NANO LETTERS
Vacancy Formation Process in Carbon Nanotubes: First-Principles Approach
2005 Vol. 5, No. 1 197-200
Jussane Rossato,† R. J. Baierle,*,† A. Fazzio,‡ and R. Mota† Departamento de Fı´sica, UniVersidade Federal de Santa Maria, 97105-900, Santa Maria, RS, Brazil, Instituto de Fı´sica, UniVersidade de Sa˜ o Paulo, CP 66318, 05315-970, Sa˜ o Paulo-SP, Brazil Received October 27, 2004
ABSTRACT The electronic and structural properties of a single-walled carbon nanotube (SWNT) under mechanical deformation are studied using firstprinciples calculations based on the density functional theory. A force is applied over one particular C-atom with enough strength to break the chemical bonds between the atom and its nearest neighbors, leading to a final configuration represented by one tube with a vacancy and an isolated C-atom inside the tube. Our investigation demonstrates that there is a tendency that the first bond to break is the one most parallel possible to the tube axis and, after, the remaining two other bonds are broken. The analysis of the electronic charge densities, just before and after the bonds breaking, helps to elucidate how the vacancy is formed on an atom-by-atom basis. In particular, for tubes with a diameter around 11 Å, it is shown that the chemical bonds start to break only when the externally applied force is of the order of 14 nN and it is independent of the chirality. The formation energies for the vacancies created using this process are almost independent of the chirality, otherwise the bonds broken and the reconstruction are dependent.
Carbon nanotubes can be considered an entirely new form of matter and are some of the first true nanomaterials engineered at the molecular level. They exhibit structural and electronic properties that have been considered breathtaking. These nanomaterials are light and flexible and, at the same time, are among the strongest known structures.1 These peculiar characteristics allow for the possibility to engineer at the ultimate level of detail, building designer materials atom-by-atom. The modification of carbon nanotube properties through deformation, doping, or by vacancy creation has attracted a lot of attention due to the possibility of functionalization by altering their transport and optical characteristics.2 In particular, concerning single vacancies or related point defects,3-7 they can be intentionally induced or occur as native defects, since it is not really possible yet to grow these structures in a totally controlled way to guarantee a defect-free product. Single vacancies have been studied6,7 and electron or ion irradiation can be efficiently used8 to artificially release C-atoms from SWNT creating vacancies. Our approach to produce the vacancy attempts to help to elucidate how the bonds are broken and how the bond reconstruction occurs around the defect. The calculations are based on first-principles density functional theory and single numerical orbitals as basis sets9 * Corresponding author. E-mail:
[email protected]. † Universidade Federal de Santa Maria. ‡ Universidade de Sa ˜ o Paulo. 10.1021/nl048226d CCC: $30.25 Published on Web 11/20/2004
© 2005 American Chemical Society
are adopted. We have used the SIESTA code,10 which solves the standard Kohn-Sham equations in a self-consistent way. The calculations are done using the local density approximation for the exchange correlation term,11 as parametrized by Perdew and Zunger,12 and the standard norm-conserving Troullier-Martins13 pseudopotentials orbitals are used to calculate the ion-electron interaction. Our study is performed using three tubes with different chirality: the (13,0) semiconductor SWNT with 10.44 Å diameter, the (10,5) mixed SWNT with 10.60 Å diameter, and the (8,8) metallic SWNT with 11.06 Å diameter. We use periodic boundary conditions and a supercell approximation with lateral separation of 18 Å between tube centers to make sure that they do not interact with each other. For the (13,0) SWNT, the supercell has three unit cells (156 atoms), for (10,5) SWNT one unit cell (140 atoms), and for (8,8) SWNT five unit cells (160 atoms). For the Brillouin zone integration we are using two Monkhorst-Pack k-points,14 along the tube axis. Here, one must note that the atom in which the force is applied is not relaxed and all other atoms, except those discussed below, are relaxed in each step of the force application. In this way, the nanotube atomic structures are fully optimized along the deformation process. To simulate a large tube over a plane surface, in our adopted model, the atoms in the supercell interface rings are assumed fixed, as well as one line of atoms parallel to the tube axis and in the opposite side of the applied force. The relaxed atomic structures of the tubes are obtained by a minimization
Figure 1. Single vacancy in a (a) (13,0) zigzag tube, (b) (10,5) mixed tube, and (c) (8,8) armchair tube. For all the nanotubes studied, reconstructions occur forming a 5-1DB defect.
Figure 2. Force versus displacement for the three nanotubes studied: (a) (13,0) zigzag, (b) (10,5) mixed, and (c) (8,8) armchair tube. The linear region is associated with elastic deformation and the horizontal region is associated with plastic deformation.
of the total energy using Hellmann-Feynman forces calculations, and the structural optimizations were performed using the conjugate gradient algorithm until the residual forces were below 0.08 nN. Initially, we have used our formalism to study the case of a single vacancy, which is created removing a C-atom from the tube wall, leaving three dangling bonds (DB) around the unrelaxed vacancy. By letting the system relax completely, our calculations show that the local structures reconstruct in such a way that two of three DBs prefer to recombine each other, forming a pentagon coupling, plus a DB remaining, in a so-called 5-1DB defect, as shown in Figure 1. This is consistent with previous theoretical predictions.5-7 To simulate the vacancy formation on an atom-by-atom basis, a systematic analysis is performed by applying a hypothetical force over one particular C atom. The final result in terms of applied force versus displacement is shown in Figure 2. The linear region, where the restoring force is proportional to the applied displacement, is associated with elastic deformation. The horizontal portion is associated to plastic deformation, which is generated once the applied force exceeds its elastic limit. In general, it is difficult to specify precisely the point at which the force-displacement curve deviates from linearity and enters the plastic region. Also, the difference in terms of elastic and plastic regimes, as shown in Figure 2, makes these materials characterized by a large elastic region and smaller plastic region. It is well established that for most materials elastic deformation persists only up to a specific limit and when the material is deformed 198
beyond that particular point the Hooke’s law ceases to be valid and permanent and nonrecoverable plastic deformation occurs.15,16 From an atomic perspective, the plastic deformation region corresponds to the breaking of the bonds between the atom where the force is applied (C0) and its neighboring atoms (C1, C2, and C3), which could reform bonds or not. For the tubes studied, we demonstrate that, from the three bonds associated with the C0 atom, where the load is applied, the first one to break is dependent on the tube chirality. For the (13,0) SWNT it is the one parallel to the tube axis (C3), as can be seen in Figure 3a; for the other two tubes, (10,5) and (8,8), the two symmetric bonds break at the same time (C1 and C2), and finally the C3 bond is broken, as shown in Figure 3b and Figure 3c, respectively. For all the tubes studied, in terms of the neighboring atoms, after the breaking of the C3-C0 bond in the (13,0) tube or C1-C0 and C2-C0 for the other two tubes, increasing the displacement, a breaking of the remaining bonds, C1-C0 and C2-C0 for the (13,0) tube, and C3-C0 bonds for the (10,5) and (8,8) tubes occur and a new bond between C1 and C2 is formed, leading to a final configuration 5-1DB defect. Figure 4 shows the contour plots for the total charge densities for the (13,0) SWNT in three different stages of the applied force. Figure 4a corresponds to 50% of the original tube radius, or 2.60 Å; in Figure 4b and 4c the corresponding displacements are 70% (3.64 Å) and 90% (4.68 Å), respectively. To display the bonds, we have adopted the plane containing the C0, C1, and C3 atoms. The C0-C2 bond, not shown in this plane, can be assumed similar to Nano Lett., Vol. 5, No. 1, 2005
Figure 3. Breaking of the bonds between the atom C0 for three different SWNTs: (a) (13,0) zigzag, (b) (10,5) mixed, and (c) (8,8) armchair tube. In (a) the C0 atom is displaced around 70% (3.6 Å), and (b) and (c) around 60% of the original tube radius (3.2 Å).
Figure 4. Contour plots of the total charge densities for (a) 50%, (b) 70%, and (c) 90% displacements of the original tube radius. The outermost and innermost contour lines correspond to 0.25 e/Å3 and 0.01 e/Å3, respectively, and the contour spacing is 0.01 e/Å3.
C0-C3. Observe that in Figure 4a the C0-C1 and C0-C3 bonds are preserved, whereas in Figure 4b the C0-C1 bond is seen more intensely than C0-C3 bond, and finally no bonds for C0 are observed in Figure 4c. For tubes with other chiralities, a similar behavior is observed. The semiconductor nanotubes studied, (13,0) and (10,5), present similar electronic behaviors when vacancies are present. The calculated band structure of the pure (13,0) SWNT (gap of approximately 0.65 eV) is presented in Figure 5a. In Figure 5b, for the single vacancy, it is shown that the corresponding band structure presents as a main feature one unoccupied level inside the band gap. The local density of charges around this level elucidates that it originates predominantly from contribution from the defect itself. The band structure for the tube with a vacancy plus one C-atom inside can be seen in Figure 5c, showing two different unoccupied levels in the gap. The local densities of charges show that the level just above the Fermi level has a character mainly associated with the C-atom inside the tube and the other level presents a character predominantly associated with the vacancy. Both these levels have a small dispersion that is related to a weak interaction between the defects in neighboring supercells. The main features present in (13,0) tubes are also present in (10,5) tubes, not explicitly presented in this paper. For the (8,8) nanotube, the band structure shows that it is really metallic. This can be observed in Figure 6a through the crossing of two levels in 2/3 of the Brillouin zone. When Nano Lett., Vol. 5, No. 1, 2005
Figure 5. Band structures for (a) (13,0) pure tube, (b) single vacancy, and (c) vacancy plus one C-atom inside the tube. The dashed line indicates the Fermi level.
Figure 6. Band structures for (a) (8,8) pure tube, (b) single vacancy, and (c) vacancy plus one C-atom inside the tube. The dashed line indicates the Fermi level.
a single vacancy is present, due the broken symmetry, there is a separation of the two electronic levels, resulting in a formation of a small gap and an unoccupied level associated with the vacancy is present in this new gap, as can be seen in Figure 6b. For the vacancy plus a C-atom inside the tube, 199
the band structure, in accordance with Figure 6c, shows that two resonant electronic levels appear in the gap, one with character associated with the vacancy and other associated with the C-atom inside the tube. In terms of energetics, the formation energy for the vacancy associated with a system consisting of a single vacancy plus one C-atom inside the tube is calculated using total energy procedure, according to eq 1: Eform[vac] ) ET[SWNT/vac/1C] + µ[C] ET[SWNT] - E[C] (1) where ET[SWNT/vac/1C] is the total energy for the tube with a vacancy and one C-atom inside, ET[SWNT] is the total energy for the pure tube, µ[C] is the chemical potential for a C-atom calculated as the energy per atom for the corresponding pure tube, and E[C] is the energy for a free C atom. The formation energies calculated using eq 1 above are 6.21 eV for the (13,0) SWNT, 6.59 eV for the (10,5) SWNT, and 7.04 eV for the (8,8) SWNT, respectively, which are consistent with 6.38 eV, 6.59 and 7.09 eV obtained for the single vacancy, without the C-atom inside the tube, using eq 2: Eform[vac] ) ET[SWNT/vac] + µ[C] - ET[SWNT] (2) where ET[SWNT/vac] is the total energy for the tube with a single vacancy. In this paper, a realistic first-principles calculation is reported for the energetics associated with the vacancy process formation based on atom-by-atom manipulation. The order of bond breaking and the reconstruction of the bonds were investigated, and it is demonstrated that they are chirality dependent. For zigzag nanotubes, initially, the bond parallel to the tube axis is broken and, after, the remaining two other bonds are broken, the resulting final system is the 5-1DB configuration with a bond reconstruction perpendicular to the tube axis. For armchair and mixed nanotubes, two bonds start to break at same time and finally the third bond is broken, with the first two bonds leading to a reconstruction leading the system to the same configuration
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of the zigzag tubes, the 5-1DB defect. In conclusion, we observed that the bonds more propitious to be broken are those along the tube axis and the reconstruction bonds have a tendency to be perpendicular to tube axis. The elasticity of the system for the initial values of the applied forces, as well as the subsequent plastic region, is observed, and it is shown that the chemical bonds start to break only when the externally applied force is of the order of 14 nN. The formation energy for the vacancy created based on this process is comparable with single vacancy formation energy and an unoccupied level associated with the defect is present in both cases. Acknowledgment. The authors are grateful for the financial support of the CNPQ and FAPERGS Brazilian agencies. The calculations were performed at the Centro Nacional de Processamento de Alto Desempenho CENAPAD/Campinas. References (1) Dai, H. Surf. Sci. 2002, 500, 218. (2) Fagan, S. B.; da Silva, L. B.; Mota, R. Nano Lett. 2003, 3, 289. Fagan, S. B.; da Silva, A. J. R.; Baierle, R. J.; Fazzio A. Phys. ReV. B 2003, 67, 33405. Baierle, R. J.; Fagan, S. B.; Mota, R.; da Silva, A. J. R.; Fazzio A. Phys. ReV. B 2001, 64, 085413. (3) Charlier, J.-C.; Ebbesen, T. W.; Lambin, Ph. Phys. ReV. B 1996, 53, 11108. (4) Hansson, A.; Paulsson, M.; Stafstrom, S. Phys. ReV. B 2000, 62, 7639. (5) Krasheninnikov, A. V.; Nordlund, K.; Sirvio, M.; Salonen, E.; Keinonen, J. Phys. ReV. B 2001, 63, 245405. (6) Ajayan, P. M.; Ravikumar, V.; Charlier, J.-C. Phys. ReV. Lett. 1998, 81, 1437. (7) Krasheninnikov, A. V.; Nordlund, K. J. Vac. Sci. Technol. B 2002, 20, 728. (8) Zhu, Y.; Yi, T.; Zheng, B.; Cao, L. Appl. Surf. Sci. 1999, 137, 83. (9) Sankey, O. F.; Nikleswsky, D. J. Phys. ReV. B 1989, 40, 3979. (10) Ordejo´n, P.; Artacho, E.; Soler, J. M. Phys. ReV. B 1996, 53, 10441. Sa´nchez-Portal, D.; Ordejo´n, P.; Artacho, E.; Soler, J. M. Int. J. Quantum Chem. 1997, 65, 453. (11) Ceperley, D. M.; Alder, B. J. Phys. ReV. Lett. 1980, 45, 566. (12) Perdew, J. P.; Zunger, A. Phys. ReV. B 1981, 23, 5048. (13) Toullier, N.; Martins, J. L. Phys. ReV. B 1993, 43, 1991. (14) Monkhorst, H. J.; Pack, J. D. Phys. ReV. B 1976, 13, 5188. (15) Sayman, O.; Kucuk, M.; Esendemir U.; Ondurucu, A. J. Reinf. Plast. Compos. 2002, 21(13), 1205. (16) Stoneham, A. M.; Godwin, P. D.; Sutton, A. P.; Bull, S. J. Appl. Phys. Lett. 1998, 72, 3142.
NL048226D
Nano Lett., Vol. 5, No. 1, 2005
