First-Principles Study on Migration and Coalescence of Point Defects

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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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The Journal of Physical Chemistry

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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

319

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

320

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

321

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

322

with the V2 (555-777) defect.

323

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

324

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,

326

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

327

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

328

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


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329 330

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)

332

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

333

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

334

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

335

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

336

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

337 338

3

Conclusion

339

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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341

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

342

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

343

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

344

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

345

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

346

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

347

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

348

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

349

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

350

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

351

vacancy defect.

352

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

353

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

354

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

355

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

356

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

357

Acknowledgement:

When the divacancy defects meet the carbon adatom defect, they

358

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

359

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

360

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

361

Project Funded by the Priority Academic Program Development of Jiangsu Higher

362

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

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TOC

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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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First-Principles Study on Migration and Coalescence of Point Defects

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