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Continuum Limit 33 and, thus, the eigenstates6 λ11 Ψk = √2 λeiφk where we have defined the angle , φk =arctanImγk. Reγk (2.30) (2.31) As one may have expected, the spinor represents an equal probability to find an electron in the state Ψλk on the A as on the B sublattice because both sublattices are built from carbon atoms with the same onsite energy ǫ(i).7 2.2 Continuum Limit In order to describe the low-energy excitations, i.e. electronic excitations the characteristic energy of which is much smaller than the band width ∼ |t|, one may concentrate on excitations at the Fermi level. This amounts to restricting the excitations to quantum states in the vicinity of the Dirac points, and one may expand the energy dispersion around ±K. The wave 6The eigenstates are defined up to a global (but k-dependent) phase, and one may also choose λ 1e−iφk Ψk = √2 λ , which is also found in the litterature. 7This is not the case for other lattices such as 2D boron nitride (BN), which also forms a hexagonal lattice. However, in the case of BN, one sublattice consists of boron atoms with an atomic (onsite) energy ǫA and the other one of nitrogen atoms with an onsite energy ǫB ̸= ǫA. The difference in the onsite energy μ = ǫA − ǫB may be accounted for in the effective Hamiltonian (2.28) if one adds a term μ10. 2 0 −1 This term opens a gap in the energy dispersion at the points K and K′, λ 2r22μ2 and it is energetically favourable to fill preferentially the sublattice with lower onsite energy. ǫk=tnnn|γk|+λ t|γk|+4,PDF Image | Physical Properties of Graphene
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