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Ab initio study on the structural, energetic and electronic features of the asymmetric armchair SWCNT junctions
Journal of Molecular Structure: THEOCHEM 861 (2008) 79–84
Contents lists available at ScienceDirect
Journal of Molecular Structure: THEOCHEM journal homepage: www.elsevier.com/locate/theochem
Ab initio study on the structural, energetic and electronic features of the asymmetric armchair SWCNT junctions Yuanfeng Ye a, Milin Zhang a, Jianwei Zhao b,*, Hongmei Liu b, Nan Wang b a b
School of Materials Science and Chemical Engineering, Harbin Engineering University, Harbin 150001, PR China School of Chemistry and Chemical Engineering, Nanjing University, Nanjing, Jiangsu 210093, PR China
a r t i c l e
i n f o
Article history: Received 31 January 2008 Received in revised form 6 April 2008 Accepted 16 April 2008 Available online 29 April 2008 Keywords: Carbon nanotubes Ab initio calculations Junctions Molecular rectifier
a b s t r a c t The structural, energetic and electronic features of asymmetric armchair single-walled carbon nanotube (SWCNT) junctions have been studied by ab initio calculations at the B3LYP/6-31G*//HF/3-21G* levels. The junctions are composed of two SWCNTs with different radius, which are connected by a set of 5-membered and 7-membered carbon rings. The results show that the metallic–metallic junction is more energetically favorable if the junction is formed with a hexagon inserted between the pentagon–heptagon (5/ 7) pair defects in the armchair nanotube. The shift of the spatial distribution of HOMO and LUMO shows that the asymmetric electronic structure of the junction could be used as a molecular rectifier. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Since the discovery by Iijima in 1991 [1], carbon nanotubes have attracted high attention regarding their unique electronic properties. The geometrical structure of a single-wall carbon nanotube (SWCNT) is determined by the circumference vector C = na1 + ma2, where a1 and a2 are translation vectors of the graphene sheet lattice. Both the armchair nanotube (n, n) and the zigzag nanotube (n, 0) have a non-chiral configuration. The (n, n) armchair nanotubes are metallic while the (n, m) tubes are semi-metallic if n–m is a non-zero multiple of three, and semi-conducting otherwise [2]. At room temperature the individual semi-conducting nanotubes can function as field-effect transistors [3], while metallic nanotubes are used as single electron transistors at low temperature [4]. When two kinds of nanotubes (one semi-conductive and the other metallic) are connected, a unique junction is formed that may behave as a rectifying diode [5]. The electronic properties of the intra-molecular junctions, have currently received lots of research interest since SWCNTs are promising candidates for wiring of individual molecules as functional components into the nanocircuits. Two nanotubes of different diameter can be connected by inserting pentagon–heptagon (5/7) pair defects into the perfect hexagonal lattice [6]. The junction varies according to the different number and location of the pentagon–heptagon pairs. In the zigzag (n, 0) SWCNT the pentagon and heptagon are close to each other * Corresponding author. Tel./fax: +86 25 83596523. E-mail addresses: [email protected], [email protected] (J. Zhao). 0166-1280/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2008.04.023
(end to end). In contrast, in the armchair (n, n) SWCNT they adopt a head to head form and are linked by a carbon–carbon bond. In the case of connecting two SWCNTs which differ so greatly in diameter, it requires more 5/7 pairs in the junction. Furthermore those 5/7 pairs are distributed in the junction and may affect the electronic structure of the junction. In the previous studies, Frontera et al. have studied the distributions of two 5/7 pair defects in the zigzag SWCNT [7]. They compared the energies of several junctions and got a specified stable structure. However, the conclusion is questionable in the armchair SWCNT junction due to the structural particularity as shown in Fig. 1. Although in experiment the existence of junction can be proven by STM in the samples produced by laser ablation [8,9], the detailed topological defects and their distributions are hard to visualize. In this paper we performed ab initio calculations to study the effect of introducing 5/7 pair defects into the network of armchair SWCNTs. We focused on the energetic, geometric and electronic features of the junctions with different distributions of 5/7 pair defects. Among the many possible joints [10], in particular four different types of joining one (6, 6) and one (4, 4) armchair nanotubes (Fig. 1) were studied and the energetic cost of each junction was analyzed by using homodesmotic equations [7]. Besides the chemical properties of the junction, the potential usage as molecular electronic device has been discussed as well. 2. Computational methods The geometries of all SWCNTs included in this study were optimized initially at the restricted Hartree-Fock level with STO-3G
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Fig. 1. (a) HF/3-21G* optimized structures of armchair nanotube junctions (I–IV) and (b) the defects in the four models which are shown in red color and the left-down is the codes of the 5/7 pair defect. (For interpretation of color mentioned in this figure the reader is referred to the web version of the article.)
basis set. These structural fragments have been used in the following connections to form junctions. The junctions were carefully connected by introducing 5/7 pair defects, and fully optimized at the HF/3-21G* level of theory using the Gaussian 03 program [11]. Previous studies have demonstrated that reliable quantitative results are obtained at this level of theory which is used to keep the size of the calculation approachable [12,13]. In order to include the electron-correlation effects and increase the accuracy in the energy prediction, single point calculations at B3LYP/6-31G* level of theory were carried out using the geometries optimized by HF/3-21G*. 3. Results and discussion 3.1. The geometric structure Fig. 1 shows the relaxed structures at HF/3-21G* level (I–IV) of different joints connecting two armchair nanotubes, (6, 6) and (4, 4). Junction I contains two 5/7 pair defects distributed equidistantly around the circumference of the tube forming a cone-like junction. Junction II contains two 5/7 pair defects aligned along the cylindrical axis. In model III, the two 5/7 pair defects are distributed around the circumference of the tube as in nanotube I, but they are placed in different rows. In model IV, a 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 defects. The difference of the distribution of the 5/7 pair defects will result in the difference of the ring-strain force and then affect the geometric structure. From the optimized structures, we can conclude that the pentagon ring is planar and the heptagon one is boat shaped which is similar to those in the zigzag SWCNT. Frontera’s work shows that the difference of sta-
bility is mainly originated from the distortion of the defect pair [7]. So the bond angles play a more important role in the stability of SWCNT. The computed bond lengths and angles of the 5/7 and 5/6/7 defects are summarized in Table 1. To ensure the accuracy of these results the geometric discussions are built on the results of HF method. The average value of bond angles in four models is very similar while it is different in the zigzag SWCNT. Then what leads to the difference of the stability? In our previous work about the chemical properties of cyclic compounds, a new method to analyze the ring-strain energy has been proposed and it can be used effectively to evaluate the stability of system [14]. Analogy to that method in which three parameters named bond length, bond angle and dihedral-angle were defined, we can infer that the bond length and dihedral-angle will play a more important role in the current models. Finally the diversity of armchair and zigzag SWCNT can be explained by the different connection of pentagon ring and heptagon ring: the bond between the two rings increases the flexibility in the present models compared with that in the zigzag ones. The mean length of the particular bond in all the models is about 1.36 Å and shows the double bond character. The separation of the pentagon and heptagon rings changes conjugated situation of the defect part and then becomes the origin of the different stability for four junctions. In conclusion, both the distribution and the connection of the 5/7 defects affect the stability of the CNT junction both in the zigzag and armchair SWCNT. 3.2. The energetic features By calculating cyclic conjugation energies, the homodesmotic reactions can be used to estimate aromaticity [10]. Here four
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Y. Ye et al. / Journal of Molecular Structure: THEOCHEM 861 (2008) 79–84 Table 1 Bond lengths (Å) and angles (°) of the 5/7 and 5/6/7 defects computed for junctions I–IV at HF level of theory (the atom code is shown in Fig. 1b) Nanotube
I II (superior) II (inferior) III (superior) III (inferior) IV
Bond length B1–2
B2–3
B3–4
B6–7
B7–8
B8–9
B9–10
Mean 5-ring
Mean 7-ring
1.436 1.435 1.441 1.431 1.447 1.425
1.431 1.431 1.433 1.431 1.417 1.431
1.403 1.401 1.419 1.397 1.44 1.40
1.462 1.463 1.465 1.455 1.473 1.466
1.426 1.423 1.423 1.437 1.413 1.429
1.486 1.477 1.481 1.463 1.483 1.482
1.391 1.391 1.401 1.444 1.391 1.396
1.423 1.422 1.431 1.420 1.435 1.419
1.441 1.438 1.442 1.450 1.440 1.443
Bond angle
I II (superior) II (inferior) III (superior) III (inferior) IV
\5–1–2
\1–2–3
\2–3–4
\12–6–7
\6–7–8
\7–8–9
\8–9–10
Mean 5-ring
Mean 7-ring
105.3 105.6 104.9 105.8 104.8 107.3
108.6 108.7 108.9 108.5 109.0 107.5
107.9 108.0 107.6 108.0 107.5 108.2
118.4 121.1 117.7 120.5 117.9 117.2
123.8 124.1 123.9 124.0 124.0 125.7
126.2 124.2 125.2 124.6 125.5 126.1
121.6 124.0 121.5 122.6 122.0 121.3
107.3 107.4 107.1 107.4 107.1 107.7
122.5 123.4 122.1 122.9 122.4 122.6
homodesmotic reactions were used to calculate the reaction energy of joints and further evaluate the stability of junctions. This method has also been adopted by Frontera to study the zigzag SWCNT junction [7]. The results of energy at HF/3-21G* and DFT/ 6-31G* levels of theory are presented in Table 2.
ð6; 6Þ5 ðC144 H24 Þ þ ð4; 4Þ5 ðC96 H16 Þ ! 2 IðC120 H20 Þ
ð1Þ
ð6; 6Þ9 ðC228 H24 Þ þ ð4; 4Þ9 ðC152 H16 Þ ! 2 IIðC190 H20 Þ
ð2Þ
ð6; 6Þ8 ðC216 H24 Þ þ ð4; 4Þ8 ðC144 H16 Þ ! 2 IIIðC180 H20 Þ
ð3Þ
ð6; 6Þ7 ðC180 H24 Þ þ ð4; 4Þ7 ðC120 H16 Þ ! 2 IVðC150 H20 Þ
ð4Þ
In the equations, the subscript number is used to define the number of rows (belts of hexagons) presented in each nanotube. The smaller the computed energy is, the more stable the nanotube junction is. At the HF/3-21G* and B3LYP/6-31G* level of theory, both the results pointed out that reaction energy of the series of models follows this order: IV > III > I > II. The reaction energy of model IV is much smaller than the others, demonstrating model IV is more energetically stable. However, it is different for the zigzag junctions in which the defect distributions prefer the order: IV > II > III > I [7]. It shows that the defects distributions of armchair SWCNT junction is favorable like model IV in which a hexagon is inserted between the pentagon and heptagon rings. Therefore, we can conclude that in the junctions of SWCNT (including the zigzag and armchair) the insertion of hexagons between the pentagon and heptagon rings is a favorable method. In the other models the defect distribution is different in armchair and zigzag SWCNTs. This perhaps can be explained by the different connection of pentagon and heptagon in the 5/7 defect: the pentagon and heptagon rings are close to each other to form a new unit in zigzag SWCNT junction while they are separated by a carbon–car-
Table 2 Reaction energies (in a.u.) obtained by means of homodesmotic equations, which are used to evaluate the energetic cost of the joints Nanotube
E(66 + 44)
E(66)
E(44)
DE(HF/3-21G*)
I II III IV
4530.2332 7166.336 6789.7595 5660.0404
5436.834 8600.4041 8148.4524 6792.6434
3623.9616 5732.6105 5431.3831 4527.6729
0.3292 0.3426 0.3165 0.2355
E(66 + 44)
E(66)
E(44)
DE(B3LYP/6-31G*)
4584.8762 7252.6107 6871.5124 5728.2032
5502.363 8703.9217 8246.4926 6874.4157
3667.6844 5801.65235 5496.8121 4582.2478
0.295 0.3526 0.2799 0.2571
I II III IV
bon bond in armchair one. In fact the carbon–carbon bond in the armchair SWCNT acts as a hexagon which can relax the tension induced by the torsion. 3.3. The electronic features The frontier orbital approach [15] is well known for providing improved understanding of chemical reactivity. It has been demonstrated that most of the frontier orbitals theory of chemical reactivity can be rationalized within the framework of density functional theory (DFT) of the electronic structure of molecules [16,17]. DFTbased frontier orbitals play an important role also in the prediction of many ground state molecular properties. It should be noted that the Kohn–Sham (KS) orbitals and eigenvalues are very difficult to extract quantitative information and often viewed as just an auxiliary construct, meanwhile the exception is that the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). On the other hand the KS eigenvalues and orbitals are very similar to the single particle energy and wave function of the real system [18,19]. At the same time that changes in the spatial distribution of molecular orbitals, especially the frontier molecular orbitals (such as HOMO and LUMO) which are excellent indicators of many molecular properties [20,21]. In this work, we focus on the spatial distribution of the frontier molecular orbitals of four SWCNT models and compare their difference induced by the different distribution of defects. As shown in Fig. 2a the HOMO orbitals corresponding to all the models are found to be nearly localized on the entire skeleton and the positions of the defects show an obvious concentration. The distribution of LUMO orbitals seems more regular. They mainly stay around the circumference of the bigger tube. Furthermore the position of defects also affects the LUMO orbitals: the closer to the defects, the more concentrated the orbitals are. From above observation we can conclude that the different distribution of the defects results in the different distribution of frontier orbitals and furthermore affects, even determinates some molecular properties. So it is very important and useful to understand the defect distribution and control them in the experiments, though it looks very difficult in the present practice. Furthermore, these molecular orbitals mainly consist of the orbitals of carbon atoms of the tube, the auxiliary hydrogen atoms used to saturate the tube ends do not contribute to the frontier orbitals of armchair nanotubes. This is because most atoms in the joints are in a high tensile state, which may greatly increase the HOMO–LUMO gap. To further study the rectified effect of different junctions, we applied a uniform electric field along the axis from two opposite
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Fig. 2. (a) The spatial distribution of molecular frontier orbitals for four junction models at the zero-electric field and (b) the shift of frontier orbitals for the model I under 1.54 109 V/m electric field (the arrow is the direction of the electric field).
directions. The strength of electric field is 1.54 109 V/m which may reasonably represent the working condition of the molecular devices [22–25]. In previous experimental work under the similar electric field, we had used a current sensing atomic force microscope (CSAFM) to prepare nanosized patterns of an alkanethiol self-assembled monolayer (SAM) within a SAM of a different alkanethiol on a gold substrate [26]. The changes of HOMO and LUMO distribution in the different junctions are similar. Fig. 2b displays the representative models (junction I), from which we can see that the HOMO always shifts from the high potential to the low potential in both electric fields directions. However the LUMO shifts differently: when the direction of electric field is from (4, 4) to (6, 6), it moves from the low potential to the high potential, and when the electric field is opposite it almost does not shift. It is
obviously related with the original distribution of LUMO in the zero-electric field, which mainly locates on the thicker (6, 6). It is known that if an orbital is delocalized across the molecule, an electron injected into the molecule at the energy of the orbital has a high probability of reaching the other end [27–29]. So in the present study, the electric field from two opposite directions changes the distribution of the HOMO and LUMO differently. One makes them more delocalized and the other makes them more localized. Then we can approximately research the asymmetric I–V behavior through the asymmetrical variation of HOMO and LUMO under the interaction of the applied electric field. Although they will really contribute to the I–V behavior only when their energy levels are located within the bias window [30,31]. This property may give a good illumination to use the junctions as a rectifier.
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Obviously the conductance of a defective nanotube can show absolute differences compared with that of the pristine one [32]. Our question now is whether the different defects distribution can also result in the change of conductance. Here we analyze the density of states (DOS) for all the models and compare the difference. In Fig. 3 the DOS of four junctions reveals resonant states, which can be associated with different peculiar sites in the five- and seven-member rings. The observation of resonant states (sharp peaks) in the density of states of nanotube junction, close to the Fermi level, would be a proof of their existence.
I
8 7
b Density of states / arb. unit
Density of states / arb. unit
a
In every structure, the presence of resonant states, coming from the pentagonal and heptagonal atomic sites, is clearly visible. One sharp peak below and above the Fermi level can be associated with one 5/7 pair defect. In fact there are only two peaks in model I because the two pairs of 5/7 defects have the same location and then the corresponding peaks overlap. In model II there are also two peaks because when 5/7 pair defects are aligned along the direction of the tube, the main contribution comes from the first pair defect and the contribution of the other one vanishes. The same phenomenon was found in Charlier’s work
6 5 4 3 2 1 0 -1 -10
II
12 10 8 6 4 2 0
-8
-6
-4
-2
0
2
-10
4
-8
-6
Energy / eV 12
III
10 8 6 4 2
-8
-6
-4
-2
0
2
4
8 6 4 2
-10
-8
-6
f
8 6 4 2 0 -8
-6
-4
-2
Energy / eV
-4
-2
0
2
4
0
2
4
Energy / eV
Density of states / arb. unit
Density of states / arb. unit
4
0
10
-10
2
IV
Energy / eV
e
0
10
0 -10
-2
d Density of states / arb. unit
Density of states / arb. unit
c
-4
Energy / eV
0
2
4
8 7 6 5 4 3 2 1 0 -1 -10
-8
-6
-4
-2
Energy / eV
Fig. 3. Density of states for the four models at the zero-electric field (the a–d is corresponding to the model I–IV) and the model I in the non-zero-electric field with two opposite directions (e and f), the arrows show the peaks which are corresponding resonant states coming from the pentagonal and heptagonal atomic sites.
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[32]. There are separately four and three peaks for model III and IV though the two peaks below and above the Fermi level are much closer. But when the external electric field is applied, the positions of sharp peaks move and the peak heights change. Furthermore the different directions of the electric field induce the corresponding change (Fig. 3e and f). 4. Conclusion In summary, we have simulated several nanotubes containing different junctions that convert a (6, 6) armchair nanotube into a (4, 4) armchair nanotube using HF and DFT calculations. These joints are structurally different and the energetic cost of each joint has been evaluated using homodesmotic equations at several levels of theory. As a result, we can conclude that the metallic–metallic junction is more favorable when a hexagon between 5/7 pair defects is inserted in the armchair nanotube. In contrast, the junction with two 5/7 pair defects distributed along the cylindrical axis is unfavorable. The results also show that the different distributions of defects can lead to the different distribution of frontier orbitals. Furthermore the properties of the SWCNT will be affected. Acknowledgments The authors thank The National Natural Science Foundation of China (NSFC) (Nos. 20503012 and 20435010) and Key laboratory of superlight materials and surface technology, Ministry of education, Harbin Engineering University for financial supports. Mr. Y. Ye thanks Mrs. Danmin. Ye for her help during the preparation of the manuscript.
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Contents lists available at ScienceDirect
Journal of Molecular Structure: THEOCHEM journal homepage: www.elsevier.com/locate/theochem
Ab initio study on the structural, energetic and electronic features of the asymmetric armchair SWCNT junctions Yuanfeng Ye a, Milin Zhang a, Jianwei Zhao b,*, Hongmei Liu b, Nan Wang b a b
School of Materials Science and Chemical Engineering, Harbin Engineering University, Harbin 150001, PR China School of Chemistry and Chemical Engineering, Nanjing University, Nanjing, Jiangsu 210093, PR China
a r t i c l e
i n f o
Article history: Received 31 January 2008 Received in revised form 6 April 2008 Accepted 16 April 2008 Available online 29 April 2008 Keywords: Carbon nanotubes Ab initio calculations Junctions Molecular rectifier
a b s t r a c t The structural, energetic and electronic features of asymmetric armchair single-walled carbon nanotube (SWCNT) junctions have been studied by ab initio calculations at the B3LYP/6-31G*//HF/3-21G* levels. The junctions are composed of two SWCNTs with different radius, which are connected by a set of 5-membered and 7-membered carbon rings. The results show that the metallic–metallic junction is more energetically favorable if the junction is formed with a hexagon inserted between the pentagon–heptagon (5/ 7) pair defects in the armchair nanotube. The shift of the spatial distribution of HOMO and LUMO shows that the asymmetric electronic structure of the junction could be used as a molecular rectifier. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction Since the discovery by Iijima in 1991 [1], carbon nanotubes have attracted high attention regarding their unique electronic properties. The geometrical structure of a single-wall carbon nanotube (SWCNT) is determined by the circumference vector C = na1 + ma2, where a1 and a2 are translation vectors of the graphene sheet lattice. Both the armchair nanotube (n, n) and the zigzag nanotube (n, 0) have a non-chiral configuration. The (n, n) armchair nanotubes are metallic while the (n, m) tubes are semi-metallic if n–m is a non-zero multiple of three, and semi-conducting otherwise [2]. At room temperature the individual semi-conducting nanotubes can function as field-effect transistors [3], while metallic nanotubes are used as single electron transistors at low temperature [4]. When two kinds of nanotubes (one semi-conductive and the other metallic) are connected, a unique junction is formed that may behave as a rectifying diode [5]. The electronic properties of the intra-molecular junctions, have currently received lots of research interest since SWCNTs are promising candidates for wiring of individual molecules as functional components into the nanocircuits. Two nanotubes of different diameter can be connected by inserting pentagon–heptagon (5/7) pair defects into the perfect hexagonal lattice [6]. The junction varies according to the different number and location of the pentagon–heptagon pairs. In the zigzag (n, 0) SWCNT the pentagon and heptagon are close to each other * Corresponding author. Tel./fax: +86 25 83596523. E-mail addresses: [email protected], [email protected] (J. Zhao). 0166-1280/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.theochem.2008.04.023
(end to end). In contrast, in the armchair (n, n) SWCNT they adopt a head to head form and are linked by a carbon–carbon bond. In the case of connecting two SWCNTs which differ so greatly in diameter, it requires more 5/7 pairs in the junction. Furthermore those 5/7 pairs are distributed in the junction and may affect the electronic structure of the junction. In the previous studies, Frontera et al. have studied the distributions of two 5/7 pair defects in the zigzag SWCNT [7]. They compared the energies of several junctions and got a specified stable structure. However, the conclusion is questionable in the armchair SWCNT junction due to the structural particularity as shown in Fig. 1. Although in experiment the existence of junction can be proven by STM in the samples produced by laser ablation [8,9], the detailed topological defects and their distributions are hard to visualize. In this paper we performed ab initio calculations to study the effect of introducing 5/7 pair defects into the network of armchair SWCNTs. We focused on the energetic, geometric and electronic features of the junctions with different distributions of 5/7 pair defects. Among the many possible joints [10], in particular four different types of joining one (6, 6) and one (4, 4) armchair nanotubes (Fig. 1) were studied and the energetic cost of each junction was analyzed by using homodesmotic equations [7]. Besides the chemical properties of the junction, the potential usage as molecular electronic device has been discussed as well. 2. Computational methods The geometries of all SWCNTs included in this study were optimized initially at the restricted Hartree-Fock level with STO-3G
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Y. Ye et al. / Journal of Molecular Structure: THEOCHEM 861 (2008) 79–84
Fig. 1. (a) HF/3-21G* optimized structures of armchair nanotube junctions (I–IV) and (b) the defects in the four models which are shown in red color and the left-down is the codes of the 5/7 pair defect. (For interpretation of color mentioned in this figure the reader is referred to the web version of the article.)
basis set. These structural fragments have been used in the following connections to form junctions. The junctions were carefully connected by introducing 5/7 pair defects, and fully optimized at the HF/3-21G* level of theory using the Gaussian 03 program [11]. Previous studies have demonstrated that reliable quantitative results are obtained at this level of theory which is used to keep the size of the calculation approachable [12,13]. In order to include the electron-correlation effects and increase the accuracy in the energy prediction, single point calculations at B3LYP/6-31G* level of theory were carried out using the geometries optimized by HF/3-21G*. 3. Results and discussion 3.1. The geometric structure Fig. 1 shows the relaxed structures at HF/3-21G* level (I–IV) of different joints connecting two armchair nanotubes, (6, 6) and (4, 4). Junction I contains two 5/7 pair defects distributed equidistantly around the circumference of the tube forming a cone-like junction. Junction II contains two 5/7 pair defects aligned along the cylindrical axis. In model III, the two 5/7 pair defects are distributed around the circumference of the tube as in nanotube I, but they are placed in different rows. In model IV, a 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 defects. The difference of the distribution of the 5/7 pair defects will result in the difference of the ring-strain force and then affect the geometric structure. From the optimized structures, we can conclude that the pentagon ring is planar and the heptagon one is boat shaped which is similar to those in the zigzag SWCNT. Frontera’s work shows that the difference of sta-
bility is mainly originated from the distortion of the defect pair [7]. So the bond angles play a more important role in the stability of SWCNT. The computed bond lengths and angles of the 5/7 and 5/6/7 defects are summarized in Table 1. To ensure the accuracy of these results the geometric discussions are built on the results of HF method. The average value of bond angles in four models is very similar while it is different in the zigzag SWCNT. Then what leads to the difference of the stability? In our previous work about the chemical properties of cyclic compounds, a new method to analyze the ring-strain energy has been proposed and it can be used effectively to evaluate the stability of system [14]. Analogy to that method in which three parameters named bond length, bond angle and dihedral-angle were defined, we can infer that the bond length and dihedral-angle will play a more important role in the current models. Finally the diversity of armchair and zigzag SWCNT can be explained by the different connection of pentagon ring and heptagon ring: the bond between the two rings increases the flexibility in the present models compared with that in the zigzag ones. The mean length of the particular bond in all the models is about 1.36 Å and shows the double bond character. The separation of the pentagon and heptagon rings changes conjugated situation of the defect part and then becomes the origin of the different stability for four junctions. In conclusion, both the distribution and the connection of the 5/7 defects affect the stability of the CNT junction both in the zigzag and armchair SWCNT. 3.2. The energetic features By calculating cyclic conjugation energies, the homodesmotic reactions can be used to estimate aromaticity [10]. Here four
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Y. Ye et al. / Journal of Molecular Structure: THEOCHEM 861 (2008) 79–84 Table 1 Bond lengths (Å) and angles (°) of the 5/7 and 5/6/7 defects computed for junctions I–IV at HF level of theory (the atom code is shown in Fig. 1b) Nanotube
I II (superior) II (inferior) III (superior) III (inferior) IV
Bond length B1–2
B2–3
B3–4
B6–7
B7–8
B8–9
B9–10
Mean 5-ring
Mean 7-ring
1.436 1.435 1.441 1.431 1.447 1.425
1.431 1.431 1.433 1.431 1.417 1.431
1.403 1.401 1.419 1.397 1.44 1.40
1.462 1.463 1.465 1.455 1.473 1.466
1.426 1.423 1.423 1.437 1.413 1.429
1.486 1.477 1.481 1.463 1.483 1.482
1.391 1.391 1.401 1.444 1.391 1.396
1.423 1.422 1.431 1.420 1.435 1.419
1.441 1.438 1.442 1.450 1.440 1.443
Bond angle
I II (superior) II (inferior) III (superior) III (inferior) IV
\5–1–2
\1–2–3
\2–3–4
\12–6–7
\6–7–8
\7–8–9
\8–9–10
Mean 5-ring
Mean 7-ring
105.3 105.6 104.9 105.8 104.8 107.3
108.6 108.7 108.9 108.5 109.0 107.5
107.9 108.0 107.6 108.0 107.5 108.2
118.4 121.1 117.7 120.5 117.9 117.2
123.8 124.1 123.9 124.0 124.0 125.7
126.2 124.2 125.2 124.6 125.5 126.1
121.6 124.0 121.5 122.6 122.0 121.3
107.3 107.4 107.1 107.4 107.1 107.7
122.5 123.4 122.1 122.9 122.4 122.6
homodesmotic reactions were used to calculate the reaction energy of joints and further evaluate the stability of junctions. This method has also been adopted by Frontera to study the zigzag SWCNT junction [7]. The results of energy at HF/3-21G* and DFT/ 6-31G* levels of theory are presented in Table 2.
ð6; 6Þ5 ðC144 H24 Þ þ ð4; 4Þ5 ðC96 H16 Þ ! 2 IðC120 H20 Þ
ð1Þ
ð6; 6Þ9 ðC228 H24 Þ þ ð4; 4Þ9 ðC152 H16 Þ ! 2 IIðC190 H20 Þ
ð2Þ
ð6; 6Þ8 ðC216 H24 Þ þ ð4; 4Þ8 ðC144 H16 Þ ! 2 IIIðC180 H20 Þ
ð3Þ
ð6; 6Þ7 ðC180 H24 Þ þ ð4; 4Þ7 ðC120 H16 Þ ! 2 IVðC150 H20 Þ
ð4Þ
In the equations, the subscript number is used to define the number of rows (belts of hexagons) presented in each nanotube. The smaller the computed energy is, the more stable the nanotube junction is. At the HF/3-21G* and B3LYP/6-31G* level of theory, both the results pointed out that reaction energy of the series of models follows this order: IV > III > I > II. The reaction energy of model IV is much smaller than the others, demonstrating model IV is more energetically stable. However, it is different for the zigzag junctions in which the defect distributions prefer the order: IV > II > III > I [7]. It shows that the defects distributions of armchair SWCNT junction is favorable like model IV in which a hexagon is inserted between the pentagon and heptagon rings. Therefore, we can conclude that in the junctions of SWCNT (including the zigzag and armchair) the insertion of hexagons between the pentagon and heptagon rings is a favorable method. In the other models the defect distribution is different in armchair and zigzag SWCNTs. This perhaps can be explained by the different connection of pentagon and heptagon in the 5/7 defect: the pentagon and heptagon rings are close to each other to form a new unit in zigzag SWCNT junction while they are separated by a carbon–car-
Table 2 Reaction energies (in a.u.) obtained by means of homodesmotic equations, which are used to evaluate the energetic cost of the joints Nanotube
E(66 + 44)
E(66)
E(44)
DE(HF/3-21G*)
I II III IV
4530.2332 7166.336 6789.7595 5660.0404
5436.834 8600.4041 8148.4524 6792.6434
3623.9616 5732.6105 5431.3831 4527.6729
0.3292 0.3426 0.3165 0.2355
E(66 + 44)
E(66)
E(44)
DE(B3LYP/6-31G*)
4584.8762 7252.6107 6871.5124 5728.2032
5502.363 8703.9217 8246.4926 6874.4157
3667.6844 5801.65235 5496.8121 4582.2478
0.295 0.3526 0.2799 0.2571
I II III IV
bon bond in armchair one. In fact the carbon–carbon bond in the armchair SWCNT acts as a hexagon which can relax the tension induced by the torsion. 3.3. The electronic features The frontier orbital approach [15] is well known for providing improved understanding of chemical reactivity. It has been demonstrated that most of the frontier orbitals theory of chemical reactivity can be rationalized within the framework of density functional theory (DFT) of the electronic structure of molecules [16,17]. DFTbased frontier orbitals play an important role also in the prediction of many ground state molecular properties. It should be noted that the Kohn–Sham (KS) orbitals and eigenvalues are very difficult to extract quantitative information and often viewed as just an auxiliary construct, meanwhile the exception is that the highest occupied molecular orbital (HOMO) and the lowest unoccupied molecular orbital (LUMO). On the other hand the KS eigenvalues and orbitals are very similar to the single particle energy and wave function of the real system [18,19]. At the same time that changes in the spatial distribution of molecular orbitals, especially the frontier molecular orbitals (such as HOMO and LUMO) which are excellent indicators of many molecular properties [20,21]. In this work, we focus on the spatial distribution of the frontier molecular orbitals of four SWCNT models and compare their difference induced by the different distribution of defects. As shown in Fig. 2a the HOMO orbitals corresponding to all the models are found to be nearly localized on the entire skeleton and the positions of the defects show an obvious concentration. The distribution of LUMO orbitals seems more regular. They mainly stay around the circumference of the bigger tube. Furthermore the position of defects also affects the LUMO orbitals: the closer to the defects, the more concentrated the orbitals are. From above observation we can conclude that the different distribution of the defects results in the different distribution of frontier orbitals and furthermore affects, even determinates some molecular properties. So it is very important and useful to understand the defect distribution and control them in the experiments, though it looks very difficult in the present practice. Furthermore, these molecular orbitals mainly consist of the orbitals of carbon atoms of the tube, the auxiliary hydrogen atoms used to saturate the tube ends do not contribute to the frontier orbitals of armchair nanotubes. This is because most atoms in the joints are in a high tensile state, which may greatly increase the HOMO–LUMO gap. To further study the rectified effect of different junctions, we applied a uniform electric field along the axis from two opposite
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Fig. 2. (a) The spatial distribution of molecular frontier orbitals for four junction models at the zero-electric field and (b) the shift of frontier orbitals for the model I under 1.54 109 V/m electric field (the arrow is the direction of the electric field).
directions. The strength of electric field is 1.54 109 V/m which may reasonably represent the working condition of the molecular devices [22–25]. In previous experimental work under the similar electric field, we had used a current sensing atomic force microscope (CSAFM) to prepare nanosized patterns of an alkanethiol self-assembled monolayer (SAM) within a SAM of a different alkanethiol on a gold substrate [26]. The changes of HOMO and LUMO distribution in the different junctions are similar. Fig. 2b displays the representative models (junction I), from which we can see that the HOMO always shifts from the high potential to the low potential in both electric fields directions. However the LUMO shifts differently: when the direction of electric field is from (4, 4) to (6, 6), it moves from the low potential to the high potential, and when the electric field is opposite it almost does not shift. It is
obviously related with the original distribution of LUMO in the zero-electric field, which mainly locates on the thicker (6, 6). It is known that if an orbital is delocalized across the molecule, an electron injected into the molecule at the energy of the orbital has a high probability of reaching the other end [27–29]. So in the present study, the electric field from two opposite directions changes the distribution of the HOMO and LUMO differently. One makes them more delocalized and the other makes them more localized. Then we can approximately research the asymmetric I–V behavior through the asymmetrical variation of HOMO and LUMO under the interaction of the applied electric field. Although they will really contribute to the I–V behavior only when their energy levels are located within the bias window [30,31]. This property may give a good illumination to use the junctions as a rectifier.
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Obviously the conductance of a defective nanotube can show absolute differences compared with that of the pristine one [32]. Our question now is whether the different defects distribution can also result in the change of conductance. Here we analyze the density of states (DOS) for all the models and compare the difference. In Fig. 3 the DOS of four junctions reveals resonant states, which can be associated with different peculiar sites in the five- and seven-member rings. The observation of resonant states (sharp peaks) in the density of states of nanotube junction, close to the Fermi level, would be a proof of their existence.
I
8 7
b Density of states / arb. unit
Density of states / arb. unit
a
In every structure, the presence of resonant states, coming from the pentagonal and heptagonal atomic sites, is clearly visible. One sharp peak below and above the Fermi level can be associated with one 5/7 pair defect. In fact there are only two peaks in model I because the two pairs of 5/7 defects have the same location and then the corresponding peaks overlap. In model II there are also two peaks because when 5/7 pair defects are aligned along the direction of the tube, the main contribution comes from the first pair defect and the contribution of the other one vanishes. The same phenomenon was found in Charlier’s work
6 5 4 3 2 1 0 -1 -10
II
12 10 8 6 4 2 0
-8
-6
-4
-2
0
2
-10
4
-8
-6
Energy / eV 12
III
10 8 6 4 2
-8
-6
-4
-2
0
2
4
8 6 4 2
-10
-8
-6
f
8 6 4 2 0 -8
-6
-4
-2
Energy / eV
-4
-2
0
2
4
0
2
4
Energy / eV
Density of states / arb. unit
Density of states / arb. unit
4
0
10
-10
2
IV
Energy / eV
e
0
10
0 -10
-2
d Density of states / arb. unit
Density of states / arb. unit
c
-4
Energy / eV
0
2
4
8 7 6 5 4 3 2 1 0 -1 -10
-8
-6
-4
-2
Energy / eV
Fig. 3. Density of states for the four models at the zero-electric field (the a–d is corresponding to the model I–IV) and the model I in the non-zero-electric field with two opposite directions (e and f), the arrows show the peaks which are corresponding resonant states coming from the pentagonal and heptagonal atomic sites.
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[32]. There are separately four and three peaks for model III and IV though the two peaks below and above the Fermi level are much closer. But when the external electric field is applied, the positions of sharp peaks move and the peak heights change. Furthermore the different directions of the electric field induce the corresponding change (Fig. 3e and f). 4. Conclusion In summary, we have simulated several nanotubes containing different junctions that convert a (6, 6) armchair nanotube into a (4, 4) armchair nanotube using HF and DFT calculations. These joints are structurally different and the energetic cost of each joint has been evaluated using homodesmotic equations at several levels of theory. As a result, we can conclude that the metallic–metallic junction is more favorable when a hexagon between 5/7 pair defects is inserted in the armchair nanotube. In contrast, the junction with two 5/7 pair defects distributed along the cylindrical axis is unfavorable. The results also show that the different distributions of defects can lead to the different distribution of frontier orbitals. Furthermore the properties of the SWCNT will be affected. Acknowledgments The authors thank The National Natural Science Foundation of China (NSFC) (Nos. 20503012 and 20435010) and Key laboratory of superlight materials and surface technology, Ministry of education, Harbin Engineering University for financial supports. Mr. Y. Ye thanks Mrs. Danmin. Ye for her help during the preparation of the manuscript.
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