Example: Square Well Potential
A square well potential can be solved by the tight binding model by looking for a solution for a
finite potential well. A finite potential well has an exponential decay at the potential wall. In a 1-
dimensional case we could take the lowest energy solution and put it at the bottom of everyone. With
the tight binding model it is possible to calculate the dispersion relationship for electrons moving in
a periodic potential like this. That problem is interesting because it is analytically solvable (Kronig
Penney Model). Therefore the tight binding solution can be compared to the the exact solution.
To get an idea about the solution, a graphically overview (fig. 32) can be quite useful. Solutions occur
that are a lot like the solutions of two wells that form a covalent bond. With a small potential well
between the two waves there is an overlap of the two wavefunctions. So you get a bonding and an
antibonding solution. We have more wells next to each other, you get the lowest total energy solution
when you take them all with a symmetric wavefunction (Take the basic solution. To get the next
solution multiply the first by one). For an antibonding solution you multiply the solution by−1(e
iπa
)
then by1(e
i 2 π
a)and so on. So the k-vector is πa. You have the basic solution and just multiply by
the phase factor. Ifkis zero you multiply by 1 , 1 , 1 ,.. and ifk=πa you get+1,− 1 ,+1,− 1 .... and in
between you get a complex factor which is not in the figure because it’s hard to draw. In fig. 32 it is
shown how the solutions look like.
Figure 32: Solution of the square well potential.