Importance of the energy gap in electronic transport properties

[Rzbs]

TL;DR Summary

Importance of energy gap in electronic transport properties

In the solid state physics by Ashcroft & Mermin, in chapter 9 there is a paragraph that I would be grateful if anyone could explain it more for me. The paragraph is:

As it said in chapter 12 it will be seen. I read chapter 12 but unfortunately I can't understand what exactly it want to say...

 

[DrDu]

Part of this we discussed in another thread. An electric field will drive the electrons through the band in k space. When they reach a zone boundary and there is a band gap, they get reflected to the opposite zone boundary. However, if there is no gap, the dynamics will be non-adiabatic and they will move on into another band. Did you study treating the crystal potential as a small perturbation?

 

[Rzbs]

An electric field will drive the electrons through the band in k space. When they reach a zone boundary and there is a band gap, they get reflected to the opposite zone boundary.

I get this but I can't understand the following

However, if there is no gap, the dynamics will be non-adiabatic and they will move on into another band.

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Did you study treating the crystal potential as a small perturbation?

I study Chapter 9 of Aschcroft & Mermin, electrons in weak periodic potential or nearly free electron approximation.

 

[DrDu]

If you studied chapter 9, then you know that free electron bands are parabolic and the weak potential introduces gaps at the BZ boundary. If these gaps are large, the electrons will remain in the same band in extended BZ. If the band gap are small, the electrons may end up in a higher band if the electric field is strong enough. After all, the electrons will follow the parabolic band if there is no potential. You may also see it like this: When you accelerate an electron, it may get reflected if the Bragg condition is fulfilled (i.e. the wavelength is equal to a BZ boundary vector). If an electron is accelerated slowly, it will fulfill this condition approximately during a long time, so that many reflections take place. However, if the electron is accelerated rapidly, the time it fulfills the Bragg condition is very short, so that reflection becomes improbable. What is short and what is long depends not only on the strength of the accelerating field, but also on the strength of the crystal potential, which also determines the width of the gap.
So finally we understand, if the gap is large, and the band is full, the occupancy of the k-levels won't change on average, hence the substance is an isolator. If the band is not full or if the band gap is 0, the occupancy of the band will change when a field is applied and the substance will behave like a metal.

 

[Rzbs]

Thanks for your explanation. It is very helpful.
Could you please introduce another book except Kittel that could help me to understand more? (because I study Ashcroft & Mermin by myself).
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I have another question. I would be grateful if you have time to answer it, as you said:

When they reach a zone boundary and there is a band gap, they get reflected to the opposite zone boundary

I don't know do electron reflections affect electronic transport properties? if yes (that I think it is so) how they affect?

 

[DrDu]

current is e*v. What happens to v upon reflection?

 

[Rzbs]

current is e*v. What happens to v upon reflection?

I think since the electron wave function is standing wave at BZ edge, v becomes zero;as this figure shows: