Quantum Hall effect - Wikipedia, the free encyclopedia

Quantum Hall effectFrom Wikipedia, the free encyclopediaThe quantum Hall effect (or integer quantum Hall effect) is a quantum-mechanical version of the Hall effect, observed in two-dimensional electron systems subjected to low temperatures and strong magnetic fields, in which the Hall conductance G undergoes certain quantum Hall transitions to take on the quantized valueswhere is the channel current, is the Hall voltage, e is the elementary charge and h is Planck's constant. The prefactor ν is known as the "filling factor", and can take on either integer (ν = 1, 2, 3, .. ) or fractional (ν = 1/3, 2/5, 3/7, 2/3, 3/5, 1/5, 2/9, 3/13, 5/2, 12/5 ...) values. The quantum Hall effect is referred to as the integer or fractional quantum Hall effect depending on whether ν is an integer or fraction respectively. The integer quantum Hall effect is very well understood, and can be simply explained in terms of single-particle orbitals of an electron in a magnetic field (see Landau quantization). The fractional quantum Hall effect is more complicated, as its existence relies fundamentally on electron–electron interactions. It is also very well understood as an integer quantum Hall effect, not of electrons but of charge-flux composites known as composite fermions. In 1988, it was proposed that there was quantum Hall effect without Landau levels. This quantum Hall effect is referred to as the quantum anomalous Hall (QAH) effect. There is also a new concept of the quantum spin Hall effect which is an analogue of the quantum Hall effect, where spin currents flow instead of charge currents.[1]Contents◾1 Applications◾2 History◾3 Integer quantum Hall effect – Landau levels◾4 Mathematics◾5 See also◾6 References◾7 Further readingApplicationsThe quantization of the Hall conductance has the important property of being incredibly precise. Actual measurements of the Hall conductance have been found to be integer or fractional multiples of e2/h to nearly one part in a billion. This phenomenon, referred to as "exact quantization", has been shown to be a subtle manifestation of the principle of gauge invariance.[2] It has allowed for the definition of a new practical standard for electrical resistance, based on the resistance quantum givenby the von Klitzing constant R K = h/e2 = 25812.807557(18) Ω.[3] This is named after Klaus von Klitzing, the discoverer of exact quantization. Since 1990, a fixed conventional value R K-90 is used in resistance calibrations worldwide.[4] The quantum Hall effect also provides an extremely precise independent determination of the fine structure constant, a quantity of fundamental importance in quantum electrodynamics.HistoryThe integer quantization of the Hall conductance was originally predicted by Ando, Matsumoto, and Uemura in 1975, on the basis of an approximate calculation which they themselves did not believe to be true. Several workers subsequently observed the effect in experiments carried out on the inversion layer of MOSFETs. It was only in 1980 that Klaus von Klitzing, working at the high magnetic field laboratory in Grenoble with silicon-based samples developed by Michael Pepper and Gerhard Dorda, made the unexpected discovery that the Hall conductivity was exactly quantized. For this finding, von Klitzing was awarded the 1985 Nobel Prize in Physics. The link between exact quantization and gauge invariance was subsequently found by Robert Laughlin. Most integer quantum Hall experiments are now performed on gallium arsenide heterostructures, although many other semiconductor materials can be used. In 2007, the integer quantum Hall effect was reported in graphene at temperatures as high as room temperature,[5] and in the oxide ZnO-Mg x Zn1-x O.[6] Integer quantum Hall effect – Landau levelsIn two dimensions, when classical electrons are subjected to a magnetic field they follow circular cyclotron orbits. When the system is treated quantum mechanically, these orbits are quantized. The energy levels of these quantized orbitals take on discrete values: , where ωc = eB/m is the cyclotron frequency. These orbitals are known as Landau levels, and at weak magnetic fields, their existence gives rise to many interesting "quantum oscillations" such as the Shubnikov–de Haas oscillations and the de Haas–van Alphen effect (which is often used to map the Fermi surface of metals). For strong magnetic fields, each Landau level is highly degenerate (i.e. there are many single particle states which have the same energy E n). Specifically, for a sample of area A, inmagnetic field B, the degeneracy of each Landau level is (where g s represents a factor of 2 for spin degeneracy, and is the magnetic flux quantum). ForHofstadter's butterfly sufficiently strong B-fields, each Landau level may have so many states that all of the free electrons in the system sit in only a few Landau levels; it is in this regime where one observes the quantum Hall effect.MathematicsThe integers that appear in the Hall effect are examples oftopological quantum numbers. They are known inmathematics as the first Chern numbers and are closelyrelated to Berry's phase. A striking model of much interest inthis context is the Azbel-Harper-Hofstadter model whosequantum phase diagram is the Hofstadter butterfly shown inthe figure. The vertical axis is the strength of the magneticfield and the horizontal axis is the chemical potential, whichfixes the electron density. The colors represent the integer Hall conductances. Warm colors represent positive integersand cold colors negative integers. The phase diagram isfractal and has structure on all scales. In the figure there is an obvious self-similarity.Concerning physical mechanisms, impurities and/or particular states (e.g., edge currents) are important for both the 'integer' and 'fractional' effects. In addition, Coulomb interaction is also essential in the fractional quantum Hall effect. The observed strong similarity between integer and fractional quantum Hall effects is explained by the tendency of electrons to form bound states with an even number of magnetic flux quanta, called composite fermions .See also◾quantum Hall transitions◾Fractional quantum Hall effect◾Composite fermions◾Hall effect◾Hall probe◾Graphene◾Quantum spin Hall effect◾Coulomb potential between two current loops embedded in a magnetic fieldReferences1.^ Ezawa, Zyun F. (2013). Quantum Hall Effects: Recent Theoretical and Experimental Developments (3rd ed.). World Scientific. ISBN 978-981-4360-75-3.2.^ Laughlin, R. (1981). "Quantized Hall conductivity in twodimensions" (/abstract/PRB/v23/i10/p5632_1). Physical Review B 23 (10): 5632–5633. Bibcode:1981PhRvB..23.5632L (/abs/1981PhRvB..23.5632L).doi:10.1103/PhysRevB.23.5632 (/10.1103%2FPhysRevB.23.5632). Retrieved 8 May 2012.3.^ Tzalenchuk, Alexander; Lara-Avila, Samuel; Kalaboukhov, Alexei; Paolillo, Sara; Syväjärvi, Mikael;Yakimova, Rositza; Kazakova, Olga; Janssen, T. J. B. M.; Fal'ko, Vladimir; Kubatkin, Sergey (2010)."Towards a quantum resistance standard based on epitaxialgraphene" (/nnano/journal/v5/n3/abs/nnano.2009.474.html). NatureNanotechnology5 (3): 186–189. arXiv:0909.1220 (https:///abs/0909.1220).Bibcode:2010NatNa...5..186T (/abs/2010NatNa...5..186T).doi:10.1038/nnano.2009.474 (/10.1038%2Fnnano.2009.474). PMID 20081845(https:///pubmed/20081845).4.^ "conventional value of von Klitzing constant" (/cgi-bin/cuu/Value?rk90). NIST.5.^ Novoselov, K. S.; Jiang, Z.; Zhang, Y.; Morozov, S. V.; Stormer, H. L.; Zeitler, U.; Maan, J. C.;Boebinger, G. S.; Kim, P.; Geim, A. K. (2007). "Room-Temperature Quantum Hall Effect in Graphene".Science315 (5817): 1379. arXiv:cond-mat/0702408 (https:///abs/cond-mat/0702408).Bibcode:2007Sci...315.1379N (/abs/2007Sci...315.1379N).doi:10.1126/science.1137201 (/10.1126%2Fscience.1137201). PMID 17303717(https:///pubmed/17303717).6.^ Tsukazaki, A.; Ohtomo, A.; Kita, T.; Ohno, Y.; Ohno, H.; Kawasaki, M. (2007). "Quantum Hall Effectin Polar Oxide Heterostructures". Science315 (5817): 1388–91. Bibcode:2007Sci...315.1388T(/abs/2007Sci...315.1388T). doi:10.1126/science.1137430(/10.1126%2Fscience.1137430). PMID 17255474(https:///pubmed/17255474).Further reading◾Ando, Tsuneya; Matsumoto, Yukio; Uemura, Yasutada (1975). "Theory of Hall Effect in a Two-Dimensional Electron System". J. Phys. Soc. Jpn.39 (2): 279–288.Bibcode:1975JPSJ...39..279A (/abs/1975JPSJ...39..279A).doi:10.1143/JPSJ.39.279 (/10.1143%2FJPSJ.39.279).◾Klitzing, K.; Dorda, G.; Pepper, M. (1980). "New Method for High-Accuracy Determination of the Fine-Structure Constant Based on Quantized Hall Resistance". Phys. Rev. Lett.45 (6): 494–497. Bibcode:1980PhRvL..45..494K(/abs/1980PhRvL..45..494K). doi:10.1103/PhysRevLett.45.494(/10.1103%2FPhysRevLett.45.494).◾Laughlin, R. B. (1981). "Quantized Hall conductivity in two dimensions". Phys. Rev. B.23(10): 5632–5633. Bibcode:1981PhRvB..23.5632L(/abs/1981PhRvB..23.5632L). doi:10.1103/PhysRevB.23.5632(/10.1103%2FPhysRevB.23.5632).◾Yennie, D. R. (1987). "Integral quantum Hall effect for nonspecialists". Rev. Mod. Phys.59(3): 781–824. Bibcode:1987RvMP...59..781Y(/abs/1987RvMP...59..781Y). doi:10.1103/RevModPhys.59.781(/10.1103%2FRevModPhys.59.781).◾Hsieh, D.; Qian, D.; Wray, L.; Xia, Y.; Hor, Y. S.; Cava, R. J.; Hasan, M. Z. (2008). "A topological Dirac insulator in a quantum spin Hall phase". Nature452 (7190): 970–974.arXiv:0902.1356 (https:///abs/0902.1356). Bibcode:2008Natur.452..970H(/abs/2008Natur.452..970H). doi:10.1038/nature06843(/10.1038%2Fnature06843). PMID 18432240(https:///pubmed/18432240).◾25 years of Quantum Hall Effect, K. von Klitzing, Poincaré Seminar (Paris-2004). Postscript (http://parthe.lpthe.jussieu.fr/poincare/textes/novembre2004.html). Pdf(/PhysicsHorizon/25yearsQHE-lecture.pdf).◾Magnet Lab Press Release Quantum Hall Effect Observed at Room Temperature (/mediacenter/news/pressreleases/2007february15.html)◾Avron, Joseph E.; Osadchy, Daniel, Seiler, Ruedi (2003). "A Topological Look at the Quantum Hall Effect" (/resource/1/phtoad/v56/i8/p38_s1).Physics Today56 (8): 38. Bibcode:2003PhT....56h..38A(/abs/2003PhT....56h..38A). doi:10.1063/1.1611351(/10.1063%2F1.1611351). Retrieved 8 May 2012.◾Zyun F. Ezawa: Quantum Hall Effects - Field Theoretical Approach and Related Topics.World Scientific, Singapore 2008, ISBN 978-981-270-032-2◾Sankar D. Sarma, Aron Pinczuk: Perspectives in Quantum Hall Effects. Wiley-VCH, Weinheim 2004, ISBN 978-0-471-11216-7◾Baumgartner, A.; Ihn, T.; Ensslin, K.; Maranowski, K.; Gossard, A. (2007). "Quantum Hall effect transition in scanning gate experiments". Physical Review B76 (8).Bibcode:2007PhRvB..76h5316B (/abs/2007PhRvB..76h5316B).doi:10.1103/PhysRevB.76.085316 (/10.1103%2FPhysRevB.76.085316).◾E. I. Rashba and V. B. Timofeev, Quantum Hall Effect, Sov. Phys. - Semiconductors v. 20, pp.617-647 (1986).Retrieved from "/w/index.php?title=Quantum_Hall_effect&oldid=605304404"Categories: Hall effect Condensed matter physics Quantum electronics SpintronicsQuantum phases Mesoscopic physics◾This page was last modified on 22 April 2014 at 14:51.◾Text is available under the Creative Commons Attribution-ShareAlike License; additional terms may apply. By using this site, you agree to the Terms of Use and Privacy Policy.Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc., a non-profitorganization.。