# effective theory of graphene

This is a question about deriving effective mass theory for graphene. For the two sub-lattice atoms, the wave equation can be written as the massless Dirac equation:

$\displaystyle -i\hbar v_F \begin{pmatrix} 0 & \partial_x -i\partial_y \\\partial_x +i\partial_y & 0 \end{pmatrix} \left(\begin{array}{c} \Psi_A \\ \Psi_B \end{array}\right)=E \left(\begin{array}{c} \Psi_A \\ \Psi_B \end{array}\right) \ \ \ \ \ (1)$ where ${A,B}$ are two subatoms.

The derivation of the equation went back to 1984, which is the paper I am trying to understand. In the article, they argued that at first order of ${\vec{\kappa}\cdot \vec{p}}$ expansion, the momentum matrix can be written in the form, under group-theoretic arguments (equation (3) in the paper):

$\displaystyle \bar{p}\begin{pmatrix} 0 & \hat{x} -i\hat{y} \\ \hat{x} +i\hat{y} & 0 \end{pmatrix} . \ \ \ \ \ (2)$

What is the argument behind it? Is there anyone read the paper or know the answer?

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If you just want to derive the massless Dirac equation, you can do it in a much more elementary way. Model graphene with a (nearest-neighbor) tight-binding model on the honeycomb lattice. Fourier-transform and expand to linear order in momentum (around the Dirac points), then you should find the Dirac Hamiltonian. –  Heidar Mar 14 '11 at 22:57
as for instance is done in this very nice paper The Electronic Spectrum of Fullerenes from the Dirac Equation. –  user346 Mar 14 '11 at 23:32

## 2 Answers

In the paper of J.M. McCLURE(1956), he showed how to directly calculate the momentum matrix. (eq.2.5,2.6)

Diamagnetism of Graphite,Phys. Rev. 104, 666–671 (1956) http://prola.aps.org/abstract/PR/v104/i3/p666_1

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Take a look at this paper http://prb.aps.org/abstract/PRB/v82/i24/e245313 .

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In the future please include arXiv abstract page if possible, e.g., arxiv.org/abs/0807.4204 –  Qmechanic Feb 19 '12 at 14:16