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22 Electronic Band Structure of Graphene 2.1 Tight-Binding Model for Electrons on the Honeycomb Lattice The general idea of the tight-binding model is to write down a trial wave- function constructed from the orbital wavefunctions, φ(a)(r − Rj), of the atoms forming a particular lattice described by the (Bravais) lattice vectors Rj =mja1+nja2,wheremj andnj areintegers.1 Inaddition,thetrialwave- function must reflect the symmetry of the underlying lattice, i.e. it must be invariant under a translation by any arbitrary lattice vector Ri. We con- sider, for simplicity in a first step, the case of a Bravais lattice with one atom per unit cell and one electron per atom. The Hamiltonian for an arbitrary electron, labelled by the integer l, is given by h ̄2 XN Hl =−2m∆l + V(rl −Rj), (2.1) j=1 where ∆l = ∇2l is the 2D Laplacian operator, in terms of the 2D gradient ∇l = ∂/∂xl + ∂/∂yl with respect to the electron’s position rl = (xl, yl), and m is the electron mass. Each ion on site Rj yields an electrostatic potential felt by the electron, and its overall potential energy PNj V (rl − Rj ), where N is the number of lattice sites, is, therefore, a periodic function with respect to an arbitrary translation by a lattice vector Ri in the thermodynamic limit N → ∞. The total Hamiltonian is the sum over all electrons, N H = XHl, (2.2) l if we suppose one electron per lattice site, as mentioned above. The tight-binding approach is based on the assumption that the electron l is originally bound to a particular ion at the lattice site Rl, i.e. it is described to great accuracy by a bound state of the (atomic) Hamiltonian a h ̄2 Hl =−2m∆l +V(rl −Rl), whereas the contributions to the potential energy ∆V = PNj̸=l V (rl−Rj) from the other ions at the sites Rj, j ̸= l, may be treated perturbatively. The 1Because we are interested in 2D lattices, we limit the discussion to two dimensions, the generalisation to arbitrary dimensions being straight-forward.PDF Image | Physical Properties of Graphene
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