Understanding Band Structure Diagrams

[Philip Land]

Hi!

I never really understood band structure diagrams. I think they represent the energy of an electron, with the given circumstances, at a k vector.

Are the electrons only allowed to be on the lines here? Or can they also be in areas enclosed under/between certain lines?

What are some interesting properties I could deduce by looking at a bandstructure? Such as if the material is a metal, semiconductor etc?

I have seen a bunch of these diagrams but this is still very fuzzy for me, grateful if someone could provide some basic clarity.

 

[ZapperZ]

First of all, have you studied reciprocal space in crystal structure? This includes crystal orientation, direction, Brillouin zone, etc...

Secondly, have you, for example, solved the free-electron (or semi-free electron) problem in metals, and arrived at the E vs. k relationship for such a case? In other words, have you seen a simple dispersion curve?

And finally, do you know what Fermi energy, Fermi level, Fermi surface are?

Zz.

 

[jihai]

I think you should find a book about solid state physics and read it. Then you will get some answers of your questions.
Basically,band structure is a conventional way to show eigenvalues of the Hamilton about solid state matters. The progress to get the eigenvalues is like that you solve the Hamilton of Hydrogen atom in Quantum Mechanics, of course more complicated.

 

[Philip Land]

First of all, have you studied reciprocal space in crystal structure? This includes crystal orientation, direction, Brillouin zone, etc...

Secondly, have you, for example, solved the free-electron (or semi-free electron) problem in metals, and arrived at the E vs. k relationship for such a case? In other words, have you seen a simple dispersion curve?

And finally, do you know what Fermi energy, Fermi level, Fermi surface are?

Zz.

Yes, I have derived the relation between E and K for free electrons in metals. I have also used Brillouin zones to draw 2-d Fermi-surfaces.

Although I just began with condensed matter, I think understanding the Bandstructure diagram would help me a great deal.

For example, here the say "The crystal behaves as a metal if one or more bands are partly filled. The crystal is a semiconductor or semimetal if one or two bands are slightly filled or slightly empty."

- Ok. This makes sense. But how does this translate into the Bandstructure? What does "filled" mean in the figure? Again, are the elections on the bands, or between bands?

By looking at the figure, how would I determine if its eg. a semi-metal/metal semiconductor etc? The background theory make some sense, the diagrams don't.

 

[ZapperZ]

Yes, I have derived the relation between E and K for free electrons in metals. I have also used Brillouin zones to draw 2-d Fermi-surfaces.

Although I just began with condensed matter, I think understanding the Bandstructure diagram would help me a great deal.

For example, here the say "The crystal behaves as a metal if one or more bands are partly filled. The crystal is a semiconductor or semimetal if one or two bands are slightly filled or slightly empty."

- Ok. This makes sense. But how does this translate into the Bandstructure? What does "filled" mean in the figure? Again, are the elections on the bands, or between bands?

By looking at the figure, how would I determine if its eg. a semi-metal/metal semiconductor etc? The background theory make some sense, the diagrams don't.

Let's use the example you wrote.

Look at all the "spaghetti lines". If there is a band that crosses the Fermi energy, then it means that there is an electronic dispersion curve that is similar to what you saw when you solved for the free-electron gas. This is a metal! If you don't see anything crossing the Fermi energy, then it is not a metal.

A semiconductor or an insulator is simply a non-Fermi crossing band structure but of varying degree. Look at the highest peak in the valence band below the Fermi energy, and then look at the lowest dip in the conduction band above the Fermi energy. That separation is the band gap. The size of the band gap will dictate if it is a semiconductor or an insulator. If the bottom of the dip lines up with the top of the peak, then you have a direct band gap. If not, it is an indirect band gap.

Etc... etc. You get group velocity, effective mass, and other parameters as well from the band structure.

Zz.

 

[Philip Land]

Let's use the example you wrote.

Look at all the "spaghetti lines". If there is a band that crosses the Fermi energy, then it means that there is an electronic dispersion curve that is similar to what you saw when you solved for the free-electron gas. This is a metal! If you don't see anything crossing the Fermi energy, then it is not a metal.

A semiconductor or an insulator is simply a non-Fermi crossing band structure but of varying degree. Look at the highest peak in the valence band below the Fermi energy, and then look at the lowest dip in the conduction band above the Fermi energy. That separation is the band gap. The size of the band gap will dictate if it is a semiconductor or an insulator. If the bottom of the dip lines up with the top of the peak, then you have a direct band gap. If not, it is an indirect band gap.

Etc... etc. You get group velocity, effective mass, and other parameters as well from the band structure.

Zz.

Thanks a lot for this!