Can we have band gap anywhere?

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    Band Band gap Gap
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Discussion Overview

The discussion revolves around the existence and location of band gaps in solid-state physics, particularly in relation to the Brillouin zone. Participants explore theoretical frameworks, models, and examples from solid-state texts, addressing the conditions under which band gaps appear and their positions within the Brillouin zone.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants reference solid-state texts, noting that band gaps are typically found at the borders of the Brillouin zone, while others point out that band gaps can also occur at the center, such as in GaAs at the Gamma point.
  • One participant mentions the derivation of band gaps from weak periodic potentials, suggesting that perturbation theory provides insights into this phenomenon.
  • Another participant questions the relationship between the mechanism of band gap formation and the specific locations of band gaps within the Brillouin zone.
  • It is noted that band gaps can be found at the center of the Brillouin zone in certain models, such as tight-binding models with specific orbital configurations.
  • Participants discuss the lifting of degeneracy at the Brillouin zone and how this affects the presence of band gaps, indicating that real solids may exhibit band gaps not aligned with Brillouin zone boundaries.

Areas of Agreement / Disagreement

Participants express differing views on the location of band gaps, with some asserting they are typically at the Brillouin zone borders while others argue they can also be found at the center. The discussion remains unresolved regarding the definitive relationship between weak periodic potentials and the locations of band gaps.

Contextual Notes

Participants reference various solid-state physics texts and models, indicating a reliance on specific definitions and assumptions that may not be universally accepted. The discussion includes unresolved questions about the implications of perturbation theory and Fermi's golden rule in relation to band gaps.

hokhani
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According to solid state texts in Brillouin zone borders where diffraction condition satisfies we have a band gap. However I usually see Band gaps which are located in the center of Brillouin zone. Please correct me if I am wrong.
 
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which text are you reading?
 
rigetFrog said:
which text are you reading?

In "solid state physics, Ashcroft - Mermin, chapter 9, electrons in a weak periodic potential", it is explained that due to weak perturbation we have energy gap at Brillouin zone border. Moreover we now for example that in GaAs the energy gap is at K=0 (Gamma point) which is not at the Brillouin zone border.
 
I'm sure where to start.

Do you understand how a weak periodic potential gives rise to a bandgap?

Liboff's quantum textbook has a nice derivation using perturbation theory.

Alternatively, you could use Fermi's golden rule using the weak periodic potential at the perturbing hamiltonian.
 
rigetFrog said:
I'm sure where to start.

Do you understand how a weak periodic potential gives rise to a bandgap?

Liboff's quantum textbook has a nice derivation using perturbation theory.

Alternatively, you could use Fermi's golden rule using the weak periodic potential at the perturbing hamiltonian.

What do you mean? Is there any direct relationship between "how a weak periodic potential gives rise to a bandgap" and "where a band gap is" ?
In many cases, the band gap is indeed not at the high symmetry points or located in the boundaries of the Brillouin zones (using vasp or wien2k, for examples),is it ?
Another, what is "Fermi's golden rule" ? Is it related to band gap ?
http://en.wikipedia.org/wiki/Fermi's_golden_rule "In quantum physics, Fermi's golden rule is a way to calculate the transition rate (probability of transition per unit time) from one energy eigenstate of a quantum system into another energy eigenstate, due to a perturbation."
 
You are perfectly right, In a solid, you can find the bandgap at the center of the BZ. It is easy to find a tight-binding model with just two orbitals per unit cell having the gap at the center of the BZ. It is also true that, starting from free electrons, with a parabolic dispersion relation, and considering the periodic potential as a perturbation, the gap shows up necessarily at the border of the BZ.
 
Starting from a free electron picture, the only region where you would expect bands to cross is at the BZ. This degeneracy is usually lifted when a periodic potential is present. So if you define the band gap as the minimal distance between two bands, you will find a band gap only, if the degeneracy at the BZ is lifted. This does not preclude the situation that in real solids with strong potentials the minimal distance between the bands is not at the BZ, but e.g. at its center.
 

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