The effect of screening on current distribution and conductance quantisation in narrow quantum Hall systems

Abstract

We summarize and critically discuss the screening theory of the integer quantized Hall effect (IQHE), which was recently developed in order to understand scanning‐force‐microscope experiments on the current distribution in narrow Hall bars. The theory is based on the combination of a self‐consistent, non‐linear screening theory with a linear, local transport theory. In addition to the explanation of the mentioned experiments, it allows a simple understanding of the IQHE, including the enormous reproducibility of the measured quantized resistance values. Both achievements result from two key mechanisms: The first, which follows from the screening theory, is the existence of incompressible strips, with integer values of the local Landau level filling factor and quantized values of longitudinal and Hall resistivity, in certain magnetic field intervals. The second, a result of the local transport theory, is the confinement of an imposed dissipative current to these strips. This confinement leads to the quantization of the global resistances, without any localization assumptions. We also discuss limitations and possible extensions as well as further applications of this theory. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Landauer-Büttiker approach for hyperfine mediated electronic transport in the integer quantum Hall regime

“…Finally, we extended our models to account for the effects of quadrupolar splitting of nuclear levels and also depict the electronic transport signatures that arise from single and multi-photon processes. We believe that this work sets stage for a more rigorous approach which will include a self-consistent solution of the potential profile [45][46][47][48][49][50][51][52][53][54][55] of the channel along with the spatial distribution the nuclear spin profile. In this section, we show that the rate of spin-flip scattering from the forward propagating up-spin (down-spin) channel originating in the source contact to the forward propagating down-spin (up-spin) channel terminating in the drain contact per unit energy at the QPC can indeed be approximated by a transmission coefficient that is dependent on the nuclear polarization.…”

Current distribution in narrow translation-invariant quantum-Hall-systems with lateral density modulation

“…The early experiments showed that, for magnetic field values B in the quantum Hall plateau (QHP) corresponding to the integer Landau-level filling factor ν=k, the ASDC flows dissipationless through so-called 'incompressible stripes' (ISs), which had been predicted by Chklovskii et al [6,7] as a consequence of the zero-temperature screening properties of a 2DES in a strong perpendicular magnetic field. Later selfconsistent thermal-equilibrium calculations [8,9] confirmed that such ISs should exist at finite, sufficiently low temperatures, and that they should be able to carry an ASDC nearly dissipationless [10][11][12].…”

“…To describe quantum Hall states with finite ASDC, the current-carrying non-equilibrium state was approximated by a local-equilibrium state with position-dependent electrochemical potential μ * (r) and conductivity tensor ŝ ( ) r , which determine the density s = ( )ˆ( ) ( ) j r r Er of the ASDC by a local version of Ohm's law, with the gradient of the electrochemical potential defining the driving electric field, * m  = ( ) ( ) e E r r [10][11][12]. These calculations revealed a strongly nonlinear response behavior under the conditions of the IQHE, with a current-induced asymmetry between the ISs near opposite sample edges.…”

Current-induced asymmetries of incompressible strips in narrow quantum Hall systems

“…The full Hartree approximation requires us to solve the eigenvalue problem with the potential V (x), but we assume that V (x) varies within the 2DES slowly on the scale of the magnetic length B = √h c/eB. This allows us to approximate the energy eigenvalue of the Landau state with center coordinate X by E n (X ) = V (X ) + E n , with E n the Landau energy without confinement, and to neglect the extent of the states in the x-direction, when calculating the density [16]. This leads in thermal equilibrium to the Thomas-Fermi approximation (TFA)…”