Free electron dispersion relation, help?

Click For Summary

Discussion Overview

The discussion revolves around the modification of the free electron dispersion relation due to the presence of a periodic potential, exploring theoretical frameworks and models such as the Kronig-Penney model. Participants express varying levels of understanding and seek clarification on the topic.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant asks for clarification on how the free electron dispersion relation is altered by a periodic potential.
  • Another participant suggests that the dispersion will resemble parabolas shifted by the reciprocal lattice vector in the repeated zone scheme, mentioning points of degeneracy and the resulting energy gap due to Pauli’s exclusion principle.
  • A later reply challenges the accuracy of the description regarding the repeated zone scheme, indicating that the parenthetical statement about its naming is incorrect.
  • Another participant reiterates the initial question and provides a link to the Kronig-Penney model as a potential resource for understanding the topic.

Areas of Agreement / Disagreement

There is no consensus on the accuracy of the descriptions provided, particularly regarding the repeated zone scheme. Multiple viewpoints and interpretations of the dispersion relation exist, indicating ongoing debate and uncertainty.

Contextual Notes

Participants express varying levels of confidence in their explanations, and there are indications of missing assumptions or definitions that could clarify the discussion further. The mathematical treatment of the problem remains a point of contention.

taffara_121
Messages
1
Reaction score
0
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!
 
Physics news on Phys.org
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:
  • Like
Likes   Reactions: 12x4
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
 

Similar threads

Graduate Understand Electronic Dispersion: Definition & Examples
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Instantaneous dipole moment and orbitals
  • · Replies 2 ·
Replies
2
Views
973
Undergrad Bragg condition and Bloch states
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Dispersion Relation in Different Media
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Fermi Energy Calculations About Non Parabolic Dispersions
  • · Replies 3 ·
Replies
3
Views
3K
Graduate Dispersion relation for the free electron model
  • · Replies 1 ·
Replies
1
Views
9K
Graduate Change in phonon dispersion with pressure
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad What is the significance of the dispersion relation?
  • · Replies 5 ·
Replies
5
Views
3K
Undergrad What does the wavenumber in a group velocity represent?
  • · Replies 2 ·
Replies
2
Views
2K
Heat capacity from dispersion relation
  • · Replies 1 ·
Replies
1
Views
2K