Are Filled Bands in Insulators Equivalent to Non-Mobile Electrons?

[Niles]

Hi all

A conductor is a conductor, because it has an unfilled energy band. Likewise, insulators have filled bands, hence they do not conduct.

Now, these two statements say nothing about the mobility of the electrons. How is it that an unfilled band is equivalent of mobile electrons, while a filled band is equivalent of non-mobile electrons? I think it has something to do with a continuous change of the momentum of the electrons by applying a field; am I correct?

 

[ZapperZ]

It is how one describes the conduction band. In the simplest case, the conduction band is described via a series of plane-wave states. If you've solved for a simple free-electron wavefunction, you'll find that the electron is non-local, i.e., try to find <x>, for example.

This isn't always true for the filled valence band.

Another way to look at this is via the "hopping" parameter. The unfilled band only requires an electron with very minimal energy above the Fermi energy to be able to hop to an empty state. In the conduction band, all the states are filled. Since a fermion cannot hop onto a state that's already filled, its ability to hop has been suppressed. Thus, no mobility.

Zz.

 

[Niles]

Another way to look at this is via the "hopping" parameter. The unfilled band only requires an electron with very minimal energy above the Fermi energy to be able to hop to an empty state.

Please take a look at this picture: http://www.all-science-fair-projects.com/science_fair_projects_encyclopedia/upload/a/a3/Semiconductor_band_structure_%28lots_of_bands%29.png

Now, the length of the rectangles span over k-vectors. How do you explain from this graph that an electron with minimal energy above the Fermi energy is able to hop to an empty state, since the empty states are along the horizontal axis?

 

[ZapperZ]

Please take a look at this picture: http://www.all-science-fair-projects.com/science_fair_projects_encyclopedia/upload/a/a3/Semiconductor_band_structure_%28lots_of_bands%29.png

Now, the length of the rectangles span over k-vectors. How do you explain from this graph that an electron with minimal energy above the Fermi energy is able to hop to an empty state, since the empty states are along the horizontal axis?

Link doesn't work.

What I mean by "minimal energy" is that since the band of states is continuous, it takes almost nothing for an electron to move into an empty state/location.

Zz.

 

[torquil]

I fixed the link for you:

http://www.all-science-fair-projects.com/science_fair_projects_encyclopedia/upload/a/a3/Semiconductor_band_structure_(lots_of_bands).png"

Torquil

 

[DrDu]

Note that there are also insulators with an unfilled energy band, namely so called "Mott insulators".

 

[Niles]

I fixed the link for you:

http://www.all-science-fair-projects.com/science_fair_projects_encyclopedia/upload/a/a3/Semiconductor_band_structure_(lots_of_bands).png"

Torquil

Thanks!

Link doesn't work.

What I mean by "minimal energy" is that since the band of states is continuous, it takes almost nothing for an electron to move into an empty state/location.

Zz.

I think there is a fundamental thing about the bands I haven't understood yet. When I apply an external E-field, then an electron in e.g. the 1. Brillouin zone goes into a state in the 2. Brillouin zone, and the electrons in the 2. Brillouin zone go the third and so on? This is what you mean by electrons moving into empty states?

 

[ZapperZ]

Thanks!

Link still doesn't work for me.

I think there is a fundamental thing about the bands I haven't understood yet. When I apply an external E-field, then an electron in e.g. the 1. Brillouin zone goes into a state in the 2. Brillouin zone, and the electrons in the 2. Brillouin zone go the third and so on? This is what you mean by electrons moving into empty states?

No, there's no need for transition into different Brillouin zones.

Look at the electrons at the Fermi energy. Add an infinitesimal amount of energy to it. Can it go "up" to that state? Sure it can! It's empty! It doesn't even change its k-momentum that much in doing that.

Zz.

 

[torquil]

Link still doesn't work for me.

Wow, that is strange. It works for me some of the times. If I right-click and choose "copy link location", and paste it manually in the address field, then it works. When I subsequently click on the lin, it works.

If I click on the link without having manually pasted it in the address field first, then it doesn't work...

Well well, you have probably seen the picture before anyway.

The same picture is here:
http://en.wikipedia.org/wiki/Conduction_band

Torquil

 

[Niles]

Look at the electrons at the Fermi energy. Add an infinitesimal amount of energy to it. Can it go "up" to that state? Sure it can! It's empty! It doesn't even change its k-momentum that much in doing that.

Zz.

Ok, I am convinced now. And the reason why this doesn't hold for insulators is that insulators do not have Fermi spheres. Correct?

I am a little worried when you say "states". We are talking about an electron moving from an ion #1 to an ion #2 in the lattice. How does that correspond to an electron gaining crystal momentum?

 

[ZapperZ]

Ok, I am convinced now. And the reason why this doesn't hold for insulators is that insulators do not have Fermi spheres. Correct?

I am a little worried when you say "states". We are talking about an electron moving from an ion #1 to an ion #2 in the lattice. How does that correspond to an electron gaining crystal momentum?

Not sure why "states" and "gaining momentum" have anything to do with each other in this context. The 'states' here also include spatial location, as in the exclusion principle. I also didn't say anything about gaining crystal momentum, did I?

Zz.

 

[Niles]

Not sure why "states" and "gaining momentum" have anything to do with each other in this context. The 'states' here also include spatial location, as in the exclusion principle. I also didn't say anything about gaining crystal momentum, did I?

Zz.

That is right, you did not. But we are applying an electric field, so the electrons gain momentum.