Graphene Dirac Model

One-Sentence Description

The graphene Dirac model is the low-energy two-sublattice Hamiltonian obtained by expanding the honeycomb tight-binding model near its two inequivalent Dirac points.

Physical Setup

Graphene has two carbon atoms per primitive unit cell, conventionally labeled $A$ and $B$. Keeping one $p_z$ orbital per atom and nearest-neighbor hopping gives a two-band Bloch Hamiltonian in sublattice space.

Spin is usually a passive degeneracy in the simplest model. Real graphene also has longer-range hopping, strain, interactions, spin-orbit effects, disorder, and substrate effects; these are not part of the minimal Dirac model unless added explicitly.

Lattice Bloch Form

The nearest-neighbor honeycomb Hamiltonian has the schematic sublattice form

$$H(\mathbf k) = -t \begin{pmatrix} 0&f(\mathbf k)\\ f^\ast(\mathbf k)&0 \end{pmatrix},$$

where $f(\mathbf k)$ is the sum of phase factors over the three nearest-neighbor bonds. The energies are

$$E_\pm(\mathbf k) = \pm t\,\lvert f(\mathbf k)\rvert.$$

At the inequivalent corners $K$ and $K'$ of the Brillouin zone, $f(\mathbf k)$ vanishes and the two bands touch in the minimal model.

Dirac Expansion

Expanding around one valley with $\mathbf k=K+\mathbf q$ gives

$$H_K(\mathbf q) = \hbar v_F \left( q_x\sigma_x+q_y\sigma_y \right),$$

up to basis and orientation conventions. Around the other valley, one component changes sign:

$$H_{K'}(\mathbf q) = \hbar v_F \left( -q_x\sigma_x+q_y\sigma_y \right),$$

in a common convention. The Pauli matrices here act on sublattice pseudospin, not on electron spin.

The low-energy spectrum is conical:

$$E_\pm(\mathbf q) = \pm\hbar v_F \sqrt{q_x^2+q_y^2}.$$

What It Teaches

The model shows how an effective Dirac Hamiltonian can emerge from a nonrelativistic lattice system. The Dirac structure is a statement about sublattice amplitudes and valley expansion, not about electrons literally moving at the speed of light.

The model also introduces Berry phase, chirality, valley degeneracy, and the idea that mass terms can open a gap by breaking or modifying the assumptions of the minimal model.

Canonical Links

• Graphene and Dirac Materials
• Tight-Binding Model
• Bloch Theorem
• Berry Phase
• Berry Curvature
• Dirac Equation
• Pauli Matrices

Common Mistakes

• Treating sublattice pseudospin as real spin.
• Applying the Dirac expansion over the whole Brillouin zone.
• Ignoring the second valley when counting degeneracies or symmetry constraints.
• Assuming the minimal model includes spin-orbit coupling, disorder, interactions, or substrate-induced gaps.
• Calling any two-band cone “graphene” without the honeycomb symmetry and valley context.

Quick Check

Why does the minimal graphene Hamiltonian use Pauli matrices even before real spin is included?

Solution

The two components of the low-energy wavefunction are amplitudes on the two sublattices. Pauli matrices act on this two-dimensional sublattice space. Real electron spin is an additional degree of freedom in the simplest model.

References

• P. R. Wallace, “The band theory of graphite,” Physical Review 71, 622-634, 1947.
• A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, “The electronic properties of graphene,” Reviews of Modern Physics 81, 109-162, 2009.
• M. I. Katsnelson, Graphene: Carbon in Two Dimensions, Cambridge University Press, 2012.