As a specific example consider a cubic lattice The nearest neighbour lattice

As a specific example consider a cubic lattice the

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As a specific example, consider a cubic lattice. The nearest neighbour lattice sites are a 2 n ( ± a, 0 , 0) , (0 , ± a, 0) , (0 , 0 , ± a ) o and the hopping parameters are the same in all directions: t a = t . The dispersion relation is then given by E ( k ) = E 0 - 2 t cos( k x a ) + cos( k y a ) + cos( k z a ) (2.37) The width of this band is Δ E = E max - E min = 12 t . Note that for small k , the dispersion relation takes the form of a free particle E ( k ) = constant + ~ 2 k 2 2 m ? + . . . where the e ective mass m ? is determined by various parameters of the underlying lattice, m ? = ~ 2 / 2 ta 2 . However, at higher k the energy is distorted away from the that of a free particle. For example, you can check that k x ± k y = /a (with k z = 0) is a line of constant energy. 2.3.5 Deriving the Tight-Binding Model Above, we have simply written down the tight-binding model. But it’s interesting to ask how we can derive it from first principles. In particular, this will tell us what physics it captures and what physics it misses. – 67 –
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To do this, we start by considering a single atom which we place at the origin. The Hamiltonian for a single electron orbiting this atom takes the familiar form H atom = p 2 2 m + V atom ( x ) The electrons will bind to the atom with eigenstates φ n ( x ) and discrete energies n < 0, which obey V Figure 33: H atom φ n ( x ) = n φ n ( x ) A sketch of a typical potential V atom ( x ) and the binding energies n is shown on the right. There will also be scattering states, with energies > 0, which are not bound to the atom. Our real interest lies in a lattice of these atoms. The resulting potential is V lattice ( x ) = X r 2 V atom ( x - r ) This is shown in Figure 34 for a one-dimensional lattice. What happens to the energy levels? Roughly speaking, we expect those electrons with large binding energies — those shown at the bottom of the spectrum — to remain close to their host atoms. But those that are bound more weakly become free to move. This happens because the tails of their wavefunctions have substantial overlap with electrons on neighbouring atoms, causing these states to mix. This is the physics captured by the tight-binding model. The weakly bound electrons which become dislodged from their host atoms are called valence electrons . (These are the same electrons which typically sit in outer shells and give rise to bonding in chemistry.) As we’ve seen previously, these electrons will form a band of extended states. Let’s see how to translate this intuition into equations. We want to solve the Hamil- tonian H = p 2 2 m + V lattice ( x ) (2.38) Our goal is to write the energy eigenstates in terms of the localised atomic states φ n ( x ). Getting an exact solution is hard; instead, we’re going to guess an approximate solution. – 68 –
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V Extended states Localised states Figure 34: Extended and localised states in a lattice potential.
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