Estimation of the Young’s modulus of single-walled carbon nanotubes under electric field using tight-binding method

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Highlights

  • A tight-binding study of Carbon nanotubes is presented.

  • The variation of Young’s modulus with the tube diameter is explained.

  • The Young’s modulus of CNT in the presence of an external electric field is predicted.

Abstract

In this paper, the Young’s modulus of single-walled carbon nanotubes (SWCNTs) under uniaxial strain is studied by using the tight-binding method in the nearest neighbor approximation while the curvature effect take into account. Also, the effect of an external electric field on the Young’s modulus of carbon nanotubes is investigated. The strain dependent tight-binding parameters are obtained by comparing the tight-binding band structure of graphene with DFT calculations. In the next step, total energy and then the Young’s modulus of single-walled carbon nanotubes are calculated. Our results indicate the dependence of Young’s modulus on the tube diameter and the external electric field. By applying the axial electric field, the dependency of Young’s modulus to the diameter becomes less important. Also, by increasing the axial electric field, the Young’s modulus decreases. The axial electric field becomes much more important in the armchair nanotubes.

Introduction

Since their discovery in 1991 [1], carbon nanotubes (CNTs) have invoked considerable interest. CNTs are ideal reinforcing materials for a new class of super strong nano-composites due to their extraordinary properties, such as the exceptionally high stiffness and strength, which are in the range of TPA, the extreme resilience, the ability to sustain large elastic strain, the high aspect ratio, and low density [2]. Many groups have calculated the strain energy and the Young’s modulus of single-wall carbon nanotubes (SWCNTs). Apparently, based on the previous studies, it is clear that the elastic parameters of the carbon nanotube depend on its structural parameters. But, there is no agreement between the given values for the Young’s modulus of CNTs. Since each group has used a different methodology to calculate the mechanical properties of CNTs, it is found out that the Young’s modulus depends on different parameters. However, all of them have the same order of magnitude in the value of the Young’s modulus. Treacy et al. [3] obtained Young’s modulus at an average value of 1.8 TPA. Krishnan et al. [4] reported an average value of 1.25 TPA, while Salvetat et al. [5] obtained an average Young’s modulus of 0.81 TPA. Theoretically, ab initio calculations have yielded the Young’s moduli of 1.06 TPA to 1.14 TPA for SWCNTs [6], [7]. The tight-binding methods also showed a significant scatter in the Young’s modulus values which varies from 0.676 TPA to 1.27 TPA [8], [9]. Kalamkarov et al. [10] obtained the Young’s modulus from 0.96 TPA to 1.04 TPA for the nanotube diameter ranging from 4 Å to 35 Å using analytical and numerical techniques. Tserpes et al. [11] used three-dimensional finite element model for armchair, zigzag, and chiral SWCNTs. This model showed that the Young’s modulus varies with tube diameter from 0.95 TPA to 1.05 TPA. It can be seen that the diameter of SWCNTs has a direct effect on the Young’s modulus of both armchair and zigzag chirality forms. This effect becomes stronger in the case of SWCNTs with small diameters. The Young’s modulus of chiral SWCNTs is also affected but not significantly. With increasing the tube diameter, the Young’s modulus of SWCNTs increases. But SWCNTs with different tube diameter follow different trends [11]. A recent research showed that the electric field has significant effects on the nanostructures [12]. In this paper, we study the Young’s modulus of SWCNTs in the nearest neighbor non-orthogonal tight-binding method and investigate the effect of the electric field on the Young’s modulus of zigzag and armchair SWCNTs.

Section snippets

Method

SWCNTs are formed from the vector Ch = ma1 + na2, where a1 and a2 are the primitive lattice vectors of a graphite sheet. SWCNTs and graphene sheets have the honeycomb structures containing two carbon atoms (called A and B). Here, we present the tight-binding Hamiltonian matrix for graphene and SWCNTs which takes into account the curvature. We use one 2s and three 2p orbitals per atom for constructing the tight-binding Hamiltonian matrix H and the overlap matrix S. The matrices may be written in

Result

We have considered the σ and π bonds in the tight-binding study of Graphene. The parameters can be calculated with different methods, which led to slightly different results [24], [25], [26]. In this work, the tight-binding results for band structure are fitted to the DFT one. Fig. 1 shows the band structure of Graphene in the DFT and tight-binding models. All the tight-binding parameters were tabulated in Table 1 in which ɛs and ɛp are the onsite energies of the 2s and 2p orbitals,

Conclusion

The electronic properties of Graphene are studied by using the DFT and tight-binding models. The tight-binding parameters are obtained by fitting the results of two models. Also, the Young’s modulus of Carbon nanotubes is calculated by using the second derivate of total energy with respect to the applied strain. Our results show the importance of an external electric field on the Young’s modulus of tubes. The variation of Young’s modulus in the armchair nano structures due to the electric field

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