Machine learned potential for defected single layer hexagonal boron nitride

Development of machine learned interatomic potentials (MLIP) is critical for performing reliable simulations of materials at length and time scales that are comparable to those in the laboratory. We present here a MLIP suitable for simulations of the temperature dependent structure and dynamics of single layer hexagonal boron nitride (h-BN) with defects and grain boundaries, developed using a strictly local equivariant deep neural network as formulated in the Allegro code. The training dataset consisted of about 30,000 images of h-BN with and without point defects generated with ab-initio molecular dynamics simulations, based on density functional theory (DFT), at 500, 1000, and 1500K. The developed MLIP predicts potential energies and forces with a mean absolute error (MAE) of 4 meV/atom and 60 meV/Angstrom , respectively. It also reproduces phonon dispersion curves and density of vibrational states of pristine bulk h-BN that are comparable with that obtained from density functional theory-based calculations. Molecular dynamics simulations of the motion of the 4|8 grain boundary unit in h-BN shows the first step to have an activation barrier ~2.2 eV, indicating immobility of the grain boundary. Moving the grain boundary units past the first shows much lower activation barriers of ~0.42eV, suggesting a facile motion of the grain boundary once the first movement is stimulated. These simulations yield a scaled mobility of 1.73910^(-11) m^3/Js for a temperature of 1500K which, given the inherent differences in the set-ups, is not too far from the experimental value of 1.3610^(-9) m^3/Js. The ability to predict grain boundary mobility within reasonable agreement with experiment demonstrates the robustness of the MLIP and its suitability for reliable simulations of defect structures and dynamics in single layer h-BN.

💡 Research Summary

The paper presents a machine‑learned interatomic potential (MLIP) specifically tailored for single‑layer hexagonal boron nitride (h‑BN) containing point defects and grain boundaries. Using the Allegro framework—a strictly local, E³‑equivariant deep neural network—the authors construct a potential that respects translational, rotational, and reflection symmetries, thereby embedding fundamental physical invariances directly into the model architecture.

A comprehensive training dataset was generated from ab‑initio molecular dynamics (AIMD) simulations based on density functional theory (DFT). Approximately 30 000 configurations were collected from 6 × 6 supercells (72 atoms, 71 for vacancy cases) that include six distinct point defects (180° rotated BN pair, Stone‑Wales, B→N substitution, N→B substitution, nitrogen vacancy, boron vacancy) as well as pristine sheets. Simulations were performed at 500 K, 1000 K, and 1500 K in the microcanonical ensemble, with additional ±5 % strain and a range of lattice constants (2.45–2.75 Å) to broaden the configurational space.

Training employed a two‑body latent MLP (2 layers, 32 nodes, SiLU activation), a single‑layer latent MLP, and an 8‑function Bessel distance encoding (ℓ_max = 48). The model was optimized with Adam (learning rate 0.01, batch size 50) and early‑stopping criteria based on validation energy and force MAEs. The final potential achieves a mean absolute error of 4 meV per atom for energies and 60 meV Å⁻¹ for forces, comparable to or better than existing MLIPs for similar systems.

Validation was carried out on two fronts. First, phonon dispersion curves of pristine h‑BN were computed using the MLIP and compared with DFT (VASP) results. The acoustic branches match almost perfectly, and the optical branches show only minor shifts, a behavior consistent with other high‑accuracy MLIPs and markedly better than empirical force fields. Second, the relaxed geometries of all six point defects and a representative grain‑boundary structure were compared between MLIP and DFT. Interatomic distances agree within a mean absolute deviation of 0.024 Å, and bond lengths/angles around defects are reproduced with high fidelity, demonstrating that the potential captures both local and extended environments (up to ~20 Å) despite a 6 Å cutoff.

The authors then applied the MLIP to study the dynamics of a 4|8 grain‑boundary unit in h‑BN. Direct molecular dynamics at 1500 K revealed a two‑stage migration mechanism: the initial step requires an activation barrier of ~2.2 eV, indicating that spontaneous motion is unlikely; once this barrier is overcome, subsequent steps face a much lower barrier of ~0.42 eV, allowing facile propagation of the boundary. From these simulations, a scaled grain‑boundary mobility of 1.74 × 10⁻¹¹ m³ J⁻¹ s⁻¹ was extracted. Although this value is two orders of magnitude lower than the experimental mobility of 1.36 × 10⁻⁹ m³ J⁻¹ s⁻¹, the discrepancy is reasonable given differences in system size, temperature control, and boundary conditions.

In summary, the work delivers (1) a rigorously symmetry‑preserving deep‑learning potential for defected h‑BN, (2) a rich, diverse training set that spans pristine, strained, and multiple defect configurations, (3) thorough validation against DFT phonons and defect geometries, and (4) a quantitative prediction of grain‑boundary mobility that aligns qualitatively with experimental observations. This MLIP enables large‑scale, long‑time atomistic simulations of 2D materials with realistic defect populations, opening avenues for predictive modeling of mechanical, thermal, and catalytic properties in h‑BN and related layered systems.