No abstract

Due to the chiral nature of electrons in a monolayer of graphite (graphene) one can expect weak antilocalisation and a positive weak-field magnetoresistance in it. However, trigonal warping (which breaks p → −p symmetry of the Fermi line in each valley) suppresses antilocalisation, while intervalley scattering due to atomically sharp scatterers in a realistic graphene sheet or by edges in a narrow wire tends to restore conventional negative magnetoresistance. We show this by evaluating the dependence of the magnetoresistance of graphene on relaxation rates associated with various possible ways of breaking a 'hidden' valley symmetry of the system. The chiral nature [1,2,3,4] of quasiparticles in graphene (monolayer of graphite), which originates from its honeycomb lattice structure and is revealed in quantum Hall effect measurements [5,6], is attracting a lot of interest. In recently developed graphene-based transistors [5,6] the electronic Fermi line consists of two tiny circles [7] surrounding corners K ± of the hexagonal Brillouin zone [8], and quasiparticles are described by 4-, which characterise electronic amplitudes on two crystalline sublattices (A and B), and the HamiltonianHere, we use direct products of Pauli matrices σ x,y,z , σ 0 ≡1 acting in the sublattice space (A, B) and Π x,y,z , Π 0 ≡1 acting in the valley space (K ± ) to highlight the form ofĤ in the non-equivalent valleys [8]. Near the center of each valley electron dispersion is determined by the Dirac-type part v σp ofĤ. It is isotropic and linear. For the valley K + the electronic excitations with momentum p have energy vp and are chiral with σp/p = 1, while for holes the energy is −vp and σp/p = −1. In the valley K − , the chirality is inverted: it is σp/p = −1 for electrons and σp/p = 1 for holes. The quadratic term in Eq. (1) violates the isotropy of the Dirac spectrum and causes a weak trigonal warping [8].Due to the chirality of electrons in a graphene-based transistor, charges trapped in the substrate or on its surface cannot scatter carriers in exactly the backwards direction [2,7], provided that they are remote from the graphene sheet by more than the lattice constant. In the theory of quantum transport [9] the suppression of backscattering is associated with weak anti-localisation (WAL) [10]. For purely potential scattering, possible WAL in graphene has recently been related to the Berry phase π specific to the Dirac fermions, though it has also been noticed that conventional weak localisation (WL) may be restored by intervalley scattering [11,12].In this Letter we show that the WL magnetoresistance in graphene directly reflects the degree of valley symmetry breaking by the warping term in the free-electron Hamiltonian (1) and by atomically sharp disorder. To describe the valley symmetry, we introduce two sets of 4×4 Hermitian matrices: 'isospin' Σ = (Σ x , Σ y , Σ z ) and 'pseudospin' Λ = (Λ x , Λ y , Λ z ). These are defined asand form two mutually independent algebras, [ Σ, Λ] = 0, The Dirac part ofĤ in Eq. (4), v Σp and potenti...

Electron-phonon scattering is studied within an effective-mass theory. A continuum model for acoustic phonons is introduced and electron-phonon interaction due to modification of band structure is derived as well as a normal deformation potential. In a metallic nanotube, the deformation potential does not participate in electron scattering and a metallic nanotube becomes nearly a one-dimensional ballistic conductor even at room temperature. A resistivity determined by small band-structure interaction depends on the chirality at low temperatures. A magnetic field perpendicular to the axis induces electron scattering by the deformation potential, giving rise to huge positive magnetoresistance.

A brief review is given of electronic and transport properties of carbon nanotubes obtained mainly in a kÁp scheme. The topics include a giant Aharonov-Bohm effect on the band gap and a Landau-level formation in magnetic fields, magnetic properties, interaction effects on the band structure, optical absorption spectra, and exciton effects. Transport properties are also discussed including absence of backward scattering except for scatterers with a potential range smaller than the lattice constant, its extension to multi-channel cases, a conductance quantization in the presence of short-range and strong scatterers such as lattice vacancies, and transport across junctions between nanotubes with different diameters. A continuum model for phonons in the long-wavelength limit and the resistivity determined by phonon scattering are reviewed as well.