Ph. D. Thesis - The University of Texas at Austin

where t(∼ 2.8eV) is the nearest neighbor hopping energy (hopping betweendifferent sublattices A and B) and γ( ⃗ k) is defined asγ( ⃗ k) = 1 + e i⃗ k·⃗a + e i⃗ k·( ⃗ b+⃗a)(2.5)The energy dispersion derived from this hamiltonian can be written as E( ⃗ k) =±t|γ( ⃗ k)|. The reciprocal lattice vectors in the hexagonal BZ are given by⃗G 1 = 2π a (1, 1 √3 ),⃗ G2 = 4π √3a(0, 1). (2.6)It is easy to see that γ( ⃗ k) and hence the energy E( ⃗ k) vanishes at two pointsK and K ′ at the corners of the Brillouin zone, these points are called Diracpoints for reasons that will become clear towards the end of this section. TheDirac points are given by⃗K = 2π a ( 1 √3,1√ ), K′ ⃗ = − 2π 3 a ( √ 1 1, √ ). (2.7)3 3Expanding γ( ⃗ k) around the Dirac point K with ⃗p = ⃗ K + ⃗q gives:γ( ⃗ k) = ⃗q · ∂γ∂p | ⃗p= ⃗K = √32 ta(q x + iq y ). (2.8)The effective hamiltonian around the K point can therefore be written asH(⃗q) = v F ⃗σ · ⃗q, (2.9)where σ i are Pauli matrices, ⃗q is momentum measured relative to the Diracpoint K and v F = ( √ 3/2)ta ≈ 1×10 6 m/s is graphene’s material specific speedof light. A similar expression can be obtained around the other Dirac point14