Quantum point contact

A Quantum Point Contact (QPC) is a narrow constriction between two wide
electrically-conducting regions, of a width comparable to the electronic wavelength (nano- to micrometer). Quantum point contacts were first reported in 1988 by a Dutch group (Van Wees et al.) and, independently, by a British group (Wharam et al.).

Fabrication

There are different ways of fabricating a QPC. It can be realised for instance in a break-junction by pulling apart a piece of conductor until it breaks. The breaking point forms the point contact. In a more controlled way, quantum point contacts are formed in 2-dimensional electron gases (2DEG), e.g. in GaAs/AlGaAs heterostructures. By applying a voltage to suitably-shaped gate electrodes, the electron gas can be locally depleted and many different types of conducting regions can be created in the plane of the 2DEG, among them quantum dots and quantum point contacts.

Another means of creating a point contact is by positioning an STM-tip close to the surface of a conductor.

Properties

Geometrically a quantum point contact is a constriction in the transverse direction which presents a resistance to the motion of electrons. Applying a voltage $V$ across the point contact a current will flow, the size given by $I=GV$, where $G$ is the conductance of the contact. This formula resembles Ohm's law for macroscopic resistors. However there is a fundamental difference here resulting from the small system size which requires a quantum mechanical point of view.

At low temperatures and voltages, electrons contributing to the current have a certain energy/momentum/wavelength called Fermi energy/momentum/wavelength. The transverse confinement in the quantum point contact results in a quantisation of the transverse motion much like in a waveguide. The electron wave can only pass through the constriction if it interferes constructively which for a given size of constriction only happens for a certainnumber of modes $N$. The current carried by such a quantum state is the product of the velocity times the electron density. These two quantities by themselves differ from one mode to the other, but their product is mode independent. As a consequence, each state contributes the same amount $e^2/h$ per spin direction to the total conductance

:$G=N\; G\_Q$.

This is a fundamental result; the conductance does not take on arbitrary values but is quantised in multiples of the conductance quantum $G\_Q=2e^2/h$ which is expressed through electron charge $e$ and Planck constant $h$. The integer number $N$ is determined by the width of the point contact and roughly equals the width divided by twice the electron wavelength. As a function of the width (or gate voltage in the case of GaAs/AlGaAs heterostructure devices) of the point contact, the conductance shows a staircase behaviour as more and more modes (or channels) contribute to the electron transport. The step-height is given by $G\_Q$.

An external magnetic field applied to the quantum point contact lifts the spin degeneracy and leads to half-integer steps in the conductance. In addition, the number $N$ of modes that contribute becomes smaller. For large magnetic fields $N$ is independent of the width of the constriction, given by the theory of the quantum Hall effect.

An interesting feature, not yet fully understood, is a plateau at $0.7G\_Q$, the so-called 0.7-structure.

Applications

Apart from studying fundamentals of charge transport in mesoscopic conductors, quantum point contacts can be used as extremely sensitive charge detectors. Since the conductance through the contact strongly depends on the size of the constriction, any potential fluctuation (for instance, created by other electrons) in the vicinity will influence the current through the QPC. It is possible to detect single electrons with such a scheme. In view of quantum computation in solid-state systems, QPCs may be used as readout devices for the state of a qubit.

Further reading

* cite journal
author=H. van Houten and C.W.J. Beenakker
year=1996
title=Quantum point contacts
journal=Physics Today
volume=49
issue=7
pages=22–27
url= http://arxiv.org/abs/cond-mat/0512609
doi=10.1063/1.881503
format=
* cite journal
author=C.W.J.Beenakker and H. van Houten
year=1991
title=Quantum Transport in Semiconductor Nanostructures
journal=Solid State Physics
volume=44
url= http://arxiv.org/abs/cond-mat/0412664
format=
* cite journal
author=B.J. van Wees et al.
year=1988
title=Quantized conductance of point contacts in a two-dimensional electron gas
journal=Physical Review Letters
volume=60
pages=848–850
doi=10.1103/PhysRevLett.60.848
* cite journal
author=D.A. Wharam et al.
year=1988
title=One-dimensional transport and the quantization of the ballistic resistance
journal=J. Phys. C
volume=21
pages=L209
doi=10.1088/0022-3719/21/8/002
* cite journal
author=J.M. Elzerman et al.
year=2003
title=Few-electron quantum dot circuit with integrated charge read out
journal=Physical Review B
volume=67
pages=161308
doi=10.1103/PhysRevB.67.161308
* cite journal
author=K. J. Thomas et al.
year=1996
title=Possible spin polarization in a one-dimensional electron gas
journal=Physical Review Letters
volume=77
pages=135
doi=10.1103/PhysRevLett.77.135
* cite journal
author=Nicolás Agraït, Alfredo Levy Yeyati, Jan M. van Ruitenbeek
year=2003
title=Quantum properties of atomic-sized conductors
journal=Physics Reports
volume=377
pages=81