Machine learning and evolutionary prediction of superhard B-C-N compounds

Superhard materials, exhibiting Vickers hardness (H) ≥ 40 GPa, are crucial for applications like abrasives and cutting tools. While diamond is the hardest known material, its limitations in oxidizing conditions and high-speed machining of ferrous alloys necessitate the search for alternatives. Light elements such as B, C, N, and O offer promise due to their ability to form multiple short covalent bonds, resulting in strong, hard structures. Cubic boron nitride (c-BN) and various boron carbides exemplify this, with reported hardness values ranging from 50–70 GPa for c-BN and varying significantly for boron carbides depending on the boron content and structure. Several superhard B-C-N ternary compounds have also been identified, yet exploring the vast phase space remains challenging. First-principles simulations based on DFT are powerful for predicting superhard compounds, but remain computationally expensive. Data-driven approaches, leveraging machine learning, offer a more efficient alternative. Previous studies have successfully used machine learning to predict material properties, often incorporating features like cohesive energy and volume per atom; however, obtaining these features for novel compounds requires additional calculations or measurements. This work focuses on developing a model that uses only features directly derivable from the chemical formula, enabling efficient large-scale prediction of superhard materials.

Existing literature demonstrates successful application of machine learning in materials science, particularly for predicting elastic properties. Studies like Meredig et al.'s work on screening ternary compositions and de Jong et al.'s gradient boosting technique using features such as volume per atom showcase the power of this approach. However, these often rely on computationally expensive or experimentally challenging-to-obtain parameters. This study addresses this limitation by using only features readily derived from chemical formulae.

This research employs a data-driven approach using random forests models to predict mechanical properties of materials from their chemical formulae. The methodology involves several key steps: 1. **Data Acquisition:** The Materials Project database was used to obtain DFT-calculated bulk (K) and shear (G) moduli for over 10,000 compounds. Data filtering criteria were applied to exclude thermodynamically unfavorable compounds (formation energy ≥ 0.2 eV per atom), layered materials (Voigt and Reuss modulus difference > 50 GPa), and those with extreme Pugh's ratios (G/K < 0.25 or > 4.0). 2. **Feature Generation:** Sixty features were generated from each compound's chemical formula, categorized as: (a) stoichiometric attributes (Lᵖ norms); (b) elemental properties (atomic number, mass, etc.); (c) orbital occupations (s, p, d, f electrons); and (d) ionic bonding levels (maximum and mean ionic character). The Matminer Python library aided in feature generation. 3. **Model Training and Validation:** Random forests were used for regression tasks to predict K and G. Ninety percent of the dataset was used for training and validation, with tenfold cross-validation to determine optimal tree depth. The remaining 10% served as a test set for unbiased evaluation using the Pearson correlation coefficient (r). 4. **Prediction and Structure Search:** The trained models were applied to a series of B-C-N compositions to predict their mechanical properties. Promising compositions with predicted hardness > 40 GPa underwent further investigation using evolutionary structure prediction (USPEX) and DFT calculations using VASP to validate the predicted properties and structural stability. 5. **DFT Calculations:** DFT calculations, employing the VASP software, were used to verify the structural, electronic, and phonon properties of selected B-C-N compounds. Elastic constants, phonon dispersion spectra, and electronic band structures were determined. The strain-stress method was used to compute elastic constants, and the PHONOPY package was used for phonon calculations.

The random forests models demonstrated high prediction accuracy for bulk (K) and shear (G) moduli (r values of 0.940 and 0.907 respectively), allowing for reliable prediction of hardness using Tian's empirical model (although with a lower r value of ~0.79). Feature importance analysis indicated that atomic radius and d electron occupation were strongly correlated with bulk modulus. Triangular graphs for B-C-N compounds revealed that a 1:1 B:N ratio resulted in several superhard compositions. BC₂N and BC₄N showed predicted hardness values consistent with experimental findings. Evolutionary structure prediction identified B₂C₃N as a dynamically stable superhard phase. Further analysis using the atomic substitution rule in diamond unit cell led to the discovery of two additional promising candidates: BC₁₀N and B₄C₅N₃. DFT calculations confirmed the superhard nature (hardness > 40 GPa) of BC₁₀N, B₄C₅N₃, and B₂C₃N. BC₁₀N exhibited a hardness of 87 GPa, comparable to diamond, and a wide band gap, indicating insulating behavior. B₄C₅N₃ and B₂C₃N were found to be metallic. All three compounds were dynamically stable, as indicated by the absence of negative phonon modes. Analysis of formation energies suggested that BC₁₀N, with a relatively low formation energy, could be synthesized using low-temperature plasma methods.

This study successfully combines machine learning and evolutionary structure prediction to identify novel superhard materials. The high accuracy of the machine learning model, utilizing only readily available features, makes it a powerful tool for high-throughput screening of materials. The validation through DFT calculations confirms the potential of these predicted compounds for practical applications. The discovery of BC₁₀N, a superhard insulator with a hardness comparable to diamond but with potentially superior performance in higher temperature and humidity environments, is a significant finding. The metallic nature of B₄C₅N₃ and B₂C₃N raises the possibility of exploring superconductivity in these materials.

This work successfully demonstrated the effectiveness of combining machine learning and first-principles calculations for discovering superhard materials. The identification of BC₁₀N, B₄C₅N₃, and B₂C₃N as potential superhard materials with distinct electronic properties is a major contribution. Future work should focus on experimental synthesis of these compounds and further exploration of their potential applications in extreme environments. The methodology presented here can be expanded to explore other material classes and properties.

The accuracy of the hardness prediction relies on the empirical Tian's model, which might have limitations, particularly in regions with low bulk and shear moduli. The study focused on dynamically stable structures at ambient pressure, and high-pressure phases were not considered. Experimental verification is necessary to confirm the synthesizability and exact properties of the predicted compounds. The model's performance might be improved by incorporating additional features, but this would reduce the efficiency of high-throughput screening.