Why do the plateaus in the Quantum Hall Effect appear phenemenologically? Something to do with the 1D transport for the edge states?
This is a citate from a 'layman' paper I wrote on QHE:
However, the story is still not complete because how can we account for the plateau nature of the QHE? How is it possible that the quantized values of the resistivity persist over (long) ranges of magnetic field? It turns out that the plateaux owe their existence to disorder, arising from the inherently dirty experimental samples. They contain impurities which can be modelled by adding a random potential V(x) to the Hamiltonian. As we introduce a random potential, the degenerate states in each Landau levels become broadened into a band of states, having a Lorentzian lineshape centered at the unperturbed energy of each Landau level. Intuitively, the further an electronic state moves away from the unperturbed energy of each Landau leven, the more affected it is by the disorder and hence the more it has a tendency to be localized. Hence, the states close to the center of the Landau level are less localized than those at the edge of the Lorentzian distrubution. From the simple picture that extended states form only at the center of each Landau level and all the other states are localized, we can explain the origin of the Quantum hall plateaux. Because the current is carried only by the states at the center of each Landau level, the current should jump discontinuously as the Fermi level is tuned through the center of each Landau level. Further, the current should remain constant if the occupation of the extended states remains unchanged.