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

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.

💡 Research Summary

This paper presents a comprehensive framework that combines polynomial machine‑learning potentials (MLPs) with stochastic self‑consistent harmonic approximation (SSCHA) to perform global crystal‑structure searches and finite‑temperature phase‑stability assessments under extreme pressure and temperature conditions. The authors focus on elemental silicon as a test case, exploring pressures up to 100 GPa and temperatures up to 1000 K.

The methodological core is a polynomial MLP built from rotationally invariant descriptors derived from radial functions and spherical harmonics. By constructing p‑order polynomial invariants of the neighbor density, the model captures many‑body interactions while remaining linear in its parameters, allowing efficient training via ordinary least‑squares regression. To increase expressive power, the authors introduce a hybrid MLP architecture that sums two independent polynomial models with different cutoff radii, maximum angular momentum, and polynomial order. This hybridization retains the computational speed of linear regression but broadens the feature space, improving accuracy for both short‑range and longer‑range interactions.

A critical bottleneck for high‑pressure applications is the availability of representative training data. The authors therefore augment an existing dataset (≈12 k structures optimized at 0 GPa) with a new high‑pressure dataset. They re‑optimize 86 prototype structures at 25, 50, 75, and 100 GPa, generate supercells, and apply random lattice expansions, distortions, and atomic displacements. After a structure‑selection procedure, 4 k high‑pressure configurations are added, yielding a combined dataset of ≈16 k structures (dataset 2). All DFT calculations use the PBE functional within the PAW formalism, a 400 eV plane‑wave cutoff, and a k‑point spacing of ~0.09 Å⁻¹, ensuring high‑quality reference energies, forces, and stresses.

Training on dataset 2, the hybrid MLP achieves markedly lower root‑mean‑square errors (RMSE) in energy, forces, and stress compared with a single‑model baseline, especially in the 50–100 GPa range where the baseline previously struggled. The model’s robustness across a wide pressure window enables its use in a Random Structure Search (RSS) workflow. The authors iteratively combine RSS (which generates random crystal candidates) with MLP‑based energy/force evaluations, feeding newly discovered low‑energy structures back into the training set. This loop converges to a set of global minima and numerous metastable structures for each pressure.

To assess finite‑temperature stability, the authors perform SSCHA calculations on all local minima obtained from the RSS. SSCHA incorporates anharmonic phonon effects by sampling stochastic atomic displacements and self‑consistently updating the harmonic reference potential. By leveraging the MLP for rapid force evaluations, the authors make SSCHA tractable for thousands of structures, even those with low symmetry that would be prohibitive with direct DFT. The resulting free‑energy landscape yields a pressure‑temperature phase diagram that reproduces all experimentally known silicon polymorphs (diamond‑Si, β‑Sn, Imma, simple hexagonal, Cmce, HCP, FCC) and predicts several new metastable candidates. Notably, the SSCHA reveals temperature‑induced dynamical stabilization of structures that are unstable at 0 K, illustrating the importance of anharmonicity at high temperature.

The complete workflow—(1) generation of a pressure‑diverse DFT dataset, (2) training of a hybrid polynomial MLP, (3) RSS‑driven global structure search, and (4) SSCHA‑based free‑energy evaluation—delivers high accuracy and efficiency. The authors benchmark computational cost, showing that the MLP‑accelerated SSCHA reduces wall‑time by roughly an order of magnitude relative to pure DFT, while maintaining sub‑meV/atom energy fidelity.

In conclusion, the study demonstrates that polynomial MLPs, especially when combined in a hybrid fashion, can serve as reliable surrogates for DFT across a broad pressure‑temperature space. Coupled with SSCHA, they enable systematic, high‑throughput prediction of phase diagrams for materials under extreme conditions. The authors suggest that the framework is readily extensible to multicomponent systems, complex alloys, and even liquid or amorphous phases, opening a pathway for large‑scale computational discovery of high‑pressure materials.