Boron Fullerenes: A First-Principles Study
- Nevill Gonzalez Szwacki^{1}Email author
DOI: 10.1007/s11671-007-9113-1
© to the authors 2007
Received: 11 November 2007
Accepted: 4 December 2007
Published: 15 December 2007
Abstract
A family of unusually stable boron cages was identified and examined using first-principles local-density functional method. The structure of the fullerenes is similar to that of the B_{12}icosahedron and consists of six crossing double-rings. The energetically most stable fullerene is made up of 180 boron atoms. A connection between the fullerene family and its precursors, boron sheets, is made. We show that the most stable boron sheets are not necessarily precursors of very stable boron cages. Our finding is a step forward in the understanding of the structure of the recently produced boron nanotubes.
Keywords
Boron clusters Boron fullerenes and nanotubes Boron sheets Quantum-mechanical modelingIntroduction
The chemistry of boron resembles that of carbon in its ability to catenate and form molecular networks. Unlike carbon, bulk boron cannot be found in nature and all known boron allotropes where obtained in the laboratory. All of them are based on different arrangements of B_{12} icosahedrons. It is very natural to believe that not only carbon but also boron posesses molecular allotropes (fullerenes and nanotubes). Experimental and theoretical research on the chemistry of boron nanomaterials is developing rapidly. The existence of quasi-planar [1] and tubular [2] boron clusters was predicted by theory and confirmed more recently by experiment [3, 4]. Up to now, however, very little is known about the properties of these novel boron nanostructures.
In this work, we will describe the properties of a family of boron nearly round cages which are built from crossing boron double-rings (DRs). The smallest members of the family, B_{12} and B_{80}, where previously studied using first-principles methods [5]. Here, we will show how to construct bigger cages with similar structural characteristics to those found in B_{12} and B_{80}. We will also show the connection between boron cages and nanotubes as well as their precursors—boron sheets.
Method
The calculations were performed within the density functional theory framework, using ultrasoft Vanderbilt pseudopotentials [6] and local-density approximation for the Perdew–Burke–Ernzerhof exchange-correlation potential [7]. Computations were done using the plane-wave-based Quantum-ESPRESSO package [8]. The optimized geometries of the structures were found by allowing the full relaxation of the atoms in the cell until the atomic forces were less than 10^{−3} and 10^{−4} Ry/Bohr for the atomic cages and sheets, respectively. A proper k-point sampling for each system together with a 35 Ry cut-off for the plane-wave basis set have been used to ensure energy convergence to less than 1 meV/atom. To study properties of the fullerenes (nanotubes) the supercell geometry was taken to be a cubic (tetragonal) cell with sufficiently large lattice constant (constants) to avoid interactions between periodic replicas of the cluster. For infinitely long structures the supercell was optimized using variable cell optimization methods included in the program package.
Boron Fullerenes
The increase of the number of atoms by 20 between DRs belonging to two consecutive members of the fullerene family can be explained using the B_{80} and B_{180} cages as follows: each of the DRs in B_{80} is adjacent to 10 pentagons (holes), so if, in order to obtain the B_{180}, we add one atom to each side of the pentagons the number of atoms of every DR increases already by 10. In addition each DR requires still another 10 atoms in order to preserve its structure, so the total number of atoms for every DR increases by 20. We highlighted this in B_{180} in Fig. 1b, where in one of the DRs the additional 20 atoms (respect to a DR in B_{80}) described above were colored in black and white.
The next in size fullerene after B_{180} is made up of 300 atoms (it is built from DRs with 70 atoms). The optimized structure of this cage is shown in the left of Fig. 3c. The total number of atoms in B_{12}, B_{80}, and B_{180} cannot be described by one general formula since the number of atoms shared by the DRs varies from one cage to another. Note in Fig. 1b that the black atoms belong only to one DR, in contrast, the white atoms are shared by two crossing DRs. However, for B_{300}and all larger fullerenes the number of shared atoms is constant and the total number of atoms in the cage can be obtained using a simple formulaN(n) = N(B_{180}) + 120(n − 3), where withn(n ≥ 4) we label the fullerenes starting from the smallest cage. The number of atoms in each of the n-fullerene DRs can be expressed by the formula N _{DR}(n) = N _{DR}(B_{12}) + 20(n − 1), where n ≥ 1.
Point symmetries, cohesive energies, and interatomic distances,d _{BB}, between neighboring boron atoms for fullerenes and sheets
Symmetry | E _{coh}(eV/atom) | d _{BB}(Å) | |
---|---|---|---|
B_{80} | I _{ h } | 5.76 | 1.67–1.73 |
B_{180} | I _{ h } | 5.77 | 1.62–1.97 |
B_{300} | I _{ h } | 5.76 | 1.57–1.91 |
α | D _{6h } | 5.94 | d _{1} = 1.68,d _{2} = 1.69 |
β | D _{3h } | 5.80 | 1.62–2.02 |
γ | D _{3h } | 5.77 | 1.62–1.94 |
Flat TS | D _{6h } | 5.62 | 1.71 |
Buckled TS | C _{2v } | 5.85 | d _{1} = 1.62,d _{2} = 1.86,h = 0.87 |
Strip | D _{2h } | 5.69 | d _{1} = 1.61,d _{2} = 1.68 |
The structure of the cages influences also their electronic properties. For B_{180} the highest occupied molecular orbital (HOMO) and lowest unoccupied molecular orbital (LUMO) are triply degenerate. (Similar result was reported previously for B_{80} [5].) For B_{300} the triple degeneracy both of HOMO and LUMO is slightly lifted (by less than 95 meV). This is most probably the result of the structural distortions mentioned above. The HOMO–LUMO energy gaps are 0.43 and 0.10 eV for B_{180} and B_{300}, respectively. We have also calculated the electronic properties of the boron strip which was found to be metallic.
Boron Sheets
Our investigation of possible candidates for the BS we restricted to those which have structural similarities with the fullerenes studied above. In Fig. 3, we have shown three cages, B_{80}, B_{180}, and B_{300}, and their corresponding sheets α β, and γ, respectively. In the top of Fig. 3a–c, we have highlighted on each cage the characteristic atomic motif of the fullerene which also will appear on the sheet corresponding to it (see the sheets in the bottom of Fig. 3a, b and the sheet in the right of Fig. 3c). Let us forget for a moment about the holes in the cages, then the characteristic motif for the B_{80} is a cluster of 7 atoms with one central atom lying almost in plain defined by a hexagonal chain of 6 atoms. This cluster has C _{3v } symmetry. An isolated neutral cluster made up of 7 atoms has C _{2v } symmetry [9]. The next two cages, B_{180} and B_{300}, have motifs which are similar in shape and consist of quasi-planar structures of 12 and 18 atoms, respectively, and C _{3v } symmetry. These clusters have 9 and 12 peripheral atoms and 3 and 6 central atoms in B_{180} and B_{300} cages, respectively. The interatomic distances between neighboring boron atoms in the fragment of B_{180} are shown in Fig. 1c. As it was mentioned in the previous section the isolated neutral cluster of 12 atoms is a quasi-planar convex structure of C _{3v } symmetry. It was experimentally determined that this cluster has unusually large HOMO–LUMO gap of 2 eV [3]. It was also suggested that this cluster must be extremely stable electrically and should be chemically inert [9]. Perhaps these unusual characteristics are responsible also for the outstanding stability of the B_{180} fullerene.
All BSs studied in this work are fully planar and have metallic properties. We have found that the α-BS which is a precursor of the B_{80} cage is the most stable sheet over all studied (in agreement with recent findings [16]). The point symmetry of that sheet is D _{6h }. Figure 3a (bottom) shows the unit cell used for calculations of the α-BS. The BSs β and γ, corresponding to B_{180} and B_{300} fullerenes, respectively, have D _{3h } symmetry. The shape of the holes in BSs will be determined by the type of atomic motif we are using to build the sheet. In the case of the α-BS the holes are hexagons and in the case of β and γ BSs the holes are distorted dodecagons and hexagons-like, respectively. It is important to observe that the BSs shown in Fig. 3 can also be seen as built from interwoven boron stripes. This observation may help to understand not only structural but also electronic properties of BSs.
TheE _{coh}for theα-BS is 5.94 eV/atom and is bigger than theE _{coh}for theβ andγ BSs by 0.14 and 0.17 eV/atom, respectively. This is an interesting result since it means that the most stable structure for the BS does not necessarily have to be a precursor of very stable (in energy) boron cages.
Although there are some discrepancies between the E _{coh} values obtained in this work (see Table 1) and reported in the literature, the differences between E _{coh} values corresponding to different sheets match very well. Indeed the E _{coh} for the buckled TS is higher in energy than E _{coh} for the flat TS by 0.24 eV/atom and this value is close to 0.21 and 0.22 eV/atom reported in Refs. [16, 12], respectively. Similarly, E _{coh} for the sheet α is higher in energy than E _{coh} for the buckled sheet by 0.08 eV/atom, what represents slightly smaller value than 0.11 eV/atom obtained in previous calculations [16].
Fullerene-derived Nanotubes
Carbon nanotubes and fullerenes are closely related structures. Capped nanotubes are elongated fullerene-like cylindrical tubes which are closed at the rounded ends [2]. To look for similar connections between boron nanostructures we have investigated boron nanotubes derived from the B_{80} cages.
The shortest nanotube is, of course, the B_{80} cage and its E _{coh} is 5.76 eV/atom [5]. We have also optimized the structures of two longer finite tubes – the B_{160} (see Fig. 4b) and the previously described B_{240} (see Fig. 4c). As expected the stability of the nanostructures increases with increasing lengths of the tube: E _{coh}(B_{160}) = 5.81 eV/atom and E _{coh}(B_{240}) = 5.84 eV/atom. The most stable is the infinite nanotube (see Fig. 4d) with E _{coh} = 5.87 eV/atom. The HOMO–LUMO energy gaps for the B_{160} and B_{240} clusters are 0.33 and 0.02 eV, respectively, and the infinite tube was found to be metallic.
Summary
We are predicting the existence of a family of very stable boron fullerenes. The cages have similar structure consisting of six interwoven boron DRs. The most stable fullerene is made up of 180 atoms and has almost perfect spherical shape. A recently proposed very stable BS of triangular and hexagonal motifs is a precursor of the B_{80}cage. However, it was shown that the most stable sheets are not necessarily the precursors of very stable boron cages. Finally, we have shown that the proposed fullerenes and novel boron nanotubes are closely related structures.
Declarations
Acknowledgment
The Interdisciplinary Centre for Mathematical and Computational Modelling of Warsaw University is thanked for a generous amount of CPU time.
Authors’ Affiliations
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