The title already gives away the question. I see that disorder effects that the Landau levels are broadened out. They allow states to be either extended through the whole solid or localized to a specific region. Why is this important for the plateaus in Hall resistance to form?


The usual plot of the IQHE has conductivity on the vertical axis and electron density on the horizontal axis. Thus, the plateaus correspond precisely to increasing the electron density yet having the conductivity remain the same.

One of the consequences of disorder is that disordered systems can have many states that are localized, hence, cannot contribute to conductivity. Thus, the plateaus precisely correspond to filling those localized states that cannot contribute to conductivity, which is why it stays the same.

If you didn't have disorder, you'd have no localized states, and then increasing the density will necessarily fill states that contribute to conductivity, giving the classical linear dependence between the two quantities.

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