Hopping conduction in the Landau level tails in GaAs-AlxGa1-xAs heterostructures at low temperatures

Ebert, G.; Klitzing, K. von; Probst, C.; Schuberth, E.; Ploog, K.; Weimann, G.

doi:10.1016/0038-1098(83)90441-6

Cited by 120 publications

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“…Using the fact that the conductivity s xx in the QHE plateaus at low temperatures is dominated by variable-range hopping (VRH) [3] a direct access to the localization length j can be gained from analyzing the dependence of s xx on temperature, current, and frequency. Experimentally mainly the temperature dependence of s xx in the VRH regime was investigated [4].…”

Section: High Frequency Conductivity In the Quantum Hall Regimementioning

confidence: 99%

High Frequency Conductivity in the Quantum Hall Regime

2001

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We have measured the complex conductivity s xx of a two-dimensional electron system in the quantum Hall regime up to frequencies of 6 GHz at electron temperatures below 100 mK. Using both its imaginary and real part we show that s xx can be scaled to a single function for different frequencies and several transitions between plateaus in the quantum Hall effect. Additionally, the conductivity in the variable-range hopping regime is used for a direct evaluation of the localization length j. Even for large filling factor distances dn from the critical point we find j~dn 2g with a scaling exponent g 2.3. DOI: 10.1103/PhysRevLett.86.5124 PACS numbers: 73.43.-f, 71.23.An, 71.30.+h, 72.20.My It is widely accepted that the understanding of the integer quantum Hall effect (QHE) is closely related to a disorder driven localization-delocalization transition occurring in two-dimensional electron systems (2DES) in high magnetic fields [1]. Many experimental and theoretical works approve the interpretation of the transition between adjacent QHE plateaus as a quantum critical phase transition. It is governed by a diverging localization length j~jE 2 E c j 2g which scales with the distance of the energy E from the critical energy E c in the center of a Landau band. The exponent g ഠ 2.3 is believed to be a universal quantity independent of disorder. For finite systems with effective size L eff theory predicts that the conductivities s ab follow scaling functions s ab f ab ͓L eff ͞j͑E͔͒ resulting in a finite width DE~L 21͞g eff of the transition region. The effective system size L eff is determined by the physical sample size, the electron temperature T , or the frequency f.The most common test of scaling uses an analysis of the temperature or frequency dependence of the conductivity peak width in the QHE plateau transition. However, lacking an exact expression for L eff ͑T , f͒, this method does not allow direct access to the scaling behavior of the localization length.An alternative approach to scaling was proposed by Polyakov and Shklovskii [2]. Using the fact that the conductivity s xx in the QHE plateaus at low temperatures is dominated by variable-range hopping (VRH) [3] a direct access to the localization length j can be gained from analyzing the dependence of s xx on temperature, current, and frequency. Experimentally mainly the temperature dependence of s xx in the VRH regime was investigated [4]. However, due to an unknown theoretical prefactor, j could only be estimated from these experiments.In contrast, the frequency driven variable-range hopping conductivity s xx ͑f͒ [in the limit s xx ͑f͒ ¿ s xx ͑0͒] is given by [2] Res xx ͑v͒ 2p 3 ee 0 jv ,linearly depending on both frequency f v͞2p and localization length j with no unknown prefactors.Here we report on measurements of the complex conductivity s xx up to frequencies f 6 GHz at low temperatures down to below 100 mK. We will show that Im͑s xx ͒ can be scaled with a single-parameter function to Re͑s xx ͒, independent of temperature, frequency, and filling factor. ...

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Section: High Frequency Conductivity In the Quantum Hall Regimementioning

confidence: 99%

High Frequency Conductivity in the Quantum Hall Regime

2001

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“…The origin of these plateaus is attributed to a position of the Fermi energy within a mobility gap. The density of localized states in the mobility gap is unimportant for a discussion of transport data at zero temperature but must be known if variable range hopping [1] (which includes the density of states at the Fermi energy) plays a role or if thermodynamic properties are studied (specific heat [2], De Haas-Van Alphen effect [3]). Moreover, all transport theories include information about the density of states (DOS), and a microscopic theory of the quantum Hall effect or the longitudinal resistivity should give first of all a correct answer for the density of states.…”

Section: Introductionmentioning

confidence: 99%

Density of states in Landau level tails of GaAsAlxGa1−xAs heterostructures

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“…VRH dominates the conductivity at low temperatures, when the localization length becomes much smaller than the effective temperature length L T . In the QHE regime the VRH conductivity is given as [22,24,25] σ xx (T ) = σ 0 exp − T 0 /T , k B T 0 = C e 2 4πǫǫ 0 ξ ,with a temperature dependent prefactor σ 0 ∝ 1/T . The characteristic temperature T 0 is determined by the Coulomb energy at a length scale given by the localization length ξ, the dimensionless constant C being in the…”

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

2002

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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 ...

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Hopping conduction in the Landau level tails in GaAs-AlxGa1-xAs heterostructures at low temperatures

Cited by 120 publications

References 5 publications

High Frequency Conductivity in the Quantum Hall Regime

High Frequency Conductivity in the Quantum Hall Regime

Density of states in Landau level tails of GaAsAlxGa1−xAs heterostructures

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

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