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

Paolo Marconcini, Massimo Macucci|arXiv (Cornell University)|May 6, 2011

Graphene research and applications79 references79 citations

TL;DR

This paper presents a comprehensive application of the k·p method to graphene, carbon nanotubes, and graphene nanoribbons, demonstrating that their low-energy electronic properties are accurately described by the Dirac equation. By deriving the probability density and current density in graphene and applying boundary conditions, the method successfully predicts the electronic band structures of nanotubes and nanoribbons with high accuracy using a continuum approach based on envelope functions and effective mass parameters.

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.

Motivation & Objective

• To provide a unified theoretical framework for the k·p method in both periodic and non-periodic systems, particularly for low-dimensional carbon nanostructures.
• To establish the connection between the k·p method and the Dirac equation in graphene, enabling a relativistic-like description of its electronic states.
• To extend the method to carbon nanotubes and graphene nanoribbons by applying appropriate boundary conditions to the envelope function formalism.
• To compare different formulations of the probability current and density in graphene, ensuring consistency with quantum mechanical expectations.
• To demonstrate the efficiency and accuracy of the k·p method in predicting key electronic properties without full ab initio calculations.

Proposed method

• The k·p method is applied at the K-point of graphene's Brillouin zone, using a 2×2 Hamiltonian derived from tight-binding parameters to yield the Dirac-like Hamiltonian.
• The method employs a multi-band envelope function approach based on Luttinger-Kohn theory, valid for heterostructures and nanostructures with spatially varying potentials.
• The probability density and current density are derived from the spinor wavefunction, with the current density showing vanishing transverse component in confined systems.
• Boundary conditions are applied to model edge states in zigzag and armchair nanoribbons, leading to quantized energy levels and band gaps.
• The effective mass approximation is used to describe electron and hole transport, with Fermi velocity vF as a key parameter.
• The formalism is extended to carbon nanotubes by imposing periodic boundary conditions along the tube axis, recovering the chiral and metallic behavior based on the tube's (n,m) index.

Experimental results

Research questions

• RQ1 How can the k·p method be systematically applied to graphene to derive its low-energy Dirac Hamiltonian?
• RQ2 What are the correct expressions for the probability density and current density in graphene within the k·p framework, and how do they compare to alternative formulations?
• RQ3 How does the envelope function method with k·p parameters reproduce the electronic band structure of carbon nanotubes and graphene nanoribbons?
• RQ4 What role do boundary conditions play in determining the band gap and edge states in zigzag and armchair graphene nanoribbons?
• RQ5 To what extent can the k·p method predict the electronic properties of carbon nanostructures without full ab initio calculations?

Key findings

• The k·p method applied to graphene at the K-point yields a Hamiltonian equivalent to the Dirac equation, with linear dispersion and Fermi velocity vF as a key parameter.
• The probability current density in graphene is shown to be proportional to the velocity operator, with the transverse component vanishing in transversely confined systems, as required by symmetry.
• For graphene nanoribbons, the method correctly predicts the existence of edge states in zigzag ribbons and the opening of a band gap in armchair ribbons based on their width and chirality.
• The band gap in armchair nanoribbons scales inversely with width, consistent with known analytical results from tight-binding models.
• Carbon nanotubes are shown to exhibit metallic or semiconducting behavior depending on their (n,m) chiral index, with the k·p method reproducing the quantized energy levels and vanishing density of states at the Fermi level in metallic tubes.
• The formalism provides a consistent and computationally efficient alternative to ab initio methods for studying electronic transport and optical properties in low-dimensional carbon systems.