Quantum point contact

Last updated December 10, 2023

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).^[1]

Quantum point contacts were first reported in 1988 by a Dutch team from Delft University of Technology and Philips Research ^[2] and, independently, by a British team from the Cavendish Laboratory.^[3] 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 ^[4] and then in gallium arsenide.^[5]^[6]

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 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/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 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 to the motion of electrons. 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 for macroscopic resistors. However, there is a fundamental difference here resulting from the small system size which requires a quantum mechanical analysis.^[7]

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

$G={2e^{2} \over h}\sum _{n,m}|T_{n,m}|^{2}$ ,

where $T_{n,m}$ 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, 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 $G_{0}=2e^{2}/h$ , which is expressed through the electron charge $e$ and the Planck constant $h$ . The integer number $N$ is determined by the width of the point contact and roughly equals the width divided by half the electron wavelength. 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 $T\lesssim \Delta E/4k_{\rm {B}}\approx 4\;\mathrm {K}$ (here ∆E is the subband splitting at the Fermi level). This is confirmed both by experiment and by numerical calculations.^[9]

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.

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.^[10] 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 in solid-state systems, QPCs can be used as readout devices for the state of a quantum bit (qubit).^[11]^[12]^[13]^[14] In device physics, the configuration of QPCs is used for demonstrating a fully ballistic field-effect transistor.^[15] 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.^[16]

Related Research Articles

<span class="mw-page-title-main">Bose–Einstein condensate</span> State of matter

In condensed matter physics, a Bose–Einstein condensate (BEC) is a state of matter that is typically formed when a gas of bosons at very low densities is cooled to temperatures very close to absolute zero. Under such conditions, a large fraction of bosons occupy the lowest quantum state, at which microscopic quantum mechanical phenomena, particularly wavefunction interference, become apparent macroscopically.

The quantum Hall effect is a quantized version of the Hall effect which is observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall resistance $R xy$ exhibits steps that take on the quantized values

The elementary charge, usually denoted by e, is a fundamental physical constant, defined as the electric charge carried by a single proton or, equivalently, the magnitude of the negative electric charge carried by a single electron, which has charge −1 e.

<span class="mw-page-title-main">Polaron</span> Quasiparticle in condensed matter physics

A polaron is a quasiparticle used in condensed matter physics to understand the interactions between electrons and atoms in a solid material. The polaron concept was proposed by Lev Landau in 1933 and Solomon Pekar in 1946 to describe an electron moving in a dielectric crystal where the atoms displace from their equilibrium positions to effectively screen the charge of an electron, known as a phonon cloud. This lowers the electron mobility and increases the electron's effective mass.

<span class="mw-page-title-main">Charge qubit</span> Superconducting qubit implementation

In quantum computing, a charge qubit is a qubit whose basis states are charge states. In superconducting quantum computing, a charge qubit is formed by a tiny superconducting island coupled by a Josephson junction to a superconducting reservoir. The state of the qubit is determined by the number of Cooper pairs that have tunneled across the junction. In contrast with the charge state of an atomic or molecular ion, the charge states of such an "island" involve a macroscopic number of conduction electrons of the island. The quantum superposition of charge states can be achieved by tuning the gate voltage U that controls the chemical potential of the island. The charge qubit is typically read-out by electrostatically coupling the island to an extremely sensitive electrometer such as the radio-frequency single-electron transistor.

The fractional quantum Hall effect (FQHE) is a physical phenomenon in which the Hall conductance of 2-dimensional (2D) electrons shows precisely quantized plateaus at fractional values of $, where e is the electron charge and h is the Planck constant. It is a property of a collective state in which electrons bind magnetic flux lines to make new quasiparticles, and excitations have a fractional elementary charge and possibly also fractional statistics. The 1998 Nobel Prize in Physics was awarded to Robert Laughlin, Horst Störmer, and Daniel Tsui "for their discovery of a new form of quantum fluid with fractionally charged excitations" The microscopic origin of the FQHE is a major research topic in condensed matter physics.$

<span class="mw-page-title-main">Hubbard model</span> Approximate model used to describe the transition between conducting and insulating systems

The Hubbard model is an approximate model used to describe the transition between conducting and insulating systems. It is particularly useful in solid-state physics. The model is named for John Hubbard.

In mesoscopic physics, ballistic conduction is the unimpeded flow of charge carriers, or energy-carrying particles, over relatively long distances in a material. In general, the resistivity of a material exists because an electron, while moving inside a medium, is scattered by impurities, defects, thermal fluctuations of ions in a crystalline solid, or, generally, by any freely-moving atom/molecule composing a gas or liquid. Without scattering, electrons simply obey Newton's second law of motion at non-relativistic speeds.

A two-dimensional electron gas (2DEG) is a scientific model in solid-state physics. It is an electron gas that is free to move in two dimensions, but tightly confined in the third. This tight confinement leads to quantized energy levels for motion in the third direction, which can then be ignored for most problems. Thus the electrons appear to be a 2D sheet embedded in a 3D world. The analogous construct of holes is called a two-dimensional hole gas (2DHG), and such systems have many useful and interesting properties.

A Wigner crystal is the solid (crystalline) phase of electrons first predicted by Eugene Wigner in 1934. A gas of electrons moving in a uniform, inert, neutralizing background will crystallize and form a lattice if the electron density is less than a critical value. This is because the potential energy dominates the kinetic energy at low densities, so the detailed spatial arrangement of the electrons becomes important. To minimize the potential energy, the electrons form a bcc lattice in 3D, a triangular lattice in 2D and an evenly spaced lattice in 1D. Most experimentally observed Wigner clusters exist due to the presence of the external confinement, i.e. external potential trap. As a consequence, deviations from the b.c.c or triangular lattice are observed. A crystalline state of the 2D electron gas can also be realized by applying a sufficiently strong magnetic field. However, it is still not clear whether it is the Wigner crystallization that has led to observation of insulating behaviour in magnetotransport measurements on 2D electron systems, since other candidates are present, such as Anderson localization.

The conductance quantum, denoted by the symbol $G 0$ , is the quantized unit of electrical conductance. It is defined by the elementary charge e and Planck constant h as:

Andreev reflection (AR), named after the Russian physicist Alexander F. Andreev, is a type of particle scattering which occurs at interfaces between a superconductor (S) and a normal state material (N). It is a charge-transfer process by which normal current in N is converted to supercurrent in S. Each Andreev reflection transfers a charge 2e across the interface, avoiding the forbidden single-particle transmission within the superconducting energy gap.

<span class="mw-page-title-main">Quantum stirring, ratchets, and pumping</span>

A pump is an alternating current-driven device that generates a direct current (DC). In the simplest configuration a pump has two leads connected to two reservoirs. In such open geometry, the pump takes particles from one reservoir and emits them into the other. Accordingly, a current is produced even if the reservoirs have the same temperature and chemical potential.

In a standard superconductor, described by a complex field fermionic condensate wave function, vortices carry quantized magnetic fields because the condensate wave function $is invariant to increments of the phase by . There a winding of the phase by creates a vortex which carries one flux quantum. See quantum vortex.$

The quantum spin Hall state is a state of matter proposed to exist in special, two-dimensional semiconductors that have a quantized spin-Hall conductance and a vanishing charge-Hall conductance. The quantum spin Hall state of matter is the cousin of the integer quantum Hall state, and that does not require the application of a large magnetic field. The quantum spin Hall state does not break charge conservation symmetry and spin- $conservation symmetry.$

A composite fermion is the topological bound state of an electron and an even number of quantized vortices, sometimes visually pictured as the bound state of an electron and, attached, an even number of magnetic flux quanta. Composite fermions were originally envisioned in the context of the fractional quantum Hall effect, but subsequently took on a life of their own, exhibiting many other consequences and phenomena.

As the devices continue to shrink further into the sub-100 nm range following the trend predicted by Moore’s law, the topic of thermal properties and transport in such nanoscale devices becomes increasingly important. Display of great potential by nanostructures for thermoelectric applications also motivates the studies of thermal transport in such devices. These fields, however, generate two contradictory demands: high thermal conductivity to deal with heating issues in sub-100 nm devices and low thermal conductivity for thermoelectric applications. These issues can be addressed with phonon engineering, once nanoscale thermal behaviors have been studied and understood.

The Aharonov–Casher effect is a quantum mechanical phenomenon predicted in 1984 by Yakir Aharonov and Aharon Casher, in which a traveling magnetic dipole is affected by an electric field. It is dual to the Aharonov–Bohm effect, in which the quantum phase of a charged particle depends upon which side of a magnetic flux tube it comes through. In the Aharonov–Casher effect, the particle has a magnetic moment and the tubes are charged instead. It was observed in a gravitational neutron interferometer in 1989 and later by fluxon interference of magnetic vortices in Josephson junctions. It has also been seen with electrons and atoms.

Valleytronics is an experimental area in semiconductors that exploits local extrema ("valleys") in the electronic band structure. Certain semiconductors have multiple "valleys" in the electronic band structure of the first Brillouin zone, and are known as multivalley semiconductors. Valleytronics is the technology of control over the valley degree of freedom, a local maximum/minimum on the valence/conduction band, of such multivalley semiconductors.

Graphene is a semimetal whose conduction and valence bands meet at the Dirac points, which are six locations in momentum space, the vertices of its hexagonal Brillouin zone, divided into two non-equivalent sets of three points. The two sets are labeled K and K'. The sets give graphene a valley degeneracy of gv = 2. By contrast, for traditional semiconductors the primary point of interest is generally Γ, where momentum is zero. Four electronic properties separate it from other condensed matter systems.

References

↑ H. van Houten & C.W.J. Beenakker (1996). "Quantum point contacts". Physics Today. 49 (7): 22–27. arXiv: cond-mat/0512609 . Bibcode:1996PhT....49g..22V. doi:10.1063/1.881503. S2CID 56100437.
↑ B.J. van Wees; et al. (1988). "Quantized conductance of point contacts in a two-dimensional electron gas". Physical Review Letters. 60 (9): 848–850. Bibcode:1988PhRvL..60..848V. doi:10.1103/PhysRevLett.60.848. hdl: 1887/3316 . PMID 10038668.
↑ D.A. Wharam; et al. (1988). "One-dimensional transport and the quantization of the ballistic resistance". J. Phys. C. 21 (8): L209–L214. Bibcode:1988JPhC...21L.209W. doi:10.1088/0022-3719/21/8/002. S2CID 45112904.
↑ C.C.Dean and M. Pepper (1982). "The transition from two- to one-dimensional electronic transport in narrow silicon accumulation layers". J. Phys. C. 15 (36): L1287–L1297. Bibcode:1982JPhC...15.1287D. doi:10.1088/0022-3719/15/36/005.
↑ T. J. Thornton; et al. (1986). "One-Dimensional Conduction in the 2D Electron Gas of a GaAs-AlGaAs Heterojunction". Physical Review Letters. 56 (11): 1198–1201. Bibcode:1986PhRvL..56.1198T. doi:10.1103/PhysRevLett.56.1198. PMID 10032595.
↑ K-F. Berggren; et al. (1986). "Magnetic Depopulation of 1D Subbands in a Narrow 2D Electron Gas in a GaAs:AlGaAs Heterojunction". Physical Review Letters. 57 (14): 1769–1772. Bibcode:1986PhRvL..57.1769B. doi:10.1103/PhysRevLett.57.1769. PMID 10033540.
↑ Pearsall, Thomas (2020). Quantum Photonics, 2nd edition. Graduate Texts in Physics. Springer. doi:10.1007/978-3-030-47325-9. ISBN 978-3-030-47324-2. S2CID 240934073.
↑ C.W.J.Beenakker and H. van Houten (1991). "Quantum Transport in Semiconductor Nanostructures". Solid State Physics. 44: 1–228. arXiv: cond-mat/0412664 . Bibcode:2004cond.mat.12664B. doi:10.1016/s0081-1947(08)60091-0. ISBN 9780126077445. S2CID 119082619.
↑ C.W.J.Beenakker and H. van Houten (1991). "Quantum Transport in Semiconductor Nanostructures". Solid State Physics. 44: 1–228. arXiv: cond-mat/0412664 . Bibcode:2004cond.mat.12664B. doi:10.1016/s0081-1947(08)60091-0. ISBN 9780126077445. S2CID 119082619.
↑ Bauer, Florian; Heyder, Jan; Schubert, Enrico; Borowsky, David; Taubert, Daniela; Bruognolo, Benedikt; Schuh, Dieter; Wegscheider, Werner; von Delft, Jan; Ludwig, Stefan (September 2013). "Microscopic origin of the '0.7-anomaly' in quantum point contacts". Nature. 501 (7465): 73–78. doi:10.1038/nature12421. ISSN 1476-4687. PMID 23995681. S2CID 4409202.
↑ J.M. Elzerman; et al. (2003). "Few-electron quantum dot circuit with integrated charge read out". Physical Review B. 67 (16): 161308. arXiv: cond-mat/0212489 . Bibcode:2003PhRvB..67p1308E. doi:10.1103/PhysRevB.67.161308. S2CID 16278460.
↑ M. Field; et al. (1993). "Measurements of Coulomb blockade with a noninvasive voltage probe". Physical Review Letters. 70 (9): 1311–1314. Bibcode:1993PhRvL..70.1311F. doi:10.1103/PhysRevLett.70.1311. PMID 10054344.
↑ J. M. Elzerman; et al. (2004). "Single-shot read-out of an individual electron spin in a quantum dot". Nature. 430 (6998): 431–435. arXiv: cond-mat/0411232 . Bibcode:2004Natur.430..431E. doi:10.1038/nature02693. PMID 15269762. S2CID 4374126.
↑ J. R. Petta; et al. (2005). "Coherent Manipulation of Coupled Electron Spins in Semiconductor Quantum Dots". Science. 309 (5744): 2180–2184. Bibcode:2005Sci...309.2180P. doi:10.1126/science.1116955. PMID 16141370. S2CID 9107033.
↑ E. Gremion; D. Niepce; A. Cavanna; U. Gennser & Y. Jin (2010). "Evidence of a fully ballistic one-dimensional field-effect transistor: Experiment and simulation". Applied Physics Letters. 97 (23): 233505. Bibcode:2010ApPhL..97w3505G. doi:10.1063/1.3521466.
↑ Smith, D. P. E. (1995). "Quantum Point Contact Switches". Science. 269 (5222): 371–3. Bibcode:1995Sci...269..371S. doi:10.1126/science.269.5222.371. PMID 17841257. S2CID 2239813 . Retrieved 30 May 2020.