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| A '''two-dimensional electron gas (2DEG)''' is a gas of [[electron]]s free to move in two dimensions, but tightly confined in the third. This tight confinement leads to quantized [[energy level]]s for motion in that 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 [[electron hole|holes]] is called a two-dimensional hole gas (2DHG), and such systems have many useful and interesting properties.
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| ==Realizations of 2DEG==
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| [[Image:FET cross section.png|thumb|right|250px|In MOSFETs, the 2DEG is only present when the transistor is in inversion mode, and is found directly beneath the gate oxide.]]
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| [[Image:HighElectronMobilityTrasistor Band Image.PNG|thumb|right|250px|[[Band edge diagram]] of a basic HEMT. Conduction band edge {{math|E<sub>C</sub>}} and [[Fermi level]] {{math|E<sub>F</sub>}} determine the electron density in the 2DEG. Quantized levels form in the triangular well (yellow region) and optimally only one of them lies below {{math|E<sub>F</sub>}}.]]
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| [[Image:Sketch of a heterostructure - 2DEG in GaAs-GaAlAs.png|thumb|Heterostructure corresponding to the band edge diagram above.]]
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| Most 2DEG are found in [[transistor]]-like structures made from [[semiconductor]]s. The most commonly encountered 2DEG is the layer of electrons found in [[MOSFET]]s. When the transistor is in [[inversion layer (semiconductors)|inversion mode]], the electrons underneath the [[gate oxide]] are confined to the semiconductor-oxide interface, and thus occupy well defined energy levels. Nearly always, only the lowest level is occupied (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.
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| Other methods for engineering 2DEGs are [[HEMT|high-electron-mobility-transistor]]s (HEMTs) and rectangular [[quantum well]]s. HEMTs are [[field-effect transistors]] that utilize the [[heterojunction]] between two semiconducting materials to confine electrons to a triangular [[quantum well]]. Electrons confined to the heterojunction of HEMTs exhibit higher [[electron mobility|mobilities]] than those in MOSFETs, since the former device utilizes an intentionally [[Induced high electron mobility transistor|undoped channel]] thereby mitigating the deleterious effect of [[ionized 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.
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| Electrons may also be confined to the surface of a material. For example, free electrons will float on the surface of [[liquid 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.<ref>W. T. Sommer ''Liquid Helium as a Barrier to Electrons'' Physical Review Letters '''12''', 271–273 (1964).[http://dx.doi.org/10.1103/PhysRevLett.12.271]</ref> Besides liquid helium, there are also solid insulators (such as [[topological insulator]]s) that support conductive surface electronic states.
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| Recently, atomically thin solid materials have been developed ([[graphene]], as well as metal dichalcogenide such as [[molybdenum disulfide]]) where the electrons are confined to an extreme degree. The two-dimensional electron system in [[graphene]] can be tuned to either a 2DEG or 2DHG by [[field effect (semiconductor)|gating]] or chemical [[doping (semiconductor)|doping]].
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| This has been a topic of current research due to the versatile (some existing but mostly envisaged) applications of graphene.<ref>K.S. Novoselov et al.: ''A roadmap for graphene'', Nature 490, 192 (2012).[http://dx.doi.org/10.1038/nature11458]</ref>
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| Another notable 2DEG was discovered in 2004 at the [[Lanthanum aluminate-strontium titanate interface|LaAlO<sub>3</sub>/SrTiO<sub>3</sub>]] interface.<ref>{{cite journal|last=Ohtomo|author2=Hwang|title=A high-mobility electron gas at the LaAlO<sub>3</sub>/SrTiO<sub>3</sub> heterointerface|journal=Nature|date=29 Jan 2004|year=2004|volume=427|doi=10.1038/nature02308}}</ref> The origin of this 2DEG is still unknown, but it may be similar to modulation doping in semiconductors, with electric-field-induced oxygen vacancies acting as the dopants.
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| ==Experiments==
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| Considerable research involving 2DEGs and 2DHGs has been done, and much continues to this day. 2DEGs offer a mature system of extremely high [[electron mobility|mobility]] electrons, especially at low temperatures. When cooled to 4 K, 2DEGs may have mobilities <math>\mu</math> of the order of 1,000,000 cm<sup>2</sup>/Vs and lower temperatures can lead to further increase of <math>\mu</math> still. Specially grown, state of the art [[heterostructure]]s with mobilities around 30,000,000 cm<sup>2</sup>/(V·s) have been made.<ref>A. Kumar, G. A. Csáthy, M. J. Manfra, L. N. Pfeiffer, and K. W. West, ''Nonconventional Odd-Denominator Fractional Quantum Hall States in the Second Landau Level'', Physical Review Letters '''105''', 246808 (2010); also arXiv:1009.0237.[http://dx.doi.org/10.1103/PhysRevLett.105.246808]</ref> These enormous mobilities offer a test bed for exploring fundamental physics, since besides confinement and [[Effective mass (solid-state physics)|effective mass]], the electrons do not interact with the semiconductor very often, sometimes traveling several [[micrometre|micrometers]] before colliding; this so-called mean free path <math>\ell</math> can be estimated in the parabolic band approximation as
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| :<math> \ell=v_F\tau = \sqrt{2\pi n} \frac{\hbar\mu}{e} \approx 5.2\ \mu\mathrm{m}\times \mu\ [10^6\ \mathrm{cm^2/Vs}]\sqrt{n\ [10^{11}\ \mathrm{cm^{-2}}]}
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| </math>
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| where <math>n</math> is the electron density in the 2DEG. Note that <math>\mu</math> typically depends on <math>n</math>.<ref>W. Pan, N. Masuhara, N. S. Sullivan, K. W. Baldwin, K. W. West, L. N. Pfeiffer, and D. C. Tsui, ''Impact of Disorder on the 5/2 Fractional Quantum Hall State'', Physical Review Letters '''106''', 206806 (2011); also arXiv:1109.6911.[http://arxiv.org/abs/1109.6911]</ref> Mobilities of 2DHG systems are smaller than those of most 2DEG systems, in part due to larger effective masses of holes (few 1000 cm<sup>2</sup>/(V·s) can already be considered high mobility<ref>M. Myronov et al.: ''Observation of high mobility 2DHG with very high hole density in the modulation doped strained Ge quantum well at room temperature'', Phys. E 40, 1935 (2008).[http://dx.doi.org/10.1016/j.physe.2007.08.142]</ref>).
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| Aside from being in practically every semiconductor device in use today, two dimensional systems allow access to interesting physics. The [[quantum Hall effect]] was first observed in a 2DEG,<ref>K. v. Klitzing, G. Dorda, and M. Pepper: ''New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance'', Physical Review Letters '''45''', 494–497 (1980).[http://dx.doi.org/10.1103/PhysRevLett.45.494]</ref> which led to two [[Nobel Prize]]s, in [http://nobelprize.org/physics/laureates/1985/index.html 1985] and [http://nobelprize.org/physics/laureates/1998/ 1998]. Spectrum of a laterally modulated 2DEG (a two-dimensional [[superlattice]]) subject to magnetic field ''B'' can be represented as the [[Hofstadter's butterfly]], a fractal structure in the energy vs ''B'' plot, signatures of which were observed in transport experiments.<ref>M. C. Geisler, J. H. Smet, V. Umansky, K. von Klitzing, B. Naundorf, R. Ketzmerick, and H. Schweizer, ''Detection of a Landau Band-Coupling-Induced Rearrangement of the Hofstadter Butterfly'', Physical Review Letters '''92''', 256801 (2004).</ref> Many more interesting phenomena pertaining to 2DEG have been studied.[[#Footnotes|[1]]]
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| ==Footnotes==
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| *1. Examples of more 2DEG physics. Full control of the 2DEG [[spin polarization]] was recently demonstrated, C. Phelps, T. Sweeney, R. Cox, and H. Wang ''Ultrafast Coherent Electron Spin Flip in a CdTe Quantum Well'', Physical Review Letters '''102''', 237402 (2009). Possibly, this could be relevant to [[quantum information science|quantum information technology]]. [[Wigner crystal]]lization in magnetic field. Microwave-induced magnetoresistance oscillations discovered by R. G. Mani, J. H. Smet, K. von Klitzing, V. Narayanamurti, W. B. Johnson, and V. Umansky, Nature (London) 420, 646 (2002). Possible existence of non-abelian quasiparticles in the fractional quantum Hall effect at filling factor 5/2.
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| ==References==
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| * ''Quantum Semiconductor Structures'' by Claude Weisbuch and Borge Vinter, ISBN 0-12-742680-9
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| * ''Physics of Low Dimensional Semiconductors'' by John H. Davies, ISBN 0-521-48148-1
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| {{reflist}}
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| [[Category:Transistors]]
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| [[Category:Quantum electronics]]
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| [[Category:Mesoscopic physics]]
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