Landau levels degeneracy

I'm studying the Landau quantization of free electron systems and a question comes to my mind when I arrive to the degeneracy of the levels. This is given by the following expression:

$$N=\frac{2e}{hc} HL^2$$

Where $$H$$ is the magnetic field and $$L^2$$ is the area of the system in the direction perpendicular to the magnetic field.

As we can see, this expression does not depend on the energy level. How is this possible? I mean, following my intuition, when we turn on the magnetic field, all the levels being $$ħ\omega_c$$ around a Landau level, collapse into this Landau level, so at the end, the number of states in a particular level should increase as we go to higher Landau levels, as well as the degeneracy, since at higher energies in the free electron model the number of available states increases. Am I wrong about this idea? I guess intuition can fail when we talk about these complex phenomena.

• Apologies if this is known to you, but the occupation of the highest Landau level ranges from completely full to entirely empty, en.m.wikipedia.org/wiki/Landau_quantization. I'm currently reading this myself, so thanks for your question. Jan 31 '20 at 13:51
• Yes I know, this is not exactly related to my question, but thank you anyway :) I try to give an answer to my own question right below. You can check it. Jan 31 '20 at 18:11

There is no $$k$$ dependence in the dispersion relation of Landau Levels. They are flat. Since degeneracy counts the number of $$\left(k\right)$$ states available for the system at a particular energy, if energy state is independent of $$k$$, then each state will have the same degeneracy.

• Ok, I'm not sure about how correct your answer is. I think that, even though there is not a dependence in the energy on the k-direction, there is a dependence on the magnitude k, which is related to E, i.e., there are discrete allowed values of k in the $k_x$-$k_y$ plane, but in any case after some studying I got some conclusions. I write them below. Jan 31 '20 at 18:07
• Yes. But the same k will be present in each energy level. So whatever dependency on k the energy has, it is level independent. Feb 3 '20 at 14:05

I think this explanation is quite cumbersome without a proper picture so I suggest you go to Kittle's Introduction to Solid State Physics, page 246, Fig. 24.

When you sketch the Landau levels in a 2 dimensional k-space with $$k_x$$ and $$k_y$$ axes, you can observe that they describe circles, and that the distance between circles corresponding to consecutive levels is not always constant, but as we increase the levels, it becomes shorter. This is because in energy, they are always separated by $$\hbar w_c$$, and the energy increases as $$k^2$$, so for keeping the energy interval constant, the change in the value of $$k$$ between one level and the next one must decrease as we go to higher levels! This happens so the area between levels is constant and the Onsager's relation fulfills.

Now, let's consider the 2DEG before the magnetic field is turned on. The k-points are distributed so that they occupied an area $$\frac{(2 \pi )^2}{A}$$. When we turn the magnetic field on, they must collapse into the closer Landau level. In the 2DEG, the number of available states, as I said in my question, is lower at lower energies. However, the distance between Landau levels is higher, and then, more states enter the area between circles and this compensates the fact that the number of states is lower and we recover the value of the degeneracy.

I hope this is clear, because it's no easy to explain it properly in a writting text.