We have studied the effect of both long-and short-range disorder on frequency scaling of the diagonal magnetoconductivity σ xx in the integer quantum Hall effect regime of two-dimensional electrons confined to Al x Ga 1-x As/Al 0.33 Ga 0.67 As single heterostructures for two Al concentrations, x. Within the frequency range 100 MHz f 20 GHz and for a temperature T = 35 mK, we found that the frequency scaling exponent c changes from 0.6 ± 0.05 for a GaAs/Al 0.33 Ga 0.67 As heterostructure, where the disorder is dominated by long-range ionized impurity potentials, to c = 0.42 ± 0.06 for Al 0.015 Ga 0.985 As/Al 0.33 Ga 0.67 As heterostructures, where the dominant contribution to the disorder is from short-range alloy potential fluctuations. This value of c allows us to estimate the dynamical scaling exponent as z = 1 ± 0.13.

We have investigated experimentally the scaling behaviour of quantum Hall transitions in GaAs/AlGaAs heterostructures of a range of mobility, carrier concentration, and spacer layer width. All three critical scaling exponents γ, κ and p were determined independently for each sample. We measure the localization length exponent to be γ ≈ 2.3, in good agreement with expected predictions from scaling theory, but κ and p are found to possess non-universal values. Results obtained for κ range from κ = 0.16 ± 0.02 to κ = 0.67 ± 0.02, and are found to be Landau level (LL) dependent, whereas p is found to decrease with increasing sample mobility. Our results demonstrate the existence of two transport regimes in the LL conductivity peak; universality is found within the quantum coherent transport regime present in the tails of the conductivity peak, but is absent within the classical transport regime found close to the critical point at the centre of the conductivity peak. We explain these results using a percolation model and show that the critical scaling exponent depends on certain important length scales that correspond to the microscopic description of electron transport in the bulk of a two-dimensional electron system.

Abstract. We present a unifying model of plateau-to-plateau transitions in the quantum Hall effect based on results from high resolution frequency scaling experiments. We show that as the frequency or quantum coherence length of the two-dimensional electron system is varied, one observes a crossover between classical percolation and quantum percolation in the measured values of the critical scaling exponents of the plateau-to-plateau transitions. This crossover is dependent on the relationship between certain relevant length scales of a twodimensional system and can be explained using a quantum percolation model. The model explains why quantum criticality can be observed in some systems, but is absent from others.