Introduction

Solar energy is the most important basic energy among all types of renewable energy1,2. The technologies that convert solar energy to electrical power (such as photovoltaic (PV) generation and photoelectrochemical generation, will receive more attention in multi-functional clean energy sources3,4,5. Practical thin-film PV cells are based on quaternary chalcogenides (I2-II-IV-X4) of sphalerite crystals such as CdTe and Cu(In, Ga)(S, Se)2 (CIGSSe), which is cheaper to process and exhibit competitive performance levels compared to conventional crystal silicon-based PVs. Their battery power conversion efficiency (PCE) exceeds 20% at present6,7. However, these materials require expensive or rare elements (In, Te), or even toxic (Cd), severely limiting their large-scale development. Kesterite Cu2ZnSn(S, Se)4 (CZTSSe), is a potential thin-film PV material, in which the In and Ga in CIGSSe are replaced with Zn and Sn, and its record PCE of 12.6% is significantly lower than that of CdTe/CIGSSe8. One possible reason for this is the antisite disorder in the kesterite structure, which significantly affects the open-circuit voltage and device performance. CZTSSe, where the smaller Zn of is replaced by Ba with a larger ionic radius (Cu2BaSnS4-xSex(CBTSSe)) to ease the antisite disorder, has demonstrated better performance in PV in comparison with CZTSSe9,10,11,12. Continuing this process, replacing Cu with Ag in CBTSSe to form Ag2BaSnSe4 (ABTSe), and replacing Ba with Sr to form Cu2SrSnSe4 (CSTSSe), yields materials that have recently shown great promise with respect to thin-film PV applications13,14.

These cases enlighten us that it is worth exploring the wider space of I2-II-IV-X4 chalcogenide semiconductors (where I-, II-, and IV-sites are occupied by the different oxidation states of the cations and X-site is a chalcogenide anion)15,16,17, to identify earth-abundant, environmentally friendly thin-film PV materials inspired by existing compounds. A possible path toward the discovery of I2-II-IV-X4 materials is to synthesize or theoretically calculate the properties of massive sets of potential structures in terms of element substitution (e.g., I = Li+, Cu+, or Ag+; II = Ba2+ or Sr2+; IV = Sn4+ or Ge4+; X = O2−, S2−, or Se2−), and then screen for materials with good electro-optical properties. For example, semiconductors with band gaps (Eg) (particularly with direct band gaps) in the visible wavelength region (0.9–1.6 eV, the range of optimum optical conversion efficiency), and strong optical responses in the visible spectrum, are considered promising thin-film PVs18,19. Unfortunately, such an approach is impractical because of the high costs and long cycles of the necessary experimentation and is also not amenable to high-throughput computing due to excessive computational costs. Finding a way to quickly discover I2-II-IV-X4 chalcogenide semiconductors is an important challenge for current research, which is of great significance for identifying thin-film PVs and further improving PCE.

With the rise of machine learning (ML) applications, data-driven approaches to material design and selection have promoted the development of materials science. ML methods extend far beyond the limitations of other current electronic structure analysis methods, to investigate novel, emergent phenomena originating from the complexity of the physical systems20,21,22,23. ML technologies, such as deep neural networks (DNNs)24,25, support vector machines (SVMs)26,27, and random forest (RF)28,29 algorithms, have made remarkable achievements in materials science. Ding et al. used over 104 catalytic samples to design non-noble metal electrocatalytic proton exchange membrane fuel cells30. Based on the ML model, Ali et al. achieved fast recovery of the cubic structure in mixed cation perovskite thin films from high-throughput calculation database31. Moreover, there are some remarkable reports on the application of ML methods in PV materials23,32,33,34. However, the applications of ML in these systems use supervised learning, the biggest imperfection of which is that it still requires an adequate data set to ensure the accuracy of predictions. According to our current knowledge base, the data set of I2-II-IV-X4 chalcogenides is still quite scarce, and existing supervised learning models are unable to predict properties based on structures, and a relevant ML model has not been reported with such a small data set. Consequently, a more sensible strategy to overcome the limited data available is urgently needed.

In this study, we proposed an unsupervised learning (UL) model with unlabeled data and apply it to a representative case of exploring the I2-II-IV-X4 chalcogenides for thin-film PV materials. Based on the structure of I2-II-IV-X4 chalcogenides and recombination of elements from the periodic table, a total of 2700 structures (containing 27 identified materials) with four different space groups were selected as the initial data set, and 1520 candidates were screened out based on the tolerance factor. We used an agglomerative hierarchical clustering (AHC) algorithm35 to accomplish UL, and proposed a descriptor representing the sums and differences of elemental properties to cluster I2-II-IV-X4 chalcogenides. Our unsupervised model clusters I2-II-IV-X4 chalcogenides into one group with suitable Eg, while the other groups of materials had larger Eg values. Based on the high-precision Heyd–Scuseria–Ernzerhof calculations (HSE06) method, we quantitatively calculated the Eg and optical absorption coefficient36,37 of the selected compounds, and successfully discovered eight I2-II-IV-X4 chalcogenides with good electro-optical properties (Ag2BaTiS4, Ag2BaTiSe4, Ag2BaCrS4, Ag2BaSiSe4, Ag2BaZrS4, Ag2BaZrSe4, Ag2BaHfSe4, and Cu2BaMnSe4). We further demonstrated that these chalcogenides have good thermal stabilities at room temperature using ab initio molecular dynamic (AIMD) simulations. The proposed AHC-UL model bypasses the challenge of data scarcity in traditional ML methods and effectively avoids extremely long computational and experimental cycles. Based on the recombination of all elements in the periodic table, the proposed model reduces the screening period of I2-II-IV-X4 chalcogenides by ~12.1 years. We hope that the eight I2-II-IV-X4 chalcogenides proposed from 2700 unknown compounds will be served as promising thin-film PVs to significantly improve PCE.

Results

Workflow of material discovery

The workflow for unsupervised discovery of I2-II-IV-X4 chalcogenides for thin-film PVs is illustrated in Fig. 1, including four modules: determination of crystal structures (Fig. 1a–d)38, element selection from the periodic table (Fig. 1e), the establishment of the ML model (Fig. 1f), and ab initio calculation (Fig. 1g). In this procedure, considering that the I2-II-IV-X4 chalcogenide has four different space groups, I222, P31, Ama2, and \(I\bar 42m\), we selected one structure from each space group as initial structures, as shown in Fig. 1a (see elemental sites and structures in Supplementary Fig. 1). Then, the proper site elements were selected according to the oxidation state and coordination number from the periodic table, where I = Li+, Na+, K+, Cu+, or Ag+; II = Ca2+, Sr2+, Ba2+, Eu2+, or Pb2+; IV = Ti4+, Zr4+, Hf4+, Cr4+, Mn4+, Co4+, Si4+, Ge4+, or Sn4+; X = O2−, S2−, or Se2−, as shown in Fig. 1e. Thus, 2700 different structures (675 compounds without considering four space groups) were generated as the initial data set for the ML module. We used the tolerance factor (Tf), to make preliminary judgments regarding the structural stabilities16, leaving 1520 I2-II-IV-X4 chalcogenides to be further studied (380 compounds without considering space groups). Next, strong relationships between compounds were established by feature engineering, and the 380 structures were clustered based on the AHC-UL algorithm. Finally, 26 candidates covering two space groups (I222 and P31) were selected from one of the ten groups, corresponding to Fig. 1f. As shown in Fig. 1g, ab initio calculations were performed to predict the electro-optical properties and evaluate the thermal stabilities of the 26 candidates, and eight I2-II-IV-X4 chalcogenides were identified as promising thin-film PV materials. The step-by-step screening process is discussed in Supplementary Fig. 2 and Supplementary Note 1.

Fig. 1: Schematic of the unsupervised discovery of I2-II-IV-X4 chalcogenides.
figure 1

a Crystallographic view of Ag2BaSnSe4 (I222). b Crystallographic view of Cu2BaSnS4 (P31). c Crystallographic view of Cu2BaSnSe4 (Ama2). d Crystallographic view of Ag2BaGeS4 (\(I\bar 42m\)). e Element selection of I (yellow), II (green), IV (purple), and X (light red) from the periodic table. f Workflow of an unsupervised guided discovery of candidate I2-II-IV-X4 chalcogenides. g Further accurate verification of the I2-II-IV-X4 chalcogenides through ab initio calculations. The gray arrows represent the sequential workflow.

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Unsupervised learning of I2-II-IV-X4 chalcogenides

The process used here for UL is shown in Fig. 1f, consisting of three parts: data set, feature engineering, and algorithm39,40. First, there are five, five, nine, and three elements to choose for the I-, II-, IV-, and X-sites, respectively, which can form 675 different compounds, and 2700 different structures with four space groups. Then, Tf (TI and TIV) were applied to make a preliminary judgment about the structural stabilities of these compounds. As shown in Fig. 2a, the Tf plot of 675 compounds distinguished by X-site is presented, where 380 compounds with 0.94 < TI < 1.22 and 0.84 < TIV < 1.11 (cyan area in Fig. 2a, see Supplementary Fig. 3) are potentially stable (see more details of Tf in Methods). Thus, a data set of 380 I2-II-IV-X4 compounds without considering space groups were selected for the next UL clustering.

Fig. 2: Unsupervised learning of I2-II-IV-X4 semiconductors.
figure 2

a Plot of tolerance factors (TI and TIV) of 675 I2-II-IV-X4 compounds, where 380 likely stable compounds are in the cyan area. b Computed features of six selected compounds based on SDEPs, dotted boxes show significant differences in features. c Bottom-up tree diagram (dendrogram) generated by the agglomerative hierarchical clustering (AHC) method. The dashed line shows the position where all compounds are partitioned into ten groups, marked as G1–G10 from left to right and distinguished by different colors. d Mapping the dendrogram to the band gaps, revealing the grouping of 27 known I2-II-IV-X4 semiconductors (Supplementary Table 1). The color bar shows the scale of band gap.

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Feature engineering, which can transform raw random data into model training data to be closely related to the output attributes, and determines the upper limit of the ML model. The goal of this work is to determine I2-II-IV-X4 chalcogenides with good electro-optical properties. The band gap Eg is a basic parameter of electronic properties, which needs to be considered first. Therefore, we need to build a feature set to create a strong relationship between the compounds and electronic properties. The factors affecting the Eg are complex, but to train the model, the feature set must be limited. From previous works21,41,42,43, we found that the elemental properties of materials have good mapping relationships with their band gaps. Therefore, nine elemental properties were selected, including the atomic number (Z), group number (g), covalent radius (Rcov), and first ionization energy (Eie). The list of all elemental properties is provided in Supplementary Table 2. To construct the feature vectors, we proposed a descriptor using the sums and differences of elemental properties (SDEPs for short) for I2-II-IV-X4 compounds, and constructed a 72-dimensional SDEP in this way for each I2-II-IV-X4 compound (see Methods and Supplementary Fig. 4). The feature plots of the six selected compounds (Cu2EuSiS4, Li2BaGeS4, Ag2BaGeS4, Cu2BaSnSe4, Cu2BaSnS4, and Ag2BaSnSe4) are shown in Fig. 2b. As the input of clustering, these feature curves look very similar and play an important role in clustering results. From Fig. 2b, the feature curves of the selected six structures are very similar, but there are still significant differences (see dotted boxes), such as the tenth and 20th features, which are also the key to clustering.

Based on SDEPs, the AHC-UL algorithm35,44 was used to cluster the 380 I2-II-IV-X4 compounds. The bottom-up tree diagram (dendrogram) generated by the AHC is presented in Fig. 2c, where an appropriate partition line is selected and the 380 compounds are classified into ten groups (from G1, G2, …, to G10). The different colors in Fig. 2c correspond to different groups. More details about the position of the partition line are discussed in Supplementary Note 2 and Supplementary Fig. 5. The grouping shows a good quality of clustering as different groups are well differentiated, and the SDEPs share similar characteristics within the same groups (Supplementary Fig. 4). Details of the 72-dimensional feature vector for each compound are provided in Supplementary Table 3. Therefore, a visible clustering of I2-II-IV-X4 compounds can be found using AHC. From Fig. 2c, d, most of the known I2-II-IV-X4 compounds with Eg < 2.0 eV are clustered into G3 in the dendrogram (Fig. 2d), including ten structures of Ag2BaGeSe4 (Eg = 0.85 eV), ABTSe (Eg = 1.42 eV), Cu2BaGeSe4 (Eg = 1.88 eV), and CBTSSe (Eg = 1.96 eV), etc. In addition, G1, including Li2BaGeSe4 (Eg = 2.4 eV) and Li2BaSnS4 (Eg = 3.07 eV), are all known compounds with Eg > 2.0 eV. For G5 and G8, each contains only one known compound with Eg > 2.0 eV, Li2EuGeSe4 (Eg = 2.54 eV), and Li2PbGeS4 (Eg = 2.41 eV), respectively. Thus, as confirmed by the above correlations between the groups and band gaps, our proposed SDEPs and UL model can capture the chemical and physical relations of the electronic properties of I2-II-IV-X4 chalcogenides. Besides, we also performed the K-means method to cluster I2-II-IV-X4 compounds, the comparison of AHC and K-means are provided in Supplementary Fig. 6, Supplementary Table 4, and Supplementary Note 3.

Physical insights from unsupervised learning

The clustering of I2-II-IV-X4 chalcogenides by SDEPs provides physical insights into the understanding of compounds exhibiting useful stabilities, proper electronic properties, and suitable crystal structures. We counted the number of compounds in each group, including both unknown compounds and 27 known compounds (Fig. 3a). G1 and G10 had the most compounds at 52 each, while G2 and G6 contained the fewest compounds at 16 each, indicating that a targeted study of these groups would significantly narrow down the initial scope, which contained 380 compounds with four space groups. In addition, we generalize and summarize the ratios of known compounds in these groups (Fig. 3a and Supplementary Table 4). Remarkably, G3 contains 36 compounds (Supplementary Table 4), of which ten are known compounds, accounting for 27.8%, which is much higher than the ratios of known compounds in other groups. In terms of stability, the structures in G3 are likely to be more stable, as the 27 known compounds have been experimentally synthesized. This is an obvious implication that the unknown 26 compounds in G3 deserve further targeted study.

Fig. 3: Further analysis of group partitioning by unsupervised learning.
figure 3

a A statistical graph of the number of compounds in each group (left), and the ratio of known compounds in each group (right). b Violin plots of Eg of 27 known compounds, ten known compounds in G3, and other 17 known compounds not clustered in G3. The outer shells of the violins bound all data, narrow horizontal lines bound 90% of the data, thick horizontal lines bound 50% of the data, and white dots represent medians. The dashed line shows the position of Eg = 2.0 eV. c Five known I2-II-IV-X4 semiconductors in G3, where I = Ag+. d Five known I2-II-IV-X4 semiconductors in G3, where I = Cu+.

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As shown in Fig. 3b, the violin plot of the ten known compounds in G3 showed a significantly lower Eg, with an average of 1.75 eV and a median of 1.80 eV, and the majority of the 17 known compounds outside of G3 have significantly higher Eg, with an average of 2.27 eV and a median of 2.18 eV. This shows great potential for the discovery of I2-II-IV-X4 compounds with lower Eg among the 26 unknown compounds in G3 with good electro-optical properties. Moreover, from Supplementary Table 5 and Supplementary Fig. 7, we found that the 36 compounds in G3 showed the same I-site of Ag+ or Cu+, II-site of Ba2+, and X-site of S2− or Se2−, revealing the dependence of their stabilities and electronic properties on their elemental properties. T hese discoveries from UL have important guiding significance for the design of I2-II-IV-X4 compounds with stable and good electronic properties.

In addition to the discovery of elemental and electronic properties, we also found patterns in the crystal structures of the four space groups. The crystal structures of five known compounds with I = Ag+ and II = Ba2+ are shown in Fig. 3c. These structures contain two kinds of similar space groups: \(I\bar 42m\) (40%) and I222 (60%), where the I-X4 tetrahedra are flattened and share edges with the II-X8 dodecahedra, do not bear any resemblance to the square antiprisms observed in the P31 and Ama2 structures (Fig. 1a and Supplementary Fig. 1). In Fig. 3d, where the other five known compounds with I = Cu+ and II = Ba2+ are presented, they happen to contain two other types of space groups, Ama2 (20%) and P31 (80%). These phenomena indicate that the ions at the I-site have an effect on the crystal structures, and the stable structures can be hopefully obtained when different chemical formulas are combined with suitable space groups (e.g., when I = Ag+, the space group of I222 is better, while for I = Cu+, the space group of P31 is better). To further study the 26 unknown compounds in G3 and narrow down the screening, we focused on two main space groups; when Ag+ is at the I-site, we selected I222, while when Cu+ is at the I-site, we selected P31. As a result, the scope of exploration narrowed from 1520 structures to 26 structures (see the list in Supplementary Table 5).

From Supplementary Fig. 4, the dashed boxes show the significant different features in G3, which are 10th, 19th, 56th features, indicating that the difference of dipole polarizability between I, II, and IV-site elements and the difference of atomic number between I, or II, or IV and X-site elements are important to electronic property of the I2-II-IV-X4 compounds (Supplementary Table 3). In addition to speeding up the screening process, UL can estimate important features for guiding the designing of high-performance I2-II-IV-X4 compounds. Therefore, the supervised learning method may be further developed to accurately analyze the importance of features.

Electro-optical properties of eight I2-II-IV-X4 chalcogenides

We performed ab initio calculations to predict the electro-optical properties of the 26 I2-II-IV-X4 structures in G3. First, geometric structure optimizations were performed, and the crystal structures remained in good geometric arrangements. To achieve high precision prediction, we used the high-level HSE06 calculation, which is considered to be close to the experimental results37,45,46,47,48, to calculate the electronic and optical properties of the screened 26 candidates. From Table 1 and Fig. 4, 20 structures were determined to be semiconductors (Eg > 0), with direct (D) or indirect (I) band gaps. The band structures and density of states (DOSs) of 20 structures are shown in Fig. 5 and Supplementary Figs. 8–23. From the predicted band structures, 15 structures have indirect band gaps, most of which come are the structures with I = Cu+. The other five structures have direct band gaps, all from the structures with I = Ag+ (1.42 eV for Ag2BaTiS4, 1.18 eV for Ag2BaTiSe4, 2.13 eV for Ag2BaHfS4, 1.76 eV for Ag2BaHfSe4, and 1.33 eV for Ag2BaSiSe4).

Table 1 Ab initio calculations for 20 semiconductors from G3.
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Fig. 4: Edges of VBMs and CBMs for 20 I2-II-IV-X4 semiconductors.
figure 4

The color bar (from purple to red) shows the scale of band gap.

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Fig. 5: Four I2-II-IV-X4 chalcogenides with Eg in the range of 0.9–1.6 eV, predicted by HSE06 calculations.
figure 5

The optimized crystal structures, band structures and density of states of Ag2BaTiS4, Ag2BaTiSe4, Ag2BaSiSe4, and Ag2BaZrSe4 are shown in a–c Ag2BaTiS4, d–f Ag2BaTiSe4, g–i Ag2BaSiSe4, J–l Ag2BaZrSe4. In the band structures, the red dots present CBMs, while the blue dots present VBMs. The electronic structures of the other four I2-II-IV-X4 chalcogenides are provided in Supplementary Figs. 13, 20, 21, and 23.

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The edges of the conduction band minimums (CBMs) and valence band maximums (VBMs) of 20 semiconductors are shown in Fig. 4. We found that when I = Ag+, CBMs and VBMs tend to be symmetric and have smaller Eg, which may lead to more information for applications such as photocatalysis and bipolar tubes, in addition to thin-film PVs. When I = Cu+, most CBMs and VBMs are asymmetric, and CBMs are generally farther from Fermi level than VBMs. For IV-site, taking Ti, Zr, and Hf as an example, we found that with the increase of atomic number, the positions of the CBMs and VBMs deviate from the Fermi level, leading to the wider band gaps. The corresponding DOSs are also presented in Fig. 5 and Supplementary Figs. 8–23, where CBMs are dominated by I- and X-site atoms, while VBMs are mainly influenced by IV-site atoms. All the CBMs and VBMs show that II-site atoms (Ba2+) do not dominate the scene, which is consistent with the fact that II-site atoms are indistinctive in the 26 unknown structures (which are all Ba2+) from G3. The above results indicate the importance of elements in regulating CBMs and VBMs to impact the Eg of I2-II-IV-X4 structures, and UL can effectively cluster similar elemental and conductivity characteristics into one group.

It is worth noting that there are four structures (1.42 eV for Ag2BaTiS4, 1.18 eV for Ag2BaTiSe4, 1.33 eV for Ag2BaSiSe4, and 1.60 eV for Ag2BaZrSe4, as shown in Fig. 4) having Eg between 0.9 and 1.6 eV, which is the range of optimal optical conversion efficiency18. In particular, Ag2BaTiS4, Ag2BaTiSe4, and Ag2BaSiSe4 have direct Eg, this means that the electron transitions do not require phonon release or absorption; As a result, electrons and holes are more likely to recombine. They may be used as potential thin-film PVs. Moreover, Ag2BaCrS4, Ag2BaZrS4, Ag2BaHfSe4, and Cu2BaMnSe4 have band gaps around 0.9 or 1.6 eV, are also likely to have high optical conversion efficiency, we also took them into account. Since PV suitability is the primary motivation for examining these properties, more analysis of the optical properties of these eight structures is necessary. The calculated absorption coefficients (α) of these eight structures based on the HSE06 functional are presented in Fig. 6 and Supplementary Fig. 24. They all show strong optical responses (α > 105 cm−1) in the visible spectrum (1.65–3.26 eV, the colorful background in Fig. 6), and the absorption coefficients are largely isotropic in this range, showing only minor variations among || a, || b, and || c directions, potentially indicating that there is only a small performance dependence on film orientation for thin-film PVs. The optimal band gaps and desired optical absorptions of the eight I2-II-IV-X4 chalcogenides show that they have great promise as thin-film PVs and exhibit good performance.

Fig. 6: Calculated optical absorption coefficients of four I2-II-IV-X4 chalcogenides.
figure 6

a Ag2BaTiS4, b Ag2BaTiSe4, c Ag2BaSiSe4, d Ag2BaZrSe4. E || a parallel to reciprocal a axis, E || b parallel to reciprocal b axis, and E || c parallel to the reciprocal c axis. The optical absorption coefficients of the other four I2-II-IV-X4 chalcogenides are provided in Supplementary Fig. 24.

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In addition to the eight outstanding candidates for thin-film PVs, the proposed method screens out 12 other I2-II-IV-X4 chalcogenide semiconductors that are far away from the range of 0.9–1.6 eV (the corresponding band structures and DOSs are provided in Supplementary Figs. 8–23), which are also likely to play important roles in PV devices, even in other fields (photocatalysis, sensors, detectors, etc.). There are many suggested improvements to properly adjust their band gaps or optical properties to achieve superior performance.

Our UL model not only overcomes the problem of data scarcity, but also greatly shortens the cycle of I2-II-IV-X4 chalcogenide discovery, positively differing high-throughput calculations by previous works49,50. In this work, 27 known structures were excluded from the initial 1520 I2-II-IV-X4 chalcogenides, and 26 candidates were finally screened out. In terms of computational screening, each structure required 260,464 s on a 24-CPU supercomputer (Supplementary Fig. 25) through the high-precision HSE06 method, meaning that the present work saves ~12.1-year computational cycles for 1467 structures. This will provide research export and method for the next generation of thin-film PVs. In addition, structural features also have an important impact on the band gap prediction51, but we did not consider them in the clustering due to the lack of data. With the further development of high-throughput computing, more and more I2-II-IV-X4 materials will be available, at which point we will be able to establish supervised learning models to accurately predict band gaps using classification or regression methods.

Thermodynamic stabilities of eight I2-II-IV-X4 chalcogenides

For the eight I2-II-IV-X4 chalcogenides with good electro-optical properties, their thermodynamic stability should be evaluated in addition to the preliminary stability determined by UL and geometric structure optimization in order to facilitate their practical application. Therefore, we performed AIMD to evaluate the thermodynamic stabilities of the screened I2-II-IV-X4 candidates. As shown in Fig. 7 and Supplementary Fig. 26, the total energies of all the systems fluctuate within a very small range without a clear drop or rise during the simulations at 300 K. The crystal structure snapshots were extracted by an interval of 1.0 ps, without obvious expansion or contraction. Moreover, formation enthalpy is an important criterion to test crystal stability34,52, thus we also calculated the formation enthalpies for eight I2-II-IV-X4 candidates with two host space groups of I222 and P31. As shown in Supplementary Fig. 27, all structures with a space group of I222 show the lower formation enthalpies than the structures with space group of P31, especially seven structures with I = Ag+ show the lower formation enthalpies than experimental Ag2BaSnS4, indicating their good stability. It is noting that the selected Cu2BaMnSe4 with a space group of P31 has a slightly higher formation enthalpy than Cu2BaMnSe4 with a space group of I222 and experimental Cu2BaSnS4, which may be metastable since the small difference in formation enthalpy and its smooth energy fluctuations during AIMD. The individual energy values (isolated atoms and bulk structures) are provided in Supplementary Table 6. The above results indicate that the eight I2-II-IV-X4 chalcogenides can maintain the integrity of their crystal structures and good thermal stabilities at room temperature. This indicates that the eight structures selected in this work are stable. We expect that they can be further widely applied in thin-film PVs with good performance.

Fig. 7: Evaluations of thermodynamic stabilities of four I2-II-IV-X4 chalcogenides by AIMD.
figure 7

a Ag2BaTiS4, b Ag2BaTiSe4, c Ag2BaSiSe4, d Ag2BaZrSe4. The thermodynamic stabilities of other four I2-II-IV-X4 chalcogenides are provided in Supplementary Fig. 26. The total energy of each step and the crystal structures of each ps are also presented.

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Discussion

I2-II-IV-X4 chalcogenides have become important materials for thin-film PVs. The discovered I2-II-IV-X4 chalcogenides meet the criteria for earth-abundance and environmental friendliness, and demonstrate great potential for improving PV performance. However, traditional approaches, such as experimental synthesis and high-throughput computing, are limited due to the long-time cycle, and even the reliable ML model is impeded by the scarcity of material property data.

In summary, our achievements are as follows: (1) We propose an accessible descriptor of SDEPs based on the isolated elemental properties to obtain the feature vector of I2-II-IV-X4 compounds, which can be expanded to other material systems. (2) eight I2-II-IV-X4 compounds (Ag2BaTiS4, Ag2BaTiSe4, Ag2BaCrS4, Ag2BaSiSe4, Ag2BaZrS4, Ag2BaZrSe4, Ag2BaHfSe4, and Cu2BaMnSe4) with optimal band gaps, desired optical absorptions, and practical thermal stabilities at room temperatures were selected out of 2700 original structures by UL. They demonstrate great potential as thin-film PVs. (3) Because each structure requires an average of 260,464 s with a 24-CPU supercomputer, our method significantly reduced the scope for screening and calculation (from 2700 structures to 1520 structures, to 26 structures), dramatically shortening the computational cycle of material discovery by ~12.1 years. (4) This study demonstrates the potential of UL in material discovery, thus surmounting the obstacle of data scarcity, which may lead to important ideas and methods for the future discovery of materials.

Furthermore, we hope that the eight candidates revealed in this work will be synthesized experimentally for the preparation and application of thin-film PVs. For the other 16 I2-II-IV-X4 semiconductors identified, they are also of high research value in different fields according to their band gaps. In our subsequent work, we will focus on finding better descriptors to explore more precise quantitative laws of I2-II-IV-X4 structures, such as the expanded Shannon radii, cell volume33. Meanwhile, this work is a typical case of UL in material discovery, we look forward to its vigorous development in materials science.

Methods

Tolerance factor

Tolerance factors serve as descriptors for phase stability with quaternary I2-II-IV-X4 semiconductors, which can be used for structure prediction in an empirically driven learning model16. For I2-II-IV-X4 materials, two dimensionless tolerance factors (TI and TIV) describing the geometric relations have been derived recently. The formula is as follows,

$$T_{{{\mathrm{I}}}} = \sqrt {\frac{{4 + \sqrt 2 }}{3}} \frac{{r_{{{\mathrm{I}}}} + r_{{{\mathrm{X}}}}}}{{r_{{{{\mathrm{II}}}}} + r_{{{\mathrm{X}}}}}}$$
(1)
$$T_{{{{\mathrm{IV}}}}} = \sqrt {\frac{{4 + \sqrt 2 }}{3}} \frac{{r_{{{{\mathrm{IV}}}}} + r_{{{\mathrm{X}}}}}}{{r_{{{{\mathrm{II}}}}} + r_{{{\mathrm{X}}}}}}$$
(2)

where rI, rII, rIV, and rX are the ionic radii of the I, II, IV, and X elements, respectively, and ideally TI = TIV = 1.0. Sun et al. calculated the tolerance factors using ionic radii, where the ranges of TI and TIV are 1.0 to 1.22 and 0.84 to 1.04. Therefore, in this work, we initially set tolerance factors ranging from 0.84 to 1.22. After calculating the 675 compounds, the ranges of TI and TIV are 0.94 to 1.48 and 0.84 to 1.11, respectively. Thus, in the further UL, we set TI ranges from 0.94 to 1.22 (the red dashed line in Fig. 2a), and TIV ranges from 0.84 to 1.11 (the blue dashed line in Fig. 2a).

Descriptor of SDEPs

According to our previous work and relevant literature reports21,41,42,43, we built descriptors from properties of isolated atoms at the I-, II-, IV-, and X-sites. The nine properties of atomic number (Z), group number (g), covalent radius (Rcov), Van der Waals radius (Rvdw), valence-electron number (Nv), electron affinity (Eea), dipole polarizability (Dp), first ionization energy (Eie), and Pauling electronegativity (X) are considered in this work. For each elemental property (φ), we calculated the minima and maxima of the absolute values of the sums and differences of elemental properties (SDEPs). Succinctly, we introduce the following notations for an elemental property φ:

$${\Delta} _\varphi ^{min \pm } = {{{\mathrm{min}}}}\left( {\left| {\varphi _{{{\mathrm{I}}}} \pm \varphi _{{{{\mathrm{II}}}}}} \right|,\left| {\varphi _{{{\mathrm{I}}}} \pm \varphi _{{{{\mathrm{IV}}}}}} \right|,\left| {\varphi _{{{{\mathrm{II}}}}} \pm \varphi _{{{{\mathrm{IV}}}}}} \right|} \right)$$
(3)
$${\Delta} _\varphi ^{max \pm } = {{{\mathrm{max}}}}\left( {\left| {\varphi _{{{\mathrm{I}}}} \pm \varphi _{{{{\mathrm{II}}}}}} \right|,\left| {\varphi _{{{\mathrm{I}}}} \pm \varphi _{{{{\mathrm{IV}}}}}} \right|,\left| {\varphi _{{{{\mathrm{II}}}}} \pm \varphi _{{{{\mathrm{IV}}}}}} \right|} \right)$$
(4)
$$\nabla _\varphi ^{min \pm } = {{{\mathrm{min}}}}\left( {\left| {\varphi _{{{\mathrm{I}}}} \pm \varphi _{{{\mathrm{X}}}}} \right|,\left| {\varphi _{{{{\mathrm{II}}}}} \pm \varphi _{{{\mathrm{X}}}}} \right|,\left| {\varphi _{{{{\mathrm{IV}}}}} \pm \varphi _{{{\mathrm{X}}}}} \right|} \right)$$
(5)
$$\nabla _\varphi ^{max \pm } = {{{\mathrm{max}}}}\left( {\left| {\varphi _I \pm \varphi _X} \right|,\left| {\varphi _{II} \pm \varphi _X} \right|,\left| {\varphi _{IV} \pm \varphi _X} \right|} \right)$$
(6)

For each of these equations, we can calculate an 18-dimensional vector based on nine properties. Therefore, a 72-dimensional feature vector can be obtained for each I2-II-IV-X4 compound (as shown in Fig. 2b, Supplementary Fig. 4, and Supplementary Table 3). Compared with the 36-dimensional feature vectors obtained from the previous nine independent elemental properties (four site elements), the 72-dimensional feature vectors contain more abundant information, especially the information of differences between elemental properties.

Unsupervised algorithm

UL is accomplished by performing AHC, and the dendrogram function used in AHC is from the SciPy package35. In AHC, the similarity between samples is calculated by a similarity measure, and each sample is reconnected step by step in order to form nodes. The nodes are organized into a bottom-up tree diagram hierarchy (dendrogram), where the leaf nodes of the tree represent a single sample, and non-leaf nodes are generally obtained by merging similar or close sample sets. The Euclidean distance (L2) between two I2-II-IV-X4 compounds was used as the similarity metric, and Ward linkage was used to measure group dissimilarity.

The advantage of AHC is that the partition can be stopped at any time, which means that the number of groups (K) can be adjusted dynamically and directly. In this study, K = 10 performs well. More details regarding K can be found in Supplementary Fig. 5 and Supplementary Note 2.

First-principles calculation

All first-principles calculations were conducted using the Vienna Ab initio Simulation Package (VASP)53. The Perdew–Burke–Ernzarhof (PBE) generalized gradient approximation (GGA) functionals54 and project-augmented wave (PAW) atom potentials are employed to perform geometric structure optimizations55,56. In this work, for the structures with a space group of I222, the experimental Ag2BaSnS4 (materials id: mp-555166) served as the starting structure for the cell optimizations; for the structures with a space group of P31, the experimental Cu2BaSnS4 (materials id: mp-17954) served as the starting structure, both of which were obtained from the Materials Project38. The cutoff energy for the plane-wave basis was set as 500 eV. The structure optimization process was ended when an energy convergence lower than 10−5 eV and atomic force less than 0.05 eV/Å. Further, the HSE06 functional was performed for electronic structure calculations and optical properties36,37, and the high symmetry points of electrons were obtained from the online tool of the seek-path. More details on the calculations are provided in Supplementary Note 4. Further, to see the effect of lattice constant on the band gap57, we also optimized eight promising I2-II-IV-X4 materials by using the high-level meta-GGA functional (strongly constrained and appropriately normed semilocal density functional, SCAN)58,59, then calculated the band gaps with HSE06 functional, and found that the differences in structures and band gaps are small (see Supplementary Table 7). This indicates that the convergence criterions (energy and atomic forces) we calculated are reasonable.

The AIMD process was used to evaluate the thermal stability, for structure with a space group of I222, a 128-atom 2 × 2 × 2 supercells was built, and for space group of P31, a 196-atom 2 × 2 × 2 supercell was built. There are enough atoms for phase transition simulations43,60,61. With a time step of 1.0 fs, a total of 5 ps of kinetic processes were performed for the structures. It is noted that a longer time scale or larger system size would help to build confidence in stability conclusion but that for now, these cell dimensions will work. During this process, the temperature was controlled at 300 K using the Nosé–Hoover thermostat62,63.