Graphene - Pseudo-relativistic Theory

Pseudo-relativistic Theory

The electrical properties of graphene can be described by a conventional tight-binding model; in this model the energy of the electrons with wave vector k is

with the nearest-neighbor hopping energy γ₀ ≈ 2.8 eV and the lattice constant a ≈ 2.46 Å. Conduction and valence band, respectively, correspond to the different signs in the above dispersion relation; they touch each other in six points, the "K-values". However, only two of these six points are independent, whereas the rest is equivalent by symmetry. In the vicinity of the K-points the energy depends linearly on the wave vector, similar to a relativistic particle. Since an elementary cell of the lattice has a basis of two atoms, the wave function even has an effective 2-spinor structure. As a consequence, at low energies, even neglecting the true spin, the electrons can be described by an equation which is formally equivalent to the massless Dirac equation. Moreover, in the present case this pseudo-relativistic description is restricted to the chiral limit, i.e., to vanishing rest mass M₀, which leads to interesting additional features:

Here v_F ~ 106 is the Fermi velocity in graphene which replaces the velocity of light in the Dirac theory; is the vector of the Pauli matrices, is the two-component wave function of the electrons, and E is their energy.