Octahedral-motif-guided design of optoelectronic semiconductors via interpretable machine learning

Abstract

Halide perovskites’ remarkable optoelectronic properties stem from their metal halide octahedra. This connection underscores the promise of systematically extracting the physical rules encoded in octahedral motifs to steer the discovery of optoelectronic materials. Here, we develop an octahedral motif-centric, data-driven framework that couples interpretable machine learning with high-throughput first-principles calculations to accelerate the discovery of optoelectronic semiconductors. We construct motif-based descriptors to train a gradient boosting regression tree model for thermodynamic stability evaluation, achieving a low mean absolute error of 83 meV per atom on datasets comprising ~10⁴ materials. Leveraging the model to accelerate materials discovery, we identify 19 unexplored thermodynamically stable semiconductors with favorable optoelectronic properties. Among them Ca₂GaCoO₅ was successfully synthesized and experimentally verified to exhibit a strong visible light photoresponse. These results support the effectiveness of the machine learning framework for octahedra-containing semiconductors and suggest its potential for extension to other motif-based materials families.

Metal-halide octahedra underpin the strong optoelectronic properties of halide perovskites, but general design rules remain difficult. Yang et al. develop an octahedral motif-based model coupled with first principles screening to discover 19 new semiconductors and validate one experimentally.

Introduction

Optoelectronic semiconductors enable solar energy conversion, photodetection, and light emission. A large and important subset of these materials is built from octahedrally coordinated units, most notably perovskite-type compounds dominated by BX₆ motifs^1–6. In halide perovskites, soft corner-sharing octahedral motifs produce strongly antibonding band edges, yielding direct gaps in the visible range, strong absorption, and small carrier effective masses^6–8. Oxide-based octahedral frameworks, in contrast, typically exhibit more ionic bonding and stiffer lattices, which often widen band gaps but improve chemical and structural robustness^9,10. More broadly, across octahedra-containing crystal families, changes in octahedral chemistry, distortion, and connectivity systematically reshape electronic structure and transport^11–14. These trends suggest a general principle: local octahedral motifs provide a common structural handle linking optoelectronic behavior with materials stability. However, first-principles understanding of octahedral stability remains largely perovskite-centric, where systematic trends are typically derived within a few closely related prototypes. Extending such prototype-bound insights to the full landscape of octahedra-containing crystals is nontrivial because the same coordination motif can host widely different bonding and connectivity environments. This limitation is acute for heterogeneous spaces spanning oxides, halides, chalcogenides^15,16, spinels¹⁷, and related frameworks. Hence, a quantitative and transferable approach is needed that captures stability trends directly from octahedral chemistry and geometry rather than relying on prototype-specific heuristics.

Data-driven machine learning (ML)^18,19 has emerged as a powerful complement to first-principles calculations for materials discovery, especially when trained on density functional theory (DFT)^20,21 datasets of crystal structures and formation energies^22–24. Most existing ML models for crystalline materials rely primarily on elemental composition, global lattice descriptors, or generic local-environment representations²⁵. For example, Voronoi-based crystal descriptors incorporate local coordination and packing information, while Smooth Overlap of Atomic Positions (SOAP)-type descriptors encode local atomic environments in a symmetry-preserving but high-dimensional form^26,27. While highly effective, these approaches generally connect elemental and whole-crystal features directly to crystal-level properties, making it difficult to extract chemically explicit motif-level factors. Recently physics-informed descriptors that encode local bonding environments have shown improved interpretability, such as physically interpretable tolerance factors²⁸ and octahedral distortion descriptors²⁹. Bartel et al.²⁸ proposed a Sure Independence Screening and Sparsifying Operator (SISSO)-derived analytical descriptor, the new octahedral factor, which provides probabilistic guidance for the stability of diverse perovskite oxides and halides. Here, we extend this perspective by treating the octahedral motif itself as the central, chemically resolved representation unit for understanding stability trends in octahedra-containing materials beyond a single perovskite-prototype family. At the same time, deep learning methods such as graph neural networks learn latent structural representations directly from crystal graphs and have shown strong predictive performance, but typically rely on much larger datasets and offer less direct chemical interpretability in the small, heterogeneous data regime considered here³⁰.

Motivated by these challenges, we seek a framework that elevates octahedral motifs from descriptive structural units to quantitative and interpretable predictors of thermodynamic stability across chemically diverse crystal families. Rather than treating stability as a black box outcome of composition or lattice statistics, our approach explicitly encodes local octahedral chemistry and coordination contrasts in a motif-based representation designed to remain transferable beyond any single structural prototype. Specifically, we construct physically motivated, motif-based descriptors in which octahedral chemistry, local coordination contrast, and short-range structural heterogeneity are encoded at the level of the motif rather than only at the level of the full crystal or a generic local environment. These descriptors are used to train a gradient boosting regression tree (GBRT) model based on the dataset (detailed in Supplementary Note 1 and Supplementary Figs. 1–4) for thermodynamic stability evaluation across chemically diverse crystal families, followed by symbolic regression (SR) to extract compact analytical expressions that expose the dominant stability factors (Fig. 1). Formation energy is chosen as the primary ML target because it provides the most general and physically consistent entry point for motif-level stability learning, whereas convex hull stability and optoelectronic properties are evaluated subsequently within the screening workflow. Beyond predictive accuracy, this framework differs from generic local environment encodings by yielding chemically interpretable motif-level stability rules, and differs from latent graph-based models by explicitly retaining a human readable connection to octahedral chemistry. It thereby enables the targeted identification of stable optoelectronic semiconductors from previously unexplored octahedra-containing chemical spaces, with experimental synthesis and optoelectronic validation demonstrating its practical effectiveness.

Fig. 1. Data-driven framework for octahedral-motif-guided discovery of optoelectronic materials.

a Overview of the three-step workflow: dataset curation, interpretable ML model construction, and optoelectronic material screening. We curate 3362 octahedral-containing crystals from ICSD under predefined chemical and structural constraints. Feature engineering is employed to build the ML stability model and to derive compact stability relations by symbolic regression. Symbols and meanings: χ (electronegativity), $R^C$ (covalent radius), $M$ (elemental mass), $L_{Octa}$ (octahedral bond length), $A_{Octa}$ (octahedral bond angle), $O_f$ (octahedral factor), $\rho$ (crystal density), $SG$ (space group), $V_{cell}$ (volume of unit cell), $F$ (elemental feature), $S_x$ (Voronoi-face area of coordinating atoms). The trained stability model, together with DFT evaluations, yields 19 thermodynamically stable optoelectronic candidates. $\mathrm{\Delta }{E}_f$ is the formation-energy change, ${\mathrm{\Delta }E}_{hull}$ is the convex hull energy, $E_g$ is the band gap, $\mu$ is the reduced effective mass, and $I_{\alpha \left(E\right)}$ is the near-edge absorption integral. b Representative octahedral building units, including single octahedra, multiple-octahedra motifs, and anti-octahedral motifs, together with typical motif connection patterns.

Results

Development of octahedral-motif-centric, data-driven framework with interpretable ML

To construct an interpretable ML model to evaluate how octahedral motifs affect crystal stability, we built a formation energy prediction model using octahedral-motif-specific feature engineering (Fig. 2a). In feature construction (Supplementary Note 2), we adopt a hierarchical representation that links elemental chemistry, local BX₆ octahedral motifs, and global crystal packings. This hierarchy makes the motif a physically meaningful bridge between elements and crystal frame, enabling interpretable rules that remain transferable across diverse prototypes. At the elemental level, we include standard atomic attributes (e.g., radius, electronegativity, electron affinity)^31–33. At the motif level, we explicitly resolve the octahedral environment using 23 physics motivated descriptors spanning Goldschmidt tolerance factor $T_f$^31,34,35, octahedral factor $O_f$^31,36, octahedral distortion measures^37,38, local chemical contrast descriptors between the B-site and coordinating anions²⁶, and indicators of octahedral connectivity. At the crystal level, we add global descriptors such as density, packing fraction, space group and Coulomb interactions³⁹. This procedure generates an initial pool of 664 candidate features.

Fig. 2. Feature engineering and octahedra-specific descriptors.

a Overview of the feature engineering workflow, comprising feature construction, iterative feature selection, and algebraic feature combination. Feature construction assembles a hierarchical descriptor pool from elemental attributes, octahedral-motif descriptors, and crystal-level metrics. Feature selection removes weakly informative descriptors. Feature combination generates new composite descriptors using algebraic operators, including addition, subtraction, multiplication, and division under descriptor-family, physical-compatibility, and complexity constraints. Feature selection and feature combination are repeated iteratively to arrive at the final compact motif-informed feature set. b The two important octahedra-specific descriptors obtained through feature engineering.

We then refine the feature space through a 3-staged GBRT^31,33,40,41 feature engineering workflow that iteratively couples feature selection with feature combination (Supplementary Note 3 and Supplementary Fig. 5). In stage 1, GBRT-based selection reduces the raw feature pool through five feature selection steps. In stages 2 and 3, octahedral motif anchors are combined by algebraic operators into level-1 and level-2 descriptors, followed by additional feature selection. This process reduces the descriptor number while enriching chemically interpretable descriptor families and lowering the model error. Subsequent recursive feature elimination (RFE)^42,43 compacts the motif-informed pool to a 38-feature error plateau (Supplementary Fig. 6), followed by a final RFE to create a representative 34-feature compact set used for our GBRT training and SR (Supplementary Note 4 and Supplementary Fig. 7). Complete workflow and the feature list are provided in methods section and Supplementary Table 1.

Using these 34 motif-informed descriptors, the GBRT model exhibits reliable predictive accuracy for DFT formation energy ($E_f^{DFT}$). Based on the constructed dataset of 3362 octahedra-containing structures, a 10-fold cross validation yields a mean absolute error (MAE) of 83 meV/atom and R² = 0.97 (Fig. 3a). Additional learning-curve and diagnostic analyses show no strong signatures of overfitting and only weak inter-feature correlations (Supplementary Figs. 8–11). A complementary error analysis on one test set (Supplementary Note 5) shows that the model remains accurate for most samples (54.8% samples have absolute error below 50 meV/atom), while the largest errors are concentrated in a few chemically atypical outliers (Supplementary Fig. 12). To place this error in context, descriptor-based ML models for crystal formation energies commonly operate in the 100 meV/atom regime, while graph network models reach much lower errors only when trained on datasets orders of magnitude larger than the present one^26,44–46. Thus, the current accuracy is appropriate for the intended role of the model in this work, namely, motif-guided ranking and preselection within a chemically and structurally heterogeneous octahedral domain rather than universal prediction across all inorganic crystals.

Fig. 3. Results of machine learning and symbolic regression.

a Comparison between GBRT-predicted formation energies and DFT-calculated formation energies. MAE, MSE, RMSE, R² are obtained from 10-fold cross-validation. b Top ten features ranked by GBRT feature importance. B and X represent the central and coordinate atoms of the octahedron. χ is electronegativity, $R^C$ is covalent radius, $NdV$ and $NuV$ are the number of d-valence electron and unfilled valence electron, respectively. $LD$ represents local differences in elemental properties. $GSbandgap$ is the elemental ground-state band gap at 0 K. c Relationship between formation energy and the most important descriptor ${\chi }_B/R_B^C-6\times {\chi }_X/R_X^C$. Points are grouped by the dominant $X_{Octa}$ species (F, O, Cl, S, and others). The box plot summarizes formation-energy distributions for each group, with mean and median indicated. d Comparison between symbolic regression-calculated formation energies and DFT-calculated formation energies.

Identification of key motif-level factors and compact analytical expressions governing material stability

Analysis of GBRT feature importance reveals that octahedral-motif-derived descriptors dominate the prediction of formation energy, providing clear physicochemical insight into stability trends. Among the 34 retained features, those explicitly encoding local octahedral environments account for a disproportionate share of model importance (Supplementary Table 1), indicating that variations in octahedral chemistry rather than global lattice attributes primarily govern $E_f$. Within this motif-focused subset, two classes of descriptors emerge as the principal stability determinants (Fig. 2b). The most influential descriptor is ${\chi }_B/R_B^C-6\times {\chi }_X/R_X^C$, which alone accounts for 29% of the total feature importance (Fig. 3b). Here, electronegativity is scaled by the covalent radius to reflect bond-forming capability, and the factor of six explicitly encodes octahedral coordination. This descriptor quantifies the balance between the electron-attracting capability of the central $B_{Octa}$ cation and that of its six coordinating $X_{Octa}$ ligands. Rather than serving as a purely statistical variable, it can be interpreted as a motif-level measure of how the octahedral coordination shell balances bond ionicity, covalency, and local bond competition. Consistent with periodic trends in electronegativity and covalent radius, it cleanly separates materials according to anion chemistry and exhibits a monotonic correlation with $E_f$ (Fig. 3c and Supplementary Fig. 13). Physically, this indicates that the thermodynamic stability of octahedral frameworks is governed by how the electron attraction of the central cation, measured on a covalent length scale, is balanced by its coordinating ligands.

The second dominant motif descriptor class is formed by local difference ($LD$) measures, which quantify chemical contrast across the octahedral coordination shell. $LD$ descriptors are defined as area-weighted averages of the absolute difference in an elemental property $F$ between $B_{Octa}$ and $X_{Octa}$ sites over the faces ($S_i$) of the octahedral Voronoi polyhedron⁴⁷:

$${LD}_F=\frac{{\sum }_i\left(S_i\times ∣F_B-F_X∣\right)}{{\sum }_i{S}_i}$$ 1

They are a family of motif-resolved quantities computed from the same Voronoi area weighted formula, so that the local chemical contrast between $B_{Octa}$ and $X_{Octa}$ is modulated by the geometry of the octahedral coordination shell through the face areas $S_i$. Descriptors such as ${LD}_{NdV}$ and ${LD}_{\chi }$ quantify the mismatches in d-electron configuration and electronegativity, respectively. Six $LD$-related descriptors were retained in the final model, indicating that the degree of chemical mismatch between the central cation and its coordinating anions is a robust, motif-level lever controlling stability across diverse octahedral chemistries. From the perspective of local bonding, these descriptors reflect differences in electronic character between the central cation and its ligands, and therefore relate to the favorability of local hybridization and bond stabilization within the octahedral shell.

In contrast, the low ranking of conventional perovskite descriptors highlights the limitations of classical rules when applied beyond fixed structural prototypes. The ionic radius-based analog ${\chi }_B/R_B^i-6\times {\chi }_X/R_X^i$ and octahedral factor $O_f$, although successful within single perovskite families^31,34,36, contribute little to stability prediction in a chemically heterogeneous octahedral space. Similarly, simple geometric metrics such as bond angles, connectivity counts, and space group identifiers show limited discriminative power and are eliminated during feature selection (Supplementary Note 6). These results demonstrate that prototype-specific heuristics fail once chemical diversity dominates, whereas motif-based descriptors remain transferable and physically interpretable.

This result does not contradict the established role of octahedral geometry in perovskite electronic structure. In halide perovskites, metal-halide-metal bond angles and octahedral tilting strongly affect the optical band gap by modulating metal-halide orbital overlap⁴⁸, while bonding analyses have shown that metal-halide orbital interactions and antibonding band-edge states are central to their optoelectronic behavior⁴⁹. These studies establish that octahedral geometry, covalency, and metal-ligand hybridization are tightly coupled in prototype perovskites⁵⁰. Our results are consistent with this picture because the leading descriptors identified here emphasize electronegativity balance, covalent length scales, and local $B_{Octa}$-$X_{Octa}$ chemical contrast. However, in the present chemically heterogeneous octahedral dataset, these effects are distributed across motif-resolved chemical and local-difference descriptors rather than being captured primarily by a single perovskite-style geometric descriptor such as a bond angle, distortion, or octahedral factor.

To provide more direct evidence, we examine whether the retained motif-resolved descriptors miss important octahedral distortion information, we augmented the final 34-feature pool with six additional features regarding octahedral distortions or coordination environments and retrained the GBRT model (Supplementary Note 7 and Supplementary Fig. 14a)⁵¹. The predictive performance changes only marginally, with the 10-fold cross-validation MAE varying from 83 to 85 meV/atom, and all added descriptors remaining in the low-importance tail of the feature ranking (Supplementary Fig. 14b, c). A surrogate analysis further indicates that the original 34-feature set already captures a substantial fraction of the information carried by these geometry-related descriptors (Supplementary Fig. 14d). These results suggest that the retained motif-resolved descriptors encode distortion- and coordination-related effects in a distributed manner, even though they are not explicit one-to-one geometric measures.

The GBRT model is used as the primary predictive model to capture nonlinear interactions among motif-resolved descriptors across a chemically heterogeneous space. To further distill these insights, we employ SR^52,53 to compress the GBRT-learned feature space into a compact analytical stability relation. This complementary use of GBRT and SR combines the flexibility of nonlinear prediction with the transparency of symbolic representation, enabling both stability-aware screening and chemically interpretable extraction of motif-level stability relations from the same descriptor space. The resulting expression achieves a Pearson correlation coefficient of r = 0.90 (Fig. 3d) and the formula is:

$$E_f^{SR}=\alpha \times EN\times {NV}_1+\beta \times {NV}_2\times E_{ewald}+\gamma$$ 2

where $EN$=${\chi }_{mean}\times \sqrt{L{D}_{\chi }}$ represents electronegativity scale that is explicitly modulated by local differences, ${NV}_1$ = $\sqrt{\frac{N_{pVal}}{N_{Val}}}$ indicates normalized p-valence electrons with respect to valence electrons, ${NV}_2={\left(\frac{N_{Unfilledavgdev}}{N_{Unfilledmean}}\right)}^2\times N_{pVal}\times N_{Unfilledrange}$ accounts for variations in unfilled electron configurations. $E_{ewald}$ refers to the energy of coulombic interactions by Ewald summation. The coefficients $\alpha$, $\beta$, $\gamma$ are constants at −0.87, 0.34, and 0.16, respectively. The leading term $\alpha \times EN\times {NV}_1$ combines an electronegativity scale modulated by $L{D}_{\chi }$ with a normalized valence-electron descriptor. This term alone already captures most of the stability trend (r = 0.82; Supplementary Fig. 15a), and $EN$ reproduces the same anion-dependent separation learned by the GBRT-leading motif descriptor (Supplementary Fig. 15b and 15c). To further assess the transferability of the SR relation beyond dataset, we directly applied the SR expression to 100 octahedral structures randomly selected from the Materials Project (MP)⁵⁴ without refitting (Supplementary Note 8). The resulting Pearson correlation remains high ($r$ = 0.89; Supplementary Fig. 15d), indicating that the compact SR descriptor retains meaningful predictive power on an independent external dataset. Taken together, these results show that SR reduces the octahedral motif feature space to a simple stability relationship. This relationship is consistent with the feature importance identified by GBRT and condenses the dominant bonding-relevant motif variables into a compact analytical form, thereby linking local octahedral chemistry to thermodynamic stability in a more chemically interpretable way.

Design of new SOMs from previously unexplored octahedra-containing chemical space

With the interpretable stability model in hand, we next use it to explore new semiconductor oxides and chalcogenides whose properties are governed by octahedral motifs. Starting from the 3362 octahedra-containing parent structures in the dataset, we generate candidates by systematically exploring chemically analogous $B_{Octa}$ and $X_{Octa}$ configurations, in which elements in octahedral motifs are chosen from the same main group and neighboring periods to ensure charge neutrality. This strategy favors motif-preserving interpolation within learned octahedral chemistries, allowing the model to serve as a more reliable and interpretable guide for stability-aware materials design. Rather than attempting unconstrained extrapolation to arbitrary crystal topologies, our workflow is designed to explore chemically coherent variations around experimentally reported octahedral frameworks, so that the learned stability rules remain applicable within the model domain. After excluding compositions already reported in Inorganic Crystal Structure Database(ICSD)⁵⁵, Open Quantum Materials Database (OQMD)⁵⁶ and the MP⁵⁴, this design space contains 2442 unexplored candidates (Fig. 4a).

Fig. 4. Stability screening workflow for motif-preserving candidates.

a Chemical diversity of the 2442 candidates, shown by their composition types and the distribution of $X_{Octa}$ species. b Distribution of $\mathrm{\Delta }{E}_f$ for 2442 candidate materials. 691 materials satisfying $\mathrm{\Delta }{E}_f$ ≤ 25 meV/atom are retained. c Comparison between ML-predicted formation energy and DFT-calculated formation energy for the 691 retained candidates. d Relationship between ${\mathrm{\Delta }E}_{hull}$ and $E_f^{DFT}$ for the 691 candidate materials. 180 materials with ${\mathrm{\Delta }E}_{hull}$ ≤ 0 eV/atom are selected.

To quantify stability trends within this space, we define the formation energy change of each candidate relative to its parent as:

$$\mathrm{\Delta }{E}_f=E_f^{cand}-E_f^{parent}$$ 3

where $E_f^{cand}$ is the ML-predicted formation energy for a generated candidate and $E_f^{parent}$ is the DFT formation energy of the corresponding prototype structure. A negative $\mathrm{\Delta }{E}_f$ indicates that the candidate is predicted to be more stable within the same octahedral motif framework. Using the thermal energy scale at room temperature ($k_{\mathrm{B}}T\approx$ 25 meV/atom) as a physically motivated and practically conservative screening threshold, we identify 691 candidates with $\mathrm{\Delta }{E}_f$ ≤ $k_{\mathrm{B}}T$ (Fig. 4b) from the 2442 candidates. The energy scale of approximately 25 meV/atom corresponds to Boltzmann thermal energy $k_{\mathrm{B}}T$ at room temperature and is used as a practical reference for prioritizing candidate compounds. Such a small near-hull energy window is widely used in computational materials screening to account for the fact that many experimentally realized inorganic compounds are slightly metastable and that screening workflows inevitably involve numerical uncertainty^57,58. Under this criterion, the ML-based screening may miss some candidates close to the stability boundary, and some discarded structures may still possess relatively high thermodynamic stability.

To evaluate the extrapolative performance of the model in the candidate materials space, we further carried out DFT calculations for these generated candidates. Within this motif-preserving design space, ML predictions remain consistent with subsequent DFT calculations (R² = 0.94 and MAE = 141 meV/atom in Fig. 4c), indicating that the learned motif-level stability rules can be transferred beyond the original training compounds to newly generated octahedral candidates. We further examined the impact of volume changes induced by variations in the generative components on stability evaluation and found that their effect is negligible (Supplementary Note 9 and Supplementary Figs. 16–18). We also performed an external evaluation on the same MP-structures set as illustrated in SR evaluation (Supplementary Note 10). As shown in Supplementary Fig. 19, the predictions of model still show agreement with the DFT values, supporting meaningful transferability across chemically diverse octahedral systems outside the original training domain.

Following ML screening, we perform first-principles calculations of convex hull stability, band gap, carrier effective mass and optical absorption to identify optoelectronic SOMs. As shown in Fig. 4d and Supplementary Note 11, convex hull analysis identifies 180 thermodynamically stable compounds ($\mathrm{\Delta }{E}_{hull}$ ≤ 0 eV/atom) from the 691 candidates. Among them, 129 materials exhibit a nonzero band gap (Supplementary Table 2). To quantify carrier transport, we use the reduced effective mass^59,60:

$$\mu =\frac{1}{1/m_h^{*}+1/m_e^{*}}$$ 4

where $m_h^{*}$ and $m_e^{*}$ represent the effective masses of holes and electrons. To quantify the absorption onset relevant for near-edge photoresponse, we define an integrated absorption coefficient:

$$I_{\alpha \left(E\right)}={\int }_{E_g}^{E_g+0.2eV}\alpha \left(E\right)dE$$ 5

where $\alpha \left(E\right)$ is the absorption spectrum, $E_g$ is the band gap and large $I_{\alpha \left(E\right)}$ represents steep absorption onset near the band edge. Similar integral measures of optical spectra have been used to characterize absorption strength and related physicochemical behavior^61,62. We set screening thresholds $\mu$ < 1.0 $m_0$ and $I_{\alpha \left(E\right)}$ > 1000 to identify candidate optoelectronic semiconductors with metrics on the same physically relevant scale as representative halide-perovskite absorbers⁶³. The chosen $\mu$ criterion is likewise consistent with common materials screening practice, where carrier effective masses below 1.0 $m_0$ are often used to exclude clearly heavy carrier systems while retaining candidates in a transport relevant regime⁶⁴. Compared to typical perovskite optoelectronic materials, the threshold of $I_{\alpha \left(E\right)}$ lies in a reasonable range, as demonstrated by benchmarks of $I_{\alpha \left(E\right)}$ = 2722 eV cm⁻¹ for MAPbI₃, 1679 for CsSnI₃, and 1740 for CsPbBr₃.

Finally, 19 SOMs satisfy our optoelectronic criteria (Fig. 5a and Supplementary Table 3). Their band gaps span a wide range from 0.03 to 5.48 eV, with the majority lying between about 1 and 3 eV. The carrier transport is generally favorable: 12 out of 19 compounds exhibit $\mu$ below 0.6 $m_0$ and 8 of them have $\mu$ below 0.5 $m_0$, indicating broadly light band-edge quasiparticles (Fig. 5b). Electron effective masses are consistently below 1.0 $m_0$, whereas hole masses exceed 1.0 $m_0$ for most systems, so transport is expected to be electron dominated except in N₆Re₂SBa₆ and InFK₆Te₄, where both electrons and holes remain relatively light. The thresholds are used as prioritization criteria rather than strict boundaries, therefore these 19 SOMs could be understood as the highest priority subset with optoelectronic metrics comparable in scale to MAPbI₃, while candidates outside the selected thresholds may still merit future investigation. All candidates show a sharp absorption onset near the band edge, though their full absorption spectra vary substantially (Fig. 5c). CoSbO is particularly notable since its absorption coefficient already exceeds 5 × 10⁵cm⁻¹ at 2 eV and increases almost monotonically with energy. Pt₂NaS₃ also shows strong visible light absorption, outperforming MAPbI₃. Several compounds, including Ca₂GaCoO₅, reach absorption coefficients on the order of 10⁵cm⁻¹ near their band edges. In contrast, Zn₄Ga₂O₇ and N₆Re₂SBa₆ display comparatively weaker overall absorption, even though they feature very small effective masses or extremely narrow band gaps.

Fig. 5. Optoelectronic properties and crystal structures of the discovered SOMs.

a Reduced effective mass $\mu$ and near-edge absorption integral $I_{\alpha \left(E\right)}$ for 129 SOMs. The 19 SOMs meeting the optoelectronic criteria are highlighted. b Electrons ($m_e^{*}$) and holes ($m_h^{*}$) effective masses for 19 SOMs. c Optical absorption spectra of 19 SOMs and MAPbI₃. d Chemical formulas and crystal structures of the 19 SOMs. Octahedral compositions are indicated in brackets.

To illustrate how these transport and absorption behaviors arise from the underlying octahedral frameworks, we examine four representative SOMs in more detail (Fig. 5d). CoSbO features a 3D network of strongly distorted CoO₃Sb₃ motifs connected by both corner and edge sharing. It has a direct band gap of 0.91 eV, with both VBM and CBM dominated by Co states and band-edge charge concentrated on Co at the junctions of the octahedra (Supplementary Fig. 20). The strong spatial overlap of band-edge states on this Co-centered framework is consistent with its exceptionally large absorption coefficient. Pt₂NaS₃ adopts a layered arrangement of edge sharing PtS₆ and PtS₄Na₂ octahedra and is a near direct semiconductor with a band gap 1.46 eV. Both VBM and CBM are derived from Pt and S states, and the band-edge charge is localized mainly on S within the octahedra, so that an extended Pt-S network supports strong optical transitions and overall absorption is higher than that of MAPbI₃. Ca₂GaCoO₅ represents a more regular 3D framework in which corner sharing CoO₆ octahedra are filled by GaO₃ tetrahedra and Ca-centered polyhedra. Its direct 2.01 eV gap at the $\mathrm{\Gamma }$-point arises from spin-polarized Co-d states localized at the connections between CoO₆ octahedra, and the band dispersion combines dispersive and more localized components along the $\mathrm{\Gamma }$-X-S-Y-$\mathrm{\Gamma }$ path. Consistent with this mixed dispersive character, the absorption coefficient is comparable to MAPbI₃ at the edge but growing more moderately at higher photon energies. Zn₄Ga₂O₇ highlights a different type of octahedral environment. Its GaO₆ octahedral layers are supported by ZnO₃ tetrahedra and GaO₄ units. The reduced effective mass $\mu$ = 0.16 $m_0$ is the smallest among all candidates, driven by a very light electron mass, and the VBM and CBM involve hybridized Zn, Ga, and O states with band-edge charge distributed over ZnO₃, GaO₄, and GaO₆ polyhedra. Despite this highly dispersive conduction band, its overall absorption is more moderate, with $\alpha \left(E\right)$ increasing in steps and exhibiting several pronounced peaks rather than a single monotonically rising profile. These cases collectively demonstrate that strongly connected and chemically active octahedral frameworks, such as the CoO₆ and PtS₆ networks, tend to combine light band-edge quasiparticles with intense, broadband absorption, whereas more complex distributions of band-edge states over multiple motifs, as in Zn₄Ga₂O₇, can decouple very light effective masses from the strong absorption responses.

To understand the nature of the chemical bonding in representative SOMs, we have performed the crystal orbital Hamiltonian population (COHP) analysis (Supplementary Note 12 and Supplementary Fig. 21). Consistent with the band structures, projected density of states, and band-edge charge densities (Supplementary Fig. 20), the chemically active interactions are concentrated on motif-resolved $B_{Octa}$-$X_{Octa}$ bonds rather than being distributed uniformly across the crystal. For example, in Ca₂GaCoO₅, although Ga-O provides the strongest overall framework bonding, the dominant near-edge antibonding contribution is localized on Co-O, consistent with the Co-derived band edges and the central role of CoO₆ octahedra in the photoresponse. In Pt₂NaS₃, Pt-S overwhelmingly dominates both the total bonding strength and the near-edge antibonding contribution, while Pt-Na is negligible, confirming that the optoelectronic functionality is controlled by the Pt-S octahedral network rather than by all structural components equally. These COHP results connect the ML-identified descriptors to the established bonding picture of octahedral semiconductors^48–50. Our representative SOMs follow the same motif-level logic: the electronically active states are concentrated on specific $B_{Octa}$-$X_{Octa}$ bonding and antibonding channels. Therefore, descriptors based on electronegativity contrast, covalency, and $B_{Octa}$-$X_{Octa}$ local differences act as physically interpretable proxies for the bonding competition and hybridization pathways that link octahedral chemistry to stability and band-edge activity.

High-performing SOMs correspond to octahedral frameworks with strong and well-defined $B_{Octa}$-$X_{Octa}$ chemical contrast, where chemically active bonds and band-edge states are concentrated on extended octahedral networks, giving rise to both light effective masses and intense optical transitions. In more motif-mixed environments, band-edge charge is distributed across multiple polyhedra, and the relevant bonding channels are correspondingly more delocalized, which decouples carrier transport from strong absorption. Octahedral-motif-based descriptors therefore provide a unified, physics-informed set of design handles that connect local chemistry, thermodynamic stability, and optoelectronic functionality, enabling the targeted discovery of new SOMs.

To further assess dynamical stability, we performed phonon calculations for all 19 SOMs identified in this work (Supplementary Note 13 and Supplementary Fig. 22). Six compounds, including Ca₂GaCoO₅, Ag₂Rb₄SbBr₉, CoSbO, O₂InNa₃, Ge₃Tl₂ZrO₉, and ZnRb₂Bi₂S₅, exhibit no imaginary phonon modes within the harmonic approximation. We therefore regard the phonon stable subset as a group of candidate SOMs, while the remaining SOMs may still warrant further investigation under finite temperature or synthesis-dependent conditions. However, phonon instability within the harmonic approximation does not necessarily mean that a material cannot be synthesized. In practice, successful synthesis still faces nontrivial barriers and depends on multiple additional factors, such as reaction pathways, nucleation, diffusion, temperature effects, and kinetic trapping.

Experimental synthesis and characterization of newly discovered SOM Ca₂GaCoO₅

To test the practical realizability of the motif-guided screening strategy, we selected Ca₂GaCoO₅ for experimental verification as a proof-of-principle case. This compound exhibits a direct band gap, large absorption coefficients comparable to that of MAPbI₃⁶⁵, phonon dynamical stability, and a practically accessible synthesis route^66,67. This experimental validation supports the feasibility of the proposed workflow, while a broader demonstration of its general applicability will require synthesis and characterization of additional predicted candidates in future work.

Its synthesis strategy can be adapted from the isostructural compound Ca₂GaFeO₅ by replacing the Fe precursor with a Co precursor⁶⁸. As a result, Ca₂GaCoO₅ was successfully synthesized using four different cobalt precursors (Supplementary Fig. 23), demonstrating its facile synthesizability. Experimental details are provided in the methods section. Figure 6a compares the measured X-ray diffraction (XRD) pattern with the simulated pattern, confirming that Ca₂GaCoO₅ crystallizes in the orthorhombic Pnma space group. Refinement of stress-free powder yields lattice parameter (a = 5.333 Å, b = 14.734 Å, c = 5.547 Å) in excellent agreement with DFT optimized values (a = 5.335 Å, b = 14.356 Å, c = 5.615 Å), with deviation below 2%. High-resolution transmission electron microscopy (HRTEM) further corroborated the Pnma crystal structure of Ca₂GaCoO₅. Lattice fringes corresponding to the (120) and (001) planes are clearly resolved, and the selected area electron diffraction (SAED) pattern matches the calculated crystallographic model (Fig. 6b). Scanning electron microscopy (SEM) in the lower panel of Fig. 6b reveals a densely sintered morphology with well-defined grain boundaries, no detectable secondary phases, and an average grain size of approximately 2.09 μm. Furthermore, complementary energy dispersive X-ray spectroscopy (EDS) mapping shows a homogeneous distribution of Ca, Ga, Co, and O elements (Supplementary Fig. 24). Together, these results verify the successful synthesis of phase pure, highly crystalline Ca₂GaCoO₅.

Fig. 6. Experimental validation of crystal Ca₂GaCoO₅.

a Experimental and simulated XRD patterns of Ca₂GaCoO₅. b HRTEM image (scale bar = 2 nm) with SAED pattern (inset, scale bar = 2 nm⁻¹) and SEM micrograph (scale bar = 5 μm) with grain size distribution (inset) of the synthesized Ca₂GaCoO₅. c ($\alpha$hν)² with absorption spectrum (inset) as a function of hν of synthesized Ca₂GaCoO₅. d Optoelectronic response at 0.1 V upon irradiation with light of various wavelengths.

The optical properties of Ca₂GaCoO₅ were examined by UV-vis absorption spectroscopy. As shown in Fig. 6c (inset), the absorption spectrum exhibits pronounced optical transitions characteristic of a semiconductor. The optical band gap was determined to be 1.6 eV (Fig. 6c) based on the Tauc transform formula: ($\alpha$hν)² = B(hν-$E_g$), where B is a constant, $\alpha$ is the absorption coefficient and hν represents photon energy. This value is slightly smaller than the DFT-calculated gap of 2.01 eV (details provided in methods section). Indeed, a sub band gap absorption tail below 1.6 eV is observed and can be associated with localized impurity levels arising from mixed Co²⁺/Co³⁺ valence states, which introduce allowed sub-gap optical transitions. To probe the optoelectronic response, time-dependent current measurements were performed under an applied bias of 0.1 V to ensure stable current readings. As shown in Fig. 6d, Ca₂GaCoO₅ exhibits a pronounced photoelectric response under illumination at both 630 nm (0.436 W/cm²) and 520 nm (0.296 W/cm²), with the current consistently returning to baseline upon cessation of light. The reversible and wavelength-dependent photocurrent suggests effective photocarrier generation in the visible range.

Discussion

In summary, we establish a motif-centric and physically interpretable machine learning framework for the rational discovery of optoelectronic semiconductors dominated by octahedral coordination. By constructing descriptors that explicitly encode local octahedral chemistry and coordination contrasts, the model connects empirical chemical intuition with data-driven prediction and enables the extraction of transferable stability rules. The learned stability trends indicate that thermodynamic stability across octahedra-containing materials is primarily governed by motif-level chemical factors, especially electronegativity balance on a covalent length scale and local coordination shell contrast. Guided by these motif-level stability principles, we systematically explored a previously unexplored octahedra-containing chemical space and identified a diverse set of thermodynamically stable compounds with promising optoelectronic properties. Phonon calculations were further carried out for the representative SOMs and six compounds exhibit dynamical stability. Experimental synthesis and optoelectronic validation of Ca₂GaCoO₅ further demonstrate the practical reach of the approach.

From a conceptual perspective, the novelty of the present motif-centric framework introduces an intermediate structural-unit layer between elemental and whole-crystal features, thereby explicitly encoding chemically and geometrically meaningful mesoscale structural patterns. In chemically diverse yet motif-constrained materials spaces, such a representation facilitates the extraction of transferable stability rules with direct chemical meaning. At the same time, because the descriptor construction is defined on local center-coordination motifs rather than on a single crystal prototype, the same strategy may in principle be extended to other motif-based materials families beyond the octahedral systems studied here. While the resulting predictions are well grounded within the space of octahedra-containing materials, the framework has not explicitly captured the full complexity of structure formation, including phase competition, kinetic effects, or the emergence of alternative coordination motifs. Overcoming these limitations will require approaches that move beyond fixed motif spaces and allow the generation of new structural candidates, rather than selection from predefined configurations. In this regard, integrating interpretable motif-based principles with structure generation models offers a promising path forward, because such models could expand the search beyond parent-derived substitutions while preserving chemically meaningful motif constraints. We therefore view the present workflow as a complementary component for future generative materials design, where physically grounded octahedral stability rules can guide exploration of broader structural landscapes.

On the experimental side, beyond the validated Ca₂GaCoO₅, the framework identifies a broader set of octahedra-containing candidates that appear chemically and synthetically accessible. Several ternary compounds, such as CoSbO, NaPt₂S₃, and Zn₄Ga₂O₇, are closely related to previously reported materials and can be understood as arising from straightforward elemental substitutions within known structural motifs, suggesting accessibility via conventional solid state or chalcogenide synthesis routes. More compositionally complex quaternary compounds, including ZnRb₂Bi₂S₅ and Ag₂Rb₄SbBr₉, extend beyond well-studied families but remain structurally analogous to established chalcogenide or pnictide frameworks. Their identification illustrates the practical reach of the motif-guided strategy and highlights clear opportunities for future experimental exploration. In this sense, the motif-based framework functions not only as a discovery tool but also as a chemically interpretable map that can help organize and prioritize experimental efforts across diverse octahedra-containing materials.

Methods

Dataset construction

The octahedral motif dataset was curated by exhaustively screening the ICSD (Data Release 2022.2) using the following criteria: presence of an octahedral motif, high crystal symmetry, inclusion of chalcogen or halogen elements, and a primitive cell with fewer than 50 atoms. As shown in the left panel of Fig. 1a, 3362 crystals featuring octahedral motifs were selected to construct the dataset. The methodology of identifying octahedral motifs and screening process are provided in Supplementary Fig. 1.

This dataset showcases extensively diverse structures that are distinct from traditional perovskite materials. These structures are distributed across 110 different space groups, which represent nearly half of all recognized space group types. In terms of composition, binary to quinary compounds make up the vast majority of the dataset (Supplementary Fig. 2a). The unit-cell sizes range from 2 to 48 atoms, with a median of 24 atoms (Supplementary Fig. 2b). The number of symmetry operations ranges from 6 to 192, with a median of 16, indicating that the dataset is not restricted only to highly symmetric structures (Supplementary Fig. 2c). In addition, dimensionality analysis shows that the dataset is dominated by 3D compounds but also contains a subset of low-dimensional octahedral materials, including 2D, 1D, and 0D systems (Supplementary Fig. 2d). Therefore, the dataset is designed to provide reliable evaluation performance for compositionally diverse and commonly encountered octahedra-containing materials. However, caution is needed when applying the present model to very low-symmetry structures, large-unit-cell compounds, or more complex multi-motif systems that are underrepresented in the current dataset.

Octahedral connectivity in the dataset is similarly varied. In addition to the common corner-sharing connections, octahedral motifs can also engage in edge-sharing, face-sharing, or a combination of multiple sharing modes when linked with themselves or other polyhedral motifs (Fig. 1b). For example, TiNi₃S₆ features both corner-sharing and face-sharing. Statistics in Supplementary Fig. 2e show that the dataset’s crystals featuring octahedral motifs predominantly exhibit corner-sharing (35.0%), edge-sharing (16.7%), and face-sharing (5.7%), with isolated configurations making up 13.5%. Mixed connectivity types are also common, including corner+edge (21.9%), corner+face (2.7%), edge+face (0.1%), and corner+edge+face (4.4%).

The dataset covers 64 elements in total (Supplementary Fig. 3a). $B_{Octa}$ sites are predominantly occupied by metals, with 63 distinct elements appearing at $B_{Octa}$ and Na, Fe, Mn, Co, Ni, Ti, Li, and Cd being most frequent (Supplementary Fig. 3b). $X_{Octa}$ sites include 40 elements and are dominated by O, F, S, and Cl, which contribute 49.9%, 18.6%, 11.1%, and 7.6%, respectively (Supplementary Fig. 3c). Notably, certain structures possess octahedral motifs with multiple central elements (e. g., LiSnS₂ contains both LiS₆ and SnS₆ motifs) as well as mixed coordinating elements (e.g., NaKCrO₂F₄ contains NaO₂F₄ motifs). The octahedral motifs also coexist with other polyhedra in some structures, for example, Na₃MoFO₄ contains both tetrahedral and octahedral motifs. A small fraction (4.1%) contains anti-octahedral motifs, where non-metals occupy $B_{Octa}$ and metals occupy $X_{Octa}$, such as the FNa₆ motif in Na₃MoFO₄. This diversity and complexity enhance the generality of our analysis, but also challenge the ML model.

High-throughput computational analysis reveals that the formation energies of the crystals in dataset approximate a normal distribution, with the mean and mode both centered around −2.0 eV/atom (Supplementary Fig. 4). In general, the stability of a compound is examined by calculating the formation energies defined as follows:

$$E_f=E_{compound}-\sum _{n=1}^T{z_{n}E}_n^{bulk}$$ 6

where the $E_{compound}$ is the calculated total energy of a given octahedral compound, T represents the count of distinct elemental types, $z_n$ and $E_n$ are the number of atoms and the total energy associated with n-th element within the unit cell, respectively.

Density functional theory calculation

The high-throughput DFT calculations were performed using the Vienna ab initio simulation package (VASP)²¹ and Jilin Artificial-Intelligence Aided Materials-Design Integrated Package (JAMIP)⁶⁹. The electron−ion interactions were carried out using plane-wave pseudopotential methods²¹. The calculations were performed using a generalized gradient approximation⁷⁰ within the framework of the Perdew-Burke-Ernzerhof (PBE)²⁰ exchange-correlation functional. The kinetic energy cutoff for the plane-wave expansion of electron wavefunctions was set to 1.3 times of the maximum default energy cutoff, and a grid spacing of 2$\mathrm{\pi }$ × 0.03 Å⁻¹ was used for Brillouin zone sampling. The crystal structures were fully optimized with the energy convergence criteria of 1 × 10⁻⁶ eV and force convergence criteria of 0.01 eV Å⁻¹. We employed the DFT + U method to accurately describe the strong correlation effects arising from the d-orbital electrons of the transition metal elements. The DFT + U method is implemented in JAMIP, which automatically identifies transition metal elements in the components and sets the U values^71–73. For crystals that do not contain transition metals, we utilized the Heyd-Scuseria-Ernzerhof (HSE06)⁷⁴ hybrid functional to correct the insufficiently accurate band gaps based on the PBE level. The obtained $E_g^{HSE}$ values were utilized as a scissor to correct the band structure and optical absorption with respect to its $E_g^{PBE}$. A similar methodology has been employed in previous studies^75,76. The transport effective mass was obtained by employing BoltzTraP⁷⁷. For crystal Ca₂GaCoO₅, the structural relaxations and electronic structure calculations of were carried out within the spin-polarized formalism to properly account for possible magnetic effects. Its band structure was obtained by considering strong correlation effects of Co d-orbital electrons under HSE06 hybrid functional.

The phonon calculations were performed with a supercell (ensuring lattice parameters larger than 10 Å) of the primitive cell using the finite-displacement method⁷⁸. In the phonon calculations, the total energy convergence threshold was fixed at 1 × 10⁻⁸eV. We also fully optimized the crystal structures with higher force and energy convergence criteria of 0.0001 eV/Å and 1 × 10⁻⁸eV before we systematically performed our phonon calculations. Chemical bonding was analyzed using the crystal orbital Hamilton population method as implemented in LOBSTER⁷⁹.

Feature engineering

We constructed an initial feature pool containing 409 elemental features, 23 local motif features, and 232 global crystal features. Elemental and statistical descriptors include maximum, minimum, mean, range, mode, average deviation, and standard deviation of elemental properties, while motif descriptors explicitly resolve the $B_{Octa}$-$X_{Octa}$ coordination environment (More details are provided in Supplementary Note 2). Feature selection was performed using GBRT feature importance. At each selection step, a GBRT model was trained on the current feature pool, features were ranked by averaged importance, and weakly contributing features were removed. This procedure was used as a greedy refinement strategy to obtain a compact and informative descriptor subset, rather than to guarantee a globally optimal subset. Feature combination was introduced iteratively after feature selection. The allowed algebraic operators are addition, subtraction, multiplication, and division. To keep the generated descriptors physically interpretable, feature combinations were constrained by descriptor family, physical compatibility, and feature complexity. Addition and subtraction were applied only to compatible feature families, division was used for physically meaningful normalization, and multiplication was mainly used in motif-template expressions such as octahedral coordination-weighted terms. The complete workflow and quantitative evaluation are provided in Supplementary Figs. 5 and 6.

Ensemble learning method

We employed the interpretable ensemble learning model GBRT in this work, which is one of the decision tree-based ML algorithms and adopts the gradient boosting method. It sequentially trains decision trees to fit the residuals from previous iterations. GBRT has been widely adopted for ML tasks in the field of materials science, such as predicting formation energy^25,31, convex hull energy⁸⁰, band gap^25,31,40 of perovskites, and the infrared, piezoelectric, and dielectric responses of materials⁸¹. The model was constructed using the Scikit-learn package⁸². 10-fold cross-validation was conducted to ensure the robustness of the model’s performance and its insensitivity to specific data subsets utilized for training and testing. The metrics for regression models include mean absolute error (MAE), mean squared error (MSE), root mean squared error (RMSE), and coefficient of determination (R²). The lower the MAE, MSE, and RMSE values of a model, the better the model’s performance. Conversely, the closer the R² value of a model is to 1, the better its performance. The hyperparameter search method, results, and the selected hyperparameters are provided in Supplementary Note 14, Supplementary Table 4 and Supplementary Table 5, respectively.

Symbolic regression method

SR was performed using the SISSO package⁵². The source code was obtained from the SISSO GitHub repository (https://github.com/rouyang2017/SISSO), and the implementation used here is SISSO.3.2, released in September 2022. The input dataset consisted of 3362 structures, with the DFT-calculated formation energy per atom as the target and the final 34 motif-resolved descriptors as the primary features. To preserve physical interpretability and avoid arbitrary addition or subtraction between descriptors of different physical types, the allowed mathematical operators for feature construction were multiplication, division, inverse, square root, and square, i.e., (×), $\left(/\right)$, ${\left(\cdot \right)}^{-1}$, $\sqrt{\cdot }$, and ${\left(\cdot \right)}^2$. The maximum feature complexity was set to 5, and SISSO generated feature spaces up to the third rung. The number of candidate features increased from 34 primary features to 1249 first-rung features, 325,585 second-rung features, and 1.23 $\times$ 10⁹ third-rung features. To avoid numerically unstable expressions, features with maximum absolute values smaller than 10⁻⁵ or larger than 10⁵ were discarded. Sure independence screening (SIS) was then used to retain the top 100 candidate features at each SIS step. Sparse regression was performed using the $L_0$ sparsifying operator, with a fitted intercept and RMSE as the model selection metric. The top 20-ranked models were saved for inspection. In general, the accuracy of the SR improves with the complexity of the equation^83,84. However, excessively complex equations may obscure the underlying relationship between features and target properties. Therefore, a balance between accuracy and interpretability is required. In this work, we selected a two-dimensional SISSO equation with seven physical quantities because it provided a compact analytical expression while substantially improving over the one-dimensional descriptor. The final two-dimensional expression gave RMSE = 0.369 eV/atom, $R^2$ = 0.815, and Pearson $r$ = 0.90 on the dataset. The parameters of SISSO are provided in Supplementary Table 6.

Experimental details

Ca₂GaCoO₅ was synthesized by solid-state reaction. In addition to CaCO₃ and Ga₂O₃, four cobalt precursors were adopted: CoO, Co₃O₄, Co₂O₃ (Rhawn Company), and Co₂O₃ (Seedior Company). Stoichiometric mixtures were wet-milled in ethanol for 12 h, dried at 80 °C, and then pre-sintered at 800 °C for 12 h in a muffle furnace. The powders obtained were re-milled for 12 h, pressed into 10 mm diameter pellets, and finally sintered at 1100 °C for 3 h with a heating rate of 5 °C/min.

The crystalline structure analysis was examined by X-ray diffraction (PANalytical Empyrean, Cu Kα radiation, λ = 1.5406 Å) operating at 40 kV and 40 mA with a step size of 0.0065°. Crystal structure was further verified through HRTEM (FEI Talos F200X) with SAED pattern analysis. Grain morphology was characterized using dual-beam FIB-SEM (TESCAN AMBER), size distribution was obtained by statistical distribution analysis. Optical absorption spectra (200–2500 nm) were recorded on a Shimadzu UV-3600i Plus spectrophotometer, with the optical band gap determined by Tauc plot extrapolation. Photoresponse properties were evaluated under 630 nm (0.436 W/cm²) and 520 nm (0.296 W/cm²) illumination using a Keysight B1500A semiconductor parameter analyzer.

Author contributions

L.Z. and Y.S. conceived the idea and led this research. X.Y. performed data curation, DFT calculation and analysis, ML model training, and developed certain tools. W.Z., W.F. and Y.Z. performed the synthetical experiments and characterizations. X.Y. and X.H. performed the materials synthesis and characterization. X.Y., W.Z., Y.S., Y.Z. and L.Z. wrote the manuscript with inputs from all the authors.

Peer review

Peer review information

Nature Communications thanks Jun Chen, Tianmin Wu and the other anonymous reviewer(s) for their contribution to the peer review of this work. A peer review file is available.

Funding

This work was supported by the National Natural Science Foundation of China (Grants Nos. 62125402, T2521005, 62321166653), the Graduate Innovation Fund of Jilin University (Grant No. 2023CX060), and the Scientific and Technological Innovation Project of Changbaishan Laboratory, Jilin Province (Grant No. CBS2026008). Y.S. acknowledges the National Natural Science Foundation of China (Grant No. 12504021) and the Basic Research Program of Jiangsu (Grant No. BK20240391).

Data availability

The data generated in this study have been deposited in the figshare repository under 10.6084/m9.figshare.32134264. The data necessary to reproduce the results reported in this study, including the descriptor data, machine-learning datasets, screening results, and related processed data, are available in this repository.

Code availability

The code used in this study has been deposited in the GitHub repository https://github.com/XiaoyuYang-Jack/OctaMotif-ML and archived in Zenodo under 10.5281/zenodo.20609442. The code includes scripts for feature engineering workflow, machine-learning model training, symbolic-regression analysis, candidate screening, and data visualization.

Competing interests

The authors declare no competing interests.

Supplementary information

The online version contains supplementary material available at 10.1038/s41467-026-75216-0.