Conservation of Energy with non-allowed states

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Discussion Overview

The discussion revolves around the conservation of energy in the context of electrons in insulators, particularly focusing on the implications of applying an electric field and the concept of band structure. Participants explore how potential energy is affected by electric fields and the conditions under which conduction occurs in materials with band gaps.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant suggests that when an electric field is applied to an electron at the Fermi level of an insulator, the electron does not flow because higher momentum states require significant energy to access due to the band gap.
  • Another participant agrees that electrons gain potential energy when an electric field is applied, but notes that all electrons and states around them experience this change, while conduction states remain distant.
  • A participant questions whether the existence of any band gap, regardless of size, inherently prevents conduction, suggesting that thermal energy can sometimes enable electrons to reach the conduction band in materials with smaller gaps.
  • There is a discussion about the nature of potential energy in an electric field, with one participant pointing out that the potential is not constant, leading to questions about why some electrons might gain more potential energy than others.
  • Another participant clarifies that while the potential may vary, all electrons at the same location in the electric field gain the same energy.

Areas of Agreement / Disagreement

Participants express differing views on the implications of band gaps for conduction and the effects of electric fields on electron potential energy. There is no consensus on whether any band gap prevents conduction or how potential energy varies among electrons in an electric field.

Contextual Notes

The discussion highlights the complexity of band structure and the role of thermal energy in enabling conduction, as well as the nuances of potential energy in non-uniform electric fields. Some assumptions about the behavior of electrons in these contexts remain unresolved.

gyroscopeq
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(This though came up while learning about band structure, so that is how I am going to explain it, but I think it applies equally well to a square well, for example).

Say you have an electron at the Fermi level of an insulator. Then, you apply an electric field. No current flows, because all of the states of higher momentum (i.e. kinetic energy) require giving the electron a bunch of energy to get to, due to the band gap. What happens to the energy that should be given to every electron, qV? It seems like applying this field needs to be giving the electrons energy. Is it the case that they are essentially just given potential energy? In that case, is it also accurate to say that there is no classical analog for the "force" that would need to be holding them in place to keep them from moving in an applied potential, given that it is just something that falls out of the SWE and not a classical potential?

Thank you!
 
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They gain potential energy (or lose - the zero-point is arbitrary!), but so do all other electrons and all states around them. The energy states that allow conduction are still far away.

There is no classical analog for band structure in general.
 
mfb said:
They gain potential energy (or lose - the zero-point is arbitrary!), but so do all other electrons and all states around them. The energy states that allow conduction are still far away.

By this argument, why would it not be that the existence of *any* gap, regardless of size, prevents conduction? (And more to the point, there is always a small gap between levels, the continuum limit is just an approximation.)

Also, in the case of an electric field, V is by definition *not* constant, right? So why wouldn't some electrons gain more potential energy than others? (I guess at a given point the potential gain would be the same and perhaps that's what's relevant).

Sorry for so many follow ups. This just seems like something that is *very* important to understand completely.
 
gyroscopeq said:
By this argument, why would it not be that the existence of *any* gap, regardless of size, prevents conduction?
Thermal energy can bring electrons to the conduction band, for smaller band gaps that happens more often.
(And more to the point, there is always a small gap between levels, the continuum limit is just an approximation.)
Between levels, yes, but not between bands. And the differences between levels are so tiny you cannot note them at all for macroscopic objects.

Also, in the case of an electric field, V is by definition *not* constant, right?
Sure.
So why wouldn't some electrons gain more potential energy than others?
All electrons at the same place gain the same energy.
(I guess at a given point the potential gain would be the same and perhaps that's what's relevant).
Right.
 

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