Raman and infrared-active modes in ${\mathrm{MS}}_2$ nanotubes $(\mathrm{M}=\mathrm{Mo},\mathrm{W})$

Lattice dynamics of all the single-wall ${\mathrm{MS}}_2$ nanotubes with a diameter range of $1–20\mathrm{nm}$ is studied within full symmetry implemented valence-force-field model. Particularly, frequencies of infrared and Raman active modes are examined to clarify possible application of these methods in the experimental sample characterization. A prescription for experimental distinction of zig-zag, armchair and chiral tubes, based on their different symmetry and resulting selection rules, is proposed. Displacements significantly depend on the chiral angle only for two ${\mathrm{MoS}}_2$ modes, namely twisting in phase (rigid layer) and breathing out of phase, and none of ${\mathrm{WS}}_2$ modes.

Figure 1

Inverse diameter dependence of $m=0$ modes. Gray stands for ${\mathrm{MoS}}_2$, while black is for ${\mathrm{WS}}_2$ tubes. Fits are given for ${\mathrm{BA}}_0$ modes.

Figure 2

Upper and middle panel: ${\mathrm{TWI}}_0$ and ${\mathrm{BO}}_0$ modes in ${\mathrm{MoS}}_2$ (10,10) tube (frequencies $392{\mathrm{cm}}^{-1}$ and $393{\mathrm{cm}}^{-1}$, respectively). Dark (light) dots are $\mathrm{Mo}\left(\mathrm{S}\right)$ atoms. Note that both modes have a significant radial, as well as circumferential component. Lower panel: Enlarged part of Fig. 1 emphasizing the crossing of the diameter dependencies of ${\mathrm{TWI}}_0$ and ${\mathrm{BO}}_0$ at approximately $2.5\mathrm{nm}$.

Figure 3

Inverse diameter dependence of $\mid m\mid =1$ (left) and $\mid m\mid =2$ (right) modes for ${\mathrm{MoS}}_2$ (solid-gray), and ${\mathrm{WS}}_2$ (dashed-black) tubes. Low-energy region: For $\mid m\mid =1$, modes are arranged as ${\mathrm{LA}}_1$, ${\mathrm{BA}}_1$; while for $\mid m\mid =2$, as ${\mathrm{TWA}}_2$, ${\mathrm{LA}}_2$, and ${\mathrm{BA}}_2$ with energy. High-energy region: The arrangements of modes for ${\mathrm{MS}}_2$ tubes are the same as for $m=0$ (Fig. 1) except that ${\mathrm{TWO}}_{1∕2}$ modes have a larger frequency than ${\mathrm{LO}}_{1∕2}$ for ${\mathrm{WS}}_2$ tubes.

Figure 4

Phonon dispersion of (17,17) ${\mathrm{WS}}_2$ tube. Branches which end at $\Gamma$ point with IR and Raman active modes are emphasized.