The quantum Hall effect in narrow quantum wires

Abstract

The quantum phase diagram of disordered quantum wires in a strong magnetic field is reviewed. For uncorrelated disorder potential the 2‐terminal conductance, as calculated with the numerical transfer matrix method, shows zero temperature discontinuous transitions between exactly integer plateau values and zero. This is explained by the dimensional crossover of the bulk localisation length, which drives a transition from delocalised to localised edge states. In the thermodynamic limit, fixing the aspect ratio of the wire, there is a transition from the one dimensional chiral metal of extended edge states to localisation along the wire. In the vicinity of this chiral metal insulator transition (CMIT), states are identified which are superpositions of edge states with opposite chirality. The bulk contribution of such states is found to decrease with increasing wire width. Based on exact diagonalisation results for the eigenstates and their participation ratios, we conclude that these states are characteristic for the CMIT, and have the appearance of nonchiral edges states. Thereby these states are distinguishable from other states in the quantum Hall wire, namely, extended edge states, two‐dimensionally (2D) localized, quasi‐1D localized, and 2D critical states. In the presence of spatially correlated random potential we find with the numerical transfermatrix method that the potential correlation results in a shift of quantized conductance plateaus in long wires proportional to the strength of the random potential. This shift is found to be insensitive to the strength of magnetic fields and the same for all plateaus. A semiclassical explanation of this effect is proposed. We conclude with an outlook on modfications of the quantum phase diagram due to the spin degree of freedom of the electrons and their interactions. We discuss the stability of the phase diagram at finite temperature. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Solutions of the Schroedinger equation for piecewise harmonic potentials: Remarks on the asymptotic behavior of the wave functions

“…These so called "bathtub" potentials have been used as confining potentials for electrons in nanostructures, in particular when analyzing the quantum Hall effect. [18][19][20][21][22][23] As the potential is an even function, it is convenient to take as independent solutions in the region −l < z < l ψ 1 (z) = cos √ Ez,…”