Edge states at high magnetic field (Quantum ballistic transport)

Question

I am learning about edge states at high magnetic field (Quantum ballistic sample). What I understand so far from it is that at high magnetic field, Landau states arise.

Then (I don't know exactly why?), edge states are formed. To enhance my understanding, can someone help me with these questions:

*Why must the chemical potential cross a Landau level?

*Why is the net current carried solely by these edge states?

(I am only interested in intuition, no derivations)

Answer 1

Here's the explanation I found very intuitive from our mesoscopic physics class.

1. Edge states form because at the boundaries the effective potential for electrons goes to infinity – these are real "walls" on the edges. Electrons cannot move outside of the sample and therefore their wave functions are zero outside the walls. We have seen in very elementary quantum mechanics the "infinite square well" model and here you see it in action.
2. Since the potential shoots up to infinity, it must cross the chemical potential somewhere near the edges.
3. The electrons available for transport all reside at or close to the Fermi level. In the bulk, electrons are in the heavily degenerate Landau levels, which are usually some distance below $E_F$. The Fermi distribution means that all these Landau levels are usually fully occupied. Therefore there is simply no empty states for the electrons to go into in the bulk, even if they'd like to scatter to the center. Thus electron transport and current only occur along the edges.

All these can be found in Cees Harman's lecture notes on mesoscopic physics.

Answer 2

A good intuitive picture of what happens when a high magnetic field is applied, can be obtained by the image posted in a related topic: Why bulk states in quantum hall effect do not contribute to electric conductivity.

When electrons are put into a magnetic field (perpendicular to the 2D plane in this case) they precess around it. In the bulk, there is sufficient space for them to complete the orbit. Furthermore, the radius of the orbit is determined by the field and only specific radii are allowed. Consequently, these states occur at specific energies. In a DOS (density of states) diagram these would appear as sharp Dirac deltas at the specific energies in the ideal case (the density at the allowed states is infinite). However, due to impurities you have a bit of symmetric broadening of the DOS around these energy levels. Now, whenever the chemical potential (Fermi Level) is near such energies, the states, which are called Landau Levels can be occuppied.

Moreover, you see that the effective movement that these electrons do is just circular orbits; they just keep turning round and round and there is not any significant propagation towards any direction. In fact, such states are the only allowed states in the bulk.

Near the edges however, the electrons cannot complete an orbit; instead they are on what they are called, skipping orbits which have a net effect of propagating towards one direction. Notice that the propagation direction at the two edges is opposite. This is due to the inversion symmetry breaking that is induced by the external magnetic field. Thus, near the edges we have states that contribute to current whereas in the bulk, this contribution is practically negligible.

This is a very intuitive picture and there are a lot of details to be seen if you want to study the subject in depth. A good lecture series pdf that I have found regarding quantum Hall effect can be found here.