Size, Shape, and Low Energy Electronic Structure of Carbon Nanotubes

Abstract

A theory of the long wavelength low energy electronic structure of graphite-derived nanotubules is presented. The propagating π 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.Since the discovery of a new family of carbon based structures formed by folding graphite sheets into compact tube-shaped objects, there has been interest in the electronic properties which can be realized with these structures [1]. It is now understood that these tubes exhibit insulating, semimetallic or metallic behavior depending on the helicity of the mapping of the graphite sheet onto the surface of the tube [2][3][4]. Discrete microscopic defects, in the form of disclination pairs, provide an interface between neighboring straight tubule segments of different helicities with different electronic gaps, providing a novel class of elemental heterojunctions [5].In this Letter we investigate the effects of shape fluctuations on the electronic properties of the carbon nanotubes. We present a new formulation of this problem which allows us to study the effects of geometry on the quantum dynamics for a π electron propagating within the surface of the wrapped graphite sheet. We show that the very rich phenomenology already well established for straight single wall cylindrical tubules can be derived directly from this geometrical theory [2][3][4]. We then extend the model to consider the effects of inhomogeneous deformations in the form of local twists and bends of the tubule on the low energy electronic structure. These are important low energy structural degrees of freedom of the tubules, and indeed one finds that these deformations are easily quenched into any three dimensional network composed of tubules. We show that these shape fluctuations also have a very strong effect on the low energy electronic and transport properties.An isolated two dimensional sheet of graphite is a semimetal, with the Fermi energy residing at a critical point in the two dimensional π electron spectrum. The Fermi surface is collapsed to a point for this system; there are two distinct Fermi points at K (K ′ ) points of the zone (±4π/3a, 0), where a is the length of the primitive translation vector (a = √ 3d where d is the nearest neighbor bond length of the graphite lattice). Expanding the π electron Hamiltonian around either of these points, and linearizing in spatial derivatives, one finds that the low energy electronic states are described by a massless two dimensional Dirac Hamiltonian, H eff = v σ · p, where p denotes a two dimensional momentum in the graphite plane, and the σ's are the 2 × 2 Pauli matrices [6]. Here the two spin polarizations of the particle refe...

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