A lattice dynamical treatment for the total potential energy of single-walled carbonnanotubes and its applications: relaxed equilibrium structure, elastic properties, andvibrational modes of ultra-narrow tubes

Abstract

In this paper, we propose a lattice dynamic treatment for the total potential energy ofsingle-walled carbon nanotubes (SWCNTs) which is, apart from a parameter for thenonlinear effects, extracted from the vibrational energy of the planar graphene sheet. Theenergetics, elasticity and lattice dynamics are treated in terms of the same set offorce constants, independently of the tube structures. Based upon this proposal,we have investigated systematically the relaxed lattice configuration for narrowSWCNTs, the strain energy, the Young’s modulus and Poisson ratio, and the latticevibrational properties with respect to the relaxed equilibrium tubule structure. Ourcalculated results for various physical quantities are nicely in consistency with existingexperimental measurements. In particular, we verified that the relaxation effectmakes the bond length longer and the frequencies of various optical vibrationalmodes softer. Our calculation provides evidence that the Young’s modulus ofan armchair tube exceeds that of the planar graphene sheet, and that the largediameter limits of the Young’s modulus and Poisson ratio are in agreement withthe experimental values of graphite; the calculated radial breathing modes forultra-narrow tubes with diameters ranging between 2 and 5 Å coincide with theexperimental results and the existing ab initio calculations with satisfaction. Fornarrow tubes with a diameter of 20 Å, the calculated frequencies of optical modes inthe tubule’s tangential plane, as well as those of radial breathing modes, arealso in good agreement with the experimental measurements. In addition, ourcalculation shows that various physical quantities of relaxed SWCNTs can actually beexpanded in terms of the chiral angle defined for the corresponding ideal SWCNTs.