Zero $m$ phonons in $\mathrm{Mo}{\mathrm{S}}_2$ nanotubes

Phonon dispersions of single-wall $\mathrm{Mo}{\mathrm{S}}_2$ nanotubes are calculated within a full symmetry implemented valence force-field model. For transversal, twisting, and breathing modes symmetry assignation, chirality, and diameter dependence of their frequencies and displacements are discussed. Tubular structure is found to be characterized by two Raman active modes: by the in-phase breathing mode (in full analogy to carbon nanotubes) with frequency approaching Brillouin scattering domain (as diameter approaches nm), and by the high-energy breathing mode with sulfur shells breathing in phase, but out of phase relative to the molybdenum atoms. Likewise, the longitudinal rigid-shell mode, where sulfur shells vibrate out of phase whilst molybdenum atoms barely move, is predicted to be a fingerprint of the cylindrical configuration in infrared spectra. It is also found that twisting rigid-layer modes characterize chirality of the tubes. Finally, the large diameter limit is discussed and related to the measured Raman and infrared spectra of the layered structure.

Figure 1

Vertical mirror plane in the zigzag (15,0) (left) and horizontal mirror plane in the armchair (10,10) tube (right). Sulfur atoms are in light gray while the molybdenum atoms are in dark gray.

Figure 2

Longitudinal (upper panels) and twisting modes (lower panels) of the tube (12,8): LO, LI and TWO, TWI from the left to the right. Sulfur atoms are in light gray while the molybdenum atoms are in dark gray.

Figure 3

Structure of $2\mathrm{H}\text{−}\mathrm{Mo}{\mathrm{S}}_2$ with trigonal prism coordination. $\mathrm{Mo}\text{−}\mathrm{S}$ bond lengths ($r$ and $r^{\prime }$), $\mathrm{S}\text{−}\mathrm{Mo}\text{−}\mathrm{S}$ bond angles ($\psi$ and $\theta$), and $\mathrm{Mo}\text{−}\mathrm{S}\text{−}\mathrm{Mo}$ bond angle $\left(\varphi \right)$ are indicated. Sulfur atoms are in white while the molybdenum atoms are in black.

Figure 4

Phonon dispersions in the tube (17,17). Zero $m$ branches are in bold. The arrows point to the modes BA $\left(21{\mathrm{cm}}^{-1}\right)$ and BI $\left(485{\mathrm{cm}}^{-1}\right)$, which are predicted to distinguish between the layered and the tubular structure in Raman measurements.