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Free electron dispersion relation, help?

  1. May 14, 2012 #1
    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.

  2. jcsd
  3. Aug 19, 2012 #2
    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.
    Last edited: Aug 19, 2012
  4. Aug 24, 2012 #3
    Ignore the part in the parenthesis; that's not true.
  5. Aug 24, 2012 #4
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