# Free electron gas band structure?

• I
How can I see, by looking at a band structure if the substance in question can be viewed as a free electron gas (FEG) or not?

What characterizes a FEG in a bandstructure plot?

Thanks in advance!

## Answers and Replies

ZapperZ
Staff Emeritus
Science Advisor
Education Advisor
How can I see, by looking at a band structure if the substance in question can be viewed as a free electron gas (FEG) or not?

What characterizes a FEG in a bandstructure plot?

Thanks in advance!

I'm rather surprised that you asked this, considering what you wrote in this post:

https://www.physicsforums.com/threads/band-structure-diagrams.966249/#post-6134363

If you have derived the dispersion relation for a free-electron gas, then what exactly is the issue here? Do you not know what
$$E(k) = \frac{\hbar^2k^2}{2m}$$
looks like graphically?

Zz.

I'm rather surprised that you asked this, considering what you wrote in this post:

https://www.physicsforums.com/threads/band-structure-diagrams.966249/#post-6134363

If you have derived the dispersion relation for a free-electron gas, then what exactly is the issue here? Do you not know what
$$E(k) = \frac{\hbar^2k^2}{2m}$$
looks like graphically?

Zz.
Well, the electrons will occupy parabolic bands, but that's true for many band structures, all through the are not free electron gases, so there must be something else than that simple argument allowing me to by looking at a plot see if its a free electron gas, such as no splitting between bands?

How can I see, by looking at a band structure if the substance in question can be viewed as a free electron gas (FEG) or not?

What characterizes a FEG in a bandstructure plot?

Thanks in advance!
I guess you are looking for the band structure of the nearly free electron in a crystal, right?
For that, you can transfer the free electron E-k relation to the first Brillouin zone.

fluidistic
Gold Member
Well, the electrons will occupy parabolic bands, but that's true for many band structures, all through the are not free electron gases, so there must be something else than that simple argument allowing me to by looking at a plot see if its a free electron gas, such as no splitting between bands?
Not really. The perfectly parabolic dispersion relation is a signature of a free electron model, at least as far as I understand. If the electrons interact weakly with the ions making the solid, a better description of the electrons can be obtained by using the nearly free electron model, that do take into account a potential of interaction between the electrons and the lattice. As a result, the dispersion relation is almost parabolic, but it has gaps, and it isn't quite parabolic due to a distorsion near the Brillouin zone (BZ). As you can imagine, if you complicate even more the description of the properties of the electrons, there is all the reasons in the world to guess that the dispersion relation will tend not to be a perfect parabola, which differs from the FEM.

@Lord Jestocost I would rather not look at the Fermi surface, because if we take a look at the one of lithium, it looks like a sphere that has no gap, i.e. it is entirely within the first BZ, even though it isn't exactly spherical. However the density of state near the Fermi energy differs somewhat compared to that of the FEM. So I wouldn't think that taking a look at the Fermi surface is a good indicator, but I may be wrong

Lord Jestocost
Gold Member
@fluidistic

You are right! The answer to the OP’s question
How can I see, by looking at a band structure if the substance in question can be viewed as a free electron gas (FEG) or not?

should be: Physically, electrons in metals can in principle not be viewed as free electron gases, as the electrons always experience the crystal potential. Some metals have nearly spherical Fermi surfaces, i.e., the crystal potential does not “distort” too much the free electron gas Fermi surface.

Philip Land
Dr_Nate
Science Advisor
One can look at the Fermi surface whether it is closely related to the free electron sphere or not.
10. Electron Dynamics and Fermi Surfaces
The FEG dispersion relation is spherically symmetric so it gives a spherical Fermi surface. Other dispersion relations that are spherically symmetric could do the same. For example, a Dirac point, which has linear dispersion, could possibly create a spherical Fermi surface.

hutchphd
Science Advisor
Homework Helper
It seems to me more important here, for pedagogical reasons, to emphasize the band gaps which follow directly from the periodicity of the background potential of the ionic cores and the coherent backscatter near the Brillouin zone edge. This precludes eigenstates that produce net current.
So for short answer the the OP, it is the gaps......