A Machine Learning Potential for Graphene

Patrick Rowe; Gábor Csányi; Dario Alfè; Angelos Michaelides

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A Machine Learning Potential for Graphene

Patrick Rowe, Gábor Csányi, Dario Alfè, Angelos Michaelides

arXiv:1710.04187v3Oct 11, 2017cond-mat.mtrl-sci

TL;DR

Graphene simulations require accurate potentials because empirical models show conflicting predictions and discrepancies against reference data. The paper constructs a GAP machine-learning potential and compares it with empirical and bond-order potentials against ab initio and experimental references. The GAP model predicts graphene’s lattice, thermal-expansion, and phonon properties quantitatively while costing much less than direct ab initio molecular dynamics.

Problem
Existing calculations and empirical potentials provide conflicting predictions for graphene properties, motivating a more accurate model.
Method
The study constructs a graphene interatomic potential using the Gaussian Approximation Potential machine-learning methodology and compares it with empirical potentials.
Results
The graphene GAP model quantitatively predicts lattice parameter, coefficient of thermal expansion, and phonon properties, whereas none of the other considered potentials predicts all of them successfully.
Takeaways & Limitations
For graphene, machine-learning potentials can approach ab initio accuracy at much lower cost, including approximately four orders of magnitude lower cost for 200 atoms.

Abstract

from arXiv · show

We present an accurate interatomic potential for graphene, constructed using the Gaussian Approximation Potential (GAP) machine learning methodology. This GAP model obtains a faithful representation of a density functional theory (DFT) potential energy surface, facilitating highly accurate (approaching the accuracy of ab initio methods) molecular dynamics simulations. This is achieved at a computational cost which is orders of magnitude lower than that of comparable calculations which directly invoke electronic structure methods. We evaluate the accuracy of our machine learning model alongside that of a number of popular empirical and bond-order potentials, using both experimental and ab initio data as references. We find that whilst significant discrepancies exist between the empirical interatomic potentials and the reference data - and amongst the empirical potentials themselves - the machine learning model introduced here provides exemplary performance in all of the tested areas. The calculated properties include: graphene phonon dispersion curves at 0 K (which we predict with sub-meV accuracy), phonon spectra at finite temperature, in-plane thermal expansion up to 2500 K as compared to NPT ab initio molecular dynamics simulations and a comparison of the thermally induced dispersion of graphene Raman bands to experimental observations. We have made our potential freely available online at [http://www.libatoms.org].

I. INTRODUCTION

Graphene research has faced conflicting predictions for fundamental properties, while existing empirical potentials impose restrictive interaction forms. This work introduces a GAP machine-learning potential and evaluates it against empirical models and reference data.

Conflicting predictions for graphene’s coefficient of thermal expansion include disagreement even over its sign.
Accurate graphene models would benefit studies of phonon-assisted diffusion, thermal transport, and nuclear quantum effects in simulations.
Empirical carbon potentials evolved from Tersoff and REBO to AIREBO and LCBOP, adding long-range interactions or improved bonding descriptions.
Empirical and bond-order potentials are limited by assumptions about closed mathematical forms for interactions and by physical bias from empirical observations.
Machine-learning approaches avoid such prescribed potential-energy decompositions beyond smoothness and vanishing interactions at large distances.
The study uses Gaussian Approximation Potentials to compare modern machine-learning and empirical many-body potentials for graphene.

II. CONSTRUCTION OF A GAUSSIAN APPROXIMATION POTENTIAL

The GAP model interpolates a DFT potential-energy surface using kernel-based local atomic environments, combining two-body, three-body, and many-body descriptors. Its formulation supports accurate energies and forces for configurations within the sampled phase space without imposing fixed analytic interaction forms.

Model construction: GAP represents the potential-energy surface through Gaussian kernel regression trained on quantum-mechanical observations.The observations include atomic forces and total energies, with virial stresses also available in the reference dataset.
Model design: Unlike empirical potentials, GAP does not require prior assumptions that microscopic interactions follow harmonic, Morse, or Lennard-Jones functional forms.Predictions can be used to generate molecular-dynamics trajectories in the manner of an empirical potential.
Model construction: The total energy is decomposed into local contributions from two-body, three-body, and many-body interactions.Each interaction class uses a distinct descriptor and associated kernel function, with contributions weighted by their measured effects on energy and forces.
Descriptors: Descriptors transform atomic coordinates into rotationally and translationally invariant inputs for machine learning.The two-body descriptor is an interatomic distance, while the many-body term uses the SOAP descriptor for general n-body configurations.
Model design: The combined descriptor model was selected because it had previously improved GAP stability and, during development, increased accuracy and training-data efficiency.The model combines descriptors rather than relying on a single descriptor method.
Kernel evaluation: Kernel functions quantify similarity between chemical environments and combine weighted basis-function contributions to predict local energies.The total system energy is obtained by summing the contributions associated with each descriptor over the atoms and descriptor types.

III. GENERATION OF TRAINING DATA

The training dataset combines tightly converged plane-wave DFT calculations with iterative sampling of molecular-dynamics configurations. The final dataset spans varied temperatures and lattice parameters, with a withheld validation subset and published model files.

Reference calculations: Training data were generated from tightly converged plane-wave DFT calculations on configurations sampled from molecular-dynamics trajectories.The reference calculations used VASP and supplied energies, forces, and virial stresses for training.
Iterative sampling: 300 initial configurations were used to generate a preliminary GAP model, which then guided further trajectory sampling and dataset expansion.Ab initio energies, forces, and virial stresses from newly sampled configurations were added through repeated improvement iterations.
Dataset coverage: The final dataset contained 1083 configurations of 200 atoms spanning 300–4000 K and lattice parameters of 2.460–2.480 Å.The initial trajectories sampled free-standing graphene at 1000, 2000, and 3000 K with a lattice parameter of 2.465 Å.
Validation and fitting: A random 5% of the configurations was withheld as a validation set for benchmarking the GAP fitting procedure.The fitting used expected errors of σE = 10^-3 eV for energies, σf = 5 × 10^-4 eV for forces, and σv = 5 × 10^-3 for virial stresses.
Availability: The training configurations and GAP model files, together with the QUIP source code required to use the model, were made freely available online.The repository is identified as http://www.libatoms.org.

IV. FORCE PREDICTION

The graphene GAP model closely reproduces DFT forces on an independent reference dataset, while several empirical potentials show substantially larger errors. Its force accuracy is also relatively insensitive to the exchange-correlation functional used as reference.

Force accuracy: The graphene GAP predictions align very closely with the reference DFT forces on an independent configuration dataset.The comparison separates in-plane and out-of-plane force components.
Force accuracy: 0.028 eV Å^-1 is the reported GAP force RMSE for the in-plane direction, compared with 0.019 eV Å^-1 out of plane.
Comparison with other potentials: 0.69 and 0.55 eV Å^-1 are the force RMSEs reported for DFTB and LCBOP, respectively, substantially above the GAP errors.
Comparison with other potentials: 3.1 eV Å^-1 is the Tersoff force RMSE, while AIREBO-Morse errors reach a maximum above 11 eV Å^-1.
Reference-method dependence: The relative ranking of the benchmarked methods and the expected GAP performance are insensitive to the exchange-correlation functional choice.

V. LATTICE PARAMETERS AND IN-PLANE THERMAL EXPANSION

The study evaluates graphene lattice parameters and thermal expansion against ab initio molecular dynamics, finding that the GAP model reproduces both the absolute lattice parameter and its temperature dependence particularly well. Other potentials show larger deviations, including incorrect expansion trends or high-temperature fragmentation.

Ground-state lattice parameter: 0.1% is the lattice-parameter error for both the Graphene GAP and ReaxFF potentials relative to the ab initio result.DFT predicts a 0 K lattice parameter of 2.464 Å.
Thermal expansion: The graphene GAP model agrees excellently with first-principles calculations in both absolute and relative lattice parameters across temperature.
Comparison with other potentials: ReaxFF predicts very strong negative thermal expansion and fragments the graphene sheet above 1500 K, below the experimentally determined melting point.
Coefficient of thermal expansion: The GAP and AIMD calculations predict weaker CTE temperature dependence, a sign change near 1000 K, and a plateau above 1500 K.
Comparison with other potentials: LCBOP, AIREBO, and AIREBO-Morse predict strongly temperature-dependent CTEs, while REBO predicts positive CTE across the measured range.

VI. PREDICTION OF PHONON SPECTRA

The graphene GAP model reproduces experimental and reference phonon behavior across zero- and finite-temperature tests, while empirical potentials show substantial branch- and wavevector-dependent errors. The model also agrees well with observed thermal dispersion of the graphene E2g mode.

0 K phonon spectra: The graphene GAP model accurately reproduces the experimentally determined phonon spectrum across the considered high-symmetry directions.Its band dispersion is accurately predicted across the sampled Brillouin-zone regions.
0 K phonon spectra: Sub-meV accuracy is achieved for phonon frequencies at almost all high-symmetry points.
Comparison with empirical potentials: The high-energy LO branch is a shared failure point for empirical potentials, whereas the graphene GAP model predicts it with negligible error.
Finite-temperature phonons: All potentials predict strong thermally induced dispersion in the highest-energy LO/TO branches, but their predictions for other modes vary substantially.AIREBO and AIREBO-Morse predict a strong temperature dependence for the ZO branch.
Finite-temperature phonons: The graphene GAP model agrees well with experimentally observed thermal dispersion of the E2g mode at Γ, whereas LCBOP qualitatively disagrees despite reproducing the zero-temperature dispersion shape.AIREBO and REBO also agree with the observed effects; AIREBO-Morse slightly overestimates them and Tersoff substantially enhances them.

VII. CONCLUSIONS AND DISCUSSION

The graphene GAP potential reproduces key structural, thermal, and vibrational properties against ab initio and experimental references while substantially reducing computational cost. Empirical potentials often describe individual properties reasonably, but none matches the full evaluated range, and the graphene-specific GAP model is not transferable across carbon phases.

Method: The GAP method constructs a machine-learning potential for graphene trained using energies, forces, and virial stresses from high-quality van der Waals-inclusive DFT.It is benchmarked alongside commonly used potentials against ab initio and experimental references.
Performance: The graphene GAP model quantitatively predicts the lattice parameter, coefficient of thermal expansion, and phonon properties.These properties are evaluated against both ab initio and experimental references.
Comparison: None of the other evaluated potentials successfully predicts the whole range of properties, although many provide reasonable predictions for individual properties.REBO is identified as the best empirical model overall for graphene lattice dynamics.
Computational cost: The GAP method costs approximately four orders of magnitude less than direct ab initio molecular dynamics for 200 atoms, with only a marginal compromise on accuracy.Including training-database generation still leaves a significant computational-cost reduction.
Computational cost: The GAP model has the same system-size cost scaling as force-field molecular dynamics rather than the O(N^3) scaling of DFT, making savings more effective for larger systems.The framework targets accuracy close to AIMD at much reduced cost rather than universal applicability.
Scope and limitations: The presented GAP model accurately treats free-standing graphene but is not transferable by construction to other carbon phases such as diamond.Machine-learning models may fail to extrapolate into unfamiliar chemical-space regions, although GAPs are systematically improvable.