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66 CHAPTER 4. ELECTRONIC STRUCTURE For the lower energy (bonding) state, the electron density has a maximum between the atoms. For the higher energy (antibonding) state, the electron density has a node between the atoms. 4.4.2 Linear chain Now let us generalise this approach to a chain of atoms, subject to periodic boundary conditions. Bloch’s theorem seriously constrains the possibilities for forming hybridised states from the atomic orbitals. If we want to make up a wave-function using only one-orbital per unit cell we now know that it must be of the form |ψ〉 = ∑ n e ikRn |n〉 , (4.28) where R n = position of atom n, |n〉 is an atomic orbital centred on atom n, so that the associated wavefunction is 〈r| n〉 = φ(r − R n ). The atomic orbitals are eigenfunctions of the single-atom Hamiltonian Ĥ(0) n = ˆp2 + V 2m atom(r − R n ): Ĥ (0) n |n〉 = E 0 |n〉, and we are looking for approximate eigenstates to the full Hamiltonian Ĥ = ˆp2 + V 2m latticer. To check that |ψ〉 = ∑ n eikRn |n〉 complies with Bloch’s theorem, we can apply a translation ˆT a to |ψ〉: ˆT a ψ(r) = ψ(r+a). This maps an atomic orbital centred on atom n (φ(r−R n )) onto an orbital centred on atom m (φ(r − R m )). ˆT a φ(r − R n ) = φ(r − (R n − a)) = φ(r − R m ) =⇒ R n − a = R mˆTa |ψ〉 = ˆT ∑ a n e ikRn |n〉 = ∑ m e ikRn |m〉 = e ika ∑ m e ikRm |m〉 = e ika |ψ〉 Now, we can apply the Hamiltonian to the state |ψ〉 constructed from the atomic orbitals: Ĥ |ψ〉 = E |ψ〉 Assuming that the orbitals are orthogonal, we left multiply with one of the basis states. It saves algebra to use the basis state 〈0|, which is centred at the origin. This gives the dispersion E(k): 〈0| Ĥ |ψ〉 = E = ∑ n e ikRn 〈0| Ĥ |n〉 If we neglect matrix elements between atomic orbitals which are not next to one another, i.e. we consider only nearest-neighbour ’hopping’, with ’hopping elements’ (or ’transfer integrals’) t ∗ = t = 〈n| Ĥ |n + 1〉, and define Ẽ0 = 〈n| Ĥ |n〉, then we obtain a particularly simple expression for the energy dispersion: E k = Ẽ0 + 2t cos (ka) (4.29) Note that there is some ambiguity in defining t. Some textbooks use a different definition of t, t := − 〈n| Ĥ |n + 1〉, in order to obtain E k = Ẽ0 − 2t cos (ka). The hopping element t = 〈n| Ĥ |n + 1〉 is < 0 between s-orbitals, giving the familiar free-electron like dispersion near k = 0, whereas it is < 0 between p x orbitals along the x− direction, for example.
4.4. TIGHT BINDING: LINEAR COMBINATION OF ATOMIC ORBITALS 67 Figure 4.4: Treating the periodic potential as a perturbation on top of the atomic potential caused by a single atomic core Figure 4.5: Eigenvalues of the 1D chain (4.29) are confined to a band in energy centred on the (shifted) atomic energy level Ẽ 0 . If N is very large, the energies form a continuous band and are periodic in m. Then we replace the index m by the continuous crystal momentum k = 2πm/Na, with a the lattice constant. So we could label the states more symmetrically by keeping a range −N/2 + 1 < m < N/2 (or −π/a < k < π/a); this is the first Brillouin zone. Because, as usual we apply periodic boundary conditions, allowed the values of k are discrete, but very close together, spaced by ∆k = 2π/L, where L = Na. The range of k must cover k max − k min = 2π/a to give N states. It is convenient to choose the range −π/a < k < π/a, the first Brillouin zone. 4.4.3 Generalised LCAO (tight binding) method We have seen that it is very easy to obtain an energy dispersion by combining a single set of atomic orbitals, because Bloch’s theorem dictates the precise form of this combination. It is straightforward to extend the calculation presented above to higher dimensions. The Bloch states is written exactly as before, in (4.28), |ψ〉 = ∑ n eikRn |n〉, but now the sum extends over all the atoms in a three-dimensional solid. Correspondingly, the dispersion given by (4.4.2), E(k) = ∑ n 〈0| Ĥ |n〉 now contains contributions from hopping processes in all eikRn three directions. For instance, in a simple cubic lattice with nearest-neighbour hoping (matrix elements 〈0| Ĥ |n〉 = 0 if atom n is not a nearest neighbour to atom 0, and = t for if atom n
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