Two points to note1. The bandwidth is proportional to the hopping term.2. It is surprisingly easy to generalize the result to 2 and 3 dimensions, becauseallwe need to do is sumover the nearest neighbours! If they are symmetrically located then the bandwidth would simply be2ztwherezis the number of nearest neighbours or the co-ordination number of the lattice.PROBLEM : Calculate the group velocity of a particle at the bottom of the band and at the corner(k=±π/a). Show that there is a point of inflection (where the second derivative changes sign) somewherebetweenk= 0 andk=±π/a.How does the Bloch function look?Here’s a plot of how the functions look.Thek= 0 wavefunction is shown for reference, because thathas the maximum resemblance with the ”atomic” wavefunctions.Here we assumed that the atomicwavefunction is a gaussian. See Fig. 5.2.
5.1. DIATOMIC MOLECULE AND LINEAR CHAIN OF ATOMS85Figure 5.2: The dots are the atomic sites, thek= 0 wavefunction shows what the atomic states are like.The other two show what the linear combination of those wavefunctions, as given by Bloch’s theorem ,would look like. See eqn. 5.13.
86CHAPTER 5. TIGHT BINDING OR LINEAR COMBINATION OF ATOMIC ORBITALS (LCAO)Generalising to 2 and 3 dimensions: with 1 oribital per siteThe generalisation is easy. To handle 2 and 3d lattices we need to write the wavefunction as|ψk)=1√NsummationdisplayReik.R|φR)(5.26)(r|ψk)=1√NsummationdisplayReik.Rφ(r−R)(5.27)Where the sum runs over all direct lattice vectorsRand|φR)is the atomic state centered atR. Whiletaking the expectation value of energy we will group the series of terms into three and ignore the interactionbetween sites which are not nearest neighbours or next-nearest-neighbours:H=T+V1+V2+V3+...+VN(5.28)E(k)=(ψk|H|ψk)(5.29)=1NsummationdisplayR=R′(φR′|H|φR)+1NsummationdisplayR,R′nearestneigh-bourseik.(R−R′)(φR′|H|φR)+a0a0a0a0a0a0a0a0a64a64a64a64a64a64a64a641NsummationdisplayR,R′furtherthannearestneigh-bours...(5.30)≈E0+summationdisplaynearestneigh-bourseik.RtR(5.31)Since all sites are identical, it is sufficient to sum over the nearest neighbours of the site atR= 0.PROBLEM : Consider a 2-d rectangular lattice with sidesaandb.1. Show that following eqn. 5.31 the bandstructure would be of the formE(kx,ky) =E0−2t1cos(akx)−2t2cos(bky)(5.32)2. What is the reciprocal lattice? Draw the first Brillouin zone.3. Plot the constant energy contours, assumingt1> t2>0 anda < b.Why is this physicallyreasonable?4. Plot some constant energy contours.How do the contours look for smallk?How do the shapeschange at slightly largerk? Do all constant energy contours close within the first Brillouin zone?Similarly for 3d lattices like BCC with 8 nearest neighbours and FCC with 12 neighbours can be summedup.