How to calculate the gap for a solid?

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SUMMARY

The energy gap for a solid can be calculated using the Kronig-Penney model, which involves setting up an infinite potential field represented by a periodic array of potential bumps. A common approach is to utilize a Delta function comb to derive a parametric equation containing sine functions. The absence of solutions in specific areas indicates the energy gaps, as the maximum value of sine is constrained to 1. References for further reading include Kittel's and Griffiths' texts on solid state physics and quantum mechanics.

PREREQUISITES
  • Understanding of the Kronig-Penney model
  • Familiarity with solid state physics concepts
  • Knowledge of quantum mechanics principles
  • Basic proficiency in solving parametric equations
NEXT STEPS
  • Study the Kronig-Penney model in detail
  • Explore the Delta function comb potential in solid state physics
  • Learn about energy band theory and its implications
  • Review Kittel's and Griffiths' textbooks for advanced concepts
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Students and researchers in solid state physics, quantum mechanics enthusiasts, and professionals involved in materials science and semiconductor research will benefit from this discussion.

gcc
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Hello everyone:

Do you know how to calculate the energy gap (the forbitten band) for a solid from the Kroning-Penney model?
 
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check any solid state book, Kittel comes to mind as does Ashcroft and Mermin.
 
It has been a while since I have done this.

But I believe the answer you are looking for is:

Set up a simple infinite potenential field consisting of a perriodic array of potential bumps. (i think the only one I have worked out for my self is the case of a Delta function comb)

Then you have a parametric equation with sines in it. Because the solution can not be greater than 1 (maximum of sin(x) = 1) you find areas where there is no solution. Those are you gaps.

I know I have seen this in Kittel, Griffiths (quantum mechanics) and a couple other places.
 

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