We show that a one-dimensional topological superconductor can be realized in carbon nanotubes, using a relatively small magnetic field. Our analysis relies on the intrinsic curvature-enhanced spin-orbit coupling of the nanotubes, as well as on the orbital effect of a magnetic flux threaded through the nanotube. Tuning experimental parameters, we show that a half-metallic state may be induced in the nanotube. Coupling the system to an Ising superconductor, with an appreciable spin-triplet component, can then drive the nanotube into a topological superconducting phase. The proposed scheme is investigated by means of real-space tight-binding simulations, accompanied by an effective continuum low-energy theory, which allows us to gain some insight on the roles of different terms in the Hamiltonian. We calculate the topological phase diagram and ascertain the existence of localized Majorana zero modes near the edges. Moreover, we find that in the absence of a magnetic field, a regime exists where sufficiently strong interactions drive the system into a time-reversal-invariant topological superconducting phase.
We present a new mechanism of carbon nanotube superconductivity that originates from edge states which are specific to graphene. Using on-site and boundary deformation potentials which do not cause bulk superconductivity, we obtain an appreciable transition temperature for the edge state. As a consequence, a metallic zigzag carbon nanotube having open boundaries can be regarded as a natural superconductor/normal metal/superconductor junction system, in which superconducting states are developed locally at both ends of the nanotube and a normal metal exists in the middle. In this case, a signal…
We have altered the superconductivity of a suspended rope of single walled carbon nanotubes, by coating it with organic polymers. Upon coating, the normal state resistance of the rope changes by less than 20 percent. But superconductivity, which on the bare rope shows up as a substantial resistance decrease below 300 mK, is gradualy suppressed. We correlate this to the suppression of radial breathing modes, measured with Raman Spectroscopy on suspended Single and Double-walled carbon nanotubes. This points to the breathing phonon modes as being responsible for superconductivity in carbon nanot…
Long linear carbon chains encapsulated inside carbon nanotubes are a very close realization of carbyne, the truly one-dimensional allotrope of carbon. Here we study individual pairs of double-walled carbon nanotubes and encapsulated linear carbon chains by tip-enhanced Raman scattering. We observe that the radial breathing mode of the inner nanotube correlates with the frequency of the carbon chain's Raman mode, revealing that the nanotube chirality determines the vibronic and electronic properties of the encapsulated carbon chain. We provide the missing link that connects the properties of th…
We propose the model of a manifold of one-dimensional interacting electron systems to account for the superconductivity observed in ropes of nanotubes. We rely on the strong suppression of single-particle hopping between neighboring nanotubes in a disordered rope and conclude that the tunnelling takes place in pairs of electrons, which are formed within each nanotube due to the existence of large superconducting correlations. Our estimate of the transition temperature is consistent with the values that have been measured experimentally in ropes with about 100 metallic nanotubes.
We present ab initio calculations on the band structure and density of states of single wall semiconducting carbon nanotubes with high degrees (up to 25%) of B, Si and N substitution. The doping process consists of two phases: different carbon nanotubes (CNTs) for a constant doping rate and different doping rates for the zigzag (8, 0) carbon nanotube. We analyze the doping dependence of nanotubes on the doping rate and the nanotube type. Using these results, we select the zigzag (8, 0) carbon nanotube for toxic gas sensor calculation and obtain the total and partial densities of states for CNT…
In carbon nanotubes, the most abundant defects, caused for example by irradiation or chemisorption treatments, are small perturbing clusters, i.e. bi-site defects, extending over both A and B sites. The relative positions of these perturbing clusters play a crucial role in determining the electronic properties of carbon nanotubes. Using bandstructure and electronic transport calculations, we find out that in the case of armchair metallic nanotubes a band gap opens up when the clusters fulfill a certain periodicity condition. This phenomenon might be used in future nanoelectronic devices in whi…
We investigate the thermal conductivity of four types of deformed carbon nanotubes by using the nonequilibrium molecular dynamics method. It is reported that various deformations have different influence on the thermal properties of carbon nanotubes. For the bending carbon nanotubes, the thermal conductivity is independent on the bending angle. However, the thermal conductivity increases lightly with XY-distortion and decreases rapidly with Z-distortion. The thermal conductivity does not change with the screw ratio before the breaking of carbon nanotubes but decreases sharply after the critica…
The entanglement of two atoms (ions) doped into a carbon nanotube has been investigated theoretically. Based on the photon Green function formalism for quantizing electromagnetic field in the presence of carbon nanotubes, small-diameter metallic nanotubes are shown to result in a high degree of the two-qubit atomic entanglement for long times due to the strong atom-field coupling.
We use numerical simulations to analyze recent experimental measurements of short-channel carbon nanotube field-effect transistors with palladium contacts. We show that the gate strongly modulates the contact properties, an effect that is distinct from that observed in Schottky barrier carbon nanotube transistors. This modulation of the contacts by the gate allows for the realization of superior subthreshold swings for short channels, and improved scaling behavior. These results further elucidate the behavior of carbon nanotube-metal contacts, and should be useful in the optimization of high-p…
The objective of the present paper is to investigate the mechanical properties of carbon nanotubes. We use classical molecular dynamics simulation in order to study the effect of external compression, bending and torsion to nanotubes. We find that in the low temperature limit, the nanotubes are resilient, sustaining extreme strain without signs of brittleness or plasticity. For high temperatures and under tension the nanotubes show brittle or plastic behavior.
We develop the Bardeen-Cooper-Schrieffer (BCS) formalism for the superconductivity of carbon nanotubes. It is found that the superconducting transition temperature Tc of single-wall carbon nanotubes decreases exponentially with the increase of the tube diameter because the density of states near the Fermi energy is inversely proportional to the tube diameter. For the multi-wall carbon nanotubes, the Cooper paring hopping between layers enhances the superconducting correlation and increases the superconducting transition temperature, which is consistent with the experimental observation.
We propose a nonequilibrium Green's function approach to calculate the ac conductance of various finite-length carbon nanotubes. The simulated ac conductance differs significantly from that of the infinite-length ones. At the low-frequency limit, the profiles of the quantized conductance are still observable in the finite-length carbon nanotubes, but many more peaks appear on the conductance curves. We also show that the conductance of finite-length carbon nanotubes will oscillate as a function of the ac frequency. The dependence of the oscillation on the lengths, helicities and defects of the…
Extensive molecular dynamics simulations of water permeation and ion selectivity of the single-walled carbon nanotubes with the radial deformation are presented . The simulated results indicate that there is a close relationship between the minor axis of deformed carbon nanotubes and the variety, density as well as the position of functional groups. The critical minor axis of different diameter carbon nanotubes exists, and the carbon nanotube whose minor axis is less than the critical minor axis owns the selectivity of chlorine and sodium ions. Meanwhile, compared with intrinsic carbon nanotub…
Photonic crystal nanocavities are used to enhance photoluminescence from single-walled carbon nanotubes. Micelle-encapsulated nanotubes are deposited on nanocavities within Si photonic crystal slabs and confocal microscopy is used to characterize the devices. Photoluminescence spectra and images reveal nanotube emission coupled to nanocavity modes. The cavity modes can be tuned throughout the emission wavelengths of carbon nanotubes, demonstrating the ability to enhance photoluminescence from a variety of chiralities.
Two paramount challenges in carbon nanotube research are achieving chirality-controlled synthesis and understanding chirality-dependent device physics. High-throughput and in-situ chirality and electronic structural characterization of individual carbon nanotubes is crucial for addressing these challenges. Optical imaging and spectroscopy has unparalleled throughput and specificity, but its realization for single nanotubes on substrates or in devices has long been an outstanding challenge. Here we demonstrate video-rate imaging and in-situ spectroscopy of individual carbon nanotubes on various…
We analyze optical absorption by atomically doped carbon nanotubes with a special focus on the frequency range close to the atomic transition frequency. We derive the optical absorbtion line-shape function and, having analyzed particular achiral nanotubes of different diameters, predict the effect of absorbtion line splitting due to strong atom-vacuum-field coupling in small-diameter nanotubes. We expect this effect to stimulate relevant experimental efforts and thus to open a path to new device applications of atomically doped carbon nanotubes in modern nanotechnologies.
We have theoretically investigated electromechanical properties of freely suspended carbon nanotubes when a current is injected into the tubes using a scanning tunneling microscope. We show that a shuttle-like electromechanical instability can occur if the bias voltage exceeds a dissipation-dependent threshold value. An instability results in large amplitude vibrations of the carbon nanotube bending mode, which modify the current-voltage characteristics of the system.
With the use of the semiclassical Boltzmann equation we have calculated a direct current (d.c) in undoped zigzag carbon nanotube (CN) by mixing two coherent electromagnetic waves with commensurate frequencies i.e and . This effect is attributed to the nonparabolicity of the electron energy band which is very strong in carbon nanotubes. We observed that the current is negative similar to that observed in superlattice. However if the phase shift lies between and there is an inversion and the current becomes positive. It is interesting to note that exhibit negative differential conductivity as ex…
We report the first observation of trions (charged excitons), three-particle bound states consisting of one electron and two holes, in hole-doped carbon nanotubes at room temperature. When p-type dopants are added to carbon nanotube solutions, the photoluminescence and absorption peaks of the trions appear far below the E11 bright exciton peak, regardless of the dopant species. The unexpectedly large energy separation between the bright excitons and the trions is attributed to the strong electron-hole exchange interaction in carbon nanotubes.
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.
We present a two-step procedure to probe hotspots of plasmon-enhanced Raman scattering with carbon nanotubes. Dielectrophoretic deposition places a small carbon nanotube bundle on top of plasmonic Au nanodimer. After 'pre-characterising' both the nanotubes and dimer structure, we subsequently use the tip of an AFM to push the bundle into the plasmonic hotspot located in the 25nm wide dimer gap, characterize its location inside the gap, and observe the onset of plasmon-enhanced Raman scattering. Evidence for the activation of the carbon nanotube's double-resonant D-mode by the near-field of the…
Deformations of single-wall carbon nanotubes are investigated within the tight-binding model with deformation-dependent hopping energies. We show that the nanotubes tend to twist and shrink spontaneously at zero temperature. The explicit values of the deformation parameters are computed for a wide range of nanotubes with varying diameter and chirality. The changes of the spectral gap associated with the spontaneous deformation are shown to depend on the chirality of the nanotubes.
A mechanism of stimulated emission of electromagnetic radiation by an electron beam in carbon nanotubes is theoretically considered. Three basic properties of carbon nanotubes, a strong slowing down of surface electromagnetic waves, ballisticity of the electron motion over typical nanotube length, and extremely high electron current density reachable in nanotubes, allow proposing them as candidates for the development of nano-scale Chernekov-type emitters, analogous to traveling wave tube and free electron laser. Dispersion equations of the electron beam instability and the threshold condition…
We present a comprehensive characterization of an individual multiwalled carbon nanotube transport device combining electron microscopy and Raman spectroscopy with electrical measurements. Each method gives complementary information that mutually help to interpret each other. A sample design that allows for combining these investigation methods on individual carbon nanotube devices is introduced. This offers a direct correlation of transport features and shifts of Raman modes with structural properties as e.g. the contact interface and the morphology of the nanotube.
The Rashba spin-orbit coupling in carbon nanotubes and its effect on spin-dependent transport properties are analyzed theoretically. We focus on clean non-interacting nanotubes with tunable number of subbands $N$. The peculiar band structure is shown to allow in principle for Datta-Das oscillatory behavior in the tunneling magnetoresistance as a function of gate voltage, despite the presence of multiple bands. We discuss the conditions for observing Datta-Das oscillations in carbon nanotubes.
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 …
The ability to tune the thermal resistance of carbon nanotube mechanical supports from insulating to conducting could permit the next generation of thermal management devices. Here, we demonstrate fabrication techniques for carbon nanotube supports that realize either weak or strong thermal coupling, selectively. Direct imaging by in-situ electron thermal microscopy shows that the thermal contact resistance of a nanotube weakly-coupled to its support is greater than 250 K*m/W and that this value can be reduced to 4.2(+5.6/-2.1) K*m/W by imbedding the nanotube in metal contacts.
We present an investigation of different thin-film evaporated ferromagnetic materials for their suitability as electrodes in individual single-wall and multi-wall carbon nanotube-based spin devices. Various electrode shapes made from permalloy (Ni_{81}Fe_{19}), the diluted ferromagnet PdFe, and PdFe/Fe bilayers are studied for both their micromagnetic properties and their contact formation to carbon nanotubes. Suitable devices are tested in low-temperature electron transport measurements, displaying the typical tunneling magnetoresistance of carbon nanotube pseudo spin valves.
Hysteretic switching in the magnetoresistance of short-channel, ferromagnetically contacted individual single wall carbon nanotubes is observed, providing strong evidence for nanotube spin transport. By varying the voltage on a capacitively coupled gate, the magnetoresistance can be reproducibly modified between +10% and -15%. The results are explained in terms of wave vector matching of the spin polarized electron states at the ferromagnetic / nanotube interfaces.
Centimeter scale aligned carbon nanotube arrays are grown from nanoparticle metal catalyst pads. We find the nanotubes grow both with and against the wind. A metal underlayer provides in-situ electrical contact to these long nanotubes with no post growth processing needed. Using the electrically contacted nanotubes, we study electrical transport of 0.4 cm long nanotubes. The source drain I-V curves are quantitatively described by a classical, diffusive model. Our measurements show that the outstanding transport properties of nanotubes can be extended to the cm scale and open the door to large …
We present atomically-resolved STM images of single-wall carbon nanotubes (SWNTs) embedded in a crystalline nanotube rope. Although they may be interpreted as of a chiral nanotube, the images are more consistently explained a an achiral armchair tube with a quenched twist distortion. The existence of quenched twists in SWNTs in ropes might explain the fact that both as-grown bulk nanotube material and individual ropes have insulator-like conductivity at low temperature.
Raman measurements in carbon allotropes are generally associated with the exploration of the vibrational modes. Here, we present a theory of the non-resonant inelastic light scattering accompanied by the excitations of intersubband electron-hole pairs in carbon nanotubes and predict the selection rules and polarization properties of the dominant intersubband Raman active modes.
We have contacted single-walled carbon nanotubes after aligning the tubes by the use of surface acoustic waves. The acoustoelectric current has been measured at 4.2 K and a probing of the low-dimensional electronic states by the surface acoustic wave has been detected. By decreasing the acoustic wavelength resulting in an adjustment to the length of the defined carbon nanotube constriction a quantization of the acoustoelectric current has been observed.
Photoluminescent carbon nanotubes are expected to become versatile room-temperature single-photon sources that have applications in quantum information processing. Quantum emission from carbon nanotubes is often induced by localization of excitons or exciton-exciton annihilation. Here, we modify the local energy landscape of excitons by decorating nanoscale pentacene particles onto air-suspended single-walled carbon nanotubes. Directional exciton diffusion from the undecorated region to the decorated site is demonstrated, suggesting exciton trapping induced by local dielectric screening from p…
We study theoretically the interactions of excitonic states with surface electromagnetic modes of small-diameter (~1 nm) semiconducting single-walled carbon nanotubes. We show that these interactions can result in strong exciton-surface-plasmon coupling. The exciton absorption lineshapes exhibit the line (Rabi) splitting $~0.1-0.3$ eV as the exciton energy is tuned to the nearest interband surface plasmon resonance of the nanotube. We expect this effect to open a path to new optoelectronic device applications of semiconducting carbon nanotubes.
Direct measurements of the diffusion length of excitons in air-suspended single-walled carbon nanotubes are reported. Photoluminescence microscopy is used to identify individual nanotubes and to determine their lengths and chiral indices. Exciton diffusion length is obtained by comparing the dependence of photoluminescence intensity on the nanotube length to numerical solutions of diffusion equations. We find that the diffusion length in these clean, as-grown nanotubes is significantly longer than those reported for micelle-encapsulated nanotubes.
We demonstrate theoretically that an off-resonant circularly polarized electromagnetic field can induce a persistent current in carbon nanotubes, which corresponds to electron rotation about the nanotube axis. As a consequence, the nanotubes acquire magnetic moment along the axis, which depends on their crystal structure and can be detected in state-of-the-art measurements. This effect and related phenomena are analyzed within the developed Floquet theory describing the electronic properties of the nanotubes irradiated by the field.
In this paper, we present the results of helium temperature transport measurements through carbon nanotubes using an AFM conductive tip as a mobile gate. In semiconducting nanotubes we observe shifting of the conductance peaks with changing of the AFM tip position. This result can be explained with a particle in the box quantum model. In high quality metallic nanotubes we observe that the local Coulomb potential does not destroy the fourfold degeneracy of the energy levels.
Recent measurements in single-walled carbon nanotubes show that, on resonance, all nanotubes display the same peak optical conductivity of approximately 8 $e^2/h$, independent of radius or chirality [Joh \emph{et al.}, \emph{Nature Nanotechnology} \textbf{6}, 51 (2011)]. We show that this uniform peak conductivity is a consequence of the relativistic band structure and strength of the Coulomb interaction in carbon nanotubes. We further construct a minimalist model of exciton dynamics that describes the general phenomenology and provides an accurate prediction of the numerical value of the peak…
Single wall carbon nanotubes cooled to cryogenic temperatures are outstanding electronic as well as nano-electromechanical model systems. To probe a largely unperturbed system, we measure a suspended carbon-nanotube device where the nanotube is grown in the last fabrication step, thus avoiding damage and residues from subsequent processing. In this ultra-clean device, we observe the transport spectrum and its interaction with nano-electromechanics over a wide gate voltage range and thereby over a wide range of coupling parameters between the quantum dot and the contact electrodes.
Magnetic response of interacting electrons in a zigzag carbon nanotube threaded by a magnetic flux is investigated within a Hartree-Fock mean field approach. Following the description of energy spectra for both non-interacting and interacting cases we analyze the behavior of persistent current in individual branches of a nanotube. Our present investigation leads to a possibility of getting a filling-dependent metal-insulator transition in a zigzag carbon nanotube.
Zitterbewegung (ZB, trembling motion) of electrons in semiconductor carbon nanotubes is described taking into account dephasing processes. The density matrix formalism is used for the theory. Differences between decay of ZB oscillations due to electron localization and that due to dephasing are discussed.
The efficiencies of photonic devices are primarily governed by radiative quantum efficiency, which is a property given by the light emitting material. Quantitative characterization for carbon nanotubes, however, has been difficult despite being a prominent material for nanoscale photonics. Here we determine the radiative quantum efficiency of bright excitons in carbon nanotubes by modifying the exciton dynamics through cavity quantum electrodynamical effects. Silicon photonic crystal nanobeam cavities are used to induce the Purcell effect on individual carbon nanotubes. Spectral and temporal b…
We investigate the operational characteristics of a nanorelay based on a conducting carbon nanotube placed on a terrace in a silicon substrate. The nanorelay is a three terminal device that acts as a switch in the GHz regime. Potential applications include logic devices, memory elements, pulse generators, and current or voltage amplifiers.
We demonstrate charge pumping in semiconducting carbon nanotubes by a traveling potential wave. From the observation of pumping in the nanotube insulating state we deduce that transport occurs by packets of charge being carried along by the wave. By tuning the potential of a side gate, transport of either electron or hole packets can be realized. Prospects for the realization of nanotube based single-electron pumps are discussed.
Effects of impurity on the spin-dependent transport in a single wall carbon nanotube spin-valve, as ferromagnetic electrode/carbon nanotube/ferromagnetic electrode model junction is numerically investigated. Using a generalized Green's function method and the Landauer-Buttiker formalism, the impurity conditions are determined by randomly substitution of carbon atoms in the honeycomb carbon nanotube lattice by nitrogen and boron atoms. We have found that transport characteristics, including the spin-dependent current and tunnel magnetoresistance are strongly influenced by the impurity effects. …
Carbon nanotubes bridge the molecular and crystalline quantum worlds, and their extraordinary electronic, mechanical and optical properties have attracted enormous attention from a broad scientific community. We review the basic principles of fabricating spin-electronic devices based on individual, electrically-gated carbon nanotubes, and present experimental efforts to understand their electronic and nuclear spin degrees of freedom, which in the future may enable quantum applications.
We demonstrate an individual single-walled carbon nanotube light emitter integrated onto a microcavity and a waveguide operating in the telecom wavelength regime. Light emission from the carbon nanotube is enhanced at the cavity resonance and is efficiently extracted from the waveguide facet. We have transferred carbon nanotubes to a nanobeam cavity with a dry process, ensuring that an individual carbon nanotube is used. The guided light emission from a chirality-identified single carbon nanotube has a narrow linewidth of less than 1.3 nm and an off-resonance rejection of $\sim$17 dB. The wave…
We demonstrate a possibility for exciton Bose-Einstein condensation in individual small-diameter (~1-2 nm) semiconducting carbon nanotubes. The effect occurs under the exciton-interband-plasmon coupling controlled by an external electrostatic field applied perpendicular to the nanotube axis. It requires fields ~1 V/nm and temperatures below 100 K that are experimentally accessible. The effect offers a testing ground for fundamentals of condensed matter physics in one dimension and opens up perspectives to develop tunable coherent polarized light source with carbon nanotubes.
The temperature distributions in current-carrying carbon nanotubes have been measured with a scanning thermal microscope. The obtained temperature profiles reveal diffusive and dissipative electron transport in multi-walled nanotubes and in single-walled nanotubes when the voltage bias was higher than the 0.1-0.2 eV optical phonon energy. Over ninety percent of the Joule heat in a multi-walled nanotube was found to be conducted along the nanotube to the two metal contacts. In comparison, about eighty percent of the Joule heat was transferred directly across the nanotube-substrate interface for…
We report the observation of thermally driven mechanical vibrations of suspended doubly clamped carbon nanotubes, grown by chemical vapor deposition (CVD). Several experimental procedures are used to suspend carbon nanotubes. The vibration is observed as a blurring in images taken with a scanning electron microscope. The measured vibration amplitudes are compared with a model based on linear continuum mechanics.
The measured electric resistance of carbon nanotubes wrapped with DNA molecules depends strongly on the spin of the injected electrons. Motivated by these experiments, we study the effect of helix-shaped potentials on the electronic spectrum of carbon nanotubes. We find that in combination with the curvature-induced spin-orbit coupling inherent to nanotubes, such a perturbation opens helicity-dependent gaps. Within these partial gaps, left-moving electrons carry a fixed spin-projection that is reversed for right-moving electrons, and the probability of electrons to transfer through the nanotub…
We find that electrons in single-wall carbon nanotubes may propagate substantial distances (tens of nanometers) under the metal contacts. We perform four-probe transport measurements of the nanotube conductance and observe significant deviations from the standard Kirchhoff's circuit rules. Most noticeably, injecting current between two neighboring contacts on one end of the nanotube, induces a non-zero voltage difference between two contacts on the other end.
We study theoretically the electrical transport between aligned carbon nanotubes in nanotube ropes, and between shells in multiwall carbon nanotubes. We focus on transport between two metallic nanotubes (or shells) of different chiralities with mismatched Fermi momenta and incommensurate periodicities. We perform numerical calculations of the transport properties of such systems within a tight-binding formalism. For clean (disorder-free) nanotubes the intertube transport is strongly suppressed as a result of momentum conservation. For clean nanotubes, the intertube transport is typically domin…
We investigate theoretically the feasibility of generation and amplification of terahertz radiation in aligned achiral carbon nanotubes (zigzag and armchair) in comparison with a superlattice in the presence of a constant (dc) and high-frequency (ac) electric fields. The electric current density expression is derived using the semiclassical Boltzmann transport equation with a constant relaxation time with the electric field applied along the nanotube axis. Our analysis on the current density versus electric field characteristics demonstrates negative differential conductivity at high frequency…
We report the first observation of photon antibunching in the photoluminescence from single carbon nanotubes. The emergence of a fast luminescence decay component under strong optical excitation indicates that Auger processes are partially responsible for inhibiting two-photon generation. Additionally, the presence of exciton localization at low temperatures ensures that nanotubes emit photons predominantly one by one. The fact that multiphoton emission probability can be smaller than 5% suggests that carbon nanotubes could be used as a source of single photons for applications in quantum cryp…
The phonon dispersion of chiral single wall carbon nanotubes has been obtained from $6\times 6$ dyanimic matrix. The present manuscript is the extension of G. D. Mahan and Gun Sang Jeon's work on armchair and zigzag nanotubes. [see P. R. B., {\bf 70}, 075405 (2004)]. We use spring and mass model with the proper phonon potential suggested by \cite{M2}. We can calculate the dispersion of single wall carbon nanotubes near $Γ$ point with arbitrary chirality. The results are compatible with Mahan et. al's results for armchair and zigzag SWNTs.
