Machine Learning Prediction of Charged Defect Formation Energies from Crystal Structures

Recent advances in materials informatics have expanded the number of synthesizable materials. However, screening promising candidates, such as semiconductors, based on defect properties remains challenging. This is primarily due to the lack of a general framework for predicting defect formation energies in multiple charge states from structural data. In this Letter, we present a protocol, namely data normalization, Fermi level alignment, and treatment of perturbed host states, and validate it by accurately predicting oxygen vacancy formation energies in three charge states using a single model. We also introduce a joint machine-learning model that integrates defect formation energies and band-edge predictions for virtual screening. Using this framework, we identify 89 hole-dopable oxides, including BaGaSbO, a potential ambipolar photovoltaic material. Our protocol is expected to become a standard approach for machine-learning studies on point defect formation energies.

💡 Research Summary

This paper addresses a critical bottleneck in computational materials discovery: the ability to predict point‑defect formation energies across multiple charge states using only crystal‑structure information. The authors focus on oxygen vacancies (V_O) in oxides as a test case and develop a comprehensive workflow that combines three key preprocessing steps—Fermi‑level alignment, data normalization, and removal of perturbed‑host‑state (PHS) defects—with a single crystal‑graph convolutional neural network (CGCNN) model.

First, they align the Fermi level of all compounds to a common reference based on core potentials (using ZnO’s VBM as the reference). By shifting the formation energies of each charge state (q = 0, +1, +2) and selecting the reference Fermi level that minimizes the mean‑difference Δ between the charge‑state distributions, the authors ensure that the target values overlap substantially. This alignment reduces the variance that would otherwise hinder gradient‑based learning.

Second, after alignment, they standardize the formation‑energy values (zero mean, unit variance) to improve training stability. Third, they identify and exclude defects that exhibit perturbed host states—defects whose occupied levels lie above the conduction‑band minimum, leading to shallow, delocalized states that are poorly described in finite supercells. By removing these PHS entries (which constitute about 10 % of the data), the model’s mean absolute error (MAE) improves by 0.02–0.03 eV, confirming that the noisy PHS data degrade predictive performance.

The CGCNN architecture encodes atomic species and bond lengths as graph nodes and edges, pools features at the oxygen site where the vacancy is created, and concatenates the explicit charge state q before passing the vector through fully connected layers. This design enables a single network to predict formation energies for all three charge states simultaneously, avoiding the need for separate models for each q. On an independent test set, the model achieves MAEs of 0.29 eV (neutral), 0.22 eV (+1), and 0.37 eV (+2), outperforming the authors’ previous random‑forest approach that required additional first‑principles descriptors.

Recognizing that defect formation energies depend linearly on the Fermi level, the authors also train a separate CGCNN model to predict valence‑band maxima (VBM) from unit‑cell structures. By feeding the VBM predictions into the defect‑energy model—a “joint model”—they can evaluate formation energies at the material‑specific VBM without performing separate DFT calculations. The joint model yields modest but consistent accuracy gains (0.02–0.07 eV) over a model trained directly on VBM‑aligned data, and it benefits from the much larger VBM datasets available in public repositories such as the Materials Project.

Armed with this joint model, the authors conduct a high‑throughput virtual screening of 1,809 theoretically stable oxides (sourced from recent stability studies). They first filter out compounds where oxygen vacancies act as “hole killers” (i.e., where V_O is stable in a negative charge state with low formation energy under O‑rich conditions). Only 89 oxides (≈5 % of the initial set) survive this filter, highlighting the intrinsic difficulty of finding p‑type oxides. After excluding toxic or expensive elements, 32 candidates are subjected to more rigorous evaluation: effective masses, dielectric constants, and optical absorption spectra are computed using dielectric‑dependent hybrid (DDH) functionals, which reliably predict band gaps.

Among these, BaGaSbO emerges as a standout candidate. It exhibits exceptionally low electron and hole effective masses (0.34 m₀ and 0.19 m₀ in the ab‑plane), a steep optical absorption onset, and an almost direct band gap (the indirect–direct difference is <0.1 eV). Hybrid‑functional defect calculations confirm that oxygen vacancies remain neutral at the VBM, meaning they do not compensate holes, and that both n‑type (via Mg or Zn doping) and p‑type (via La or Y doping) behavior is achievable—making BaGaSbO a promising ambipolar photovoltaic material.

The paper concludes by emphasizing four contributions: (1) a protocol for normalizing defect formation energies across charge states; (2) a single, structure‑only CGCNN capable of multi‑charge prediction with MAEs below 0.4 eV; (3) a joint model that couples defect‑energy and band‑edge predictions to enable rapid screening; and (4) a demonstration of large‑scale virtual screening that identifies 89 hole‑dopable oxides, with BaGaSbO highlighted for solar‑energy applications. Limitations are acknowledged: the current dataset focuses on oxygen vacancies, PHS defects are excluded rather than modeled, and transition‑level predictions remain qualitative. Future work should expand the defect database to include a broader variety of point defects, develop methods to treat PHS quantitatively, and integrate more accurate transition‑level predictions. Overall, the study provides a practical, scalable framework that can become a standard tool for machine‑learning‑driven defect engineering in semiconductors and related materials.