le than itsanosad-e energy

Physics Letters A 337 (2005) 431–434

www.elsevier.com/locate/pla

Stability of disclinations in carbon monolayer

Sérgio Azevedo

Departamento de Física, Universidade Estadual de Feira de Santana km-03, Br-116 Norte, 44031-460 Feira de Santana, Ba, Brazil

Received 17 December 2004; received in revised form 31 January 2005; accepted 2 February 2005

Communicated by R. Wu

Abstract

We apply first-principles calculations to study the stability of carbon structures with disclinations of anglesθ = ±π3 .

Nanocones and saddle-like compounds are investigated. It is shown that a layer with negative disclination is more stabconverse. We have also investigated the effect of introducting boron or nitrogen as substitutional impurity in carbon ndles. Nitrogen, out and on the defect, is shown to increase the formation energy, while boron, on heptagon, lowers thfor formation of the heptagonal defect. In addition, we have found that such structures display spin polarization. 2005 Elsevier B.V. All rights reserved.

PACS: 71.20.Tx; 71.15.Mb

Keywords: Negative disclination; Stability; Saddle-like

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1. Introduction

Different kinds of nanoscale structures, like fulleenes and carbon nanotubes, have been observetheir unique properties have attracted great atten[1,2]. The deviation from the planar topology hasstrong effect on electronic properties. Also, thecorporation of heptagons, pentagons and otherfects into a hexagonal network of carbon nanotuincreases the local curvature and it can lead to thesure of the tubes[3]. The structure of the cap depenon the included specific defect, but generally it h

E-mail address: sazevedo@mail.uefs.br(S. Azevedo).

0375-9601/$ – see front matter 2005 Elsevier B.V. All rights reserveddoi:10.1016/j.physleta.2005.02.010

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the aspect of a conical surface from the bulk mate[4,5]. On the other hand, in nanotubes that are opeduring growth, the introduction of heptagon leadschanges in nanotubes size and orientation. Theoical calculation has been carried out to investigthe electronic structure and stability of carbon strtures with positive and negative disclinations[6–8].Recently, boron nitride structures have been invegated[4,5,9]. In carbon, it was shown that energy miimization leads to 60◦ disclination structures (conicasheets) with isolated pentagon at the apex. In repapers[10,11], theoretical investigations of a generclass of carbon nitride materials have been develoIt was found that the presence of a nitrogen atomdefect, lowers the energy responsible for the penta

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432 S. Azevedo / Physics Letters A 337 (2005) 431–434

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nal defect, and in contrast, out of defect, it increathis energy. Besides, it was shown that such subtutional impurity increased the energy to produceheptagon, but decreased it for a pentagon. Therefowas concluded that the role of nitrogen is to lowerformation energy for pentagons. Using first-principcalculations, we address the relative stability of potive and negative disclinations with pentagon and htagon, respectively, in the present study. We haveinvestigated the role of boron impurities in a carbmonolayer with−60◦ disclination. By the results othese calculations we can conclude that the role ofboron on heptagon is analogous to that of the nitroon pentagon.

A conical surface is obtained by a “cut and gluprocess, characteristic of the formation of topologidefects. For instance, to create a single five-membring in a carbon sheet, a sector of angle2π

6 is cut outfrom the center of any hexagon (so that one ofedges of the hexagon is removed) and the looseare joined together. The result is a conical structwith a pentagon at its (truncated) apex. The remoof a sector of angle4π

6 results in a square or two petagons at the joining together the remaining carbstructure. Conversely, a seven-membered ring maformed by inserting a wedge of angle2π

6 , adding anextra edge to one of the hexagons. In this case, thsult is a saddle-like structure with a heptagon atcenter.

Examples of structures built in this way and usin the calculations are shown inFig. 1. The first one,Fig. 1a, is a carbon nanocone with a pentagonthe apex. The second is carbon saddle-like geotry, with 196 atoms, 161 carbons and 35 hydrogeFigs. 1c to d display saddle-like structures with N(Bas substitutional impurity on and out of heptagoFigs. 1e to f display heptagonal structures with boras substitutional impurity on and out of defect, resptively.

Our calculations are based on the density futional theory[12] as implemented in the SIESTA program[13]. We make use of non-conserving TroullieMartins pseudopotencials[14] in the Kleinman–Bylander factorized form[15] and a double-ζ basisset composed of numerical atomic of finite range.larization orbitals are included for nitrogen, boron, acarbon atoms, and we use the generalized gradienproximation[16] (GGA) for the exchange-correlatio

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Fig. 1. Five saddle-like structures and one nanocone: carbontrogen and boron atoms are represented by gray, black and wrespectively: (a) carbon nanocone with a pentagon, disclinatio60◦, in the tip; (b) carbon negative disclination with a heptagon;and (d) saddle-like structure with substitutional nitrogen on andof defect, respectively; (e) and (f) boron substitutional on defectout of defect.

potential. All the geometries are fully relaxed, wiresidual forces smaller than 0.1 eV/Å.

2. Formation energy

From now on we proceed to a comparative analyof the formation energy of the several carbon molayers with disclination described inFig. 1. The ap-proach used here is based on prior determinatiochemical potential for nitrogen, boron, and carbonaddress this issue. Such approach was used in slar calculations in the literature[17–20]. Therefore, wewill make a brief description in this section.

We introduce the theoretically calculated chemipotentialsµ , µ , and µ for nitrogen, boron and

N B C

S. Azevedo / Physics Letters A 337 (2005) 431–434 433

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carbon, respectively. The chemical potentialµN is ob-tained from solid nitrogen in theα phase, while ametallicα–β is used to obtain the chemical potentµB [19]. The chemical potentialµC is obtained froma carbon planar sheet. Since we are dealing withnite clusters, we used hydrogen atoms to saturategling bonds at the edges. The chemical potentialthe carbon–hydrogen bonds (µCH) is obtained, usingµC from a total energy calculation for a finite carbsheet. Such considerations allow us to write formatenergy of the carbon monolayer with disclination a

(1)Eform = Etot − nCµC − nB(N)µB(N) − nHµH,

whereEtot is the calculated total energy of the cabon structure with disclination from SIESTA. In thexpressions above,nB(N) is the number of boron or nitrogen atoms,nC is the number of carbon atoms,nHis the number of hydrogen–carbon atoms. Note thaEq. (1) we are using a finite monolayer of carbonreference, and ascribing a null value to its formatenergy. The values for chemical potential areµH =−15.46 eV,µC = −154.86 eV,µB = −77.20 eV, andfinally µN = −270.21 eV.

Our first-principles results for the formation enerof carbon structures with 60◦ and−60◦ disclinationsare shown inTable 1. The underlined numbers in thtable indicate the most stable structures. The lastcolumns ofTable 1show that the structure with negtive disclination, heptagon, is more stable than thewith positive disclination, pentagon. Namely, the eltic energy required to fold a finite sheet into a conelarger than energy for formation of saddle-like monlayer.

Table 1Formation energies (in eV) of the carbon saddle-like structures60◦ angle nanocone shown inFig. 1

Disclination Impurity Position Eform (eV)

60◦ no no 5.0−60◦ no no 4.6−60◦ N on heptagon 4.8−60◦ N out heptagon 4.8−60◦ B on heptagon 4.3−60◦ B out heptagon 5.2

The first column indicates the disclination for each structure.second column indicates the type of impurity. The third colushows the position of boron and nitrogen in structure with disclition. The fourth column shows the formation energies as discuin the text. Underlined values indicate the most stable structure

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In the case of substitutional nitrogen, on defect,found that it presents an increase in the formationergy by 0.2 eV. This result agrees with that obtainin [10], which uses semi-empirical PM3 Hamiltoniafor geometry optimizations. The nitrogen as impurout of heptagon, on hexagon, increased the energ0.3 eV. On the other hand, the presence of boronstitutional on heptagon lowers the formation eneby 0.3 eV. However, the substitutional impurity boroout of heptagon, increases the energy by 0.6 eV. Thresults were obtained by calculating the differencetotal energy between a carbon monolayer with disnation in which the boron or nitrogen is placeddefect or out of defect, and pure carbon sheet wdisclination.

In a recent paper it has been shown that theof nitrogen in the formation of the CNx material is tolower the energy cost for formation of pentagons[10].In another recent work[11], it was shown that boronincreased the energy for formation of pentagons. Four results described inTable 1, we can see that therexists a gain in energy by placing the boron inheptagon instead of hexagon. Therefore, these reclearly show that boron plays the same role as thanitrogen for pentagon. Namely, the presence of bolowers the energy to form a heptagon, negative cuture, but it increases the energy formation to formpentagon, positive curvature.

We investigate the case of polarization spin for cbon monolayer with 60◦ and−60◦ disclinations. Wehave found spin splitting for structures with pentagoand heptagons, since such structures display annumber of electrons. We can conclude thereforestructures with odd electron number display net spTherefore 60◦, 120◦, and 240◦ disclinations presenspin polarization.

3. Conclusions

In summary, we used first-principles calculationsaddress the question of the stability of carbon molayer with disclinations. We showed that the structuwith heptagon, in this case with negative curvatuare more stable that the ones with pentagon, poscurvature. We also investigated the case of a pof boron or nitrogen, as substitutional impurities,structures with negative disclination, on heptagon

434 S. Azevedo / Physics Letters A 337 (2005) 431–434

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out of heptagon. It was seen that substitutional nigen, out of defect, increased the formation energy0.3 eV. The substitutional boron on heptagon, caua decrease of 0.3 eV. Therefore, we can concludethe role of boron in the formation of saddle-like carbstructures, with boron as substitutional impurity, islower the energy cost for formation of heptagons.the other hand, the substitutional nitrogen enhanthe energy cost for formations of heptagons and loers the energy of formation for pentagons. In additiit is shown that structure that displays an odd numof electrons, for example, disclinations with angles−60◦ and 60◦, has spin polarization.

Acknowledgements

This work has been supported by CNPq.

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