Size, Shape and Low Energy Electronic Structure of Carbon Nanotubes

A theory of the long wavelength low energy electronic structure of graphite-derived nanotubules is presented. The propagating $\pi$ electrons are described by wrapping a massless two dimensional Dirac Hamiltonian onto a curved surface. The effects of the tubule size, shape and symmetry are included through an effective vector potential which we derive for this model. The rich gap structure for all straight single wall cylindrical tubes is obtained analytically in this theory, and the effects of inhomogeneous shape deformations on nominally metallic armchair tubes are analyzed.Comment: 5 pages, 3 postscript figure

Electric Polarization of Heteropolar Nanotubes as a Geometric Phase

The three-fold symmetry of planar boron nitride, the III-V analog to graphene, prohibits an electric polarization in its ground state, but this symmetry is broken when the sheet is wrapped to form a BN nanotube. We show that this leads to an electric polarization along the nanotube axis which is controlled by the quantum mechanical boundary conditions on its electronic states around the tube circumference. Thus the macroscopic dipole moment has an {\it intrinsically nonlocal quantum} mechanical origin from the wrapped dimension. We formulate this novel phenomenon using the Berry's phase approach and discuss its experimental consequences.Comment: 4 pages with 3 eps figures, updated with correction to Eqn (9

Electronic Structure of Carbon Nanotube Ropes

We present a tight binding theory to analyze the motion of electrons between carbon nanotubes bundled into a carbon nanotube rope. The theory is developed starting from a description of the propagating Bloch waves on ideal tubes, and the effects of intertube motion are treated perturbatively in this basis. Expressions for the interwall tunneling amplitudes between states on neighboring tubes are derived which show the dependence on chiral angles and intratube crystal momenta. We find that conservation of crystal momentum along the tube direction suppresses interwall coherence in a carbon nanorope containing tubes with random chiralities. Numerical calculations are presented which indicate that electronic states in a rope are localized in the transverse direction with a coherence length corresponding to a tube diameter.Comment: 15 pages, 10 eps figure

The Electronic Spectrum of Fullerenes from the Dirac Equation

The electronic spectrum of sheets of graphite (plane honeycomb lattice) folded into regular polihedra is studied. A continuum limit valid for sufficiently large molecules and based on a tight binding approximation is derived. It is found that a Dirac equation describes the flat graphite lattice. Curving the lattice by insertion of odd numbered rings can be mimicked by coupling effective gauge fields. In particular the $C_{60}$ and related molecules are well described by the Dirac equation on the surface of a sphere coupled to a color monopole sitting at its center.Comment: 29 pages, 7 figures. IASSNS-HEP-92/5

Dimerization structures on the metallic and semiconducting fullerene tubules with half-filled electrons

Possible dimerization patterns and electronic structures in fullerene tubules as the one-dimensional pi-conjugated systems are studied with the extended Su-Schrieffer-Heeger model. We assume various lattice geometries, including helical and nonhelical tubules. The model is solved for the half-filling case of $\pi$-electrons. (1) When the undimerized systems do not have a gap, the Kekule structures prone to occur. The energy gap is of the order of the room temperatures at most and metallic properties would be expected. (2) If the undimerized systems have a large gap (about 1eV), the most stable structures are the chain-like distortions where the direction of the arranged trans-polyacetylene chains is along almost the tubular axis. The electronic structures are ofsemiconductors due to the large gap.Comment: submitted to Phys. Rev. B, pages 15, figures 1

Electronic states of metallic and semiconducting carbon nanotubes with bond and site disorder

Disorder effects on the density of states in carbon nanotubes are analyzed by a tight binding model with Gaussian bond or site disorder. Metallic armchair and semiconducting zigzag nanotubes are investigated. In the strong disorder limit, the conduction and valence band states merge, and a finite density of states appears at the Fermi energy in both of metallic and semiconducting carbon nanotubes. The bond disorder gives rise to a huge density of states at the Fermi energy differently from that of the site disorder case. Consequences for experiments are discussed.Comment: Phys. Rev. B: Brief Reports (to be published). Related preprints can be found at http://www.etl.go.jp/~harigaya/NEW.htm

Van Hove Singularities in disordered multichannel quantum wires and nanotubes

We present a theory for the van Hove singularity (VHS) in the tunneling density of states (TDOS) of disordered multichannel quantum wires, in particular multi-wall carbon nanotubes. We assume close-by gates which screen off electron-electron interactions. Diagrammatic perturbation theory within a non-crossing approximation yields analytical expressions governing the disorder-induced broadening and shift of VHS's as new subbands are opened. This problem is nontrivial because the (lowest-order) Born approximation breaks down close to the VHS. Interestingly, compared to the bulk case, the boundary TDOS shows drastically altered VHS, even in the clean limit.Comment: 4 pages, 2 figures, accepted with revisions in PR

Coherent Control of Photocurrents in Graphene and Carbon Nanotubes

Coherent one photon ($2 \omega$) and two photon ($\omega$) electronic excitations are studied for graphene sheets and for carbon nanotubes using a long wavelength theory for the low energy electronic states. For graphene sheets we find that coherent superposition of these excitations produces a polar asymmetry in the momentum space distribution of the excited carriers with an angular dependence which depends on the relative polarization and phases of the incident fields. For semiconducting nanotubes we find a similar effect which depends on the square of the semiconducting gap, and we calculate its frequency dependence. We find that the third order nonlinearity which controls the direction of the photocurrent is robust for semiconducting t ubes and vanishes in the continuum theory for conducting tubes. We calculate corrections to these results arising from higher order crystal field effects on the band structure and briefly discuss some applications of the theory.Comment: 12 pages in RevTex, 6 epsf figure

Electronic instabilities in 3D arrays of small-diameter (3,3) carbon nanotubes

We investigate the electronic instabilities of the small-diameter (3,3) carbon nanotubes by studying the low-energy perturbations of the normal Luttinger liquid regime. The bosonization approach is adopted to deal exactly with the interactions in the forward-scattering channels, while renormalization group methods are used to analyze the low-energy instabilities. In this respect, we take into account the competition between the effective e-e interaction mediated by phonons and the Coulomb interaction in backscattering and Umklapp channels. Moreover, we apply our analysis to relevant experimental conditions where the nanotubes are assembled into large three-dimensional arrays, which leads to an efficient screening of the Coulomb potential at small momentum-transfer. We find that the destabilization of the normal metallic behavior takes place through the onset of critical behavior in some of the two charge stiffnesses that characterize the Luttinger liquid state. From a physical point of view, this results in either a divergent compressibility or a vanishing renormalized velocity for current excitations at the point of the transition. We observe anyhow that this kind of critical behavior occurs without the development of any appreciable sign of superconducting correlations.Comment: 10 pages, 12 figure