Introduction

Metal halide perovskites (MHPs) are a class of materials with the chemical formula of ABX3, where A, B, and X represent monovalent organic/inorganic cations, divalent metal cations, and monovalent halide anions, respectively1 MHPs have received considerable research efforts1 owing to their outstanding optoelectronic properties, low cost, and easy fabrication, which make them promising materials for photovoltaics2, light-emitting diodes3, lasers4, and photodetectors5. Among the various MHPs, compounds containing methylammonium (MA) or formamidinium (FA) at the A-site and Pb at the B-site have been extensively studied and have demonstrated the best optoelectronic properties5. However, commercialization of the MHPs is still challenging because of several obstacles, such as the toxicity of Pb and instability under adverse conditions such as high temperature and humidity, which primarily originate from organic cations6. Therefore, to facilitate their commercialization, MHPs with better stability while containing less Pb should be developed without compromising their optoelectronic properties6.

To overcome these challenges, several researchers have suggested substitutional alloying (i.e., doping or mixing with different elements or chemicals at each site in an MHP)7 as a promising approach for tuning the stability and optoelectronic properties of MHPs8, and thus, the Pb-derived toxicity of MHPs can be effectively mitigated by partially replacing Pb with other elements. In the field of metallurgy, the concept of high-entropy alloys has recently demonstrated that mixing multiple elements can improve thermodynamic stability by exploiting the large configurational entropy9. This concept has also been applied in MHPs10. For example, the A-site mixing entropy improved the stability of perovskite solar cells11,12 and light-emitting diodes13. Moreover, high-entropy alloying of the B-site of MAPbBr3 improved the colloidal stability and optical performance while reducing the Pb content14. In this study, we primarily focused on B-site alloying to achieve both improved stability and lower Pb content.

However, because the compositional space of MHPs with multiple alloying elements is virtually infinite, utilizing an experimental design approach is not practical for exploring the chemical space of element-mixed MHPs. Even the use of high-throughput experiments only achieved the synthesis of a few tens of binary and ternary alloys of MHPs15,16,17. In this regard, several computational approaches utilizing either density functional theory (DFT) calculations18,19,20,21 or a combination of DFT with data-driven machine learning (ML) models22,23,24 have been proposed for the rapid screening of the compositional space of MHPs. For example, Wang et al.18 investigated the effect of entropy on the thermodynamic stability and optoelectronic properties of Cs2B+B3+Cl6 double-perovskite alloys using DFT calculations. They used the special quasi-random structure (SQS)25,26 approach to model disordered double-perovskite alloys, showing entropy-driven stabilization. Moreover, Yang et al.19 presented generalized gradient approximation and hybrid-level computations on a DFT dataset of 495 ABX3 perovskite alloys, performed modeling mixed perovskites using the SQS approach, and suggested 32 compounds showing promising stability and photovoltaic efficiency from the screening.

For exploration using ML, Choubisa et al.23 developed a “crystal site feature embedding” (CSFE) representation, which achieves low errors when predicting DFT energies and bandgaps of mixed MHPs. They used CSFE to explore the chemical space of MHPs and discovered how a small amount of Cd doping (i.e., ~1.5%) can change the bandgap to 1.1 eV, which is desirable for the active layers of solar cells23. Moreover, Mannodi-Kanakkithodi et al.24 proposed an ML-driven high-throughput screening framework based on the stability, bandgap, and defect tolerance of mixed MHPs, where the tabulated elemental properties of each species of the MHPs were used as inputs for ML (i.e., neural networks). Using the proposed screening framework, 392 out of 17955 MHPs were identified as promising candidates for photon absorbers24.

Despite the practical applicability of the aforementioned computational design framework, most previous studies do not guarantee the most stable atomic configuration for each composition. That is, previous works18,19,20,21,23,24 utilized random sampling or SQS methods to estimate the properties of random alloys and did not explore all possible configurations exhaustively. Understanding the properties of the ground-state atomic configuration is important because the atomic configuration of an alloy is likely to be the most stable configuration, and the properties of alloys, such as the bandgap, can be significantly altered by configuration changes, even at the same composition27,28. In this regard, Yamamoto et al.22 utilized the cluster expansion approach to identify the ground state of B-site mixed iodide perovskites (i.e., ABxB’1-xI3); however, the authors only investigated the effect of B-site mixing on thermodynamic stability.

In this work, we propose a DFT/ML-combined framework (Fig. 1) for the design of B-site-alloyed MHPs with improved stability and optoelectronic properties by accessing all possible atomic configurations. To achieve this, we employed a crystal graph convolution neural network (CGCNN)29 as a surrogate model for predicting the stability and the electronic band structure computed at the Perdew–Burke–Ernzerhof functional revised for solids (PBEsol) level30. For the construction of the DFT-calculated database used for training the CGCNN, a 20-atom unit cell was used, in which B-site alloyed MHPs were modeled with a compositional step of 1/4. The trained CGCNN is then used to exhaustively explore the enlarged chemical space of B-site alloyed MHPs with the increased compositional resolution by four times (i.e., 1/16 compositional step) remaining the A- and X-sites as unary (i.e., A: Cs, K, and Rb; X: Br, Cl, and I). Through ML-assisted screening, 110 compounds were predicted to be thermodynamically stable and exhibited promising electronic band structures, and validations on the thermodynamic stability and electronic band structures were further conducted using DFT calculations with the semi-local PBEsol and hybrid PBE031 functionals, respectively. From this, we identified 10 promising candidates for solar-cell application after examining the PBE0-calculated bandgaps. To this end, the main contributions of our work can be summarized as follows.

  • We explored the whole possible atomic configurational space using the trained CGCNN model to identify the most stable atomic configurations within the given B-site mixed MHP composition which is not explicitly identified in the previous work as mentioned earlier.

  • We expanded the explored B-site mixed MHP domain up to the quaternary phase including the mixing entropy effect in terms of both thermodynamic stability and the electronic properties compared to the previous works which explored up to the ternary phase for B-site mixing without the mixing entropy effect23,24.

  • We proposed to use PBE031 combined with the spin-orbit coupling (SOC) correction32 term as a more accurate approach for estimating experimental bandgaps of MHPs exhibiting lower prediction error compared to the other hybrid functionals such as HSE0633 and B3LYP34.

Fig. 1: Overview of the present study.
figure 1

a Schematic representation of crystal structure for ABX3 metal halide perovskites (MHPs). B1, B2, B3, and B4 represent elements positioned in the B-site of MHPs. b Mixing of entropic contribution \((-T\Delta {S}_{{\rm{mix}}})\) in AB1xB21-xX3 system at varying temperature. c Histogram of decomposition enthalpy (∆Hdecomp) with and without the mixed entropic contribution (at T = 298 K) in our Perdew–Berke–Ernzerhof functional revised for solids (PBEsol) dataset. d Schematic workflow of the present study.

Full size image

Results

Design strategy

Figure 1d illustrates the overall workflow of this study. The generated PBEsol data were used to train the CGCNN models (see details in Training data generation in METHODS section). Specifically, three CGCNN models were independently trained to predict three target properties (see details in Crystal Graph Convolutional Neural Networks training in METHODS section): regression of the decomposition enthalpy (∆Hdecomp) and bandgap (Egap) and binary classification for band type (see also Thermodynamic stability and bandgap of B-site mixed metal halide perovskites in METHODS section for details on calculating each target property). For the band type classification, the non-indirect and indirect band types were labeled as positive and negative classes, respectively.

To explore the chemical space of multi-element MHPs, we used a four-fold enlarged structure compared to the training data, containing 16 B-sites (i.e., 80 atoms per unit cell). This allows a finer resolution of the compositional step (i.e., 6.25 at.%) and we considered B-site alloying up to the quaternary system. For each composition of the MHP, CGCNN-predicted ∆Hdecomp (i.e., \(\Delta {H}_{{\rm{decomp}}}^{{\rm{CGCNN}}}\)) was used to identify the most stable atomic configurations, and the mixing entropy term \((-T\Delta {S}_{{\rm{mix}}})\) was added to \(\Delta {H}_{{\rm{decomp}}}^{{\rm{CGCNN}}}\) at a temperature of 298 K. The entropic contribution, \(-T\Delta {S}_{{\rm{mix}}}\), is proportional to temperature, as can be seen in Fig. 1b. Therefore, the entropy-driven stabilization is more effective at higher temperatures. Even at 298 K, the distribution of \(\Delta {H}_{{\rm{decomp}}}-T\Delta {S}_{{\rm{mix}}}\) in our training data shifted negatively when compared to that of ∆Hdecomp, as shown in Fig. 1c.

Next, we considered Bartel’s tolerance factor35, τ, which is a new data-driven tolerance factor, to classify whether an arbitrary compound is a perovskite (see details in Tolerance factor of B-site mixed metal halide perovskites in METHODS section). τ exhibited better classification accuracy than Goldschmidt’s tolerance factor36. We also compared the classification accuracy of τ with that of Filip’s geometric limits for the formability of perovskites37. After a comparison of classification accuracy for eight materials that were experimentally confirmed to exist as perovskites, we concluded that τ predicts closer to the experimental results (see details in Supplementary Note 2).

For solar-cell applications, CGCNN predicts the band type and bandgap (\({E}_{{\rm{gap}}}^{{\rm{CGCNN}}}\), hereafter) of the atomic configuration with the lowest ∆Hdecomp at each composition. Because the CGCNN model is trained to predict PBEsol-calculated bandgap \(({E}_{{\rm{gap}}}^{{\rm{PBEsol}}})\), we selected the compositions showing direct bandgaps with the condition \({E}_{{\rm{gap}}}^{{\rm{CGCNN}}}\) < 0.5 eV as promising compounds for photovoltaics. Considering that PBEsol underestimates bandgap by approximately 1.0–1.5 eV when compared to PBE0 (refer to Supplementary Table 5), we expect that the candidates satisfying the latter condition (\({E}_{{\rm{gap}}}^{{\rm{CGCNN}}}\) < 0.5 eV) will show a PBE0-calculated bandgap \(({E}_{{\rm{gap}}}^{{\rm{PBE}}0})\) of approximately 1.0–2.0 eV, which is close to an ideal bandgap range of approximately 1.2–1.4 eV for single-junction photovoltaics according to the Shockley–Queisser limit38. Moreover, this bandgap range (1.0 < \({E}_{{\rm{gap}}}^{{\rm{PBE}}0}\) < 2.0 eV) is close to the ideal bandgap for a top cell in a tandem solar cell (1.72 eV), with silicon as the bottom cell, which has a bandgap of 1.1 eV17.

The compounds selected based on the stability and band structure criteria were further validated for their stability and band structure through PBEsol and PBE0 calculations, respectively, to examine their potential application in photovoltaics. We optimized the geometry of the selected compounds using PBEsol and then calculated the band structure using PBE0. We calculated the carrier effective mass, optical absorption spectra, and spectroscopic limited maximum efficiency (SLME). The SLME was calculated according to a previous study by Yu et al.39 using the open-source SL3ME code (https://github.com/ldwillia/SL3ME). In the SLME calculations, the optical absorption spectra and bandgap were used as inputs, assuming an air mass 1.5 global solar spectrum (https://www.nrel.gov/grid/solar-resource/spectra-am1.5.html).

Analysis of the training dataset

The distribution of the thermodynamic stability of the 3,159 DFT (PBESol)-calculated training data was first analyzed, as shown in Fig. 2. As shown in Fig. 2a, we observed that ∆Hdecomp distribution moves toward a lower energy regime (i.e., orange color in Fig. 2a) by employing the mixing entropy term (i.e., \(-T\Delta {S}_{{\rm{mix}}}\)). As expected, a greater shift to the stable region can be obtained by mixing more elements. In fact, without considering the mixing entropy, CsGeBr3 showed the lowest ∆Hdecomp value among the 3159 DFT-calculated data; however, we obtained more stable data for the composition of CsGe0.25Hg0.25Pb0.25Sn0.25Cl3 corresponding to the lowest \({\Delta H}_{{\rm{decomp}}}-T\Delta {S}_{{\rm{mix}}}\) value among the entire 3159 data. Additionally, we note that alloys with Ge lead to ∆Hdecomp distribution in the lower energy range when compared to that of alloys without Ge (Fig. 2b), which is consistent with the results of previous computational studies19,24 (refer to Supplementary Fig. 6 for the effects of including other elements at the B-site). As shown in Supplementary Fig. 6, alloys with Zn result in ∆Hdecomp distribution in a higher energy range when compared to alloys without Zn. Consequently, ∆Hdecomp becomes lowest (highest) when the B-site contains only Ge (Zn). Therefore, when the B-site contains a single element, the ∆Hdecomp distribution is widest, while the ∆Hdecomp distribution narrows as the number of elements in the B-site increases.

Fig. 2: Analysis of the training dataset.
figure 2

a Decomposition enthalpy (∆Hdecomp) distribution of 3159 density functional theory (DFT)-calculated data with respect to the number of unique elements in the B-site before (blue) and after (orange) including the mixing entropy term (at T = 298 K), respectively. b Difference in distribution of the decomposition energy with the existence of Ge element in the training dataset. Dashed lines indicate the mean value of each decomposition energy distribution.

Full size image

We analyzed the correlation between perovskite formability as described by Bartel’s tolerance factor (τ) and thermodynamic stability (Fig. 3a). Fig. 3a shows a positive correlation between τ and \(\Delta {H}_{{\rm{decomp}}}-T\Delta {S}_{{\rm{mix}}}\), as both values indicate that compounds with lower values are more stable. However, a region showing inconsistency between the two metrics exists at τ > 4.18, which might be due to the imperfectness of τ. The dataset used to derive τ does not contain B-site alloyed perovskites35, thereby resulting in an inaccurate prediction for B-site alloyed perovskites. Moreover, τ ignores the mixing entropy-driven stabilization (Eq. (3)); thus, when we compare ∆Hdecomp with τ, less data exists in the inconsistent region (τ > 4.18 and ∆Hdecomp < 0), as shown in Supplementary Fig. 7. Specifically, the data deviating from the positive correlation (i.e., τ > 4.18 and ∆Hdecomp-T∆Smix < −85 meV atom−1) are all Ge-containing compounds, namely, CsGeBr3, CsGeCl3, CsCd0.25Ge0.75Br3, and CsGe0.75Hg0.25Br3; however, experimental results have confirmed that CsGeBr3 and CsGeCl3 exist as perovskite structures40. Despite this imperfection of τ, we can safely apply the condition, τ < 4.18, to screen stable compounds if combined with the condition of maintaining the lowest possible value of \(\Delta {H}_{{\rm{decomp}}}-T\Delta {S}_{{\rm{mix}}}\), because the two metrics show a positive correlation. For electronic band-related properties, as shown in Fig. 3b, the amount of indirect-bandgap data was 4.68 times larger than that of the non-indirect-bandgap data. Moreover, of the 905 data with a bandgap of less than 0.5 eV, 654 data were indirect.

Fig. 3: Visualization of the training data set.
figure 3

a Decomposition enthalpy with mixing entropy term \((\Delta {H}_{{\rm{decomp}}}-T\Delta {S}_{{\rm{mix}}})\) versus Bartel’s tolerance factor (τ) and b histogram of bandgap with the number of data for indirect and non-indirect bandgap displayed. “Non-indirect” includes direct, metallic, and semi-metallic materials. c Pearson correlation coefficient between the four output properties and fractions of elements.

Full size image

Next, we calculated the Pearson correlation coefficient between the elemental fractions and the four output properties, ∆Hdecomp, τ, bandgap, and band type, and the results are illustrated in Fig. 3c. ∆Hdecomp and τ tend to be lower (more stable) when the A-site is occupied by Cs, whereas I in the X-site increases ∆Hdecomp and τ (more unstable). An increase in the fraction of Ge tends to decrease ∆Hdecomp, whereas increasing the fraction of Zn tends to increase ∆Hdecomp. The bandgap tends to decrease when the A- and X-sites are occupied by Cs and I, respectively. Notably, the correlation trend between ∆Hdecomp and bandgap is opposite in the X-site: I (Cl) tends to increase (decrease) ∆Hdecomp but decrease (increase) the bandgap. Increasing the Ge, Hg, and Sn fractions decreases the bandgap, whereas increasing the Cd, Pb, and Zn fractions increases it. In this correlation analysis of the band type, we represented the indirect band type as zero and the non-indirect band type as one. Thus, a positive correlation implies that increasing the fraction of certain elements tends to change the compounds into non-indirect-bandgap materials and vice versa. The Cs fraction showed the highest positive correlation (0.21), whereas the fractions of all other elements exhibited a relatively low correlation with the band type.

Validation of the trained CGCNN models

To validate the prediction performance of the trained CGCNN models, we plotted the test set prediction results, as shown in Fig. 4. When predicting the thermodynamic stability (Fig. 4a), the trained CGCNN model showed promising prediction accuracy with a mean absolute error (MAE) of 0.45 meV atom−1, indicating that the model could effectively capture the quantitative relation between the crystal structure and thermodynamic stability.

Fig. 4: Validation of trained models.
figure 4

Parity plot between crystal graph convolution neural network (CGCNN)-predicted and DFT (PBESol)-calculated (a) ∆Hdecomp and b bandgap on the test set. c: Confusion matrix of classification test. “Non-indirect” includes direct, metallic, and semi-metallic materials.

Full size image

The trained models also showed highly promising prediction performance for the prediction of the bandgap (Fig. 4b) and band type (Fig. 4c). Specifically, the prediction performance of the trained CGCNN model for bandgap prediction (Fig. 4b, MAE of 0.037 eV and root mean squared error of 0.061 eV) was superior to that of a previous study24.

In the classification of the band type (Fig. 4c), our model showed promising classification performance with a low inclusion of misclassified data (i.e., false-negative and false-positive data). Only 3 and 10% of the negative and positive classes, respectively, were misclassified with a high accuracy of 0.96 and recall value of 0.90. In other words, our model could effectively recover the true-positive distribution with a relatively low inclusion of false-positive data from the negative class. Here, we noted that the relatively low precision value of 0.84 was due to the class distribution imbalance of the negative data (Fig. 3b). Despite the large class imbalance between the positive and negative data, the model shows a similar amount of misclassified positive and negative data, as shown in Fig. 4c. Therefore, we expect no significant reduction in sampling efficiency due to the inclusion of false-positive data.

Design of B-site mixed MHPs for solar-cell application

As described previously, we applied the trained CGCNN models to explore the chemical space of the B-site-alloyed MHPs with a four-fold enlarged unit cell consisting of 80 atoms. For A- and X-sites, we considered the chemical elements of {Cs, K, Rb} and {Br, Cl, I}, respectively. For B-site alloying, we utilized the chemical elements of {Ge, Sn, Pb, Zn, Cd, and Hg} under binary, ternary, and quaternary alloying conditions. The compositional ratio of the alloying was varied from zero to one in 0.0625 steps. Then the search space consisted of 2025 chemical compositions for binary alloying and 18,900 for ternary alloying. For quaternary alloying, we considered only Cs for the A-site to reduce the computational cost of the search procedure. Through the exploration of binary and ternary alloying systems, we determined that ∆Hdecomp becomes lower when Cs is in the A-site. Thus, the search space for quaternary alloying was reduced to 20,475 compositions. For each chemical composition, the CGCNN model was applied to all possible atomic configurations to determine the lowest \(\Delta {H}_{{\rm{decomp}}}-T\Delta {S}_{{\rm{mix}}}\). In total, 41,400 compositions and approximately 5.6 × 1011 atomic configurations were explored for alloyed MHPs using the CGCNN model during the search process. The details of computing 5.6 × 1011 atomic configurations are described in the Supplementary Note 3.

Figure 5 shows the distributions of \(\Delta {H}_{{\rm{decomp}}}^{{\rm{CGCNN}}}-T\Delta {S}_{{\rm{mix}}}\) of CsGexSn1-xBr3 and CsGexHgySn1-x-yCl3 systems as case examples, showing the lowest \(\Delta {H}_{{\rm{decomp}}}^{{\rm{CGCNN}}}-T\Delta {S}_{{\rm{mix}}}\) in binary and ternary systems, respectively. As mentioned previously, the CGCNN covers the compositions beyond the chemical space of the training data because the unit cell size is four times larger than that of the training data. We also observed large variations in the \(\Delta {H}_{{\rm{decomp}}}^{{\rm{CGCNN}}}-T\Delta {S}_{{\rm{mix}}}\) for compositions with a high ∆Smix (i.e., close to equi-atomic ratio). In particular, at the fraction of 0.5 in Fig. 5a, three inequivalent atomic configurations exhibit different ∆Hdecomp in the training data, and the energy deviation lies on the prediction range of our CGCNN models. The lowest \(\Delta {H}_{{\rm{decomp}}}^{{\rm{CGCNN}}}-T\Delta {S}_{{\rm{mix}}}\) values predicted by the CGCNN model (x = 0.75 for Fig. 5a and x = 0.5625, y = 0.25 for Fig. 5b) were validated once again using DFT calculations, as shown in Fig. 5. The error between the CGCNN predictions and the DFT data (80 atoms) was below 10 meV atom−1.

Fig. 5: ML-enabled search of alloyed perovskite systems.
figure 5

a CGCNN-predicted \(\Delta {H}_{{\rm{decomp}}}-T\Delta {S}_{{\rm{mix}}}\) of a CsGexSn1-xBr3 and b CsGexHgySn1-x-yCl3 systems in comparison with the training and DFT data (80 atoms). The variance of CGCNN data is represented as a line at each composition. In b, only the lowest \(\Delta {H}_{{\rm{decomp}}}-T\Delta {S}_{{\rm{mix}}}\) value is plotted at each composition for clearance. In b, the validation and CGCNN data at the same composition (Ge: Hg: Sn = 0.5625: 0.25: 0.1875) are −107.60 and −103.79 meV atom−1, respectively.

Full size image

With the most stable atomic configurations for each composition, we applied the following four screening criteria to discover potentially promising candidates for solar-cell application: (1) non-indirect bandgap, (2) \({E}_{{\rm{gap}}}^{{\rm{CGCNN}}}\) < 0.5 eV, (3) τ value that is as low as possible (i.e., τ < 4.18), and (4) \(\Delta {H}_{{\rm{decomp}}}^{{\rm{CGCNN}}}-T\Delta {S}_{{\rm{mix}}}\) that is as low as possible (i.e., \(\Delta {H}_{{\rm{decomp}}}^{{\rm{CGCNN}}}-T\Delta {S}_{{\rm{mix}}} < 0\)). However, increasing the fraction of Ge tends to decrease ∆Hdecomp, as shown in Fig. 2b; therefore, we selected candidates with the top three low values of \(\Delta {H}_{{\rm{decomp}}}^{{\rm{CGCNN}}}-T\Delta {S}_{{\rm{mix}}}\) within each region divided into five intervals for τ lower than 4.18. We additionally selected candidates containing at least 50% Sn or Pb, because Sn- and Pb-containing MHPs are the most promising materials for solar-cell applications41. Throughout the screening procedure, 110 compounds were selected.

Subsequently, the 110 selected compounds were validated using DFT calculations with the PBEsol functional. We determined that the error between \(\Delta {H}_{{\rm{decomp}}}^{{\rm{CGCNN}}}-T\Delta {S}_{{\rm{mix}}}\) and \(\Delta {H}_{{\rm{decomp}}}^{{\rm{PBEsol}}}-T\Delta {S}_{{\rm{mix}}}\) did not exceed 6 meV atom−1 (view Supplementary Fig. 8a for the error distribution). However, in the case of the bandgap and band type, the predictions from the CGCNN were significantly inaccurate. That is, the CGCNN predicted that all 110 compounds would have a non-indirect bandgap; however, 79 compounds exhibited an indirect bandgap. Moreover, the MAE for bandgap predictions was 0.038 for systems with 20 atoms, whereas it increased to 0.14 for systems with 80 atoms (Supplementary Fig. 8b). We assume that these results were obtained as the band structure-related properties do not change continuously with compositional variations when compared to the thermodynamic stability, causing high prediction uncertainty for unseen compositional space. The detailed analysis of increased bandgap prediction error can be found in the Supplementary Note 4, Supplementary Table 1, Supplementary Fig. 2, and Supplementary Fig. 3. One practically applicable strategy to enhance the prediction reliability could be adding training data randomly selected from the target domain after the DFT calculations as similarly done in the active learning strategy. A detailed description of the mitigation strategy can also be found in the Supplementary Note 5 and Supplementary Fig. 4.

We calculated the PBE0 bandgap for only those compounds that were calculated to have a direct bandgap using PBEsol, yielding 31 candidates (refer to Supplementary Table 5 for the full list of the selected 31 candidates) with \({E}_{{\rm{gap}}}^{{\rm{PBE}}0}\) close to the Shockley–Queisser limit38 (1.2–1.4 eV) or the optimal bandgap of the top cell of a tandem solar cell17 (1.73 eV). From among these 31 compounds, 10 compounds showing an optimal bandgap were selected for each system, as summarized in Table 1. All selected compounds have direct bandgaps, as shown in Supplementary Fig. 9.

Table 1 Selected 10 perovskite compounds
Full size table

Previous experimental results have been reported for compositions similar to those listed in Table 1. We compared experimental and PBE0-calculated bandgaps of those compounds in Table 2. The difference between experimentally reported band gaps and DFT-calculated band gaps for 19 reported perovskite compounds is summarized in Supplementary Table 4. Previous experimental studies have reported CsGexSn(1-x)Br342, CsGe0.5Sn0.5I343, CsPbxSn(1-x)Br344, and CsGexPb(1-x)I345 as light absorbers for perovskite solar cells. Combining Table 2 and Supplementary Table 4, the root mean squared error (RMSE) between the experimentally reported bandgaps and our PBE0-calculated bandgaps is approximately 0.30 eV (see also Supplementary Fig. 5). Specifically, in the CsGexSn(1-x)Br3 system, the PBE0 band gap underestimates the experimental band gap by approximately 0.5 ~ 0.6 eV, while for other systems, it underestimates approximately 0.1 eV.

Table 2 Comparison between PBE0-calculated and experimental bandgap (\({E}_{{\rm{gap}}}^{{\rm{PBE}}0}\) and \({E}_{{\rm{gap}}}^{\exp }\), respectively) of compounds in Table 1
Full size table

The effective mass of the carriers listed in Table 1 does not exceed 1 me (mass of free electrons), except for CsGe0.5625Hg0.3125Sn0.125Cl3 and CsCd0.125Ge0.5625Hg0.1875Sn0.125Cl3. The carrier effective mass is known to directly affect the mobility of electrons in the electron transport layer, which in turn affects the power conversion efficiency of solar cells. The optical absorption spectra and SLME versus the sample thickness are shown in Supplementary Fig. 10. Note that the SLME in Table 1 is a convenient metric for comparing the suitability of the compounds for photon absorbers; however, it does not represent a realistic power conversion efficiency. Considering that our \({E}_{{\rm{gap}}}^{{\rm{PBE}}0}\) value underestimates \({E}_{{\rm{gap}}}^{\exp }\) by approximately 0.5–0.6 eV in the CsGexSn1-xBr3 system, the SLME of CsGe0.5625Sn0.4375Br3 and CsGe0.4375Pb0.0625Sn0.5Br3 will be lower than that of CsCd0.125Ge0.5625Hg0.1875Sn0.125Cl3. Thus, considering both the stability and SLME, we suggest that CsGe0.3125Sn0.6875I3 and CsGe0.0625Pb0.3125Sn0.625Br3 are promising single-junction and tandem perovskite solar cells, respectively.

Discussion

Although the DFT/ML design framework presented here shows several promising aspects, substantial challenges still exist, which must be overcome for the experimental achievement of this framework. A discrepancy exists between the \({E}_{{\rm{gap}}}^{{\rm{PBE}}0}\) and \({E}_{{\rm{gap}}}^{\exp }\) values. Owing to the significant discrepancies between CGCNN predictions and DFT results regarding the band type, certain compounds that were predicted as indirect-bandgap materials by the CGCNN may be direct bandgap materials in reality. The mitigation strategy for bandgap prediction is described in the Supplementary Note 5. Furthermore, our metric of thermodynamic stability, \(\Delta {H}_{{\rm{decomp}}}-T\Delta {S}_{{\rm{mix}}}\), ignores the fact that Ge and Sn readily oxidize to a + 4 state during the fabrication procedure, resulting in a significant loss of open-circuit voltage46,47. Further extension should be made to consider other important properties such as defect formation energy, electronic levels of such defects, charge carrier transport properties, and stability of surfaces and interfaces of perovskites for photovoltaics and other optoelectronic applications. Finally, A- and X-site alloying should also be considered in future studies.

This study explored the chemical space (compositional and configurational) of B-site-mixed all-inorganic perovskites by combining DFT and ML. A total of 3,159 PBEsol data points were generated for the thermodynamic stability, bandgap, and bandgap type. The CGCNN models were trained on the PBEsol data using the crystalline structures as inputs to predict the aforementioned properties. The trained CGCNN models screened 110 of the 41,400 compounds with desirable properties. Of the 110 compounds, 31 were selected after examining their band types and PBE0-calculated bandgaps. Furthermore, the carrier effective masses, optical absorption spectra, and SLME of 10 of the 31 compounds were calculated. Among the 10 compounds, CsGe0.3125Sn0.6875I3 and CsGe0.0625Pb0.3125Sn0.625Br3 were suggested as promising single-junction and tandem perovskite solar cells, respectively, considering their stability and SLME. We have also presented the limitations of this study and perspectives for future research. The DFT/ML design framework proposed herein forms the basis for designing perovskite alloys with desirable properties.

Methods

Training data generation

Training data were generated through DFT calculations using PBEsol30. As shown in Fig. 1a, we employed a crystal structure consisting of four formula units of ABX3 as a computational unit cell (i.e., the number of atoms in the unit cell = 20), with a focus on multi-element mixing at the B-site. For the A- and X-sites in ABX3, we used one of the three monovalent alkali cations (Cs, K, and Rb) and one of the three halogen anions (Br, Cl, and I), respectively. We excluded organic cations such as methylammonium (MA) and formamidinium (FA) from the A-site for two reasons: (1) instability issues of perovskite devices that primarily arise from organic components6 and (2) higher positional degrees of freedom than alkali elements, which cause additional energy contributions from various molecular configurations. For the B-site, we considered multi-element mixing of six metal cations (Cd, Ge, Hg, Pb, Sn, and Zn) up to the quaternary system because four available atomic positions of the B-site exist within the considered computational unit cell, giving 126 (= 6H4, combination with repetition) compositions. From this, 1134 (= 3 × 126 × 3) unique ABX3 compositions were obtained, and by considering all possible atomic configurations within each composition, 3159 unique B-site-mixed ABX3 structures were obtained for the training data. The details of obtaining 3159 unique structures are described in Supplementary Note 1 and Supplementary Fig. 1. The aforementioned six metal elements for the B-site were chosen to minimize any significant perturbation of the electronic structure of the pristine Pb-based halide perovskite, which is good for applications in solar cells. To achieve this, we intentionally chose cations that exhibit electronic similarities to Pb2+ ions. This selection was based on two key conditions: 1) maintaining the same charged state (2+) and 2) ensuring filled d-electron orbitals. The idea behind these conditions originated from the understanding that considering a cation with a different charge state can result in charge imbalances, potentially leading to the formation of metallic materials. Additionally, the presence of unfilled d-orbitals tends to introduce complex interactions with anions, leading to a markedly different electronic structure.

Crystal Graph Convolutional Neural Networks training

For all CGCNN models, an unrelaxed input geometry was used as the input to predict the target properties of the DFT-relaxed structures, which allowed us to predict the target properties of the relaxed structures without the costly DFT structural relaxations. Here, the experimentally well-known structure of CsPbI3 was used as the input for the CGCNN models; moreover, we present the crystal structure of CsPbI3 in the CIF format in Supplementary Note 6. The atomic features of the original CGCNN paper29 and default hyperparameters of the CGCNN code distributed in GitHub (https://github.com/txie-93/cgcnn) were used. The hyperparameters used in this study are listed in Supplementary Table 3. A total of 3159 PBEsol data points were randomly divided in a ratio of 7:1:2 as training, validation, and test data. The validation data were used to determine the model parameters using an early-stopping approach.

Thermodynamic stability and bandgap of B-site mixed metal halide perovskites

To estimate the thermodynamic stabilities of the compounds, we used the previously defined decomposition enthalpy, ∆Hdecomp19,20,21,22,24. Decomposition enthalpy is defined as the difference between the energy of ABX3 and the sum of the energies of the most stable AX and BX2 phases. Thus, ∆Hdecomp of the B-site mixed MHPs (i.e., AB*X3, where B* represents mixed B-site species such as Cd0.25Ge0.75 and Hg0.25Pb0.25Sn0.5) can be defined as follows:

$$\Delta {H}_{{\rm{decomp}}}\left({\rm{AB}}^* {\rm{X}}_3\right)=E\left({\rm{AB}}^* {\rm{X}}_3\right)-E\left({\rm{AX}}\right)-\sum _{i}{x}_{i}E\left({{\rm{B}}}_{i}{{\rm{X}}}_{2}\right),$$
(1)

where E is the DFT-calculated total energy of the relaxed geometries and xi is the fraction of the element Bi. From Eq. (1), we can observe that a more negative ∆Hdecomp indicates that the system is more stable when compared to the other binary metal halide phases (i.e., AX and BX2). The energetically most stable structures for the AX and BX2 phases were obtained from Materials Project (MP), an open-access database that offers material properties48. Conversely, the structure of GeCl2 was retrieved from the Open Quantum Materials Database (OQMD)49 because the structure of GeCl2 is not available in the MP. We present a list of MP or OQMD IDs of AX and BX2 phases and their DFT (PBESol)-calculated total energies in Supplementary Table 2.

The mixing entropy, ∆Smix, is defined as the configurational entropy based on the ideal solid solution model (Eq. (2)):

$$\Delta {S}_{{\rm{mix}}}={-k}_{B}\mathop{\sum }\limits_{i=1}^{N}{x}_{i}\,\mathrm{ln}{x}_{i},$$
(2)

where kB is the Boltzmann constant, N is the number of components at the B-site, and xi is the atomic fraction of component i at the B-site.

In addition to the thermodynamic stability, the band type and bandgap (Egap) were calculated to estimate the possibility of application in photovoltaics at the PBEsol level. Because MHPs with indirect bandgaps are not usually suitable for photovoltaics, we classified the band types into two categories: indirect and non-indirect (i.e., direct, metallic, and semi-metallic). Materials determined to have a metallic or semi-metallic bandgap using PBEsol can be identified as having a direct bandgap using PBE0; thus, we did not exclude them from the PBE0 calculations. This approach assumes that the band types calculated using PBEsol and PBE0 are consistent with each other, although the bandgap calculated using PBEsol (\({E}_{{\rm{gap}}}^{{\rm{PBEsol}}}\)) is underestimated when compared to that calculated using PBE0 (\({E}_{{\rm{gap}}}^{{\rm{PBE}}0}\)).

Tolerance factor of B-site mixed metal halide perovskites

According to Bartel et al.35, the probability of a compound being a perovskite increases as τ becomes smaller than 4.18, and vice versa. τ is calculated using Eq. (3):

$$\tau =\frac{{r}_{X}}{{r}_{B}}-{n}_{A}\left({n}_{A}-\frac{\frac{{r}_{A}}{{r}_{B}}}{\mathrm{ln}\left(\frac{{r}_{A}}{{r}_{B}}\right)}\right),$$
(3)

where nA is the oxidation state of A and ri is the ionic radius of ion i. Here, rB is the composition-weighted average of the ionic radius of each B element (i.e., \({r}_{{\rm{B}}}={\sum }_{i=1}^{N}{x}_{i}{r}_{{B}_{i}}\)). We selected the compositions showing \(\Delta {H}_{{\rm{decomp}}}-T\Delta {S}_{{\rm{mix}}} \,<\, 0\) and τ < 4.18 as stable compounds.

Density functional theory calculations

All DFT calculations were performed using the Vienna Ab initio Simulation Package (VASP) 6.1.2.50, and projector augmented wave pseudopotentials51. The pseudopotentials of all elements were chosen according to the official recommendations of the VASP. For training data generation and final DFT validation of the selected compounds, the projection operators of the nonlocal part of the pseudopotential were calculated in reciprocal and real spaces, respectively. The energy cut-off for a plane-wave representation was 500 eV. All the cell parameters and atomic coordinates are relaxed using a convergence criterion of 10−5 eV on the energies of the self-consistent electronic step and 0.01 eV/Å on the atomic force. The Brillouin zone was sampled using a Γ-centered automatic k-points generation scheme that was implemented in VASP, with a length of 50 Å for determining the subdivisions. Cell relaxation was performed first; then, the energy was obtained from the consequent fixed-cell relaxation.

The band structure was calculated for the training data and for the selected compounds through screening. For training data generation, the band structure was calculated using PBEsol at high-symmetry k-points in the line mode. For the final DFT validation of the selected compounds, the band structures were calculated using PBE0 with Γ-centered 2 × 2 × 2 Monkhorst–Pack k-points. According to PBEsol, the band edges of all the selected compounds were predicted to lie on the high-symmetry k-points contained in the Γ-centered 2 × 2 × 2 Monkhorst–Pack k-point. Therefore, the Γ-centered 2 × 2 × 2 Monkhorst–Pack k-points were used in the PBE0 calculation to calculate the bandgap and reduce the computational time. Both band structure calculations used the PBEsol-optimized structure as the input and SOC corrections32. We compared the bandgaps calculated using PBE0, HSE0632, and B3LYP33 functionals for 19 representative perovskite materials with known experimental bandgaps, as shown in Supplementary Table 4 and Supplementary Fig. 5. After comparison, we found that PBE0 best aligns with the experimental bandgaps. The VASPKIT code52 was used to determine the PBEsol band type and bandgap from the data calculated using VASP.

The effective masses of electrons and holes were calculated based on the PBEsol-calculated band structures using the sumo53 software, which is a set of command-line tools for plotting and analysis of periodic ab initio calculations. The optical absorption spectra were calculated using the LOPTICS tag54, setting the number of grid points at which the density of states (NEDOS tag) was set to 2000 for each structure. We obtained optical absorption spectra on Γ-centered 4 × 4 × 4 Monkhorst–Pack k-points with the k-point down-sampling scheme using the PBE0 functional with SOC corrections. Sumo53 was also used to post-process the optical absorption spectra.