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Crystal orbital study on carbon chains encapsulated in armchair carbon nanotubes with various diameters
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Crystal orbital study on carbon chains encapsulated in armchair carbon nanotubes with various diameters Yang Wanga, Yuanhe Huangb,*, Baohua Yangb, Ruozhuang Liub a
Departamento de Quı´mica, C-9, Universidad Autono´ma de Madrid, 28049 Madrid, Spain Department of Chemistry, Beijing Normal University, 100875 Beijing, China
b
A R T I C L E I N F O
A B S T R A C T
Article history:
Theoretical studies are presented on carbon nanowires (CNWs) made of linear carbon
Received 29 May 2007
chains encapsulated inside armchair carbon nanotubes with various tube diameters. The
Accepted 16 November 2007
structural and electronic properties as well as bonding features were investigated system-
Available online 3 December 2007
atically by using ab initio self-consistent-field crystal orbital method based on density functional theory. The interaction between the tube and the chain becomes more obvious as the diameter of the CNW decreases and even weaker chemical bonds are formed between the tube and the chain in the smallest CNWs. The comparison of the elastic moduli between CNWs and CNTs supports that the mechanically unstable carbon chain is indeed protected by the CNT shell upon the encapsulation. All the CNWs we calculated are metals with zero band gaps. The encapsulation of the carbon chain may modulate the electronic properties for the CNWs depending on the tube size and the filling density of carbon atoms. Ó 2007 Elsevier Ltd. All rights reserved.
1.
Introduction
Filling and doping carbon nanotubes (CNTs) [1] with various atoms or molecules to get materials with novel structures and fascinating properties have attracted more and more attention [2–4]. Recent experiments [5,6] reported the existence of carbon nanowires (CNWs)1 made of linear carbon atom chains encapsulated inside the innermost tube of multiwalled CNTs. These CNWs are new one-dimensional (1D) carbon allotropes and expected to have novel electronic properties, which may lead to the practical applications in nanoelectronics. More recently, single-wall CNTs (SWCNTs) containing the long-chain C2nH2 (n = 4–6) polyyne molecules have been successfully synthesized and were confirmed by Raman spectroscopy and X-ray diffraction measurement [6–8]. Theoretical studies were performed on various properties of CNWs, including the thermodynamic and mechanical stabilities [5], geometric structures [9–12], vibration frequen-
cies [10,13], heat conductions [13,14], magnetic properties [15], as well as electronic and transport properties [11,12,16]. However, further study is necessary for understanding more structure–property relationships of the CNWs. For example, for the CNWs with the armchair CNT walls (simply called ‘armchair CNWs’ in the following), there have been no systematical studies on the effect of the diameter of the CNWs. Most of the previous studies focus on the CNWs with relatively larger tube diameters; how about the situation for the CNWs with relatively small tube diameters? Therefore, one of our aims is to explore the properties of the CNWs varying with the tube diameter. As the diameter of CNW decreases, the orbital interaction between the CNT and the chain is expected to be stronger. In fact we have investigated size effect on the CNWs made of linear carbon chains inserted inside zigzag CNTs (hereafter simply called ‘zigzag CNWs’) [12], and found that the interaction between the tube and the chain becomes more obvious when the tube size decreases,
* Corresponding author: Fax: +86 10 5880 0567. E-mail address: [email protected] (Y. Huang). 1 According to its original definition [6], a CNW here consists of an outer CNT wall and an inner linear carbon chain, although some authors have referred to a CNW as the in-tube carbon chain only. 0008-6223/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.carbon.2007.11.043
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leading to the change of geometric structures and energy bands upon encapsulation. Thus, as for the armchair CNWs with various diameters, we would like to know the influence of the orbital interaction on the geometric and electronic structures and the possibility of formation of chemical bonds between the chain and the tube. In this work, the geometric structures and electronic properties of both the isolated infinite carbon chain and armchair CNWs are calculated systematically by self-consistent-field crystal orbital (SCF-CO) method based on density functional theory (DFT). To the best of our knowledge, there has been no ab initio SCF-CO calculation reported by far on the armchair CNWs. Based on our computational results, the structural and electronic properties are discussed in detail for armchair CNWs with various diameters and filling atom densities. Finally, chemical bonding features between the chain atoms and the tube atoms (especially for the small CNWs) are investigated by using the crystal orbital overlap population (COOP) [17] analyses.
2.
Models and methods
2.1.
Computational models
Since the carbon atom chains enclosed inside CNTs found in experiments [5,6] have linear structures, our studies are restricted to filling linear chains into CNTs. According to the calculations [9] by the Lennard–Jones interaction potential method, the inner chains inside CNTs keep linear until the diameters of the outer tubes become larger than about ˚ , the size between the diameters of (6,6) and (7,7). There8.6 A fore, the armchair CNTs selected in this work to accommodate linear carbon chains range from (2,2) to (6,6). Note that (2,2) is the smallest CNT found in experiments so far [18]. Two series of armchair CNW models, C2@(n,n) and d-C@(n,n) (see Fig. 1 for n = 4), were proposed by filling a linear carbon chain into the armchair CNT (n,n) (n = 2–6). Here, a commensurability condition of the 1D periodicity between the outer CNT and that of the inner chain was imposed in order to reduce the computational effort. Hence, each unit cell of C2@(n,n) contains one unit cell of the CNT (n,n) plus two atoms of the in-tube chain (denoted as ‘C2-Chain’), and there are thus totally 4n + 2 carbon atoms in a unit cell of C2@(n,n) (see Fig. 1(a)). The dC@(n,n) model is the dimerized model of the corresponding C@(n,n) model so that we can investigate the possibility of the
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dimerization of carbon chain. Each unit cell of C@(n,n) contains one unit cell of the CNT (n,n) plus one atom of the inner chain (denoted as ‘C-Chain’), and there are thus totally 4n + 1 carbon atoms in the unit cell of C@(n,n). Then the d-C@(n,n) model is constructed by doubling the repeat unit of C@(n,n). In the dC@(n,n) model, the two repeat unit cells of C@(n,n) were taken as a super cell which includes two carbon atoms of the in-tube chain (denoted as ‘d-C-Chain’), and there are thus totally 8n + 2 carbon atoms in each unit cell of d-C@(n,n) (see Fig. 1(b)). It is necessary to point out that the commensurate conditions have been applied for the previous calculations on the studies of CNW [10–12,14–16], although the commensurate conditions will bring an artificial effect to some extent. To reduce this artificial effect from the commensurability, we need a very large super cell for the combined system [10], which is not feasible by using the existing computational resources. Moreover, in our C2@(n,n) model with larger CNTs, the average ˚ , which is C–C bond length of the inner chain is about 1.3 A quite close to the calculated C–C separation for the isolated carbon chain (which will be shown below). This is reasonable due to weak interaction between the inner chain and tube wall. Therefore, the commensurate conditions are advisable. As for our d-C@(n,n) model, although the average C–C separation is much larger than the calculated isolated carbon chain, it provides a theoretical simulation for the CNWs with sparser chain atoms. Since there have been no experimental data by far about the distance between chain atoms in the CNW [14], theoretical exploration would be helpful to understand more possible structures and properties of the CNWs. In the C2@(n,n) models, the average distance between the nearest neighboring atoms of the inner C2-Chain is about ˚ , a distance so close enough to form chemical bonds. In 1.3 A the C@(n,n) and d-C@(n,n) models, the average separation between the carbon atoms of the inner C-Chain (or d-C-Chain) is equal to the 1D translational length of the armchair CNT ˚ ), still shorter than the Van der Waals spacing (2.5 A ˚ ). The C2-Chain and the C-Chain (or d-C-Chain) repre(3.4 A sent the denser and the sparser density of carbon encapsulations, respectively, allowing us to investigate the effect of the filling density of the carbon chain on the properties of CNWs.
2.2.
Computational methods
All the optimizations of geometries and calculations of properties were performed by using ab initio SCF-CO method based
Fig. 1 – Models of CNWs: (a) C2@(4,4) and (b) d-C@(4,4). Dashed rectangles denote the 1D unit cells. D is the tube diameter and a is the translational length. C1 and C2 in (b) are the reference distances, the difference of which indicates the dimerization of the CNT wall (see text).
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on DFT with the Perdew–Burke–Ernzerhof (PBE) [19–21] functional. All computations were done by the CRYSTAL03 program [22] with the 6-21G* basis set, which is implemented in the program and has been proved to be appropriate for periodic solid-state calculations [22]. In the SCF-CO calculations, tolerances on change in eigenvalues and total energy were set to 109 (the default value is 106) and 108 (default 105) hartree/cell, respectively. In the geometric optimizations, the default thresholds were adopted for the maximum and the root mean square (RMS) forces, as well as the maximum and the RMS atomic displacements on all the atoms. The geometric structures were fully optimized for all the CNWs C2@(n,n) and C@(n,n), as well as the corresponding pristine CNTs (n,n) and the isolated infinite carbon atom chain. It means that both the atomic positions in the unit cell and the cell parameters of the 1D structures were optimized with the total energy minima. The geometries of d-C@(n,n) (except dC@(5,5)), were also fully optimized on the basis of the optimized geometries of the corresponding C@(n,n). Since the five-folded rotation symmetry is not utilizable for periodic calculations by the CRYSTAL03 program [22], it is quite difficult to perform fully geometric optimization for d-C@(5,5). Thus, we only relaxed the geometries of the inner d-C-Chain of d-C@(5,5) and fixed the geometries of the CNT wall, which was obtained from the optimized geometries of the CNT wall of C@(5,5). In fact, the differences of the optimized bond lengths of the CNT wall between C@(n,n) and d-C@(n,n) are ˚ and 0.009 A ˚ for n = 4 and 6, respectively. less than 0.007 A Therefore, our treatment mentioned above for the optimization of d-C@(5,5) is reliable. Based on the optimized geometries, we calculated the electronic properties, such as total energies, Mulliken populations, band structures, densities of states (DOS) and COOP. Young’s moduli were evaluated for the isolated carbon chain, C2@(4,4) and C2@(6,6) CNWs, and the corresponding pristine (4,4) and (6,6) CNTs. The C2@(5,5) CNW is not investigated because the five-folded rotation symmetries cannot be applied with CRYSTAL03 program [22], which will require much computational effort. The Young’s modulus Y is obtained from the secondary derivative of energy E with respect to strain e along the tube axis. Namely, 1 o2 E Y¼ 2 V oe 0
e¼0
where V0 is the equilibrium volume of the system. Apart from the singlet states, we also performed geometric optimizations and total energy calculations for the triplet states of these armchair CNWs. Our results demonstrate that the singlet state of C2@(n,n) is lower in energy than the triplet state by 0.45, 0.80, 0.91, 0.71 and 1.07 eV per chain atom for n = 2–6, respectively. In the case of d-C@(n,n), the singlet state is still lower in energy than the triplet state by 0.21, 0.01 and 0.02 eV per chain atom for n = 2–4, respectively. It is found that the singlet states of d-C@(5,5) and d-C@(6,6) are slightly higher in energy than the corresponding triplet states by 0.06 and 0.09 eV per chain atom, respectively. However, the magnetic properties of CNWs have not been observed yet in 2
the experiments [5,6]. Therefore, in the present work, we focus mainly on the singlet states for these armchair CNWs.
3.
Results and discussion
3.1.
Geometric structures and energies
Firstly, we focus on the possibility of dimerization of the carbon chain encapsulated inside the CNT. Our calculations show that the free isolated carbon chain prefers a nearly undimerized structure, in accordance with recent LDA and GGA results [11,23,24]. In order to analyze the effect of the CNT wall on the inner chain, we take the naked inner carbon chain of C2@(n,n) or d-C@(n,n) as the reference isolated chain. Hence, the reference isolated chain (denoted as ‘isolated C2-Chain’ or ‘isolated d-C-Chain’) has exactly the same translational length as the corresponding C2@(n,n) or d-C@(n,n). According to our calculations, both the isolated C2-Chain and the inner C2-Chain inside C2@(n,n) prefer the nearly equidistant configuration, similar to the free isolated chain. For the C2-Chain inside C2@(n,n) (n = 2–6), the maximum dimerization energy2 of the chain is less than 0.6 meV/atom and the maximum difference between the two alternative C–C ˚ . In contrast, both the isolated bonds is smaller than 0.006 A d-C-Chain and the inner d-C-Chain inside d-C@(n,n) (n = 3–6) are significantly dimerized. For the d-C-Chain inside C@(n,n) (n = 3–6), the difference between the longer and shorter C–C ˚ and the dimerization energy is 0.9– distance is about 2.4 A 3 eV/atom. However, d-C@(2,2) is an exception in the series of C@(n,n) CNWs. The dimerization of the inner d-C-Chain inside d-C@(2,2) can be almost neglected for the alternation of ˚ and the dimerization enthe C–C bond lengths is just 0.0004 A ergy is just 0.11 meV/atom. This reflects significant interaction between the CNT wall and the inner chain of d-C@(2,2), which will be discussed in detail later. Next, let us pay attention to the geometric structures of the CNTs before and after the encapsulation of the carbon chain. The encapsulation of the chain leads to a small structural deformation of the CNT walls for the CNWs C2@(n,n) with lager diameters (n = 4–6). The differences of C–C bond lengths between the CNT walls of the CNWs and the corre˚ . The corresponding pristine CNTs are all smaller than 0.03 A sponding diameter and cell parameter differences are all ˚ (see Table 1). For the smaller C2@(n,n) smaller than 0.07 A (n = 2, 3), more significant changes of the CNT structure are observed. The elongation of the cell parameters upon encap˚ and 0.41 A ˚, sulation for C2@(3,3) and C2@(2,2) is 0.007 A respectively. As we may expect, the smaller the CNW is, the larger the structural deformations would be. However, we note that the expansion of the tube diameter for C2@(2,2) ˚ ) in˚ ) is a little bit smaller than that for C2@(3,3) (0.36 A (0.31 A stead. This would be ascribed mainly to the weak chemical bonds formed between the CNT wall and the in-tube chain of C2@(2,2), since the distance between the wall and the chain ˚ ). In fact, the structure of C2@(2,2) is is close enough (1.59 A
Note that the cohesive energy is obtained by subtracting the energies of the isolated carbon atoms in the singlet state (rather than the triplet state, which is the ground state for the isolated carbon atom) from the total energy of the chain. This definition of cohesive energy is the same as that in [23], for the convenience of comparison.
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Table 1 – Computational results for armchair CNTs and CNWs Model
(2,2) (3,3) (4,4) (5,5) (6,6) C@(2,2) d-C@(3,3) d-C@(4,4) d-C@(5,5) d-C@(6,6) C2@(2,2) C2@(3,3) C2@(4,4) C2@(5,5) C2@(6,6) a b c d e f g
˚) aa (A
2.543 2.469 2.467 2.469 2.470 2.562 4.955 4.926 4.906 4.934 2.953 2.476 2.487 2.493 2.538
˚) Db (A
2.874 4.217 5.545 6.883 8.229 3.151 4.306 5.573 6.883 8.235 3.183 4.574 5.613 6.862 8.187
Ec (eV)
1.13 0.52 0.29 0.18 0.13 1.76 1.05 0.56 0.40 0.32 2.40 1.22 0.45 0.31 0.24
qd (jej)
Overlap population
– – – – – 0.052 0.014 0.257 0.230 0.195 0.048 0.066 0.044 0.060 0.052
OPcce
OPcwf
– – – – – 0.017 0.275 0.874 0.935 0.950 0.176 0.608 0.619 0.610 0.632
– – – – –
OPwwg 0.510 0.447 0.461 0.462 0.460 0.441 0.462 0.466 0.462 0.460 0.349 0.451 0.454 0.460 0.462
0.135 0.021 0.004 0.001 0.000 0.251 0.002 0.004 0.001 0.000
Optimized cell parameter. Optimized tube diameter. Average energy per carbon atom. Mulliken charge on each chain atom. Orbital overlap population between the two neighboring chain atoms. Orbital overlap population between the nearest chain atom and wall atom. Orbital overlap population between the two nearest wall atoms.
2.5
Fig. 2 – The structure of the smallest CNWs C2@(2,2).
rather unique in that the network formed by the wall atoms is broken down due to the encapsulation of the C2-Chain, as shown in Fig. 2. It is surprising to see that the carbon atom on the C2-Chain inside C2@(2,2) has the coordination of six. In other words, there are hexacoordinate carbon atoms [25– 32] in C2@(2,2). This provides an example of hypercoordinate carbon atoms in a 1D periodic structure. The chemical bonds in C2@(2,2) will be discussed in detail later. Similarly, for the large d-C@(n,n) (n = 4–6) CNWs, only minor structural changes on the CNT walls are observed upon the encapsulation of the chain, while for the small d-C@(3,3) and C@(2,2) CNWs, the changes are more remarkable. In addition, the CNT wall of d-C@(n,n) is also found to be slightly dimerized. The degree of the dimerization of the CNT wall can be indicated by the parameter d = c1–c2; c1 and c2 are described in Fig. 1(b). The d values for d-C@(6,6), d-C@(4,4) and d˚ , respectively. Hence, the C@(3,3) are 0.005, 0.014 and 0.063 A dimerization of the CNT wall becomes more obvious with the decreasing tube diameter of d-C@(n,n), indicating the increasing interaction between the tube and the chain. The average energies per atom relative to that of 2D graphene for the armchair and zigzag CNWs and the pristine CNTs
E (ev/atom)
2
(n ,n ) d-C@(n ,n ), C@(2,2) C2 @(n ,n ) (m,0) C @(m,0) C2 @(m,0)
1.5
1
0.5
0
C60
3
4
5
6
7
8
Tube diameter (Å) Fig. 3 – The average energies per atom relative to that of graphene for pristine CNTs (n,n) and (m,0) and CNWs C@(2,2), d-C@(n,n), C2@(n,n), C@(m,0) and C2@(m,0), where m = 6–10, n = 3–6 for d-C@(n,n) and n = 2–6 for other models. are shown in Fig. 3. Although all the armchair and zigzag CNWs are less stable energetically than the corresponding pristine CNTs, some CNWs are still more stable energetically than the C60 molecule (see Fig. 3), such as C2@(5,5), C2@(6,6), dC@(5,5), d-C@(6,6), C2@(10,0), C2@(9,0) and C@(10,0). The average energies of CNWs and the pristine CNTs increase remarkably as the tube diameters decrease. The large C2@(n,n) (n = 4– 6) is more stable energetically than the corresponding d-C@(n,n), while the small C2@(n,n) (n = 2, 3) is much more unstable energetically than the corresponding d-C@(3,3) or C@(2,2). This indicates that the large diameter CNWs would
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be more favorable to accommodate the chains with relatively larger filling density of carbon atoms, whereas small CNWs would be favorable to relatively less filling density of carbon atoms. These results are in analogy with those for zigzag CNWs, C2@(n,0) and C@(n,0) [12]. We would like to point out that all the above discussions are based only on the energies, but the experimental structures may not be absolutely dependent on their energetic stabilities. It has been shown experimentally that some of the energetically unfavorable structures could form under conditions far from equilibrium [10], such as the quenching process [33,34] from the extremely high temperatures used to generate CNTs [35]. The experiments for carbon clusters [33] and sulfur clusters [34] have clearly shown that the energetically unfavorable structures can form by the rapid cooling of the plasma, because there is too short time for the atoms to rearrange to form the energetically most stable structures. As to the elastic properties, the calculated Young’s modulus for the pristine (4,4) and (6,6) CNTs is 1.06 and 0.96 TPa, respectively, in agreement with the experimental [36] and theoretical [37,38] values for SWCNTs, which are evaluated to be around 1 TPa. According to our calculations, the Young’s modulus for the C2@(4,4) and C2@(6,6) CNWs is 1.25 and 0.99 TPa, respectively. Thus the elastic modulus of the combined CNW system is a little bit larger than that of the corresponding pristine CNT. This is probably due to the interaction between the tube and the inner chain. It is interesting to see that the calculated Young’s modulus for the isolated carbon chain in the free space is only 0.46 TPa, which is much smaller than the elastic modulus for the CNWs. Hence, in this sense, our results support that the CNT shell indeed ‘protects’ the mechanically unstable carbon chain by encapsulating [5].
3.2.
Electronic properties
The calculated band structures and DOS for several representative armchair CNWs are shown in Fig. 4. All armchair CNWs studied here are metallic, the same as the corresponding pristine CNTs. Moreover, the band structures are kept almost unchanged for the rigid sliding motion of the carbon chain enclosed inside the relatively large CNWs, C2@(n,n) and dC@(n,n) (n = 3–6). We discuss CNWs d-C@(n,n) firstly. In Fig. 4, the d-C-Chainderived bands are depicted in dashed lines and are labeled according to their symmetries. As for the large diameter CNWs, d-C@(n,n) (n = 4–6), the characteristics of band structures for the two constituent parts (CNT and carbon chain) are still maintained, but the band levels shift, following the rigid-band shift approximation (RBSA) [39,40]. Such a rigid overlap of band structures of the two components indicates weak orbital interaction between the CNTwall and the d-C-Chain in the larger CNWs, which is in analogy with the results for the larger zigzag CNWs [10]. It is worthy to mention that the flat d-C-Chain-derived p1 bands give rise to high DOS near the Fermi level (see Fig. 4(a)), which contributes most of the DOS (about 71–76% ) to the total DOS at the Fermi level and makes the total DOS much greater than that of the corresponding pristine CNT. Since the DOS near the Fermi level has the contribution from both components, both the CNT wall and the in-tube d-C-Chain can act as the pathway of
charge carriers in d-C@(n,n) (n = 4–6), which is similar to the large diameter zigzag CNWs, C2@(9,0) and C2@(10,0) [10]. As shown in Fig. 4(e) and (h), the shapes of the bands originated from both components change remarkably for dC@(3,3) and C@(2,2), and so do the positions of these bands. As for d-C@(3,3), the bandwidth of r2 band is 1.78 eV and the p1 bands become much wider with bandwidth of about 1.01 eV. In the band structure of C@(2,2), the C-Chain-derived r1, r2 and p bands are all elevated above the Fermi level due to the strong orbital interaction between the CNT and the CChain. In fact, from the projected DOS shown in Fig. 4(h), we can see that these C-Chain-derived bands are hybridized much with the CNT-derived bands, which also reflects strong orbital interaction between the CNT and the C-Chain. The large hybridization indicates that the charge carriers are easy to move from CNT to the C-Chain, and vice versa. Unlike the large d-C@(n,n) (n = 4–6) CNWs, for the small C@(2,2) and dC@(3,3) CNWs, the C-Chain or d-C-Chain contributes less to the total DOS at the Fermi level, about 9% and 19% for C@(2,2) and d-C@(3,3), respectively. Since there is no flat band near the Fermi level, no higher DOS would be expected. Now we take a look at the band structures of CNWs C2@(n,n). The RBSA works also well for the band structures of the large diameter CNWs, C2@(n,n) (n = 4–6). However, in the band structure of C2@(2,2), it is hard to discern which component a band is derived from, because the orbital interaction and hybridization between the C2-Chain and the CNT (2,2) are so remarkable. Since the C2-Chain-derived p1 and p2 bands of C2@(n,n) are much wider than the d-C-Chain-derived bands of d-C@(n,n), the DOS contributed from C2-Chain is relatively lower and spreads over a broad energy range. The C2Chain contributes 4–33% to the total DOS of C2@(n,n) (n = 2–6) at the Fermi level. After all, in all the C2@(n,n) CNWs that we studied, the charge carriers can move through the pathway of both the chain and the CNT. Compared with the considerable electron transfer (about 0.25–0.45 electron/cell) from the CNT to the carbon chain in the zigzag CNWs C2@(n,0) (n = 6–10) [10], remarkable electronic charge transfer (0.2–0.25 electron/cell) from the CNT to the chain is also observed in large diameter armchair CNWs C@(n,n) (n = 4–6), as shown in Fig. 5. In other armchair CNWs, C@(n,n) (n = 2, 3) and C2@(n,n) (n = 2–6), the charge transfer between the CNT and the chain is much less (less than 0.07 electron/cell).
3.3.
COOP and bonding features in armchair CNWs
The COOP analysis was first proposed Hughbanks and Hoffmann [17] to investigate chemical bonding features in periodic systems. The positive and negative COOP peaks for the specified bond correspond to bonding and anti-bonding characteristics, respectively. The integral of the COOP curve up to the Fermi level gives the total overlap population (OP) [41] between the two given atoms. The OP scales like bond order [42] and thus can be used as a measure of bond strength [17]. At the level of theory in this work, the OP for the C–C single bond of diamond, the C@C double bond of ethylene and the C „ C triple bond of acetylene is 0.34, 0.68 and 0.93, respectively. The OP values for the C–C bonds of the pristine CNTs (n,n) fall in the range of 0.4–0.5 (see Table 1).
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a
b
c
d
e
f
h
i
Fig. 4 – Plots of band structures, DOS and COOP for (a) (6,6); (b) d-C@(6,6); (c) C2@(6,6); (d) (3,3); (e) d-C@(3,3); (f) C2@(3,3); (g) (2,2); (h) C@(2,2) and (i) C2@(2,2). Dashed–dotted lines indicate Fermi levels and dashed lines denote the chain-derived bands. In the DOS plots, solid line represents the total DOS, while the shadowed area enveloped by dashed line represents the projected DOS from the in-tube chain. In the COOP plots, the COOP between the two neighboring chain atoms is depicted in solid line and the COOP between the nearest chain atom and wall atom is depicted in dashed line. The value of DOS and COOP is measured in states eV1 cell1 and eV1 cell1, respectively.
Firstly, we consider the chemical bonding between the carbon chain and the CNT wall of CNWs. From Table 1, the largest OPcw is just 0.021 for CNWs d-C@(n,n) and C2@(n,n) (n = 3–6), indicating that no chemical bond is formed between the chain and the wall in the relatively larger CNWs. This is coincident with the fact that the closest distance between ˚ for the chain atom and the tube atom is not less than 2.2 A
these CNWs. Thus the atoms are not close enough to form chemical bonds. For the smallest CNWs, C@(2,2) and C2@(2,2), the closest distance between the chain atom and the tube atom is ˚ and 1.59 A ˚ , respectively, which is a little bit larger than 1.70 A ˚ ). The value of the ordinary C–C single bond length (1.54 A OPcw for C@(2,2) and C2@(2,2) is 0.135 and 0.251, respectively.
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Fig. 5 – Isodensity surface plots of the electron accumulation (white) and depletion (black) are shown for d-C@(6,6). The values for the white and black surfaces are 104e/(a.u.)3. The gray circles indicate the positions of the carbon atoms.
Therefore, the chain atom and wall atom form a chemical bond to some extent, though it is weaker than the typical C–C single bond (OP = 0.34). These weak chemical bonds formed between the chain atom and wall atom in C@(2,2) and C2@(2,2) can also be recognized in the COOP plots (see Fig. 4(h) and (i)). In Fig. 4(h), the negative peaks of COOP between the nearest chain atom and wall atom for C@(2,2) above the Fermi level fall in the region where the C-Chain-derived r1, r2 and p bands are located. These negative peaks indicate the anti-bonding interaction between the chain atom and the tube atom, leading to the above-mentioned elevation of the r1, r2 and p bands of C@(2,2). The positive COOP peaks corresponding to the bonding interaction between the chain atom and the tube atom should be expected below the Fermi level at lower energy level (not plotted in Fig. 4(h)). Therefore, these COOP peaks correspond to the effective chemical bond formed between the nearest chain atom and wall atom in C@(2,2). Similarly, the negative peaks above the Fermi level and the positive peaks below the Fermi level for the COOP between the chain atom and the wall atom in C2@(2,2) indicate the effective chemical bonds formed between the chain and the wall in C2@(2,2). Now we turn our attention to the chemical bonding of the in-tube chains. For the large CNWs d-C@(n,n) (n = 4–6), the d˚ , which is C-Chain has the nearest C–C distance of about 1.3 A slightly shorter than the ordinary C@C double bond length ˚ ). It is found that the OP between the nearest two (1.33 A atoms on the d-C-Chain, OPcc, is rather large (about 0.9, see Table 1) and close to the value for the C „ C triple bond of acetylene (OP = 0.93). The positive peaks of COOP below the Fermi level (see Fig. 4(b)) further confirm these strong C–C bonds. As for d-C@(3,3), the value of OPcc decreases remarkably to 0.275, which can be explained by the COOP plots. As shown in Fig. 4(e), the positive peaks of the COOP for C–C bonds of the d-C-Chain of d-C@(3,3) are mainly above the Fermi level, leading to the remarkable decrease of the total OP,
OPcc. The value of OPcc for C@(2,2) is only 0.017, which manifests that chemical bonds are not formed on the C-Chain inside C@(2,2). Such a small value of OPcc is due mainly to the ˚ on the C-Chain inrelatively large C–C distance of about 2.6 A side C@(2,2). In addition, the decrease of the total OP is also caused by the unoccupied positive COOP peaks for the C–C bond of the C-Chain, as shown in Fig. 4(h). The C–C bond length for the C2-Chain inside CNWs ˚ and the corresponding OPcc C2@(n,n) (n = 3–6) is about 1.25 A is 0.6, the value close to that for ethylene (OP = 0.68), indicating that C–C bonds on the C2-Chain inside these CNWs are even stronger than C–C bonds on the CNT wall (OPww 0.46). These C–C bonds can be confirmed by the positive peaks below Fermi level in the COOP plots, as shown in Fig. 4(c) and (f). As for the C2-Chain inside C2@(2,2), the C–C bond length ˚ , which is still a little shorter than the is elongated to 1.48 A ˚ ). However, length of the ordinary C–C single bond (1.54 A the OPcc for this C–C bond is just 0.176, which is 1/4–1/3 of the OPcc for CNWs C2@(n,n) (n = 3–6). This can also be understood from the COOP plot. In Fig. 4(i), there is a positive bonding peak above the Fermi level, which means that some bonding orbitals become unoccupied and the corresponding C–C bond is thus weakened. Although the OPcc for C–C bond of the C2-Chain inside C2@(2,2) is relatively small, it still indicates the chemical bonding between the carbon atoms of the C2-Chain to some extent. Note that the bond strength is not always consistent with the bond length. For example, there have been many very long C–C single bonds with the bond ˚ [43–47]. On the other hand, length in the range of 1.6–1.9 A the length of a C–C single bond can also be squeezed to be ˚ [48]. In addition, the atom of the C2-Chain and 1.32–1.51 A the atom of the wall in C2@(2,2) form a stronger C–C bond with OPcw = 0.251. Therefore, each carbon atom of the C2Chain inside C2@(2,2) forms six weaker chemical bonds with six neighboring carbon atoms, two on the C2-Chain and four on the CNT wall. Hence, as mentioned above, there are hexacoordinate carbon atoms in CNW C2@(2,2). At the same time, C–C bonds on the CNTwall of C2@(2,2) is also weakened due to the chemical bonding between the chain atom and the wall atom. From Table 1, we can see that OPww for C2@(2,2) is just 0.349, which is obviously smaller than that for CNWs C2@(n,n) (n = 3–6) (OPww = 0.45–0.46).
4.
Conclusion
In summary, we investigated the structural and electronic properties as well as chemical bonding features for CNWs made of linear carbon chains encapsulated inside armchair CNTs. Our crystal orbital calculations show that the interaction between the tube and the in-tube chain is mainly Van der Waals interaction in the armchair CNWs with relatively large tube diameters. The C2-Chain inside the C2@(n,n) is undimerized, while the d-C-Chain inside d-C@(n,n) is dimerized, with the exception of the carbon chain inside C@(2,2), which prefers an undimerized configuration. For the smallest CNWs, C@(2,2) and C2@(2,2), weak chemical bonds are formed to some extent between the tube and the chain. In addition, the chain atoms inside C2@(2,2) have an abnormal coordination of 6, giving an example of hypercoordinate carbons in a
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4 6 (2 0 0 8) 2 7 6–28 4
1D periodic structure from a point of view of calculations. Among all the CNWs we considered here, the most stable ones are C2@(5,5) and C2@(6,6). The former has the diameter of about 0.7 nm, which is close to the value observed in experiments [5,6]. The elastic modulus along the tube axis for the CNW is expected to be a little larger than that for the corresponding pristine CNT and much larger than that for the isolated carbon chain in the free space. This demonstrates that the mechanically unstable carbon chain is indeed protected by the CNT shell upon the encapsulation. All armchair CNWs are metallic. Both the tube and the chain can act as the pathway of charge carriers in these CNWs. The band structure for the larger diameter CNW can be understood within the framework of RBSA, indicating the weak orbital interaction between the tube and the chain. As the diameter of CNW decreases, the orbital interaction and hybridization becomes more remarkable between the tube and the chain, leading to obvious changes of the shapes and positions of some bands derived from both components. Considerable electron transfer (about 0.20–0.25 electron/cell) is observed from the CNT wall to the in-tube chain in the larger diameter CNWs d-C@(n,n) (n = 4–6), while there is much less charge transfer in the other armchair CNWs. It seems that the d-C-Chain inside d-C@(n,n) (n = 3–6) can obviously modulate the electronic properties of the CNT and give much higher DOS at the Fermi level. In conclusion, our results suggest that filling carbon atom chains with different atom densities into the CNTs with different chiralities and diameters will produce various CNWs with different electronic properties.
Acknowledgements This work is supported by the Research Fund for the Doctoral Program of Higher Education (20060027001), the National Natural Science Foundation of China (No. 20373008) and Major State Basic Research Development Programs (Grant No. 2002CB613406).
R E F E R E N C E S
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4 6 ( 2 0 0 8 ) 2 7 6 –2 8 4
available at www.sciencedirect.com
journal homepage: www.elsevier.com/locate/carbon
Crystal orbital study on carbon chains encapsulated in armchair carbon nanotubes with various diameters Yang Wanga, Yuanhe Huangb,*, Baohua Yangb, Ruozhuang Liub a
Departamento de Quı´mica, C-9, Universidad Autono´ma de Madrid, 28049 Madrid, Spain Department of Chemistry, Beijing Normal University, 100875 Beijing, China
b
A R T I C L E I N F O
A B S T R A C T
Article history:
Theoretical studies are presented on carbon nanowires (CNWs) made of linear carbon
Received 29 May 2007
chains encapsulated inside armchair carbon nanotubes with various tube diameters. The
Accepted 16 November 2007
structural and electronic properties as well as bonding features were investigated system-
Available online 3 December 2007
atically by using ab initio self-consistent-field crystal orbital method based on density functional theory. The interaction between the tube and the chain becomes more obvious as the diameter of the CNW decreases and even weaker chemical bonds are formed between the tube and the chain in the smallest CNWs. The comparison of the elastic moduli between CNWs and CNTs supports that the mechanically unstable carbon chain is indeed protected by the CNT shell upon the encapsulation. All the CNWs we calculated are metals with zero band gaps. The encapsulation of the carbon chain may modulate the electronic properties for the CNWs depending on the tube size and the filling density of carbon atoms. Ó 2007 Elsevier Ltd. All rights reserved.
1.
Introduction
Filling and doping carbon nanotubes (CNTs) [1] with various atoms or molecules to get materials with novel structures and fascinating properties have attracted more and more attention [2–4]. Recent experiments [5,6] reported the existence of carbon nanowires (CNWs)1 made of linear carbon atom chains encapsulated inside the innermost tube of multiwalled CNTs. These CNWs are new one-dimensional (1D) carbon allotropes and expected to have novel electronic properties, which may lead to the practical applications in nanoelectronics. More recently, single-wall CNTs (SWCNTs) containing the long-chain C2nH2 (n = 4–6) polyyne molecules have been successfully synthesized and were confirmed by Raman spectroscopy and X-ray diffraction measurement [6–8]. Theoretical studies were performed on various properties of CNWs, including the thermodynamic and mechanical stabilities [5], geometric structures [9–12], vibration frequen-
cies [10,13], heat conductions [13,14], magnetic properties [15], as well as electronic and transport properties [11,12,16]. However, further study is necessary for understanding more structure–property relationships of the CNWs. For example, for the CNWs with the armchair CNT walls (simply called ‘armchair CNWs’ in the following), there have been no systematical studies on the effect of the diameter of the CNWs. Most of the previous studies focus on the CNWs with relatively larger tube diameters; how about the situation for the CNWs with relatively small tube diameters? Therefore, one of our aims is to explore the properties of the CNWs varying with the tube diameter. As the diameter of CNW decreases, the orbital interaction between the CNT and the chain is expected to be stronger. In fact we have investigated size effect on the CNWs made of linear carbon chains inserted inside zigzag CNTs (hereafter simply called ‘zigzag CNWs’) [12], and found that the interaction between the tube and the chain becomes more obvious when the tube size decreases,
* Corresponding author: Fax: +86 10 5880 0567. E-mail address: [email protected] (Y. Huang). 1 According to its original definition [6], a CNW here consists of an outer CNT wall and an inner linear carbon chain, although some authors have referred to a CNW as the in-tube carbon chain only. 0008-6223/$ - see front matter Ó 2007 Elsevier Ltd. All rights reserved. doi:10.1016/j.carbon.2007.11.043
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leading to the change of geometric structures and energy bands upon encapsulation. Thus, as for the armchair CNWs with various diameters, we would like to know the influence of the orbital interaction on the geometric and electronic structures and the possibility of formation of chemical bonds between the chain and the tube. In this work, the geometric structures and electronic properties of both the isolated infinite carbon chain and armchair CNWs are calculated systematically by self-consistent-field crystal orbital (SCF-CO) method based on density functional theory (DFT). To the best of our knowledge, there has been no ab initio SCF-CO calculation reported by far on the armchair CNWs. Based on our computational results, the structural and electronic properties are discussed in detail for armchair CNWs with various diameters and filling atom densities. Finally, chemical bonding features between the chain atoms and the tube atoms (especially for the small CNWs) are investigated by using the crystal orbital overlap population (COOP) [17] analyses.
2.
Models and methods
2.1.
Computational models
Since the carbon atom chains enclosed inside CNTs found in experiments [5,6] have linear structures, our studies are restricted to filling linear chains into CNTs. According to the calculations [9] by the Lennard–Jones interaction potential method, the inner chains inside CNTs keep linear until the diameters of the outer tubes become larger than about ˚ , the size between the diameters of (6,6) and (7,7). There8.6 A fore, the armchair CNTs selected in this work to accommodate linear carbon chains range from (2,2) to (6,6). Note that (2,2) is the smallest CNT found in experiments so far [18]. Two series of armchair CNW models, C2@(n,n) and d-C@(n,n) (see Fig. 1 for n = 4), were proposed by filling a linear carbon chain into the armchair CNT (n,n) (n = 2–6). Here, a commensurability condition of the 1D periodicity between the outer CNT and that of the inner chain was imposed in order to reduce the computational effort. Hence, each unit cell of C2@(n,n) contains one unit cell of the CNT (n,n) plus two atoms of the in-tube chain (denoted as ‘C2-Chain’), and there are thus totally 4n + 2 carbon atoms in a unit cell of C2@(n,n) (see Fig. 1(a)). The dC@(n,n) model is the dimerized model of the corresponding C@(n,n) model so that we can investigate the possibility of the
277
dimerization of carbon chain. Each unit cell of C@(n,n) contains one unit cell of the CNT (n,n) plus one atom of the inner chain (denoted as ‘C-Chain’), and there are thus totally 4n + 1 carbon atoms in the unit cell of C@(n,n). Then the d-C@(n,n) model is constructed by doubling the repeat unit of C@(n,n). In the dC@(n,n) model, the two repeat unit cells of C@(n,n) were taken as a super cell which includes two carbon atoms of the in-tube chain (denoted as ‘d-C-Chain’), and there are thus totally 8n + 2 carbon atoms in each unit cell of d-C@(n,n) (see Fig. 1(b)). It is necessary to point out that the commensurate conditions have been applied for the previous calculations on the studies of CNW [10–12,14–16], although the commensurate conditions will bring an artificial effect to some extent. To reduce this artificial effect from the commensurability, we need a very large super cell for the combined system [10], which is not feasible by using the existing computational resources. Moreover, in our C2@(n,n) model with larger CNTs, the average ˚ , which is C–C bond length of the inner chain is about 1.3 A quite close to the calculated C–C separation for the isolated carbon chain (which will be shown below). This is reasonable due to weak interaction between the inner chain and tube wall. Therefore, the commensurate conditions are advisable. As for our d-C@(n,n) model, although the average C–C separation is much larger than the calculated isolated carbon chain, it provides a theoretical simulation for the CNWs with sparser chain atoms. Since there have been no experimental data by far about the distance between chain atoms in the CNW [14], theoretical exploration would be helpful to understand more possible structures and properties of the CNWs. In the C2@(n,n) models, the average distance between the nearest neighboring atoms of the inner C2-Chain is about ˚ , a distance so close enough to form chemical bonds. In 1.3 A the C@(n,n) and d-C@(n,n) models, the average separation between the carbon atoms of the inner C-Chain (or d-C-Chain) is equal to the 1D translational length of the armchair CNT ˚ ), still shorter than the Van der Waals spacing (2.5 A ˚ ). The C2-Chain and the C-Chain (or d-C-Chain) repre(3.4 A sent the denser and the sparser density of carbon encapsulations, respectively, allowing us to investigate the effect of the filling density of the carbon chain on the properties of CNWs.
2.2.
Computational methods
All the optimizations of geometries and calculations of properties were performed by using ab initio SCF-CO method based
Fig. 1 – Models of CNWs: (a) C2@(4,4) and (b) d-C@(4,4). Dashed rectangles denote the 1D unit cells. D is the tube diameter and a is the translational length. C1 and C2 in (b) are the reference distances, the difference of which indicates the dimerization of the CNT wall (see text).
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on DFT with the Perdew–Burke–Ernzerhof (PBE) [19–21] functional. All computations were done by the CRYSTAL03 program [22] with the 6-21G* basis set, which is implemented in the program and has been proved to be appropriate for periodic solid-state calculations [22]. In the SCF-CO calculations, tolerances on change in eigenvalues and total energy were set to 109 (the default value is 106) and 108 (default 105) hartree/cell, respectively. In the geometric optimizations, the default thresholds were adopted for the maximum and the root mean square (RMS) forces, as well as the maximum and the RMS atomic displacements on all the atoms. The geometric structures were fully optimized for all the CNWs C2@(n,n) and C@(n,n), as well as the corresponding pristine CNTs (n,n) and the isolated infinite carbon atom chain. It means that both the atomic positions in the unit cell and the cell parameters of the 1D structures were optimized with the total energy minima. The geometries of d-C@(n,n) (except dC@(5,5)), were also fully optimized on the basis of the optimized geometries of the corresponding C@(n,n). Since the five-folded rotation symmetry is not utilizable for periodic calculations by the CRYSTAL03 program [22], it is quite difficult to perform fully geometric optimization for d-C@(5,5). Thus, we only relaxed the geometries of the inner d-C-Chain of d-C@(5,5) and fixed the geometries of the CNT wall, which was obtained from the optimized geometries of the CNT wall of C@(5,5). In fact, the differences of the optimized bond lengths of the CNT wall between C@(n,n) and d-C@(n,n) are ˚ and 0.009 A ˚ for n = 4 and 6, respectively. less than 0.007 A Therefore, our treatment mentioned above for the optimization of d-C@(5,5) is reliable. Based on the optimized geometries, we calculated the electronic properties, such as total energies, Mulliken populations, band structures, densities of states (DOS) and COOP. Young’s moduli were evaluated for the isolated carbon chain, C2@(4,4) and C2@(6,6) CNWs, and the corresponding pristine (4,4) and (6,6) CNTs. The C2@(5,5) CNW is not investigated because the five-folded rotation symmetries cannot be applied with CRYSTAL03 program [22], which will require much computational effort. The Young’s modulus Y is obtained from the secondary derivative of energy E with respect to strain e along the tube axis. Namely, 1 o2 E Y¼ 2 V oe 0
e¼0
where V0 is the equilibrium volume of the system. Apart from the singlet states, we also performed geometric optimizations and total energy calculations for the triplet states of these armchair CNWs. Our results demonstrate that the singlet state of C2@(n,n) is lower in energy than the triplet state by 0.45, 0.80, 0.91, 0.71 and 1.07 eV per chain atom for n = 2–6, respectively. In the case of d-C@(n,n), the singlet state is still lower in energy than the triplet state by 0.21, 0.01 and 0.02 eV per chain atom for n = 2–4, respectively. It is found that the singlet states of d-C@(5,5) and d-C@(6,6) are slightly higher in energy than the corresponding triplet states by 0.06 and 0.09 eV per chain atom, respectively. However, the magnetic properties of CNWs have not been observed yet in 2
the experiments [5,6]. Therefore, in the present work, we focus mainly on the singlet states for these armchair CNWs.
3.
Results and discussion
3.1.
Geometric structures and energies
Firstly, we focus on the possibility of dimerization of the carbon chain encapsulated inside the CNT. Our calculations show that the free isolated carbon chain prefers a nearly undimerized structure, in accordance with recent LDA and GGA results [11,23,24]. In order to analyze the effect of the CNT wall on the inner chain, we take the naked inner carbon chain of C2@(n,n) or d-C@(n,n) as the reference isolated chain. Hence, the reference isolated chain (denoted as ‘isolated C2-Chain’ or ‘isolated d-C-Chain’) has exactly the same translational length as the corresponding C2@(n,n) or d-C@(n,n). According to our calculations, both the isolated C2-Chain and the inner C2-Chain inside C2@(n,n) prefer the nearly equidistant configuration, similar to the free isolated chain. For the C2-Chain inside C2@(n,n) (n = 2–6), the maximum dimerization energy2 of the chain is less than 0.6 meV/atom and the maximum difference between the two alternative C–C ˚ . In contrast, both the isolated bonds is smaller than 0.006 A d-C-Chain and the inner d-C-Chain inside d-C@(n,n) (n = 3–6) are significantly dimerized. For the d-C-Chain inside C@(n,n) (n = 3–6), the difference between the longer and shorter C–C ˚ and the dimerization energy is 0.9– distance is about 2.4 A 3 eV/atom. However, d-C@(2,2) is an exception in the series of C@(n,n) CNWs. The dimerization of the inner d-C-Chain inside d-C@(2,2) can be almost neglected for the alternation of ˚ and the dimerization enthe C–C bond lengths is just 0.0004 A ergy is just 0.11 meV/atom. This reflects significant interaction between the CNT wall and the inner chain of d-C@(2,2), which will be discussed in detail later. Next, let us pay attention to the geometric structures of the CNTs before and after the encapsulation of the carbon chain. The encapsulation of the chain leads to a small structural deformation of the CNT walls for the CNWs C2@(n,n) with lager diameters (n = 4–6). The differences of C–C bond lengths between the CNT walls of the CNWs and the corre˚ . The corresponding pristine CNTs are all smaller than 0.03 A sponding diameter and cell parameter differences are all ˚ (see Table 1). For the smaller C2@(n,n) smaller than 0.07 A (n = 2, 3), more significant changes of the CNT structure are observed. The elongation of the cell parameters upon encap˚ and 0.41 A ˚, sulation for C2@(3,3) and C2@(2,2) is 0.007 A respectively. As we may expect, the smaller the CNW is, the larger the structural deformations would be. However, we note that the expansion of the tube diameter for C2@(2,2) ˚ ) in˚ ) is a little bit smaller than that for C2@(3,3) (0.36 A (0.31 A stead. This would be ascribed mainly to the weak chemical bonds formed between the CNT wall and the in-tube chain of C2@(2,2), since the distance between the wall and the chain ˚ ). In fact, the structure of C2@(2,2) is is close enough (1.59 A
Note that the cohesive energy is obtained by subtracting the energies of the isolated carbon atoms in the singlet state (rather than the triplet state, which is the ground state for the isolated carbon atom) from the total energy of the chain. This definition of cohesive energy is the same as that in [23], for the convenience of comparison.
CARBON
279
4 6 (2 0 0 8) 2 7 6–28 4
Table 1 – Computational results for armchair CNTs and CNWs Model
(2,2) (3,3) (4,4) (5,5) (6,6) C@(2,2) d-C@(3,3) d-C@(4,4) d-C@(5,5) d-C@(6,6) C2@(2,2) C2@(3,3) C2@(4,4) C2@(5,5) C2@(6,6) a b c d e f g
˚) aa (A
2.543 2.469 2.467 2.469 2.470 2.562 4.955 4.926 4.906 4.934 2.953 2.476 2.487 2.493 2.538
˚) Db (A
2.874 4.217 5.545 6.883 8.229 3.151 4.306 5.573 6.883 8.235 3.183 4.574 5.613 6.862 8.187
Ec (eV)
1.13 0.52 0.29 0.18 0.13 1.76 1.05 0.56 0.40 0.32 2.40 1.22 0.45 0.31 0.24
qd (jej)
Overlap population
– – – – – 0.052 0.014 0.257 0.230 0.195 0.048 0.066 0.044 0.060 0.052
OPcce
OPcwf
– – – – – 0.017 0.275 0.874 0.935 0.950 0.176 0.608 0.619 0.610 0.632
– – – – –
OPwwg 0.510 0.447 0.461 0.462 0.460 0.441 0.462 0.466 0.462 0.460 0.349 0.451 0.454 0.460 0.462
0.135 0.021 0.004 0.001 0.000 0.251 0.002 0.004 0.001 0.000
Optimized cell parameter. Optimized tube diameter. Average energy per carbon atom. Mulliken charge on each chain atom. Orbital overlap population between the two neighboring chain atoms. Orbital overlap population between the nearest chain atom and wall atom. Orbital overlap population between the two nearest wall atoms.
2.5
Fig. 2 – The structure of the smallest CNWs C2@(2,2).
rather unique in that the network formed by the wall atoms is broken down due to the encapsulation of the C2-Chain, as shown in Fig. 2. It is surprising to see that the carbon atom on the C2-Chain inside C2@(2,2) has the coordination of six. In other words, there are hexacoordinate carbon atoms [25– 32] in C2@(2,2). This provides an example of hypercoordinate carbon atoms in a 1D periodic structure. The chemical bonds in C2@(2,2) will be discussed in detail later. Similarly, for the large d-C@(n,n) (n = 4–6) CNWs, only minor structural changes on the CNT walls are observed upon the encapsulation of the chain, while for the small d-C@(3,3) and C@(2,2) CNWs, the changes are more remarkable. In addition, the CNT wall of d-C@(n,n) is also found to be slightly dimerized. The degree of the dimerization of the CNT wall can be indicated by the parameter d = c1–c2; c1 and c2 are described in Fig. 1(b). The d values for d-C@(6,6), d-C@(4,4) and d˚ , respectively. Hence, the C@(3,3) are 0.005, 0.014 and 0.063 A dimerization of the CNT wall becomes more obvious with the decreasing tube diameter of d-C@(n,n), indicating the increasing interaction between the tube and the chain. The average energies per atom relative to that of 2D graphene for the armchair and zigzag CNWs and the pristine CNTs
E (ev/atom)
2
(n ,n ) d-C@(n ,n ), C@(2,2) C2 @(n ,n ) (m,0) C @(m,0) C2 @(m,0)
1.5
1
0.5
0
C60
3
4
5
6
7
8
Tube diameter (Å) Fig. 3 – The average energies per atom relative to that of graphene for pristine CNTs (n,n) and (m,0) and CNWs C@(2,2), d-C@(n,n), C2@(n,n), C@(m,0) and C2@(m,0), where m = 6–10, n = 3–6 for d-C@(n,n) and n = 2–6 for other models. are shown in Fig. 3. Although all the armchair and zigzag CNWs are less stable energetically than the corresponding pristine CNTs, some CNWs are still more stable energetically than the C60 molecule (see Fig. 3), such as C2@(5,5), C2@(6,6), dC@(5,5), d-C@(6,6), C2@(10,0), C2@(9,0) and C@(10,0). The average energies of CNWs and the pristine CNTs increase remarkably as the tube diameters decrease. The large C2@(n,n) (n = 4– 6) is more stable energetically than the corresponding d-C@(n,n), while the small C2@(n,n) (n = 2, 3) is much more unstable energetically than the corresponding d-C@(3,3) or C@(2,2). This indicates that the large diameter CNWs would
280
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4 6 ( 2 0 0 8 ) 2 7 6 –2 8 4
be more favorable to accommodate the chains with relatively larger filling density of carbon atoms, whereas small CNWs would be favorable to relatively less filling density of carbon atoms. These results are in analogy with those for zigzag CNWs, C2@(n,0) and C@(n,0) [12]. We would like to point out that all the above discussions are based only on the energies, but the experimental structures may not be absolutely dependent on their energetic stabilities. It has been shown experimentally that some of the energetically unfavorable structures could form under conditions far from equilibrium [10], such as the quenching process [33,34] from the extremely high temperatures used to generate CNTs [35]. The experiments for carbon clusters [33] and sulfur clusters [34] have clearly shown that the energetically unfavorable structures can form by the rapid cooling of the plasma, because there is too short time for the atoms to rearrange to form the energetically most stable structures. As to the elastic properties, the calculated Young’s modulus for the pristine (4,4) and (6,6) CNTs is 1.06 and 0.96 TPa, respectively, in agreement with the experimental [36] and theoretical [37,38] values for SWCNTs, which are evaluated to be around 1 TPa. According to our calculations, the Young’s modulus for the C2@(4,4) and C2@(6,6) CNWs is 1.25 and 0.99 TPa, respectively. Thus the elastic modulus of the combined CNW system is a little bit larger than that of the corresponding pristine CNT. This is probably due to the interaction between the tube and the inner chain. It is interesting to see that the calculated Young’s modulus for the isolated carbon chain in the free space is only 0.46 TPa, which is much smaller than the elastic modulus for the CNWs. Hence, in this sense, our results support that the CNT shell indeed ‘protects’ the mechanically unstable carbon chain by encapsulating [5].
3.2.
Electronic properties
The calculated band structures and DOS for several representative armchair CNWs are shown in Fig. 4. All armchair CNWs studied here are metallic, the same as the corresponding pristine CNTs. Moreover, the band structures are kept almost unchanged for the rigid sliding motion of the carbon chain enclosed inside the relatively large CNWs, C2@(n,n) and dC@(n,n) (n = 3–6). We discuss CNWs d-C@(n,n) firstly. In Fig. 4, the d-C-Chainderived bands are depicted in dashed lines and are labeled according to their symmetries. As for the large diameter CNWs, d-C@(n,n) (n = 4–6), the characteristics of band structures for the two constituent parts (CNT and carbon chain) are still maintained, but the band levels shift, following the rigid-band shift approximation (RBSA) [39,40]. Such a rigid overlap of band structures of the two components indicates weak orbital interaction between the CNTwall and the d-C-Chain in the larger CNWs, which is in analogy with the results for the larger zigzag CNWs [10]. It is worthy to mention that the flat d-C-Chain-derived p1 bands give rise to high DOS near the Fermi level (see Fig. 4(a)), which contributes most of the DOS (about 71–76% ) to the total DOS at the Fermi level and makes the total DOS much greater than that of the corresponding pristine CNT. Since the DOS near the Fermi level has the contribution from both components, both the CNT wall and the in-tube d-C-Chain can act as the pathway of
charge carriers in d-C@(n,n) (n = 4–6), which is similar to the large diameter zigzag CNWs, C2@(9,0) and C2@(10,0) [10]. As shown in Fig. 4(e) and (h), the shapes of the bands originated from both components change remarkably for dC@(3,3) and C@(2,2), and so do the positions of these bands. As for d-C@(3,3), the bandwidth of r2 band is 1.78 eV and the p1 bands become much wider with bandwidth of about 1.01 eV. In the band structure of C@(2,2), the C-Chain-derived r1, r2 and p bands are all elevated above the Fermi level due to the strong orbital interaction between the CNT and the CChain. In fact, from the projected DOS shown in Fig. 4(h), we can see that these C-Chain-derived bands are hybridized much with the CNT-derived bands, which also reflects strong orbital interaction between the CNT and the C-Chain. The large hybridization indicates that the charge carriers are easy to move from CNT to the C-Chain, and vice versa. Unlike the large d-C@(n,n) (n = 4–6) CNWs, for the small C@(2,2) and dC@(3,3) CNWs, the C-Chain or d-C-Chain contributes less to the total DOS at the Fermi level, about 9% and 19% for C@(2,2) and d-C@(3,3), respectively. Since there is no flat band near the Fermi level, no higher DOS would be expected. Now we take a look at the band structures of CNWs C2@(n,n). The RBSA works also well for the band structures of the large diameter CNWs, C2@(n,n) (n = 4–6). However, in the band structure of C2@(2,2), it is hard to discern which component a band is derived from, because the orbital interaction and hybridization between the C2-Chain and the CNT (2,2) are so remarkable. Since the C2-Chain-derived p1 and p2 bands of C2@(n,n) are much wider than the d-C-Chain-derived bands of d-C@(n,n), the DOS contributed from C2-Chain is relatively lower and spreads over a broad energy range. The C2Chain contributes 4–33% to the total DOS of C2@(n,n) (n = 2–6) at the Fermi level. After all, in all the C2@(n,n) CNWs that we studied, the charge carriers can move through the pathway of both the chain and the CNT. Compared with the considerable electron transfer (about 0.25–0.45 electron/cell) from the CNT to the carbon chain in the zigzag CNWs C2@(n,0) (n = 6–10) [10], remarkable electronic charge transfer (0.2–0.25 electron/cell) from the CNT to the chain is also observed in large diameter armchair CNWs C@(n,n) (n = 4–6), as shown in Fig. 5. In other armchair CNWs, C@(n,n) (n = 2, 3) and C2@(n,n) (n = 2–6), the charge transfer between the CNT and the chain is much less (less than 0.07 electron/cell).
3.3.
COOP and bonding features in armchair CNWs
The COOP analysis was first proposed Hughbanks and Hoffmann [17] to investigate chemical bonding features in periodic systems. The positive and negative COOP peaks for the specified bond correspond to bonding and anti-bonding characteristics, respectively. The integral of the COOP curve up to the Fermi level gives the total overlap population (OP) [41] between the two given atoms. The OP scales like bond order [42] and thus can be used as a measure of bond strength [17]. At the level of theory in this work, the OP for the C–C single bond of diamond, the C@C double bond of ethylene and the C „ C triple bond of acetylene is 0.34, 0.68 and 0.93, respectively. The OP values for the C–C bonds of the pristine CNTs (n,n) fall in the range of 0.4–0.5 (see Table 1).
CARBON
g
281
4 6 (2 0 0 8) 2 7 6–28 4
a
b
c
d
e
f
h
i
Fig. 4 – Plots of band structures, DOS and COOP for (a) (6,6); (b) d-C@(6,6); (c) C2@(6,6); (d) (3,3); (e) d-C@(3,3); (f) C2@(3,3); (g) (2,2); (h) C@(2,2) and (i) C2@(2,2). Dashed–dotted lines indicate Fermi levels and dashed lines denote the chain-derived bands. In the DOS plots, solid line represents the total DOS, while the shadowed area enveloped by dashed line represents the projected DOS from the in-tube chain. In the COOP plots, the COOP between the two neighboring chain atoms is depicted in solid line and the COOP between the nearest chain atom and wall atom is depicted in dashed line. The value of DOS and COOP is measured in states eV1 cell1 and eV1 cell1, respectively.
Firstly, we consider the chemical bonding between the carbon chain and the CNT wall of CNWs. From Table 1, the largest OPcw is just 0.021 for CNWs d-C@(n,n) and C2@(n,n) (n = 3–6), indicating that no chemical bond is formed between the chain and the wall in the relatively larger CNWs. This is coincident with the fact that the closest distance between ˚ for the chain atom and the tube atom is not less than 2.2 A
these CNWs. Thus the atoms are not close enough to form chemical bonds. For the smallest CNWs, C@(2,2) and C2@(2,2), the closest distance between the chain atom and the tube atom is ˚ and 1.59 A ˚ , respectively, which is a little bit larger than 1.70 A ˚ ). The value of the ordinary C–C single bond length (1.54 A OPcw for C@(2,2) and C2@(2,2) is 0.135 and 0.251, respectively.
282
CARBON
4 6 ( 2 0 0 8 ) 2 7 6 –2 8 4
Fig. 5 – Isodensity surface plots of the electron accumulation (white) and depletion (black) are shown for d-C@(6,6). The values for the white and black surfaces are 104e/(a.u.)3. The gray circles indicate the positions of the carbon atoms.
Therefore, the chain atom and wall atom form a chemical bond to some extent, though it is weaker than the typical C–C single bond (OP = 0.34). These weak chemical bonds formed between the chain atom and wall atom in C@(2,2) and C2@(2,2) can also be recognized in the COOP plots (see Fig. 4(h) and (i)). In Fig. 4(h), the negative peaks of COOP between the nearest chain atom and wall atom for C@(2,2) above the Fermi level fall in the region where the C-Chain-derived r1, r2 and p bands are located. These negative peaks indicate the anti-bonding interaction between the chain atom and the tube atom, leading to the above-mentioned elevation of the r1, r2 and p bands of C@(2,2). The positive COOP peaks corresponding to the bonding interaction between the chain atom and the tube atom should be expected below the Fermi level at lower energy level (not plotted in Fig. 4(h)). Therefore, these COOP peaks correspond to the effective chemical bond formed between the nearest chain atom and wall atom in C@(2,2). Similarly, the negative peaks above the Fermi level and the positive peaks below the Fermi level for the COOP between the chain atom and the wall atom in C2@(2,2) indicate the effective chemical bonds formed between the chain and the wall in C2@(2,2). Now we turn our attention to the chemical bonding of the in-tube chains. For the large CNWs d-C@(n,n) (n = 4–6), the d˚ , which is C-Chain has the nearest C–C distance of about 1.3 A slightly shorter than the ordinary C@C double bond length ˚ ). It is found that the OP between the nearest two (1.33 A atoms on the d-C-Chain, OPcc, is rather large (about 0.9, see Table 1) and close to the value for the C „ C triple bond of acetylene (OP = 0.93). The positive peaks of COOP below the Fermi level (see Fig. 4(b)) further confirm these strong C–C bonds. As for d-C@(3,3), the value of OPcc decreases remarkably to 0.275, which can be explained by the COOP plots. As shown in Fig. 4(e), the positive peaks of the COOP for C–C bonds of the d-C-Chain of d-C@(3,3) are mainly above the Fermi level, leading to the remarkable decrease of the total OP,
OPcc. The value of OPcc for C@(2,2) is only 0.017, which manifests that chemical bonds are not formed on the C-Chain inside C@(2,2). Such a small value of OPcc is due mainly to the ˚ on the C-Chain inrelatively large C–C distance of about 2.6 A side C@(2,2). In addition, the decrease of the total OP is also caused by the unoccupied positive COOP peaks for the C–C bond of the C-Chain, as shown in Fig. 4(h). The C–C bond length for the C2-Chain inside CNWs ˚ and the corresponding OPcc C2@(n,n) (n = 3–6) is about 1.25 A is 0.6, the value close to that for ethylene (OP = 0.68), indicating that C–C bonds on the C2-Chain inside these CNWs are even stronger than C–C bonds on the CNT wall (OPww 0.46). These C–C bonds can be confirmed by the positive peaks below Fermi level in the COOP plots, as shown in Fig. 4(c) and (f). As for the C2-Chain inside C2@(2,2), the C–C bond length ˚ , which is still a little shorter than the is elongated to 1.48 A ˚ ). However, length of the ordinary C–C single bond (1.54 A the OPcc for this C–C bond is just 0.176, which is 1/4–1/3 of the OPcc for CNWs C2@(n,n) (n = 3–6). This can also be understood from the COOP plot. In Fig. 4(i), there is a positive bonding peak above the Fermi level, which means that some bonding orbitals become unoccupied and the corresponding C–C bond is thus weakened. Although the OPcc for C–C bond of the C2-Chain inside C2@(2,2) is relatively small, it still indicates the chemical bonding between the carbon atoms of the C2-Chain to some extent. Note that the bond strength is not always consistent with the bond length. For example, there have been many very long C–C single bonds with the bond ˚ [43–47]. On the other hand, length in the range of 1.6–1.9 A the length of a C–C single bond can also be squeezed to be ˚ [48]. In addition, the atom of the C2-Chain and 1.32–1.51 A the atom of the wall in C2@(2,2) form a stronger C–C bond with OPcw = 0.251. Therefore, each carbon atom of the C2Chain inside C2@(2,2) forms six weaker chemical bonds with six neighboring carbon atoms, two on the C2-Chain and four on the CNT wall. Hence, as mentioned above, there are hexacoordinate carbon atoms in CNW C2@(2,2). At the same time, C–C bonds on the CNTwall of C2@(2,2) is also weakened due to the chemical bonding between the chain atom and the wall atom. From Table 1, we can see that OPww for C2@(2,2) is just 0.349, which is obviously smaller than that for CNWs C2@(n,n) (n = 3–6) (OPww = 0.45–0.46).
4.
Conclusion
In summary, we investigated the structural and electronic properties as well as chemical bonding features for CNWs made of linear carbon chains encapsulated inside armchair CNTs. Our crystal orbital calculations show that the interaction between the tube and the in-tube chain is mainly Van der Waals interaction in the armchair CNWs with relatively large tube diameters. The C2-Chain inside the C2@(n,n) is undimerized, while the d-C-Chain inside d-C@(n,n) is dimerized, with the exception of the carbon chain inside C@(2,2), which prefers an undimerized configuration. For the smallest CNWs, C@(2,2) and C2@(2,2), weak chemical bonds are formed to some extent between the tube and the chain. In addition, the chain atoms inside C2@(2,2) have an abnormal coordination of 6, giving an example of hypercoordinate carbons in a
CARBON
4 6 (2 0 0 8) 2 7 6–28 4
1D periodic structure from a point of view of calculations. Among all the CNWs we considered here, the most stable ones are C2@(5,5) and C2@(6,6). The former has the diameter of about 0.7 nm, which is close to the value observed in experiments [5,6]. The elastic modulus along the tube axis for the CNW is expected to be a little larger than that for the corresponding pristine CNT and much larger than that for the isolated carbon chain in the free space. This demonstrates that the mechanically unstable carbon chain is indeed protected by the CNT shell upon the encapsulation. All armchair CNWs are metallic. Both the tube and the chain can act as the pathway of charge carriers in these CNWs. The band structure for the larger diameter CNW can be understood within the framework of RBSA, indicating the weak orbital interaction between the tube and the chain. As the diameter of CNW decreases, the orbital interaction and hybridization becomes more remarkable between the tube and the chain, leading to obvious changes of the shapes and positions of some bands derived from both components. Considerable electron transfer (about 0.20–0.25 electron/cell) is observed from the CNT wall to the in-tube chain in the larger diameter CNWs d-C@(n,n) (n = 4–6), while there is much less charge transfer in the other armchair CNWs. It seems that the d-C-Chain inside d-C@(n,n) (n = 3–6) can obviously modulate the electronic properties of the CNT and give much higher DOS at the Fermi level. In conclusion, our results suggest that filling carbon atom chains with different atom densities into the CNTs with different chiralities and diameters will produce various CNWs with different electronic properties.
Acknowledgements This work is supported by the Research Fund for the Doctoral Program of Higher Education (20060027001), the National Natural Science Foundation of China (No. 20373008) and Major State Basic Research Development Programs (Grant No. 2002CB613406).
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CARBON
4 6 ( 2 0 0 8 ) 2 7 6 –2 8 4
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