The Strain Energy and Young’s Moduli of Single-Wall Carbon Nanotubules Calculated from the Electronic Energy-Band Theory

The strain energies in straight and bent single-wall carbon nanotubules (SWNTs) are calculated by taking account of the total energy of all the occupied band electrons. The obtained results are in good agreement with previous theoretical studies of different approaches and experimental observations. We calculate the Young’s moduli and wall thickness of SWNT from the bending strain energy for the first time. The obtained results reveal that the continuum elasticity theory may not be directly applicable to SWNT, although it serves well to describe the elasticity of multi-wall carbon nanotubules. PACS numbers: 61.48.+c,63.20.Dj,71.20.HK,71.25.-s Typeset using REVTEX 1 Since the discovery of carbon nanotubules (CNTs) [1], it has been excited a considerable interest in the last few years [2]. There are many works on both the theoretical [3–5] and experimental [6] studies about the electronic structure of the CNTs, and many unique and novel properties have been discovered. For example, the insulating, semimetallic, or metallic behavior is should to be dependent on the radius and the helicity of CNT [3]. On the thermal and mechanical properties, the tubes are significantly stiffer than any presently know materials [7]. To understand the intriguing properties, many groups have calculated the strain energy [8–13] and the Young’s moduli [14–16] of the single-wall carbon nanotubules (SWNTs). Among these calculations, many approaches are dependent on an empirical potential between carbon atoms, such as the Tersoff-Brenner potential [17] or the continuum elasticity model deduced by an empirical model [11,13]. Although the continuum elasticity model serves well to describe the deformation of multi-wall carbon nanotubules [13], in resent theoretical studies about the Young’s moduli of SWNTs [14–16], there are some discrepancies which are merely due to adopting different empirical potentials and different relations of the continuum elasticity theory (CET). Therefore, how to calculate the Young’s moduli of SWNT is still an open question. Considering the fact that the mechanical properties are determined by the bonds between atoms, i.e., by the electronic structure, we can pose the general questions as follows: Can one calculate both the strain energy and the elastic moduli of the SWNTs without empirical potentials, but derive them directly from microscopic electronic energy bands? Can one use directly CET in SWNTs? In this Letter, we present a new method to compute the strain energy of the straight SWNTs directly from electronic properties without introducing the empirical potentials. We then extend our method to calculate the strain energy of the bending SWNTs, and find the bending energy caused by the π-band electrons increases as the bending curvature of the SWNT increases, but the σ-band electrons energy decreases. The obtained bending elastic energy is far smaller than the prediction from CET. Our results as have been suggested in some previous studies reveal that the CET may not be applicable to the single-atom layer structure. 2 Generally, the total energy of carbon system is given by the sum [18,19]: Etotal = Eel + Erep (1) where Eel is the sum of energy band electrons for the occupied states, Erep is given by a repulsive pair potential only depending on the distance between two carbon atoms. They are given by