Principal component analysis enables the design of deep learning potential precisely capturing LLZO phase transitions

The study addresses the need for accurate, transferable, and computationally efficient interatomic potentials to simulate complex solid-state electrolyte materials, particularly Li₇La₃Zr₂O₁₂ (LLZO), a garnet electrolyte with high Li-ion conductivity in its cubic phase but stability challenges at room temperature. Traditional DFT-based molecular dynamics is too costly for large-scale models, while constructing diverse, non-redundant training sets for machine-learning potentials remains difficult. The authors propose a deep learning interatomic potential for LLZO and introduce a convergence strategy based on principal component analysis (PCA) coverage between training and test (MD-generated) datasets. The objective is to obtain a potential that accurately reproduces energies, forces, structures, and dynamics, including LLZO phase transitions, while enabling large-scale, cost-effective simulations relevant to interfacial phenomena and dendrite growth in solid-state batteries.

Background on LLZO: LLZO exists in tetragonal and cubic phases; the cubic phase offers higher Li-ion conductivity yet is less stable at room temperature. Solid-state batteries with LLZO face interfacial contact issues, phase transitions, structural disorder, chemical segregation, and dendritic Li growth along grain boundaries, which increase interfacial impedance and can cause short circuits. Many of these microscale mechanisms are difficult to probe experimentally, motivating atomistic simulation. Machine-learning interatomic potentials: Building on the neural network potential concept by Behler and Parrinello (2007), subsequent models include GAP, MTP, GDML, and Deep Potential Molecular Dynamics (DeePMD). DeePMD-kit trains deep neural networks to reproduce ab initio energies, atomic forces, and virials with linear scaling in system size, enabling accurate and efficient MD. A critical challenge remains the efficient construction of compact yet comprehensive training sets that cover relevant atomic environments, especially in multi-component solid electrolytes with complex interfaces. The paper situates its contribution as a method to systematically expand and verify training set coverage using PCA, targeting robust performance across crystalline, amorphous, and interfacial LLZO configurations.

Dataset construction and iteration: - Initial training set composed of three parts: 1) Elemental and compound structures of Li, La, Zr, O from Materials Project and structures generated by scaling lattice constants across compositions and space groups (diverse structural motifs as LLZO building blocks). 2) First-principles molecular dynamics snapshots for LLZO crystals at 400, 800, 1200, 1600 K; amorphous LLZO generated by melting at 3000 K then cooling to 300 K; additional structures at 3000, 2000, 1000, 500, and 300 K to capture kinetic information of crystalline and amorphous phases. 3) Two-body potential data to constrain interatomic distances and handle cases with atoms too close or too far during MD, thereby reducing unphysical interactions. - Iterative refinement: • Train a preliminary DP on the initial set; run MD; extract local structural feature matrices via PCA from MD trajectories; compute coverage of test (MD-generated) structures by the training set. • If coverage not converged, perform DFT on MD trajectory structures; identify structures with DP–DFT energy error >1% and add them to the training set; retrain and repeat. • For amorphous validation, generate a 3×3×3 supercell (5184 atoms), melt and cool; sample 10 structures at 1000, 2000, 3000 K; partition each into 64 smaller blocks placed in a 40 Å cube (to avoid periodic interactions) for DFT energy/force evaluation and error-driven augmentation. PCA coverage calculation: - Construct standardized local-structure descriptor matrix X, compute covariance matrix, and obtain eigenvectors/eigenvalues; select m principal components such that cumulative contribution ≥95% (m < p). Project training and test sets into the reduced space T(n×m). - For coverage, decompose into 2D projections for all component pairs; discretize each plane into grids; mark grid occupancy for test (1/−1) and training (1/0) sets; compute T_cover = T_train × T_test and the coverage rate P = NUM(T_cover=1)/NUM(T_test=1). DFT settings: - VASP with PAW pseudopotentials; spin-polarized GGA-PBE exchange–correlation; Gaussian smearing 0.1 eV for relaxations; plane-wave cutoff 500 eV; k-point sampling via VASP k-spacing sk=0.25; convergence criteria reported as 10 eV for total energy and 0.1 eV/Å for forces. Deep potential training (DeePMD-kit): - Descriptor: se_e2_a (DeepPot-SE) using both angular and radial information; cutoff radius 6.0 Å with inverse-distance smoothing from 0.5 to 6.0 Å. - Networks: filter (10, 20, 40); fitting (120, 120, 120). - Training: 6,000,000 steps with Adam optimizer; initial learning rate 0.001, exponential decay (decay step 2000, decay rate 0.996). - Loss function includes MSE of energies and forces; energy prefactor decreases 0.02→1, force prefactor 1000→1; virial data not included. Validation analyses: - DP vs DFT comparisons for energies and forces across crystalline, amorphous, and slab LLZO datasets; RMSE tracked across iterations. - Structural validation via RDF comparisons between AIMD and DPMD for cubic (1200 K) and amorphous (3000 K) LLZO. - Phase transition simulations: NPT MD on 3×3×3 supercell of t-LLZO (5148 atoms); analyze lattice parameters, volume vs temperature, XRD patterns vs temperature, RDF vs temperature; PCA coverage of crystal and amorphous structures against final training set.

- Iterative training markedly improves training set coverage and accuracy: • Initial coverage of test by training set: 75.34%; after four iterations: 99.51% average coverage, indicating convergence. • Energy RMSE reductions (final potential): - Crystalline LLZO: 2.04 meV/atom (initial reported ~6.13 meV/atom) - Amorphous LLZO: 3.68 meV/atom (initial ~11.71 meV/atom) - Slab LLZO: 3.50 meV/atom (initial ~29.01 meV/atom) • Force RMSE (final potential): per component ~107 meV/Å (x), 109 meV/Å (y), 107 meV/Å (z) for crystal; ~202–204 meV/Å for amorphous; ~210–235 meV/Å for slab. - Error-driven data augmentation: using >1% DP–DFT energy-error criterion and four iterations reduced energy errors from eV/atom scale (first iteration for 1/64 amorphous blocks) to meV/atom scale; final error within 1%. - Structural dynamics fidelity: • RDFs from DPMD closely match AIMD for cubic LLZO at 1200 K and amorphous LLZO at 3000 K across Li–O, La–O, Zr–O, and O–O pairs. • Two-body analysis helps identify problematic short-range interactions and mitigate unphysical artifacts from close contacts. - Phase transition behavior captured: • Tetragonal-to-cubic transition temperature ≈900 K from NPT MD (close to experimental 923 K), with temperature-dependent XRD showing disappearance of tetragonal peaks and emergence of cubic peaks near 900 K. • Predicted thermal expansion coefficient for cubic LLZO between 1000–1500 K: 5.48 × 10⁻⁵ K⁻¹ (reported as similar to experimental 1.30 × 10⁻⁵ K⁻¹). • Large volume change between 1900–2100 K accompanied by randomization of lattice parameters and RDF signatures indicative of melting to liquid/amorphous LLZO. - PCA-based coverage correlates with error reductions, supporting its use as a convergence and sufficiency metric for training datasets.

The work demonstrates that using PCA-derived coverage between test (MD-generated) and training sets as an iterative convergence criterion yields a compact yet comprehensive dataset for training deep interatomic potentials in complex multi-component systems. This strategy directly addresses the challenge of ensuring sufficient sampling of diverse atomic environments without excessive redundancy. The resulting DP accurately reproduces DFT energies and forces across crystalline, amorphous, and slab LLZO, and closely matches AIMD RDFs, indicating fidelity in structural and dynamical properties. Importantly, the model captures a key materials behavior—the tetragonal-to-cubic phase transition near 900 K—and reproduces temperature-dependent XRD features and thermal expansion trends, thus validating its predictive power for phase behavior. The approach enhances confidence in the transferability of potentials to large-scale simulations by quantifying the similarity (coverage) of large-system local environments to those in the training data via PCA. Overall, the findings confirm that PCA-based coverage is a practical and reliable metric to guide data augmentation and assess convergence. This improves the robustness of machine-learning potentials applied to solid-state electrolytes, enabling efficient exploration of interfacial processes and phase transformations that are otherwise challenging to probe experimentally.

By combining a diverse training dataset (databases, first-principles MD snapshots across phases and temperatures, and two-body interaction data) with an iterative, PCA-based coverage convergence strategy, the authors developed a deep learning interatomic potential for LLZO that achieves DFT-level accuracy for energies, forces, and structural dynamics at a fraction of the computational cost. The model accurately reproduces RDFs, predicts the tetragonal-to-cubic transition temperature (~900 K) consistent with experiments, and yields a reasonable thermal expansion coefficient for cubic LLZO. The PCA coverage metric proves effective for assessing training set sufficiency and guiding data augmentation, thereby enhancing potential transferability to large-scale simulations. Potential future directions include extending the coverage-guided training approach to other complex solid-state electrolytes and interfaces, incorporating additional observables (e.g., virials/stress) to improve pressure- and stress-dependent behavior, and applying the method to study interfacial transport and dendrite phenomena in realistic battery architectures.

- Transferability concerns from small, DFT-trained systems to large-scale simulations are acknowledged; the study mitigates this via PCA coverage analysis but ultimate generalizability across unseen chemistries and defect landscapes may require further validation. - The training did not include virial/stress data, which may limit accuracy for properties sensitive to pressure/stress or elastic responses. - Thermal expansion coefficient differs from experimental value (predicted 5.48×10⁻⁵ K⁻¹ vs experimental 1.30×10⁻⁵ K⁻¹), indicating potential quantitative discrepancies in some thermophysical properties. - Two-body potential augmentation was needed to handle close-contact artifacts, suggesting that short-range interactions can be challenging and may require careful curation. - Reported DFT convergence settings include a relatively loose energy convergence (10 eV as stated), which could affect the reference data quality if not a typographical issue.