Development of a machine learning potential for graphene

Patrick Rowe, Gábor Csányi, Dario Alfè, and Angelos Michaelides

Phys. Rev. B 97, 054303 – Published 5 February 2018

Abstract

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].

Received 4 October 2017

DOI:https://doi.org/10.1103/PhysRevB.97.054303

Physics Subject Headings (PhySH)

Electronic structure Interatomic & molecular potentials

2-dimensional systems Equilibrium lattice models Graphene Honeycomb lattice

Band structure methods Density functional calculations Density functional theory First-principles calculations Machine learning Molecular dynamics Tight-binding model

Atomic, Molecular & OpticalStatistical Physics & ThermodynamicsCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Patrick Rowe¹, Gábor Csányi², Dario Alfè³, and Angelos Michaelides¹

¹Thomas Young Centre, London Centre for Nanotechnology, and Department of Physics and Astronomy, University College London, Gower Street, London, WC1E 6BT, United Kingdom
²Engineering Laboratory, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, United Kingdom
³Thomas Young Centre, London Centre for Nanotechnology and Department of Earth Sciences, University College London, Gower Street, London WC1E 6BT, United Kingdom

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Issue

Vol. 97, Iss. 5 — 1 February 2018

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Images

Figure 1
Force correlations (left) and associated force errors (right) on an independent reference data set of configurations for the graphene GAP model, DFTB, LCBOP, and Tersoff potentials as compared to the reference DFT method, the plots for all methods considered can be found in the SM. Black points indicate forces perpendicular to the plane of the graphene sheet (out-of-plane) while red points indicate forces oriented in the plane. The inset in the graphene GAP plot has a different scale on the $y$ axis to show more clearly the distribution of force errors, which are smallest for large forces with a Gaussian distribution.
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Figure 2
(a) Thermal dependence of the lattice parameter of graphene between 60 and 2500 K, for a range of potentials as compared to the reference value calculated from ab initio molecular dynamics calculations. (b) Thermal dependence of lattice parameter, $a$ , normalized according to the predicted value at 60 K, emphasizing the relative behavior of the different methods—a range of predictions is observed, from monotonically increasing or decreasing lattice parameters to more complex nonmonotonic behavior in the case of GAP, LCBOP, and AIMD calculations. (c) Computed thermal expansion coefficients for graphene as a function of temperature calculated using Eq. (12), for DFT and various potentials.
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Figure 3
Comparison of model predictions using the finite displacement method [75] to phonon dispersion from XRD [76, 77]. Black lines represent the calculated phonon spectrum and red is the reference XRD. The GAP model accurately reproduces the experimentally determined phonon spectrum over all of the high-symmetry directions considered. Labels for branches are shown on the Graphene GAP plot (left) along with symmetry labels at the $Γ$ point (right). Note that the highest energy LO branch is not shown for the Tersoff potential in this figure—this branch crosses the $Γ$ point at approximately 350 meV.
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Figure 4
Absolute errors in prediction of phonon band frequencies along the high-symmetry directions in the graphene Brillouin zone, separated by phonon branch type. The thick red line denotes the mean absolute error (MAE) summed across all bands. Notable similarities in the error predicting the character of the LO branch can be seen across the LCBOP, REBO, and AIREBO(-Morse) potentials (black line).
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Figure 5
Finite temperature phonon calculations for graphene simulations between 60 and 2500 K derived directly from molecular dynamics simulations. Strong thermally induced dispersion is seen for the highest energy $E_{2g}$ symmetry phonon modes across all potentials, corresponding to the observed thermally induced dispersion of the Raman G band of graphene. Varying predictions are made for the transverse optical (ZO) branch's dependence on temperature: the AIREBO(-Morse) potentials predict this to have a strong thermal dispersive character. Blue corresponds to simulations at 60 K, through to 2500 K for red in a linear scale.
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Figure 6
Change in $Γ$ point frequencies for graphene $E_{2g}$ symmetry vibrational mode in the region of 150–1400 K. Compared with results from variable temperature Raman spectroscopy, which have been corrected for the strain induced by the adsorption of the graphene sheet onto the SiN substrate.
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