Linearized augmented cylindrical wave method for chiral nanotubes

Abstract

The linearized augmented cylindrical wave (LACW) method has been developed for calculation of the electronic structure of carbon nanotubes [1‐3]. The LACW method is an extension of the well-known linearized augmented plane wave method (LAPW), one of the most exact methods in the band theory of crystals [4‐6], to cylindrical molecules. The major argument for the use of cylindrical waves to describe nanotubes is that such a choice of a basis set allows explicit consideration of the cylindrical geometry of nanotubes, which, in particular, ensures fast convergence of a computational procedure. However, all previous calculations of single- and double-walled carbon and non-carbon nanotubes by the LACW method dealt only with achiral nanotubes, ( n , n ) armchair and ( n , 0 ) zigzag nanotubes with a small number of atoms per translational unit cell [1‐3, 7‐12]. For even small-diameter chiral nanotubes, the number of atoms in the translational unit cell can be very large. For example, the translational cell of the achiral (10,10) nanotube comprises 40 atoms, whereas the translational cell of the chiral (10,9) tube of somewhat smaller diameter comprises 1084 atoms. As is known, the basis set required for convergence rapidly increases with an increase in the number of atoms in the unit cell, which renders impracticable calculations of chiral tubes. It is worth noting that alternative ab initio quantum-chemical calculations of the electronic structure of nanotubes, such as the tight binding method with an atomic basis set and the pseudopotential method with a basis set of plane waves, face the same problem as the LACW method. Therefore, all ab initio band calculations of nanotubes have been restricted to armchair and zigzag tubes or, at best, have dealt with some of the simplest chiral tubes of the smallest diameter and, thus, with a minimal number of atoms in the translational cell. These facts indicate that all rather than only translational symmetry properties of nanotubes should be considered in development of the theory of their electronic structure.