Localization of the electronic excitations in single-walled carbon nanotubes with embedded line impurities

A matrix operator formalism is used to study the excitations in long, single-walled carbon nanotubes with the dynamic electronic properties described by a tight-binding model where the interactions between atoms take place via nearest-neighbour hopping. Defects in the form of substitutional impurity atoms are introduced to study the localized electronic modes of the nanotube as well as the propagating modes of the pure (host) material. The impurities are assumed to have the form of one or more line defects parallel to the nanotube axis. Two geometric configurations are investigated corresponding to the longitudinal axis of the nanotube being parallel to either a zigzag or an armchair direction of the graphene lattice. A tridiagonal matrix technique is employed to solve the electronic operator equations that provide a description of the frequencies of the discrete modes of the system and their spatial amplitudes. Numerical examples are presented for different nanotube diameters and spatial configurations of the impurity lines.