Article pubs.acs.org/JPCA
Geometry Dependence of Electronic and Energetic Properties of One-Dimensional Peanut-Shaped Fullerene Polymers Yusuke Noda, Shota Ono, and Kaoru Ohno* Department of Physics, Graduate School of Engineering, Yokohama National University, 79-5 Tokiwadai, Yokohama 240-8501, Japan S Supporting Information *
ABSTRACT: In the present study, we investigate different types of 1D peanut-shaped fullerene polymers (PSFPs) using density functional theory to understand the electronic states and the energetic stability of curved carbon nanomaterials. We generated 53 different models of the 1D PSFPs by means of the generalized Stone−Wales transformations and performed structural optimization for each model. Band structures of the 1D PSFPs exhibit either metallic or semiconducting property according to the geometrical structures. We find that the energetic stability of the 1D PSFPs depends on the geometry: the more octagon and pentagon−octagon pairs (heptagons and hexagon−heptagon pairs) in their geometrical structures, the more stable (unstable) the 1D PSFPs. changes in the electronic structure of monolayer graphene.13−15 It transforms the four-ring cluster into a different carbon ring cluster: for example, a cluster composed of four hexagons is transformed into a cluster composed of two pentagons and two heptagons by a single bond rotation. A single vacancy defect also has been considered; however, the activation energy of a single vacancy defect on graphene is ∼ 20 eV. The activation energy of the Stone−Wales defect on graphene is about 5−7 eV.16 It indicates that careful experimental techniques would enable one to generate desired carbon materials through GSW transformations. It has been theoretically suggested that such a defect created by the GSW transformation produces a defect band above the Fermi level, an opening of a band gap, and a shift of the Dirac cone.17 The concentration of the defect is an important quantity to dominate the electronic properties. Such a GSW defect was found to markedly affect conducting electrons in CNTs. It has been reported that the defect produces quasi-bound states and thus reduces the quantized conductance,18 enhances the reactivity compared with the pristine CNTs,19 and opens a band gap in metallic zigzag CNTs.20 The GSW transformation has been used for studying the microscopic fusion mechanism of two C60s,21 and the 1D PSFPs can be created by several times of the GSW transformation between C60s. We start from the T3 model (a semiconducting PSFP with energy gap ∼1.17 eV) given in ref 22, apply the GSW transformations several times to the model, and create 53 different 1D PSFPs. We study the correlation between their geometrical structures and the total energy or electronic structures of the resulting 1D PSFPs. We find that the energy band gap decreases or vanishes by the GSW transformation. Through an analysis of the total energy, we find that the stability of the 1D PSFPs with the GSW defect depends on the
I. INTRODUCTION Not only eminent carbon allotropes with sp2 bonds such as fullerenes (C60s),1 carbon nanotubes (CNTs),2 and graphene,3 but also many types of curved carbon nanomaterials have attracted much interest in recent years because of their unique physical properties. Mackay and Terrones proposed a carbon crystal structure in 1991, which is often called the Mackay crystal.4 It includes only hexagons and octagons in the curved surface with a negative Gaussian curvature. Onoe et al. have synthesized carbon polymers, which are generated by electronbeam irradiation to C60 films and called 1D peanut-shaped C60 polymers (PSFPs).5 The 1D PSFPs have positive and negative Gaussian curvatures and have shown interesting phenomena such as Tomonaga−Luttinger liquid states6 and anomalous electron transport properties.7,8 Although they have used the infrared (IR) spectroscopy,9−11 their geometrical structure has not been clarified so far. Quite recently, we have studied the electronic band structures of several 1D PSFPs and corrugated graphene systems with pentagons, hexagons, heptagons, and octagons using density functional theory (DFT).12 An important finding of this study is that the presence of an octagon adjoined to two pentagons in a supercell brings about the simultaneous occurrence of flat and dispersive bands at the Fermi level, quite similar to the band structure of precious metals. Although the study on the electronic states and the energetic stability of such curved carbon nanomaterials with the trivalent (sp2 bonding) structure and the negative Gaussian curvature is very important, there is no systematic investigation regarding such curved carbon nanomaterials. The main purpose of this study is to give a systematic study on the electronic states and the energetic stability of such curved carbon nanomaterials. The negatively curved surface consists of heptagons and octagons in terms of atomic structures, so we use a generalized Stone−Wales (GSW) transformation, a 90° rotation of a carbon−carbon bond, for generating heptagons and octagons in this study. The GSW transformation gives rise to significant © 2015 American Chemical Society
Received: December 15, 2014 Revised: March 4, 2015 Published: March 4, 2015 3048
DOI: 10.1021/jp512451u J. Phys. Chem. A 2015, 119, 3048−3055
Article
The Journal of Physical Chemistry A
indicating that there are only sp2 bonds. (There is no twocoordinated sp bond and no four-coordinated sp3 bond.) Figure 2 shows all of these optimized structures. In the 1D PSFPs, there are not only pentagons and hexagons but also heptagons a or octagons generated by GSW transformations. We name these 1D PSFPs FPnX (n = 1−6 and X = A, B, C, ...), where n stands for the number of GSW trials and X stands for a symbol to classify these models. (See Figure 2.) These models include heptagons or octagons as well as pentagons and hexagons in the waist part, resulting in the generation of the curved geometry. They can be roughly divided into two types: a 1D PSFP model with a fully waist part and one with a partly waist part. In the former, there is only a concave surface on the fully waist part like FP4K, FP5N, and FP6L. In the latter, there is not only a concave surface but also a flat surface like CNTs on the partly waist part like FP3A, FP4A, and FP5B. The appearance of the partly waist part is related to how many hexagons are densely located in the part. As we know, CNTs have only hexagons to form a cylindrical structure. Therefore, hexagons prefer to form a CNT surface because the geometry is similar to that of the pristine graphene. Hexagons adjacent to each other in the 1D PSFPs form a simple cylindrical surface the same as CNTs. However, in contrast with the stability of hexagons forming a cylindrical CNT surface, instabilities of such a simple cylindrical form occur when the heptagons and octagons exist. Table 1 lists the electronic and geometrical properties of all 1D PSFPs considered here. The point group symmetry, P, lattice parameter along the tube axis, L, total energy per unit cell, Etot, energy band gap, Eg, and the number of faces, Fs, and edges (carbon−carbon bonds), Est (s,t = 5−8), are listed in this Table. The results of the 1D PSFPs are arranged in order of increasing total energy. As listed, the 1D PSFPs exhibit D5d, D5h, C2h, C2, Cs, and C1 symmetry. The lattice parameter is about L = 8 Å, which is slightly longer than the diameter of the C60 molecule (∼7 Å). From this Table and Figure 2, one may see that almost all of the 1D PSFPs with a partly waist part are relatively unstable in our calculations. Later, we discuss the other properties in more detail. FP4K, FP3F, FP4I, FP5F, FP5J, and FP6C have isolated octagons with two pentagons in the unit cell and exhibit flat and dispersive bands at the Fermi level, whose geometrical and electronic are the same as Figure 2 and Figure S1a−e (Supporting Information) of our previous paper,12 respectively. a. Electronic Band Structures. Figure 3 shows the band structures of the 54 1D PSFPs including original T3. T3 and FP5N have nondegenerate and doubly degenerate band dispersions because of their high symmetry (D5d and D5h, respectively). The other models have only nondegenerate bands. All panels in Figure 3 have different band structures due to their different geometrical structures. However, similar characteristics of band dispersions appear around the Fermi levels. For valence bands, there are three nearly flat bands at −1.0 eV. This characteristic seems to be independent of their geometrical structures as long as in our calculations. A widely dispersive band appears in the vicinity of the Fermi level in each band structure. The bandwidth is >1.0 eV. In some cases, the widely dispersive band is divided into two or three parts because of their low symmetry. For conduction bands, the band dispersions are more complex. There are a few flat and dispersive bands, and they sometimes come across each other. As listed in Table 1, the maximum of Eg is 1.17 eV in the T3 model (see Figure 1a), from which we have generated all
number of heptagons and octagons. In Section II, we explain how to make 1D PSFP models and calculate their electronic states from first-principles. Then, in Section III, we show the results of our calculations and give some discussion. Finally, we summarize this paper in Section IV.
II. METHODOLOGY We performed first-principles calculations using Vienna ab initio simulation package (VASP)23 based on DFT.24 We used an exchange-correlation functional of generalized gradient approximation (GGA) proposed by Perdew, Burke, and Ernzerhof25 and projector augmented wave (PAW) method26 with a plane-wave cutoff energy of 500 eV. The 8×1×1 Monkhorst−Pack k-point grid was used for geometrical optimizations of the 1D PSFPs. We set the periodic unit cell of rectangular cuboid shape with L×H×H Å3 (L is lattice parameter of the primitive PSFP region and H = 15.000 Å is fixed) because we have to set vacuum space in the directions that are perpendicular to the PSFP region. All of the lattice parameters are also optimized with fixed H. All of the 1D PSFP models were optimized until atomic forces are 2.0 eV. Interestingly, FP5N has no heptagons in the waist part. Instead, the waist part is covered with five octagons. The total energy of 1D PSFPs seems to be independent of the lattice parameter, L, and the point group symmetry, P. To investigate the correlation between the value of Etot and the number of polygons, we show the tree diagram of the 1D PSFPs in Figure 4: the value of Etot is indicated by triangle, circle, and square
ρ (x , y ) =
∑i (xi − x ̅ )(yi − y ̅ ) {∑i (xi − x ̅ )2 }{∑i (yi − y ̅ )2 }
where x ̅ and y ̅ stand for the means of two variables xi and yi, respectively. In general, values between 0 and 0.3 (0 and −0.3), values between 0.3 and 0.7 (−0.3 and −0.7), and values between 0.7 and 1.0 (−0.7 and −1.0) indicate a weak, moderate, and strong positive (negative) linear relationship, respectively. For the number of s-sided polygons, Fs, ρ(Etot, F7) is 0.67, indicating that Etot increases with increasing F7, whereas ρ(Etot, F8) is −0.73, indicating that Etot decreases with increasing F8. They show a strong dependence of energetic stability (instability) on the number of octagons (heptagons) in the 1D PSFPs. However, ρ(Etot, F5) and ρ(Etot, F6) are −0.36 and 0.07, respectively. They indicate that there is no relation between the total energy and the number of pentagons or the number of hexagons. Moreover, we found that ρ(Etot, E67) is 0.71 and ρ(Etot, E58). There is a strong dependence in the energetic stability (instability) on the number of pentagon− octagon (hexagon-heptagon) pairs in the 1D PSFPs. Incidentally, ρ(Etot, L) is 0.01, and thus there is no relation between the total energy and the lattice parameter of the 1D PSFPs. In summary, heptagons and hexagon−heptagon pairs give rise to instability, and octagons and pentagon−octagon pairs give rise to stability. Finally, we further comment on the correlation between the geometry of the waist part and the total energy. For simple explanation, we consider the FP5A model shown in Figure 2. The outline along the tube axis in the side view of the FP5A model shows both straight line (upper side) and wiggling line (lower side). In the upper side, the surface geometry of the FP5A consists of only hexagons, which is similar to that of
Figure 4. Tree diagram of the 1D PSFPs. The value of the vertical axis is the total energy, Etot, listed in Table 1. Orange triangles, blue circles, and green squares stand for the models with the following conditions: F7 > F8, F7 = F8, and F7 < F8, respectively. Each model can change the geometrical structure through GSW transformations according to black solid lines.
when F7 > F8, F7 = F8, and F7 < F8, respectively. At n ≤ 4, almost all of the models are less stable than T3, and there are more heptagons than octagons in the models (triangles in Figure 4). It means that a few of GSW rotations lead to the instability of the 1D PSFPs because of the large number of heptagons. In the case of n = 5 and 6, some of the 1D PSFP models are likely to become more stable than the models with n ≤ 4 and there are more octagons than heptagons (squares in
Table 2. CCs of Total Energy, Etot, and the Number of Faces, Fs, or the Number of Edges, Est CC
F5
F6
F7
F8
E56
E57
E58
E66
E67
E68
E77
E78
E88
−0.36
0.07
0.67
−0.73
−0.31
0.52
−0.76
0.04
0.71
−0.60
0.43
−0.19
−0.60
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DOI: 10.1021/jp512451u J. Phys. Chem. A 2015, 119, 3048−3055
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The Journal of Physical Chemistry A
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CNTs, relatively stable nanocarbons. Although the presence of such a surface covered with hexagons may stabilize the FP5A, the corrugated surface in the lower side destabilizes it. This is because in the lower side heptagons are strongly distorted. A similar relation can be found in FP3A, FP4A, and FP6A. Thus, the energetic stability in the 1D PSFPs is determined by the geometry of the surface as well as the number of octagons previously mentioned.
IV. CONCLUSIONS Many of the 1D PSFP models obtained from the polymerization of C60 molecules are investigated systematically using a first-principles calculation based on DFT with GGA functional and PAW method with plane-wave basis set. Both geometrical structures and periodic unit cells of the 1D PSFPs are optimized. We confirm the transition of electronic states of the 1D PSFPs through GSW transformations owing to band structure calculations. There is no dependence of the transition and band gaps of the 1D PSFPs on the number of GSW rotations. It is found that the electronic properties and the stability of the 1D PSFPs are dependent on the patterns of polymerization from our statistical analysis. The structures with octagons and pentagon−octagon pairs are energetically favorable, and FP5N is the most stable energetically because it has five octagons in the unit cell. Heptagon and hexagon− heptagon pairs lead to the instability of the 1D PSFPs. While it is not easy to experimentally clarify the geometrical structure of the 1D PSFP, our discussion presented here would be used for the material design of defect-induced carbon nanomaterials.
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ASSOCIATED CONTENT
S Supporting Information *
Atomic coordinates of all 1D PSFPs and the strategy of generating 1D PSFPs, which are newly generated in this study. This material is available free of charge via the Internet at http://pubs.acs.org.
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AUTHOR INFORMATION
Notes
The authors declare no competing financial interest.
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ACKNOWLEDGMENTS We thank J. Onoe for helpful discussions. This study was supported by Grant-in-Aids for Scientific Research (B) (grant no. 25289218) and a Japan Society for the Promotion of Science (JSPS) Fellowship from the JSPS, and a Grant-in-Aid for Scientific Research on Innovative Areas (grant no. 25104713) from The Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.
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REFERENCES
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DOI: 10.1021/jp512451u J. Phys. Chem. A 2015, 119, 3048−3055
