Free electron dispersion relation, help?

[taffara_121]

Hi there,

Could anybody explain how the free electron dispersion relation would be modified by the presence of a periodic potential..? I'm struggling to get my head around it.

Thanks!

 

[tejas777]

You will find figure 9.4 from Ashcroft and Mermin very useful in understanding how the dispersion of the free electron is modified in the presence of a weak periodic potential. In the so called repeated zone scheme (also sometimes called the extended zone scheme) the free electron dispersion in the periodic potential will look like a set of parabolas shifted by the reciprocal lattice vector along the k-axis. You can see that the parabolas will intersect a certain points. These are the points of degeneracy. According to Pauli’s exclusion principle you cannot have two electrons in the same state. As a result, this degeneracy will be split by the opening of a gap. Using degenerate perturbation theory you can show that the magnitude of this gap is twice the amplitude of the periodic potential.

Disclaimer: Please do not take what I have said here too literally. I strongly recommend you take a look at chapter 8 and 9 of Ashcroft and Mermin. The mathematical treatment of this problem presented in this book is, in fact, the simplest one you can use without getting into any trouble. The verbose description I gave above is so that you can develop an intuition for this phenomenon; I understand how overwhelming (and frustrating) it can be to get lost in the mathematical formalism.

 

[tejas777]

In the so called repeated zone scheme (also sometimes called the extended zone scheme) the free electron dispersion in the periodic potential will look like a set of parabolas shifted by the reciprocal lattice vector along the k-axis.

Ignore the part in the parenthesis; that's not true.

 

[sokrates]

Hi there,

Could anybody explain how the free electron dispersion relation would be modified by the presence of a periodic potential..? I'm struggling to get my head around it.

Thanks!

This could be helpful:
Specifically: Kronig-Penney Model
http://en.wikipedia.org/wiki/Particle_in_a_one-dimensional_lattice