Zigzag Tubes Research Articles

A comparison of different methods of Young’s modulus determination for single-wall carbon nanotubes (SWCNT) using molecular dynamics (MD) simulations

The computed values of Young’s modulus ( Y) of single-wall carbon nanotubes given by four common methods based on (i) the determination of stress for a fixed value of strain, (ii) the determination of strain energy for a fixed value of strain, (iii) the longitudinal vibrations, and (iv) the transverse vibrations, and a new method (v) based on the determination of strain for a fixed value of stress have been compared to check the consistency of different methods. The computed values of Y are found to be in agreement with each other with the exception that results of the transverse vibration method differ from those given by other methods when the aspect ratio, namely, the ratio of length to the radius of the tube, is small; the results of the transverse vibration method for the tubes of small diameter are also found to differ from those given by other methods when the commonly used value of thickness of the tube, 3.4 Å, is assumed. The solutions of these problems are discussed in terms of an appropriate consideration for the value of thickness and diameter dependent end-correction in the length of the tube. Effect of defects in the form of vacancies, van der Waals (VDW) interactions, chirality, and diameter of the carbon nanotubes on Y has also been investigated. Y is found to be sensitive to the number of vacancies. Y is found to decrease by ∼1% when VDW interactions between carbon atoms are ignored. Y is also found to be lower for an armchair tube compared to a zigzag tube of the same diameter. As regards the dependence of Y on diameter, we found that as the diameter increases from ∼7 Å to ∼25 Å, Young’s modulus drops by 4% and 8%, respectively, for armchair and zigzag tubes. These results are discussed and compared with other experimental and computed results reported in the literature.

Plasmons in isolated single-walled carbon nanotubes

The collective excitation modes of individual single-walled carbon nanotubesare studied theoretically in the nearest-neighbour tight-binding approximation.Umklapp terms are taken into account and local-field effects are studied.Plasmon mode dispersion relations for carbon nanotubes with radiusR∼7 Å are obtained. We report a strong dependence of the plasmon dispersion on the chirality ofthe carbon nanotubes. In particular, armchair and zigzag tubes have dispersive plasmons,but some of the chiral tubes have essentially dispersionless plasmons. Our results offer analternative explanation to the origin of the experimentally observed low-energydispersionless modes found in momentum-dependent EELS experiments. Importantdifferences between a tight-binding model and free electron gas model are discussed and itis shown that some important qualitative features cannot be captured by a freeelectron gas model. An acoustic plasmon mode is found in all metallic carbonnanotubes, giving support to the experimental findings in Raman scattering.

Theoretical Study of Boron Nitride Nanotubes with Defects in Nitrogen-Rich Synthesis

On the basis of calculations using the density functional theory, it is shown that BNNT synthesis could produce tubes deprived of one (B1 hole) or two (B2 hole) boron atoms under the condition where nitrogen atoms exist in excess throughout this study. The relative populations of various isomers of defective tubes will depend on the chirality of the tube. Interestingly, calculations show that B2 holes are much more favored than B1 holes, particularly in armchair tubes. Electronic properties are modified in such a way that the band gap is decreased through the introduction of defect states inside the gap. Magnetic properties will also be dependent on the chirality. The majority of armchair tubes with B2 holes will be nonmagnetic, while the majority of zigzag tubes with defects will exhibit magnetism. Contrary to the case of defect-free BNNT, the defective tubes are expected to be easily subject to reduction by accommodating excess electrons in the presence of Li atoms. In addition, the defect sites will show a higher affinity toward hydrogenation than the defect-free sites.

Electronic band structure, electron-phonon interaction, and superconductivity of (5,5), (10,10), and (5,0) carbon nanotubes

The electronic band structure, electron-phonon interaction, and superconducting transition are investigated for the (5,5), (10,10) armchair and (5,0) zigzag single walled carbon nanotubes (SWCNT). While the electron-phonon interaction and superconducting transition temperature are very small in (5,5) and negligibly small in (10,10) armchair tubes, they are considerable in the (5,0) zigzag tube. The (5,0) tube, which is a semiconductor with $\ensuremath{\pi}\text{\ensuremath{-}}\mathrm{bands}$ alone contributing to the band structure, exhibits metallic character with $\ensuremath{\pi}\text{\ensuremath{-}}\ensuremath{\sigma}$ bands hybridization. As a result the van Hove like singularity in the density of states moves towards the top of the valance band enhancing the effective density of states near the Fermi energy. Consequently the electron-phonon interaction increases and the (5,0) tube turns out to be a superconductor with appreciable transition temperature and found to be the most probable member of the family of $4\phantom{\rule{0.3em}{0ex}}\mathrm{\AA{}}$ diameter carbon nanotubes for which the superconducting transition temperature is experimentally measured.

Mechanical properties of single- and double-walled carbon nanotubes under hydrostatic pressure

Based on a molecular mechanics coupled with atomistic-based continuum theory, a closed-form formula is presented to examine the elastic properties of single- and double-walled carbon nanotubes subjected to hydrostatic pressure. Following the present model, the effects of the armchair and zigzag CNT structures on the pressure behavior are theoretically investigated. The computational result indicates that the bulk modulus is less sensitive to the chiral structures except for very small tube diameters. Moreover, closed-end nanotubes under hydrostatic pressure exhibit a larger bulking modulus than open ended nanotubes. The cap of the zigzag tubes has a larger effect on the bulk modulus when compared to the armchair tubes, especially in small diameter nanotubes. The predicted strain and the bulk modulus are in good agreement with existing theoretical results.

Size effect of X-shaped carbon nanotube junctions

Systematic calculations have been performed for X-shaped junctions formed between two crossed identical carbon nanotubes (armchair or zigzag tubes) with diameters ranging from 3.92 Å to 9.49 Å using an empirical potential method. The formation energy of X junctions is found to be a function of the diameter of the tubes. The formation temperature is dependent on the curvature of tubes. The calculations show that it is energetically favorable to form X junctions by the direct heating method if the tube diameter is less than 0.85 nm.

Reentrant Semiconducting Behavior of Zigzag Carbon Nanotubes at Substitutional Doping by Oxygen Dimers

The electronic structures of carbon nanotubes doped with oxygen dimers are studied using the ab initio pseudopotential density functional method. The fundamental energy gap of zigzag semiconducting nanotubes exhibits a strong dependence on both the concentration and configuration of oxygen-dimer defects that substitute for carbon atoms in the tubes and on the tube chiral index. For a certain type of zigzag nanotube when doped with oxygen dimers, the energy gap is closed and the tube becomes semimetallic. At higher oxygen-dimer concentrations the gap reopens, and the tube exhibits semiconducting behavior again. The change of the band gap of the zigzag tube is understood in terms of their response to the strains caused by the dimer substitutional doping.

First-principles calculation of the electronic structure and energy loss near edge spectra of chiral carbon nanotubes

We present first principles calculations of the electronic structure of small carbon nanotubes with different chiral angles θ and different diameters (d<1nm). Results are obtained with a full potential method based on the density functional theory (DFT), with the local density approximation (LDA). We compare the band structure and density of states (DOS) of chiral nanotubes with those of zigzag and armchair tubes with similar diameters. The carbon K-edge energy loss near edge structures (ELNES) have been studied and π* and σ* contributions have been evaluated. These contributions give information on the degree of hybridization for the small chiral nanotubes.

Tubular Configurations and Structure-Dependent Anisotropic Strains in GaS Multi-Walled Sub-Microtubes

Bending strains and bonding structures of GaS multiwalled submicrotubes have been examined by transmission electron microscopy. Experimental observations reveal that the strain involved in building the GaS tubes is more complicated than the theoretical prediction and appears anisotropic and dependent on the tubular configurations. The armchair tube bears a larger lattice compression than the zigzag tube. The hexagonal GaS semiconducting compound degrades its crystal symmetry upon bending, leading to the anisotropy of the bonding structures. Curving of GaS sheets to form tubes is found to be dominated by the structures and electrostatic fields on the sheet surface, which eventually gives a well-controlled tubular structure showing preferred zigzag and close-to-zigzag configurations. Finally, interlayer packing structures of multiwalled tubes are examined.

Elastic moduli of single-walled carbon nanotubes and their ropes

We report the results of a model calculation for studying the effects of pressure on a bunch of carbon nanotubes as well as individual nanotubes. Carbon nanotubes are extremely rigid in the axial direction. At pressures that we work with, the deformation in the axial direction comes out to be negligibly small in comparison to that in the transverse direction. We use the six-exponential and Brenner potentials to account for inter- and intratube interactions, respectively. Using second derivatives of potential, Young's modulus for single-walled armchair, zigzag and chiral tubes of different radii have been calculated. The values found by us in this simple model turn out to be in good agreement with other theoretical and experimental values. The strain dependence of Young's modulus has also been studied. We have also calculated the Poisson ratio and shear modulus of various single-walled nanotubes. We find that hydrostatic pressure is an ideal probe to study the radial deformations of the nanotubes. The nanotubes are considered to be flexible, identified by a flattening of cylinders under pressure through a parameter $f$. We calculate the total energy of the bunches having faceted tubes. The free energy thus calculated enables us to calculate phase changes at various pressures. From our calculations, we find the phase transformation to occur at about $5\phantom{\rule{0.3em}{0ex}}\mathrm{GPa}$. Young's modulus of nanoropes has also been calculated at various pressures and at the phase transition we obtain a discontinuity in the curve. A much simplified form of the Brenner potential has been suggested and results are compared with those obtained from original form of Brenner potential.

Silicon and III-V compound nanotubes: Structural and electronic properties

Unusual physical properties of single-wall carbon nanotubes have started a search for similar tubular structures of other elements. In this paper, we present a theoretical analysis of single-wall nanotubes of silicon and group III-V compounds. Starting from precursor graphene-like structures we investigated the stability, energetics and electronic structure of zigzag and armchair tubes using first-principles pseudopotential plane wave method and finite temperature ab-initio molecular dynamics calculations. We showed that (n,0) zigzag and (n,n) armchair nanotubes of silicon having n > 6 are stable but those with n < 6 can be stabilized by internal or external adsorption of transition metal elements. Some of these tubes have magnetic ground state leading to spintronic properties. We also examined the stability of nanotubes under radial and axial deformation. Owing to the weakness of radial restoring force, stable Si nanotubes are radially soft. Undeformed zigzag nanotubes are found to be metallic for 6 < n < 11 due to curvature effect; but a gap starts to open for n > 12. Furthermore, we identified stable tubular structures formed by stacking of Si polygons. We found AlP, GaAs, and GaN (8,0) single-wall nanotubes stable and semiconducting. Our results are compared with those of single-wall carbon nanotubes.

Exciton Resonances Quench the Photoluminescence of Zigzag Carbon Nanotubes

We show that the photoluminescence intensity of single-walled carbon nanotubes is much stronger in tubes with large chiral angles--armchair tubes--because exciton resonances make the luminescence of zigzag tubes intrinsically weak. This exciton-exciton resonance depends on the electronic structure of the tubes and is found more often in nanotubes of the +1 family. Armchair tubes do not necessarily grow preferentially with present growth techniques; they just have stronger luminescence. Our analysis allows us to normalize photoluminescence intensities and find the abundance of nanotube chiralities in macroscopic samples.

Electronic structure properties of carbon nanotubes obtained by density functional calculations

The total energies, atomic charges, and energy band gaps of (3,3) and (7,7) armchair and (6,0) and (12,0) zig-zag carbon nanotubes (CNTs) of various lengths were determined using density functional ab initio (DFT) calculations. Armchair CNTs were found to be less reactive than zig-zag CNTs. The calculated band gap, which accounts for metallic character, was smaller in (3 n,0) zig-zag tubes than in armchair CNTs and in both cases the band gaps decreased with increasing lengths of the tubes. However, frontier orbitals, which account for conductivity, were localized in zig-zag but delocalized in armchair tubes.

Band Theory of Single-Walled Carbon Nanotubes

In this paper, a curvilinear coordinate system is used in space and in k-space to study the energy band of single-walled carbon nanotubes wrapped at a helical angle. Using this method, a general function of the bandgap associated with the radius of the tube and the helical angle is derived based on the tight-binding theory. The three-dimensional hexagonal Brillouin zone of the tube is on the surface of cylinder in the k-space. For two tubes with different diameters, there is a distance between the cylindrical Brillouin zones in the radial direction. The Brillouin zone varies with the radius of the tube and the number of cells on the circumference. For the metallic zigzag tubes, the bandgaps decrease discretely to zero at the corners of the Brillouin zones, and those corners are singular points of zero gaps. With the transformation of coordinates, the metallic zigzag type is proven to be equivalent to an armchair configuration. Electrical characteristics of the chiral effects are briefly highlighted.

First Principles Study of Work Functions of Single Wall Carbon Nanotubes

We perform first principles calculations on work functions of single wall carbon nanotubes, which can be divided into two classes according to tube diameter (D). For class I tubes (D > 1 nm), work functions lie within a narrow distribution (approximately 0.1 eV) and show no significant chirality or diameter dependence. For class II tubes (D < 1 nm), work functions show substantial changes, with armchair tubes decreasing monotonically with diameter, while zigzag tubes show the opposite trend. Surface dipoles and hybridization effects are shown to be responsible for the observed work function change.

Formation and electronic properties of double-walled boron nitride nanotubes

The electronic and structural properties of double-walled boron nitride (BN) nanotubes are studied using the first principle pseudopotential density functional method. It is shown that zigzag-type tubes have a larger formation energy for the double-walled configuration than the armchair-type structure, and that interwall stacking plays a significant role for intertube interactions and in the formation of multiwall BN nanotubes. The fundamental energy gap of double-walled BN nanotubes was found to be smaller than that of single-walled tubes mostly due to band shift. It is shown that the electronic properties of double-walled BN tubes exhibit a slight but noticeable difference for the zigzag and armchair type tubes studied.

The radial breathing mode of carbon nanotubes

We report an extensive set of results for the radial breathing modes (RBM) of infinite and finite length single walled carbon nanotubes using the second generation reactive empirical bond order potential (REBO) developed by Brenner et al. As expected, the frequency, ν of the RBM is inversely proportional to the nanotube radius, R 0, We find two different linear fits to the data, one for zigzag tubes (3.25 THz nm) and one for armchair tubes For finite tubes, the RBM rapidly approaches the infinite length value for nanotubes greater than 5 nm in length.

Strength of radial breathing mode in single-walled carbon nanotubes

We show by ab initio calculations that the electron-phonon coupling matrix element ${\mathcal{M}}_{e\text{\ensuremath{-}}\mathit{ph}}$ of the radial breathing mode in single-walled carbon nanotubes depends strongly on tube chirality. For nanotubes of the same diameter the coupling strength ${\ensuremath{\mid}{\mathcal{M}}_{e\text{\ensuremath{-}}\mathit{ph}}\ensuremath{\mid}}^{2}$ is up to one order of magnitude stronger for zigzag tubes than for armchair tubes. For $({n}_{1},{n}_{2})$ tubes ${\mathcal{M}}_{e\text{\ensuremath{-}}\mathit{ph}}$ depends on the value of $({n}_{1}\ensuremath{-}{n}_{2})\phantom{\rule{0.3em}{0ex}}\mathrm{mod}\phantom{\rule{0.3em}{0ex}}3$, which allows us to discriminate semiconducting nanotubes with similar diameter by their Raman scattering intensity. We show measured resonance Raman profiles of the radial breathing mode which support our theoretical predictions.

Quantum interferences in the Raman cross section for the radial breathing mode in metallic carbon nanotubes

The lineshapes of the Raman excitation profiles for radial breathing modes in carbon nanotubes are shown to be strongly affected by interference effects that arise whenever strong optical transitions are separated by a small energy. This is the case in metallic zigzag and chiral tubes, where one-dimensional singularities in the electronic joint density of states are split due to the trigonal warping of the electronic band structure of two-dimensional graphene. It is shown that the proper modeling of these interferences is crucial for the identification of the (n,m) indices using Raman spectroscopy.

Electron transport properties of the finite double-walled carbon nanotubes

Based upon the Landauer formula,we have studied electron transport properties of finite incommensurate and commensurate double_walled carbon nanotubes(DWNTs).The results obtained show that the geometrical structures of the tube may produce significant effects on their electron transport properties:Electron transport of the incommensurate DWNTs may be either ballistic or non_ballistic,depending on the energy region.The commensurate DWNTs formed by the armchair tubes may displa y quick and slow oscillation;however,those formed by the zigzag tubes have no re gular slow oscillation.