As a specific example, consider a cubic lattice. The nearest neighbour lattice sitesarea2n(±a,0,0),(0,±a,0),(0,0,±a)oand the hopping parameters are the same inall directions:ta=t. The dispersion relation is then given byE(k) =E0-2t⇣cos(kxa) + cos(kya) + cos(kza)⌘(2.37)The width of this band isΔE=Emax-Emin= 12t.Note that for smallk, the dispersion relation takes the form of a free particleE(k) = constant +~2k22m?+. . .where the e↵ective massm?is determined by various parameters of the underlyinglattice,m?=~2/2ta2. However, at higherkthe energy is distorted away from the thatof a free particle. For example, you can check thatkx±ky=⌥⇡/a(withkz= 0) is aline of constant energy.2.3.5 Deriving the Tight-Binding ModelAbove, we have simply written down the tight-binding model. But it’s interesting toask how we can derive it from first principles.In particular, this will tell us whatphysics it captures and what physics it misses.– 67 –
To do this, we start by considering a single atom which we place at the origin. TheHamiltonian for a single electron orbiting this atom takes the familiar formHatom=p22m+Vatom(x)The electrons will bind to the atom with eigenstatesφn(x) and discrete energies✏n<0,which obeyVFigure 33:Hatomφn(x) =✏nφn(x)A sketch of a typical potentialVatom(x) and the binding energies✏nis shown on the right. There will also be scattering states, withenergies✏>0, which are not bound to the atom.Our real interest lies in a lattice of these atoms. The resultingpotential isVlattice(x) =Xr2⇤Vatom(x-r)This is shown in Figure34for a one-dimensional lattice. What happens to the energylevels?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. Butthose that are bound more weakly become free to move. This happens because the tailsof 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 calledvalence electrons. (These are the same electrons which typically sit in outer shells andgive rise to bonding in chemistry.) As we’ve seen previously, these electrons will forma band of extended states.Let’s see how to translate this intuition into equations. We want to solve the Hamil-tonianH=p22m+Vlattice(x)(2.38)Our goal is to write the energy eigenstates in terms of the localised atomic statesφn(x).Getting an exact solution is hard; instead, we’re going to guess an approximate solution.– 68 –
VExtended statesLocalised statesFigure 34:Extended and localised states in a lattice potential.