Computing the size of a band gap between different bands (Minitest 3 2019)

Physics Q&A
1

Question

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Using the following expression to calculate band gaps:

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My questions

  1. Where does the factor 2 in front of the expression in the answer model come from?
  2. Why do you integrate from -a+E to a+E instead of 0 to a like in the lecture notes?
  3. How does the sum in the integral reduce to 1/a(A-B) in the first case and 1/a(A+B) in the second case?
  4. I understand that to find the size of the first band gap you evaluate in between -\pi/2a and \pi/2a, since a band gap opens at these locations. The second band gap opens at k=0, but to calculate the size you evaluate in between -\pi/a and \pi/a. Why is that?

Answer

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2
  1. The gap is given by 2W (see the NFEM lecture notes)
  2. \epsilon is a small number here that can be set to zero after the integration. The important thing is to integrate over a full period of V
  3. Note that the exponent in the integrand is different.
  4. When folded back into the Brillouin zone, this is the same k-point as you describe. This is where the two parabolas of interest cross.