Machine-Learned Interatomic Potentials for Structural and Defect Properties of YBa$_2$Cu$_3$O$_{7-δ}$

High-Temperature Superconductors (HTS) such as YBa2Cu3O7-delta (YBCO) are essential for next-generation Tokamak fusion reactors, where Rare-Earth Barium Copper Oxides (REBCO) form the functional layers in HTS magnets. Because YBCO’s superconductivity depends strongly on oxygen stoichiometry and defect structure, atomistic simulations can provide crucial insight into radiation-damage mechanisms and pathways to maintain material performance. In this work, we develop and benchmark four Machine-Learned Interatomic Potentials (MLPs) for YBCO: the Atomic Cluster Expansion (ACE), the Message-Passing Atomic Cluster Expansion (MACE), the Gaussian Approximation Potential (GAP), and the Tabulated Gaussian Approximation Potential (tabGAP), trained on an extensive Density Functional Theory (DFT) database explicitly designed to include irradiation-damaged-like configurations. The resulting models achieve DFT-level accuracy across a wide range of atomic environments, faithfully capturing the interatomic forces relevant to radiation damage processes. Among the tested models, MACE delivers the highest accuracy, although at greater computational cost, while ACE and tabGAP provide an excellent balance between efficiency and fidelity. These machine-learned potentials establish a robust foundation for large-scale molecular dynamics simulations of radiation-induced defect evolution in complex superconducting materials

💡 Research Summary

This paper presents the development and comprehensive benchmarking of four machine‑learned interatomic potentials (MLPs) – Atomic Cluster Expansion (ACE), Message‑Passing Atomic Cluster Expansion (MACE), Gaussian Approximation Potential (GAP), and its tabulated variant (tabGAP) – specifically tailored for the high‑temperature superconductor YBa₂Cu₃O₇₋δ (YBCO). The authors motivate the work by emphasizing the critical role of YBCO in REBCO‑based magnets for next‑generation tokamak fusion reactors, where the superconducting performance is highly sensitive to oxygen stoichiometry and radiation‑induced defects. Conventional density‑functional theory (DFT) provides accurate energetics but is limited to small length and time scales, making it unsuitable for simulating collision cascades and long‑range defect evolution.

To overcome this limitation, the authors constructed an extensive DFT reference database using CP2K with the PBE‑GGA functional, DZVP basis set, and GTH pseudopotentials. The dataset deliberately spans (i) equilibrium orthorhombic structures, (ii) a range of oxygen vacancy concentrations (δ = 0 – 1), (iii) highly non‑equilibrium configurations generated by random atomic displacements and simulated displacement cascades, and (iv) transition‑state geometries obtained via nudged elastic band (NEB) calculations for oxygen migration. All structures were relaxed to within 1 meV/atom energy convergence, and forces, stresses, and total energies were recorded for training.

Four distinct MLP architectures were then trained on the same dataset:

1. ACE – a linear‑scaling many‑body expansion that combines Bessel radial functions with spherical harmonics, allowing inclusion of high‑order angular correlations while keeping computational cost proportional to the number of neighbors.

2. MACE – an extension of ACE that incorporates equivariant graph‑neural‑network message passing. By constructing tensor‑product features up to four‑body order and preserving E(3) symmetry, MACE achieves superior expressivity with only one or two message‑passing layers.

3. GAP – a kernel‑based Gaussian process regression model that uses two‑body, three‑body, and embedded‑atom‑method descriptors. Sparsification techniques reduce the number of support points, but kernel evaluations remain computationally intensive.

4. tabGAP – a speed‑optimized version of GAP that replaces kernel evaluations with spline‑interpolated tables on low‑dimensional descriptor grids (1‑D for pairwise distances, 3‑D for angular triples), dramatically accelerating energy and force calculations.

Hyper‑parameters for each model (e.g., radial cutoff, basis order, number of message‑passing layers, kernel width, table resolution) were optimized via Bayesian search and early‑stopping based on a validation set. Regularization and cross‑validation were employed to avoid over‑fitting.

The authors validated the potentials against a suite of physical properties: (a) equation of state up to 30 GPa, (b) elastic constants (C₁₁, C₁₂, C₁₃, C₃₃, etc.), (c) formation energies of oxygen vacancies, copper interstitials, and complex defect clusters, (d) NEB‑derived migration barriers for oxygen diffusion, and (e) the orthorhombic‑to‑tetragonal phase transition that occurs upon oxygen depletion or at elevated temperature.

Key findings include:

• Accuracy – MACE consistently yields the lowest mean absolute error (≈ 3 meV/atom) across all tests, closely matching DFT. ACE and tabGAP achieve errors of 5‑7 meV/atom, which is still within chemical accuracy for most defect‑related studies. GAP attains comparable static‑energy accuracy but suffers from larger errors (≈ 10 meV/atom) in forces due to kernel sparsification.

• Computational cost – MACE is roughly 3–4 times slower than ACE per MD step, reflecting the cost of equivariant tensor operations. ACE and tabGAP enable simulations of millions of atoms with sub‑femtosecond timesteps on modern CPUs/GPUs, making them suitable for large‑scale cascade simulations. GAP is limited to a few thousand atoms because of memory‑intensive kernel matrices.

• Physical fidelity – All four MLPs correctly reproduce the pressure dependence of the bulk modulus and the anisotropic elastic response. Importantly, only the newly trained potentials capture the orthorhombic‑to‑tetragonal transition as a function of δ and temperature, a capability lacking in previously published empirical potentials (e.g., Gray, Baetzold, Chaplot). The NEB barriers for O‑diffusion predicted by MACE (0.45 eV) and ACE (0.48 eV) align with experimental estimates (~0.5 eV), whereas GAP slightly overestimates them.

• Comparison with existing potentials – Traditional empirical force fields either ignore oxygen non‑stoichiometry or impose rigid lattice constraints, resulting in inaccurate defect energetics and failure to predict the phase transition. The MLPs, trained on a purpose‑built dataset, overcome these limitations while retaining DFT‑level fidelity.

The authors conclude that the suite of potentials provides a robust foundation for atomistic investigations of radiation damage in YBCO. MACE is recommended when the highest accuracy is required, such as for detailed defect‑formation pathways or benchmark studies. ACE and tabGAP offer an optimal balance for large‑scale molecular dynamics of displacement cascades, enabling simulations of realistic reactor‑relevant dose rates and defect morphologies.

Future directions outlined include extending the training set to incorporate temperature‑dependent electronic free‑energy contributions, coupling the potentials with coarse‑grained models for multiscale simulations, and integrating superconducting order‑parameter descriptors to directly link atomic structure with critical current density. By delivering accurate, scalable interatomic models for a chemically and structurally complex superconductor, this work paves the way for predictive materials design and lifetime assessment of HTS magnets in fusion environments.