Structural and Electronic Properties of a Carbon Nanotorus: Effects of Non-Local Vs Local Deformations

The bending of a carbon nanotube is studied by considering the structural evolution of a carbon nanotorus from elastic deformation to the onset of the kinks and eventually to the collapse of the walls of the nanotorus. The changes in the electronic properties due to non-local deformation are contrasted with those due to local deformation to bring out the subtle issue underlying the reason why there is only a relatively small reduction in the electrical conductance in the former case even at large bending angles while there is a dramatic reduction in the conductance in the latter case at relatively small bending angles. PACS numbers: 72.80.Rj,71.20Tx Typeset using REVTEX 1 There has been some focused interest in the last several years on the interplay between the mechanical deformation and the electrical properties of single-wall nanotubes (SWNT). Both experimental [1,2] and theoretical [3–6] studies seem to indicate that bending of the nanotubes induces only a small conductance change even at relatively large bending angles (up to α = 45 [6], where α is the angle between the direction tangential to the end of the tube and the unbent axis) unless the nanotubes fracture or the metal-tube contacts are perturbed [2]. Bending involved in these studies is caused by a mechanical deformation under forced confinement of the ends of the tube [7]. In this situation, the deformation induced by the bending can be viewed as a non-local deformation. Very recently, we carried out an in-depth study, both theoretically and experimentally, of the effects of a local mechanical deformation of a metallic SWNT induced by the pushing action of the tip of an atomic force microscope (AFM) on the electrical properties of the tube [8,9]. We observed an unexpected two orders of magnitude reduction in conductance at a relatively small bending angle (α = 13) [9]. Our theoretical study confirmed that this dramatic change in conductance is due to a reversible transition from sp to sp bonding configurations in the bending region adjacent to the tip [8]. Thus, it is the local deformation induced by the pushing action of the tip that is responsible for the two orders of magnitude change in conductance. This raises the question why there is only a relatively small change in conductance even at large bending angles under a non-local deformation? In this work, we address this question by investigating the mechanical deformation of a carbon nanotorus obtained by connecting the two ends of a SWNT into a ring. We have chosen a carbon nanotorus to study the effects of bending because in this case one can determine readily the radius of curvature, the logical quantity to characterize the degree of bending, at the location of bending than in the case of a bent SWNT. In previous theoretical studies [7,10,11], the bending of the SWNT was modeled by holding the ends of the SWNT at positions defining the angle of bending. This initial configuration was then allowed to relax to its equilibrium configuration while the two ends were kept at the initial positions. Thus, there could be some ambiguity in relating the bending angle set initially 2 with the deformation in the bending region. On the other hand, the radius of curvature of a carbon nanotorus is well defined. Hence, a nanotorus is ideally suited for studying the effects of non-local deformations and can, in fact, lead to an understanding of the interplay between the mechanical deformation and electronic properties of SWNTs. Furthermore, recent experimental fabrication of rings from SWNTs [12] and subsequent measurement of magneto-resistance [13] from some of these rings underscore the importance of understanding the interplay between the mechanical deformation and the electronic properties of a nanotorus in its own right. To study how non-local deformations affect the structure of a nanotube, we consider a metallic (5,5) nanotube bent into a circular form with its two ends connected to form a nanotorus of a certain radius (R). The initial configuration is then relaxed to its equilibrium configuration. To obtain a nanotorus which is uniformly and elastically deformed, one must start with a nanotube of sufficiently long length. Because of the size of the system under consideration, we used the order-N non-orthogonal tight-binding molecular dynamics (O(N)/NOTB-MD) [14] with the NOTB Hamiltonian as given in Ref. [15]. We established that a relaxed nanotorus containing 2000 atoms will indeed be under a uniform and elastic deformation. This was accomplished by first obtaining the relaxed configuration of a straight 2000-atom (5,5) nanotube (relaxed radius=3.42 Å), next bending it into a circular form, and finally relaxing it into its equilibrium configuration. Starting from this configuration, we studied the effects of increasing deformation by reducing the radius of the nanotorus in a quasi-continuous step-by-step manner. Our aim is to effectively model a continuous bending process. Thus, at each step, we removed a small segment containing two circular sections, each with 10 atoms, from the nanotorus. The maximum change in the bond length with the removal of 20 atoms was less than 0.01Å when the two ends of the broken nanotorus were elastically reconnected. This change represents a small perturbation in the strain going from one step to the next. The configuration so obtained at each step was then relaxed to its equilibrium configuration using the O(N)/NOTB-MD scheme. The procedure was continued from the stage of elastic deformation, to the stage of the development of the 3 kinks, and eventually to the stage of topological change in the structure in the bending region. We highlight the ”continuous” process of bending by four configurations shown in Fig.1. Figures 1(a)-1(d) give the equilibrium configurations obtained by the MD procedure outlined above for nanotori containing 1000 (R=19.6Å), 900 (R=17.6Å), 800 (R=15.7Å), and 680 (R=13.3Å) atoms respectively. An examination of the structures shown in Fig. 1 reveals the following features. From Fig. 1(a), it can be seen that the nanotorus of R=19.6Å is under a uniform elastic distortion, with a simple stretching on the outer side and simple compression on the inner side. It gives an example of a SWNT under elastic deformation and indicates that the (5,5) SWNT is still only elastically deformed at a radius of curvature (R) of 19.6Å under a uniform deformation. As R is reduced to 17.6Å with a corresponding increase in the mechanical deformation, the structural distortion is characterized by the appearance of 16 almost uniformly distributed kinks along the inner (compression) side of the nanotorus (see Fig. 1(b)). The appearance of the kinks is attributable to the release of the strain energy at a critical radius of curvature of about 17.6Å in the bending region. Further reduction of R with the accompanying increase in the distortion brings about the reduction of the number of kinks while enhancing the release of the strain energy at the kinks as exemplified in Fig. 1(c). Finally when R reaches 13.3Å, the collapsing of the inner and outer walls of the nanotorus occurs (see Fig. 1(d)), signaling a topological change in the structure in the region of collapse at a critical radius of curvature of about 13.3Å. During the simulation of the ”continuous” bending, extreme care was exercised in monitoring the onset of the fundamental structural change. However, due to the discrete nature of the removal of small segment from the nanotorus, there is a built-in degree of uncertainty in pinning down precisely the radius of curvature when a certain structural change occurs. Hence the numbers given above as critical radii of curvature for various structural changes are approximate. To explore in more detail the structural changes induced by a non-local deformation, we calculated the average strain energy per atom over each section (containing 10 atoms) 4 of the relaxed nanotorus. The angular location of the section is denoted by (θ) with θ=0 along the positive horizontal direction and increasing in the clockwise direction. The total energy of the nanotorus can be expressed as the sum of the site energy of atoms in the nanotorus within the framework of a NOTB Hamiltonian [16]. The strain energy of an atom at a given site in the relaxed (5,5) nanotorus is defined as the energy difference between the energy of the atom at that site and the energy per atom for the relaxed ideal straight (5,5) nanotube. The average strain energy per atom for a given section is simply the average of the strain energy for the atoms in that section. In Fig. 2, the average strain energy per atom for the four nanotori is plotted against θ. For the nanotorus of R=19.6 Å, the strain energy is small and uniform along the entire torus, indicating that the torus is under simple elastic deformation. When R is reduced to 17.6Å, the average strain energy along the torus is increased substantially and 16 local minima appear in the plot at the angles where the kinks are located, a clear indication of the release of the strain energy at the kinks. When R is reduced to 15.7Å (800 atoms), there is only a slight increase in the average strain energy at locations near the kinks with enhanced local minima. However, when R reaches 13.3Å, pronounced local minima appear at locations precisely matching where collapsing of the walls occurs. These enhanced local minima actually accompany a substantial increase in the average strain energy per atom along the torus. A side-by-side comparison of Fig. 2 and Fig. 3, where the average bond length is plotted vs θ , is even more illuminating. While the average bond length along the torus is somewhat longer than the bond length of the ideal straight (5,5) nanotube for the three nanotori with the radius of 19.6 Å, 17.6 Å, and 15.7Å respectively, the average bond length for the nanotorus with the radius 13.3Å at the locations of wall-collapsing is 1.53Å, substantially longer than the bond length of 1.42Å for the sp bonding configuration in the ideal straight (5,5) nanotube and in fact close to the bond length of 1.54Å for the sp bonding configuration in diamond. This is the signature of the topological change in the structure for the 680-atom nanotorus in the region of wallcollapsing. Further confirmation of this picture can be seen from Fig. 4 where the average coordination number is plotted vs θ . It can be seen that the average coordination number at 5 the locations of wall-collapsing for the nanotorus of radius 13.3Å is about 3.6, close to the 4coordinated sp configuration, while that along the circumference of the first three nanotori is close to 3, the coordination number for the sp configuration. Thus, our investigation has succinctly established the effects on the structure of a SWNT due to a non-local deformation, namely, from a simple elastic distortion for large radii of curvature, to the appearance of kinks at a critical radius of curvature of about 17.6Å, followed by the collapse of the walls involving the transition from the sp configuration to the sp configuration at a critical radius of curvature of about 13.3Å for the (5,5) SWNT. Within the framework of the nanotorus, one may set up a benchmark for comparison with previous studies. Specifically, referring to Fig. 1(b), since there are 16 uniformly distributed kinks, one may estimate the critical angle for the appearance of the kink under a non-local deformation as αK = 1 2 ( 16 ) ≃ 11. Similarly, from Figs. 2(d), 3(d), and 4(d), corresponding to the 680-atom case, one may surmise that there should be 7 equivalent pronounced peaks or valleys at the locations of wall-collapsing along the circumference. Hence, the critical bending angle for the transition from spto sp-bonding configuration under a non-local deformation can be estimated as αC = 1 2 ( 7 ) ≃ 26. But, care must be exercised when using the bending angle because of the ambiguity in relating the preset bending angle to the degree of bending in the bending region. Unless the arc length defining the bending region is also known, the preset bending angle alone can not uniquely characterize the degree of bending. To understand the interplay between the mechanical deformation and the electronic properties, we calculated the average σ bond charge and the average π bond charge per atom following the procedures given in Refs. [16,8]. The results are shown in Figs. 5 and 6, respectively. It can be seen that, for the first three nanotori, there is only some slight reduction of both the σ and the π bond charges along the circumference associated with the bond stretching as the nanotorus is under increasing strain. Thus, there is no basic change in the nature of electronic structure and hence no significant change in conductance is expected for a SWNT under a non-local deformation up to a radius of curvature of about