Atwo-dimensional electron gas (2DEG) is ascientific model insolid-state physics. It is anelectron gas that is free to move in two dimensions, but tightly confined in the third. This tight confinement leads to quantizedenergy 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 ofholes is called a two-dimensional hole gas (2DHG), and such systems have many useful and interesting properties.

Most 2DEGs are found intransistor-like structures made fromsemiconductors. The most commonly encountered 2DEG is the layer of electrons found inMOSFETs (metal–oxide–semiconductorfield-effect transistors). When the transistor is ininversion mode, the electrons underneath thegate oxide are confined to the semiconductor-oxide interface, and thus occupy well defined energy levels. For thin-enough potential wells and temperatures not too high, only the lowest level isoccupied (see the figure caption), and so the motion of the electrons perpendicular to the interface can be ignored. However, the electron is free to move parallel to the interface, and so is quasi-two-dimensional.

Other methods for engineering 2DEGs arehigh-electron-mobility-transistors (HEMTs) and rectangularquantum wells. HEMTs arefield-effect transistors that utilize theheterojunction between two semiconducting materials to confine electrons to a triangularquantum well. Electrons confined to the heterojunction of HEMTs exhibit highermobilities than those in MOSFETs, since the former device utilizes an intentionallyundoped channel thereby mitigating the deleterious effect ofionized impurity scattering. Two closely spaced heterojunction interfaces may be used to confine electrons to a rectangular quantum well. Careful choice of the materials and alloy compositions allow control of the carrier densities within the 2DEG.

Electrons may also be confined to the surface of a material. For example, free electrons will float on the surface ofliquid helium, and are free to move along the surface, but stick to the helium; some of the earliest work in 2DEGs was done using this system.^[1] Besides liquid helium, there are also solid insulators (such astopological insulators) that support conductive surface electronic states.

Recently, atomically thin solid materials have been developed (graphene, as well as metal dichalcogenide such asmolybdenum disulfide) where the electrons are confined to an extreme degree. The two-dimensional electron system ingraphene can be tuned to either a 2DEG or 2DHG (2-D hole gas) bygating or chemicaldoping. This has been a topic of current research due to the versatile (some existing but mostly envisaged) applications of graphene.^[2]

A separate class of heterostructures that can host 2DEGs are oxides. Although both sides of the heterostructure are insulators, the 2DEG at the interface may arise even without doping (which is the usual approach in semiconductors). Typical example is a ZnO/ZnMgO heterostructure.^[3] More examples can be found in a recent review^[4] including a notable discovery of 2004, a 2DEG at theLaAlO₃/SrTiO₃ interface^[5] which becomes superconducting at low temperatures. The origin of this 2DEG is still unknown, but it may be similar tomodulation doping in semiconductors, with electric-field-induced oxygen vacancies acting as the dopants.

Considerable research involving 2DEGs and 2DHGs has been done, and much continues to this day. 2DEGs offer a mature system of extremely highmobility electrons, especially at low temperatures. When cooled to 4 K, 2DEGs may have mobilities${\displaystyle \mu }$ of the order of 1,000,000 cm²/Vs and lower temperatures can lead to further increase of${\displaystyle \mu }$ still. Specially grown, state of the artheterostructures with mobilities around 30,000,000 cm²/(V·s) have been made.^[6] These enormous mobilities offer a test bed for exploring fundamental physics, since besides confinement andeffective mass, the electrons do not interact with the semiconductor very often, sometimes traveling severalmicrometers before colliding; this so-called mean free path${\displaystyle \ell }$ can be estimated in the parabolic band approximation as

${\displaystyle \ell =v_{\mathrm{F}}\tau =\sqrt{2\pi n}\frac{\hslash \mu }{e}\approx 5.2\mu \mathrm{m}\times \mu [{10}^6\mathrm{c}{\mathrm{m}}^2/\mathrm{V}\mathrm{s}]\sqrt{n[{10}^{11}\mathrm{c}{\mathrm{m}}^{-2}]}}$

where${\displaystyle n}$ is the electron density in the 2DEG. Note that${\displaystyle \mu }$ typically depends on${\displaystyle n}$.^[7] Mobilities of 2DHG systems are smaller than those of most 2DEG systems, in part due to larger effective masses of holes (few 1000 cm²/(V·s) can already be considered high mobility^[8]).

Aside from being in practically every semiconductor device in use today, two dimensional systems allow access to interesting physics. Thequantum Hall effect was first observed in a 2DEG,^[9] which led to twoNobel Prizes in physics, ofKlaus von Klitzing in 1985,^[10] and ofRobert B. Laughlin,Horst L. Störmer andDaniel C. Tsui in 1998.^[11] Spectrum of a laterally modulated 2DEG (a two-dimensionalsuperlattice) subject to magnetic fieldB can be represented as theHofstadter's butterfly, a fractal structure in the energy vsB plot, signatures of which were observed in transport experiments.^[12] Many more interesting phenomena pertaining to 2DEG have been studied.[A]

• Two-dimensional gas

• A. Examples of more 2DEG physics. Full control of the 2DEGspin polarization was demonstrated.^[13] Possibly, this could be relevant toquantum information technology.Wigner crystallization in magnetic field. Microwave-induced magnetoresistance oscillations discovered by R. G. Mani et al.^[14] Possible existence of non-abelian quasiparticles in the fractional quantum Hall effect at filling factor 5/2.

• Weisbuch, C.; Vinter, B. (1991).Quantum Semiconductor Structures: Fundamentals and Applications.Academic Press.ISBN 0-12-742680-9.
• Davies, J. H. (1997).The Physics of Low-dimensional Semiconductors: An Introduction.Cambridge University Press.ISBN 0-521-48148-1.

• Transistors
• Quantum electronics
• Mesoscopic physics
• MOSFETs
• Surfaces

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