Hopping Conductivity in the Quantum Hall Effect: Revival of Universal Scaling

Abstract

We have measured the temperature dependence of the conductivity σxx of a two-dimensional electron system deep into the localized regime of the quantum Hall plateau transition. Using variable-range hopping theory we are able to extract directly the localization length ξ from this experiment. We use our results to study the scaling behavior of ξ as a function of the filling factor distance |δν| to the critical point of the transition. We find for all samples a power-law behavior ξ ∝ |δν| −γ with a universal scaling exponent γ = 2.3 as proposed theoretically.PACS numbers: 73.40. Hm, 71.23.An, 72.20.My, 71.30.+h The prominent feature of the quantum Hall effect (QHE) is the emergence of quantized plateaus in the Hall resistance of a two-dimensional electron system (2DES) reflecting the localization of states at the Fermi edge. A deeper understanding of this fascinating phenomenon is gained from regarding the transition region between adjacent QHE plateaus [1,2]. In this regime the dissipative transport is governed by the delocalized states in the vicinity of the Landau level center. The degree of localization is expressed by the localization length ξ denoting the typical extension of the electron wave function. For any sample with a finite size L theory predicts the conductivity tensor in the plateau transition to follow a scaling function f (L/ξ) with a diverging localization length ξ = |δν| −γ [3] where δν = ν − ν c denotes the filling factor distance to the critical point ν c of the transition. The critical exponent γ = 2.3 was predicted to be universal, its value determined numerically [4] and validated by variation of the sample size [5]. Experimentally the scaling of the conductivity near the critical point was verified with great success for different samples by temperature, current, and frequency dependent measurements of the transition width [6,7,8]. These new parameters introduce effective lengths, and L f ∝ f 1/z and thereby add additional exponents z and p. These effective lengths replace the physical sample size L in the scaling functions f (L/ξ). Recent experiments extended the focus to the transition from the quantum Hall state to the Hall insulator [9,10,11]. In these investigations striking similarities to the scaling behavior in the transition between different QH states were observed [12] hinting to the same universality class for both types of transitions [13,14].In spite of the great success of scaling theory in the QHE there are still some experiments not fitting into the picture of universality. Non-universal exponents were observed in the dependence of the transition width on temperature [15], current [16] and frequency [17]. Other experiments seem to contradict scaling theory at all, both for the Hall plateau-insulator transition [18] and the transition between QHE plateaus [19,20].However, before making conclusions on a general failure of scaling theory it has to be considered that nearly all experiments do not measure the localization length ξ directly. Therefore an assumption about the ...

Nonequilibrium mesoscopic conductance fluctuations as the origin of 1/f noise in epitaxial graphene

“…As a result of variable range hopping, the longitudinal resistivity is known to increase nonlinearly with temperature, which has been shown for GaAs [74] as well as for epitaxial graphene devices [75]. Furthermore, the electric field strength causes the same effect as an effective temperature in the variable range hopping mechanism [74,76]. Thus, the large increase of 1/f noise PSD with increasing current could result from a strongly nonlinear current dependence of the longitudinal resistivity due to variable range hopping.…”

Spin-zero anomaly in the magnetic quantum oscillations of a two-dimensional metal

“…Although the nature of this crossover is unclear, it may be caused by intra-and inter-layer hopping. The VRH conductivity (10) is well documented in semiconductor 2D integer quantum Hall effect (IQHE) systems (see [36,37] and references therein). In β -(BEDT-TTF) 2 SF 5 CH 2 CF 2 SO 3 , the conducting layers are weakly coupled with the CP nearly fixed which favors the IQHE within the layers.…”

Possible existence of a band of extended states induced by inter-Landau-band mixing in a quantum Hall system

“…Scaling behaviors of plateau-plateau transitions are observed in recent experiments 29,30,31,32,33 . At a first glance, it seems that these results conflict with both non-scaling experiments and our numerical results.…”

“…In the non-scaling experiments 21 , it is known that data not too close to a transition point follow a scaling law. The other is that the samples used in all scaling experiments 29,30,31,32,33 are clean with very high mobility while non-scaling behavior was observed in relative dirty samples 21,22,23 . In fact, the mobility in recent scaling experiments is at least one order of magnitude larger than that in early non-scaling experiments.…”