Two-dimensional elasticity determines the low-frequency dynamics of single- and double-walled carbon nanotubes

We develop a continuous theory of low-frequency dynamics for nanotubes with walls constituted by single-atom monolayer, the topological elasticity of which is not related to its vanishing macroscopic thickness. The applicability region of the theory proposed includes all truly two-dimensional materials such as graphene and MoS${}_2$. New comprehensive interpretation and analytical expressions for low-frequency modes in single-walled carbon nanotube (SWCNT) are given. The theory unambiguously relates the radial breathing modes of SWCNT and breathinglike modes of the double-walled carbon nanotube (DWCNT). The existing Raman data on DWCNTs are fitted better than in the frame of previous models.

Figure 1

Displacement fields for low-frequency modes (with $n$ = 0,1,2) and zero wave vector. Modes (a)–(f) are the only RA low-frequency modes in chiral SWCNT. In achiral nanotubes the only RA modes are limited to $g$ modes [panels (a), (c), (d), and (e)]. (a) RBM; (b) stretching with $n$ = 1; (c), (d) bending and stretching with $n=2$, respectively (note that the displacement field of the bending mode involves practically no membrane stretching); and (e), (f) shear modes with $n$ = 1 and $n=2$, corresponding displacement fields are perpendicular to the figure plane, displacement amplitude is given by the circle size, and open or full circles correspond to up or down displacements, respectively.