Abstract
The symmetry properties of the singlewalled BC_{2}N nanotubes were investigated. All the BC_{2}N nanotubes possess nonsymmorphic line groups. In contrast with the carbon and boron nitride nanotubes, armchair and zigzag BC_{2}N nanotubes belong to different line groups, depending on the index n (even or odd) and the vector chosen. The number of Raman active phonon modes is almost twice that of the infraredactive phonon modes for all kinds of BC_{2}N nanotubes.
Keywords:
BC_{2}N nanotubes; Symmetry; Group theoryIntroduction
Carbon nanotubes have been extensively studied because of their interesting physical properties and potential applications. Motivated by this success, scientists have been exploring nanotubes and nanostructures made of different materials. In particular, boron carbon nitride (B_{x}C_{y}N_{z}) nanotubes have been synthesized [1,2]. Theoretical studies have also been carried out to investigate the electronic, optical and elastic properties of BC_{2}N nanotubes using the firstprinciples and tightbinding methods, respectively [36].
Besides the elastic and electronic properties, theoretical and experimental research on phonon properties of BC_{2}N nanotubes is also useful in understanding the properties of the nanotubes. For example, the electron–phonon interaction is expected to play crucial roles in normal and superconducting transition. Furthermore, symmetry properties of nanotubes have profound implications on their physical properties, such as photogalvanic effects in boron nitride nanotubes [7]. Studies on the symmetry properties of carbon nanotubes predicted the Raman and infraredactive vibrations in the singlewalled carbon nanotubes [8], which are consistent with the experimental data [9] and theoretical calculations [10]. A similar work was carried out by Alon on boron nitride nanotubes [11], and the results were later confirmed by firstprinciples calculations [12]. And the symmetry of BC_{2}N nanotube was reported [13]. The purpose of this study is to extend the symmetry analysis to BC_{2}N nanotubes and to determine their line groups. The vibrational spectra of BC_{2}N nanotubes are predicted based on the symmetry. The number of Raman and infrared (IR)active vibrations of the BC_{2}N nanotubes is determined accordingly.
Structures of BC_{2}N Nanotubes
Similar to carbon or boron nitride nanotubes [14,15], a singlewalled BC_{2}N nanotube can be completely specified by the chiral vector which is given in terms of a pair of integers (nm) [3]. However, compared to a carbon and boron nitride nanotubes, different BC_{2}N nanotubes can be obtained by rolling up a BC_{2}N sheet along different directions, as shown in Fig. 1a, because of the anisotropic geometry of the BC_{2}N sheet. If we follow the notations for carbon nanotubes [14], at least two types of zigzag BC_{2}N nanotubes and two types of armchair nanotubes can be obtained [6]. For convenience, we refer the two zigzag nanotubes obtained by rolling up the BC_{2}N sheet along the a_{1} and the a_{2} directions as ZZ1 and ZZ2, respectively, and two armchair nanotubes obtained by rolling up the BC_{2}N sheet along the R_{1} and R_{2} directions as AC1 and AC2, respectively. The corresponding transactional lattice vectors along the tube axes are T_{a1}, T_{a2}, T_{R1}, and T_{R2}, respectively, as shown in Fig. 1a. It is noted that T_{a2} is parallel to R_{2}, T_{R1} to b_{1}, and T_{R2} to a_{2}. An example of each type of BC_{2}N nanotubes is given in Fig. 1b–f.
Figure 1. Atomic configuration of an isolated BC_{2}N sheet. Primitive and translational vectors are indicated
Symmetry of BC_{2}N Nanotubes
We first consider the achiral carbon nanotubes with the rotation axis of order n, i.e., zigzag (n, 0) or armchair (nn). The nonsymmorphic linegroup [16] describing such achiral carbon nanotubes can be decomposed in the following way [17]:
where is the 1D translation group with the primitive translation T_{z} = T_{z}, and E is the identity operation. The screw axis involves the smallest nonprimitive translation and rotation [11].
The corresponding BC_{2}N sheet of the zigzag (n, 0) BC_{2}N nanotubes (ZZ1) (Fig. 1b) is shown in Fig. 2. They have vertical symmetry planes as indicated byg. In this case, theD_{nh}andD_{nd}point groups reduce toC_{nv}due to the lack of horizontal symmetry axis/plane, andS_{2n}vanishes for the lack of the screw axis. Thus,
The point group of the line group is readily obtained from Eq. 2,
To determine the symmetries at the Γ point of the 12 N (Nis the number of unit cells in the tube andN = nfor ZZ1 BC_{2}N nanotubes) of phonons inZZ1 BC_{2}N nanotubes and the number of Raman or IRactive modes, we have to associate them with the irreducible representations (irrep’s) ofC_{nv}. Here, two cases need to be considered.
Figure 2. 2D projections of zigzag BC2N nanotubes (ZZ1).zis a glide plane
Case 1
nis odd (orn = 2m + 1, m is an integer)
The character table of C_{(2}_{m+1)v} possesses m + 2 irrep’s [18], i.e.,
The 12 N phonon modes transform according to the following irrep’s:
where
stands for the reducible representation of the atom positions inside the unit cell. The prefactor of 4 in reflects the four equivalent and disjoint sublattices made by the four atoms in the ZZ1 BC_{2}N nanotubes. is the vector representation. Of these modes, the ones that transform according to (the tensor representation) or are Raman or IRactive, respectively. Out of the 12 N modes, four have vanishing frequencies [19], which transform as and corresponding to the three translational degrees of freedom giving rise to null vibrations of zero frequencies, and one rotational degree about the tube’s own axis, respectively.
Case 2
nis even (orn = 2m,mis an integer)
The character table of C_{2mv} possesses m + 3 irrep’s [18], i.e.,
The 12 N phonon modes transform according to the following irrep’s:
where
is the vector representation. Of these modes, the ones that transform according to (the tensor representation) or are Raman or IRactive, respectively. Out of the 12 N modes, four (which transform as and ) have vanishing frequencies [16].
The numbers of Raman and IR active modes are 30 and 18, respectively, for ZZ1 BC_{2}N nanotubes irrespective n.
The armchair (n,n) BC_{2}N nanotubes (AC1) (Fig. 1d), corresponding to the BC_{2}N sheet shown in Fig. 3, have horizontal planes as indicated byg. TheD_{nh}andD_{nd}point groups reduce toC_{nh}owing to the lack ofC_{2}axes andS_{2n}vanishes for the lack of the screw axis.
The point group of the line group is readily obtained from Eq. 2,
To determine the symmetries (at the Γ point) of the 12 N (N = n) phonons in AC1 BC_{2}N nanotubes and the number of Raman or IRactive modes, two cases need consideration, by associating them with the irrep’s of C_{nh}.
Figure 3. 2D projections of armchair BC2N nanotubes (AC1).zis a glide plane
Case 1
nis odd (n = 2 m + 1)
The character table of C_{(2m+1)h} possesses 4m + 2 irrep’s [18], i.e.,
The 12 N phonon modes transform according to the following irrep’s:
where
and is the vector representation. Of these modes, the ones that transform according to (the tensor representation) or are Raman or IRactive, respectively. Out of the 12 N modes, four (which transform as and ) have vanishing frequencies [19].
Case 2
nis even(n = 2m)
The character table of C_{2mh} possesses 4m irrep’s [18], i.e.,
The 12 N phonon modes transform according to the following irrep’s:
where
is the vector representation. Of these modes, the ones that transform according to (the tensor representation) or are Raman or IRactive, respectively. Out of the 12 N modes, four (which transform as and ) have vanishing frequencies [19].
The numbers of Raman and IR active modes are 19 and 10, respectively, for AC1 BC_{2}N nanotubes in irrespective of n. The numbers of Raman and IR active phonon modes for ZZ1 BC_{2}N nanotubes are almost twice as for AC1 BC_{2}N nanotubes, which is similar to boron nitride nanotubes [11].
The nonsymmorphic line group describing the ()chiral carbon nanotubes can be decomposed as follows:
where; whered_{R}is the greatest common divisor of and ; dis the greatest common divisor of and ; S_{N/d}and S_{N}are the screwaxis operations with the orders ofN/dandN, respectively. The point group of the line group is obtained from Eq. 26,
where and are the rotations embedded in S_{N/d}and S_{N}, respectively.
For chiral (nm) BC_{2}N nanotubes, the point group D_{N} reduces to C_{N} due for the lack of C_{2} axes. Here, , where d_{R} is the greatest common divisor of and ; d is the greatest common divisor of and . The BC_{2}N sheets corresponding to ZZ2 and AC2 are shown in Fig. 4a and b, which are chiral in nature. The σ_{v} and σ_{h} vanish in Fig. 4a and b, respectively, for ZZ2 and AC2 BC_{2}N nanotubes, N = 4n. The point group corresponding to the two models is expressed as:
The character table of C_{N} has N irrep’s, i.e.,
The 12 N phonon modes transform according to the following irrep’s:
where and . Of these modes, the ones that transform according to and/or are Raman and/or IR active, respectively. Out of the 24 N modes, four (which transform as and ) have vanishing frequencies [19].
Experimentally, only several Raman/IRactive modes can be observed. The observable Ramanactive modes are with the range of 0–2000 cm^{−1}. The E_{2g} mode around 1580 cm^{−1} is related to the stretching mode of C–C bond. The E_{2g} mode around 1370 cm^{−1} is attributed to B–N vibrational mode [20,21]. The experimental Raman spectra between 100 and 300 cm^{−1} should be attributed to E_{1g} and A_{1g} modes [22].
Figure 4. 2D projections of BC2N nanotubesaZZ2 andbAC2.zis a glide plane
Conclusions
In summary, the symmetry properties of BC_{2}N nanotubes were discussed based on line group. All BC_{2}N nanotubes possess nonsymmorphic line groups, just like carbon nanotubes [8] and boron nitride nanotubes [11]. Contrary to carbon and boron nitride nanotubes, armchair and zigzag BC_{2}N nanotubes belong to different line groups, depending on the index n (even or odd) and the vector chosen. By utilizing the symmetries of the factor groups of the line groups, it was found that all ZZ1 BC_{2}N nanotubes have 30 Raman and 18 IR active phonon modes; all AC1 BC_{2}N nanotubes have 19 Raman and 10 IRactive phonon modes; all ZZ2, AC2, and other chiral BC_{2}N nanotubes have 33 Raman and 21 IRactive phonon modes. It is noticed that the numbers of Raman and IR active phonon modes in ZZ1 BC_{2}N nanotubes are almost twice as in AC1 BC_{2}N nanotubes, but which is almost the same as those in chiral, ZZ2, and AC2 BC_{2}N nanotubes. The situation in BC_{2}N nanotubes is different from that in carbon or boron nitride nanotubes [8,11].
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