Quantum Hall Plateaus Research Articles

Upshift of the fractional quantum Hall plateaux: evidence for repulsive scattering for composite fermions

It is now established that if there are an equal number of positively and negatively charged impurities, the quantum Hall plateaux in ρ xy ( B) are expected to be centred about the classical value ρ xy = B/( en) where n is the 2D carrier density. It has also been shown that when repulsive or attractive scattering centres are deliberately introduced close to a two-dimensional electron gas (2DEG), the centres of the quantum Hall plateaux are shifted to higher or lower magnetic field, respectively. In this paper, we present magneto transport measurements on a high-quality GaAs electron system. We extend this concept to the composite fermions where each electron is bound to two magnetic flux quanta. We provide compelling evidence that the dominant scattering for the composite fermions is different from that for electrons. This is supported by studying the upshift of the ν=1 quantum Hall plateau within the composite fermion picture.

On the Theory of Electric DC‐Conductivity: Linear and Non‐linear Microscopic Evolution and Macroscopic Behavior

We consider the Schrödinger time evolution of charged particles subject to a static substrate potential and to a homogeneous, macroscopic electric field (a magnetic field may also be present). We investigate the microscopic velocities and the resulting macroscopic current. We show that the microscopic velocities are in general non‐linear with respect to the electric field. One kind of non‐linearity arises from the highly non‐linear adiabatic evolution and (or) from an admixture of parts of it in so‐called intermediate states, and the other kind from non‐quadratic transition rates between adiabatic states. The resulting macroscopic dc‐current may or may not be linear in the field. Three cases can be distinguished: (a) the microscopic non‐linearities can be neglected. This is assumed to be the case in linear response theory (Kubo formalism, …). We give arguments which make it plausible that often such an assumption is indeed justified, in particular for the current parallel to the field. (b) The microscopic non‐linearities lead to macroscopic non‐linearities. An example is the onset of dissipation by increasing the electric field in the breakdown of the quantum Hall effect. (c) The macroscopic current is linear although the microscopic non‐linearities constitute an essential part of it and cannot be neglected. We show that the Hall current of a quantized Hall plateau belongs to this case. This illustrates that macroscopic linearity does not necessarily result from microscopic linearity. In the second and third cases linear response theory is inadequate. We elucidate also some other problems related to linear response theory.

Spatial distribution of edge states in the quantum Hall plateaus investigated by magnetocapacitance

The spatial distribution of the edge states in the quantum Hall (QH) regime has been investigated by the magnetocapacitance measured with the multigate of thin wires along the sample boundary. We have observed the compressible electronic states which exist within about $1\ensuremath{\mu}\mathrm{m}$ along the sample boundary in the QH regime. The edge states become broad by more than a few microns when the negative side gate bias is applied to the outermost gate wire.

Critical currents of ideal quantum Hall superfluids

Filling factor $\nu=1$ bilayer electron systems in the quantum Hall regime have an excitonic-condensate superfluid ground state when the layer separation $d$ is less than a critical value $d_c$. On a quantum Hall plateau current injected and removed through one of the two layers drives a dissipationless edge current that carries parallel currents, and a dissipationless bulk supercurrent that carries opposing currents in the two layers. In this paper we discuss the theory of finite supercurrent bilayer states, both in the presence and in the absence of symmetry breaking inter-layer hybridization. Solutions to the microscopic mean-field equations exist at all condensate phase winding rates for zero and sufficiently weak hybridization strengths. We find, however, that collective instabilities occur when the supercurrent exceeds a critical value determined primarily by a competition between direct and exchange inter-layer Coulomb interactions. The critical current is estimated using a local stability criterion and varies as $(d_c-d)^{1/2}$ when $d$ approaches $d_c$ from below. For large inter-layer hybridization, we find that the critical current is limited by a soliton instability of microscopic origin.

Phase diagram for quantum hall bilayers at nu=1.

We present a phase diagram for a double quantum well bilayer electron gas in the quantum Hall regime at a total filling factor nu=1, based on exact numerical calculations of the topological Chern number matrix and the (interlayer) superfluid density. We find three phases: a quantized Hall state with pseudospin superfluidity, a quantized Hall state with pseudospin "gauge-glass" order, and a decoupled composite Fermi liquid. Comparison with experiments provides a consistent explanation of the observed quantum Hall plateau, Hall drag plateau, and vanishing Hall drag resistance, as well as the zero-bias conductance peak effect, and suggests some interesting points to pursue experimentally.

Deconstructing the Liouvillian approach to the quantum Hall plateau transition

We examine the Liouvillian approach to the quantum Hall plateau transition, as introduced recently by Sinova, Meden, and Girvin [Phys. Rev. B {\bf 62}, 2008 (2000)] and developed by Moore, Sinova and Zee [Phys. Rev. Lett. {\bf 87}, 046801 (2001)]. We show that, despite appearances to the contrary, the Liouvillian approach is not specific to the quantum mechanics of particles moving in a single Landau level: we formulate it for a general disordered single-particle Hamiltonian. We next examine the relationship between Liouvillian perturbation theory and conventional calculations of disorder-averaged products of Green functions and show that each term in Liouvillian perturbation theory corresponds to a specific contribution to the two-particle Green function. As a consequence, any Liouvillian approximation scheme may be re-expressed in the language of Green functions. We illustrate these ideas by applying Liouvillian methods, including their extension to $N_L > 1$ Liouvillian flavors, to random matrix ensembles, using numerical calculations for small integer $N_L$ and an analytic analysis for large $N_L$. We find that behavior at $N_L > 1$ is different in qualitative ways from that at $N_L=1$. In particular, the $N_L = \infty$ limit expressed using Green functions generates a pathological approximation, in which two-particle correlation functions fail to factorize correctly at large separations of their energy, and exhibit spurious singularities inside the band of random matrix energy levels. We also consider the large $N_L$ treatment of the quantum Hall plateau transition, showing that the same undesirable features are present there, too.

Dependence of widths of the integer quantum Hall plateau on quantumlifetime

In this report we demonstrate that the width of the spin-split plateaus of theinteger quantum Hall effect are determined by the quantum lifetime of theelectrons via the exchange interaction. The widths of the adjacent Landau-splitplateaus are a consequence of the pronounced changes of the spin-splitplateaus. In contrast, the transport lifetime and thus the electron mobility arenot directly related to the width of the quantum Hall plateaus.

Conservation laws in the quantum Hall Liouvillian theory and its generalizations

It is known that the localization length scaling of noninteracting electrons near the quantum Hall plateau transition can be described in a theory of the bosonic density operators, with no reference to the underlying fermions. The resulting “Liouvillian” theory has a U(1|1) global supersymmetry as well as a hierarchy of geometric conservation laws related to the noncommutative geometry of the lowest Landau level (LLL). Approximations to the Liouvillian theory contain quite different physics from standard approximations to the underlying fermionic theory. Mean-field and large- N generalizations of the Liouvillian are shown to describe problems of noninteracting bosons that enlarge the U(1|1) supersymmetry to U(1|1)× SO( N) or U(1|1)× SU( N). These noninteracting bosonic problems are studied numerically for 2⩽ N⩽8 by Monte Carlo simulation and compared to the original N=1 Liouvillian theory. The N>1 generalizations preserve the first two of the hierarchy of geometric conservation laws, leading to logarithmic corrections at order 1/ N to the diffusive large- N limit, but do not preserve the remaining conservation laws. The emergence of nontrivial scaling at the plateau transition, in the Liouvillian approach, is shown to depend sensitively on the unusual geometry of Landau levels.

Network models for localization problems belonging to the chiral symmetry classes

We consider localisation problems belonging to the chiral symmetry classes, in which sublattice symmetry is responsible for singular behaviour at a band centre. We formulate models which have the relevant symmetries and which are generalisations of the network model introduced previously in the context of the integer quantum Hall plateau transition. We show that the generalisations required can be re-expressed as corresponding to the introduction of absorption and amplification into either the original network model, or the variants of it that represent disordered superconductors. In addition, we demonstrate that by imposing appropriate constraints on disorder, a lattice version of the Dirac equation with a random vector potential can be obtained, as well as new types of critical behaviour. These models represent a convenient starting point for analytic discussions and computational studies, and we investigate in detail a two-dimensional example without time-reversal invariance. It exhibits both localised and critical phases, and band-centre singularities in the critical phase approach more closely in small systems the expected asymptotic form than in other known realisations of the symmetry class.

Dynamical scaling of the quantum Hall plateau transition.

Using different experimental techniques, we examine the dynamical scaling of the quantum Hall plateau transition in a frequency range f=0.1-55 GHz. We present a scheme that allows for a simultaneous scaling analysis of these experiments and all other data in literature. We observe a universal scaling function with an exponent kappa=0.5+/-0.1, yielding a dynamical exponent z=0.9+/-0.2.

The ν=5/2 quantum Hall state revisited: spontaneous breaking of the chiral fermion parity and phase transition between Abelian and non-Abelian statistics

We review a recent development in the theoretical understanding of the ν=5/2 quantum Hall plateau and propose a new conformal field theory, slightly different from the Moore–Read one, to describe another universality class relevant for this plateau. The ground state is still given by the Pfaffian and is completely polarized, however, the elementary quasiholes are charge 1/2 anyons with Abelian statistics θ= π/2, which obey complete spin–charge separation. The physical hole is represented by two such quasiholes plus a free neutral Majorana fermion. We also compute the periods and amplitudes of the chiral persistent currents in both states and show that they have different temperature dependence. Finally, we find indications of a classical two-step phase transition between the new and the Moore–Read states, through a compressible state, which is characterized by the spontaneous breaking of a hidden Z 2 symmetry corresponding to the conservation of the chiral fermion parity. We believe that this transition could explain the “kink” observed in the activation experiment for ν=5/2.

A generalized treatment of the dynamical scaling of the quantum Hall plateau transition

Using different experimental techniques we deduce the dynamical scaling exponent z for the quantum Hall effect plateau transition. Comparing our results to other data in the literature we find z=0.9±0.2, a value hinting to a strongly interacting electron system.

INCREASE OF QUANTUM HALL PLATEAU WIDTHS DUE TO ELECTRON-PHONON INTERACTION

We calculate the conductivities for a model system of the integer quantum Hall effect including electron-phonon interactions, with a substrate potential chosen such that microscopic processes can be understood in detail. We find that in general the macroscopic Hall current is not only composed of velocities arising from the Schrödinger time evolution but also of velocities generated by electron-phonon interactions and that the latter can contribute to the formation of the quantized Hall plateaus. We show for typical quantum Hall systems the phonon induced contributions lead to Hall plateaus which are somewhat larger than the plateaus of the dissipative conductivity. Experimentally this difference has been known for a long time, but it seemed unexplained so far. In the case where the substrate potential has no spatial fluctuations in the direction of the macroscopic electric field, all states are conducting and the dissipative conductivity vanishes only near integer filling factors. Nevertheless, the calculated Hall conductivity shows broad quantized plateaus, which here are entirely generated by velocities arising from phonon induced relaxation processes. These results show that absence of dissipation is not indispensable for the existence of quantized Hall plateaus.

Conductance fluctuations at the quantum Hall plateau transition

We analyze the conductance fluctuations observed in the quantum Hall regime for a bulk two-dimensional electron system in a Corbino geometry. We find that characteristics like the power spectral density and the temperature dependence agree well with simple expectations for universal conductance fluctuations in metals, while the observed amplitude is reduced. In addition, the dephasing length ${L}_{\ensuremath{\Phi}}\ensuremath{\propto}{T}^{\ensuremath{-}1/2},$ which governs the temperature dependence of the fluctuations, is surprisingly different from the scaling length ${L}_{\mathrm{sc}}\ensuremath{\propto}{T}^{\ensuremath{-}1}$ governing the width of the quantum Hall plateau transition.

Dynamics of the far-infrared photoresponse in quantum Hall systems

We have studied the time evolution of the far-infrared photoresponse of two-dimensional electron systems in the quantum Hall (QH) regime. We have identified different contributions to the photoresponse by using different sample geometries and by changing the direction of the magnetic field. In general, we have found a fast response (microseconds) related to photoinduced Hall currents, and longer response times (up to some hundreds of milliseconds) related to the photoinduced longitudinal resistance. Both types of the response are present in the bolometric response bound to the flanks of the QH plateau, and in the cyclotron resonance signal bound to the photon energy of the probe radiation. We give a qualitative explanation of our results on the basis of impurity-induced potentials for the localization of excited carriers.

Hopping conductivity in the quantum Hall effect: revival of universal scaling.

We have measured the temperature dependence of the conductivity sigma(xx) of a two-dimensional electron system deep into the localized regime of the quantum Hall plateau transition. Using variable-range hopping theory we extract directly the localization length xi from this experiment. We use our results to study the scaling behavior of xi as a function of the filling factor distance /deltanu/ to the critical point of the transition. We find for all samples a power-law behavior xi equivalent to /deltanu/(-gamma) in agreement with the theoretically proposed universal exponent gamma = 2.35.

Connecting polymers to the quantum Hall plateau transition

A mapping is developed between the quantum Hall plateau transition and two-dimensional self-interacting lattice polymers. This mapping is exact in the classical percolation limit of the plateau transition, and diffusive behavior at the critical energy is shown to be related to the critical exponents of a class of chiral polymers at the $\theta$-point. The exact critical exponents of the chiral polymer model on the honeycomb lattice are found, verifying that this model is in the same universality class as a previously solved model of polymers on the Manhattan lattice. The mapping is obtained by averaging analytically over the local random potentials in a previously studied lattice model for the classical plateau transition. This average generates a weight on chiral polymers associated with the classical localization length exponent $\nu = 4/3$. We discuss the differences between the classical and quantum transitions in the context of polymer models and use numerical results on higher-moment scaling laws at the quantum transition to constrain possible polymer descriptions. Some properties of the polymer models are verified by transfer matrix and Monte Carlo studies.

Growth and characterization of p-type HgTe/Hg1−xCdxTe single quantum wells using nitrogen and arsenic

p-type HgTe/Hg0.3Cd0.7Te(001) quantum wells (QWs) have been grown with molecular-beam epitaxy (MBE) on Cd0.96Zn0.04Te substrates using modulation doping techniques. Both plasma-excited nitrogen and evaporated cadmium arsenide have been utilized for in situ doping during MBE growth. A comparison of the electrical and structural properties of QWs fabricated by the two doping techniques has been made. Two-dimensional hole concentrations in nitrogen-doped QWs (up to 1.0×1012 cm−2) were significantly higher than in arsenic-doped QWs (below 0.5×1012 cm−2). However, by means of a gate-controlled Hall bar, hole densities up to 1.1×1012 cm−2 have been achieved in the latter system. Hall mobilities up to 1.0×105 cm2/(V s) have been measured. Whereas all samples exhibit pronounced although irregular Shubnikov–de Haas oscillations, quantum Hall plateaus in the arsenic-doped samples are broader and better defined.

Magnetic field effects on the ferromagnetism and transport properties of Kagome dot superlattices

Magnetic field effects on a Kagome dot superlattice are studied. The Hofstadter butterfly has wide gap structures where a wide quantized Hall plateau should be observable in experiments. A magnetic-field-induced ferromagnetic–paramagnetic transition and a metal–insulator one are predicted. All of these magnetic field effects should be observable in small magnetic fields (of the order of 0.1 T ).

Frequency scaling of microwave conductivity in the integer quantum Hall effect minima

We measure the longitudinal conductivity ${\ensuremath{\sigma}}_{\mathrm{xx}}$ at frequencies $1.246 \mathrm{GHz}<~f<~10.05 \mathrm{GHz}$ over a range of temperatures $235 \mathrm{mK}<~T<~4.2$ K with particular emphasis on the quantum Hall plateaus. We find that $\mathrm{Re}({\ensuremath{\sigma}}_{\mathrm{xx}})$ scales linearly with frequency for a range of magnetic field around the center of the plateaus, i.e., where ${\ensuremath{\sigma}}_{\mathrm{xx}}(\ensuremath{\omega})\ensuremath{\gg}{\ensuremath{\sigma}}_{\mathrm{xx}}^{\mathrm{dc}}.$ The width of this scaling region decreases with higher temperature and vanishes by 1.2 K altogether. Comparison between localization length determined from ${\ensuremath{\sigma}}_{\mathrm{xx}}(\ensuremath{\omega})$ and dc measurements on the same wafer show good agreement.