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Anisotropic mechanical properties and Stone–Wales defects in graphene monolayer: A theoretical study
Physics Letters A 374 (2010) 2781–2784
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Anisotropic mechanical properties and Stone–Wales defects in graphene monolayer: A theoretical study B.B. Fan a , X.B. Yang b , R. Zhang a,c,∗ a b c
School of Materials Science & Engineering, Zhengzhou University, Henan 450001, PR China Department of Physics, South China University of Technology, Guangzhou 510640, PR China Zhengzhou Institute of Aeronautical Industry Management, Henan 450046, PR China
a r t i c l e
i n f o
Article history: Received 20 February 2010 Received in revised form 16 April 2010 Accepted 25 April 2010 Available online 1 May 2010 Communicated by R. Wu
a b s t r a c t We investigate the mechanical properties of graphene monolayer via the density functional theoretical (DFT) method. We find that the strain energies are anisotropic for the graphene under large strain. We attribute the anisotropic feature to the anisotropic sp2 hybridization in the hexagonal lattice. We further identify that the formation energies of Stone–Wales (SW) defects in the graphene monolayer are determined by the defect concentration and also the direction of applied tensile strain, correlating with the anisotropic feature. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Graphene [1] has been attracting much attention due to its fascinating properties such as extremely high mobility [2], high elasticity [3], quantum electronic transport [4,5], a tunable band gap [6] and electromechanical modulation [7]. It shows potential applications in future solid devices, providing an alternative to the traditional silicon semi-conductors [8,9]. Strain and defects are always inevitable in nanostructures, which could dominate the electronic properties. One of the most important defects in graphene is the Stone–Wales defect (SW defect), in which the rotation of a single pair of carbon atoms creates adjacent pairs of pentagonal and heptagonal rings [10]. This imperfection in graphene is introduced either during the synthesis or due to stress applied [11]. The defects may also be generated during the process of positioning graphene with either scanning tunneling microscopy (STM) or atomic force microscopy (AFM) tips due to the mechanical exploitation involved [12,13]. The SW defect plays an important role during the growth of carbon nanostructures. Once formed, the pentagon/heptagons could move along the structure, creating dislocation centers in regions of positive (pentagons) or/and negative (heptagons) Gaussian curvature of the deformed graphene sheet. A positive curvature of graphene sheets is produced by the presence of pentagons, and negative curvature appears with a heptagons. The curvature ultimately leads to the closing of nanostructured graphene [14].
*
Corresponding author at: Zhengzhou Institute of Aeronautical Industry Management, Henan 450046, PR China. Tel.: +86 371 66002013; fax: +86 371 68889638. E-mail address: [email protected] (R. Zhang). 0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.04.066
Many researchers have reported the anisotropic nature of graphene and the SW defect in single-walled carbon nanotubes (SWCNT). Tombros et al. [15] had studied the anisotropic spin relaxation properties in graphene, and Dinadayalane et al. [16] found that the reactivity of the C–C bond shared by two heptagons in the SW defect of SWCNTs depends on the direction of the SW defect. Zhou et al. [17] found that the formation energy of the SW defect is related with the radius of CNT and the direction in the tube. A detailed study on the anisotropic mechanical properties and the SW defect in graphene sheet, however, has not been performed to date. In this work, we studied the mechanical responses under tensile strain along several directions of graphene and the formation energy of SW defect using a density-functional tight-binding method (DFTB), and elucidated how the formation energy of SW defect changes with the direction of the applied tensile stress, and what is the most preferential direction for the SW defects under the applied tensile strain. 2. Computational details Our calculations were performed using the DFT method implemented in the DFTB+ code [18]. The force convergence criterion was set to 10−6 au and a k-point mesh of 8 × 8 × 1 was adopted with Monkhorst–Pack sampling. Atom positions were fully optimized using a conjugate gradient (CG) algorithm until the interaction forces were less than 10−8 au/atom. Periodic boundary conditions were employed, and 10 Å of a vacuum separation along the c axis was used, which is enough wide to eliminate interactions between the graphene layers in neighboring supercells.
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B.B. Fan et al. / Physics Letters A 374 (2010) 2781–2784
In the calculation on the strain energy and the formation energy of SW defects in graphene along different directions, consecutive studies were conducted as follows: (1) Generating the strained graphene by extending the lattice constant along different directions. Three symmetrical strained directions were considered as (n, 0), (n, n), and (2n, n) directions. Using the rectangular cell, we extended one of the lattice vectors to induce the tensile strain and search the optimized value for the other lattice vector according to the total energies. (2) Calculating the strain energy of graphene under the abovementioned situations by E |SE = ( E strain − E perf )/n, where, n is the number of carbon atoms; E strain and E perf corresponds to the total energy of the systems with or without tensile strain, respectively. (3) Calculating the SW formation energy by E sw = ( E def − E perf )/n, where, n is the number of carbon atoms, and E def is the strain energy of the SW defect imbedded in a flat graphene monolayer. 3. Results and discussion Fig. 1 shows the representative directions in the graphene with sp2 hybridizations, in which the C–C bond length is 1.420 Å with the bond angle of 120◦ . Figs. 2(a) and (b) show the geometry of graphene along (n, n) direction and that under symmetrically applied 12% stretching stress, respectively. Compared to the initial structure without stress applied, the bond length has changed to 1.536 Å and 1.418 Å, with the bond angle of 115.75◦ and 128.49◦ . Fig. 2(c) is the structure of graphene embedded with a SW defect, and the corresponding bond parameters are shown in Table 1. Fig. 3 shows the strain energy varying as a function of the tensile strain along different directions. The general trend of strain energy change is a smooth increase as the tensile strain increases. It illustrates that, asymmetrical strain distributions in graphene lead to diverse results, as reflected by the direction dependent strain energies. There is a larger strain energy change in the (n, 0) direction than that in the other two directions, although the curves for the (n, n) and (2n, n) directions almost completely overlapped
Fig. 1. The representative directions in the graphene.
when the tensile strain is under 8%, indicating that the graphene monolayer is anisotropic under large deformation. This behavior is consistent with the calculated results using the first-principles methods [19] and also experimental results [20]. It has been pointed out that the properties of graphene under finite deformations can be anisotropic, according to the study via the Brenner’s potential theory [21]. Another study based on molecular dynamics simulations also found the anisotropic material behavior [22]. We conjecture that the anisotropic mechanical property might be due to the charge distributions of π orbits, which requires more detailed investigations. Fig. 4 is the energy variations of perfect graphene and the one with SW defects versus the tensile strain along selected directions. Both the strain energies of perfect graphene and the one with defects increase as the tensile strain increases. It is shown that the strain energy of the SW defect is much higher than the perfect structure under lower strain. The result indicates that the perfect structure is much more stable than the SW defect structure under small tensile strain. As shown in Fig. 4(a), the strain energy of the SW defect structure is always higher than the perfect structure when the tensile is along the (n, n) direction, where the difference decreases as the strain energy increases. Along the (2n, n) direction (shown in Fig. 4(b)), when the tenslie strain is larger than 8%, the structure with the SW defects is more stable than the prefect one. The existence of this critical points indicate that the SW defect can be induced within graphene monolayer through the apTable 1 The bond parameters of the graphene structure embedded with a SW defect. Bond angle (◦ )
Bond length (Å) AB BC CD DE BF
1.459 1.450 1.466 1.338 1.461
BAH ABC BCD CDE FBC
121.82 126.25 140.33 122.51 115.48
Fig. 3. Strain energy versus tensile strain in the various representative directions. The points represent actual DFTB data; the curves are a guide to the eye.
Fig. 2. (a) The initial structure of graphene along (n, n) direction; (b) the structure of graphene under symmetrically applied 12% stretching stress perpendicular to the C–C bonds; and (c) the structure of graphene embedded with a SW defect.
B.B. Fan et al. / Physics Letters A 374 (2010) 2781–2784
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Fig. 5. Strain energy as a function of SW defect concentration: (a) in the (2n, n) direction; and (b) in the (n, 0) direction.
graphene under large strain is evidenced in the different strain energies under the same tensile along different directions. Consequently, the formation of SW defects with the same concentration depends on the direction of applied tensile stress. Our finding is useful for promoting nanoscale device fabrications and for developing strain-based devices. Acknowledgements Fig. 4. Strain energy as a function of the tensile strain in the representative directions: (a) in the (n, n) direction; (b) in the (2n, n) direction; and (c) in the (n, 0) direction.
The work described in this paper is supported by the National Natural Science Foundation [project No. 50972132].
plication of tensile stress, consistent with the previous literature [23]. And along the (n, 0) direction, the energy exchange point is at about 2%, much smaller than that in the (2n, n) direction. Comparing the three cases in Fig. 3, we can conclude that, when the tensile stress is applied on the graphene monolayer, the structure in (n, n) direction is the most stable one in the above three cases, while along the (n, 0) direction the SW defect is most easily to form. Moreover, the concentration of SW defect is also important. There is a general trend for strain energy to decrease as the concentration increases, as shown in Fig. 5. However, the formation energy is different in different directions. Along the (2n, n) direction, when the concentration decreases to 0.24%, i.e. one SW defect embedded in 420 carbon atoms, the formation energy of the critical point is 183 meV; while in the (n, 0) direction, at the same concentration, the formation energy of that point is about 18.2 meV, as shown in Fig. 5(b). It indicates that the SW defect is easier to form when the applied tensile strain is along the (n, 0) direction.
References
4. Summary In summary, the mechanical properties of graphene monolayer were investigated using DFT method. The anisotropic nature of
[1] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A.A. Firsov, Science 306 (2004) 666. [2] K.I. Bolotin, K.J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, H.L. Stormer, Solid State Communications 146 (2008) 351. [3] C. Lee, X. Wei, J.W. Kysar, J. Hone, Science 321 (2008) 385. [4] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M.I. Katsnelson, I.V. Grigorieva, S.V. Dubonos, A.A. Firsov, Nature 438 (2005) 197. [5] Y. Zhang, Y.-W. Tan, H.L. Stormer, P. Kim, Nature 438 (2005) 201. [6] M.Y. Han, B. Ozyilmaz, Y. Zhang, P. Kim, Physical Review Letters 98 (2007) 206805. [7] J.S. Bunch, A.M. van der Zande, S.S. Verbridge, I.W. Frank, D.M. Tanenbaum, J.M. Parpia, H.G. Craighead, P.L. McEuen, Science 315 (2007) 490. [8] P.G. Silvestrov, K.B. Efetov, Physical Review Letters 98 (2007) 016802. [9] H. Terrones, M. Terrones, E. Hernadez, N. Grobert, J.C. Charlier, P.M. Ajayan, Physical Review Letters 84 (2000) 1716. [10] A.J. Stone, D.J. Wales, Chemical Physics Letters 128 (1986) 501. [11] M. Buongiorno Nardelli, B.I. Yakobson, J. Bernholc, Physical Review B 57 (1998) R4277. [12] W. Fa, F. Hu, J. Dong, Physics Letters A 331 (2004) 99. [13] H. Kim, J. Lee, S.J. Kahng, Y.W. Son, S.B. Lee, C.K. Lee, J. Ihm, Y. Kuk, Physical Review Letters 90 (2003) 216107. [14] Y. Miyamoto, A. Rubio, S. Berber, M. Yoon, D. Tomaek, Physical Review B 69 (2004) 121413. [15] N. Tombros, S. Tanabe, A. Veligura, C. Jozsa, M. Popinciuc, H.T. Jonkman, B.J. van Wees, Physical Review Letters 101 (2008) 046601. [16] T.C. Dinadayalane, J. Leszczynski, Chemical Physics Letters 434 (2007) 86.
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B.B. Fan et al. / Physics Letters A 374 (2010) 2781–2784
[17] L.G. Zhou, S.-Q. Shi, Applied Physics Letters 83 (2003) 1222. [18] B. Aradi, B. Hourahine, T. Frauenheim, The Journal of Physical Chemistry A 111 (2007) 5678. [19] G. Gui, J. Li, J. Zhong, Physical Review B (Condensed Matter and Materials Physics) 78 (2008) 075435.
[20] K.S. Kim, Y. Zhao, H. Jang, S.Y. Lee, J.M. Kim, K.S. Kim, J.-H. Ahn, P. Kim, J.-Y. Choi, B.H. Hong, Nature 457 (2009) 706. [21] M. Arroyo, T. Belytschko, Physical Review B 69 (2004) 115415. [22] C.D. Reddy, S. Rajendran, K.M. Liew, Nanotechnology (2006) 864. [23] M.T. Lusk, L.D. Carr, Carbon 47 (2009) 2226.
Contents lists available at ScienceDirect
Physics Letters A www.elsevier.com/locate/pla
Anisotropic mechanical properties and Stone–Wales defects in graphene monolayer: A theoretical study B.B. Fan a , X.B. Yang b , R. Zhang a,c,∗ a b c
School of Materials Science & Engineering, Zhengzhou University, Henan 450001, PR China Department of Physics, South China University of Technology, Guangzhou 510640, PR China Zhengzhou Institute of Aeronautical Industry Management, Henan 450046, PR China
a r t i c l e
i n f o
Article history: Received 20 February 2010 Received in revised form 16 April 2010 Accepted 25 April 2010 Available online 1 May 2010 Communicated by R. Wu
a b s t r a c t We investigate the mechanical properties of graphene monolayer via the density functional theoretical (DFT) method. We find that the strain energies are anisotropic for the graphene under large strain. We attribute the anisotropic feature to the anisotropic sp2 hybridization in the hexagonal lattice. We further identify that the formation energies of Stone–Wales (SW) defects in the graphene monolayer are determined by the defect concentration and also the direction of applied tensile strain, correlating with the anisotropic feature. © 2010 Elsevier B.V. All rights reserved.
1. Introduction Graphene [1] has been attracting much attention due to its fascinating properties such as extremely high mobility [2], high elasticity [3], quantum electronic transport [4,5], a tunable band gap [6] and electromechanical modulation [7]. It shows potential applications in future solid devices, providing an alternative to the traditional silicon semi-conductors [8,9]. Strain and defects are always inevitable in nanostructures, which could dominate the electronic properties. One of the most important defects in graphene is the Stone–Wales defect (SW defect), in which the rotation of a single pair of carbon atoms creates adjacent pairs of pentagonal and heptagonal rings [10]. This imperfection in graphene is introduced either during the synthesis or due to stress applied [11]. The defects may also be generated during the process of positioning graphene with either scanning tunneling microscopy (STM) or atomic force microscopy (AFM) tips due to the mechanical exploitation involved [12,13]. The SW defect plays an important role during the growth of carbon nanostructures. Once formed, the pentagon/heptagons could move along the structure, creating dislocation centers in regions of positive (pentagons) or/and negative (heptagons) Gaussian curvature of the deformed graphene sheet. A positive curvature of graphene sheets is produced by the presence of pentagons, and negative curvature appears with a heptagons. The curvature ultimately leads to the closing of nanostructured graphene [14].
*
Corresponding author at: Zhengzhou Institute of Aeronautical Industry Management, Henan 450046, PR China. Tel.: +86 371 66002013; fax: +86 371 68889638. E-mail address: [email protected] (R. Zhang). 0375-9601/$ – see front matter © 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.physleta.2010.04.066
Many researchers have reported the anisotropic nature of graphene and the SW defect in single-walled carbon nanotubes (SWCNT). Tombros et al. [15] had studied the anisotropic spin relaxation properties in graphene, and Dinadayalane et al. [16] found that the reactivity of the C–C bond shared by two heptagons in the SW defect of SWCNTs depends on the direction of the SW defect. Zhou et al. [17] found that the formation energy of the SW defect is related with the radius of CNT and the direction in the tube. A detailed study on the anisotropic mechanical properties and the SW defect in graphene sheet, however, has not been performed to date. In this work, we studied the mechanical responses under tensile strain along several directions of graphene and the formation energy of SW defect using a density-functional tight-binding method (DFTB), and elucidated how the formation energy of SW defect changes with the direction of the applied tensile stress, and what is the most preferential direction for the SW defects under the applied tensile strain. 2. Computational details Our calculations were performed using the DFT method implemented in the DFTB+ code [18]. The force convergence criterion was set to 10−6 au and a k-point mesh of 8 × 8 × 1 was adopted with Monkhorst–Pack sampling. Atom positions were fully optimized using a conjugate gradient (CG) algorithm until the interaction forces were less than 10−8 au/atom. Periodic boundary conditions were employed, and 10 Å of a vacuum separation along the c axis was used, which is enough wide to eliminate interactions between the graphene layers in neighboring supercells.
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B.B. Fan et al. / Physics Letters A 374 (2010) 2781–2784
In the calculation on the strain energy and the formation energy of SW defects in graphene along different directions, consecutive studies were conducted as follows: (1) Generating the strained graphene by extending the lattice constant along different directions. Three symmetrical strained directions were considered as (n, 0), (n, n), and (2n, n) directions. Using the rectangular cell, we extended one of the lattice vectors to induce the tensile strain and search the optimized value for the other lattice vector according to the total energies. (2) Calculating the strain energy of graphene under the abovementioned situations by E |SE = ( E strain − E perf )/n, where, n is the number of carbon atoms; E strain and E perf corresponds to the total energy of the systems with or without tensile strain, respectively. (3) Calculating the SW formation energy by E sw = ( E def − E perf )/n, where, n is the number of carbon atoms, and E def is the strain energy of the SW defect imbedded in a flat graphene monolayer. 3. Results and discussion Fig. 1 shows the representative directions in the graphene with sp2 hybridizations, in which the C–C bond length is 1.420 Å with the bond angle of 120◦ . Figs. 2(a) and (b) show the geometry of graphene along (n, n) direction and that under symmetrically applied 12% stretching stress, respectively. Compared to the initial structure without stress applied, the bond length has changed to 1.536 Å and 1.418 Å, with the bond angle of 115.75◦ and 128.49◦ . Fig. 2(c) is the structure of graphene embedded with a SW defect, and the corresponding bond parameters are shown in Table 1. Fig. 3 shows the strain energy varying as a function of the tensile strain along different directions. The general trend of strain energy change is a smooth increase as the tensile strain increases. It illustrates that, asymmetrical strain distributions in graphene lead to diverse results, as reflected by the direction dependent strain energies. There is a larger strain energy change in the (n, 0) direction than that in the other two directions, although the curves for the (n, n) and (2n, n) directions almost completely overlapped
Fig. 1. The representative directions in the graphene.
when the tensile strain is under 8%, indicating that the graphene monolayer is anisotropic under large deformation. This behavior is consistent with the calculated results using the first-principles methods [19] and also experimental results [20]. It has been pointed out that the properties of graphene under finite deformations can be anisotropic, according to the study via the Brenner’s potential theory [21]. Another study based on molecular dynamics simulations also found the anisotropic material behavior [22]. We conjecture that the anisotropic mechanical property might be due to the charge distributions of π orbits, which requires more detailed investigations. Fig. 4 is the energy variations of perfect graphene and the one with SW defects versus the tensile strain along selected directions. Both the strain energies of perfect graphene and the one with defects increase as the tensile strain increases. It is shown that the strain energy of the SW defect is much higher than the perfect structure under lower strain. The result indicates that the perfect structure is much more stable than the SW defect structure under small tensile strain. As shown in Fig. 4(a), the strain energy of the SW defect structure is always higher than the perfect structure when the tensile is along the (n, n) direction, where the difference decreases as the strain energy increases. Along the (2n, n) direction (shown in Fig. 4(b)), when the tenslie strain is larger than 8%, the structure with the SW defects is more stable than the prefect one. The existence of this critical points indicate that the SW defect can be induced within graphene monolayer through the apTable 1 The bond parameters of the graphene structure embedded with a SW defect. Bond angle (◦ )
Bond length (Å) AB BC CD DE BF
1.459 1.450 1.466 1.338 1.461
BAH ABC BCD CDE FBC
121.82 126.25 140.33 122.51 115.48
Fig. 3. Strain energy versus tensile strain in the various representative directions. The points represent actual DFTB data; the curves are a guide to the eye.
Fig. 2. (a) The initial structure of graphene along (n, n) direction; (b) the structure of graphene under symmetrically applied 12% stretching stress perpendicular to the C–C bonds; and (c) the structure of graphene embedded with a SW defect.
B.B. Fan et al. / Physics Letters A 374 (2010) 2781–2784
2783
Fig. 5. Strain energy as a function of SW defect concentration: (a) in the (2n, n) direction; and (b) in the (n, 0) direction.
graphene under large strain is evidenced in the different strain energies under the same tensile along different directions. Consequently, the formation of SW defects with the same concentration depends on the direction of applied tensile stress. Our finding is useful for promoting nanoscale device fabrications and for developing strain-based devices. Acknowledgements Fig. 4. Strain energy as a function of the tensile strain in the representative directions: (a) in the (n, n) direction; (b) in the (2n, n) direction; and (c) in the (n, 0) direction.
The work described in this paper is supported by the National Natural Science Foundation [project No. 50972132].
plication of tensile stress, consistent with the previous literature [23]. And along the (n, 0) direction, the energy exchange point is at about 2%, much smaller than that in the (2n, n) direction. Comparing the three cases in Fig. 3, we can conclude that, when the tensile stress is applied on the graphene monolayer, the structure in (n, n) direction is the most stable one in the above three cases, while along the (n, 0) direction the SW defect is most easily to form. Moreover, the concentration of SW defect is also important. There is a general trend for strain energy to decrease as the concentration increases, as shown in Fig. 5. However, the formation energy is different in different directions. Along the (2n, n) direction, when the concentration decreases to 0.24%, i.e. one SW defect embedded in 420 carbon atoms, the formation energy of the critical point is 183 meV; while in the (n, 0) direction, at the same concentration, the formation energy of that point is about 18.2 meV, as shown in Fig. 5(b). It indicates that the SW defect is easier to form when the applied tensile strain is along the (n, 0) direction.
References
4. Summary In summary, the mechanical properties of graphene monolayer were investigated using DFT method. The anisotropic nature of
[1] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, Y. Zhang, S.V. Dubonos, I.V. Grigorieva, A.A. Firsov, Science 306 (2004) 666. [2] K.I. Bolotin, K.J. Sikes, Z. Jiang, M. Klima, G. Fudenberg, J. Hone, P. Kim, H.L. Stormer, Solid State Communications 146 (2008) 351. [3] C. Lee, X. Wei, J.W. Kysar, J. Hone, Science 321 (2008) 385. [4] K.S. Novoselov, A.K. Geim, S.V. Morozov, D. Jiang, M.I. Katsnelson, I.V. Grigorieva, S.V. Dubonos, A.A. Firsov, Nature 438 (2005) 197. [5] Y. Zhang, Y.-W. Tan, H.L. Stormer, P. Kim, Nature 438 (2005) 201. [6] M.Y. Han, B. Ozyilmaz, Y. Zhang, P. Kim, Physical Review Letters 98 (2007) 206805. [7] J.S. Bunch, A.M. van der Zande, S.S. Verbridge, I.W. Frank, D.M. Tanenbaum, J.M. Parpia, H.G. Craighead, P.L. McEuen, Science 315 (2007) 490. [8] P.G. Silvestrov, K.B. Efetov, Physical Review Letters 98 (2007) 016802. [9] H. Terrones, M. Terrones, E. Hernadez, N. Grobert, J.C. Charlier, P.M. Ajayan, Physical Review Letters 84 (2000) 1716. [10] A.J. Stone, D.J. Wales, Chemical Physics Letters 128 (1986) 501. [11] M. Buongiorno Nardelli, B.I. Yakobson, J. Bernholc, Physical Review B 57 (1998) R4277. [12] W. Fa, F. Hu, J. Dong, Physics Letters A 331 (2004) 99. [13] H. Kim, J. Lee, S.J. Kahng, Y.W. Son, S.B. Lee, C.K. Lee, J. Ihm, Y. Kuk, Physical Review Letters 90 (2003) 216107. [14] Y. Miyamoto, A. Rubio, S. Berber, M. Yoon, D. Tomaek, Physical Review B 69 (2004) 121413. [15] N. Tombros, S. Tanabe, A. Veligura, C. Jozsa, M. Popinciuc, H.T. Jonkman, B.J. van Wees, Physical Review Letters 101 (2008) 046601. [16] T.C. Dinadayalane, J. Leszczynski, Chemical Physics Letters 434 (2007) 86.
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B.B. Fan et al. / Physics Letters A 374 (2010) 2781–2784
[17] L.G. Zhou, S.-Q. Shi, Applied Physics Letters 83 (2003) 1222. [18] B. Aradi, B. Hourahine, T. Frauenheim, The Journal of Physical Chemistry A 111 (2007) 5678. [19] G. Gui, J. Li, J. Zhong, Physical Review B (Condensed Matter and Materials Physics) 78 (2008) 075435.
[20] K.S. Kim, Y. Zhao, H. Jang, S.Y. Lee, J.M. Kim, K.S. Kim, J.-H. Ahn, P. Kim, J.-Y. Choi, B.H. Hong, Nature 457 (2009) 706. [21] M. Arroyo, T. Belytschko, Physical Review B 69 (2004) 115415. [22] C.D. Reddy, S. Rajendran, K.M. Liew, Nanotechnology (2006) 864. [23] M.T. Lusk, L.D. Carr, Carbon 47 (2009) 2226.