Electromagnetic characteristics of single-walled finite-length carbon nanotubes - absorption cross-section and field enhancement in the near zone - are theoretically studied in a wide frequency range from terahertz to visible. The analysis is based on the impedance-type effective boundary conditions and the integral equation technique. Comparison with experimental results is carried out allowing qualitative physical interpretation of low-frequency (far-IR and terahertz) absorption band observed in experiments. Potentiality of CNTs for the IR photothermolysis of living cells is discussed. Strong local field enhancement is predicted to be inherent to metallic CNTs in the near-field zone providing necessary mechanism for far-IR and terahertz near-field optics.
We present an approach for time-dependent quantum transport based on a self-consistent non-equilibrium Green function formalism. The technique is applied to a ballistic carbon nanotube transistor in the presence of a time harmonic signal at the gate. In the ON state the dynamic conductance exhibits plasmonic resonant peaks at terahertz frequencies. These vanish in the OFF state, and the dynamic conductance displays smooth oscillations, a signature of single particle quantum effects. We show that the nanotube kinetic inductance plays an essential role in the high-frequency behavior.
The controlled introduction of covalent sp$^{3}$ defects into semiconducting single-walled carbon nanotubes (SWCNTs) gives rise to exciton localization and red-shifted near-infrared luminescence. The single-photon emission characteristics of these functionalized SWCNTs make them interesting candidates for electrically driven quantum light sources. However, the impact of sp$^{3}$ defects on the carrier dynamics and charge transport in carbon nanotubes remains an open question. Here, we use ultrafast, time-resolved optical-pump terahertz-probe spectroscopy as a direct and quantitative technique …
We describe a film of highly-aligned single-walled carbon nanotubes that acts as an excellent terahertz linear polarizer. There is virtually no attenuation (strong absorption) when the terahertz polarization is perpendicular (parallel) to the nanotube axis. From the data we calculated the reduced linear dichrosim to be 3, corresponding to a nematic order parameter of 1, which demonstrates nearly perfect alignment as well as intrinsically anisotropic terahertz response of single-walled carbon nanotubes in the film.
We use electrostatic force microscopy and scanned gate microscopy to probe the conducting properties of carbon nanotubes at room temperature. Multi-walled carbon nanotubes are shown to be diffusive conductors, while metallic single-walled carbon nanotubes are ballistic conductors over micron lengths. Semiconducting single-walled carbon nanotubes are shown to have a series of large barriers to conduction along their length. These measurements are also used to probe the contact resistance and locate breaks in carbon nanotube circuits.
The stabilization of carbon nanotubes via the filling with inner tubes is demonstrated by probing the optical transitions in double-walled carbon nanotube bundles under hydrostatic pressure with optical spectroscopy. Double-walled carbon nanotube films were prepared from fullerene peapods and characterized by HRTEM and optical spectroscopy. In comparison to single-walled carbon nanotubes, the pressure-induced redshifts of the optical transitions in the outer tubes are significantly smaller below $\sim$10 GPa, demonstrating the enhanced mechanical stability due to the inner tube already at low …
We formulate and justify several proposals utilizing the unique electronic properties of different types of carbon nanotubes for a broad range of applications to THz optoelectronics, including THz generation by hot electrons in quasi-metallic nanotubes, frequency multiplication in chiral-nanotube-based superlattices controlled by a transverse electric field, and THz radiation detection and emission by armchair nanotubes in a strong magnetic field.
At any metal-carbon nanotube interface there is charge transfer and the induced interfacial field determines the position of the carbon nanotube band structure relative to the metal Fermi-level. In the case of a single-wall carbon nanotube (SWNT) supported on a gold substrate, we show that the charge transfers induce a local electrostatic potential perturbation which gives rise to the observed Fermi-level shift in scanning tunneling spectroscopy (STS) measurements. We also discuss the relevance of this study to recent experiments on carbon nanotube transistors and argue that the Fermi-level al…
Based on the similarity between the chiral nanotube and the classical solenoid, we study chiral transport along the circumferential direction in a carbon nanotube. We calculate the chiral conductivity, representing a circumferential current induced by an electric field along the nanotube axis for various chiralities of carbon nanotubes. We find that the chiral conductivity in a chiral nanotube is in general non zero, and oscillates as a function of the Fermi energy. This oscillating behavior is attributed to the subband structure of the nanotubes and the warping of the Fermi surfaces.
Electronic properties of metal-finite semiconducting carbon nanotube interfaces are studied as a function of the nanotube length using a self-consistent tight-binding theory. We find that the shape of the potential barrier depends on the long-range tail of the charge transfer, leading to an injection barrier thickness comparable to half of the nanotube length until the nanotube reaches the bulk limit. The conductance of the nanotube junction shows a transition from tunneling to thermally-activated transport with increasing nanotube length.
We present a detailed study of the static enhancement effects of electric charges in micron-length single-walled carbon nanotubes, using theoretically an atomic charge-dipole model and experimentally electrostatic force microscopy. We demonstrate that nanotubes exhibit at their ends surprisingly weak charge enhancements which decrease with the nanotube length and increase with the nanotube radius. A quantitative agreement is obtained between theory and experiments.
Electroabsorption spectroscopy of well-identified index-defined semiconducting carbon nanotubes is reported. The measurement of high definition electroabsorption spectra allows direct indexation with unique nanotube chirality. Results show that at least for a limited range of diameters, electroabsorption is directly proportional to the exciton binding energy of nanotubes. Electroabsorption is a powerful technique which directly probes into carbon nanotube excitonic states, and may become a useful tool for in situ study of excitons in future nanotube-based photonic devices such as electroabsorp…
The possibility for double-wall carbon nanotube to operate as the bolt and nut pair is studied. The barriers for relative motions of walls along the helical "thread" line and for jumps on neighbour helical lines are calculated as functions of wall lengths for the set of double-wall carbon nanotubes. The dynamics of relative motion of carbon nanotube walls along the helical line under the action of external forces is considered. Perforated nanodrill, variable nanoresistor and other principally new nanotube based mechanical nanodevices using these motion are proposed. Possible operation modes of…
The random phase approximation (RPA) theory is used to derive the Luttinger parameter g for metallic carbon nanotubes. The results are consistent with the Tomonaga-Luttinger models. All metallic carbon nanotubes, regardless if they are armchair tubes, zigzag tubes, or chiral tubes, should have the same Luttinger parameter g. However, a (10,10) carbon peapod should have a smaller g value than a (10,10) carbon nanotube. Changing the Fermi level by applying a gate voltage has only a second order effect on the g value. RPA theory is a valid approach to calculate plasmon energy in carbon nanotube s…
Spin transport in carbon-based materials has stimulated much interest due to their ballistic conductance and a long phase coherence length. While much research has been conducted on individual carbon nanotubes, current growth and placement techniques are incompatible with large-scale fabrication. Here we report nonlocal spin injection and detection in single wall carbon nanotube networks. We observe spin transport over a distance of 1 um, and extract a spin diffusion length of 1.6 - 2.4 um with an injected spin polarization from CoFe into nanotube network of 18 - 41%. Our observations demonstr…
We demonstrate that in a densely packed aligned array of single walled carbon nanotubes, the breakdown of one nanotube leads to a highly correlated breakdown of neighboring nanotubes, thereby producing a nano-fissure. We show that the origin of the correlation is the electrostatic field of the broken nanotubes that produces locally inhomogeneous current and Joule heating distributions in the neighboring intact nanotubes triggering their breakdowns in the vicinity of the broken nanotubes. Our results suggest that the densely aligned array behaves like a correlated solid.
Prompted by recent reports on $\sqrt{3} \times \sqrt{3}$ graphene superlattices with intrinsic inter-valley interactions, we perform first-principles calculations to investigate the electronic properties of periodically nitrogen-doped graphene and carbon nanotube nanostructures. In these structures, nitrogen atoms substitute one-sixth of the carbon atoms in the pristine hexagonal lattices with exact periodicity to form perfect $\sqrt{3} \times \sqrt{3}$ superlattices of graphene and carbon nanotubes. Multiple nanostructures of $\sqrt{3} \times \sqrt{3}$ graphene ribbons and carbon nanotubes ar…
We used tip-enhanced Raman spectroscopy to study the diameter-dependent Raman modes in a contacted carbon nanotube rope. We show that with the near-field tip enhancement a large number of nanotubes within a rope can be identified, even if the nanotube modes can not be distinguished in the far-field signal. Several metallic and semiconducting nanotubes can be identified and assigned to nanotube families. Additionally, we provide a tentative chiral index assignment.
Using single-nanotube absorption microscopy, we measured the absorption cross section of (6,5) carbon nanotubes at their second-order optical transition. We obtained a value of 3.2 10-17 cm2/C atom with a precision of 15% and an accuracy below 20%. This constitutes the first metrological investigation of the absorption cross section of chirality-identified nanotubes. Correlative absorption-luminescence microscopies performed on long nanotubes reveal a direct manifestation of exciton diffusion in the nanotube.
The unique topology and exceptional properties of carbon nanoscrolls (CNSs) have inspired unconventional nano-device concepts, yet the fabrication of CNSs remains rather challenging. Using molecular dynamics simulations, we demonstrate the spontaneous formation of a CNS from graphene on a substrate, initiated by a carbon nanotube (CNT). The rolling of graphene into a CNS is modulated by the CNT size, the carbon-carbon interlayer adhesion, and the graphene-substrate interaction. A phase diagram emerging from the simulations can offer quantitative guideline toward a feasible and robust physical …
We compute electrostatic fields induced deformations of cantilevered finite-length metallic carbon nanotubes, using an energy minimization method based on a charge-dipole moment interaction potential combined with an empirical many-body potential. The influence of field strength, field direction and tube geometry on the electrostatic deflection is investigated for both single and double walled tubes. These results could apply to nanoelectromechanical devices based on cantilevered carbon nanotubes.
The radial-breathing-like phonon modes (RBLMs) of the double-walled carbon nanotubes are studied in a simple analytical model, in which the interaction force constants (FCs) can be obtained analytically from the continuous model. The RBLMs frequencies are obtained by solving the dynamical matrix, and their relationship with the tube radii can be obtained analytically, offering a powerful experimental tool for determining precisely the radii of the multi-walled carbon nanotubes.
Although carbon nanotubes consist of honeycomb carbon, they have never been fabricated from graphene directly. Here, it is shown by quantum molecular-dynamics simulations and classical continuum-elasticity modeling, that graphene nanoribbons can, indeed, be transformed into carbon nanotubes by means of twisting. The chiralities of the tubes thus fabricated can be not only predicted but also externally controlled. This twisting route is an opportunity for nanofabrication, and is easily generalizable to ribbons made of other planar nanomaterials.
Recent years witnessed a rapid growth of interest of scientific and engineering communities to thermal properties of materials. Carbon allotropes and derivatives occupy a unique place in terms of their ability to conduct heat. The room-temperature thermal conductivity of carbon materials span an extraordinary large range - of over five orders of magnitude - from the lowest in amorphous carbons to the highest in graphene and carbon nanotubes. I review thermal and thermoelectric properties of carbon materials focusing on recent results for graphene, carbon nanotubes and nanostructured carbon mat…
We study the binding of chlorine atoms to carbon nanotubes and graphene at a Stone-Wales defect and to the sidewalls of pristine nanotubes. We show using ab initio thermodynamics that if the environment is chlorine-rich enough, the unzipping of carbon nanotubes into graphene nanoribbons with chlorinated edges may be thermodynamically feasible. By controlling the Cl chemical potential through temperature and pressure, opening selectively tubes below a threshold diameter might be possible. Additionally, we find increased binding energies for chlorine atoms bound to Stone-Wales defects as compare…
Semiconducting single-walled carbon nanotubes are studied in the diffusive transport regime. The peak mobility is found to scale with the square of the nanotube diameter and inversely with temperature. The maximum conductance, corrected for the contacts, is linear in the diameter and inverse temperature. These results are in good agreement with theoretical predictions for acoustic phonon scattering in combination with the unusual band structure of nanotubes. These measurements set the upper bound for the performance of nanotube transistors operating in the diffusive regime.
We report the realization and characterization of independently controllable tunnel barriers within a carbon nanotube. The nanotubes are mechanically bent or kinked using an atomic force microscope, and top gates are subsequently placed near each kink. Transport measurements indicate that the kinks form gate-controlled tunnel barriers, and that gates placed away from the kinks have little or no effect on conductance. The overall conductance of the nanotube can be controlled by tuning the transmissions of either the kinks or the metal-nanotube contacts.
Ultrafast terahertz spectroscopy accesses the {\em dark} excitonic ground state in resonantly-excited (6,5) SWNTs via internal, direct dipole-allowed transitions between lowest lying dark-bright pair state $\sim$6 meV. An analytical model reproduces the response which enables quantitative analysis of transient densities of dark excitons and {\em e-h} plasma, oscillator strength, transition energy renormalization and dynamics. %excitation-induced renormalization. Non-equilibrium, yet stable, quasi-1D quantum states with dark excitonic correlations rapidly emerge even with increasing off-resonan…
We report on helicity sensitive photovoltaic terahertz radiation response of a carbon nanotube made in a conguration of a field effect transistor. We find that the magnitude of the rectified voltage is different for clockwise and anticlockwise circularly polarized radiation. We demonstrate that this effect is a fingerprint of the plasma waves interference in the transistor channel. We also find that that the presence of the helicity- and phase-sensitive interference part of the photovoltaic response is a universal phenomenon which is obtained in the systems of different dimensionality with dif…
We develop a new chemical route to prepare carbon nanotubes at room temperature. Graphite powder is immersed in a mixed solution of nitric and sulfuric acid with potassium chlorate. After heating the solution up to 70°C and leaving them in the air for 3 days, we obtained carbon nanotube bundles. This process could provide an easy and inexpensive method for the preparation of carbon nanotubes.
We report on the nano-electron beam assisted fabrication of atomically sharp iron-based tips and on the creation of a nano-soldering iron for nano-interconnects using Fe-filled multiwalled carbon nanotubes (MWCNTs). High energy electron beam machining has been proven a powerful tool to modify desired nanostructures for technological applications and to form molecular junctions and interconnections between carbon nanotubes. Recent studies showed the high degree of complexity in the creation of direct interconnections between multiwalled and CNTs having dissimilar diameters. Our technique allows…
Using the first-principles spin density functional approach, we have studied magnetism of a new type of all-carbon nanomaterials, i.e., the carbon nanowires inserted into the single-walled carbon nanotubes. It is found that if the 1D carbon nanowire density is not too higher, the ferromagnetic ground state will be more stable than the antiferromagnetic one, which is caused by weak coupling between the 1D carbon nanowire and the single-walled carbon nanotube. Also, both dimerization of the carbon nanowire and carbon vacancy on the tube-wall are found to enhance the magnetic moment of the compos…
We show that the presence of electrically insulating graphite oxide (GO) within a single wall carbon nanotube network strongly enhances electrical conductivity, whereas reduced graphite oxide, even though electrically conductive, suppresses electrical conductivity within a composite network with single wall carbon nanotubes. Measurements of Young modulus and of Raman spectra strongly support our interpretation of the indirect role of the oxide groups, present in graphite oxide within the single wall carbon nanotubes/graphite oxide composite, through electronic doping of metallic single wall ca…
In this paper the new concept of super-bridges, i.e. kilometre-long bridges suspended over carbon nanotube cables, is introduced. The analysis shows that the use of realistic (thus defective) carbon nanotube bundles as suspension cables can enlarge the current limit main span by a factor of 3.
First-principles simulations were performed to investigate reaction of carbon nanotubes with adsorbed metal atoms. Mechanical modification of their structures enhances chemical reactivity of carbon nanotubes. Adsorption of a tungsten, tantalum, or niobium atom on a (5, 0) nanotube with a Stone-Wales defect was shown to have characteristically strong chemisorption. Bond-breaking in the carbon-carbon network and formation of a local metal-carbon complex were observed during the simulation. Adsorption of W, Ta, Ni or Mo on a twisted (5, 0) nanotube showed a preferred breaking of several bonds, ev…
Excitons play major roles in optical processes in modern semiconductors, such as single-wall carbon nanotubes (SWCNTs), transition metal dichalcogenides, and 2D perovskite quantum wells. They possess extremely large binding energies (>100~meV), dominating absorption and emission spectra even at high temperatures. The large binding energies imply that they are stable, that is, hard to ionize, rendering them seemingly unsuited for optoelectronic devices that require mobile charge carriers, especially terahertz emitters and solar cells. Here, we have conducted terahertz emission and photocurrent …
We show that separating metallic from semiconducting carbon nanotubes by dielectrophoresis is developing towards a bulk separation method, which allows for the first time to produce thin films of only metallic single-walled carbon nanotubes and to measure their optical absorption spectra. The data proofs that the selectivity of the separation scheme is independent from the nanotube diameter.
We have fabricated field effect transistors from carbon nanotubes using a novel selective placement scheme. We use carbon nanotubes that are covalently bound to molecules containing hydroxamic acid functionality. The functionalized nanotubes bind strongly to basic metal oxide surfaces, but not to silicon dioxide. Upon annealing, the functionalization is removed, restoring the electronic properties of the nanotubes. The devices we have fabricated show excellent electrical characteristics.
Fundamentally new families of carbon single walled nanotubes are proposed. These nanotubes, called graphynes, result from the elongation of covalent interconnections of graphite-based nanotubes by the introduction of yne groups. Similarly to ordinary nanotubes, arm-chair, zig-zag, and chiral graphyne nanotubes are possible. Electronic properties, predicted using tight-binding and ab initio density functional methods, show a rich variety of metallic and semiconducting behaviors.
The weak-chemisorption/charge-transfer picture for adsorption of aromatic molecules over carbon nanotubes, proposed in the commented paper, is criticized.
We calculate the van der Waals energy of a C60 molecule when it is encapsulated in a single-walled carbon nanotube with discrete atomistic structure. orientational degrees of freedom and longitudinal displacements of the molecule are taken into account, and several achiral and chiral carbon nanotubes are considered. A comparison with earlier work where the tube was approximated by a continuous cylindrical distribution of carbon atoms is made. We find that such an approximation is valid for high and intermediate tube radii; for low tube radii, minor chirality effects come into play. Three molec…
Electronic and optoelectronic devices based on thin films of carbon nanotubes are currently limited by the presence of metallic nanotubes. Here we present a novel approach based on nanotube alkyl functionalization to physically remove the metallic nanotubes from such network devices. The process relies on preferential thermal desorption of the alkyls from the semiconducting nanotubes and the subsequent dissolution and selective removal of the metallic nanotubes in chloroform. The approach is versatile and is applied to devices post-fabrication.
Metal catalysts supporting the growth of Single Wall Carbon Nanotubes display different carbon solubilities and chemical reactivities. In order to specifically assess the role of carbon solubility, we take advantage of the physical transparency of a tight binding model established for Ni-C alloys, to develop metal carbon models where all properties, except carbon solubility, are similar. These models are used to analyze carbon incorporation mechanisms, modifications of metal / carbon wall interfacial properties induced thereby, and the associated nanotube growth mechanisms. Fine tuning carbon …
Carboxylic functionalization (-COOH groups) of carbon nanotubes is known to improve their dispersion properties and increase the electrical conductivity of carbon-nanotube - polymer nanocomposites. We have studied experimentally the effects of this type of functionalization on the thermal conductivity of the nanocomposites. It was found that while even small quantities of carbon nanotubes (~1 wt%) can increase the electrical conductivity, a larger loading fraction (~3 wt%) is required to enhance the thermal conductivity of nanocomposites. Functionalized multi-wall carbon nanotubes performed th…
MD simulations based on an empirical potential energy surface were used to study the nucleation of bamboo-like carbon nanotubes (BCNTs). The simulations reveal that inner walls of the bamboo structure start to nucleate at the junction between the outer nanotube wall and the catalyst particle. In agreement with experimental results, the simulations show that BCNTs nucleate at higher dissolved carbon concentrations (i.e., feedstock pressures) than those where non-bamboolike carbon nanotubes are nucleated.
Large-scale fully interconnected ring carbon nanotube (CNT) networks were first prepared using thermo chemical vapor deposition in nano-channel network templates of porous anodic alumina. This conductive CNT network film consists of billions of nanotube segments with a single layer, and could be made as large as the centimeter size of the template, with a uniform two-dimensional ring topological structure. These CNT networks could be grown from tiny Fe/Co catalyst particles which were electro-deposited into the templates, or totally without additional metallic catalysts. The common interconnec…
A simple method to disperse individual single walled carbon nanotubes (SWCNT ) on an atomically flat substrate is presented. Proper tuning of ac modes of atomic force microscopes(AFM) is discussed. This is needed to discriminate between individual nanotubes and very small bundles. The distribution of lengths of the nanotubes measured by these methods is reported.
The lattice specific heat of carbon nanotubes is evaluated within the microscopic model proposed by Mahan and Jeon, published in the Physical Review B, in 2004. Phonons are considered for a single wall carbon nanotube in the armchair configuration. As expected, low temperature and high temperature regions show different behavior of specific heat. Carbon nanotubes are also displaying a very interesting lattice transport depending on the diameter, with a rather high thermal conductivity for small diameters.
Classical molecular dynamics (MD) simulations employing Brenner potential for intra-nanotube interactions and Van der Waals forces for polymer-nanotube interfaces are used to invetigate the thermal expansion and diffusion characteristics of carbon nanotube-polyethylene composites. Additions of carbon nanotubes to polymer matrix are found to increase the glass transition temperature Tg, and thermal expansion and diffusion coefficients in the composite above Tg. These findings could have implications in CNT composite processing, coating and painting applications.
This letter gives a credit to a pioneering paper that is almost unknown to scientific community. On the basis of Transmission Electron Microscopy images and X-ray Ray Diffraction patterns of carbon multi-layer tubular crystals the authors suggested a model of nanotube structure formation and hypothesis on various chirality of carbon nanotubes.
A novel method is developed to enable the formation, positioning and patterning of individual metal nano-clusters with controllable and monodispersed sizes down to 1-2 nm-scale. Among wide range of potential applications, the nanoparticle arrays are used for deterministic synthesis of monodispersed single walled carbon nanotubes. The carbon nanotubes are grown at individually controlled locations with near one-to-one yield, reaching one of the ultimate goals of nanotube synthesis.
Exploring the synthesis of novel molecular nanostructures has been in the forefront of material research in the last decade. One of the most interesting nanostructures are single wall carbon nanotubes (SWCNTs). Their catalyst free growth, however, remains an elusive goal. Here, we present the growth of single wall carbon nanotubes from organic solvents such as benzene and toluene in a confined environment, inside a host SWCNT. The solvents encapsulated in SWCNTs are transformed to an inner tube when subject to a heat treatment under dynamic vacuum at 1270 $^{\circ}$C. We used isotope labeling …
We predict new forms of carbon consisting of one and two dimensional networks of interlinked single wall carbon nanotubes, some of which are energetically more stable than van der Waals packing of the nanotubes on a hexagonal lattice. These interlinked nanotubes are further transformed with higher applied external pressures to more dense and complicated stable structures, in which curvature-induced carbon sp$^{3}$ re-hybridizations are formed. We also discuss the energetics of the bond formation between nanotubes and the electronic properties of these predicted novel structures.
Novel applications in nanotechnology rely on the design of tailored nano-architectures. For this purpose, carbon nanotubes and nanoparticles are intensively investigated. In this work we study the influence of non-functionalized carbon nanotubes on the synthesis of CdSe nanoparticles by means of organometallic colloidal routes. This new synthesis methodology does not only provide an effective path to attach nanoparticles non-covalently to carbon nanotubes but represents also a new way to control the shape of nanoparticles.
We use a multiscale procedure to derive a simple continuum model of multiwalled carbon nanotubes that takes into account both strong covalent bonds within graphene layers and weak bonds between atoms in different layers. The model predicts polygonization of crossections of large multiwalled nanotubes as a consequence of their curvature-induced turbostratic structure.
Persistent currents in rings constructed from carbon nanotubes are investigated theoretically. After studying the contribution of finite temperature or quenched disorder on covalent rings, the complexity due to the bundle packing is addressed. The case of interacting nanotori and self-interacting coiled nanotubes are analyzed in details in relation with experiments.
We theoretically compute the interface thermal resistance between crossing single walled carbon nanotubes of various chiralities, using an atomistic Green's function approach with semi-empirical potentials. The results are then used to model the thermal conductivity of three dimensional nanotube pellets in vacuum. For an average nanotube length of 1 $μ$m, the model yields an upper bound for the thermal conductivity of densely compacted pellets, of the order of a few W/m-K. This is in striking contrast with the ultra-high thermal conductivity reported on individually suspended nanotubes. The re…
A new group-theory method to relate a spectrum of carbon nanotubes with that of a graphene monolayer is elaborated. The spectrum reconstruction is performed using a virtual intermediate planar periodic structure. Selected irreducible representations of its space symmetry group and all those of the nanotube one are isomorphic and therefore span the correlated excitations. The method is applied to study the origin of the zone-center phonon modes of chiral and achiral carbon nanotubes. The structure of the G-band for particular types of nanotubes is determined. The results obtained could be usefu…
The analytical expressions for the van der Waals potential energy and force between two crossed carbon nanotubes are presented. The Lennard-Jones potential for two carbon atoms and the method of the smeared out approximation suggested by L.A. Girifalco were used. The exact formula is expressed in terms of rational and elliptical functions. The potential and force for carbon nanotubes were calculated. The uniform potential curves for single- and multi- wall nanotubes were plotted. The equilibrium distance, maximal attractive force, and potential energy have been estimated.
