Composition and Structure Based GGA Bandgap Prediction Using Machine Learning Approach

Machine Learning (ML) has emerged as a powerful tool in the field of data analytics, enabling academia and industry professionals to glean insights and make predictions based on complex and varied data sources Sarker (2021). Both academia and industry professionals share a common aim of more accurate and effortless implementation of ML techniques to design advanced functional materials. sch ; Lee and Lim (2021); Ahmad et al. (2022); Guan et al. (2020). In academia, researchers are exploring new ML techniques and algorithms, while in industry, there is a growing demand for professionals with expertise in ML. But both face challenges as inadequacy of large amounts of high$-$quality data and ensuring ML models are transparent and explainable, particularly in applications such as healthcare where biased or incomplete data can lead to inaccurate predictions and serious aftereffects Esmaeilzadeh (2020); Reddy et al. (2020).

At its core, ML involves the use of algorithms that can learn from data and improve over time, without requiring explicit programming. The importance of ML lies in its ability to process and analyze vast amounts of data and identifying patterns as well as anomalies that may not be apparent to humans. This makes it a valuable tool in many industries, from healthcare to finance to manufacturing, where it can be used to optimize processes, improve decision$-$making, uncover new business opportunities, and identifying materials with superior properties. However, while ML has the potential to revolutionize many aspects of modern life, it is not a simple process. Developing effective ML models requires careful consideration of data quality, relevance, and cleanliness. In order to leverage ML, one must first identify the right data sources that can be applied to the learning process. This may involve collecting data from disparate sources and integrating them into a unified data model. Once data has been collected, it must be prepared for analysis by cleansing and tidying it. This process involves transforming the data into a format that can be understood by ML algorithms, which often require numerical data rather than textual information.

Materials science is an interdisciplinary field that focuses on the study of the properties, structure, and synthesis of materials. However, traditional methods of materials discovery and design are often time$-$consuming and costly, which has hindered accelerated progress in this field. The use of ML has rapidly transformed materials science by providing a powerful tool for accelerating the design and discovery of advanced functional materials and optimizing their synthesis process. Bishara et al. (2023); Panchal et al. (2019); Ramprasad et al. (2017); Lee et al. (2016); Oliynyk et al. (2016); Butler et al. (2018).

In materials science, ML algorithms can analyze vast amounts of data from experiments and/or simulations stored in databases to identify patterns and relationships those may be difficult to detect using traditional approaches. The collaboration of ML and computational materials science offers numerous advantages. One of the most significant benefits is the ability to predict the properties of materials from a wide chemical space before they are synthesized. By predicting the properties of materials based on their chemical composition, crystal structure, and other properties, ML algorithms can significantly reduce the time and cost associated with experimental synthesis and testing of materials Morgan and Jacobs (2020); Suh et al. (2020); Liu et al. (2017). The ML algorithms can also aid in the discovery of new materials with desired properties. By analyzing extensive databases of materials, ML algorithms can identify materials with properties similar to the desired ones and propose modifications to optimize their properties. This approach has the potential to accelerate the materials discovery and design process Juan et al. (2021); Cole (2020); Oganov et al. (2019); Pulido et al. (2017).

There are several comprehensive materials databases available to researchers those provide access to an extensive list of data for materials, which is essential for the discovery, design, and optimization of new materials. Most of these databases support Application Programming Interface (API) based data retrieval, allowing users to efficiently mine, sort, and, retrieve specific data they need. Web based applications such as aflowlib Curtarolo et al. (2012), JARVIS Choudhary et al. (2020), novel materials discovery laboratory Draxl and Scheffler (2019) (NOMAD) encyclopedia, crystals AI Zheng et al. (2020), citrine informatics (thermoelectric predictor) Gaultois et al. (2016), NIMS tools Tanifuji et al. (2019), SUNCAT Winther et al. (2019) catalysis property predictor, and matlearn Peterson and Brgoch (2021) use advanced ML algorithms and artificial intelligence techniques to predict various properties of materials. These tools can help researchers to screen and design new materials with desirable properties, such as thermoelectric transport properties, electronic structure, catalytic activity, etc. JARVIS, crystals AI, thermoelectric predictor, NIMS tools, SUNCAT, and aflow library are materials simulation software and they use density functional theory (DFT) and ML algorithms to predict properties of materials. Inorganic crystal structure database (ICSD) Belsky et al. (2002), open crystal database (OCD), Pearson crystal structure data (PCXD), and Cambridge Structure Database (CSD) are some of the crystal structure databases those possess experimentally measured crystal structures data of materials. While the open quantum materials database (OQMD) Saal et al. (2013) is a database of quantum mechanical calculations for materials properties. Materials project Jain et al. (2013) is a web$-$based platform that provides materials data and computational tools for materials scientists and engineers. Citrine informatics is a materials informatics company that uses ML algorithms to analyze and predict material’s properties. Some of the other databases are the materials web online data base Friedman (2009), computational materials repository Rasmussen and Thygesen (2015), materials cloud platform for two$-$dimensional (2D) materials Mounet et al. (2018), Harvard clean energy project (CEP) for organic photo$-$voltaics materials Hachmann et al. (2011), and thermoelectrics design lab for thermoelectric materials Gorai et al. . All these tools and resources are important for materials scientists and engineers to design and discover new materials with desired properties. Each of these web applications specializes in predicting a specific set of material properties and they often include databases of material’s properties those can be used to search for and compare materials with specific properties.

The use of ML has significantly decreased the computational time required for calculating materials properties, as compared to first principles DFT calculations Hohenberg and Kohn (1964); Kohn and Sham (1965). Moreover, this has been achieved without compromising accuracy and has enabled fast descriptive and predictive learning. Further, ML is particularly useful in handling multidimensional data, which is critical for accelerating materials development. Photovoltaics, piezoelectrics, magnetocalorics, magnetoelectrics, and thermoelectrics are some functional materials systems those have benefited from this revolution with demonstrated success in screening and designing efficient materials using ML algorithms. In materials science, selecting the most suitable ML algorithm for a particular application in a specific domain can also be a challenging task. This is because, different algorithms are designed to solve different types of problems and the effectiveness of a particular algorithm can depend on the specific characteristics of the materials data and the features used to train the model. One important consideration when selecting a ML algorithm is the type of materials data that is being analyzed. For example, if the data involves identifying patterns in X$-$ray diffraction spectra or crystallographic data, then ML algorithms such as clustering and pattern recognition may be appropriate. On the other hand, if the data involves predicting mechanical properties or optimizing material processing conditions, then regression and optimization algorithms may be more suitable. If the database has a huge number of data for a specific property then more advanced deep learning techniques may be suitable to employ.

Another factor to consider when selecting a ML algorithm in materials science is the size and complexity of the data. For instance, in the case of high$-$throughput materials screening, where large volumes of data are generated from diverse materials properties, unsupervised learning algorithms such as principal component analysis (PCA) or clustering may be useful for reducing the dimensionality of the data and identifying important features. Additionally, the desired outcome of the analysis is also a crucial factor in selecting an appropriate ML algorithm. For example, if the goal is to identify new materials with specific properties, then unsupervised learning algorithms such as clustering or generative adversarial networks (GANs) may be more useful Tshitoyan et al. (2019); Jang et al. (2020); Graser et al. (2018); Wei et al. (2019). Finally, the computational resources available and the level of interpretability required are also important considerations when selecting a particular ML algorithm in materials science. Deep learning algorithms such as convolutional neural networks (CNNs) and recurrent neural networks (RNNs) may be more computationally expensive, but they can often provide highly accurate predictions. On the other hand, simpler algorithms such as Decision tree (DT) or Linear regression may be more interpretable and easier to implement. Anyway, whatever model we adopt, the quality of the dataset and appropriate features are key to predict the properties reliably. Here we have adopted regression methods because they are relatively less complex and easy to interpret and evaluate. The motivation behind selectively choosing the models used in this study is their proven superiority in performance on various datasets from diverse fields.

Priyanga et al. Mattur et al. (2022) compared the performance of different ML classifier models for predicting the nature of bandgap. She compared Logistic regression, K$-$Nearest Neighbors (KNN), Support vector clustering, Random Forest (RF) Breiman (2001); Liaw et al. (2002); Svetnik et al. (2003); Li et al. (2013), DT Kushwah et al. (2022), Light Gradient Boosting Machine (LGBM) Ke et al. (2017) and eXtreme Gradient Boosting (XGB) Chen and Guestrin (2016); Chen et al. (2015) for their predictability of nature of bandgap for perovskite materials. Wang et al. trained a stacking model Pavlyshenko (2018); Meharie et al. (2022) for bandgap value prediction with LGBM, RF, XGB, and Gradient Boost Decision Tree (GBDT) as base models Wang et al. (2022). Further ML models such as CatBoost Prokhorenkova et al. (2018), Gradient Boosting (GB) Friedman (2001, 2002); Guelman (2012), HistGradientBoosting, ExtraTrees, XGB, DT, bagging Breiman (1996); Bühlmann (2012); Sutton (2005); Friedman and Hall (2007), LGBM, GaussianProcess, artificial neural network (ANN), and light long short$-$term memory (LightLSTM) were compared for predictability of photocatalytic degradation of malachite green dye Jaffari et al. (2023). These are few latest studies on material properties using ML models.

The bandgap is a fundamental material property that plays a critical role in different optoelectronic applications. The range of bandgap value required depends on the application and also different materials with varying bandgap values are necessary for various purposes. Materials with a small bandgap value are useful in electronics and detectors, while materials with a large bandgap value are useful in solar cells, LED, and transparent conducting applications. Materials with a narrow bandgap value are useful in thermoelectric devices. Estimating the bandgap value accurately is crucial in Materials Science and can lead to the development of new materials with enhanced properties for various applications, especially, for green energy technologies. Predicting the bandgap value of materials is an important task in Materials Science and has been the subject of many published papers Elsheikh et al. (2014); Radecka et al. (2008); Okumura (2015); Takahashi et al. (2007). To choose the correct model for bandgap prediction, it is important to consider several factors, including the choice of ML algorithm, the quality and size of the dataset, feature selection, model validation, and model interpretability. There are many ML algorithms those can be used for predicting bandgap values, including regression models and neural networks. Each algorithm has its own strengths and weaknesses and the choice of algorithm will depend on the characteristics of the dataset as well as the specific problem being addressed. The quality of the data used to train the model is critical to its accuracy and generalizability.

The another important aspect is that the dataset should be large enough to capture the full range of variability in the material properties and should be representative of the materials being studied to avoid biases. The dataset should also be clean and free of errors or outliers those could skew the predicting capability. The selection of features or inputs those are used to predict bandgap value are important factors in determining the accuracy of the model. Relevant features may include elemental composition, crystal structure, and other material properties. It is important to select features those are relevant to the problem being addressed and that have a strong correlation with the targeted property. Once a model has been trained with the datasets and appropriate features, it is important to validate its performance on a separate test dataset that has not been used for training. This helps to ensure that the model is able to generalize to new data and is not overfitting to the training dataset. The ability to interpret the results of a model is also important for understanding the factors those are driving the predictions. It may be noted that, the models those are more interpretable, such as DT or Linear regression models, can provide insights into the underlying relationships between the input features and the targeted property.

In their study, Heng et al. utilized neural networks and leveraged available experimental data from the literature to predict the bandgap of binary semiconductor alloys Heng et al. (2000). They focused on the elemental properties of the constituents and observed a continuous variation of bandgap values in A_iB_1-i alloys as the mole fractions of the binary constituents changed. This observation led them to conclude that the bandgap value is influenced by the interplay between attractive and repulsive forces of the constituents as well as the diffusion rate of atoms during the formation of the semiconductor.

Zhuo et al. employed variety of ML methods to model the bandgap energies of 3896 inorganic solids those were reported in experimental studies Zhuo et al. (2018). Their analysis covered a wide range of materials and aimed to accurately predict their bandgap values. By incorporating different ML techniques they sought to improve the understanding and prediction of bandgap values in diverse inorganic solids. Zeng et al. Zeng et al. (2002) extended this analysis to ABC₂ type chalcopyrites. They employed neural network techniques to explore the relationship between elemental properties and bandgaps values in this class of materials. By investigating a different class of compounds, they aimed to expand the understanding of the factors influencing bandgap energies. In a different approach, Weston and Stampfl Weston and Stampfl (2018) combined DFT calculations with multiple ML models to predict the bandgap value of Kesterite I₂-II-IV-V₄ semiconductors. They trained their models using 200 bandgap values calculated using the hybrid HSE functional and focused on classifying these bandgap values as either direct or indirect. This approach allowed them to gain insights into the bandgap characteristics of the Kesterite materials.

These studies collectively demonstrate the application of neural networks, DFT calculations, and ML models in predicting and understanding the bandgap behavior of various semiconductor alloys and inorganic solids. Each study contributed to advancing the knowledge in this field and provided valuable insights into the factors influencing bandgap value and its nature. However, while ML holds enormous potential, it is not a simple process. One of the biggest challenges is collecting and preparing data for analysis. In order to train a ML model, one should first identify the right data sources and ensure that the data is accurate, relevant, and clean. This may involve combining data from multiple sources, such as open databases, literature’s etc and transforming them into a format that can be understood by ML algorithms. Another challenge is selecting the right ML algorithm for a given task. There are many different algorithms to choose from, each with its own strengths and weaknesses. Some algorithms are better suited for classification tasks, while others are better suited for regression tasks. Some algorithms are more robust to noise and outliers, while others are more sensitive to them. Selecting the right algorithm requires careful consideration of the data and the problem at hand. Despite these challenges, ML has become an essential tool for one looking to leverage their data assets and gain a deeper understanding of their task and to accelerate it. By following best practices for data preparation and algorithm selection, one can develop robust and accurate ML models that drive innovation and create new opportunities.

In this section, we will provide a comprehensive discussion on how to achieve accurate predictions for the bandgap values for HH compounds. We would like to stress the importance of feature engineering in ML methods, as it can significantly affect the predicting capability of the ML models. The selection and engineering of relevant features for a given dataset are crucial for achieving the best possible model accuracy. However, featurizing and selecting specific features with good correlation with targeted properties for a dataset can be a challenging and time$-$consuming task. Therefore, we aim to use easily accessible information with the input parameters as simple as possible. We have used the featurization method provided in matminer based on both the composition and structure of the materials, We have used 8 ML models based on the selection of input features, which we will explain in detail in the following sections.

Our objective is to develop appropriate ML models those can accurately predict the bandgap values of large number of HH compound compositions derived based on empirical 18 VEC rule those are expected to be semiconductors, which is essential for identifying new materials with desired bandgap value. We can determine the most effective set of input features for predicting the bandgap values of HH compounds. Through our analysis and discussion, we hope to provide valuable insights into the process of accurately predicting the bandgap values of HH compounds using ML techniques and it can be generalized to predict bandgap values for all inorganic solids. To identify the most relevant features, we used the permutation of features for feature importance and correlation matrix heatmap analysis, which is shown in Fig.2. To assess the performance accuracy of each model, two standard performance metrics were considered and those are RMSE (Root Mean Square Error) and R² (coefficient of determination). Among these two performance matrices, the R² metric is a dimensionless statistical measure that indicates how well the data fits a regression line, typically ranging from 0 to 1 and RMSE provides a measure of the average deviation from the actual value. The RMSE expresses the average model predictions in the units (here eV) of the targeted property (here bandgap) used to train the data and is a negatively oriented score, meaning that a lower score indicates better performance of the model. These two performance matrices are defined as.

and

, where y_i represents the true value, $\hat{y}_{i}$ represents the predicted value, and $\bar{y_{i}}$ represents the average value. Additionally,$i$ ranges from 1 to N, where N represents the total number of targeted property data present in the test dataset.

To enhance the quality of our feature set, we employed the Pearson correlation method using the pandas library. This technique allowed us to identify features closely related to our targeted property. We then examined a heatmap illustrating the Pearson correlation coefficients (PCC). Only PCC values surpassing the absolute threshold of 0.2 were visualized, as shown in Figure 4(a). These values ranged from -1 to 1, with a higher absolute value indicating a stronger correlation between the target and features. Analyzing the PCC heatmap, we were able to eliminate some features with little or no correlation to the targeted property. For example, a feature with a PCC value close to 0 was likely to be removed, as it would contribute minimally to the model’s predictive ability. Conversely, features with high PCC values were retained, as they were likely to be important predictors of the output variable. The correlation matrix among features as shown in figure2b is also obtained using the Pearson correlation method, which calculates pairwise correlations between features in the columns of a data frame. The resulting correlation values range from -1 to 1, with a higher absolute value signifying a stronger correlation between features. By analyzing the correlation heat map, we effectively eliminated highly correlated features mitigating issues like multicollinearity and overfitting. Through these steps, we reduced the dimensions of our feature space while maintaining its informative aspects.

Our feature selection and processing methods played a crucial role in developing our predictive models. They helped us identify critical features and thus enhance the model’s predictive capacity.

Composition dependent descriptors are a type of material properties those describe how the properties of a material depend on its composition or the relative amounts of the different elements in the material. One such descriptor is the stoichiometric average of the elemental properties, which is a weighted average of the elemental properties based on their stoichiometry in the material. The stoichiometric average is a useful descriptor because, it takes into account the contributions of all the elements in the material, rather than just one or two dominant elements. The elemental properties those are typically used to calculate the stoichiometric average include the atomic number, ionic radius, electronegativity, and electron configuration of the constituent elements. To calculate the stoichiometric average, the elemental properties are first weighted by their stoichiometric coefficients, which are the numbers those represent the relative amounts of each element in the material. The stoichiometric average can be used to predict the properties of materials based on their composition. For example, it has been found that the stoichiometric average of the electronegativity of the elements in a material can predict its bandgap value if it originates from the ionicity, which is an important property for electronic and optoelectronic applications. Other properties those can be predicted using the stoichiometric average include the thermal and mechanical properties of materials.

We have used the matminer library to obtain a dataset of materials with known bandgap values obtained from GGA$-$PBE calculations and their corresponding composition as well as structure features. In addition to these features, we incorporated the mean, maximum, and standard deviation of atomic numbers as well as the structural features of the corresponding materials by accounting the GGA$-$PBE calculated lattice parameter values $a$, $b$, $c$, $\alpha$, $\beta$ and $\Gamma$ obtained from the corresponding calculated crystal structure.

We have split the generated dataset into a training set and a testing set with 8:2 ratio. The training set was used to train our models and the testing set was used to evaluate their performance. We have employed the following eight regression models such as RF, GB, XGB, LGBM, DT, AdaBoost Freund et al. (1996); Solomatine and Shrestha (2004); Collins et al. (2002); Shrestha and Solomatine (2006) to identify the best performing model. For the performance analysis, we have considered R² and RMSE performance metrices to select the best algorithm for our dataset. After assessing the performance of the considered regression models on test data, we found that the RF regressor exhibited the best performance among the considered ML models followed by that the bagging regressor performed well. We have also considered the following seven models as a base for the bagging ensemble model analysis such as DT, CatBoost, GB, LassoCV Tibshirani (1996), LGBM, RidgeCV, XGB With AdaBoost as booster. Among the various models considered in the bagging ensemble model analysis, we found that the bagging with DT and AdaBoost showed the best performance.

The ML regression models were trained using available bandgap values from matminer database obtained from DFT calculations based on GGA$-$PBE functional , excluding the well studied compounds those such as Si, GaAs, TiO₂ etc were used for validation. Subsequently, we have performed band structure calculations using the GGA$-$PBE functional to cross validate the ML predicted bandgap value of newly identified HH compounds. Further improve the DFT predicted bandgap value we have also used the TB$-$mBJ functional and the corresponding calculated bandgap values are compared in table4. This combination of ML and DFT hybrid approach holds promising potential for accelerated and reliable bandgap predictions in new semiconducting materials.

The DFT calculations for structural optimization and the electronic structure calculations were performed using projector$-$augmented plane$-$wave (PAW) Kresse and Joubert (1999) method, as implemented in the Vienna ab$-$initio simulation package (VASP) Kresse and Furthmüller (1996). The generalized gradient approximation (GGA$-$PBE) Perdew et al. (1996) proposed by Perdew$-$Burke
$-$Ernzerh has been used for the exchange$-$correlation potential to compute the equilibrium structural parameters and electronic structure. The irreducible part of the first Brillouin zone (IBZ) was sampled using a Monkhorst pack scheme Monkhorst and Pack (1976) and employed a 12$\times$12$\times$12 k$-$mesh for geometry optimization.

A plane$-$wave energy cutoff of 600 eV is used for geometry optimization. The convergence criterion for energy was taken to be 10^-6 eV/cell for electronic minimisation step and less than 1 meV/Å  is used for Hellmann$-$Feynman force acting on each atom to find the equilibrium atomic positions. Our previous study Choudhary and Ravindran (2020) shows that the computational parameters used for the present study are sufficient enough to accurately predict the equilibrium structural parameters for HH compounds. In our electronic structure calculations we focused on the high symmetry directions of the first IBZ of face centered cubic systems. Considering that the GGA$-$PBE method tends to underestimate bandgap values, we have also included results from higher level calculations, such as the TB$-$mBJ method in tablexx though it cannot be compared with ML prediction values. The other theoretical details of these calculations can be found elsewhere Choudhary et al. (2023); Tran and Blaha (2009); Koller et al. (2011, 2012). In order to see the dynamical stability of the predicted four HH alloys based on 18 VEC, we have made lattice dynamic calculation to check for their dynamic stability. The phonon dispersion curves are generated using the finite displacement method implemented in the VASP$-$phonopy Togo and Tanaka (2015) interface with the supercell approach. We have used the relaxed primitive cells to create supercells of dimension 3 × 3 × 3 with the displacement distance of 0.01 Å for our finite displacement calculations to estimate the dynamical matrix.

In Table 4 we have listed out the predicted bandgap values from our ML regression models discussed in section III along with those obtained from DFT calculation mentioned above. It may be noted that, the HH compounds listed in Table 4 are predicted based on 18 VEC rule and their corresponding bandgap values obtained from GGA$-$PBE calculations are directly compared with those predicted from our best performing ML models. This comparison is made to validate our ML trained models performance and also their reliability to predict GGA bandgap value of any unknown new compounds. Additionally, we have predicted the GGA$-$PBE bandgap values for well$-$studied compounds such as Si, C, SiC, LiF, BN, TiO₂, GaS, SiGe and SiSn (see Fig. 7(a)) using the best performing ML models to show their reliability to predict GGA bandgap value of wide range of semiconductors. The overall observation is that, we have obtained minimal error between the bandgap values calculated using the GGA$-$PBE method and those predicted based on our trained ensemble ML models for well known semiconductors.

The calculated electronic band structures of newly identified HH compounds ZrFeTe, LuNiSb, ZrSbRh and LaNiBi are displayed in Fig.7 as (a),(b), (c), (d), respectively. From these figures it is clear that all these compounds possess indirect bandgap behavior with the valence band maximum (VBM) located at the $\Gamma$ point and the conduction band minimum (CBM) at the X point. At the $\Gamma$ point, the bands near the VBM show a triply degenerate state. The degenerate bands near the VBM create a well dispersed band structure that favors hole conductivity in these systems. However, the flat band dispersion near the CBM along the $\Gamma$ $-$X direction suggests low electron conductivity due to the high effective mass of electrons.

In order to check the phase stability of our selected compounds, we have made enthalpy of formation ($\Delta H_{f}$) calculations. Our calculated $\Delta H_{f}$ is negative for ZrFeTe, LuNiSb, ZrSbRh and LaNiBi with the value of -0.76 eV/atom, -0.88 eV/atom, -1.0 eV/atom and -0.77 eV/atom, respectively. The negative $\Delta H_{f}$ values confirm that these materials are stable at ambient conditions and also can be synthesised experimentally. Additionally, to check the dynamical stability of those compounds we have calculated the phonon dispersion curves along high symmetry directions within the fcc BZ. We used the Phonopy code with the GGA$-$PBE functional and the finite difference method to compute the phonon dispersion at equilibrium lattice parameters under harmonic approximation. The phonon dispersion curve provides valuable information about the dynamical stability of a crystal. A dynamically stable crystal exhibits only positive (real) phonon frequencies throughout the phonon dispersion curve. In contrast, a dynamically unstable crystal also display negative (imaginary) phonon frequencies. The calculated phonon dispersion curves for predicted new HH alloys are given in Fig.8. From these phonon dispersion curves we found that no one of these compounds exhibit a dynamical instability and hence no negative frequency is found in any of these compounds. This result suggests that all these compounds will be stable under the ambient condition. The observation of high stability in these newly identified HH alloys is consistent with the alloy theory of solids that 18 VEC rule brings extra stability to the system.

In summary, our trained best performing machine learning models exhibited minimal deviations from actual values when predicting GGA$-$PBE bandgap values using structural and composition based features. When compared to DFT calculations, ML models trained on existing DFT data can provide comparable prediction accuracy in significantly less time. These ML models can effectively aid in the search for compositions with desirable bandgap value across a wide chemical space, which accelerate material discovery with desired properties. Among the 8 stand-alone ML regression models we have studied, Adaboost performed the poorest. Conversely, the DT model showed signs of over-fitting on the training data, as indicated by its performance matrices. The other stand-alone models consistently improved their performance when we increase the dataset size, emphasising their robustness, accuracy, and generalization capabilities. Notably, CatBoost regression excelled on larger datasets, while LGBM regression demonstrated strong performance on smaller datasets. RF regression consistently delivered reliable results on smaller and medium$-$sized datasets. GB performed well across datasets of varying sizes, while XGB exhibited mixed performance depending on the dataset size. It is important to note that these results depend on the specific dataset characteristics, feature engineering techniques, and pre$-$processing steps used. More exploration and experimentation with different combinations of features and pre$-$processing techniques can further enhance the predictive capabilities of these models.

The selection of an appropriate model should be based on a comprehensive understanding of the dataset and the specific requirements of the prediction task. Subsequently, we employed above mentioned stand alone ML models to predict the bandgap values of new HH compounds designed based on 18 VEC. Through this process, we identified several new materials with significant potential for applications in thermoelectrics, solar cells, and photocatalysis. In addition, we employed ensemble model such as stacking and bagging models to enhance the predicting capability for obtaining the GGA$-$PBE bandgap values for unseen datasets. It’s important to take into account that in this study, we adopted GGA$-$PBE bandgap values to train our ML models. For attaining a superior level of accuracy to predict the bandgap value on par with that measured experimentally, we have to train our ML models with experimental bandgap values which is available only for very limited number of systems. Also, the best performing ML models can be trained using the bandgap values obtained from more advanced DFT calculations such as hybrid functionals, GW method etc. However the bandgap values obtained from such advanced DFT methods are very limited owing to the fact that such calculation are computationally very intensive. Moreover, the calculated phonon dispersion curve and negative $\Delta H_{f}$ in all the selected compounds considered in this study confirmed their phase and dynamic stability at ambient condition. We hope that the present approach hold the potential to greatly expedite the discovery of new materials possessing desirable properties for various energy$-$related applications.

There are no conflicts to declare.

The authors are grateful to the SCANMAT Centre, Central University of Tamil Nadu, Thiruvarur for providing the computer time at the SCANMAT supercomputing facility. The authors would also like to acknowledge the SERB$-$Core Research Grant (CRG) vide file no.CRG/2020/001399.