Brillouin zones and energy bands

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    Brillouin Energy
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Discussion Overview

The discussion revolves around the relationship between Brillouin zones and energy bands in the context of solid-state physics, particularly within the semi-classical model and the Nearly Free Electron model. Participants explore the implications of band filling, the nature of energy gaps, and the behavior of electrons across different Brillouin zones.

Discussion Character

  • Technical explanation
  • Conceptual clarification
  • Debate/contested

Main Points Raised

  • One participant notes that in the semi-classical model, electrons with k values in the same Brillouin zone are considered to be in the same energy band, seeking clarification on this concept.
  • Another participant mentions that there are 2N states in each band and that for a single k-vector, there are infinitely many energy bands, indicating a potential misunderstanding of band filling.
  • A later reply challenges the idea that bands fill sequentially, pointing out that bands can overlap, leading to partially filled bands across multiple Brillouin zones.
  • One participant explains that in the Nearly Free Electron model, energy gaps arise due to Bragg reflections, and describes how energy values in different Brillouin zones correspond to different energy bands, while also noting the equivalence of k values across zones through the subtraction of reciprocal lattice vectors.
  • Another participant expresses a preference for Kittel's treatment over Ashcroft's, suggesting a divergence in perspectives on the material.

Areas of Agreement / Disagreement

Participants express differing views on the filling of energy bands and the relationship between Brillouin zones and energy bands. There is no consensus on the explanation of why electrons in a certain Brillouin zone are considered to be at the same band, indicating ongoing debate and exploration of the topic.

Contextual Notes

Participants reference specific pages and figures from Ashcroft and Mermin's text, indicating that their understanding is based on particular interpretations of the material. There are also mentions of overlapping bands and the implications of energy gaps, which may require further clarification or context.

cavalier3024
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In the semi-classical model, i noticed that all the electrons with values of k that are in the same brillouin zone are considered to be at the same energy band, but i can't quite understand why it is so.
i know that in each brillouin zone the number of allowed states (of k) is the same as the number of states at each band (N), but that doesn't really provide a full explanation.

so if you can briefly explain the idea behind this (or refer me to the relevant pages at Ashcroft) i would be thankful.
 
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There are 2N states in each band, and for a single k-vector, there are infinitely many energy bands (this can e.g. be seen when looking at the energy bands in the reduced zone scheme).

So when one energy band is filled, electrons start filling the next one, and so on. On what page in A&M have you read this?
 
you said that 'when one energy band is filled, electrons start filling the next one', but that's not always the case. sometimes bands overlap and one band may start filling before the previous one finished, giving two partially filled bands.
you can see what i mean at page 224 (chapter 12), figure 12.3. they show that 2N electrons fill part of the first brillouin zone and also part of the second BZ. then they say that this produces two partially filled bands. so what i understand is that all occupied states that are in the first brillouin zone are filling one band, and the ones at the second zone fill another band.
what i don't understand is why the brillouin zones determine the bands. why all the electrons that occupy a certain BZ are considered to be at the same band?
 
In the Nearly Free Electron model energy gaps appear due to bragg reflections with atoms in the Brillouin zone limit.

In a one dimensional model you have then an energy gap at K=+-pi/a, K=+-2pi/a etc... Biger k values give biger energy values. So those energy values in the first BZ are the less energetic values, the 1st band. The energy values in the second BZ correspond to the 2nd band etc...

But every k value in the second BZ can be taken to the first BZ by substracting a G vector from the reciprocal lattice, every vector in the third BZ can be taken to the 1st one by substracting 2 G vectors etc... So they are "the same" k vector and every k value has an energy value in each energy band.

I hope this messy explanation clears your ideas. I like more Kittel than Ashcroft (chapter 7)
 

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