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?

  • 4
    $\begingroup$ 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. $\endgroup$
    – Heidar
    Mar 14, 2011 at 22:57
  • 3
    $\begingroup$ as for instance is done in this very nice paper The Electronic Spectrum of Fullerenes from the Dirac Equation. $\endgroup$
    – user346
    Mar 14, 2011 at 23:32
  • $\begingroup$ What do $\Psi_{A,B}$ represent? @Z.Sum $\endgroup$
    – SRS
    Jul 12, 2018 at 6:47

1 Answer 1


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) Link


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