Degeneracy breaking and intervalley scattering due to short-ranged impurities in finite single-wall carbon nanotubes

TL;DR

This paper theoretically investigates how short-range impurities in finite metallic carbon nanotubes cause degeneracy breaking and intervalley scattering, using an effective mass model and boundary conditions to analyze energy shifts and pseudospin effects.

Contribution

It introduces a detailed theoretical framework for understanding impurity-induced degeneracy breaking and intervalley scattering in finite single-wall carbon nanotubes.

Findings

Impurities cause sinusoidal energy level shifts depending on their position.

Impurity position within the unit cell affects the effective Hamiltonian structure.

Symmetries relate to degeneracy breaking and pseudospin mixing in armchair nanotubes.

Abstract

We present a theoretical study of degeneracy breaking due to short-ranged impurities in finite, single-wall, metallic carbon nanotubes. The effective mass model is used to describe the slowly varying spatial envelope wavefunctions of spinless electrons near the Fermi level at two inequivalent valleys (K-points) in terms of the four component Dirac equation for massless fermions, with the role of spin assumed by pseudospin due to the relative amplitude of the wave function on the sublattice atoms (``A'' and ``B''). Using boundary conditions at the ends of the tube that neither break valley degeneracy nor mix pseudospin eigenvectors, we use degenerate perturbation theory to show that the presence of impurities has two effects. Firstly, the position of the impurity with respect to the spatial variation of the envelope standing waves results in a sinusoidal oscillation of energy level shift as a function of energy. Secondly, the position of the impurity within the hexagonal graphite unit cell produces a particular 4 by 4 matrix structure of the corresponding effective Hamiltonian. The symmetry of this Hamiltonian with respect to pseudospin flip is related to degeneracy breaking and, for an armchair tube, the symmetry with respect to mirror reflection in the nanotube axis is related to pseudospin mixing.