Curvature effect on the radial breathing modes of single-walled carbon nanotubes

Based on the energy of a relaxed carbon nanotube relative to the graphite sheet (graphene), we construct an analytical correction to the carbon-carbon force constants. Then, the radial breathing modes (RBM) of a number of single-walled carbon nanotubes (SWNTs) are calculated. It is found that for small diameter SWNTs, the curvature effect becomes significant and chirality dependent. Our calculated RBMs are in very good agreement with experimental measurements, especially for ultrasmall diameter SWNTs.

Figure 1

The first and second nearest neighboring bonds (a) for zigzag, (b) for armchair, and (c) for chiral SWNTs.

Figure 2

The value of $A$ as a function of tube diameter. The open circles represent the value of $A$ for (5, 0), (4, 2), and (3, 3) tubes.

Figure 3

Frequencies of the RBM of a number of SWNTs for (a) zigzag and (b) armchair tubes. The solid lines are a linear fit to the data for the large diameter SWNTs.