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

In summary, the quantum hall effect is observed when the highest occupied Landau level is completely filled at an integer filling factor. This leads to plateaus and zeros in the resistances due to the electrons being unable to scatter and localized states being present in the gaps between Landau levels. Disorder, in the form of impurities, contributes to these plateaus by creating localized states that do not contribute to current flow. This effect can also be understood by considering the classical solutions of a particle in an electromagnetic field, where the centers of circular trajectories travel on equipotential surfaces.
  • #1
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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!
 
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  • #2
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.
 

What is the Quantum Hall Effect?

The Quantum Hall Effect is a phenomenon in condensed matter physics where the Hall resistance of a two-dimensional electron gas exhibits quantized plateaus at certain values of magnetic field and electron density.

What causes the Quantum Hall Effect?

The Quantum Hall Effect is caused by the interaction between electrons and a magnetic field in a two-dimensional system. As the magnetic field increases, the electrons become confined to specific energy levels, resulting in the quantized plateaus in the Hall resistance.

How is the Quantum Hall Effect measured?

The Quantum Hall Effect is typically measured by passing a current through a sample in the presence of a magnetic field and measuring the voltage perpendicular to the current. The resulting Hall resistance is then compared to the quantized values to determine the electron density.

What is the practical application of the Quantum Hall Effect?

The Quantum Hall Effect has several practical applications, including precise measurements of fundamental constants such as the fine structure constant and the electron charge and mass. It is also used in the development of new technologies such as quantum computers and sensors.

What are the different types of Quantum Hall Effect?

There are two main types of Quantum Hall Effect: integer and fractional. The integer Quantum Hall Effect occurs at specific values of the magnetic field and electron density, while the fractional Quantum Hall Effect occurs at fractional values and is observed in systems with strong electron-electron interactions.

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