Fermi energy - Quantum mechanics.

In summary, this conversation discusses the calculation of the fermi energy for N non interacting electrons inside a 3D box with unequal side lengths. The fermi energy is calculated as a function of N, Lx, Lz, and me. When taking the limit of Lz-->0 for a 2D system, there seems to be a conflict where the fermi energy approaches infinity. Upon closer examination, it is revealed that the problem statement is misleading and the true physical meaning of the fermi energy must be considered in order to resolve the conflict.
  • #1
cosmic_tears
49
0

Homework Statement


(I copy-pasted the question)
Consider N non interacting electrons inside a 3D box similar to the one we saw
in class, but this time it is not cubical, i.e. Lx = Ly [tex]\neq[/tex] Lz.
1. Calculate the fermi energy as a function of N;Lx;Lz;me...
2. What happens if we take Lz --> 0, i.e. the limit of a 2D system. Is there
a problem?
3. Calculate the fermi energy in a 2D system (Lx = Ly). Do you obtain a
reasonable result?
4. It would be a great conflict in case the 2D result cannot be obtained by
taking the limit Lz-->0. Did you nd such a conflict? If so, explain how
it can be avoided.


Homework Equations


I solved the 3 first questions. I hope I didn't make a calculation mistake, but anyhow, I don't think it'll fundimentally change the problem of question 4...
So here are the answers I've got:

For the 3D box, I marked Lx=Ly=L, and Lz stayed...
So:
Ef = (1/2m)*((6N((hbar)^2)*pi)/(Lz*L^2))^3/2

For the 2D problem I got:

Ef = (4Npi*(hbar)^2)/(mL^2)


The Attempt at a Solution



Problem is, obviously, it seems that Ef-->infinity as Lz --->0. I have no idea how to resolve this conflict! :)

Thank you very very very much for reading and helping!

Love :)
Tomer.
 
Last edited:
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  • #2
cosmic_tears said:
Consider N non interacting electrons inside a 3D box ... Lx = Ly [tex]\neq[/tex] Lz.
1. Calculate the fermi energy as a function of N;Lx;Lz;me...
2. What happens if we take Lz --> 0, i.e. the limit of a 2D system. Is there
a problem?
3. Calculate the fermi energy in a 2D system (Lx = Ly). Do you obtain a
reasonable result?
4. It would be a great conflict in case the 2D result cannot be obtained by
taking the limit Lz-->0. Did you nd such a conflict? If so, explain how
it can be avoided.

Problem is, obviously, it seems that Ef-->infinity as Lz --->0. I have no idea how to resolve this conflict! :)
I did not read your solutions, but I think I know what the problem is. My main hint would be that this seems to be a trick question. Actually, I would say that the problem statement flat-out lies about something, and you, the trusting student, are not well-equipped to combat such subtle chicanery. The resolution can be approached from the more distorted pose of the problem statement as given; however, this envolves the realization of several (or at least a few) quite subtle facts about the physical significance of the Fermi energy.

I recommend that you try to figure out why I say that this is a trick question (i.e. that the problem statement is actually a lie). Another hint for this is to consider how the density of states changes when you cross over from Lz>L to Lz<L, especially in terms of the contribution in the z-direction.

If you really want to approach the resolution from the dark side, you must think outside the box ... literally. This gets at the true physical meaning of Fermi energy. Consider the environment of lowest energy electrons outside the box, and how much energy must be added to one of them in order to stuff it inside the box. Is this equal to the energy that the electron must have when it is inside the box? (I told you it was subtle.) If you are able to resolve the disconnect between the 2-D system and 2-D limit of the 3-D system in this way, then you will see that the problem statement is quite misleading, if not a flat-out-lie.

Please ask for clarification on any of this. This envolves a very subtle issue, and it is very difficult to give hints that do not spoil the ending.
 

1. What is Fermi energy in quantum mechanics?

Fermi energy is a concept in quantum mechanics that represents the highest energy level occupied by electrons in a solid at absolute zero temperature. It is also known as the Fermi level and is a measure of the energy required to remove an electron from a solid.

2. How is Fermi energy related to the electronic structure of a solid?

Fermi energy is directly related to the electronic structure of a solid, as it represents the energy level at which the highest energy electrons reside. It is determined by the number of electrons in a solid and the energy levels available for their occupation.

3. Why is Fermi energy important in materials science and engineering?

Fermi energy is important in materials science and engineering because it helps us understand the electronic properties of solids and how they behave in different conditions. It also plays a crucial role in the study of electrical and thermal conductivity, as well as in the design and development of electronic devices.

4. Can Fermi energy be changed or controlled?

Yes, Fermi energy can be changed or controlled by altering the number of electrons in a solid or by changing the temperature. The energy level can also be modified by applying an external electric or magnetic field.

5. How is Fermi energy different from other energy levels in a solid?

Fermi energy is the highest occupied energy level in a solid at absolute zero temperature, while other energy levels represent the energy required to excite an electron above the Fermi level. It is also unique in that it is a collective property of a solid, rather than belonging to individual electrons.

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