Global structure searches under varying temperatures and pressures using polynomial machine learning potentials: A case study on silicon

Abstract

Polynomial machine learning potentials (MLPs) based on polynomial rotational invariants have been systematically developed for various systems and applied to efficiently predict crystal structures. In this study, we propose a robust methodology founded on polynomial MLPs to comprehensively enumerate crystal structures under high-pressure conditions and to evaluate their phase stability at finite temperatures. The proposed approach involves constructing polynomial MLPs with high predictive accuracy across a broad range of pressures, conducting reliable global structure searches, and performing exhaustive self-consistent phonon calculations. We demonstrate the effectiveness of this approach by examining elemental silicon at pressures up to 100 GPa and temperatures up to 1000 K, revealing stable phases across these conditions. The framework established in this study offers a powerful strategy for predicting crystal structures and phase stability under high-pressure and finite-temperature conditions.

Figures (18)

• Figure 1: Overview of the current procedure in this study. The dashed boxes and corresponding section numbers indicate the sections in which the respective topics are discussed.
• Figure 2: (a) Volume distributions for the structures in datasets 1 and 2, displayed on a logarithmic scale for better visibility. (b) Distributions of the coordination numbers around atoms in the optimized prototype structures at 0, 25, 50, 75, and 100 GPa. The distributions for structures optimized under higher pressures are shown with darker lines.
• Figure 3: (a) Distribution of the cohesive energy for all prototype structures optimized at pressures ranging from 0 to 100 GPa. Distributions of (b) cohesive energy values, (c) forces, and (d) stress tensor components for dataset 2 are presented. They are calculated using the optimal MLP. The numerical values enclosed in squares represent the RMSEs for energy, force, and stress tensor components, expressed in units of meV/atom, eV/Å, and meV/atom, respectively. In panels (b), (c), and (d), the RMSEs are computed using the test dataset.
• Figure 4: (a) Distributions of $E + \bar{\sigma}V$ for local minimum structures, computed using both the DFT calculation and the polynomial MLP, at the first iteration of the iterative MLP update procedure. (b) Distributions of $E + \bar{\sigma}V$ for local minimum structures, computed using the DFT calculation and the polynomial MLP, at the final iteration. For comparison, distributions computed using various interatomic potentials are also shown, including the MEAM potential MEAM_potential, the Tersoff potential Tersoff_potential, the quadratic SNAP Zuo2020, and the GAP PhysRevX.8.041048. The local minimum structures used in these evaluations are obtained from the RSS with the polynomial MLP at the final iteration. In the figure, “Poly. MLP” and “Quad. SNAP” denote the polynomial MLP and the quadratic SNAP, respectively.
• Figure 5: (a) Relative enthalpy values of local minimum structures computed using the MLP for elemental Si. In the left and central panels, the maximum values of the vertical axis are set to 1.0 and 0.1 eV/atom, respectively. The orange open squares represent structures with relative enthalpy values below 30 meV/atom, as computed by the MLP. (b) Relative enthalpy values of the structures indicated by the orange open squares in (a), obtained through geometry optimizations using DFT calculations.