Berry-Phase-Effects-on-Electronic-Properties

BerryPhaseEffectsonElectronicPropertiesDiXiaoMaterialsScience&TechnologyDivision,OakRidgeNationalLaboratory,OakRidge,TN37831,USAMing-CheChangDepartmentofPhysics,NationalTaiwanNormalUniversity,Taipei11677,TaiwanQianNiuDepartmentofPhysics,TheUniversityofTexasatAustin,Austin,TX78712,USA(Dated:January5,2010)Eversinceitsdiscovery,thenotionofBerryphasehaspermeatedthroughallbranchesofphysics.Overthelastthreedecades,itwasgraduallyrealizedthattheBerryphaseoftheelectronicwavefunctioncanhaveaprofoundeffectonmaterialpropertiesandisresponsibleforaspectrumofphenomena,suchaspolarization,orbitalmagnetism,various(quantum/anomalous/spin)Halleffects,andquantumchargepumping.Thisprogressissummarizedinapedagogicalmannerinthisreview.Westartwithabriefsummaryofnecessarybackground,andgiveadetaileddiscussionoftheBerryphaseeffectinavarietyofsolidstateapplications.Acommonthreadofthereviewisthesemiclassicalformulationofelectrondynamics,whichisaversatiletoolinthestudyofelectrondynamicsinthepresenceofelectromagneticfieldsandmoregeneralperturbations.Finally,wedemonstrateare-quantizationmethodthatconvertsasemiclassicaltheorytoaneffectivequantumtheory.ItisclearthattheBerryphaseshouldbeaddedasanessentialingredienttoourunderstandingofbasicmaterialproperties.ContentsI.Introduction2A.Topicaloverview2B.Organizationofthereview3C.BasicConceptsofTheBerryphase41.Cyclicadiabaticevolution42.Berrycurvature53.Example:Thetwo-levelsystem6D.BerryphaseinBlochbands7II.Adiabatictransportandelectricpolarization8A.Adiabaticcurrent8B.Quantizedadiabaticparticletransport91.Conditionsfornonzeroparticletransportincyclicmotion102.Many-bodyinteractionsanddisorder103.AdiabaticPumping11C.ElectricPolarizationofCrystallineSolids111.TheRice-Melemodel12III.Electrondynamicsinanelectricfield13A.Anomalousvelocity13B.Berrycurvature:Symmetryconsiderations14C.ThequantumHalleffect15D.TheanomalousHallEffect161.Intrinsicvs.extrinsiccontributions162.AnomalousHallconductivityasaFermisurfaceproperty18E.ThevalleyHalleffect18IV.Wavepacket:ConstructionandProperties19A.Constructionofthewavepacketanditsorbitalmoment19B.Orbitalmagnetization20C.Dipolemoment21D.Anomalousthermoelectrictransport22V.Electrondynamicsinelectromagneticfields23A.Equationsofmotion23B.Modifieddensityofstates241.Fermivolume242.StredaFormula25C.Orbitalmagnetization:Revisit25D.Magnetotransport261.Cyclotronperiod262.Thehighfieldlimit263.TheLowFieldLimit27VI.Electrondynamicsundergeneralperturbations27A.Equationsofmotion27B.Modifieddensityofstates28C.DeformedCrystal29D.Polarizationinducedbyinhomogeneity301.Magneticfieldinducedpolarization31E.SpinTexture31VII.Quantizationofelectrondynamics32A.Bohr-Sommerfeldquantization32B.Wannier-Starkladder33C.deHaas-vanAlphenoscillation33D.Canonicalquantization(Abeliancase)34VIII.MagneticBlochbands35A.Magnetictranslationalsymmetry35B.BasicsofmagneticBlochband36C.Semiclassicalpicture:hyperorbits38D.Hallconductivityofhyperorbit38IX.Non-Abelianformulation39A.Non-Abelianelectronwavepacket40B.SpinHalleffect41C.Quantizationofelectrondynamics41D.Diracelectron42E.Semiconductorelectron43F.IncompletenessofeffectiveHamiltonian44G.Hierarchystructureofeffectivetheories44X.Outlook452Acknowledgments45A.AdiabaticEvolution45References46I.INTRODUCTIONA.TopicaloverviewIn1984,MichaelBerrywroteapaperthathasgen-eratedimmenseintereststhroughoutthedifferentfieldsofphysicsincludingquantumchemistry(Berry,1984).Thisisabouttheadiabaticevolutionofaneigenenergystatewhentheexternalparametersofaquantumsys-temchangeslowlyandmakeupaloopintheparameterspace.Intheabsenceofdegeneracy,theeigenstatewillsurelycomebacktoitselfwhenfinishingtheloop,buttherewillbeaphasedifferenceequaltothetimeintegraloftheenergy(dividedby¯h)plusanextra,whichisnowcommonlyknownastheBerryphase.TheBerryphasehasthreekeypropertiesthatmaketheconceptimportant(Bohmetal.,2003;ShapereandWilczek,1989a).First,itisgaugeinvariant.Theeigen-wavefunctionisdefinedbyahomogeneouslinearequation(theeigenvalueequation),soonehasthegaugefreedomofmultiplyingitwithanoverallphasefactorwhichcanbeparameterdependent.TheBerryphaseisunchanged(uptointegermultipleof2π)bysuchaphasefactor,providedtheeigen-wavefunctioniskepttobesinglevaluedovertheloop.ThispropertymakestheBerryphasephysical,andtheearlyexperimentalstudieswerefocusedonmeasuringitdirectlythroughinterferencephenomena.Second,theBerryphaseisgeometrical.Itcanbewrit-tenasaline-integralovertheloopintheparameterspace,anddoesnotdependontheexactrateofchangealongtheloop.ThispropertymakesitpossibletoexpresstheBerryphaseintermsoflocalgeometricalquantitiesintheparameterspace.Indeed,BerryhimselfshowedthatonecanwritetheBerryphaseasanintegralofafield,whichwenowcallastheBerrycurvature,overasurfacesuspendingtheloop.AlargeclassofapplicationsoftheBerryphaseconceptoccurwhentheparametersthem-selvesareactuallydynamicalvariablesofslowdegreesoffreedom.TheBerrycurvatureplaysanessentialroleintheeffectivedynamicsoftheseslowvariables.Thevastmajorityofapplicationsconsideredinthisreviewareofthisnature.Third,theBerryphasehascloseanalogiestogaugefieldtheoriesanddifferentialgeometry(Sim