Instituto de Física ‘Gleb Wataghin’
Tiago Botari
Atomistic Simulations of
Two-dimensional Materials:
Silicene, Graphene, and Carbon Nitrides
(Simulações Atomísticas de Materiais
Bidimensionais: Siliceno, Grafeno e Nitreto de
Carbono)
Campinas
2016
Atomistic Simulations
of Two-dimensional Materials:
Silicene, Graphene, and Carbon
Nitrides
(Simulações Atomísticas de Materiais Bidimensionais:
Siliceno, Grafeno e Nitreto de Carbono)
Thesis presented to the Institute of Physics ‘Gleb Wataghin’ of the University of Campinas in partial fulfilment of the requirements for the degree of Doctor in Science. Tese apresentada ao Instituto de Física ‘Gleb Wataghin’ da Universidade Estadual de Campinas como parte dos requisitos à obtenção do título de Doutor em Ciências.
Supervisor:Douglas Soares Galvão
ESTE EXEMPLAR CORRESPONDE À VERSÃO FINAL DA TESE DEFENDIDA PELO ALUNO
TIAGO BOTARI, E ORIENTADA
PELO PROF. DOUGLAS SOARES GALVÃO.
Campinas
2016
Ficha catalográfica
Universidade Estadual de Campinas Biblioteca do Instituto de Física Gleb Wataghin Lucimeire de Oliveira Silva da Rocha - CRB 8/9174
Botari, Tiago,
B657a BotAtomistic simulation of two-dimensional materials: silicene, graphene and carbon nitrides / Tiago Botari. – Campinas, SP : [s.n.], 2016.
BotOrientador: Douglas Soares Galvão.
BotTese (doutorado) – Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin.
Bot1. Simulação computacional. 2. Dinâmica molecular - Métodos de simulação. 3. Termodinâmica - Propriedades mecânicas. 4. Estabilidade térmica. 5. Processos químicos. 6. Diagrama de fases. I. Galvão, Douglas Soares,1961-. II. Universidade Estadual de Campinas. Instituto de Física Gleb Wataghin. III. Título.
Informações para Biblioteca Digital
Título em outro idioma: Simulações atomísticas de materiais bidimensionais: siliceno,
grafeno e nitreto de carbono
Palavras-chave em inglês:
Computational simulation
Molecular dynamics - Simulation methods Termodynamics - Mechanical properties Thermal stability
Chemical processes Phase diagrams
Área de concentração: Física Titulação: Doutor em Ciências Banca examinadora:
Douglas Soares Galvão [Orientador] Luiz Eduardo Moreira Carvalho de Oliveira Antonio Riul Júnior
Ado Jorio de Vasconcelos José Arruda de Oliveira Freire
Data de defesa: 29-02-2016
Acknowledgements
I would like to thank:
• Professor Douglas Soares Galvão for advising me during this PhD project. • GSONM’s members for fruitful discussions and collaborations.
• The professors from IFGW for contributing to my academic formation
• IFGW’s staff for the indispensable help and support regarding bureaucratic issues. • Professor Volker Blum and the members of AIMS’s group for teaching and
sup-porting me during the internship at Duke university. • My friends for all the support and help.
• FAPESP, for the financial support of the project through grants #2012/12448-3 and #2014/11986-7.
Resumo
Apresentamos três trabalhos distintos em materiais bidimensionais. Nos-sas investigações foram feitas utilizando os seguintes métodos computacionais: Den-sity Functional Theory (DFT), Density Functional Tight-Binding (DFTB) e simulação por dinânica molecular clássica reativa. Nós dividimos essa tese em três partes: I. Pro-priedades Mecânicas e a Dinâmica de Fratura do Siliceno; II. Investigação do Mecanismo de Cura do Grafeno, e; III. Estabilidade Termodinâmica de Materiais de Nitreto de Car-bono.
Na primeira parte (I), investigamos as propriedades mecânicas e o padrão de fratura de membranas de siliceno através de simulações computacionais. O modelo es-tudado consiste de membranas suspensas do tipo armchair e zigzag, sendo as estruturas maiores com aproximadamente 1000 átomos. Inicialmente, investigamos a estrutura de energia mínima do siliceno usando os métodos DFT, DFTB e ReaxFF. Em seguida, inves-tigamos o processo de estiramento das membranas utilizando dinâmica molecular com os métodos DFTB e ReaxFF. Então, a partir da dinâmica molecular, calculamos o módulo de Young, coeficiente de Poisson e o valor da deformação crítica na qual a fratura do material acontece, entre outras. Nós obtivemos que o módulo de Young é 10 vezes menor que o do grafeno em condições similares. Também, analisando o processo de fratura, observa-mos algumas reconstruções para as membranas armchair. Investigaobserva-mos também questões como distribuição do stress, diminuição das ondulações fora do plano da estrutura, con-hecida como buckling), e a dependência da fratura com a variação da temperatura.
Na parte II, utilizando dinâmica molecular clássica reativa, investigamos o processo de cura do grafeno. Empregando um modelo computacional, simulamos as condições dos recentes experimentos de reconstrução de membranas defeituosas de gra-feno, incluindo a dependência da temperatura e os efeitos de aquecimento devido às inter-ações de um feixe de elétrons com o sistema. Nossos resultados mostram que para que o mecanismo de cura do grafeno seja iniciado é necessária a presença de um reservatório de átomos de carbono e energia disponível para esses átomos vencerem a barreira de energia das bordas do defeito. Obtivemos reconstruções perfeitas em duas condições: altas tem-peraturas ou com o uso da fonte de calor (para certas taxas de energia) quando o sistema é mandido em temperatura ambiente.
Na parte III, nós investigamos diferentes fases dos materiais de nitreto de car-bono: oligômeros, cristais moleculares, cristais poliméricos e estrutura tipo grafite. Nós realizamos cálculos usando DFT e outros métodos que vão além do DFT. Primeiramente, otimizamos todas as estruturas estudadas e então, investigamos a estabilidade termod-inâmica e as propriedades eletrônicas. Nossos resultados mostram que a fase polimérica melondeve ser a fase obtida nas sínteses comumente reportadas na literatura. Entretanto, diferentes grupos de maneira equivocada têm identificado a fase g-heptazina como o ma-terial sintetizado. Também investigamos as propriedades eletrônicas para as fases melon e g-heptazina.
Abstract
We present here three works on distinct two-dimensional materials. Our investigations were carried out using the following computational simulation methods: Density Functional Theory (DFT), Density Functional Tight-Binding and classical reac-tive molecular simulations. We divided the thesis to three parts: I. Mechanical Properties and Dynamical Fracture of Silicene; II. Investigation of Graphene Healing Mechanisms, and; III. Thermodynamical Stability of Carbon Nitride Materials.
In part I, mechanical properties and dynamical fracture of silicene membranes were investigated via computational simulations. The studied models consist of arm-chair/zigzag suspended silicene membranes. The larger structures were chosen to have around 1000 atoms. Initially, we investigated the minimum energy structure using DFT, DFTB and ReaxFF. We found a good agreement of minimum energy structures com-paring the three methods and other results reported in the literature. Afterwards, we investigated the stretching process using molecular dynamic simulations with DFTB and ReaxFF methods. Then, using the data from these molecular dynamics simulations, we calculated the Young’s modulus, Poisson’s ratio and critical strain values among others. We obtained that the Young’s modulus of silicene is 10 times smaller than the one for graphene under similar conditions. Also, by analysing the fracturing process, we ob-served some boundary reconstructions for the armchair membranes. We also addressed the stress distributions, unbuckling mechanisms, and the fracture dependence on the tem-perature. We also analysed the differences due to distinct edge morphologies, namely zigzag and armchair.
In part II, classical reactive molecular dynamic simulations were carried out to investigate the graphene healing mechanism. Using a computational model, we simulated the conditions of recent experiments of reconstruction of defective graphene, including the dependence of temperature and heating effects due to electron beam interactions with the system. Our results show that it is necessary a source of carbon atoms to trigger the graphene healing mechanism. Besides that, these atoms must have sufficient energy to overcome the energy barrier of the defect boundaries. We obtained perfect reconstruction of a defect under two conditions: high temperature through the whole system or with the aid of a local heating source at room temperature, for certain energy rates.
In part III, different carbon nitride structures were investigated, such as oli-gomers, molecular crystals, polymeric crystalline phases and graphene-like phases. We carried out calculations using density functional theory (DFT), as well as methods be-yond DFT. We optimized all structures, and investigated their thermodynamic stability and electronic properties. Our results showed that the polymeric melon phase should be obtained in the common synthesis process reported in the literature. Many of these ex-perimental reports incorrectly identified the synthesized material as g-heptazine. We also investigated the electronic properties for the melon structure and g-heptazine.
Contents
1 Introduction 1
1.1 Graphene . . . 4
1.2 Silicene . . . 5
1.3 Carbon Nitride Materials . . . 6
1.4 Objectives . . . 8
1.5 Thesis Layout . . . 8
2 Mechanical Properties and Fracture Dynamics of Silicene 11 2.1 Introduction . . . 12
2.2 Methodology . . . 14
2.3 Results and Discussion . . . 17
2.3.1 Structural investigation . . . 17
2.3.2 Mechanical Properties and Fracture Patterns . . . 18
2.4 Summary and Conclusions . . . 23
3 Investigation of Graphene Healing Mechanisms 27 3.1 Introduction . . . 28
3.2 Methods and Model . . . 29
3.3 Results and Discussion . . . 32
3.3.1 Potential Energy Landscape . . . 32
3.3.2 Temperature Dependence. . . 34
3.3.3 Electron Beam Effects . . . 35
4 Thermodynamic Stability of Carbon Nitride Materials 43
4.1 Introduction . . . 44
4.2 Methodology . . . 46
4.3 Results and Discussions. . . 49
4.3.1 Optimization of the Structures . . . 49
4.3.2 Predicted Kohn-Sham Electronic Band Structures of the Materials 52 4.3.3 Thermodynamic Stability . . . 53
4.4 Conclusions . . . 59
5 Conclusions 61
6 Perspectives 65
References 67
CHAPTER
1
Introduction
In his lecture "Plenty of Room at the Bottom" [1], Richard P. Feynman de-scribed how wide are the possibilities when matter can be controlled at the atomic scale. In special, the possibilities to study phenomena that arise from complexity as well as the strong potential for technological applications. This particular lecture in 1959 is consid-ered the birth of the nanotechnology field.
Feynman was not only talking about simple miniaturization processes. He was rather wondering about how to go to the limits of the known physics, where matter cannot be considered as a continuum medium but should be considered as an arrangement of small pieces: molecules and atoms. This would demand novel approaches, instead of just simple improvements of the pre-existing ones. Looking back, 57 years after this seminal lecture, we have been able to achieve technological developments that go beyond those predicted by Feynman. Nowadays, it is possible to synthesize so tiny materials that we can consider them as a close realization of distinct dimensionalities. However, the nanotechnology field is still very young and we continue to have plenty of room for novel developments.
The properties of a material can be highly dependent on its dimensionality, i.e, the same chemical compound can present distinct characteristics when its crystalline
structure is arranged in 0-D, 1-D, 2-D or 3-D [2,3,4].
Figure 1.1: Carbon materials in different dimensionalities (a) Graphene 2-D; (b) Graphite 3-D; (c) Fullerenes 0-D and; (d) Nanotubes 1-D. Adptaded from [5]
For instance, carbon, a very versatile element, can be found in different forms and multiple dimensionalities, as shown in Figure1.1. In this context, different carbon structures were already synthesized, such as, fullerenes [6, 7] (Figure1.1 (c)) that rep-resent the realization of 0-D structures, linear atomic chain (LAC) and nanotubes [6, 8] (Figure 1.1 (d)) representing 1-D structures, graphene [9] (Figure 1.1 (a)) for 2-D and diamond [10, 11] and graphite for 3-D (Figure 1.1 (b)). The realization and controlled synthesis of these different carbon structures opened new frontiers in materials science.
Fullerenes, one of those structures that started with this revolution, are molec-ular closed-cages made of carbon. These spherically shaped carbon molecules were first reported by Kroto et al. in 1985 [12]. They found a molecule containing 60 carbon atoms and proposed a geometry similar to a soccer ball, i.e, both pentagonal and hexagonal rings forming a closed cage. The large production of fullerenes was achieved five years later in 1990 [13] and, in 1992, it was first found in nature in a Precambrian Russian rock [14]. Fullerenes are promising for different applications, such as, solar cell [15,16], medicinal diagnosis and therapy [17], hydrogen storage [18], and others. [19]
wit-nessed a new and important discovery with the synthesis of carbon nanotubes, tubular structures formed by hexagonal rings. These structures present a high length-to-diameter ratio and represent a close realization of one dimensional structures. Carbon nanotubes were first reported in 1991 by Iijima [20] and their large scale synthesis was demon-strated in 1992 [21,22]. Nowadays, it has already been synthesized with lengths of up to a half-meter [23]. Nanotube properties are highly dependent on its diameter and helicity which can be engineered for particular applications. The most common synthesis tech-nique during the early years was the arc discharge method, which generates a distribution of different families of high quality carbon nanotubes which need of post-synthesis sep-aration [24, 22]. The advent of chemical vapour deposition (CVD) techniques [25, 26] allowed better control over the produced nanotubes’ characteristics, such as length, diam-eter and chirality. There are many possible applications of carbon nanotubes. They are, for instance, widely used in the nanocomposites in order to obtain light and resistant ma-terials [27]. Also, the electronic and transport properties of carbon nanotubes [28] make them suitable for electronic applications, such as flexible electronics [29].
For a long time the realization of two-dimensional materials was put in doubt. Theoretical works had pointed out that two dimensional materials should be thermo-dynamically unstable [30]. Additionally, it was known experimentally that the melting temperature of thin films decrease fast diminishing the material thickness and then they became unstable typically in the order of 12 atomic layers [31,32,9].
In order to obtain single layers, many unsuccessful attempts of splitting lay-ered material were tried. This laylay-ered materials have strong in-plane and weak interaction among the layers. In the first attempts [33], chemical exfoliation was used in order to ob-tain single layer graphene from graphite. However, separation of a single layer from the liquid mixture proved to be very difficult and no single layer was observed. Other works showed that by using chemical exfoliation it is possible to obtain scrolls from multilayer membranes instead of single layers [34,35].
The successful isolation of a two-dimensional structure was reported in 2004 [36]. Novoselov et al., using mechanical exfoliation in place of chemical exfoliation, obtained single layer from graphite, the so-called graphene structure. This process was done by repeated peeling of commercially available highly oriented pyrolytic graphite
(HOPG) using a scotch tape. In the following year, experimental isolation of different mono-layer materials was also reported by Novoselov et al.[3]. They showed isolation of single layers of boron nitride (BN), BSCCO (Bi2Sr2CaCu2Ox) and 2D dicalcogenites
(MoS2, NbSe2) using a mechanical exfoliation technique. These materials opened new
frontiers in materials science, which inspired further investigations on two-dimensional materials such as silicene, germanene, phosphorene, etc [37].
In the next section, we describe in more details the materials that were subject of research in this thesis: graphene, silicene and carbon nitride materials.
Graphene
Graphene is a single layered material with one atom thickness arranged in a honeycomb crystal structure. It can be obtained by different processes, such as mechan-ical exfoliation or chemmechan-ical vapor deposition (CVD) [38]. Early studies on the graphene structure were theoretical investigations of the graphite framework. Graphene was used as a first theoretical approximation for studying the properties of graphite [39, 40, 41]. Although graphene was referenced as an academic material in the past, nowadays it is one of the most active scientific fields [42].
Such interest in graphene materials is related to their properties, which in-clude: (i) presence of a linear dispersion in the electronic band structure, which causes the charge carrier to behave like relativistic mass-less particles [43]; (ii) quantum Hall effect (QHE) even at room temperature [44,45]; (iii) high electronic conductivity [9]; (iv) high thermal conductivity; (v) high mechanical strength and flexibility [46,47,48].
Concerning to graphene applications, there are many possibilities. Graphene high electronic conductivity and chemical stability make graphene-based electronics very interesting [9]. On the other hand, graphene behaves as a semi-metal and some engi-neering in its band gap is needed for many electronic applications. Examples include integration of graphene in electrocatalysis [49], fuel cells [50], batteries [51,52], devices for hydrogen generation [53], field-effect devices [54, 55, 56, 57], and a realization of integrated circuits by IBM researchers reported in 2011 [58]. Also, the mechanical
pro-Figure 1.2: (a) silicene membrane top view; (b) a hexagonal ring of silicene; (c) silicene membrane lateral. Adptaded from [65]
prieties of graphene have been exploited, such as in light composite materials [59]
Silicene
Silicene is a silicon-based single layered material that has the same honey-comb crystal structure of graphene. Takeda et al. using theoretical calculations showed that silicene can be stable [60]. However, while graphene is planar, silicene has a buckled structure, i.e, the adjacent atoms present an out-of-plane displacement, as shown in Fig-ure1.2. Inspired by the graphene realization, other theoretical works have investigated silicene stability and electronic band structure [61,62]. Today different groups have syn-thesized silicene membranes on different substrates [63,64].
Unlike carbon, silicon has a preference for sp3 hybridization over sp2. This preference is evident when we look for stable crystalline structure for silicon atoms is the diamond structure (sp3), while for carbon it is graphite (sp2). This chemical difference leads to distinct features for silicene when compared with graphene. One such feature is the presence of some level of buckling in the crystalline structure of silicene, as demon-strated by theoretical calculations [66]. Despite the presence of buckling in the structure, the Dirac cone in the Brillouin zone (BZ) can still be found [66]. Signatures of Dirac cones were claimed to have been experimentally observed for silicene over silver surface
[67]. However, later, ab initio calculations suggested that the observed conical features in the angle-resolved photoelectron spectroscopy measurements are in fact due to the silver substrate [68,69].
Silicene can be produced in one-dimensional arrays of nano-ribons over a Ag(110) surface [70,71,72,73], while epitaxial growth can be obtained over a Ag(111) surface [74,75,76,77]. Additionally, it can be grown in Ir(111) [78], Au(110) [79], and on non-metallic surfaces of zirconium diboride, ZrBr2(0001) [80]. Evidences of possible
silicene growth over an insulating material, lanthanum aluminate (LaAlO3) surface, were
also reported [81,82].
Despite the reported synthesis of silicene, it is very expensive and difficult to be produced. Some of the necessary conditions for silicene growth are ultra-high vacuum, controlled silicon atom deposition, and control over the substrate temperature [64]. Nev-ertheless, silicene can expand the frontiers of technological applications for nanomaterials [63, 83, 64]. Different theoretical predictions were made for its properties, for instance: quantum spin Hall effect [84], electric field-dependent states [85], giant magnetoresitance [86], and superconductivity [87]. Technological applications of silicene were already the-orized for Li-ion batteries (LiBs) [88, 89, 90, 64] and FET devices, that were recently produced [91].
Carbon Nitride Materials
The interest in carbon nitride (CN) materials [92] is connected to recent ad-vances in their synthesis and technological applications [93, 94, 95, 96]. Many studies have shown that carbon nitride materials are very promising for a broad range of appli-cations such as fuel cells [97], batteries [98], hydrogen storage [99], metal-free photo-catalysis [100], electrocatalysis [101], molecules sieves [102], tribology [103], and others [104]. The possible different carbon and nitrogen arrangements can be exploited to better fit specific applications. There are many theoretical investigations studing various carbon nitride structures [92, 105, 106, 107, 108, 109, 110]. As an example, the triazine and heptazine are some of the structures used to build these materials (Figure1.3).
Figure 1.3: (a) triazine molecule; (b) heptazine molecule the R is arbitrary substituents.
Figure 1.4: Graphitic form of carbon nitride materials. (a) g-h-triazine; and (b) g-h-heptazine.
The triazine molecule (C3H3N3, as shown in Figure1.3a) could polymerize to
form the graphitic phases g-h-triazine and g-o-triazine, as shown in Figure1.4(a). Also, the heptazine molecule (C6N7, as shown in Figure 1 b) is the building unit to form
g-h-heptazine, as shown in Figure1.4 (b). Other structures could also be derived from the graphitic form or from combination of molecular precursors. The melon compound [111] is one of those (see Figure1.5). As a matter of fact, there may be a variety of different sizes and polymeric architectures [112]. The graphitic phase was synthesized by different groups [113,114]. Also, 2D arrangements of heptazines, melon and other CN compounds were synthesized [92].
Despite their importance, theoretical investigations of this class of materials to date have only covered a limited range of the structures of experimental interest. A number of carbon nitride structures have their electronic and mechanical properties re-ported, in most part of graphitic structures [115, 116,117,118, 117,119] and molecular precursors [120].
Figure 1.5: Possible structures of melon polymer: (a) compact structure model (b) linear form; (c) symmetric triangular form.
Objectives
The constant emergence of new and intriguing experimental results in the field of two-dimensional materials creates the need for theoretical investigations and explana-tions in materials science. The theoretical approach can be made at different levels, either supporting previous interpretations from experiment or providing novel explanations for intriguing phenomena. In this sense, we investigated three different materials for which novel experimental results opened new questions: I) mechanical properties and dynami-cal fractures for silicene membrane, II) investigation of graphene healing mechanism, and III) the thermodynamical stability of carbon nitride materials.
Thesis Layout
This thesis is organized into six chapters. In chapter 1 we present an intro-duction where we give special attention to two-dimensional materials such as graphene, silicene and those based on carbon nitride. In chapter 2 we present the mechanical and fracture properties of silicene. In chapter 3 we present the investigation of graphene heal-ing mechanisms. In chapter 4, the investigation of thermodynamics and stability of carbon nitride materials is presented. In chapter 5 we present general conclusions and finally, in
CHAPTER
2
Mechanical Properties and
Fracture Dynamics of Silicene
As graphene became one of the most important materials today, there is a renewed interest on other similar structures. One example is silicene, the silicon ana-logue of graphene. It shares some remarkable graphene properties, such as the Dirac cone, but presents some distinct ones, such as a pronounced structural buckling. We have investigated, through density functional based tight-binding (DFTB), as well as reactive molecular dynamics (using ReaxFF), the mechanical properties of suspended single-layer silicene. We calculated the elastic constants, analyzed the fracture patterns and edge re-constructions. We also addressed the stress distributions, unbuckling mechanisms and the fracture dependence on the temperature. We analysed the differences due to distinct edge morphologies, namely zigzag and armchair. The work of this chapter was published in Physical Chemistry Chemical Physics journal [121].
Introduction
Carbon nanostructures have been proposed as the structural basis for a series of new technological applications. The versatility that carbon exhibits in forming differ-ent structures can be attributed to its rich chemistry, reflected on the fact that it can as-sume three quite distinct and different hybridization states: sp3(diamond), sp2(graphite, graphene, fullerenes and nanotubes [122]) an sp (graphynes [123, 124, 125]). Carbon based structures of low dimensionality exhibit extraordinary structural, thermal [126] and electronic [127] properties. Among these structures, graphene (see Figure 2.1) has been considered one of the most promising [128,129,130] due to its unique electronic and me-chanical properties. However, its zero bandgap value hinders transistor applications [130]. As a consequence, there is a renewed interest in other possible graphene-like structures, based on carbon or in other chemical elements. Other group IV elements, such as silicon and germanium, present a chemistry similar to that of carbon in some aspects, although the number of known carbon structures surpasses those based on silicon or germanium. A natural question is whether these elements could also form two dimensional honeycomb arrays of atoms, similar to graphene [131]. The corresponding silicon and germanium structures were named silicene (see Figure2.1) and germanene [132], respectively. Sil-icene was firstly predicted to exist based on ab initio calculations in 1994 [133] and has been recently synthesized by different groups [134,135,76].
Silicene presents some properties that makes it very promising to electronic applications. The electronic Dirac cone exhibited by graphene is also found in silicene [132]. A notable difference between graphene and silicene is that while the former is completely planar, the latter presents a significant level of buckling, meaning that in sil-icene atoms are not in purely sp2hybridized states. This is due to the pseudo-Jahn Teller effect [136, 137], which introduces instability in high symmetry configurations, and can be exploited in some electronic applications [137]. It has been pointed out that pucker-ing causes loss of the sp2 character, lowering the plane stiffness and the fract that linear atomic chains (LACs) may be formed during the fracturing process [138]. It is expected that some level of buckling should be always present in silicene, independently of the strain value [139]. For hydrogenated silicenes (the so-called silicanes), it has been
pro-Figure 2.1: Schematic view of graphene and silicene membranes, in the same scale. (a/c) and (b/d) refer to frontal and lateral view of graphene and silicene, respectively.
posed that the buckling should decrease linearly with the strain [139]. In the last years sil-icene has been object of many experimental and theoretical investigations [63,140,141]. Silicene nanoribbons have been experimentally produced over Ag(110) surface [140]. Larger silicene nanosheets have also been synthesized [142]. Some of the theoretical aspects investigated include tuning of electronic properties under stress load [143, 138], transitions from semimetal to metal [144], bandgap dependence on buckling geometries [145], mechanical properties [138,65,146,147], formation of silicene between graphene layers [148], the influence of defects [149] and chemical functionalizations [150]. How-ever, most studies in the literature have been based on small structures.
There are several studies regarding fracture mechanisms on silicene mem-branes under strain[151,152, 146]. The contribution of the present work comes from an investigation of the relative importance of aspects such as edge terminations (armchair and/or zigzag), membrane size and temperature effects. We have carried fully atomistic molecular dynamics (MD) simulations of silicene under dynamical strain at finite temper-atures using reactive classical molecular dynamics in association with ab initio density functional theory (DFT) and tight binding methods.
Table 2.1: Comparison between our data and available results in the literature. a0 is the lattice
parameter, ∆ is the buckling value, dSi−Siis the Silicon bond distance, C is the plane stiffness, ν
is the Poisson ratio and εc is the critical strain. (ZZ) and (AC) stand for Zigzag and Armchair
directions, respectively. ’*’ means this value was estimated from the curve in Fig. 1 (g), from Topsakal and Ciraci [138].
Method Structure a0 ∆ dSi−Si C ν εc
Ref. - Å Å Å N/m -DFT-LDA[131] Silicene 3.83 0.44 2.25 62 0.30 -DFT - LDA[143] Silicene 3.83 0.42 2.25 63.0 0.31 20 DFT-GGA-ours Silicene 3.83 0.48 2.28 - - -ReaxFF-ours Silicene 3.80 0.67 2.3 - - -SCC-DFTB-ours Silicene 3.87 0.59 2.32 - - -DFT-GGA[153] Silicene - - 62.4(ZZ)/59.1(AC) - -DFT-GGA[138] Silicene - - - 62.0 - -DFT-GGA[139] Silicene - 0.45 2.28 60.06(ZZ)/63.51(AC) 0.41(ZZ)/0.37(AC) 14(ZZ)/18(AC) MD-EDIP[153] ACM/ZZM - - - 64.6/65.0 19.5/15.5 SCC-DFTB-ours ACM/ZZM - 0.59 2.32 62.7/63.4 0.30/0.30 17/21
ReaxFF-ours ACM/ZZM - 0.67 2.3 43.0 0.28/0.23 15/30 DFT-GGA[138] ACM - - - 51.0 - 23* DFT-GGA[139] Silicane - 0.72 2.36 54.50(ZZ)/54.79(AC) 0.25(ZZ)/0.23(AC) 33(ZZ)/23(AC) DFT-GGA[150] Silicane 3.93 0.72 2.38 52.55 0.24
-Methodology
We studied the structural and dynamical aspects of silicene membranes under strain and their fracture patterns using classical and quantum methods. Equilibrium ge-ometries were studied with three different methods, DFT, with the code Dmol3[154,155], density functional based tight-binding method, with DFTB+ [156] and reactive classical molecular dynamics, via ReaxFF[157]. DFT calculations offer higher accuracy, however, in order to reliably simulate the rupturing dynamics of silicene membranes we need to use large systems, precluding the use of DFT due to the high computational costs. Thus, for the dynamical studies we used only tight-binding and reactive classical molecular dy-namics calculations. The structural calculations with DFT were used in order to validate the accuracy of the other used methods.
For the DFT calculations, we used the Dmol3package as implemented on the Accelrys Materials Studio suite [154,155]. We carried out geometry optimization calcu-lations with the Perdew-Burke-Ernzerhof (PBE) functional under the generalized gradient approximation (GGA), with all atoms free to move and full cell optimizations. The con-vergence criteria were 10−4eV in energy, 0.05eV / Å for the maximum force and 0.005 Å as the maximum displacement. Core electrons were explicitly treated and a double
nu-merical plus polarization (DNP) basis set was used. Since the largest silicene membranes studied in this work contain approximately 1600 atoms, far beyond the reasonable size for a long-time all electron dynamical calculation using DFT methodology, we also used the density functional based tight-binding method (DFTB) for systems of intermediate size (hundreds of atoms) as well as a reactive force field method for systems of large size (∼ 1600) atoms.
The tight-binding calculations were carried out using the Self-Consistent Charge Density Functional based Tight-Binding (SCC-DFTB) [158,159] method, as implemented on DFTB+ [156]. The Density Functional based Tight-Binding (DFTB) is a DFT-based approximation method and can treat systems composed by a large number of atoms. SCC-DFTB is an implementation of DFTB approach and has the advantage of using self-consistent redistribution of Mulliken charges (SCC) that corrects some deficiencies of the non-SCC standard DFTB methods [159]. Dispersion terms are not, by default, considered in any DFTB method and were included in this work via Slater-Kirkwood Polarizable atomic model, as implemented in the DFTB+ package [156].
Reactive classical molecular dynamics simulations were carried using the ReaxFF method [157]. ReaxFF is a reactive force field developed by van Duin, Goddard III and co-workers for use in MD simulations of large systems. It is similar to standard non-reactive force fields, like MM3[160] in which the system energy is divided into par-tial energy contributions associated with, amongst others; valence angle bending, bond stretching, and non-bonded van der Waals and Coulomb interactions. A major difference between ReaxFF and usual, non-reactive force fields, is that it can handle bond formation and dissociation. It was parameterized using density functional theory (DFT) calcula-tions, being the average deviations between the heats of formation predicted by ReaxFF and the experiments equal to 2.8 and 2.9 kcal/mol, for non-conjugated and conjugated systems, respectively [157]. We use this force field as implemented in the Large-scale atomic/molecular massively parallel simulator (LAMMPS) code [161]. The ReaxFF force field was recently used to investigate several chemical reactions and mechanical proper-ties of systems containing silicon atoms, such as the oxidation of silicon carbide [162] as well as silicene stabilized by bilayer graphene[148].
con-ditions for both edge morphologies, i.e., zigzag and armchair membranes (ZZM and ACM), were used to study the dynamical aspects of fracturing processes. Typical size of these membranes for ReaxFF simulations were 95Å× 100 Å, for armchair and zigzag edge terminated structures, respectively. Smaller structures were considered for DFTB+ calculations, in which membrane sizes were 28× 28Å2, for armchair and zigzag edge ter-minated membranes, respectively. All structures were initially thermalized using molecu-lar dynamics (MD), in a NPT ensemble with the external pressure value set to zero along the periodic direction before the stretching process is started. This procedure guaran-teed the initial structures were at equilibrium dimensions and temperature, thus excluding any initial stress stemming from thermal effects. In order to simulate this stretching two different temperatures were considered, 10K and 150 K, controlled either by a Nosé-Hoover[163] chain or an Andersen[164] thermostat as implemented on LAMMPS and DFTB+, respectively. Strain was generated by the gradual increase of the unit cell value along the periodic direction. We have used time-steps of 0.05 f s and a constant strain rate of 10−6/ f s was applied for the ReaxFF simulations. For the SCC-DFTB we used time-steps of 1 f s and applied a strain equal to 10−5 at intervals of 10 f s, resulting in a strain rate of 10−6/ f s as in the ReaxFF case. These conditions were held fixed until the complete mechanical rupture of the membranes. Other strain rate values were tested, ranging from 10−7/ f s to 10−3/ f s. It was verified that for a value of 10−5/ f s or lower the results were equivalent. This strain rate is comparable to the ones used in previous studies [151,153,146]. Repeated runs under same conditions yielded equivalent results.
In order to obtain useful information regarding the dynamics of deformation and rupturing throughout the simulations, we calculated the virial stress tensor[165,166] which can be defined as
σi j=
∑Nk mkvkivkj
V +
∑Nk rki· fkj
V , (2.1)
where N is the number of atoms, V is the volume, m the mass of the atom, v is the velocity, r is the position and f the force acting on the atom. Stress-strain curves were obtained considering the relation between the uniaxial component of stress tensor in a specific direction, namely σii, and the strain defined as a dimensionless quantity which is
direction[166]
εi=
∆Li
Li , (2.2)
where i= 1, 2 or 3. Using this quantity it is also usefull to define the Young Modulus, Y = σii/εi, and the Poisson ratio, which is the negative ratio between a transverse and an
axial strain
ν=−dεi dεj
, (2.3)
where i6= j. We also calculated a quantity which is related to the distortion state of the system, known as von Mises stress[166], defined as
σvm=
s
(σ11− σ22)2+ (σ22− σ33)2+ (σ11− σ33)2+ 6 σ122 + σ232 + σ312
2 , (2.4)
components σ12, σ23 and σ31 are called shear stresses. von Mises stress
pro-vides very helpful information on fracturing processes because, by calculating this quan-tity for each timestep, it is possible to visualize the time evolution and localization of stress on the structure. This methodology was successfully used to investigate the mechanical failure of carbon-based nano structures such as graphene, carbon nanotubes [167] and also silicon nanostructures[166].
Results and Discussion
Structural investigation
We first obtained the minimized geometries for silicene by utilizing the three methods described above: DFT, SCC-DFTB and ReaxFF. Graphene and silicene struc-tures, as optimized by the ReaxFF method, can be compared at the same scale as presented in Figure2.1. The calculated values for silicene, using the ReaxFF method, were d= 2.3 Å for the Si-Si bond length, ∆= 0.67 Å for the buckling value and α = β = 112◦for the angle value (see Figure2.1). DFT and SCC-DFTB calculations resulted, respectively, in values of d= 2.28 and d = 2.32 Å for the Si-Si bond length, ∆ = 0.48 and ∆ = 0.59
Figure 2.2: Stress versus strain curves for zigzag and armchair edge terminated structures. Re-sults for the temperature of 150K and for both ReaxFF and SCC-DFTB methods. See text for discussions.
Å for the buckling and 116◦ and 113◦ for both angles α and β . There is a good agree-ment between these values and those reported in the literature, see table4.2. The ReaxFF results for graphene are dc= 1.42 Å for the C-C bond length, no buckling and αc= 120◦
for the bond angle values.
Mechanical Properties and Fracture Patterns
Typical stress-strain curves can be divided into 3 different regions: (i) the har-monic region, where the stress-strain curve is linear and the Young’s Modulus is defined; (ii) the anharmonic region, where the stress increases non-linearly with the increasing strain; and (iii) the plastic region, where the structure undergoes irreversible structural changes. The point at which mechanical failure happens defines two quantities, the final stress, which is the maximum stress value reached before rupturing, and the critical strain
εc, which is the strain value at the moment of rupture. The value of εcis taken as the point
after which the stress decreases abruptly.
The stress versus strain curves were calculated using both ReaxFF and SCC-DFTB methods, at 150 K for both zigzag and armchair membranes, as shown in Figure 2.2. The harmonic region is easily identified as the region where the behavior is linear. This behavior is observed only for sufficiently small strain values and is gradually changed as we move towards the plastic region. As the structure reaches the critical strain value εc, rupture happens, causing an abrupt fall on the stress values.
Young’s Modulus values for armchair and zigzag membranes were obtained by fitting the linear region. We found very small differences between the values for mem-branes of different edge terminations. For the armchair memmem-branes we found the values of 43 N/m (0.043 TPa.nm) with ReaxFF and of 62.7 N/m (0.0627 TPa.nm) with SCC-DFTB. For the zigzag membranes we found the values of 43 N/m (0.43 TPa.nm) with ReaxFF and 63.4 N/m (0.0634 TPa.nm) with SCC-DFTB. Comparison between the re-sults obtained with SCC-DFTB and values published in the literature shows a very good agreement[131, 153, 143,138, 139]. Young’s moduli calculated using ReaxFF present a discrepancy of around 30% when compared with these results. However, the qualitative behaviour described by both methods is in very good agrement, as further discussed be-low. Estimating the thickness of silicene as the van der Waals diameter of 4.2 Å we obtain a value of 0.149 TPa for the Young’s Modulus in the SCC-DFTB and 0.102 TPa in the ReaxFF calculations. It is interesting to note that these values are 7 up to 10 times smaller than the corresponding of graphene ones under similar conditions [168, 169]. The ob-tained values for the Poisson ratios were 0.30 using SCC-DFTB for both ACM and ZZM, and 0.28 and 0.23 using ReaxFF for ACM and ZZM, respectively, as shown in Table4.2. Despite presenting similar Young’s Modulus values, zigzag and armchair mem-branes exhibit a notable difference in their critical strain values, εc, as shown in Table4.2.
The εcvalue is highly dependent on temperature, going from εc= 0.20 and 0.35 (armchair
and zigzag, respectively) at 10 K to εc= 0.15 and 0.30 at 150 K. In order to explain this
dependence, we stress that kinetic energy fluctuations of atoms in the structure increase with the temperature. These fluctuations allow the crossing of the energy barrier for the creation of defects at lower strain values.
There is also a notable dependence on the edge morphology, εcdiffering by a
factor of up to 2 if we compare an armchair and a zigzag membrane. In order to understand this different behaviour of εc, we have to consider the direction of applied strain in relation
to the hexagonal atomic arrangement.
Figure 2.3: Bonds length and angles values values for ACM. d1is represented by red line, d2is
represented by green line, α orange line and β is represented by violet line.
With the application of strain in the system, the hexagonal symmetry is broken and thus two different angles can be defined for each hexagon (Figure 2.1), α and β , that can either increase or decrease during the deformation process, depending on the direction of applied strain. As shown in Figures 2.3 and 2.4, the dependence of these angles with strain is almost linear for ε< εc. The same symmetry breaking is evidenced
by the appearance of two distinct bond values, also shown in Figures2.3and2.4. When strain is applied to armchair membranes, the strain has the same direction of some of the chemical bonds of the structure (d1as defined in Figure2.1), but this is not true in the case
Figure 2.4: Bonds length and angles values for ZZM. d1 is represented by red line, d2 is
repre-sented by green line, α orange line and β is reprerepre-sented by violet line.
of the structure, so, the relative increase of global strain is not the same as the relative increase of the chemical bond length, while in the case of armchair membranes this can happen for some chemical bonds (d1). This means that, comparing both structures being
deformed until they reach the critical chemical bond length value, one can see that zigzag structures must be more strained than their armchair counterparts. This effect redistributes the applied force making zigzag structures more resilient to mechanical deformation. The curves of the bond lengths versus strain also show clearly the fact that it takes higher strain values for zigzag membranes to reach the same bond lengths as the armchair membranes. This analysis can be extended to graphene as both graphene and silicene share the same honeycomb structure.
The stretching dynamics in the plastic region is also dependent on membrane type. For armchair membranes, edge reconstructions are present when it reaches the plastic region. As shown in Figure 2.5 (a) and (b), hexagonal rings are rearranged into
pentagonal and triangular ones. Square rings are formed at higher strain levels, as shown in Figure2.5 (c) and (d). These reconstructions results are consistent for both methods. Triangular and pentagonal rings have been observed in fracture patterns by Topsakal and Ciraci [138]. In the case of zigzag membranes no reconstructions were observed (see Figure2.6).
Another unique aspect of silicene under strain is the unbuckling process. We observed the decrease of buckling, ∆, with increasing strain, using both methods. This decrease is almost linear with angular coefficient of −0.276 for armchair and −0.283 for zigzag using SCC-DFTB and −1.522 for both types of membranes using ReaxFF. We observed a continuous buckling decrease during the stretching, however, the buckling continues to exist and the structure breaks before its disapperance.
We also analysed the von Mises stress distribution, which is defined by equa-tion2.4. Using the ReaxFF method we calculated this distribution along the whole stretch-ing process. Representative snapshots of this process are shown in Figures2.6and2.7.
For the zigzag membranes the von Misses stress are uniformly distributed before the fracture (Figure2.6(a)). When the membrane fracture starts, stress decreases in regions close to the fracture, as shown in Figure 2.6 (b). The rupture creates clean and well-formed armchair edged structures, with only very few pentagon and heptagon reconstructed rings, as shown in Figure2.6(c).
The corresponding results for the armchair structures present a significant number of edge reconstructions (see Figure 2.7(a)), with the formation of mostly pen-tagon and heppen-tagon rings. As we can see in Figure2.7(b) and (c), in this case the fractured structure presents less clear and more defective zigzag edge terminated structures. It can also be seen that the von Mises stress distribution is much less uniform during the whole process, even after the fracture starts. This local stress concentration leads to more recon-structed rings in this case. Similar fracture patterns have been observed in graphene [169], most notably that fractured armchair structures produce zigzag edge terminated ones and vice-versa and with the formation of pentagon and heptagon reconstructed rings.
Summary and Conclusions
We investigated, by means of fully atomistic molecular dynamics simulations under two different methods, ReaxFF and SCC-DFTB, the structural and mechanical properties of single-layer silicene membranes under mechanical strain. There is a qualita-tive agreement between the results obtained with both methods. Young’s modulus values obtained were 43.0 N/m (for both ACM and ZZM) and 62.7 N/m (ACM) and 63.4 N/m (ZZM) using the ReaxFF and the SCC-DFTB methods, respectively. These values present good agreement with those found in the literature. The critical strain and final stress values were shown to be highly dependent on both temperature and edge morphology, the latter being explained by simple geometric arguments. Temperature also plays a fundamental role in the fracture and reconstruction process. When the system is heated, fracture for-mation barrier can be transposed and critical strains are lowered. The critical strain value, εc, goes from 0.20 and 0.35 (armchair and zigzag, respectively) at 10 K to 0.15 and 0.30
at 150 K.
Silicene fracture patterns are similar in some aspects to those observed on graphene, but important differences were also noted, such as, the presence of buckling due to a pseudo Jahn-Teller effect. Although the buckling value was progressively reduced during strain application, it was not eliminated, even when significant stress was imposed to the structure, as complete rupture happened before this value could reach zero.
Our results show that, while the Young’s moduli values are virtually isotropic for silicene membranes, the critical strain is not. Also, under similar conditions, graphene is many times ( 10 times) tougher than silicene.
Figure 2.5: Detailed view of the edge reconstructions for both ReaxFF and SCC-DFTB methods at (a) and (b) low strain and at (c) and (d) high strain.
Figure 2.6: Typical snapshots from MD simulations showing different stages of the mechanical failure of a zigzag silicene membrane under mechanical strain. The scale goes from low stress (yellow/lighter) to high stress (red/darker).
Figure 2.7: Typical snapshots from MD simulations showing different stages of the mechanical failure of an armchair silicene membrane under mechanical strain. The scale goes from low stress (yellow/lighter) to high stress (red/darker).
CHAPTER
3
Investigation of Graphene Healing
Mechanisms
Large holes in graphene membranes were recently shown to heal, either at room temperature during a low energy STEM experiment, or by annealing at high tem-peratures. However, the details of the healing mechanism remain unclear. We carried out fully atomistic reactive molecular dynamics simulations in order to address these mech-anisms under different experimental conditions. Our results show that, if a carbon atom source is present, high temperatures can provide enough energy for the carbon atoms to overcome the potential energy barrier and to produce perfect reconstruction of the graphene hexagonal structure. At room temperature, this perfect healing is only possi-ble if the heat effects of the electron beam from STEM experiment are explicitly taken into account. The reconstruction process of a perfect or near perfect graphene structure involves the formation of linear carbon chains, as well as rings containing 5, 6, 7 and 8 atoms with planar (Stone-Wales like) and non-planar (lump like) structures. These results shed light on the healing mechanism of graphene when subjected to different experimental conditions. Additionally, the methodology presented here can be useful for investigating the tailoring and manipulations of other nano-structures. This work of this chapter is
published in Carbon journal [170].
Introduction
Graphene, a two-dimensional carbon allotrope that has unique electronic, thermal and mechanical properties [9]. Originally, this material was obtained from graphite using an exfoliation process called the “scotch tape” method [36]. Although this method yields pristine graphene samples, such a process is very costly and not easily scalable. Presently, chemical vapor deposition (CVD) has been the most widely used process to grow graphene samples on a diverse set of substrates, such as steel [171], Ni [172,173], and Cu [174].
Graphene properties are extremely sensitive even to small modifications in its honeycomb structure [175]. Inherent defects from CVD growth processes represent ob-stacles to some technological applications, because they can degrade graphene electronic and mechanical properties. On the other hand, defects can be usefully exploited to obtain different properties for specific applications [176].
Recently, it was demonstrated that etched nanoholes of up to 100 vacancies on graphene membranes can be healed under low power STEM observation, even at room temperature [177]. The healing effect consists of the reconstruction (knitting) of the graphene structure. Carbon atoms from external sources near the hole region, eventu-ally interact with its edges and can fill the vacancies. Hydrocarbon impurities near the membrane can also serve as a source for these extra carbon atoms. These nanohole fill-ings can occur with the formation of either non-hexagonal, near-amorphous, or perfectly hexagonal structures [177].
Other experimental works have also addressed the reconstruction of mono-vacancies and holes in graphene. Chen et al. [178] demonstrated the effectiveness of ther-mal annealing up to 900oC (1173.15 K) to heal defects in graphene membranes. Khol-monov et al. [179] demonstrated defect healing in the top layer of multilayer graphene via CVD techniques using acetylene as a carbon feedstock and iron (Fe) as a catalyst at 900oC (1173.15 K). The evolution and control of nanoholes in graphene by carbon atom
thermal-induced migrations were studied by Xu et al. [116]. For temperatures around 525oC (798.15K), graphene oxide (GO) can be healed and simultaneously reduced by methane plasma resulting in high quality graphene [180]. Moreover, self-repair mecha-nism during graphene sculpting by a focused electron beam were observed by Song et al. [181], while transformation of amorphous carbon into graphene was investigated by Barreiro et al. [182].
To address these healing and self-repair mechanisms, we carried out fully atomistic molecular dynamics simulations using the ReaxFF reactive force field[183]. Initially, we investigated the potential energy landscape near a graphene nanohole (defec-tive regions) using a carbon atom as probe to estimate the energies involved during the healing process. Then, we carried out simulations at different temperatures (controlled by thermostats) to investigate the role of thermal energy in these healing mechanisms. Finally, we simulated the heating effects of an electron beam scanning to address the so called self-healing mechanism at room temperature reported by Zan et al. [177]. Our sim-ulations show that graphene healing can be obtained by a simple annealing at high temper-ature in the presence of a carbon source. However, we could not observe the holes being filled at room temperature because of the energy barriers involved in the process. But if we take into account the electron beam heating effects in the simulations, the graphene healing at room temperature can be observed. In this case, either an imperfect or even a perfect hexagonal structure, depending on the specific energy rate for the beam heating, are obtained.
Methods and Model
We carried out fully atomistic reactive molecular dynamics simulations (MD) using the ReaxFF force field [183], as implemented in the Large-scale atomic/molecular massively parallel simulator (LAMMPS) package [184, 185]. ReaxFF is a reactive force field that allows the study of formation and dissociation of chemical bonds with lower computational cost in comparison to ab initio methods. ReaxFF parametrization is based on density functional theory (DFT) calculations and was successfully used to investigate many dynamical and chemical processes [121, 186, 187]. In the present work, we used
Figure 3.1: Schematic representation of the computational model considered in our simulations. Local heating is represented by the yellow region, while added atoms (carbon atoms deposited) are represented in red.
the Chenoweth et al. (2008) C/H/O ReaxFF parametrization[188] and simulations were carried out with a time steps of 0.1 f s, with temperatures controlled through a Nosé-Hoover chain thermostat [163,189].
For comparison with ReaxFF results, we have also carried out some calcula-tions using the Self-Consistent Charge Density Functional Tight-Binding (SCC-DFTB) [158, 159] method, as implemented on DFTB+ package [156]. DFTB is a DFT-based method and can handle large systems. SCC-DFTB is an implementation of DFTB ap-proach that has the advantage of using Mulliken self-consistent charge redistribution (SCC), which corrects some deficiencies of standard DFTB methods [159]. In general, dispersion terms are not considered in DFTB methods and were included here via Slater-Kirkwood Polarizable atomic model [156].
The computational model used in our calculations consisted of a single-layer graphene membrane (aligned along the xy plane) with a hole (3.2 Å of radius) in its cen-ter. We considered two simulation scenarios: (i) the healing mechanism dependence on temperature and; (ii) mimicking effects of an electron beam scanning to trigger the heal-ing mechanism at room temperature. Scenario (i) was implemented usheal-ing temperatures ranging from 300K up to 2000K. For scenario (ii), in order to simulate the heat effect in-duced by the electron beam interaction with the system [190,191,192,193,194], we have applied a local heating protocol in which a rescaling of atomic velocities inside a cylin-drical region is performed (see Figure3.1). The position of the heated region was varied during the simulations, thus mimicking the STEM experiments. Also, for both scenarios, we restricted the movement of atoms located at the edges of the graphene membrane by using virtual springs with elastic constants K= 30.0 Kcal/mol.Å. For scenario (ii) we also fixed the temperature in these atoms in order to dissipate the accumulation of en-ergy. In our simulations, additional carbon atoms (called “added atoms” in the text and colored in red in all figures) with random kinetic energy values were deposited at random positions and at regular intervals of 500 f s.
Carbon depositions were made using single atoms, but we need to emphasise that depending on specific experimental conditions, several small hydrocarbons or other carbon species (C2and C3) can be present. For instance, in a methane plasma [195] it is
expected the presence of CH, CH2and CH3species in the environment [196]. However,
in order to focus the investigation of the healing mechanism and speed up the molecu-lar dynamics simulations, we opted to use only single carbon atoms for the deposition processes. The inclusion of other hydrocarbons, which would brings necesity to con-sider other complicated questions, such as the activation energies and heat of formation for each considered species. These questions, despite of being interesting, would bring unnecessary complications to our analysis.
The local heating protocol adds a non-translational kinetic energy to the atoms in a cylindrical region centered along the z axis perpendicular to the graphene membrane (Figure3.1). The local heating scanned the membrane along the x and y directions, sim-ilarly to what is done by an electron beam in a STEM experiment. The effects of local heating with a cylindrical radius of 3.5 Å and energy rates between 0−3.0 kcal/(mol. f s)
were investigated. If the energy rate per area is maintained constant, we can expect that changes in the cylindrical radius would not change the necessary energy rate to trigger the healing process. On the other hand, we expect that a small cylindrical radius would increase the necessary time of scanning to obtain a complete healing. All results pre-sented throughout the text are representative simulations that illustrate typical results for the different processes of the graphene healing mechanisms.
Results and Discussion
Potential Energy Landscape
First, we will discuss the potential energy experienced by a carbon atom probe placed near to a defective graphene structure containing a large hole, i.e., a energy land-scape mapping. These mappings were generated for the xy plane while considering differ-ent fixed out-of-plane probe distances, i.e., differdiffer-ent z-coordinates values measured from the graphene basal plane reference (see Figure3.1). From these mappings it is possible to identify the most reactive and repulsive regions experienced by the probe atom. Through this analysis we can obtain a reasonable evaluation for the threshold energy values in-volved in the healing processes.
For a free atom on the membrane surface (i.e., an atom not covalently bonded to graphene), the equilibrium distance is around 3.2 Å. At this distance, it is energetically favorable for an atom to be placed above the hollow site of a hexagonal ring rather than above a carbon atom. In the case of a hole etched on the membrane, the probe atom starts to be repelled as it approaches the hole borders, because of the decrease on the van der Waals interaction between the atom and the membrane. This interpretation is supported by results shown in Figure4.2(a). If we consider, for instance, that a free atom is at a z distance 2.2 Å above the membrane, the lattice becomes highly repulsive and a pronounced minimum energy takes place near the hole edges indicating highly chemical reactive regions, as shown in Figure4.2(b).
Figure 3.2: Potential energy (landscape energy) for a probe atom placed above the graphene mem-brane with a hole of radius 4.5 Å; (a) potential energy for probe atom at a distance of 3.2 Å from graphene; (b) potential energy for a probe atom placed at 2.2 Å ; and (c) Minimum potential en-ergy map for a probe atom near a hole of radius 3.2 Å etched in a graphene membrane. For each (x,y) position we represent the height (z direction) position in which the free atom potential energy reaches a minimum, while the color represents the value of this potential energy for that specific point.
Depending on the z distance between the free atom and the membrane surface, repulsive or attractive regions exist. Therefore, to start filling the hole with free atoms on the membrane surface, the free atoms must be able to easily migrate to the defective
re-gions, i.e., these free atoms must have sufficient energy to overcome the energetic barrier found at the hole edges. These barrier energies can be visualized by mapping the min-imum landscape potential energy over the z distance for each planar coordinate of the probe atom, as shown in Figure4.2 (c). The energetic barrier near the edge of the hole as well as the potential energy inside the hole region is significantly higher than that on the surface, as highlighted in reddish colors. However, if the free atom reaches the empty region (inside the hole), there is a high probability for it to be trapped. As soon as the atom is trapped the reaction with the edge of the hole is facilitated.
The energy mapping profiles indicate that the healing mechanism is not effec-tive at room temperature because the probability for a free atom to overcome the energetic barrier and enter into the hole region is quite low. In fact, for a distance of 3.2 Å from the membrane, the difference in the potential energy between the hole center and a hol-low site has a value around 40 meV considering a hole with a radius of 4.5Å , while the thermal energy provided at room temperature is about 25 meV . For large holes this en-ergy difference can reach values around 150 meV. Thus, higher temperatures are needed to increase the healing probability. Additionally, we also generated energy mappings us-ing the SCC-DFTB method [158, 159], in order to contrast with the data obtained from ReaxFF simulations. We considered the case of a hole with radius of 3.2 Å and a carbon probe atom placed at a distance of 3.2 Å from the membrane. As expected, although the intrinsic barrier values are different (30 and 80 meV for ReaxFF and SCC-DFTB, respectively), the general trends of the maps are qualitatively similar.
Temperature Dependence
In order to investigate the temperature dependence on the graphene healing mechanism, we performed molecular dynamics simulations in the temperature range of 300 K up to 2000 K. The model system consists of a graphene membrane in which a hole is etched in its center. Also, carbon atoms are added at regular time intervals to mimic the presence of a carbon source (see methodology section for a complete description). The filling of holes is expected to depend mainly on the presence of thermal fluctuations that should be high enough to allow the added atoms to cross the energy barrier and enter into
the defective region, in addition to the thermal fluctuations of the atoms from the defect itself.
For 300 K, the added atoms do not have the necessary energy to reach the chemically active region, located at the edges of the defect, and no complete filling was observed, thus confirming the trends obtained from the energy map analysis. The obtained final structure is shown in Figure3.3 (a). Increasing the temperature enhances the reac-tivity of the added atoms and of the graphene atoms near the hole region, but the healing is also incomplete for temperatures up to 800 K. In this range of temperature, some linear atomic chains (LACs) can be found, but their inter-conversion into ring structures was not observed, see Figure3.3(b). In addition, we observed some reconstruction of hole edges. For 1000 K, the thermal fluctuation reaches a point in which the energy is sufficiently large to convert these LACs into more stable structures (5 up to 8 member rings) and an imperfect healing takes place, as shown in Figure3.3(c). For 1500 K, a perfect healing was observed, since the thermal fluctuations allow the conversion of non-hexagonal to hexagonal rings, as shown in Figure3.3(d). The graphene healing induced by annealing at high temperature was obtained by different experimental investigations, for instance, for temperatures of 800oC(1073.15 K) [174] and 900oC (1173.15 K) [178,179], in good agreement our results.
Electron Beam Effects
To analyze the healing mechanism in similar conditions to those reported by Zan et al. [177], we have investigated other possibilities that could trigger hole filling mechanisms. Zan et al. [177] observed the healing of large holes in graphene at room temperature during low energy STEM observations. They suggest that the STEM elec-tron beam could act as a local heating mechanism, thus allowing the carbon atoms from impurities to come closer to the hole and to fill it.
The mechanisms of electron beam interactions and manipulations of nanos-tructures are well-know [191, 192, 193, 194]. Banhart [193] emphasizes that when en-ergetic electrons or ions strike a target, different mechanisms of energy or momentum transfer can take place. The main contributions for radiation effects are due to, amongst
Figure 3.3: Final obtained structures from molecular dynamics simulations at different tempera-tures: (a) T= 300 K; (b) T = 800 K; (c) T = 1000 K; (d) T = 1500 K. The added carbon atoms are indicated in red. The formation of rings containing 5 and 7 atoms are highlighted.
others: electronic excitations or ionization of individual atoms, creation of collective elec-tronic excitations, bond breaking, phonon generation and atom displacements in the inte-rior of the sample [193]. The secondary effects are emission of photons and emission of Auger electrons [193]. Moreover, electron beams of low energy present a more intense interaction with the target [191], thus transferring energy more efficiently to the lattice than a higher energy beam. Thus, elastic collisions and some mechanisms pointed out by Banhart [193] could constitute the main contributions for heating of the target.
In order to mimic these experimental conditions, we incorporated into the simulations the main effects of the interaction between the electron beam and the target, by using a local heating source in a cylindrical region of space, as shown in Figure3.1. This local heating provides energy at a constant rate to a specific spot. Also, this spot moves through the membrane mimicking the movement of the electron beam scanning in the STEM experiments. With this simulation protocol, it is possible to investigate different conditions of the electron beam by controlling the energy rate values. The local
Figure 3.4: Sequence of different configurations from MD simulations leading to a perfect healing of a defective graphene membrane at room temperature. Local heating areas (for the case of energy rate of 0.5 kcal/(mol. f s)), are indicated in yellow color. Graphene atoms and added carbon ones are displayed in grey and red colors, respectively.
heating approach can be useful in tailoring complex structures or even to sculpt free-standing graphene [197]. In some cases, applying high temperature through the whole system could cause the destruction of the complete structure, which can be prevented by the use of heating spots. For this kind of situation, a local heating source as the one we are using here, can work as a precision tool for fixing local defects or even to produce local chemical modifications.
Here, we have analyzed the effect of a simulated heating source with an en-ergy rate in the range of 0 to 3.0 kcal/(mol. f s), in the presence of a thermostat set to maintain a fixed temperature of 300 K on atoms located outside the heated region. For energy rates above 3.0 kcal/(mol. f s), we observed structural damages with an increase of the hole size.
For a local heating spot with energy rates up to 0.25 kcal/(mol. f s) (10.08 meV / f s), we observed no hole filling. Added atoms did not receive the necessary amount of energy to overcome the energy barriers in the interface between the membrane and the defective region. However, an energy rate of 0.5 kcal/(mol. f s) (21.68 meV / f s) can lead to a per-fect healing. Some typical simulation snapshots of this regime are presented in Figure3.4
Figure 3.5: Snapshots from MD simulations showing the final structures for imperfect healing of a hole in a graphene membrane: (a) local heating of 1.0 kcal/(mol. f s), resulting in the formation of a flat defect (FD) structure; (b) local heating of 1.5 kcal/(mol. f s), resulting in the formation of a Lump Defect (LD) structure; (c) local heating of 3.0 kcal/(mol. f s), resulting in a larger hole. See text for discussions.
Figure 3.6: Structural transformation in a defective graphene membrane driven by local heating (indicated in yellow color) of 2.0 kcal/(mol. f s): (a) Lump defect generated from high energy flux; (b) Lump defect interconversion to a LAC (linear atomic chain) structure; (c) LAC structure detachment from the defective graphene.
In this case, the added atoms acquire the necessary energy to approach the hole edge and to react with it, filling and reconstructing the hole. With this energy rate, the local heat was able to provide the necessary amount of kinetic energy to facilitate the absorption of added atoms. Another effect of this local heating is the increase of local chemical reactiv-ity at the hole edges, caused by out-of-plane thermal fluctuations of atoms in that region. When the hole is completely filled, new coming atoms are deflected (bounced off) by the healed membrane.
Increasing the energy rate of local heating above 0.5 kcal/(mol. f s) is likely to be effective in filling the hole, but it could also generate structures that are not perfectly hexagonal. We also observed the incorporation of more atoms than necessary to obtain perfect healing, forming a defective re-knit structure. We named these defective struc-tures Flat (FD) and Lump (LD) Defects. FD strucstruc-tures consist of planar or quasi-planar structures containing non-hexagonal rings (5, 7 and 8 atom ring), while the LD defects are those which deviate from the membrane plane rendering the “lump” membrane with a 3D structure similar to that proposed by Lusk and Carr [198]. FD and LD structures can be seen in Figure3.5(a) and Figure3.5(b), respectively.
For an energy rate of 1.0 kcal/(mol. f s) (43.36 meV / f s), the local heating is more likely to generate FD structures observed in the final healing processes, as shown in Figure3.5(a). LD structures appear with the increase of the local heating energy rate, for instance, for 1.5 kcal/(mol. f s) (65.05 meV / f s), as shown in Figure3.5(b). For this energy value, the local heating eases the incorporation of added atoms causing some of
