The k.p method and its application to graphene, carbon nanotubes and graphene nanoribbons: the Dirac equation

TL;DR

This paper reviews the k.p method, a semi-empirical approach for band structure calculation, and applies it to graphene, carbon nanotubes, and nanoribbons, deriving their electronic properties via the Dirac equation.

Contribution

It provides a comprehensive overview of the k.p method and demonstrates its application to graphene-related materials, deriving their electronic properties from the Dirac equation.

Findings

Derivation of the Dirac equation for graphene properties

General expressions for probability density and current in graphene

Extension of the method to carbon nanotubes and nanoribbons

Abstract

The k.p method is a semi-empirical approach which allows to extrapolate the band structure of materials from the knowledge of a restricted set of parameters evaluated in correspondence of a single point of the reciprocal space. In the first part of this review article we give a general description of this method, both in the case of homogeneous crystals (where we consider a formulation based on the standard perturbation theory, and Kane's approach) and in the case of non-periodic systems (where, following Luttinger and Kohn, we describe the single-band and multi-band envelope function method and its application to heterostructures). The following part of our review is completely devoted to the application of the k.p method to graphene and graphene-related materials. Following Ando's approach, we show how the application of this method to graphene results in a description of its properties in terms of the Dirac equation. Then we find general expressions for the probability density and the probability current density in graphene and we compare this formulation with alternative existing representations. Finally, applying proper boundary conditions, we extend the treatment to carbon nanotubes and graphene nanoribbons, recovering their fundamental electronic properties.