Forbidden Electron Energies in Band Theory of Solids

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Discussion Overview

The discussion revolves around the behavior of electrons at the edges of the Brillouin zone in the band theory of solids, particularly focusing on forbidden energy regions and their implications for electron conduction. Participants explore concepts related to wave vectors, standing wave functions, and the relationship between band gaps and Bragg's reflection.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that electrons at the Brillouin zone border cannot occupy certain energy levels due to the Bragg condition, leading to questions about their fate—whether they are scattered or lose energy.
  • Others explain that at the Brillouin zone, interference between wave vectors k and -k alters the energy of the electrons, depending on the alignment of density maxima and the crystal potential.
  • A participant questions whether electrons at the Brillouin zone edge have standing wave functions, suggesting this may inhibit their role in conduction.
  • Another participant agrees with the standing wave function perspective and introduces the idea that the group velocity, represented by dE/dk, becomes zero at the Brillouin zone, indicating a lack of conduction capability.
  • A separate inquiry is made regarding the relationship between band gaps and Bragg's reflection, indicating a potential connection that remains unexplored in depth.

Areas of Agreement / Disagreement

Participants express some agreement on the implications of standing wave functions at the Brillouin zone edge for electron conduction. However, there remains uncertainty about the exact consequences for electrons in forbidden energy regions and the relationship between band gaps and Bragg's reflection, suggesting multiple competing views and unresolved aspects.

Contextual Notes

The discussion includes assumptions about the behavior of electrons in solids, the definitions of wave vectors, and the implications of group velocity, which are not fully resolved. The relationship between band gaps and Bragg's reflection is also mentioned but lacks detailed exploration.

Who May Find This Useful

Researchers and students interested in solid-state physics, band theory, and the behavior of electrons in crystalline materials may find this discussion relevant.

hokhani
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In band theory of solids, when an electron's wave vector lies at the first brillouin zone border, it satisfies the bragg condition and there is some forbidden region for that wave vector. I like to know what happens for such these electrons that they can not have some energies in the forbidden region? are they scattered out of crystal? or are they lose their energy?
 
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At the Brillouin zone, a wave with wavevector k can interfere with the scattered wave with wavevector -k. Instead of solutions exp (ikx) and exp (-ikx) one has to consider cos(kx) and sin(kx). This interference either lowers or rises the energy depending on whether the resulting density maxima of the sin and cos, respectively, coincide with the minima or maxima of the crystal potential.
 
By this you mean that an electron with wave vector at brillouin zone edge has standing wave function and hence has no role in conduction?
 
hokhani said:
By this you mean that an electron with wave vector at brillouin zone edge has standing wave function and hence has no role in conduction?

Yes, entirely correct. Another argument is the following: As dE/dk is the group velocity, and the energy has either a minimum or maximum at the Brillouin zone, the group velocity vanishes there.
 
Band gap and diffraction

Is there any relations between band gap and Bragg's reflection?
 

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