First-Principles Study on Migration and Coalescence of Point

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First-Principles Study on Migration and Coalescence of Point Defects in Mono-layer Graphene Liang Wu, Tingjun Hou, Youyong Li, Kwok Sum Chan, and Shuit-Tong Lee J. Phys. Chem. C, Just Accepted Manuscript • DOI: 10.1021/jp405130c • Publication Date (Web): 22 Jul 2013 Downloaded from http://pubs.acs.org on July 24, 2013

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First-Principles Study on Migration and Coalescence of Point Defects in Mono-layer Graphene 1 Liang Wu , Tingjun Hou1, Youyong Li1*, K. S. Chan2*, Shuit-Tong Lee1 1. Institute of Functional Nano & Soft Materials (FUNSOM) and Jiangsu Key Laboratory for Carbon-Based Functional Materials & Devices, Soochow University, Suzhou, Jiangsu 215123, China 2. Center for Functional Photonics, Department of Physics and Materials Science, City University of Hong Kong, Hong Kong. Email: [email protected], [email protected]

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Graphene is a promising material due to its outstanding properties. Point defects

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can be created artificially and tailor / improve the relative properties of pristine

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graphene. Defective graphene is potential material for electronic devices and sensors.

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Under irradiation or heat treatment, defects may diffuse and aggregate together. Here

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we perform density functional theory (DFT) to illustrate the migration and

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coalescence processes of the point defects. We find that the presence of

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single-vacancy (SV) defect stimulates the migration of another SV defect to bring

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them together and form the adjacent single vacancy defects. The adjacent single

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vacancy defects can combine into a divacancy defect and we study the path. We also

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study the structural rearrangement of divacancy defect and conclude the relative

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stability of different types of divacancy defects. In addition, we find that divacancy

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defect [V2 (5-8-5) defect] is ready to be healed by a neighboring adatom defect. In

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comparison, divacancy defect [V2 (555-777) defect] cannot be healed by an adatom

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defect directly. Our results provide the mechanism of migration and coalescence

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processes of point defects in graphene, which is useful for nano-engineering of

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graphene with defect.

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Abstract

Keywords: DFT; graphene, defect, vacancy, transition state.

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1. Introduction

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Graphene, one of the carbon allotropes, has attracted extensive attentions when

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isolated by mechanical exfoliation firstly.1 Ideal graphene has outstanding

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properties.2-6 However, structural defects can alter the mechanical and electronic

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properties dramatically.7-13 Defective graphene shows promising application in

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electronic devices and sensors due to its unique properties.7, 14-15 Atomistic simulation

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of graphene reveals unexpected effect caused by defects.16-17 Unlike other bulk

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materials, the sp2-hybridized carbon atoms can reorganize themselves and then form

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different polygons, which causes the curvature of the graphene layer.18-19 Combined

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with heat treatment, irradiation is an effective way to introduce defect in graphene

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layer artificially.14, 20-23 These make defect a powerful approach to tailor / improve

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properties of graphene. Defects are building blocks for the promising graphene-based

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materials.

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Intrinsic point defects (such as topological defects, vacancies and adatoms) of

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graphene have been experimentally proved to be numerous. And these defects have a

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strong impact on the mechanical, electronic and magnetic properties.13, 19, 24-29 With

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the help of high-resolution transmission electron microscopy (HRTEM) and scanning

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tunneling microscope (STM), structural defects in graphene can be recognized with

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atomic resolution.14,

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distortion which saturates two of the three dangling bonds toward the missing atom,

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and this defect can induce intrinsic magnetism in graphene-based materials.9,

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Divacancy [V2 (5-8-5) defect] will also undergo a structural reconstruction and it can

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further rearrange its structure by a Stone-Wales type bond rotation.14 Calculated

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results found that it is much easier for single vacancy diffusing on the graphene plane

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than the divacancy.7 With tight-binding molecular dynamics (TBMD) simulations,

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Lee et al.9 showed that two single vacancies may coalesce into V2 (5-8-5) defect at

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3000 K then further reconstruct into a new defect structure [V2 (555-777) defect] at

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higher temperatures. Under aberration-corrected HRTEM, it is found that V2 (555-777)

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defect can also convert to V2 (5555-6-7777) defect by a bond rotation process

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Single-vacancy [V1 (5-9)] undergoes a Jahn-Teller

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similarly.14 A band gap of more than 0.3 eV is opened at the Fermi level in V2

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(5555-6-7777) defect, which show the potential application in field-effect

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transistors.33

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generated accompanying with the vacancies. It has been experimentally observed in

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the periphery of the vacancies.27 It can heal single-vacancy defect in graphite and the

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energy barrier is 1.3 eV.34 The migration barrier of adatom is low and two

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neighboring adatoms can form an Inverse Stone-Wales (ISW) defect.7, 35 Moreover,

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the reaction barriers to construct the ISW defect are sufficiently low in the vicinity of

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vacancies, another important ‘blister’ defect pattern could be achieved through the

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thermally activated restructuring of coalesced adatoms.18 Actually, the adatom may

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also coalesce with the vacancy defect while the interaction mechanisms are scarcely

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understood at a basic level.

The adatom defect, formed by an additional carbon atom, is usually

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Here we study the migration and coalescence of the common adatom defect and

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the vacancy defect. We use density-functional theory (DFT) to make a systematical

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study on the structural evolution of the migration and coalescence processes for these

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point defects. Our results provide valuable information for the application of defect

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nano-engineering and understanding the growth and annealing mechanisms.

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2. Computational methods and details

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Our calculations are performed using the density-functional theory (DFT)

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framework, implemented in the DMOL3 package36-37. The calculations are performed

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with a double numeric plus polarization (DNP) basis set.35 Generalized gradient

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approximation (GGA) with the Perdew-Wang parameterization (PW91) is used to

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describe the exchange-correlation energy functional.38 The simulated supercell

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contains 128 carbon atoms of the mono-layer pristine graphene and the vacuum

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region is selected to be 15Å to avoid the interaction between neighboring graphene

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layers.9,

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convergence of 1×10−5 hartree on the total energy. The calculated C-C bond length is

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1.42Å, which agrees with previous researches.40-41

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39-40

The Kohn–Sham equations are solved self-consistently with a

The formation energy Ef is defined as:

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Ef=Ed+NμC- Ep

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In this formula, Ed and Ep represent the total energies of defective and pristine

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graphene, positive N represents the number of carbon atoms removed from the pristine

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graphene (negative N represents the number of carbon atoms added to the pristine

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graphene), andμC is the chemical potential of carbon atom.

eq. (1)

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The linear or quadratic synchronous transit (LST/QST) transition state (TS)

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search algorithm combined with conjugate gradient refinements and TS optimization

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techniques are used to construct a reaction pathway and explore the energy barriers of

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the structural evolution processes.42 A vibrational analysis is carried out to confirm

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that the transition state configuration is stationary after a successful TS search

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calculation. A true transition state will have exactly one mode with a negative

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vibrational frequency (imaginary vibrational) while all other frequencies will be

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real.43 Meanwhile, the confirmation calculations with the nudged-elastic band (NEB)

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algorithm44 are performed to ensure the direct connection of transition states with the

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respective reactant and product. This TS search method has been checked and made

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comparisons with experimental results.35, 39

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3. Results and discussion

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3.1 Migration and coalescence of two single vacancies

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It is well-known that defects are not always stationary and that their migration

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has an important influence on the properties of a defective crystal. In graphene, each

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defect shows certain mobility parallel to the graphene plane. The mobility might be

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immeasurably low for extended vacancy complexes, or very high for adatoms on a

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pristine graphene lattice. The migration is generally governed by an activation barrier

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which depends on the defect type. Thus the migration increases exponentially with

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temperature.

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The simplest defect in any material is the missing lattice atom. As shown in

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Figure 1, single-vacancy (SV) in graphene undergoes a Jahn-Teller distortion which

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leads to the saturation of two of the three dangling bonds toward the missing atom.

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One dangling bond always remains owing to geometrical reasons. This leads to the

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formation of a five-membered and a nine-membered ring [V1(5-9) defect]. We

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determine that the formation energy of the SV defect is 7.73eV and the migration

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energy barrier of a SV in pristine graphene is 1.33eV. These calculated results are

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consistent with previous estimation.39

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Previous studies showed that carbon implantation can produce single-vacancy

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defects and vacancy clusters which induces the ferromagnetism of the system. When

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the samples are annealed at 200 °C, the ferromagnetism disappears simultaneously.45

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The migration and coalescence of single-vacancy is achievable by the tight-binding

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molecular dynamics method.9

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Here we calculate the energy barriers for each step of the migration, coalescence

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and rearrangement processes by the transition state (TS) search method. We compare

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different reaction paths and determine the most favorable one. The study of the

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structural evolution processes of SV defects is essential for understanding the

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annealing mechanisms and the influence on magnetic properties.

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As shown in Figure 1, the migration processes of two neighboring SV defects

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are investigated. Defect structure A (shown in Figure 1a) is defined as the initial

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structure, then one SV defect V0 migrates toward carbon atom1. The migration

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process of V0 can also be regarded as the hopping of carbon atom1. The energy barrier

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is 1.26eV and it is a little lower than that of a SV defect migrates in pristine graphene.

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Then the SV defect V1 (structure B) is formed as shown in Figure 1b. Similarly, defect

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V1 can also migrate toward carbon atom2 and the two adjacent single vacancies

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(Di-SV) defect is achieved finally (Figure 1c) with an energy barrier of 0.67eV. The

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formation energy of the Di-SV defect is 12.74eV, and it is 2.28eV lower than that of

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the two neighboring single vacancies. Figure 1d is the calculated energy profiles of

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the migration processes of the two neighboring single vacancies. TA-B, TB-C is the

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transition state of the two steps of the migration processes. The energy barrier of the

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second migration step is much lower. These results suggest that the presence of SV

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defect accelerates the migration of the neighboring SV defect to come together.

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Figure 1 The migration processes of two neighboring single vacancies are studied by

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the TS search method: (a) initial structure A in our calculation; (b) structure B; (c)

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structure C; (d) Calculated energy profiles of the migration processes of the two

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neighboring single vacancies, TA-B, TB-C is the transition state of the defect structure

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A→B and B→C reaction path, respectively.

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As shown in Figure 2a and 2b, the Di-SV defect can coalesce together and

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become divacancy V2 (5-8-5). The formation energy of V2 (5-8-5) defect is 7.41eV,

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which is much lower than Di-SV defect. Thus the energy release is abundant for the

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coalescence process.

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with tight-binding molecular dynamics (TBMD) simulation, and the calculated energy

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barrier of the coalescence is 1.60eV with the TB method (1.52eV with LDA method).

Lee et al.9 presented the promising coalescence mechanisms

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With the same reaction path (reaction path Ⅰ shown in Figure 2a), Zhang et al.39

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show that the energy barrier is 2.17eV with the TS search method. However, we

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propose two comparable reaction paths for the coalescence process of Di-SV defect to

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the V2 (5-8-5) defect. In reaction path Ⅰ, carbon atom4 (colored in red) moves to the

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site of vacancy V2 and carbon atom3 (colored in blue) occupies the site of atom4

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simultaneously. TS1 is the transition state of the coalescence process and the energy

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barrier of this reaction path is 1,97eV.

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In the other reaction path (path II as shown in Figure 2b), the carbon atom4 is

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rigid and then carbon atom3 moves to the site of vacancy V2 via a bond rotation. The

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calculated energy barrier of this reaction path is 1.17eV, and TS2 is the transition state

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of the coalescence process. The calculated results is obvious that the energy barrier of

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reaction path Ⅱ is much lower than reaction path I. In a word, the energy barrier is

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not high for the Di-SV defects coalesce to the divacancy [V2 (5-8-5)] defect and the

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energy release is abundant. After the coalescence, the dangling bond of the SV defect

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is saturated, so the V2 (5-8-5) defect is nonmagnetic. Our calculation results can

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explain the disappearance of ferromagnetism after annealing.45

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As shown in Figure 2c, the gray atoms represent TS1 structure and the brown

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atoms represent the TS2 structure. The structural difference between the two

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transition states is not obvious. The angle α is 132.79o for TS1 and 146.32o for TS2.

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Thus we perform further calculations to compare the two transition states.

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As shown in Figure 2c, the distribution of the bond lengths for the two transition

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state structures is presented. The equilibrium C-C bond length of pristine graphene is

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1.420Å (dashed line). We can clearly find that most of the bond lengths are near the

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dashed line (the equilibrium C-C bond length) and the distribution of the bond lengths

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for TS2 structure is closer to the dashed line. The calculated average bond length of

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TS1 and TS2 is 1.426Å and 1.423Å, respectively. Since the existence of vacancy

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defect, the average bond length is extended compared to the pristine graphene and the

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TS2 shows a shorter average bond length. The shorter average bond length in TS2

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explains the lower energy barrier of reaction path II.

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In previous studies, reaction path Ⅰ is taken as the structural evolution process

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for the coalescence of Di-SV defects. But our calculation results indicate that reaction

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path II is more favorable due to lower energy barrier. Moreover, the coalescence of

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Di-SV defects (1.17eV) also has a lower energy barrier than the migration of SV

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(1.33eV) on graphene layer. Thus, the nearing SV defects will aggregate together

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when they migrate on the graphene layer under irradiation heat treatment.

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Figure 2 (a) Energy profiles for the coalescence of the Di-SV defect with the

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transition path I;(b) Energy profiles for the coalescence of Di-SV with the transition

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path II. (c) The distribution of the bond lengths of the two transition state structures,

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the solid square represents the distribution of bond length for TS1 and the open circle

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represents the distribution of bond length for TS2.

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3.2 Structural rearrangement of divacancy defect

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Divacancy defect can reconstruct itself and form different defect configurations.

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The V2 (5-8-5) defect can become the V2 (555-777) defect via a Stone-Wales like

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bond rotation. Similarly, the V2 (555-777) defect can also reconstruct itself and form

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the V2 (5555-6-7777) defect. Three different configurations of the divacancy defect

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and the structural reconstruction processes are illustrated in Figure 3. The transition

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state (TS3) of the Stone-Wales type transformation for the V2 (5-8-5) defect is

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determined and the calculated energy barrier of the structural rearrangement is 5.1eV.

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The formation energy of the V2 (555-777) defect is 6.79eV and it is the lowest among

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the three configurations of the divacancy defect. The further structural reconstruction

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process of the V2 (555-777) defect is similar with an energy barrier of 6.28eV and

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TS4 is the transition state. In addition, the formation energy of the V2 (5555-6-7777)

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defect is 7.08eV, which is between the V2 (555-777) defect and the V2 (5-8-5) defect.

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The energy barriers of the reconstruction processes are much higher than the

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migration and coalescence processes of SV defects, and the structural rearrangements

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are difficult to occur unless under particular conditions. Thus, the reconstructed

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divacancy configurations are stable at room temperature and we can engineer

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patterned defects with different configurations for interesting properties.

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Figure 3 Energy profiles for the structural rearrangement of the divacancy defects,

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TS3 is the transition state of the reconstruction process for V2 (5-8-5) defect to V2

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(555-777) defect and TS 4 is the transition state of the reconstruction process for V2

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(555-777) defect to V2 (5555-6-7777) defect.

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3.3 Interaction of adatom defect and divacancy defect

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The adatom defect is usually generated accompany with the vacancy defect and

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it can heal the vacancy defect under radiation or heat treatment. Moreover, we can

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engineer different defect patterns with the adatom defect artificially. V2 (5-8-5) defect

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and V2 (555-777) defect are the most commonly studied divacancy defects in

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graphene and carbon nanotube. We are interested in the migration and coalescence of

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adatom defect in the periphery of the two common divacancy defects. Experimental

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and calculated results show that the adatom defect prefers the bridge state structure on

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graphene layer.27,

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between the adjacent bonds. Once the adatom defect approaches near the divacancy

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defect, the migration and coalescence processes become more complicated.

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The adatom migrates toward the vacancy defect by hopping

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In order to find the possible reaction paths of these structural evolution processes,

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we perform the adsorption of carbon adatom (Cad) on the defective graphene firstly.

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The adsorption model is illustrated in Figure 4. The Cad is introduced on the top of the

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V2 (5-8-5) defect, and the distance is 3Å. Then the structure is optimized without any

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constrains. The final defect structure of Cad on defective graphene is achieved after

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geometrical optimization. A series of different adsorption sites near the divacancy

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defect (in the red dashed circle shown in Figure 4a) are performed with the same

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method. By analyzing the optimized final structures of the Cad adsorption, we can find

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out the most possible migration and coalescence processes of the adatom defect. The

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energy barriers of the structural evolution processes are determined by the TS search

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method.

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Figure 4 Adsorption model of carbon adatom (Cad) on the V2 (5-8-5) defect. (a) Top

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view of the adsorption model and the adsorption area (in the red dashed circle) is

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around the divacancy; (b) Side view and the distance between the Cad and the

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defective graphene is 3Å.

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After the calculation of Cad adsorption, we present the most possible migration

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and coalescence processes of adatom defect with V2 (5-8-5) defect. The defect

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configuration shown in Figure 5a (structure A1) is defined as the initial structure of

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the calculation. Then the carbon adatom defect (Cad) migrates toward the V2 (5-8-5)

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defect and structure B1 is formed. Afterwards, the Cad defect continues migrating and

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there are three possible C-C bonds (b1, b2, and b3) where the Cad defect may hop

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toward. The energy barriers of the three possible migration paths are determined. The

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location of bond b1 is found to be the most favorable one. As shown in Figure 5c,

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structure C1 is formed after the hopping of the Cad to bond b1. It is more complicated

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containing a tetragon, a pentagon, a heptagon and a decagon, which cause the

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curvature of the graphene. The side view of structure C1 is shown in Figure 5d. The

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Cad defect is almost in the plane of the graphene layer in structure C1.

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Structure C1 can be transformed into a SV defect (shown in Figure 4e and f).

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Each possible recombination paths are considered and two reasonable ones are

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presented. In one recombination path, as shown in Figure 5c, the carbon atom1

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migrates to the site of the vacancy V1, meanwhile the carbon atom2 connects to the

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adatom Cad and the recombination energy barrier is 1.36eV. Finally, the SV defect

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structure E1 is achieved. In the other recombination path, the carbon atom2 migrates

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to the site of the vacancy V2 with a C-C bond rotation and the other single vacancy

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configuration (structure F1) is achieved. The recombination energy barrier is 1.95eV,

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which is higher than that of the former reaction path. Thus the recombination path C1

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to E1 is more energetically favorable. The calculated energy profiles are shown in

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figure 5g. The SV defects formed in the two recombination paths are different, but the

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sites of Cad defect in the two SV defects are consistent with each other.

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In previous experimental researches, it is found that the adatoms appear mostly

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in the vicinity of the vacancies and the vacancy and neighboring adatom has been

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predicted to have a recombination barrier.27 As shown in Figure 5g, our results show

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that the migration energy barrier of the Cad defect is 0.91eV and 0.73eV respectively,

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but the recombination energy is 1.36eV, which is higher than the migration process.

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Our calculation results are consistent with the experimental results. In summary, the

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Cad defect migrates in graphene by hopping between the adjacent bonds, and when it

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approaches the V2 (5-8-5) defect it can heal V2 (5-8-5) defect into a SV defect with an

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energy release of 5.79eV. Thus, in order to get the designed defect pattern in the

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vicinity of V2 (5-8-5) defect with adatom defect, it is essential to avoid the

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recombination of the defects.

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Figure 5 The migration and recombination processes of adatom defect and V2 (5-8-5)

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defect: (a) initial structure A1 in our calculation; (b) structure B1; (c) Structure C1; (d)

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Side view of structure C1; (e) Structure E1, it is a SV defect and the missed carbon

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atom is in the site of vacancy V1; (f) Structure F1, it is another SV defect, and the

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missed carbon atom is in the site of vacancy V2; (g) Calculated energy profiles of the

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migration and recombination processes of Cad defect and V2 (5-8-5) defect, TA1-B1,

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TB1-C1, TC1-E1, and TC1-F1 are the transition states of the structural evolution processes.

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Similarly, we also study the migration and coalescence processes of Cad defect

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and V2 (555-777) defect. As we discussed above, the Cad adsorption on V2 (555-777)

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defect is studied firstly with the same method as the Cad adsorption on V2 (5-8-5)

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defect. The most possible reaction paths of the migration and coalescence processes

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are shown in Figure 6.

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Structure A2 is defined as the initial structure. We find that Cad defect migrates

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toward the V2 (555-777) defect by hopping between the adjacent bonds. Unlike the

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migration processes toward V2 (5-8-5) defect, structure B2, C2, D2, E2, and F2 is

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formed with the structural evolution processes of the Cad migration, respectively. The

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energy barrier of the migration processes for A2→B2, B2→C2, C2→D2, D2→E2,

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and E2→F2 is 0.43eV, 1.00eV, 0.78eV, 0.39eV and 0.56eV, respectively. As shown in

313

Figure 6e, two adjacent heptagon of the V2 (555-777) defect become two octagons

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due to the presence of Cad defect in structure E2. Finally, Cad defect migrates to the top

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of carbon atom Cm which is in the middle of the V2 (555-777) defect. Cad is connected

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to the three adjacent carbon atoms and Cm, and the calculated bond length is 1.590 Å

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and 1.565 Å (as shown in Figure 6f and g). The two carbon atoms (Cad and Cm) form a

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dumbbell-like configuration. The calculated energy profiles are concluded in Figure

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6h. The energy release of the recombination of the adatom defect and the V2 (555-777)

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defect is 1.8eV, which is much lower than that of the adatom defect and the V2 (5-8-5)

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defect. Our calculation results indicate that the adatom defect does not coalescence

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with the V2 (555-777) defect.

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V2(555-777) defect is more stable than V2(5-8-5), which makes V2(555-777)

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harder to be healed. Our results indicate that V2(555-777) is a symmetrical structure

325

(Figure 6) and doesn’t show a suitable vacancy to incorporate the adatom. Moreover,

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the adatom defect can migrate out of the V2(555-777) defect via reversed structural

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evolution processes. However, V2(555-777) could rearrange to V2(5-8-5) with an

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energy barrier 5.1eV (section 3.2) and can be healed by adatom indirectly.

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Figure 6 The migration and recombination processes of adatom defect and V2

331

(555-777) defect: (a) initial structure A2 in our calculation; (b) structure B2; (c)

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structure C2; (d) structure D2; (e) structure E2, two adjacent heptagon is formed; (f)

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structure F2, the Cad defect is in the middle of the V2 (555-777) defect; (g) side view

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of structure F2; (h) Calculated energy profiles of the migration and coalescence

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processes of Cad defect and V2 (555-777) defect, TA2-B2, TB2-C2, TC2-D2, TD2-E2, and

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TE2-F2 are the transition states of the structural evolution processes.

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3

Conclusion

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We systematically study the migration and coalescence processes of the common

340

point defects with the transition state search method. The migration energy barrier of

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SV defect or Cad defect in pristine graphene is not high. The presence of SV defect

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promotes the migration processes of another SV defect to come together and form the

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Di-SV defect. The Di-SV defects can transform into a divacancy [V2 (5-8-5)] defect,

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and we compare two possible reaction paths. Calculated energy barrier of the reaction

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path II for the coalescence process is 1.17eV, and the reaction path Ⅱ is more

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favorable. In addition, our calculation explains the disappearance of ferromagnetism

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after annealing. V2 (5-8-5) defect can reconstruct itself to a more stable structure [V2

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(555-777) defect], and V2 (555-777) defect can also reconstruct to V2 (5555-6-777)

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defect. The energy barriers of the reconstruction processes are 5.1eV and 6.28eV,

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which is much higher than the migration and coalescence processes of the single

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vacancy defect.

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show different behaviors. V2 (5-8-5) defect will be healed to a SV defect by the

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adatom defect while the V2 (555-777) defect won’t be healed. The recombination

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energy barrier of the carbon adatom defect and the V2 (5-8-5) defect is 1.36eV. Our

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results provide the mechanism of migration and coalescence of point defects in

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graphene, which is useful for nano-engineering of graphene with defect.

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Acknowledgement:

When the divacancy defects meet the carbon adatom defect, they

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The work is supported by the National Basic Research Program of China (973

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Program, Grant No. 2012CB932400 and 2010CB934500), the National Natural

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Science Foundation of China (Grant No. 91233115, 21273158, and 91227201), and a

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Project Funded by the Priority Academic Program Development of Jiangsu Higher

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Education Institutions (PAPD). This is also a project supported by the Fund for

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Migration and coalescence of point defects in mono-layer graphene studied by DFT

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calculations

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