Quantum point contact
Encyclopedia
A Quantum Point Contact 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.). They are based on earlier work by the British group which showed how split gates could be used to convert a two-dimensional electron gas into one-dimension, first in Silicon (Dean and Pepper) and then in Gallium Arsenide (Thornton et al., Berggren et al.)
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 Ga
As
/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 dot
s and quantum point
contacts.
Another means of creating a point contact is by positioning an STM
-tip close to the surface of a conductor.
to the motion of electron
s. Applying a voltage
across the point contact induces a current to flow, the magnitude of this current is given by 
, where 
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 analysis.
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 certain number of modes
. 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 
per spin direction to the total conductance
.
This is a fundamental result; the conductance does not take on arbitrary values but is quantised in multiples of the conductance quantum
which is expressed through electron charge 
and Planck constant 
. The integer number 
is determined by the width of the point contact and roughly equals the width divided by half 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
.
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
of modes that contribute becomes smaller. For large magnetic fields 
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
, the so-called 0.7-structure.
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
.
Wavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
(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.). They are based on earlier work by the British group which showed how split gates could be used to convert a two-dimensional electron gas into one-dimension, first in Silicon (Dean and Pepper) and then in Gallium Arsenide (Thornton et al., Berggren et al.)
Fabrication
There are different ways of fabricating a QPC. It can be realised for instance in a break-junctionBreak junction
A break junction is a electronic device which consists of two metal wires separated by a very thin gap, on the order of the inter-atomic spacing . This can be done by physically pulling the wires apart or through chemical etching or electromigration...
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 Ga
Gallium
Gallium is a chemical element that has the symbol Ga and atomic number 31. Elemental gallium does not occur in nature, but as the gallium salt in trace amounts in bauxite and zinc ores. A soft silvery metallic poor metal, elemental gallium is a brittle solid at low temperatures. As it liquefies...
As
Arsenic
Arsenic is a chemical element with the symbol As, atomic number 33 and relative atomic mass 74.92. Arsenic occurs in many minerals, usually in conjunction with sulfur and metals, and also as a pure elemental crystal. It was first documented by Albertus Magnus in 1250.Arsenic is a metalloid...
/AlGaAs heterostructures. By applying a voltage
Voltage
Voltage, otherwise known as electrical potential difference or electric tension is the difference in electric potential between two points — or the difference in electric potential energy per unit charge between two points...
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 dot
Quantum dot
A quantum dot is a portion of matter whose excitons are confined in all three spatial dimensions. Consequently, such materials have electronic properties intermediate between those of bulk semiconductors and those of discrete molecules. They were discovered at the beginning of the 1980s by Alexei...
s and quantum point
contacts.
Another means of creating a point contact is by positioning an STM
Scanning tunneling microscope
A scanning tunneling microscope is an instrument for imaging surfaces at the atomic level. Its development in 1981 earned its inventors, Gerd Binnig and Heinrich Rohrer , the Nobel Prize in Physics in 1986. For an STM, good resolution is considered to be 0.1 nm lateral resolution and...
-tip close to the surface of a conductor.
Properties
Geometrically a quantum point contact is a constriction in the transverse direction which presents a resistanceElectrical resistance
The electrical resistance of an electrical element is the opposition to the passage of an electric current through that element; the inverse quantity is electrical conductance, the ease at which an electric current passes. Electrical resistance shares some conceptual parallels with the mechanical...
to the motion of electron
Electron
The electron is a subatomic particle with a negative elementary electric charge. It has no known components or substructure; in other words, it is generally thought to be an elementary particle. An electron has a mass that is approximately 1/1836 that of the proton...
s. Applying a voltage
across the point contact induces a current to flow, the magnitude of this current is given by , where is the conductance of the contact. This formula resembles Ohm's lawOhm's law
Ohm's law states that the current through a conductor between two points is directly proportional to the potential difference across the two points...
for macroscopic resistors. However there is a fundamental difference here resulting from the small system size which requires a quantum mechanical analysis.
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
Waveguide
A waveguide is a structure which guides waves, such as electromagnetic waves or sound waves. There are different types of waveguides for each type of wave...
. The electron wave can only pass through the constriction if it interferes constructively which for a given size of constriction only happens for a certain number of modes
. 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 per spin direction to the total conductance.This is a fundamental result; the conductance does not take on arbitrary values but is quantised in multiples of the conductance quantum
Conductance quantum
The conductance quantum is the quantized unit of conductance. It is defined as G0 = 2e2/h = ≈ Ω−1. It appears when measuring the conductance of a quantum point contact....
which is expressed through electron charge and Planck constant . The integer number is determined by the width of the point contact and roughly equals the width divided by half the electron wavelengthWavelength
In physics, the wavelength of a sinusoidal wave is the spatial period of the wave—the distance over which the wave's shape repeats.It is usually determined by considering the distance between consecutive corresponding points of the same phase, such as crests, troughs, or zero crossings, and is a...
. 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
.An external magnetic field
Magnetic field
A magnetic field is a mathematical description of the magnetic influence of electric currents and magnetic materials. The magnetic field at any given point is specified by both a direction and a magnitude ; as such it is a vector field.Technically, a magnetic field is a pseudo vector;...
applied to the quantum point contact lifts the spin degeneracy and leads to half-integer steps in the conductance. In addition, the number
of modes that contribute becomes smaller. For large magnetic fields 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
, 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 ispossible 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
Qubit
In quantum computing, a qubit or quantum bit is a unit of quantum information—the quantum analogue of the classical bit—with additional dimensions associated to the quantum properties of a physical atom....
.