Where are the edge-states near the quantum point contacts? A self-consistent approach

Abstract

In this work, we calculate the current distribution, in the close vicinity of the quantum point contacts (QPCs), taking into account the Coulomb interaction. In the first step, we calculate the bare confinement potential of a generic QPC and, in the presence of a perpendicular magnetic field, obtain the positions of the incompressible edge states (IES) taking into account electron-electron interaction within the Thomas-Fermi theory of screening. Using a local version of the Ohm's law, together with a relevant conductivity model, we also calculate the current distribution. We observe that, the imposed external current is confined locally into the incompressible strips. Our calculations demonstrate that, the inclusion of the electron-electron interaction, strongly changes the general picture of the transport through the QPCs.

Modeling of quantum point contacts in high magnetic fields and with current bias outside the linear response regime

“…Then we can replace the wave functions in both directions by wave packets centered at X and at y, and the center coordinate dependent eigenenergy E n ͑X͒ can be approximated to E n + V tot ͑X , y͒. It follows that the spatial distribution of the electron density within the TFA is given by the expression 31,32,71 n el ͑x,y͒ = ͵ D͓E,͑x,y͔͒f͓E + V tot ͑x,y͒ − ‫ء‬ ͔dE, ͑A5͒…”

“…There exist many theories which take the potential profile from the frozen charge model as an initial condition, and provides explicit calculation schemes to obtain charge, current, and potential distributions. [29][30][31][32][33] One of the most complete schemes, even in the presence of an external B field, is the local spin-density approximation ͑LSDA͒ within the density-functional theory ͑DFT͒. 34 The LSDA+ DFT ͑Refs.…”

Distribution of an Ohmic current in the close vicinity of a quantum point contact

“…However, one can not add more electrons to the IS since there are no states available. The only way to screen the Landau tilting is to increase the widths of the ISs on one side where the Hall potential is larger [9,19]. This effect clearly demonstrates the importance of e − e interactions.…”

“…self-consistent Born approximation (SCBA) [18], provided that local electron distribution is known. Moreover, the position dependent electrochemical potential, thereby the local current distribution, was obtained within a local version of the Ohm's law under the condition of a local equilibrium [9,19]. This self-consistent theory was implemented to many interesting 2DEG systems successfully explaining the relevant experiments [20,21,22,23,24,25].…”

The self-consistent calculation of the edge states at quantum Hall effect (QHE) based Mach–Zehnder interferometers (MZI)

“…We believe that, if (not only if) the nondissipative current is confined to the IES, where no backscattering takes place, the observed amplitude variation of the visibility as a function of B field at the experiments can simply be explained by an emerging IES at the QPCs. This claim also promotes the fact that, in such sensitive experiments the geometrical shape of the QPCs may play an important role [4,10], although the transmission amplitude remains unchanged. In summary, the spatial distribution of the backscattering free IES at a MZI (topologically equivalent geometry) is studied, exploiting the smooth variation of the external potential, within the Thomas-Fermi approximation.…”