On the validity of empirical potentials for simulating ... - IOPscience

Journal of Physics: Condensed Matter (2015)

On the validity of empirical potentials for simulating radiation damage in graphite Supplementary information C. D. Latham1 , A. J. McKenna1 , T. P. Trevethan1 , M. I. Heggie1 , M. J. Rayson1 , P. R. Briddon2 1

Department of Chemistry, Faculty of Engineering and Physical Sciences, University of Surrey, Guildford, GU2 7XH, UK 2

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School of Electrical and Electronic Engineering, Newcastle University, Newcastle upon Tyne, NE1 7RU, UK

Structures

The atomic coordinates for all the optimized structures used in the present work are provided in the form of a compressed zip-format archive, containing xyzcoordinate files within a simple directory structure. The files take their names from their descriptions in the main article. It should also be possible, using a suitable viewing application, to identify each structure from the illustrations provided in this document and the main article. The units used for the atomic coordinates are Ångströms (1 × 10−10 m). The supercells are orthorhombic, comprising 6 × 3 × 2 eight-atom unit cells when no defect is present. Fixed lattice parameters are used for the supercells. The dimensions of the unit cells √ for the DFT calculations are a = 2.445821 Å, b = a 3 Å, and c = 6.5186297 Å, which represent the values optimized using the local density approximation (LDA) with the aimpro program package [1, 2, 6, 7]. For the EDIP calculations, only the prismatic parameter c = 6.4 Å is different from the LDA; a and b are the same. The calculations using the AIREBO potential have a = 2.418 Å, √ b = a 3 Å, and c = 6.738 Å. It is necessary here to specify these lengths with more digits than is physically significant, in order to keep the lattice vectors numerically consistent with the atomic coordinates given in the supplementary data. Unless stated otherwise, all of the following illustrations are generated from the atomic coordinates calculated using the LDA. The exceptions are the structures that are found by the lammps [5] and edip [3] packages, using either the EDIP (Environment-Dependent Interatomic Potential) or AIREBO (Adaptive Intermolecular Reactive Empirical Bond Order) potential, to be local minima in energy, yet are highly unstable with density functional theory (DFT). 1

Close examination of the structures show that the LDA and the generalized gradient approximation (GGA) yield similar results. The main difference is that the graphene sheets making up the host crystal tend to be slightly more distorted by the presence of defects for the GGA than the LDA.

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Self-interstitial atoms

The structures depicted here are generated from supercells containing four graphene sheets, meaning that their thickness is 2c. Thus, the left-hand image of each set, which represents the view directly along the prismatic direction of the host crystal, include all four sheets. The oblique views are cropped so they show only two or three of the four sheets.

Figure 1: The spiro interstitial. DFT predicts this to be the most stable selfinterstitial state.

Figure 2: The β-split interstitial. The α-split interstitial is a similar defect.

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Figure 3: The grafted interstitial. DFT predicts this to be the least stable of the four self-interstitial states.

Figure 4: The Y-lid transition state for spiro interstitial reorientation.

Figure 5: The canted interstitial. DFT predicts this to be a highly unstable configuration; however, it is the most stable state for the EDIP.

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Figure 6: The bridge interstitial. This structure is another minimum on the EDIP energy surface that is not stable for DFT. It lies in between the canted interstitial and the grafted interstitial.

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Self interstitial pairs

Thirteen structures have been identified for closely-bound self-interstitial pairs. DFT predicts that these have binding energies in the range 0.7–3.1 eV, approximately. The structures depicted here are generated in a similar way to those for single interstitial atoms, shown in section 2. For the sake of clarity, it is necessary to include part of the neighbouring supercell in some of the images. Atoms that form the main part of each defect structure are coloured yellow.

Figure 7: The bipentagon grafted intralayer bridge. DFT predicts this to be the most stable state for a pair of interstitial atoms.

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Figure 8: The bent twin-triangle interlayer bridge.

Figure 9: The flat twin-triangle interlayer bridge.

Figure 10: The αβ double interlayer bridge.

Figure 11: The ββ bent interlayer bridge. 5

Figure 12: The bipentagon interlayer bridge.

Figure 13: The αβ bent interlayer bridge.

Figure 14: The twisted twin-triangle interlayer bridge.

Figure 15: The αα arch bridge. 6

Figure 16: The ββ arch bridge.

Figure 17: The skew bipentagon interlayer bridge.

Figure 18: The αβ double split pair.

Figure 19: The isolated pentagons intralayer defect.

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Vacancies

Isolated lattice vacancies are expected to experience a Jahn-Teller distortion, that results in a pairwise reconstruction for two of the three atoms neighbouring the empty lattice site. This is confirmed by DFT calculations. During vacancy migration in graphite, the moving atom passes near to, but slightly offset from the perfect, C2v -symmetry, split vacancy state. Although the reconstructed α and β vacancies are stable energy minima for the AIREBO potential, the lowest energy vacancy is the split vacancy configuration.

Vα (LDA)

Vβ (LDA)

Transition state (LDA)

C2v split V (AIREBO)

Figure 20: Isolated lattice vacancies in graphite, viewed along the prismatic direction. Each image represents one pair of neighbouring graphene sheets taken from supercell models of the defects that have four sheets. The atoms are highlit in contrasting colours to aid identification, where green are in the front sheet, red are in the sheet behind, and yellow are defect-related. The upper pair of images represent α and β vacancies, according DFT calculations, lower left is the transition state for migration, also according to DFT, and lower right is the lowest energy vacancy for the AIREBO potential.

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Coplanar vacancy pairs

The images for coplanar vacancy pairs follow a similar pattern to the isolated vacancies (section 4) where each one shows one pair of neighbouring graphene sheets from a supercell containing four sheets.

Figure 21: The haeckelite structure divacancy.

Figure 22: The butterfly defect divacancy.

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Figure 23: The nearest neighbour αβ divacancy.

Figure 24: The ‘second neighbour’ divacancy.

Figure 25: The twin pentagon-heptagon divacancy.

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Figure 26: The offset trans third neighbour divacancy.

Figure 27: The cis third neighbour divacancy.

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Cross-layer vacancy pairs

In the images of cross-layer vacancy pairs, defect-related atoms in one graphene sheet are coloured yellow, and are coloured dark pink in the neighbouring sheet. Other atoms from the host crystal are coloured green. In contrast to the images in the main article, which are made from the pair of graphene sheets containing the defect, each image shown here is generated from all four graphene sheets that the supercell models contain.

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ββ∗

Figure 28: The single-cross second-neighbour ββ divacancy, V2,1 .

ββ

Figure 29: The double-cross close ββ divacancy, V2,2 .

ββ

Figure 30: The single-cross close ββ divacancy, V2,1 .

αβ

Figure 31: The single-cross αβ divacancy, V2,1 . 12

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Displacement defects

The illustration of the Stone-Wales defect (figure 32) is constructed in a similar way to that for the isolated vacancies (section 4) and coplanar vacancy pairs (section 5) i.e. it uses two sheets, one of them containing the defect, taken from the supercell model. The intimate Frenkel defect images and the ‘pinch’ defect image follow a similar pattern to the illustrations of interstitial defects (sections 2 and 3).

Figure 32: The Stone-Wales defect.

Figure 33: The α intimate Frenkel defect.

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Figure 34: The β intimate Frenkel defect.

Figure 35: The ‘pinch’ defect. This structure is an artefact of both the EDIP and the AIREBO potential.

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Parameters

The parameters used for the AIREBO potential are those provided with the lammps program package [5], which is available from hrefhttp://lammps.sandia.govhttp://lammps.sandia.gov. The parameters used by the edip program package for the EDIP are given in [3, 4].

References [1] P. R. Briddon and R. Jones. LDA calculations using a basis of Gaussian orbitals. Phys. Status Solidi B, 217(1):131–171, January 2000. [2] J. P. Goss, M. J. Shaw, and P. R. Briddon. Marker-method calculations for electrical levels using Gaussian-orbital basis sets. In D. A. Drabold and S. K. Estreicher, editors, Theory of defects in semiconductors, volume 104 of Topics Appl. Phys., chapter 3, pages 69–94. Springer-Verlag, Berlin Heidelberg, 2007. [3] N. A. Marks. Generalizing the environment-dependent interaction potential for carbon. Phys. Rev. B, 63(3):035401, December 2000. [4] N. A. Marks. Modelling diamond-like carbon with the environmentdependent interaction potential. J. Phys: Condens. Matter., 14(11):2901–2927, March 2002. [5] S. Plimpton. Fast parallel algorithms for short-range molecular dynamics. J. Comp. Phys., 117(1):1–19, March 1995. [6] M. J. Rayson. Lagrange-Lobatto interpolating polynomials in the discrete variable representation. Phys. Rev. E, 76(2):026704, August 2007. [7] M. J. Rayson and P. R. Briddon. Rapid iterative method for electronicstructure eigenproblems using localised basis functions. Computer Phys. Commun., 178(2):128–134, January 2008.

University of Surrey Guildford GU2 7XH United Kingdom

Newcastle University Newcastle upon Tyne NE1 7RU United Kingdom 24th July 2015

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