Symmetry Properties of Single-Walled BC2N Nanotubes
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The symmetry properties of the single-walled BC2N nanotubes were investigated. All the BC2N nanotubes possess nonsymmorphic line groups. In contrast with the 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. The number of Raman- active phonon modes is almost twice that of the infrared-active phonon modes for all kinds of BC2N nanotubes.
KeywordsBC2N nanotubes Symmetry Group theory
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 . Studies on the symmetry properties of carbon nanotubes predicted the Raman- and infrared-active vibrations in the single-walled carbon nanotubes , which are consistent with the experimental data  and theoretical calculations . A similar work was carried out by Alon on boron nitride nanotubes , and the results were later confirmed by first-principles calculations . And the symmetry of BC2N nanotube was reported . 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
Symmetry of BC2N Nanotubes
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 .
n is odd (orn = 2m + 1, m is an integer)
n is even (orn = 2m,m is an integer)
The numbers of Raman- and IR- active modes are 30 and 18, respectively, for ZZ-1 BC2N nanotubes irrespective n.
n is odd (n = 2 m + 1)
n is even(n = 2m)
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 .
where and are the rotations embedded in S N/d and S N , respectively.
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  and boron nitride nanotubes . 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].