Theory of quantized Hall effect at low temperatures

At millikelvin temperatures the quantized Hall effect (QHE) is characterized by sharp steps connecting the quantized Hall resistance plateaus. We explain this behavior on the basis of the single-electron approximation and continuum percolation theory. It is shown that even when the magnetic field corresponds to partially filled Landau level on average, locally the sample breaks into patches having the occupation numbers 0 or 1. This represents a peculiar type of the metal-insulator transition, driven by disorder. Extended (global) electron states may not exist at equilibrium or arbitrarily small applied voltage $V$. Global states begin to appear at a certain critical voltage $V_{\mathrm{cr}}$, which is of the order of characteristic magnitude of potential fluctuations on the scale of the sample size. For small $V>\mathrm{}{V}_{\mathrm{cr}}$ the fraction of global states is still small owing to the smallness of the average electric field compared with the fluctuating field in the inversion layer. Because of this, transitions between the QHE plateaus require only a minor change in the density of states and at $T=0\mathrm{}$ they occur in small intervals of the magnetic field. Owing to the tail of the Fermi-Dirac distribution at nonzero $T$ the deviation of the Hall conductivity from its nearest quantized value, $\left(\frac{e^2}{h}\right)\times \mathrm{integer}$, is activated. This explains the plateau flatness and the high precision of the QHE measurements of the fine-structure constant.