Localized modes in capped single-walled carbon nanotubes

It is well established that a special class of spatially localized modes can be generated at surfaces. Such surface states have been studied in many branches of physics, including electrons in crystals, surface phonons, surface polaritons, and optical surface modes in waveguide arrays. However, a direct observation of highly localized excitations at the atomic level is rather difficult. For example, it took almost 60 years to observe the electronic Tamm states that were originally suggested in 1932. Carbon nanotubes have attracted considerable attention in recent years, and many of their properties have been observed experimentally. The growing interest in carbon nanotubes can be explained by their unique physical properties and their potential for a wide range of applications. In particular, carbon nanotubes are known for their superior mechanical strength and good heat conductance. Thus, it seems natural to explore a variety of surface-mediated effects in carbon nanotubes, which may acquire additional features in such highly discrete systems with curved geometry and can be subsequently measured experimentally. In particular, the recent experimental studies of the edge structures of thermally treated graphite opened the road for the experimental probing of surface effects. In this letter we study numerically the surface vibrations of capped singlewalled carbon nanotubes and predict the existence of highly localized modes at the tips. These linear localized modes are somewhat similar to the well-known surface Tamm states known in solids. We consider four types of single-walled carbon nanotubes SWCNs , which are known to have the smallest diameters. The structure of the nanotubes is shown in Fig. 1. A zigzag m ,0 SWCN with the indices m=5,6 has L 2 transverse segments consisting of N=m+2mL+m carbon atoms each tip has m atoms, and each segment has 2m atoms . The structures with the smallest number of atoms N=30 for m=5 and N=36 for m=6 at L=2 transform into fullerenes C30 C30.1-D5h and C36 No. 15-D6h , respectively. The armchair nanotube m ,m with the indices m=5,6 has L 4 transverse segments with N=2m+2mL+2m carbon atoms with 2m atoms in the tip and the segment . The structure with the smallest number of atoms N=60 for m=5 and N=72 for m=6 for L=4 is transformed into fullerenes C60 bucky-ball C60-Ih and C72 C72-D6d , respectively. To describe oscillations of the nanotube, we present the system Hamiltonian in the form