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34 Electronic Band Structure of Graphene vector is, thus, decomposed as k = ±K + q, where |q| ≪ |K| ∼ 1/a. The small parameter, which governs the expansion of the energy dispersion, is, therefore, |q|a ≪ 1. It is evident from the form of the energy dispersion (2.22) and the effective Hamiltonian that the basic entity to be expanded is the sum of the phase factors γk. We need to distinguish the sum at the K point from that at the K′ point, 2 q ±K 1 + e±iK·a2 eiq·a2 + e±iK·a3 eiq·a3 ≃ 1+e±i2π/3 1+iq·a2 − 1(q·a2)2 γq± ≡ γk=±K+q = = γ±(0) + γ±(1) + γ±(2) +e∓i2π/3 1+iq·a3 − 1(q·a3)2 2 qqq By definition of the Dirac points and their position at the BZ corners K and K′, we have γ±(0) = γ = 0. We limit the expansion to second order in |q|a. First order in |q|a The first order term is given by γ±(1) = i√3a h(q + √3q )e±i2π/3 + (−q + √3q )e∓i2π/3i q2xyxy = ∓3a(qx ±iqy), (2.32) which is obtained with the help of sin(±2π/3) = ±√3/2 and cos(±2π/3) = 2 −1/2. This yields the effective low-energy Hamiltonian H e ff , ξ = ξ h ̄ v ( q σ x + ξ q σ y ) , qFxy where we have defined the Fermi velocity8 vF ≡ −3ta = 3|t|a 2h ̄ 2h ̄ ( 2 . 3 3 ) (2.34) 8The minus sign in the definition is added to render the Fermi velocity positive because the hopping parameter t ≃ −3eV happens to be negative, as mentioned in the last section.PDF Image | Physical Properties of Graphene
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