First principles calculations using density functional theory were carried out to obtain the geometrical properties and the radial breathing mode (RBM) frequency of 40 different single-walled carbon nanotubes with small diameter. Fourteen chiral nanotubes with diameters less than 0.8 nm were considered, for which the number of atoms in the unit cell is not larger than 200. From the achiral (zigzag, armchair) nanotubes all those having a diameter less than 1.6 nm were considered. The geometrical parameters deviate from the values obtained from simple wrapping of a perfect hexagonal sheet. The deviation from the ideal behaviour increases with increasing curvature. The most prominent change is the increase of the diameter with respect to its ideal value. The lattice constant along the tube axis exhibits a slight shrinking. The RBM frequency does not follow the usually assumed 1/d behaviour; there is a general softening with increasing curvature. The softening of the RBM frequency cannot be described by a simple function of the diameter; it also depends on the chiral angle and the metallicity of the tube. In addition to this, the coupling of the totally symmetric radial motion with the totally symmetric tangential motion(s) has a non-negligible effect.
ASJC Scopus subject areas
- Physics and Astronomy(all)