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Quantum Magnetooscillations near Classical and Quantum Phase Transitions

Permanent Link: http://ufdc.ufl.edu/UFE0043755/00001

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Title: Quantum Magnetooscillations near Classical and Quantum Phase Transitions
Physical Description: 1 online resource (87 p.)
Language: english
Creator: Wang, Chungwei
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: ferromagnetic -- magnetooscillations -- phase -- quantum -- quasiclassical
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We study the density of states (DOS) and quantum magnetooscillations both in a three-dimensional (3D) and in quasi-two-dimensional (2D) strongly correlated systems near the ferromagnetic-type quantum and classical critical points. Recent experiments on magnetooscillations in the Nernst coefficient of bismuth have reignited interest to the DOS and magnetooscillations in a 3D disordered system, which, to the best of our knowledge, have not been investigated thoroughly yet. We adopt the self-consistent Born approximation for short-range disorders, while using the quasiclassical path integral approach for long-range disorders. In the case of short-range disorders, we generalize the Doman's approach, who considered weak disorders and found the self-energy self-consistently by taking into account scattering only within a single Landau level, to the case of the inter-Landau level scattering, and obtain an expression for the density of state valid in the regimes of both weak and strong damping. In the case of long-range disorders, we demonstrate using the path-integral approach that there is an inhomogeneous broadening of Landau levels in 3D. We also study many-body effects in quantum magnetooscillations of a quasi-two-dimensional strongly correlated system near the critical point. The amplitude of magnetooscillations is determined by the electron self-energy, which is of the non-Fermi-liquid form near the quantum critical point (QCP). We demonstrate, however, that the correct result cannot be obtained simply by substituting the T = 0 self-energy into the Lifshitz-Kosevich formula. Our finding is that the divergence of the correlation length near a critical point implies necessarily that static fluctuation has a dominant effect on magnetooscillations, and the damping takes a Gaussian form in the inhomogeneous broadening regime. This leads to strong deviations of the oscillation amplitude from the Lifshitz-Kosevich form. This very different from previous studies that considered only the dynamic part of the self-energy. Taking the temperature dependence of the correlation length into account, we analyze the temperature dependence of thermal damping as well.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Chungwei Wang.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Maslov, Dmitrii.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-06-30

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System ID: UFE0043755:00001

Permanent Link: http://ufdc.ufl.edu/UFE0043755/00001

Material Information

Title: Quantum Magnetooscillations near Classical and Quantum Phase Transitions
Physical Description: 1 online resource (87 p.)
Language: english
Creator: Wang, Chungwei
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: ferromagnetic -- magnetooscillations -- phase -- quantum -- quasiclassical
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: We study the density of states (DOS) and quantum magnetooscillations both in a three-dimensional (3D) and in quasi-two-dimensional (2D) strongly correlated systems near the ferromagnetic-type quantum and classical critical points. Recent experiments on magnetooscillations in the Nernst coefficient of bismuth have reignited interest to the DOS and magnetooscillations in a 3D disordered system, which, to the best of our knowledge, have not been investigated thoroughly yet. We adopt the self-consistent Born approximation for short-range disorders, while using the quasiclassical path integral approach for long-range disorders. In the case of short-range disorders, we generalize the Doman's approach, who considered weak disorders and found the self-energy self-consistently by taking into account scattering only within a single Landau level, to the case of the inter-Landau level scattering, and obtain an expression for the density of state valid in the regimes of both weak and strong damping. In the case of long-range disorders, we demonstrate using the path-integral approach that there is an inhomogeneous broadening of Landau levels in 3D. We also study many-body effects in quantum magnetooscillations of a quasi-two-dimensional strongly correlated system near the critical point. The amplitude of magnetooscillations is determined by the electron self-energy, which is of the non-Fermi-liquid form near the quantum critical point (QCP). We demonstrate, however, that the correct result cannot be obtained simply by substituting the T = 0 self-energy into the Lifshitz-Kosevich formula. Our finding is that the divergence of the correlation length near a critical point implies necessarily that static fluctuation has a dominant effect on magnetooscillations, and the damping takes a Gaussian form in the inhomogeneous broadening regime. This leads to strong deviations of the oscillation amplitude from the Lifshitz-Kosevich form. This very different from previous studies that considered only the dynamic part of the self-energy. Taking the temperature dependence of the correlation length into account, we analyze the temperature dependence of thermal damping as well.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Chungwei Wang.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Maslov, Dmitrii.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2013-06-30

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0043755:00001


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QUANTUMMAGNETOOSCILLATIONSNEARCLASSICALANDQUANTUMPHASETRANSITIONSByCHUNGWEIWANGADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

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c2011ChungweiWang 2

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Idedicatethistoeveryonewhoisinterestedinthistopic. 3

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ACKNOWLEDGMENTS IwouldliketogivethankstoallthehelpandguideIhavereceivedfromDr.MaslovduringmyPhDresearchcareer.IalsowanttothanktoHridisPal,AliAshra,VivekMishra,andallothercolleaguesfordiscussions. 4

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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

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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

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LISTOFTABLES Table page 2-1Orderparametersandtheirconjugateelds .................... 21 4-1AmplitudeCoefcientsIsandIcwithdifferentscatteringstrength ........ 59 7

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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

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LISTOFSYMBOLS BmagneticeldsstaticspinsusceptibilityEelectronicenergyEFFermienergy"electronicenergywithrespecttothechemicalpotential,i.e."=E)]TJ /F3 11.955 Tf 11.96 0 Td[(g(")densityofstatesatenergy"GMatsubaraGreen'sfunctionGRretardedGreen'sfunctionJnBesselfunctionoftherstkindFFermiwavelengthlBmagneticlengthLnmassociatedLaguerrepolynomialmelectronbandmassMmagnetizationmccyclotronmassmrenormalizedelectronmasschemicalpotentialkdirectionparalleltotheappliedmagneticeld,oralongthezdirection?directionperpendiculartotheappliedmagneticeld,orinthexyplane 9

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thermodynamicpotentialNeelectronnumberdensity!ccyclotronfrequencyRccyclotronradiusMatsubaraself-energyRretardedself-energySwdampingfactorofmagnetooscillationsinthequasiclassicalregimetkhoppingintegralinaquasi-two-dimensionalsystemalongthezdirectionwherethemagneticeldisappliedV(r)order-parameterdisorderatthepositionrW(q)correlationfunctioninthemomentumspacefW(r)correlationfunctioninthepositionspacecorrelationlengthofinteractions 10

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LISTOFABBREVIATIONS dHvAdeHaas-vanAlphen(effect)DOSdensityofstatesFLFermiliquidGLGinzburg-Landau(theory)LKLifshitz-KosevitchNFLnon-FermiliquidQCPquantumcriticalpointSCBAself-consistentBornapproximationSdHShubnikov-deHaas(effect)3Dthree-dimensional2Dtwo-dimensional 11

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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

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formula.Ourndingisthatthedivergenceofthecorrelationlengthnearacriticalpointimpliesnecessarilythatstaticuctuationhasadominanteffectonmagnetooscillations,andthedampingtakesaGaussianformintheinhomogeneousbroadeningregime.ThisleadstostrongdeviationsoftheoscillationamplitudefromtheLifshitz-Kosevichform.Thisverydifferentfrompreviousstudiesthatconsideredonlythedynamicpartoftheself-energy.Takingthetemperaturedependenceofthecorrelationlengthintoaccount,weanalyzethetemperaturedependenceofthermaldampingaswell. 13

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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

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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

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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

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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

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Figure1-2. TemperaturedependenceofthedeHaas-vanAlphenamplitudeinCeCoIn5withthemagneticeldat6-7T.ThedatashowthatabettertisobtainedbyusingaNFLformbelow20mK,whichisnotdescribedbytheLKformula(inset).However,theextendedLKexpressionwithaNFLformofselfenergy/p T,neartheantiferromagneticQCP,tswell(mainplot).Thisresultissuggestiveonly.(ReprintedgurewithpermissionfromRef.[ 18 ].CopyrightbytheAmericanPhysicalSociety,2005) 18

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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

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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

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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

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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,whereT0.ThecoefcientbispositivetoguaranteeisrealwhenTTc.Inthemean-eldapproximation, L=ZdDxa 22(r)+c 4(r(r))2+J(r)(r)=Xka 2+c 4k2eke)]TJ /F19 7.97 Tf 6.59 0 Td[(k)]TJ /F10 11.955 Tf 14.38 3.02 Td[(eJke)]TJ /F19 7.97 Tf 6.58 0 Td[(k;(2) 22

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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

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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

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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

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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

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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

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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!
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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

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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

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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
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WhileapproachingtheQCP,thecorrelationlengthgetslongerandlonger,eachelectronstartstoseethecurvatureofitstrajectoryoncethecorrelationlengthisofsameorderofthemagneticlengthlB.IfthecorrelationlengthislongerthanthecyclotronlengthRc,theeffectofmagneticeldonthetrajectoryofelectronsareevenmorepronouncedandthereshowsaninhomogeneousbroadening[ 30 43 ].Theself-energywithouttakingthemagneticeldintoaccountisthususefulformagnetooscillationsonlywhenthecorrelationlengthissufcientlyshort. 32

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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

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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

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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

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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

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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

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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

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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

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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

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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=max(n;n0),andLjn)]TJ /F4 7.97 Tf 6.59 0 Td[(n0jN
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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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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

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BIOGRAPHICALSKETCH ChungweiWangwasborninTaiwanin1976.Afterreceivinghisbachelor'sdegreeinphysics(1998)andmasterofscienceinphysics(2000)fromNationalTsing-HuaUniversity,heholdajobofdoingconstructionandmaintenanceofexperimentalstationsinNationalSynchrotronRadiationResearchCenterfortwoyearsanddevelopedalgorithmsofcomputersimulationsofnon-crystallinex-raydiffractionmicroscopyatInstituteofPhysicsinAcademiaSinicafor3yearsasanalternativeofthemandatorymilitaryservice.In2005,hecametotheUniversityofFloridaforthephysicsdoctoralprogram.HedefendedhisPh.D.dissertationonDecember1,2011. 87