On the structure of strips exhibiting the integral quantum Hall effect in an inhomogeneous 2D electron system

A formalism is developed to generalize the results obtained for “incompressible” strips exhibiting the integral quantum Hall effect in a spatially inhomogeneous 2D electron system to the cases of finite temperatures, significant electron density gradients, etc. Specifically, the concept of the “quality” of a given integer quantum Hall effect strip (channel) is introduced; the quality is proportional to the derivative dn(x)/dx in the central part of the channel [n(x) is the electron density distribution over the channel]. For a well-defined channel, this derivative tends to zero. If a noticeable gradient arises in the n(x) distribution, the channel does not exhibit the quantum Hall effect and ceases to exist. The conditions are determined under which a channel exhibiting the integral quantum Hall effect breaks down. The results of calculations are used to interpret the available experimental data.     

Inhomogeneous Hall-geometry sample in the quantum Hall regime

A generalization of the known theory describing the Hall channels with integer filling factors in inhomogeneous 2D electronic samples to the case of a stationary nonequilibrium state (with a nonzero Hall voltage V _H across the 2D system) is proposed. For the central strip located near the extremum of the electron density, the theory predicts a change in its width and a shift of the whole strip from the equilibrium position as functions of V _H . The theoretical results are used to interpret recent experiments on measuring the local electric fields along the Hall samples both in equilibrium conditions and in the presence of transport in the quantum Hall regime.     

Theory of the Half-integer Quantum Hall Effect in Graphene

The unusual quantum Hall effect (QHE) in graphene is described in terms of the composite (c-) bosons, which move with a linear dispersion relation. The “electron” (wave packet) moves easier in the direction [1 1 0 c-axis] ≡ [1 1 0] of the honeycomb lattice than perpendicular to it, while the “hole” moves easier in [0 0 1]. Since “electrons” and “holes” move in different channels, the particle densities can be high especially when the Fermi surface has “necks”. The strong QHE arises from the phonon exchange attraction in the neighborhood of the “neck” surfaces. The plateau observed for the Hall conductivity and the accompanied resistivity drop is due to the superconducting energy gap caused by the Bose-Einstein condensation of the c-bosons, each forming from a pair of one-electron–two-fluxons c-fermions by phonon-exchange attraction. The half-integer quantization rule for the Hall conductivity: (1/2)(2P?1)(4e²/h), P=1,2,..., is derived.     

Single-electron transport through quantum point contact

Here, we employ a numerical approach to investigate the transport and conductance characteristics of a quantum point contact. A quantum point contact is a narrow constriction of a width comparable to the electron wavelength defined in a two-dimensional electron gas (2DEG) by means of split-gate or etching technique. Their properties have been widely investigated in the experiments. In our study, we define a quantum Hall based split-gate quantum point contact with standard gate geometry. Firstly, we obtain the spatial distribution of incompressible strips (current channels) by applying a self consistent Thomas-Fermi method to a realistic heterostructure under quantized Hall conditions. Later, time-dependent Schrödinger equation is solved for electrons injected in the current channels. The transport characteristics and time-evolutions are analyzed in the integer filling factor regime (ν = 1) with the single electron density. The results confirm that the current direction in a realistic quantum point contact can be controllable with the external interventions.     

Current breakdown of the integer and fractional quantum Hall effects detected by torque magnetometry

We have developed a novel technique that enables measurements of the breakdown of both the integer and fractional quantum Hall effects in a two-dimensional electron system without the need to contact the sample. The critical Hall electric fields that we measure are significantly higher than those reported by other workers, and support the quasi-elastic inter-Landau-level tunnelling model of breakdown. Comparison of the fractional quantum Hall effect results with those obtained on the integer quantum Hall effect allows the fractional quantum Hall effect energy gap to be determined and provides a test of the composite-fermion theory. The temperature dependence of the critical current gives an insight into the mechanism by which momentum may be conserved during the breakdown process.     

Critical behaviour of SU(n) quantum chains and topological non-linear σ-models

The critical behaviour of SU(n) quantum “spin” chains, Wess-Zumino-Witten σ-models and grassmanian σ-models at topological angle θ = π (of possible relevance to the quantum Hall effect) is reexamined. It is argued that an additional Z_n symmetry is generally necessary to stabilize the massless phase. This symmetry is not present for the σ-models for n > 2 and is only present for certain representations of “spin” chains.     

Reprint of : Correlated voltage probe model of relaxation in two Coulomb-coupled edge channels

A phenomenological correlated voltage probe model is introduced to mimic the effects of inelastic scattering between particles in different conduction channels of a phase coherent conductor. As an illustration, the non-equilibrium distribution functions of two noisy co-propagating chiral edge channels of the integer quantum Hall effect are calculated and compared with recent experiments. The method is further applied to calculate the linear response current noise through an interacting Mach–Zehnder interferometer.     

Correlated voltage probe model of relaxation in two Coulomb-coupled edge channels

A phenomenological correlated voltage probe model is introduced to mimic the effects of inelastic scattering between particles in different conduction channels of a phase coherent conductor. As an illustration, the non-equilibrium distribution functions of two noisy co-propagating chiral edge channels of the integer quantum Hall effect are calculated and compared with recent experiments. The method is further applied to calculate the linear response current noise through an interacting Mach–Zehnder interferometer.     

Concentration dependence of the anomalous Hall effect in Fe/SiO2 granular films below the percolation threshold

The anomalous Hall effect is studied on Fe_x(SiO₂)_1?x nanocomposite films with x<0.7 in the vicinity of the percolation transition (x _c ≈0.6). It is found that, as the transition is approached from the side of metallic conduction, the Hall angle nonmonotonically varies, passing through a minimum. A qualitative model for describing the concentration dependence of the anomalous Hall effect is proposed. The model is based on that of the conductivity of a two-phase system near the percolation threshold [9, 10]. The anomalous Hall effect is governed by two conduction channels: one of them (a conducting network) is formed by large metal clusters that are separated by narrow dielectric interlayers below the percolation threshold, and the other is represented by the dielectric part of the medium containing Fe grains; in this part of the medium, the anomalous Hall effect occurs through the interference of amplitudes from the tunneling junctions in a set of three grains. It is shown that, at x<x _c , the network may give rise to a “shunting” effect, which makes the effective Hall voltage even less than the Hall voltage of the dielectric component.     

Is the peak value of σ xx at the quantum Hall transition universal?

The question of the universality of the longitudinal peak conductivity at the integer quantum Hall transition is considered. For this purpose, a system of 2D Dirac fermions with random mass characterised by variance g is proposed as a model which undergoes a quantum Hall transition. Whilst for some specific models the longitudinal peak conductivity σ _xx was found to be universal (in agreement with the conjecture of Lee et al. as well as with some numerical work), we find that σ _xx is reduced by a factor (1 + g/2π)^?1, at least for small g. This provides some theoretical evidence for the non-universality of σ _xx , as observed in a number of experiments.     

Classical and quantum hall effect measurements in GaInNAs/GaAs quantum wells

We have performed magneto-transport experiments in modulation-doped Ga_0.7In_0.3N_yAs_1−y/GaAs quantum wells with nitrogen mole fractions 0.4%, 1.0% and 1.5%. Classical magnetotransport (resistivity and low-field Hall effect) measurements have been performed in the temperatures between 1.8 and 275 K, while quantum Hall effect measurements in the temperatures between 1.8 and 47 K and magnetic fields up to 11 T.The variations of Hall mobility and Hall carrier density with nitrogen mole fractions and temperature have been obtained from the classical magnetotransport measurements. The results are used to investigate the scattering mechanisms of electrons in the modulation-doped Ga_0.7In_0.3N_yAs_1−y/GaAs quantum wells. It is shown that the alloy disorder scattering is the major scattering mechanism at investigated temperatures.The quantum oscillations in Hall resistance have been used to determine the carrier density, effective mass, transport mobility, quantum mobility and Fermi energy of two-dimensional (2D) electrons in the modulation-doped Ga_0.7In_0.3N_yAs_1−y/GaAs quantum wells. The carrier density, in-plane effective mass and Fermi energy of the 2D electrons increases when the nitrogen mole fraction is increased from y=0.004 to 0.015. The results found for these parameters are in good agreement with those determined from the Shubnikov-de Haas effect in magnetoresistance.     

Integer and fractional quantum Hall effect in a strip of stripes

We study anisotropic stripe models of interacting electrons in the presence of magnetic fields in the quantum Hall regime with integer and fractional filling factors. The model consists of an infinite strip of finite width that contains periodically arranged stripes (forming supercells) to which the electrons are confined and between which they can hop with associated magnetic phases. The interacting electron system within the one-dimensional stripes are described by Luttinger liquids and shown to give rise to charge and spin density waves that lead to periodic structures within the stripe with a reciprocal wavevector 8k _F in a mean field approximation. This wavevector gives rise to Umklapp scattering and resonant scattering that results in gaps and chiral edge states at all known integer and fractional filling factors ν. The integer and odd denominator filling factors arise for a uniform distribution of stripes, whereas the even denominator filling factors arise for a non-uniform stripe distribution. We focus on the ground state of the system, and identify the quantum Hall regime via the quantized Hall conductance. For this we calculate the Hall conductance via the Streda formula and show that it is given by σ _H = νe ²/h for all filling factors. In addition, we show that the composite fermion picture follows directly from the condition of the resonant Umklapp scattering.     

Exact-exchange density functional theory of the integer quantum Hall effect: strict 2D limit

A strict bidimensional (strict-2D) exact-exchange (EE) formalism within the framework of density-functional theory (DFT) has been developed and applied to the case of an electron gas subjected to a strong perpendicular magnetic field, that drives the system to the regime of the integer quantum Hall effect (IQHE). As the filling of the emerging Landau levels proceeds, two main features results: i) the EE energy minimizes with a discontinuous derivative at every integer filling factor ν; and ii) the EE potential display sharp discontinuities at every integer ν. The present contribution provides a natural improvement as compared with the widely used local-spin-density approximation (LSDA), since the EE energy functional fully contains the effect of the magnetic field, and includes an inter-layer exchange coupling for multilayer systems. As a consistency test, the LSDA is derived as the leading term of a low-field expansion of the EE energy and potential.     

Anomalous fluctuations of 2D-electron recombination radiation intensity in the quantum Hall regime

Anomalous intensity fluctuations are observed in the spectrum of radiative ecombination of quasi-two-dimensional (2D) electrons with photoexcited holes in a single quantum well. The fluctuations are observed exclusively under the conditions of the quantum Hall effect (QHE). It is shown that, if the QHE conditions are not fulfilled, the radiation intensity fluctuates strictly following the Poisson distribution 〈δN ²〉/〈N〉= 1), whereas in the QHE regime the fluctuation amplitude increases by several orders of magnitude (〈δ N ²〉/〈N〉～10²). It is demonstrated that the maxima of the emission noise amplitude coincide with the maxima of inverse magnetoresistance of 2D electrons in the QHE regime and correspond to establishing an anomalously high uniformity of the system.     

Giant hysteresis of magnetoresistance in the quantum hall effect regime

A simple system consisting of a two-dimensional electron gas with a narrow conducting wire is studied. In this system, a giant hysteresis of both longitudinal and Hall magnetoresistances in the quantum Hall effect regime is observed for even and odd filling factors v of the Landau levels. At v = 1 and v = 2, the giant hysteresis occurs in the background of the zero-resistance plateau, and the width of the hysteresis loop in a magnetic field is comparable to the plateau width. At the entry to the hysteresis region, the magnetoresistance varies in a threshold manner; i.e., a magnetically induced breakdown of the quantum Hall effect takes place. It is shown that the system under study reflects the relaxation processes in the two-dimensional electron gas adjacent to the wire and, therefore, represents an effective instrument for investigating the hysteresis phenomena in the two-dimensional electron gas itself. An unusual “anticoercive” behavior of the hysteresis is revealed. A comparative analysis of the results obtained and the experimental data on the long relaxation of eddy currents and on the ferromagnetic state of the quantum Hall liquid indicates the common physical origin of these effects.     

Topological invariants for fractional quantum hall states

We calculate a topological invariant, whose value would coincide with the Chern number in the case of integer quantum Hall effect, for fractional quantum Hall states. In the case of Abelian fractional quantum Hall states, this invariant is shown to be equal to the trace of the K-matrix. In the case of non-Abelian fractional quantum Hall states, this invariant can be calculated on a case by case basis from the conformal field theory describing these states. This invariant can be used, for example, to distinguish between different fractional Hall states numerically even though, as a single number, it cannot uniquely label distinct states.     

Is there a quantization condition for the classical problem of charge and pole?

In elementary derivations of the quantization of azimuthal angular momentum the eigenfunction is determined to be exp(im φ), which is “oversensitive” to the rotation φ → φ+2π, unlessm is an integer. In a recent paper Kerner examined the classical system of charge and magnetic pole, and expressed Π, a vector constant of motion for the system, in terms of a physical angle ψ, to deduce a remarkable paradox. Kerner pointed out that Π(ψ) is “oversensitive” to ψ → ψ+2π unless a certain charge quantization condition is met. Our explicandum of this paradox highlights the distinction between coordinates in classical and quantum physics. It is shown why the single-valuedness requirement on Π(ψ) is devoid of physical significance. We are finally led to examine the classical analog of the quantum mechanical argument that demonstrates the quantization of magnetic charge, to show that there is “no hope” of a classical quantization condition.     

Scaling in the quantum Hall effect regime in n-InGaAs/GaAs nanostructures

The longitudinal ρ _xx (B) and Hall ρ _xy (B) magnetoresistances are investigated experimentally in the integer quantum Hall effect (QHE) regime in n-InGaAs/GaAs double quantum well nanostructures in the range of magnetic fields B = (0–16) T and temperatures T = (0.05–70) K before and after IR illumination. The results are evaluated within the scaling hypothesis with regard to electron-electron interaction.