Quantum Point Contact
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A quantum point contact (QPC) is a narrow constriction between two wide
electrically conducting Electrical resistivity (also called specific electrical resistance or volume resistivity) is a fundamental property of a material that measures how strongly it resists electric current. A low resistivity indicates a material that readily allows ...
regions, of a width comparable to the electronic
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
(nano- to micrometer). The importance of QPC lies in the fact that they prove quantisation of ballistic conductance in mesoscopic systems. The conductance of a QPC is quantized in units of 2e^2/h, the so-called
conductance quantum The conductance quantum, denoted by the symbol , is the quantized unit of electrical conductance. It is defined by the elementary charge The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalent ...
. Quantum point contacts were first reported in 1988 by a Dutch team from
Delft University of Technology Delft University of Technology ( nl, Technische Universiteit Delft), also known as TU Delft, is the oldest and largest Dutch public technical university, located in Delft, Netherlands. As of 2022 it is ranked by QS World University Rankings among ...
and Philips Research and, independently, by a British team from the
Cavendish Laboratory The Cavendish Laboratory is the Department of Physics at the University of Cambridge, and is part of the School of Physical Sciences. The laboratory was opened in 1874 on the New Museums Site as a laboratory for experimental physics and is name ...
. 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 Silicon is a chemical element with the symbol Si and atomic number 14. It is a hard, brittle crystalline solid with a blue-grey metallic luster, and is a tetravalent metalloid and semiconductor. It is a member of group 14 in the periodic ...
and then in gallium arsenide. This quantisation is reminiscent of the quantisation of the Hall conductance, but is measured in the absence of a magnetic field. The zero-field conductance quantisation and the smooth transition to the quantum Hall effect on applying a magnetic field are essentially consequences of the equipartition of current among an integer number of propagating modes in the constriction.


Fabrication

There are several different ways of fabricating a quantum point contact. It can be realized in a
break-junction A break junction is an electronic device which consists of two metal wires separated by a very thin gap, on the order of the inter-atomic spacing (less than a nanometer). This can be done by physically pulling the wires apart or through chemical etc ...
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 a two-dimensional electron gas (2DEG), e.g. in
GaAs Gallium arsenide (GaAs) is a III-V direct band gap semiconductor with a zinc blende crystal structure. Gallium arsenide is used in the manufacture of devices such as microwave frequency integrated circuits, monolithic microwave integrated ...
/ AlGaAs heterostructures. By applying a
voltage Voltage, also known as electric pressure, electric tension, or (electric) potential difference, is the difference in electric potential between two points. In a static electric field, it corresponds to the work needed per unit of charge t ...
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 dots (QDs) are semiconductor particles a few nanometres in size, having optical and electronic properties that differ from those of larger particles as a result of quantum mechanics. They are a central topic in nanotechnology. When the q ...
s and quantum point contacts. Another means of creating a QPC is by positioning the tip of a scanning tunneling microscope close to the surface of a conductor.


Properties

Geometrically, a quantum point contact is a constriction in the transverse direction which presents a
resistance Resistance may refer to: Arts, entertainment, and media Comics * Either of two similarly named but otherwise unrelated comic book series, both published by Wildstorm: ** ''Resistance'' (comics), based on the video game of the same title ** ''T ...
to the motion of
electron The electron (, or in nuclear reactions) is a subatomic particle with a negative one elementary electric charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary partic ...
s. Applying a voltage V across the point contact induces a current to flow, the magnitude of this current is given by I=GV, where G is the conductance of the contact. This formula resembles
Ohm's law Ohm's law states that the current through a conductor between two points is directly proportional to the voltage across the two points. Introducing the constant of proportionality, the resistance, one arrives at the usual mathematical equatio ...
for macroscopic resistors. However, there is a fundamental difference here resulting from the small system size which requires a quantum mechanical analysis. It is most common to study QPC in two dimensional electron gases. This way the geometric constriction of the point contact turns the conductance through the opening to a one dimensional system. Moreover, it requires a quantum mechanical description of the system that results in the quantisation of conductance. Quantum mechanically, the current through the point contact is equipartitioned among the 1D subands, or transverse modes, in the constriction. It is important to state that the previous discussion does not take into account possible transitions among modes. The Landauer formula can actually be generalized to express this possible transitions
G= \sum_ , T_, ^2,
where T_ is the transition matrix which incorporates non-zero probabilities of transmission from mode ''n'' to ''m''. At low temperatures and voltages, unscattered and untrapped electrons contributing to the current have a certain energy/momentum/wavelength called Fermi energy/momentum/wavelength. Much like in a
waveguide A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting the transmission of energy to one direction. Without the physical constraint of a waveguide, wave intensities de ...
, the transverse confinement in the quantum point contact results in a "quantization" of the transverse motion—the transverse motion cannot vary continuously, but has to be one of a series of discrete modes. The waveguide analogy is applicable as long as coherence is not lost through scattering, e.g., by a defect or trapping site. The electron wave can only pass through the constriction if it interferes constructively, which for a given width of constriction, only happens for a certain number 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 of e^2/h per spin direction to the total conductance G=NG_0. This is a fundamental result; the conductance does not take on arbitrary values but is quantized in multiples of the
conductance quantum The conductance quantum, denoted by the symbol , is the quantized unit of electrical conductance. It is defined by the elementary charge The elementary charge, usually denoted by is the electric charge carried by a single proton or, equivalent ...
G_0=2e^2/h, which is expressed through the electron charge e and the
Planck constant The Planck constant, or Planck's constant, is a fundamental physical constant of foundational importance in quantum mechanics. The constant gives the relationship between the energy of a photon and its frequency, and by the mass-energy equivalen ...
h. The integer number N is determined by the width of the point contact and roughly equals the width divided by half the electron
wavelength In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats. It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tr ...
. As a function of the width of the point contact (or gate voltage in the case of GaAs/AlGaAs heterostructure devices), the conductance shows a staircase behavior as more and more modes (or channels) contribute to the electron transport. The step-height is given by G_Q. On increasing the temperature, one finds experimentally that the plateaux acquire a finite slope until they are no longer resolved. This is a consequence of the thermal smearing of the Fermi-Dirac distribution. The conductance steps should disappear for T\lesssim\Delta E/4k_\approx 4\;\mathrm (here ∆''E'' is the subband splitting at the Fermi level). This is confirmed both by experiment and by numerical calculations. An external
magnetic field A magnetic field is a vector field that describes the magnetic influence on moving electric charges, electric currents, and magnetic materials. A moving charge in a magnetic field experiences a force perpendicular to its own velocity and t ...
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.


The 0.7 anomaly

Anomalous features on the quantized conductance steps are often observed in transport measurements of quantum point contacts. A notable example is the plateau at 0.7G_Q, the so-called 0.7-structure, arising due to enhanced electron-electron interactions arising from a smeared van Hove singularity in the local 1D density of states in the vicinity of the charge constriction. Unlike the conductance steps, the 0.7-structure becomes more pronounced at higher temperature. 0.7-structure analogues are sometimes observed on higher conductance steps. Quasi-bound states arising from impurities, charge traps, and reflections within the constriction may also result in conductance structure close to the 1D limit.


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 Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Thoug ...
in
solid-state Solid state, or solid matter, is one of the four fundamental states of matter. Solid state may also refer to: Electronics * Solid-state electronics, circuits built of solid materials * Solid state ionics, study of ionic conductors and their ...
systems, QPCs can be used as readout devices for the state of a
quantum bit In quantum computing, a qubit () or quantum bit is a basic unit of quantum information—the quantum version of the classic binary bit physically realized with a two-state device. A qubit is a two-state (or two-level) quantum-mechanical system ...
(qubit). In device physics, the configuration of QPCs is used for demonstrating a fully ballistic field-effect transistor. Another application of the device is its use as a switch. A nickel wire is brought close enough to a gold surface and then, by the use of a piezoelectric actuator, the distance between the wire and the surface can be changed and thus, the transport characteristics of the device change between electron tunneling and ballistic.


References


Further reading

* * * *{{Cite book , last1 = Timp , first1 = G. , chapter = Chapter 3: When Does a Wire Become an Electron Waveguide , doi = 10.1016/S0080-8784(08)62393-5 , title = Semiconductors and Semimetals Volume 35 , series = Semiconductors and Semimetals , volume = 35 , pages = 113–190 , year = 1992 , isbn = 9780127521350 Quantum mechanics Nanoelectronics Quantum electronics Mesoscopic physics