 48 kinds of Ge-based candidates were subsequently validated using CGCNN and UPET models, leading to the identification of (CH)4NH2GeGeCl6, C3H4NSGeGeBr6, C5NH12GeGeBr6 and C6OH13GeGeBr6 as promising photovoltaic materials, with HSE + SOC-calculated bandgaps of 1.5137, 1.7515, 1.6778 and 1.1680 eV, respectively. These results demonstrate that integrating data-driven prediction with first-principles validation provides an effective strategy for accelerating the discovery of next-generation photovoltaic materials.

Here an ML-DFT workflow screened 180 000 lead-free hybrid perovskites using XGBoost, evaluated 48 Ge-based candidates with CGCNN and UPET, and identified four promising photovoltaic candidates with HSE + SOC bandgaps of 1.17–1.75 eV.graphic file with name d6ra07191k-ga.webp

1. Introduction

The increasing global demand for clean and sustainable energy has intensified the search for the next-generation photovoltaic technologies capable of replacing conventional fossil fuel-based energy sources.1 Although silicon solar cells currently dominate the photovoltaic market, their manufacturing processes remain energy-intensive and costly.2 Metal halide perovskite solar cells have emerged as promising alternatives because of their high power conversion efficiencies, low fabrication costs, and tunable optoelectronic properties.3,4 Since their introduction in photovoltaic applications, perovskite materials have demonstrated rapid improvements in device performance, making them among the most actively studied materials for renewable energy applications.3–5

Despite these advantages, the widespread commercialization of perovskite solar cells remains limited by several fundamental challenges.4 Most high-efficiency perovskite absorbers contain lead (Pb), which raises environmental and health concerns because of its toxicity.6 In addition, many perovskite materials suffer from poor thermal and environmental stability, resulting in performance degradation under practical operating conditions.4,7 Consequently, the discovery of stable, lead-free perovskite materials with suitable electronic properties has become an important research objective.8

Traditional materials discovery relies heavily on experimental synthesis and density functional theory (DFT) calculations to evaluate candidate materials.9,10 Although DFT provides accurate predictions of material properties, it is computationally expensive and impractical for screening hundreds of thousands of candidate compounds.11 Recent advances in machine learning (ML) and deep learning (DL) methods have provided an alternative approach by learning relationships between crystal descriptors and material properties from existing datasets.11 High-throughput ML/DL models can rapidly predict formation energies and electronic properties while substantially reducing computational cost.11,12 However, ML/DL predictions alone often require further validation using physics-based computational methods to ensure reliability.11

Recent ML/DL-assisted photovoltaic screening studies have demonstrated the potential of combining machine learning with first-principles calculations to accelerate the exploration of complex perovskite-related materials.13–16 Hu et al. developed a ML-assisted workflow to accelerate the screening of thousands of photovoltaic Ruddlesden–Popper perovskites structures.17 Sa et al. systematically studied the structural, electronic, and optical properties of hybrid MAGeX3 (X = Cl, Br, I) perovskites and demonstrated the tunability of their electronic properties through structural modulation.18 Liu et al. investigated the stability and optoelectronic properties of CsGeX3, Cs2GeX6, and Ge–Sn mixed perovskites, showing that selected Ge-containing compounds can exhibit suitable bandgaps and strong optical absorption for photovoltaic applications.19

In this work, an integrated computational framework combining ML algorithms, DL models, and first-principles calculations is developed for the discovery of Ge-based lead-free perovskite materials. Nine ML regression algorithms were evaluated using elemental employed to screen approximately 180k charge-neutral perovskite candidates. The SHAP feature analysis was used to interpret feature importance,20 while Crystal Graph Convolutional Neural Networks (CGCNN)21 and the Universal Pretrained Electronic Transformer (UPET)22 provided additional predictive validation. Promising candidates were subsequently examined using DFT calculations. By integrating data-driven prediction models with first-principles validation, this work establishes an efficient framework for accelerating the discovery of environmentally friendly Ge-based perovskite materials for future photovoltaic applications.

2. Numerical methods

As shown in Fig. 1, the workflow for the discovery of the perovskites comprises of six stages. In the data preparation stage, the dataset containing formation energies and bandgap values for 4456 hybrid perovskites is retrieved from previously published literature.13 A total of 96 elemental feature combinations, generated from physicochemical properties of constituent elements in the target crystal structures, were used to develop models for formation energies and bandgaps, which enable identification of stable and high-performance perovskite materials. To minimize the effects of differing feature scales and improve model performance, all input variables were normalized prior to the model training. For model training, the dataset was then randomly divided into training and testing subsets using an 80 : 20 ratio. For model-performance evaluation, the coefficient of determination (R2), mean squared error (MSE), and mean absolute error (MAE) were used. These metrics compared the predictive accuracy and generalization capabilities of the different algorithms.

Fig. 1. Overall workflow of the proposed computational framework for the discovery and validation of Ge-based lead-free perovskite materials.

Fig. 1

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Initially, nine classical machine learning algorithms were used for models selection, which include Gradient Boosting Regressor (GBR), Kernel Ridge Regression (KRR), Random Forest Regressor (RFR), Decision Tree Regressor (DTR), Multi-Layer Perceptron Regressor (MLP), Extra Trees Regressor (ETR), K-Nearest Neighbors Regressor (KNR), Support Vector Regression (SVR), and Extreme Gradient Boosting (XGBoost).23 Separate models were developed for the prediction of formation energies and bandgaps. The training and prediction dataset were aligned to ensure identical feature structures during model development. Hyperparameter optimization was performed using the Hyperopt library with cross-validation to determine the optimal model configurations.24 Hyperparameter optimization employed the tree-structured Parzen estimator implemented in Hyperopt24 with 300 trials, were then used to train the final models. Each configuration was evaluated using five-fold cross-validation on the training subset, without additional shuffling of the folds. The optimization objective was to maximize the coefficient of determination (R2). The discrete search spaces, recorded optimum values, and parameters used for the final XGBoost model training are summarized in Table 1 as an example.

Table 1. Hyperparameter optimization results for the XGBoost model. The recorded optimum values from grid search and the final parameters used for model training are listed.

Parameter Search values Recorded optimum Final fit
Max_depth {4, 6, …, 28} 6 8
Learning_rate {0.01, 0.06, …, 0.46} 0.31 0.21
Gamma {0, 0.2, …, 0.8} 0 0
n_Estimators {80, 84, …, 148} 140 144
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Since the classical ML algorithms only consider the elemental features when building structure-properties models, which neglect the structural information, generally leading to poor precision in predicting properties such as bandgaps, carrier transport properties and etc. Therefore, herein we also used the Crystal Graph Convolutional Neural Networks (CGCNN) algorithm,25 which is a deep learning (DL) model specifically designed for materials property prediction considering crystal structure information. A crystal structure is represented as a graph in which atoms are treated as nodes and interatomic connections are treated as edges in CGCNN. Through a series of graph convolution operations, the model aggregates information from neighboring atoms and automatically learns structural features relevant to the target material properties. The CGCNN model was trained using datasets containing 3489 bandgap and 1346 formation energy values.

Furthermore, for comparison, we also used the Universal Pre-trained Electronic Transformer (UPET) model26 to predict the bandgaps of perovskites candidates, which is a transformer-based DL framework developed for predicting electronic properties of crystalline materials. Unlike traditional ML methods that rely on handcrafted descriptors and graph neural networks that require explicit crystal graph construction, UPET leverages large-scale pretraining to learn representations of crystal structures and their associated electronic properties, which means that, it can efficiently predict properties like bandgap across diverse material systems, since it utilizes knowledge learned from extensive materials datasets.

Feature importance analysis was conducted to evaluate the contribution of individual elemental descriptors to the prediction of formation energy and bandgap, providing insights into the factors that influence perovskite stability and electronic properties. To understand the factors governing the stability and electronic properties of perovskites, feature importance analysis was performed using SHapley Additive exPlanations (SHAP) values generated from the optimized XGBoost models.20 SHAP analysis quantifies the contribution of each input feature to the model predictions and provides insights into the physicochemical descriptors that most strongly influence formation energy and bandgap.

For the structural relaxation of perovskite crystals and the bandstructure calculations stages as shown in Fig. 1, the ab initio calculations were carried out within the framework of density functional theory (DFT) using the Vienna Ab initio Simulation Package (VASP).27 The ion–electron interaction is described by atomic pseudopotentials within the plane-basis projector augmented wave (PAW) method.28 The Perdew–Burke–Ernzerhof (PBE) functional within the generalized gradient approximation (GGA) was used to treat the exchange–correlation energy.29 A plane-wave energy cutoff of 600 eV was applied consistently across all calculations. K-Point sampling for all calculations was generated using the VASPKIT package,30 specifying a k-point spacing of 0.04 Å−1, resulting in a dense and uniform Monkhorst–Pack mesh suitable for accurate Brillouin Zone integration. Structural optimization was performed by fully relaxing both lattice parameters and atomic positions. The convergence criteria were set to 10−8 eV for total energy and 0.05 eV Å−1 for the Hellmann–Feynman forces on all atoms. We also used the all-electron FHI-aims code for the precise calculations of bandgaps for the selected perovskite candidates,31 by employing the Heyd–Scuseria–Ernzerhof (HSE06) hybrid exchange–correlation functionals.32,33

To further characterize the band-edge carrier transport, electron and hole effective masses were calculated using the AMSET framework.34 The electronic structures used for the effective-mass calculations were obtained from PBE calculations without spin–orbit coupling on a dense 6 × 6 × 6 uniform Monkhorst–Pack k-point mesh. AMSET was then used to interpolate the band structures and to obtain the band-edge effective masses by nonparabolic fitting of the energy dispersion near the CBM and VBM along the available high-symmetry directions.

Thermodynamic stability was further evaluated using reference convex-hull phase diagrams constructed from Materials Project entries.35 The target energies were processed using the MaterialsProject2020Compatibility scheme,36 and Ehull was calculated using pymatgen.37 The principal decomposition phases were recalculated using PBE/PAW single-point calculations for comparison.

3. Numerical results and discussions

3.1. Machine-learning modeling and feature-importance analysis

Based on the performance comparison of ML models (GBR, KRR, RFR, DTR, SVR, MLR, XGBoost, and ETR) for predicting formation energies and bandgaps, as shown in Fig. 2, we selected XGBoost as the machine-learning model used in the screening workflow (Fig. 1), because it provides an effective balance between prediction accuracy and computational efficiency. Following the hyperparameter optimization process described in Table 1, the optimized XGBoost models were applied to predict the formation energies and bandgaps of approximately 180 000 charge-neutral perovskite compounds.13

Fig. 2. Performance comparison of machine-learning models (GBR, KRR, RFR, DTE, SVR, MLR, XGBoost and ETR) for predicting (a) formation energy and (b) bandgap.

Fig. 2

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Candidate materials were screened using predefined formation energy and bandgap criteria to identify promising perovskite compositions with favorable stability and photovoltaic characteristics. Further selections for appropriate bandgaps were conducted using CGCNN and UPET models. Model hyperparameters, including learning rate, batch size, and feature dimensions, were optimized during training. The final trained XGBoost models achieved MAE of 0.111 eV for bandgap prediction and 0.014 eV per atom for formation energy prediction.

As shown in Fig. 3(a), the SHAP analysis of the XGBoost model for formation energy revealed that, the ten most influential factors include the polarizability of the A-site organic molecule, the atomic-radius ratio of X to B2, the ionic radius of the X-site atom, the first ionization energies and Mendeleev numbers of the B1- and B2-site atoms, and the Pauling electronegativity and proton affinity of the B1-site species. These descriptors are primarily associated with the molecular and atomic characteristics of the A-site molecule and the B–X framework, highlighting the critical role of the B–X framework in governing the formation energy and, consequently, the chemical stability of organic–inorganic hybrid perovskites. Specifically, the X-site ionic radius and polarization-related descriptors reflect the size matching and electrostatic interactions between the halide anions and metal cations. Variations in these descriptors directly affect B–X bond strength, octahedral distortion, and lattice strain, which collectively determine the thermodynamic stability of the perovskite framework.13,38

Fig. 3. SHAP feature importance analysis for (a) formation energies, (b) bandgaps.

Fig. 3

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As shown in Fig. 3(b), the SHAP analysis of the XGBoost model for band gaps revealed that, the ten most influential factors include the polarizability of the A-site organic molecule, the ionic radius of the X-site atom, the difference between the radius of the A-site molecule and the mean radius of the B1- and B2-site atoms, i.e. rA − (rB1 + rB2)/2, sum of radius of X-site atom and the mean radius of the B1- and B2-site atoms, i.e. rA + (rB1 + rB2)/2, heat of formation of B2- and B1-site atoms, electron affinity energy for B2-site atom, melting points of B1- and B2-site atoms, LUMO of A-site molecule. Similarly, these ten descriptors are also associated with the molecular and atomic characteristics of the A-site molecule and the B–X framework, highlighting the critical role of the B–X framework in governing the bandgaps of hybrid perovskites. Specifically, the X-site ionic radius and A/B/X geometric descriptors regulate the local coordination environment, octahedral distortion, and B–X bond lengths, thereby modifying orbital overlap and electronic-band dispersion. Since the valence-band maximum (VBM) and conduction-band minimum (CBM) of halide perovskites are primarily determined by the hybridization between metal-cation orbitals and halogen p orbitals,13 structural variations in the B–X framework can directly alter the band-edge positions and bandgap values.

To further elucidate how the features of the A-site molecules and B-site atoms contribute to the band gaps, which play a dominant role in determining the optoelectronic properties of perovskites, we performed Pearson correlation analysis on the ten most influential descriptors. The results are shown in Fig. 4, revealing strong correlations among the heats of formation and melting points of the B1- and B2-site atoms, as well as among rA and the derived descriptors rA ± (rB1 + rB2)/2, with absolute Pearson correlation coefficients exceeding 0.8. Therefore, the key factors governing the band gaps are primarily the heats of formation of the B-site atoms, the molecular and atomic radii of the A-, B-, and X-site constituents, and the LUMO energy of the A-site molecule, which are in good agreement with previous reports based on the ab initio calculations.39–41

Fig. 4. Correlation analysis for important features extracted from XGBoost model for bandgaps of hybrid perovskites.

Fig. 4

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3.2. High-throughput materials screening: selection of Ge-based perovskites

The optimized XGBoost models were applied to predict the formation energies and bandgaps of approximately 180 000 charge-neutral perovskite compositions after model selection and evaluation. Candidate materials were screened according to criteria designed to identify compounds with favorable chemical stability and photovoltaic performance identified by appropriate bandgaps. To reduce the computational cost of subsequent analyses, we selected candidates with predicted bandgaps between 0.5 and 3.5 eV and formation energies below −0.1 eV per atom for further investigation. These criteria were used as initial screening filters to identify perovskite candidates with suitable electronic properties and thermodynamic stability. The bandgap window of 0.5–3.5 eV was defined for the PBE-level ML predictions because the training dataset used in this work contains bandgaps calculated within the PBE framework.13 In addition to the well-known tendency of PBE to underestimate the bandgaps of semiconductors, the XGBoost predictions are also subject to finite model errors. We therefore adopted a deliberately broad bandgap window to account for both the systematic uncertainty associated with the PBE reference data and the statistical uncertainty of the ML model, thereby reducing the likelihood of false-negative screening. This range encompasses the Shockley–Queisser optimum bandgap for single-junction photovoltaic devices (∼1.34 eV). Similar bandgap criteria have been widely adopted in previous high-throughput and machine-learning-based screening studies of photovoltaic materials.13,14,42

Furthermore, compositions containing highly toxic, environmentally hazardous, or economically impractical elements were excluded from further consideration. For example, materials containing elements such as Pb or other toxic heavy metals and scarce and expensive elements such as Au were removed from the candidate pool to prioritize compounds with greater potential for large-scale and sustainable deployment. Among the screened candidates, compounds containing identical B1- and B2-site cations were isolated for additional analysis.

Based on the XGBoost-predicted formation energies and bandgaps within the screening ranges described above, and after excluding compositions containing toxic or prohibitively expensive elements, more than 36 000 compositions were retained. Their distribution with respect to the B1- and B2-site elements is shown in Fig. 5, which reveals that, a substantial fraction of the B-site elements are either scarce or supply-constrained, such as Hf, Ta, W, and several rare-earth elements, or toxic, such as Pb, Cd, Hg, and Tl, thereby substantially narrowing the range of practically viable candidates.

Fig. 5. Distribution of (a) B1- and (b) B2-site elements.

Fig. 5

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Herein, particular attention was given to Ge-based perovskites, with a focus on identifying environmentally friendly compositions that exhibit promising stability and photovoltaic performance while remaining compatible with established semiconductor manufacturing processes. We selected germanium as a lead-free alternative to Pb because of its potential to maintain favorable optoelectronic properties while reducing toxicity concerns associated with conventional lead-containing perovskites. Candidate materials with Ge occupying the B-site positions were extracted from the screened dataset for further analysis because they satisfied the predefined formation-energy and band-gap criteria, as listed in Table 2. Then, the selected Ge-based perovskite candidates were converted into crystal structure files and used as inputs for the trained CGCNN and pre-trained UPET models. Subsequently, the formation energies and bandgaps for these selected Ge-based perovskites were predicted by CGCNN and UPET models. By incorporating structural information directly into the prediction process, CGCNN enabled a more comprehensive assessment of the stability and electronic properties of the candidate perovskites, therefore, the CGCNN predictions provided an additional validation of the trends identified by the XGBoost models. UPET predictions provided an independent evaluation of the electronic properties of the screened materials.

Table 2. Comparison of bandgap predictions (in eV) generated by CGCNN, UPET, and XGBoost.

Chemical formula CGCNN UPET XGBoost Chemical formula CGCNN UPET XGBoost
(CH3)2CHNH3GeGeCl6 4.17 1.17 3.00 CH3CH2CH2NH3GeGeI6 4.28 1.49 2.11
C6OH13GeGeI6 3.68 3.81 2.01 C2OH5GeGeBr6 3.45 1.45 2.06
(CH3)2CHNH3GeGeBr6 3.95 1.29 2.21 (CH3)4NGeGeBr6 4.91 3.83 2.25
C2OH5GeGeI6 2.10 0.10 1.30 C6NH14GeGeI6 3.59 3.92 2.01
NH2NH3GeGeCl6 4.43 1.01 2.71 CH3CH2CH2NH3GeGeBr6 4.55 1.55 2.58
C5OH12GeGeCl6 4.11 4.10 3.07 C7H7GeGeI6 3.44 0.24 2.07
C(CH)5NH3GeGeBr6 3.59 0.98 2.67 C(CH)5NH3GeGeI6 3.06 0.58 1.97
C4NH10GeGeBr6 3.88 2.68 2.16 (CH3)2(CH2)2NH2GeGeBr6 3.57 0.19 2.56
C3H4NSGeGeI6 1.82 0.10 1.80 C(CH3)2CH2NH2GeGeBr6 4.65 3.53 3.05
C6OH13GeGeBr6 3.94 4.58 2.68 C3H4NSGeGeBr6 2.13 0.51 2.46
NH2NH3GeGeBr6 4.03 1.19 2.35 C5NH12GeGeBr6 3.97 4.27 2.25
C3H5N2GeGeI 2.23 0.39 1.54 C3H4NSGeGeCl6 2.35 0.24 3.24
C4OH9GeGeBr6 3.95 1.82 2.27 C3H5N2GeGeBr6 3.33 1.15 2.02
NC4H8GeGeCl6 4.04 0.31 3.44 C2OH5GeGeCl6 3.72 1.05 2.56
C4NH10GeGeCl6 4.11 3.28 3.10 C(CH3)2CH2NH2GeGeI6 4.42 3.03 2.35
C6NH14GeGeCl6 4.04 4.57 3.48 CH3CH2CH2NH3GeGeCl6 4.75 2.27 3.36
CH3NH3GeGeBr6 3.53 1.86 2.00 C3H5N2GeGeCl6 3.58 0.79 2.96
(CH)4NH2GeGeBr6 3.60 0.50 2.28 C7H7GeGeBr6 3.73 0.43 2.82
NH4GeGeCl6 4.63 0.30 2.04 C5NH12GeGeCl6 4.17 4.62 3.12
C5OH12GeGeBr6 3.91 3.86 2.28 C6OH13GeGeCl6 4.13 4.75 3.48
(CH3)4NGeGeCl6 5.10 4.38 3.12 C4OH9GeGeCl6 4.19 1.64 3.21
(CH3)3NHGeGeCl6 4.09 2.63 2.73 (CH3)2(CH2)2NH2GeGeCl6 3.91 0.64 3.40
(CH)4NH2GeGeCl6 3.84 0.53 3.17 NC4H8GeGeBr6 3.81 0.31 2.52
C6NH14GeGeBr6 3.84 4.49 2.68 CH3NH3GeGeCl6 3.90 1.28 2.45
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The three models utilize fundamentally different representations of materials: XGBoost relies on engineered elemental descriptors, CGCNN learns directly from crystal graph structures, and UPET employs transformer-based representations learned through large-scale pretraining. Therefore, we assess the consistency of these prediction approaches by comparing bandgap values for the selected Ge-based perovskites obtained from XGBoost, CGCNN, and UPET. Each model captures distinct aspects of material behavior, and their comparison provides a more comprehensive evaluation of candidate compounds. XGBoost emphasizes relationships derived from elemental and physicochemical descriptors, whereas CGCNN incorporates local atomic environments through graph-based learning. UPET further extends this capability by leveraging knowledge acquired during large-scale pretraining on diverse materials datasets. Agreement among these methods suggests that the predicted bandgap values are robust and less dependent on the assumptions of any single modeling approach. Furthermore, consistency across the three frameworks increases confidence in the predicted electronic properties of candidate materials.

Conversely, discrepancies between models may indicate compositions that require further investigation using first-principles calculations. The discrepancies among the three models probably originate from the different levels of structural information of perovskites encoded in their representations. As mentioned above, the XGBoost method, as a classical machine-learning algorithm, uses only elemental and compositional descriptors, without incorporating any structural information. The CGCNN method incorporates crystal structures by representing them as graphs.21 However, these graphs are constructed solely based on interatomic distances between neighboring atoms as edge features and atomic properties as node features. Consequently, CGCNN does not explicitly encode crystallographic symmetries or detailed local coordination environments. As a more advanced GNN architecture, UPET captures complete local geometric information, including directional features beyond pairwise distances, and further benefits from large-scale pretraining.22 Herein, we combined these three methods to increase the likelihood of identifying perovskites with suitable band gaps.

As shown in Fig. 6(a), for these 48 Ge-based perovskites, CGCNN predicts larger band gaps, with a mean value of 3.96 eV, and exhibits the narrowest prediction distribution. In contrast, UPET produces the broadest range of predictions, with a mean value of 2.16 eV and a standard deviation of 1.55 eV, indicating a stronger sensitivity to compositional variations. While the XGBoost predictions lie primarily between those of CGCNN and UPET, with a mean band gap of 2.70 eV. In addition, as shown in Fig. 6(b), the band-gap predictions obtained using these three methods exhibit relatively low pairwise correlations, likely because the models capture different aspects of the composition–property relationships discussed above. Their combined use therefore provides complementary information and can increase the likelihood of identifying promising candidates, although materials showing large intermodel discrepancies should be further validated through first-principles calculations.

Fig. 6. (a) Distribution and (b) correlation analysis on the bandgaps predicted by the XGBoost, CGCNN and UPET models.

Fig. 6

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3.3. DFT-verified Ge-based perovskites candidates

Based on the combined results of XGBoost, CGCNN, and UPET, a subset of Ge-based perovskite materials showing favorable stability and bandgap characteristics was selected for DFT calculations using VASP. Using multiple predictive frameworks prior to DFT validation reduces the number of candidate materials that require computationally expensive first-principles calculations while increasing confidence in selected compounds. The candidates were then subsequently analyzed through electronic structure calculations to further evaluate their suitability for photovoltaic applications. Here, four hybrid perovskites from Table 2, namely (CH)4NH2GeGeCl6, C3H4NSGeGeBr6, C5NH12GeGeBr6 and C6OH13GeGeBr6, were selected for further structural relaxation and precise bandgap calculations.

Accurate prediction of material properties requires crystal structures that are close to their equilibrium configurations. In this work, the initial crystal structures were generated using the experimentally reported cubic α-FAPbI3 structure as the parent prototype.43 Starting from this prototype, candidate structures were generated through systematic chemical substitution. Specifically, Pb sites were replaced by Ge-containing species, iodine sites were substituted by different halide anions, and the FA cation was replaced by more than ten small organic cations. To ensure a consistent initialization protocol for high-throughput structure generation, the molecular center of mass of each substituted organic cation was aligned with that of the original FA molecule. For the Ge-based compositions, charge neutrality was explicitly considered during structure generation. The resulting double-Ge perovskite compositions follow the general formula (A)2Ge2X6, where A represents a monovalent organic cation and X represents a monovalent halide anion.

Subsequently, the structural optimization for the generated structures is necessary to account for lattice distortions and atomic rearrangements that minimize the total energy of the system. A series of DFT calculations were performed over a range of unit-cell volumes to identify the equilibrium lattice volume. For each candidate material, the lattice volume was systematically varied while preserving the overall crystal symmetry. The total energy corresponding to each volume was calculated, generating an energy–volume relationship for the structure. The resulting energy–volume data was fitted using the Birch–Murnaghan equation to determine the volume associated with the minimum total energy.44 The volume corresponded to the thermodynamically preferred structure at 0 K and served as the starting point for subsequent structural optimization. While maintaining the optimized lattice parameters, structural relaxation calculations were performed after determining the equilibrium volume. During relaxation, atomic positions were allowed to be adjusted until the forces acting on all atoms satisfied the convergence criteria specified in the calculation settings. The optimized structures for the four hybrid perovskites are shown in Fig. 7.

Fig. 7. Relaxed crystal structures for (a) (CH)4NH2GeGeCl6, (b) C3H4NSGeGeBr6, (c) C5NH12GeGeBr6 and (d) C6OH13GeGeBr6.

Fig. 7

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To further assess the thermodynamic stability of the four selected Ge-based hybrid perovskites, a reference convex-hull analysis was performed. The calculated Ehull values for (CH)4NH2GeGeCl6, C3H4NSGeGeBr6, C5NH12GeGeBr6, and C6OH13GeGeBr6 are 0.59, 0.47, 0.55, and 0.55 eV per atom, respectively. The principal decomposition pathways were further checked using PBE/PAW calculations, yielding decomposition energies of 0.58, 0.49, 0.44, and 0.43 eV per atom, respectively. These results indicate that the four structures lie above the reference convex hull at 0 K.

We compared the DFT-calculated band structures with the bandgap values predicted by XGBoost, CGCNN, and UPET. While differences between the methods were expected due to variations in model assumptions and computational approaches, overall agreement among the prediction frameworks provided additional confidence in the identified candidate materials. Materials exhibiting favorable bandgaps across both machine learning predictions and DFT calculations were considered promising for photovoltaic applications. The comparison further demonstrated the effectiveness of combining data-driven screening approaches with first-principles calculations for accelerated perovskite discovery.

Initially, the values of bandgaps of (CH)4NH2GeGeCl6, C3H4NSGeGeBr6, C5NH12GeGeBr6 and C6OH13GeGeBr6, highlighted in bold in Table 2, were predicted by XGBoost, CGCNN, and UPET, yielding values of 3.84/0.53/3.17 eV, 2.13/0.51/2.46 eV, 3.97/4.27/2.25 eV and 3.94/4.58/2.68 eV, respectively. The bandgaps calculated using the PBE functional with spin–orbit coupling (SOC) for these four hybrid perovskites are 0.3193, 0.4121, 0.3212, and 0.8646 eV, respectively, which are significantly smaller than those predicted by the XGBoost, CGCNN, and UPET models. This discrepancy is partially attributed to the neglect of SOC interactions in the training datasets of perovskites, which can lead to a systematic overestimation of the bandgaps for perovskites containing heavy elements.

To accurately calculate the bandgaps of these four hybrid perovskites, which are critical for evaluating their potential in photovoltaic applications, we performed HSE calculations using the FHI-aims software. FHI-aims employs an all-electron approach with HSE functionals, which generally provides reliable bandgap predictions for semiconductors. By using FHI-aims, the calculated HSE + SOC band structures for (CH)4NH2GeGeCl6, C3H4NSGeGeBr6, C5NH12GeGeBr6 and C6OH13GeGeBr6, are shown in Fig. 8. The results indicate that all four hybrid perovskites exhibit indirect bandgaps, with HSE + SOC-calculated bandgaps of 1.5137, 1.7515, 1.6778, and 1.1680 eV, respectively, which are larger than the corresponding PBE + SOC values of 1.1944, 1.3394, 1.3566, 0.3034 eV, respectively. The substantial difference between the HSE + SOC and PBE + SOC bandgaps not only indicates that semilocal PBE functionals significantly underestimate the electronic bandgaps of these hybrid perovskites, even when SOC effects are included, but also highlights the importance of accurately treating exchange–correlation effects and many-body interactions in hybrid perovskites.

Fig. 8. Calculated HSE + SOC bandgaps for (a) (CH)4NH2GeGeCl6, (b) C3H4NSGeGeBr6, (c) C5NH12GeGeBr6 and (d) C6OH13GeGeBr6.

Fig. 8

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To provide additional insight into the band-edge carrier transport of the four selected Ge-based hybrid perovskites, we further calculated their electron and hole effective masses using AMSET. The calculated electron effective masses are 11.58, 0.78, 1.63, and 1.60m0 for (CH)4NH2GeGeCl6, C3H4NSGeGeBr6, C5NH12GeGeBr6, and C6OH13GeGeBr6, respectively, and the corresponding hole effective masses are 45.78, 13.49, 2.32, and 22.66m0, respectively. These band-edge effective-mass results provide transport-related information complementary to the calculated band gaps; however, carrier mobility also depends on carrier-scattering mechanisms and cannot be inferred from the effective mass alone.

4. Conclusions

In this work, we present an integrated computational framework that combines machine learning, deep learning, and first-principles calculations to accelerate the discovery of lead-free Ge-based hybrid perovskites for photovoltaic applications. The optimized XGBoost model enabled the screening of approximately 180 000 perovskite compositions, while SHAP analysis identified the key physicochemical descriptors governing formation energy and bandgap. Subsequent evaluation using CGCNN, UPET, and DFT calculations further refined the candidate set and increased confidence in the predicted electronic properties of selected Ge-based compounds. Four Ge-based perovskites, namely (CH)4NH2GeGeCl6, C3H4NSGeGeBr6, C5NH12GeGeBr6 and C6OH13GeGeBr6, were selected, and their bandgaps were accurately calculated using the HSE + SOC method, yielding values of 1.5137, 1.7515, 1.6778, and 1.1680 eV, respectively, which are suitable for photovoltaic applications. Beyond the specific Ge-based candidates identified in this study, the proposed hierarchical workflow provides a scalable framework for accelerated materials discovery.

Author contributions

Q. J. conceived the project and supervised the research. C. S. developed and trained the models and analyzed the results. C. S. wrote the original manuscript. R. L. contributed to the DFT calculations and revisions. C. S., R. L. and H. Z. contributed to the discussion of the results.

Conflicts of interest

The authors declare no competing interests.

Acknowledgments

The authors acknowledge the use of ChatGPT (OpenAI) for assistance in improving the language and readability of the manuscript.

Data availability

The datasets generated and/or analysed during this current study are available from the corresponding author upon reasonable request.

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Associated Data

This section collects any data citations, data availability statements, or supplementary materials included in this article.

Data Availability Statement

The datasets generated and/or analysed during this current study are available from the corresponding author upon reasonable request.

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