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Carbon Nanotube Superarchitectures: An Ab Initio Study Rulong Zhou,†,‡ Rui Liu,†,§ Lei Li,† Xiaojun Wu,|| and Xiao Cheng Zeng*,† †
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Department of Chemistry and Nebraska Center for Materials and Nanoscience, University of Nebraska—Lincoln, Lincoln, Nebraska 68588, United States ‡ School of Science and Engineering of Materials, Hefei University of Technology, Hefei, Anhui 230009, China § Department of Physics, Tsinghua University, Beijing 100084, China Department of Materials Science and Engineering and Hefei National Lab for Physical Materials at Microscale, University of Science and Technology of China, Hefei, Anhui 230026, China ABSTRACT: An advantage of using single-walled carbon nanotubes (SWCNTs) as building blocks in tailoring materials functionality is that many unique properties of SWCNTs can be captured. We present an ab initio study of covalent assembly of SWCNTs into a variety of functional carbon superarchitectures, including two-dimensional (2D) hexagonal and orthogonal and 3D simple cubic (or orthogonal), stacked hexagonal, diamond-like superarchitectures. To achieve a sensible design of SWCNT-based functional materials, we have explored the nodal structures and connectivity, particularly, relative stability of various topological defect-containing junctions at the nodal (or joint) region. The quantumchemical calculations suggest that the 2D hexagonal SWCNT superarchitectures are universally semiconducting, while 3D SWCNT superarchitectures are mostly metallic, regardless of whether the constituent SWCNTs are metallic or semiconducting. In particular, for the metallic SWCNT superarchitectures, their electronic properties are insensitive to the electronic properties of constituent SWCNTs. This remarkable property of SWCNT superarchitectures may be exploited to dodge the experimental subtlety for separation of metallic from semiconducting SWCNTs in CNT bundles. We have also calculated elastic constants of the SWCNT superarchitectures. We predict that SWCNT superarchitectures can be mechanically as robust as many solid semiconductors and metals. In view of their high specific surface area for materials functionality, the metallic SWCNT superarchitectures may find applications in fuel cells, battery electrodes, or nanoelectronic devices.
’ INTRODUCTION Single-walled carbon nanotubes (SWCNTs)1 3 display a combination of remarkable properties such as ultrahigh Young’s modulus and tensile modulus, high thermal conductivity, ballistic electron transport, and high aspect ratio. These properties render SWCNTs a unique one-dimensional (1D) material for applications in nanotechnology. Using SWCNTs as building blocks, many unique properties of SWCNTs can be captured in tailoring materials functionality. Previous theoretical and experimental studies have shown that by introducing large rings more than six carbon atoms4 8 it is possible to form negative-curvature structures from graphite. These early works suggest the possibility of fabrication of stable superstructures of SWCNTs. Nevertheless, over the past two decades, major research efforts have been devoted to isolate CNTs.9 16 Recently, several studies have been reported on the materials design of 2D and 3D nanotube superarchitectures, e.g., supergraphene.17 23 Such SWCNT superarchitectures not only can retain some of the extraordinary properties of 1D SWCNTs in 3D structures but also can offer a class of 2D and 3D trusslike materials with unprecedented architectures and open-ended materials functionality and tunability. Furthermore, covalent assembly of carbon nanostructures has been demonstrated in polymerization of fullerenes for the formation of 1D 3D polymerized-fullerene crystals.24,25 For r 2011 American Chemical Society
example, 2D and 3D carbon architectures via covalent assembly of smaller-sized fullerene Cn15 Å were added into the supercell to avoid interaction between the image and the system. For k-point sampling, because the lattice constants of all superarchitectures considered are larger than 20 Å, only the Γ point was adopted during the structural optimization. After geometric optimization, the electronic structures were computed with denser meshes to obtain more accurate band structures; 10 10 1 and 3 3 3 in the Monkhorst Pack special k-point scheme40 were employed for 2D and 3D superarchitectures, respectively. All atomic positions and the length of the supercell were fully relaxed using the conjugate-gradient algorithm until the maximum force on each atom and the maximum component of the stress tensors were less than 0.02 eV/Å and 0.2 GPa, respectively. As a test, the optimized lattice constant of diamond, (8,0) and (6,6) SWCNT, is 3.577 Å (the experimental value is 3.567 Å), 4.282 Å, and 2.474 Å, respectively. The calculated bond length of two inequivalent C C bonds (B1,B2) of (8,0) and (6,6) SWCNTs are (1.423, 1.440) Å
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and (1.431, 1.433) Å, respectively. According to our calculations, (8,0) SWCNT is a semiconductor with a direct band gap of 0.56 eV, while (6,6) SWCNT is metallic, consistent with previous calculations. Elements of elastic constant tensor are defined as ∂σkl/∂εij, where σkl and εij are the elements of stress and strain tensors, respectively. To compute elastic constants of SWCNT superarchitectures, the lattices along different directions are deformed with various strains, and then the atomic positions are fully optimized. For each type of deformation (along any direction), strains from 0.015 to 0.015 with the increment of 0.006 were used. Lastly, elastic constants were obtained by taking the average of results of Δσkl/Δε for each type of deformation, where Δσkl is the stress element difference between two consecutively deformed structures. The increment of strains is chosen carefully using bulk silicon and diamond as a benchmark test. Our calculated C11, C12, C44, and bulk moduli of silicon and diamond are 151, 59, 73, 89 and 1086, 155, 586, 466 GPa, respectively, in good agreement with the experimental values.
’ RESULTS AND DISCUSSION A. 2D Superarchitectures. A1. Structures and Stabilities. We consider two crystalline forms of planar SWCNT superarchitectures, namely, hexagonal and orthogonal, built on (6,6) or (8,0) SWCNT, respectively. For the hexagonal superarchitecture, each node is a Y-junction of SWCNTs, and the angle between two nearest-neighbor SWCNT branches (stemming from the same node) is 120°. The nearest nodes are akin to the nearest sites in the carbon graphene. For orthogonal superarchitectures, each node is a cross junction, and the angle between two nearestneighbor SWCNT branches is 90°. The cross junction originates from the T-junction, which can be constructed by merging one additional SWCNT to the T-junction, where the newly added branch is opposite to the vertical branch of the T-junction. In some sense, the Y-junction and T-junction can be viewed as a structural motif for the planar hexagonal and orthogonal SWCNT superarchitecture, respectively. Using DFT calculations, we have also examined relative stability of various Y- and T-junctions containing different topological defects. We note that Xue et al. have previously studied relative stabilities of Y- and T-junctions using a tight-binding potential.32 They show that for (6,6) SWCNTs there are two most stable forms of Y-junctions, one with two heptagonal rings in the joint region and another with an octagonal ring in the joint region. The cohesive energies of both Y-junctions are nearly degenerate, which is also confirmed by our DFT calculations. We consider the Y-junctions as the structural motif for constructing the (6,6) hexagonal superarchitecture, one named as (6,6)H-2D-A and another as (6,6)H2D-B (see Figure 1). Similarly, for (8,0) SWCNTs, there are two most stable Y-junctions, both having a pair of heptagonal rings in the joint region but one with the pair aligned normal to the plane of the Y-junction and another with the pair aligned parallel to plane of the Y-junction. The two (8,0) hexagonal superarchitectures constructed from the two (8,0) Y-junctions are named as (8,0)H-2D-A and (8,0)H-2D-B, respectively (Figure 1). To build the orthogonal superarchitectures, we have used the most stable (6,6) T-junction proposed by Xue et al.32 and a (8,0) T-junction whose structure is similar to that of the most stable (9,0) T-junction proposed by Xue et al. as the building unit. The (6,6) and (8,0) orthogonal superarchitectures constructed are named as (6,6)O-2D-A and (8,0)O-2D, respectively (Figure 1). 18175
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Figure 1. Two representative planar hexagonal and orthogonal SWCNT superarchitectures (top panel) and optimized structural units (top and side views) based on the (6,6) and (8,0) Y- and T-junctions, respectively (middle and bottom panel). Heptagonal and pentagonal rings in the joint regions are highlighted in red and light blue, respectively, and octagonal rings are highlighted in dark blue.
Note that both (6,6) and (8,0) T-junctions contain eight heptagonal rings and two pentagonal rings in the joint regions. Two representative structures of 2D SWCNT superarchitectures are shown in Figure 1 (top panel). The computed cohesive energies, the lattice constants, and the surface areas of all 2D SWCNT superarchitectures considered are listed in Table I. The cohesive energy per atom is defined as Ecoh = (nEC Etotal)/n, where Etotal and EC are the total energy per supercell of a superarchitecture and a single carbon atom, respectively, and n is the number of carbon atoms in a supercell. The surface areas are calculated by using the Connolly algorithm,41 where the Connolly radius is set to be 2.0 Å, the same as that used in ref 19. As shown in Figure 1, the structural unit (or supercell) of (8,0)H2D-A, (8,0)H-2D-B, and (6,6)H-2D-A contains 12 heptagonal rings, respectively, while that of (6,6)H-2D-B contains 6 octagonal rings. The topological structure of (6,6)H-2D-A is identical to that of the (6,6) supergraphene reported by Romo-Herrera et al.,19 and that of (8,0)H-2D-A is similar to the (10,0) supergraphene.19 The cohesive energy of (8,0)H-2D-B is 2 meV greater than that of (8,0)H-2D-A, while the cohesive energy of (6,6)H-2D-A is 3 meV greater than that of (6,6)H-2D-B. The structural unit of the planar orthogonal superarchitecture (8,0)O-2D or (6,6)O-2D-A contains 16 heptagonal and 4 pentagonal rings. Note that (6,6)O-2D-A differs from the supersquare proposed by Roma-Herrera et al.19 whose structural unit contains 8 heptagonal, 4 octagonal, and 4 pentagonal rings.
The latter is also constructed and named as (6,6)O-2D-B. The calculated cohesive energy of (6,6)O-2D-B is 14 meV/atom greater than that of (6,6)O-2D-A, which suggests that in the joint region of the (6,6) SWCNT junction the octagonal rings are perhaps more favorable. The local structure of (8,0)O-2D is also different from the (10,0) supersquare proposed by RomoHerrera et al.19 whose structural unit contains 8 octagonal, 8 pentagonal, and 4 heptagonal rings. Overall, as shown in Table I, the hexagonal superarchitectures possess greater cohesive energies than the orthogonal superarchitectures. This is understandable because the orthogonal superarchitectures have higher density of topological defects than the hexagonal superarchitectures. One may also notice from Figure 1 that after structural relaxation the cross section of SWCNT branches of (8,0)H-2D-B and (6,6)H-2D-B is no longer circular in shape but becomes elliptical. We will see later that such a structural deformation in the SWCNT branches will have important effects on the overall superarchitectures’ electronic and mechanical properties. A2. Electronic Structures. The computed electronic band structures of the seven 2D SWCNT superarchitectures considered are shown in Figure 2. Apparently, the four planar hexagonal superarchitectures are all semiconducting regardless of whether the constituent SWCNT is metallic or semiconducting. The band gaps of the (8,0)H-2D-A, (8,0)H-2D-B, (6,6)H-2D-A, and (6,6,)H-2D-B superarchitectures are 0.17, 0.31, 0.71, and 0.42 eV, respectively. Except (6,6)H-2D-B, three other hexagonal 18176
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The Journal of Physical Chemistry C Table I. Cohesive Energies Per Atom, Numbers of Atoms Per Supercell, Lattice Constants, Surface Areas, and Electronic Band Gaps of All Seven 2D and Seventeen 3D SWCNT Superarchitectures Considered in the Study
superarchitectures all possess a direct band gap. It is known that the DFT/GGA (see Computational Method) tends to underestimate the band gap of semiconductors. Hence, in reality the band gaps of the four hexagonal superarchitectures are likely wider than the computed values. Among the planar orthogonal superarchitectures, (8,0)O-2D and (6,6)O-2D-A are metallic, while (6,6)O-2D-B is a semiconductor. Note that the (8,0) SWCNT is semiconducting, while the (6,6) SWCNT is metallic. Apparently, the electronic structures of SWCNT superarchitectures are not strongly correlated with the electronic structures of constituent SWCNTs. This conclusion is contrary to a previous study which found the electronic properties of SWCNT superarchitectures were very similar to the constituent SWCNTs.19 For example, Romo-Herrera et al.19 predicted that the (6,6) SWCNT superarchitectures were always metallic, while the (10,0) SWCNT superarchitectures were always semiconducting. One possible reason for this qualitative difference in the predicted electronic properties is that here the superarchitecture structures are relaxed based on the DFT theory rather than the empirical force field. As can be seen in Figure 2, for planar hexagonal superarchitectures, several energy bands near the Fermi level are quite flat, and they distribute within a narrow energy range, thereby
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resulting in a high density of states (DOS). Other bands are separated clearly from these bands. These bands near the Fermi level appear to be similar to the impurity (defect) bands or surface bands normally observed in semiconducting materials. The high DOS for the impurity bands or surface bands allows possible generation of high density electron or hole carriers, which is important for applications in electronic devices. To determine which section in the SWCNT superarchitectures contributes to these bands, we plot the wave functions of these bands at the Γ point for every planar hexagonal superarchitecture. The wave functions of the first three bands below the Fermi Level (EF 1 to EF 3) and above the Fermi level (EF +1 to EF +3) for each hexagonal superarchitecture are shown in Figure 3. For (8,0)H-2D-A, the states of all the bands near the Fermi level are mainly contributed by the atoms near the topological defects. Atoms in the branches also have minor contribution to some bands. For (8,0)H-2D-B, however, besides those atoms near the defects, atoms in the sides of the branches also contribute significantly to the states associated with the EF 3 to EF 1 bands due to larger deformation of the SWCNTs (circular to elliptical shape in the cross section of SWCNTs) in the (8,0)H-2D-B superarchitecture. Note that the wave functions of the EF 1 to EF 3 bands of (8,0)H-2D-B delocalize extensively due to the side atoms on the high-curvature region of the elliptical-shaped SWCNTs in the superarchitecture. An implication of this result is that the electron could transport along the superarchitecture if an appropriate bias voltage is applied. In summary, for the (8,0) planar hexagonal superarchitectures, the energy bands near the Fermi level indeed belong to the defect bands or the surface bands as expected. For the (6,6) planar hexagonal superarchitectures, however, the physics is somewhat different. As shown in Figure 3, although the atoms near the defects still contribute mainly to the states of the EF 3 to EF +3 bands, the atoms in the branches, including those not only on the side but also on top and at bottom, have notable contribution. This is also true for other bands near the Fermi level. The difference in wave function distribution between (8,0) and (6,6) hexagonal superarchitectures is possibly due to the difference in the electronic properties of constituent (8,0) and (6,6) SWCNTs. For (8,0) and (6,6) planar orthogonal superarchitectures, the wave functions of those bands that cross or near the Fermi level are plotted in Figure 3. Clearly, the wave function distribution for the orthogonal superarchitectures is markedly different from that for the hexagonal superarchitectures. In the orthogonal superarchitectures, the wave functions delocalize extensively in all the branches, implying that electrons could easily transport through the planar orthogonal superarchitectures under fairly appropriate bias voltages. A3. Elastic Properties. We have also computed elastic constants of the six 2D SWCNT superarchitectures (see Table II). To this end, we implement a thick vacuum slab parallel to the plane of a superarchitecture in the simulation cell so that the interaction between the superarchitecture and its periodic images is negligible. We set the 2D superarchitecture in the xy plane and set the vector B a along the x axis and vector B c along the z axis. The stresses within the xy plane (i.e., σ11, σ22, and σ12) are in inverse proportion to the lattice constant in the z direction (c) because the forces within the xy plane are constant while the area of the side of the simulation cell (normal to the xy plane) is proportional to the lattice constant in the z direction (c). As such, the stresses calculated in the xy plane should multiply a scaling factor c/d where d is thickness of the 2D superarchitecture. The calculated 18177
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Table II. Calculated Elastic Constants of Six 2D and Eight 3D SWCNT Superarchitectures elastic constant (GPa) C11
C22
(8,0)H-2D-A
172.3
172.3
108.2
(8,0)H-2D-B
200.0
200.0
112.1
45.6
(6,6)H-2D-A
196.0
196.0
73.9
60.5
(6,6)H-2D-B
266.8
266.8
81.8
98.8
(8,0)O-2D
238.2
268.9
2.9
12.9
(6,6)O-2D
C12
C13
C23
C44
C55
10.5
C66
296.6
252.6
(8,0)H-3D-A1
45.6
45.6
75.1
27.5
20.3
20.3
5.6
5.6
9.5
(8,0)H-3D-B1 (6,6)H-3D-A1
41.3 64.4
41.3 64.4
75.8 118.3
22.5 22.0
18.6 35.4
18.6 35.4
4.7 14.2
4.7 14.2
10.5 21.4
35.7
19.5
(6,6)H-3D-B1
68.0
68.0
47.9
29.9
22.6
22.6
8.9
8.9
(8,0)O-3D-A
45.6
45.5
107.3
19.9
33.9
33.8
8.8
9.0
1.7
(6,6)O-3D-A
59.6
98.8
49.7
38.5
28.1
33.2
15.0
5.9
12.6
(9,0)D-A
44.2
44.2
48.5
23.4
19.0
19.0
6.7
7.3
10.7
(6,6)D-A
38.4
38.4
43.4
18.0
14.4
14.4
6.9
7.5
diamond SWCNT(6,6)a bulk silicon a
C33
1086
1086
1086
151
151
996 151
156
156
58.5
58.5
156 58.5
586 72.7
586 72.7
10.3 586 72.7
Wall thickness is 0.34 nm.
elastic constants C11, C22, and C12 should also multiply the scaling factor c/d. Our DFT calculations predict that values of C11, C12, and C66 of planar hexagonal superarchitectures are in the range of 150 300, 70 120, and 50 100 GPa, respectively, which are about 1/5, 1/2, and 1/8 of the corresponding elastic constants of the graphite41 (C11, 1000 1200 GPa; C12, 100 200 GPa; and C66, 400 500 GPa). For the planar orthogonal superarchitectures, C11 and C22 are within the range of 200 300 GPa, whereas C12 and C66 are much smaller than those of the planar hexagonal superarchitectures. Notice that C12 of the two planar orthogonal superstructures is negative, suggesting that the 2D orthogonal superstructures are not stable under shear stresses. This is consistent with their lower energetic stability compared to the planar hexagonal superarchitectures. The elastic constants of (6,6) SWCNT superarchitectures are systematically larger than those of (8,0) SWCNT superarchitectures, and the elastic constants of hexagonal superarchitectures are higher than those of silicon (Table II) or many metals. So, as far as mechanical properties are concerned, the planar hexagonal and orthogonal superarchitectures can be alternative nanostructures in place of silicon and copper nanofilms in electronic devices. B. 3D SWCNT Superarchitectures. B1. Structures and Stabilities. On the basis of the optimized planar hexagonal and orthogonal superarchitectures, we construct 3D hexagonal and orthogonal superarchitectures by covalently connecting two SWCNTs at each node in the direction perpendicular to the planar superarchitecture. For the (8,0) planar hexagonal superarchitecture, the (9,0) SWCNTs are selected as the vertical branches in light of C3 symmetry at each node of the 2D hexagonal superarchitecture. Taking into account differences in topological defects in the joint region, we construct two 3D superarchitectures based on each of the (8,0) and (6,6) planar hexagonal and orthogonal superarchitectures. The obtained 13 3D superarchitectures (see Figure 4) are named as (8,0)H-3DA1 and (8,0)H-3D-A2 [based on (8,0)H-2D-A], (8,0)H-3D-B1 and (8,0)H-3D-B2 [based on (8,0)H-2D-B], (6,6)H-3D-A1 and
(6,6)H-3D-A2 [based on (6,6)H-2D-A], (6,6)H-3D-B1 and (6,6) H-3DB2 [based on (6,6)H-2D-B], (8,0)O-3D-A and (8,0)O-3DB [based on (8,0)O-2D], (6,6)O-3D-A and (6,6)O-3D-B [based on (6,6)O-2D-A], and (6,6)O-3D-C [based on (6,6)O-2D-B], respectively. Besides the 3D hexagonal and orthogonal superarchitectures, we also construct diamond-like 3D superarchitectures (see Figure 4). The building block for diamond-like superarchitectures is the tetrahedral SWCNT junction. Because of the C3 symmetry along any branch of a tetrahedral CNT junction, we select the (9,0) and (6,6) SWCNTs to construct two diamond-like superarchitectures. Considering two different topological defects in the joint region, four diamond-like superarchitectures, namely, (9,0)D-A, (9,0)D-B and (6,6)D-A, (6,6)D-B, respectively, are built. The optimized structural units for all 17 3D superarchitectures are shown in Figure 4, and their cohesive energies, lattice constants, and surface areas are listed in Table I. As mentioned above, considering structural compatibility, an optimal way to build a stacked hexagonal superarchitecture is by connecting two vertical (9,0) SWCNTs to each node of the (8,0) planar hexagonal superarchitecture or by connecting two vertical (6,6) SWCNTs to each node of the (6,6) planar hexagonal superarchitecture. As such, the pattern of topological defects in the original planar hexagonal superarchitectures is maintained, thereby retaining high stability of the basal planar structures. In (8,0)H-3D-A1, (8,0)H-3D-B1, (6,6)H-3D-A1, and (6,6)H-3DB1, six additional octagonal rings are required at each node for the nodal connectivity of two vertical SWCNT branches with the planar basal layer. Hence, each structural unit (supercell) of (8,0)H-3D-A1, (8,0)H-3D-B1, and (6,6)H-3D-A1 superarchitectures contains 12 heptagonal and 12 octagonal rings, and the structural unit of (6,6)H-3D-B1 contains 18 octagonal rings (Figure 4). After structural relaxation, the cross section of vertical SWCNT branches is still nearly circular in shape, and the structure of the planar basal layer is little changed. As shown in Table I, the (8,0)H-3D-A1 and (8,0)H-3D-B1 superarchitectures are nearly 18178
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Figure 2. Computed band structures of the seven 2D SWCNT superarchitectures, along with those of the constituent (8,0) and (6,6) SWCNTs. The symmetric points are K( 1/3,2/3,0) and M(0,1/2,0) for planar hexagonal superarchitectures and X(0,1/2,0) and M(1/2,1/2,0) for planar orthogonal superarchitectures, all in units of reciprocal lattice vectors.
degenerate in energy. However, the (6,6)H-3D-A1 superarchitecture is more stable than (6,6)H-3D-B1, and the relatively large difference in cohesive energy (12 meV/atom) between them is likely due to the larger number of octagonal rings per node presented in (6,6)H-3D-B1. In general, the 3D stacked hexagonal superarchitectures all possess notably less cohesive energies than the constituent planar hexagonal superarchitectures, with energy difference ranging from 15 to 30 meV/atom. It is known that an octagonal ring is energetically less favorable compared to one or two heptagonal (pentagonal) rings in any SWCNT. However, whether this rule of thumb is still applicable to SWCNT superarchitectures requires further study. To this end, we consider a new group of 3D stacked hexagonal superarchitectures, namely, (8,0)H-3D-A2, (8,0)H-3D-B2, (6,6)H3D-A2, and (6,6)H-3D-B2 (Figure 4). These four superarchitecture structures are attained by transforming the octagonal rings in the joint regions (the “elbow” section between the vertical SWCNT branches and the planar basal layer) into heptagonal or pentagonal rings via adding or removing atoms. For example, in (8,0)H-3D-A2, there are three heptagonal rings in each “corner” section between the vertical SWCNT branch and any two lateral SWCNT branches in the planar basal layer. The total
number of heptagonal rings in a single structural unit (Figure 4) is 36. In (8,0)H-3D-B2, three additional heptagonal rings and one pentagonal ring are created in each corner section. There are in total 48 heptagonal and 12 pentagonal rings in each structural unit. For (6,6)-3D-A2 and (6,6)H-3D-B2, six additional heptagonal rings are created to merge a (6,6) SWCNT with the planar basal layer. One structural unit of (6,6)H-3D-A2 contains 36 heptagonal rings, and that of (6,6)H-3D-B2 contains 6 octagonal and 24 heptagonal rings. The topological defects in the planar basal layers of (6,6)H-3D-A2 and (6,6)H-3D-B2 are the same as those in the corresponding planar superarchitectures [(6,6)H2D-A and (6,6)H-2D-B], whereas those in (8,0)H-3D-A2 and (8,0)H-3D-B2 change dramatically. The variation of topological defects in the planar basal layers of these (8,0) 3D hexagonal superarchitectures, relative to the corresponding 2D superarchitectures, can lead to marked changes in their electronic structures. After structural relaxation, the structures of (8,0)H-3D-A2, (8,0)H-3D-B2, (6,6)H-3D-A2, and (6,6)H-3D-B2 deform significantly, especially those of the added vertical SWCNT branches whose cross section turns into a triangular shape. The deformation of cross section of SWCNTs in (8,0) superarchitectures is even more dramatic. Due to the increase of the 18179
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Figure 3. Wave function iso-surfaces (in green and blue) of the bands near the Fermi level for 2D SWCNT superarchitectures. The isovalue is 0.05.
number of topological defects, (8,0)H-3D-A2, (8,0)H-3D-B2, (6,6)H-3D-A2, and (6,6)H-3D-B2 possess much less cohesive energy (about 30 meV/atom in difference) than (8,0)H-3D-A1, (8,0)H-3D-B1, (6,6)H-3D-A1, and (6,6)H-3D-B1, respectively. This result implies that the octagonal rings in the 90°-angle junctions may be more favorable for connecting vertical SWCNT branches with the planar basal layer. For 3D orthogonal superarchitectures, each structural unit of (8,0)O-3D-A and (8,0)O-3D-B contains 24 heptagonal rings, respectively (Figure 4). A pair of heptagonal rings appears in the elbow section between any two SWCNT branches. In (8,0)O3D-B, the heptagonal rings are evenly distributed among all elbow sections in each node, whereas in (8,0)O-3D-A the distribution of heptagonal rings is not even (Figure 4). As a result, the structure of (8,0)O-3D-A deforms strongly after the structural relaxation. The cross section of four SWCNT branches turns into elliptical shape, while that of two remains circular. (8,0)O-3D-B deforms less strongly, and the cross section of all six SWCNT branches remains circular in shape due to the even distribution of the topological defects. The cohesive energy of (8,0)O-3D-B is 11 meV/atom more than that of (8,0)O-3D-A due to the weaker deformation in (8,0)O-3D-B. Compared to the (8,0) stacked hexagonal superarchitectures, the (8,0) orthogonal superarchitectures are much less stable (30 meV/atom difference in cohesive energy), similar to the 2D counterparts. In (6,6)O-3D-A, one octagonal ring exists in the elbow section between any two SWCNT branches (Figure 4). Each structural unit contains eight octagonal and eight heptagonal rings. As mentioned
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above, each octagonal ring can be transformed into two heptagonal rings through adding two atoms in the center of the octagonal ring. As such, a highly symmetric structure (6,6)O3D-B is created, where a pair of heptagonal rings exist in the elbow section. Each structural unit of (6,6)O-3D-B contains 24 heptagonal rings. After structural relaxation, the cross section of SWCNTs in (6,6)O-3D-A deforms more strongly than that of (6,6)O-3D-B due to uneven distribution of topological defects (Figure 4). The cross section of two added vertical SWCNT branches in (6,6)O-3D-A remains circular in shape, while that of the other four SWCNT branches turns into elliptical shape. Unlike (8,0) 3D orthogonal superarchitectures, surprisingly, the higher symmetric (6,6)O-3D-B is much less stable (30 meV/ atom difference in cohesive energy) than (6,6)O-3D-A. This result reinforces our conclusion that an octagonal ring in a 90°angle junction is energetically more favorable than a pair of heptagonal rings. To provide more evidence, we construct a new superarchitecture (6,6)O-3D-C by transforming all pairs of heptagonal rings in (6,6)O-3D-A into octagonal rings. Each structural unit of (6,6)O-3D-C contains 12 octagonal rings, the same as studied by Romo-Herrera et al.19 The cohesive energy of (6,6)O-3D-C is 5 meV/atom greater than that of (6,6)O-3D-A. Note that the (6,6) 3D orthogonal superarchitectures have close cohesive energies to the (6,6) stacked hexagonal superarchitectures (Table I). For diamond-like (9,0) superarchitectures, a pair of heptagonal rings are needed to connect any two neighbor SWCNT branches (Figure 4). In (9,0)D-A, every pair of heptagonal rings is connected with each other, while in (9,0)D-B every pair of heptagonal rings is separated by a hexagonal ring. The latter superarchitecture is the same as reported by Romo-Herrera et al.19 Each structural unit of (9,0)D-A or (9,0)D-B contains 24 heptagonal rings. We find that (9,0)D-A is much more stable than (9,0)D-B (85 meV/atom difference in cohesive energy; see Table I). This can be understood by the fact that in (9,0)DB most bonds in the heptagonal rings are elongated more strongly than those in (9,0)D-A. In addition, we build two diamond-like (6,6) superarchitectures, (6,6)D-A19 and (6,6)DB. Each structural unit of (6,6)D-A contains 12 octagonal rings, that is, one octagonal ring per elbow section. By transforming every octagonal ring into a pair of heptagonal rings, another diamond-like superarchitecture (6,6)D-B is attained. The cohesive energy of (6,6)D-A is 13 meV/atom greater than that of (6,6)D-B, again suggesting that the octagonal rings are more favorable in the joint region of (6,6) SWCNT superarchitectures. In fact, among all the 3D SWCNT superarchitectures, the diamond-like superarchitectures are predicted to be the most stable (see Table I). B2. Electronic Structures. The computed band structures of 16 3D superarchitectures are shown in Figure 5, and the band gaps are listed in Table I. Contrary to previous results,19 we find that most 3D superarchitectures are metallic. Four exceptions are: the three stacked hexagonal superarchitectures (8,0)H-3D-B1, (6,6)H-3D-A1, and (6,6)H-3D-B2 and one 3D orthogonal (or simple cubic) superarchitecture (6,6)O-3D-A, which are all semiconducting with the band gap of 0.14, 0.16, 0.21, and 0.035 eV, respectively. It is worthy of noting that in these four semiconducting superarchitectures the topological defects that connect the vertical SWCNT branches and planar basal layer differ from those in the planar basal layer, whereas in the metallic superarchitectures the topological defects are still the same (i.e., either heptagonal or octagonal). 18180
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Figure 4. Three representative 3D hexagonal, orthogonal (simple cubic), and diamond-like SWCNT superarchitectures (top panel) and optimized structural units (in a supercell) of 16 3D SWCNT superarchitectures. Heptagonal and pentagonal rings at the joint regions are highlighted in red and light blue, respectively, and octagonal rings are highlighted in dark blue. Two different views are displayed for three 3D orthogonal superarchitectures.
The band structures of different types of superarchitectures differ dramatically from each other. For the stacked hexagonal superarchitectures, many bands are fairly flat in the region of Γ f K f M, which are contributed by the added vertical SWCNT branches. The flat bands suggest very weak hybridization between the vertical SWCNT branches and planar basal layers. Besides the flat bands, some bands of 3D hexagonal superarchitectures show features in the region of Γf M similar to those of the planar hexagonal superarchitectures, which also supports that little hybridization occurs between the vertical SWCNT branches and the planar basal layers. For 3D orthogonal superarchitectures, the energy bands disperse much more strongly compared to the 3D hexagonal superarchitectures, while for the diamond-like superarchitectures many flat bands arise in the k-space especially for diamond-like (6,6) superarchitectures. The band structures of the diamond-like (6,6) and (9,0) superarchitectures are distinctly different from each other. For the diamond-like (6,6) superarchitectures, several densely distributed
bands are separated by wide gaps within the energy window of ( 1.0 eV, 1.0 eV), whereas for the diamond-like (9,0) superarchitectures, the bands disperse more strongly and the gaps among them are much narrower (Figure 5). More interestingly, the DOS near the Fermi level is very high for all the diamond-like superarchitectures, while the DOS is fairly low for the 3D hexagonal and orthogonal superarchitectures. Hence, the diamond-like superarchitectures would possess higher electron conductivity than 3D hexagonal and orthogonal superarchitectures. To gain more insight into electronic structures of different 3D superarchitectures, especially those near the Fermi levels, we plot iso-surfaces of the wave functions for the five bands above and below the Fermi levels (Figure 6a and 6b). For the 3D hexagonal superarchitectures, there are some common features in the isosurfaces of the wave functions. Taking (6,6)H-3D-A1 as an example, the first two bands below the Fermi level (EF 1, EF 2) and the third and fourth bands above the Fermi level (EF +3, EF +4) counted from the Γ point are flat within the range 18181
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Figure 5. Computed band structures of the 16 3D SWCNT superarchitectures. The symmetric points are K( 1/3,2/3,0), M(0,1/2,0), F(0,1/2,1/2) for the stacked hexagonal superarchitectures, X(0,1/2,0), R(1/2,1/2,1/2), M(1/2,1/2,0) for the orthogonal superarchitectures, and W(1/2,1/4,3/4), L(1/2,1/2,1/2), X(1/2,0,1/2), K(3/8,3/8,3/4) for the diamond-like superarchitectures, all in units of reciprocal lattice vectors.
of Γ f M (Figure 5), and they are entirely contributed by the vertical SWCNT branches (Figure 6a). The wave functions of other bands (EF 3, EF 4, EF 5) and (EF +1, EF +2, EF +5) are mainly contributed by the planar basal layers, which are very similar to those bands near the Fermi level for the planar superarchitecture (6,6)H-2D-A. The plotted wave functions demonstrate that the vertical SWCNT branches have little coupling with the planar basal layers in (6,6)H-3D-A1. The case of (6,6)H-3DB1, however, is slightly different; i.e., some couplings between the vertical branches and the planar basal layers can be discerned. In
view that (6,6)H-3D-B1 only contains octagonal rings, we attribute this coupling to the hybridization among the octagonal rings. For the 3D orthogonal superarchitectures such as (8,0)O-3DB and (6,6)O-3D-C, the bands near the Fermi level are mainly contributed by nodal regions of the superarchitectures. The SWCNT branches of (8,0)O-3D-B have almost no contribution to the first three bands below and above the Fermi level, whereas the SWCNT branches of (6,6)O-3D-C do contribute to these bands. The differences between the wave functions of (8,0)O-3D-B and (6,6)O-3D-C are also reflected in the band 18182
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Figure 6. Iso-surfaces (green and blue lobes) of wave function for the bands near the Fermi level of various 3D superarchitectures. The iso-value is 0.05. (a) Legend in blue denotes contribution mainly from the vertical SWCNT branches (perpendicular to the plane of the paper).
structures of the two superarchitectures (Figure 5). Apparently, the bands closest to the Fermi level of (8,0)O-3D-B are very flat, reflected by localized distribution of the wave functions for these bands. On the other hand, the bands of (6,6)O-3D-C are much more dispersive in the k-space region. The wave function distribution for the diamond-like (9,0) superarchitectures is also much different from that for the diamond-like (6,6) superarchitectures. As shown in Figure 6b, most bands near the Fermi level of (9,0)D-A are mainly contributed by local defects in the joint region, except the EF + 3 band which is much more dispersive (as shown by green and blue lobes on the branch section), compared to other bands. There are many other highly dispersed bands, contributed by the atoms in the SWCNT branches, below and above the densely distributed bands near the Fermi level (Figure 5). In the (6,6)DA structure, although the bands close to the Fermi level are highly localized in a narrow energy window (see Figure 5), the wave functions of these bands distribute quite extensively (see Figure 6b). The first four bands below and above the Fermi level are contributed by different SWCNT branches of the tetrahedral junction. The defect rings in the joint region do not show significant
contribution to these bands, but their contributions are apparent in the bands EF 5 and EF + 5, which are separated from the bands near the Fermi level (Figure 5). From the band structures and the distributions of the wave functions, we expect that the (6,6)D-A superarchitecture may possess much higher electric conductivity than any other 3D SWCNT superarchitectures. B3. Elastic Properties. Elastic constants of several 3D superarchitectures are also calculated, as listed in Table II. Values of C11, C22, and C33 are typically in the range of 40 100 GPa, and those of C12, C13, and C23 are in the range of 20 40 GPa. As discussed above, the calculated elastic constant of the 3D superarchitectures should depend on the length of the SWCNT branches in the superarchitectures. For example, for the 3D hexagonal superarchitectures (the vertical branches are along the B c vector direction), the normal stresses σ11 and σ22 are inversely proportional to the lattice constant c (Table I) when the lattice constant a (b = a for hexagonal superarchitectures) keeps invariant, and σ33 is inversely proportional to the square of a when the lattice constant c stays constant. For (8,0)H-3D-A1, the smallest values of a and c can be about 22.0 and 12.0 Å, respectively. Hence, the upper limit of the elastic constant of C11 and C33 is 80 and 200 GPa, 18183
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The Journal of Physical Chemistry C respectively. If both a and c adopt their smallest value, the elastic constants of C11 and C33 can be larger. Hence, the 3D superarchitectures with short length in SWCNT branches may possess high elastic constants, higher than many semiconductors and metals. On the other hand, if the length of SWCNT branches is long, the connection in the SWCNT superarchitecture will have little influence on the elastic properties. In this case, the superarchitectures may have properties similar to constituent SWCNTs.
’ CONCLUSIONS We have investigated optimal covalent assembly of 2D and 3D SWCNT superarchitectures and their electronic and mechanical properties. Various local structures at the joint region and their relative stabilities are discussed in detail. For planar hexagonal and orthogonal superarchitectures, the local structures at the joint region are consistent with those of the most stable Y- and T-shaped SWCNT junction, respectively. For 3D stacked hexagonal superarchitectures, the predicted most stable structures are obtained by connecting two vertical SWCNT branches with the most stable planar hexagonal superarchitectures at each node through the creation of octagonal rings. For 3D orthogonal superarchitectures, highly symmetric structures are more favored, where every pair of SWCNT branches is connected by the creation of a pair of heptagonal rings at each node for (8,0) SWCNT superarchitectures or one octagonal ring at each node for (6,6) SWCNT superarchitectures. The predicted most stable structure of diamond-like (8,0) or (6,6) superarchitectures is obtained by connecting each pair of SWCNT branches through a pair of heptagonal rings or one octagonal ring. We have shown that in zigzag SWCNT-based superarchitectures pairs of heptagonal rings are generally favored, while in armchair SWCNTbased superarchitectures octagonal rings are generally preferred. Our study suggests that the planar hexagonal structures and diamond-like structures are likely the most stable 2D and 3D SWCNT superarchitectures, respectively. The calculated band structures suggest that the most stable 2D and 3D hexagonal SWCNT superarchitectures are semiconducting, while the orthogonal and diamond-like SWCNT superarchitectures are metallic, regardless of electronic properties of constituent SWCNTs. Hence, the challenge for the separation of metallic and semiconducting SWCNTs can be averted when the specific form of SWCNT architecture is fabricated. On the electronic structures, we find that the energy bands near the Fermi levels for the zigzag SWCNT superarchitectures are mainly contributed by atoms in the joint region, whereas for armchair SWCNT superarchitectures the SWCNT branches contribute significantly to the bands. The DOS near the Fermi level for the diamond-like superarchitectures is particularly high, implying that diamond-like SWCNT superarchitectures can possess high electric conductivity. On the mechanical properties, the elastic constants, C11, C22, and C33, of 2D and 3D SWCNT superarchitectures can be as large as 300 GPa. Hence, SWCNT superarchitectures are mechanically robust as many semiconductors and metals. In view of their high electric conductivity, high mechanical stability, and large specific surface areas, SWCNT superarchitectures may find applications, among others, in ion storage, fuel cell, or nanoelectronics. ’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected].
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’ ACKNOWLEDGMENT X.C.Z. is grateful to Dr. Yongfeng Lu and Dr. Ignacio Perez for helpful discussions. This work is supported by grants from the ONR (N00014-09-1-0943), ARL (W911NF1020099), NSF (CMMI-0709333), and the Nebraska Research Initiative and by the University of Nebraska’s Holland Computing Center. ’ REFERENCES (1) Hirsch, A. Nat. Mater. 2010, 9, 868–871. (2) Iijima, S. Nature 1991, 354, 56–58. (3) Jorio, A; Dresselhaus, G.; Dresselhaus, M. S. Carbon Nanotubes: Advanced Topics in the Synthesis, Structure, Properties and Applications; Springer: Gemany, 2008. (4) Mackay, A. L.; Terrones, H. Nature 1991, 352, 762–762. (5) Lenosky, T.; Gonze, X.; Teter, M.; Elser, V. Nature 1992, 355, 333–335. (6) Vanderbilt, D.; Tersoff, J. Phys. Rev. Lett. 1992, 68, 511–514. (7) Huang, M-Z; Ching, W. Y.; Lenosky, T. Phys. Rev. B 1993, 47, 1593–1606. (8) Iijima, S.; Ichihashi, T.; Ando, Y. Nature 1992, 356, 776–778. (9) Terrones, M. Annu. Rev. Mater. Sci. 2003, 33, 419–501. (10) Burghard, M. Surf. Sci. Rep. 2005, 58, 1–109. (11) Tasis, D.; Tagmatarchis, N.; Bianco, A.; Prato, M. Chem. Rev. 2006, 106, 1105–1136. (12) Sinha, N.; Ma, J.; Yeow, J. T. W. J. Nanosci. Nanotechnol. 2006, 6, 573–590. (13) Vairavapandian, D; Vichchulada, P; Lay, M. D. Anal. Chim. Acta 2008, 626, 119–129. (14) Hu, Y.; Shenderova, O. A.; Brenner, D. W. J. Comput. Theory Nanosci. 2007, 4, 199–221. (15) Dervishi, E.; Li, Z; Xu, Y.; Saini, V.; Biris., A. R.; Lupu, D.; Biris, A. S. Part. Sci. Technol. 2009, 27, 107–125. (16) Eder, D. Chem. Rev. 2010, 110, 1348–1385. (17) Coluci, V. R.; Galvao, D. S.; Jorio, A. Nanotechnology 2006, 17, 617–621. (18) Coluci, V. R.; Pugno, N. M.; Dantas, S. O.; Galvao, D. S.; Jorio, A. Nanotechnology 2007, 18, 335702. (19) Romo-Herrera, J. M.; Terrones, M.; Terrones, H.; Dag, S.; Meunier, V. Nano Lett. 2007, 7, 570–576. (20) Romo-Herrera, J. M.; Terrones, M.; Terrones, H.; Meunier, V. ACS Nano 2008, 2, 2585–2591. (21) Romo-Herrera, J. M.; Terrones, M.; Terrones, H.; Meunier, V. Nanotechnology 2008, 19, 315704. (22) Ivanovskaya, V. V.; Ivanovskii, A. L. J. Superhard Mater. 2010, 32, 67–87. (23) Hernandez, E.; Meunier, V.; Smith, B. W.; Rurali, R.; Terrones, H.; Nardelli, M. B.; Terrones, M.; Luzzi, D. E.; Charlier, J.-C. Nano Lett. 2003, 3, 1037–1042. (24) Wei, D.; Liu, Y. Adv. Mater. 2008, 20, 2815–2841. (25) Ismach, A.; Kantorovich, D.; Joselevich, E. J. Am. Chem. Soc. 2005, 127, 11554–11555. (26) Terrones1, M.; Terrones, H.; Banhart, F.; Charlier, J. C.; Ajayan, P. M. Science 2000, 288, 1226–1229. (27) Terrones1, M.; Banhart, F.; Grobert, N.; Charlier, J. C.; Terrones, H.; Ajayan, P. M. Phys. Rev. Lett. 2002, 89, 075505. (28) Krasheninnikov, A. V.; Nordlund, K.; Keinonen, J.; Bahhart, F. Nucl. Instrum. Methods Phys. Res., Sect. B 2003, 202, 224–229. (29) Jin, C. H.; Suenaga, K.; Iijima, S. Nat. Nano. 2008, 3, 17–21. (30) Xue, B.; Shao, X.; Cai, W. Comput. Mater. Sci. 2008, 43, 531– 539. (31) Tsai, P.-C.; Jeng, Y.-R.; Fang, T.-H. Phys. Rev. B 2006, 74, 045406. (32) Ponomareva, I.; Chernozatonskii, L. A.; Andriotis, A. N.; Menon, M. New J. Phys 2003, 5, 119. (33) Li, Y.; Qiu, X.; Yang, F.; Wang, X.; Yin, Y. Nanotechnology 2008, 19, 225701. 18184
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