Quantum Hall Effect - Wikipedia the Free Encyclopedia

6/25/2015 QuantumHalleffectWikipedia,thefreeencyclopedia

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QuantumHalleffectFromWikipedia,thefreeencyclopedia

ThequantumHalleffect(orintegerquantumHalleffect)isaquantummechanicalversionoftheHalleffect,observedintwodimensionalelectronsystemssubjectedtolowtemperaturesandstrongmagneticfields,inwhichtheHallconductanceundergoescertainquantumHalltransitionstotakeonthequantizedvalues

where isthechannelcurrent, istheHallvoltage,eistheelementarychargeandhisPlanck'sconstant.Theprefactorisknownasthe"fillingfactor",andcantakeoneitherinteger(=1,2,3,...)orfractional(=1/3,2/5,3/7,2/3,3/5,1/5,2/9,3/13,5/2,12/5,...)values.ThequantumHalleffectisreferredtoastheintegerorfractionalquantumHalleffectdependingonwhetherisanintegerorfraction,respectively.TheintegerquantumHalleffectisverywellunderstood,andcanbesimplyexplainedintermsofsingleparticleorbitalsofanelectroninamagneticfield(seeLandauquantization).ThefractionalquantumHalleffectismorecomplicated,asitsexistencereliesfundamentallyonelectronelectroninteractions.AlthoughthemicroscopicoriginsofthefractionalquantumHalleffectareunknown,thereareseveralphenomenologicalapproachesthatprovideaccurateapproximations.ForexampletheeffectcanbethoughtofasanintegerquantumHalleffect,notofelectronsbutofchargefluxcompositesknownascompositefermions.In1988,itwasproposedthattherewasquantumHalleffectwithoutLandaulevels.ThisquantumHalleffectisreferredtoasthequantumanomalousHall(QAH)effect.ThereisalsoanewconceptofthequantumspinHalleffectwhichisananalogueofthequantumHalleffect,wherespincurrentsflowinsteadofchargecurrents.[1]

Contents

1Applications2History3IntegerquantumHalleffectLandaulevels4Mathematics5Seealso6References7Furtherreading

Applications

ThequantizationoftheHallconductancehastheimportantpropertyofbeingincrediblyprecise.ActualmeasurementsoftheHallconductancehavebeenfoundtobeintegerorfractionalmultiplesofe2/htonearlyonepartinabillion.Thisphenomenon,referredtoas"exactquantization",hasbeenshowntobeasubtlemanifestationoftheprincipleofgaugeinvariance.[2]Ithasallowedforthedefinitionofanewpracticalstandardforelectricalresistance,basedontheresistancequantumgivenbythevonKlitzingconstantRK=h/e2=25812.807557(18).[3]ThisisnamedafterKlausvonKlitzing,thediscovererof

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exactquantization.Since1990,afixedconventionalvalueRK90isusedinresistancecalibrations

worldwide.[4]ThequantumHalleffectalsoprovidesanextremelypreciseindependentdeterminationofthefinestructureconstant,aquantityoffundamentalimportanceinquantumelectrodynamics.

History

TheintegerquantizationoftheHallconductancewasoriginallypredictedbyAndo,Matsumoto,andUemurain1975,onthebasisofanapproximatecalculationwhichtheythemselvesdidnotbelievetobetrue.SeveralworkerssubsequentlyobservedtheeffectinexperimentscarriedoutontheinversionlayerofMOSFETs.Itwasonlyin1980thatKlausvonKlitzing,workingatthehighmagneticfieldlaboratoryinGrenoblewithsiliconbasedsamplesdevelopedbyMichaelPepperandGerhardDorda,madetheunexpecteddiscoverythattheHallconductivitywasexactlyquantized.Forthisfinding,vonKlitzingwasawardedthe1985NobelPrizeinPhysics.ThelinkbetweenexactquantizationandgaugeinvariancewassubsequentlyfoundbyRobertLaughlin.MostintegerquantumHallexperimentsarenowperformedongalliumarsenideheterostructures,althoughmanyothersemiconductormaterialscanbeused.In2007,theintegerquantumHalleffectwasreportedingrapheneattemperaturesashighasroomtemperature,[5]andintheoxideZnOMgxZn1xO.[6]

IntegerquantumHalleffectLandaulevels

Intwodimensions,whenclassicalelectronsaresubjectedtoamagneticfieldtheyfollowcircularcyclotronorbits.Whenthesystemistreatedquantummechanically,theseorbitsarequantized.Theenergylevelsofthesequantizedorbitalstakeondiscretevalues:

wherec=eB/misthecyclotronfrequency.TheseorbitalsareknownasLandaulevels,andatweakmagneticfields,theirexistencegivesrisetomanyinteresting"quantumoscillations"suchastheShubnikovdeHaasoscillationsandthedeHaasvanAlpheneffect(whichisoftenusedtomaptheFermisurfaceofmetals).Forstrongmagneticfields,eachLandaulevelishighlydegenerate(i.e.therearemanysingleparticlestateswhichhavethesameenergyEn).Specifically,forasampleofareaA,inmagneticfieldB,thedegeneracyofeachLandaulevelis

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Hofstadter'sbutterfly

wheregsrepresentsafactorof2forspindegeneracy,and0=21015Wbisthemagneticfluxquantum.ForsufficientlystrongBfields,eachLandaulevelmayhavesomanystatesthatallofthefreeelectronsinthesystemsitinonlyafewLandaulevelsitisinthisregimewhereoneobservesthequantumHalleffect.

Mathematics

TheintegersthatappearintheHalleffectareexamplesoftopologicalquantumnumbers.TheyareknowninmathematicsasthefirstChernnumbersandarecloselyrelatedtoBerry'sphase.AstrikingmodelofmuchinterestinthiscontextistheAzbelHarperHofstadtermodelwhosequantumphasediagramistheHofstadterbutterflyshowninthefigure.Theverticalaxisisthestrengthofthemagneticfieldandthehorizontalaxisisthechemicalpotential,whichfixestheelectrondensity.ThecolorsrepresenttheintegerHallconductances.Warmcolorsrepresentpositiveintegersandcoldcolorsnegativeintegers.Thephasediagramisfractalandhasstructureonallscales.Inthefigurethereisanobviousselfsimilarity.

Concerningphysicalmechanisms,impuritiesand/orparticularstates(e.g.,edgecurrents)areimportantforboththe'integer'and'fractional'effects.Inaddition,CoulombinteractionisalsoessentialinthefractionalquantumHalleffect.TheobservedstrongsimilaritybetweenintegerandfractionalquantumHalleffectsisexplainedbythetendencyofelectronstoformboundstateswithanevennumberofmagneticfluxquanta,calledcompositefermions.

Seealso

QuantumHalltransitionsFractionalquantumHalleffectCompositefermionsHalleffectHallprobeGrapheneQuantumspinHalleffectCoulombpotentialbetweentwocurrentloopsembeddedinamagneticfield

References1. Ezawa,ZyunF.(2013).QuantumHallEffects:RecentTheoreticalandExperimentalDevelopments(3rded.).

WorldScientific.ISBN9789814360753.2. Laughlin,R.(1981)."QuantizedHallconductivityintwodimensions"

(http://prb.aps.org/abstract/PRB/v23/i10/p5632_1).PhysicalReviewB23(10):56325633.Bibcode:1981PhRvB..23.5632L(http://adsabs.harvard.edu/abs/1981PhRvB..23.5632L).doi:10.1103/PhysRevB.23.5632(https://dx.doi.org/10.1103%2FPhysRevB.23.5632).Retrieved8May2012.

3. Tzalenchuk,AlexanderLaraAvila,SamuelKalaboukhov,AlexeiPaolillo,SaraSyvjrvi,MikaelYakimova,RositzaKazakova,OlgaJanssen,T.J.B.M.Fal'ko,VladimirKubatkin,Sergey(2010)."Towardsaquantumresistancestandardbasedonepitaxialgraphene"(http://www.nature.com/nnano/journal/v5/n3/abs/nnano.2009.474.html).NatureNanotechnology5(3):186189.arXiv:0909.1220(https://arxiv.org/abs/0909.1220).Bibcode:2010NatNa...5..186T(http://adsabs.harvard.edu/abs/2010NatNa...5..186T).doi:10.1038/nnano.2009.474

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(https://dx.doi.org/10.1038%2Fnnano.2009.474).PMID20081845(https://www.ncbi.nlm.nih.gov/pubmed/20081845).

4. "conventionalvalueofvonKlitzingconstant"(http://physics.nist.gov/cgibin/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)."RoomTemperatureQuantumHallEffectinGraphene".Science315(5817):1379.arXiv:condmat/0702408(https://arxiv.org/abs/condmat/0702408).Bibcode:2007Sci...315.1379N(http://adsabs.harvard.edu/abs/2007Sci...315.1379N).doi:10.1126/science.1137201(https://dx.doi.org/10.1126%2Fscience.1137201).PMID17303717(https://www.ncbi.nlm.nih.gov/pubmed/17303717).

6. Tsukazaki,A.Ohtomo,A.Kita,T.Ohno,Y.Ohno,H.Kawasaki,M.(2007)."QuantumHallEffectinPolarOxideHeterostructures".Science315(5817):138891.Bibcode:2007Sci...315.1388T(http://adsabs.harvard.edu/abs/2007Sci...315.1388T).doi:10.1126/science.1137430(https://dx.doi.org/10.1126%2Fscience.1137430).PMID17255474(https://www.ncbi.nlm.nih.gov/pubmed/17255474).

Furtherreading

Ando,TsuneyaMatsumoto,YukioUemura,Yasutada(1975)."TheoryofHallEffectinaTwoDimensionalElectronSystem".J.Phys.Soc.Jpn.39(2):279288.Bibcode:1975JPSJ...39..279A(http://adsabs.harvard.edu/abs/1975JPSJ...39..279A).doi:10.1143/JPSJ.39.279(https://dx.doi.org/10.1143%2FJPSJ.39.279).Klitzing,K.Dorda,G.Pepper,M.(1980)."NewMethodforHighAccuracyDeterminationoftheFineStructureConstantBasedonQuantizedHallResistance".Phys.Rev.Lett.45(6):494497.Bibcode:1980PhRvL..45..494K(http://adsabs.harvard.edu/abs/1980PhRvL..45..494K).doi:10.1103/PhysRevLett.45.494(https://dx.doi.org/10.1103%2FPhysRevLett.45.494).Laughlin,R.B.(1981)."QuantizedHallconductivityintwodimensions".Phys.Rev.B.23(10):56325633.Bibcode:1981PhRvB..23.5632L(http://adsabs.harvard.edu/abs/1981PhRvB..23.5632L).doi:10.1103/PhysRevB.23.5632(https://dx.doi.org/10.1103%2FPhysRevB.23.5632).Yennie,D.R.(1987)."IntegralquantumHalleffectfornonspecialists".Rev.Mod.Phys.59(3):781824.Bibcode:1987RvMP...59..781Y(http://adsabs.harvard.edu/abs/1987RvMP...59..781Y).doi:10.1103/RevModPhys.59.781(https://dx.doi.org/10.1103%2FRevModPhys.59.781).Hsieh,D.Qian,D.Wray,L.Xia,Y.Hor,Y.S.Cava,R.J.Hasan,M.Z.(2008)."AtopologicalDiracinsulatorinaquantumspinHallphase".Nature452(7190):970974.arXiv:0902.1356(https://arxiv.org/abs/0902.1356).Bibcode:2008Natur.452..970H(http://adsabs.harvard.edu/abs/2008Natur.452..970H).doi:10.1038/nature06843(https://dx.doi.org/10.1038%2Fnature06843).PMID18432240(https://www.ncbi.nlm.nih.gov/pubmed/18432240).25yearsofQuantumHallEffect,K.vonKlitzing,PoincarSeminar(Paris2004).Postscript(http://parthe.lpthe.jussieu.fr/poincare/textes/novembre2004.html).Pdf(http://hrma.physics.sjtu.edu.cn/PhysicsHorizon/25yearsQHElecture.pdf).MagnetLabPressReleaseQuantumHallEffectObservedatRoomTemperature(http://www.magnet.fsu.edu/mediacenter/news/pressreleases/2007february15.html)Avron,JosephE.Osadchy,Daniel,Seiler,Ruedi(2003)."ATopologicalLookattheQuantumHallEffect"(http://www.physicstoday.org/resource/1/phtoad/v56/i8/p38_s1).PhysicsToday56(8):38.Bibcode:2003PhT....56h..38A(http://adsabs.harvard.edu/abs/2003PhT....56h..38A).doi:10.1063/1.1611351(https://dx.doi.org/10.1063%2F1.1611351).Retrieved8May2012.ZyunF.Ezawa:QuantumHallEffectsFieldTheoreticalApproachandRelatedTopics.WorldScientific,Singapore2008,ISBN9789812700322SankarD.Sarma,AronPinczuk:PerspectivesinQuantumHallEffects.WileyVCH,Weinheim2004,ISBN9780471112167Baumgartner,A.Ihn,T.Ensslin,K.Maranowski,K.Gossard,A.(2007)."QuantumHalleffecttransitioninscanninggateexperiments".PhysicalReviewB76(8).Bibcode:2007PhRvB..76h5316B(http://adsabs.harvard.edu/abs/2007PhRvB..76h5316B).



doi:10.1103/PhysRevB.76.085316(https://dx.doi.org/10.1103%2FPhysRevB.76.085316).E.I.RashbaandV.B.Timofeev,QuantumHallEffect,Sov.Phys.Semiconductorsv.20,pp.617647(1986).

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Quantum Hall Effect - Wikipedia the Free Encyclopedia