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Prediction model of band-gap for AX binary compounds by combination of density functional theory calculations and machine learning techniques
Joohwi Lee, Atsuto Seko, Kazuki Shitara, Isao Tanaka
TL;DR
Accurate band-gap prediction is difficult because standard DFT gaps can be inaccurate, while higher-accuracy methods are computationally expensive. The paper develops OLSR, LASSO, and SVR models using Kohn-Sham gaps and elemental and structural descriptors to predict G0W0 gaps for 156 AX binary compounds. The best SVR model achieves an RMSE of 0.18 eV, and the resulting corrected band-gaps are intended to support materials exploration.
Problem
Standard DFT band-gaps can be inaccurate, while more accurate hybrid-functional and GW calculations have high computational costs that limit large databases.
Method
The study predicts G0W0 band-gaps for 156 AX binary compounds using OLSR, LASSO, and SVR with Kohn-Sham gaps plus elemental and crystal-structure predictors.
Results
The best SVR model yields an RMSE of 0.18 eV, while adding mBJ gaps and varied elemental and structural predictors significantly decreases RMSE.
Takeaways & Limitations
The estimated, well-corrected band-gaps should be useful as predictors for high-throughput materials screening and exploration.
Abstract
from arXiv · showhide
Machine learning techniques are applied to make prediction models of the G0W0 band-gaps for 156 AX binary compounds using Kohn-Sham band-gaps and other fundamental information of constituent elements and crystal structure as predictors. Ordinary least square regression (OLSR), least absolute shrinkage and selection operator (LASSO) and non-linear support vector regression (SVR) methods are applied with several levels of predictor sets. When the Kohn-Sham band-gap by GGA (PBE) or modified Becke-Johnson (mBJ) is used as a single predictor, OLSR model predicts the G0W0 band-gap of a randomly selected test data with the root mean square error (RMSE) of 0.54 eV. When Kohn-Sham band gap by PBE and mBJ methods are used together with a set of various forms of predictors representing constituent elements and crystal structures, RMSE decreases significantly. The best model by SVR yields the RMSE of 0.18 eV. A large set of band-gaps estimated in this way should be useful as predictors for materials exploration.
I. Introduction
Accurate band-gap estimation is limited by systematic errors in standard DFT approximations and the high cost of more accurate methods. This study addresses the gap with machine-learning models that use calculated gaps and constituent and structural information to predict G0W0 gaps for AX compounds.
- Motivation: PBE Kohn-Sham gaps underestimate band-gaps, with an RMS difference of 2.25 eV from experiment for 32 AX compounds.The Kohn-Sham gap is the difference between the lowest unoccupied and highest occupied eigenvalues.
- Motivation: Hybrid-functional and GW approaches can improve band-gap accuracy but have high computational costs, limiting large band-gap databases.The paper therefore motivates computationally affordable alternatives for accurate estimation.
- Approach: OLSR, LASSO, and SVR are evaluated with several predictor-set levels for estimating band-gaps from more affordable calculations and descriptors.Regression methods using GGA-level datasets and fundamental information about constituent elements and crystal structures had been applied to materials properties and band-gaps.
- Approach: The study constructs prediction models for G0W0 quasi-particle gaps of AX binary compounds using DFT Kohn-Sham gaps, elemental information, and crystal-structure predictors.The target is the G0W0 gap obtained with HSE06 rather than the experimental band-gap.
A. Regression methods
The study compares linear and nonlinear regression methods for relating target properties to predictors, emphasizing predictor selection and flexible nonlinear modeling.
- OLSR, LASSO, and nonlinear SVR are compared as regression methods for prediction models.The study frames reliability as dependent on training-set quality, predictor selection, and regression method.
- OLSR determines predictor coefficients by minimizing the squared residual norm over n observations.X is the n×p predictor matrix, and y is the n-dimensional target-property vector.
- LASSO combines linear regression with a sparse representation containing few nonzero regression coefficients.Its penalty uses the L1-norm, with λ controlling the trade-off between sparsity and accuracy.
- SVR uses the kernel trick to map features into a high-dimensional space and simplify nonlinear learning.The implementation uses a Gaussian kernel, with internal parameters optimized by minimizing cross-validation score.
B. Computational detail of first principles calculation
The calculations cover AX binary compounds across selected element families and four crystal structures, producing PBE, mBJ, and G0W0 band-gap data for modeling.
- 216 combinations of constituent elements and crystal structures were considered across wurtzite, cesium chloride, zincblende, and rocksalt structures.Transition-metal compounds were excluded, and the cation and anion sets are explicitly enumerated.
- 156 compounds with positive Eg (PBE) and Eg (mBJ) were selected from the 216 calculated compounds for regression.Eg (PBE) and Eg (mBJ) were calculated first, followed by G0W0 calculations for the selected dataset.
- Cohesive energy calculated with PBE was included as a predictor for Eg (G0W0).The dataset was composed of AX binary compounds from I-VII, II-VI, III-V, and IV-element families.
- First-principles calculations used the PAW method implemented in VASP, with 500 eV cutoffs for PBE and mBJ calculations.Structures were optimized using PBEsol until residual forces fell below 0.005 eV/Å, with structure-dependent Γ-centered meshes.
- G0W0 calculations used GW-updated pseudopotentials, a 600 eV cutoff, and reduced k-point meshes.The reduced meshes were specified separately for WZ, RS, ZB, and CC structures.
A. Comparison of theoretical band-gaps
The comparison shows that PBE and mBJ differ substantially from experimental band-gaps, while mBJ improves agreement and G0W0 provides a better description overall.
- Experimental band-gaps were available for 32 of the 216 AX binary compounds, limiting the experimental comparison.Differences between experimental groups can be scattered, especially for indirect band-gaps.
- PBE KS-gap differences from experiment average -2.11 eV for direct compounds and -1.38 eV for indirect compounds.The corresponding RMS differences are 2.45 eV and 1.77 eV, respectively.
- mBJ notably improves theoretical band-gaps relative to PBE, although it still tends to underestimate them.The remaining underestimation is explicitly noted for mBJ calculations.
- G0W0 calculations show a much better description of band-gaps in both mean value and RMS difference.The comparison is based on theoretical and experimental band-gaps for the available compounds.
- The authors state that the calculated results may be applicable for estimating real band-gaps.
B. Correction of band-gaps of PBE and mBJ
The correction models use PBE and mBJ Kohn-Sham gaps to predict G0W0 gaps, with combined predictors improving linear-model performance over single-gap predictors.
- B. Correction of band-gaps of PBE and mBJ: Eg (G0W0) shows almost linear dependence on Eg (PBE) and Eg (mBJ) in the plotted relationships.The models are constructed using OLSR from these Kohn-Sham gaps.
- B. Correction of band-gaps of PBE and mBJ: The evaluation randomly assigns three quarters of the dataset to training and the remainder to testing, repeating selection 100 times.Test RMSE, 10-fold CV score, and test MAPE are evaluated.
- B. Correction of band-gaps of PBE and mBJ: 0.52 eV and 0.57 eV RMSE are obtained by OLSR using Eg (PBE) and Eg (mBJ), respectively, as single predictors.
- B. Correction of band-gaps of PBE and mBJ: 0.36 eV RMSE is obtained when Eg (PBE) and Eg (mBJ) are used together as OLSR predictors.This is smaller than the RMSE values of the corresponding single-predictor models, with similar behavior for CV score and MAPE.
- B. Correction of band-gaps of PBE and mBJ: The physical reason for the improvement from two predictors is not clear, although the higher correlation offers a phenomenological explanation.
C. Comparison of different prediction models
The study compares OLSR, LASSO, and SVR across predictor sets that combine band-gaps with elemental, structural, and DFT-derived information. Adding predictors improves performance, with SVR achieving the lowest reported RMSE and no serious over-fitting indicated.
- Predictor construction: Predictor sets include Eg (PBE), Eg (mBJ), cohesive energy, crystalline volume per atom, elemental properties, and coordination number.Elemental candidates include ionic charge, period, atomic number, atomic mass, van der Waals radius, electronegativity, and first ionization energy.
- Predictor construction: Element-specific predictors are converted into symmetric sums, absolute differences, and products before candidate selection.Original, inverse, and symmetric forms are examined across three predictor sets, with a maximum of 41 predictors in set (3).
- Regression-model comparison: 0.36 eV is the OLSR RMSE when both Eg (PBE) and Eg (mBJ) are used as predictors.This result precedes the further improvement obtained with larger predictor sets.
- Regression-model comparison: 0.18 eV is the best RMSE achieved by SVR with predictor set (3).SVR performs better than OLSR and LASSO when the same predictor set is used.
- Validation: Training and test errors are similar, and RMSE and cross-validation scores are very close, indicating no serious over-fitting problem.The same tendency is observed for CV score and MAPE, while no notable outlier has averaged RMSE above 1 eV across all three models.
IV. Conclusion
The study develops models for predicting G0W0 band-gaps of 156 AX binary compounds from Kohn–Sham gaps and constituent and structural information. Combining Eg (mBJ) with broader predictors substantially reduces error, and the best SVR model reaches 0.18 eV RMSE without serious over-fitting.
- IV. Conclusion: 0.52 eV is the OLSR RMSE when Eg (PBE) is used as a single predictor on randomly selected test data.Adding Eg (mBJ) and varied elemental and structural predictors significantly decreases RMSE.
- IV. Conclusion: 0.18 eV is the RMSE of the best SVR model, with no serious over-fitting problem reported.The model uses the expanded predictor strategy described in the conclusion.
- IV. Conclusion: The established band-gap estimates should be useful as predictors for high-throughput materials screening.
Appendix A. The band-gaps of the list of materials which were used at this study.
The appendix lists materials, crystal structures, experimental gaps, gap type, and calculated PBE and G0W0 target gaps. It distinguishes direct, indirect, and metallic entries and notes that only positive band-gaps were used.
- Appendix A: The material records report crystal structure, experimental band-gap, gap type, Eg (PBE), and Eg (G0W0, Target).The listed structures include rocksalt, zinc-blende, wurtzite, and related entries shown in the appendix table.
- Appendix A: D, L, and M denote direct band-gap, indirect band-gap, and metallic materials, respectively.
- Appendix A: Only positive band-gaps were used for the study, while metallic materials were confirmed using PBE calculations rather than GW.
- Appendix A: The appendix includes literature sources for selected experimental values through references 37, 38, and 39.