Edge and Impurity Eects on Quantization of Hall Currents (original) (raw)
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Equality of the Bulk and Edge Hall Conductances in a Mobility Gap
Communications in Mathematical Physics, 2005
We consider the edge and bulk conductances for 2D quantum Hall systems in which the Fermi energy falls in a band where bulk states are localized. We show that the resulting quantities are equal, when appropriately defined. An appropriate definition of the edge conductance may be obtained through a suitable time averaging procedure or by including a contribution from states in the localized band. In a further result on the Harper Hamiltonian, we show that this contribution is essential. In an appendix we establish quantized plateaus for the conductance of systems which need not be translation ergodic.
Journal of Physics-condensed Matter, 2003
Charge transport of a two-dimensional electron gas in the presence of a magnetic field is studied by means of the Keldysh-Green function formalism and the tightbinding method. We evaluate the spatial distributions of persistent (equilibrium) and transport (nonequilibrium) currents, and give a vivid picture of their profiles. In the quantum Hall regime, we find exact conductance quantization both for persistent currents and for transport currents, even in the presence of impurity scattering centres and moderate disorder.
Edge Currents for Quantum Hall Systems I: One-Edge, Unbounded Geometries
Reviews in Mathematical Physics, 2008
Devices exhibiting the integer quantum Hall effect can be modeled by one-electron Schrödinger operators describing the planar motion of an electron in a perpendicular, constant magnetic field, and under the influence of an electrostatic potential. The electron motion is confined to unbounded subsets of the plane by confining potential barriers. The edges of the confining potential barrier create edge currents. In this, the first of two papers, we prove explicit lower bounds on the edge currents associated with one-edge, unbounded geometries formed by various confining potentials. This work extends some known results that we review. The edge currents are carried by states with energy localized between any two Landau levels. These one-edge geometries describe the electron confined to certain unbounded regions in the plane obtained by deforming half-plane regions. We prove that the currents are stable under various potential perturbations, provided the perturbations are suitably small ...
Theory of the quantized hall effect (II)
Nuclear Physics B, 1984
We review the derivation that the effective lagrangian describing the critical fluctuations in a two-dimensional disordered electronic system in a transverse magnetic field contains a novel, topological term. We extend this result in several directions. We show how the importance of topological concepts can be seen by examining in detail the nature of the boundary current whenever the Fermi energy lies within a localized state region. This insight allows us to construct a field theoretic quantization argument. Our argument is reminiscent of the Laughlin-Halperin quantization approach, in that we make use of the response of the system to sources with nontrivial gauge topology. This then leads to a discussion of how to use the effective field theory to actually compute the response, and of why localization must break down somewhere within the Landau band. Our methodology unifies the results of Laughlin, Halperin and Thouless with the field theoretic approach to localization pioneered by Wegner.
Edge Currents for Quantum Hall Systems, II. Two-Edge, Bounded and Unbounded Geometries
Annales Henri Poincaré, 2008
Devices exhibiting the integer quantum Hall effect can be modeled by oneelectron Schrödinger operators describing the planar motion of an electron in a perpendicular, constant magnetic field, and under the influence of an electrostatic potential. The electron motion is confined to bounded or unbounded subsets of the plane by confining potential barriers. The edges of the confining potential barriers create edge currents. This is the second of two papers in which we review recent progress and prove explicit lower bounds on the edge currents associated with one-and two-edge geometries. In this paper, we study various unbounded and bounded, two-edge geometries with soft and hard confining potentials. These two-edge geometries describe the electron confined to unbounded regions in the plane, such as a strip, or to bounded regions, such as a finite length cylinder. We prove that the edge currents are stable under various perturbations, provided they are suitably small relative to the magnetic field strength, including perturbations by random potentials. The existence of, and the estimates on, the edge currents are independent of the spectral type of the operator.
On transport in quantum Hall systems with constrictions
Europhysics Letters (EPL), 2007
Motivated by recent experimental findings, we study transport in a simple phenomenological model of a quantum Hall edge system with a gate-voltage controlled constriction lowering the local filling factor. The current backscattered from the constriction is seen to arise from the matching of the properties of the edge-current excitations in the constriction (ν2) and bulk (ν1) regions. We develop a hydrodynamic theory for bosonic edge modes inspired by this model, finding that a competition between two tunneling process, related by a quasiparticle-quasihole symmetry, determines the fate of the low-bias transmission conductance. In this way, we find satisfactory explanations for many recent puzzling experimental results.
Transport through constricted quantum Hall edge systems: Beyond the quantum point contact
Physical Review B, 2008
Motivated by surprises in recent experimental findings, we study transport in a model of a quantum Hall edge system with a gate-voltage controlled constriction. A finite backscattered current at finite edge-bias is explained from a Landauer-Buttiker analysis as arising from the splitting of edge current caused by the difference in the filling fractions of the bulk (ν1) and constriction (ν2) quantum Hall fluid regions. We develop a hydrodynamic theory for bosonic edge modes inspired by this model. The constriction region splits the incident long-wavelength chiral edge density-wave excitations among the transmitting and reflecting edge states encircling it. The competition between two interedge tunneling processes taking place inside the constriction, related by a quasiparticlequasihole (qp-qh) symmetry, is accounted for by computing the boundary theories of the system. This competition is found to determine the strong coupling configuration of the system. A separatrix of qp-qh symmetric gapless critical states is found to lie between the relevant RG flows to a metallic and an insulating configuration of the constriction system. This constitutes an interesting generalisation of the Kane-Fisher quantum impurity model. The features of the RG phase diagram are also confirmed by computing various correlators and chiral linear conductances of the system. In this way, our results find excellent agreement with many recent puzzling experimental results for the cases of ν1 = 1/3, 1. We also discuss and make predictions for the case of a constriction system with ν2 = 5/2.
Theory of the quantized Hall effect (I)
Nuclear Physics B, 1984
This is the first of a series of three papers presenting a field theoretic approach to the (integrally) quantized Hall effect. The basic idea is that the transverse conductivity o,.y directly couples to a topological quantum number characterizing the phase relationship bctween advanced and retarded electron propagators. This allows us to present a reformulation of the Laughlin quantization argument as well as a direct demonstration of the breakdown of the two-dimensional scaling theory of localization. This paper summarizes all our results and discusses a physical picture of the emergence of extended states.
Nonequilibrium distribution of edge and bulk current in a quantum Hall conductor
Physical review. B, Condensed matter, 1991
A quantitative model is presented that accounts for the experimental observation that fourterminal resistances of a high-mobility quantum Hall conductor cannot be related directly to a single resistivity tensor. The key ingredient is that the highest (partly occupied) Landau level is completely decoupled from the other levels except at the contacts. The current carried by the top level alone is related through a resistivity tensor to the longitudinal and the transverse electric-field components. Deviations from a homogeneous current distribution close to the contacts are taken into account. For the other {fully occupied) Landau levels the usual edge-channel description is used. It is shown how the electric field can be incorporated into this description in order to obtain a consistent model. A mechanism for the onset of nonlinear behavior is given.
Theory of the quantized Hall effect (III)
Nuclear Physics B, 1984
In the previous paper, we have demonstrated the need for a phase transition as a function of 0 in the non-linear o-model describing the quantized Hall effect. In this work, we present arguments for the occurrence of exactly such a transition. We make use of a dilutc gas instanton approximation as well as present a more rigorous duality argument to show that the usual scaling of the conductivity_ to zero at large distances i~ altered whenever o~l~ u~-2te-/t,~ n integer. This then completes our theo~ of the quantized Hall effect,