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32 Electronic Band Structure of Graphene solution of ǫk = 0, so is −kD. In graphene, there is one pair of Dirac points, and the zero-energy states are, therefore, doubly degenerate. One speaks of a twofold valley degeneracy, which survives when we consider low-energy electronic excitations that are restricted to the vicinity of the Dirac points, as is discussed in section 2.2. Effective tight-binding Hamiltonian Before considering the low-energy ex1citations and the continuum limit, it is useful to define an effective tight-binding Hamiltonian, 1 0 =. This Hamiltonian effectively omits the problem of non-orthogonality of the wavefunctions by a simple renormalisation of the nnn hopping amplitude, as alluded to above. The eigenstates of the effective Hamiltonian (2.28) are the spinors Ψ λ = a λk , ( 2 . 2 9 ) k bλk the components of which are the probability amplitudes of the Bloch wave- function (2.5) on the two different sublat1tices A and B. They may be deter- mined by considering the eigenvalue equation Hk(tnnn = 0)Ψλk = λt|γk|Ψλk, which does not take into account the nnn hopping correction. Indeed, these eigenstates are also those of the Hamiltonian with tnnn ̸= 0 because the nnn Here, H≡t |γ| 1 represents the 2 × 2 on1e-matrix k nnnk2 0γk∗ term is proportional to the one-matrix equation (2.29) yields λ γk∗ λ −iφ λ ak =λ|γk|bk =λe kbk +t . (2.28) γk 0 01 . The solution of the eigenvaluePDF Image | Physical Properties of Graphene
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