Understanding the Kronig-Penney Model in Solid State Physics"

[maria clara]

I'm trying to understand the idea behind the Kronig-Penney model, and its relevance to solid state physics. I understand that the model refers to a particle in a periodic potential. Using Bloch's theorem, and regular boundary conditions the following equation is obtained:
$$\frac{P}{ka}$$sin(ka)+cos(ka)=cos(qa)
(a is the period of the potential)
Since the expression on the right-hand side of the equation takes only values between -1 and 1, and the function on left-hand side might get values outside this range, energy gaps (and "energy bands") are created.
So this is a good description of the energy of an electron in a periodic potential. But is this description relevant to all solids? In conductors there is no forbidden gap between the valence band and the conduction band; In semiconductors there is a gap but it's relatively small, and in insulators this gap is relatively large. Does that mean that the difference between these situations is the parameter "P" in the equation above?

Thanks:shy:

 

[xboy]

Maria, I suspect this question would be better answered were you to post it in the Solid State forum. After all, this is not a homework question. Anyways, here's what I can tell you:

We do know that in solids ions are periodically arranged, and we can expect that their electrostatic potentials would vary periodically. So the presence of periodic potential is relevant to all solids.

Then we usually make a further assumption, that the conduction electron is nearly free. That is, we assume it to be a otherwise free, except for being in a weak periodic potential. This is not generally true, but it turns out that predictions are made from such models are good approximations to actual behavior of solids.

To get to the 'Kronig Penny Model' we make one more assumption, the periodic potential is in the form of square wells. Whereas in reality we should expect something a bit more parabolic( electrostatic potential falls off as 1/r from the source). So the Kronig Penny Model is even less general, but simpler for calculations, but we still see the qualitative features of solids, like band structure, emerge from it.

To get more accurate quantitative predictions, I suppose solid state physicists use more realistic potentials.