Modern materials science focuses on intelligently designing materials with specific properties. Two-dimensional (2D) materials offer significant potential due to their susceptibility to property alteration through controlled defect introduction. However, modeling these designer materials is challenging. While density functional theory (DFT) can calculate individual defect properties, it isn't scalable for large-scale exploration. The research utilizes machine learning to overcome this limitation. Existing material databases are limited in their applicability to defect properties. This paper presents a new database for 2D materials and their defects, analyzing the electronic properties of defects in MoS2 as a case study. The database facilitates analysis using machine learning methods, promising to accelerate materials discovery and design. Defects significantly modify the mechanical, thermal, electronic, optical, and other properties of solids, finding applications in areas like single-photon emitters, qubits, and catalysis. The ability to controllably engineer defects requires understanding the structure-property relationship, a complex task given the vast possibilities of host materials and defect configurations. Existing computational databases like Material Projects and OQMD, along with machine learning methods such as graph neural networks, are valuable but haven't sufficiently addressed the challenge of predicting defect properties. This is partly due to limited defect datasets and the difficulty of predicting quantum states. The current study aims to address this gap by providing a comprehensive dataset to improve machine learning models for defect prediction.

The introduction thoroughly reviews existing computational material databases such as Material Projects, OQMD, AFLOWLIB, and NOMAD, highlighting their contributions to materials discovery. It also discusses various machine-learning approaches used in materials science, including graph neural networks (e.g., MEGNet, CGNN, SchNet, GemNet). The limitations of applying machine learning to predict defect properties are discussed, emphasizing the lack of suitable datasets and the challenges in predicting quantum states and nonlinear quantum properties. Finally, the review summarizes previous studies on machine learning applied to defects in solids, focusing on the prediction of point defect properties in 2D materials, vacancy migration and formation energies in alloys, and defect dynamics in 2D TMDCs. The authors also mention existing work by Fabian et al. on a quantum point defect database, noting that despite its comprehensiveness, the size and data density remain limited for efficient AI prediction. The need for a larger, more comprehensive dataset of defect properties is thus established as crucial for both materials engineering and machine-learning model improvement.

The researchers created two types of datasets: structured and dispersive. The structured dataset consists of DFT-computed properties of 5933 defect configurations for MoS2 and 5933 for WSe2, focusing on single, double, and triple-site defects with various components (vacancies and substitutions). This detailed dataset provides fine-tuned features correlated with crystal lattices and quantum mechanical properties. The dispersive dataset aims for broader coverage of defect space by calculating properties across a wide concentration range of defects in several 2D materials (MoS2, WSe2, hBN, GaSe, InSe, and black phosphorus), each with 500 configurations. This approach, though leading to sparser data, allows for the development of more universal machine learning algorithms. Density functional theory (DFT) calculations using the PBE functional were performed with the Vienna Ab Initio Simulation Package (VASP), employing the projector-augmented wave (PAW) method. A plane-wave energy cutoff of 500 eV was used, and spin polarization was included. The supercell sizes and computational parameters are detailed in Supplementary materials. Formation energy was calculated using the standard formula (E_D - E_pristine - Σn_iμ_i), and interaction energy for defect complexes was calculated (E_D - ΣE_i). Electronic properties (HOMO, LUMO, and band gap) were also characterized. The authors explain their choice to use the deepest Kohn-Sham orbital as a reference due to the finite cell effect and its impact on the calculated valence band edge.

The structured dataset revealed a correlation between band gap and formation energy: band gaps decrease with increasing formation energy, converging near 0.3 eV. This suggests that achieving deep defect levels requires higher formation energy. The property maps (band gap vs. formation energy) show distinct characteristics for different 2D materials, serving as potential material fingerprints. The study also found that defects with unpaired electrons were present in GaSe, InSe, BP, and C-doped hBN, showing an asymmetric distribution of band gaps. For V2 defects (one Mo vacancy and one S vacancy), the interaction energy, HOMO, and LUMO showed oscillatory behavior with the distance between vacancies. This oscillation is attributed to the resonance of the electron wave function with the honeycomb lattice. A simplified two-orbital model is used to explain the observed quantum fluctuations in the interaction energy and the HOMO and LUMO levels. The model considers the coupling between localized defect orbitals at lattice sites A and B, and explains that the stabilization energy arises from the competition between exchange and direct Coulomb interactions. This model successfully explains the oscillatory behavior observed in the interaction energy and the localization of the wave functions at different sites. Detailed wave function plots for different configurations of the V2 defect are provided to visually demonstrate the effects of hybridization on the HOMO energy levels.

The results demonstrate that properly structured datasets can unveil complex structure-property correlations, providing guidance for materials engineering. The identified trends in the band gap versus formation energy plots offer insights into manipulating defects in TMDCs for targeted properties. The distinct property distribution maps for different 2D materials could be useful in identifying materials with desired properties and acting as fingerprints for materials characterization. The detailed study of V2 defects highlights the importance of considering quantum mechanical effects in understanding defect interactions. This study's database provides a platform for training and validating machine learning models to predict defect properties, thereby accelerating materials design and discovery. The oscillatory behavior of the interaction energy, HOMO, and LUMO energies as a function of the distance between the defects highlights the complex quantum nature of the interactions involved.

This research successfully created a comprehensive database of defect properties in 2D materials. The analysis revealed crucial structure-property correlations that can be used to guide materials design. The creation of both structured and dispersive datasets is valuable for training different types of machine learning models. Future work could involve expanding the database to include more materials and defect types, exploring different DFT functionals, and incorporating advanced machine-learning techniques to refine prediction models for more accurate materials design.

The study primarily uses the PBE functional for DFT calculations, which is known to underestimate band gaps. While the authors argue that the general trends observed should be transferable to results from other methods, using more advanced hybrid functionals or many-body methods could improve accuracy. The high computational cost of these methods limited their use in this high-throughput study. Further, the current database is limited to a subset of 2D materials and defect types; expanding this database will be essential for more comprehensive analysis and model training.