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I think I can understand hall effect using the right hand rule, the electrons experience lorentz force in the presence of a magnetic field but I don't get why quantum? I saw the resistivity looks like a stepped graph why do electrons behave in such a manner like a current flowing through a multi stages zener diode? (OK I made up the multi stages)

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The simplest explanation requires basic knowledge about quantum point contacts. QPCs are narrow openings between two electron gases, where transverse motion of electrons is quantized. The conductance through such openings is guantized in the units of the conductance quantum $$G_0 = \frac{2e^2}{h}.$$ This happens, because for a one-dimensional motion the group velocity exactly cancels out with the density of states (the former is proportional to $\partial_k\epsilon(k)$, whereas the latter is inversely proportional to this quantity). The conductance is therefore proportional to the number of conducting transverse energy levels, each level contributing $G_0$. More formally this physics is expressed by the Landauer formula.

In quantum Hall effect the magnetic field is so strong that the electrons are confined to the walls of the conductor, moving in effectively one-dimensional channels, and thus exhibiting conductance quantization (which means that the resistance scales as $1/n$ with the number of conducting sub-bands.)

What I described here is the Integer quantum Hall effect (IQHE), which is essentially a one-particle phenomenon. Fractional quantum Hall effect has to do with electron-electron interactions and by far more complex.

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