A novel choice of the graphene unit vectors, useful in zone-folding computations

TL;DR

This paper introduces a new choice of graphene lattice vectors that simplifies zone-folding calculations for carbon nanotube dispersion relations, making the process more efficient and easier to interpret.

Contribution

It proposes a specific selection of graphene unit vectors based on nanotube symmetry and translation vectors, enhancing zone-folding computations.

Findings

Simplifies the identification of equivalent wave vectors in the reciprocal space.

Reduces computational effort by focusing on relevant energy bands.

Facilitates understanding of the relationship between graphene and nanotube reciprocal spaces.

Abstract

The dispersion relations of carbon nanotubes are often obtained cross-sectioning those of graphene (zone-folding technique) in a rectangular region of the reciprocal space, where it is easier to fold the resulting relations into the nanotube Brillouin zone. We propose a particular choice of the unit vectors for the graphene lattice, which consists of the symmetry vector and the translational vector of the considered carbon nanotube. Due to the properties of the corresponding unit vectors in the reciprocal space, this choice is particularly useful for understanding the relationship between the rectangular region where the folding procedure is most easily applied and the overall graphene reciprocal space. Such a choice allows one to find, from any graphene wave vector, the equivalent one inside the rectangular region in a computationally inexpensive way. As an example, we show how the use of these unit vectors makes it easy to limit the computation to the bands nearest to the energy maxima and minima when determining the nanotube dispersion relations from those of graphene with the zone-folding technique.