Comparison of elastic constants and surface energies of β-Sn from density functional theory, universal machine learning potential, and empirical potentials
Abstract
The full elastic constant tensor and the surface energies of five low-index planes—(100), (101), (110), (111), and (001)—of β-Sn (tetragonal, I41/amd) are calculated using density functional theory (DFT) with the Perdew–Burke–Ernzerhof (PBE) functional and a universal machine-learning interatomic potential (MLIP; Preferred Potential, PFP v8), which provides both PBE and regularized-restored strongly constrained and appropriately normed (r2SCAN) calculation modes, each with and without Grimme’s D3 dispersion correction. Results are compared with three modified embedded-atom method (MEAM) potentials and experimental data. Both DFT and PFP resolve the long-standing order-of-magnitude C44 deficiency of MEAM potentials (1.5–7.9 GPa against the experimental 22.0 GPa), yielding C44 = 17.9–34.3 GPa. In contrast, C12 is systematically underestimated by all first-principles-based methods (15.2–41.5 GPa against 59.4 GPa) regardless of the choice of exchange–correlation functional, indicating a limitation of current functionals rather than an artifact of MLIP training. For surface energies, DFT/PBE predicts (100) to be the lowest-energy plane of β-Sn, and six of the seven non-DFT methods reproduce this. The equilibrium Wulff shapes of all methods except one MEAM potential retain a rounded polyhedral character qualitatively similar to that of DFT/PBE. Among the non-DFT methods, PFP/r2SCAN reproduces the DFT/PBE surface energies most closely. No single method simultaneously reproduces both the full elastic constant tensor and the surface-energy anisotropy of β-Sn, and the choice of computational method should be guided by the specific phenomenon of interest; among the methods tested, PFP/PBE + D3 offers the smallest mean absolute percentage error (MAPE) of 25.1% for the elastic constants with respect to experiment, while PFP/r2SCAN gives the smallest MAPE of 5.0% for the surface energies with respect to DFT/PBE.