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Continuum Limit 41 Figure 2.5: Schematic view of an ARPES measurement (from Ref. [21]) The analyser detects the energy Ef of the photoemitted electron as a function of the angles θ and φ, which are related to the momentum of the electronic state. lines in Fig. 2.4 (a) at energy |t+tnnn| the crossing points of which correspond to the M points. As mentioned above, the dispersion relation has saddle points at these points at the border of the first BZ, which yield the van- Hove singularities in the density of states (see Fig. 2.3). In Fig. 2.4 (b), we compare constant-energy contours of the full dispersion relation to those obtained from Eq. (2.44) calculated within a second-order expansion. The contours are indistinguishable for an energy of ǫ = |t|/3 ≃ 1 eV, and the continuum limit yields rather accourate results up to energies as large as 2 eV. Notice that, in today’s exfoliated graphene samples on a SiO2 substrates, one may probe, by field-effect doping of the graphene sheet, energies which are on the order of 100 meV. Above these energies the condenser breaks down, and Fig. 2.4 (a) indicates that the continuum limit (2.44) yields extremely accurate results at these energies. 2.3 Experimental Characterisation of the Elec- tronic Band Structure The energy dispersion relation of solids may be determined by ARPES (an- gle resolved photoemission spectroscopy). A typical ARPES measurement consists of the detection of the direction and energy of a photoemitted elec-PDF Image | Physical Properties of Graphene
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