J. Phys. Chem. C 2007, 111, 1625-1630
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Structural and Electronic Study of Nanoscrolls Rolled up by a Single Graphene Sheet Yu Chen,*,† Jing Lu,*,†,‡ and Zhengxiang Gao† Mesoscopic Physics Laboratory, Department of Physics, Peking UniVersity, Beijing 100871, P.R. China, and Department of Physics, UniVersity of Nebraska at Omaha, Omaha, Nebraska 68182-0266 ReceiVed: September 15, 2006; In Final Form: NoVember 5, 2006
With special topology differing from nanotubes, carbon nanoscrolls (CNSs) rolled up by a single graphene sheet show unique properties and have potential applications in hydrogen storage and energy storage. We studied various types of nanoscrolls by using first principle calculation. The minimum innermost radius of armchair-type nanoscrolls is achieved. The electronic structures of nanoscrolls are greatly related to chirality (n,m). Armchair nanoscrolls are metallic or semimeatllic depending on their sizes and those metallic ones have larger density of states at the Fermi level than metallic single-walled nanotubes (SWNTs). Zigzag nanoscrolls are semiconductors with energy gaps much smaller than corresponding zigzag SWNTs; moreover, they get small gaps at (16,0) and (19,0), while zigzag SWNTs have small gaps at (3n,0).
Carbon nanotubes (CNTs), which were first discovered in 1991,1 have unique mechanical and electronic properties that fascinate many researchers, resulting in great quantities of both experimental and theoretical work. Nowadays people can obtain nanotubes via the laser-vaporization method, the electric-arc discharge technique, or chemical vapor deposition (CVD).2-4 According to the calculation based on the tight-binding zonefolding (TBZF) approximation,5-7 single-walled nanotubes (SWNTs) were predicted to be probably metallic or semiconductive depending on their chiral vectors C Bh ) na b1 + ma b2, which alone can determine the structures of the tubes. This was confirmed by direct measure of densities of states via STM in 1998.8,9 Multiple-walled nanotubes (MWNTs) are also exhaustively studied with most energy devoted to Russian dolls, which consist of nested SWNTs. However, there is another type of MWNTs, Swiss roll, being called nanoscrolls here, which will be our protagonist in this article. Nanoscrolls, much like SWNTs, are rolled up by a single layer of graphene. Unlike SWNTs, which are wound into close cylinders, nanoscrolls are open. They cannot be determined uniquely by chiral vectors. Silent for many years, however, a new route of easy synthesis10 brought the renaissance of nanoscrolls. People found the special topological structure of a nanoscroll may lead to potential use in hydrogen storage and energy storage in supercapacitors or batteries.11 Theoretically, they were first studied via continuum elasticity theory,12-15 which failed to tell us the details of structures at the atomic level. Then Setton performed a molecular mechanics calculation within a pairwise interaction,16 which became the main limitation of the work. Later on, Braga et al.17 directed molecular dynamics simulations to fill us with the knowledge of formation and stability of the scrolls. They found scrolls coil automatically when the critical overlap is achieved, also they found they unwind at the charge injection. Recently, Pan et al.18 gave us the first principle calculations on the electronic structure and optical properties of nanoscrolls. They found two specified * Address correspondence to this author. E-mail:
[email protected] (J.L.);
[email protected] (Y.C.). † Peking University. ‡ University of Nebraska at Omaha.
Figure 1. Illustration for definition of parameters. The curve γ stands for the side view of (24,24)-I. l1 is the normal of γ at point A, crosses γ at B. l2 is the tangent of γ at point A.
armchair nanoscrolls are metal or semimetal within the LDA approximation. The energy bands are nondegenerate and a small energy gap appeared that they attribute to interlayer interaction. However, when the CNSs are more stable than the corresponding unrolled graphene and how the different structure parameters work on the electronic structures of CNSs are questions that are still unanswered. In this paper, we found the minimum radius of stable armchair nanoscrolls and obtained the electronic structure dependence on structural parameters of nanoscrolls via first principle calculation based on density functional theory (DFT) and local density approximation (LDA). The plane-wave basis with ultrasoft pseudopotential proposed by Vanderbilt19 is used for C and H, and CASTEP (CAmbridge Serial Total Energy Package) code20 is also employed. The parametrization for exchange-correlation energy by Perdew and Zunger21 is enlisted
10.1021/jp066030r CCC: $37.00 © 2007 American Chemical Society Published on Web 01/05/2007
1626 J. Phys. Chem. C, Vol. 111, No. 4, 2007
Chen et al. TABLE 1: Structural Parameters for Initial and Optimized Nanoscrolls and Relative Energies Compared with the Energy of the Corresponding Praphene Sheet for Some Nanoscrollsa σ/Å
d/Å
(n,m)
initial
optimized
initial
optimized
E/eV
(10,10) (11,11) (20,20) (22,22) (24,24)-I (24,24)-II (25,25) (28,28) (15,0) (16,0) (17,0) (18,0) (19,0)
12.794 10.582 16.882 25.593 19.579 27.786 20.673 22.729 6.981 8.193 8.173 10.633 11.055
10.361 8.689 16.300 25.259 19.586 27.381 20.269 22.542 5.735 7.002 7.107 8.418 9.805
8.092 9.908 19.079 19.031 23.388 20.814 23.942 27.003 9.302 9.586 10.495 10.495 11.043
8.998 10.716 19.565 19.344 23.404 21.155 23.293 27.090 9.692 10.160 11.209 11.223 11.648
13.378 1.089 0.886 -0.127 0.013 -0.503 -0.731
a d is the innermost diameter of nanoscrolls. σ is the overlap length of nanoscrolls. E is the energy of nanoscrolls relative to the corresponding graphene sheet. (n,m) represents the spiral vector that defines the minimum periodic cell.
Figure 2. Initial and optimized structures of nanoscrolls. Every upper graph is the initial structure of the nanoscroll while the lower one is the optimized structure. The first two are armchair nanoscrolls (10,10) and (24,24)-I, and the following two are zigzag nanoscrolls (17,0) and (18,0).
too. Supercells we used are required to have the distance of the nearest carbon atoms of two separated nanoscrolls (graphene) maintained above 7-8 Å so that the nanoscrolls (graphene) could be thought to be independent. The cutoff energy is 240 eV and the convergence in energy and force is set at 1 × 10 -5 eV and 0.03 eV/Å. We investigated a series of nanoscrolls which are periodic along the axis direction. The structure of these nanoscrolls could be determined by the side-view curves truncated by the planes perpendicular to the axes of the scrolls and the smallest periodical cells of the corresponding unrolled graphene sheets, which can be described uniquely via the chiral vectors. We call nanoscrolls of type (n,n) armchair nanoscrolls and those of type (n,0) zigzag nanoscrolls. The automatic formation of nanoscrolls is due to the competition between adhere energy and elastic energy. Among them, one comes from the Van de Wass interaction while the other comes from the bend of the graphene sheet. As is shown in Figure 1, let γ be the truncated curve, A and C are two ends of γ, while the normal l1 of the curve γ at A intersects γ at B. To implicate adhere energy, we define the
length of BC k to be the overlapped length σ. To depict the bend extent, we employ the average radius (inverse of the curvature) of the innermost layer of nanoscrolls, represented by R. Then, d ) 2R can represent CNSs’ innermost diameter. We studied armchair nanoscrolls (10,10), (20,20), (22,22), (24,24), (25,25), and (28,28) and zigzag nanoscrolls (15,0), (16,0), (17,0), (18,0), and (19,0), whose ends are saturated by H. We labeled them by chiral numbers (n,m) and distinguished isomeric structures by roman numbers. The initial and optimized structures are shown in Figure 2 for some of them. In the initial structures, we fixed R ) 60°, β ) 90°, and γ ) 90°, which are angels between every two primitive vectors. The lengths of two primitive vectors a and b are fixed too, putting them around 28-38 Å, according to the size of the scrolls. We laid the axes of the scrolls along the c direction, and the length of c was optimized and was initially set to be 2.45 or 4.25 Å for armchair scrolls or zigzag scrolls, respectively. We also employed 5 k-points for our geometry optimization. The optimized length of c was changed within 3‰ in each optimized scroll. In the optimized structures, we also found the outermost and innermost carbon-carbon bonds are shrunk to around 1.40 Å and then the next outermost and innermost bonds are elongated to 1.44 Å while the others remained unchanged at 1.42 Å. In Braga’s letter they argued there should be a minimum innermost radius for stable nanoscrolls, which is supposed to be around 10 Å. We noticed that in Pan’s letter, the radius of their nanoscrolls is much smaller than 10 Å. That is too small for scrolls to be really stable. With the length of curve γ fixed, on the condition that the innermost radius of the scroll is less than the minimum of the stable radius, the shrinkage in the radius makes elastic energy’s transcend adhere energy and would raise the energy of the scroll. In Table 1 we have shown every parameter before and after optimizations; ground state energies are given relative to corresponding unrolled graphene sheets. We noticed two facts: first, the relative ground state energy does not go down below zero until the chain length transcends that of (24,24); second, for those with n being less than 24, their radius after optimization obviously increases while for the others, their radius is not obviously changed. Furthermore, we can see more directly that two (24,24) type nanoscrolls with different initial radii achieved different stable points and the nanosrolls with larger radii have a lower ground state energy. We can conclude that the smallest inner diameter of stable CNSs is around 23 Å.
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Figure 3. Energy band structures of armchair CNS (10,10), (11,11), (24,24)-I, and (24,24)-II. The horizontal line characterizes the Fermi level. For smaller CNS (10,10) and (11,11), there is only a tiny pseudogap at X and the two bands are not crossing over each other, thus they are semimetals. For CNS (24,24), which is a metal, the two energy bands do cross each other.
This result confirms the previously reported value obtained from the molecular dynamics simulations17 and is compared with the experimental value of 20 Å22 to 50 Å.23 Energy band structures of (10,10) and (11,11), two configurations of (24,24), and a series of zigzag nanoscrolls, that is, (15,0), (16,0), (17,0), (18,0), and (19,0), are calculated in our work and are given in Figures 3 and 4, respectively. In the calculation of electronic properties for most scrolls, we employed 69 k-points, which can describe the band structure very well. An exception was made for armchair (24,24) and (15,0) with k-points being taken as 49. Compared with SWNTs, straightforward, the degeneration at high symmetry points is disabled. The symmetry of a zigzag or an armchair nanotube is Dn X Ci4, where Dn is Dnd for odd n and Dn is Dnh for even n. So most of nanotubes’ energy bands are doubly degenerate. However, nanoscrolls have a much lower symmetry of Cs, then the nanoscrolls’ energy bands are all nondegenerate. Nanoscrolls of type (24,24) are metallic while (10,10) and (11,11) are semimetallic with a small gap between the highest occupied band and the lowest unoccupied band. This is consistent with Pan’s work. In Pan’s letter, the pseudogap was explained to be the result of interlayer interactions. Comparing the band structures between the two configurations of (24,24), we noticed that they are much alike. This implies that chirality plays a great role in the electronic structure of scrolls. The density of states for armchair nanoscrolls (10,10), (11,11), (24,24)-I, and (24,24)-II is shown in Figure 5. The density of states at the Fermi level is quite a bit larger than those of SWNTs. We know, for metallic SWNTs, the density of states at the Fermi level per unit cell is N(EF) ) 8c/x3πa|t|, despite their diameters and helicity.24 Here c is the length of the unit cell along the axial direction, a is the lattice constant of the grphene layer, and |t| is the nearest-neighbor C-C tight bonding overlap energy. In our paper, we chose t to be 2.5 eV to be consistent with the first principle calculations.25 This means all metallic armchair SWNTs have a density of states of 1 state/ eV‚cell at the Fermi level. While in the case of armchair nanoscrolls, their density of states at the Fermi level is much
higher with 6.5 states/eV‚cell for CNS (10,10), 6.5 states/eV‚ cell for CNS (11,11), and 7.0 states/eV‚cell for CNS (24,24)-I and (24,24)-II. This very fact implies that metallic nanoscrolls are much better conductors than are SWNTs. Zigzag nanoscrolls are semiconductors according to our calculation. There are also small gap semiconductors in zigzag scrolls. Unlike zigzag SWNTs’ getting them at (3n,0), small gap semiconductor scrolls show up at (16,0) and (19,0). The reason for the small gap SWNTs is attributed to the special energy band of graphene and the periodical boundary condition of the TBZF theory purposed by Dressellhaus et al.5 The periodical boundary condition arouses quantization of k-vectors that are perpendicular to the axes, and only when n - m is a multiple of 3 could these quantized k-lines pass through k-points in 2D graphene energy bands, where band gaps are zero. So when the chirality (n,m) satisfies n - m being a multiple of 3, the SWNT will be a small gap semiconductor or metal. However, the result is not strict. For small nantubes, σ-π rehybridiztion26 may break the simple rule. In Figure 6, we present energy gaps for zigzag nanoscrolls and zigzag nanotubes. Data for zigzag nanotubes come from first principle calculations in ref 27. We can see for nanoscrolls that small gap semicoductors are (16,0) and (19,0). We also expect (22,0) to be a small gap semiconductor. Here we try to explain why small gap semiconductors appear at these locations. Nanoscrolls are not wound into cylinders as nanotubes are, so they do not share the periodical boundary condition for SWNTs. We also noticed that there are bond length changes in optimized structures, which can cause band distortion. This distortion can be the result of the boundary effect, interlayer interaction, or σ-π rehybridization. Here we only take part of the effects that the boundary condition brought into consideration. That is, we suppose the boundary condition would not change the energy eigen states very much but only quantize k-vectors in a different manner. Then still we can view nanoscrolls’ energy bands being cut from graphene layer’s π and π* energy bands. Phenomenally, we restrict B k by a phase shift that is required by periodic boundary condition: B kC Bh ) 2πj - θ, j ) 1, 2, ..., n for (n,0). We get
1628 J. Phys. Chem. C, Vol. 111, No. 4, 2007
Chen et al.
Figure 4. Energy band structures for zigzag CNS (15,0), (16,0), (17,0), (18,0), and (19,0). They are all semiconductors, among which CNS (16,0) and (19,0) are small gap semiconductors. This is quite different from those small gap semiconductors of SWNT (3n,0). The band gaps are smaller than those of corresponding carbon nanotubes.
nky,ja ) 2πj - θ and insert it into the graphene’s band dispersive relation equation:28
{
Eg2D(kx,ky) ) (t 1 + 4 cos
( ) ( )
x3kxa kya cos + 2 2
( )}
kya 4 cos 2 2
We then get
{
Ejz ) (t 1 + 4 cos
( )
x3kxa jπ θ cos + 2 n 2n
(
)
1/2
(1)
θ jπ n 2n
(
4 cos2
)}
1/2
(2)
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J. Phys. Chem. C, Vol. 111, No. 4, 2007 1629
Figure 6. Band gap dependency of chiral number (n,m). The square data are energy band gaps for zigzag nanoscrolls, the circle data are band gaps for SWNTs, and the upward triangles and downward triangles are fitting gaps for zigzag CNSs according to our approximations.
gaps of the corresponding SWNTs in Figure 6. Our approximation could give those locations for small gap semiconductors. This approximation, however, predicts that when n is large, the small gap semiconductors will again shift to the type (3n,0). If this is true, it suggests that the quantization condition for B k‚C Bh being only a phase shift from a multiple of π is a reasonable phenomenal approximation. But we can see the energy gaps we fitted were far from the values achieved by our first principle calculation, so this is very rough. As we explained before, the energy band distortion we neglected can be the main contribution for the difference of the two calculations. In conclusion, below 23 Å , the critical value of the innermost diameter, the nanoscrolls are not likely to remain in this configuration naturally. The most related factor to the electronic structure of nanoscrolls is chirality of (n,m). We found armchairtype nanoscrolls to be semimetallic or metallic. Compared with SWNTs, armchair nanoscrolls may have better conductivity for their greater values of density of states at the Fermi level. The energy gaps of zigzag nanoscrolls change quite periodically within our calculation but are quite different from SWNTs. CNSs (16,0) and (19,0) are small gap semiconductors with energy gaps of 0.046 and 0.097 eV, not (15,0) or (18,0). Moreover, zigzag nanoscrolls have smaller energy gaps on average than the corresponding carbon nanotubes.
Figure 5. Density of states for armchair nanoscrolls (10,10), (11,11), and (24,24)-I. The dashed line represents the Fermi level.
The only parameter here is θ, then we can fit θ from one energy gap of a zigzag nanoscroll. We set t ) 2.5 eV.25 Because energy gaps of zigzag nanoscrolls are at Γ points, the energy gap is the minimum of 2t|1 + 2 cos(jπ/n - θ/2n)|. We fit θ by the energy gap of CNS (16,0). With restriction of θ being positive and less than π, θ could be 1.9242 or 2.2641. We present the energy gaps given by these two values of θ with an upward triangle line and a downward triangle line respectively in Figure 6. We also compared our calculated CNSs’ gaps to those energy
Acknowledgment. This work was supported by the NSFC (Grant Nos. 10474123, 10434010, and 20131040), national 973 projects (No. 2002CB613505, MOST of China), 985 Project and Creative Team Project of MOE of China, and Nebraska Research Initiative (No. 4132050400) of the U.S.A. Our calculations were partially carried out in the HP Cluster of the Calculation Center of Science and Engineering of Peking University. References and Notes (1) Ijima, S. Nature (London) 1991, 354, 56. (2) Bethune, D. S.; Kiang, C. H.; Devries, M.; Gorman, G.; Savoy, R.; Vazquez, J.; Beyers, R. Nature (London) 1993, 363, 605. (3) Thess, A.; Lee, R.; Nikolaev, P.; Dai, H. J.; Petit, P.; Robert, J.; Xu, C. H.; Lee, Y. H.; Kim, S. G.; Rinzler, A. G.; Colbert, D. T.; Scuseria, G. E.; Tomanek, D.; Fisher, J. E.; Smalley, R. E. Science 1996, 273, 483.
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