Free electron dispersion relation, help?

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SUMMARY

The free electron dispersion relation is significantly altered by the presence of a periodic potential, as illustrated in figure 9.4 of Ashcroft and Mermin. In the repeated zone scheme, the dispersion appears as parabolas shifted by the reciprocal lattice vector along the k-axis, leading to points of degeneracy. According to Pauli’s exclusion principle, these degeneracies result in a gap, the magnitude of which is twice the amplitude of the periodic potential, as demonstrated through degenerate perturbation theory. For a comprehensive understanding, chapters 8 and 9 of Ashcroft and Mermin provide the necessary mathematical framework.

PREREQUISITES
  • Understanding of free electron theory
  • Familiarity with periodic potentials
  • Knowledge of the Kronig-Penney model
  • Basic principles of quantum mechanics and Pauli’s exclusion principle
NEXT STEPS
  • Study the Kronig-Penney model in detail
  • Explore the mathematical treatment of periodic potentials in Ashcroft and Mermin
  • Learn about degenerate perturbation theory and its applications
  • Investigate the implications of band theory in solid-state physics
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Students and researchers in solid-state physics, quantum mechanics enthusiasts, and anyone looking to deepen their understanding of electron behavior in periodic potentials.

taffara_121
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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!
 
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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.
 
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tejas777 said:
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.
 
taffara_121 said:
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
 

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