Fermi Energy in a 3D Box: What Happens in the 2D Limit?

[cosmic_tears]

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 \neq 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.

 

[turin]

Consider N non interacting electrons inside a 3D box ... Lx = Ly \neq 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.