Figure 29: Dispersion relation and the density of states over the energy.7.2.3 Tight Binding Model
The tight binding model is another method for band structure calculations. Starting with the atomic
wavefunction we imagine a crystal where the atoms are much further apart than in a normal crystal
(the internuclear distances are big). They are still in the same arrangement (for example an fcc
structure) but the bond lengths are shorter. Therefore they don’t really interact with each other. The
atomic wavefunctions are the electron states. If you bring the atoms together they will start to form
bonds. The energies will shift. In fig. 30 at the left side the atoms are far apart (they are all in the
ground state, all same energy, 1s) and to the right side the atomic wavefunctions start to overlap and
they start to form bonds. The states split into bands which we are calculating. If the lattice constant
gets smaller the band gets wider and there is more overlap between the wavefunctions (a little bit like
making a covalent bond). If two atoms with given wavefunctions are brought together, they interact
in a way that the wavefunction splits into a bonding state (goes down in energy) and an antibonding
state (goes up in energy). We consider a 1-dimensional crystal and look at the Coulomb potential of
every atom in it. The total potential is the sum over all potentials. Adding a lot of Coulomb potentials
together gives a periodic potential:
V(r) =∑
nν(r−nax) n=.....− 1 , 0 ,+1....The Hamiltonian then is
H=
−~^2
2 m∇^2 +
∑
nν(r−nax).The atomic wavefunction is chosen as a basis. The wavefunction one on every lattice site is taken as a
basis to do the calculation. The wavefunctionΨqnis labeled by two letters.nlabels the position and
qlabels which excited state it is (1s,2p,..). Knowing the wavefunction and the Hamiltonian we can
form the Hamiltonian matrix. If we solve every matrix element numerically we get the eigenfunctions
(of the Hamiltonian) and the eigenvalues (energies).