Fermi Radius for 2D Rectangular Lattice?

[ThereIam]

Homework Statement

Hi all,

This is Kittel 9.2. This problem has been asked about before, but the people asking found solutions.

I'm trying to find the free electron Fermi radius for a 2D metal with a rectangular primitive cell (a = 2 Ang, b = 4 Ang).

Homework Equations

Please forgive my lack of latex skills:

E_f = hbar^2/(2m) * k_f^2
N = 2 * "volume" of Fermi sphere / volume element size.

For 3D this is N = V / (3 pi^2) * k_f^3 = 4 pi k_f^3 / 3 *(L /2 pi) ^3

The Attempt at a Solution

Since this is a 2D problem I take the volume of the Fermi sphere to be pi*k_f^2 (the area of a Fermi circle.

I am confused about the volume element. My lattice has different periodicities in each direction. I would assume based on the derivation of the volume element for phonons that this means my volume element should reflect this. I am tempted to take it to be (4 pi^2)/(a*b) = (2 pi^2)/(a^2). This is just the area of the first Brillouin zone.

I am not sure if this makes sense. I'm requiring that the wavefunctions be periodic like traveling plane waves in the respective directions k_x, k_y, as on page 137. Do I need to modify my expression for the Fermi energy to reflect my rectangular lattice in any way?

Anyway, I then get:
N = V * 1/(volume element) = a^2*k_f^2/pi

Then I assume N = 1, for valency 1? Though I just calculated the area of the occupied states...

 

[mark726]

Hello!

Based on your equations, it looks like you are on the right track. However, there are a few things that need to be clarified.

Firstly, for a 2D system, the Fermi energy should be calculated using the area of the Fermi circle, as you have correctly done. However, the volume element should be the area of the first Brillouin zone, which in this case would be (2pi/a)*(2pi/b) = 4pi^2/(ab). This is because in a 2D system, the wavefunctions are not periodic in all three dimensions, so the volume element should only include the two dimensions that are periodic.

Secondly, the N value that you are calculating is the total number of electrons in the system. This value should be equal to the number of valence electrons in the system, not necessarily 1. So if the system has multiple valence electrons, you would need to modify your equation accordingly.

Finally, for a rectangular lattice, the expression for the Fermi energy would be slightly different than for a square lattice. It would be E_f = hbar^2/(2m)*(k_x^2 + k_y^2), where k_x and k_y are the components of the wavevector in the x and y directions, respectively.

I hope this helps clarify things for you. Let me know if you have any further questions. Good luck with your calculations!