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Machine-learning based interatomic potential for amorphous carbon

Volker L. Deringer, Gábor Csányi

arXiv:1611.03277v1cond-mat.mtrl-sci

TL;DR

The paper addresses the difficulty of simulating structurally diverse liquid and amorphous carbon with both DFT-level accuracy and tractable system sizes. It develops a hierarchical GAP using two-, three-, and many-body descriptors, and reports accurate energies, structural properties, liquid structures, and ta-C surface behavior relative to DFT benchmarks. The resulting model is presented as an intermediate-cost approach for large-scale simulations of amorphous carbon and its surfaces.

  • Problem

    Amorphous carbon requires large cells and long relaxation times, while DFT methods are limited in practice to a few hundred atoms and prior crystalline-carbon ML potentials do not cover amorphous configurations.

  • Method

    The paper constructs a Gaussian approximation potential combining two-body, three-body, and many-body descriptors within one machine-learning framework.

  • Results

    The GAP predicts energies largely within tens of meV/atom and faithfully recovers sp3 counts, medium-range order, liquid structures, surface energies, and surface reconstructions against DFT benchmarks.

  • Takeaways & Limitations

    GAP models can provide accurate large-scale atomistic simulations of amorphous materials and surfaces at costs intermediate between DFT and empirical potentials.

Abstract

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We introduce a Gaussian approximation potential (GAP) for atomistic simulations of liquid and amorphous elemental carbon. Based on a machine-learning representation of the density-functional theory (DFT) potential-energy surface, such interatomic potentials enable materials simulations with close-to DFT accuracy but at much lower computational cost. We first determine the maximum accuracy that any finite-range potential can achieve in carbon structures; then, using a novel hierarchical set of two-, three-, and many-body structural descriptors, we construct a GAP model that can indeed reach the target accuracy. The potential yields accurate energetic and structural properties over a wide range of densities; it also correctly captures the structure of the liquid phases, at variance with state-of-the-art empirical potentials. Exemplary applications of the GAP model to surfaces of "diamond-like" tetrahedral amorphous carbon (ta-C) are presented, including an estimate of the amorphous material's surface energy, and simulations of high-temperature surface reconstructions ("graphitization"). The new interatomic potential appears to be promising for realistic and accurate simulations of nanoscale amorphous carbon structures.

I. INTRODUCTION

Amorphous carbon spans diverse local structures, but existing simulation methods trade accuracy against computational cost. The paper introduces a hierarchical GAP framework intended to approach DFT accuracy while enabling larger simulations of liquid and amorphous carbon.

  • Motivation: Amorphous carbon contains density-dependent mixtures of sp2 and sp3 carbon, while tetrahedral amorphous carbon is a dense, sp3-rich form of technological interest.Low- and high-density forms are only loosely reminiscent of graphite and diamond because their actual structures are more complex.
  • Motivation: Empirical carbon potentials enable large-scale molecular-dynamics simulations but can underestimate sp3 concentrations, describe surfaces poorly, and compromise accuracy across properties.Examples include applications to fracture, friction, wear, and thin-film deposition, alongside reported shortcomings.
  • Motivation: DFT-based methods provide atomistic models and benchmarks but are practically limited to systems of a few hundred atoms.This system-size limitation makes several practically relevant amorphous-carbon scenarios inaccessible to predictive DFT-quality simulations.
  • Motivation: Machine-learning potentials map atomic environments to energies and forces using accurate quantum-mechanical reference data, reducing computational effort by many orders of magnitude after training.The approach is designed to bridge the gap between empirical potentials and DFT-based simulations.
  • Contribution: The paper introduces a GAP for liquid and amorphous carbon, determines the accuracy limit of finite-range potentials, and tests the model on bulk structures and ta-C surfaces.The GAP combines two-body, three-body, and many-body descriptors within one machine-learning framework.
  • Method: Combining two-body, three-body, and SOAP many-body descriptors improves carbon-dimer behavior, avoiding the short-range unphysical behavior seen with a many-body-only model.The combined model gives a highly satisfactory potential-energy curve, whereas simpler descriptor sets miss aspects of the longer-range behavior or extrapolate poorly below 0.8 Å.

A. General protocol for melt–quench simulations

The melt–quench protocol generates amorphous-carbon structures by equilibrating carbon at high temperatures, quenching exponentially, and annealing at 300 K. The same temperature protocol is used in both DFT and GAP simulations.

  • Structural data were obtained from established melt–quench molecular-dynamics protocols for amorphous carbon.
  • The isolated-dimer potential-energy scan is used to assess structural descriptors against DFT reference data.
  • Carbon was held at 9000 K for 3 ps, then at 5000 K for 3 ps, before exponential quenching over 0.5 ps and annealing at 300 K for 3 ps.
  • DFT and GAP simulations employed the same temperature protocol, although the initial simulations used DFT and subsequent simulations used GAP.

B. DFT-based (“ab initio”) molecular dynamics

The training and benchmark structures are generated with DFT-based ab initio molecular dynamics and iteratively expanded with GAP-driven simulations. DFT-LDA provides the common reference for energies and forces, while the workflow emphasizes amorphous structures across densities.

  • Initial training structures and amorphous-carbon benchmarks were generated using DFT-based ab initio molecular dynamics.
  • The calculations used the local-density approximation, which the authors describe as the de facto standard for many current amorphous-carbon applications.
  • All training structures were subjected to single-point DFT-LDA calculations to obtain well-converged energies and forces.
  • An initial GAP reproduced the 9000 K liquid well and the 5000 K liquid satisfactorily, but not the amorphous phase.
  • Ten uncorrelated structures at 2.0, 3.0, 3.25, and 3.5 g cm−3 were reequilibrated and quenched with ab initio MD to emphasize high-density amorphous phases.
  • The database was iteratively expanded with GAP-driven melt–quench simulations using the same temperature profiles as the DFT simulations.
  • Amorphous surfaces were added by generating tetrahedral-amorphous-carbon structures with GAP, creating slabs with vacuum, and including heated amorphous slabs from ab initio MD.
  • The combined database included liquid, amorphous, crystalline, and isolated-dimer data, then was split into 90:10 training and test sets.

D. GAP model fitting

The GAP model combines hierarchical two-body, three-body, and many-body descriptors while testing the locality limits of finite-range potentials. These tests establish density- and structure-dependent force-locality bounds that guide the chosen cutoff and target accuracy.

  • Regularization errors were set to 0.002 eV for energies and 0.2 eV Å−1 for forces in liquid and amorphous structures.
  • The GAP model uses two-body, three-body, and SOAP terms, with the potential files made freely available.
  • Locality tests: Locality tests fix atoms inside a sphere, randomly displace atoms outside it, and measure the central atom’s force standard deviation as a function of rfix.
  • Locality tests: Diamond forces become practically local at rfix = 5.5 Å, whereas graphite shows larger deviations that decay less rapidly.
  • Locality tests: At 2.0 g cm−3, amorphous carbon is less local than at 3.0 g cm−3, although dense ta-C retains notable nonlocality because sp2 and sp3 motifs coexist.
  • Target accuracy: For a-C with a 3.7 Å cutoff, the lowest achievable force-component standard deviation is estimated at approximately 1 eV Å−1.
  • Target accuracy: Increasing rcut to 7.0 Å could lower the standard deviation to approximately 0.7 eV Å−1, but the added computational expense was not considered worthwhile.

B. Energies and forces

The GAP’s accuracy improves systematically as increasingly complex structural descriptors are added, with the full hierarchical model reaching the finite-range accuracy limit. Across melt–quench trajectories, force errors remain similar while force magnitudes vary substantially, and SOAP is needed for physically meaningful structural predictions.

  • Descriptor hierarchy: Adding 3b descriptors improves over 2b alone, while adding SOAP enables the GAP to reach the accuracy limit imposed by nonlocality.The 2b-only model shows little force correlation, the 2b+3b model is better, and the combined 2b, 3b, and many-body model reaches the locality-based target.
  • Configuration dependence: Crystalline configurations have the lowest fitting errors, whereas disordered liquid carbon is more difficult for the heterogeneous training database.The lowest-energy training points correspond to crystalline structures and show the smallest errors.
  • Melt–quench forces: Force errors are similar across melt–quench stages, but their absolute magnitudes differ substantially between liquid and amorphous phases.Relative performance is therefore better for liquid forces than for amorphous-phase forces when errors are normalized by the force distribution.
  • Crystalline benchmarks: The GAP reproduces diamond’s lattice parameter accurately and graphite’s c parameter as 6.518 Å versus 6.625 Å from DFT, despite a slight −1.6% overbinding.Its transferability to amorphous and crystalline phases comes at a small accuracy cost for some diamond elastic properties, while short-ranged Tersoff and Brenner cannot describe graphite interlayer spacing.
  • Melt–quench structure: At 3.0 g cm−3, approximately one-third of liquid-carbon atoms are fourfold coordinated, rising to approximately 70% after quenching and annealing at 300 K in DFT.The combined 2b, 3b, and SOAP model reproduces the liquid sp3 concentrations, although it underestimates the quenched amorphous value.
  • Structural properties: The GAP closely matches DFT radial distribution functions for liquid and amorphous carbon across densities, while screened Tersoff lowers the first liquid RDF minimum nearly to zero.The GAP also reproduces density-dependent angular distributions, including features near 120°, 109°, and 180°; screened Tersoff deviates significantly and lacks 60° and 180° features.

D. Coordination statistics and medium-range order

The GAP reproduces key density-dependent coordination and medium-range structural features of liquid and amorphous carbon, while larger cells reveal ring sizes inaccessible to small simulations.

  • Coordination statistics: The GAP tracks DFT fourfold-coordinated atom counts across densities more closely than empirical Brenner and Tersoff potentials, especially at lower densities.Its largest residual error occurs at 3.0 g cm−3.
  • Medium-range order: DFT ring distributions center near six-membered rings at high density but broaden toward larger rings at low density, indicating structural voids.No large-membered rings occur in ta-C.
  • Medium-range order: The GAP recovers three- and four-membered rings in liquid and low-density amorphous carbon, whereas screened Tersoff overestimates ring sizes and omits these rings.For ta-C at 300 K, all three methods produce very similar results.
  • System-size effects: 8,000-atom and averaged 216-atom structures agree for small and medium rings, but only the larger system contains rings with n ≥18.The smaller cells are too small to reproduce these very large rings.
  • System-size effects: Realistic studies of amorphous-carbon voids and porosity require structures on the order of at least several thousand atoms.The limitation arises because 216-atom cells cannot reproduce very large rings.
  • System-size effects: An 8,000-atom ta-C model contains no voids and forms a dense diamond-like network without rings larger than n = 15.The 216-atom simulations already estimate its medium-range structural order well.

E. Elastic properties

The study evaluates Young’s modulus from relaxed fixed-density structures and finds that GAP agrees well with experiments, while screened Tersoff overestimates absolute stiffness.

  • Method: Young’s modulus is obtained by relaxing 216-atom structures at fixed density, computing the full 6 × 6 elastic-constant matrix, and inverting it to obtain compliance.The modulus is averaged over the three spatial directions and independent structures.
  • Results: The GAP results agree very well with experiments across all relevant densities.The screened Tersoff potential predicts the same density-dependent trend but significantly overestimates absolute values, especially at high densities.
  • Results: Young’s modulus increases with density and sp3 concentration, reflecting greater diamond-likeness.The comparison includes structures generated by both GAP and screened Tersoff, including DFT-generated structures.

F. From the bulk to surfaces

GAP simulations extend amorphous-carbon modeling to ta-C surfaces, where surface structure, energy, and temperature-driven graphitization are examined in large slabs.

  • Motivation: Surface simulations are motivated by the importance of dense diamond-like carbon in coatings and by the difficulty of modeling amorphous surfaces.Earlier DFT studies of ta-C surfaces were restricted to very small systems, while empirical potentials can mispredict surface energies.
  • Surface energies: Amorphous surface energies require averaging because amorphous structures lack unique cleavage planes; the study compares GAP estimates with DFT values.Five surfaces were cleaved from a 1,000-atom bulk ta-C structure.
  • Annealing protocol: The simulations use 1,000-atom slabs heated for 10 ps, annealed for 10 ps, cooled for 20 ps, and sampled across ten independent structures.Target temperatures include 1000, 2000, and 3000 K.
  • Surface relaxation: Freshly cleaved ta-C surfaces contain dangling bonds and low-coordinated atoms, whose defects largely disappear during annealing.The top-view analysis focuses on atoms within the outermost 3 Å and colors them by coordination number.
  • Graphitization: At 1000 K the surface remains strongly disordered, while 2000 K produces graphitized layers several Å thick that remain connected to subsurface sp3 atoms.Sixfold rings and likely metastable five- and sevenfold ring pairs appear at 2000 K.
  • Graphitization: At 3000 K the entire slab graphitizes, a defective graphene sheet begins detaching, and near-surface sp3 atoms disappear.The figure caption also notes disintegration of ta-C regions in the slab center.

V. CONCLUSIONS

The paper presents a GAP model for liquid and amorphous carbon that recovers energetic, structural, and surface properties while occupying an intermediate accuracy–cost regime between DFT and empirical potentials.

  • Conclusions: The GAP predicts energies largely within tens of meV/atom and faithfully recovers sp3 counts, ring-statistics-based medium-range order, surface energies, and reconstructions.These capabilities span liquid, amorphous, and ta-C surface simulations.
  • Conclusions: GAP is many orders of magnitude faster than DFT but slower than state-of-the-art empirical potentials, while retaining similarly linear scaling.The authors position it as an intermediate tool for accurate large-scale simulations of amorphous materials and surfaces.

COMPOSITION OF TRAINING DATABASE

The database combines bulk liquids, amorphous structures, surfaces, crystalline references, dimers, and iteratively generated configurations. Training data are randomly split, with 90% used for training and the remainder for testing.

  • Database organization: The database stores atomic positions, forces, total energies, and virials in concatenated extended xyz configurations.Configurations are grouped using custom config type and detailed ct keywords.
  • Bulk liquid and amorphous carbon: 9000 K liquid carbon was sampled at five densities from 1.5 to 3.5 g cm−3, with snapshots taken throughout five trajectories.The step0 set contains 462 training structures from 500 total computed configurations.
  • Bulk liquid and amorphous carbon: Additional low-density liquid data at 1.50 and 2.00 g cm−3 were added at 9000 and 5000 K to prevent excessive sp chains.Other liquid datasets included preliminary-GAP melt–quench structures at 2.0 and 3.0 densities and high-density tetrahedral structures at 3.25 and 3.5 g/cm3.
  • Reference structures: Reference configurations also included distorted diamond and graphite structures plus isolated C2 dimers spanning bond lengths of 0.8–3.7 Å.The dimer set used 30 DFT configurations in a 20 × 15 × 15 Å3 cell with Γ-point sampling.

ERRORS FOR TEST VERSUS TRAINING SET

The randomly held-out test set has error distributions that closely match those of the training set for both energies and force components. This supports both the potential’s quality and the database’s structural completeness.

  • Error comparison: Training and test errors are practically superimposable for the randomly drawn structural subsets.The test set contains 450 structures excluded from training, while the training set contains 4050 structures.
  • Energy errors: Energy cumulative error distributions are very close between the 4050 training structures and 450 test structures.The scatterplot compares DFT-computed with GAP-predicted total energies.
  • Force errors: Force-component cumulative error distributions are practically indistinguishable between training and test sets.The figure reports this comparison across all structural models in the two subsets.

TRIMER POTENTIALS

Isolated-trimer tests expose how training data affect GAP extrapolation beyond extended structures. Including dimer data removes major qualitative errors, while adding trimer data further improves the potential-energy surface.

  • Test setup: The trimer test fixes the dimer spacing at 1.5, 2.0, or 2.5 Å while scanning a third carbon atom across finely meshed grid points.Interaction energies are evaluated relative to a free dimer at the given spacing plus a free atom.
  • Model comparisons: A GAP trained without dimer or trimer data produces erroneous extrapolation, including an intermediate-distance energy barrier absent from DFT.The comparison is shown for isolated carbon trimer potential-energy scans.
  • Model comparisons: A GAP fitted to dimer but not trimer data reproduces the trimer potential-energy surface qualitatively well, with small deviations remaining.The line cuts compare DFT data points with GAP results for the scanned configurations.
  • Model comparisons: Including DFT trimer datapoints during training further improves the trimer potential-energy surface.This final model gives the best description among the tested GAP variants.
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