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Tight-Binding Model for Electrons on the Honeycomb Lattice 29 The nnn hopping amplitudes yield the diagonal elements of the hopping matrix, AA BB ik·a1 −ik·a1 ik·a2 −ik·a2 ik·a3 −ik·a3 tk =tk = tnnn e +e +e +e +e +e X 3 = 2tnnn cos(k · ai) = tnnn |γk|2 − 3 , i=1 and one obtains, thus, the secular equation tAA−ǫk (t−sǫk)γ∗ detk AAk=0 (2.20) (t − sǫk)γk tk − ǫk with the two solutions (λ = ±) ǫ λk = k . ( 2 . 2 1 ) tAA + λt|γk| 1 + λs|γk| This expression may be expanded under the reasonable assumptions s ≪ 1 and tnnn ≪ t, which we further justify at the end of the paragraph, ǫλ = tAA +λt|γ |−st|γ |2 =vt′ |γ |2 +λt|γ | k k k k nnnk k ′ X3 ut X3 = 2tnnn cos(k·ai)+λt 3+2 cos(k·ai) (2.22) i=1 i=1 where we have defined the effective nnn hopping amplitude t′nnn ≡ tnnn − st , (2.23) and we have omitted the unimportant constant −3tnnn in the last equation. One, therefore, notices that the overlap corrections simply yield a renor- malisation of the nnn hopping amplitudes. The hopping amplitudes may be determined by fitting the energy dispersion (2.22) obtained within the tight- binding approximation to those calculated numerically in more sophisticated band-structure calculations. These yield a value of t ≃ −3 eV for the nn hopping amplitude and t′nnn ≃ 0.1t, which justifies the above-mentioned ex- pansion for t′nnn/t ≪ 1. Notice that this fitting procedure does not allow for a distinction between the “true” nnn hopping amplitude tnnn and the contri- bution from the overlap correction −st. We, therefore, omit this distinction in the following discussion and omit the prime at the effective nnn hopping amplitude, but one should keep in mind that it is an effective parameter with a contribution from nn overlap corrections.PDF Image | Physical Properties of Graphene
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