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Continuum Limit 37 in terms of the total surface A.9 The factor g takes into account the de- generacy due to internal degrees of freedom – g = 4 in the case of graphene because of the twofold valley degeneracy and the electron spin. The calculation of the density of states may be rather involved because one needs to calculate a 2D integral over the wave vector as a function of energy, by inverting the energy dispersion. In the vicinity of the Dirac points, however, it may be calculated quite easily because of the isotropy of the energy dispersion (2.35). One may identify Ag Z q(ǫ) X ≃2π dqq, k / ǫ +k ≤ ǫ 0 where we have profited from the isotropy of the energy dispersion in the vicinity of the Dirac point, with the right hand side of Eq. (2.39), and one obtains By inverting the energy dispersion (2.35), one obtains ρ(ǫ ≥ 0) = gǫ , 2 π h ̄ 2 v F2 and for the full (positive and negative) energy range10 ρ(ǫ) = g|ǫ| , ρ(ǫ) = g q(ǫ) . 2π ∂ǫ/∂q (2.41) (2.42) 2 π h ̄ 2 v F2 as long as |ǫ| ≪ |t|. The density of states, therefore, vanishes linearly at the Dirac points, at zero energy. This is a direct consequence of the linearity of the energy dispersion in the vicinity of the Dirac points. This particular situation needs to be contrasted to the conventional case of electrons in 2D metals, with an energy dispersion of ǫ = h ̄2q2/2m∗, in terms of the band mass m∗, where one obtains a constant density of states, ρ2D(ǫ) = gm∗/2πh ̄2. The full density of states, in the absence of nnn hopping corrections, has been calculated by Hobson and Nierenberger in 1953 [20]. It is schematically 9The density of states defined in this manner is indeed the density of states per unit surface, or unit volume in 3D. 10The negative branch of the density of states may be calculated in the same manner.PDF Image | Physical Properties of Graphene
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