What Causes the Plateaus and Zeros in the Quantum Hall Effect?

[*FaerieLight*]

Hi! I'm having trouble understanding the quantum hall effect, that is, the fact that the Hall resistance versus magnetic field curve has regions where it drops to zero, and the longitudinal resistance versus magnetic field curve features plateaus.

When the filling factor is an integer, this corresponds to the situation where the highest occupied Landau level is completely filled, which means that the Fermi energy lies between an occupied Landau level and an unoccupied Landau level. How is it that this leads to the plateaus and zeros in the resistances? Is it as simple as electrons not being able to be scattered because there is nowhere for them to scatter to, at low temperatures and high B fields? How do the impurities in the lattice, which trap electrons in their potentials, contribute to the Hall effect?

I also don't get the fractional quantum hall effect. All the sources I've been looking at say that it is the result of electron correlation or interaction, and it corresponds to the case where the Fermi energy is partway up a Landau level. But how does this electron interaction give rise to the plateaus in the longitudinal resistance, and the zeros in the transverse resistance?

Thanks a lot!

 

[explainplease]

I will only answer your question about the integer quantum Hall effect because that's what I have knowledge about:

So to understand how disorder gives rise to plateau, you need to understand what disorder does first. So disorder (impurities) introduce add a random potential term, we require that this potential varies slowly over the characteristic length scale of the sample (we call it the magnetic length), you can from here prove that the electrons travel on equipotential surfaces.*
Now near a maxima or a minima of the potential, the equipotential contours are loops (imagine hills and valleys), so electrons at that energy get trapped, these are called localized states and they do not contribute to the current because they are unable to cross the sample, electrons that can cross the sample though are on equipotential surfaces that cut the whole sample and that is at V=0, these are extended states, that was the crucial point. Now localized states are present where there was a gap in the landau levels before, and when you fix the electron density and decrease the magnetic field, you're effectively decreasing the degeneracy in each Landau level, meaning that electrons need to occupy more states now, but instead of jumping right to the next landau level, these localized states get filled, but they do not contribute to the current so you get your plateau.

*Here is guideline for this exercise if you're interested: We use the classical solutions of a particle in an electromagnetic field, which was circular trajectories, promote the variables to operators and put them in the Heisenberg equation of motion, and they tell us how the centers of those circles evolve, you'll find they are travel on equipotential surfaces.