The intrinsic quantum Hall effect
We first consider an interacting two-dimensional electron gas in a ballistic quantum wire in an external magnetic field. Self-consistent calculations are made of the electrostatic Hall potential (EHP), the local chemical potential (LCP), and current density in a uniform ballistic quantum wire containing two-dimensional electrons in a perpendicular magnetic field B when either one or two subbands are occupied. The corresponding Hall resistances, REHP and RLCP, are also calculated. The former is nearly linear in B in spite of subband depopulation. The latter is quantised but the quantisation steps are rounded because of overlap of the forward and backward going wave functions. Secondly, self-consistent calculations are also made of wave functions and the two kinds of Hall resistances for the same system in a weak perpendicular magnetic field when several subbands are occupied. We find intermittent quenching of the Hall resistance associated with the local chemical potential as the electron density varies. The quenching is due to the overlap of opposite-going wave functions in the same subband, which is enhanced significantly by the singularity of the density of states at the subband minima aa well as by Coulomb interactions between the electrons. Finally, with a, model calculation, we demonstrate that a non-invasive measurement of intrinsic quantum Hall effect defined by the local chemical potential in a ballistic quantum wire can be achieved with the aid of a pair of voltage leads which are separated 9Y potential barriers from the wire. Biittiker's formula is used to determine the chemical potential being measured and is shown to reduce exactly to the local chemical potential in the limit of strong potential confinement in the voltage leads. Conditions for quantisation of Hall resistance and measuring local chemical potential are given.