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# Symmetry Properties of Single-Walled BC_{2}N Nanotubes

*Nanoscale Research Letters*
**volume 4**, Article number: 498 (2009)

## Abstract

The symmetry properties of the single-walled 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 infrared-active phonon modes for all kinds of BC_{2}N nanotubes.

## Introduction

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 first-principles and tight-binding methods, respectively [3–6].

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 infrared-active vibrations in the single-walled 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 first-principles 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 single-walled BC_{2}N nanotube can be completely specified by the chiral vector which is given in terms of a pair of integers (*n* *m*) [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 *ZZ*-1 and *ZZ*-2, respectively, and two armchair nanotubes obtained by rolling up the BC_{2}N sheet along the R_{1} and R_{2} directions as AC-1 and AC-2, 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.

## 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 (*n* *n*). The nonsymmorphic line-group [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 (*ZZ*-1) (Fig. 1b) is shown in Fig. 2. They have vertical symmetry planes as indicated by**g**. In this case, the*D*_{
nh
}and*D*_{
nd
}point groups reduce to*C*_{
nv
}due to the lack of horizontal symmetry axis/plane, and*S*_{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 (*N* is the number of unit cells in the tube and*N* = *n* for ZZ-1 BC_{2}N nanotubes) of phonons in*ZZ*-1 BC_{2}N nanotubes and the number of Raman- or IR-active modes, we have to associate them with the irreducible representations (irrep’s) of*C*_{
nv
}. Here, two cases need to be considered.

### Case 1

*n* is odd (or*n* = 2*m* + 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 *ZZ*-1 BC_{2}N nanotubes. is the vector representation. Of these modes, the ones that transform according to (the tensor representation) or are Raman- or IR-active, 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

*n* is even (or*n* = 2*m*,*m* is 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 IR-active, 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 *ZZ*-1 BC_{2}N nanotubes irrespective *n*.

The armchair (*n*,*n*) BC_{2}N nanotubes (AC-1) (Fig. 1d), corresponding to the BC_{2}N sheet shown in Fig. 3, have horizontal planes as indicated by**g**. The*D*_{
nh
}and*D*_{
nd
}point groups reduce to*C*_{
nh
}owing to the lack of*C*_{2}axes and*S*_{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 AC-1 BC_{2}N nanotubes and the number of Raman- or IR-active modes, two cases need consideration, by associating them with the irrep’s of C_{
nh
}.

### Case 1

*n* is odd (*n* = 2 m + 1)

The character table of *C*_{(2m+1)h} possesses 4*m* + 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 IR-active, respectively. Out of the 12 N modes, four (which transform as and ) have vanishing frequencies [19].

### Case 2

*n* is even(*n* = 2*m*)

The character table of *C*_{2mh} possesses 4*m* 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 IR-active, 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 AC-1 BC_{2}N nanotubes in irrespective of n. The numbers of Raman- and IR- active phonon modes for *ZZ*-1 BC_{2}N nanotubes are almost twice as for AC-1 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; where*d*_{
R
}is the greatest common divisor of and ; *d* is the greatest common divisor of and ; S_{
N/d
}and S_{
N
}are the screw-axis operations with the orders of*N/d* and*N*, 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 (*n* *m*) 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 *ZZ*-2 and *AC*-2 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 *ZZ*-2 and *AC*-2 BC_{2}N nanotubes, *N* = 4*n*. 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/IR-active modes can be observed. The observable Raman-active 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].

## 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 *ZZ*-1 BC_{2}N nanotubes have 30 Raman- and 18 IR- active phonon modes; all *AC*-1 BC_{2}N nanotubes have 19 Raman- and 10 IR-active phonon modes; all *ZZ*-2, *AC*-2, and other chiral BC_{2}N nanotubes have 33 Raman- and 21 IR-active phonon modes. It is noticed that the numbers of Raman- and IR- active phonon modes in *ZZ*-1 BC_{2}N nanotubes are almost twice as in *AC*-1 BC_{2}N nanotubes, but which is almost the same as those in chiral, *ZZ*-2, and *AC*-2 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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Pan, H., Feng, Y. & Lin, J. Symmetry Properties of Single-Walled BC_{2}N Nanotubes.
*Nanoscale Res Lett* **4, **498 (2009) doi:10.1007/s11671-009-9272-3

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### Keywords

- BC
_{2}N nanotubes - Symmetry
- Group theory