Accurate, transferable, and verifiable machine-learned interatomic potentials for layered materials

TL;DR

This work tackles the challenge of predicting moiré reconstructions in multilayer 2D materials by introducing a split machine-learned interatomic potential (MLIP) framework that separately models intralayer and interlayer interactions, each with tailored cutoffs. A physically grounded mean disregistry error (MDE) metric based on Voronoi-centered disregistry vectors is proposed to evaluate large-scale moiré structures, addressing the shortcomings of conventional energy/force RMSE metrics. The authors also demonstrate that quasi-one-dimensional (Q1D) moiré surrogates provide a computationally feasible and predictive validation pathway for 2D moiré relaxations, with strong correlation between Q1D and 2D performance. The approach is validated on HfS2/GaS bilayers, achieving accurate relaxations and electronic structure predictions, and is designed to be model-agnostic and applicable to complex multilayer systems beyond TMDs, enabling scalable exploration of moiré phenomena with rigorous accuracy.</p>

Abstract

Twisted layered van-der-Waals materials often exhibit unique electronic and optical properties absent in their non-twisted counterparts. Unfortunately, predicting such properties is hindered by the difficulty in determining the atomic structure in materials displaying large moiré domains. Here, we introduce a split machine-learned interatomic potential and dataset curation approach that separates intralayer and interlayer interactions and significantly improves model accuracy -- with a tenfold increase in energy and force prediction accuracy relative to conventional models. We further demonstrate that traditional MLIP validation metrics -- force and energy errors -- are inadequate for moiré structures and develop a more holistic, physically-motivated metric based on the distribution of stacking configurations. This metric effectively compares the entirety of large-scale moiré domains between two structures instead of relying on conventional measures evaluated on smaller commensurate cells. Finally, we establish that one-dimensional instead of two-dimensional moiré structures can serve as efficient surrogate systems for validating MLIPs, allowing for a practical model validation protocol against explicit DFT calculations. Applying our framework to HfS2/GaS bilayers reveals that accurate structural predictions directly translate into reliable electronic properties. Our model-agnostic approach integrates seamlessly with various intralayer and interlayer interaction models, enabling computationally tractable relaxation of moiré materials, from bilayer to complex multilayers, with rigorously validated accuracy.

Figures (5)

• Figure 1: Separation of length scales for designing accurate MLIPs for layered materials. a Total energy for a layered system can be separated into an intralayer term involving non-interacting layers and a residual interlayer interaction, which naively requires models with longer truncation distances. b Distributions of intralayer and interlayer energies within our generated datasets. The distribution of intralayer energies resembles that in the training of MLIPs with standard approaches, but typical interlayer energies are much smaller -- on the order of the error$\sigma_\mathrm{intra}$ of the intralayer MLIP model ($\sim$100 meV/atom) --, motivating the split model approach.
• Figure 2: Performance of classical interatomic potentials and both unified and split MLIPs on our hold-out test datasets. a Absolute residual energy, relative to DFT, for various intralayer models. Error bars mark the interquartile range, and dots mark the median absolute residuals. Horizontal lines are the maximum desired model errors (see text). Split and unified models perform similarly on the intralayer predictions when using similar model parameters. b Similar comparison for interlayer models. Here, even small split MLIPs (6080 trainable parameters) outperform large unified MLIPs (109824 trainable parameters).
• Figure 3: Validation metrics for moiré structural effects in twisted $\mathrm{MoS}_2$/$\mathrm{WSe}_2$ heterostructures. a Atomic configurations at high-symmetry stacking positions (AA, AB, and BA), showing W atoms (coloured hexagons), reference Mo atoms, and disregistry vectors (arrows). b Interlayer potential energy landscape per atom versus displacement ($\delta r_y$) for a ground-truth density functional theory (DFT) prediction and two machine-learned interatomic potential (MLIP) models trained on a unified dataset: Unified Model 1 (UM1) with a neighbour cutoff radius of 6Å, and Unified Model Biased 2 (UM1) with a neighbour cutoff radius of 10Å. Vertical lines indicate high-symmetry stacking configurations: AA (red), AB (blue), and BA (green). c Voronoi-centered disregistry vectors during continuous displacement of top $\mathrm{WSe}_2$ layer, mapped onto the pristine $\mathrm{MoS}_2$ lattice (blue background cell). d Atomic relaxation patterns in a 1.1$^{\circ}$ twisted $\mathrm{MoS}_2$/$\mathrm{WSe}_2$ heterostructure predicted using UM1 and UM2. The colour scale indicates local disregistry magnitude at Mo sites, revealing domain formation through regions of constant disregistry. e Distribution of Voronoi-centered disregistry vectors of the same twisted structures in 3d, highlighting domain asymmetry through the density of vector clustering near high-symmetry positions. f Structural validation of moiré geometry using the mean disregistry error (MDE) metric versus model energy prediction RMSE for MLIPs trained with varying degrees of model corruption. Shaded regions denote the desired target MDE (grey, below $0.01$ Å), and the maximum interlayer energy RMSE (green, $\leq 0.4$ meV/atom) and maximum intralayer energy RMSE (orange, $\leq200$ meV/atom) to achieve that target MDE.
• Figure 4: One-dimensional periodic moiré bilayers as a surrogate system for two-dimensional twisted structures. a Quasi-1D (Q1D) heterostructure: 24-unit-cell strip of $\mathrm{WSe}_2$ ($a=3.319$Å) aligned with 25-unit-cell strip of $\mathrm{MoS}_2$ ($a=3.184$Å). The lattice mismatch requires $\sim$0.07% strain in the $\mathrm{WSe}_2$ layer to achieve periodicity. The initial, unrelaxed structure in transparency is overlaid onto the relaxed structure. Shifting of local disregistry and out-of-plane buckling is observed. b Top view comparing unrelaxed (top) and DFT-relaxed (bottom) structures. Insets show local stacking configurations at three positions, with hexagons marking top-layer W atoms indicating relative disregistry from bottom-layer Mo atoms (coloured by indexed position along the strip). c Voronoi-centered disregistry analysis of unrelaxed (top) and relaxed (bottom) structures. Coloured hexagons represent disregistry vectors of W atoms, demonstrating relaxation-induced displacement along the short axis that reduces stacking energy while increasing strain. d Validation of structural predictions, using the mean disregistry error (MDE), comparing Q1D and 2D moiré structures. These MLIPs were trained against classical force fields to allow 2D moiré structures to be validated against a ground truth (see text). Models are corrupted following the same scheme as in Fig. 3, and error bars represent the 25th and 75th quartiles across multiple noise seed initializations. Notably, the performance of the split MLIP on the 1D structure is a good predictor of the quality of the MLIP on the 2D moiré structure.
• Figure 5: a Q1D structures of bilayer of HfS$_2$/GaS before (transparent circles) and after (solid circles) relaxation by our split MLIP model. b Band structure of the Moiré system unfolded onto a strained GaS unit cell (see SI Fig. 3 SI), computed within DFT at the initial (green), MLIP-relaxed (red), and DFT-relaxed (blue) structures. Continuous curves denote the band structure of a single strained GaS monolayer. c Similar to b, but unfolded onto a strained HfS$_2$ cell. d MLIP-relaxed 2D moiré cell of GaS/HfS$_2$, along with the magnitude of the displacement vectors, showcasing the significant reconstruction and formation of triangular domains.