Quantum Hall edge state

A quantum Hall edge state is confined across a boundary but can extend along it and carry electric current. B. I. Halperin related these current-carrying boundary states to quantized Hall transport in a disordered two-dimensional system.^[ref]

A skipping orbit is the semiclassical path of a charged particle whose cyclotron arcs are repeatedly interrupted by a reflecting boundary. The centre of successive arcs advances along the edge. For electrons in a perpendicular positive $B_z$, the opposite sides of a strip carry opposite directed motion. A smooth boundary replaces hard reflections by guiding-centre drift; the quantum counterpart is a quantum Hall edge state.

For locally linear electron confinement potential energy $U(y)$, two Zeeman splitting branches cross the same Fermi energy at positions differing by the energy splitting divided by $|U^{\prime }|$. The expression neglects self-consistent electrostatic screening and exchange interaction enhancement. If the quoted slope is an electrostatic voltage gradient instead of an electron-energy gradient, its magnitude must be multiplied by the elementary charge. A screened compressible quantum Hall edge strip is not a single sharp crossing with an imposed bare slope.

Self-consistent electrostatics allows a partially filled edge Landau level to screen strongly and flatten its effective energy. Such compressible regions alternate with incompressible strips whose chemical potential lies in a gap and across which the potential changes. An externally prescribed unscreened smooth edge misses this redistribution. Temperature, disorder and interactions beyond the electrostatic approximation can change the strip profile.

An incompressible quantum Hall edge strip is an edge region where the local chemical potential lies in a gap between available electron states. Small potential changes then alter the local electron density only weakly, so that region cannot screen an external field as effectively as a neighboring compressible quantum Hall edge strip.

In the self-consistent electrostatic picture, incompressible regions alternate with compressible ones as the confining potential varies toward an edge. Much of the potential change can occur across the incompressible regions. Their widths depend on the gap and density profile; a prescribed unscreened edge potential misses the redistribution that produces the strips.^[ref]

A compressible quantum Hall edge strip is an edge region where a Landau level is partly occupied and nearby states permit the local electron density to adjust. This adjustment screens the confining field and tends to flatten the effective energy of that level near the chemical potential.

It contrasts with an incompressible quantum Hall edge strip, where a gap suppresses the corresponding density response. Their alternating arrangement is a self-consistent property of electrons and the confining electrostatics, rather than simply a drawing of unperturbed levels through a fixed potential. Finite temperature, disorder and interactions can modify the ideal strip structure.^[ref]

A depleted potential hill inside a two-dimensional electron gas. In a strong perpendicular magnetic field, closed quantum Hall edge states circulate around it. Coupling these states to propagating edge channels can produce resonant backscattering; Coulomb blockade can substantially alter the single-particle resonance pattern.

A quantum dot is an electron-confining potential well, whereas an antidot is a depleted potential hill. At the same magnetic field, their closed guiding-centre drift has opposite circulation. A series dot needs transmission through two barriers; a side-coupled antidot can connect already propagating quantum Hall edge states and induce backscattering. Both can have discrete energies and charging effects, but a electrical conductance peak or dip depends on lead connectivity rather than solely on the existence of a closed orbit.

A closed antidot edge orbit can tunnel-couple extended channels on opposite edges. When its electrochemical energy aligns with the reservoirs, resonant backscattering reduces conductance from an otherwise transmitting integer plateau. Different connectivity can instead add a transmission path and produce peaks. Gate voltage changes both the orbit and tunnel couplings; charging and spin alter the simplest fixed-area Aharonov-Bohm resonance sequence.

A spin-resolved resonant path adds at most $e^2/h$ to the electrical conductance of directly transmitted spectator channels. Conversely, resonant transfer into a returning edge removes at most $e^2/h$ from an otherwise open channel. For an isolated level, $T(E)={\Gamma }_L{\Gamma }_R/[(E-{ϵ}_d{)}^2+({\Gamma }_L+{\Gamma }_R{)}^2/4]$ reaches one only for equal couplings. Integer offsets follow the channel connectivity and do not follow from a local filling factor alone. temperature, asymmetry, coherent background paths and inelastic processes can change the apparent resonance height and shape.

In an integer-filling charging model, $f$ occupied spin-resolved branches can drive approximately $f$ charge rearrangements per added $h/e$ flux. Two coupled spin branches can therefore yield near-$h/(2e)$ oscillations. This does not change the elementary single-electron flux quantum and is not guaranteed by counting two independent noninteracting spin spectra alone.

For a fixed effective enclosed area $S$, changing the magnetic flux by $h/e$ changes the single-electron Aharonov-Bohm phase by $2\pi$. Gate-dependent orbit area, spin branches and charging can modify the observed conductance-resonance pattern; the lithographic depleted area need not equal the exact enclosed edge-orbit area.