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PAGE 1 QUANTUMMAGNETOOSCILLATIONSNEARCLASSICALANDQUANTUMPHASETRANSITIONSByCHUNGWEIWANGADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011 PAGE 2 c2011ChungweiWang 2 PAGE 3 Idedicatethistoeveryonewhoisinterestedinthistopic. 3 PAGE 4 ACKNOWLEDGMENTS IwouldliketogivethankstoallthehelpandguideIhavereceivedfromDr.MaslovduringmyPhDresearchcareer.IalsowanttothanktoHridisPal,AliAshra,VivekMishra,andallothercolleaguesfordiscussions. 4 PAGE 5 TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 LISTOFSYMBOLS .................................... 9 LISTOFABBREVIATIONS ................................ 11 ABSTRACT ......................................... 12 CHAPTER 1INTRODUCTION ................................... 14 2CRITICALPHENOMENA .............................. 19 2.1PhaseTransitionsandOrderParameters .................. 19 2.2ClassicalCriticalPhenomena ......................... 21 2.3QuantumCriticalPhenomena ......................... 23 2.4Spin-FermionModel .............................. 25 2.5AnExampleandStartingPoint:Self-EnergyNearTheTwo-DimensionalFerromagneticCriticalPoint .......................... 26 2.5.1IssueNumberOne:DynamicorStaticSelf-Energy? ........ 30 2.5.2IssueNumberTwo:Self-EnergywithorwithoutMagneticFields? 31 3MAGNETOOSCILLATIONSANDDENSITYOFSTATES ............. 33 3.1IntroductiontoMagnetooscillations ...................... 33 3.2TheoryofMagnetooscillations-Overview .................. 36 3.2.1Introduction ............................... 36 3.2.2FirstApproach:theLuttinger-WardFormula ............. 38 3.2.3SecondApproach:DensityofStates ................. 38 3.3DensityofStatesinHighLandauLevelsofaTwo-DimensionalDisorderedSystem ..................................... 39 3.3.1DensityofStatesforShort-RangeCorrelations,withlB .... 40 3.3.2DensityofStatesforLong-RangeCorrelations,withlB .... 43 4DENSITYOFSTATESANDDEHAAS-VANALPHENOSCILLATIONSINADISORDEREDTHREE-DIMENSIONALELECTRONSYSTEM ......... 45 4.1DOSforshort-rangecorrelationslB ................... 46 4.2DOSforLong-RangeCorrelationslB .................. 52 4.3DHvAOscillations ............................... 56 5 PAGE 6 5MAGNETOOSCILLATIONSNEARTHEQUASI-TWO-DIMENSIONALQUANTUMCRITICALITY ..................................... 60 5.1TheoryofQuantumMagnetooscillations .................. 61 5.1.1SCBARegime:Short-RangecorrelationslB .......... 62 5.1.2QuasiclassicalRegime:Long-RangeCorrelationslB ..... 66 5.2QuantumMagnetooscillationsNeartheFerromagneticCriticalPoint ... 68 5.2.1SCBAregime:Short-RangeCorrelationslB .......... 68 5.2.2QuasiclassicalRegime:Long-RangeCorrelationslB ..... 71 5.2.2.1lBRc ......................... 71 5.2.2.2Rcv?=p tk ..................... 72 5.2.2.3v?=p tk ......................... 72 5.3TemperatureDependenceoftheMagnetooscillationAmplitude ...... 73 5.4Conclusions ................................... 76 APPENDIX ASTATICSPINSUSCEPTIBILITYINTHESPIN-FERMIONMODEL ....... 78 BDENSITYOFSTATESIN3DSHORT-RANGEIMPURITIES .......... 81 CDENSITYOFSTATESIN3DLONG-RANGEIMPURITIES ........... 83 REFERENCES ....................................... 84 BIOGRAPHICALSKETCH ................................ 87 6 PAGE 7 LISTOFTABLES Table page 2-1Orderparametersandtheirconjugateelds .................... 21 4-1AmplitudeCoefcientsIsandIcwithdifferentscatteringstrength ........ 59 7 PAGE 8 LISTOFFIGURES Figure page 1-1SpinSusceptibilityandEffectiveMassofElectronsinSiInversionLayer .... 17 1-2TemperaturedependenceofthedeHaas-vanAlphenamplitudeinCeCoIn5 .. 18 2-1Phasediagramsofaferromagnet .......................... 20 2-2Phasediagramnearthequantumcriticalpoint .................. 24 2-3Thelowest-orderdiagramforthefermionicself-energy(!;T;k) ........ 27 3-1Pictorialexplanationofmagnetooscillations .................... 34 3-2Feynmandiagramsofself-energyinthemagneticeld .............. 41 4-1Three-dimensionalDensityofStateFromaGivenLandauLevelintheShort-RangeCorrelations ...................................... 51 4-23-dimensionaldensityofstatesfromagivenLandaulevelinthequasiclassicalregime ......................................... 55 5-1Thelowest-orderdiagramforthefermionicself-energy(!;n;kz).ThevertexfunctioninthemagneticeldisgJjn)]TJ /F4 7.97 Tf 6.58 0 Td[(n0j(q?Rc). ................... 70 5-2Temperaturedependenceofmagnetooscillationamplitudesnearthecriticality 74 8 PAGE 9 LISTOFSYMBOLS BmagneticeldsstaticspinsusceptibilityEelectronicenergyEFFermienergy"electronicenergywithrespecttothechemicalpotential,i.e."=E)]TJ /F3 11.955 Tf 11.96 0 Td[(g(")densityofstatesatenergy"GMatsubaraGreen'sfunctionGRretardedGreen'sfunctionJnBesselfunctionoftherstkindFFermiwavelengthlBmagneticlengthLnmassociatedLaguerrepolynomialmelectronbandmassMmagnetizationmccyclotronmassmrenormalizedelectronmasschemicalpotentialkdirectionparalleltotheappliedmagneticeld,oralongthezdirection?directionperpendiculartotheappliedmagneticeld,orinthexyplane 9 PAGE 10 thermodynamicpotentialNeelectronnumberdensity!ccyclotronfrequencyRccyclotronradiusMatsubaraself-energyRretardedself-energySwdampingfactorofmagnetooscillationsinthequasiclassicalregimetkhoppingintegralinaquasi-two-dimensionalsystemalongthezdirectionwherethemagneticeldisappliedV(r)order-parameterdisorderatthepositionrW(q)correlationfunctioninthemomentumspacefW(r)correlationfunctioninthepositionspacecorrelationlengthofinteractions 10 PAGE 11 LISTOFABBREVIATIONS dHvAdeHaas-vanAlphen(effect)DOSdensityofstatesFLFermiliquidGLGinzburg-Landau(theory)LKLifshitz-KosevitchNFLnon-FermiliquidQCPquantumcriticalpointSCBAself-consistentBornapproximationSdHShubnikov-deHaas(effect)3Dthree-dimensional2Dtwo-dimensional 11 PAGE 12 AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyQUANTUMMAGNETOOSCILLATIONSNEARCLASSICALANDQUANTUMPHASETRANSITIONSByChungweiWangDecember2011Chair:DmitriiMaslovMajor:PhysicsWestudythedensityofstates(DOS)andquantummagnetooscillationsbothinathree-dimensional(3D)andinquasi-two-dimensional(2D)stronglycorrelatedsystemsneartheferromagnetic-typequantumandclassicalcriticalpoints.RecentexperimentsonmagnetooscillationsintheNernstcoefcientofbismuthhavereignitedinteresttotheDOSandmagnetooscillationsina3Ddisorderedsystem,which,tothebestofourknowledge,havenotbeeninvestigatedthoroughlyyet.Weadopttheself-consistentBornapproximationforshort-rangedisorders,whileusingthequasiclassicalpathintegralapproachforlong-rangedisorders.Inthecaseofshort-rangedisorders,wegeneralizetheDoman'sapproach,whoconsideredweakdisordersandfoundtheself-energyself-consistentlybytakingintoaccountscatteringonlywithinasingleLandaulevel,tothecaseoftheinter-Landaulevelscattering,andobtainanexpressionforthedensityofstatevalidintheregimesofbothweakandstrongdamping.Inthecaseoflong-rangedisorders,wedemonstrateusingthepath-integralapproachthatthereisaninhomogeneousbroadeningofLandaulevelsin3D.Wealsostudymany-bodyeffectsinquantummagnetooscillationsofaquasi-2Dstronglycorrelatedsystemnearthecriticalpoint.Theamplitudeofmagnetooscillationsisdeterminedbytheelectronself-energy,whichisofthenon-Fermi-liquidformnearthequantumcriticalpoint(QCP).Wedemonstrate,however,thatthecorrectresultcannotbeobtainedsimplybysubstitutingtheT=0self-energyintotheLifshitz-Kosevich 12 PAGE 13 formula.Ourndingisthatthedivergenceofthecorrelationlengthnearacriticalpointimpliesnecessarilythatstaticuctuationhasadominanteffectonmagnetooscillations,andthedampingtakesaGaussianformintheinhomogeneousbroadeningregime.ThisleadstostrongdeviationsoftheoscillationamplitudefromtheLifshitz-Kosevichform.Thisverydifferentfrompreviousstudiesthatconsideredonlythedynamicpartoftheself-energy.Takingthetemperaturedependenceofthecorrelationlengthintoaccount,weanalyzethetemperaturedependenceofthermaldampingaswell. 13 PAGE 14 CHAPTER1INTRODUCTIONThequantummagnetooscillationsareoscillationsinphysicalparametersasafunctionoftheappliedmagneticeld.ThosecanbedividedintothedeHaas-vanAlphen(dHvA)effect,whichistheoscillationsinthermodynamicquantities,e.g.magnetizationsorspecicheat,andtheShubnikov-deHaas(SdH)effect,whichistheoscillationintransportquantities,e.g.resistivityandthermopower.ofresistivity.Theoriginoftheseeffectsisrelatedtotheoscillatorystructureoftheelectronicdensityofstates(DOS),whichoccursbecauseoftheLandauquantizationoftheenergylevelsinamagneticeld.MagnetoosillationshaveproventobeaninvaluabletoolforstudyingthegeometryoftheFermisurfaceinmetals.Onecanalsoextracttherenormalizedeffectivemassfromthetemperaturedependenceoftheoscillationamplitude.Therstobservationofmagnetooscillationsdatesbackto1930,whenanoscillatorymagnetic-elddependencewasobservedinboththeresistivity[ 1 ]andmagnetization[ 2 ]ofbismuth.ItwaspredictedindependentlyinthesameyearbyLandau,whoaccountedforthemagneticoscillationsinthefreeelectrontheory[ 3 ].In1952,OnsagergeneralizedtheLandau'sfreeelectrontheorytoBlochelectronsbyconsideringhighLandaulevelsneartheFermisurface[ 4 ].In1956,LifshitzandKosevitch(LK)gaveadetailedquantitativeexplanationofoscillationsinacleanthree-dimensional(3D)system[ 5 ].Originally,theeffectofmagnetooscillationsfocusedonthelinkbetweenoscillationfrequencyandtheFermisurfacegeometry.Lateron,theoryalsotookintoaccounttheeffectonelectroninteractionswithimpurities,phonons,andotherelectrons[ 6 10 ],allofwhichresultinthedampingofoscillations.Initially,experimentalinvestigationsofmagnetooscillationswerefocusedon3Dsystems.Later,theprogressingrowingoftwo-dimensional(2D)conductorsattractedboththeexperimental[ 11 14 ]andthetheoretical[ 15 16 ]attentiontomagnetooscillationsin2Delectrongases.Oneofthewell-knownexampleswasgiven 14 PAGE 15 byPudalovetal.[ 14 ].BystudyingSdHoscillationsusingindependentlycontrolledparallelandperpendicularmagneticelds,theseauthorsdeterminedtheelectronspinsusceptibility,theeffectivemassm,andrenormalizedLandefactorgofmobileelectronsinSiinversionlayers,asafunctionofthenumberdensity.ThisstudywasanimportantsteptowardsquantifyingtheFermi-liquid(FL)parametersof2Dsystems.Figure 1-1 givesmoredetailsonhowthreeparametersareextractedfromthedata.Recently,peoplearemoreinterestedinhowmagneticoscillationswillbehaveinquasi-2Dheavy-fermionsystemsnearthequantumcriticalpoints(QCP)[ 17 20 ],high-Tccuprates[ 21 25 ],andironpnictidesuperconductors[ 26 27 ].Sofar,mostoftheexperimentaldatawereconsistentlytusingtheFLmodel,whichleadstotheLKform.However,inquasi-2Dsystemswhereinteractionsorcorrelationsarestrong,theLKtheorymustbreakdown.Evenina3Dsystem,theLKtheoryisnotstrictlyvalidaswell.TheinteractionscanbetakenintoaccountintheextendedLKtheory[ 46 49 50 ],wherethedampingofmagnetooscillationsarerelatedtotheelectronicself-energy.In2005,McCollametal.[ 18 ]diddeHaas-vanAlphen(dHvA)measurementsonCeCoIn5andclaimedthattheself-energyofnon-Fermi-liquid(NFL)formtwellintheextendedLKtheory.In2010,FritzandSachdevinvestigatedmagnetooscillationsbyconsideringfermionscoupledtoaninternalU(1)gaugeeldinadisordered2Dsystem,whichwasshowntohaveadeviationfromtheFLprediction[ 28 ].Inthesameyear,ThompsonandStampaddressedtheproblembyconsideringthespin-fermionmodel[ 29 ].Theycalculatedself-energyintheframeofself-consistentBornapproximation(SCBA)inaquasi-2DandanalyzeditsFLandNFLbehavior.Bothapproachesonlytookdynamicuctuationsintoconsideration.However,WendthatstaticuctuationscanhaveanimportanteffectonmagnetooscillationsneartheQCP.Besides,neartheQCPwherethecorrelationlengthislongerthanthemagneticlength 15 PAGE 16 lB,SCBAisnotapplicablesincetheself-energyfromcrosseddiagramsisofthesameorderasfromSCBAdiagrams[ 30 ].Althoughmostofrecentworkwasfocusedon2Delectronsystemswheretheydisplayfascinatingquantumphenomena,likethequantumHalleffect,andstrongcorrelationsneartheQCP,magnetooscillationsina3DsystemhaveattractedrenewedinterestsincerecentexperimentalworksonoscillationsoftheNernstcoefcientinbismuthbyBehniaetal.[ 31 32 ],whoextendedmeasurementstoquantumlimit.Toourbestknowledge,thereisnotyetacompletediscussionoftheDOSordHvAoscillationsina3Dsystemasthatina2Dsystem,whereRaichetal.studiedtheDOSdiagrammaticallydifferentlengthsofcorrelations.Therefore,magnetooscillationsina3Dsystemarealsodiscussed.InChapter 2 ,Igiveareviewoncriticalphenomena,anddiscussthespin-fermionmodel,whichiswidelyappliedtoasystemnearaferromagneticQCP.InChapter 3 ,basicprinciplesofmagneticoscillationsarediscussed.WediscusstheformulationofthetheoryofmagnetooscillationsbycomparingtheconventionalLuttinger-wardformulaandthecalculationofthedensityofstates.Weconsidera2DsysteminboththeSCBAregimes(lB)andthequasiclassicalregime(lB).InChapter 4 ,wemakecalculationsofthedensityofstatesandmagnetooscillationswithstaticuctuationsina3Dsystem.InChapter 5 ,wecalculatetheself-energyandthedampingfactorofmagnetooscillationsneartheQCP.Bytakingthetemperaturedependenceofcorrelationlengthsintoaccount,ityieldsthethermaldampingofoscillationamplitudes.Forsimplicity,thePlanckconstant~issettobe1throughoutthethesis. 16 PAGE 17 AShubnikov-deHaasoscillationsasafunctionofB)]TJ /F16 6.974 Tf 6.22 0 Td[(1?forsampleSi6-14/10 BShubnikov-deHaasoscillationsasafunctionofB?forsampleSi6-14/10 CParametersgm,m,andgasafunctionofrsFigure1-1. SpinSusceptibilityandEffectiveMassofElectronsinSiInversionLayer.B?isthemagneticeldappliednormaltothesample.Fig. 1-1A :(a)Electrondensityisn=10:61011cm)]TJ /F6 7.97 Tf 6.59 0 Td[(2,temperatureisT=0:35K,andthemagneticparalleltothesampleisBk=4:5T.;(b)n=9:751011cm)]TJ /F6 7.97 Tf 6.59 0 Td[(2,T=0:35K,andBk=1:5T;(c)n=2:021011cm)]TJ /F6 7.97 Tf 6.58 0 Td[(2,T=0:2K,andBk=0:34T.Thedataareshownasthesolidlines,whilethedashedlinesarethets.AllcurvesarenormalizedbytherstharmonicamplitudeA1.Awell-pronouncedbeatingpatternwasobservedatanonzeroBk.Fromthebeatfrequency,therenormalizedspinsusceptibility/gmcanbedetermined.Fromthedampingofamplitude,mcanbedeterminedusingtheLifshitz-Kosevitchformula.Fig. 1-1B :TheamplitudeofSdHoscillationscanbesignicantlyenhancedbyapplyingBkatsmallB?.Theinsetsshowthetemperaturedependencesoftting(T+TD)m,whereTDistheDingletemperature,asdiscussedinChapter 3 .Fromtheinterceptandslopeofthelineargraph,mandTDcanbedetermined.From(a)and(b),itisalsoseenthattheeffectivemassmisnoteffectedbyBk.Fig. 1-1C :Astheelectrondensityndecreases(rsincreases),g,m,andgmincrease.Thesolidlinein(a)showsthedatabyOkamotoetal.[ 13 ].Nearthe2Dmetal-insulatortransitionwhichoccursatrc8:2,thespinsusceptibility/gmisenhanced.Theopendotsin(b)and(c)wereobtainedbyassumingthatTDisTindependent.Thesoliddotswereobtainedbyattributingthechangeinthermaldampingsolelytothetemperaturedependenceoftheshort-rangescattering,andthereforetheextractedm(rs)isweaker.Thevaluesofgin(c)wereobtainedbydividinggmbym(rs).(ReprintedgurewithpermissionfromRef.[ 14 ].CopyrightbytheAmericanPhysicalSociety,2002) 17 PAGE 18 Figure1-2. TemperaturedependenceofthedeHaas-vanAlphenamplitudeinCeCoIn5withthemagneticeldat6-7T.ThedatashowthatabettertisobtainedbyusingaNFLformbelow20mK,whichisnotdescribedbytheLKformula(inset).However,theextendedLKexpressionwithaNFLformofselfenergy/p T,neartheantiferromagneticQCP,tswell(mainplot).Thisresultissuggestiveonly.(ReprintedgurewithpermissionfromRef.[ 18 ].CopyrightbytheAmericanPhysicalSociety,2005) 18 PAGE 19 CHAPTER2CRITICALPHENOMENA 2.1PhaseTransitionsandOrderParametersMatterwhichexistsindifferentstatesofphasesshowsdifferentsymmetryandhasdifferentmechanical,thermal,orelectromagneticproperties.Thosepropertiesaredeterminedbysomeexternalconditions,suchastemperature,pressure,magneticeld,andelectriceld.Ifthoseexternalconditionsarechanged,atparticularvaluesthesystemcanundergoatransitionfromonephasetoanother,whichiscalledaphasetransition.Thesymmetryofphasesisamacroscopiccollectiveproperty,usuallydescribedbytheorderparameter,whichvanishesinonephase(thedisorderedphase),andisnon-zerointheotherphase(theorderedphase).Anexternaleldwhichcoupleslinearlytothemicroscopicvariable,whoseexpectationvalueistheorderparameteris,calledaconjugateeldtotheorderparameter.AccordingtotheLandauclassication,aphasetransitionisrstorderiftheorderparameterisdiscontinuousatthetransitionpoint,whileitissecondorderiftheorderparameteriscontinuousatthetransition(critical)point.Thesecondorderphasetransitionsaresometimesalsocalledcontinuousphasetransitions.Intheexampleofaparamagnet-ferromagnettransition,theorderparameteristhemagnetizationM,whichisdenedasthethermodynamicaverageofthespinSiby M=hXiSii:(2)TheconjugateeldtotheorderparameterMisthemagneticeldB.ThephasediagramisdepictedinFig.( 2-1 ).AtB=0,ifthetemperatureisloweredtobebelowthecriticaltemperatureTc,thesystemisspontaneouslymagnetizedanditundergoesacontinuousphasetransition.BelowthecriticaltemperatureTc,byapplyingthemagneticeld,thesystemcanbebroughtfromthe"/#statetothe#/"statediscontinuously 19 PAGE 20 AThephasediagramintheB-Tplane BThephasediagramintheM-HplaneFigure2-1. Phasediagramofaferromagnet.AtB=0,ifthetemperatureisloweredtobebelowthecriticaltemperatureTc,thesystemisspontaneouslymagnetizedanditundergoesacontinuousphasetransition,whichischaracterizedbytheorderparameterM(magnetization),theaverageofthemicroscopicspins.Therecanbetwopossibleferromagneticstate,either"stateor#state.Theyareseparatedbythephaseboundary,indicatedbythethicksolidlineinFig. 2-1A .ByapplyingthemagneticeldbelowTc,thesystemcanbebroughtfromthe"/#statetothe#/"statediscontinuouslyandthesystemundergoesarstorderphasetransition. andthesystemundergoesarstorderphasetransition.OtherexamplesoforderparametersaresummarizedinTable( 2-1 ).Inthevicinityofcriticalpoints,thecriticalbehaviorofdifferentsystemsaresimilar.Thisisbecausethecriticalbehaviorofasystemisdescribedbyanintegraloveranorderparameterofanactionwhichitselfisafunctionaloftheorderparameter.Thisremarkablefeatureistermeduniversality.Althoughtheorderparameteriszerointhedisorderedstate,itscorrelation(oructuation)isnon-zeroandbecomelong-rangednearthecriticalpoint,wherethecorrelationlengthandcorrelationtimediverge. 20 PAGE 21 Table2-1. Orderparametersandtheirconjugateelds PhasetransitionOrderparameterConjugateeld Paramagnet-ferromagnetMagnetizationMagneticeldM=hPiSiiBParamagnet-antiferromagnetStaggeredmagnetizationStaggeredeldN=M1)]TJ /F8 11.955 Tf 11.96 0 Td[(M2BsParaelectric-ferroelectricPolarizationElectriceldPEGas-liquidDensitydifferencePressureliquid)]TJ /F3 11.955 Tf 11.96 0 Td[(gasPSuperuidtransitionBosecondensateCondensatesourcehayk=0iSuperconductingCooper-pairamplitudeCooper-pairsourcehck"c)]TJ /F19 7.97 Tf 6.58 0 Td[(k#i 2.2ClassicalCriticalPhenomenaTheclassicalphasetransitionsaredrivenbyacompetitionbetweentheenergyofasystemanditsthermaluctuations.Nearaclassicalcriticalpoint,thermaluctuationsdivergeandphasetransitionsoccur.ThephasetransitioncanbegenerallydescribedbythephenomenologicalGinzburg-Landau(GL)theory[ 33 ].Theapproachisverygeneralandcanbeappliede.g.tomagnetictransitions,superuidity,andsuperconductivity.Itisalsonotnecessarytoattemptamicroscopicderivation.Inmostcases,theformofGLfunctionalisdeterminedfromtheknowledgeofthesymmetryoforderparameters,anditisnotnecessarytoattemptamicroscopicderivation.Givenalocalorderparameter(r),possessingsymmetrywithrespectto(r)!)]TJ /F3 11.955 Tf 9.3 0 Td[((r),andwithtranslationalandrotationalinvariance,theGLfunctionalcanbewrittenas L[(r)]=ZdDrc 2(r(r))2+a 22(r)+b 44(r))]TJ /F3 11.955 Tf 11.95 0 Td[(J(r)(r);(2)wherea,b,andcare-independentconstants,andJ(r)istheconjugateeldof(r).TheintegrationextendsoveraD-dimensionalvolumeLD.Ithastobenoticedthattheorderparameter(r)iswrittenasascalar;however,thiscanbealsoappliedtoavector 21 PAGE 22 orderparameter.Forexample,ifwestartfromaO(3)Heisenbergmodel,wesimplyneedtoreplace(r)byS(r).Thepartitionfunctionofthesystemhastheform Z=Z0(T)ZD[(r)]eL[(r)]=T(2)Thefunctionalintegralreferstoasumoverallpossiblecongurationsof(r)withweightingeL[(r)]=T.ThefactorZ0(T)isduetotheshort-wavelengthcongurationswhichdonotcontributeto(r).Themostprobablecongurationof(r)isgivenbythestationary-phaseapproximationwhichisdeterminedby 0=L (r)=a+b2(r))]TJ /F3 11.955 Tf 13.56 8.08 Td[(c 2r2(r))]TJ /F3 11.955 Tf 11.95 0 Td[(J(r);(2)and 2L [(r)]2=a+3b2(r)>0:(2)IntheabsenceoftheexternalconjugateeldJ(r),theuniformsolution(r=0)ofEq.( 2 )is =0;r )]TJ /F3 11.955 Tf 9.29 0 Td[(a b:(2)Thesolution=0standsforthedisorderedstate,whereT>Tc.Insertionof=0intoEq.( 2 )yieldsa>0whenT>Tc.Ontheotherhand,thesolutions=p )]TJ /F3 11.955 Tf 9.3 0 Td[(a=bstandfortheorderedstates,whereT PAGE 23 whereekandeJkaretheFouriertransformsof(r)andJ(r).ThesummationoverkisrestrictedtotherstBrillouinzone.Thequartictermb4=4isconsideredsmallandneglectedinthemean-eldapproximation.Therefore,thepartitionasafunctionoftheconjugateeldtakestheformZ[eJ]=Z0ZYkdekeL=T=Z0Ykdeke)]TJ /F20 7.97 Tf 6.59 11.2 Td[((a 2+c 4k2)eke)]TJ /F23 5.978 Tf 5.76 0 Td[(k)]TJ /F20 7.97 Tf 8.33 2.01 Td[(eJke)]TJ /F23 5.978 Tf 5.76 0 Td[(k=T=Cexp"Xk1 2a+ck2eJkeJk#; (2)whereCisaeJk-independentconstant.Hence,thecorrelationfunctionW(k)readsW(k)heke)]TJ /F19 7.97 Tf 6.59 0 Td[(k0i=@2lnZ[eJ] @eJ)]TJ /F19 7.97 Tf 6.59 0 Td[(k@eJk0J=0=kk02 2a+ck2/kk01 )]TJ /F6 7.97 Tf 6.59 0 Td[(2+k2; (2)whereisthecorrelationlength.Therefore,/p c=a/1=p T)]TJ /F3 11.955 Tf 11.95 0 Td[(Tc.Thecorrelationfunctiondivergesatsmallkasthetemperatureapproachesthecriticaltemperature,whichmeansthatthecorrelationsbecomemoreandmorelong-rangedasTcisapproached.Foramagneticphasetransition,thecorrelationfunctionisthespinsusceptibility. 2.3QuantumCriticalPhenomenaAquantumphasetransitionisthecontinuousphasetransitionthattakesplaceatzerotemperature.Incontrasttoclassicalphasetransitionsthataredrivenbythermaluctuations,thequantumphasetransitionsaredrivenbyquantumuctuationsassociatedwiththeHeisenberg'suncertaintyprinciple.Themean-eldGLtheoryisnotvalidneartheQCP,Besides,statisticsanddynamicsareinextricablyconnectedandshouldbetreatedtogetherneartheQCP.SomeresultsaboutcorrelationlengthsnearaD-dimensionalitinerantferromagneticorantiferromagneticcriticalityweredevelopedbyHertz,Millis[ 39 ],andMoriya[ 40 ]withtherenormalizationgroupapproach 23 PAGE 24 Figure2-2. PhasediagramneartheQCP.NeartheQCP,therecreatesa`V-shaped'phaseinthephasediagram,calledthequantumcriticalregime,wherethemassisstronglyrenormalizedandthesystemmayshowaNFLbehavior. .Forexample,nearaferromagneticcriticalityina2Dsystem,thecorrelationlength/1=p TlnT.Nearantiferromagneticcriticalityin2D,/p lnT=T[ 39 ].ThetypicalphasediagramnearthequantumcriticalityinanitinerantelectronsystemisshowninFig.( 2.3 ).Theorderedphasecanferromagnetism,antiferromagnetism,etc.Unliketheclassicalcriticalpoint,wherethethermalcriticaluctuationstakeplaceinanarrowregionaroundthephasetransition,therecreatesa`V-shaped'phasenearQCPinthephasediagram,calledthequantumcriticalregime,andtheeffectofquantumcriticalityisfeltevenifthetemperatureisnotreachingabsolutezero.Alotofmaterials,suchasferromagnetsandantiferromagnets,havebeenobservedtoshowquantumcriticalbehavior,whereitinerantelectronsinthesystemshowNFLbehavior.Ina2Dsystem,thosequantumuctuationsdecaymoreslowlywithdistancesandthereforeinteractionsarestronger.Asaresult,manyoftheexperimentsonquantumcriticalphenomenaareonlayeredcompounds. 24 PAGE 25 2.4Spin-FermionModelTounderstandthephysicalpropertynearthequantumcriticality,Weconsiderthephenomenologicalspin-fermionmodel,whichdescribeslow-energyitinerantelectronsinteractingwithlong-wavelengthcollectivebosonicexcitationsineitherspinorchargechannels.ThegeneralapproachtoderiveistostartwiththeHubbardmodel,inwhichelectronsinteractwithashortrangefour-fermioninteractions,decouplethefour-fermioninteractionsinauxiliaryeldsbyperformingtheHubbard-Stratonovichtransformation,andintegrateoutthehigh-energyfermions[ 34 ].Neartheferromagneticquantumcriticality,theHamiltonianinzeromagneticeldcanbedescribedby[ 35 ] H=Hf+Hb+Hint;(2)whereHf=Xk;"kcyk;ck;Hb=X)]TJ /F6 7.97 Tf 6.58 0 Td[(1sSqS)]TJ /F19 7.97 Tf 6.58 0 Td[(qHint=gXk;q;;cyk;;ck+q;Sq: (2)HfistheelectronicHamiltonian,HbisthebosonicHamiltonianwhichdescribesspinuctuations,andHintdescribesfermionsinteractingwiththecollectivespinexcitationsSq.Thismodelisnotrestrictedtotheferromagneticinstabilitywithvectorspinuctuations.ItcanbealsoappliedtoaQCPinthechargechannel(e.g.aPomeranchukinstabilitytowardsnematicordering)oraferromagneticinstabilityinsystemswithIsingsymmetry,byreplacingthebosonicvectoreldSwithascalareld.AsAPPENDIX A shows,thestaticspinsusceptibilitysofaquasi-2Delectronsystemisgivenby s(q)=0 )]TJ /F6 7.97 Tf 6.58 0 Td[(2+q2?+tk v?2(1)]TJ /F7 11.955 Tf 11.96 0 Td[(cos(qzb));(2) 25 PAGE 26 whereisaconstantwhichdependsonhowtheDOSvarieswiththeenergy,isthecorrelationlength,andtkistheinter-planecoupling.Thespin-fermioncouplingg1===mnearthecriticalpoint.Ina3Dhomogeneoussystem, s(q)=0 )]TJ /F6 7.97 Tf 6.59 0 Td[(2+q2:(2)Thespin-fermioncouplingg1==2=p m3EF=2nearthecriticalpoint.Aftertakingthedynamicuctuationsintoaccount,thefullbosonicpropagatoratthelowestordertothespin-fermioninteractionneartheQCPisdescribedby(q;)=0=(0)]TJ /F6 7.97 Tf 6.59 0 Td[(1s+d(q;jj)),wherethedynamicpolarizationdisgivenbyd(q;)=g20((q;))]TJ /F7 11.955 Tf 11.96 0 Td[((q;=0))=)]TJ /F7 11.955 Tf 9.71 0 Td[(gX!Zd3k (2)3[G0(k+q;!+)G0(k+q;!))]TJ /F3 11.955 Tf 11.96 0 Td[(G0(k+q;!)G0(k+q;!)]; (2)whereg=g20,andisthebosonicMatsubarafrequency.Inaquasi-2Dsystem,thedynamicpolarizationbubbleiscomplicated.Itdependson,q?,qk,andtk.Thestrengthofinter-planecouplingtkdetermineswhetherthesystemis2D-likeor3D-like. 2.5AnExampleandStartingPoint:Self-EnergyNearTheTwo-DimensionalFerromagneticCriticalPointBeforeconsideringmagnetooscillations,itisinstructivetogiveashortsurveyoftheself-energyofa2Dferromagneticsystemintheabsenceofthemagneticeld.ThedynamicbubbleEq.( 2 )reads[ 35 37 ]d(q;)=)]TJ /F7 11.955 Tf 9.72 0 Td[(gZd2k (2)2i(n("k+q))]TJ /F3 11.955 Tf 11.96 0 Td[(n("k)) ("k+q)]TJ /F3 11.955 Tf 11.95 0 Td[("k)]TJ /F3 11.955 Tf 11.95 0 Td[(i)("k+q)]TJ /F3 11.955 Tf 11.96 0 Td[("k)=)]TJ /F3 11.955 Tf 9.3 0 Td[(img Zd"kZd 2("k) "k+q)]TJ /F3 11.955 Tf 11.95 0 Td[("k)]TJ /F3 11.955 Tf 11.95 0 Td[(i=mg jj p (qvF)2+2: (2) 26 PAGE 27 Figure2-3. Thelowest-orderdiagramforthefermionicself-energy(!;T;k) SubsequentanalysiswillshowthatqvF/jj1=3;)]TJ /F6 7.97 Tf 6.59 0 Td[(1jj,andtherefore, (q;)=jj q;(2)where=mg=vF.ThisiscorrectintheballisticregimewherethetypicalvalueofqvFismuchlargerthantheimpurityscatteringrate1=.Inthediffusiveregime,qvF1=,thedynamictermjj=qisreplacedby2jj=vFq2,andthespin-fermioncouplingisstronglydressedbytheimpurityscattering[ 41 42 ],whichisnotconsideredbelow.However,itwillbeshowninthenextchapterthatonlythestaticuctuationsmakecontributionstotheeffectivedeHaas-vanAlphen(dHvA)mass,andthereforeitisnotanissuewhetherthesystemisdiffusiveorballistic.TothelowestorderinFig.( 2-3 ),thefermionicself-energyatMatsubarafrequency!isgivenby1(!;T;"k)=3gT 0XZd2q (2)2G0(k+q;!+)(q;)=3gTXZd2q 421 i(!+))]TJ /F3 11.955 Tf 11.95 0 Td[("k)]TJ /F8 11.955 Tf 11.95 0 Td[(qvF1 q2+)]TJ /F6 7.97 Tf 6.59 0 Td[(2+jj q=)]TJ /F3 11.955 Tf 9.3 0 Td[(i3gT 2XZ10dqq p (qvF)2+(!++i"k)2sgn(!+) q2+)]TJ /F6 7.97 Tf 6.59 0 Td[(2+jj q; (2)wherethesubscriptoftheself-energymeansthatonlyonebosonicpropagatorisincluded. 27 PAGE 28 Sinceweareconsideringlowenergyexcitations,itisinstructivetoseparatetheself-energyintothefrequency-andmomentum-dependentpartsas1(!;T;"k)=1(!;T)+("k)where1(!;T)=1(!;T;"k=0)and1("k)=1(!=0;T;"k) (2)Inthezero-temperaturelimit,theMatsubarasumcanbereplacedbythefrequencyintegral.At)]TJ /F6 7.97 Tf 6.58 0 Td[(1(!)1=3(or!1=3),thesystemisinthequantum-criticalregime.Thetypicalbosonicmomentumtransferq()1=3ismuchlargerthanthefermionicfrequency!,since!=vF(!)1=3(or!p g"Fequivalently)isalwaysassumedforlowenergyexcitation.Thismakesthefrequency-dependentpartoftheself-energyas[ 35 37 ]1(!)=3g 42vFZ!)]TJ /F4 7.97 Tf 6.58 0 Td[(!dZ10dqq q3+jj=)]TJ /F3 11.955 Tf 9.3 0 Td[(i!1=30j!j2=3sgn(!);where!0=3p 3g2 162EF: (2)ThisresultshowsthattheinteractionsbetweenfermionsandbosonicuctuationsleadtoabreakdownoftheFLbehavior.Itcanbeobservedthat1issmallerthanthebarefermionicfrequency!if!>!0,wherethesystembehavesasanearlyfreeFermigas.Therefore,theNFLformofself-energyinEq.( 2 )isonlyrelevantforfrequencies! PAGE 29 Weareinterestedinthequantumcriticalregime,wheretheself-energyhasaNFLform.Bytakingthetemperaturedependenceintoaccount,thedynamicpartofEq.( 2 )canbegeneralizedas 1;dyn(!;T)=)]TJ /F3 11.955 Tf 9.29 0 Td[(i!1=30T2=3f(!=T)sgn(!);(2)wherethescalingfunctionfsatisestheboundaryconditionsthatfj!=T!1=(j!j=T)2=3andfj!=T!0.ThelatterconditioncanberoughlyseenfromEq.( 2 )thatdynamiccontributionsfromeachpositivebosonicfrequencymostlycancelwiththeircorrespondingnegativefrequencycontributionas!=T.Thesubscriptdynofmeansthatthisresultexcludesthestaticterm,=0,ofthebosonicpropagator.Themomentum-dependentpartofself-energycanbeobtainedbytakingtheTaylorexpansionofEq.( 2 )[ 35 ].NeartheQCP,!1=3,and)]TJ /F6 7.97 Tf 6.58 0 Td[(2isnegligibleinthebosonicpropagator.TheMatsubarasumcanbealsoapproximatedbythefrequencyintegral.Therefore,tothelinearorderin"k,1("k)=)]TJ /F3 11.955 Tf 9.3 0 Td[(i3g (2)2ZdZ10dqsgn() p (vFq)2+(+i"k)2q q2+jj q=)]TJ /F7 11.955 Tf 10.5 8.08 Td[(3g"k 22Z10dZ10dqjj ((vFq)2+2)3=2q2 q3+jj: (2)Theresultdependsontheinterplaybetween=vFand(jj)1=3.Therefore,itisconvenienttointroduceascale!max=p v3F=p 2gEF=,denedasthefrequencyatwhich=vF=(jj)1=3.Thisfrequencywellexceeds!0g2=EF,whichsetstheupperlimitoffrequencyinthequantumcriticalregime.Therefore,at!=!max,thesystembehaveslikeanearlyidealFermigasand1(!)=!1.Introducingtherescaledvariablesasy=q=(!max)1=3andx==!max,Eq.( 2 )yields1("k)=)]TJ /F7 11.955 Tf 9.3 0 Td[(3g!max 22v3F"kZ10dxZ10dyxy2 (x2+y2)3=2(x+y3))]TJ /F7 11.955 Tf 21.92 0 Td[(0:087p g="F"k: (2) 29 PAGE 30 Thus,themomentum-dependentpartofself-energyonlyleadstoaniteregularrenormalizationoftheeffectivemass.Sofar,wecomputedthefermionicandbosonicself-energiesusingaperturbationexpansionaroundfreefermions.However,itturnsouthigherorderexpansionsaroundfreefermionsdonotconvergeneartheQCP.Therefore,acontrollableandconvergentexpansionhastobedeveloped.ThefullconsiderationswerediscussedbyRechetal.[ 35 ]withtheEliashbergprocedurebysolvingtheself-consistentDyson'sequations.Inthistheory,boththevertexcorrectionsandthemomentum-dependentpartofthefermionicself-energyareneglected,andthefreefermionicGreen'sfunctionG0isreplacedbythefullGreen'sfunctionGas G)]TJ /F6 7.97 Tf 6.59 0 Td[(1(k;!)=i!)]TJ /F8 11.955 Tf 11.96 0 Td[(vF(k)]TJ /F8 11.955 Tf 11.96 0 Td[(kF))]TJ /F7 11.955 Tf 11.95 0 Td[((!):(2)Thebosonicpropagator(q;)isalsocalculatedwithfullfermionicGreen'sfunctions.Theyshowedthevertexcorrectionstotheself-energyareoforderO(1)andarenotparametricallysmall.However,ifthenumberoffermionicavorsNisconsideredlarge,thosecorrectionsbecomeO(1=N)1andtheEliashbergtheoryworks(1=Nexpansion).AnotherconditionforthevalidityoftheEliashbergtheoryistorequireg=NEF1,whichisquitegenericforlow-energyexcitations.Itturnsoutthatthelowestorderexpansionoffermionicself-energyintheEliashbergtheorycoincideswiththeone-loopfermionicself-energy1,whichisexpandedaroundthenon-interactingfermions. 2.5.1IssueNumberOne:DynamicorStaticSelf-Energy?Sofar,weonlyfocusedonthedynamicpartoftheself-energy.Normally,thisisthestartingpointthatpeopleusedtostudymagnetooscillationsneartheQCP[ 18 29 ].ThesystemisthenconsideredtoshowtheNFLbehaviornearthePomeranchukorferromagneticinstabilityiftheoscillationamplitudesdecaywiththeexponent2i(!;T)=!c/T2=3.Thestaticpart(=0)ofself-energy1;st,however,canbe 30 PAGE 31 veryimportantduetothestrongdivergenceofstaticuctuationsneartheQCP,where!1.Thismakestheself-energyatnitetemperatureverydifferentfromthatatzerotemperature.Inthefollowingchapters,itisshownthatthethermaldampingofmagnetooscillationamplitudesindeeddependsonthestaticpartofself-energyratherthanitsdynamicpart,insometemperaturerange.Evenmore,intemperatureswhere22T&!c,thethermaldampingofmagnetooscillationsispurelydeterminedby1;steventhoughthedynamicself-energycanbelarger.Itisapparentlythatthestaticpartofself-energy1;sttakestheformofEq.( 2 ).However,thetemperature-dependentcorrelationlengthshouldbetakenintoaccountaswell.Inthequantumcriticalregimeneara2DferromagneticorPomeranchukQCP,thecorrelationlengthhasbeenshownbytherenormalizationgroupapproachtobe)]TJ /F6 7.97 Tf 6.59 0 Td[(1(T)=p g0T=vF,whereg0=g0(T)=~g0lnT0 T[ 39 ],~g0andT0areconstantwithdimensionsofenergy,andthisrelationholdsonlyforT PAGE 32 WhileapproachingtheQCP,thecorrelationlengthgetslongerandlonger,eachelectronstartstoseethecurvatureofitstrajectoryoncethecorrelationlengthisofsameorderofthemagneticlengthlB.IfthecorrelationlengthislongerthanthecyclotronlengthRc,theeffectofmagneticeldonthetrajectoryofelectronsareevenmorepronouncedandthereshowsaninhomogeneousbroadening[ 30 43 ].Theself-energywithouttakingthemagneticeldintoaccountisthususefulformagnetooscillationsonlywhenthecorrelationlengthissufcientlyshort. 32 PAGE 33 CHAPTER3MAGNETOOSCILLATIONSANDDENSITYOFSTATES 3.1IntroductiontoMagnetooscillationsThemagnetooscillations,deHaas-vanAlphen(dHvA)oscillationsorShubnikov-deHaas(SdH)oscillations,describethephenomenonthatthemagnetizationofasystemoscillatesasafunctionof1=Bwiththeperiod 1 B=2e Ae;(3)whereAeisthecross-sectionalareaoftheextremalorbitsontheFermisurfaceinaplanenormaltothemagneticeld[ 4 ].ThephenomenonispurelyquantumandiscloselyrelatedtotheoscillatorystructureoftheelectronicDOS,whichisattributedtotheLandauquantizationofenergylevelsinamagneticeld.Inthefree-electronmodel,thespacingbetweenenergylevelsinamagneticeldis!c=eB=m,where!cisthecyclotronfrequency(energy).Thetypicalvalueof!cis10)]TJ /F6 7.97 Tf 6.59 0 Td[(4eVifB=1Tesla.However,typicalchemicalpotentialisseveralelectronvolts.Therefore,thequantumnumbernneartheFermisurfaceis n=!c104=B(Tesla)1:(3)Onsager[ 4 ]thereforegeneralizedthefreeelectronresultstoBlochelectronsbyconsideringtheBohr'scorrespondenceprinciple,whichisvalidforlargequantumnumbers.Whenauniformmagneticeldisappliedalongthezdirection,theelectronicenergytakestheform "n(kz)="n+"z;(3)where"nistheenergyoftheLandaulevel,and"zistheenergyduetheelectronicmotionalongthezdirection.Forexample,"n=(n+1=2)!c)]TJ /F3 11.955 Tf 12.48 0 Td[(and"z=k2z=2minahomogeneous3Dfree-electronsystem.TheLorentzforceequation_k=)]TJ /F3 11.955 Tf 9.3 0 Td[(evByields_k?B,whichmeanskzisaconstantwhiletheelectrontraversestheorbit.Sincekzis 33 PAGE 34 A B CFigure3-1. Pictorialexplanationofmagnetooscillations:Energylevelsarequantizedintheplaneperpendiculartotheappliedmagneticeld,whichisalongthezdirectioninthegure.Foraagivenquantumnumbern,theclassicalorbitsofdifferentkzformaLandautube(red)ofcross-sectionalarea(n+)Ainthemomentumspace.TuningthemagneticeldcanchangethecrosssectionsofLandautubes.WheneveraLandautubestronglyoverlapswiththeFermisurfaceofthickness",theDOSwillhaveasharppeak.In( 3-1A )and( 3-1C ),theorbitsofconstantenergyontheLandautubeareextremalorbits.Therefore,thereisalargeportionofoverlapbetweentheLandautubeandtheshellaroundtheFermisurface.In( 3-1B ),theorbitsarenotextremalandtheportionofoverlapisnotenhancedasin( 3-1A )and( 3-1C ).Thiscanbealsounderstoodasfollows.Ontheextremalorbits,@"n(kz)=@kz=0,andtherefore,kz=p 2mz",wheremzisthebandmassdenedby1=mz=@2"=@k2z.Ontheotherhand,fornon-extremalorbits,kz="=vzp 2mz". aconstantofthesemiclassicalmotion,weapplythecorrespondenceprincipletolevelswithaspeciedkz,andquantumnumbersnandn+1.Itgives "n(kz)"n+1(kz))]TJ /F3 11.955 Tf 11.95 0 Td[("n(kz)=1 T("n(kz);kz);(3) 34 PAGE 35 whereT(";kz)istheperiodofthesemiclassicalorbitalmotiongivenby[ 44 ]T(";kz)=1 eB@A(";kz) @"; (3)andA(";kz)isthecross-sectionalareaoftheclosedorbit.Therefore,forgivenkz,thedifferenceoftheenclosedareasfortwoadjacentLandaulevelsisA("n(kz);kz)A("n+1(kz);kz))]TJ /F3 11.955 Tf 11.96 0 Td[(A("n(kz);kz)("n+1)]TJ /F3 11.955 Tf 11.96 0 Td[("n)@A(";kz) @""="n=2eB: (3)Asaresult,theareaenclosedbythesemiclassicalorbitatlargendependsonnaccordingto A("n(kz);kz)=(n+)A;(3)whereisroughlyaconstantbetween0and1.Inafreeelectronsystem=1=2.ItcanbeseenfromEq.( 3 )thatthespacingbetweenadjacentLandaulevelsincreaseswhentheappliedmagneticeldincreases.Foraagivenquantumnumbern,theclassicalorbitsofdifferentkzformaLandautubeofcross-sectionalareasatisfyingEq.( 3 ).Whenever"n(kz)isclosetotheenergyofanextremalorbit,theLandautubewillhavealargeoverlapwiththeFermisurface,andthereforetheDOSwillhaveasharppeak.Ontheotherhand,forthenon-extremalorbits,theoverlapissmallandthereforetheenhancementisweak.ThoseareillustratedinFig.( 3-1 ).Givenxedchemicalpotential,thecross-sectionalareaisxed.However,fordifferentmagneticelds,Aaredifferent.Therefore,wecanformulatearelationbetweendifferentmagneticeldsB1andB2,withquantumnumbersnandn+1ontheextremalorbitrespectively.Itreads Ae=(n+)A(B1)=(n++1)A(B2):(3) 35 PAGE 36 UsingEq.( 3 ),thisyields 1 B=1 B2)]TJ /F7 11.955 Tf 17.04 8.09 Td[(1 B1=2e Ae:(3)Theextremalcross-sectionalareaoftheFermisurfaceinaplaneperpendiculartothemagneticeldcanbethereforedeterminedbymeasuringtheperiodofmagnetooscillationswithrespectto1=B.Byrotatingthesampleoralteringthemagneticelddirection,theextremalareasoftheFermisurfacealongalldifferentdirectionscanbemappedout,andthereforetheshapeofFermisurfacecanbereconstructed. 3.2TheoryofMagnetooscillations-Overview 3.2.1IntroductionIntheprevioussection,weusethesemiclassicaltreatmenttoexplaintheoscillationsofmagnetization.ThedHvAeffectisacommontoolforstudyingthegeometryofFermisurfacesinmetals.Moreover,italsoallowsonetomeasuretherenormalizedeffectivemassviathethermaldampingofoscillationamplitudes.TherstquantitativetheoryofthedHvAeffectwasgivenbyLifshitzandKosevitch(LK)in1956[ 5 ].Theyconsideredanarbitraryenergyspectrumina3Dsystem.ThemainresultoftheLKtheoryisthattheoscillatorypartofthethermodynamicpotentialinamagneticeldBcanbeexpressedas =e 23=2eB5=2 mc2@2Ae @k2z)]TJ /F6 7.97 Tf 6.58 0 Td[(1=21Xr=1r)]TJ /F6 7.97 Tf 6.58 0 Td[(5=2RTRcos2rAe 2eB)]TJ /F3 11.955 Tf 11.95 0 Td[( 4;(3)where RT=22rT=!c sinh(22rT=!c)(3)isathermaldampingfactor,!c=eB=mcisthecyclotronfrequencywiththecyclotronmassmc=(@Ae=@")=2,andtheZeemantermis R=cosrmc me;(3) 36 PAGE 37 wheremeisthefree-electronmass.ThistermcomesfromtheenergydifferenceeB=meofspin-upandspin-downelectronsinamagneticeld.Inafree-electronsystem,Ae=k2F=2meandthereforemc=me.Rthusbecomes()]TJ /F7 11.955 Tf 9.3 0 Td[(1)r.ThemagnetizationcanbefoundusingathermodynamicrelationM=)]TJ /F3 11.955 Tf 9.29 0 Td[(@=@B.ItcanbeclearlyseenintheLKformula(Eq.( 3 ))thattheperiodofdHvAoscillationsmeasuredwithrespectto1=Bis2e=Ae.Thisisconsistentwiththeargumentgivenintheprevioussection.However,LKformuladoesnottakeintoaccounttheinteractionsofelectronswithimpurities,phonons,andotherelectrons.ThedHvAeffectwaslaterappliedtoinvestigatetheeffectofelectroninteractionswithimpuritiesandphonons,whichresultinthedampingofoscillations.Interactionswithimpuritiesgaveanadditionaldampingterm RD=exp)]TJ /F3 11.955 Tf 13.1 8.09 Td[(r !c=exp)]TJ /F7 11.955 Tf 10.49 8.09 Td[(22rTD !(3)intotheLKformula,where1=2wassimplytheimaginarypartofretardedself-energy,andTD=1=2iscalledDingletemperature[ 6 ].Theeffectofelectron-phononinteractions,however,turnedouttobemoresubtle.WilkinsandWoo[ 7 ]demonstratedthatthecyclotroneffectivemasswhichappearsinthethermaldampingfactorwillberenormalizedbytheelectron-phononinteraction.Contrarytothesimplelifetimeargument,whichsuggeststhattheDingletemperatureisstronglydependentontemperatureas1=2/T3forelectron-phononinteractions.However,thiscontradictedtoexperimentsonsomestronglyelectron-phononcouplingsystem,suchasmercury[ 45 ],theDingletemperatureofwhichwasfoundtobetemperatureindependent.Thisindicatesthatthesimplelifetimeargumentsarenotsatisfying.AmoresatisfactorytreatmentistoapplytheeldtheoreticapproachwithMatsubaraGreen'sfunctions[ 8 46 50 ].Thisapproachusuallystartsfromtheso-calledLuttinger-Wardformula[ 46 ].Forafree-electronenergyspectrum"k=k2=2m)]TJ /F3 11.955 Tf 12 0 Td[(withMatsubaraself-energy(!)dueto 37 PAGE 38 interactions,itcanbeshownthat[ 48 50 ] =)]TJ /F7 11.955 Tf 10.49 8.09 Td[(2(m!c)3=2 2T1X!=T1Xr=1r)]TJ /F6 7.97 Tf 6.59 0 Td[(3=2Rexp)]TJ /F7 11.955 Tf 10.5 8.09 Td[(2r !c[!)]TJ /F26 11.955 Tf 11.96 0 Td[(Im(!)]cos2r !c[)]TJ /F26 11.955 Tf 11.96 0 Td[(Re(!)])]TJ /F7 11.955 Tf 13.15 8.09 Td[(1 4;(3)where!=(2n+1)TareMatsubarafrequencies.Ifallofself-energycorrectionsareneglected,Eq.( 3 )recoverstheLKformulaexactly.Therefore,Eq.( 3 )issometimesreferredtoastheextendedLifshitz-Kosevich(ELK)formulafordHvAoscillations. 3.2.2FirstApproach:theLuttinger-WardFormulaTheLuttinger-Wardformula[ 46 50 ]gives =)]TJ /F3 11.955 Tf 9.3 0 Td[(TX!Trln)]TJ /F2 11.955 Tf 5.48 -9.68 Td[()]TJ /F3 11.955 Tf 9.3 0 Td[(G(!;))]TJ /F6 7.97 Tf 6.59 0 Td[(1(3)fortheoscillatorypartofthethermodynamicpotential.Thetraceistakenoverallquantumnumbers,suchasmomentum,Landaulevels,etc..ThisformulahasbeenwidelyappliedtostudythedHvAeffect.However,theelectronself-energyinthistreatmentiscalculatedintheabsenceofthemagneticeld[ 46 50 ].Intheforthcomingdiscussion,itwillbeshownthattheself-energyisindependentofthemagneticeldonlyifthecorrelationlengthoftheinteractionisshorterthanthemagneticlength.Thiscanbeunderstoodasfollows.Acharacteristicenergyscaleassociatedwiththecorrelationlengthis1=m2,andthespacingofLandaulevelenergyis!c=1=ml2B.Therefore,electronswillfeeltheenergyquantizationduetothemagneticeldis1=m2<1=ml2B,whichis>lB.Asaresult,theself-energydependsonthemagneticeldforlongcorrelationlengthandtheLuttinger-Wardformulabreaksdownif>lB.Nearaquasiclassicalorquantumphasetransition,theLuttinger-Wardformulaisguaranteedtofailandanewapproachisrequired. 3.2.3SecondApproach:DensityofStatesAnotherapproachtothedHvAeffectistocalculatetheDOSandthereforetheelectrondensityinamagneticeld.OncetheelectrondensityNisfound,boththe 38 PAGE 39 thermodynamicpotential=)]TJ /F10 11.955 Tf 11.29 9.64 Td[(RNdandthemagnetizationM=)]TJ /F3 11.955 Tf 9.3 0 Td[(@=@Barenaturallygiven.Itcanbeobservedthat=)]TJ /F10 11.955 Tf 11.29 9.63 Td[(RNd=)]TJ /F3 11.955 Tf 9.3 0 Td[(TP"TrRGd=)]TJ /F3 11.955 Tf 9.3 0 Td[(TP"Trln()]TJ /F3 11.955 Tf 9.3 0 Td[(G)]TJ /F6 7.97 Tf 6.59 0 Td[(1),whichisactuallythesameastheLuttinger-Wardformula.However,forexample,inthecaseoflong-rangepotentialwithlB,theGreen'sfunctionofelectronsinamagneticeldisverycomplicated,andtoourbestofknowledge,nobodyhasevercalculateditasyet.Therefore,itisimpossibletoapplytheLuttinger-Wardformulatostudythemagnetooscillationswithlong-rangeinteractions.However,theDOS,orthetraceoftheimaginarypartofGreen'sfunctionscanbederivedbyresummationofFeynmandiagrams[ 30 ]orbyapath-integraltreatment[ 43 ].Thus,tondtheDOSofasystemisaneasierandmoreapplicableapproachtostudymagnetooscillations. 3.3DensityofStatesinHighLandauLevelsofaTwo-DimensionalDisorderedSystemAsmentionedintheprevioussection,tostudymagnetooscillations,itiseasierandmoreapplicabletostartbyconsideringtheDOSofthesystem.ThissectiongivesasketchyoverviewofDOSinhighLandaulevelsofa2Ddisorderedelectronsystem.Ithastobeemphasizedwhatitmeansbydisorderhereisnotrestrictedtothepotentialdisorder,whichisnotofourinterest.Weareinterestedintheorder-parameterdisorder.Forclassicalphasetransitions,theorder-parameterdisorderisstatic.Forquantumphasetransitions,theorder-parameterdisorderhasbothstaticanddynamicpart.Inthissection,wearedescribingthestaticpartofdisorders.Whenamagneticeldisappliedtoanelectronsystemalongthezdirection,theenergyisquantizedinthexyplanewhilethespectrumalongthezdirectionisnotchanged.Physically,thisisbecausethemagneticforceisalwaysperpendiculartothemagneticeld,andcannotchangethemotionalbehavioralongthemagneticeld.Therefore,tostudytheeffectofmagneticeldsinanelectronsystem,itissufcienttolookata2Dsystem.A3Doraquasi-2Dsystemcanbetheneasilygeneralizedby 39 PAGE 40 addingtermswhicharerelatedtothespectrumofasystemonly,butarenotaffectedbythemagneticeld.Weconsideradisorderedsystemasanexample,butthiscanbeappliedtoanysystemswhichhasstaticcorrelationasthedominantinteractions.Throughoutthearticle,forsimplicity,wedonottakespinsofelectronsintoaccountbecausethosenormallyjustcaseanphasefactorR=()]TJ /F7 11.955 Tf 9.3 0 Td[(1)rasshowninEq.( 3 ),whichisnotofourinterest.Whenauniformmagneticeldisappliednormaltoa2Dhomogeneouselectronsystem,theelectronicenergytakestheform "n=n+1 2!c)]TJ /F3 11.955 Tf 11.96 0 Td[(;(3)where!cisthecyclotronfrequencyeB=m,andisthechemicalpotential.Thediscussioncanbeseparatedintotworegimes.Intheshort-rangecorrelationregimelB,theSCBAisconsidered[ 30 54 ].Forthelong-rangescatterers,SCBAfailsandtheDOScanbederivedbyresummingleadingsequencesofFeynmandiagrams[ 30 ]orbythequasiclassicalpathintegraltreatment[ 43 ]. 3.3.1DensityofStatesforShort-RangeCorrelations,withlBThissectionbasicallyfollowstheproceduregivenbyRaikhandShahbazyan[ 30 ].Theone-loopself-energyisshownintherstdiagraminFig.( 3-2 ).WiththechoiceofLandaugaugeA=(0;xB;0),theretardedself-energyreads 1;R(!;n;ky)=hnkyjVG0Vjnkyi=1 VsXn0;k0yXqW(q)hn;kyjeiqrjn0;k0yi2 !)]TJ /F3 11.955 Tf 11.96 0 Td[("n0;(3)whereVsisthesamplesize,W(q)istheFouriertransformofthecorrelatoroftheorder-parameterdisorderfW(jr)]TJ /F8 11.955 Tf 11.95 0 Td[(r0j)=hV(r)V(r0)i,andqisthemomentumtransferofelectronsinthexyplane.Thematrixelementhn;kyjeiqrjn0;k0yi2hastheformhn;kyjeiqrjn0;k0yi2=2ky;k0y+qyjAnn0(q?lB)j2; (3) 40 PAGE 41 wherejAnn0(qlB)j2=e)]TJ /F4 7.97 Tf 6.59 0 Td[(q2l2B=2N!q2l2B 2jn)]TJ /F4 7.97 Tf 6.58 0 Td[(n0jLjn)]TJ /F4 7.97 Tf 6.59 0 Td[(n0jN PAGE 42 ForLandaulevelsneartheFermisurface,withfn;n0g1,Ann0(qlB)canbeapproximatedbytheBesselfunctionJjn)]TJ /F4 7.97 Tf 6.59 0 Td[(n0j(qRc)[ 30 ].NotethatthisasymptoticapproximationisvalidiftheargumentoftheBesselfunctionislessthann[ 30 ].Therefore,onegetsqn=Rc1=F,whereFistheFermiwavelength.Sincethetypicalvalueofqisoftheorderof)]TJ /F6 7.97 Tf 6.59 0 Td[(1,itdemandsthatF.Equation( 3 )thustakestheformR(";n)=Xn0Zd2q (2)2W(q)J2jn)]TJ /F4 7.97 Tf 6.59 0 Td[(n0j(qRc) ")]TJ /F3 11.955 Tf 11.96 0 Td[("n0)]TJ /F7 11.955 Tf 11.96 0 Td[(R(";n0): (3)ThisisgenerallythestartingpointofndingtheDOSintheSCBAregime.Afterndingtheself-energy,theDOSg(")isgivenbyg(")=Xngn(")=)]TJ /F7 11.955 Tf 21.77 8.09 Td[(1 22l2BXnImGR(";n)=)]TJ /F7 11.955 Tf 21.78 8.09 Td[(1 22l2BXnIm1 ")]TJ /F3 11.955 Tf 11.96 0 Td[("n)]TJ /F7 11.955 Tf 11.96 0 Td[(R(";n): (3)IthastobeemphasizedthattheSCBAonlyworksinshort-rangeimpuritieswithlB.Forexample,inFig.( 3-2 ),theself-interactingcrossdiagramisnotcontainedintheSCBA.ItwasshownbyRaikhandShahbazyan[ 30 ]thatthosediagramswithself-interactionsmakethesamecontributionofself-energyastheSCBAdiagramsinlong-rangimpurities,wherelB.Therefore,aresummationofdiagramsarerequiredtoyieldtheexpressionfortheDOS.Physically,thiscanbeunderstoodasfollows.Inthecaseofshort-rangeimpurities,thephasespaceofthecrossingdiagramsisstronglysuppressedrelativetothenon-crossingdiagrams.ThereforethosecrossingdiagramsarenegligibleandtheSCBAworks.However,inthecaseoflong-rangimpuritieslB,themomentumtransferisq)]TJ /F6 7.97 Tf 6.59 0 Td[(1,whichismuchsmallerthanthereciprocalofthetypicalmagneticlengthscalel)]TJ /F6 7.97 Tf 6.59 0 Td[(1B.Therefore,bothcrossingandnon-crossingdiagramsareconnedtoasmallanglescatteringandtheyhavethesamevolumeofphasespace.Thus,non-crossingdiagramscannotbeneglectedintheresummation. 42 PAGE 43 3.3.2DensityofStatesforLong-RangeCorrelations,withlBAsdiscussedinSec. 3.3.1 ,SCBAfailstheregimewherelB,becausediagramsfromcross-interactionsareofthesameorderofthosefromSCBA[ 30 ].Inthiscase,theLandaulevelsplitting!c=eB=mismuchlargerthanthetypicalenergylevelscaleduetoscattering.Therefore,almostnovirtualtransitionbetweenLandaulevelsoccurs(on-mass-shellprocess).Thus,theGreen'sfunctioncanbegivenbythequasiclassicalpathintegraltreatment.ThetraceofitsGreen'sfunctionin(E;p)-spaceisgivenby[ 55 57 ]GR(E)=Z1GR(r0=r;t)ei(E+i)tdt=m 2(ln()]TJ /F3 11.955 Tf 9.29 0 Td[(E)]TJ /F3 11.955 Tf 11.95 0 Td[(i))]TJ /F7 11.955 Tf 11.95 0 Td[(2i1Xr=1exp(irS)exp()]TJ /F3 11.955 Tf 9.3 0 Td[(ir)): (3)whereE="+.Thewindingnumberrrepresentsthenumberoftimestheorbitistraversed.Theimaginarypartofthersttermdividedby)]TJ /F3 11.955 Tf 9.3 0 Td[(leadstom=2,whichistheDOSinahomogeneous2Dsystem,whilethesecondtermcontributestooscillationsofDOS.Sistheeikonalfunction,whichisgivenbyS=Ir=Rcpdr=Ir=Rcp 2m(E)]TJ /F3 11.955 Tf 11.96 0 Td[(V(r))^)]TJ /F3 11.955 Tf 11.96 0 Td[(eAdr2E !c)]TJ /F10 11.955 Tf 11.95 16.27 Td[(Ir=RcjdrjV(r) v; (3)wherev=p 2E=m.Bytakingtheaverageofcorrelations,Eq.( 3 )readsGR(E)=m 2(ln()]TJ /F3 11.955 Tf 9.3 0 Td[(E)]TJ /F3 11.955 Tf 11.96 0 Td[(i))]TJ /F7 11.955 Tf 11.96 0 Td[(2i1Xr=1exp2irE !c)]TJ /F7 11.955 Tf 13.15 8.09 Td[(1 2)]TJ /F3 11.955 Tf 11.96 0 Td[(r2Sw(E)): (3)ThedampingfactorisSw=1 2v2Ir=RcjdrjIr0=Rcjdr0jfW(jr)]TJ /F8 11.955 Tf 11.96 0 Td[(r0j;z=z0)=1 163!2cZ20dZ20d0Z10dq?q?Z20dW(q)expiqRcsin)]TJ /F3 11.955 Tf 11.96 0 Td[(0 2cos=1 2!2cZ10dqqW(q)J20(qRc); (3) 43 PAGE 44 whereJ0istheBesselfunctionoftherstkind.Equation( 3 )recoverstoGR(E)ofa2DfreeelectronsystemifthedampingSwiszero.TheDOSisgivenbyg(")=)]TJ /F7 11.955 Tf 9.29 0 Td[(1 ImGR(")=m 21Xr=exp2irE !c)]TJ /F7 11.955 Tf 13.15 8.09 Td[(1 2)]TJ /F3 11.955 Tf 11.96 0 Td[(r2Sw: (3)UsingthePoissonformula1Xr=f(r)=1Xn=Z1drf(r)exp()]TJ /F7 11.955 Tf 9.3 0 Td[(2irn);ityieldsg(")=Xngn(")=m 2Xnr Swexp )]TJ /F3 11.955 Tf 12.75 8.09 Td[(2 Sw")]TJ /F3 11.955 Tf 11.95 0 Td[("n !c2!: (3)Equation( 3 )showsthatLandaulevelsacquireaGaussianform.Thedetailedcalculationisdiscussedlaterinthequasi-2Dsystem. 44 PAGE 45 CHAPTER4DENSITYOFSTATESANDDEHAAS-VANALPHENOSCILLATIONSINADISORDEREDTHREE-DIMENSIONALELECTRONSYSTEMTheinuenceofimpuritiesontheoscillatoryamplitudeofDOSandmagnetizationsof3Dmetalshavebeenstudiedsince1952byDingle[ 6 ],whoassumedconstantcollisiontime.Later,Bychkov[ 51 ]andDoman[ 52 ]treatedthisproblembyndingthesingleparticleGreen'sfunctionofelectrongas.DomancalculatedtheDOSwithintheself-consistentBornapproximation(SCBA),inwhichmixingofLandaulevelsbydisorderscatteringisneglected.Hesolvedacubicequationfortheself-energyforshortrangecorrelations,whenthecorrelationlengthismuchshorterthanthemagneticlengthlB.In1969,Dyakonovetal.studiedastronglycompensatedsemiconductorandderivedanexpressionfortheDOSwithinthestatisticalapproachforasystemwithultra-long-rangecorrelations,whenthecorrelationlengthismuchlongerthanthecyclotronradiusRc[ 53 ].After,mostofworkwasfocusedon2Delectronsystemswhichdisplayfascinatingquantumphenomena,suchasthequantumHalleffect.AdetaileddescriptionoftheDOSofa2Dsysteminasemi-classicalregimewasprovidedbyRaikhandShahbazyan[ 30 ],whoresummedleadingsequencesofFeynmandiagramsforalltypesofdisorder.LaterthesameresultswerederivedbyMirlinetal.[ 43 ]inapath-integralapproach.RecentexperimentsonmagnetooscillationsintheNernstcoefcientofbismuth[ 31 32 ]havere-ignitedinteresttomagnetooscillationsin3D.Tothebestofourknowledge,acompletedescriptionofthedeHaas-vanAlphen(dHvA)effectin3DsimilartothatgiveninRefs.[ 30 ]and[ 43 ]for2Dislacking.Weconsiderahomogeneous3Delectronsysteminthepresenceofweakdisorder,suchthattheLandaulevelsarewell-separated.Forsimplicity,thespinsofelectronsareneglected.Althoughweareinterestedinorder-parameterdisorders,thetheoreticalconstructionisthesameasthethoseforimpurityscatterings.Whenauniformmagnetic 45 PAGE 46 eldisappliedalongthezdirection,theelectronicenergytakestheform "(n;kz)="n+k2z 2m;(4)where "n=n+1 2!c)]TJ /F3 11.955 Tf 11.96 0 Td[(:(4)!cisthecyclotronfrequencyeB=m,andisthechemicalpotential.Intheshort-rangecorrelationregimelB,theSCBAisconsidered[ 30 54 ].Forthelong-rangescatterers,SCBAfailsandthequasiclassicalpathintegraltreatmentworks[ 30 43 ]. 4.1DOSforshort-rangecorrelationslBInshort-rangecorrelationslB,theself-energycanbederivedbytheSCBA.WiththechoiceoftheLandaugaugeA=(0;xB;0),itisgivenbyEq.( 3 ),exceptthat"nshouldbereplacedby"(n;kz)andthereshouldbeanintegraloverkzaswell.Theretardedself-energyin3DthereforereadsR(";n;kz)=hnkykzjVG0Vjnkykzi=Xn0Zdqz 2Zd2q? (2)2W(q)J2jn)]TJ /F4 7.97 Tf 6.58 0 Td[(n0j(q?Rc) ")]TJ /F3 11.955 Tf 11.96 0 Td[("n0)]TJ /F6 7.97 Tf 13.15 5.7 Td[((kz+qz)2 2m)]TJ /F7 11.955 Tf 11.96 0 Td[(R(";n0;kz+qz); (4)whereq?isthemomentumtransferofelectronsinthexyplane.AsmentionedinSec. 3.3.1 ,thevalidityofthisapproximationrequiresthattheLandaulevelsareneartheFermisurface,withfn;n0g1,andtheargumentoftheBesselfunctionislessthann[ 30 ].Thelatterconditionmeansq?n=Rc1=F,whereFisthedeBrogliewavelength.Sincethetypicalvalueofqisoftheorderof)]TJ /F6 7.97 Tf 6.59 0 Td[(1,itdemandsthatF.Aftershiftingtheqzvariable,Eq.( 4 )takestheformR(";n;kz)=Xn0Zdqz 2Zd2q? (2)2W(q?;qz)]TJ /F3 11.955 Tf 11.96 0 Td[(kz)J2jn)]TJ /F4 7.97 Tf 6.59 0 Td[(n0j(q?Rc) ")]TJ /F3 11.955 Tf 11.96 0 Td[("n0)]TJ /F4 7.97 Tf 14.91 5.7 Td[(q2z 2m)]TJ /F7 11.955 Tf 11.96 0 Td[(R(";n0;qz): (4) 46 PAGE 47 Thesumovern0canbeseparatedintothediagonalterm(n0=n)andtheoff-diagonalterms(n06=n).ForhighLandaulevelswithenergiesneartheFermienergy,typicalvaluesofkzaredeterminedfromthecondition!ck2z=2m,i.e.kz1=lB)]TJ /F6 7.97 Tf 6.58 0 Td[(1forshort-rangedisorder.NeglectingbothqzandkzinW(q?;qz)]TJ /F3 11.955 Tf 12.7 0 Td[(kz),weseethattheqzintegralinthediagonalpart(n=n0)iscontrolledbyqzp mj")]TJ /F3 11.955 Tf 11.96 0 Td[("nj.l)]TJ /F6 7.97 Tf 6.58 0 Td[(1B.Therefore,thecorrelatorWinthediagonalpartcanbeconsideredasafunctionofq?only,andthekzdependenceofRcanbeneglectedaswell.Thecontributionoftheoff-diagonalterms,ontheotherhand,canbecalculatedbyapplyingthePoissonformula,andaposterioriargumentjustiestheyyieldaconstantcontribution)]TJ /F3 11.955 Tf 9.3 0 Td[(i=2.Explicitly,Eq.( 4 )thenreadsR(";n)=Zdqz 2Zd2q? (2)2W(q?;0)J20(q?Rc) ")]TJ /F3 11.955 Tf 11.96 0 Td[("n)]TJ /F4 7.97 Tf 14.9 5.7 Td[(q2z 2m)]TJ /F7 11.955 Tf 11.95 0 Td[(R(";n))]TJ /F3 11.955 Tf 17.29 8.09 Td[(i 2=)]TJ /F3 11.955 Tf 9.3 0 Td[(i)]TJ ET q .478 w 150.23 -295.35 m 250.2 -295.35 l S Q BT /F10 11.955 Tf 150.23 -297.98 Td[(p ")]TJ /F3 11.955 Tf 11.95 0 Td[("n)]TJ /F7 11.955 Tf 11.95 0 Td[(R(";n))]TJ /F3 11.955 Tf 17.29 8.09 Td[(i 2; (4)where)-277(=r m 2Zd2q? (2)2W(q?;0)J20(q?Rc): (4)Typicalvalueofq?RcisaboutRc=1intheSCBAregime.Therefore,theBesselfunctioncanbeasymptoticallyapproximatedbyJ20(q?Rc)(2=q?Rc)cos2(q?Rc)]TJ /F3 11.955 Tf 12.06 0 Td[(=4).Equation( 4 )thusbecomes)-278(=r m 21 RcZ10dq? 22W(q?;0): (4)Thistermisproportionaltoboththecyclotronfrequency!candthescatteringstrength.Theoff-diagonalterm)]TJ /F3 11.955 Tf 9.3 0 Td[(i=2is)]TJ /F3 11.955 Tf 9.3 0 Td[(i 2=XrZdqz 2Zd2q? (2)2Zdn0W(q)J2jn)]TJ /F4 7.97 Tf 6.59 0 Td[(n0j(q?Rc)e2irn0 ")]TJ /F3 11.955 Tf 11.96 0 Td[("n0)]TJ /F4 7.97 Tf 14.91 5.7 Td[(q2z 2m)]TJ /F7 11.955 Tf 11.96 0 Td[(R(";n0): (4)Typicalvaluesofqzforthehoppingisoftheorderofp n)]TJ /F3 11.955 Tf 11.96 0 Td[(n0=lB.Ontheotherhand,typicalvaluesofqzcontrolledbyW(q)isoftheorderof)]TJ /F6 7.97 Tf 6.59 0 Td[(1.Whenbothmeet, 47 PAGE 48 i.e.p n)]TJ /F3 11.955 Tf 11.96 0 Td[(n0=lB)]TJ /F6 7.97 Tf 6.59 0 Td[(1,thereisamajorcontributiontotheintegral.Thissetsthetypicalvaluesofn)]TJ /F3 11.955 Tf 13.25 0 Td[(n0toben)]TJ /F3 11.955 Tf 13.25 0 Td[(n0l2B=2RcF=2Rc=.ThismakesJ2jn)]TJ /F4 7.97 Tf 6.59 0 Td[(n0j(q?Rc)1=q?Rc.Therefore,byconsideringonlythenon-oscillatorytermr=0,whichdominatesovertheoscillatorypart,Eq.( 4 )yields1 21 vFZdqzZ10dq?W(q): (4)ThisissimplythescatteringrateatB=0.Bytakingthetypicalvalueofqz)]TJ /F6 7.97 Tf 6.58 0 Td[(1,Eq.( 4 )becomes 1 2)]TJ ET q .478 w 229 -200.23 m 266.35 -200.23 l S Q BT /F2 11.955 Tf 229 -202.86 Td[(p m!c:(4)IfW(q)obeystheLorentzianformW(q)=W0=(q2+)]TJ /F6 7.97 Tf 6.59 0 Td[(2),representinge.g.ascreenedCoulombpotentialorstaticuctuationsoftheorderparameternearaphasetransition,becomes)-277(=r m 2W0 4Rc: (4)However,inthiscase,theintegralofEq.( 4 )becomeslogarithmicdivergentandq?shouldbecutoffatq?F.Therefore,Eq.( 4 )yields1 2W0 vFZ)]TJ /F22 5.978 Tf 5.75 0 Td[(1F0dq? p q2?+)]TJ /F6 7.97 Tf 6.59 0 Td[(2W0 vFln(=F); (4)andtherelationbetweenand)]TJ /F1 11.955 Tf 10.64 0 Td[(is 1 2)]TJ ET q .478 w 205.87 -481.65 m 243.22 -481.65 l S Q BT /F2 11.955 Tf 205.87 -484.28 Td[(p m!cln(=F):(4)ifthecorrelationpotentialtakesaLorentzianform.ThishasalogarithmicdifferencefromEq.( 4 ),whichrequirestheconvergenceofintegral.Equation( 4 )canbereducedtoadimensionlessform x=)]TJ /F3 11.955 Tf 9.3 0 Td[(i p (bn+ia))]TJ /F3 11.955 Tf 11.96 0 Td[(x;(4) 48 PAGE 49 withbn=(")]TJ /F3 11.955 Tf 12.32 0 Td[("n)=)]TJ /F6 7.97 Tf 7.32 4.34 Td[(2=3,a=1=(2)]TJ /F6 7.97 Tf 7.31 4.34 Td[(2=3),andx=(R+i=2)=)]TJ /F6 7.97 Tf 7.32 4.34 Td[(2=3.Ithastobenoticedthat,accordingtoEq.( 4 ), a)]TJ /F6 7.97 Tf 7.31 4.34 Td[(1=3 p m!c=lB )]TJ /F6 7.97 Tf 7.32 4.34 Td[(1=3 p !c=)]TJ /F6 7.97 Tf 7.31 4.34 Td[(1=3 p ~!c;(4)where~!c(=lB)2!c,whichisshowntobeausefulscalewhilemakingaDOSplotorcomparingthediagonaltermsandoff-diagonaltermsofself-energy.FromEq.( 4 ),itisclearthata1correspondstoastronglycorrelatedsystemwhilea1correspondstoaweaklycorrelatedsystem.Forconvenience,wesetcn=bn+ia.Equation( 4 )isequivalenttoacubicequation x3)]TJ /F3 11.955 Tf 11.95 0 Td[(cnx2)]TJ /F7 11.955 Tf 11.95 0 Td[(1=0:(4)ThisisageneralizationoftheDoman's[ 52 ]equationwhichdoesnottakeintoaccountinter-Landauleveltransitions.thatonlyconsideredthediagonaltermofself-energy.Theirarethreesolutions.However,onlyoneofthemisconsistentwiththeoriginalEq.( 4 ).Itisgivenby x=cn 3)]TJ /F10 11.955 Tf 13.15 18.53 Td[()]TJ /F7 11.955 Tf 5.48 -9.68 Td[(1)]TJ 11.96 9.89 Td[(p 3ic2n 33p 4(cn))]TJ /F10 11.955 Tf 13.16 18.53 Td[()]TJ /F7 11.955 Tf 5.47 -9.68 Td[(1+p 3i(cn) 63p 2;(4)where (cn)=27+2c3n+3p 3p 27+4c3n1=3(4)TheDOSisthusgivenbyg(")=Xngn(")=)]TJ /F7 11.955 Tf 21.78 8.08 Td[(1 22l2BXnZdqz 2ImGR(";n;qz)=p 2m 42l2BXnRe1 p ")]TJ /F3 11.955 Tf 11.96 0 Td[("n)]TJ /F7 11.955 Tf 11.95 0 Td[(R(";n): (4)UsingEq.( 4 ),g(")isreducedto g(")=)]TJ 9.3 9.89 Td[(p 2m 42l2B)]TJ /F10 11.955 Tf 10.51 19.56 Td[(XnImR(";n)+i 2:(4) 49 PAGE 50 Further,byusingEq.( 4 ),gn(")isthengivenby g(")=)]TJ 9.3 9.89 Td[(p 2m 42XnImx(cn) a:(4)Therefore,gn(")isproportionalto)]TJ /F26 11.955 Tf 11.3 0 Td[(Imx=a,whichisthediagonaltermofself-energyoveroff-diagonaltermsofself-energy.ToseehowthestrengthofscatteringaffectstheproleoftheDOS,itisinstructivetoinvestigatehowgn(")changeswithrespecttothevalueofa,whichsignieswhetherthecorrelationscatteringisstrongorweak.Asaplot,anabscissaaxishastobecarefullychosentobea-independent.Therefore,bn=(")]TJ /F3 11.955 Tf 12.76 0 Td[("n)=)]TJ /F6 7.97 Tf 7.32 4.34 Td[(2=3isnotaproperchoiceinthat)]TJ /F1 11.955 Tf 10.64 0 Td[(isrelatedtheparametera.ByEq.( 4 ),bncanberewrittenas bn=")]TJ /F3 11.955 Tf 11.95 0 Td[("n )]TJ /F6 7.97 Tf 7.31 3.45 Td[(2=3")]TJ /F3 11.955 Tf 11.96 0 Td[("n ~!ca2:(4)Thus,itbecomescleartoseetherelationbetweengn(")andaiftheplotismadeasgn(")withrespectton,wheren=(")]TJ /F3 11.955 Tf 11.96 0 Td[("n)=~!c.TheresultcanbeseeninFig.( 4-1 ).AsAppendix B shows,theasymptoticbehaviorofgn(")isgivenby gn(")p 2m 428><>:1=p n;ifnmaxfa3;a2ga3=2jnj3=2;if)]TJ /F3 11.955 Tf 9.3 0 Td[(nmaxfa3;a2g(4)Asnmaxfa3;a2g,itcanbeseenthattheDOSapproachesthefreeelectronlimit,whichreads gn(")=(2m)3=2!c 82p ")]TJ /F3 11.955 Tf 11.95 0 Td[("n;if")]TJ /F3 11.955 Tf 11.96 0 Td[("nmax1=2;)]TJ /F6 7.97 Tf 7.31 4.34 Td[(2=3.(4)Themaximumofgn(")isgivenby n8><>:p 3=2a1ifa1(p 3=2a)3=21ifa1(4) 50 PAGE 51 Figure4-1. DensityofStatesgn(")asafunctionof=(")]TJ /F3 11.955 Tf 11.96 0 Td[("n)=~!cfordifferentvaluesofa=1=2)]TJ /F6 7.97 Tf 7.32 4.34 Td[(2=3=)]TJ /F6 7.97 Tf 19.74 4.34 Td[(1=3=p ~!c,foragivenLandaulevel.Theinsetshowsgn(")forlargea.SymbolsOindicatethemaximaofgn("),locatedatna3=p 3.Thepeaksdecrease,broaden,andmoveawayfrom"="nasaincreases.TheDOSgn(")approachesthe3Dfreeelectronlimitinamagneticledasgn(")1=p n(intheunitsofp 2m=42),whennmaxfa;a3g.Forweakscatterings(a1),themaximalvalueofgn(")isgivenbygn(")p 3=2a1,andthediagonaltermofself-energydominatesovertheoff-diagonaltermsif)]TJ /F3 11.955 Tf 9.3 0 Td[(a2n1(or)]TJ /F7 11.955 Tf 9.3 0 Td[()]TJ /F6 7.97 Tf 7.31 4.34 Td[(2=3")]TJ /F3 11.955 Tf 11.95 0 Td[("n~!c).Inthisregion,gn(")hasapronouncedpeak.Inthecaseofstrongcorrelations,gn(")(p 3=2a)3=21,andtheoff-diagonaltermsofself-energyalwaysdominateoverthediagonalterms. Thismeansgn(")issmearedbyincreasingthestrengthofthecorrelator.Itsmaximumispositionedat na3 p 3;(4)whichisshiftedmoreandmoreabovethebottomoftheLandaulevelastheinteractionstrengthgetsstrongerandstronger. 51 PAGE 52 Instrongcorrelationinteractions(a1),themaximalvalueofgn(")(intheunitofp 2m=42)is(p 3=2a)3=2,whichisalreadymuchsmallerthan1.Therefore,gn(")1oncea1,andoff-diagonaltermsofself-energydominate.Inthiscase,R(";n)1=2,whichisjusttheself-energyintheabsenceofmagneticeld.Inweakcorrelationinteractions(a1),thediagonaltermdominatesif)]TJ /F3 11.955 Tf 9.3 0 Td[(a2n1(or)]TJ /F7 11.955 Tf 9.3 0 Td[()]TJ /F6 7.97 Tf 7.32 4.33 Td[(2=3")]TJ /F3 11.955 Tf 13.16 0 Td[("n~!c).Theoff-diagonaltermsdominateeitherfor")]TJ /F3 11.955 Tf 12.62 0 Td[("n~!c)]TJ /F6 7.97 Tf 7.32 4.34 Td[(2=3=a2or")]TJ /F3 11.955 Tf 12.63 0 Td[("n)]TJ /F7 11.955 Tf 26.25 0 Td[()]TJ /F6 7.97 Tf 7.32 4.34 Td[(2=3.Forbothregimes,j")]TJ /F3 11.955 Tf 12.63 0 Td[("nj1=2sincea=1=2)]TJ /F6 7.97 Tf 7.32 4.34 Td[(2=31.Therefore,eveniftheoff-diagonaltermsofself-energydominateinthoseregimes,theDOSgn(")(/ImGR),behavesasafreeelectronDOS.Therefore,onlythenon-hoppingdiagonalpartofself-energyneedstobeconsideredinweakinteractionswherea1.ThisisconsistentwithDoman'sresult[ 52 ]. 4.2DOSforLong-RangeCorrelationslBAsisdiscussedinSec. 3.3.2 ,theSCBAfailsintheregimewherelB.ThisisbecausethecrosseddiagramsareofthesameorderofthosefromtheSCBA[ 30 ].Inthiscase,theGreen'sfunctioncanbegivenbythequasiclassicalpathintegraltreatment.TheapproachisthesameasthatisdiscussedinSec. 3.3.2 ,exceptthatthereshouldbeakzintegralandEshouldbereplacedbyE)]TJ /F3 11.955 Tf 12.14 0 Td[(k2z=2m,theenergyinthecyclotronplane.Therefore,thetraceoftheretardedGreen'sfunctionin(E;p)-spaceisgivenbyG0R(rb;ra;t)=)]TJ /F3 11.955 Tf 9.29 0 Td[(i(t)Zdkz 2Zdky 2e)]TJ /F4 7.97 Tf 6.59 0 Td[(ik2z 2mteikz(zb)]TJ /F4 7.97 Tf 6.58 0 Td[(za)+iky(yb)]TJ /F4 7.97 Tf 6.59 0 Td[(ya)1Xn=0 n(xb)]TJ /F3 11.955 Tf 11.96 0 Td[(kyl2B) n(xa)]TJ /F3 11.955 Tf 11.95 0 Td[(kyl2B)e)]TJ /F4 7.97 Tf 6.59 0 Td[(i(n+1 2)!ct; (4) 52 PAGE 53 where nisthe2DelectronwavefunctioninamagneticelddescribedintheLandaugauge.ThetraceofitsFouriertransformin(E;p)-spacethustakestheformG0R(E)=Z1G0R(rb=ra;t)ei(E+i)tdt=1 2l2BZ1dkz 21Xn=01 E)]TJ /F10 11.955 Tf 11.96 9.68 Td[()]TJ /F3 11.955 Tf 5.48 -9.68 Td[(n+1 2!c)]TJ /F4 7.97 Tf 14.63 5.7 Td[(k2z 2m+i=m 2Z1dkz 2(ln)]TJ /F3 11.955 Tf 9.29 0 Td[(E+k2z 2m)]TJ /F3 11.955 Tf 11.95 0 Td[(i)]TJ /F7 11.955 Tf 11.95 0 Td[(2i1Xr=1exp 2ir"E)]TJ /F4 7.97 Tf 14.62 5.7 Td[(k2z 2m !c)]TJ /F7 11.955 Tf 13.15 8.08 Td[(1 2#!E)]TJ /F3 11.955 Tf 15.58 8.08 Td[(k2z 2m); (4)whereE="+.Thewindingnumberrrepresentsthenumberoftimestheorbitistraversed.Thersttermleadstomp 2mE=22,whichistheDOSinahomogeneous3Dsystem,whilethesecondtermcontributestooscillationsofDOS.Inasystemwithdisorder,itiseasiertostartwithG0R(E),consideringdisordercorrelationscatteringasaperturbation.Itcanbecheckedtheexponent2i(E)]TJ /F3 11.955 Tf 12.36 0 Td[(k2z=2m)=!ccorrespondstotheeikonalfunctioniS=iHr?=Rcpdr?,whichisconsistentwiththepathintegralresultofthexed-energyGreen'sfunction[ 56 57 ].Therefore,theeikonalfunctionS,aftertakingdisordercorrelationsreducestoS=Ir?=Rcpdr?=Ir?=Rcp 2m(E?)]TJ /F3 11.955 Tf 11.96 0 Td[(V(r))^?)]TJ /F3 11.955 Tf 11.96 0 Td[(eAdr?2E? !c)]TJ /F10 11.955 Tf 11.95 16.27 Td[(Ir?=Rcjdr?jV(r) v?; (4)whereE?=E)]TJ /F3 11.955 Tf 12.12 0 Td[(k2z=2mandv?=p 2E?=m.AfteraveragingoverthedisorderV(r),theGreen'sfunctionreducestoGR(E)=m 2Z1dkz 2ln)]TJ /F3 11.955 Tf 9.3 0 Td[(E+k2z 2m)]TJ /F3 11.955 Tf 11.95 0 Td[(i)]TJ /F7 11.955 Tf 9.3 0 Td[(2i1Xr=1exp 2ir"E)]TJ /F4 7.97 Tf 14.62 5.7 Td[(k2z 2m !c)]TJ /F7 11.955 Tf 13.15 8.09 Td[(1 2#)]TJ /F3 11.955 Tf 11.96 0 Td[(r2Sw!E)]TJ /F3 11.955 Tf 15.59 8.09 Td[(k2z 2m): (4) 53 PAGE 54 ThedampingfactorisSw=1 2v2?Ir?=Rcjdr?jIr0?=Rcjdr0?jfW(jr?)]TJ /F8 11.955 Tf 11.96 0 Td[(r0?j;z=z0)=1 163!2cZ20dZ20d0Z1dqzZ10dq?q?Z20dW(q?;qz)expiq?Rcsin)]TJ /F3 11.955 Tf 11.96 0 Td[(0 2cos=1 2!2cZ1dqzZ10dq?q?W(q?;qz)J20(q?Rc): (4)Atypicalvalueofqzis)]TJ /F6 7.97 Tf 6.58 0 Td[(1l)]TJ /F6 7.97 Tf 6.58 0 Td[(1B,whichmeansthatthisrepresentsaclassicalorbitandnohoppingbetweenLandaulevelsoccurs.Therefore,Sw)]TJ /F3 11.955 Tf 7.31 0 Td[(=p m!2c,where)]TJ /F1 11.955 Tf 10.64 0 Td[(isdenedinEq.( 4 ).Furthermore,SwincreasesasincreasesandsaturatesasRc,since)]TJ /F1 11.955 Tf 10.63 0 Td[(isproportionalto.IfthecorrelationfunctionfollowstheOrnstein-ZernikeformW(q)=W0=(q2+)]TJ /F6 7.97 Tf 6.59 0 Td[(2),whichisrelevantforphasetransitionsasisdiscussedinSec. 2.2 ,theintegralisdivergent,andthemomentumshouldbecutoffat1=lB,tomakesurethequasiclassicalapproachworks.Therefore,Sw=1 2!2cZ1dqzZ10dq?q?W0 q2?+q2z+)]TJ /F6 7.97 Tf 6.59 0 Td[(2J20(q?Rc)=1 2!2cZ10dq?q?W0 p q2?+)]TJ /F6 7.97 Tf 6.59 0 Td[(2J20(q?Rc)=W0 2v?!cln(L=lB); (4)whereL=minf;Rcg.Afterresummation(Appendix C )bythePoissonformula,theDOSatEisgivenbyg(E)=)]TJ /F7 11.955 Tf 9.3 0 Td[(1 ImGR(E)=m 221Xn=r SwZp 2mE0dkzexp8<:)]TJ /F3 11.955 Tf 12.76 8.09 Td[(2 Sw"E)]TJ /F4 7.97 Tf 14.62 5.7 Td[(k2z 2m !c)]TJ /F10 11.955 Tf 11.96 16.86 Td[(1 2+n#29=; (4)Inthecasep SwE=!c,whichisgenerallycorrectforweakscattering,thiscanbefurtherapproximatedas 54 PAGE 55 Figure4-2. DimensionlessdensityofstatesasinEq.( 4 )=S1=4wvs.Hn=(")]TJ /F3 11.955 Tf 11.96 0 Td[("n)=!c,foragivenLandaulevel.=S1=4w=R10dy p yexph)]TJ /F4 7.97 Tf 12.98 4.7 Td[(2 Sw(y)]TJ /F3 11.955 Tf 11.96 0 Td[(Hn)2i.ThepeaksoftheLandaulevelhere,labeledby5arelocatedatHn0:17p Sw.AsSw!0,thecurvecrossesovertotheDOSofafree3Dsysteminauniformmagneticeld. g(E)=Xngn(E)=(2m)3=2 82!2c Sw1=41Xn=0(hn); (4)where(hn)=Z10dy p yexp)]TJ /F7 11.955 Tf 11.29 0 Td[((y)]TJ /F3 11.955 Tf 11.95 0 Td[(hn)2=8><>:p hne)]TJ /F21 5.978 Tf 5.76 0 Td[(h2n=2 2hI1 4h2n 2+I)]TJ /F22 5.978 Tf 5.76 0 Td[(1 4h2n 2i;ifhn0p )]TJ /F4 7.97 Tf 6.58 0 Td[(hne)]TJ /F21 5.978 Tf 5.76 0 Td[(h2n=2K1 4h2n 2 p 2;ifhn<0 (4) 55 PAGE 56 andhn= p SwE !c)]TJ /F10 11.955 Tf 11.95 16.86 Td[(n+1 2= p SwHn: (4)I(z)andK(z)aremodiedBesselfunctionsoftherstandsecondkinds,respectively.HnisdenedbyHn=(")]TJ /F3 11.955 Tf 12.84 0 Td[("n)=!c.Equation( 4 )andEq.( 4 )areconsistentwiththeresultsgivenbyDyakonovetal.[ 53 ],whostudiedastronglycompensatedsemiconductorandderivedanexpressionfortheDOSwithinthestatisticalapproach.ForagivenLandauleveln,gn(")isshowninFig.( 4-2 ).Themaximumof(hn)isathn0:54,whichmeans")]TJ /F3 11.955 Tf 11.95 0 Td[("n0:17p Sw!c.TheasymptoticbehaviorofmodiedBesselfunctionsisgivenby 8><>:I(z)ez p 2z1+(1)]TJ /F6 7.97 Tf 6.58 0 Td[(2)(1+2) 8z+:::K(z)p 2ze)]TJ /F4 7.97 Tf 6.59 0 Td[(z;(4)forzj2)]TJ /F7 11.955 Tf 11.95 0 Td[(1=4j.Therefore, (hn)8><>:p =hn;ifhn13=2 2p jhnjexp()]TJ /F3 11.955 Tf 9.3 0 Td[(h2n);ifhn<0andjhnj1.(4)Thus,theDOSgn(E)behavesas gn(E)8><>:(2m)3=2!c 82p ")]TJ /F4 7.97 Tf 6.58 0 Td[("n;if")]TJ /F3 11.955 Tf 11.96 0 Td[("np Sw (2m)3=2!c 16p ")]TJ /F4 7.97 Tf 6.59 0 Td[("n;if"n)]TJ /F3 11.955 Tf 11.95 0 Td[("p Sw (4)TheDOScorrespondstoa3Dfreeelectronsystemifhn1,anditdropsquicklytozeroinaGaussianformfor)]TJ /F3 11.955 Tf 9.3 0 Td[(hn1. 4.3DHvAOscillationsThemagnetizationcanbecalculatedbytakingthederivativeofthermodynamicpotentialwithrespecttomagneticeldB.Thiscanbeeasilydonebystartingwiththe 56 PAGE 57 numberdensityofelectronsNe=1Xn=0Z1d!gn(!)nF(!) (4)wherenF(!)istheFermi-Diracdistribution.Inprevioussections,ithasbeenshownthatgn(!)canbewrittenasafunctionf(!)]TJ /F4 7.97 Tf 6.59 0 Td[("n !c)inboththeSCBAandquasiclassicalregimes.Therefore,afterapplyingthePoissonformula,theoscillatorypartofNecanbeexpressedasNe=Xr6=0Z10dnZ1d!nF(!)f!)]TJ /F3 11.955 Tf 11.95 0 Td[("n !cexp(2irn): (4)Usingthecondition!candchangingofvariablez=(!)]TJ /F3 11.955 Tf 12.34 0 Td[("n)=!c,itcanbeshownthatNe=Xr6=0Z1d!nF(!)exp2ir! !cexp2ir !c)]TJ /F7 11.955 Tf 13.15 8.08 Td[(1 2F(r); (4)where F(r)=Z1dzf(z)exp()]TJ /F7 11.955 Tf 9.3 0 Td[(2irz):(4)Bytheintegrationidentity Z1eiy ey+1dy=)]TJ /F3 11.955 Tf 9.3 0 Td[(i sinh();(4)andthepropertyF(r)=F(r),NecanbefurtherreducedtoNe=Xr>0Im (22rT=!c) rexp2ir !c)]TJ /F7 11.955 Tf 13.15 8.09 Td[(1 2F(r); (4)where (z)=z=sinh(z),andF(r)=Z1dzf(z)exp()]TJ /F7 11.955 Tf 9.29 0 Td[(2irz)=8><>:)]TJ /F5 7.97 Tf 12.14 11.32 Td[(p 2m 42aR1dzImx)]TJ /F3 11.955 Tf 5.48 -9.69 Td[(ia+z!c )]TJ /F22 5.978 Tf 5.29 2.35 Td[(2=3e)]TJ /F6 7.97 Tf 6.58 0 Td[(2irz;iflB.(2m)3 2 82!2c Sw1=4R1dz(z=p Sw)e)]TJ /F6 7.97 Tf 6.59 0 Td[(2irz;iflB. (4) 57 PAGE 58 ThemagnetizationisM=)]TJ /F3 11.955 Tf 9.3 0 Td[(@=@B,where=)]TJ /F10 11.955 Tf 11.29 9.64 Td[(RNedisthethermodynamicpotential.ItyieldsM=e mXr>0 (22rT=!c) rImexp2ir !c)]TJ /F7 11.955 Tf 13.15 8.09 Td[(1 2F(r): (4)ThisformulageneralizestheLKresultdHvAoscillations[ 5 ],exceptforthatF(r)containstheeffectofscattering.Ingeneral,F(r)iscomplex,andthereforescatteringcausesaphaseshiftaswellasdampingofthedHvAoscillations.Inthequasiclassicalregime,thesimpleGaussianformoftheintegrand(Eq.( 4 ))makesitpossibletogettheanalyticresult,whichreadsM=Xr>0()]TJ /F7 11.955 Tf 9.3 0 Td[(1)re3=2p B 43 (22rT=!c) r3=2sin2r !c)]TJ /F3 11.955 Tf 13.15 8.09 Td[( 4exp)]TJ /F2 11.955 Tf 5.48 -9.69 Td[()]TJ /F3 11.955 Tf 9.29 0 Td[(r2Sw;iflB. (4)IntheSCBAregime,F(r)cannotbeobtainedanalyticallyandshouldbesolvednumerically.InsertingEq.( 4 )intoEq.( 4 ),themagnetizationyieldsM=p 2e 42lBXr>0()]TJ /F7 11.955 Tf 9.3 0 Td[(1)r (22rT=!c) r)]TJ /F3 11.955 Tf 9.3 0 Td[(Iscos2r !c+Icsin2r !c=p 2e 42lBXr>0()]TJ /F7 11.955 Tf 9.3 0 Td[(1)r (22rT=!c) rp I2s+I2csin2r !c)]TJ /F3 11.955 Tf 11.96 0 Td[(; (4)where Is=)]TJ /F3 11.955 Tf 9.3 0 Td[(lB aZ1dzsin(2rz)Im lB 2z a2+ia!;(4)and Ic=)]TJ /F3 11.955 Tf 9.3 0 Td[(lB aZ1dzcos(2rz)Im lB 2z a2+ia!(4)areFresnelintegrals.Thephaseisgivenby =arctan(Is=Ic):(4)Consideringtherstharmonicr=0,andtakinglB==10asexample,IsandIcwithdifferentvaluesofaarecalculatedasshowninTable 4-1 .Itcanbeseenthatnotonly 58 PAGE 59 Table4-1. AmplitudeCoefcientsIsandIcwithdifferentscatteringstrength,givenbyEq.( 4 )andEq.( 4 ).Therstharmonicr=1andlB==10areconsidered. lB=aIsIc 1000:50:5=4=0:7850:50:4900:4960:7801:00:4550:4700:7701:50:3730:4040:7462:00:2510:3010:6952:50:1280:1840:6083:00:04330:08680:463 theamplitudesdecreasesastheinteractionstrengthincreases,butthephasesalsoshiftwithrespecttoa. 59 PAGE 60 CHAPTER5MAGNETOOSCILLATIONSNEARTHEQUASI-TWO-DIMENSIONALQUANTUMCRITICALITYRecently,therehasbeenasurgeofinteresttomagnetooscillationswillbehaveinquasi-two-dimensional(2D)heavy-fermionsystemsneartheQCP[ 17 20 ]andhigh-Tccuprates[ 21 25 ].Sofar,mostoftheexperimentaldatawereconsistentlytusingtheFLmodel,whichleadstotheLKform.However,inquasi-2Dsystemswhereinteractionsorcorrelationsarestrong,theLKtheorymustbreakdown.Evenina3Dsystem,theLKtheoryisnotstrictlyvalidaswell.TheinteractionscanbetakenintoaccountintheextendedLKtheory[ 10 46 48 ],wheredampingofmagnetooscillationsisrelatedtotheelectronself-energy.In2005,McCollametal.[ 18 ]performeddeHaas-vanAlphen(dHvA)measurementsonCeCoIn5andfoundthattheself-energyofNFLformisneededtotintotheextendedLKtheory.Onthetheoreticalside,in2010,FritzandSachdev[ 28 ]investigatedmagnetooscillationsinasystemoffermionscoupledtoaninternalU(1)gaugeeldinadisordered2DsystemthatthemagnetooscillationpatterndeviatesfromtheFLprediction.Inthesameyear,ThompsonandStampaddressedtheproblembyconsideringthespin-fermionmodel[ 29 ].Theycalculatedself-energyintheframeofSCBAinaquasi-2DandanalyzedboththeFLandNFLregimes.Bothapproachestookonlydynamicuctuationsintoconsideration.However,wendthatstaticuctuationscanhaveanimportanteffectonmagnetooscillationsneartheQCP.Besides,neartheQCPwherethecorrelationlengthislongerthanthemagneticlengthlB,SCBAisnotapplicablesincetheself-energyfromcrosseddiagramsisofthesameorderasfromSCBAdiagrams[ 30 ].Inthischapter,wedevelopthetheoryofdHvAoscillationsnearaQCPinaquasi-2Dsystem.Forsimplicity,thespindependenceofeffectivemassisneglectedaswell.WerstdiscusshowtheextendedLKformulaisgeneralizedforaquasi-2DinteractingelectronsystembothintheSCBA(lB)andinthequasiclassical 60 PAGE 61 regimes(lB).InSec. 5.2 ,wecalculatetheself-energyandthedampingfactorofmagnetooscillationsinthespin-fermionmodel.Thetemperatureisassumedtobehigherthanthecyclotronfrequencytojustifytheneglectoftheoscillatorypartofself-energy.InSec. 5.3 ,thetemperaturedependenceofcorrelationlengthistakenintoaccount.Thedampingoftheoscillationamplitudeisshowntoscalewithtemperatureaseitherase)]TJ 6.59 6.76 Td[(p T,e)]TJ /F4 7.97 Tf 6.59 0 Td[(Tln(1=T),ore)]TJ /F4 7.97 Tf 6.59 0 Td[(T,dependingoninterplayofthecyclotronlength,thecorrelationlength,andthestrengthofinterplanarcoupling.Wewillshowthatstaticuctuationsplayanimportantroleindeterminingdampingmagnetooscillations.Thelowtemperaturelimit,wheredynamicuctuationsmayplaythemajorrole,isalsodiscussed.Sec. 5.4 concludesthethesisandsummarizesourndings. 5.1TheoryofQuantumMagnetooscillationsThedHvAoscillationsinaFermiliquidareusuallydescribedquantitativelybytheLuttinger-Wardformula[ 46 ]fortheoscillatorypartofthethermodynamicpotential=)]TJ /F3 11.955 Tf 9.3 0 Td[(TP!Trlnf)]TJ /F7 11.955 Tf 15.28 0 Td[((G)]TJ /F6 7.97 Tf 6.58 0 Td[(1g,whereGistheelectronMatsubaraGreen'sfunction.ThemagnetizationMisrelatedtothethermodynamicpotentialbyM=)]TJ /F3 11.955 Tf 9.3 0 Td[(@=@H.Thus,thedeterminationoftheGreen'sfunctionoroftheself-energybecomestherststepinndingthemagnetooscillationamplitude.Theself-energycanbefoundedwithintheSCBA.AlthoughtheSCBAworkswellforshort-rangecorrelations,itisnotapplicableforasystemneartheQCPwithalongcorrelationlength.RaikhandShahbazyanshowedthattheSCBAisjustiedonlyforlB,wherelB=1=p eBisthemagneticlength[ 30 ].WhenlB,diagramsfrombothself-interactionsandSCBAareequallyimportant.Inthiscase,itisdifculttondeachGreen'sfunctionforgivenLandauleveln.Besides,thereisalsoapossibilitythattheself-energydependsonmagneticeld.ThosecircumstancesmaketheLuttinger-Wardformulainapplicable.Alternatively,onecanndtheGreen'sfunctionandtraceittoobtainaclosedexpressionoftheDOS[ 30 ].Therefore,itiseasiertostartbyndingtheDOSg(!)andthenumberofelectronsNe 61 PAGE 62 bytakingchemicalpotentialasconstant,whichcorrespondstoworkinginagrandcanonicalensemble.Thethermodynamicpotentialcanthenbefoundby=)]TJ /F10 11.955 Tf 11.29 9.63 Td[(RNed.Weconsideraquasi-2Dsystem.Whenauniformmagneticeldisappliedalongthezdirection,theelectronenergytakestheform "(n;kz)="n+"z;(5)where 8><>:"n=)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(n+1 2!c)]TJ /F3 11.955 Tf 11.96 0 Td[(;"z=2tk(1)]TJ /F7 11.955 Tf 11.96 0 Td[(cos(kzb))(5)!cisthecyclotronfrequencyeB=m,andtkandbarethehoppingintegralandthelatticeconstantalongthezdirection,respectively.Forsimplicity,thespinresponseofelectronsisneglected.Intheshort-rangecorrelationregimewhenlB,theSCBAworkswell.Inthelong-rangecasewhenlB,theSCBAfailsandaresummationofFeynmandiagramsisrequired[ 30 ].Inthiscase,aneasierapproachistousethequasiclassicalpathintegraltreatment[ 43 ],whichisadoptedinthisthesis. 5.1.1SCBARegime:Short-RangecorrelationslBTheconventionalexpressionfortheelectronnumberdensityis Ne=TX!XnZ=b)]TJ /F4 7.97 Tf 6.59 0 Td[(=bdkz 2D(n)G(!;"n;kz):(5)D(n)=m!c=isthedegeneracyoftheLandaulevelsinaquasi-2Dfreeelectronsystem.WiththehelpofthePoissonformula1Xn=0f(n)=Z1)]TJ /F6 7.97 Tf 6.58 0 Td[(1=2f(n)dn+Xr6=0Z1)]TJ /F6 7.97 Tf 6.58 0 Td[(1=2f(n)e2irndn;theoscillatorypartofNecanbeexpressedasNe=Tm X!Xr6=0Z=b)]TJ /F4 7.97 Tf 6.58 0 Td[(=bdkz 2exp2ir !c)]TJ /F7 11.955 Tf 13.15 8.09 Td[(1 2Z1d"nG(!;n;kz)e2ir"n=!c; (5) 62 PAGE 63 wheref!;gistakenintoaccounttoextendthelowerlimitof"ntoinnity.Therefore,thethermodynamicpotential=)]TJ /F10 11.955 Tf 11.3 9.63 Td[(RNedisobtainedas=)]TJ /F3 11.955 Tf 10.5 8.09 Td[(Tm!c X!Xr6=0Z b)]TJ /F21 5.978 Tf 7.78 3.26 Td[( bdkz 21 2irexp2ir !c)]TJ /F7 11.955 Tf 13.15 8.09 Td[(1 2Z1d"nG(!;n;kz)e2ir"n=!c: (5)WiththepropertyG(!)=G()]TJ /F3 11.955 Tf 9.3 0 Td[(!),thepotentialcanbefurthersimpliedas=)]TJ /F3 11.955 Tf 9.3 0 Td[(mT!c 2X!Xr>0Z b)]TJ /F21 5.978 Tf 5.75 0 Td[( bdkz 2rImexp2ir !c)]TJ /F7 11.955 Tf 13.15 8.09 Td[(1 2Z1G(!;n;kz)e2ir"n=!cd"n: (5)TheelectronGreen'sfunctioncanbewrittenas G)]TJ /F6 7.97 Tf 6.59 0 Td[(1(!;n;kz)=i!)]TJ /F3 11.955 Tf 11.95 0 Td[("n)]TJ /F3 11.955 Tf 11.96 0 Td[("z)]TJ /F7 11.955 Tf 11.95 0 Td[((!;n;kz);(5)Iftheself-energycanbewrittenas(!;"n;kz)(!),asitwillbeshownlater,isnonzeroonlyfor!withthesamesignasraccordingtoJordan'slemma.Thecontourintegralover"nandintegrationoverkzthenyields=2mT!c bX!>0Xr>01 rJ04rtk !cRe(exp2ir !c)]TJ /F7 11.955 Tf 13.15 8.09 Td[(1 2exp )]TJ /F7 11.955 Tf 10.49 8.85 Td[(2r!+i(!)+2itk !c!)=2mT!c bX!>0Xr>0()]TJ /F7 11.955 Tf 9.3 0 Td[(1)r rexp)]TJ /F7 11.955 Tf 10.5 8.09 Td[(2r[!+i(!)] !cJ04rtk !ccos 2r)]TJ /F3 11.955 Tf 5.48 -9.68 Td[()]TJ /F7 11.955 Tf 11.96 0 Td[(2tk !c!; (5)whereJn(z)istheBesselfunctionoftherstkind.ThemagnetizationM=)]TJ /F3 11.955 Tf 9.29 0 Td[(@=@Bisthengivenby M=4eT)]TJ /F3 11.955 Tf 5.48 -9.68 Td[()]TJ /F7 11.955 Tf 11.96 0 Td[(2tk b!cX!>0Xr>0()]TJ /F7 11.955 Tf 9.3 0 Td[(1)r+1exp)]TJ /F7 11.955 Tf 10.49 8.09 Td[(2r[!+i(!)] !cJ04rtk !csin 2r)]TJ /F3 11.955 Tf 5.48 -9.68 Td[()]TJ /F7 11.955 Tf 11.95 0 Td[(2tk !c!(5) 63 PAGE 64 If4tk!c,theBesselfunctiontakestheasymptoticformJ0(z)p 2=zcos(z)]TJ /F3 11.955 Tf 12.11 0 Td[(=4).ThemagnetizationbecomesM=4eT)]TJ /F3 11.955 Tf 5.48 -9.68 Td[()]TJ /F7 11.955 Tf 11.95 0 Td[(2tk b!cX!>0Xr>0()]TJ /F7 11.955 Tf 9.3 0 Td[(1)r+11 2r !c 2rtkexp)]TJ /F7 11.955 Tf 10.49 8.09 Td[(2r[!+i(!)] !c(sin 2r)]TJ /F3 11.955 Tf 5.48 -9.68 Td[()]TJ /F7 11.955 Tf 11.96 0 Td[(4tk !c+1 4!+sin2r !c)]TJ /F7 11.955 Tf 13.15 8.09 Td[(1 4): (5)Therefore,therearetwotermsofoscillations.Theformercorrespondstotheextremalorbitwithkz==b,whilethelattercorrespondstotheextremalorbitwithkz=0.Ontheotherhand,if4tk!c,J0(z)1andthemagnetizationreads M=4eT)]TJ /F3 11.955 Tf 5.48 -9.69 Td[()]TJ /F7 11.955 Tf 11.95 0 Td[(2tk b!cX!>0Xr>0()]TJ /F7 11.955 Tf 9.29 0 Td[(1)r+1exp)]TJ /F7 11.955 Tf 10.5 8.08 Td[(2r[!+i(!)] !csin 2r)]TJ /F3 11.955 Tf 5.48 -9.69 Td[()]TJ /F7 11.955 Tf 11.96 0 Td[(2tk !c!:(5)Thereisonlyonetermofoscillationsforagivenwindingnumberr.Itcorrespondstotheorbitatkz==2b,whichistheaverageoftheextremalorbitsatkz=0andkz==b.ThisisbecauseallorbitswithdifferentkzcorrespondstothesameLandaulevel(2Dlimit),if4tk!c.Sincetkinquasi-2D,2tk=!cof()]TJ /F7 11.955 Tf 12.41 0 Td[(2tk)=!cinEq.( 5 )onlygivesasmallcorrectionandisneglected.Ifthecondition22T=!c&1,thedominantamplitudecomesfromtherstharmonicr=1withthelowestelectronMatsubarafrequency!=T.Therefore,theoscillationamplitudeinquasi-2Disapproximatedby A1=A10exp)]TJ /F7 11.955 Tf 10.5 8.08 Td[(22T !c;(5)where A10=8><>:4eT b!c;if4tk!c,1 2q !c 2tk4eT b!c;if4tk!c,(5)antherenormalizedcyclotronfrequencyisdenedas !c=eB=mdHvA;(5) 64 PAGE 65 wheremdHvA=m=1+i(T;T) T: (5)Thismaybeverydifferentfromthezero-temperaturemassm=m=@ReR(!)=@!.Furthermore,inelasticquasiparticlerelaxationduetoaselectron-electronorelectron-phononinteractionshasnocontributionto(T;T)andthereforedoesnotentertheamplitudeofmagneto-oscillations[ 58 ].Oneimportantissueiswhethertheoscillatorypartofself-energycanbeneglectedwhencalculatingtheamplitudes.Lettheself-energybedecomposedintotwopartsas=0+osc,where0maydependontheeldmonotonicallybutnotoscillatory,andthesecondterm,osc,isthepartoftheself-energythatcontainsdHvAoscillations.Thethermodynamicpotentialcanbeexpandedaround0inTaylorseriesas=(0)+O2() 2=02oscwherethestationaryproperty==0at=0hasbeenused.Luttinger[ 47 ]showedthatosc/(!c=)3=20ina3-dimensionalsystem.ThisresultisquitegeneralandcanbeextendedtotheD-dimensionalsystem[ 59 ]asosc/(!c=)D=20.Therefore,()=(0)=1+O)]TJ /F7 11.955 Tf 5.48 -9.69 Td[((!c=)D.However,theoscillatorypartofthethermodynamicpotentialobtainedintheabsenceofoscscalesasosc(0)/(!c=)1+D 2(0)inthelow-temperatureregime,whereT!c[ 47 50 59 ].Hence,theoscillatorypartofthethermodynamicpotentialcomingfromosccomparedtoosc(0)is(!c=)D(0)=osc(0)=(!c=)D 2)]TJ /F6 7.97 Tf 6.59 0 Td[(1.Ina3Dsystem,thisratioismuchsmallerthan1andthecontributionfromosctoosccanbethereforeneglected.Ontheotherhand,ina2Dsystem,(!c=)2(0)=osc1,theoscillationsintheself-energyoscandinthethermodynamicpotentialoscitselfareequallyimportant.Therefore,theoscillatorypartofself-energyingeneralcannotbeneglectedina2Dsystem.However,wehaveshownthattheoscillatorypartofthethermodynamicpotentialoscisexponentiallysmall,asexp()]TJ /F7 11.955 Tf 9.3 0 Td[(22[T+i=]=!c)ifthetemperatureissufcientlyhigh,i.e.22T=!c&1.Thisexponentialdependenceisvalidfortheamplitudesof 65 PAGE 66 alloscillatoryquantities,includingtheself-energyosc.Therefore,withexponentialaccuracy,O(2osc)exp()]TJ /F7 11.955 Tf 9.3 0 Td[(42[T+i=]=!c),whichismuchsmallerthanoscandcanbeneglected[ 58 ].Thismeansthatitisunnecessarytotaketheoscillatorypartofself-energyintoaccountwhileconsideringtheamplitudesofmagnetooscillations. 5.1.2QuasiclassicalRegime:Long-RangeCorrelationslBThepreviousdiscussionisrelevantonlytoSCBAintheregime,whenlB,wheretheSCBAworks.IntheoppositecaselB,theself-energyfromSCBAdiagramsareofthesameorderasthosefromcrossedones[ 30 ].ResummationoftheseriesoftheGreen'sfunctionisquitecomplicated.However,itdescribesasimplequasiclassicalpictureandcanbemoreelegantlytreatedbythepathintegralformalism[ 43 ].Thequasiclassicalrequirementcanbeunderstoodfromtheenergyargument.Thecharacteristicenergyofthescatteringis"1=m2whiletheLandaulevelspacingis!c=eB=m=1=ml2B.ElectronsstayalmostinthesameLandaulevelif"!c,whichisjustlB.Inthiscase,electronshavenotransitionsbetweenLandaulevelsandthereforecanbeconsideredastomovingonclassicalcircularorbits.Thepathintegralapproachthusworks.ThetraceofretardedGreen'sfunctionisgivenby[ 43 56 57 ]GR(")=m Z b)]TJ /F21 5.978 Tf 7.78 3.26 Td[( bdkz 2(ln()]TJ /F3 11.955 Tf 9.3 0 Td[(E+"z)]TJ /F3 11.955 Tf 11.95 0 Td[(i))]TJ /F7 11.955 Tf 11.96 0 Td[(2i1Xr=1exp2irE)]TJ /F3 11.955 Tf 11.95 0 Td[("z !c)]TJ /F7 11.955 Tf 13.15 8.09 Td[(1 2)]TJ /F3 11.955 Tf 11.96 0 Td[(r2Sw(E)]TJ /F3 11.955 Tf 11.95 0 Td[("z)); (5)whereE="+,and"+iistheanalyticcontinuationofMatsubarafrequencyi!.ThersttermleadstotheDOSinaquasi-2Dsystemfreeofmagneticeld,whilethesecondtermcontributestooscillationsofDOSduetoquantizationofenergylevelsinmagneticelds.ThedampingfactorisalreadygivenbyEq.( 4 )asSw=1 2!2cZ b)]TJ /F21 5.978 Tf 7.78 3.26 Td[( bdqzZ10dq?q?W(q?;qz)J20(q?Rc); (5) 66 PAGE 67 wherev?=p 2m(E)]TJ /F3 11.955 Tf 11.96 0 Td[("z)p 2mE,Rc=v?=!c,andfW(r)istheorder-parametercorrelationfunction.Intheconstruction,thecorrelationisstatic.Anaposterioriargumentshowsthatthedynamiccorrelationsarenegligibleformagnetooscillationsif22T&!c.TheDOSat"isthereforeg(")=)]TJ /F7 11.955 Tf 9.3 0 Td[(1 ImGR(")=m 221Xr=Z b)]TJ /F21 5.978 Tf 7.78 3.26 Td[( bdkzexp2irE)]TJ /F3 11.955 Tf 11.96 0 Td[("z !c)]TJ /F7 11.955 Tf 13.15 8.08 Td[(1 2)]TJ /F3 11.955 Tf 11.95 0 Td[(r2Sw: (5)UsingthePoissonformula1Xr=f(r)=1Xn=Z1drf(r)exp()]TJ /F7 11.955 Tf 9.3 0 Td[(2irn);weobtaing(")=1Xn=gn("); (5)wheregn(")=m 22Z b)]TJ /F21 5.978 Tf 7.78 3.26 Td[( bdkzr Swexp )]TJ /F3 11.955 Tf 12.76 8.09 Td[(2 Sw")]TJ /F3 11.955 Tf 11.96 0 Td[("n)]TJ /F3 11.955 Tf 11.95 0 Td[("z !c2!: (5)TheelectronnumberdensityisthengivenbyNe=1Xn=Z1d"nF(")gn(")=XrZ1)]TJ /F22 5.978 Tf 7.79 3.26 Td[(1 2dnZ1d"nF(")gn(")exp(2irn); (5)wherenF(")=1=(e"=T+1)istheFermi-Diracdistribution.TheoscillatorypartofNeyieldsNe=Xr>0Im8<:2T sinh22rT !cexp2ir !c)]TJ /F7 11.955 Tf 13.15 8.09 Td[(1 2Fr(Sw)9=;; (5) 67 PAGE 68 whereFr(Sw)=m 22Z1dyZ b)]TJ /F21 5.978 Tf 7.78 3.26 Td[( bdkzr Swexp")]TJ /F3 11.955 Tf 12.76 8.09 Td[(2 Swy)]TJ /F3 11.955 Tf 13.75 8.09 Td[("z !c2)]TJ /F7 11.955 Tf 11.95 0 Td[(2iry#=m 22Z b)]TJ /F21 5.978 Tf 7.79 3.26 Td[( bdkzexp)]TJ /F2 11.955 Tf 5.48 -9.69 Td[()]TJ /F3 11.955 Tf 9.3 0 Td[(r2Sw)]TJ /F7 11.955 Tf 11.96 0 Td[(2ir"z=!c=m bexp)]TJ /F7 11.955 Tf 10.49 8.52 Td[(4irtk !cJ04rtk !cexp)]TJ /F2 11.955 Tf 5.48 -9.69 Td[()]TJ /F3 11.955 Tf 9.29 0 Td[(r2Sw: (5)Inthelastline,thekzdependenceofSwwasneglectedbecause"z.Therefore,themagnetooscillationtakestheformM=)]TJ /F3 11.955 Tf 11.01 8.09 Td[(@ @B=@RNed @B=2eT)]TJ /F3 11.955 Tf 5.48 -9.68 Td[()]TJ /F7 11.955 Tf 11.96 0 Td[(2tk b!cXr>0()]TJ /F7 11.955 Tf 9.3 0 Td[(1)r+1e)]TJ /F4 7.97 Tf 6.59 0 Td[(r2Sw sinh22rT !cJ04rtk !csin 2r)]TJ /F3 11.955 Tf 5.48 -9.68 Td[()]TJ /F7 11.955 Tf 11.95 0 Td[(2tk !c!=8>><>>:2eT()]TJ /F6 7.97 Tf 6.59 0 Td[(2tk) b!cPr>0()]TJ /F6 7.97 Tf 6.59 0 Td[(1)r+1e)]TJ /F21 5.978 Tf 5.76 0 Td[(r2Sw sinh22rT !csin2r()]TJ /F6 7.97 Tf 6.58 0 Td[(2tk) !c;if4tk!c.2eT()]TJ /F6 7.97 Tf 6.59 0 Td[(2tk) b!cPr>01 2q !c 2rtk()]TJ /F6 7.97 Tf 6.58 0 Td[(1)r+1e)]TJ /F21 5.978 Tf 5.76 0 Td[(r2Sw sinh22rT !csin2r()]TJ /F6 7.97 Tf 6.59 0 Td[(4tk) !c+1 4+sin2r !c)]TJ /F6 7.97 Tf 13.15 4.7 Td[(1 4;if4tk!c. (5)If22T=!c&1,thedominantamplitudecomesfromtherstharmonicr=1,whichreads A1=A10exp)]TJ /F7 11.955 Tf 9.29 0 Td[(22T !c)]TJ /F3 11.955 Tf 11.95 0 Td[(Sw;(5)whereA10isgivenbyEq.( 5 ). 5.2QuantumMagnetooscillationsNeartheFerromagneticCriticalPointNeartheclassicalcriticalpoint,thespinuctuationtakestheOrnstein-Zernikeform,whileneartheQCP,Weconsiderthespin-fermionmodel.ThosehavebeendiscussedinChapter 2 5.2.1SCBAregime:Short-RangeCorrelationslBWestartbyconsideringtheone-loopself-energy1(!;n;kz).WiththechoiceofLandaugaugeA=(0;xB;0),thediagramfor1canbeshowninFig.( 5-1 ),the 68 PAGE 69 expressionofwhichreads1(!;n;kz)=3g2bTXn0;Z b)]TJ /F21 5.978 Tf 5.76 0 Td[( bdqz 2Zd2q? (2)2(;q?;qz)jhnjeiqxxjn0ij2 i(!+))]TJ /F3 11.955 Tf 11.96 0 Td[("n0)]TJ /F3 11.955 Tf 11.96 0 Td[("z(kz+qz); (5)IntheSCBAapproach,i(!+))]TJ /F3 11.955 Tf 12.23 0 Td[("n0)]TJ /F3 11.955 Tf 12.23 0 Td[("zshouldbereplacedbyi(!+))]TJ /F3 11.955 Tf 12.22 0 Td[("n0)]TJ /F3 11.955 Tf 12.23 0 Td[("z)]TJ /F7 11.955 Tf -449.09 -23.9 Td[((!+;n0;kz)inthedenominator.SinceweareinterestedinLandaulevelsneartheFermisurfacewherefn;n0g1,andtheregimelB=p nFwhereFistheelectronFermiwavelength,Eq.( 5 )intheframeofSCBAcanbefurthersimpliedas[ 30 ](!;n;kz)=3g2bTXn0;Z b)]TJ /F21 5.978 Tf 5.76 0 Td[( bdqz 2Zd2q? (2)2(;q?;qz)J2jn)]TJ /F4 7.97 Tf 6.59 0 Td[(n0j(q?Rc) i(!+))]TJ /F3 11.955 Tf 11.95 0 Td[("n0)]TJ /F3 11.955 Tf 11.95 0 Td[("z(kz+qz))]TJ /F7 11.955 Tf 11.95 0 Td[((!+;n0;kz+qz): (5)Thecouplingvertexinamagneticeldcanbethereforeconsideredasg2Jjn)]TJ /F4 7.97 Tf 6.59 0 Td[(n0j(q?Rc).Anaposterioriargumentshowsthatq?RcRc==p nlB=p n.ThisallowsonetoreplaceJ2jn)]TJ /F4 7.97 Tf 6.59 0 Td[(n0jbyJ2jn)]TJ /F4 7.97 Tf 6.58 0 Td[(n0j(q?Rc)=(2=q?Rc)cos2(q?Rc)]TJ /F7 11.955 Tf 12.04 0 Td[((n)]TJ /F3 11.955 Tf 12.04 0 Td[(n0)=2)]TJ /F3 11.955 Tf 12.05 0 Td[(=4)forrelevantnandn0.Theoscillatorypartafterintegrationoverq?isnegligiblysmall.Thisyields(!;n;kz)=3g2bT 22RcXn0;Z b)]TJ /F21 5.978 Tf 5.76 0 Td[( bdqz 2Z10dq?(;q?;qz) i(!+))]TJ /F3 11.955 Tf 11.96 0 Td[("n0)]TJ /F3 11.955 Tf 11.96 0 Td[("z(kz+qz))]TJ /F7 11.955 Tf 11.95 0 Td[((!+;n0;kz+qz): (5)Thesumovern0canbecalculatedwiththehelpofthePoissonformula.Theoscillatorypartoftheself-energyisexponentiallysmallif22T&!c,oraresmallerthanthenonoscillatorypartbyafactorp !c=4tkif4tk!c.Thus,neglectingtheoscillatorypartisjustiedinbothcases,andweobtain(!;T))]TJ /F7 11.955 Tf 9.3 0 Td[(3ig2bT 2v?XZ b)]TJ /F21 5.978 Tf 5.75 0 Td[( bdqz 2Z10dq?sgn(!+)(;q?;qz): (5) 69 PAGE 70 Theself-energydoesnotdependonthequantumnumbernandkz.NeartheQCP,=v?=p g0T.TheSCBAcriterionlBimpliesTv2?=g0l2B!c=g0!c.Therefore,onlythelowestMatsubarafrequency!=Tisofinterestashigherharmonicsofmagnetizationsareexponentiallysmallerandarethereforenegligible.Asaresult,onlystaticuctuationswith=0givecontributionstotheself-energy.Thisyields(T;T)st(T;T))]TJ /F7 11.955 Tf 9.3 0 Td[(3igbT 2v?Z b)]TJ /F21 5.978 Tf 5.75 0 Td[( bdqz 2Z10dq? q2?+tk v?2(1)]TJ /F7 11.955 Tf 11.96 0 Td[(cos(qzb))+)]TJ /F6 7.97 Tf 6.58 0 Td[(2; (5)whereststandsforthestaticpartofself-energy.Inaquasi-2Dsystem,p tk=v?l)]TJ /F6 7.97 Tf 6.59 0 Td[(1B,whichismuchsmallerthan)]TJ /F6 7.97 Tf 6.59 0 Td[(1andnegligibleintheSCBAregime.Therefore,typicalmomentumtransferisq?)]TJ /F6 7.97 Tf 6.59 0 Td[(1.Thesystemshowsapurely2Dbehaviorandtheself-energyneartheQCPcanbethuswrittenas (T;T))]TJ /F7 11.955 Tf 23.12 8.09 Td[(3igT 4v?=)]TJ /F7 11.955 Tf 10.49 8.09 Td[(3ig 4s T g0:(5)Nearaclassicalcriticalpoint,=v?=p g0(T)]TJ /F3 11.955 Tf 11.96 0 Td[(Tc),whereTcisthecriticaltemperature.Thisyields (T;T))]TJ /F7 11.955 Tf 23.11 8.08 Td[(3igT 4v?)]TJ /F7 11.955 Tf 45.35 8.08 Td[(3igTc 4p g0(T)]TJ /F3 11.955 Tf 11.96 0 Td[(Tc)(5)neartheclassicalphasetransition.NoticethatthetemperaturedependenceinEq.( 5 )andEq.( 5 )doesnotcomefromfermionicfrequency!=T.Therefore, Figure5-1. Thelowest-orderdiagramforthefermionicself-energy(!;n;kz).ThevertexfunctioninthemagneticeldisgJjn)]TJ /F4 7.97 Tf 6.59 0 Td[(n0j(q?Rc). 70 PAGE 71 theself-energydonotcontainanydynnamocpropertyoffermions.Furthermore,thisself-energyisthesameasthatintheabsenceofthemagneticeld.ThisisreasonablesinceintheconditionlBimpliestheLandaulevelspacingismuchsmallerthanthecharacteristicenergyofscattering. 5.2.2QuasiclassicalRegime:Long-RangeCorrelationslBIntheregimewherelB,theSCBAisnotapplicable.Thequasiclassicaltreatment,however,willbevalidandgivesanelegantapproachtotheproblem.Tondtheoscillationamplitude,weneedtondthedampingfactorSw,wherethecorrelationfunctionW(q)is3g2bT(q).Thedynamicpartofuctuationsisneglectedaswell,forthereasonsexplainedinthepreviouschapter.Afterintegrationoverqz,Eq.( 5 )becomesSw=3gT !2cZ10dq?q?J20(q?Rc) s (q2?+)]TJ /F6 7.97 Tf 6.59 0 Td[(2)q2?+)]TJ /F6 7.97 Tf 6.59 0 Td[(2+2tk v?2: (5)IfRcv?=p tk,namelyp tk!c,thesystemreachesthe2Dlimit.Furthermore,inthequantumcriticalregime,)]TJ /F6 7.97 Tf 6.59 0 Td[(1=v?=p g0T.Therefore,theuppercutoffofthecorrelationlengthneartheQCPisp !c=g0Rc,whichisdeterminedbythecondition22T=!c&1.Theresultscanbediscussedinthreedifferentregimes. 5.2.2.1lBRcInthisregime,p tk=v?muchsmallerthan)]TJ /F6 7.97 Tf 6.59 0 Td[(1andcanbeneglected.Thetypicalmomentumtransferisq?)]TJ /F6 7.97 Tf 6.58 0 Td[(1.Thereforeq?RcRc=1,andtheBesselfunctioncanbeapproximatedasJ20(q?Rc)(2=q?Rc)cos2(q?Rc)]TJ /F3 11.955 Tf 12.78 0 Td[(=4).Thus,Eq.( 5 )becomesSw3gT Rc!2cZ10dq?1 q2?+)]TJ /F6 7.97 Tf 6.59 0 Td[(2=3gT 2!cv?: (5)Thedampingincreaseslinearlywiththecorrelationlength.Therefore, A1=A10exp)]TJ /F7 11.955 Tf 9.29 0 Td[(22T !c1+3g 4v?:(5) 71 PAGE 72 ThisisthesameastheamplitudeintheSCBAregime.ItshouldbeemphasizedthatthoseresultsfromtheSCBAandthequasiclassicalpathintegralapproacharethesameintheregimelBRconlyiftheoscillationsofself-energyareexponentiallyweeksothatthedifferencebetweenthesecondharmonicr=2termsinEq.( 5 )andEq.( 5 )differexponentiallysmall.Thisissatisedif22T!c. 5.2.2.2Rcv?=p tkInthisregime,q?RcRc=1,andthereforeJ01.Equation( 5 )thusbecomesSw3gT !2cZR)]TJ /F22 5.978 Tf 5.76 0 Td[(1c)]TJ /F22 5.978 Tf 5.76 0 Td[(1dq? q?=3gT !2cln Rc: (5)Thedampingfactorincreaseswiththecorrelationlengthlogarithmically.Theamplitudebehavesas A1A10exp)]TJ /F7 11.955 Tf 10.49 8.09 Td[(22T !c Rc)]TJ /F6 7.97 Tf 6.58 0 Td[(3gT=!2c:(5) 5.2.2.3v?=p tkInthisregime,typicalmomentumtransferisq?p tk=v?R)]TJ /F6 7.97 Tf 6.58 0 Td[(1c.Thereforeq?Rc1,andtheBesselfunctionisJ01.Therefore,Eq.( 5 )becomesSw3gT !2cZR)]TJ /F22 5.978 Tf 5.76 0 Td[(1cp 2tk v?dq? q?3gT !2cln !c p 2tk!: (5)Thedampingfactorsaturatesasthecorrelationlengthincreases.Theamplitudetakestheform A1A10exp)]TJ /F7 11.955 Tf 10.5 8.09 Td[(22T !c !c p 2tk!)]TJ /F6 7.97 Tf 6.59 0 Td[(3gT=!2c;(5)whichdecreasesastemperatureincreases.ThedampingfactorsforRcareshowntobeinverselyproportionalto!2c,whichisasignatureoftheinhomogeneousbroadening.Inthisregime,thedampingdependsontheinterplanecouplingtk,whichmeansitisalreadyinthe3Dregime. 72 PAGE 73 5.3TemperatureDependenceoftheMagnetooscillationAmplitudeIntheprevioussection,wefoundthedampingfactorsinthethreedifferentregimes,whereRc,Rcv?=p tk,orv?=p tk.Intheclassicalcriticalregime,=v?=p g0(T)]TJ /F3 11.955 Tf 11.96 0 Td[(Tc),whereTcisthecriticaltemperature.NeartheclassicalcriticalpointwhereTTc!c,thedampingfactorsofEq.( 5 ),Eq.( 5 ),andEq.( 5 )canberewrittenas Sw=8>>>>><>>>>>:3gTc 2!cp g0(T)]TJ /F4 7.97 Tf 6.58 0 Td[(Tc);ifT)]TJ /F3 11.955 Tf 11.96 0 Td[(Tc!2c g03gTc 2!2cln!2c g0(T)]TJ /F4 7.97 Tf 6.58 0 Td[(Tc);ift2k g0T)]TJ /F3 11.955 Tf 11.96 0 Td[(Tc!2c g03gTc 2!2cln!2c 2t2k;ifT)]TJ /F3 11.955 Tf 11.96 0 Td[(Tct2k g0(5)Thedampingfactorincreasesortheamplitudeofoscillationsdecreasesastemperaturedecreases,andsaturatesnearthecriticalpoint.NeartheQCPwhereTc=0,thedampingfactorsbecome Sw=8>>>>><>>>>>:3gp T 2!cp g0;ifT!2c g03gT 2!2cln!2c g0T;ift2k g0T!2c g03gT 2!2cln!2c 2t2k;ifTt2k g0(5)Thedampingfactordecreasesortheamplitudeofoscillationsincreasesastemperaturedecreases.IntheFLregime,thecorrelationlengthcanbeapproximatedasatemperature-independentconstant.Therefore,dampingfactorsaredescribedbyEq.( 5 ),Eq.( 5 ),andEq.( 5 ),wheredependsondoping.ThoseresultsaredepictedinFig.( 5-2 ).Sofar,weconsideredonlytheregimesof22T&!c,whenstaticuctuationsoftheorderparametergiveadominantcontributiontotheself-energyandmagnetooscillationamplitudes.However,toobservethedHvAoscillationsexperimentally,theoppositecondition22T.!cisusuallyrequired.Inthiscase,thedynamicuctuationsoftheorderparametermayplayaroleinmagnetooscillations.Inthelowtemperature 73 PAGE 74 Figure5-2. Temperaturedependenceofmagnetooscillationamplitudesnearthecriticality.Neartheclassicalcriticalpoint,thedampingfactorbehavesasEq.( 5 ).Theoscillationamplitudesdecreaseasthetemperaturedecreases.NeartheQCP,thedampingfactorbehavesasEq.( 5 ).Theoscillationamplitudesincreaseasthetemperatureisdecreased.IntheFLregime,thecorrelationlengthistemperature-independent.ThedampingfactorsaregivenbyEq.( 5 ),Eq.( 5 ),andEq.( 5 ).Atgiventemperature,theyincreaseasweapproachtheQCP. limitneartheQCPwherelB,SCBAfailsandthequasiclassicaltreatmentisrequired.However,thequasiclassicaltreatmentdoesnotincludedynamicuctuations.Nevertheless,itcanbecheckedfromEq.( 5 )andEq.( 5 )that2i(T;T)=!c=Swinthelong-rangecorrelationregimelBiftheconditioni!cisfullled.ThistellsusthattheSCBAcanbeextendedtothequasiclassicalregimetoincludethedynamicuctuationsifi!candifonlytherstharmonicofoscillationsneedstobeconsidered.Thiscanbeachievedbyconsideringtheeffectofreal(potential)disorders,whichisstrongenoughsothatthecondition!c=.1issatised.Thus,alreadytherstharmoniccontainsasmallDinglefactorexp()]TJ /F3 11.955 Tf 9.3 0 Td[(=!c),andosccanbeneglected.Higherharmonicswithr2areexponentiallysmallerthantherstonebyandcanbe 74 PAGE 75 neglectedaswell.Therefore,itissufcienttotakeonlytherstharmonicamplitudeintoaccount.ThisgivesA1(T)=A1;T(T)exp()]TJ /F3 11.955 Tf 9.3 0 Td[(=!c)whereA1;T(T)=A10X!>0exp)]TJ /F7 11.955 Tf 9.3 0 Td[(2 !c[!+i(!;T)]: (5)Now,allofthefermionicMatsubarafrequencieshavetobetakenintoconsideration.Inthequantumcriticalregimewhere(T)1=3)]TJ /F6 7.97 Tf 6.59 0 Td[(1,namelyTg2"2F=g30,thedynamicpartofself-energytakestheformdyn(!;T))]TJ /F7 11.955 Tf 9.3 0 Td[(3igbT 2!cXsgn(!+)Z b)]TJ /F21 5.978 Tf 5.75 0 Td[( bdqz 2Z10dq?q?J20(q?Rc) q2?+tk v?2(1)]TJ /F7 11.955 Tf 11.95 0 Td[(cos(qzb))+jj q?=)]TJ /F7 11.955 Tf 9.3 0 Td[(3igT 2!cXsgn(!+)Z10dq?q?J20(q?Rc) s q2?+jj q?q2?+jj q?+2tk v?2 (5)If(T)1=3R)]TJ /F6 7.97 Tf 6.59 0 Td[(1c,namelyT!3c=g,Eq.( 5 )issimpliedthesameformasinEq.( 2 ),i.e.dyn(!;T)=)]TJ /F3 11.955 Tf 9.3 0 Td[(i!1=30T2=3f(!=T): (5)Itcanbeshownthatthetemperature-dependentpartofdynamicself-energy!1=30T2=3ismuchsmallerthanthestaticpartofself-energyist=!cSw=2,whereSwisgivenbyEq.( 5 ).Therefore,thethermaldampingduetodynamicpartofself-energycanbeneglectedifT!3c=g.Eq.( 5 )thusbecomesA1;T(T)=A10 2TZ10d!exp)]TJ /F7 11.955 Tf 10.49 8.08 Td[(2 !ch!+!1 30!2 3i)]TJ /F3 11.955 Tf 11.95 0 Td[(Sw!cA10e)]TJ /F4 7.97 Tf 6.58 0 Td[(Sw 42Tminf1;p !c=!0g!cA10 42T(1)]TJ /F3 11.955 Tf 11.95 0 Td[(Sw)minf1;p !c=!0g (5) 75 PAGE 76 Again,onlythestaticpartofself-energycontributestothethermaldampingofoscillations.If(T)1=3R)]TJ /F6 7.97 Tf 6.59 0 Td[(1c,namelyT!3c=g,thedynamicpartofself-energystarttoplayarole.However,itcanbefoundthattypical!rangesoverregimesthatshowdifferenttemperaturedependences.Therefore,thereseemsnosimpleformtorepresentthethermaldampingiftemperatureislowerthan!3c=g,whichisonlyabouttheorder0:01K,givenB10T,1eV,andg!c.Itisknownthattheself-energy(!;T)hasaneffectonthecorrectiontothespecicheatcoefcient,whichendstobeproportionaltoT)]TJ /F6 7.97 Tf 6.59 0 Td[(1=3neartheQCPina2Dsystem.Thedivergenceofthespecicheatcoefcientisdeterminedbythedynamiccontributionandisrelatedtothedivergentrenormalizationoftheeffectivemassby@ReR1;dyn=@",where"istheanalyticallycontinuatedrealfrequency.AsforconductivityneartheQCP,theinverseofthetransporttimeis1=trPRd2q(!+;T;q)q2=k2F,where(!+;T;q)isthepartofself-energythatcomesfromthebosonicmomentumtransferqandenergytransfer.Anextrafactorofq2=k2F,whichisrelatedtothescatteringangle,eliminatesthedivergenceofstaticuctuationsinthemomentumintegralatq=kF!0,andtherefore,staticuctuationsaresuppressed.Sinceq!1=3,1=tr!2=3!2=3!4=3.Hence,theconductivity,orthetransporttime,neartheferromagneticQCPpurelycomesfromthedynamicuctuationsandisproportionaltoT4=3.OnlythedHvAeffectmeasuresthestaticuctuationsandhasaneffectivemassmdHvA=mist=T=!cSw=22T,whereSwisgivenbyEq.( 5 ). 5.4ConclusionsInsummary,wedevelopedthetheoryofmagnetooscillationsinaquasi-2DsystembothintheSCBAregime(lB)andinthequasiclassicalregime(lB)nearaquantumorclassicalcriticalpointwithinthespin-fermionmodel,wefoundexplicitexpressionsfortheamplitudesofthequantummagnetooscillationsneartheQCP.Thethermaldampingoftheoscillationamplitudesresultsfromthecouplingbetweenelectronsandstaticspinuctuationsinawideintervalofeldsandtemperature. 76 PAGE 77 Obviously,staticuctuationsdominateinthehightemperatureregime,i.e.when22T&!c.However,theyalsodominateinthelowtemperatureregimewhen!3c=gT.!c=22(Anexplicitcalculationisperformedunderanassumptionofsufcientlystrongdisorder!c.1).ThismakesdHvAmassverydifferentfromthespecicheateffectcoefcient,whichcomesfromthedynamicuctuations.Intheclassicalcriticalregime,/1=p T)]TJ /F3 11.955 Tf 11.96 0 Td[(Tcneartheclassicalcriticalpoint,while/p T=lnTneartheQCP,ifRc.BothoftheformsshowtheNFLbehavior.IfthetemperatureisfurtherdecreasedsothatRc,electronscanfeelthecurvatureofthetrajectoryduetothemagneticeld,duringtheprocessoffermion-uctuationcoupling.Thethermaldampingofoscillationamplitudesthusbehavesasexp()]TJ /F3 11.955 Tf 9.3 0 Td[(Sw),whereSw/Tln(1=T)=!2corT=!2c,whichshowsboththeNFLbehaviorandinhomogeneousbroadening.Theself-energycanbeconsideredas!cSw=2whichisinverselyproportionaltothecyclotronfrequencyifonlytherstharmonicofoscillationsistakenintoaccount.Oncethecorrelationislongerthanv?=tk,thesystemismore3D-like,wherethedampingfactorSwdoesnotdependonthecorrelationlengthandislinearlyproportionaltotemperature. 77 PAGE 78 APPENDIXASTATICSPINSUSCEPTIBILITYINTHESPIN-FERMIONMODELForasufcientlylong-rangedinteractionkFa1,wherethelengthaisradiusofinteractions,thestaticspinsusceptibilityscanbegivenbytherandomphaseapproximation(RPA)by s= 1)]TJ /F3 11.955 Tf 11.96 0 Td[(g(q);(A)whereisthedensityofstates(DOS),and(q)isthepolarizationbubble.Considerasystemwhichisquasi-isotropicinthexyplane,withaweakhoppingtkinthezdirection.Theenergyofanelectroncanbewrittenas "k="k;k+"k;?;(A)where "k;k=2tk(1)]TJ /F7 11.955 Tf 11.96 0 Td[(cos(kzb));(A)wherebisthelatticeconstantalongthezdirection,andtkinaquasi-2Dsystem."k;?isthe2Delectronicenergy,whichreads "k?=k2? 2m+O)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(k?4)]TJ /F3 11.955 Tf 11.95 0 Td[(;(A)whereO)]TJ /F8 11.955 Tf 5.48 -9.68 Td[(k?4comesfromaweakin-planelatticeeffects.Thepolarizationbubble(q)thustakestheform(q)=)]TJ /F3 11.955 Tf 9.3 0 Td[(bTX!Zd3k (2)3G0(k+q;!)G0(k+q;!)=)]TJ /F3 11.955 Tf 9.3 0 Td[(bZ)]TJ /F4 7.97 Tf 6.59 0 Td[(=b=bdkz 2Zd2k? (2)2n("k+q))]TJ /F3 11.955 Tf 11.96 0 Td[(n("k) "k+q)]TJ /F3 11.955 Tf 11.96 0 Td[("k; (A) 78 PAGE 79 wherebisthelatticeconstantalongthezdirection.ByexpandingtheFermifunctionwithrespectto"?,weobtainn("k)=n("?)+"k;kn0("?)+1 2"2k;kn00("?)+1 6"3k;kn000("?)+:::n("k+q)=n("?)+)]TJ /F3 11.955 Tf 5.48 -9.69 Td[("k+q;k+"?n0("?)+1 2)]TJ /F3 11.955 Tf 5.48 -9.69 Td[("k+q;k+"?2n00("?)+1 6)]TJ /F3 11.955 Tf 5.48 -9.69 Td[("k+q;k+"?3n000("?)+:::; (A)where"?="k+q;?)]TJ /F3 11.955 Tf 11.95 0 Td[("k;?=q2? 2m+v?q?+O(k4?): (A)Therefore,n("k+q))]TJ /F3 11.955 Tf 11.96 0 Td[(n("k)=n0("?)"+n00("?) 2("))]TJ /F7 11.955 Tf 5.48 -9.68 Td[("+2"k;k+n000("?) 6(")(")2+3"k;?"?+3"k+q;?"k;?+::: (A)where"="?+"z=q2? 2m+v?q?+2tk(cos(kzb))]TJ /F7 11.955 Tf 11.96 0 Td[(cos((kz+qz)b))+O(k4?) (A)Thus,(q)bZ b)]TJ /F21 5.978 Tf 7.78 3.26 Td[( bdkz 2Zd"??("?)Z20d 2n0("?)+n00("?) 2"+n000("?) 6(")2+:::=bZ b)]TJ /F21 5.978 Tf 7.78 3.26 Td[( bdkz 2Z20d 2?)]TJ /F3 11.955 Tf 13.15 8.09 Td[(0? 2"+00? 6(")2+:::; (A)where?isthedensityofstatesofthe2DsystemontheFermisurface.Thisyields (q)+)]TJ /F7 11.955 Tf 9.3 0 Td[(30 2+00v2?q2? 12+200t2k 3(1)]TJ /F7 11.955 Tf 11.95 0 Td[(cos(qzb)):(A) 79 PAGE 80 Inanexactlyhomogeneous2Dsystem,=m=.Thisgives(q)=m=,whichdoesnotdependonqatall.However,ifthelatticeistakenintoaccount,canweaklydependontheenergy,andtherefore0and00arenonvanishing.InsertionofEq.( A )inEq.( A )yields s(q)=0 )]TJ /F6 7.97 Tf 6.58 0 Td[(2+q2?+tk v?2(1)]TJ /F7 11.955 Tf 11.96 0 Td[(cos(qzb));(A)whereisaconstantwhichdependsonhowthedensityofstatesvarieswiththeenergy,andisthecorrelationlength,whichisdeterminedby )]TJ /F6 7.97 Tf 6.59 0 Td[(2=12(1)]TJ /F3 11.955 Tf 11.96 0 Td[(g?) )]TJ /F6 7.97 Tf 6.58 0 Td[(30 2+00v2?:(A)g=1(g==m)correspondsto)]TJ /F6 7.97 Tf 6.58 0 Td[(1=0,wherethesystemshowsstronginstability.Ina3Dhomogeneoussystemwhere"k=k2=2m)]TJ /F3 11.955 Tf 11.96 0 Td[(,(q)takestheform (q)=Z10d"k("k)Z20d 2n0("k)+n00("k) 2"+n000("k) 6(")2+:::;(A)where "="k+q)]TJ /F3 11.955 Tf 11.95 0 Td[("k=kq 2m+q2 2m;(A)and=p m3("k+)=2=2istheDOSofthe3Dsystem.Eq.( A )yieldsaresult+O(q2).Therefore, s(q)=0 )]TJ /F6 7.97 Tf 6.59 0 Td[(2+q2:(A)Asg!1(g!2=p m3=2),thecorrelationlength!1. 80 PAGE 81 APPENDIXBDENSITYOFSTATESIN3DSHORT-RANGEIMPURITIESItisshowninChapter 4 thatthedensityofstates(DOS)in3Dshort-rangeimpuritiesisgivenbyg"=Pngn("),where gn(")=)]TJ 9.3 9.89 Td[(p 2mImx(cn) 42a;(B)and x=cn 3)]TJ /F10 11.955 Tf 13.15 18.53 Td[()]TJ /F7 11.955 Tf 5.48 -9.68 Td[(1)]TJ 11.96 9.89 Td[(p 3ic2n 33p 4(cn))]TJ /F10 11.955 Tf 13.16 18.53 Td[()]TJ /F7 11.955 Tf 5.47 -9.68 Td[(1+p 3i(cn) 63p 2;(B)and (cn)=27+2c3n+3p 3p 27+4c3n1=3:(B)Thoseparametersaredenedby x=(R+i=2)=)]TJ /F6 7.97 Tf 7.31 4.34 Td[(2=3cn=bn+iabn=(")]TJ /F3 11.955 Tf 11.96 0 Td[("n)=)]TJ /F6 7.97 Tf 7.31 4.34 Td[(2=3a=1=(2)]TJ /F6 7.97 Tf 7.31 4.33 Td[(2=3))]TJ /F6 7.97 Tf 7.32 4.33 Td[(1=3=p ~!c~!c=(2=l2B)!cn=bna2=(")]TJ /F3 11.955 Tf 11.95 0 Td[("n)=~!c:(B)Byexpandingxwithrespectto1=cnorcn,itcanbeshownthat x=8><>:)]TJ /F4 7.97 Tf 6.59 0 Td[(i p cn+O1 c2n;ifjcnj1:)]TJ /F6 7.97 Tf 10.49 4.72 Td[(1+p 3i 2+cn 3)]TJ /F6 7.97 Tf 13.15 4.72 Td[(1)]TJ 6.59 6.6 Td[(p 3i 18c2n+O(c3n);ifjcnj1:(B)Thiscanbeseparatedintofourdifferentcases. 1. fa;1gbn: )]TJ /F26 11.955 Tf 11.96 0 Td[(Imx=1 p bn+O)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(b)]TJ /F6 7.97 Tf 6.58 0 Td[(5=2n:(B) 2. fa;1g)]TJ /F3 11.955 Tf 33.88 0 Td[(bn: )]TJ /F26 11.955 Tf 11.96 0 Td[(Imx=a 2jbnj3=2+O)]TJ /F3 11.955 Tf 5.48 -9.68 Td[(b)]TJ /F6 7.97 Tf 6.59 0 Td[(3:(B) 81 PAGE 82 3. fa;jbnjg1:)]TJ /F26 11.955 Tf 11.29 0 Td[(Imx=p 3 2)]TJ /F3 11.955 Tf 13.15 8.09 Td[(a 3)]TJ 13.15 17.98 Td[(p 3b2n)]TJ /F7 11.955 Tf 11.96 0 Td[(2ab)]TJ 11.95 9.89 Td[(p 3a2 18+O(c3): (B) 4. 1a:)]TJ /F26 11.955 Tf 11.29 0 Td[(Imx=Imi p bn+ia+O(c)]TJ /F6 7.97 Tf 6.58 0 Td[(7=2)1 (a2+b2n)1=4cos)]TJ /F7 11.955 Tf 9.3 0 Td[(1 2arctana bn: (B)InsertionofthoseparametersinEq.( B )intoEq.( B )andEq.( B )yieldtheresultthat )]TJ /F26 11.955 Tf 11.96 0 Td[(Imx=a8><>:1=p nifnmaxfa3;a2ga3=2jnj3=2if)]TJ /F3 11.955 Tf 9.29 0 Td[(nmaxfa3;a2g(B)TherstoneapproachestotheDOSinthefreeelectronlimit.BytakingthederivativeofEq.( B )orEq.( B ),itcanbealsofoundthatthemaximumofmaxf)]TJ /F26 11.955 Tf 17.27 0 Td[(Imx3=agispeakedatbna=p 3inbotha1anda1cases.Itmeansthatgn(")ispeakedat na3 p 3;(B)whichisshiftedmoreandmoreawayfromthebottomoftheLandaulevelastheimpurityscatteringgetsstrongerandstronger.InsertionofEq.( B )intoEq.( B )andEq.( B ),themaximumof)]TJ /F26 11.955 Tf 11.29 0 Td[(Imx=aisfoundtobep 3=2aifa1,and(p 3=2a)3=2ifa1.anddecreasesasaisincreased.Thismeansgn(")issmeareddownbyincreasingthestrengthoftheimpurityscattering.Instrong-impurityscatterings(a1),themaximalvalueof)]TJ /F26 11.955 Tf 11.29 0 Td[(Imx=ais(p 3=2a)3=2,whichisalreadymuchsmallerthan1.Therefore,)]TJ /F26 11.955 Tf 11.29 0 Td[(Imx=a1oncea1,andoff-diagonaltermsofself-energydominate.Inweak-impurityscatterings(a1),thediagonaltermdominatesif)]TJ /F3 11.955 Tf 9.3 0 Td[(a2n1,andotherwisetheoff-diagonaltermsdominate,ascanbecheckedfromtheasymptoticbehaviorof)]TJ /F26 11.955 Tf 11.29 0 Td[(Imx3=agiveninEq.( B )andEq.( B ). 82 PAGE 83 APPENDIXCDENSITYOFSTATESIN3DLONG-RANGEIMPURITIESThetraceofretardedGreen'sfunctionin3Dlong-rangimpuritiesisgivenbyEq.( 4 ).Thedensityofstates(DOS)readsg(E)=)]TJ /F7 11.955 Tf 9.3 0 Td[(1 ImGR(E)=m 22(p 2mE+Z1dkz1Xr=1exp)]TJ /F2 11.955 Tf 5.48 -9.69 Td[()]TJ /F3 11.955 Tf 9.3 0 Td[(r2Swcos2rE? !c)]TJ /F7 11.955 Tf 13.15 8.09 Td[(1 2(E?))=m 221Xr=Zp 2mE0dkze)]TJ /F4 7.97 Tf 6.58 0 Td[(r2Swexp2irE? !c)]TJ /F7 11.955 Tf 13.15 8.08 Td[(1 2; (C)whereE?=E)]TJ /F3 11.955 Tf 11.95 0 Td[(k2z=2m.BythePossionsummationformula,1Xr=exp)]TJ /F2 11.955 Tf 5.48 -9.69 Td[()]TJ /F3 11.955 Tf 9.3 0 Td[(r2Sw+2irA=1Xn=Z1drexp)]TJ /F3 11.955 Tf 9.3 0 Td[(r2Sw+2i(A)]TJ /F3 11.955 Tf 11.95 0 Td[(n)r=1Xn=r Swexp)]TJ /F3 11.955 Tf 10.5 8.09 Td[(2(A)]TJ /F3 11.955 Tf 11.96 0 Td[(n)2 Sw; (C)withA=E?=!c)]TJ /F7 11.955 Tf 11.95 0 Td[(1=2,ityieldsg(E)=m 221Xn=r SwZp 2mE0dkzexp8<:)]TJ /F3 11.955 Tf 12.75 8.09 Td[(2 Sw"E)]TJ /F4 7.97 Tf 14.63 5.69 Td[(k2z 2m !c)]TJ /F10 11.955 Tf 11.96 16.86 Td[(1 2+n#29=; (C)Bychangeofvariabley=k2z=(2m!cp Sw),Eq.( C )becomes g(E)=(2m)3=2 82r !c Sw1Xn=ZE !cp Sw0dy p ySw 21=4exp )]TJ /F10 11.955 Tf 11.29 16.86 Td[(y)]TJ /F3 11.955 Tf 22.76 8.09 Td[( p SwE !c)]TJ /F10 11.955 Tf 11.96 16.86 Td[(n+1 22!:(C)ForhighLandaulevels,E=!cn1.Therefore,theupperlimitofintergralE=!cp SwcanbeextendedtoinnityifSwn2.Therefore, g(E)=Xngn(E)=(2m)3=2 82!2c Sw1=4XnZ10dy p yexp)]TJ /F7 11.955 Tf 9.29 0 Td[((y)]TJ /F3 11.955 Tf 11.95 0 Td[(hn)2;(C)where hn= p SwE !c)]TJ /F10 11.955 Tf 11.96 16.86 Td[(n+1 2:(C) 83 PAGE 84 REFERENCES [1] W.ShubnikovandW.J.deHaas,Proc.Neth.R.Acad.Sci.,33,130(1930). [2] W.J.deHaasandP.M.vanAlphen,Proc.Neth.R.Acad.Sci.,33,1106(1930). [3] L.D.Landau,Z.Phys.64,629-637(1930). [4] L.Onsager,Phil.Mag.43,1006(1952). [5] I.M.LifshitzandA.M.Kosevich,Sov.Phys.JETP2,636(1956).Peierls,Z.Phys.1,186(1933). [6] R.B.Dingle,Proc.Roy.Soc.A211,517(1952). [7] J.W.WilkinsandJ.W.F.Woo,Phys.Lett.17,89(1965). [8] M.FowlerandR.E.Prange,Physics1,315(1965). [9] F.Pelzer,Phys.Rev.B44,293(1991). [10] A.Wasserman,M.Springford,andF.Han,J.Phys.Condens.Matter3,5335(1991). [11] H.L.Stormer,T.Haavasoja,V.Narayanamurti,A.C.Gossard,andW.Wiegman,J.Vac.Sci.Technol.B2,423(1983). [12] E.Gornik,R.Lassnig,G.Strasser,H.L.Stormer,A.C.Gossard,andW.Wiegmann,Phys.Rev.Lett.54,1820(1985). [13] T.Okamoto,K.Hosoya,S.Kawaji,andA.Yagi,Phys.Rev.Lett.82,3875(1999). [14] V.M.Pudalov,M.E.Gershenson,H.Kojima,N.Butch,E.M.Dizhur,G.Brunthaler,A.Prinz,andG.Bauer,Phys.Rev.Lett.88,196404(2002). [15] V.Sa-yakanit,N.Choosiri,andHenryR.Glyde,Phys.Rev.B38,1340(1988). [16] K.Jauregui,V.I.Marchenko,andI.D.Vagner,Phys.Rev.B41,12922(1990). [17] N.Kimura,M.Endo,T.Isshiki,S.Minagawa,A.Ochiai,H.Aoki,T.Terashima,S.Uji,T.Matsumoto,andG.G.Lonzarich,Phys.Rev.Lett.92,197002(2004). [18] A.McCollam,S.R.Julian,P.M.C.Rourke,D.Aoki,andJ.Flouquet,Phys.Rev.Lett.94,186401(2005). [19] C.Capan,L.Balicas,T.P.Murphy,E.C.Palm,R.Movshovich,D.Hall,S.W.Tozer,M.F.Hundley,E.D.Bauer,J.D.Thompson,J.L.Sarrao,J.F.DiTusa,R.G.Goodrich,andZ.Fisk1,Phys.Rev.B80,094518(2009). 84 PAGE 85 [20] C.Capan,Y.-J.Jo,L.Balicas,R.G.Goodrich,J.F.DiTusa,I.Vekhter,T.P.Murphy,A.D.Bianchi,L.D.Pham,J.Y.Cho,J.Y.Chan,D.P.Young,andZ.Fisk,Phys.Rev.B82,035112(2010). [21] N.Doiron-Leyraud,C.Proust,D.LeBoeuf,J.Levallois,J-B.Bonnemaison,R.Liang,D.A.Bonn,W.N.Hardy,andLTaillefer,Nature447,565(2007). [22] CyrilJaudet,DavidVignolles,AlainAudouard,JulienLevallois,D.LeBoeuf,NicolasDoiron-Leyraud,B.Vignolle,M.Nardone,A.Zitouni,RuixingLiang,D.A.Bonn,W.N.Hardy,L.Taillefer,andCyrilProust,Phys.Rev.Lett.100,187005(2008). [23] B.Vignolle,A.Carrington,R.A.Cooper,M.M.J.French,A.P.Mackenzie,C.Jaudet,D.Vignolles,CyrilProust,andN.E.Hussey,Nature455,952-955(2008). [24] AlainAudouard,CyrilJaudet,DavidVignolles,RuixingLiang,D.A.Bonn,W.N.Hardy,LouisTaillefer,andCyrilProust,Phys.Rev.Lett.103,157003(2009). [25] J.M.Tranquada,D.N.Basov,A.D.LaForge,andA.A.Schafgans,Phys.Rev.B81,060506(R)(2010). [26] C.Putzke,A.I.Coldea,I.Guillamon,D.Vignolles,A.McCollam,D.LeBoeuf,M.D.Watson,I.I.Mazin,S.Kasahara,T.Terashima,T.Shibauchi,Y.Matsuda,andA.Carrington,cond-mat.supr-con/arXiv:1107.4375. [27] A.Carrington,cond-mat.supr-con/arXiv:1110.1326v1. [28] LarsFritzandSubirSachdev,Phys.Rev.B82,045123(2010). [29] L.ThompsonandP.C.E.Stamp,Phys.Rev.B80,100514(2010). [30] M.E.RaikhandT.V.Shahbazyan,Phys.Rev.B47,1522(1992). [31] K.Behnia,M.A.Measson,andY.Kopelevich,Phys.Rev.Lett.98,166602(2007). [32] K.Behnia,L.Balicas,andY.Kopelevich,Science317,1729(2007). [33] V.L.Ginzburg,andL.D.Landau,Sov.Phys.JETP20,1064(1950). [34] A.V.Chubukov,D.L.Maslov,S.Gangadharaiah,andL.I.Glazman,Phys.Rev.B71,205112(2005). [35] J.Rech,C.Pepin,andA.V.Chubukov,Phys.Rev.B74,195126(2006). [36] P.A.Lee,Phys.Rev.Lett.63,680(1989). [37] L.B.IoffeandA.I.Larkin,Phys.Rev.B39,8988(1989) [38] J.A.Hertz,Phys.Rev.B14,1165(1976). 85 PAGE 86 [39] A.J.Millis,Phys.Rev.B48,7183(1993). [40] T.Moriya,SpinFluctuationsinItinerantElectronMagnetism,Springer-Verlag(1985) [41] I.Paul,C.Pepin,B.N.Narozhny,D.L.Maslov,Phys.Rev.Lett.95,017206(2005). [42] I.Paul,Phys.Rev.B77,224418(2008). [43] A.D.Mirlin,E.Altshuler,andP.Wole,AnnalenderPhysik5,281(1996). [44] NeilW.AshcroftandN.DavidMermin,SolidStatePhysics,SaundersCollegePublishing(1976). [45] C.J.Palin,Proc.R.Soc.A329,17(1972). [46] J.M.Luttinger,Phys.Rev.119,1153(1960). [47] J.M.LuttingerandJ.C.Ward,Phys.Rev.118,1417(1961). [48] S.EngelsbergandG.Simpson,Phys.Rev.B2,1657(1970). [49] S.Engelsberg,Phys.Rev.B16,966(1978). [50] A.WassermanandM.Springford,Adv.Phys.45,471(1996). [51] Yu.A.Bychkov,SovietPhys.JETP12,977(1961). [52] B.G.S.Doman,J.Phys.Chem.Solids27,233-1236(1966). [53] M.I.Dyakonov,A.L.Efros,andD.L.Mitchell,Phys.Rev.180,813(1969). [54] T.AndoandY.Uemura,J.Phys.Soc.Jpn.36,959(1974). [55] HagenKleinert,PathIntegralsinQuantumMechanics,Statistics,andPolymerPhysics,3rdEd.,WorldScientic(2004). [56] MatrinC.Gutzwiller,J.Math.Phys.8,1979(1967). [57] MartinC.Gutzwiller,J.Math.Phys.12,343(1971). [58] G.W.Martin,D.L.Maslov,andM..Y.Reizer,Phys.Rev.B68,241309(2003). [59] S.Curnoe,andP.C.E.Stamp,Phys.Rev.Lett.80,3312(1998). 86 PAGE 87 BIOGRAPHICALSKETCH ChungweiWangwasborninTaiwanin1976.Afterreceivinghisbachelor'sdegreeinphysics(1998)andmasterofscienceinphysics(2000)fromNationalTsing-HuaUniversity,heholdajobofdoingconstructionandmaintenanceofexperimentalstationsinNationalSynchrotronRadiationResearchCenterfortwoyearsanddevelopedalgorithmsofcomputersimulationsofnon-crystallinex-raydiffractionmicroscopyatInstituteofPhysicsinAcademiaSinicafor3yearsasanalternativeofthemandatorymilitaryservice.In2005,hecametotheUniversityofFloridaforthephysicsdoctoralprogram.HedefendedhisPh.D.dissertationonDecember1,2011. 87 |