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Experiments on Electron Interaction and Localization in Disordered Magnetic Thin Films

Permanent Link: http://ufdc.ufl.edu/UFE0024640/00001

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Title: Experiments on Electron Interaction and Localization in Disordered Magnetic Thin Films
Physical Description: 1 online resource (122 p.)
Language: english
Creator: Misra, Rajiv
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: anomalous, disorder, ferromagnetism, gadolinium, hall, insulator, interactions, itinerant, localization, magnetoresistance, metal, spinwave, thinfilm
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

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Abstract: We present results of experimental in-situ magneto-transport studies on thin ferromagnetic films for a range of disorder values, characterized by sheet resistance R_{0}=R_{xx}(T = 5 K). In the limit of weak disorder, R_{0} < hbar/e^{2}, studies of quantum corrections to the conductivity tensor in thin films of iron, an itinerant ferromagnet, reveals evidence for a disorder-dependent localization corrections to the anomalous Hall (AH) conductivity. For low values of disorder (R_{0} < 150 Ohms), AH conductivity receives no quantum corrections (delta sigma_{xy} / sigma_{xy} =0). As disorder increases, a finite logarithmic temperature dependence to AH conductivity appears and finally evolves towards a universal weak localization correction (delta sigma_{xy} /sigma_{xy}=-delta R_{xy} / R_{xy}). The studies on a series of gadolinium thin films, in the weak disorder limit, shows the existence of a new type of quantum correction to conductivity, which has an approximately linear temperature dependence. We attribute it to spin-wave-mediated quantum corrections to conductivity, analogous to the Altshuler-Aronov electron-electron contribution in disordered systems. Finally, with the advancement into the strong disorder regime (R_{0} > hbar/e^{2}), an Anderson localization transition is seen.
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 Rajiv Misra.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Hebard, Arthur F.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-08-31

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024640:00001

Permanent Link: http://ufdc.ufl.edu/UFE0024640/00001

Material Information

Title: Experiments on Electron Interaction and Localization in Disordered Magnetic Thin Films
Physical Description: 1 online resource (122 p.)
Language: english
Creator: Misra, Rajiv
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2009

Subjects

Subjects / Keywords: anomalous, disorder, ferromagnetism, gadolinium, hall, insulator, interactions, itinerant, localization, magnetoresistance, metal, spinwave, thinfilm
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 present results of experimental in-situ magneto-transport studies on thin ferromagnetic films for a range of disorder values, characterized by sheet resistance R_{0}=R_{xx}(T = 5 K). In the limit of weak disorder, R_{0} < hbar/e^{2}, studies of quantum corrections to the conductivity tensor in thin films of iron, an itinerant ferromagnet, reveals evidence for a disorder-dependent localization corrections to the anomalous Hall (AH) conductivity. For low values of disorder (R_{0} < 150 Ohms), AH conductivity receives no quantum corrections (delta sigma_{xy} / sigma_{xy} =0). As disorder increases, a finite logarithmic temperature dependence to AH conductivity appears and finally evolves towards a universal weak localization correction (delta sigma_{xy} /sigma_{xy}=-delta R_{xy} / R_{xy}). The studies on a series of gadolinium thin films, in the weak disorder limit, shows the existence of a new type of quantum correction to conductivity, which has an approximately linear temperature dependence. We attribute it to spin-wave-mediated quantum corrections to conductivity, analogous to the Altshuler-Aronov electron-electron contribution in disordered systems. Finally, with the advancement into the strong disorder regime (R_{0} > hbar/e^{2}), an Anderson localization transition is seen.
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 Rajiv Misra.
Thesis: Thesis (Ph.D.)--University of Florida, 2009.
Local: Adviser: Hebard, Arthur F.
Electronic Access: RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2011-08-31

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2009
System ID: UFE0024640:00001


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EXPERIMENTSONELECTRONINTERACTIONANDLOCALIZATIONIN DISORDEREDMAGNETICTHINFILMS By RAJIVMISRA ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2009 1

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c r 2009RajivMisra 2

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Tomyparents 3

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ACKNOWLEDGMENTS IwouldliketoexpressmysinceregratitudeandthanktoArtH ebard,myresearch Advisor.Hehasbeenaverykindheartedpersonfullofyouthf ulvigourforthepursuitof science.Hisexperienceandguidancehaveprovedinvaluabl einmyresearcheorts.His positive,open-mindedattitudetowardresearchcreatesau niquelaboratoryenvironment fullofencouragement.IamthankfulandgratefultoKhandka rMuttalibandPeter Woelreforwonderfuldiscussionsandaveryfruitfulcollab oration. IwouldliketothankAmIanBiswasforvaluableexperimental suggestionsandhelp. Discussionwithhimalwaysledtoabetterunderstandingofe xperiments.Ialsothank DimitriMaslovforthewonderfulsolidstatephysicshetaug ht.Iamthankfultomy committeemembersforgivingmetheirinvaluabletimeandsu ggestion.Iwouldliketo thankallofmylabmembersandfriendsfortheirsupportande ncouragement.Ialso thanksJayfrompumpshop,GregandJohnfromcryogenicservi ces,Ed,Marc,Bill,John, SkipandMikefrommachineshop,Pete,LarryandRobfromelec tronicshopfortheir invaluableservices. Iamindebtedtomyparentsfortheirsupportandconstantenc ouragement.I appreciatethewarmthandaectionofmybrothersandsister .Finally,Ithankmyloving wife,Preeti,forherunconditionallove,supportandunder standing. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................4 LISTOFTABLES .....................................8 LISTOFFIGURES ....................................9 ABSTRACT ........................................12 CHAPTER 1INTRODUCTION ..................................13 2HALLEFFECTINFERROMAGNETICMETALS ...............18 2.1Ferromagnetism .................................18 2.1.1ItinerantElectronMagnetism .....................18 2.1.2FerromagnetismofLocalizedMoments ................19 2.2NormalHallEect ...............................21 2.3AnomalousHallEect .............................22 2.3.1SkewScatteringMechanism ......................23 2.3.2SideJumpScatteringMechanism ...................24 2.3.3AnomalousHallEectinSystemswithLocalizedMagnet icCarriers 25 3EFFECTOFDISORDERONTRANSPORTPROPERTIES ..........27 3.1WeakLocalization ...............................27 3.2EectofMagneticFieldonWeakLocalization ................29 3.3Antilocalization .................................31 3.4TheEectofElectron-ElectronInteraction ..................32 3.5QuantumCorrectionstoHallConductivity ..................36 3.5.1NormalHallConductivity .......................36 3.5.2AnomalousHallConductivity .....................37 3.5.2.1QuantumCorrectionsfromElectron-ElectronInter actions 37 3.5.2.2QuantumCorrectionsfromWeakLocalization .......37 3.6AndersonLocalizationTransition .......................38 4EVOLUTIONOFQUANTUMCORRECTIONSWITHDISORDER ......40 4.1Introduction ...................................40 4.2ExperimentalDetails ..............................41 4.2.1ContactPadPreparation ........................41 4.2.2ThinFilmDeposition ..........................41 4.2.3MeasurementTechniques ........................42 4.3MeasurementsandAnalysis ..........................45 4.4Conclusions ...................................52 5

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5QUAUNTUMCORRECTIONSTOCONDUCTIVITYINTHINFILMSOFGD 54 5.1Introduction ...................................54 5.2ExperimentalDetails ..............................55 5.3MeasurementsandAnalysis ..........................55 5.4Conclusions ...................................62 6ANDERSONTRANSITIONINTHINFILMSOFGADOLINIUM .......63 6.1Introduction ...................................63 6.2ExperimentalDetails ..............................64 6.2.1ContactPadPreparation ........................64 6.2.2ThinFilmDeposition ..........................65 6.3MeasurementsandAnalysis ..........................67 6.3.1TransportPropertiesatZeroMagneticField .............67 6.3.1.1SamplesinSeries1 ......................67 6.3.1.2SamplesinSeries2 ......................73 6.3.2TransportPropertiesinthePresenceofMagneticFiel d .......82 6.3.2.1EectsonLongitudinalConductivity ............82 6.3.2.2BehaviorofHallResistancewithTemperature .......85 6.4Conclusions ...................................91 7SUMMARYANDFUTUREWORK ........................92 7.1Summary ....................................92 7.2ProposedFutureDirections ..........................94 APPENDIX AELECTRICFIELDGATINGWITHIONICLIQUIDS ..............96 A.1Introduction ...................................96 A.2SampleFabrication ...............................97 A.3MeasurementandAnalysis ...........................98 A.4Conclusions ...................................103 BANOMALOUSHALLEFFECTINIRONANDGADOLINIUMFILMS ....105 B.1Introduction ...................................105 B.2ExperimentalDetailsandMeasurementTechniques .............105 B.3MeasurementsandAnalysis ..........................106 B.4Conclusions ...................................111 COBSERVATIONOFAPOWERLAWBEHAVIORINTHINFILMS ......113 C.1Introduction ...................................113 C.2ExperimentalDetailsandMeasurementTechniques .............113 C.3ResultsandAnalysis ..............................114 C.4Conclusions ...................................116 6

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REFERENCES .......................................118 BIOGRAPHICALSKETCH ................................122 7

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LISTOFTABLES Table page 4-1Growthparametersforironthinlms ........................41 4-2Halleectmeasurementbypermutation ......................45 5-1Growthparametersforgadoliniumlms ......................55 6-1Growthparametersforgadoliniumlms ......................65 6-2Fittingparameters ..................................78 C-1Growthparametersforironlms ..........................113 C-2Growthparametersforcopperlms .........................113 8

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LISTOFFIGURES Figure page 1-1PictureofSHIVA(SampleHandlingInVAcuum). ................16 2-1Typicalplotoftransverseresistance R xy asafunctionofmagneticeld .....22 3-1Closedelectrontrajectoriesinwhichtheycantravelin oppositedirections. ...28 4-1Goldcontactpads(yellowincolour)onsapphiresubstra te. ...........41 4-2Hallbargeometrywithtwocurrentleadsandfourvoltage leads. .........42 4-3Measurementsusingaclock-intechnique ......................43 4-4Measurementsusingfourterminaldctechnique ..................44 4-5Evolutionofquantumcorrectionswithdisorder ..................47 4-6Evolutionofquantumcorrectionswithdisorder(contin ued) ............48 4-7Evolutionofquantumcorrectionswithdisorder(contin ued). ............49 4-8Coecients A R and A AH foraseriesofironthinlms ...............50 5-1Temperaturedependenceonalogarithmicscaleof xx fortwoGdthinlms ..57 5-2Dependenceofthettingparametersonthedisorderpara meter R 0 .......58 5-3Normalizedrelativechange, N ( xy ),fortwogadoliniumlms ..........61 5-4Spin-wavecontributionstothelongitudinalconductiv ity .............62 6-1Fingeredcontactpad. ................................65 6-2Hallcrossgeometryusedforgadoliniumlms. ...................66 6-3Temperaturedependenceofconductivityforsamplesins eries1 .........68 6-4Temperaturedependenceofconductivityforsamplesins eries1(continued) ...69 6-5Temperaturedependenceofconductivityforsamplesins eries1(continued). ..70 6-6Behavioroftheparameter A forallthesamplesinseries1. ............71 6-7Behaviorofthepower p forallthesamplesinseries1. ..............72 6-8Behavioroftheprefactor B forallthesamplesinseries1. ............72 6-9Temperaturedependenceofconductivityforsamplesins eries2 .........74 6-10Temperaturedependenceofconductivityforsamplesin series2(continued) ...75 9

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6-11Temperaturedependenceofconductivityforsamplesin series2(continued). ..76 6-12BehavioroftheconstantAasafunctionof R 0 forbothseries1andseries2. .77 6-13BehavioroftheconstantAasafunctionof R 0 forseries2. ...........77 6-14BehavioroftheconstantAasafunctionof R 0 ...................79 6-15BehaviorofthecoecientBandpowerpasafunctionof R 0 (disorder) .....79 6-16Scalingregimeasafunctionofdisorder. ......................80 6-17Signicanceoftheparameter A inthemetallicstate. ...............81 6-18Signicanceoftheparameter A atthecriticalpoint. ...............81 6-19Signicanceoftheparameter A intheinsulatingstate. .............82 6-20Plotofconductivityversustemperaturefor R 0 =23295n. ............83 6-21Plotofconductivityversustemperaturefor R 0 =27872n. ............83 6-22Plotofconductivityversustemperaturefor R 0 =31010n. ............84 6-23Plotofconductivityversustemperaturefor R 0 =34819n. ............84 6-24BehavioroftheconstantAwithmagneticeld. ..................86 6-25Plotof R xy versustemperaturefor R 0 =23295n. .................86 6-26Plotof R xy versustemperaturefor R 0 =27872n. .................87 6-27Plotof R xy versustemperaturefor R 0 =31010n. .................87 6-28Plotof R xy versustemperaturefor R 0 =34819n. .................88 6-29Plotof R xy versustemperaturefor R 0 =37918n. .................88 6-30Plotof R xy versustemperaturefor R 0 =43792n. .................89 6-31PlotofanomalousHallconductivityvstemperature ................90 A-1Congurationsandcircuitmodelfortheeldgatingexpe riment .........98 A-2Frequencydependenceofthesheetimpedanceattheindic atedgatevoltages ..100 A-3Temperaturedependenceoftheimpedanceatdierentgat evoltages .......102 B-1SchematicdiagramofFe/C 60 deposition. .....................106 B-2HalldataforFe/C 60 lmat T =50K. .......................107 B-3 R xx vstemperatureplotforaFe/C 60 lm .....................107 10

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B-4HalldataforFe/C 60 lmatseveraltemperature. .................108 B-5Plotof r ( T )vs T forFe/C 60 lms. .........................109 B-6Plotof 0 R s M s =d vs T foraFe/C 60 lm ......................109 B-7HalldataforaGd( R 0 =428n)lmat T =5 K alongwithaLangevint. ...110 B-8HalldataforaGd( R 0 =428n)lmatdierenttemperature. ..........110 B-9Plotof r ( T )vs T foraseriesofGdlms. .....................111 B-10Plotof 0 R s M s =d vs T foraseriesofGdlms ...................112 C-1 R xx asafunctionoftemperatureforaseriesofironthinlms. ..........114 C-2Plotofparameter W asafunctionoftemperatureforthinlmofiron .....115 C-3 R xx asafunctionoftemperatureforaseriesofcopperthinlms. ........116 C-4Plotofparameter W asafunctionoftemperatureforthinlmofcopper ....117 11

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy EXPERIMENTSONELECTRONINTERACTIONANDLOCALIZATIONIN DISORDEREDMAGNETICTHINFILMS By RajivMisra August2009 Chair:ArthurF.HebardMajor:Physics Wepresentresultsofexperimental in-situ magneto-transportstudiesonthin ferromagneticlmsforarangeofdisordervalues,characte rizedbysheetresistance R 0 R xx ( T =5K).Inthelimitofweakdisorder, R 0 < ~ =e 2 ,studiesofquantum correctionstotheconductivitytensorinthinlmsofiron, anitinerantferromagnet, revealsevidenceforadisorder-dependentlocalizationco rrectionstotheanomalousHall (AH)conductivity.Forlowvaluesofdisorder( R 0 < 150n),AHconductivityreceives noquantumcorrections( xy = xy =0).Asdisorderincreases,anitelogarithmic temperaturedependencetoAHconductivityappearsandnal lyevolvestowardsa universalweaklocalizationcorrection( xy = xy = R xy =R xy ).Thestudiesonaseries ofgadoliniumthinlms,intheweakdisorderlimit,showsth eexistenceofanewtype ofquantumcorrectiontoconductivity,whichhasanapproxi matelylineartemperature dependence.Weattributeittospin-wave-mediatedquantum correctionstoconductivity, analogoustotheAltshuler-Aronovelectron-electroncont ributionindisorderedsystems. Finally,withtheadvancementintothestrongdisorderregi me( R 0 > ~ =e 2 ),anAnderson localizationtransitionisseen. 12

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CHAPTER1 INTRODUCTION Attemptstounderstandtheelectricalconductivityofamet alstartedalmosta centuryago.Oneoftheimportantstepsinthedevelopmentof thisunderstandingisthe pioneeringworkofBlochin1928.TheBlochtheoremgovernst hebehaviorofelectron inametalwithaperfectperiodicarrangementofatoms.Theq uantummechanical wavefunctionoftheelectronisinanextendedstate.Thisim pliesthattheelectroncan movefreelythroughoutthelatticeresultinginhighelectr icalconductivity. Thenextimportantstepinthisunderstandingistheintrodu ctionofdisorderor deviationsfromperiodicity.Thiscanbeunderstoodintwol imitingcases.Firstisthe weakdisorderlimit.Theelectronwavefunctionremainsext endedthroughoutthesample butitlosesitsphasecoherenceonthelengthscaleoftheord erofmeanfreepath l .Inthis limit,withtheoccurenceofaseriesofscatteringevents,t heinterferenceofelectronwith itself(weaklocalization)andwithotherelectrons(elect ron-electroninteractions)presents quantumcorrectionstotheconductivity.Insimplewords,t heresistanceofthemetal increases.Manyexperimentshavebeenperformedonnon-mag neticmetalstostudythese quantumcorrectiontoconductivity,butthecaseofelectro ntransportinferromagnetic metalsisstillunresovedandneedsalotofexperimentation Thesecondlimitingcaseintheunderstandingofdisorderoc curswhendisorder isstrong.Onewouldnowthinkthattheeigenfunctionofthee lectronwouldhave variationinamplitudeaboutaconstantvalue,butwouldrem ainextendedinnature. Surprizingly,thisisnotthecase.Rather,thewavefunctio noftheelectroninthe presenceofstrongdisorderdecaysexponentially.Thisphe nomenoniscalledAnderson localization.Theremarkablepredictionofthescalingthe oryoflocalization,theexistence ofametal-insulatortransitionfordimension d> 2,ledtoanextensiveresearchinthe eldofcondensedmatterphysics.Itisstillnotfullyunder stoodandexamplesofsystems showingsuchtransitionsareneeded. 13

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Magnetismisaphenomenonthathasbeenstudiedforthousand sofyears.Although theuseofmagneticneedlecompassdatesbacktotheeleventh century,itwasnot untillaround1600thatscienticprinciplebehindthework ingofthecompasswas proposed.Studyoftherelationshipbetweenelectricityan dmagnetismbeganwiththe accidentaldiscoverybyOerstedin1819.Thissparkedaseri esoffamousexperiments byAmpere,Gauss,Faradayandothers.Maxwellextendedthes einsightsintoaeldof electromagnetismbyacarefulderivationofasetoffourequ ations.Thisunderstanding wasfurtherdevelopedbyEinsteininhistheoryofrelativit y,whichgaveanewperspective tounderstandmagnetism. Thepresenceofmagnetisminasystemmakestheunderstandin goftheelectronic transportevenmorechallengingbypresentinganextradegr eeoffreedominterms oflongrangeorder.Animportanttoolthathasbeenusedtost udyferromagnetism inmagneticmaterialsistheanomalousHall(AH)eect.Rece ntly,therehasbeena considerablegrowthofinterestintheAHeectduetotheimp ortanceassociatedwiththe understandingofspinpolarizationandspin-orbitinterac tionsforelectronictransportin materialsandstructuresforspinelectronics.Thespinoft heelectronwasusuallyignored inconventionalelectronics.Asthetransportofelectronc reatesachargecurrent,spinof theelectroncancreateaspincurrent,thusaddinganewdegr eeoffreedomtoelectronics andcreationofaneweldof\spintronics".Thisisbasedont hecontrolandmanipulation ofthespinoftheelectroninadditiontoitscharge.Concept sofseveralnewdeviceshave emergedwhichincludespintransistors,spinmemorydevice s,quantumcomputersetc. Inthisdissertation,weaddresstheeectofdisorderonthe electronictranport propertiesofthinlmsofferromagneticmaterials.Theexp erimentalstudyisdividedinto twocategories-weakdisorderlimit,wherequantumcorrect ionstotheBoltzmann (classical)conductivityareseen,andthenearlystrongdi sorderlimit.Intherst categoryofweakdisorder,westartoutwiththinlmsofiron ,whichisanitinerant ferromagnet.Thelongrangeorderisduetothecompetitionb etweentheexchange 14

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andthekineticenergyoftheconductionelectrons.Inthese lms,westudytheeect ofdisorderonthequantumcorrectionstothelongitudinala ndanomalousHall(AH) conductivity.Wedemonstrateevidenceforadisorder-depe ndentlocalizationcorrection toAHconductivity.Wethenquestionedourselvesaboutwhat wouldbethebehavior inasystemwithlocalizedmoments,likegadolinium.Toaddr essthisissue,westudied thelowtemperaturetransportpropertiesofgadoliniumthi nlms.Wendanewtype ofquantumcorrectiontotheconductivityinthissystemwit hanapproximatelylinear dependenceontemperature.Thisisattributedtospin-wave -mediatedquantumcorrections totheconductvity,analogoustotheAltshuler-Aronovelec tron-electroncontributionin disorderedsystems. Wethenaddresstheissueofwhathappenswhenwegointoaregi meofdisorder beyondthatofquantumcorrections.Weseekanswerstothequ estionabouthowthe systemgoesfromweaktostrongdisorder.Forthispurpose,w estudyultrathinlms ofgadolinium.Ourresultsindicatethatindoingso(fromwe aktostrongdisorder),the systemgoesthrougharegionwheretheconductivityhasapow erlawdependenceon temperature.WeseeevidenceforanAndersonlocalizationt ransitioningadoliniumthin lms.Also,weseeaninterestingbehaviorofanomalousHall conductivityinthesame regime. Alltheexperimentsthathavebeendiscussedabovehavebeen performed insitu in aspecialsetupcalledSHIVA(SampleHandlinginVAcuum),sh owninFig. 1-1 .Ithasa cleverdesignthatcombineadepositionchamber,aloadlock andacryostatintoasingle vacuumsystemwiththevacuumoftheorderoflow10 9 Torr.Thesystemhasspecial magneticarmsthatareusedtomovethesamplebetweenthedep ositionchamberand thecryostatviatheloadlockwithoutbreakingvacuum.Thec ryostathasa7Tmagnet andcangodownto 4 : 5Ktemperature.Otherinterestingcapabilitiesofthissys tem arethatwecangrowsamplesattemperaturesaslowas130Kand canmonitorthesheet resistanceofthesampleduringgrowth.Insummary,wehavea uniquesetupwherewecan 15

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studyultrathinlmsofmaterialswhichareextremelysensi tivetoair,withoutworrying abouttheunwantedeectsofoxidationandcorrosion. Figure1-1.PictureofSHIVA(SampleHandlingInVAcuum). Thisdissertationisorganizedasfollows.Chapter2descri besthegeneralideasof ferromagnetismwithaspecialdescriptionofanomalousHal leectasatooltostudy magneticmaterials.Variousscatteringmechanismrespons ibleforthiseectarediscussed. Chapter3focussesonthediscussionofquantumcorrections totheclassicalelectronic transport.Thephenomenonofweaklocalizationandinterac tioncorrectionsaredescribed. WealsogiveabriefdescriptionofAndersonlocalizationtr ansition.Chapter4describes 16

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theexperimentonthestudyoftheeectofdisorderonthequa ntumcorrectionsto theconductivitytensorofthinlmsofiron.Theexperiment onthestudyofquantum correctiontoconductivityinthinlmsofgadoliniumispre sentedinChapter5.The unexpectedexistenceofanewtypeofquantumcorrection,wi thanapproximatelylinear temperaturedependence,hasbeenshown.Withtheadvanceme nttowardstrongdisorder, anAndersonlocalizationtransitionisseeningadoliniumt hinlms.Thisexperimentis describedinChapter6.Summaryofalltheresultsanddiscus sionofsomepossiblefuture experimentsispresentedinChapter7.Wehavethreeappendi cesintheend.InAppendix A,wediscusstheexperimentofelectriceldgatingofindiu moxidethinlmsusingionic ruidasdielectric.WepresentresultsonHalldataofironan dgadoliniumthinlmwith theiranalysisusingaLangevintinAppendixB.Finally,in AppendixC,wepresent resultsontheobservationofapowerlawdependenceofcondu ctivityontemperaturefor ironandcopperthinlms. 17

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CHAPTER2 HALLEFFECTINFERROMAGNETICMETALS 2.1Ferromagnetism Thephenomenonofspontaneousalignmentofthespinsbelowa certaintemperature, inzeromagneticeldiscalledferromagnetism.Thistemper atureiscalledtheCurie temperature, T c .Above T c ,thealignmentislostandparamagneticsusceptibilityis observed.Inthisdissertation,wearepresentingresultsa boutironandgadoliniumthin lms.Ironbelongstothe3 d transitionmetalsgroupwherethemagnetismarisesfromthe spinoftheitinerantelectronsinthe3 d band.Incontrast,themagnetismingadolinium isduetothelocalizedmoments.Thefollowingsubsectionsp resentabriefdiscussionon thesetwomodelsofmagnetism.2.1.1ItinerantElectronMagnetism Itinerantelectronmagnetismarisesbecauseofthecompeti tionbetweentheexchange andthekineticenergyoftheelectrons.Inametallicelectr onsysteminaparamagnetic state,thereareanequalnumberofupanddownspinelectrons .Themagnetizationofthis systeminzeromagneticeldiszero.Nowimagineasituation whereweputsomespin downelectrons,withinanenergyrange oftheFermienergy E F ,inthespinupband. Thechangeinthetotalenergy,indoingthiscanbewrittenas E = E kin + E ex (2{1) where E kinetic and E exchange arethedierencesintheexchangeandkineticenergies betweenthetwostates.If N ( E F )isthedensityofstatesattheFermilevel,wecansee that E kinetic = 1 2 ( N ( E F ) ) "> 0(2{2) where(1 = 2) N ( E F ) representsthenumberofspindownelectronstransferredto thespin upelectronband.Thechangeintheexchangeenergyisgivenb y, E exchange = 1 2 J ( N ( E F ) ) 2 (2{3) 18

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where J istheexchangeinteractionbetweentheelectronswhichlow erstheenergyif thespinsarealignedparalleltoeachother.WecanseefromE qns. 2{2 and 2{3 that thekineticenergyandtheexchangeenergyplayoppositerol es.Theformeropposesa ferromagneticorder,whilethelatterfavorsit.Thustheco nditionfortheoccurrenceof ferromagnetismisgivenas E =[1 JN ( E F )] N ( E F )( ) 2 0 : (2{4) Thisinequality,theStonercondition,isquitesuccessful inexplainingtheferromagnetic behaviorofthe3 d transitionelementssuchasiron,nickelandcobalt. 2.1.2FerromagnetismofLocalizedMoments Theinteractionbetweentheatomiclocalizedspinsisdescr ibedbyaHamiltonianof thefollowingform[ 1 ], H = X i 6 = j J ij ~ S i ~ S j : (2{5) Inthemeaneldapproximation, H = g B X i ~ H mf ~ S i (2{6) where g istheLandefactor, B istheBohrmagnetonand ~ H mf ,calledthemoleculareld orthemeaneld,isgiven ~ H mf = 2 g B X j ( 6 = i ) J ij D ~ S j E : (2{7) Nowinthepresenceofanexternalmagneticeld(z-directio n),themagnitudeofthe eectivemagneticeldfeltbyaspin ~ S i isgivenby H effective = H + H mf = H 2 g B zJ h S z i ; (2{8) 19

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withtheassumptionthat J ij = J isnonzeroonlyfornearestneighborsandeachspinhas z nearestneighbors.Sowecanwritethemagnetizationas h M z i = c H effective = c H + 2 zJ ( g b ) 2 h M z i N = H; (2{9) where N isthetotalnumberofspinsand isthesusceptibilitydenedwithrespectto theexternaleld,writtenas[ 1 ] = c 1 2 z N 1 ( g B ) 2 J c : (2{10) Wecanseethatthissusceptibility,knownasCurie-Weisssu sceptibility,isenhancedby thepresenceofexchangeinteractionsbetweenthespins.It divergestoinnityatthecurie point,whichimpliesthat h M z i6 =0for H =0.Inotherwords,thereisalongrange orderingofspinsintheabsenceofmagneticeld. Inthecaseofgadolinium,thedirectoverlapofthelocalize d4 f electronsisnegligible. Thecouplingbetweenthe4 f electronsisdominatedbytheindirectRKKYinteractionin themetal[ 2 ].The4 f electronspolarizetheconductionelectrons,intheirvici nity,through adirectintra-atomicexchange.Thesepolarizedconductio nelectronsthenmediatethe interactionsbetweenthe4felectronsofdierentatoms.Th einteractionscanbedescribed bythefollowingHamiltonian: H 4 f 4 f = X i;j ( i 6 = j ) J ( ~ R ij ) ~ S i ~ S j ; (2{11) where ~ S i and ~ S j arethetotalangularmomentumofthe4 f electronsattheatoms i and j and J ( ~ R ij )istheinteractionconstantdependingonthedistance ~ R ij betweentheatoms i and j .IntheframeworkofRudermann-Kittelcalculations,theco nstant J ( ~ R ij )[ 2 ]isequal to: J ( ~ R ij )= (3 N ) 2 2 E F j (0) j 2 ( g J 1) 2 F 2 k F j ~ R i ~ R j j (2{12) 20

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whereisainteractionconstantforcouplingbetween4 f electronsandtheconduction electrons,whichdependsonthewavevectoroftheconductio nelectrons, N isthetotal numberofconductionelectronsperunitvolume, E F istheFermienergy, g J istheLande factorand F ( x )= x cos x sin x x 4 : (2{13) TheRKKYinteractionistheprincipleinteractionmechanis mbetweenthelocalized4 f spinsinarareearthmetal. 2.2NormalHallEect In1879,Hallobservedthatwhenamagneticeldisappliedat rightanglestothe directionofcurrentrow,anelectriceldissetupwhichisp erpendiculartoboththe directionofcurrentrowandthemagneticeld.Thedrifting chargecarriersexperience aLorentzforce,resultinginabuildupofchargesontheside softheconductor,untilthe transverseelectriceldjustcompensatestheeectofLore ntzforce.TheHallcoecient R H isgivenby E y = R H Bj x (2{14) sothat R H =1 =ne (2{15) where E y isthetransverseelectriceldinthe y direction, B isthemagneticeldin z direction, j x isthecurrentdensityinthe x directionand n isthenumberofcharge carriersperunitvolumeofthemetal.Inamodelwithtwotype sofcarriers, R H isgiven by R H =( 2 1 =n 1 e 1 + 2 2 =n 2 e 2 ) = 2 (2{16) where 1 and 2 aretheconductivities(totalconductivity = 1 + 2 ), n 1 and n 2 arethe numberofchargecarriersperunitvolume,and e 1 and e 2 arethechargesforthetwotypes ofcarriers,respectively. R H ispositiveforholesandnegativeforelectrons. 21

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2.3AnomalousHallEect TheHalleectinmagneticmetalsexhibitssomeunusualfeat ures.Theplotof Hallresistivity xy asafunctionofmagneticeldshowsaninitialrapidriseof xy with increasing ~ B followedbyasecondlinearportionhavingarelativelysmal lergradient.It canbeexpressedbythefollowingphenomenologicalexpress ion[ 3 ] xy = R n B + 0 R s M (2{17) where ~ B istheappliedmagneticinduction, R n istheordinaryornormalHallcoecient, R s isthespontaneousHallcoecient, 0 isthepermeabilityoffreespaceand M willbe thespontaneousmagnetization M s belowthecurietemperature.Thus,thesecondterm isasignatureofamagneticmaterialandcanbepresentinafe rromagneticdomainat zeroeld.ItdoesnotarisesfromtheapplicationofLorentz forceontheelectrons,and isthereforeknownas anomalousHalleect .Figure 2-1 showsthetypicalHalleectin 0 10 20 30 40 50 0 20 40 60 R x y ( )B(KOe)T=5K R s M s /d Saturation field Figure2-1.Typicalplotoftransverseresistance R xy asafunctionofmagneticeldfora thinlmofiron(thickness d ),showingthepresenceofanomalousHalleect. thinlmsofIron.Therearevariousscatteringmechanismst hathavebeeninvokedto 22

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explaintheoccurrenceofanomalousHalleect.Inthenexts ectionwewillgiveabrief introductiontothesemechanisms.2.3.1SkewScatteringMechanism Weconsiderthescatteringofanelectronbyanionfromastat e ~ k to ~ k 0 .Ifthe probabilityof W ( ~ k; ~ k 0 )isdierentfromtheprobability W ( ~ k 0 ; ~ k )ofthetheinverseprocess, wesaythattheelectronisskewscattered.Thisscatteringc anbeexplainedbyspinorbit interactions.Anelectronrotatingaboutanucleusseesama gneticeld ~ B whenviewedin aframeofreferenceinwhichtheelectronisatrest. ~ B = ~p ~ E= ( mc )(2{18) where ~p isthemomentumoftheelectronand ~ E istheelectriceldstrengthatthe electron.Thisresultsinacouplinginteractionbetweenor bitalandspinangular momentumandaddsatermtotheHamiltoniangivenby[ 3 ] H so = 1 4 m 2 c 2 ~ ( ~ 5 V ~p )(2{19) where ~ isthePaulispinoperatorandVistheelectricpotentialtha ttheelectronnds. Smit[ 4 5 ]consideredtheeectoftheperiodicspin-orbitinteracti oninaferromagnet andfoundthatitleadstoamodicationoftheBlochfunction s.Theperturbationsfrom perfectperiodicitygivesrisetoscatteringoftheelectro nandthisscattering,togetherwith spinorbitinteractions,isasymmetricandgivesrisetothe anomalousHalleect.The totalscatteringamplitude A total oftheelectronisgivenasthesumofscatteringamplitude duetotheelectron-ionpotential A ion andthatduetospin-orbitcoupling A so A total = A ion + A so (2{20) Theprobabilityofscatteringoftheelectronisgivenby W ( ~ k; ~ k 0 )= vnS ( q ) j A total j 2 (2{21) 23

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where v isspeedoftheelectron, S ( q )isthestructurefactorandnisthenumberof scatteringunitperunitvolume.Thiscanbewrittenas W ( ~ k; ~ k 0 )= vnS (2 k F sin( = 2)) f ( A ion ) 2 +2 A ion A so +( A ion ) 2 g (2{22) TherstterminEqn. 2{22 correspondstotheusualionicscatteringandislarge,the secondresultsinskewscatteringoftheelectronandthethi rdisasecondordereect resultinginthechangeofthespindirectionoftheelectron .Thesecondterminvolvesthe followingquantities: ~ z ~ k ~ k 0 (2{23) where ~ z isaunitvectorinthedirectionoftheappliedeld.Thepres enceofthevector productinthistermshowsthattheinterchangeof ~ k and ~ k 0 leadstoachangeinthesign oftheterm,leadingtoskewscatteringoftheelectron.2.3.2SideJumpScatteringMechanism ToexplaintheanomalousHalleect,Berger,in1970,propos edthesidejump scatteringmechanism[ 6 ]forasystemofferromagneticallypolarizedfreeelectron s.Inthis scattering,anelectronundergoesasmallsidejumpduetosp in-orbitinteractionduring eachcollision.TheHamiltonian[ 7 ]withspin-orbitinteractionisgivenby, H = V ( ~r )+( ~ = 4 m 2 c 2 ) ~ ~ 5 V ( ~r ) ~p (2{24) wherethersttermrepresentsthesingleimpuritypotentia landthesecondtermthe spin-orbitinteraction; ~ isthePaulimatrix.Thesidejumpexperiencedbytheelectro nis givenby 4 ~r k 1 4 ( ~ =mc ) 2 ~ k ~ (2{25) anditiscalculatedusingtherelation ~! k = 4 ~r k = tr (2{26) 24

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where ~! k istheanomalousvelocity,whichistheexpectationvalueof thespin-dependent partofthevelocityoperator,and1 = tr isthetransportcollisionrate. 2.3.3AnomalousHallEectinSystemswithLocalizedMagnet icCarriers Kondo[ 8 ],in1962,investigatedtheanomalousHalleectofferroma gneticmetals onthebasisofthelocalizedd-electron(orf-electron)mod el.Hefoundthatthedirects-d interaction,betweenthespinoftheconductionelectronan dthespinangularmomentum oftheincompleted(orf)shell,cannotbyitselfgiveriseto anomalousHalleect.Itis necessarytoinvoketheintrinsicspin-orbitcouplingofth elocalizedmagneticelectrons. TheHamiltonian,whichisvalidwhentheorbitalangularmom entaoflocalizedelectrons arenotquenched,isexpressedas H = H s d + H s o (2{27) wherethersttermistheusuals-dinteractionandthesecon dcomesfromthesecond-order perturbationandislinearinthespin-orbitcouplingconst ant andisproportionalto ~ S n ( ~ k ~ k 0 )(2{28) Here ~ S n isthespinangularmomentumofthen-thion.Thisimpliestha tthetransition probabilityfromastate ~ k toanotherstate ~ k 0 isnotequaltothatfrom ~ k 0 to ~ k andthis factleadstotheanomalousHalleect.However,iftheintri nsicspin-orbitcouplingis quenched,thenthereisnoskewscatteringandthereforenoa nomalousHalleect.Kondo [ 9 ]pointedoutthatGadoliniumisknowntobeinanSstateandth ereforeithasno orbitalangularmomentum,butitexhibitsalargeanomalous Halleect.Toexplainthis, Kondosuggestedthattheintrinsicspin-orbitinteraction isnowdominatedbythecovalent mixinginteraction,whichisaresonantmixinginteraction betweenalocalizedmagnetic stateandaconductionelectronstate.Thismixinggivesris etoskewscatteringofthe s electrons,resultinginananomalousHalleect. 25

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Anotherspin-orbitinteraction[ 3 10 ],thathasbeensuggestedtoexplainthe anomalousHalleect,istheinteractionbetweenthemagnet iceldduetolocalized moment(duetoincompletedorfshell)andthatduetocurrent carrying s electrons temporarilylocatedinthevicinityoftheions.Toseehowsk ewscatteringresultsfrom this,weconsideralocalizedmoment ~ M attheoriginoftherectangularcoordinateframe. Thislocalizedmomentsetsupavectorpotential ~ A givenby ~ A = ( ~ M ~r ) r 3 (2{29) where ~r isthepositionvector.Thisvectorpotentialaddsanewterm totheHamiltonian inthefollowingway, H = ( ~p e c ~ A ) 2 2 m = 1 2 m ( p 2 + e 2 c 2 A 2 ) e 2 mc ( ~p: ~ A + ~ A:~p ) : (2{30) Usingthefactthat ~ r ~ A =0,wehavethespin-orbittermintheHamiltonianas, H so = e mc ( ~p: ~ A )= e mc ~p: ( ~ M ~r r 3 )= e mcr 3 ~ M: ( ~r ~p ) : Clearly, H so changessignwhenthepositionofthechargecarriersisrere ctedintheplane denedby ~ M andtheprimarycurrentdirection,resultinginskewscatte ringandhencean anomalousHalleect. 26

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CHAPTER3 EFFECTOFDISORDERONTRANSPORTPROPERTIES Inanidealperiodiclattice,theBlochtheoremdescribesth ebehavioroftheelectrons. Asweallowdeviationsfromperfectperiodicity,thephenom enonofelectronscattering fromimpurities,staticdefects,phonons,magnons,etccom esintothepicture.Next, comestheappearanceofquantumcorrectionstoconductivit ywiththeoccurrenceofa seriesofscatteringevents.Therearevarioustimescalesw hichplayanimportantrole. Thefrequencyoftheelasticscatteringeventsinwhichthee lectronpreservesitsenergy j andthereforeitstemporalvariationisdenotedby1 = .Theinelasticscatteringevents inwhichanelectronchangesitsenergy j andforgetsitsphaseafterthecollisionare characterizedbythefrequency1 = .Inthenextsection,wewillpresentthevarious quantumcorrectionstoconductivity. 3.1WeakLocalization Themanifestationofthewavepropertiesoftheelectronaga instthebackgroundof diusivemotioninthepresenceofalargenumberofelastics catteringcentersresultsin quantumcorrectionstoconductivity.Toseehowthishappen s,weconsideramodelof noninteractingelectronswhicharescatteredfromimpurit ies.Weconsiderallthepossible pathsthathavetheformofaloopandbringstheelectronback tothepointr=0andthen groupthemintopairswithequivalentsetofscatteringcent ersbutwithoppositedirection ofelectrontravel.Thesearecalledself-intersectingtra jectories[ 11 ],showninFig. 3-1 .In theclassicalpicture,theprobability W (0 ;t )isthesumoftheprobabilitiesalongdierent paths.Accordingtoquantummechanics,thetotalprobabili tyisequaltothesquareof themodulusofthesumofallamplitudesfordierentpaths.S incetheself-intersecting loopshaveidenticalphasesandif 1 and 2 representtheprobabilityamplitudesforthe anticlockwiseandtheclockwiseself-intersectingloops, wehavetheprobabilityas: classicalprobability j 1 j 2 + j 2 j 2 =2 j 1 j 2 (3{1) quantumprobability j 1 + 2 j 2 = j 1 j 2 + j 2 j 2 + 1 2 + 1 2 =4 j 1 j 2 (3{2) 27

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Eqn. 3{2 showsthattheprobabilityofndingtheelectronattheorig inisdoubled r=0 } Scattering center Electron path Figure3-1.Closedelectrontrajectoriesinwhichtheycant ravelinoppositedirections. becauseofquantuminterference,leadingtoanincreaseinr esistivityoradecreaseof conductivity.Thisphenomenoniscalled weaklocalization Tocalculatetherelativecorrectiontoconductivity, = ,onehastodeterminethe probabilityofobtainingaself-intersectingtrajectory. Theelectrontrajectoryistreated asatubeofnitethicknessoftheorderofwavelength ~ =p 0 .Thevolumeaccessible totheelectronatanymomenttisoftheorderof( p x 2 ) 3 =( Dt ) 3 2 ,wherethediusion constant D lv F .Now,inagiventime dt ,thevolumefromwhichtheelectroncanreach theoriginisoftheorderof v F dt 2 ,where isthedeBrogliewavelengthoftheelectron. Sotheprobabilitythattheelectronrevisitstheoriginwit hinatime dt isgivenbythe ratioofthesetwovolumesandtoobtainthetotalprobabilit y,wehavetointegrateover time.Integrationisperformedwiththelowerlimitas l=v F ,fordiusionconcepttobe applicable,andtheupperlimittobetheinelasticscatteri ngtime ,afterwhichthephase relaxesandtheamplitudecoherencebreaks.Wehave[ 11 12 ]for d =3: Z v F 2 dt ( Dt ) 3 2 v F 2 D 3 2 ( 1 1 2 1 1 2 )= ( k 2 F l ) 1 ( 1 l 1 L )(3{3) d =2: Z v F 2 dt ( Dt ) b v F 2 Db ln ( = ) 1 ( k F l )( k F b ) ln ( = )(3{4) 28

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d =1: Z v F 2 dt ( Dt ) 1 2 b 2 2 v F 2 Db 2 ( L l ) 1 ( k F b ) 2 ( L l 1)(3{5) where L p D l p =>>l iscalledthephase-breakinglengthandbis thethicknessofalmordiameterofawire.Theabsolutevalu esofthechangesofthe conductivity d = d b 3 d aregivenby[ 13 ]: d =3: 3 e 2 ~ l + e 2 ~ L e 2 ~ l 1 1 2 1 2 (3{6) d =2: 2 2( e 2 ~ ) ln ( L l ) e 2 ~ ln (3{7) d =1: 1 e 2 ~ L (3{8) wherewehaveusedthefact( =l ) 2 ( ~ =p 0 l ) 2 andtheconductivityisestimatedasfollows: n e e 2 =m n e e 2 l=p 0 p 20 e 2 l= ~ 3 (3{9) Nowinalltheaboverelations,phasebreakingtime hasastrongtemperature dependenceandisdeterminedbytheinelasticscatteringme chanismwhichcouldbe eitherscatteringofelectronsfromelectronsorfromphono ns.Theformerisdominantat lowtemperatures.Thetemperaturedependenceof indierentdimensionis d =3: 1 = AT 2 + BT 3 (3{10) d =2: 1 = AT + BT 3 (3{11) Ineachofthesebehaviorof withtemperature,thersttermisduetothescattering ofelectronsfromelectronsandthesecondcomesfromthesca tteringofelectronsfrom phonons. 3.2EectofMagneticFieldonWeakLocalization Theapplicationofmagneticeldhasaveryinterestingeec tonthebehaviorof theinterferencecorrection.Sonow,whilecalculatingthe phasedierencethatappears betweentheinterferingamplitudes, ~p hastobereplacedby ~p ( e=c ) ~ A ,wherethevector 29

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potential ~ A retainsitssignforthereversetraversaloftheloop.Theph asedierenceis givenby H = 1 ~ Z ~p e ~ A c ( dl ) 1 ~ Z ~p e ~ A c dl = 2 e c ~ I ~ A: ~ dl =2 0 (3{12) whereisthemagneticruxacrosstheloopand 0 = hc= 2 e istheruxquantum.The appearanceofaphasedierenceresultsindestructionofth einterference,leadingtoa decreaseintheresistivity.Tondthisnegativemagnetore sistance,weintroduceanew timescalecalled H .Thediusionlength( Dt ) 1 2 denesthecharacteristicsizeofthe loopandthemagneticruxacrosstheloopis HDt .Thetime H issodenedthat H 2 andis H 0 HD (3{13) Wecalculatetheessentialeldbythecriterion H ,whichgives H 0 D : (3{14) Substituting D lv v 2 E F =m andusingn= eH=mc ,weseethatn << 1,which meansthatweareconsideringeldsmuchsmallerthanthecla ssicallystrongelds.To calculatethechangeinconductivityinthepresenceofmagn eticeld[ 13 ],wereplacethe by H inEqn. 3{3 ,sothatwehave d =3: Z H v F 2 dt ( Dt ) 3 2 : (3{15) PerformingasimilarexercisewithEqn. 3{4 wehavethefollowingresults: d =3: e 2 ~ eH ~ c 1 2 (3{16) d =2: e 2 ~ ln eHD ~ c (3{17) 30

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3.3Antilocalization Thepresenceofspin-orbitinteractioneectsthequantumc orrectionstothe conductivity[ 14 ].Now,theelectronscanripthespininthecourseofelastic scattering andthetwopairofwavesmixwitheachother.Thespin-orbits catteringtime so satises thefollowingrelation, so (3{18) Toseehowthewavefunctionsofelectronsofdierentspinsm ix,weconsideracompound particlemadeupoftwoelectrons.Thetotalspinofthispart iclecanbeinoneofthefour possiblestates.Asaresult,thewavefunctionofthecompo siteparticleorthepairhas fourcomponents[ 11 ] = 0BBBBBBB@ 0 1 ; 1 1 ; 0 1 ; 1 1CCCCCCCA = 0BBBBBBBB@ 1 p 2 ( (1)+ (2) (1) (2)+ ) (1) (2) 1 p 2 ( (1)+ (2) + (1) (2)+ ) (1)+ (2)+ 1CCCCCCCCA (3{19) where (1) and (2) denotethewavefunctionsoftherstandthesecondelectron s, respectively,andthesubscripts+and-denotethespinpart ofthewavefunction andindicatedierentspinprojections.Aftertraversingt heself-intersectingloop,the compositeparticleinterfereswithitselfandtheinterfer encetermisthesumofthefour terms.Thestates 1 ;m carryinformationabouttheelectronspinandthereforethe yare dampedwithtime so whilethestate 0 isdampedwithtime .Thecorrectiontothe conductivityisnowgivenas: Z v F 2 dt b (3 d ) ( Dt ) d 2 ( 3 2 e t= so 1 2 )(3{20) 31

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Wehavethefollowingresultfordierentdimensions[ 11 14 ]: 2 v F 8>>>><>>>>: D 3 = 2 ( 3 2 1 = 2 so 1 2 1 = 2 1 = 2 d =3 ; ( Db ) 1 3 2 ln so + 1 2 ln d =2 ; D 1 = 2 b 2 3 2 1 = 2 so + 1 2 1 = 2 d =1 : 9>>>>=>>>>; (3{21) WeobservethatwhentheconditioninEqn. 3{18 issatised,thequantumcorrectionto conductivity,duetoweaklocalization,ispositive.Thisi salsocalledantilocalization. 3.4TheEectofElectron-ElectronInteraction Theweaklocalizationeectdescribedintheearliersectio ndealtwithinterference ofanelectronwavewithitselfandwouldexisteveniftheele ctronswerenon-interacting. Nowifweconsidertheinteractionsbetweentheelectrons,w ehaveneweectswhich,also, givequantumcorrectionstotheconductivity.Inthissecti on,wewillbrierydiscussabout thesecorrections. Inadisorderedmetal,themotionoftheelectronsisdiusiv eduetorepeated scatteringfromimpurities.Thediusivemotionaltersthe timedependenceofthemotion oftheelectronsandhencetheenergydependenceoftheirpro perties.Thisisrerectedasa changeinthedensityofelectronstatesandtheconductivit y. Weconsideranelectroninaperfectmetalinastate ~ k 1 =n 1 = 2 e i ~ k 1 :~r beingscattered byanotherelectrontoastate ~ k 2 =n 1 = 2 e i ~ k 2 :~r withmomentumtransfer q = ~ k 2 ~ k 1 Inadisorderedmetal,thereisanuncertainty 1 =l in ~ k becauseofscattering,where l isthemeanfreepath.Theabovemomentumconservationwillb reakdownif ql< 1.This suggeststhatunusualeectswilloccurintheregimewhere ql< 1andthatthestrengthof suchscatteringprocesseswillbeenhancedsincethenumber ofsucheventsisnotrestricted bymomentumconservation. Inadisorderedsystem,theeectofelectron-electroninte ractionistoshifttheenergy E ofanelectroninastate E toanewenergy E ,givenby E = E + E ; (3{22) 32

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where E iscalledtheself-energy.Themanifestationofinteractio nsthroughthe self-energyleadstotwoimportantconsequences.Theseare : (a)Thedensityofstatesischangedbyanamount N ( E ), N ( E ) N ( E ) = @ < E @E (3{23) (b)Thestate E hasanitelifetime ee ( E ),givenby ~ 1 ee ( E )= = E (3{24) Now,tocalculate E ,weconsidertheparticle-holechannelmadeupofasingleel ectron atanenergy abovetheFermilevelandasinglevacantstateorholeintheF ermisea.If theenergyofinteractionbetweenanelectronat 1 andasecondelectronatenergy 2 is s ( 1 ; 2 ),thetotalselfenergyisgivenby: E = Z 0 1 N ( 2 ) s ( 1 ; 2 ) d 2 = Z 1 1 = ~ N ( 1 ~ ) s ( ) ~ d! (3{25) wheretheintegrationisoveralloccupiedstatesbelowtheF ermilevelandtheenergy dierence ~ = 1 2 .Therelativechangeinthedensityofstatesatenergy abovethe Fermilevelisgivenby[ 15 ]: N N (0) = s ( ; 0) N (0)(3{26) where s ( ; 0)istheenergyofinteractionbetweenanelectronat above E F andtheother at E F Thespacepartofthewavefunctionofaparallelspinpairofe lectronscanbewritten as: ( "" )= E ( ~r ) E 0 ( ~ r 0 ) E 0 ( ~r ) E ( ~ r 0 )(3{27) ThisisantisymmetricandobeysthePauliexclusionprincip le.Theinteractionenergy (rstorderinperturbationtheory)is: 1 2 ZZ h ( "" ) V ( ~r ~ r 0 ;t ) ( "" ) i d 3 ~rd 3 ~ r 0 (3{28) 33

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wheretheintegrationisoverthevolumeofthematerial, ~r ~ r 0 istheseparationofthe twoelectronsandthefactor1/2avoidsdoublecountingwhen summedoverallelectrons. UsingEqn. 3{27 ,wegettwotypesofterms: (a)Hartreeterms,oftheform ZZ E ( ~r ) E ( ~r ) V ( ~r ~ r 0 ;t ) E 0 ( ~ r 0 ) E 0 ( ~ r 0 ) d 3 ~rd 3 ~ r 0 (3{29) whichcanbeinterpretedastheenergyofinteractionbetwee nthechargedensity E E at ~r and E 0 E 0 at ~ r 0 (b)Exchangeterms,oftheform ZZ E 0 ( ~r ) E ( ~ r 0 ) V ( ~r ~ r 0 ;t ) E 0 ( ~r ) E ( ~ r 0 ) d 3 ~rd 3 ~ r 0 (3{30) whicharisefrommakingthetotalwavefunctionoftheelectr onwavefunctionantisymmetric anddoesnothaveanyclassicalcounterpart. Thecontributionofanexchangetermtotheselfenergyisgiv enby: E = X E 0 ZZ E 0 ( ~r ) E ( ~ r 0 ) V ( ~r ~ r 0 ;t ) E 0 ( ~r ) E ( ~ r 0 ) d 3 ~rd 3 ~ r 0 (3{31) Foragivendimensionalityd,thiscanbewrittenas[ 16 ] E = 1 (2 ) d Z dE 0 N ( E 0 ) Z ql< 1 d d ~q V b ( ~q ) ( ~q;E E 0 ) jh E j exp ( i~q:~r ) j E 0 ij 2 (3{32) where V b ( ~q )istheFouriertransformofthebareCoulombpotential e 2 =r and ( ~q;E E 0 ) isthedynamicscreeningfunctionofanelectrongas. jh E j exp ( i~q:~r ) j E 0 ij 2 isrelatedto thetimeandspaceFouriertransformof j ( ~r;t ) j 2 ,where ( ~r;t )isthetimedependent wavefunctionofanelectroninarandomsystem.Foranelectr onundergoingmanyelastic collisions,wecanassumethat j ( ~r;t ) j 2 isthesolutionofadiusionequation.Wecan write j ( ~r;t ) j 2 = 1 (4 Dt ) d= 2 exp ( r 2 = 4 Dt ) ; (3{33) 34

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whichleadsto[ 16 ] jh E j exp ( i~q:~r ) j E 0 ij 2 = 1 ~ N ( E ) Dq 2 ( Dq 2 ) 2 +( E E 0 ) 2 = ~ 2 : (3{34) Sothedensityofstatesaswellas ee dependonthedegreeofdisorderthroughthe diusionconstant.PluggingEqn. 3{34 intoEqn. 3{32 ,wegettheself-energyas[ 16 ] E = 1 ~ N ( E F )(2 ) d Z dE 0 N ( E 0 ) Z ql< 1 d d ~q V b ( ~q ) ( ~q;E E 0 ) Dq 2 ( Dq 2 ) 2 +( E E 0 ) 2 = ~ 2 : (3{35) Now,thecalculationofthedensityofstates,givenbyEqn. 3{23 ,canbedoneusingthe staticscreeningfunctioncomingfromThomas-Fermiscreen ing.Theapproximationof usingThomas-Fermiscreeninghasbeenjustifedeveninthep resenceofstaticdisorder andforthisapproximationthefactor V b ( ~q ) ( ~q;E E 0 ) 1 =N ( E F )withsmallvaluesof ~q [ 16 ]. Also,ithasbeenshownthattheenergyderivativeoftheself energy E isessentiallythe integrandofEqn. 3{35 ,leadingto[ 16 ] N ( E ) N ( E ) = 1 ~ N ( E F )(2 ) d Z ql< 1 d d ~q Dq 2 ( Dq 2 ) 2 +( E E F ) 2 = ~ 2 (3{36) UsingtheEinsteinrelation,thechangeinconductivitydue totheelectron-electron interactionisapproximatelyrelatedtothechangeinthede nsityofstatesbytherelation = N N (3{37) Forsmall = E E F andusingEqn. 3{37 ,Eqn. 3{36 yieldsthefollowing[ 16 ] / 8>>>><>>>>: T 1 = 2 d =3 ; lnTd =2 ; T 1 = 2 d =1 : 9>>>>=>>>>; (3{38) Equations 3{36 and 3{38 impliesareductionofconductivityduetoexchangeterm.In contrasttothenegativeExchangecontribution,thereisal soapositivecontributionfrom theHartreecorrection,whichisusuallysmall,butunderce rtainconditionsitmaybelarge enoughtoleadtoanoverallincreasein N ( E F ). 35

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3.5QuantumCorrectionstoHallConductivity 3.5.1NormalHallConductivity FukuyamastudiedthequantumcorrectionstonormalHallcon ductivitydueto weaklocalizationintwodimensionaldisorderedsystems[ 17 ]andshowedthatthenormal Hallcoecient R n = E y =Bj x isconstantasthetemperatureisvaried,sothattheHall resistance R n xy = R n B follows R n xy R n xy =0 : (3{39) Usingthefollowingequation, xx 1 R xx ; xy R xy R 2 xx : (3{40) thebehaviorofthenormalHallconductivitycanbededuceda s: n xy n xy = R n xy R n xy 2 R xx R xx : (3{41) UsingEqn. 3{39 ,wehave n xy n xy = 2 R xx R xx =2 xx xx (3{42) whichimpliesthatthenormalHallconductivityreceivesqu antumcorrectionsfromweak localizationandhasatemperaturedependencewithaslopet wicethatofthelongitudinal conductivity. Altshuler et.al. studiedthequantumcorrectionstonormalHallconductivit ydue toelectron-electroninteractioneectinadisorderedtwo dimensionalelectrongas[ 18 ] andfoundthattheHallconstantincreaseslogarithmically atlowtemperature,butat arateequaltotwicethatofresistivity.Thisimpliesthatt henormalHallconductivity isindependentoftemperature(doesnotreceiveanyquantum correctionsfrome-e interactions)andgivenbythefollowingrelation: n xy n xy = R n xy R n xy 2 R xx R xx =0 : (3{43) 36

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3.5.2AnomalousHallConductivity QuantumcorrectionstotheanomalousHallconductivitydue toelectron-electron interactionsandweaklocalizationhavebeencalculatedth eoreticallywithintheframework ofskewandsidejumpscatteringmechanisms.Followingaret heimportantresultsforeach case.3.5.2.1QuantumCorrectionsfromElectron-ElectronInter actions CalculationsofLangenfeld et.al. [ 19 ]showedthatthereisnoquantumcorrectionto theanomalousHalleectduetoelectroninteractionsinthe skewscatteringregime, i.e ., AH xy AH xy = R AH xy R AH xy 2 R xx R xx =0 : (3{44) TheanomalousHalleectbehavesinasimilarmannertotheno rmalHalleect.The experimentalresultsonamorphousthinlmsofironbyBergm annandYe[ 20 ]werefound tobeingoodagreementwiththeabovetheoreticalpredictio n.TheAHconductivitywas foundtobeindependentoftemperature.3.5.2.2QuantumCorrectionsfromWeakLocalization Dugaev et.al. [ 21 ]calculatedthelocalizationcorrectionstotheanomalous Hall conductivityintheframeworkofsidejumpmechanismandfou ndthatintwodimensions AH ( SJM ) xy AH ( SJM ) xy 1 ( F ) 3 ; (3{45) whereas xx xx 1 F : (3{46) SothecorrectionstoAHconductivityduetolocalizationin theframeworkofsidejump mechanismisnegligible.TheexperimentofBergmannandYec analsobeexplainedusing theaboveresult. Dugaev et.al. [ 21 ]alsocalculatedthelocalizationcorrectionstotheanoma lous Hallconductivityintheframeworkofskewscatteringmecha nismandfoundthatintwo dimensionsthelocalizationcorrectiontotheAHconductiv ityduetotheskewscatteringis 37

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nonzeroandisgivenby SS xy = e 2 36 ~ 20 k 2 F r 3 r 2 ln 1 so + 1 k 2 F # # r 3 r 2 ln # 1 so # + 1 # : (3{47) WeseefromEqn. 3{47 thattheanomalousHallconductivitycanreceivequantum corrections,withalogarithmictemperaturedependencein twodimensions,fromweak localizationwithintheskewscatteringregime.Thisisinc ontrasttothenegligible correctionofEqn. 3{45 totheanomalousHallconductivityfromweaklocalizationw ithin thesidejumpmodel. 3.6AndersonLocalizationTransition In1979,afundamentalstudybyAbrahams. et.al. proposedthescalingtheoryof localization[ 22 ].Oneoftheremarkablepredictionsofthescalingtheoryis theexistence ofacriticalpointin d> 2wherethesystemundergoesametal-insulatortransitionw ith increasingdisorder,incontrasttohavingaminimummetall icconductivity[ 23 ].Themain propertythatdistinguishesametalfromaninsulatoristhe conductivity.Thetestof whethertheconductivityiszeroornothastobedonebyextra polatingtheconductivity ( T )tothetemperature T =0becauseatnitetemperature T 6 =0,aninsulatorcanhave anitehoppingconductivity.Sotheconceptofametal-insu latortransitionmakessense onlyat T =0.Inametal-insulatortransition,thewavefunctionofth egroundstateofthe electronschangesandthischangecanbebroughtaboutbyvar yingcertainparameters. Atransitiontunedbydisorderinasystemofnoninteracting electronsinadisordered potentialiscalledanAndersontransition[ 24 ]. Atthecriticalpoint,thetransitionischaracterizedbyce rtaincriticalexponents. Forexample,thedcconductivity ( ),where isameasureofdisorder,ischaracterized nearthecriticaldisorder c bytheconductivityexponent s givenby t s where t =(1 = c ).Thedynamicalconductivityatthecriticalpoint,ontheo therhand,is characterizedbythedynamicalexponent z givenby ( ; c ) 1 =z .Thecorrelation 38

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lengthexponent isdenedby = t .Inthreedimensions, d =3,theconductivity exponent s isequaltothecorrelationlengthexponent NeartheAndersontransitionin3 D ,theconductivityisgivenbythescalingform [ 25 26 ] ( T ; )= 1 G 1 3 = B! 1 3 + A ( ) 3 + :::::: (3{48) where isaconstantwhichvaluelessthan1.Theconductivityatni te T isobtained from ( )byreplacing bythe T -dependentphaserelaxationrate1 = andwecanwrite ( T ; )= B 1 1 3 + A ( ) 1 3 ; (3{49) where A ( )= 8>>>><>>>>: a 1 ( c ) (1 ) < c ; 0 = c ; a 01 ( c ) 0 (1 ) > c : 9>>>>=>>>>; (3{50) 39

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CHAPTER4 EVOLUTIONOFQUANTUMCORRECTIONSWITHDISORDERINIRONTHIN FILMS 4.1Introduction SincethediscoveryofAndersonlocalization,ithasbeenaw ellknownfactthatthe electronicpropertiesofasolidareprofoundlyaectedbyt hepresenceofdisorder.Various excitingphenomenonstepinwhenweintroducedisorderinap erfectlattice.Thestudy oftheroleplayedbydisorderinaectingthequantumcorrec tionstoconductivitytensor inthinlmsofferromagneticmaterialsisaquiteunexplore dandaninterestingeld.The anomalousHalleectarisingevenintheabsenceofanapplie dmagneticeldresultsfrom acombinationofspin-polarizedcurrentcarriersandspinorbitinteractions.Weknow thatthespin-orbitinteractionsaectthequantumcorrect ioninnonmagneticmaterials resultinginaweakanti-localization.Alsothereisaninte rnalmagneticinduction B int ina ferromagnet.Onecanexpectasuppressionorreductionofwe aklocalizationbythiseld. Ithasbeenshowntheoretically[ 27 ]thattheprocesses,leadingtoweakantilocalizationin nonmagneticsystems,aretotallysuppressedinferromagne ts.Currently,themechanisms proposedfortheunderstandingofanomalousHalleectaret heskewscattering(SSM) [ 28 ],sidejump(SJM)[ 6 ]andtheBerryphasemechanism[ 29 ].Theroleplayedbydisorder intheselectionofthedominantscatteringmechanismposes animportantquestion.The factthatthenormalHallconductivityreceivesquantumcor rectionsfromweaklocalization [ 17 ],characterizedby WL xy xy = 2 WL xx xx (4{1) whereastheelectron-electroninteractiondonotproducea nyquantumcorrections[ 18 ] makesuswonderabouttheeectofelectron-electronintera ctionsandlocalizationin ferromagneticthinlms.Toaddressthestatusofquantumco rrectionstotheanomalous Hallconductivityandtheeectofdisorder,wepresentasys tematicstudyofthecharge transportinthinlmsofiron[ 30 ],whichisaferromagnetbyvirtueofspin-polarizationof itinerantelectrons. 40

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4.2ExperimentalDetails 4.2.1ContactPadPreparation Figure4-1.Goldcontactpads(yellowincolour)onsapphire substrate. Theextremelyairsensitivenatureofthethinlmsofironre quires insitu deposition andcharacterizationtechniques.Tomakeelectricalconta ctstosuchthinlms,predeposited contactpadsareneeded.Thefollowingstepsaretakentopre paresuchcontactpads. Sapphiresubstratesarecleanedinasonicatorwithdeioniz edwater,acetone,isopropanol andmethanolatleast10minuteseachinasequentialmanner. Thesubstratesarethen driedwithanitrogenblow.Theyareinspectedwithanoptica lmicroscopeforcleanliness. Afterthoroughcleaningofthesapphiresubstrates,contac tpadswiththegeometryshown inFig. 4-1 werepreparedbydepositing50 Aofchromiumfollowedby250 Aofgold throughashadowmaskusingthermalevaporation.4.2.2ThinFilmDeposition Aseriesofironthinlmsweregrownonhighlypolishedsapph iresubstrates(0001) atroomtemperatureusingr.f.magnetronsputteringwithth egrowthparameterslistedin Table 4-1 .ThelmsweredepositedintheHallbargeometryusingashad owmask.The Table4-1.Growthparametersforironthinlms ParametersValues Rfpower30W DCbias-180V Argonrowrate10sccm 41

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Figure4-2.Hallbargeometrywithtwocurrentleadsandfour voltageleads. aspectratioofthethevariousdimensionsintheHallbar,sh owninFig. 4-2 ,are: p c;c w= 3 ;l 4 w: (4{2) Thisyieldsmeasurementwhichhaveverysmalldeviationfro mtheideal[ 31 { 34 ].The currentandvoltageleadsofthedepositedsampleoverlappe dwiththepredeposited goldcontacts,thusensuringreliableelectricalconnecti on.Immediatelyafterdeposition, samplesweretransferredwithoutexposuretoairfromthehi ghvacuumdeposition chambertothecryostatformagnetotransport. Duringdeposition,sampleresistanceandthicknessarecon tinuouslymonitored insitu Toparameterizetheamountofdisorderinagivenlm[ 35 ],weusethesheetresistance R 0 R xx ( T =5K)where R xx isthelongitudinalresistance.Inthisstudy, R 0 spansthe rangefrom140n(60 Athick)to6250n( < 20 Athick). 4.2.3MeasurementTechniques The insitu characterizationsofthethinlmsweredoneusingthestand ardfour probelock-intechniqueforlowresistancelmsandthefour -terminald.c.measurements forlmsofhighsheetresistances.Four-terminalmeasurem entsarehelpfulinremovingthe contactresistance.Figure 4-3 showsthesetupusedfora.c.measurementofthevoltages 42

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Figure4-3.Simultaneousmeasurementoflongitudinalandt ransverseresistanceusingac lock-intechnique.TwoStanfordResearchSR830lock-inamp lierhavebeen usedinaconstantcurrentcongurationatafrequencyof17H z. inthinlmsoflowsheetresistances.WehaveusedtwoStanfo rdResearchSystemSR830 lockinampliersforsimultaneousmeasurementoflongitud inalandHallvoltageinthe Hallbarshapedthinlmsamples.Themeasurementsweredone atafrequencyof17Hz inaconstantcurrentcongurationachievedwiththehelpof aballastresistorconnected inserieswiththesample.Becausetheinputimpedanceofthe SR830amplierisonly 1 M n,weusethed.c.measurementtechniquesforhighersheetre sistancelms.Figure 4-4 showsthed.c.circuitdiagramforlongitudinalandHallres istancemeasurementfora thinlmintheHallbarconguration.TheKeithley236isaso urcemeasureunit(SMU) withaveryhighinputimpedance > 10 14 nandweuseitformeasuringthelongitudinal 43

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Figure4-4.Simultaneousmeasurementoflongitudinalandt ransverseresistanceusingfour terminaldctechnique.WehaveusedKeithley236SMUandKeit hley182 nanovoltmeterforthispurpose. voltageina sourceImeasureV conguration.TheKeithley182nanovoltmeterisa verysensitivedeviceformeasuringvoltageswithaninputi mpedance > 10 11 nandwe useitformeasuringthevoltagedevelopedacrosstheHallle adssimultaneouslywiththe longitudinalvoltagemeasurement. TheHallvoltageleadsinmostofthesampleshaveanunavoida blesmallmisalignment [ 33 ].Nowtheexcitationcurrentrowingthroughthesampleprod ucesavoltagegradient paralleltotherow.Thisgivesrisetoaspuriouscontributi oninthemeasuredHall voltage,eveninzeromagneticeld.Toeliminatethismisal ignmentvoltage,theHall 44

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Table4-2.Halleectmeasurementvoltagesobtainedbyfour permutationsofcurrentand magneticelddirection. IBV H V M V 1 ++++ V 2 -+-( V 1 V 2 )2 V H 2 V M V 3 +--+ V 4 --+( V 3 + V 4 )2 V H 2 V M ( V 1 V 2 V 3 + V 4 )4 V H 0 voltageisobtainedbytakingthemeanofallfourpermutatio nsoftheappliedmagnetic andelectricelddirectionsasshownintheTable 4-2 4.3MeasurementsandAnalysis Inthequesttondthequantumcorrectionstotheconductivi tytensorinironlms, longitudinalandtransversevoltagesweresimultaneously measuredunderconstantcurrent conditionsinmagneticeldsof 4 T whilethetemperatureisvariedslowly.Thenthe symmetricresponse R xx isextractedusingtherelation: R xx ( T;B )= ( V xx ( T;B )+ V xx ( T; B )) 2 I (4{3) whiletheantisymmetricresponse R xy iscalculatedusingrelation: R xy ( T;B )= ( V xy ( T;B ) V xy ( T; B )) 2 I : (4{4) ThecontributionfromthenormalHalleectisnegligible,a sisthemagnetoresistance.To studythescalingbehaviorofvariousquantitiesinadimens ionlesspicture,wedenethe normalizedrelativechange (NRC), N ( Q ij )= 1 L 00 R 0 Q ij Q ij = 1 L 00 R 0 Q ij ( T ) Q ij ( T 0 ) Q ij ( T 0 ) (4{5) withrespecttoourreferencetemperature T 0 =5K
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wherethenormalizedrelativechangesin R xx R xy and xy hasalogarithmictemperature dependence.WethenfollowtheBergmannnotation[ 20 ]toexpressthetemperature dependenceofvariousquantitiesasfollows: N ( R xx )= A R ln T T 0 ; N ( R xy )= A AH ln T T 0 (4{6) Now,thelongitudinalandanomalousHallconductivityareg ivenbythefollowingrelations xx = R xx ( R 2 xx + R 2 xy ) ; xy = R xy ( R 2 xx + R 2 xy ) (4{7) Usingthefactthat j R xx j R 0 and R xy ( T ) R xx ( T )forourlms,wehave xx 1 R xx ; xy R xy R 2 xx : (4{8) Sotheexpressionsforthetemperaturedependenceofthecon ductivitiesintheBergmann notationare N ( xx )= A R ln T T 0 ; N ( xy )=(2 A R A AH )ln T T 0 (4{9) InEqn. 4{9 ,wehaveused xy xy = R xy R xy 2 R xx R xx (4{10) ShowninFigs. 4-5 4-6 and 4-7 arethenormalizedrelativechangesin R xx R xy and xy asafunctionofsheetresistance R 0 (disorder)fordierentsheetresistancesstudiedat B=4Tesla.We,now,tthelowtemperaturedatatoEqn. 4{6 toobtainthecoecients A R and A AH forallthelms.Fig. 4-8 showstheplotofthecoecients A R and A AH for allthelmsstudied.Therearesomeimportantpointsaboutt hegraphsthatareworth mentioning.InFig. 4-5 ,theplotfor R 0 =140nshowsthattheslopeofthecurvefor N ( R xy )istwicethatof N ( R xx )( A AH =A R =2).TheapplicationofEqn. 4{9 tothis resultshowsthat N ( xy )=0.Inotherwords,theanomalousHallconductivitydoesno t receiveanyquantumcorrection.ThisissimilartoBY's[ 20 ]resultwheretheyobserved A R =1and A AH =2forarangeof R 0 valuesinasimilartemperaturerange. 46

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1 2 3 4 -3 -2 -1 0 1 T0=5K (R xx ) (R xy ) ( xy ) Normalized relative changeT / T0Rxx(T=5K)=140 Ohm 1 2 3 4 -2 -1 0 1 (Rxx) (Rxy) ( xy)Rxx(T=5K) = 475 Normalized relative changeT/ T0 T 0 =5K 1 2 3 4 -2 -1 0 1 (R xx ) (R xy ) ( xy ) Rxx(T=5K) = 720 Normalized relative changeT / T 0 T 0 =5K 1 2 3 4 -2 -1 0 1 (Rxx) (Rxy) ( xy)R xx (T=5K) = 990 Normalized relative changeT / T0T 0 =5K 1 2 3 4 -1 0 1 (R xx ) (R xy ) ( xy )R xx (T=5K) = 1427 Normalized relative changeT/T 0 T 0 =5K 1 2 3 4 -2 -1 0 1 (R xx ) (R xy ) ( xy )R xx (T=5K) = 1562 Normalized relative changeT / T 0 T 0 =5K Figure4-5.Evolutionofquantumcorrectionswithdisorder 47

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1 2 3 4 -2 -1 0 1 (Rxx) (Rxy) ( xy) R xx (T=5K) = 2250 Normalized relative changeT/ToT0=5K 1 2 3 4 -1 0 1 2 (Rxx) (Rxy) ( xy) R xx (T=5K) = 2740 Normalized relative changeT / To T0=5K 1 2 3 4 -1 0 1 2 (R xx ) (R xy ) ( xy )Rxx(T=5K) = 3100 Normalized relative changeT / T o T 0 =5K 1 2 3 4 -1 0 1 2 (Rxx) (Rxy) ( xy) R xx (T=5K) = 3375 Normalized relative changeT / To T 0 =5K 1 2 3 4 -1 0 1 2 (Rxx) (Rxy) ( xy)R xx (T=5K) = 3705 Normalized relative changeT / T oT0=5K 1 2 3 4 -1 0 1 2 (R xx ) (R xy ) ( xy )Rxx(T=5K)= 4511 Normalized relative changeT / T0T 0 =5K Figure4-6.Evolutionofquantumcorrectionswithdisorder (continued) 48

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1 2 3 4 -1 0 1 (Rxx) (Rxy) ( xy)Rxx(T=5K)= 5304 Normalized relative changeT / T0T 0 =5K Figure4-7.Evolutionofquantumcorrectionswithdisorder (continued). Asdisorder( R 0 )increases,anitelogarithmictemperaturedependenceto N ( xy ) appears.ThiscanbeseeninFigs. 4-5 and 4-6 for R 0 valuesfrom475nto2250n.For alltheseplots,weseefromFig. 4-8 that1
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0.0 0.5 1.0 1.5 0.0 1.0 2.0 10 2 10 3 10 4 0.0 0.5 1.0 0.0 0.5 1.0 1.5 AR AAH/AR (a)(b) R 0 ( )2AR-AAH(c)(d) AAH Figure4-8.Coecients A R and A AH asdenedinEqn. 4{6 foraseriesofironthinlms. Thisgurehasbeenadoptedfromourpaper(ref.[ 30 ])publishedinPhysical ReviewLettersandAmericanPhysicalSocietyhasacopyrigh tforit. 50

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energies F dependingonthespinindex = ; # ,andwithspin-orbitcoupling g .The totalimpuritypotentialismodeledasasumoveridenticals ingleimpuritypotentials V ( ~r ~ R j )atrandompositions ~ R j .ThelongitudinalandtheAHconductivitiesofthis modelaregivenintheweakscatteringlimitby SSM xy = 0 xx Mg so tr = ; SJM xy = e 2 Mg so tr =; (4{11) where 0 xx = e 2 ( n=m ) tr g so = g = 4 n M = n n # isthenetspindensity, isthesingle particlerelaxationtimeand t N 0 V N 0 istheaverageDOSattheFermileveland V is theimpuritypotential.ItcanbeseenfromEqn. 4{11 thattheratio SJM xy = SSM xy canin factdecreasewithincreasing R 0 ,if increasessucientlyrapidlywithdisorder,especially sincedisorderinourthinlmsischaracterizedbythesheet resistance R 0 ratherthanthe resistivity Someoftheimportantresultsofthetheoreticalcalculatio nsareasfollows: 1.Thereisnointeractioninducedln T correctionto xy .Thisholdsforboth exchangeandHartreeterms,forboththeskewscatteringand theside-jumpmodels [ 37 ]. 2.Theweaklocalization(WL)conditionmax(1 = s ; 1 = so ;! H ) 1 = 1 = tr issatisedinthetemperatureregimeunderstudy.1 = turnsouttobelargelydueto spin-conservinginelasticscatteringospinwaveexcitat ions[ 30 ]. 3.Calculationsshowthat[ 30 ] WL xx = L 00 ln( = ); WL xy = WL xx Mg so tr =: (4{12) 4.TheWLcontributionto xy fromtheside-jumpmodelisshowntobenegligible [ 21 ].Using 1 =T ,itfollowsfromEqns. 4{11 and 4{12 thatthenormalizedWL correctiontotheHallconductivityisgivenas N WL xx =ln T T 0 ; N WL xy = SSM xy ln( T=T 0 ) ( SSM xy + SJM xy ) ; (4{13) 51

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where 0 xx =1 =R 0 .ComparingwithEqn. 4{9 ,thiscorrespondsto A R =1and2 A R A AH =[ SSM xy = ( SSM xy + SJM xy )].Assumingthat SJM xy = SSM xy isafunctionof R 0 decreasing fromvalues 1at R 0 150ntovalues 1at R 0 3 k n,asdiscussedearlier,wend goodagreementwithourexperimentaldata.Inotherwords,s idejumpisthedominant mechanisminlmswithlowdisorder,asthismechanismgives negligibleweaklocalization correctiontoAHconductivity.Asthedisorderincreases,s kewscatteringmechanismstarts tobecomethedominantmechanism,resultinginanon-zerolo garithmictemperature dependenceofweaklocalizationcorrectionstoAHconducti vity. Thedataforhigher R 0 areexplainedusingagranularmodelforpolycrystalline lms.As R 0 increasesabove3 K n,thegrainsbecomemoreweaklycoupled.Nowthe longitudinalresistivity R xx = R g xx + R T xx ,whichisdominatedby R g xx arisingfromthe scatteringatthegrainboundariesatlow R 0 ,isdominatedbythetunnelingprocess R T xx at higher R 0 sothat R xx R T xx .TheAHresistivity,beinganintragranularskewscatterin g process,isindependentof R T xx .ThisresultsindeviationsfromRRscalingathigher R 0 4.4Conclusions Transportmeasurementshavebeenmadetostudyquantumcorr ectiontothe conductivitytensorinthinlmsofironinSHIVA,wherethee ectsofunwanted oxidationorcontaminationcanbeeliminated.Fortemperat ures T< 20K,alogarithmic temperaturedependenceofthelongitudinal R xx andanomalousHallresistances R xy is observed.Inthelowdisorderlimit( R 0 < 150n),wendthatrelativechangesinthe anomalousHallconductivity, xy = xy ,exhibitatemperatureindependentbehavior implyingthattherearenoquantumcorrectionsto xy .Asdisorderincreases,anite logarithmictemperaturedependenceto xy = xy appearsandthenevolvestowarda universalweaklocalizationcorrectiondenedbytheequal ity xy = xy = R xy =R xy Thuswithincreasingdisorder,weseeacrossoverfromaregi onwherethereareno quantumcorrectionsto xy toaregiondominatedbyweaklocalizationcorrections. Inotherwords,ourexperimentalresultsprovideanevidenc eforadisorder-dependent 52

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localizationcorrectionstoanomalousHallconductivityi nthinferromagneticlmsofiron [ 30 ]. 53

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CHAPTER5 QUANTUMCORRECTIONSTOCONDUCTIVITYTENSORINTHINFILMSOF GADOLINIUM 5.1Introduction Gadolinium(Gd)belongstothefamilyofrareearthmetalswi ththegroundstate electroniccongurationof[Xe]4 f 7 6 s 2 5 d .The4 f shellofGdishalf-lledwithallseven spinsalignedparallel.Thesespinsformalocalizedmagnet icmomentwithacontribution of7 B peratom.Thepolarizationofthes-dconductionelectronsd uetoexchangewith thelocalized4 f electronsgivesanadditionalcontributionof0 : 63 B peratom[ 38 ].So thetotalmagneticmomentperatomofGdaddsupto7 : 63 B .Gdisferromagneticwith acurietemperatureof293 K[ 39 ].Ithasahexagonalclosepackedstructurewiththe metalradiusof1 : 802 A.ThereportedDebyetemperatureforGdis195 K [ 40 ].TheFermi energy,forthreeelectronperatom,is E F =3 : 4eVmeasuredfromthebottomoftheband [ 2 ].AttheFermienergy,thecalculateddensityofstatesis N ( E F )=1.8electronperatom pereV[ 41 ]. ThestudyofanomalousHalleectinthinlmsofiron[ 30 ],describedinChapter4, providedastrongevidenceforthephenomenonofweaklocali zation(WL)indisordered ferromagneticlms.Thecontributionofthescatteringos pin-waves[ 42 43 ]tothe phasebreakingrate1 = ismuchlargerthanthatfromelectron-electroninteractio ns. ThefactthatGadoliniumhasamuchlargermagneticmomentan daspinwavegap ofabout30mK[ 2 44 ]raisesone'sexpectationforamuchbiggereect.Thequant um correctionstoconductivityduetoscatteringothespinwa vesshouldbeobservable, justlikethequantumcorrectionsduetoelectron-electron coulombinteraction.Inthe followingsections,weconrmthisexpectationbypresenti ngasystematicstudyofthe in situ magnetotransportonaseriesofGdlms[ 45 ]. 54

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5.2ExperimentalDetails AseriesofGadoliniumthinlmshavebeengrownonhighlypol ishedsapphire substratesheldatatemperatureof250Kusingr.f.magnetro nsputteringwithgrowth parameterslistedinTable 5-1 Table5-1.Growthparametersforgadoliniumlms ParametersValues Rfpower15W DCbias-31V Argonrowrate10sccm Weobservedgadoliniumlmsdepositedat250Ktobemorestab lewithlesscontact resistancethanthosegrownatroomtemperature.Weoptimiz edthechoiceofmetal forcontactpadsandfoundthatpalladiummakesagoodcontac tpad.Thelmswere depositedintheHallbargeometryusingshadowmasks.Theas pectratioofvarious dimensionsintheHallbarsatisfythecriteriondiscussedi nChapter4.Thecurrent andvoltageleadsofthedepositedsampleoverlappedwithth epredepositedpalladium contacts,thusensuringreliableelectricalconnection.I mmediatelyafterdeposition, samplesweretransferredwithoutexposuretoairfromthehi ghvacuumdeposition chambertothecryostatformagnetotransport. Duringdeposition,sampleresistanceandthicknessarecon tinuouslymonitored insitu Toparameterizetheamountofdisorderinagivenlm[ 35 ],weusethesheetresistance R 0 R xx ( T =5K)where R xx isthelongitudinalresistance.Inthisstudy, R 0 spansthe rangefrom370n(150 Athick)to2840n( < 35 Athick). 5.3MeasurementsandAnalysis The insitu characterizationofthethinlmswasperformedinthecryos tatofSHIVA. Measurementsaremadeusingthestandardfour-probelock-i ntechnique,asshownin Fig.4-3inChapter4.WeusetheStanfordResearchSR830lock -inampliersforthis purposeandoperatethematafrequencyof17Hzinaconstantc urrentconguration achievedwiththehelpofaballastresistorconnectedinser ieswiththesample.Theinput 55

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impedanceoftheSR830amplieris1Mn.TheyareallGPIBaddr essable,sothatall themeasurementscanbeautomated.Wemeasurethelongitudi nalconductivityforallthe samplesinthetemperaturerange5K 4 k n,thepower p decreasessignicantly.) Second,thecoecient P 3 decreaseswithincreasingdisorderbyalmostafactoroftwo and thensaturates.Third,theprefactor P 2 ofthelogarithmicterm,whichisattributedtoa combinationoflocalizationandinteractioncorrections, isconstantnearunityoverthe displayedrange. Ithasbeenshown[ 45 ]thattheunusualtemperaturedependencerevealedbythese dataisconsistentwithasumofcontributionsfromwell-kno wnquantumcorrectionsin two-dimensionsandanovelspin-wavemediatedcorrectiona nalogoustotheAltshuler-Aronov electron-electroncontributionindisorderedsystems[ 13 ].WhiletheAltshuler-Aronov contributiongivesrisetoalogarithmictemperaturedepen denceintwodimensions,the spin-wavemediatedcontributioncanbelinearintemperatu rewithincertainrangesof 56

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5 10 15 202530 2.32 2.36 2.40 0.36 0.38 0.40 xx for R 0 = 428 Fit Temp(K) xx(x 10-3 -1) xx for R 0 =2840 Fit Figure5-1.Temperaturedependenceonalogarithmicscaleo f xx fortwoGdthinlms havingsheetresistances R 0 =428n(opensquares,lefthandaxis)and R 0 =2840n(opentriangles,righthandaxis).Thesolid-linet sforeachcurve areobtainedusingEqn.( 5{1 ).Thisgurehasbeenadoptedfromourpaper (ref.[ 45 ])publishedinPhysicalReviewB(RapidCommunications)an d AmericanPhysicalSocietyhasacopyrightforit. 57

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theparameters,consistentwiththeexperiments.Thetheor yceasestobevalidforlarge disorder( R 0 > 4 k n),wherethetemperaturedependenceisnolongerlinear.To make 0 75 150 225 0.5 0.8 1.0 500 1000 1500 2000 2500 3000 0.0 0.5 1.0 0.0 0.5 1.0 P 1 P 3 (a) (b) R0( )power p(c) (d) P 2 Figure5-2.Dependenceofthettingparameters( P 1 ;P 2 ;P 3 ;p )denedinEq.( 5{1 )onthe disorderparameter R 0 forsixteenthinlmsamples.Thehorizontaldashed linesinpanels(b)and(d)representrespectivelytheavera gedvaluesof P 2 and p whereasthecurveddashedlineinpanel(c)isaguidetotheey e.Thisgure hasbeenadoptedfromourpaper(ref.[ 45 ])publishedinPhysicalReviewB (RapidCommunications)andAmericanPhysicalSocietyhasa copyrightfor it. comparisonswithearlierstudiesonFelms[ 20 30 ],wehavealsomeasuredtheanomalous Hall(AH)resistancesoftheGdlmsat7Teslamagneticeld. Boththelongitudinal andtransversevoltagesweresimultaneouslymeasuredunde rconstantcurrentconditions 58

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inmagneticeldsof 7Twhilethetemperatureisvariedslowly.Thenthesymmetri c response R xx isextractedusingtherelation: R xx ( T;B )= ( V xx ( T;B )+ V xx ( T; B )) 2 I (5{2) whiletheantisymmetricresponse R xy iscalculatedusingrelation: R xy ( T;B )= ( V xy ( T;B ) V xy ( T; B )) 2 I : (5{3) FollowingRef.[ 20 ],wedenenormalizedrelativechanges N ( Q ij ) 1 L 00 R 0 Q ij Q ij (5{4) withrespecttoareferencetemperature T 0 =5 K> 1istrueforlmsdepositedonsapphire[ 30 ] oronantimony[ 20 ].Ourcurrentresults N xy 0forGddepositedonsapphireagree 59

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withthoseofFelmsdepositedonthesamesubstrate.Asfort helongitudinalpart,the coecient A R denedby N xx = A R ln( T=T 0 )(5{6) isgivenas A R = A WLR + A IR =1+ h xx wheretherstterm( A WLR =1)isduetoWLand thesecondterm( A IR = h xx )istheexchangeplusHartreeinteractioncontribution,wi th h xx =(1 3 ~ F ) = 4where ~ F istheHartreeterm.IthasbeenarguedthatatleastforFe lms,thetotalinteractioncorrection h xx isverysmallduetoanearcancellationofthe exchangeandHartreeterms,whichisexpectedduetostrongs creening.Thisresultsin A R 1forFelms.Figure 5-2 showsthat A R 0 : 75forourGdlms.Thissuggests that h xx mayactuallybenegative,duetoanevenlargerHartreecontr ibution.Sinceboth r xy and h xx arenon-universalquantities,weonlynotethatalarge r xy (similartoFelms onsapphire)andasmallnegative h xx (largeHartreeterm,againsimilartoFelms)are consistentwiththecurrentexperimentalobservations(on Gdlmsonsapphire). Tounderstandthelinear T -dependenceofthelongitudinalconductivity,thespinwav e contributionshavebeenevaluatedwithinthestandarddiag rammaticperturbationtheory [ 45 ].Thetotalspinwavecontribution,comingfromthediagram s[ 45 ]showninFig. 5-4 ,is givenby xx L 00 Jk 2 F 2 B 2 ( F ) T Ak 2 F (5{7) forsmalldampingwhere J istheeectivespinexchangeinteractionand B istheexchange splitting[ 47 ].WecanseefromEqn. 5{7 thatthereisaapproximatelylinearcorrection toconductivitycomingfromspin-wavemediatedAlshuler-A ronovcorrectionstothe conductivity.Thiscompareswellwithourexperimentalnd ingofquantumcorrections thatareapproximatelylinearintemperatureasshowninFig s. 5-1 and 5-2 .FromEqn. 5{7 ,thedisorderdependenceofthelinear T contributionisgivenby P 3 / F ,which decreaseswithincreasingdisorder.Experimentally,assh owninpanel(c)ofFig. 5-2 P 3 doesindeeddecreaseweaklywithdisorderuptoasheetresis tance R 0 2000nandthen 60

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1 2 3 4 -6 -5 -4 -3 -2 -1 0 N ( R xx ) N ( R xy ) N ( xy ) N ( R xx ) N ( R xy ) N ( xy ) Normalized relative changeT / T 0 T0=5KB=7Tesla solid928 open2613 Figure5-3.Normalizedrelativechanges N ( xy ),denedinEq.( 5{4 ),fortwodierent sheetresistances.Forcomparisonwealsoshow N ( R xx )and N ( R xy ).This gurehasbeenadoptedfromourpaper(ref.[ 45 ])publishedinPhysical ReviewB(RapidCommunications)andAmericanPhysicalSoci etyhasa copyrightforit. appearstosaturateataxedvalue.Thusthedisorderdepend enceoftheprefactor P 3 of thelineartermisinaccordancewiththetheoreticalpredic tions.Wenotethatwhilethe linear T behaviorisobservedtobequiterobustforweakdisorder,it cannotexplainthe datafor R 0 > 4 k n. 61

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(a) (b) Figure5-4.Spin-wavecontributionstothelongitudinalco nductivity.Solidlinesare impurityaveragedGreen'sfunctions,brokenlinesarediu sons,wavyline representstheeectivespin-wavemediatedinteractions. Therearetwo diagramsoftype(a)andfourdiagramsoftype(b).Thisgure hasbeen adoptedfromourpaper(ref.[ 45 ])publishedinPhysicalReviewB(Rapid Communications)andAmericanPhysicalSocietyhasacopyri ghtforit. 5.4Conclusions Inconclusion,wehavestudiedchargetransportinultrathi nlmsofGdgrownusing in-situ techniqueswhichexcludeinparticularunwantedoxidation orcontamination.In additiontothelogarithmictemperaturedependenceexpect edfromtheweaklocalization eectsinthelongitudinalconductivityaspreviouslyseen inFelms,weobservean additionalcontributiontotheconductivitythathasanapp roximatelylinear T -dependence forsheetresistances370n R 0 2840nandtemperatures5K
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CHAPTER6 OBSERVATIONOFANDERSONLOCALIZATIONTRANSITIONINTHINFI LMSOF GADOLINIUM 6.1Introduction In1979,afundamentalstudybyAbrahams,Anderson,Licciar delloandRamakrishnan proposedthescalingtheoryoflocalization[ 22 ].Oneoftheremarkablepredictionsofthe scalingtheoryistheabsenceoftruemetallicbehavior,ins ystemsintwodimensions ( d =2)(andofcourseinall d< 2aswell).Thishasbeenveriedinnumerous experiments[ 25 48 ].Theotherpredictionistheexistenceofacriticalpointi n d> 2 wherethesystemundergoesametal-insulatortransitionwi thincreasingdisorder,in contrasttohavingaminimummetallicconductivity[ 23 ].Atransitioninducedbydisorder inasystemofnon-interactingelectronsinadisorderedpot entialiscalledanAnderson transition[ 24 ].Ithasbeenoneofthemostextensivelystudiedcasesofqua ntumphase transitions,bothexperimentallyandtheoretically.Atth ecriticalpoint,thetransitionis characterizedbycertaincriticalexponents.Forexample, thedcconductivity ( ),where isameasureofdisorder,ischaraterizednearthecriticald isorder c bytheconductivity exponent s givenby t s where t =(1 = c ).Thedynamicalconductivityatthe criticalpoint,ontheotherhand,ischaracterizedbythedy namicalexponent z given by ( ; c ) 1 =z .Thecorrelationlengthexponent isdenedby = t .Inthree dimensions, d =3,theconductivityexponent s isequaltothecorrelationlengthexponent .Whileallexperimentsconrmthecontinuousnatureofthet ransition,thevaluesof thecriticalexponentsremaincontroversialdespiteexten siveeortsoverseveraldecades. Publishedexperimentalvaluesof s and z varyfrom s 0 : 5[ 49 { 51 ], s 1[ 52 { 54 ]to s 1 : 6and z 2[ 55 ]to z 2 : 94[ 56 ]. Therearetwomajorfactorsthatcontributetothedicultyi ndeterminingthe criticalexponentsoftheAndersontransitionexperimenta lly.First,itisdicultto increasethedisordersystematicallyforagivensystemtos ucientlylargevaluesinorder todrivethesystemthroughthetransitionpoint.Thishasre strictedmostoftheearlier 63

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experimentstobedoneondopedsemiconductorsystems,such asSi:P,Si:B,Si:Asand Ge:Sbandmetal-semiconductoralloyssuchasNb 1 x Si x andAu 1 x Ge x .Theproblem withdopedsystemsisthatinteractionsmaybecomeveryimpo rtantandacomparison withthenon-interactingtheoryoftheAndersontransition maynotbepossible.The experimentaldataontheinversionlayersinsiliconshowst hatinteractionsin2 D allows metallicbehavioragainstthepredictionsofthescalingth eoryoflocalization. Inallthepreviouswork,theeortwasmadetogodowntoaslow temperaturesas possibleandstudytheconductivityasafunctionoftempera ture.Theobservationofa temperaturepowerlawnearthecriticalpointintheseexper imentdoesnotnecessarily implythatitistheleadinglowtemperaturebehavior,asthe behaviorof ( )inthe non-universalregime !
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of3500rpmfor30seconds.Theprebakingofthephotoresisti sdoneonahotplateata temperatureof110 C.Thisresultsinaphotoresist(PR)lmofthicknessaround 0.5 m. AfterprebakingthePRlm,itisexposedtoultravioletligh tthroughatransparency maskfor20seconds.AZ300orMIF319developeristhenusedto developthelmfor30 seconds,followedbycleaningwithdeionizedwater.Wenowh avethepatterntransferred ontothesubstrate.Ontothispattern,wedeposit50 Aofchromiumfollowedby200 A ofpalladiumusingthermalevaporation.Theliftoisthenp erformedbysonicatingthe sampleinacetonefor10minutes.Finallythesapphiresubst rateiscleanedasdescribedin Chapter4. Pd Sapphire subs tra te (si ze 0.35 inches squ are)Pd Pd Pd Figure6-1.Fingeredcontactpad.6.2.2ThinFilmDeposition Usingthesapphiresubstrateswiththepredepositedcontac tpad,twoseriesof gadoliniumthinlmshavebeengrownat130Kusingr.f.magne tronsputteringwith growthparameterslistedinTable 6-1 Table6-1.Growthparametersforgadoliniumlms(withdiso rderbeyondtheregimeof quantumcorrections) ParametersValues Rfpower15W DCbias-31V Argonrowrate10sccm ThelmsweredepositedintheHallcrossgeometryusingasha dowmask.The 65

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(Alldimensionsininches) Figure6-2.Hallcrossgeometryusedforgadoliniumlms.aspectratioofthevariousdimensionsintheHallcross,sho wninFig. 6-2 ,satisesthe followingrelationship, c c +2 a = 1 6 (6{1) Thedimensionshowninthisgureareininches.Thisyieldsm easurementswhich haveverysmalldeviationsfromtheideal[ 31 { 34 ].Thecurrentandvoltageleadsofthe depositedsampleoverlappedwiththepredepositedpalladi umcontacts,thusensuring reliableelectricalconnection.Immediatelyafterdeposi tion,samplesweretransferred withoutexposuretoairfromthehighvacuumdepositioncham bertothecryostatfor magnetotransportandheldatatemperatureof77Korbelow.A tthesetemperatures thesamplesarestableanddonotundergoanytime-dependent changesinresistance.If howeverthetemperatureistemporarilyraisedbacktothede positiontemperature(130 K),annealingmarkedbyaslowirreversibleincreaseinresi stanceoccurs. Duringdepostion,sampleresistanceandthicknessarecont inuouslymonitored insitu Toparameterizetheamountofdisorderinagivenlm[ 35 ],weusethesheetresistance R 0 R xx ( T =5K)where R xx isthelongitudinalresistance.Inthisstudy, R 0 spansthe rangefrom4 k n(35 Athick)to72 k n( < 20 Athick).Controlledthermalannealingthus 66

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allowsustoadvantageouslytuneasinglesamplethroughsuc cessivestagesofincreased disorder. 6.3MeasurementsandAnalysis The insitu characterizationofthethinlmshasbeendoneinthecryost atofSHIVA. Four-terminald.c.measurementsaremadeusingtheKeithle y236SMU,Keithley182 nanovoltmeterandKeithleyswitchingcard7012S.Theyarea llGPIBaddressable,so thatallthemeasurementscanbeautomated.Thelongitudina landtheHallresistance areextractedusingthereversemagneticeldreciprocity( RFMR)theorem[ 57 ].Useof RFMRtheoremallowseldsweepsofonepolaritytoextractlo ngitudinalandtransverse resistances.Wepreparedtwoseriesofsamplesatdierentp eriodoftime.Wewillpresent theresultsforeachseriescallingthembythenames-series 1andseries2. 6.3.1TransportPropertiesatZeroMagneticField6.3.1.1SamplesinSeries1 ShowninFigs. 6-3 6-4 and 6-5 aretheplotsofconductivityasafunctionof temperatureforallthesamplesstudiedinseries1.Samples with R 0 6 23295n weredierentlmspreparedunderidenticalgrowthconditi ons,whereasthosewith R 0 > 23295nareobtainedbythermalannealingofthelmwith R 0 =23295n. Everytimethelmisheatedtoatemperaturenearitsgrowtht emperature,thesheet resistanceofthelmincreases.Weattributeittomovement andclusterformationof thegadoliniumatomsathighertemperatures.Thesampleist hencooleddownandthe temperaturedependenceofthenowtime-stableconductivit yisstudiedinatemperature rangeof5Kto50K,whereweobtainreproduceabledata.Thete mperaturerangeis dividedintodiscretestepsandelectricalmeasurementare donewhilethesampleissitting ataconstanttemperature.Wehaveusedthefundamentalcons tant h=e 2 =25813nto normalizeourdata. 67

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0 5 10 15 20 25 30 35 40 45 6.0 6.5 7.0 7.5 8.0 A = 5.49 (1) B = 0.454 (6) p = 0.451 (3) 2d / (e2/h) ( 2D/ (e2/h)) = A + B T P Fit R0= 4011 Temp(K) 2D / (e2/h) 0 51015202530354045505560 3.5 4.0 4.5 5.0 5.5 A = 2.37 (2) B = 0.58 (1) p = 0.405 (4) 2d / (e2/h) ( 2D/ (e2/h)) = A + B T P Fit 2D / (e 2 /h) Temp(K)R0= 7377 0 5 10152025303540455055 2.0 2.5 3.0 3.5 4.0 A = 1.216 (8) B = 0.538 (5) p = 0.417(1) 2d / (e2/h) ( 2D/ (e2/h)) = A + B T P Fit 2D / (e 2 /h) Temp(K)R0= 11445 0 5 10152025303540455055 1.5 2.0 2.5 3.0 3.5 A = 0.29 (2) B = 0.64 (1) p = 0.381 (4) 2d / (e2/h) ( 2D/ (e2/h)) = A + B T P Fit 2D / (e 2 /h) Temp(K)R0= 17483 0 5 10152025303540455055 1.0 1.5 2.0 2.5 3.0 A = -0.08 (2) B = 0.63 (1) p = 0.386 (4) 2d / (e2/h) ( 2D/ (e2/h)) = A + B T P Fit 2D / (e 2 /h) Temp(K)R0= 23295 0 5 10 15 20 25 30 35 40 45 50 1.0 1.5 2.0 2.5 3.0 A = -0.18 (2) B = 0.65 (1) p = 0.382 (5) 2d / (e2/h) ( 2D/ (e2/h)) = A + B T P Fit 2D / (e 2 /h) Temp(K)R0= 25173 Figure6-3.Temperaturedependenceofconductivityforsam plesinseries1 68

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0 5 10152025303540455055 1.0 1.5 2.0 2.5 3.0 A = -0.23 (3) B = 0.61 (2) p = 0.390 (6) 2d / (e2/h) ( 2D/ (e2/h)) = A + B T P Fit 2D / (e 2 /h) Temp(K)R0= 27872 0 5 10152025303540455055 0.5 1.0 1.5 2.0 2.5 A = -0.46 (2) B = 0.71 (2) p = 0.367 (4) 2d / (e2/h) ( 2D/ (e2/h)) = A + B T P Fit 2D / (e 2 /h) Temp(K)R0= 31010 0 5 10152025303540455055 0.5 1.0 1.5 2.0 2.5 A = -0.57 (2) B = 0.73 (2) p = 0.362 (4) 2d / (e2/h) ( 2D/ (e2/h)) = A + B T P Fit 2D / (e 2 /h) Temp(K)R0= 34819 0 5 10152025303540455055 0.5 1.0 1.5 2.0 2.5 A = -0.57 (2) B = 0.69 (1) p = 0.371 (4) 2d / (e2/h) ( 2D/ (e2/h)) = A + B T P Fit 2D / (e 2 /h) Temp(K)R0= 37918 0 5 10152025303540455055 0.5 1.0 1.5 2.0 2.5 A = -0.67 (2) B = 0.70 (2) p = 0.368 (5) 2d / (e2/h) ( 2D/ (e2/h)) = A + B T P Fit 2D / (e 2 /h) Temp(K)R0= 43792 0 5 10152025303540455055 0.5 1.0 1.5 2.0 2.5 A = -0.77 (2) B = 0.73 (2) p = 0.362 (5) 2d / (e2/h) ( 2D/ (e2/h)) = A + B T P Fit 2D / (e 2 /h) Temp(K)R0= 48208 Figure6-4.Temperaturedependenceofconductivityforsam plesinseries1(continued) 69

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0 5 10152025303540455055 0.5 1.0 1.5 2.0 2.5 A = -0.78 (2) B = 0.71 (1) p = 0.365 (4) 2d / (e2/h) ( 2D/ (e2/h)) = A + B T P Fit 2D / (e 2 /h) Temp(K)R0= 50950 0 5 10152025303540455055 0.5 1.0 1.5 2.0 2.5 A = -0.83 (2) B = 0.73 (1) p = 0.362 (4) 2d / (e2/h) ( 2D/ (e2/h)) = A + B T P Fit 2D / (e 2 /h) Temp(K)R0= 53832 0 5 10152025303540455055 0.5 1.0 1.5 2.0 2.5 A = -0.83 (2) B = 0.72 (1) p = 0.362 (4) 2d / (e2/h) ( 2D/ (e2/h)) = A + B T P Fit 2D / (e 2 /h) Temp(K)R0= 55862 0 5 10152025303540455055 0.5 1.0 1.5 2.0 A = -0.86 (2) B = 0.70 (1) p = 0.367 (4) 2d / (e2/h) ( 2D/ (e2/h)) = A + B T P Fit 2D / (e 2 /h) Temp(K)R0= 61942 0 5 10152025303540455055 0.0 0.5 1.0 1.5 2.0 A = -0.87 (3) B = 0.67 (2) p = 0.375 (5) 2d / (e2/h) ( 2D/ (e2/h)) = A + B T P Fit 2D / (e 2 /h) Temp(K)R0= 71226 0 5 10152025303540455055 0.0 0.5 1.0 1.5 2.0 A = -0.90 (3) B = 0.65 (2) p = 0.379 (6) 2d / (e2/h) ( 2D/ (e2/h)) = A + B T P Fit 2D / (e 2 /h) Temp(K)R0= 81167 Figure6-5.Temperaturedependenceofconductivityforsam plesinseries1(continued). 70

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Forallthelmsstudied,wendthatthetemperaturedepende nceoftheconductivity isgivenby 2 D L 0 = A + BT p ; (6{2) where L 0 = e 2 =h .ThesmoothlineinFigs. 6-3 6-4 and 6-5 representsatusingEqn. 6{2 .Wehaveincludedthegraphsforallthesamplesstudiedinth eseries1toshow thequalityofthedataandthet.ThettingparametersinEq n. 6{2 namelythe dimensionlessquantity A ,theprefactor B ofthetemperaturedependenttermandthe powerpdisplayaninterestingbehavior.Thebehaviorofthe parameter A asafunction of R 0 isshowninFig. 6-6 .Wecanseethattheparameter A isdisorder-dependent.It 0 20000 40000 60000 80000 -1 0 1 2 3 4 5 6 A R 0 ( ) Figure6-6.Behavioroftheparameter A inEqn. 6{2 forallthesamplesinseries1. startsoutwithapositivevalueatlowvaluesof R 0 inthebeginning,goesthroughzero andthenbecomesnegativewithincreasingdisoder(higher R 0 ).Thisextrapolationofthe conductivitydatato T =0isindicativeofametal-insulatortransition.Thesyste mallows onetochangedisordersystematicallytodrivethesystemth roughthetransition,where A goesthroughzero.Thepower p andtheprefactor B inEqn. 6{2 forallthesamples 71

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0 20000 40000 60000 80000 0.0 0.2 0.4 0.6 0.8 1.0 power p R 0 ( ) Figure6-7.Behaviorofthepower p inEqn. 6{2 forallthesamplesinseries1. 01020304050607080 0.0 0.2 0.4 0.6 0.8 1.0 Coefficient B ( K p )R0 (K ) Figure6-8.Behavioroftheprefactor B inEqn. 6{2 forallthesamplesinseries1. intheseries1areshowninFig. 6-7 and 6-8 .Itisevidentfromtheplotthatthepowerp displaysafairlyconstantvalue,asthedisorderistunedac rossthetransition. Thestudyofthesampleinthisseries1raisessomeimportant questions.Canwe getclosertothetranstionwheretheparameter A goesthroughzero?Whatwouldbe thebehavioroftheparameter A asafunctionofdisorder,veryclosetothetransition? 72

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Theanswertoallthesequestionsdemandedfurtherstudies. We,therefore,studiedmore samplesandwecallthesesetofsamplesasseries2.Thefollo wingelaboratestheresultsof thesestudiesontheseries2samples.6.3.1.2SamplesinSeries2 Toinvestigatetheregionclosetothecriticalregime,wher ethemetal-insulator transitionoccurs,wedepositedanothergadoliniumlmwit hasheetresistanceof 14995nandstudiedthedependenceofconductivityontemper ature.Thetechnique ofthermalannealingisagainutilizedtopreparesampleswi thhighersheetresistance, R 0 ,usingthesamelm.Weheatthelmtoatemperature,closeto thetemperatureof deposition.Eortsweremadetoincreasethedensityofdata pointaroundthetransition pointwheretheparameterAchangessign.Inthisseries2,l mswithsheetresistance rangingfrom14.9 k nto39.9 k nhavebeenstudied.Thetemperaturedependenceof theconductivityisstudiedinatemperaturerangeof5Kto50 K,whereweobtain reproduceabledata.ShowninFigs. 6-9 6-10 and 6-11 aretheplotsofconductivityas afunctionoftemperatureforallthesamplesstudiedinseri es2.Wehaveincludedthe graphsforallthesamplesstudiedintheseriestoshowthequ alityofthedataandthet. Itisevidentfromtheplotsthatthetemperaturedependence oftheconductivityisstill givenby 2 D L 0 = A + BT p ; (6{3) where L 0 = e 2 =h .Table 6-2 liststhettingparametersforallthesamplesinseries2. Toseehowwellwecanreproducethedataofseries1,wehavepl ottedtheparameter A forbothseries1andseries2together.Fig. 6-12 showsthedependenceoftheparameter AinEqn. 6{3 asafunctionofsheetresistance(measureofdisorder)forb othseriesof samples.Weseethatthedataofseries2areveryclosetothat ofseries1.Now,toget acloseupviewofwhatishappeningnearthecriticalregion, weplotanexpandedviewof Fig. 6-13 ,asshownintheFig. 6-14 .WendfromtheFig. 6-14 ,thataroundthecritical 73

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0 5 10152025303540455055 1.5 2.0 2.5 3.0 3.5 R 0 = 14995 A = 0.64 (1) B = 0.561 (9) p = 0.409 (2) 2d / (e2/h) ( 2D/ (e2/h)) = A + B T P Fit 2D / (e 2 /h) Temp(K) 0 5 10 15 20 25 30 35 40 45 50 1.5 2.0 2.5 3.0 3.5 2D / (e2/h) A = 0.55 (1) B = 0.576 (7) p = 0.403 (2) R 0 = 15588 2d / (e 2 /h) ( 2D / (e 2 /h)) = A + B T P Fit Temp(K) 0 5 10 15 20 25 30 35 40 45 50 1.5 2.0 2.5 3.0 3.5 R0= 16559 2d / (e 2 /h) ( 2D / (e 2 /h)) = A + B T P Fit A = 0.54 (1) B = 0.521 (8) p = 0.422 (3) 2D / (e2/h) Temp(K) 0 5 10152025303540455055 1.5 2.0 2.5 3.0 3.5 A = 0.46 (1) B = 0.54 (1) p = 0.415 (3) Temp(K) 2D / (e 2 /h) 2d / (e2/h) ( 2D/ (e2/h)) = A + B T P Fit R 0 = 17008 0 5 10152025303540455055 1.5 2.0 2.5 3.0 A = 0.295 (1) B = 0.593 (7) p = 0.399 (2) Temp(K) 2D / (e 2 /h) 2d / (e 2 /h) ( 2D / (e 2 /h)) = A + B T P Fit R 0 = 18296 0 5 10152025303540455055 1.5 2.0 2.5 3.0 A = 0.27 (1) B = 0.566 (8) p = 0.407 (2) 2d / (e 2 /h) ( 2D / (e 2 /h)) = A + B T P Fit 2D / (e 2 /h) Temp(K) R0= 19077 Figure6-9.Temperaturedependenceofconductivityforsam plesinseries2 74

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0 5 10152025303540455055 1.5 2.0 2.5 3.0 A = 0.23 (1) B = 0.561 (7) p = 0.410 (2) 2d / (e2/h) ( 2D/ (e2/h)) = A + B T P Fit 2D / (e 2 /h) Temp(K) R0 = 19682 0 5 10152025303540455055 1.0 1.5 2.0 2.5 3.0 A = 0.16 (1) B = 0.569 (9) p = 0.407 (3) 2d / (e2/h) ( 2D/ (e2/h)) = A + B T P Fit R0= 20628 Temp(K) 2D / (e2/h) 0 5 10152025303540455055 1.0 1.5 2.0 2.5 3.0 2D / (e 2 /h) Temp(K)R 0 = 21537 A = 0.10 (1) B = 0.576 (9) p = 0.405 (2) 2d / (e 2 /h) ( 2D / (e 2 /h)) = A + B T P Fit 0 5 10152025303540455055 1.0 1.5 2.0 2.5 3.0 2D / (e 2 /h) Temp(K)R0 = 22667 A = -0.01 (1) B = 0.622 (9) p = 0.390 (2) 2d / (e2/h) ( 2D/ (e2/h)) = A + B T P Fit 0 5 10152025303540455055 1.0 1.5 2.0 2.5 3.0 2D / (e2/h) Temp(K)R 0 = 23774 A = -0.07 (1) B = 0.628 (8) p = 0.388 (2) 2d / (e 2 /h) ( 2D / (e 2 /h)) = A + B T P Fit 0 5 10152025303540455055 1.0 1.5 2.0 2.5 3.0 A = 0.17 (1) B = 0.662 (9) p = 0.379 (2) 2d / (e2/h) ( 2D/ (e2/h)) = A + B T P Fit R0= 25012 Temp(K) 2D / (e2/h) Figure6-10.Temperaturedependenceofconductivityforsa mplesinseries2(continued) 75

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0 5 10152025303540455055 1.0 1.5 2.0 2.5 A = 0.35 (1) B = 0.68 (1) p = 0.372 (2) 2d / (e 2 /h) ( 2D / (e 2 /h)) = A + B T P Fit R 0 = 28898 Temp(K) 2D / (e 2 /h) 0 5 10152025303540455055 1.0 1.5 2.0 2.5 A = 0.40 (1) B = 0.68 (1) p = 0.373 (2) 2d / (e 2 /h) ( 2D / (e 2 /h)) = A + B T P Fit R 0 = 30834 Temp(K) 2D / (e 2 /h) 0 5 10152025303540455055 0.5 1.0 1.5 2.0 2.5 A = 0.56 (1) B = 0.733 (9) p = 0.361 (2) 2d / (e2/h) ( 2D/ (e2/h)) = A + B T P Fit R0= 34908 Temp(K) 2D / (e2/h) 0 5 10152025303540455055 0.5 1.0 1.5 2.0 2.5 A = 0.67 (1) B = 0.749 (8) p = 0.357 (1) 2d / (e2/h) ( 2D/ (e2/h)) = A + B T P Fit R0= 39924 Temp(K) 2D / (e 2 /h) Figure6-11.Temperaturedependenceofconductivityforsa mplesinseries2(continued). region,theparameter A hasanearlylineardependenceondisorderandwecanwrite A /j ( R 0 R c ) L 0 j s ;s =0 : 98 0 : 07(6{4) where L 0 = e 2 =h R 0 isthemeasureofdisorder(characterizedbythesheetresis tance R = R 0 at T =5K)and R c isthevalueofcriticaldisorder.Wendthatthevalueof criticaldisorder R c =22741( 50)n.Thiscorrespondstothedisorderwherethevalueof theparameter A =0.Thebehaviorofparameter A showsthatwearecharacterizinga3 D Andersonlocalizationtransitioninourthinferromagneti cgadoliniumlms.Theeective dimensionalityofthesystemis d =3ifthetemperaturedependentcorrelationlength 76

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0 15 30 45 60 75 -1 0 1 2 3 4 5 6 series 1 series 2 A R 0 ( ) Figure6-12.BehavioroftheconstantAasafunctionof R 0 forbothseries1andseries2. 15 20 25 30 35 40-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 Samples in series 2 A R 0 ( K ) Figure6-13.BehavioroftheconstantAasafunctionof R 0 forseries2. 77

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Table6-2.FittingparametersdenedinEqn. 6{3 R (n) BpA 149950 : 561(9)0 : 409(2)0 : 64(1) 155880 : 576(7)0 : 403(2)0 : 55(1) 165590 : 521(8)0 : 422(3)0 : 54(1) 170080 : 54(1)0 : 415(3)0 : 46(1) 182960 : 593(7)0 : 399(2)0 : 295(1) 190770 : 566(8)0 : 407(2)0 : 27(1) 196820 : 561(7)0 : 410(2)0 : 23(1) 206280 : 569(9)0 : 407(3)0 : 16(1) 215370 : 576(9)0 : 405(2)0 : 10(1) 226670 : 622(9)0 : 390(2) 0 : 01(1) 237740 : 628(8)0 : 388(2) 0 : 07(1) 250120 : 662(9)0 : 379(2) 0 : 17(1) 288980 : 68(1)0 : 372(2) 0 : 35(1) 308340 : 68(1)0 : 373(2) 0 : 40(1) 349080 : 733(9)0 : 361(2) 0 : 56(1) 399240 : 749(8)0 : 357(1) 0 : 67(1) ( T ) b ,where b isthethicknessofthelm.Accordingtotheoreticalconsid erations [ 26 ],the3 d conditionisseentobemetinourexperimentsfor T> 1 K .Thebehaviorof thecoecient B andthepowerpasafunctionofsheetresistanceisdepictedi nFig. 6-15 Weseethatwithinthecriticalregime,theparameter B increasesweaklywithincreasing disorder,whilethepower p decreasesweaklyaswegothroughthecriticalregionwherei t is0.390(2). NeartheAndersontransitionin3 D ,theconductivityisgivenbythescalingform[ 25 26 ] ( T ; )= 1 G 1 3 = B! 1 3 + A ( ) 3 + :::::: (6{5) Theconductivityatnite T isobtainedfrom ( )byreplacing bythe T -dependent phaserelaxationrate1 = andwecanwrite ( T ; )= B 1 1 3 + A ( ) 1 3 ; (6{6) 78

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19202122232425 -0.3 -0.2 -0.1 0.0 0.1 0.2 0.3 Series 2 A R0 (K ) Figure6-14.BehavioroftheconstantAasafunctionof R 0 expandedaroundthecritical region. 15 20 25 30 35 40 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 Power p Coefficient B ( K p ) Power p R 0 (K ) Critical region Coefficient B Figure6-15.BehaviorofthecoecientBandpowerpasdened inEquation 6{3 asa functionof R 0 (disorder). 79

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where A ( )= 8>>>><>>>>: a 1 ( c ) (1 ) < c ; 0 = c ; a 01 ( c ) 0 (1 ) > c : 9>>>>=>>>>; (6{7) Ourexperimentalresults,obtainedatnitetemperatures, lieinaregimewherethescaling behaviorisexactlygivenbyEqn. 6{5 .ThiscanbeunderstoodwiththehelpofFig. 6-16 Alsoweknowthat iscutobythetemperatuedependentphaserelaxationrate. The valueof A ( R 0 )isobtainedbyextrapolating ( T )inthescalingregimeto T =0. Wecanunderstandthemeaningoftheparameter A withthehelpofFigs. 6-17 6-18 and 6-19 (guresaredrawnwiththeconsiderationthatthefrequency iscutobythephase relaxationrate1 = / T ). T insulator c metal TScalingregime Figure6-16.Scalingregimeasafunctionofdisorder. Wend,bycomparisonwiththeobservedtemperaturepowerla wof p =0 : 39atthe criticalpoint,that 1 / T 1 : 19 ; (6{8) 80

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0 T 1/3 T 1/3 A Figure6-17.Signicanceoftheparameter A inthemetallicstate. 0 T 1/3 T 1/3A= Figure6-18.Signicanceoftheparameter A atthecriticalpoint. 81

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0 T 1/3 T 1/3A Figure6-19.Signicanceoftheparameter A intheinsulatingstate. Thisisinagreementwiththethephaserelaxationrateinfer romagneticlms[ 30 ],which aredominatedbythescatteringospinwaveexcitations,gi venby 1 / T pluscorrections fromsubleadingterms.Withtheassumptionthat =0,Eqns. 6{5 and 6{6 explainsthe experimentaldatatorstorderwiththedisorderdependenc eof A givenby 6{4 (taking thesheetresistance R 0 at T =5Kasthedisorderparameter).Howevertoexplainthe weakvariationofpower p aswegothroughthecriticalregime,showninFig. 6-15 ,we needtoincludethesubleadingterm 3 .Workisinprogressinthisdirection. 6.3.2TransportPropertiesofSamplesinSeries1inthePres enceofMagneticField 6.3.2.1EectsonLongitudinalConductivity Untilnow,wehavepresentedexperimentalresultsobtained intheabsenceof magneticeld.Theunknowneectsofmagneticeldonsuchdi sorderedlmsledus 82

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0 5 10152025303540455055 1.0 1.5 2.0 2.5 3.0 A = 0.20 (2) B = 0.64 (2) p = 0.366 (5) B = 0 Tesla B = 7 Tesla ( 2D/ (e2/h)) = A + B T P Fit ( 2D/ (e2/h)) = A + B T P Fit A = -0.08 (2) B = 0.63 (1) p = 0.386 (4) 2D / (e 2 /h) Temp(K)R0= 23295 Figure6-20.Plotofconductivityversustemperaturefor R 0 =23295n. 0 5 10152025303540455055 1.0 1.5 2.0 2.5 3.0 A = 0.23 (3) B = 0.61 (2) p = 0.390 (6) B = 0 Tesla B = 7 Tesla ( 2D/ (e2/h)) = A + B T P Fit ( 2D/ (e2/h)) = A + B T P FitA = -0.10 (2) B = 0.72 (2) p = 0.345 (5) 2D / (e 2 /h) Temp(K)R0= 27872 Figure6-21.Plotofconductivityversustemperaturefor R 0 =27872n. 83

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0 5 10152025303540455055 1.0 1.5 2.0 2.5 A =-0.16 (2) B = 0.71 (2) p = 0.346 (5) A =-0.31 (2) B = 0.75 (1) p = 0.343 (3) A =-0.36 (2) B = 0.72 (1) p = 0.355 (4) B = 0 Tesla B = 3 Tesla B = 5 Tesla B = 7 Tesla A =-0.46 (2)B =0.71 (2)p = 0.367 (4) 2D / (e 2 /h) Temp(K)R0= 31010 smooth lines represent fitsusing ( 2D / (e 2 /h)) = A + B T P Fit Figure6-22.Plotofconductivityversustemperaturefor R 0 =31010n. 0 5 10152025303540455055 1.0 1.5 2.0 2.5 A =-0.34 (2) B = 0.78 (1) p = 0.334 (3) A =-0.35 (2) B = 0.73 (1) p = 0.348 (4) A =-0.47 (2) B = 0.75 (1) p = 0.349 (3) B = 0 Tesla B = 3 Tesla B = 5 Tesla B = 7 Tesla A =-0.57 (2)B =0.73 (2)p = 0.362 (4) 2D / (e 2 /h) Temp(K)R0= 34819 smooth lines represent fitsusing ( 2D/ (e 2 /h)) = A + B T P Fit Figure6-23.Plotofconductivityversustemperaturefor R 0 =34819n. 84

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tofurtherexperimentalinvestigationsinthepresenceofc onstantmagneticeldupto7 Tesla.ShowninFigs. 6-20 6-21 6-22 and 6-23 areplotsofthetemperaturedependence ofconductivitystudiedinthepresenceofconstantmagneti celdforsamplesinseries1. Eachoftheplotscontainsdatasetstakenatdierentmagnet icelds.Wendthatthe temperaturedependenceoftheconductivityisagaingivenb y, 2 D L 0 = A + BT p ; (6{9) where L 0 = e 2 =h Thesalientfeatureinferredfromtheseplotsisthat,forag ivensample,theparameter A increaseswithmagneticeld.Infact,ifexaminetheplotfo r R 0 =23295n,wendthat atzeromagneticeld,thevalueof A isnegative.Thiscorrespondstoaninsulatingstate at T =0.Whenwestudythesamesampleinamagneticeld B =7T,wendthatthe valueof A becomespositive.Thisimpliesanextendedstate(metallic state)atT=0for B =7T.Figure 6-24 showstheplotoftheparameter A asafunctionofmagneticeld fordierentsamples.Itpointstowardsthefactthatthemet al-insulatortransitioncanbe probedbytheapplicationofmagneticeldinthissystem,pr ovidedweareverycloseto thecriticaldisorderandontheinsulatingsideofthetrans ition.Wehavea\eld-tuned insulator-to-metaltransition."6.3.2.2BehaviorofHallResistancewithTemperature Simultaneousmeasurementsofthetransverseresistance R xy alongwith R xx discussed intheprevioussectionwerealsoperformedonthesamplesin series1.ShowninFig. 6-25 6-26 6-27 6-28 6-29 and 6-30 aretheplotsoftemperaturedependenceof R xy fordierentsamples.Weobservedthatthetemperaturedepe ndenceof R xy obeysthe followingrelationship, R xy = A + BT p : (6{10) 85

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0 1 2 3 4 5 6 7 8 -0.6 -0.4 -0.2 0.0 0.2 23295 27872 31010 34819 A B ( Tesla) Figure6-24.BehavioroftheconstantAwithmagneticeld. 0 10 20 30 40 50 -180 -160 -140 -120 -100 -80 -60 -40 R0= 23295 A = 56 (2) B = -350 (1) p = -0.292 (3) R xy ( B=7 Tesla) R xy = A + B T p Fit R xy ( )Temp(K) power p = 0.29 Figure6-25.Plotof R xy versustemperaturefor R 0 =23295n. 86

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0 10 20 30 40 50 -180 -160 -140 -120 -100 -80 -60 -40 power p= 0.32 Temp(K)Rxy ( ) R xy ( B=7 Tesla) R xy = A + B T p Fit A = 55 (1) B = -388.9 (5) p = -0.317 (2) R 0 = 27872 Figure6-26.Plotof R xy versustemperaturefor R 0 =27872n. 0 10 20 30 40 50 -200 -180 -160 -140 -120 -100 -80 -60 -40 -20 A = 53 (2) B =-407.1 (7)p = -0.328 (4) A = 51 (2) B =-372.5 (7)p = -0.334 (4) R xy ( B=3 Tesla) R xy ( B=5 Tesla) R xy ( B=7 Tesla) Rxy = A + B T p Fit Rxy = A + B T p Fit Rxy = A + B T p Fit Temp(K)R xy ( )A = 47 (3) B = -311 (1) p = -0.331 (8)R0= 31010 Figure6-27.Plotof R xy versustemperaturefor R 0 =31010n. 87

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0 10 20 30 40 50 -200 -180 -160 -140 -120 -100 -80 -60 -40 -20 Temp(K)Rxy ( )R 0 = 34819 R xy ( B=3 Tesla) R xy ( B=5 Tesla) R xy ( B=7 Tesla) Rxy = A + B T p Fit Rxy = A + B T p Fit Rxy = A + B T p FitA = 55 (3) B =-425.0 (9)p = -0.334 (5) A = 50 (3) B =-392.1 (9)p = -0.347 (6) A = 53 (3) B = -326 (1) p = -0.324 (8) Figure6-28.Plotof R xy versustemperaturefor R 0 =34819n. 0 10 20 30 40 50 -220 -200 -180 -160 -140 -120 -100 -80 -60 -40 R xy ( B=7 Tesla) R xy = A + B T p Fit Temp(K)R xy ( )R 0 = 37918 A = 55 (3) B =-436 (1)p = -0.338 (7) Figure6-29.Plotof R xy versustemperaturefor R 0 =37918n. 88

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0 10 20 30 40 50 -220 -200 -180 -160 -140 -120 -100 -80 -60 Temp(K)R 0 = 43792 R xy ( B=7 Tesla) R xy = A + B T p Fit R xy ( ) A = 65 (3) B =-462 (1)p = -0.330 (6) Figure6-30.Plotof R xy versustemperaturefor R 0 =43792n. Theinterestingthingaboutthisbehavioristhatthepowerp ofthetemperature dependenceissaturatingtoavaluecloseto-0.33.Thisisin closeresemblencetothe temperaturedependenceofthelongitudinalconductivityf orthesamesetofsamples, whereweseeadependenceoftheform 6{2 withthepower p closeto0.4. WecanndtheanomalousHallconductivityusingtherelatio n xy = R xy R 2 xx + R 2 xy : (6{11) Figure 6-31 showsthedependenceoftheanomalousHallconductivityasa functionof temperatureforsamplesinseries1.Toaccomodatetheresul tsforallthesampes,the conductivityhasbeenplottedonalogarithmicscalewhilet hetemperatureaxisisona linearscale.Beforeweemphasizetheimportantpointsabou tthisgraph,wewillrefresh ourmemoriesaboutthedatapresentedintheearlierchapter .InChapter5,wepresented Halldataforsampleswithquantumcorrrectiontoconductiv itythathaveapproximately lineartemperaturedependence.Forthosesamples(lower R 0 ),itwasalsoobservedthat anomalousHallconductivityexhibitednearlytemperature independentbehavior.In 89

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otherwords,thequantumcorrectionstoanomalousHallcond uctivitywerenegligible. Now,whenwestudysampleswithhigher R 0 (disorder)inthischapter,wendthatthe anomalousHallconductivityexhibitsatemperaturedepend entbehavior.Wearetryingto understandthisunexpectedtemperaturedependenceofthea nomalousHallconductivity. 10 20 30 40 50 0.3 0.6 0.9 1.2 1.5 3.5 4e-6 R0 4011 27872 7377 31010 11445 34819 17451 37918 19800 43792 23295 48208 xy ( -1)Temp(K) Figure6-31.PlotofanomalousHallconductivityvstempera turefordierentsamplesin series1. 90

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6.4Conclusions Inthecurrentwork,wehavestudiedthe3 D Andersonlocalizationtransitioninthin ferromagneticlmsofgadolinium.Ourexperimentaldatali einthecriticalregimeatnite temperatures,forwhichtherelevantdynamicalscalingbeh aviorisexactlyknowntobe ( ) 1 3 .Atnitetemperatures, iscutoatlowfrequenciesbythetemperature dependentphaserelaxationrateat 1 = .Wendthedominantcontributionto1 = tobegivenbyscatteringospinwaves,leadingto1 = / T pluscorrections.Asthe datashow ( T )tofollowatemperaturepowerlawwithexponent p 0 : 39,weidentify thisbehaviorwiththecriticalbehavior(modiedsomewhat bythehightemperature correctionsto1 = ).Thevalueof A ( )obtainedbyextrapolating ( T )inthescaling regimeto T =0showscriticalbehavior A ( ) / ( c ). A ( )isnotequaltothe conductivity ( T )at T =0,butisanewexperimentallyaccessibleobservablewhich describesthephysicsofscalingatnite(non-zero)temper atures. Theapplicationofmagneticelddrivesthesystemtowardsa metallicbehavior, exhibitedbytheincreaseoftheparameter A towardsamorepositivevalue.Wecanprobe themetal-insulatortransitionverycloselybyapplyingma gneticeldinsmallstepson asampleclosetothecriticaldisorder(ontheinsulatingsi de).Wehavea\eld-tuned insulator-to-metaltransition."Thetransverseresistan ce R xy alsodisplaysabehaviorof thetype R xy = A + BT p withtheexponentsaturatingclosetoavalue p -0.33,and thistranslatesintoatemperaturedependentanomalousHal lconductivity,whichdemands theoreticalinvestigations. 91

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CHAPTER7 SUMMARYANDFUTUREWORK Inthisdissertation,weinvestigatedferromagneticthin lmsunderavarietyof conditionstoextendourunderstandingofelectrontranspo rtinsystemswithferromagnetic longrangeorder.Wedrewmanyconclusions.Thoughthischap terconcludesthe dissertation,thecollectionofresultsandanalysesprese ntedinthisdisserationrepresent thebeginningofaricheldofexperimentalcondensedmatte rphysics.Thefollowing sectionprovidesthesummaryofimportantresultsandconcl usions.Finally,wewill provideaproposalofthefuturedirctionofthiswork. 7.1Summary Insummary,wehavepresentedasystematic insitu magnetotransportstudyona seriesofferromagneticthinlmscoveringabroadrangeofs heetresistances.Specically, wehaveperformeddetailedinvestigationonthefollowings ystems:namely;Fe( R 0 spans from140n(60 Athick)to6250n( < 20 A)),Gd( R 0 spansfrom370n(150 Athick)to 72 K n( < 15 A))andtwoFe/C60lms. Thestudyofquantumcorrectionstotheconductivitytensor inthinlmsofironin SHIVArevealsevidencefordisorder-dependentlocalizati oncorrectionstotheanomalous Hall(AH)conductivity[ 30 ].Fortemperatures T< 20K,alogarithmictemperature dependenceofthelongitudinal R xx andanomalousHallresistances R xy isobserved.In thelowdisorderlimit( R 0 < 150n),wendthatrelativechangesintheanomalousHall conductivity, xy = xy ,exhibitatemperatureindependentbehaviorimplyingthat there arenoquantumcorrectionsto xy .Asdisorderincreases,anitelogarithmictemperature dependenceto xy = xy appearsandthenevolvestowardauniversalweaklocalizati on correctiondenedbytheequality xy = xy = R xy =R xy .Thuswithincreasingdisorder, weseeacrossoverfromaregionwheretherearenoquantumcor rectionsto xy toaregion dominatedbyweaklocalizationcorrections. 92

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Wealsostudiedthechargetransportinferromagneticthin lmsofgadolinium,a localizedmomentsystem,inthelimitofweakdisorder.Wen dasurprisingbehaviorof thequantumcorrectionstotheconductivity.Inadditionto thelogarithmictemperature dependenceexpectedfromtheweaklocalizationeectsinth elongitudinalconductivityas previouslyseeninFelms,weobserveanadditionalcontrib utiontotheconductivitythat hasanapproximatelylinear T -dependenceforsheetresistances370n R 0 2840nand temperatures5K
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p 0 : 39,weidentifythisbehaviorwiththecriticalbehavior(mo diedsomewhatbythe hightemperaturecorrectionsto1 = ).Thevalueof A ( )obtainedbyextrapolating ( T ) inthescalingregimeto T =0showscriticalbehavior A ( ) / ( c ). A ( )isnotequal totheconductivity ( T )at T =0,butisanewexperimentallyaccessibleobservable. Theapplicationofmagneticelddrivesthesystemtowardsa metallicbehavior, exhibitedbytheincreaseoftheparameter A towardsamorepositivevalue.Wecanprobe themetal-insulatortransitionverycloselybyapplyingma gneticeldinsmallstepson asampleclosetothecriticaldisorder(ontheinsulatingsi de).Thetransverseresistance R xy alsodisplaysabehaviorofthetype R xy = A + BT p withthepowersaturatingclose toavalue p -0.33andthistranslatesintoatemperaturedependentanom alousHall conductivity,whichdemandstheoreticalinvestigations. 7.2ProposedFutureDirections Inthisdissertation,theexperimentalresultoftransport measurementsonferromagnetic thinlmsweredevotedtounderstandingoftheeectofdisor der.Westudiedbothband ferromagnetismandferromagnetismduetolocalizedmoment s.Aninterestingexperiment toextendtheunderstandingwouldbetostudytheeectofcha ngingthedensityof chargecarriersinthelmsbytheeldgatingtechniqueando bservethebehaviorofAH resistance.Astepinthedirectionofelectriceldgatingh asalreadybeentakenbyusin anexperiment[ 58 ],describedinappendix1,wherewereportontheuseofionic liquids (ILs)aseldgatedielectricsandshowthatasignicantel d-gateeectonthinlmsof amorphouscompositeInO x canbeobtainedutilizingrathersimplegatedsource-drain congurations.Wedemonstrateeld-inducedresistancech angesontheorderofafactor of10 4 forthinconductingInO x lms.Thearealcapacitancesandeld-eectmobilities noticeablyexceedthosethatcanbeachievedusingAlO x dielectrics.Inaddition,the chargestatecanbefrozeninbyreducingthetemperature,th usprovidinganopportunity forelectriceldtuningofmetal-insulatortransitionsin avarietyofnovelthin-lm 94

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systems.Withthehelpofthistechnique,weproposetogatef erromagnetisminlow electrondensitymagneticmaterialssuchasdilutemagneti csemiconductors. InChapter6,ourstudyofhighdisorderlmsofgadolinium,w hereanAnderson localizationtransitionisseen,inspiredustothinkabout theobservationofsucha transitioninothermaterials.Ourexperimentalstudieson ironandcopperthinlms, discussedinAppendixC,showstheexistenceofadisorderwh eretheconductivity hasapowerlawdependenceontemperature.Itwouldbeintere stingtodoadetailed experimentalstudiesontheselmswithacarefultuningofd isorder,sothatwecan capturethephysicsofthecriticalregime.Also,itwouldbe interestingtostudysuch transitionsinthinlmsofantiferromagneticmaterialssu chaschromium.Alsoeortsare neededtounderstandthebehavioroftheanomalousHallresi stancewithtemperature, R xy = A + BT p ,inthecriticalregimewheretheAndersonlocalizationtra nstionisseen. Finally,weneedtounderstandthebehaviorofanomalousHal lresistance,discussedin Appendix.B.Weseeasurprisingdependenceoftheparameter r ( T )= n B =k B T withthe reciprocaltemperature. 95

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APPENDIXA ELECTRICFIELDGATINGWITHIONICLIQUIDS A.1Introduction Fieldeecttransistors(FETs)playanimportantroleinour presentdayelectronic devices.Thesemiconductorsiliconanditsnativeoxide[ 59 ]arethecorecomponentsof suchdevices.Therearehoweverimportantnicheapplicatio nsinwhichitisdesirableto electriceldgateorganics,polymers,nanocomposites,co mplexoxidesand/orlowcarrier densitymetalsforamultiplicityofapplicationsincludin gdisplays,sensors,actuators, lowcostmemory,etc.Dependingontheapplication,thechoi ceofcomponentsfora eld-gateddevicecandependcriticallyoncost,durabilit y,speedofoperation,impedance, transconductanceandextentofmodulationofthelmbetwee nthesourceanddrain. Inresponsetothesechallenges,onepromisingandwellstud iedareaofinvestigationis theuseofelectrochemicaltechniquestostudysurfaceresi stancechangesattheinterface betweenaconductinglmandanelectrolyte[ 60 ].Fieldgatingeectsutilizingelectrolytes asthegatedielectrichavebeenobservedinnanotubebasedF ETs[ 61 62 ],organic electrochemicaltransistors[ 63 64 ]andelectrochemicallyinducedresistancechangesof porousnanocrystallinePt[ 65 ].Inthisappendix,wereportontheuseofionicliquids (ILs)ratherthanionicruidelectrolytesaseldgatediele ctricsandshowthatasignicant eld-gateeectonthinlmsofamorphouscompositeInO x canbeobtainedutilizing rathersimplegatedsource-draincongurations[ 58 ].Ionicliquidsarehighlypolar low-melting-temperaturebinarysaltstypicallycomprisi ngnitrogen-containingorganic cationsandinorganicanions.Sincethereisnosolvent,ILs aredistinctlydierentfrom aqueous,organic,gelorpolymerelectrolytes.Ionicliqui dsadvantageouslyhavehigh thermalstability,theyarenonvolatile,theyarecompatib lewithmostmaterialssystems, theycanbeexposedtomoderatepotentialdierenceswithou tundergoingredoxreactions, andtheyareruidoverawidetemperaturerange.Ionicliquid shavealsobeenshown tomanifestnotableperformanceadvantageswhenusedasele ctrochemicalmechanical 96

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actuators,electrochromicwindowsandnumericdisplays[ 66 ].Withrespecttoelectric eldgating,Wuetal.havesuccessfullydemonstratedtheus eofILstomodulateboth thespectraltransmittanceandtheresistanceofsinglewal lcarbonnanotubelms[ 67 ]. Forporousmaterialswithhighsurfacetovolumeratios,suc hasnanoporousmetals[ 65 ] ornanotubemats[ 67 ],liquiddielectricscanprovideecientcouplingtomosto fthe gatedmaterial.Ontheotherhand,ifthegatedmaterialisan on-porouscontinuousthin metallmsuchaslowcarrierdensityInO x [ 68 ],thenexcesschargeattheelectrochemical interfaceisscreenedwithinthemetalonadistanceontheor derofatomicdimensions, andtheresistancechangeisthereforeonlysignicantwhen thecarrierdensityislow andthethicknessofthesampleisontheorderofthemeanfree pathoftheconduction electrons.Aswewillseebelow,theadvantageofusingILdie lectricscomparedtosolid statedielectricsisasignicantlyenhancedchargemodula tionforthesamegatevoltage. Thisenhancement,whichisduetothelargedoublelayercapa citancethatoccursat theinterfaceofthemetalwiththeIL,isfoundtobesignica ntlylargerthanobtained inpreviouswork[ 68 ]usingsimilarInO x lmsincombinationwiththin-lmAlO x gate dielectrics.Thecompensatingdisadvantageisasignican tlyslowerrelaxationtime. A.2SampleFabrication Thesourceanddraincontactsofourteststructures,separa tedby1mmand comprising300 A-thickAudepositedon50 A-thickCr,wererstevaporatedonto cleanglasssubstrates.TheamorphouscompositeInO x lmswerethendepositedata rateof1 A/sthroughshadowmasksusingreactiveionbeamsputterdep osition[ 68 ].The InO x lmsarestableinair,smoothtextured,andsemitransparen twithcarrier(electron) densitiesontheorderof5 10 19 cm 3 .Foreachdepositiontwosamplesweremade: onefortheeldgatingexperimentsandtheotherforHallmea surements.Bothcoplanar (Fig. A-1 a)andoverlay(Fig. A-1 b)gatecongurationswereused.TheILchosenforour experimentwas99.5%pure1-ethyl-3-methylimidazoliumbi s(triruoromethylsulfonyl)imide (EMI-Beti)purchasedfromCovalentAssociates.Nosystema ticattemptwasmadeto 97

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optimizethechoiceofIL,althoughcarefuleortsweremade tokeepthesubstratesand liquidcontainerscleanandfreeofcontaminants. FigureA-1.Coplanar(a)andoverlay(b)gatecongurations showingtheplacementofthe gateelectrode(G),theionicliquid(IL),andtheactiveInO x thinlm connectedbetweensource(S)anddrain(D)terminals.(c)Ci rcuitmodel describingthecapacitivecouplingbetweentheInO x lmwithresistance R andtheionicliquidwithresistance R IL .Thisgurehasbeenadoptedfrom ourpaper(ref.[ 58 ])publishedinAppliedPhysicsLettersandAmerican InstituteofPhysicshasacopyrightforit. A.3MeasurementandAnalysis ThedielectricrelaxationsofILs,whichtypicallycoveral argerangeoffrequencies [ 69 ],canbecomeparticularlysluggishasthetemperatureislo wered.Westudythe trade-obetweenchargingandrelaxationtimebyusingaSol artronModel1260frequency 98

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responseanalyzer(FRA)tomeasurethegatevoltagedepende nceofthecomplex impedanceofa40 A-thickInO x lmoverthefrequencyrange10 2 -10 6 Hz.Sincethe pressedindiumcontactstothesourceanddrainterminalsha dalowcontactresistance, theFRAwasusedinatwo-terminalmodeinwhichanacvoltageo fxedamplitudeis appliedbetweenthesourceandthedrainandtheresultingph ase-shiftedcurrentdetected withasensitivecurrentamplier.Forthesemeasurementsw eusedthecoplanargate congurationofFig. A-1 a,whichhastheconvenienceofneedingonlyasmallamountof ruidtocoverboththegateandthe3mm 2 activeareabetweenthesourceanddrain.The mostecientchargecouplingoccurswhentheareasofthegat eandtheareaoftheactive lmbetweensourceanddrainareapproximatelyequal[ 67 ].Thefrequencydependence ofthemagnitudeoftheimpedanceisshowninFig. A-2 fortheindicatedgatevoltages V g .Linearityofthesource-drainvoltagewiththesource-dra incurrentwasveriedat1 Hz.Thedatashowanincrease/decreaseofimpedancefornega tive/positivegatevoltages, thusindicatingthattheInO x lmhasnegativecarriers.Thisiswellknownfromprevious work[ 68 ]butwasalsocheckedforthe40 A-thicklmunderconsiderationherebyHall measurementswheretheroom-temperaturecarrierdensityn =5 : 6 10 19 cm 3 foreld sweepsupto7Twasfound. Twoadditionalaspectsofthesedatashouldbenoted.First, thereisalarge asymmetrymanifestedbythesignicantlylargerchangesin impedancefornegative V g (electrondepletion)comparedtopositive V g (electronenhancement).Second,thereis pronouncedfrequencydependenceduetothelowionicmobili tiesinthedielectricruid; equilibriumforelectronenhancementoccursin100ms(10Hz )butforelectrondepletion (withconcomitantlylargechangesinimpedance)equilibri umoccursin100s(0.01Hz). Insightintothisbehaviorisgainedbyanalyzingthecircui tmodelshowninFig. A-1 c. Themodeltreatsinarathersimpliedmannerthecapacitive couplingbetweentheInO x lmwithresistanceRandtheionicliquidwithresistance R IL .Themagnitudeofthe impedancebetweenthesourceandthedrainforthismodelcan bereadilycalculatedas 99

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10 -1 10 1 10 3 10 5 10 3 10 4 10 5 10 6 10 7 -1.0-0.50.0 0.5 1.0 10 4 10 5 10 6 10 7 Gate Voltage (V)Sheet Resistance ( ) Sheet Impedance ( )Frequency (Hz) -1.03 V -0.55 V 0 V 0.55 V 1.03 V FigureA-2.(a)Frequencydependenceofthemagnitudeofthe sheetimpedanceatthe indicatedgatevoltages.Thecoplanarcongurationshowni nFig. A-1 awas used.Eachsweepstartingatthelowfrequencyendtakesabou t30minof measurementtimeforaconstantsource-drainvoltageampli tudeof0.5Vrms. ThesolidcurvesarethetsusingthecircuitmodelofFig. A-1 c.(b)Inset showstheplotofthelowfrequency(0.02Hz)resistancevsga tevoltagefor thesame40Athicklm.Thisgurehasbeenadoptedfromourpaper(ref. [ 58 ])publishedinAppliedPhysicsLettersandAmericanInstit uteofPhysics hasacopyrightforit. 100

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j Z ( ) j = vuut ( RR IL 2 C 2 ( R + R IL )+ R ) 2 +( R 2 !C ) 2 2 C 2 ( R + R IL ) 2 +1 2 (A{1) whichreducesto R inthelowfrequencylimitandto RR IL = ( R + R IL ),theparallel combinationof R and R IL ,inthehighfrequencylimit.Usingthe V g =0data(solid circlesofFig. A-2 ),wendagoodt(solidline)toEq. A{1 for C =0 : 178 F.We thenmakethereasonableassumptionthatCisindependentof V g andndthatEq. A{1 describeswelltheshapeof j Z ( ) j forallofthecurvesshowninthemainpanel ofFig. A-2 .Fromthesets, R IL isfoundtobenearconstantat1520 80n.Itis somewhatsurprisingandyetsatisfyingthatthesimplemode ldescribedbyEq.1works sowell:thedistributedcapacitancealongtheIL/InO x interfaceisrepresentedbyjusttwo series-connectedcapacitors, R IL isindependentof V g ,negligiblecurrentisdrawnbythe gatevoltage,anddielectricdispersionintheILhasbeenig nored. IntheFig. A-2 insetweplotthegatevoltagedependenceoftheequilibrium lowfrequency(0.02Hz)valuesofRextractedfromthedatain themainpanel.The asymmetryofthecurvearisesbecausethedepletionofcharg e(negative V g )close tothemetal-insulatortransitionhasalargereectonther esistancethandoesthe accumulationofanequivalentamountofcharge(positive V g ).Wecalculatefromthe modeltvalue C =0 : 178 Fandthe3mm 2 activeareabetweenthesourceanddrainan arealcapacitance C=A =5 : 9 F/cm 2 .For V g =1V,thiscorrespondstoanarealcharge density N FE = CV g =eA =3 : 7 10 13 cm 2 ,whereeisthechargeofanelectron.In contrasttothiseldeect(FE)measurement,theHallmeasu rementresult, n =5 : 6 10 19 cm 3 ,forthe40 A-thicklmimplies N Hall =2 : 2 10 13 cm 2 .Thus V g =-1Vacross theILissucienttodrivetheInO x lmnearlyintothefullyinsulatingstateasseen experimentally. CompatibilityoftheFEandHallmobilities, FE and Hall ,isalsoquitesatisfactory. Tocalculate FE ,weusetheBoltzmannexpression[ 68 ] FE =3 A ( @G=@V ) = 2 C and evaluate @G=@V =1 : 47 10 4 S/Vbytakingtheoddpartofthedependenceofthe 101

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conductance1 =R on V g where1 =R iscalculateddirectlyfromtheresistancesofthe Fig. A-2 inset.Theresult, FE =37 : 3cm 2 V 1 s 1 ,comparesfavorablywith Hall = 1 =eN Hall R =20 : 6cm 2 V 1 s 1 50 100 150 200 250 300 104105106107108 -1.03V -.549V 0V .549V 1.03V Sheet Impedance ( )Temperature (K) FigureA-3.Temperaturedependenceonslowcoolingofthema gnitudeoftheimpedance atthegatevoltagesshowninthelegend.The40 Athicklmwasmountedin theoverlaycongurationshowninFig. A-1 b.Measurementsweretakenat10 Hzforaconstantsource-drainvoltageamplitudeof0.05Vrm s.Thisgure hasbeenadoptedfromourpaper(ref.[ 58 ])publishedinAppliedPhysics LettersandAmericanInstituteofPhysicshasacopyrightfo rit. 102

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ThelargegateinducedchargedensitiesassociatedwithILd ielectricscanbefrozen intoplacebysimplycoolingthesamplethroughitsglasstra nsition T g withthegate voltageon.Wedemonstratethiselectriceldpolinginthe j Z ( ) j plotsofFig. A-3 for asecond40 A-thicksample.TheoverlaygatecongurationofFig.1bpre ventedany accidentalspillageofILinthecryostatandalsoaccommoda tedthermalcontraction.We usedameasuringfrequencyof10Hzduringthetemperaturesw eeps,afrequencywhich waslowenoughtoassurethatmostofthechargewastransferr edandfastenoughto followtheresistancechangesasthesamplecoolsthrough T g 250K.Thejumpsin j Z ( ) j whichareespeciallypronouncedfornegative V g ,occurastheILfreezesand R IL increases toalargevaluesothatbyEq., j Z ( ) j! R .Atthispointthemeasuredresistanceofthe InO x lmisunaectedbyacshuntsthroughthecapacitativelycou pledIL.Attheselow temperaturesremovingthegatevoltagehasnoeectonthech argestate,andthesample remainspoledforall T
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correlatedsystems,suchasmanganitesandhigh-Tcsuperco nductors,whereelectrostatic elddopingcansubstituteforchemicaldopinginthevicini tyofthemetal-insulatoror superconductor-insulatortransition.Advancesinthisdi rectionhavealreadybeenmade withnoveldielectricssuchasmicro-machinedSrTiO 3 singlecrystals[ 71 ]. 104

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APPENDIXB SOMEOBSERVATIONSABOUTANOMALOUSHALLEFFECTINIRONAND GADOLINIUMTHINFILMS B.1Introduction Animportanttoolthathasbeenusedtostudyferromagnetism inmagneticmaterials istheanomalousHall(AH)eect.Theunderlyingphysicsdes cribingtheAHeectis basedonseveralproposedmechanisms:theskew-scattering model[ 4 ],thesidejump model[ 6 ]andamorerecently,aBerryphasemodel[ 29 ].TheAHeectalsohaspotential technologicalapplicationsformagneticsensorsandmemor iesifthechangein R xy can bemadelargewithrespecttosmallchangesinappliedeld.R ecently,therehasbeena considerablegrowthofinterestintheAHEduetotheimporta nceassociatedwiththe understandingofspinpolarizationandspin-orbitinterac tionsforelectronictransportin materialsandstructuresofspinelectronics.Thespinofth eelectronwasusuallyignored inconventionalelectronics.Asthetransportofelectronc reatesachargecurrent,spinof theelectroncancreateaspincurrent,thusaddinganewdegr eeoffreedomtoelectronics andcreationofaneweldof\spintronics."Inthefollowing sections,wepresentsome observationsonAHeectinthinlmsofiron/C 60 andgadolinium. B.2ExperimentalDetailsandMeasurementTechniques WehavefoundthatSHIVAdepositedFelmsgrownatroomtempe raturewith R 0 > 50 K nareunstableintimebutthatthisinstabilitycanbebypass edbypredepositionof amonolayerofC 60 priortotheFedeposition.Inagreementwithpastexperimen tson Cu/C 60 bilayers[ 72 ],theC 60 providesashuntingpathforelectrontransport,sinceless Feisneededtogetthesamesheetresistance.Apparentlythe C 60 stabilizesthelmand doesnotgiverisetoadditionalspinripscattering[ 73 ].Usingthistechnique,wedeposited twoiron/C 60 lmsonasapphiresubstrateatroomtemperatureusingRFmag netron sputtering.Thegrowthparameterswerethesameasforiron lmsdiscussedinChapter4. WealsostudiedtheAHeectongadoliniumlmdiscussedinCh apters5and6. TheGdlmswith R 0 < 3 K nweredepositedintheHallbargeometrywhereasthe 105

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FigureB-1.SchematicdiagramofFe/C 60 deposition. higher R 0 lmsweredepositedintheHallcrossgeometryusingshadowm ask.Thecurrent andvoltageleadsofthedepositedsampleoverlappedwithth epredepositedpalladium contacts,thusensuringreliableelectricalconnection.I mmediatelyafterdeposition, samplesweretransferredwithoutexposuretoairfromthehi ghvacuumdeposition chambertothecryostatformagnetotransport.Duringdepos tion,sampleresistanceand thicknessarecontinuouslymonitored insitu .Toparameterizetheamountofdisorder inagivenlm[ 35 ],weusethesheetresistance R 0 R xx ( T =5K)where R xx isthe longitudinalresistance.The insitu characterizationofthethinlmsweredoneusing thestandardfourprobelock-intechniqueforlowresistanc elms( R 0 3 K n)andthe four-terminald.c.measurementsforlmsofhighsheetresi stances( R 0 > 3 K n). B.3MeasurementsandAnalysis ShowninFig. B-2 istheplotof R xy asafunctionofmagneticeldforaniron/C 60 lmatatemperature T =50K.Theresistanceofthesampleishigh,sotheHalldata canbetakenreliablyfortemperatureatandabove T =50K.Fig. B-3 showsthe semilogarithmicplotofthelongitudinalresistanceversu stemperature.Wecanseefrom thisplotthatweareinaregimeofhoppingtransport.Wecans eethatourHalldatacan bemodeledusingtheexpression, R xy ( T;B )= 0 R s ( T ) M ( r ( T ) B ) d + R n B d ; (B{1) where d isthethicknessofthelmandthemagnetization M ( r ( T ) B )ispresumedtohave aLangevindependencewith r ( T )= n B =k B T forsuperparamagneticgrainswith n Bohr 106

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0 10 20 30 40 50 0 50 100 150 R xy data at T=50 K Langevin fit R XY ( )B(KG) T=50KR xx (T=50K)=100K FigureB-2.HalldataforFe/C 60 lmat T =50K. 0.0 0.1 0.2 0.3 0.4 0.5 10 5 10 6 Rxx(T=5K)=1.2 M Fe/C 60 sample RXX( )T-1/2 (K -1/2) FigureB-3. R xx vstemperatureplotforaFe/C 60 lm,showingthatthesampleisinthe hoppingregime(squareroot). 107

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magnetons B pergrain.WestudiedtheHalldataforthislmatvarioustem peratures. Similarly,westudiedtheHalldataofthesecondiron/C 60 .Fig. B-4 showstheHalldata, takenatseveraltemperatures,forthesecondlm.ForeachH allcurvestudiedfora 0 10 20 30 40 50 0 20 40 60 80 R XY ( )B(KG) 10 K 15 K 20 K 25 K 50 K 75 K 125 K 150 K 200 K Fe/C60 R 0 (T=5K)=1.2 M FigureB-4.HalldataforFe/C 60 lmatseveraltemperature. lmataparticulartemperature,weextractthevalueof r ( T )forthattemperatureusing theEqn. B{1 .Thetemperature-dependentparameter r ( T )showsarathersurprising dependence.ShowninFig. B-5 isaplotofthetemperaturedependenceof r ( T )fortwo Fe/C 60 samples.Wecanseethat r ( T )increaseswithreciprocaltemperatureuptoa certainvalue,dependingonthestrengthofdisorderinthe lm,andthenbeginstorollo. Toinvestigatethisfurther,westudiedtheHalldatainGdth inlmsinthesame manner.Figs. B-7 and B-8 showthetypicalHallcurvealongwiththeLangevintand Halldataatvarioustemperatures,respectively,foraGdl m.Weextractthe r ( T )at dierenttemperatureforeachGdlm.Fig. B-9 showthebehaviorof r ( T )asafunction ofreciprocaltemperatureforaseriesofGdthinlms.Again ,weseethat r ( T )increases withreciprocaltemperatureuptoacertainvalue,dependin gonthestrengthofdisorder inthelm,andthenbeginstorollo.WecanseefromFig. B-10 thattheprefactorof 108

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0.000.020.040.060.080.10 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 R XX (5K)=1.2 M R XX (25K)=300 K (Tesla -1 )T -1 (K -1 ) Fe/C 60 samples FigureB-5.Plotof r ( T )vs T forFe/C 60 lms. 0 50 100 150 200 50 55 60 65 70 75 80 Fe/C60 R0(T=5K)=1.2 M Temp(K)( 0 R s M s ) /d ( ) FigureB-6.Plotofthetemperaturedependentprefactor 0 R s M s =d vs T foraFe/C 60 lm,showingitsdecreasewithincreasingtemperature. 109

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0 10 20 30 40 50 60 70 0 2 4 6 8 R xy data at T=5 K Langevin fit RXY( )B(KG)T = 5 KR xx (T=5K)= 428 FigureB-7.HalldataforaGd( R 0 =428n)lmat T =5 K alongwithaLangevint. -60-40-200 204060 -8 -6 -4 -2 0 2 4 6 8 B(KG)R XY ( ) 150K 30K 125K 20K 100K 10K 75K 7K 50K 5K 40K FigureB-8.HalldataforaGd( R 0 =428n)lmatdierenttemperature. 110

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0.00 0.05 0.10 0.15 0.20 0.5 1.0 1.5 2.0 2.5 Gd samples T -1 (K -1 ) (Tesla -1 ) 428 928 2203 4011 11445 17451 19800 FigureB-9.Plotof r ( T )vs T foraseriesofGdlms. theLangevinterm, 0 R s M s =d ,decreasesinmagnitudewithincreasingtemperature.The secondtermin B{1 isapproximatelyindependentoftemperature. B.4Conclusions Weobserveasurprizingtemperaturedependenceof r ( T )asafunctionofreciprocal temperature(1 =T ).Thepositiveslopeofthisgraphinthehightemperaturera nge isconsistentwiththesuperparamagneticbehaviorasseeni nthinFelmsgrown onZnSesubstrates[ 74 ]andinquench-condensedNilms[ 75 ].Asthetemperature decreases,weseeacrossoverinthisbehaviorand r ( T )startstodecreasewithdecreasing temperatures.Thisdecreasecanbeassociatedwithadisord er-induceddecreasein magneticsusceptibility.Wearetryingtounderstandthisb ehavior. 111

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0 20406080100120140160 -35 -30 -25 -20 -15 -10 -5 0 ( 0 R sMs) /d ( )Temp(K) R 0 =428 R 0 =928 R 0 =2203 R 0 =4011 Gd thin films FigureB-10.Plotofthetemperaturedependentprefactor 0 R s M s =d vs T foraseriesof Gdlms. 112

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APPENDIXC OBSERVATIONOFAPOWERLAWBEHAVIORINTHINFILMSOFIRONAND COPPER C.1Introduction AftertheobservationofAndersonlocalizationtransition inthinlmsofgadolinium, asdescribedinChapter6,wedecidedtopursuesimilarstudi esinthinlmsofiron, anitinerantferromagnet,andcopper,anon-magneticmater ial.Thefollowingsections describetheexperimentaldetailsandresultsforthisstud y. C.2ExperimentalDetailsandMeasurementTechniques Aseriesofironthinlmsweregrownonhighlypolishedsapph iresubstrateata temperatureof150Kusingr.f.magnetronsputteringwithgr owthparameterslistedin Table C-1 TableC-1.Growthparametersforironlms ParametersValues Rfpower20W DCbias-125V Argonrowrate10sccm Also,aseriesofcopperthinlmsweregrownonhighlypolish edsapphiresubstrateat atemperatureof150Kusingd.c.magnetronsputteringwithg rowthparameterslistedin Table C-2 TableC-2.Growthparametersforcopperlms ParametersValues power10W Argonrowrate10sccm ThelmsweredepositedintheHallcrossgeometryusingasha dowmask,with aspectratiodescribedinChapter6.Thecurrentandvoltage leadsofthedeposited sampleoverlappedwiththepredepositedpalladiumcontact spads,intheserpentineedge geometrydescribedinChapter6,thusensuringreliableele ctricalconnection.Immediately 113

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afterdeposition,samplesweretransferredwithoutexposu retoairfromthehighvacuum depositionchambertothecryostatformagnetotransport. The insitu characterizationofthethinlmshasbeendoneinthecryost atofSHIVA. Four-terminald.c.measurementsaremadeusingtheKeithle y236SMU,Keithley182 nanovoltmeterandKeithleyswitchingcard7012S.Theyarea llGPIBaddressable,sothat allthemeasurementscanbeautomated.Thelongitudinaland theHallresistanceare extractedusingthereversemagneticeldreciprocity(RFM R)theorem[ 57 ]. C.3ResultsandAnalysis 10 100 10 3 10 4 10 5 10 6 10 7 R 0 = 5 K R 0 = 56 K R 0 = 90 K R 0 = 145 K R 0 = 439 K R 0 = 576 K R(T=14K) = 2.3 M R xx ( ) Temperature(K) FigureC-1. R xx asafunctionoftemperatureforaseriesofironthinlms. ShowninFig. C-1 isthedependenceofthelongitudinalresistanceontempera turefor aseriesofironthinlms.Wecanseefromthisgurethatweha veawiderangeofsheet 114

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resistances.Toanalysethisdata,weplotaquantityW,whic hisgivenby, W = T d ln ( T ) dT : (C{1) WecanseefromEqn. C{1 thataconductivitybehaviorof = BT p givesavalueof W = p .Thusaconstantvalueofparameter W inaplotof W versus T impliesapower lawbehavior.Usingthisdenationof W forthedatashowninFig. C-1 ,weobtainthe plot C-2 .Weobservethat,fortwolmswith R 0 =439nand R 0 =576 K n,wehavearat W regionforatemperaturerangefrom5 20 K .Thiswouldimplytheexistenceofpower lawbehaviorinthinlmsofiron.Also,wecanseethatthepow er p 0 : 8. 10 100 0.1 1 10 Temperature(K)W = T {d ln ( T ) / d T } R0= 5 K R0= 56 K R0= 90 K R0= 145 K R0= 439 K R0= 576 K R(T=14K) = 2.3 M FigureC-2.Plotofparameter W ,asdenedinEqn. C{1 asafunctionoftemperaturefor thinlmofiron.Notetheexistenceofaregionofconstant W fortwolms. 115

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Wedidasimilarstudyonthinlmsofcoppertoseeifweseeifw egetapowerlaw behavior.Fig. C-3 showsthebehaviorofthelongitudinalresistanceasafunct ionof temperatureforaseriesofcopperthinlms.Thecorrespond ing W plotfortheselmsis showninFig. C-4 .Again,wecanseethatfortwolmswith R 0 =14nand R 0 =16 K n, wehavearat W regionforatemperaturerangefrom6 25 K withthepower p 0 : 2. 10 100 10 3 10 4 10 5 10 6 R 0 = 4 K R 0 = 14 K R 0 = 16 K R 0 = 35 K R 0 = 200 K R(T=7.3K) = 372 K R xx ( ) Temperature(K) FigureC-3. R xx asafunctionoftemperatureforaseriesofcopperthinlms. C.4Conclusions Inthinlmsofironandcopper,weobservearegionoftempera turewhere ( T )has thepowerlawdependenceontemperatureforsomevaluesofdi sorder.Acloserlookat theregionwherethepowerlawbehavioroccurswouldgiveani nsightaboutthecritical behavioraroundthemetal-insulatortransitionforthese lms. 116

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10 100 0.1 1 W = T {d ln (T) / d T } R 0 = 4 K R 0 = 14 K R 0 = 16 K R 0 = 35 K R 0 = 200 K R(T=7.3K) = 372 K Temperature(K) FigureC-4.Plotofparameter W ,asdenedinEqn. C{1 ,asafunctionoftemperaturefor thinlmofcopper.Notetheexistenceofaregionofconstant W fortwo lms. 117

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BIOGRAPHICALSKETCH RajivMisrawasborninAllahabad,UttarPradesh,India.Raj ivisthesecondchild ofMr.BhawaniPrasadMisraandMrs.UshaMisra.Hehasthrees iblings:anolder sisterandtwoyoungerbrothers.Rajivdidhisundergraduat estudiesatUniversity ofDelhi,wherehereceivedtheBachelorofScienceinPhysic s.Hethenreceivedhis MasterofPhysicsdegreeatIndianInstituteofTechnology, Delhi.Atthesameplace,he receivedhisMasterofTechnologyinComputerApplications ,inwhichhespecializedin supercomputingforscienticapplications.Hethenjoined thegraduateprogramatthe UniversityofFloridaintheFallof2004andinthesummerof2 005beganworkingfor ProfessorArthurHebard,averykindheartedandanenthusia sticperson.Rajivmethis wonderfulwife,Preeti,inthesummerof2008. 122