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 BC2N 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 BC2N 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 BC2N nanotube was reported [13]. The purpose of this study is to extend the symmetry analysis to BC2N nanotubes and to determine their line groups. The vibrational spectra of BC2N nanotubes are predicted based on the symmetry. The number of Raman- and infrared (IR)-active vibrations of the BC2N nanotubes is determined accordingly.

Structures of BC2N Nanotubes

Similar to carbon or boron nitride nanotubes [14, 15], a single-walled BC2N 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 BC2N nanotubes can be obtained by rolling up a BC2N sheet along different directions, as shown in Fig. 1a, because of the anisotropic geometry of the BC2N sheet. If we follow the notations for carbon nanotubes [14], at least two types of zigzag BC2N nanotubes and two types of armchair nanotubes can be obtained [6]. For convenience, we refer the two zigzag nanotubes obtained by rolling up the BC2N sheet along the a1 and the a2 directions as ZZ-1 and ZZ-2, respectively, and two armchair nanotubes obtained by rolling up the BC2N sheet along the R1 and R2 directions as AC-1 and AC-2, respectively. The corresponding transactional lattice vectors along the tube axes are Ta1, Ta2, TR1, and TR2, respectively, as shown in Fig. 1a. It is noted that Ta2 is parallel to R2, TR1 to b1, and TR2 to a2. An example of each type of BC2N nanotubes is given in Fig. 1b–f.

Figure 1
figure 1

Atomic configuration of an isolated BC2N sheet. Primitive and translational vectors are indicated

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Symmetry of BC2N 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]:

(1)

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 BC2N sheet of the zigzag (n, 0) BC2N nanotubes (ZZ-1) (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, andS2nvanishes for the lack of the screw axis. Thus,

(2)

The point group of the line group is readily obtained from Eq. 2,

(3)

To determine the symmetries at the Γ point of the 12 N (N is the number of unit cells in the tube andN = n for ZZ-1 BC2N nanotubes) of phonons inZZ-1 BC2N nanotubes and the number of Raman- or IR-active modes, we have to associate them with the irreducible representations (irrep’s) ofC nv . Here, two cases need to be considered.

Figure 2
figure 2

2D projections of zigzag BC2N nanotubes (ZZ-1).z is a glide plane

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Case 1

n is odd (orn = 2m + 1, m is an integer)

The character table of C(2m+1)v possesses m + 2 irrep’s [18], i.e.,

(4)

The 12 N phonon modes transform according to the following irrep’s:

(5)

where

(6)

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 BC2N 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.

(7)
(8)

Case 2

n is even (orn = 2m,m is an integer)

The character table of C2mv possesses m + 3 irrep’s [18], i.e.,

(9)

The 12 N phonon modes transform according to the following irrep’s:

(10)

where

(11)

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].

(12)
(13)

The numbers of Raman- and IR- active modes are 30 and 18, respectively, for ZZ-1 BC2N nanotubes irrespective n.

The armchair (n,n) BC2N nanotubes (AC-1) (Fig. 1d), corresponding to the BC2N sheet shown in Fig. 3, have horizontal planes as indicated byg. TheD nh andD nd point groups reduce toC nh owing to the lack ofC2axes andS2nvanishes for the lack of the screw axis.

(14)

The point group of the line group is readily obtained from Eq. 2,

(15)

To determine the symmetries (at the Γ point) of the 12 N (N = n) phonons in AC-1 BC2N nanotubes and the number of Raman- or IR-active modes, two cases need consideration, by associating them with the irrep’s of C nh .

Figure 3
figure 3

2D projections of armchair BC2N nanotubes (AC-1).z is a glide plane

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Case 1

n is odd (n = 2 m + 1)

The character table of C(2m+1)h possesses 4m + 2 irrep’s [18], i.e.,

(16)

The 12 N phonon modes transform according to the following irrep’s:

(17)

where

(18)

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].

(19)
(20)

Case 2

n is even(n = 2m)

The character table of C2mh possesses 4m irrep’s [18], i.e.,

(21)

The 12 N phonon modes transform according to the following irrep’s:

(22)

where

(23)

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].

(24)
(25)

The numbers of Raman- and IR- active modes are 19 and 10, respectively, for AC-1 BC2N nanotubes in irrespective of n. The numbers of Raman- and IR- active phonon modes for ZZ-1 BC2N nanotubes are almost twice as for AC-1 BC2N nanotubes, which is similar to boron nitride nanotubes [11].

The nonsymmorphic line group describing the ()-chiral carbon nanotubes can be decomposed as follows:

(26)

where; whered 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 ofN/d andN, respectively. The point group of the line group is obtained from Eq. 26,

(27)

where and are the rotations embedded in S N/d and S N , respectively.

For chiral (n m) BC2N nanotubes, the point group D N reduces to C N due for the lack of C2 axes. Here, , where d R is the greatest common divisor of and ; d is the greatest common divisor of and . The BC2N 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 BC2N nanotubes, N = 4n. The point group corresponding to the two models is expressed as:

(28)

The character table of C N has N irrep’s, i.e.,

(29)

The 12 N phonon modes transform according to the following irrep’s:

(30)

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].

(31)
(32)

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 E2g mode around 1580 cm−1 is related to the stretching mode of C–C bond. The E2g 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 E1g and A1g modes [22].

Figure 4
figure 4

2D projections of BC2N nanotubesa ZZ-2 andb AC-2.z is a glide plane

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Conclusions

In summary, the symmetry properties of BC2N nanotubes were discussed based on line group. All BC2N 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 BC2N 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 BC2N nanotubes have 30 Raman- and 18 IR- active phonon modes; all AC-1 BC2N nanotubes have 19 Raman- and 10 IR-active phonon modes; all ZZ-2, AC-2, and other chiral BC2N 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 BC2N nanotubes are almost twice as in AC-1 BC2N nanotubes, but which is almost the same as those in chiral, ZZ-2, and AC-2 BC2N nanotubes. The situation in BC2N nanotubes is different from that in carbon or boron nitride nanotubes [8, 11].