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Structural and energetic features of single-walled carbon nanotube junctions: a theoretical ab initio study
Chemical Physics 303 (2004) 265–270 www.elsevier.com/locate/chemphys
Structural and energetic features of single-walled carbon nanotube junctions: a theoretical ab initio study ~onero, Antoni Costa, Pablo Ballester, Carolina Garau, Antonio Frontera *, David Quin Pere M. Dey a* Department of Chemistry, Universitat de les Illes Balears, Crta. de Valldemossa km 7.5, 07122 Palma de Mallorca, Spain Received 20 April 2004; accepted 9 June 2004 Available online
Abstract Theoretical investigations on single-walled carbon nanotubes (SWCNT) have been carried out using ab initio calculations at the HF/4-31G and B3LYP/3-21G//HF/3-21G levels of theory. Several junctions between different size nanotubes have been constructed with the insertion of pentagon–heptagon pairs into the perfect hexagonal lattice. The structural and energetic characteristics of the junctions are reported. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Carbon nanotubes; Ab initio calculations; Nanotube junctions; Homodesmotic reactions
1. Introduction Since their discovery in 1991 by Iijima [1], carbon nanotubes have attracted much attention, principally due to their electronic properties, which are directly related to their geometry. Their unusual electronic properties could lead to nanometer-sized electronic devices. A single-walled carbon nanotube (SWCNT) can be either metallic or semiconducting material [2], depending on its diameter and chirality. Individual semiconducting nanotubes function as field-effect transistors at room temperature [3], while metallic nanotubes are singleelectron transistors at low temperature [4,5]. Whether two nanotubes, one semiconductor and the other metallic, are connected a junction is formed and it behaves like a rectifying diode [6]. The electronic properties of intramolecular junctions are currently focus of considerable interest because the definitive device efficiency consists in using individual molecules as functional devices and SWCNTs are promising candidates for achieving this fascinating goal.
*
Corresponding authors. Tel.: +34971173498; fax: +34971173426. E-mail address: [email protected] (A. Frontera).
0301-0104/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2004.06.022
Structurally, a SWCNT can be viewed as a seamless cylinder obtained by rolling-up a section of a two-dimensional graphene sheet (see Fig. 1). It is uniquely characterized by the roll-up vector C ¼ na1 þ ma2 ¼ ðn; mÞ, where a1 and a2 are the primitive lattice vectors of the graphene and n, m are integers. There are two achiral directions termed ‘‘zigzag’’ and ‘‘armchair’’, and they are designated by (n; 0) and (n; n), respectively. Theoretical studies [7,8] have shown that SWCNTs are metals when n 2m ¼ 0, narrow-gap semiconductors when ðn 2mÞ=3 is an integer and they are moderategap semiconductors otherwise. As predicted theoretically [9], two nanotubes with different diameter and helicities can be connected by inserting a pentagon and a heptagon in the otherwise perfect hexagonal lattice. Depending on the number and location of the pentagon–heptagon pair defects it is possible to join two perfect nanotubes. In Fig. 2 it is shown the effect of introducing a single pentagon–heptagon (5/7) defect (bold line) in the honeycomb lattice. There are eight hexagons in each row above the pentagon and just seven below the heptagon. In this manuscript, we use ab initio calculations to study the effect of introducing 5/7 pair defects in the network of carbon nanotubes. We investigate the
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C=na1+ma2=(n,m)
a1
ZIGZAG
(0,0) a2
(1,0)
(2,0) C (1,1)
(3,0)
(2,1)
(4,0)
(3,1)
(2,2)
(6,0)
(5,0)
(4,1)
(5,1) (4,2)
(3,2)
(3,3)
(6,1)
(7,1)
(6,2)
(5,2)
(5,3)
(4,3)
(8,0)
(7,0)
(9,2)
(8,2)
(7,2)
(7,3)
(8,3)
(7,4)
(6,4)
(6,5)
(5,5)
(10,0)
(9,1)
(8,1)
(6,3)
(5,4)
(4,4)
(9,0)
(8,4)
(7,5)
(6,6)
(7,6)
AR MC HA (7,7)
IR
Fig. 1. Schematics of the generation of a carbon nanotube by folding a section of a graphene sheet and the resulting nanotube is characterized by the chirality vector C. In the example shown here (dashed vector) C ¼ 3a1 + 2a2 and the tube is labeled (3,2).
1
1
3
2
3
2
4
5
4 a
7
6
8
5
6
7
8
4
5
6
7
a'
b c d e f
1
1
3
2
2
3
f' Chep
4
5
6
7
Chep tilt angle
Fig. 2. Top: schematic representation of a 5/7 pair defect (bold lines), which is the responsible of the nanotube diameter change. The labels used to define the different bonds of the defect that are used in Table 2 are shown. Bottom: schematic representation of the 5/7 pair defect, with reference to the boat-like conformation that the seven-membered ring adopts in the nanotubes.
energetic and geometric features of different distributions of 5/7 pair defects that allow the connection of two zigzag nanotubes. There are several possible joints. In particular, we study four different ways of joining one (8,0) and one (6,0) zigzag nanotubes analyzing the energetic cost of each junction by means of using homodesmotic equations. The purpose of this letter is to
learn how to add and distribute 5/7 pair defects in an energetically favorable manner in order to facilitate the future synthesis of nanotube junctions, and in contrast with previous studies [10], we analyze the energetic characteristics of the joints using homodesmotic reactions. These type of reactions, which have not been used before in evaluating the energetic features of nanotube
C. Garau et al. / Chemical Physics 303 (2004) 265–270
junctions, will allow to compare a wide variety of junctions with different number of atoms using an efficient and unambiguous criterion.
2. Computational methods Initially, the geometries [11] of all SWCNTs included in this study were optimized at the restricted Hartree– Fock level using the PM3 semiempirical SCF-MO method [12,13] as implemented in the MOPAC-93 package [14] by means of the Eigenvector Following routine [15]. These structures have been used as starting points for the optimization at the HF/3-21G and HF/431G levels of theory using the Gaussian 98 program [16]. Single point calculations at B3LYP/3-21G level of theory were carried out using the HF/3-21G optimized geometries in order to include the electron-correlation effects and to increase the accuracy in the energetic calculations which have been performed using homodesmotic equations [17]. Previous studies [18,19] have demonstrated that reliable quantitative results are obtained at this level of theory which is used to keep the size of the calculation approachable.
267
are present in Fig. 3. In nanotube 1, the two 5/7 pair defects are distributed equidistant around the circumference of the tube forming a cone-like junction, see Fig. 3. In 2, the two 5/7 pair defects are aligned along the cylindrical axis. In 3, the two 5/7 pair defects are distributed around the circumference of the tube like in nanotube 1, but they are placed in different rows, see Fig. 3. Finally, in nanotube 4 an hexagon is inserted between the pentagon and heptagon rings, giving rise to a reduction in the diameter of the nanotube equivalent to a double 5/7 pair defect. In order to evaluate the energy cost of the joints, we have used four homodesmotic reactions. Latter reactions have been successfully used to estimate aromaticity by calculating cyclic conjugation energies [25]. The energetic results at three levels of theory are present in Table 1. In the equations, the superscript number is used to define the number of rows (belts of hexagons) present in each nanotube. It is worth mentioning that the energies present in Table 1 are negative because the nanotubes 1–4 are on the left side of the reaction. The more negative the computed energy is, the more unstable the nanotube junction is. The computed energies present in Table 1 are divided by a factor of two, because they are given per mole of nanotube.
3. Results and discussion It is worth mentioning that the present work deals mainly with the energetic features of different nanotube junctions that have been scarcely studied in the literature [10]. In contrast, the electronic properties of nanotube junctions, including the tunneling conductance of the joint, have been widely studied [20–22] and they are included in several books [23,24]. The HF/4-31G relaxed structures (1–4) of different joints connecting two zigzag nanotubes, (8,0) and (6,0)
Table 1 Reaction energies (in kcal/mol) obtained by means of homodesmotic equations, which are used to evaluate the energetic cost of the joints Nanotube
1 2 3 4
DE HF/3-21G
HF/4-31G
B3LYP/3-21G//HF/3-21G
)60.25 )46.89 )53.82 )33.87
)65.60 )49.12 )56.99 )39.11
)54.07 )49.46 )73.70 )34.35
The more positive the energy is, the more stable the joint is.
Fig. 3. HF/4-31G optimized structures (1–4) of nanotube junctions. The pentagon and heptagon rings, which are responsible for the change in nanotube diameter, are visualized in black.
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C. Garau et al. / Chemical Physics 303 (2004) 265–270 6
6
2 1ðC98 H14 Þ ! ð8; 0Þ ðC112 H16 Þ þ ð6; 0Þ ðC84 H12 Þ 7
ð1Þ
5
2 2ðC100 H14 Þ ! ð8; 0Þ ðC128 H16 Þ þ ð6; 0Þ ðC72 H12 Þ ð2Þ 7
5
2 3ðC100 H14 Þ ! ð8; 0Þ ðC128 H16 Þ þ ð6; 0Þ ðC72 H12 Þ ð3Þ 7
7
2 4ðC112 H14 Þ ! ð8; 0Þ ðC128 H16 Þ þ ð6; 0Þ ðC96 H12 Þ ð4Þ At the HF/3-21G and HF/4-31G levels of theory, the results point out that the peculiar case of a single 5/6/7 defect (nanotube 4) is more stable than any other combination of two 5/7 pair defects distributed around the circumference of the nanotubes (1–3). The results also show that the situation where the two 5/7 pair defects are aligned along the cylindrical axis of the nanotube (2) is more stable than the two 5/7 pair defects distributed around the axis (1 and 3); and the least stable nanotube is 1 where the 5/7 pair defects are symmetrically distributed around the circumference. In terms of the qualitative stabilities of the joints, these results are in agreement with a previous study [10] where the calculations were performed using tight-binding molecular dynamics [26], however in that study [10] the atomic structure of the joint represented in nanotube 3 was not considered. Additionally, the energetic results presented in [10] are not intuitive, since they are given per atom and they are relative to the energy of the ideal unperturbed cylindrical nanotube, which is not described. Thus, a quantitative direct comparison between both studies cannot be performed. Both studies predict that the single 5/6/7 defect gives the most stable junction and that 5/7 pair defects are more stable when they are aligned along the nanotube axis than placed around the cylindrical circumference. The energies present in Table 1 at both HF/4-31G and HF/3-21G levels of theory are comparable, being more negative (less favorable junctions) the energies computed at the HF/4-31G level of theory. Finally, we have evaluated the inclusion of electron correlation in the calculations by using the non local hybrid threeparameter (B3LYP) exchange-correlation density functional [27]. At the B3LYP/3-21G//HF/3-21G level of theory the computed energies derived from homodesmotic equations corresponding to nanotubes 2 and 4 are very similar to the ones computed at the HF/4-31G level. In contrast, a reduction of the computed energy is observed in nanotube 1 and an important change in the computed energy is observed in nanotube 3, being the more unfavorable junction. Again, nanotube 4 (single 5/ 6/7 defect) is the most stable junction. At this level, nanotube 2 is also more stable than either 1 or 3, confirming that the best combination regarding the location
of two 5/7 pair defects is achieved when they are placed along the cylindrical axis and any other combination of two defects is energetically more unfavorable. Table 2 shows some geometrical features of nanotubes 1–4. The computed bond lengths and angles of the 5/ 7 and 5/6/7 defects are summarized in Table 2 together with the data of distances and angles computed at the same level for two model compounds, azulene as a model defect for nanotubes 1–3 and cyclohepta[f]indene as a model for nanotube 4, see Fig. 4. The bond lengths and angles obtained for the defects are similar for all nanotubes studied here, and they do not justify the difference in energy. When comparing the computed distances of the defects with those of the model compounds, an acceptable agreement is found. For instance, there are not important differences between the mean distances calculated for the five- and seven-membered rings of the nanotubes and the corresponding to the model compounds. The same behavior is observed for the angles, apart from angle ff0 (see Table 2), which is much greater in the model compounds than in the defects for all nanotubes and in one case (3, inferior defect) the difference is very important, approximately 24°. As a consequence the average value calculated for the angles of the heptagon ring is higher for the model compounds than for the nanotubes. In contrast to the pentagon shape, which is essentially planar, the heptagon is boat shaped (see Fig. 2, bottom) and this distortion causes the reduction of the ff0 angle from the 128–130° range of the models to the 106–122° range of the nanotubes. In order to measure this distortion, we give in Table 2 the tilt angle of the carbon labeled Chep in Fig. 2 measured for the optimized nanotubes 1–4 and the models. The most stable junction corresponds to nanotube 4 probably because just one defect is inserted in the otherwise perfect hexagonal lattice. The more tilted nanotubes are 1 and 3. The former has two tilt angles of 25.8° and the latter has a tilt angle of 19.7° and other of 35.5° (superior and inferior defects, respectively). Finally, in agreement with the energetic results, the nanotube 2 presents a more modest distortion (tilt angles of 20.4° and 16.9°) in comparison to nanotubes 1 and 3, this is also reflected in the average value of the heptagon angles of both defects (123.9° and 123.4°, superior and inferior, respectively), which are close to the average value of the model (128.5°). As suggested by one of the referees, in order to confirm that the different stability between the four junctions considered in the present study is related with the distortion of the 5/7 and 5/6/7 defects, we have computed the energetic strain in nanotubes 1–4 caused by the distortion comparing the energy of the fully relaxed geometries of azulene and cyclohepta[f]indene with the geometries found in the nanotubes. The results are shown in Table 3 and they are in qualitative agreement with the homodesmotic energies. The least distorted
0.0 0.0 128.5 128.6 108.1 108.0 130.0 127. 9 128.7 128.6
25.8 20.4 16.9 19.7 35.5 25.4 122.5 123.9 123.4 124.0 119.1 122.0 107.5 107.7 107.4 107.6 107.4 107.5 111.0 119.0 122.0 118.9 106.3 110.5 126.1 124.9 125.4 126.7 119.6 125.5
Tilt Mean 5-ring ff0 ef
Mean 7-ring
C. Garau et al. / Chemical Physics 303 (2004) 265–270
azulene
Table 3 Distortion energies (DEdis , kcal/mol) obtained by difference between the fully relaxed geometries of the model compounds azulene and cyclohepta[f]indene and the geometries found in nanotubes 1–4 at three levels of theory
129.1 132.3 126.9 125.1
1 2 3 4
109.9 110.3 1.398 1.404
HF/3-21G
HF/4-31G
B3LYP/3-21G//HF/3-21G
139.92 110.36 155.91 94.97
134.66 109.22 148.85 92.83
108.28 84.88 122.03 68.61
4. Conclusion
a
See Fig. 2 for the definitions of bond labels a–f. This distance corresponds to the five-membered ring of the 5/6/7 defect. b This distance corresponds to the seven-membered ring of the 5/6/7 defect.
1.413 1.416 1.385 1.387 1.383 1.386 1.383 1.402 1.397 1.402 1.393 1.403 Azulene Cyclohepta[f]indene
DEdis
nanotube is 4, which only has a single 5/6/7 defect and it is confirmed that nanotube 2 presents a modest distortion in comparison with nanotubes 1 and 3. These results are in agreement with the geometrical features (the tilt angles of the defects) of nanotubes 1–4.
108.6 107.6
106.7 107.3
123.0 122.9 122.5 122.1 120.6 125.2 124.3 126.5 122.9 125.7 123.4 121.2 105.3 105.8 104.0 105.0 105.9 106.2 1.406 1.445 1.391 1.437 1.406 1.426 1 2 2 3 3 4
(superior) (inferior) (superior) (inferior)
1.430 1.412 1.449 1.416 1.425 1.425
1.451 1.364 1.511 1.383 1.476 1.413a 1.407b 1.487 1.470a 1.481b
1.403 1.445 1.392 1.426 1.392 1.463
1.441 1.390 1.450 1.411 1.464 1.407
1.473 1.490 1.417 1.464 1.476 1.493
1.425 1.416 1.438 1.418 1.428 1.423
1.441 1.431 1.433 1.426 1.449 1.448
108.3 108.5 110.4 108.8 109.1 107.9
107.8 107.9 106.1 107.8 106.5 107.7
de cd bc ab aa0 Mean 7-ring b a
c
d
e
f
Mean 5-ring
Angles Distances Nanotube
cyclohepta[f]indene
Fig. 4. Chemical structures of azulene and cyclohepta[f]indene.
Nanotube
Table 2 and angles (°) of the 5/7 and 5/6/7 defects computed for nanotubes 1–4 at the HF/4-31G level of theory Distances (A)
269
In summary, we have minimized several nanotubes containing different junctions that convert a (8,0) zigzag nanotube into a (6,0) zigzag nanotube at three ab initio levels of theory. These joints are structurally different and the energetic cost of each joint has been evaluated using homodesmotic equations at several levels of theory. This method has not been used before to evaluate the energetic characteristics of nanotube junctions. As a result, we can conclude that the metallic-semiconducting junction is more favorable if it is carried out by introducing a single 5/6/7 defect into the cylindrical surface of the zigzag nanotube. Two 5/7 pair defects aligned along the cylindrical axis is also a favorable situation. In contrast, the introduction of two 5/7 pair defects distributed around the circumference of the nanotube are unfavorable junctions.
Acknowledgements We thank the DGICYT and Conselleria d’Innovaci o i Energia (Govern Balear) of Spain (projects BQU200204651 and PRDIB-2002GC1-05, respectively) for financial support. We thank the Centre de Supercomputaci o de Catalunya (CESCA) for computational facilities. C.G. thanks the MECD for a predoctoral
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fellowship. A.F. thanks the MCyT for a ‘‘Ram on y Cajal’’ contract. References [1] S. Iijima, Nature 354 (1991) 56. [2] M. Ouyang, J.-L. Huang, C.M. Lieber, Acc. Chem. Res. 35 (2002) 1018. [3] S.J. Tans, A.R.M. Verschueren, C. Dekker, Nature 393 (1998) 49. [4] M. Berkrath et al., Science 279 (1997) 1922. [5] S.J. Tans et al., Nature 386 (1997) 474. [6] Z. Yao, H.W.C. Postma, L. Balents, C. Dekker, Nature 402 (1999) 273. [7] J.W. Mintmire, B.I. Dunlap, C.T. White, Phys. Rev. Lett. 68 (1992) 631. [8] N. Hamada, S. Sawada, A. Oshiyama, Phys. Rev. Lett. 68 (1992) 1579. [9] B.I. Dunlap, Phys. Rev. B 49 (1994) 5643. [10] J.-C. Charlier, T.W. Ebbesen, Ph. Lambin, Phys. Rev. B 53 (1996) 11108. [11] Geometries are available upon request by contacting.
[12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24]
[25] [26] [27]
J.J.P. Stewart, J. Comput. Chem. 10 (1989) 209. J.J.P. Stewart, J. Comput. Chem. 10 (1989) 221. J.J.P. Stewart, MOPAC-93, Fujitsu Ltd., 1993. J. Baker, J. Comput. Chem. 7 (1986) 385. M.J. Frisch et al., Gaussian 98, Revision A.7, Gaussian, Inc., Pittsburgh, PA, 1998. P. George, M. Trachtman, C.W. Bock, A.M. Brett, Theor. Chim. Acta 38 (1975) 121. C. Garau et al., Chem. Phys. Lett. 374 (2003) 548. C. Garau et al., Chem. Phys. 297 (2004) 85. R. Saito, G. Dresselhaus, M.S. Dresselhaus, Phys. Rev. B 53 (1996) 2044. R. Tamura, M. Tsukada, Phys. Rev. B 55 (1997) 4991. A.N. Andriotis, M. Menon, D. Srivastava, L. Chernozatonskii, Phys. Rev. B 65 (2002) 165416. R. Saito, G. Dresselhaus, M.S. Dresselhaus, Physical Properties of Carbon Nanotubes, Imperial College Press, London, 1998. M.S. Dresselhaus, G. Dresselhaus, P. Avouris (Eds.), Carbon Nanotubes: Synthesis, Structure, Properties, and Applications, Springer-Verlag, New York, 2000. D.B. Chesnut, K.M. Davis, J. Comput. Chem. 18 (1996) 584. S. Goedecker, L. Colombo, Phys. Rev. Lett. 73 (1994) 122. A.D. Becke, J. Chem. Phys. 98 (1993) 5648.
Structural and energetic features of single-walled carbon nanotube junctions: a theoretical ab initio study ~onero, Antoni Costa, Pablo Ballester, Carolina Garau, Antonio Frontera *, David Quin Pere M. Dey a* Department of Chemistry, Universitat de les Illes Balears, Crta. de Valldemossa km 7.5, 07122 Palma de Mallorca, Spain Received 20 April 2004; accepted 9 June 2004 Available online
Abstract Theoretical investigations on single-walled carbon nanotubes (SWCNT) have been carried out using ab initio calculations at the HF/4-31G and B3LYP/3-21G//HF/3-21G levels of theory. Several junctions between different size nanotubes have been constructed with the insertion of pentagon–heptagon pairs into the perfect hexagonal lattice. The structural and energetic characteristics of the junctions are reported. Ó 2004 Elsevier B.V. All rights reserved. Keywords: Carbon nanotubes; Ab initio calculations; Nanotube junctions; Homodesmotic reactions
1. Introduction Since their discovery in 1991 by Iijima [1], carbon nanotubes have attracted much attention, principally due to their electronic properties, which are directly related to their geometry. Their unusual electronic properties could lead to nanometer-sized electronic devices. A single-walled carbon nanotube (SWCNT) can be either metallic or semiconducting material [2], depending on its diameter and chirality. Individual semiconducting nanotubes function as field-effect transistors at room temperature [3], while metallic nanotubes are singleelectron transistors at low temperature [4,5]. Whether two nanotubes, one semiconductor and the other metallic, are connected a junction is formed and it behaves like a rectifying diode [6]. The electronic properties of intramolecular junctions are currently focus of considerable interest because the definitive device efficiency consists in using individual molecules as functional devices and SWCNTs are promising candidates for achieving this fascinating goal.
*
Corresponding authors. Tel.: +34971173498; fax: +34971173426. E-mail address: [email protected] (A. Frontera).
0301-0104/$ - see front matter Ó 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.chemphys.2004.06.022
Structurally, a SWCNT can be viewed as a seamless cylinder obtained by rolling-up a section of a two-dimensional graphene sheet (see Fig. 1). It is uniquely characterized by the roll-up vector C ¼ na1 þ ma2 ¼ ðn; mÞ, where a1 and a2 are the primitive lattice vectors of the graphene and n, m are integers. There are two achiral directions termed ‘‘zigzag’’ and ‘‘armchair’’, and they are designated by (n; 0) and (n; n), respectively. Theoretical studies [7,8] have shown that SWCNTs are metals when n 2m ¼ 0, narrow-gap semiconductors when ðn 2mÞ=3 is an integer and they are moderategap semiconductors otherwise. As predicted theoretically [9], two nanotubes with different diameter and helicities can be connected by inserting a pentagon and a heptagon in the otherwise perfect hexagonal lattice. Depending on the number and location of the pentagon–heptagon pair defects it is possible to join two perfect nanotubes. In Fig. 2 it is shown the effect of introducing a single pentagon–heptagon (5/7) defect (bold line) in the honeycomb lattice. There are eight hexagons in each row above the pentagon and just seven below the heptagon. In this manuscript, we use ab initio calculations to study the effect of introducing 5/7 pair defects in the network of carbon nanotubes. We investigate the
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C. Garau et al. / Chemical Physics 303 (2004) 265–270
C=na1+ma2=(n,m)
a1
ZIGZAG
(0,0) a2
(1,0)
(2,0) C (1,1)
(3,0)
(2,1)
(4,0)
(3,1)
(2,2)
(6,0)
(5,0)
(4,1)
(5,1) (4,2)
(3,2)
(3,3)
(6,1)
(7,1)
(6,2)
(5,2)
(5,3)
(4,3)
(8,0)
(7,0)
(9,2)
(8,2)
(7,2)
(7,3)
(8,3)
(7,4)
(6,4)
(6,5)
(5,5)
(10,0)
(9,1)
(8,1)
(6,3)
(5,4)
(4,4)
(9,0)
(8,4)
(7,5)
(6,6)
(7,6)
AR MC HA (7,7)
IR
Fig. 1. Schematics of the generation of a carbon nanotube by folding a section of a graphene sheet and the resulting nanotube is characterized by the chirality vector C. In the example shown here (dashed vector) C ¼ 3a1 + 2a2 and the tube is labeled (3,2).
1
1
3
2
3
2
4
5
4 a
7
6
8
5
6
7
8
4
5
6
7
a'
b c d e f
1
1
3
2
2
3
f' Chep
4
5
6
7
Chep tilt angle
Fig. 2. Top: schematic representation of a 5/7 pair defect (bold lines), which is the responsible of the nanotube diameter change. The labels used to define the different bonds of the defect that are used in Table 2 are shown. Bottom: schematic representation of the 5/7 pair defect, with reference to the boat-like conformation that the seven-membered ring adopts in the nanotubes.
energetic and geometric features of different distributions of 5/7 pair defects that allow the connection of two zigzag nanotubes. There are several possible joints. In particular, we study four different ways of joining one (8,0) and one (6,0) zigzag nanotubes analyzing the energetic cost of each junction by means of using homodesmotic equations. The purpose of this letter is to
learn how to add and distribute 5/7 pair defects in an energetically favorable manner in order to facilitate the future synthesis of nanotube junctions, and in contrast with previous studies [10], we analyze the energetic characteristics of the joints using homodesmotic reactions. These type of reactions, which have not been used before in evaluating the energetic features of nanotube
C. Garau et al. / Chemical Physics 303 (2004) 265–270
junctions, will allow to compare a wide variety of junctions with different number of atoms using an efficient and unambiguous criterion.
2. Computational methods Initially, the geometries [11] of all SWCNTs included in this study were optimized at the restricted Hartree– Fock level using the PM3 semiempirical SCF-MO method [12,13] as implemented in the MOPAC-93 package [14] by means of the Eigenvector Following routine [15]. These structures have been used as starting points for the optimization at the HF/3-21G and HF/431G levels of theory using the Gaussian 98 program [16]. Single point calculations at B3LYP/3-21G level of theory were carried out using the HF/3-21G optimized geometries in order to include the electron-correlation effects and to increase the accuracy in the energetic calculations which have been performed using homodesmotic equations [17]. Previous studies [18,19] have demonstrated that reliable quantitative results are obtained at this level of theory which is used to keep the size of the calculation approachable.
267
are present in Fig. 3. In nanotube 1, the two 5/7 pair defects are distributed equidistant around the circumference of the tube forming a cone-like junction, see Fig. 3. In 2, the two 5/7 pair defects are aligned along the cylindrical axis. In 3, the two 5/7 pair defects are distributed around the circumference of the tube like in nanotube 1, but they are placed in different rows, see Fig. 3. Finally, in nanotube 4 an hexagon is inserted between the pentagon and heptagon rings, giving rise to a reduction in the diameter of the nanotube equivalent to a double 5/7 pair defect. In order to evaluate the energy cost of the joints, we have used four homodesmotic reactions. Latter reactions have been successfully used to estimate aromaticity by calculating cyclic conjugation energies [25]. The energetic results at three levels of theory are present in Table 1. In the equations, the superscript number is used to define the number of rows (belts of hexagons) present in each nanotube. It is worth mentioning that the energies present in Table 1 are negative because the nanotubes 1–4 are on the left side of the reaction. The more negative the computed energy is, the more unstable the nanotube junction is. The computed energies present in Table 1 are divided by a factor of two, because they are given per mole of nanotube.
3. Results and discussion It is worth mentioning that the present work deals mainly with the energetic features of different nanotube junctions that have been scarcely studied in the literature [10]. In contrast, the electronic properties of nanotube junctions, including the tunneling conductance of the joint, have been widely studied [20–22] and they are included in several books [23,24]. The HF/4-31G relaxed structures (1–4) of different joints connecting two zigzag nanotubes, (8,0) and (6,0)
Table 1 Reaction energies (in kcal/mol) obtained by means of homodesmotic equations, which are used to evaluate the energetic cost of the joints Nanotube
1 2 3 4
DE HF/3-21G
HF/4-31G
B3LYP/3-21G//HF/3-21G
)60.25 )46.89 )53.82 )33.87
)65.60 )49.12 )56.99 )39.11
)54.07 )49.46 )73.70 )34.35
The more positive the energy is, the more stable the joint is.
Fig. 3. HF/4-31G optimized structures (1–4) of nanotube junctions. The pentagon and heptagon rings, which are responsible for the change in nanotube diameter, are visualized in black.
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6
2 1ðC98 H14 Þ ! ð8; 0Þ ðC112 H16 Þ þ ð6; 0Þ ðC84 H12 Þ 7
ð1Þ
5
2 2ðC100 H14 Þ ! ð8; 0Þ ðC128 H16 Þ þ ð6; 0Þ ðC72 H12 Þ ð2Þ 7
5
2 3ðC100 H14 Þ ! ð8; 0Þ ðC128 H16 Þ þ ð6; 0Þ ðC72 H12 Þ ð3Þ 7
7
2 4ðC112 H14 Þ ! ð8; 0Þ ðC128 H16 Þ þ ð6; 0Þ ðC96 H12 Þ ð4Þ At the HF/3-21G and HF/4-31G levels of theory, the results point out that the peculiar case of a single 5/6/7 defect (nanotube 4) is more stable than any other combination of two 5/7 pair defects distributed around the circumference of the nanotubes (1–3). The results also show that the situation where the two 5/7 pair defects are aligned along the cylindrical axis of the nanotube (2) is more stable than the two 5/7 pair defects distributed around the axis (1 and 3); and the least stable nanotube is 1 where the 5/7 pair defects are symmetrically distributed around the circumference. In terms of the qualitative stabilities of the joints, these results are in agreement with a previous study [10] where the calculations were performed using tight-binding molecular dynamics [26], however in that study [10] the atomic structure of the joint represented in nanotube 3 was not considered. Additionally, the energetic results presented in [10] are not intuitive, since they are given per atom and they are relative to the energy of the ideal unperturbed cylindrical nanotube, which is not described. Thus, a quantitative direct comparison between both studies cannot be performed. Both studies predict that the single 5/6/7 defect gives the most stable junction and that 5/7 pair defects are more stable when they are aligned along the nanotube axis than placed around the cylindrical circumference. The energies present in Table 1 at both HF/4-31G and HF/3-21G levels of theory are comparable, being more negative (less favorable junctions) the energies computed at the HF/4-31G level of theory. Finally, we have evaluated the inclusion of electron correlation in the calculations by using the non local hybrid threeparameter (B3LYP) exchange-correlation density functional [27]. At the B3LYP/3-21G//HF/3-21G level of theory the computed energies derived from homodesmotic equations corresponding to nanotubes 2 and 4 are very similar to the ones computed at the HF/4-31G level. In contrast, a reduction of the computed energy is observed in nanotube 1 and an important change in the computed energy is observed in nanotube 3, being the more unfavorable junction. Again, nanotube 4 (single 5/ 6/7 defect) is the most stable junction. At this level, nanotube 2 is also more stable than either 1 or 3, confirming that the best combination regarding the location
of two 5/7 pair defects is achieved when they are placed along the cylindrical axis and any other combination of two defects is energetically more unfavorable. Table 2 shows some geometrical features of nanotubes 1–4. The computed bond lengths and angles of the 5/ 7 and 5/6/7 defects are summarized in Table 2 together with the data of distances and angles computed at the same level for two model compounds, azulene as a model defect for nanotubes 1–3 and cyclohepta[f]indene as a model for nanotube 4, see Fig. 4. The bond lengths and angles obtained for the defects are similar for all nanotubes studied here, and they do not justify the difference in energy. When comparing the computed distances of the defects with those of the model compounds, an acceptable agreement is found. For instance, there are not important differences between the mean distances calculated for the five- and seven-membered rings of the nanotubes and the corresponding to the model compounds. The same behavior is observed for the angles, apart from angle ff0 (see Table 2), which is much greater in the model compounds than in the defects for all nanotubes and in one case (3, inferior defect) the difference is very important, approximately 24°. As a consequence the average value calculated for the angles of the heptagon ring is higher for the model compounds than for the nanotubes. In contrast to the pentagon shape, which is essentially planar, the heptagon is boat shaped (see Fig. 2, bottom) and this distortion causes the reduction of the ff0 angle from the 128–130° range of the models to the 106–122° range of the nanotubes. In order to measure this distortion, we give in Table 2 the tilt angle of the carbon labeled Chep in Fig. 2 measured for the optimized nanotubes 1–4 and the models. The most stable junction corresponds to nanotube 4 probably because just one defect is inserted in the otherwise perfect hexagonal lattice. The more tilted nanotubes are 1 and 3. The former has two tilt angles of 25.8° and the latter has a tilt angle of 19.7° and other of 35.5° (superior and inferior defects, respectively). Finally, in agreement with the energetic results, the nanotube 2 presents a more modest distortion (tilt angles of 20.4° and 16.9°) in comparison to nanotubes 1 and 3, this is also reflected in the average value of the heptagon angles of both defects (123.9° and 123.4°, superior and inferior, respectively), which are close to the average value of the model (128.5°). As suggested by one of the referees, in order to confirm that the different stability between the four junctions considered in the present study is related with the distortion of the 5/7 and 5/6/7 defects, we have computed the energetic strain in nanotubes 1–4 caused by the distortion comparing the energy of the fully relaxed geometries of azulene and cyclohepta[f]indene with the geometries found in the nanotubes. The results are shown in Table 3 and they are in qualitative agreement with the homodesmotic energies. The least distorted
0.0 0.0 128.5 128.6 108.1 108.0 130.0 127. 9 128.7 128.6
25.8 20.4 16.9 19.7 35.5 25.4 122.5 123.9 123.4 124.0 119.1 122.0 107.5 107.7 107.4 107.6 107.4 107.5 111.0 119.0 122.0 118.9 106.3 110.5 126.1 124.9 125.4 126.7 119.6 125.5
Tilt Mean 5-ring ff0 ef
Mean 7-ring
C. Garau et al. / Chemical Physics 303 (2004) 265–270
azulene
Table 3 Distortion energies (DEdis , kcal/mol) obtained by difference between the fully relaxed geometries of the model compounds azulene and cyclohepta[f]indene and the geometries found in nanotubes 1–4 at three levels of theory
129.1 132.3 126.9 125.1
1 2 3 4
109.9 110.3 1.398 1.404
HF/3-21G
HF/4-31G
B3LYP/3-21G//HF/3-21G
139.92 110.36 155.91 94.97
134.66 109.22 148.85 92.83
108.28 84.88 122.03 68.61
4. Conclusion
a
See Fig. 2 for the definitions of bond labels a–f. This distance corresponds to the five-membered ring of the 5/6/7 defect. b This distance corresponds to the seven-membered ring of the 5/6/7 defect.
1.413 1.416 1.385 1.387 1.383 1.386 1.383 1.402 1.397 1.402 1.393 1.403 Azulene Cyclohepta[f]indene
DEdis
nanotube is 4, which only has a single 5/6/7 defect and it is confirmed that nanotube 2 presents a modest distortion in comparison with nanotubes 1 and 3. These results are in agreement with the geometrical features (the tilt angles of the defects) of nanotubes 1–4.
108.6 107.6
106.7 107.3
123.0 122.9 122.5 122.1 120.6 125.2 124.3 126.5 122.9 125.7 123.4 121.2 105.3 105.8 104.0 105.0 105.9 106.2 1.406 1.445 1.391 1.437 1.406 1.426 1 2 2 3 3 4
(superior) (inferior) (superior) (inferior)
1.430 1.412 1.449 1.416 1.425 1.425
1.451 1.364 1.511 1.383 1.476 1.413a 1.407b 1.487 1.470a 1.481b
1.403 1.445 1.392 1.426 1.392 1.463
1.441 1.390 1.450 1.411 1.464 1.407
1.473 1.490 1.417 1.464 1.476 1.493
1.425 1.416 1.438 1.418 1.428 1.423
1.441 1.431 1.433 1.426 1.449 1.448
108.3 108.5 110.4 108.8 109.1 107.9
107.8 107.9 106.1 107.8 106.5 107.7
de cd bc ab aa0 Mean 7-ring b a
c
d
e
f
Mean 5-ring
Angles Distances Nanotube
cyclohepta[f]indene
Fig. 4. Chemical structures of azulene and cyclohepta[f]indene.
Nanotube
Table 2 and angles (°) of the 5/7 and 5/6/7 defects computed for nanotubes 1–4 at the HF/4-31G level of theory Distances (A)
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In summary, we have minimized several nanotubes containing different junctions that convert a (8,0) zigzag nanotube into a (6,0) zigzag nanotube at three ab initio levels of theory. These joints are structurally different and the energetic cost of each joint has been evaluated using homodesmotic equations at several levels of theory. This method has not been used before to evaluate the energetic characteristics of nanotube junctions. As a result, we can conclude that the metallic-semiconducting junction is more favorable if it is carried out by introducing a single 5/6/7 defect into the cylindrical surface of the zigzag nanotube. Two 5/7 pair defects aligned along the cylindrical axis is also a favorable situation. In contrast, the introduction of two 5/7 pair defects distributed around the circumference of the nanotube are unfavorable junctions.
Acknowledgements We thank the DGICYT and Conselleria d’Innovaci o i Energia (Govern Balear) of Spain (projects BQU200204651 and PRDIB-2002GC1-05, respectively) for financial support. We thank the Centre de Supercomputaci o de Catalunya (CESCA) for computational facilities. C.G. thanks the MECD for a predoctoral
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