Raman modes of the deformed single-wall carbon nanotubes

With the empirical bond polarizability model, the nonresonant Raman spectra of the chiral and achiral single-wall carbon nanotubes (SWCNTs) under uniaxial and torsional strains have been systematically studied by ab initio method. It is found that both the frequencies and the intensities of the low-frequency Raman-active modes almost do not change in the deformed nanotubes, while their high-frequency part shifts obviously. Especially, the high-frequency part shifts linearly with the uniaxial tensile strain, and two kinds of different shift slopes are found for any kind of SWCNTs. More interestingly, Raman peaks are found in the nonresonant Raman spectra under torsional strain, which are explained by (a) the symmetry breaking and (b) the effect of bond rotation and the anisotropy of the polarizability induced by bond stretching.

Figure 1

(Color online) Calculated phonon dispersion curves and density of states are shown in the left and right panels, respectively, in each subfigure. For (10, 0) zigzag nanotube under the tensile strains of 0%, 2%, and 4%.

Figure 2

(Color online) Frequencies of Raman-active modes for $(n,0)$ zigzag tubes ($n=6$, 8, and 10) under tensile strain.

Figure 3

(Color online) The GTEs for the highest three Raman-active modes of zigzag tubes $(n,0)$, $(n=6,\dots ,11)$. The lines are fitting results (see text), and other markers (circles, diamonds, and squares) are GTEs obtained by first-principle calculations.

Figure 4

(Color online) Frequencies of Raman-active modes for some armchair and chiral SWCNTs under tensile strain. (a) (5, 5), (b) (6, 6), (c) (12, 4).

Figure 5

(Color online) The atomic displacements of three Raman-active tangential modes of (6, 6) tube. ${\omega }_0$ is the frequency of Raman-active modes of undeformed tube.

Figure 6

(Color online) 2D projection of the nanotubes under the torsional strain, where $\gamma$ is the torsion angle and $\vec{C}$ is the chiral vector. ${\vec{T}}_0$ and ${\vec{T}}_{\text{new}}$ are the translational vectors for the undeformed and the deformed tube.

Figure 7

(Color online) The Raman spectra of different SWCNTs under torsional strain. (a) (5, 5), (b) (10, 0), (c) (12, 4).

Figure 8

(Color online) Frequencies of Raman-active modes vs torsional strains for three SWCNTs. (a) (5, 5), (b) (10, 0), (c) (12, 4).

Figure 9

(Color online) Raman peak intensities of deformed (10, 0) tube with torsional angle of 2.2047°, contributed from the three individual terms in Eq. (5). The asterisk and pentagram indicate the positions of Raman-active modes. Panels (I), (II), and (III) give the contributions from the first, the second, and the last term in Eq. (5), respectively. Notice that the total intensity is not the sum of those in the three panels due to interference effects between them.