Integer quantum Hall effect in a PbTe quantum well

Transport measurements have been carried out on a 10 nm n-type PbTe/Pb_0.9Eu_0.1Te quantum well at millikelvin temperatures. The Hall and longitudinal resistances are measured in a Van der Pauw geometry under high magnetic fields up to 23 T. A robust signature of the integer quantum Hall effect is observed without any sign of parasitic parallel conduction. The unconventional sequence of filling factors associated with the integer quantum Hall effect is discussed in terms of the occupancy of multiple valleys.     

Anyons and the quantum Hall effect—A pedagogical review

The dichotomy between fermions and bosons is at the root of many physical phenomena, from metallic conduction of electricity to super-fluidity, and from the periodic table to coherent propagation of light. The dichotomy originates from the symmetry of the quantum mechanical wave function to the interchange of two identical particles. In systems that are confined to two spatial dimensions particles that are neither fermions nor bosons, coined “anyons”, may exist. The fractional quantum Hall effect offers an experimental system where this possibility is realized. In this paper we present the concept of anyons, we explain why the observation of the fractional quantum Hall effect almost forces the notion of anyons upon us, and we review several possible ways for a direct observation of the physics of anyons. Furthermore, we devote a large part of the paper to non-abelian anyons, motivating their existence from the point of view of trial wave functions, giving a simple exposition of their relation to conformal field theories, and reviewing several proposals for their direct observation.     

A classification of quantum Hall fluids

In this paper, the key ideas of characterizing universality classes of dissipationfree (incompressible) quantum Hall fluids by mathematical objects called quantum Hall lattices are reviewed. Many general theorems about the classification of quantum Hall lattices are stated and their physical implications are discussed. Physically relevant subclasses of quantum Hall lattices are defined and completely classified. The results are carefully compared with experimental data and also with other theoretical schemes (the hierarchy schemes). Several proposals for new experiments are made which could help to settle interesting issues in the theory of the (fractional) quantum Hall effect and thus would lead to a deeper understanding of this remarkable effect.     

Localization length exponent,critical conductance distribution and multifractality in hierarchical network models for the quantum Hall effect

We study hierarchical network models which have recently been introduced to approximate the Chalker-Coddington model for the integer quantum Hall effect (A.G. Galstyan and M.E. Raikh, PRB 5 (1997) 1422; Arovas et al., PRB 56 (1997) 4751). The hierarchical structure is due to a recursive method starting from a finite elementary cell. The localization-delocalization transition occurring in these models is displayed in the flow of the conductance distribution under increasing system size. We numerically determine this flow, calculate the critical conductance distribution, the critical exponent of the localization length, and the multifractal exponents of critical eigenstates.     

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.     

An alternative formulation of Hall effect and quantum phases in noncommutative space

A recent method of constructing quantum mechanics in noncommutative coordinates, alternative to implying noncommutativity by means of star product is discussed. Within this approach we study Hall effect as well as quantum phases in noncommutative coordinates. The θ-deformed phases which we obtain are velocity independent.     

Three-dimensional quantum Hall effect in Weyl and double-Weyl semimetals

The well-known quantum Hall effect (QHE) was usually studied in 2D systems. In this work, we investigate the integer QHE in 3D Weyl and double-Weyl semimetals. Based on the lattice models of Weyl and double-Weyl semimetals subjected to a uniform magnetic field, we derive the generalized 3D spinfull Hofstadter Hamiltonians and Harper equations for the two systems, and obtain their corresponding energy spectra. Furthermore, we show that for proper hopping parameters and rational magnetic fluxes, both systems exhibit the 3D QHE when the Fermi level lies in some band gaps. The 3D QHE is topologically characterized by three Chern numbers with one or two nonzero Chern values which are respectively defined for three crystal planes. The possible experimental realization and detection of the 3D QHE are also discussed.     

Quantum Hall effect: The resistivity of a 2D electron gas—a thermodynamic approach

Based on a thermodynamic approach, we have calculated the resistivity of a 2D electron gas, assumed dissipationless in a strong quantum limit. Standard measurements, with extra current leads, define the resistivity caused by a combination of Peltier and Seebeck effects. The current causes heating (cooling) at the first (second) sample contacts, due to the Peltier effect. The contact temperatures are different. The measured voltage is equal to the Peltier effect-induced thermoemf which is linear in current. As a result, the resistivity is non-zero as I→0. The resistivity is a universal function of magnetic field and temperature, expressed in fundamental units h/e². The universal features of magnetotransport data observed in the experiment confirm our predictions.     

Wave function multifractality and dephasing at metal–insulator and quantum Hall transitions

We analyze the critical behavior of the dephasing rate induced by short-range electron–electron interaction near an Anderson transition of metal–insulator or quantum Hall type. The corresponding exponent characterizes the scaling of the transition width with temperature. Assuming no spin degeneracy, the critical behavior can be studied by performing the scaling analysis in the vicinity of the non-interacting fixed point, since the latter is stable with respect to the interaction. We combine an analytical treatment (that includes the identification of operators responsible for dephasing in the formalism of the non-linear sigma-model and the corresponding renormalization-group analysis in 2 + ? dimensions) with numerical simulations on the Chalker–Coddington network model of the quantum Hall transition. Finally, we discuss the current understanding of the Coulomb interaction case and the available experimental data.     

Bulk and edge properties of the Chern-Simons Ginzburg-Landau theory for the fractional quantum Hall effect

The Chern-Simons Ginzburg-Landau theory for the fractional quantum Hall effect is studied in the presence of a confining potential We review the bulk properties of the model and discuss how the plateau formation emerges without any impurity potential. The effect is related to changes, by accumulation of charge, at the edge when the chemical potential is changed. Fluctuations about the ground state are examined and an expression is found for the velocity of the massless edge mode in terms of the confining potential. The effect of including spin is examined for the case when the system is fully polarized in the bulk. In general a spin texture may appear at the edge, and we examine this effect in the case of a small spin-down component. The low-frequency edge modes are examined and a third-order equation is found for velocities which indicates the presence of three different modes. The discussions are illustrated by numerical studies of the ground states, both for the one- and two-component cases.     

Elliptic theta functions and the fractional quantum Hall effect

Algebro-geometric methods are applied to the theoretical understanding of the fractionary quantum Hall effect on a periodic lattice. The fermionic Fock space of the many-electron system is precisely identified, and as a consequence, the variational Haldane-Rezayi ground state is decomposed in terms of one-particle wave functions at the first Landau level; the filling factor is thus analytically computed. Quasi-hole and quasi-particle excitations are also analyzed. The center of mass dynamics is described in terms of a section in a very subtle stable vector bundle. The Hall conductance arises as a topological invariant; namely, the slope of the vector bundle previously mentioned.     

Geometry of quantum Hall states: Gravitational anomaly and transport coefficients

We show that universal transport coefficients of the fractional quantum Hall effect (FQHE) can be understood as a response to variations of spatial geometry. Some transport properties are essentially governed by the gravitational anomaly. We develop a general method to compute correlation functions of FQH states in a curved space, where local transformation properties of these states are examined through local geometric variations. We introduce the notion of a generating functional and relate it to geometric invariant functionals recently studied in geometry. We develop two complementary methods to study the geometry of the FQHE. One method is based on iterating a Ward identity, while the other is based on a field theoretical formulation of the FQHE through a path integral formalism.     

Collapse of quantized Hall resistance and breakdown of dissipationless state in the integer quantum Hall effect: filling factor dependence

Magnetic field dependence of critical current for collapse of quantized Hall resistance I_cr(collapse) and critical current for breakdown of dissipationless state I_cr(breakdown) have been measured near the filling factor ν=4 of Landau levels in a GaAs/AlGaAs heterostructure Hall bar. The difference I_cr(breakdown)−I_cr(collapse) decreases against the increase and the decrease in ν from 4 and the critical behavior disappears outside of the region 3.85<ν<4.15.     

Spatial evolution of the generation and relaxation of excited carriers near the breakdown of the quantum Hall effect

We have measured the generation and relaxation of excited carriers along their drift direction near the breakdown of the quantum Hall effect (QHE). The dissipative resistivity ρ_xx(x) at current densities close to the critical value for the QHE breakdown was measured as a function of the distance x from the electron injection at x=0. By injecting “cold” electrons into constrictions at supercritical current levels, the evolution of the breakdown along the drift direction was monitored. After a smooth increase of the resistivity with the drifting distance, an avalanche-like rise towards a saturation value occurs. Drastic changes of the resistivity profiles with the applied current were found in a narrow range around the critical current. The observed behavior is attributed to impurity-assisted tunneling between Landau levels. By injecting hot electrons (excited in a periodic set of constrictions) into a region with subcritical current density, the relaxation process was analyzed. Inelastic relaxation lengths with typical values in the range from 0.3 to 4 μm were found, which agree within 10% with the elastic mean free path determined from the Hall mobility at zero magnetic field. We conclude that the energy relaxation process is triggered by scattering at impurity potentials.     

Intrinsic Hall effect and separation of Rashba and Dresselhaus spin splittings in semiconductor quantum wells

We have proposed a method to separate Rashba and Dresselhaus spin splittings in semiconductor quantum wells by using the intrinsic Hall effect. It is shown that the interference between Rashba and Dresselhaus terms can deflect the electrons in opposite transverse directions with a change of sign in the macroscopic Hall current, thus providing an alternative way to determine the different contributions to the spin--orbit coupling.     

Spin Chern numbers and time-reversal-symmetry-broken quantum spin Hall effect

The quantum spin Hall （QSH） effect is considered to be unstable to perturbations violating the time-reversal （TR） symmetry. We review some recent developments in the search of the QSH effect in the absence of the TR symmetry. The possibility to realize a robust QSH effect by artificial removal of the TR symmetry of the edge states is explored. As a useful tool to characterize topological phases without the TR symmetry, the spin-Chern number theory is introduced.     

Plateau transitions in fractional quantum Hall liquids

Effects of backward scattering between fractional quantum Hall (FQH) edge modes are studied. Based on the edge-state picture for hierarchical FQH liquids, we discuss the possibility of the transitions between different plateaux of the tunneling conductance G. We find a selection rule for the sequence which begins with a conductance (m: integer, p: even integer) in units of e ²/h. The shot-noise spectrum as well as the scaling behavior of the tunneling current is calculated explicitly. Received 5 October 1999 and Received in final form 19 November 1999     

AlO_x/LiF composite protection layer for Cr-doped(Bi,Sb)_2Te_3 quantum anomalous Hall films

We have realized robust quantum anomalous Hall samples by protecting Cr-doped(Bi,Sb)_2Te_3 topological insulator films with a combination of LiF and A1O_x capping layers.The AlO_x/LiF composite capping layer well keeps the quantum anomalous Hall states of Cr-doped(Bi,Sb)_2Te_3 films and effectively prevent them from degradation induced by ambient conditions.The progress is a key step towards the realization of the quantum phenomena in heterostructures and devices based on quantum anomalous Hall system.     

Gated spin relaxation in (1 1 0)-oriented quantum wells

Growth, photoluminescence characterisation and time-resolved optical measurements of electron spin dynamics in (1 1 0)-oriented GaAs/AlGaAs quantum wells are described. Conditions are given for MBE growth of good-quality quantum wells, judged by the width of low-temperature excitonic photoluminescence. At 170 K the electron spin relaxation rate in (1 1 0)-oriented wells shows a 100-fold reduction compared to equivalent (1 0 0)-oriented wells and also a 10-fold increase with applied electric field from 20 to 80 kV cm⁻¹. There is evidence for similar dramatic effects at 300 K. Spin relaxation is field independent below 20 kV cm⁻¹ reflecting quantum well asymmetry. The results indicate the achievability of voltage-gateable quantum well spin memory time longer than 10 ns at room temperature simultaneously with high electron mobility.