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Dispersion-corrected Machine Learning Potentials for 2D van der Waals Materials

Mikkel Ohm Sauer, Peder Meisner Lyngby, Kristian Sommer Thygesen

arXiv:2504.05754v1cond-mat.mtrl-sci

TL;DR

The enormous space of possible vdW heterostructures, along with the size of their moiré cells, makes computational exploration challenging. This work benchmarks six D3-corrected MLIPs against DFT on 336 heterobilayers and evaluates structural and electronic properties. Some models reach accuracies comparable to DFT uncertainty, with MatterSim performing best in the reported benchmark.

  • Problem

    The millions of possible vdW heterostructures and moiré cells containing hundreds or thousands of atoms make computational exploration difficult, while the effect of geometry inaccuracies on electronic structure remains relevant.

    Source: §I · p. 1

  • Method

    The authors benchmark six MLIPs augmented with D3 on 336 heterobilayers, comparing structural metrics and electronic bands with PBE-D3 calculations.

    Source: §I · p. 1

  • Results

    Some MLIPs reach accuracies comparable to DFT uncertainty; MatterSim performs best, with average errors of 23 meV for band gaps and 35 meV for general band energies.

    Source: §IV · pp. 6–7

  • Takeaways & Limitations

    The authors conclude that dispersion-corrected MLIPs are ready for large-scale structural simulations of the non-magnetic and non-metallic 2D vdW materials considered.

    Source: §IV · p. 7

    The supported scope is the studied class of non-magnetic and non-metallic materials.

    Source: §IV · p. 7

Abstract

from arXiv · show

Machine-learned interatomic potentials (MLIPs) based on message passing neural networks hold promise to enable large-scale atomistic simulations of complex materials with ab initio accuracy. A number of MLIPs trained on energies and forces from density functional theory (DFT) calculations employing semi-local exchange-correlation (xc) functionals have recently been introduced. Here, we benchmark the performance of six dispersion-corrected MLIPs on a dataset of van der Waals heterobilayers containing between 4 and 300 atoms in the moir\'e cell. Using various structure similarity metrics, we compare the relaxed heterostructures to the ground truth DFT results. With some notable exceptions, the model precisions are comparable to the uncertainty on the DFT results stemming from the choice of xc-functional. We further explore how the structural inaccuracies propagate to the electronic properties, and find excellent performance with average errors on band energies as low as 35 meV. Our results demonstrate that recent MLIPs after dispersion corrections are on par with DFT for general vdW heterostructures, and thus justify their application to complex and experimentally relevant 2D materials.

I. INTRODUCTION

The vast and atomically large space of van der Waals heterostructures makes computational exploration challenging, while the electronic consequences of geometry errors matter for applications. The paper benchmarks dispersion-corrected MLIPs and defines structural and electronic measures for comparing their relaxations with DFT.

I. INTRODUCTION

  • Millions of possible heterostructures and moiré cells that can contain hundreds or thousands of atoms make systematic exploration difficult with ab initio methods.The relevant question is how geometry inaccuracies affect electronic structure, since applications depend on material properties rather than detailed atomic structure.Source: §I · p. 1
  • The study benchmarks six D3-corrected MLIPs on 336 heterobilayers from 44 monolayers, comparing relaxed structures and electronic bands with PBE-D3 calculations.The selected moiré cells contain fewer than 300 atoms and have in-plane strain below 1%.Source: §I · p. 1

II. DEFINITION OF METRICS

  • Structural metrics cover interlayer distance, intralayer atomic rearrangement, and in-plane stress.Intralayer rearrangement is measured by the RMSD of atomic positions relative to the PBE-D3(PW) ground truth.Source: §II · p. 2
  • Electronic metrics assess band-gap changes and band-energy differences, with the latter computed as an RMSD over selected valence and conduction bands.The selected bands are the 10 highest valence and 10 lowest conduction bands.Source: §II · p. 3

A. Machine learning models

The six benchmarked MLIPs were trained primarily on bulk-crystal data, with little exposure to isolated 2D materials or vdW heterostructures and no prior fine-tuning for this study. They differ in architecture and pretraining or fine-tuning datasets.

A. Machine learning models

  • All six MLIPs were trained mainly on inorganic bulk-crystal PBE energies and forces, had limited exposure to isolated 2D materials and vdW heterostructures, and were used without prior fine-tuning.The models were augmented with D3 dispersion correction to yield PBE-D3 energies and forces.Source: A. Machine learning models · p. 3
  • The two MACE models use atomic cluster expansion descriptors: L-MACE-MP uses equivariance order l = 2 and Materials Project relaxation trajectories, while M-MACE-MPA uses l = 1 and Alexandria and Materials Project data.MatterSim uses three-body interactions and atomic positions and is pretrained on the proprietary MatterSim dataset.Source: A. Machine learning models · p. 3
  • DPA3 is a message-passing large atomic model pretrained on OpenLAM and fine-tuned on Alexandria and Materials Project data.GRACE uses a graph atomic cluster expansion and is fine-tuned on Alexandria and Materials Project data after pre-fitting on OMat24.Source: A. Machine learning models · p. 3
  • ORB combines a graph network simulator with smooth overlap of atomic positions, and predicts forces separately from the energy surface rather than as its gradient.It is pretrained on DFT relaxation trajectories from the Materials Project and Alexandria datasets.Source: A. Machine learning models · p. 3

III. RESULTS

The results section first evaluates relaxed atomic structures, then examines how structural differences affect electronic bands for two selected MLIPs, and concludes with the HetDB database.

III. RESULTS

  • The results first compare the relaxed atomic structures produced by the six dispersion-corrected MLIPs.The section then investigates electronic-band effects for two selected models and describes HetDB.Source: §III · p. 3
  • The electronic-band analysis examines how structural differences propagate to band structures for two selected MLIPs.Source: §III · p. 3
  • The results section also describes the 2D-vdW heterostructure database HetDB.Source: §III · p. 3

A. Atomic structures

The dispersion-corrected MLIPs show varying accuracy across interlayer spacing, intralayer structure, and stress. L-MACE-MP’s interlayer deviations are comparable to or smaller than the stated DFT uncertainty from the choice of exchange-correlation functional.

A. Atomic structures

  • 0.11 Å MADs for ORB, MatterSim, and GRACE approach the 0.09 Å PBE-D3(LCAO) result, while DPA3 and M-MACE-MPA reach 0.21 Å and 0.25 Å.MatterSim and GRACE have symmetric error distributions around zero; M-MACE-MPA and DPA3 have maximum outliers of 2.07 Å and 2.34 Å, respectively.Source: A. Atomic structures · p. 3, Figure 2 · p. 4
  • 0.31 meV/Å^2 MAD and 1.73 meV/Å^2 maximum outlier make MatterSim the best-performing model on the stress metric.The moiré supercell is fixed during relaxation, and the two MACE models have stress MADs above 5 meV/Å^2.Source: A. Atomic structures · pp. 4–5, Figure 4 · p. 4
  • 13 mÅ intralayer MADs make MatterSim and GRACE the most accurate; DPA3 reaches 15 mÅ, while large MACE reaches 33 mÅ.DPA3 is most consistent, with a 121 mÅ maximum outlier; all models except L-MACE-MP outperform PBE-D3(LCAO) at 21 mÅ.Source: Figure 3 · p. 4, A. Atomic structures · p. 4
  • PBE-D3(LCAO) stress results are absent because GPAW does not provide the stress tensor in LCAO mode.The stress metric measures mechanical energy due to internal strain.Source: Figure 4 · p. 4, A. Atomic structures · p. 5
  • 2.07% and 2.40% MARDs for PBE-D3 versus MACE-D3 interlayer distances with PW and LCAO bases are below the 3.1% xc-functional uncertainty.The paper concludes that L-MACE-MP reaches DFT accuracy for interlayer distance and then specializes to L-MACE-MP and MatterSim.Source: A. Atomic structures · p. 5, Figure 5 · p. 5
  • PBE-D3(LCAO) interlayer distances are systematically about 0.1 Å smaller than PW-basis values, possibly because finite-range basis functions underestimate Pauli repulsion.The compared calculations include the D3 dispersion correction.Source: A. Atomic structures · p. 5

B. Electronic structure

The electronic-property comparison tests how structural relaxation differences affect band gaps and band energies. MatterSim has the lowest reported errors, while L-MACE-MP has similar error distributions for most systems but a few significant outliers.

B. Electronic structure

  • The band-gap and band-energy comparisons use PBE(LCAO) electronic calculations on structures relaxed by different methods, so differences arise from the atomic structures.Figure 6 encodes band-gap differences against PBE(PW); its full, dashed, and dotted lines mark percentiles, with the full line showing the median.Source: Figure 6 · p. 5, B. Electronic structure · p. 6
  • MatterSim-relaxed structures yield a band-gap MAD of 22.8 meV, below the 36.5 meV for PBE(LCAO); L-MACE-MP's 50.5 meV MAD reflects a few significant outliers.For the vast majority of systems, L-MACE-MP's error distribution is very similar to MatterSim's.Source: B. Electronic structure · p. 6
  • MatterSim's band-energy MAD is 34.5 meV, compared with 59.8 meV for L-MACE-MP and 40.8 meV for PBE-D3(LCAO).The L-MACE-MP distribution is similar to MatterSim for most systems, but a few significant outliers raise its MAD.Source: B. Electronic structure · p. 6

C. The vdW heterostructure database: HetDB

HetDB provides relaxed heterobilayer structures and calculated band structures alongside comparisons of MLIP and DFT results. The study reports low MatterSim electronic-structure errors and illustrates both qualitative agreement and specific band-structure features across example systems.

C. The vdW heterostructure database: HetDB

  • HetDB makes structures and PBE(LCAO) band structures available for 336 PBE-D3(PW)-relaxed heterobilayers, with layer projections and integration with C2DB and BiDB.The authors plan to expand the database with additional structures and properties.Source: C. The vdW heterostructure database:… · p. 6
  • Three example band structures compare PBE-D3(PW)-relaxed reference bands projected onto the layers with bands from L-MACE-MP-relaxed structures; agreement is qualitative, with discrepancies especially away from valence bands.Spin-orbit coupling is omitted in these examples.Source: C. The vdW heterostructure database:… · p. 6, Figure 8 · p. 7
  • Graphene-ZnI2 has a roughly 0.3 eV direct gap along the moiré-cell band path, but a roughly 20 meV gap over the full Brillouin zone because the path misses graphene's K point.The bilayer's moiré cell contains 105 atoms.Source: C. The vdW heterostructure database:… · p. 6
  • Si2H2-PbI2 has a 1.5 eV direct interlayer gap at Γ, making it a candidate for long-lived, tunable interlayer excitons in the optical regime.Its moiré cell contains 240 atoms.Source: C. The vdW heterostructure database:… · p. 6
  • MgI2-PbBr2 has an indirect interlayer gap around 2.5 eV, with flat bottom-layer valence bands and more dispersive top-layer conduction bands.The authors suggest the distinct band character presumably reflects the bands residing on different materials; the moiré cell contains 159 atoms.Source: C. The vdW heterostructure database:… · p. 6

IV. DISCUSSION

  • MatterSim's average errors are as low as 23 meV for band gaps and 35 meV for general band energies, supporting large-scale simulations within the studied non-magnetic, non-metallic class.The benchmark covers 336 heterostructures with 4–300 atoms per unit cell; the authors report that some MLIPs reach accuracy comparable to DFT uncertainty from xc-functional choice.Source: §IV · pp. 6–7

V. METHODS

  • The workflow selects monolayers, constructs moiré supercells, relaxes heterostructures, and calculates their electronic band structures.The calculations were managed with the MyQueue task scheduler.Source: §V · p. 7

A. Selection of monolayers

  • The 44 selected monolayers include stable, non-magnetic, non-metallic materials with fewer than seven primitive-cell atoms, low hull energy, available GW bands, and a band extremum at Γ, plus eight named materials added for relevance.The selection uses an energy above the convex hull below 10 meV/atom; the authors say several criteria are intended for a future study and are of little relevance here.Source: A. Selection of monolayers · p. 7

B. Finding the moir´e supercells

Moiré supercells are selected to keep strain and cell size limited, then fine-tuned by minimizing mechanical energy. The resulting heterostructures are relaxed and evaluated for electronic band structures using DFT and dispersion-corrected models.

B. Finding the moir´e supercells

C. Relaxation and band structure calculations

  • An initial interlayer spacing is found by scanning and spline-interpolating binding energy, after which atomic positions are relaxed to a maximum force of 50 meV/Å.The scan evaluates 25 evenly spaced distances from 2 to 5 Å, adding points outside that interval when needed; the study then calculates band structures on relaxed structures with PBE(LCAO).Source: C. Relaxation and band structure… · p. 8

Appendix A: Finding moir´e supercells

The algorithm searches combinations of each layer’s lattice vectors to construct a common moiré supercell. It filters candidate cells by physical and numerical constraints, minimizes strain, and selects a solution with the fewest atoms.

Appendix A: Finding moir´e supercells

  • A common supercell is chosen by minimizing mechanical energy with the layers’ elasticity tensors, then evaluating maximum strain from the strain-tensor eigenvalues.The algorithm uses the stress-minimized common cell and discards cells whose maximum strain exceeds the limit.Source: Appendix A · pp. 9–10
  • Candidates are bounded by 1% monolayer strain, 0°–90° twist, at most 300 atoms, 15°–165° internal angle, norm ratio at most 10, and coefficients below 25.Source: Appendix A · p. 9
  • The search generates layer-specific lattice-vector combinations, matches pairs with low approximate uniaxial strain, and requires their relative twist to lie within the specified interval.Source: Appendix A · p. 9
  • Duplicate supercells are identified through their deformation tensors, retaining those with the fewest atoms, then lowest norm ratio, then smallest internal angle.Source: Appendix A · p. 9
  • From the remaining cells, the algorithm selects the first minimum-atom solution, transforms both layers’ atomic positions into the common cell, and extracts the exact twist from polar decomposition.Source: Appendix A · p. 10
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