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38 Electronic Band Structure of Graphene −t 0 t Energy Figure 2.3: Schematic plot of the density of states for electrons in graphene in the absence of nnn hopping. The dashed line indicates the density of states obtained at linear order in the continuum limit. shown in Fig 2.3. In agreement with the result (2.42) obtained at linear order in the continuum limit, it vanishes linearly at zero energy. The divergencies at ±t, called van-Hove singularities, are due to the saddle points of the energy dispersion at the M points at the borders of the first BZ (see Fig. 2.2). This divergency may be understood qualitatively from Eq. (2.42) – roughly speaking, the derivative of the energy dispersion occurs in the denominator of the expression for the density of states and, thus, causes a divergency whenever it becomes zero, such as at the extrema or saddle points of the energy dispersion. Second order in |q|a Although most of the fundamental properties of graphene are captured within the effective model obtained at first order in the expansion of the energy dispersion, it is useful to take into account second-order terms. These cor- rections include nnn hopping corrections and off-diagonal second-order con- tributions from the expansion of γk. The latter yield the so-called trigonal warping, which consist of an anisotropy of the energy dispersion around the Dirac points. The diagonal second-order correction, which stems from the nnn hopping, Density of statesPDF Image | Physical Properties of Graphene
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