Free electron gas band structure?

In summary, the dispersion relation for a free electron gas is spherically symmetric, and the Fermi surface is closely related to the free electron sphere.
  • #1
Philip Land
56
3
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!
 
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  • #2
Philip Land said:
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
[tex]E(k) = \frac{\hbar^2k^2}{2m}[/tex]
looks like graphically?

Zz.
 
  • #3
ZapperZ said:
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
[tex]E(k) = \frac{\hbar^2k^2}{2m}[/tex]
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?
 
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  • #5
Philip Land said:
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.
 
  • #6
Philip Land said:
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
 
  • #7
@fluidistic

You are right! The answer to the OP’s question
Philip Land said:
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.
 
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  • #8
Lord Jestocost said:
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.
 
  • #10
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...
 

1. What is a free electron gas band structure?

A free electron gas band structure is a model used in solid state physics to describe the energy levels of electrons in a material. It assumes that the electrons are not bound to any particular atom and are free to move throughout the material.

2. How is the free electron gas band structure different from the band structure of a regular solid?

In a regular solid, the energy levels of electrons are determined by the specific atoms and their arrangement in the material. In a free electron gas, the energy levels are determined by the overall properties of the material, such as its density and temperature.

3. What is the significance of the Fermi level in the free electron gas band structure?

The Fermi level represents the highest energy level occupied by electrons at absolute zero temperature. It is an important parameter in understanding the electrical and thermal conductivity of materials.

4. How does the free electron gas band structure explain the electrical conductivity of metals?

The free electron gas model explains the high electrical conductivity of metals by the presence of a large number of free electrons that are able to move easily in response to an electric field. This is due to the overlapping energy levels in the conduction band, which allows for a continuous flow of electrons.

5. Can the free electron gas band structure be applied to all materials?

No, the free electron gas model is most applicable to metals and some semiconductors. Insulators, on the other hand, have a large band gap between the valence and conduction bands, making the free electron gas model less relevant.

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