We investigate the ballistic electron transport in a two dimensional Quantum Wire under the action of an electric field ($E_y$). We demonstrate how the presence of a Spin Orbit coupling, due to the uniform electric confinement field gives a non-commutative effect as in the presence of a transverse magnetic field. 
We discuss how the non commutation implies an edge localization of the currents depending on the electron spins also giving a semi-classical spin dependent Hall current. 
We also discuss how it is possible obtain a quantized Spin Hall conductance in the ballistic transport regime by developing the Landauer formalism and show the coupling between the spin magnetic momentum and the orbital one due to the presence of a circulating current.

http://arxiv.org/pdf/cond-mat/0512543.pdf

Integer spin Hall effect in ballistic quantum wires

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Nanostructure physics and fabrication : proceedings of the international symposium, College Station, Texas, March 13-15, 1989

The electronic states of GaAs/AlGaAs quantum well wires under an applied magnetic field are studied by using the empirical pseudopotential method. The energy spectra are calculated as a function of the magnetic field strength. The results show a strong non-parabolicity and field-dependent subband effective mass. The formation of bulk-like Landau levels and edge (skipping) states results in a rapid increasing of density of states at the subband bottom reflecting the change of the 1D subband structures to 2D-like Landau levels. The asymptotic behaviours of the Landau levels in the weak and strong magnetic field limits are also obtained by a simple effective mass analysis and used to interpret the results of the microscopic calculation.

A microscopic study of Landau level states in quantum wires

This paper uses a tight-binding scattering formalism to calculate the spatial distribution of the electric current and field in small phase-coherent conductors. The calculation is simple and efficient, allowing one to obtain simultaneous pictures of the current density and the local potential and field in a wide range of atomic-scale structures. The use of the method is illustrated by two examples. The first shows the formation of a resistivity dipole across an extended obstacle. The second is conduction through a double constriction, where we compare the current and potential distributions at and off resonance. The current pattern at resonance is related to the wavefunction of the quasibound state responsible for the resonance. Off resonance, large internal closed currents are found inside the double constriction. At resonance the potential drops are concentrated at the entrances to the constriction, while off resonance essentially all of the potential drop occurs in the interior of the double c...

Spatial distribution of the electric current and field in atomic-scale conductors

A thin-film geometry of nonmagnetic or magnetic topological insulator (TI) alters its topological properties depending on the thickness of the film and the strength of the magnetic coupling, and can exhibit the quantum spin Hall (QSH) effect as well as the quantum anomalous Hall (QAH) effect. In this study, we pursue a quantitative investigation of effects of nonmagnetic disorder on thin-film TIs in all the possible topological phases. The results show that strong enough disorder induces a topological phase transition and generically creates metallic helical edge states featuring the QSH phase before all electronic states are eventually localized. In the thermodynamic limit, the localization–delocalization transition in the disorder-induced QSH phase is in the unitary universality class with nontrivial Chern numbers. In a finite-size system, topologically protected edge states persist unless the mobility gap collapses. It is also demonstrated that edge states extend over a macroscopic distance in the midd...

Edge States in a Thin Film of Disordered Topological Insulator

Self-consistent calculations are made of the electrostatic Hall potential (EHP), local chemical potential (LCP), and current density in a 100-nm-wide wire containing two-dimensional electrons in a perpendicular magnetic field B when either one or two subbands are occupied. The corresponding Hall resistances, R EHP and R LCP , are also calculated. The former is nearly linear in B in spite of subband depopulation. The latter is quantized but the quantization steps are rounded because of overlap of the forward and backward wave functions

Intrinsic integer quantum Hall effect in a quantum wire.

Self-consistent calculations are made of wave functions and two kinds of Hall resistance for a 2DEG confined in a uniform ballistic quantum wire in a weak perpendicular magnetic field when several subbands are occupied. The authors 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 minimum as well as by Coulomb interactions between electrons.

Quenching of the quantum Hall effect in a uniform ballistic quantum wire

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 by potential barriers from the wire. B\"uttiker'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.

https://arxiv.org/pdf/cond-mat/9405090.pdf

Noninvasive measurement of the intrinsic quantum Hall effect.

A new definition is given for the local chemical potential mu L in a semiconductor nanostructure which is transmitting currents. It is determined by the equation n(r, mu 1, mu 2,...)=n(r, mu L, mu L,...) where n denotes the electron density, r is the point at which mu L is required and mu t denotes the chemical potential in the reservoir feeding terminal. This equation for mu L avoids any reference to non-invasive voltage probes which have been used in previous definitions. It is used to discriminate between previous formulae which use the ideas.

A new definition of the local chemical potential in a semiconductor nanostructure

We have recently made calculations of the Hall resistance of a ballistic two-dimensional electron gas confined in a quantum wire by hard walls (Chu and Butcher 1993 a and b). The calculations are made in the linear transport regime at low temperatures and are based on a formula for the local chemical potential given by Imry (1989). Other authors have sometimes used a different formula which gives different results (Peeters 1988, Akera and Ando 1989 and 1990). Here we outline our calculations and present some of our results. We also give a new definition of the local chemical potential which can be evaluated exactly in the linear transport regime at low temperatures. The outcome is identical to Imry’s formula. An introduction to the theory of electron transport in low-dimensional semiconductor structures is given by Butcher (1993).

P N Butcher

Papers

Intrinsic integer quantum Hall effect in a quantum wire.

Quenching of the quantum Hall effect in a uniform ballistic quantum wire

Noninvasive measurement of the intrinsic quantum Hall effect.

A new definition of the local chemical potential in a semiconductor nanostructure

Application of the Local Chemical Potential to the Quantum Hall Effect in a Ballistic Quantum Wire