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Continuum Limit 39 is readily obtained from Eq. (12.32), 1 9a2 1 Hξ =t |γξ|2 ≃t |γξ(1)|2 = t |q|2 . (2.43) nnn nnn q nnn q 4 nnn The off-diagonal second-order corrections are somewhat more involved because we need to calculate the modulus of γqξ, |γξ | ≃ |γξ(1)|2 + 2Reγξ(1)∗γξ(2)1/2 ≃ |γξ(1)| 1 + Reγq γq . q where we have used the parametrisation qx = |q| cos φq and qy = |q| sin φq. Finally, the energy dispersion (2.22) expanded to second order in |q|a reads λ9a22 |q|a ǫq,ξ = 4 tnnn|q| + λh ̄vF |q| 1 − ξ 4 cos(3φq) . (2.44) As we have already mentioned in Sec. 2.1.3, it is apparent from Eq. (2.44) that the nnn correction (which also accounts for nn overlap correc- tions) breaks the electron-hole symmetry ǫ−λ = −ǫλ . This correction is, q,ξ q,ξ however, a rather small correction, of order |q|atnnn/t, to the first-order ef- fective Hamiltonian (2.36). The second-order expansion of the phase factor sum γq yields a more relevant correction – the third term in Eq. (2.44), of order |q|a ≫ |q|atnnn/t – to the linear theory. It depends explicitely on the valley isospin ξ and renders the energy dispersion anisotropic in q around the K and K′ point. The tripling of the period, due to the term cos(3φq), is a consequence of the symmetry of the underlying lattice and is also called trigonal warping, as mentioned above. The trigonal warping of the dispersion relation is visualised in Fig. 2.4, where we have plotted the contours of constant (positive) energy in Fourier space. The closed energy contours around the K and K′ points at low energy are sparated by the high-energy contours around the Γ point by the dashed " ξ(1)∗ ξ(2)# q q qq q |γξ(1)|2 Because of one obtains at second order ξ(1)∗ ξ(2) 9a2 2 2 Reγq γq = −ξ 16 qx(qx − 3qy) , | γ qξ | ≃ 3 a | q | 1 − ξ | q | a c o s ( 3 φ q ) , 24PDF Image | Physical Properties of Graphene
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