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Micromechanical Force Magnetometers for Measuring Magnetization at High Magnetic Fields and Low Temperatures

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Title:
Micromechanical Force Magnetometers for Measuring Magnetization at High Magnetic Fields and Low Temperatures
Creator:
Ninios, Konstantinos D
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[Gainesville, Fla.]
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Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Physics
Committee Chair:
Takano, Yasumasa
Committee Co-Chair:
Chan, Ho Bun
Committee Members:
Ingersent, J. Kevin
Hebard, Arthur F
Arnold, David
Graduation Date:
12/17/2011

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Physics -- Dissertations, Academic -- UF
dimpy -- force -- luttinger -- magnetization -- magnetometers -- mems -- tll -- wilson
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bibliography ( marcgt )
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government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Physics thesis, Ph.D.

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Abstract:
Quantum magnets and quantum phase transitions are two of the most interesting research topics in condensed matter physics nowadays. Their study often requires magnetization measurements at low temperatures (down to < 100 mK) and high magnetic fields (up to 65T). Most of the existing magnetization measurement devices can only be operated over a small fraction of the parameter space. A flexible and cost efficient device that can measure magnetization in the aforementioned conditions is highly desirable. This dissertation describes the development of new magnetization measurement devices, which we call micromechanical magnetometers, using silicon surface micromachining. These new magnetometers are superior to conventional magnetometers in terms of resolutions, versatility to operate in a variety of experimental conditions, as well as design flexibility and cost efficient fabrication. Using micromechanical magnetometers we have measured the magnetization of the strong-leg spin-1/2 ladder compound (C7H10N)2CuBr4 at temperatures down to 45 mK. Low-temperature magnetic susceptibility as a function of field exhibits a maximum near the critical field H_c, at which the magnon gap closes, as expected for a gapped one-dimensional antiferromagnet. Above H_c, a clear minimum appears in the magnetization as a function of temperature as predicted by theory. In this field region, the susceptibility in conjunction with specific heat data yields the Wilson ratio, the key parameter of the Tomonaga-Luttinger spin liquid. In addition, we have measured the magnetization of Ba3Cr2O8 as a function of the magnetic field down to 0.6 K. These measurements reveal a magnetic behavior consistent with BEC of triplets at fields higher than the critical field H_c. Hysteretic behavior in the magnetization at the vicinity of the saturation field H_s suggests that the phase transition at H_s is first order. Using the magnetization data in conjunction with other experimental methods the phase boundary of Ba3Cr2O8 was mapped at temperatures up to 2.3 K. ( en )
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In the series University of Florida Digital Collections.
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Thesis (Ph.D.)--University of Florida, 2011.
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Adviser: Takano, Yasumasa.
Local:
Co-adviser: Chan, Ho Bun.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2014-12-31
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by Konstantinos D Ninios.

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12/31/2014
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MICROMECHANICALFORCEMAGNETOMETERSFORMEASURINGMAGNETIZATIONATHIGHMAGNETICFIELDSANDLOWTEMPERATURESByKONSTANTINOSD.NINIOSADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2011

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c2011KonstantinosD.Ninios 2

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Tomyfamily 3

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ACKNOWLEDGMENTS Firstofall,IwouldlliketothankmyPh.DsupervisorProf.HoBunChan.ProfessorChanhasspentunlimitedtimewithmeinthelab,teachingmeandprovidinghishelpwitheverythingIneeded.Heofferedallkindsofassistanceandsupport,necessaryforthisresearchtobecompleted.InadditionheisoneofthefairestpeopleIhaveevermet.IwouldalsoliketothankProf.YasumasaTakanowhoisamemberofmycommittee.HiscontributiontotheworkonDIMPYwasprecious.IalsothankProfessorsArthurHebard,KevinIngersentandDavidArnoldtheothermembersofmycommitteefortheirmentoring.IowealargeamountofthankstoLuisBalicasandFedorBalakirev,mycollaboratorsfromtheNationalHighMagneticFieldLaboratory.DuringourcollaborationIhadthegreatexperiencetoperformexperimentsatthehighestmagneticeldsintheworld.IwouldalsoliketothankthestaffofthePhysicsDepartment.ThepeopleofthephysicsmachineshopMarcLink,EdStorchandBillMalphurshavealwaysbeenhelpful,andtheybuiltallthepartsIneededfast.GregLabbeandJohnGrahamhaveprovidedmewithtonsofhelponcryogenics.ThephysicselectronicshopstaffLarryPhelpsandPeteAxsonprovidedmewiththeirpreciousexperiencetohelpmebuildmyprobes.JayHortonhelpedmeminimizethemechanicalnoiseinseveralexperiments.MylifeasastudentwouldhavebeenmuchmoredifcultwithoutthenancialsupportfromAlexanderS.OnassisPublicBenetFoundation,whichIdeeplythank.Inaddition,IthankmylabmatesCoreyStambaugh,ZsoltMarcet,YiliangBaoandJieZoufortheirconstanthelpandthefuntimesinthelab.IwouldalsoliketosaythankstoallmyfriendsfromHongKong:TangLu,FengpeiSun,PhillipForsythandallthepeopleImetonthehikesforthegreattimeswhichgavemeenergytokeepworkinginthelab.IalsoowespecialthankstomyroommateDimitriosKoukiswhoisaphysicist,too,forhavingbeenabigsupportalltheseyears. 4

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FinallyIwouldliketoexpressmylovetomyfamily:myparentsandmysister.Evenfromsuchalongdistancetheyalwaystriedtoencourageandsupportmeondifculttimes.Myparentsalsotaughtmetobehonestandhumble,andIwanttosincerelythankthemforthatlessonwhichIwillfollowinmylife. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 13 2THEORETICALBACKGROUND .......................... 20 2.1PropertiesofaMagneticMoment ....................... 20 2.2SpinExcitations ................................ 21 2.2.1FerromagneticMagnons ........................ 22 2.2.2AntiferromagneticMagnons ...................... 25 2.2.3Spinons ................................. 27 2.3HaldaneChains ................................ 27 2.4SpinDimers ................................... 29 2.5Bose-EinsteinCondensation ......................... 31 2.6WilsonRatioinFermiLiquidsandTomonaga-LuttingerLiquids ...... 32 3CONVENTIONALMAGNETOMETERS ...................... 36 3.1Susceptometers ................................ 36 3.2SQUIDs ..................................... 38 3.3CantileverMagnetometers ........................... 41 3.4FaradayBalanceMagnetometers ....................... 43 4MICROMECHANICALMAGNETOMETERS .................... 46 4.1FabricationandPreparationoftheDevices ................. 46 4.2DesignoftheMicromechanicalMagnetometers ............... 50 4.3Operation .................................... 52 4.4DetectionSchemeforMagnetizationMeasurements ............ 55 4.5Calibration ................................... 59 4.5.1AbsoluteMagnetizationoftheSample ................ 59 4.5.2ExtractionoftheMagneticMomentperUnitFormula ........ 62 4.6ComparisonwithOtherMagnetizationMeasurementMethods ....... 64 5(C7H10N)2CuBr2 ................................... 68 5.1PreviousWorkonDIMPY ........................... 68 5.2Experimental .................................. 70 5.3Results ..................................... 72 5.4ProceduresFollowedforAnalyzingtheData ................. 82 6

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5.4.1ForceMeasurementandMagnetizationCalibration ......... 82 5.4.2CorrectionsontheData ........................ 87 5.5Conclusions ................................... 88 6Ba3Cr2O8 ....................................... 89 6.1BasicPropertiesofBa3Cr2O8 ......................... 89 6.2Results ..................................... 93 6.3MagneticForceandMagnetizationCalibration ............... 98 6.4Conclusions ................................... 99 7SUMMARY ...................................... 101 APPENDIX AEXPERIMENTINPULSEDMAGNETICFIELDS ................. 103 BFINITEELEMENTANALYSIS ............................ 107 CSAMPLEPREPARATIONPROCEDURE ...................... 110 REFERENCES ....................................... 111 BIOGRAPHICALSKETCH ................................ 116 7

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LISTOFFIGURES Figure page 1-1Ferromagneticandantiferromagneticspinorder. ................. 13 1-2Aladderofspindimers. ............................... 15 1-3Scanningelectronmicrographofatypicalmicromechanicalmagnetometer. .. 17 1-4Thetrampoline-likedesignofthemicromechanicalmagnetometers. ...... 17 2-1Acurrentloopinamagneticeld. ......................... 21 2-2Spinwave. ...................................... 22 2-3Thespinwavedispersionrelationforachainofferromagneticallycoupledspins. ............................................. 24 2-4Thespinwavedispersionrelationforachainofantiferromagneticallycoupledspins. ......................................... 26 2-5Thedispersionrelationforaspin-1=2HAFchain. ................. 28 2-6Thedispersionrelationforaspin-1HAFchain. .................. 28 2-7Zeemaneffectforisolateddimers. ......................... 30 2-8Zeemaneffectforinteractingdimers. ........................ 31 3-1Schematicofanacsusceptometer. ......................... 37 3-2Schematicofavibratingsamplemagnetometer. ................. 39 3-3SQUIDmagnetometer. ................................ 40 3-4FluxchangethroughaSQUID. ........................... 40 3-5Principleofoperationofthecantilever. ....................... 42 3-6PrincipleofoperationoftheFaradaybalancemagnetometer. .......... 44 4-1MEMSfabrication. .................................. 48 4-2Release. ....................................... 50 4-3SEMimageofamicromechanicalmagnetometer. ................ 52 4-4Dimples. ........................................ 52 4-5Principleofoperationofthemicromechanicalmagnetometer. .......... 54 4-6CapacitiveresponseasafunctionoftheappliedDCvoltage. .......... 56 8

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4-7Schematicofthedetectioncircuit. ......................... 57 4-8Ratiotransformerinthedetectioncircuit. ..................... 58 4-9Calibrationofthemicromechanicalmagnetometers. ............... 60 4-10Electrostaticforceappliedtothemovableplateofamagnetometerasafunctionoftime. ........................................ 61 4-11Resonanceshiftcausedbytheadditionalmassofasamplemountedontopofthemagnetometerplate. ............................. 63 4-12Achipthatcontainsfourmagnetometers. ..................... 66 5-1CrystalstructureofDIMPY. ............................. 69 5-2Specic-heatof67%deuteratedDIMPY. ...................... 70 5-3MagneticsusceptibilityofDIMPYasafunctionoftemperaturedownto2Kandatappliedmagneticeldof1T. ........................ 72 5-4MagnetizationofDIMPYasafunctionofmagneticeldatxedtemperatures. 74 5-5MagneticsusceptibilityofDIMPYasafunctionofyhemagneticeld. ...... 75 5-6MagnetizationofDIMPYasafunctionoftemperature. .............. 76 5-7Thepositionofthemagnetizationminimumasafunctionofmagneticeld. .. 77 5-8Comparisonofthemagnetizationasafunctinoftheeldat300mKwithDMRGresults. ........................................ 78 5-9Magneticspecicheat,Cm,offullydeuteratedDIMPY,plottedasCm=T. .... 79 5-10PhasediagramoffullydeuteratedDIMPY. ..................... 81 5-11DependenceoftheWilsonratioRWofDIMPYonthenormalizedmagnetizationm. ........................................... 82 5-12Rawsignalofthemagnetometer. .......................... 84 5-13Testusingelectrostaticforce. ............................ 85 5-14ThemagnetizationperformulaunitofDIMPYat1.8Kand4.3K. ........ 86 6-1Geometricfrustration. ................................ 90 6-2CrystalstructureofBa3Cr2O8. ........................... 91 6-3HeatcapacityofBa3Cr2O8. ............................. 92 9

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6-4The20pinberglasspackageusedformagnetizationmeasurementsattheNHMFLfacilityinTallahassee. ........................... 93 6-5MagnetizationofasinglecrystalofBa3Cr2O8asafunctionofmagneticeldatconstanttemperatures. .............................. 94 6-6HystereticbehavioratHs. .............................. 95 6-7CriticaleldsdeterminationforBa3Cr2O8. ..................... 96 6-8PhasediagramofBa3Cr2O8. ............................ 97 6-9Rawsignalofthemagnetometerat0.6K. ..................... 98 6-10Forceandtorquesignalsofthemagnetometerat0.6K. ............. 100 A-1Thetimeproleofthemagneticeldinashotofthe65TshortpulsemagnetatNHMFLinLosAlamos. .............................. 104 A-2SmFeAsOresponseinapulsedmanget. ..................... 105 B-1Magnetometersuitableforanisotropicsamples. .................. 108 B-2COMSOLsimulationofthemovableplateofatypicalmagnetometersubjectedtoa2.5Npointforceatthecenter. ........................ 109 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyMICROMECHANICALFORCEMAGNETOMETERSFORMEASURINGMAGNETIZATIONATHIGHMAGNETICFIELDSANDLOWTEMPERATURESByKonstantinosD.NiniosDecember2011Chair:YasumasaTakanoCochair:HoBunChanMajor:PhysicsQuantummagnetsandquantumphasetransitionsaretwoofthemostinterestingresearchtopicsincondensedmatterphysicsnowadays.Theirstudyoftenrequiresmagnetizationmeasurementsatlowtemperatures(downto<100mK)andhighmagneticelds(upto92.5T).Mostoftheexistingmagnetizationmeasurementdevicescanonlybeoperatedoverasmallfractionoftheparameterspace.Aexibleandcostefcientdevicethatcanmeasuremagnetizationintheaforementionedconditionsishighlydesirable.Thisdissertationdescribesthedevelopmentofnewmagnetizationmeasurementdevices,whichwecallmicromechanicalmagnetometers,usingsiliconsurfacemicromachining.Thesenewmagnetometersaresuperiortoconventionalmagnetometersintermsofresolutions,versatilitytooperateinavarietyofexperimentalconditions,aswellasdesignexibilityandcostefcientfabrication.Usingmicromechanicalmagnetometerswehavemeasuredthemagnetizationofthestrong-legspin-1/2laddercompound(C7H10N)2CuBr2attemperaturesdownto45mK.Low-temperaturemagneticsusceptibilityasafunctionofeldexhibitsamaximumnearthecriticaleldHc,atwhichthemagnongapcloses,asexpectedforagappedone-dimensionalantiferromagnet.AboveHc,aclearminimumappearsinthemagnetizationasafunctionoftemperatureaspredictedbytheory.Inthiseldregion, 11

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thesusceptibilityinconjunctionwithspecicheatdatayieldstheWilsonratio,thekeyparameteroftheTomonaga-Luttingerspinliquid.Inaddition,wehavemeasuredthemagnetizationofBa3Cr2O8asafunctionofthemagneticelddownto0.6K.ThesemeasurementsrevealamagneticbehaviorconsistentwithBose-EinsteincondensationoftripletsateldshigherthanthecriticaleldHc.HystereticbehaviorinthemagnetizationatthevicinityofthesaturationeldHssuggeststhatthephasetransitionatHsisrstorder.UsingthemagnetizationdatainconjunctionwithotherexperimentalmethodsthephaseboundaryofBa3Cr2O8wasmappedattemperaturesupto2.3K. 12

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CHAPTER1INTRODUCTIONQuantummagnetismisoneofthemostactiveresearchtopicsincondensedmatterphysicstoday.Thematerialsofstudy,whicharecalledquantummagnets,aresystemsconsistingofmagneticionsthatcarryspin.Thetwomostcommontypesofmagneticsolidsaretheferromagnets(Figure 1-1 A)systemswhereallthespinsliealongasingledirectionevenintheabsenceofanappliedeldandtheantiferromagnets(Figure 1-1 B)materialswhoseneighboringspinsalignantiparalleltoeachother.Aftermuchprogressinthelastcentury,ferromagnetismisreasonablywell-understoodandferromagneticmaterialsofferimportanteverydayapplications.Mostoftheinterestinmagnetismnowadaysisfocusedonantiferromagnetism. Figure1-1. Ferromagneticandantiferromagneticspinorder.(a)Ferromagneticallyorderedspinsinatwodimensionallattice.(b)Antiferromagneticallyorderedspinsinatwodimensionallattice. Ofspecialinterestarespindimermaterials.Aspindimerispairofspinsthatinteractantiferromagnetically.TheHamiltonian(bH)ofaspindimerisbH=J~S1~S2, (1) 13

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where~S1and~S2aretheoperatorsofthespinsandJisaconstant,positiveforantiferromagneticinteractions.Thetotalspin~Softhespindimerisgivenby~S=~S1+~S2, (1)forwhich(~S)2=(~S1)2+(~S2)2+2~S1~S2. (1)WiththeuseofEq. 1 theHamiltonianbecomesbH=J 2(~S)2)]TJ /F4 11.955 Tf 11.96 0 Td[((~S1)2)]TJ /F4 11.955 Tf 11.96 0 Td[((~S2)2. (1)IfthespinquantumnumbersofthespinsinthedimerareS1=S2=1=2thenthespinquantumnumber(S)ofthetotalspin(~S)canbeeitherS=0orS=1.Keepinginmindthattheeigenvalueof(~S)2isS(S+1)andwiththeuseofEq. 1 weseethatthesystemhastwoenergylevelswhichdependonthevalueofS.ForS=0theenergyis)]TJ /F4 11.955 Tf 9.29 0 Td[(3J=4andforS=1theenergyisJ=4.ItisthereforeclearthatthesystemhasanenergygapwhosevalueisequaltothemagnitudeoftheinteractionJ.AlthoughtheHamiltonianofonespindimerissimpletosolvequantummechanically,understandingthepropertiesofsuchdimersforminganarrayinarealsolidisnottrivialandrequirestheuseofmanybodyphysicstechniques.Evenforsimple-lookingsystemslikeaspin1/2twoleggedladder(Figure 1-2 )achainofspindimersthereexistsnoexactsolutionfortheHamiltonianandnumericalmethodsareemployedtopredicttheirbehavior.Inaddition,realizationofantiferromagneticspindimersystemsinrealmaterialsisnon-trivial.Typicallysuchcompoundsdonotexistinnatureandtheyarefabricatedinlaboratories.Theyaregrownassmallsinglecrystalswithmassesofmilligramsorless,andinvestigationoftheirmagneticpropertiesoftenrequiresextremeconditionssuch 14

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Figure1-2. Aladderofspindimers. ashighmagneticeldsandlowtemperatures.Thesefactsmakequantummagnetismexperimentsverychallenging.Oneofthemostimportantpropertiesmeasuredinmagnetismexperimentsismagnetization,aquantityrelatedtothenumberofspinsalignedtoanexternalmagneticeld.Variousdevices(magnetometers)havebeendevelopedovertheyearsformeasuringthemagnetizationofsmallcrystals.Theuseofeachmagnetometerislimitedtocertainrangesofmagneticeldsandtemperatureswhiletheircostisdependentonthecomplexityofthedevice.Inaddition,thesignalofsomeofthesedevicesisnotproportionaltothemagnetizationofthesample,butvarieswithsharpchangesinthemagnetization.Forthisreason,amagnetometerthatisabletomeasureabsolutemagnetizationandcombineshighresolutionwithlowcostinavarietyofexperimentalconditionsishighlydesirable.Adevicewithsuchqualicationshadbeenpreviouslyusedinapulsedeldexperiment[ 1 ]withtheparticipationofoneofourcollaborators.However,afteranunsuccessfuldceldexperimentthisprojectwasabandonedandleftintheinitialstagesofitsdevelopment.Thisdissertationdescribesthedevelopmentofnovelmagnetometers(Figure 1-3 ),themicromechanicalforcemagnetometers,whichmeasureabsolutemagnetization.Theyaresuperiortoconventionalmagnetometersintermsofhighresolutions, 15

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compatibilitywithalargevarietyofexperimentalconditions,aswellasdesignexibilityandcostefcientfabrication.Thefabricationprocessthatweusedforthedevelopmentofthenewmagnetometerscomesfromintegratedcircuit(IC)manufacturing.Smallstructuresarecreatedontopofasiliconwaferbydepositionofmateriallayerswhicharepatternedusingphotolithographyandplasmaetching.Usingthistechnologythesizeofthesmallestfeatureofthemagnetometers(suchasthewidthofaspring)isabout2m.Thisenablesmeasurementsoftheabsolutemagnetizationoftinysamples(massesofg),whichwerenotpossiblebefore.Inaddition,thematerialusedforthestructuralpartsofthemagnetometersispolysilicon,whosemechanicalpropertiesdependonlyveryweaklyontemperature.Asaconsequence,thesedevicescanbeusedattemperaturesrangingfrommKallthewaytoroomtemperature.Furthermore,theICfabricationprocessoffershighdesignexibilityandcosteffectiveness.Thesemagnetometerslooklikeatrampoline.Amovableplateisattachedbyspringsaboveaxedelectrode.Whenaforceactsontheplate,thespringswillbeextendedtogeneratearestoringforcetocounteracttheexternalforce.Asaconsequence,theplateisdisplacedparalleltothexedelectrode(Figure 1-4 ).Thissimpledesign,whichresemblesaparallelplatecapacitor,makesthecharacterizationofthemagnetometersaneasyprocess.Asaresult,theoutputsignalisaccuratelycalibratedusingelectrostaticsandthemagnetizationcanbepreciselyextracted.ThisdesignisbasedonthedeviceofRef.[ 1 ].Usingthemicromechanicalmagnetometersweperformedexperimentsonseveralspinsystemsunderawiderangeofexperimentalconditions.Inonesuchexperimentthecompound(C7H10N)2CuBr2,abbreviatedDIMPY,wasused.PreviousheatcapacityresultsperformedbysomeofourcollaboratorsshowedthatDIMPYisoneoftheonlytwoidealonedimensional(1D)spinsystemsexistingtoday[ 2 ].DIMPY,similarlytoaspindimer,hasagapintheexcitationspectrum.Applicationofamagneticeld 16

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Figure1-3. Scanningelectronmicrographofatypicalmicromechanicalmagnetometer.Thesampleontopofthedeviceisanickelspherewithamassof4g. Figure1-4. Thetrampoline-likedesignofthemicromechanicalmagnetometers.Whenaforceactsonthemovableplate,theseparationbetweentheplateandthexedelectrodechanges. 17

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closesthisgapatsomecriticaleldHc,abovewhichthesystemisdescribedasaTomonaga-Luttingerliquid(TLL).AlthoughspecicheatandmagnetocaloriceffectmeasurementshavebeenpreviouslyperformedonaTLL[ 2 4 ],nobodyhasmeasuredthemagnetizationandthemagneticsusceptibilityofsuchasystemwithhighaccuracy.Asaresult,theWilsonratio,acrucialquantityin1D,hadnotbeendeterminedforaTLLpriortothisstudy.DIMPYwasanexcellentmaterialofstudyforthenewlydevelopedmicromechanicalmagnetometersbecausenoothermagnetizationmeasurementdeviceisabletomeasuretheabsolutemagnetizationatsolowtemperatureswewentdownto45mKandwithsuchasmallsample(micrograms).Ourmagnetizationmeasurementsasafunctionofthemagneticeldinconjunctionwithspecic-heatmeasurementsperformedbyourcollaboratorsenabledforthersttimethedeterminationoftheWilsonratioinareal1Dsystem.Additionally,thebehaviorofthemagnetizationofDIMPYasafunctionoftemperaturewasmeasuredandfoundtobeingoodagreementwiththeory.InadifferentexperimentthemagnetizationofBa3Cr2O8wasmeasuredasafunctionofthemagneticeldateldsupto26T.Ba3Cr2O8isasystemofinteractingspindimersarrangedina3Dlattice.Thismaterialalsohasagapintheexcitationspectrum.ThissystemisexpectedtoundergoeBose-Einsteincondensationofmagnonsatmagneticeldshigherthantheeldthatclosesthespingap12.5T.Thecompletestudyofthegaplessphaserequiredeldsupto26TmakingmagnetizationmeasurementsonBa3Cr2O8challenging.However,usingthemicromechanicalmagnetometerswemeasuredthemagnetizationofthiscompoundintheNHMFLatTallahasseeandtheresultswerecompatiblewithspecic-heatandtorquemagnetometryexperiments[ 5 ].Finally,wetestedthedevicesona65TpulsedmagnetusingasinglecrystalofSmFeAsO.Themeasuredmagnetizationasafunctionoftheeldwasingood 18

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agreementwithresultsfromtorquemagnetometryandmuonspinrotationexperiments[ 6 ].Thestructureofthisdissertationisasfollows.Chapter 2 providesthetheoreticalbackgroundonwhichthisdissertationisbased.InChapter 3 ,abriefreviewofothertypesofmagnetometersispresented(conventionalmagnetometers).InChapter 4 ,thefabricationprocess,theprinciplesofoperation,andthecalibrationofthemicromechanicalmagnetometersaredescribed.Inadditionthemicromechanicalmagnetometersarecomparedtoothermagnetizationmeasurementdevices.Chapter 5 summarizestheresultsoftheexperimentonDIMPY.InChapter 6 ,themagnetizationmeasurementsonBa3Cr2O8arepresented.Finally,theresultsoftheSmFeAsOexperimentinthepulsedmagnetarepresentedinAppendix A 19

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CHAPTER2THEORETICALBACKGROUNDThischaptergivesanintroductiontothebasictheoreticalconceptsonwhichthisdissertationisbased. 2.1PropertiesofaMagneticMomentTheelementarymagneticquantityisthemagneticmoment,ofwhichthesimplestexampleisaclosedloopofcurrentIandareaA.Itsmagneticmoment~misdenedas~m=I~A, (2)wherethevector~Aisnormaltotheloopanditsdirectionisdeterminedbytheruleofthethumbdependingonthedirectionofthecurrentaroundtheloop.Analogoustoanelectricdipoleinanelectriceld,auniformmagneticeld~Hexertsatoque~Tmonthemagneticmoment[ 7 ]expressedbytheequation~Tm=~m~H. (2)Thetorqueonthemagneticmomenttendstoalignthemomentwiththemagneticeld~H(Figure 2-1 ),minimizingtheenergyEofthesystemE=)]TJ /F5 11.955 Tf 10.44 .5 Td[(~m~H. (2)Intheexistenceofamagneticeldgradient,amagneticforce~Fmisalsoexertedonthemagneticmoment[ 7 ]~Fm=~r(~m~H), (2)andintheabsenceofexternalcurrents,~r~H=0,theforceis~Fm=(~m~r)~H.TheStern-Gerlachexperimentconnectedthemagneticmomentwiththespin[ 8 ].Foranelectrontherelationbetweenitsspinandtheassociatedmagneticmomentmis~m=)]TJ /F6 11.955 Tf 9.3 0 Td[(gB ~~S, (2) 20

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Figure2-1. Acurrentloopinamagneticeld.Thetorqueonthemagneticmomenttriestoalignthemomentwiththemagneticeld. wheregistheg-factorthattakesavaluecloseto2forelectrons,BistheBohrmagneton,~isPlanck'sconstantdividedby2,and~Sisthespinoftheelectron.Therefore,accordingtoEq. 2 ,theenergyEofanelectroninamagneticeld~HwillbeE=gB ~~S~H,whichisE=g 2BHforspinupandspindownrespectively.ThisiscalledtheZeemansplitting. 2.2SpinExcitationsTheorderofthecrystallatticeinasolidatnon-zerotemperatureisdisruptedbythermalexcitationswhicharequantizedasphonons.Inasimilarwaythecollectiveexcitationsoftheelectronspinsinamagneticsolid(calledspinwaves)arequantizedasmagnons[ 9 11 ].Inthissectionwewillderivethespinwavedispersionrelationforferromagnetic(Figure 2-2 )andantiferromagneticsolids,usingasemiclassicalapproach. 21

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Figure2-2. Spinwave.(a)Spinwavepictureofa1Dferromagnet.(b)Topviewofthespinwave. WhenspinsinamagneticsolidinteractwitheachothertheirHamiltoniancanbewrittenasbH=Xi>jJij~Si~Sj, (2)whereSiisthespinatsiteiandJijistheexchangeinteractionbetweenthespinatthesiteiandthespinatthesitej.Inthisderivation,thespinsSiwillbetreatedasclassicalvectorsanditwillbeassumedthatallmagneticatomshavethesamespinquantumnumberS.ThesignofJijplaysaveryimportantrole.IfJijisnegativetheinteractionbetweenthespinsisferromagneticand,ifitispositive,theinteractionisantiferromagnetic. 2.2.1FerromagneticMagnonsForsimplicityletusconsidera1DsolidwheretheJijcanbetakentobeaconstantJ<0ifiandjarenearestneighborsandzeroinallothercases.Thissystemisa1DHeisenbergferromagnetanditsHamiltoniancanbewrittenas[ 12 ]bH=JXi~Si~Si+1, (2) 22

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ThetimedependenceofD~SjEisgivenby dD~SjE dt=1 i~Dh~Sj,bHiE (2a)=J i~Dh~Sj,~Sj)]TJ /F3 7.97 Tf 6.59 0 Td[(1~Sji+h~Sj,~Sj~Sj+1iE (2b)=J i~D~Sj~Sj)]TJ /F3 7.97 Tf 6.59 0 Td[(1~Sj)]TJ /F8 11.955 Tf 11.95 13.27 Td[(~Sj)]TJ /F3 7.97 Tf 6.59 0 Td[(1~Sj~Sj+~Sj~Sj~Sj+1)]TJ /F8 11.955 Tf 11.95 13.27 Td[(~Sj~Sj+1~SjE (2c)=J ~D~Sj~Sj)]TJ /F3 7.97 Tf 6.59 0 Td[(1+~Sj+1E. (2d) WeconsideronlysmallexcitationswhereSzjSandSxj,Syj<
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Fornontrivialsolutioni~!JS(1)]TJ /F4 11.955 Tf 11.95 0 Td[(coska)JS(1)]TJ /F4 11.955 Tf 11.96 0 Td[(coska))]TJ /F6 11.955 Tf 9.3 0 Td[(i~!=0. (2)SolvingEq. 2 resultsin=~!=2jJjS(1)]TJ /F4 11.955 Tf 11.96 0 Td[(coska), (2)whichisthedispersionrelationofmagnonsina1Dferromagnet(Figure 2-3 ).hereistheenergyofthemagnon. Figure2-3. Thespinwavedispersionrelationforachainofferromagneticallycoupledspins. Forka<<1thedispersionrelationbecomes~!=jJjSa2k2. (2)Fork=0theenergyiszero,meaningthatthemagnonsaregaplessexcitationsfora1DHeisenbergferromagnet.Inaddition,itisknownthatmagnonsarebosonswithspinquantumnumbersS=1andms=1. 24

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2.2.2AntiferromagneticMagnonsInthissection,wecalculatethespinwavedispersionrelationfora1DHeisenbergantiferromagnet(HAF)(J>0).Inasemiclassicalapproach,theneighboringspinsarealignedantiparalleltooneanotherinthegroundstate.Itisconvenienttoconsiderthesystemtobemadeoftwosublatticeswithoppositespins.LetAbethesublatticewithspinup(Sz=S)andBthesublatticewithspindown(Sz=)]TJ /F6 11.955 Tf 9.3 0 Td[(S).Weuseevenindicesforthespinspointingupandoddindicesforthespinspointingdown.ForevenjEq. 2 becomes dSxj dt=JS ~)]TJ /F2 11.955 Tf 5.47 -9.69 Td[()]TJ /F4 11.955 Tf 9.3 0 Td[(2Syj)]TJ /F6 11.955 Tf 11.95 0 Td[(Syj)]TJ /F3 7.97 Tf 6.59 0 Td[(1)]TJ /F6 11.955 Tf 11.95 0 Td[(Syj+1 (2a)dSyj dt=)]TJ /F6 11.955 Tf 10.5 8.09 Td[(JS ~)]TJ /F2 11.955 Tf 5.48 -9.68 Td[()]TJ /F4 11.955 Tf 9.3 0 Td[(2Sxj)]TJ /F6 11.955 Tf 11.95 0 Td[(Sxj)]TJ /F3 7.97 Tf 6.59 0 Td[(1)]TJ /F6 11.955 Tf 11.95 0 Td[(Sxj+1. (2b) Foroddj dSxj dt=JS ~)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(2Syj+Syj)]TJ /F3 7.97 Tf 6.59 0 Td[(1+Syj+1 (2a)dSyj dt=)]TJ /F6 11.955 Tf 10.5 8.08 Td[(JS ~)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(2Sxj+Sxj)]TJ /F3 7.97 Tf 6.59 0 Td[(1+Sxj+1. (2b)UsingthecreationoperatorS+=Sx+iSy,werewriteEqs. 2 and 2 ~dS+j dt=iJS)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(2S+j+S+j)]TJ /F3 7.97 Tf 6.59 0 Td[(1+S+j+1forjeven (2a)~dS+j dt=)]TJ /F6 11.955 Tf 9.29 0 Td[(iJS)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(2S+j+S+j)]TJ /F3 7.97 Tf 6.58 0 Td[(1+S+j+1forjodd. (2b)Welookagainforsolutionoftheform S+j=uei(kja)]TJ /F11 7.97 Tf 6.58 0 Td[(!t)forjeven (2a)S+j=vei(kja)]TJ /F11 7.97 Tf 6.59 0 Td[(!t)forjodd. (2b) 25

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SubstitutionofEq. 2 into 2 gives~!0B@u)]TJ /F6 11.955 Tf 9.3 0 Td[(v1CA=2JS0B@1coskacoska11CA0B@uv1CA. (2)Andfornontrivialsolution)]TJ /F10 11.955 Tf 9.3 0 Td[(~!+2JS2JScoska2JScoska~!+2JS=0. (2)whichgivesusthedispersionrelationofmagnonsina1Dantiferromagneticchain=~!=2JSjsinkaj. (2) Figure2-4. Thespinwavedispersionrelationforachainofantiferromagneticallycoupledspins. Figure 2-4 plotsEq. 2 .SimilartotheferromagneticcasethemagnonsofaHAFaregaplessexcitations.Howeverforka<<1theirdispersionrelationislinearincontrasttothequadraticdependencefortheferromagneticexcitations.ThemagnonsofaHAFareknowntobebosonswithquantumnumbersS=1andms=1.Magnonscanbeobservedininelasticneutronscatteringwithasinglemagnoninteractingwithaneutron. 26

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2.2.3SpinonsThesemi-classicalmodeloftheantiferromagnetconsideredintheprevioussectionisvalidonlywhenthetwosublatticesarespontaneouslyordered,whichisnottruefor1Dquantumantiferromagnets.True1DHAFsdonotorderevenatzerotemperature[ 12 ].Asaresult,thedispersionrelationofsuchaquantummagnetdiffersfromEq. 2 anddependsonwhetherthespinsoftheantiferromagneticchainareintegerorhalfinteger.BethegavetheexactsolutionforanS=1/2HAFchain[ 13 ].ThedispersionrelationextractedfromBethe'ssolutionis[ 14 ]=~!= 2Jjsinkaj. (2) ThisdispersionrelationmaylooksimilartoEq. 2 ;however,theseexcitationswhicharecalledspinonshavespinquantumnumberS=1=2insteadof1.Figure 2-5 plotsthespinondispersionrelationforaspin-1=2HAFchain.Theshadedregioncorrespondstotwospinexcitations(twospinons)onthespinchain.ItliesbetweenEq. 2 and~!=Jjsinka=2jasshownbyneutronscatteringexperiments[ 12 ].Atthelongwavelengthlimitspinonshavevanishinglysmallenergyandtheyareconsequentlygapless.ThisresultappliesingeneralforhalfintegerHAFchainswhoseexcitationsarealsogapless. 2.3HaldaneChainsForanintegerspinHAFchain(Haldanechain)thereexistsnoexactsolution.However,Haldanepredictedthattheexcitationspectrumwillhaveagap[ 15 ],whichisknownastheHaldanegap.TheHaldanegapisshowninFig. 2-6 .Thispredictionhasbeenconrmedbyanumberofexperimentsforthespecialcaseofspin-1chain[ 16 ].InaHaldanechaintheexcitationsarebosonswithspinS=1andms=0or1andtheyareusuallycalledtriplons.Similarly,foratwo-leggedspin-1=2antiferromagneticladderthereexistsnoexactsolution,butithasbeenshownthatthereisanenergygapintheexcitationspectrum 27

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Figure2-5. Thedispersionrelationforaspin-1=2HAFchain.Thecontinuumoftwo-spinonexcitationsisshownastheshadedregion. Figure2-6. Thedispersionrelationforaspin-1HAFchain.TheexcitationsaregappedandistheHaldanegap. 28

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(magnongap)[ 17 ].Ingeneral,evenleggedspin-1=2laddersareknowntohaveamagnongapincontrasttooddleggedspin-1=2ladderswhicharegapless[ 12 ].Inthelatter,foragivenrungtwospinswillpairupleavingasinglespinaloneresultinginasystemanalogoustoaspin-1=2chainwhichisgapless.InthenextSectionwewillexaminethelimitingcaseofatwolegspin-1=2ladderwherethereisnointeractionalongthelegs,anarrayofspindimers. 2.4SpinDimersLetusconsideraspindimer,apairofspinsthatarecoupledantiferromagnetically.InthesimplestcasewhereeachspinhasS=1=2,thegroundstateofasystemofnon-interactingspindimersisasingletwithS=0andms=0jS=0,ms=0i=1 p 2(j"ij#i)-222(j#ij"i), (2) andthethreeexcitations(triplet)are jS=1,ms=1i=j"ij"i (2a)jS=1,ms=0i=1 p 2(j"ij#i+j#ij"i) (2b)jS=1,ms=)]TJ /F4 11.955 Tf 9.3 0 Td[(1i=j#ij#i, (2c)whicharedegenerateintheabsenceofexternalmagneticeld.Whenamagneticeldisapplied,theenergyofthesingletremainsconstantwhilethedegeneratetripletsplitsduetotheZeemaneffect(Figure 2-7 A).Inparticular,theenergyofthems=)]TJ /F4 11.955 Tf 9.3 0 Td[(1decreaseslinearlywiththeappliedmagneticeld.AtsomecriticaleldHc= gB, (2)theenergyofthems=)]TJ /F4 11.955 Tf 9.3 0 Td[(1tripletstatebecomesequaltotheenergyofthesingletgroundstate.hereistheenergygapbetweenthesingletstateandthetripletexcitedstatesatzeroappliedeld.AtthecriticaleldHcthegroundstatechangesfromthesingletstatetothetripletstatewithS=1andms=)]TJ /F4 11.955 Tf 9.3 0 Td[(1.Asaresult,atT=0and 29

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H=Hcallthespinsofasystemofnon-interactingS=1=2dimersarealignedwiththemagneticeldandthemagnetizationofthesystemlookslikeastepfunction(Figure 2-7 B). Figure2-7. Zeemaneffectforisolateddimers.a)WhenamagneticeldisappliedthedegeneratetripletsplitsduetotheZeemaneffect.(b)Magnetizationasafunctionofthemagneticeld However,wheninterdimerinteractionsexist,notallthespinswillalignwiththeeldatH=Hc.Instead,onlyasmallnumberofdimersarefoundinthems=)]TJ /F4 11.955 Tf 9.29 0 Td[(1tripletstate.AtT=0andHHc,thegroundstateofthesystemisasuperpositionofms=)]TJ /F4 11.955 Tf 9.3 0 Td[(1tripletandS=0singletstates.WhentheeldiskeptxedatsomevalueH>Hc,thetripletstatescanhopfromonedimertoanotherbecauseoftheinterdimerinteractions.Consequently,thedelocalizedtripletstates,whicharebosons,canbetreatedasmagnonsinageneralsense.Ifthemagneticeldisfurtherincreasedthenumberofms=)]TJ /F4 11.955 Tf 9.3 0 Td[(1tripletstatesincreasesuptoasecondcriticaleld(Hs)whereallthedimersbecomems=)]TJ /F4 11.955 Tf 9.3 0 Td[(1tripletstates,meaningthatallthespinsarealignedwiththeeld.TheenergydiagramisshowninFigure 2-8 A.ThemagnetizationofthesystemstartsincreasingatH=HcanditsaturatesatH=Hs.ThesystematH
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Figure2-8. Zeemaneffectforinteractingdimers.(a)Thetripletstatenowisreplacedbyatripletbandduetotheinterdimerinteractions.(b)Magnetizationasafunctionofthemagneticeld magneticpropertiesofthesystemisdescribedasasecondorderphasetransitionwhichoccursatT=0.Thisphasetransitionisreferredtoasaquantumphasetransition(QPT)becauseitisnotdrivenbytemperature(itoccursatT=0)butbyquantumuctuations.ThepointonthephasediagramwheretheQPToccursiscalledthequantumcriticalpoint(QCP).FortheinteractingdimersystemtherearetwoQCPsatHcandHs. 2.5Bose-EinsteinCondensationTheBose-Einsteincondensateisastateofmatterinwhichasystemofbosonscondensesintoamacroscopicallyoccupiedsingle-bosongroundstate.Itwasobservedforthersttimeinliquid4He(whichisaboson)[ 18 19 ]andhadearlierbeensuggestedastheexplanationofitssuperuidity[ 20 21 ].Similarly,inasystemofinteractingspindimers,thebosonexcitationsundergoaBose-Einsteincondensation(BEC)attheQCP(H=Hc)andthemagneticallyorderedstate(Hc
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whereHisthetransitioneldatnon-zerotemperatureT,disthespatialdimensionofthesystemandMisthemagnetizationalongtheeld.TheBECofmagnonswasrstobservedinTlCuCl3[ 22 26 ],whichconsistsofCu2+anionsthatformspinS=1=2dimers.AdditionalevidenceforBECwasfoundinBaCuSi2O6[ 25 27 ]whereadimensionalreductionfromthree-dimensionalBECtotwo-dimensionalBECwasobservedduetogeometricfrustration.InChapter 6 theeldinducedBECofmagnonsinthegeometricallyfrustratedcompoundBa3Cr2O8isstudiedanditsphasediagramisextractedusingmicromechanicalforcemagnetometry.Inasystemcomprisedoftwo-legS=1=2HeisenbergladdersthestateabovethecriticaleldisnotalwaysaBECofmagnons.Thebehaviorofthesystemdependsontheexistenceorabsenceofinterladderinteractions.WheninterladderinteractionsexistthesystemundergoesaBECinanalogytoasystemofinteractingspindimers.However,thistimeawholeladderplaystheroleofonespindimerandtheinterdimerinteractionsarereplacedbytheinterladderinteractions.Intheabsenceofinterladderinteractionsthesystemisideally1DanditsstateisdescribedasaTomonaga-Luttingerliquid. 2.6WilsonRatioinFermiLiquidsandTomonaga-LuttingerLiquidsIdeal1DsystemssuchasspinS=1HaldanechainsandS=1=2two-legspinladdersdonotshowlongrangemagneticorderevenateldshigherthanthecriticaleldHc.ThesesystemsundergoaQPTfromagappedquantumdisorderedphasetoagaplessphaseateldHc.However,theirgaplessphase,duetotheabsenceofinterladderorinterchaininteractions,isdescribedasaTomonaga-Luttingerliquid(TLL),incontrasttothehigherdimensionalsystems,whosegaplessstateisaBECofmagnons.ATLListhe1DanalogyofaFermiliquidin2Dor3D.Indimensionshigherthanone,well-knownexamplesofFermiliquidsincludeliquid3Heandconductionelectronsinmetalsatlowtemperatures.LandaudevelopedthetheoryofFermiliquidsstatingthatthereisaone-to-onecorrespondencebetweenthe 32

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excitationsofaninteractingsystemoffermions(Fermiliquid)andtheexcitationsinaFermigas.HethusreplacedtheinteractingfermionsinaFermiliquidbyweakly-interactingquasiparticlesthatcarrythesamespin,chargeandmomentumastheoriginalparticlesdo.Thesequasiparticleshaveaneffectivemassm,andtheirlifetime()is~=(E)]TJ /F6 11.955 Tf 11.96 0 Td[(EF)2[ 28 ].AsaconsequencethedescriptionofaFermiliquidisqualitativelyanalogoustoaFermigas,andquantitiessuchasthespecicheatandthemagneticsusceptibilityhavethesamequalitativebehaviorasthoseofaFermigas.Morespecically,thespecicheat(Cfree)andthemagneticsusceptibility(free)ofaFermigasaregivenbyCfree=mk2BkF 3Tandfree=)]TJ /F7 7.97 Tf 6.67 -4.43 Td[(gB 22mkFrespectively,wheremisthemassofthefermionsandkFistheFermiwavenumber.InthecaseofFermiliquidsthesamequantitiesarenowgivenby[ 29 ]C=m mCfreeand=1 1+Fa0m mfree,wheremistheeffectivemassofthequasiparticlesandFa0isoneoftheLandauFermiliquidparameters.Itisimportanttomentionthatinbothcasesthelowtemperaturespecicheatislinearlydependentontemperatureandthemagneticsusceptibilityistemperatureindependent.OneofthemostimportantparametersthatcharacterizesFermiliquidsistheWilsonratio[ 30 ]RW=4 3kB gB2 C=T. (2)Bydividingoutthecontributionoftheeffectivemass,whichentersboththesusceptibilityandspecicheat,theWilsonratioquantiestheenhancementofthesusceptibilityduetospinuctuations.Asaresult,RWisaneffectivetoolofclassifyingheavyfermionsystems[ 31 ].Forinstance,RWis2fortheS=1=2Kondolatticeinthesingle-impuritylimit[ 30 ]and4fortheso-calledBrinkman-Rice-Gutzwillerliquid[ 32 ].Inliquid3He,oneofthemostbestknownexamplesofFermiliquids,RWvariesweaklywithpressureincontrasttothestronglypressure-dependenteffectivemass.Ittakesavaluecloseto4,theapproximatelimitingvaluefortheHubbardmodelwithcriticalon-siterepulsion,i.e., 33

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theBrinkman-Rice-Gutzwillerliquid1,providingevidencethatquasiparticlesinliquid3Hearenearlylocalized[ 33 ].In1D,anarbitrarilyweakinteractionbetweenthequasiparticlesmakestheirlifetimeevenshorterthan~=(E)]TJ /F6 11.955 Tf 13.09 0 Td[(EF)becauseoftightconstraintsthatthespatialdimensionimposesonenergyandmomentumconservationinscatteringprocesses.Asaconsequence,theFermiliquidtheorybreaksdownentirelyin1D,givingwaytotheTLLasthecorrectlow-energydescriptionoffermions[ 34 ].InaTLL,low-lyingexcitationsaremassless,collectivebosonicmodesinsteadoffermionicquasiparticles.However,thelowtemperaturemagneticsusceptibilityandspecicheatinaTLLarelikethoseofaFermiliquidisindependentoftemperatureandCislinearinT[ 35 ] =(gB)2K=(v) (2a)C=k2BT=(3v), (2b)whereKistheTLLparameterandvistheFermivelocity.Asaresult,theWilsonratiomustobeytherelationRW=4K. (2)Eq. 2 makesRWacrucialparameterin1D,evenmorecrucialthanin3D,sinceeachbranchofbosonicmodesinagivenTLLiscompletelyspeciedbyjusttwoparameters,thevelocityvandtheTLLparameterK[ 34 35 ],andsincealargevarietyofinteracting1DsystemsfallintotheTLLuniversalityclass.Despiteitssignicance,theWilsonratiohasneverbeendeterminedexperimentallyinaTLLbecauseofthelackofagoodmaterial.CandidatematerialswhosegaplessregimecouldbeaTLLatlowtemperaturesarespin-1linearchainsorspin-1=2ladders 1For3He,theBohrmagnetonBinEq. 2 mustbereplacedbythenuclearmagnetonn 34

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inmagneticeldslargerthanthemagnongaporspin-1=2linearchainsinzeroeldaswellasinmagneticelds.Nevertheless,suchsystemsarehardtondinrealcompoundsbecauseinterchainorinterladderinteractionsmustbesufcientlyweaktoensurethataBECdoesnotdestroytheTLL.InChapter 5 wedeterminetheWilsonratioofanidealstrong-legS=1=2Heisenbergspin-ladderantiferromagnet,(C7H10N)2CuBr2,forthersttimeina1Dsystem,usingthemicromechanicalforcemagnetometerswhichwehavedeveloped. 35

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CHAPTER3CONVENTIONALMAGNETOMETERSThegoalofmyresearchwastodevelopnewmagnetometersthatovercomethelimitationsofconventionalmagnetizationmeasurementdevices.Conventionalmagnetometersaredividedintotwocategories.Intherstcategorybelongdevicesthatusetheinductionmethod.Usingtheinductionmethod,themagneticeldgeneratedbythesampleismeasuredandthenthesample'smagneticmomentisextracted.Themostpopularmagnetometersofthisgroupareacsusceptometers,vibratingsamplemagnetometers(VSMs)andsuperconductingquantuminterferencedevices(SQUIDs).Inthesecondgroupofdevices,theforceorthetorqueonasampleduetoanexternalmagneticeldismeasured.ThiscategoryincludescantilevertorquemagnetometersandFaradaybalancemagnetometers.Thedevicesthatwedevelopedaremicromachinedversionsofthelatter. 3.1SusceptometersThemagnetometersthatarebasedontheinductionmethodmeasurethevoltageinducedinadetectioncoilbyauxchange.Theuxchangecanariseduetochangeinthemagnetizationofthesampleorthepositionofthesamplewithrespecttothecoil.Themagnetizationofthesampleisinducedbyanexternalmagneticeldandchangesinaccordancetotheexternaleld.Ferromagneticsamplesdonotrequireexternalmagneticeldsbecausechangesintherelativepositionbetweenthesampleandthecoilcanchangethemagneticux.Anacsusceptometerconsistsofaprimarycoilandtwocounter-woundsecondarycoilswhicharemadetobeidenticalandareconnectedinseries(Figure 3-1 ).Thesecondarycoilsareplacedinsidetheprimarycoilwhichisfedwithanaccurrentthatproducesanoscillatingmagneticeld.Thevoltageinducedinthesecondarycoilsbytheacmagneticeldoftheprimarycoilispracticallyzeroduetotheiroppositehelicity.Theacmagneticeldisusedtoinducemagnetizationonasamplethatisplacedinside 36

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oneofthesecondarycoils(detectioncoil).Avoltageproportionaltothechangeinthemagneticmomentofthesampleisinducedacrossthesecondarycoils.Thisvoltageisdetectedusingalock-inamplierlockedatthefrequencyoftheaceld.Thetypicalmagneticmomentresolutionofanacsusceptometerrangesfrom10)]TJ /F3 7.97 Tf 6.59 0 Td[(4to10)]TJ /F3 7.97 Tf 6.58 0 Td[(6emu/p Hz[ 36 ].Theacsusceptometersdonotmeasureabsolutemagnetizationandcalibrationwiththeuseofamagnetizationstandardisusuallyrequired.Inaddition,ifthetwosecondarycoilsarenotidentical,aconsiderablebackgroundappearsinthesignal,reducingsignicantlythesensitivity.Acsusceptometerscanbeusedinawiderangeoftemperatures,typicallydownto300mKbutsometimesevenatsub-millikelvintemperaturesatspecializedlabs.Howevertheyrequirerelativelylargesampleswithmassesofseveralmg. Figure3-1. Schematicofanacsusceptometer.Thetwocounter-woundcoilsareconnectedinseries.Theprimarycoilisnotshown. Counter-woundcoilmagnetometersarethemostwidelyusedwayofmeasuringmagnetizationofsamplesinpulsedmagneticelds.Thesampleisplacedintoacoil 37

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(detectioncoil),whileanothercounter-woundcoilisusedforbackgroundcompensation.Asaconsequence,duringamagneticeldpulsetherapidlychangingmagnetizationofthesampleinducesavoltageinthedetectioncoil.Sincetheinducedvoltageisproportionaltotheratethattheuxchangeshighresolutionscanbeachievedinpulsedmagnets.However,thismagnetometercannotmeasureabsolutemagnetizationandrequireslargesamples.Additionally,abigdisadvantageofthisdeviceisthebackgroundsignal.Thetwocoilsarehand-woundandthismakesitalmostimpossibletobalancetheirsignals.Thebackgroundisusuallysubtractedundertheassumptionthatitdependslinearlyontheeld.Thisassumptionisnotalwayscorrectandasaresultthenalsignalcontainssystematicerrors.Inavibratingsamplemagnetometer(VSM)theexternaleldisconstantandthesampleisoscillatedinasinusoidalwaywithrespecttostationarydetectioncoils(Figure 3-2 ).InatypicalVSMsetup,aloudspeakerisusedtovibratearodwhichisattachedtoitononeside.Thesampleismountedattheotherendoftherodandtheloudspeakervibratesthesamplealongtheaxisofthedetectioncoils.Thechanginguxinducesavoltageacrossthecoils[ 37 ],whichisproportionaltothemagneticmomentofthesampleaswellastotheamplitudeandfrequencyoftheoscillation[ 38 ].TheVSMsemployedattheNationalHighMagneticFieldLab(NHMFL)inTallahasseehavemagneticmomentresolutionof10)]TJ /F3 7.97 Tf 6.59 0 Td[(3emu/p Hzusingsamplesofseveralhundredmgandtheyoperateattemperaturesrangingfrom0.7to300Kandeldsupto33T[ 36 ].Typically,theVSMsarenotusedinverylowtemperaturemeasurementsindilutionrefrigeratorsduetotheheatdissipationresultingfromthesample'svibration. 3.2SQUIDsTheSQUIDmagnetometersareregardedasbeingamongthemostsensitivedevicesformeasuringmagnetization.TheoperationoftheSQUIDmagnetometersis 38

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Figure3-2. Schematicofavibratingsamplemagnetometer. 39

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Figure3-3. SQUIDmagnetometer.AdcSQUIDconsistsofasuperconductingringwithtwoJosephsonjunctionsthatareconnectedinparallel.IisthebiascurrentandVistheoutputvoltage.VoscillatesbetweentwovaluesdeterminedbyI.Theoscillationperiodisdeterminedbytheuxthatpassesthroughthesuperconductingring. Figure3-4. FluxchangethroughaSQUID.ThevoltageacrosstheSQUIDoscillatesbetweentwovaluesdeterminedbythebiascurrent.Theperiodisequaltotheuxquantum0=h=2e.Countingthenumberofoscillationsintheinducedvoltageenablesthecalculationofthechangeinthemagneticux. 40

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basedontheinductionmethod,butincontrasttotheacsusceptometersandtheVSMsthemeasurementsusingSQUIDSaredcinnature.AdcSQUIDcomprisesofasuperconductingringwithtwoJosephsonjunctionsthatareconnectedinparallel(Figure 3-3 ).AJosephsonjunctioniscreatedwhentwosuperconductorsareseparatedbyathinlayerofinsulator.WhenaSQUIDisplacedinamagneticeldthetotalcurrent(Itot)owingthroughitisproportionalto[ 39 ]Itot/cose ~c, (3)whereisthemagneticuxthroughthering,eisthechargeoftheelectron,and~isPlanck'sconstantdividedby2.SmallchangesinthemagneticuxthatpassesthroughtheringcausethevoltageacrosstheSQUIDtooscillatebetweentwovalueswithperiodequaltotheuxquantum0=h=2e,whenanappropriatebiascurrentisappliedtothering.Asaconsequence,countingthenumberofoscillationsintheinducedvoltageenablesthecalculationofthechangeinthemagneticux(Figure 3-4 ).ThemagneticmomentresolutionsofcommercialdcSQUIDsaretypicallybetterthan10)]TJ /F3 7.97 Tf 6.59 0 Td[(8emu/p Hz,oneofthebestresolutionscommercialmagnetometerscanoffer.Ontheotherhand,thetemperatureandmagneticeldrangesthattheSQUIDscanbeusedarelimited.ThepickupcoiloftheSQUIDhastoremainsuperconductingandthemagneticeldcannotexceedthecriticaleldofthematerialthatthecoilismadeof.Asaresult,SQUIDScannotbeusedinpulsedandhigheldresistivemagnetsandtheiruseislimitedtomoderatedcelds,upto7TformostcommercialSQUIDs.Inaddition,whenthestudyofthemagnetizationasafunctionoftheeldisrequired,themagneticeldhastobesweptveryslowlyanditmustbestabilizedateachpointforthemeasurementtobeperformed. 3.3CantileverMagnetometersCantilevermagnetometry[ 40 ]isarelativelynewmethodofmeasuringsharpchangesinthemagnetizationofanisotropicsamplesatphasetransitionsundervarious 41

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experimentalconditions.Thesampleisattachedonaexiblebeamandexperiencesatorque~Tm=~m~H,duetotheinteractionofthemoment~mwiththemagneticeld~H.Ifthereisamagneticeldgradient,thesampleisalsosubjectedtoaforce~Fm=~m~r~H.Themagnetictorqueonthesamplecausesthecantilevertodeect(Figure 3-5 ).Thedeectionisoftenmeasuredcapacitively,usingacapacitancebridge. Figure3-5. Principleofoperationofthecantilever.Thesampleisplacedwithasmallanglebetweenthehard-axis(c)andtheappliedeldH.Thetorqueonthesamplebendsthecantilever. Inorderforthismethodtowork,theexistenceofamisalignmentbetweenthemagnetizationaxisandthemagneticeldisnecessary.Thismisalignmentcanbeachievedinanumberofways.First,thesamplemayhaveananisotropicgtensororsingleionanisotropyduetospin-orbitinteraction.Additionally,theexchangeinteractionofthemagneticionsmaycontainananisotropicDzyaloshinskii-Moriyaterm[ 41 42 ].Second,ifthesampleisisotropicbutitsshapeisnotanellipsoid,anon-uniformdemagnetizationeldwillcausemisalignmentofthemagnetization.Itisalsoroutinetomountananisotropicsamplewithitsmagnetichardaxisatasmallanglewiththemagneticeld(<10). 42

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Eventhoughcantilevermagnetometerscanbeusedwithsmallsinglecrystalsamples,theirsignalconsistsofbothtorqueandforcecontributionsthatcannotbeseparated.Usingthismethod,sharpchangesinthemagnetizationofthesamplecanbeeasilydetected.However,occasionallyunwantedsignalsmaybegeneratedwhenthemagnetichardaxisalignswiththeeld,makingitdifculttointerprettheresults.Finally,theextractionoftheabsolutemagnetizationofthesamplefromthetorquesignalisnottrivialandisnotfeasibleinmostcases.Ontheotherhand,cantileverscanbeusedbothinpulsedandhigheldresistivemagnetsandoverawiderangeoftemperatures,makingthemaveryusefulqualitativetoolforthestudyofphasetransitions. 3.4FaradayBalanceMagnetometersAFaradaybalancemagnetometermeasurestheforceexertedonasamplefromaspatiallyvaryingmagneticeld.Itismadeasaparallelplatecapacitor,whereoneplateisaxedelectrodeandtheotherissuspendedbysprings(Figure 3-6 ).Thesampleismountedontothemovableplateandithasamagnetizationmthatiseitherintrinsicorinducedbyanexternalmagneticeld[ 43 ].Aspatiallyvaryingmagneticeldexertsaforceonthesample~Fm=~m~r~H,whichchangesthecapacitanceofthedevice.Thesedevicesareusuallyhome-madeandtypicalplatematerialsincludealuminum[ 44 ]ormetalizedepoxies[ 43 ].Thespringsaremadeofclampedmetalwiresormetalrodsprotrudingfromthemovableplate.Thesizesoftheplatesareonthescaleofcmandthegapthatseparatesthemovableplatefromthexedelectrodeisintherangeofseveralhundredmicrons[ 43 ].ObtaininganabsolutecalibrationoftheFaradaymagnetometersiseasyduetoitsinherentparallelplatedesign.Anelectrostaticforceisappliedtothedevicetocalibratethecapacitancesignal.Asaresult,theabsolutemagneticmomentofthesamplecanbeobtained,ifthegradientofthemagneticeldisknown.ThetypicalmagneticmomentresolutionsoftheFaradaybalancemagnetometersareinthe10)]TJ /F3 7.97 Tf 6.59 0 Td[(7 43

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emu/p Hzrange[ 45 ]andtheyarebestsuitedforbigsampleswithmassesofhundredmg.Theirresolutiondependsdrasticallyontheeldgradientandthestiffnessofthesprings.Whilesoftspringsincreasethesensitivityofthemeasurement,theytendtoreducetheresonancefrequencyofthedevicesresultinginlongresponsetimesandincreasedcouplingtomechanicalnoise.Additionally,thesemagnetometersareratherhardtofabricatebecausetheyaremadebymanualassemblyofcomponentsthatareindividuallymachined. Figure3-6. PrincipleofoperationoftheFaradaybalancemagnetometer.Themagneticforceonthesampleduetothespatiallyvaryingmagneticeldchangesthecapacitanceoftheparallelplatecapacitor. Inthischapter,wereviewedthecommonlyusedmagnetometers.Eachparticulardevicecanonlybeusedunderspecicconditions.Additionally,thecostofeachmethodisaveryimportantparameterofitsusefulness.Asanexamplewementionthatpulsedmagnetsoccasionallyfail,underthehugeaxialstressescreatedduringapulse,destroyingboththemagnetometerandtheexperimentalprobe.Itisthereforeparticularlyimportantthatthedetectionschemeiscostefcientinpulsedeldexperiments.Asaconclusion,amagnetizationmethodthatcombineshighresolutionswithlowcostinavarietyofexperimentalconditionsishighlydesirable.Inthenextchapter,wewilldescribethemicromechanicalforcemagnetometersthatwedeveloped. 44

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Theyofferanewwayofmeasuringtheabsolutemagnetizationofsmallsamples.Themicromechanicalforcemagnetometersarethemicro-sizedversionoftheFaradaybalancemagnetometers.IncontrasttothemacroscopicFaradaymagnetometers,theyyieldhighresonancefrequencieswithfastresponsetimesandnegligiblecouplingtomechanicalnoisemakingthemcapabletobeusedinpulsedanddcresistivemagnets.Sincealargenumberofdevicesarefabricatedinparallel,theyarecosteffectiveandtheirmagneticmomentresolutionrivalsthatoftheSQUIDmagnetometers. 45

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CHAPTER4MICROMECHANICALMAGNETOMETERSMicro-electro-mechanicalsystems(MEMS)orjustmicromechanicalsystemsaresmalldevicesthatconsistofstructureswithtypicalsizesvaryingfrommicronsallthewaytoseveralmillimeters.Theyareusuallymadeusingsemiconductorfabricationtechnology.MEMShavebeenwidelyusedassensorsandactuators.Somewell-knownexamplesincludeaccelerometersthatareusedincellphonesasmotionsensorsandtheheadsofinjectprinters.OverthepastdecadestheconceptofMEMShasbeenexpandedtoincludenotonlyelectro-mechanicalbutalsomanyothertypesofdevices,includingmagnetic,thermal,opticandchemicalsystems.Allthemagnetizationmeasurementsdiscussedinthisdissertationwereperformedusingmicromechanicalforcemagnetometers.Inthischapter,thetechniquesthatareusedforthefabricationofMEMSarediscussed.Then,thedesignandtheprinciplesofoperationofthemicromechanicalmagnetometersareintroduced,alongwiththedetectionschemewhichwasusedduringthemeasurements.Finally,acomparisonofMEMSmagnetometerswithothermagnetizationmeasurementdevicesispresented. 4.1FabricationandPreparationoftheDevicesAvarietyoffabricationtechniquesareusedtoproduceMEMS.Mostofthetechniquesareborrowedfromintegratedcircuitmanufacturing.Asaresult,semiconductorsarethemostcommonlyusedfabricationmaterialsforMEMS.Thestructuralpartsofamicromechanicaldeviceareconstructedonasubstrateeitherbydepositionofnewmaterialorbyetchingthesubstrateitself.Therstcase,wherefabricationoccursbydepositingnewlayersofmaterialontopofthesubstrate,iscalledsurfacemicromachiningwhilethesecondiscalledbulkmicromachining.AllthedevicesusedinthisdissertationweredesignedinhouseandfabricatedbythecommercialfoundryMEMSCAP.MEMSCAPfabricatesthesamplesusingasiliconsurfacemicromachiningprocesscalledPolyMUMPs[ 46 ].ThisprocessisaMultiUser 46

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MEMSProcess(MUMPs),wheremanydevicesfordifferentusersarefabricatedonthesamewafer.Consequently,thenalusersmustconformtotheprocessparameters.Polycrystallinesilicon(polysilicon)isusedasthestructuralmaterialandphosphosilicateglass(PSG)asthesacricialmaterial.Theprocessstartswithasiliconsubstrate,whichisheavilydopedwithphosphorusforincreasedconductivity.Next,a600nmsiliconnitridelayerisdepositedonthewaferforelectricalisolation(Figure 4-1 A).Thisisfollowedbythedepositionofa500nmlayerofpolysilicon(Poly0)directlyonthenitride(Figure 4-1 B).ThePoly0layerispatternedusingphotolithographyandetchedusingreactiveionetch(RIE).AlayerofphotoresistisspunonPoly0(Figure 4-1 C)andthenthephotoresistisexposedtoUVlightusingaphotomask(Figure 4-1 D).TheareasexposedtotheUVlightarewashedawayusingadevelopersolution(positivephotoresist)(Figure 4-1 E).Attheexposedareas,Poly0isnolongercoveredbyphotoresistandisleftunprotectedagainsttheRIEetchthatfollows(Figure 4-1 F).Thisrstlayerofpolysiliconispermanentlyxedandisusedasanelectricalinterconnectionlayer.TherestofthefabricationprocessisshowninFigure 4-2 .AfterPoly0isetchedusingRIE,a2mlayerofphosphosilicateglass(PSG),therstoxide,isdeposited.Thislayerispatternedandetchedtwice.Therstetchisusedtoproducedimplesinthepolysiliconlayerthatisdepositednext(Poly1).ThedimplesareusedtoreducethecontactareawhenamovablePoly1component,afterrelease(seebelow),comesintocontactwiththePoly0layer.Theirnominaldepthis750nm.ThesecondetchcompletelyetchesthroughtherstoxideandcreatesanchorsforthePoly1layertothesiliconsubstrate,Poly0orthenitride.Therstoxidelayerisasacriciallayerandistotallyremovedlaterinthefabricationprocess.Afterthesecondetchoftherstoxideiscompleted,therststructuralpolysiliconlayer(Poly1)isdepositedatathicknessof2mandthenpatternedandetched.ThisstepisimmediatelyfollowedbythedepositionofthesecondsacricialPSGlayer 47

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ADepositnitride BDepositPoly0 CSpinphotoresist DExposephotoresist EDevelopphotoresist FEtchPoly0usingRIEFigure4-1. MEMSfabrication.Basicstepsofthefabricationprocess 48

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(secondoxide),750nmthick.Similartotherstoxide,thesecondoxideispatternedandetchedtwice.Therstetchisusedtoprovideconnectionofthenextpolysiliconlayer(Poly2)toPoly1,whilethesecondetchetchesthroughbothoxidestoprovideanchortothePoly0orthenitride.Thesecondstructuralpolysiliconlayer(Poly2)isthendepositedandpatterned.Finallya0.5mmetallayer(chromiumandgold)isdepositedandpatternedusinglift-off.Thismetallayeristypicallyusedforprovidinglowresistancebond-padsforwirebonding.ThepreviousstepsofthefabricationareperformedbyMEMSCAP.Afteranalthicklayerofphotoresistisspunontopofthewafer,thedevicesaredicedandsenttothecustomer.Thestructurallayersofthedevicesareprotectedbyboththethickphotoresistandthesacricialoxidelayers.Theoxidelayerskeepthepolysiliconstructuresxedinplaceandthephotoresistprotectsthemicromachinesfromdust,humidityandcontaminationduringlaterstepsofthepreparation.Thedevicesreachthelabindies.Inourdesign,eachdietypicallycontains16chips,2.52.5mmeach,whichconsistofseveralmicromachines.Eachdieisdicedintochipsusingadicingsaw.Immediatelybeforemeasurement,thechipsarereleased.Thereleasereferstothewetisotropicetchofthesacricialoxidelayers.Thechipsareplacedinhydrouoricacid(HF)(49%)solutionforseveralminutesuntilthePSGlayersaretotallyremoved(Figure 4-2 ).TheetchingprocessisstoppedbymovingthedevicesfromHFtodeionized(DI)water.Afterthewetetch,partsofthemicromachinesarefreetomovebutnowtheyaresubmergedinwater.Iftheyareallowedtodryintheair,thesurfacetensionofthewatercanoccasionallypullthepolysiliconlayerstogether,causingthemtoadherestrongly.Therestoringforceduetotheelasticdeformationmaynotbesufcienttoovercomethisadhesion.Thisphenomenoniscalledstiction[ 47 ].Toavoidstictionacriticalpointdryerisused.Thechipsareplacedinachamberofthecriticalpointdryerthatislledwithmethanol.Insidethischamber,themethanolisdisplacedbycooledandpressurized 49

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Figure4-2. Release.(a)AcrosssectionviewofallthelayersinthePolyMUMPSprocess(Nottoscale).(b)Thesamedeviceafterrelease.ThePSGlayersareremovedandthemovablecomponentisfree. liquidcarbondioxide(CO2)inseveralrinsecycles.WhenmethanolistotallyreplacedbyCO2,thetemperatureisincreaseduntilasupercriticaltransitionoccursinwhichtheliquidturnsintogaswithoutcrossingaphaseboundary.ThesupercriticalCO2isslowlyventedtotheair.Criticalpointdryingsignicantlyincreasestheyieldofusabledevices. 4.2DesignoftheMicromechanicalMagnetometersThemaingoalofthisresearchistodesignmicromechanicalmagnetometersthatwillbeabletomeasuretheabsolutemagnetizationoftinysamples.Itisimportantforthemagnetometerstomeasurethemagneticforceonthesampleandminimizetorquecontributions.ThemagnetometersarebasedontheFaradaybalanceprinciple[ 43 44 ],thatwasreviewedinthelastchapter. 50

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Wedesignedseveraldifferentversionsofthemicromechanicalmagnetometers.However,themostsuccessfuloneconsistsofamovablesquarepolysiliconplate,500by500mand3.5mthick,suspendedbyfoursprings,oneateachcorner,whichareattachedtotheplateattheoneendandanchoredtothesubstrateattheother.Axedpolysiliconelectrode,ofthesamesizeasthemovableplate,islocated2mbelowthemovableplatecreatingaparallelplatecapacitor.Thespacebetweenthetwoplatesiscreatedbyetchingawaytherstsacricialoxidelayer.Thespringshavetypicallengthsbetween20and200m,givingawiderangeofspringconstantsthatcanbeusedfordifferentkindofsamples.InFigure 4-3 Aascanningelectronmicrographofatypicaldeviceisshown.Thebrightsquareandcircularstructuresarethebond-padsusedforelectricalinterconnection.Figure 4-3 Bshowsaclose-upviewofaspringwhichismadeoutofPoly1.ThelayerbelowthespringispartofthexedelectrodewhichismadeoutofPoly0.Furthermore,themovableplateismadeofbothPoly1andPoly2forincreasedthicknessandrobustness.Thesmallholesonthetopplate(etchingholes)arefabricatedtoallowHFtopenetrateunderthemovableplateduringthereleaseandconsiderablyacceleratetheetchingprocess.FinallythesquarenotchneartheetchingholesisthedimpledescribedinSection 4.1 .AsshowninFigure 4-4 ,immediatelyunderneaththedimples,thePoly0isremovedinordertopreventthetwoplatesfromelectricallyshortingincasethemovableplatesnapsdown.Somelimitationsonthedesignofthemagnetometersarethesmallestandthelargeststructuresthatcanbeconstructed,aswellasconstraintsposedbythefactthedevicesarefabricatedusingaMUMPsprocessThesmallestfeaturesizeisabout2m,determinedbytheresolutionandthealignmenttolerancesofthelithographysystemusedduringthePolyMUMPsfabricationprocess.Althoughphotolithographydoesnotsetlimitationsonhowlargeamicromachinecanbe(aslongasitcantonasinglewafer),theresidualstressesofthedepositedlayersleadtodeformationof 51

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Figure4-3. SEMimageofamicromechanicalmagnetometer.(a)Scanningelectronmicroscopeimageofatypicalmagnetometer.(b)Close-upviewofaspring Figure4-4. Dimples.(a)Crosssectionofthedevicewherethedimplesonthetopplateareshown.ThePoly0layerbelowthedimpleisremovedinordertopreventthetwoplatesfromelectricallyshortingincasethetopplatesnapsdown.(b)Thedimplesarealsousedtoreducethecontactareawhenamovableplatesnapsdown. releaseddevices[ 48 ].Basedonourexperience,releasedpolysiliconstructureslargerthan750mtendtoboworbucklewhichmakesthemunabletooperate.Finally,therestrictionsposedbytheMUMPsprocessarethenumberandthematerialofthelayersthatcanbedepositedaswellastheirthickness.Asanexample,thetopmovableplatecanbeconstructedofeitherPoly1orPoly2orboth,withthicknesses2,1.5,or3.5mrespectively. 4.3OperationThedeviceinamagneticeldoperatesasaFaradaybalance[ 43 ](Figure 4-5 A).Asampleofthemagneticmaterialisplacedandgluedtothemiddleofthemovable 52

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plate.Typicalmaterialsarenotmagneticintheabsenceofmagneticelds,whicharerequiredtoperformmeasurements.Whenamagneticeldandaeldgradientareapplied,themagneticeldmagnetizesthesampleandtheeldgradientgeneratesaforce~Fm=~m~r~Honthesamplethatchangesthedistancebetweenthemovableplateandthexedelectrode(Figure 4-5 B).Dependingontheparticularsofthecryostat,theeldgradientcanbeprovidedbyseparategradientcoilsoritcanbethenaturalgradientoftheeldwhenthesampleisdisplacedfromthecenterofthemagnet.ThemagnetometerbehaveselectricallyasparallelplatecapacitorwithcapacitanceC=0A x0)]TJ /F6 11.955 Tf 11.95 0 Td[(x, (4)whereAistheareaoftheplate,x0isthegapbetweenthetwoplateswithnoforceexertedonthemovableplate,xistheplatedisplacementwhenaforceisexertedonthemovableplateand0isthepermittivityoffreespace.ApotentialdifferencebetweenthetwoplatesofaparallelplatecapacitorgeneratesanattractiveelectrostaticforceFel=1 2dC dxV2=0A 2(x0)]TJ /F6 11.955 Tf 11.96 0 Td[(x)2V2, (4)whereVistheapplieddcvoltage.Thepreciseknowledgeofthedevicedimensionsallowsaccuratecalculationoftheforce.Anappliedelectrostaticforcedisplacesthemovableplatetoanewequilibriumposition.AtthispositiontheelectrostaticforceisbalancedbytherestoringforceofthespringsFnet=0A 2(x0)]TJ /F6 11.955 Tf 11.95 0 Td[(x)2V2)]TJ /F6 11.955 Tf 11.96 0 Td[(kx=0, (4)khereisthetotalspringconstantofallfoursprings.Tocheckwhetherthisisastableequilibriumpoint(xeq)weneedtocheckthesignoftheforce's(Eq. 4 )rstderivativedFnet dxx=xeq=0A (xeq)]TJ /F6 11.955 Tf 11.95 0 Td[(x0)3V2)]TJ /F6 11.955 Tf 11.96 0 Td[(k, (4) 53

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Figure4-5. Principleofoperationofthemicromechanicalmagnetometer.Theeldgradientgeneratesmagneticforceonthesample,thatchangesthegapbetweenthetwoplates(FaradayBalance). Forastableequilibriumthederivativeshouldbenegativeandasaresultk>0A (xeq)]TJ /F6 11.955 Tf 11.96 0 Td[(x0)3V2. (4)Asthedcvoltageincreasesandthegapbetweentheplatesdecreases,therightsideofEq. 4 becomeslargerwhilethespringconstantremainsthesame.Atthedistancewherek0A (xeq)]TJ /F7 7.97 Tf 6.58 0 Td[(x0)3V2,thespringcannotbalancetheelectrostaticforceanymoreandthetopplatecollapsesonthebottomplate.Thisphenomenoniscalledpull-in[ 49 ]anditoccurswhenthegapbetweenthetwoplatesbecomes2=3oftheirinitialseparation(x0).ThedisplacementfromtheneutralpositionisgivenbyxPI=1 3x0. (4)Thevoltagewhichcausesthissnapdowntohappeniscallthepull-involtage,VPI=s 8kx30 270A. (4) 54

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ThiseffectisclearlyshowninFigure 4-6 wherethecapacitanceofamicromechanicalmagnetometerisplottedasafunctionoftheapplieddcvoltage.Abovethepull-involtagethecapacitancejumpsdiscontinuouslytoahighervalueanditremainsconstantforhigherdcvoltages.Atthispoint,thedimplesthatarefabricatedatthebottomofthemovableplateprohibitthetwoplatesfromcomingintocontact.AsthedcvoltageisdecreasedthecapacitanceremainsatthehighvalueevenifthevoltageisdecreasedpastVPI.Then,asthevoltageisfurtherdecreased,thecapacitancesuddenlyjumpsbacktonormal.IncontrasttothedeviceshowninFigure 4-6 ,wherethetopplatereturnedtoitsneutralpositionwhenthedcvoltageissettozero,softerspringdesignsdonotalwaysprovideenoughrestoringforceandtheplatesmayremainpermanentlystuck.Thiseffectcanbeduetoanumberofforcessuchasforcesduetocapillarycondensation,molecularvanderWaalsforces,andelectrostaticforcesduetoparasiticelectricalcharges[ 50 ].Thedimplesminimizetheseforcesbyreducingthecontactarea.Ifpull-inhappensandthespringscannotrestorethemovableplatetooriginalposition,itwillbenecessarytouseaglasscapillarywithasharptiponamicromanipulatortoseparatethetwoelectrodes. 4.4DetectionSchemeforMagnetizationMeasurementsInthissectionthedetectionschemeoftheforcemeasurementsisdescribed.Figure 4-7 showstheschematicofthedeviceandthedetectioncircuitry.Thecircuitissimilartoacapacitancebridge.Twoidenticalmicromachinedcapacitorsareexcitedbyacvoltageswiththesameamplitude(Vac)but180outofphase.Bothcapacitorsarefabricatedonthesamechipbuttypicallyonlyoneofthemisabletomove.Thesampleismountedonthemovablecapacitorwhilethexedcapacitorisusedforbalancingtheoutputsignal.Theoutputvoltage(Vout)ofthebridgecircuitisVout=VacC1)]TJ /F6 11.955 Tf 11.96 0 Td[(C2 C1+C2. (4) 55

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Figure4-6. CapacitiveresponseasafunctionoftheappliedDCvoltage.WhentheDCvoltagebecomesgreaterthanthepull-involtagethetopplatecollapsesonthexedelectrode.Asthevoltageisloweredthespringpullsbackthemovableplateinplace.Thelinesareguidestotheeye. whereC1andC2arethecapacitancesofthemovableandthebalancingcapacitor.Intheidealcasescenario,whenthereisnoforceonthemovableplate,thetwocapacitancesareexactlythesameandtheoutputsignaliszero.However,inrealsystems,thecapacitancesdifferbyasmallamountandaratiotransformerisusedtocontrolthephaseandtheamplitudeofthedrivingvoltageofeachcapacitor.Asaresult,afteradjustingtheratiotransformer,thebridgecircuitbecomesbalancedandthebackgroundisnegligible.Whenamagneticoranelectrostaticforceactsonthemovableplateofthemagnetometer,thecapacitancechangesandthecircuitbecomesunbalanced.TheoutputsignalgeneratedisdescribedbyEq. 4 56

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Duringexperiments,afunctiongeneratorisusedtocreatethedrivingacvoltagewhichisthenfeedintotheratiotransformerasshowninFigure 4-8 .Theout-comingvoltagesoftheratiotransformerareappliedatthebottomelectrodesofthetwodevicesthroughblockingcapacitors.Adcvoltageisappliedtothedevicesthrougharesistor.Theoutputofthecircuitisconnectedtoachargesensitivepre-amplier,whoseoutputisthenfeedintoalock-in.Thereferenceofthelock-inissettobethesameastheacdrivingfrequency. Figure4-7. Schematicofthedetectioncircuit.Twoidenticalmicromachinedcapacitorsareexcitedbyacvoltageswiththesameamplitudebut180outofphase.Thesampleismountedonthemovablecapacitorwhiletheotherisusedforbalancingthesignal.Ifthetwocapacitancesarethesametheoutputsignaliszero.Amagneticforcechangesthecapacitanceofthecapacitorinwhichthesampleismounted.Asaresulttheoutputsignalchanges. Thedrivingfrequencyofthedeviceisusuallyselectedtobeclosetothehighestfrequencyofthelock-ininusetypicallyseveralhundredkHzforreduced1=fnoise.Inaddition,inthisdetectionschemeitisdesirablethedrivingfrequencytobefarfromtheresonancefrequencyofthemagnetometercommonresonancefrequenciesofthedevicesaretensofkHz. 57

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Figure4-8. Ratiotransformerinthedetectioncircuit.Theuseofaratiotransformerhelpsbalancingthecircuitandconsiderablyreducingthebackground. Thesensitivityofthemeasurementsdependsontheparasiticcapacitanceofcoaxialcables.Theoutputofthemeasurementcircuitisconnectedtothechargesensitivepreamplierthroughcoaxialcables.Thesecablesintroduceaparasiticcapacitancetothecircuitthatiscalledstraycapacitance(Cstray)andisshowninFigure 4-8 .Typicalstraycapacitancesperlengthforcoaxialcablesare100pF=m,whiletypicalcapacitancesofthemagnetometersare1or2pF.AsshowninFigure 4-8 ,thestraycapacitanceactsasadividerinthecircuit.Theratioofthedividerisdeterminedbytheratioofthedevice'scapacitancetothestraycapacitance.Consequently,abigportionofthesignalleakstogroundbeforeitreachesthechargepreamplierandtheresolutionofthemeasurementisreduced.However,achievingoptimalsensitivityiscrucialforsomeexperiments.Toeliminatetheparasiticcapacitanceofthecoaxialcableminiaturizedhighelectronmobilitytransistors(HEMT)areplacedincloseproximitytothemagnetometerandtheyare 58

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wirebonded,bygoldwire,directlytotheoutputelectrodeofthedevice.Forexperimentsthatdidnotinvolvemagneticeldsthismethodprovedtobeveryuseful.Weachieveda10-foldincreaseinthesensitivityofthedevice.Ontheotherhand,theoperationofHEMTsinmagneticeldsisnotstraightforward.TheorientationoftheHEMTmustbechosensothattheelectronsowinaplaneparalleltothemagneticeld.FormagnetizationexperimentsdescribedinthenextchapterstheuseofHEMTswasnotnecessarybecausetheresolutionthatweachievedwassufcientforthepurposeofthemeasurements. 4.5Calibration 4.5.1AbsoluteMagnetizationoftheSampleThemicromechanicalmagnetometersarecalibratedbytheelectrostaticforce.AccordingtoEq. 4 ,theelectrostaticforcedependsonlyonthegeometryofthedeviceandtheappliedvoltagecanbeaccuratelycalculated.Beforemagnetizationmeasurementsareperformed,theelectrostaticforceisusedtocalibratetheoutputsignalofthedeviceatzeromagneticeld.Whenthemagnetizationmeasurementsareperformedtheelectrostaticforceisturnedoffandthemagneticforcealoneisresponsibleforthechangeinthesignal.AnexampleofacalibrationprocedureisshownonFigure 4-9 .Figure 4-9 Aplotsthelock-insignalasafunctionoftheapplieddcvoltage.Usingthedevice'sdimensionsandthedcvoltage,thex-axiscanbeconvertedintoforcebyEq. 4 .ThentheproportionalityconstantbetweentheoutputsignalandtheforceisdeterminedbythelineartinFigure 4-9 B.Later,whenthemagneticforcealonewillbeactingonthedevice,thelock-insignalwillbemappedbacktoaforceusingthelineinFigure 4-9 B.Extractionofthemagneticforcefromtheoutputsignalenablesthecalculationoftheabsolutemagneticmomentofthesampleprovidedthatthemagneticeldgradientis 59

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known,usingtherelation~m=~F ~r~H. (4) Figure4-9. Calibrationofthemicromechanicalmagnetometers.(a)Thelock-insignalasafunctionoftheapplieddcvoltageThelineisaguidetotheeye.(b)UsingEq. 4 thelock-insignalismappedtoforce.Thelineisalineart. TheforceresolutionofthemagnetometersistypicallyafewpN,limitedbytheelectronicnoiseofthereadoutcircuitandthethermalnoiseofthemagnetometeritself.Thethermalnoisespectraldensityofthedeviceisgivenbyq F2n f=q 4kBT!0m Q,wheremisthemassofthemovableplate,!0istheresonancefrequencyofthedevice,Qisthequalityfactor,kBistheBoltzamnconstantandTisthetemperature.Figure 4-10 plotstheelectrostaticforceappliedonthemovableplateofamagnetometerasafunctionoftime.Theforceisincreasedby20pNevery50s.Theresolutionofthismagnetometerwas2.410)]TJ /F3 7.97 Tf 6.58 0 Td[(12N/p Hz.Asaresult,theforceincrementsof20pNwereeasilydetected.Themagnetizationresolutiondependsonboththesmallestforcethatthemagnetometerscandetectandthevalueoftheeldgradient.Largereldgradientsallowhighermagneticmomentresolutions.TypicaleldgradientsofgradientcoilsusedwithsuperconductingmagnetsareontheorderofT/m(2T/mfortheexperiment 60

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Figure4-10. Electrostaticforceappliedtothemovableplateofamagnetometerasafunctionoftime.Theelectrostaticforceisincreasedby20pNevery50s.Theforceresolutionofthismagnetometeris2.410)]TJ /F3 7.97 Tf 6.59 0 Td[(12N/p Hz. describedinChapter 5 .Indcresistiveandpulsedmagnetsthesampleisdisplacedbyasmalldistancefromtheeldcenterandthenaturaleldgradientofthemagnetisused.Typicalgradientsare20T/mfordcresistiveand200T/mforpulsedmagnetsintheNHMFL.Althoughthegradientincreasesbyafactorof10fromonemagnettypetotheother,thisisnotthecasefortheresolution,whichremainsalmostthesameindifferentmagnettypes.Thishappensduetoseveralreasons.Onepossibilityistheincreasedmechanicalnoiseintheenvironmentsofthedcresistiveandpulsedmagnets.Additionally,whenhighmagneticeldsareusedtheelectronicequipmenthastobemovedfarawayfromthemagnet.Asaresult,thelongcablesthatarerequiredincreasethestraycapacitanceandreducethesignaltonoiseratio.Finally,insuchenvironmentsthegroundofthedetectioncircuitisdirtyandunwantedelectricalnoiseisaddedtothedeviceresponse. 61

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4.5.2ExtractionoftheMagneticMomentperUnitFormulaEventhoughthegeneralbehaviorofthemagnetizationinaspinsystemgivesimportantinformationsuchasthecriticalpointwhereaQPToccurs,theknowledgeofthemagnetizationperunitformulaisnecessarytoextractconclusionsaboutindividualspinsinthemoleculesofthecompound.Tobeabletoextractthemagnetizationperunitformulathemassofthesampleisrequired.Samplesthatareuseonthemicromechanicalmagnetometershavedimensionsrangingfrom10to300m.Themassofsuchsamplesistoosmalltobemeasuredusingamicrobalance.Instead,theresonanceofthemagnetometersisusedtoestimatethesamplemass.Themicromechanicalmagnetometersbehaveasmass-springoscillatorswhenthedrivingforceisperiodic.Theresonancefrequencyofaspring-massharmonicoscillatoris!0=q k m,wheremisthemassandkisthespringconstant.Ifthemasschangestheresonancefrequencyalsochanges.Asaresult,themassofasamplewhichhasbeenmountedonthemovableplateofamagnetometercanbedeterminedfromtheshiftoftheresonancefrequency(Figure 4-11 ),ifthemassoftheplateisknown.Ifanadhesiveisusedtostickthesampleonthemovableplate,itsmassisalsodeterminedusingthismethod.Theresonancefrequencyismeasuredthreetimesbeforeplacingtheglue,afterputtingtheglueandaftermountingthesample.Thismethodrequiresknowledgeofthedensityofthepolysiliconplate.Unlikesinglecrystalsilicon,polysiliconproducedbydifferentfabricationfacilitiesmayshowwidevariationsindensity.Topreciselydeterminethedensityofpolysiliconinourmicromechanicalmagnetometers,aSQUIDmagnetometercanbeusedasdescribedbelow.Oncethedensityofpolysiliconisdeterminedforonedevice,wecanadoptthisvalueforotherdevicesmadeinthesameprocessrun.Fordevicesmadeinotherprocessruns,itmightbenecessarytorepeatthecalibrationifthemassofthesamplemustbeknownaccurately. 62

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Figure4-11. Resonanceshiftcausedbytheadditionalmassofasamplemountedontopofthemagnetometerplate.Thisshiftcorrespondstoamassof21g(a)Theresonancepeakbeforemountingthesample.(b)Theresonancepeakisshiftedtolowerfrequenciesafterthesampleismountedontothemovableplate.Thelinesareguidestotheeye. ASQUIDcanbeusedonlywithmaterialswhicharemagneticinlowelds,meaningthattheircriticaleldneedstobelessthan5T(Chapter 5 ).AcalibrationsampleismeasuredusingaSQUIDmagnetometerinsuchexperimentalconditionsthatbothSQUIDsandmicromechanicalmagnetometerscanoperate(e.g.at4Kand<5T).Then,asampleofthesamematerialismeasuredusingamicromechanicalmagnetometer.Thecalibrationsamplechosenisabouttwoordersofmagnitudeheavierthantheoneusedonthemicromechanicalmagnetometers.Themassoftheheavysample,usuallyseveraltensofmg,canbeaccuratelymeasuredbyasensitivemicrobalance.Attheend,themicromechanicalmagnetometer'sdataarecomparedtotheSQUID'sdata,bothmeasuredunderthesameexperimentalconditions,andthemassofthesampleusedonthemicromechanicalmagnetometeraswellasthedensityofthepolysiliconplatecanbeextracted.However,mostofthequantumspinsystemsknownnowadayshavecriticaleldshigherthan10T,andtheresonanceshiftisthebestwaytohaveanestimateofthesamplemass.Typicalmassesofcrystalsusedwiththemicromechanical 63

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magnetometersrangefrom1to50g.Themassresolutionoftheresonancemethoddependsonhowaccuratelytheresonancefrequencycanbedetermined,andistypically100ng. 4.6ComparisonwithOtherMagnetizationMeasurementMethodsThepurposeofdesigningthemicromechanicalforcemagnetometersistocombinetheadvantagesoftheothermagnetizationmeasurementdevices,whilekeepingthecostlow.Inthissectionwewillcompareourmethodwiththetwomostsensitivedevicesavailable,theSQUIDmagnetometersandthecantilevermagnetometers.SQUIDmagnetometersareusedtomeasuretheabsolutemagnetizationofsmallsampleswithmassesoftheorderofmgatmoderateeldsandlowtemperatures.CommercialSQUIDshavemagneticmomentresolutionsattheorderof10)]TJ /F3 7.97 Tf 6.59 0 Td[(8emu/p Hz.Theyoperateattemperaturesdownto1.8Kandmagneticeldsupto7T.AlthoughitispossibletousehomemadeSQUIDsindilutionrefrigerators[ 51 52 ],thisiscostlyandrequiresextensivemodicationoftheequipment.Moreover,inorderforthemeasurementtobeperformedthesamplehastomoveinandoutofthedetectioncoil.Asaresult,thetemperatureandtheeldhavetobestabilizedateachpoint,makingthemeasurementstimeconsuming.Especiallymeasurementsthatrequiretemperaturesweepsareconsiderablyslowandtedious.Incontrast,themicromechanicalmagnetometershavebeensuccessfullyusedindilutionrefrigerators,requiringonlyminormodicationsofthesystemandjustthreecoaxialcablestooperate.Furthermore,thetypicalmagnetizationresolutionofthemicromechanicalmagnetometersis10)]TJ /F3 7.97 Tf 6.59 0 Td[(8emu/p Hzwithconsiderableroomforfurtherimprovement,whilethetypicalmagnetizationresolutionofaSQUIDmagnetometerissimilaratlowelds.Asthemagneticeldisincreased,however,thesensitivityofaSQUIDmagnetometerdecreases.Besidestheirhighmagnetizationresolution,themicromechanicalmagnetometersoperateunderconditionsnotaccessiblebySQUIDs.Inthenextchapters,wewill 64

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describeexperimentsinwhichourdeviceswereabletooperatesuccessfullyintheenvironmentsofthe35TdcresistivemagnetsatNHMFLinTallahassee(Chapter 6 )andthe65TpulsedmagnetsatNHMFLinLosAlamos(Appendix A ).Performingexperimentsinpulsedmagnetscanbeverychallengingbecausethemeasurementneedstobequick(thepulselasts25ms),theeldscanbeveryhigh(upto92.5T)andextensivenoiseisgeneratedduringthepulse.Furthermore,pulsedmagnetsoccasionallyfail,underthehugeaxialstressescreatedduringapulse,andtheexperimentalsetupgetsdestroyed.Thehighmechanicalresonancefrequencies,usuallyhigherthan15kHz,andthesymmetricdesignofthemicromechanicalmagnetometersmakethemrobustagainstvibrationsandmechanicalnoisethatusuallyoccuratalotlowerfrequencies.Inaddition,theirfastmechanicalresponsemakesthemeasurementsveryquick.Moreover,themicromechanicalmagnetometersarefabricatedusingsiliconsurfacemicromachining,allowingawiderangeofdesignssuitablefordifferentsamplesatareasonablecost.Cantilevermagnetometersmeasuresharpchangesinthemagnetizationofsmallanisotropicsamplesinawidevarietyofexperimentalconditions.However,occasionallyunwantedsignalsmaybegeneratedwhenthemagnetichardaxisofthesamplealignswiththeeld,makingitdifculttointerprettheresults.Moreoverthedifcultytodistinguishtorquefromforcesignalsmakesitimpossibletoextracttheabsolutemagnetizationofthesampleofinterestusingcantilevers.Eveninthecasethatthesignalconsistsonlyoftorquecontributions,theextractionoftheabsolutemagnetizationisnotstraightforward.Ontheotherhand,themicromechanicalmagnetometersmeasurethemagneticforceonthesample.Thecalibrationofthemagneticforceusingadcvoltageenablestheprecisemeasurementofthemagnetizationandifthemassofthesampleisknownthemagnetizationperunitformulacanbeextracted.Inaddition,theirsymmetricdesign 65

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makesitpossibletosubtractsmalltorquecontributionsthatoccasionallyappearinthesignal,asisdescribedindetailinChapter 5 .MagnettimeispreciousatmultiuserfacilitiessuchasthepulsedmagnetfacilityatNHMFLinLosAlamosandthedcresistivemagnetfacilityatNHMFLinTallahassee.Typicallytheavailabletimeforexperiments(magnettime)islimitedto20-30hourseveryseveralmonths,andmakingthemostoutofthemagnettimeishighlydesirable.Thesmallsizeoftheforcemagnetometersmakesitpossibletohavemorethanonedeviceonthesamechip(Figure 4-12 ).Asaresult,morethanoneexperimentcanbeperformedatthesametime,providedthatenoughelectronicsareavailable,foroptimizeduseofthemagnettime. Figure4-12. Achipthatcontainsfourmagnetometers.Inthisdesignfourexperimentscanbeconductedsimultaneously. Usingcommercialfabricationprocesswecreatedmicromechanicalforcemagnetometers.Theirdesign,basedontheFaradaybalanceprinciple,enablesthemeasurementoftheabsolutemagneticmomentoftinysamplesatenvironmentswhereothermagnetometersareunabletooperate.Inthenextchapters,experimentsusingmicromechanical 66

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magnetometersunderseveralexperimentalconditions,includingdilutionrefrigerators,33Tdcresistivemagnetsand65Tshortpulsepulsedmagnets,willbedescribed. 67

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CHAPTER5(C7H10N)2CUBR2Quantumspinsystemshavebeenanimportanteldofstudyformanyyearsandcontinuetogeneratefascinatingphysics.Ofspecialinterestarematerialswithantiferromagneticexchangeinteractions,whereanenergygapduetolowdimensionalityorspindimerizationseparatesthesingletgroundstatefromtheexcitedtripletbands.TheenergygapclosesatacriticaleldHc= gB.AteldshigherthanHcthesystemisgapless.Themagneticpropertiesofthisgaplessphasedependstronglyonthedimensionalityofthesystem.Inthecaseoftwoorhigherdimensionslong-rangemagneticorderdominatestheregionabovethecriticaleld.ThisorderedstatecanbedescribedasaBECofmagnonsandanexampleofsuchasystemisstudiedinthenextChapter.Ontheotherhand,longrangemagneticorderdoesnotoccurin1Dsystemssuchaseven-legspin-1/2ladders.Incontrasttohigherdimensionalsystems,theirgaplessphaseisdescribedasaTLL.Eventhoughgapped1Dsystemshavebeenthoroughlyinvestigatedtheoretically[ 53 55 ],ideal1Dsystemswithcriticaleldsthatareaccessibleinthelaboratoryarerare.Forinstance,CuHpCl[ 56 ]wasinitiallythoughttobeaspin-1/2systembutturnedouttobeafrustratedthree-dimensionalsystem[ 57 ].AsecondexampleisNTEMP[ 58 ],aS=1chainwithlargeinterchaininteractionsthatdominatethelowtemperaturespecicheat.AnothercandidateisIPA-CuCl3[ 59 60 ]wheretheinterladderinteractionsleadtolong-rangemagneticorderinthegaplessphase.Theonly1DsystemsthatshowpromiseinexhibitingTLLbehaviorare(C5H12N)2CuBr4,atwo-legspin-1/2systemwithstrong-runginteractions[ 3 4 ],and(C7H10N)2CuBr2(DIMPY)[ 2 61 ]. 5.1PreviousWorkonDIMPY(C7H10N)2CuBr2,abbreviatedDIMPY,formsamonocliniclattice(P21/nspacegroup)withroomtemperaturelatticeparametersa=7.50A,b=31.61A,c=8.20A,and=98.97[ 61 ].ThemagnetismofDIMPYarisesfromS=1=2Cu2+ions.CuBr4)]TJ /F3 7.97 Tf -4.43 -7.98 Td[(2 68

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radicalsformtwo-legspinladdersalongthecrystallographicaaxis(Figure 5-1 ).TheBrBrdistancealongthelegsoftheladdersis3.905(1)A,whilethedistancealongtherungsissignicantlylonger,4.328(1)A[ 61 ].Therefore,inDIMPYthemagneticexchangealongthelegs(Jleg)isstrongerthantherungexchange(Jrung). Figure5-1. CrystalstructureofDIMPY.(a)CuBr4)]TJ /F3 7.97 Tf -4.43 -7.97 Td[(2anionsformtwo-legspinladdersalongthecrystallographicaaxis(viewedparalleltothecaxis).(b)PackingdiagramofDIMPYviewedparalleltotheaaxis.BluelinesrepresenttheBrBrdistancealongtherungsofaladder.Greenlinesindicatetheshortestinterladderdistance.(H-atomshavebeenremovedforclarity.)Reprintedwithpermissionfrom[ 61 ],Copyright2007AmericanChemicalSociety. Inelasticneutron-scatteringandcalorimetricmeasurements,on67%deuteratedDIMPYcrystals,byHongetal.shownolong-rangemagneticorderdownto150mK[ 2 ].Atlowmagneticelds,exponentialbehaviorofthespecic-heat(Figure 5-2 A)indicatesthatDIMPYisagapped1DHeisenbergantiferromagnet(HAF)withcriticaleldHc=3.0(3)T.AboveHc,asthetemperatureisdecreased,thespecicheatbecomesasymptoticallylinearintemperature(Figure 5-2 B),characteristicofaTLL(Section 2.6 ).Inaddition,thereisno-likeanomaly[ 62 ],indicativeofaphasetransition,attemperaturesdownto150mKandmagneticeldsupto18T.Asaconsequence,DIMPYisanexcellent1DsystemwithagaplessTLLphaseabovethecriticaleldHc, 69

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atwhichthegap=3.7(1)Kvanishes.Furthermore,usingthespecic-heatdatainconjunctionwithdensitymatrixrenormalizationgroup(DMRG)calculationstheratiox=Jleg=Jrungbetweenthelegexchangeandtherungexchangewasfoundtobe2.2(2)[ 2 ]. Figure5-2. Specic-heatof67%deuteratedDIMPY.(a)Specic-heatCmofDIMPYasafunctionoftemperatureT,forHHc.Reprintedwithpermissionfrom[ 2 ],copyright2010bytheAmericanPhysicalSociety. 5.2ExperimentalLowtemperaturemagnetizationmeasurementswereperformedusingmicromechanicalforcemagnetometersinadilutionrefrigeratorwitha9TsuperconductingmagnetinthelaboratoryofProf.HoBunChanattheUniversityofFlorida.Thesuperconductingmagnetwasequippedwithaseparategradientcoilwithaeldgradientof2T/m. 70

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Achipcontaining4micromechanicalmagnetometers1wasreleasedanddriedusingtheprocessdescribedinSection 4.1 .Then,thechipwasmountedona32pinceramicpackage.Goldwireswerebondedbetweenthebondingpadsofthedeviceandgoldpadsonthepackageusingawirebonder.Atinysinglecrystalof67%deuterated(C7H10N)2CuBr2(DIMPY)wasgluedontothemovableplateofamicromechanicalmagnetometer,withspringconstantof28N/m,usingUVglue.Thepackagewasinsertedintoasocketlocatedatthecoldngerofthedilutionrefrigerator.Thesystemwasthenloadedintoa4Hedewarsothatthemagnetometerwaspositionedatthecenterofthemagnetandthecrystallographiccaxisofthesamplewasparalleltothemagneticeld.Additionalmagnetizationmeasurementsattemperaturesdownto1.8Kandeldsupto5TwereperformedusingtheQuantumDesignSQUIDmagnetometerofthePhysicsDepartmentattheUniversityofFlorida.A67%deuteratedsinglecrystalofDIMPYwithamassof30.48mgwasloadedintotheprobeoftheSQUIDmagnetometerusingagelatincapsule.Themagnetizationwasmeasuredbothwiththecrystallographiccaxisofthesampleparallelandnormaltothemagneticeld.Magnetocaloric-effectandspecic-heatmeasurementsofa6.6mgsinglecrystaloffullydeuteratedDIMPYwereperformedbyDr.TaoHongandProf.YasumasaTakano.Thespecic-heatmeasurementsweredoneusingrelaxationcalorimetry[ 63 ],withthemagneticeldappliedalongthecaxisasinthelowtemperaturemagnetizationmeasurements.Thephononcontributionwasdeterminedfromthezero-eldentropyandwassubtractedfromthedataatallelds.Finally,thenuclear-spincontributionwas 1Thischip(named20M2)wasfabricatedbyMEMSCAPPolyMUMPsRun85.Themovableplateofthemagnetometerwhichwasusedforthisexperiment,wassupportedby8springstwoateachcorner.Thespringslookedlikeserpentineswithoneturn,andtheirdimensionswere7032m. 71

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alsosubtractedthroughasimultaneousttothedataforalleldsattemperaturesbelow700mK. 5.3ResultsFigure 5-3 showsthemagneticsusceptibilityofDIMPYasafunctionoftemperaturedownto2Kinaconstantmagneticeldof1Tappliedalongthecrystallographiccaxis.ThismeasurementwasperformedusingtheSQUIDmagnetometer(Section 5.2 ).Thesusceptibilityshowsamaximumat11.5Kanddecreasesrapidlytowardszeroasthetemperatureislowered,inagreementwithanearlierresult[ 61 ].ThismaximumischaracteristicofalowdimensionalHAFwhiletherapiddecreaseindicatestheexistenceofasingletgroundstateseparatedbyanenergygapfromtherstexcitedstate. Figure5-3. MagneticsusceptibilityofDIMPYasafunctionoftemperaturedownto2Kandatappliedmagneticeldof1T.ThemaximumischaracteristicofalowdimensionalHAF.Therapiddecreaseafterthemaximumindicatestheexistenceofaspingap. UsingmicromechanicalforcemagnetometersthemagnetizationofDIMPYwasmeasuredasafunctionoftheappliedmagneticelddownto45mK(Figure 5-4 ).Attemperatureslowerthanthemagnongapandeldslowerthanthecriticaleld,the 72

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magnetizationisnearlyzeroexceptforacontributionofparamagneticimpurities,whoseconcentrationisabout0.9%.AteldshigherthanthecriticaleldHcthemagnetizationincreasesrapidly.ThissignatureofHcbecomeslesspronouncedasthetemperaturerises.Attemperatureshigherthanthemagnongap=3.7(1)K[ 2 ],themagnetizationcurveisfeatureless,withanapproximatelyconstantincreaseratethroughouttheeldrangeofourexperiment.Inatruly1DgappedantiferromagnetandateldsnearHc,wherethedensityofexcitationsissmall,magnonscanbemappedtofree1Dfermions[ 64 65 ].Insuchasystem,thezero-temperaturemagneticsusceptibilityexhibitsasquare-rootdivergenceatHc.Asthetemperatureisraised,thissingularityreducestoaroundedpeaknearHc[ 55 ].Figure 5-5 plotsthemagneticsusceptibilityofDIMPYasafunctionofthemagneticeld.Thesusceptibilityhasbeencalculatedbydifferentiatingthemagnetizationwithrespecttotheeld.Thereisnotemperaturedependenceinthesusceptibilitywithinourresolutionatleastupto300mK.ApeaknearHcclearlyappearsinourlow-temperaturedata,providingstrongevidencefortheexcellent1DcharacterofDIMPYinsupportofheat-capacityandinelasticneutron-scatteringresults[ 2 ].Aminimuminthemagnetizationasafunctionoftemperatureisanothercharacteristicfeatureofagapped1DantiferromagnetandmarkstheupperlimitoftheTLLtemperatureregime[ 55 64 66 ].SuchminimahavebeenobservedinNDMAP[ 67 ],whichisanantiferromagneticS=1linear-chaincompound.Therehasbeennoevidenceofminimainanyother1Dantiferromagnetsbecauseofmaterialproblemssuchastheexistenceofeld-inducedstaggeredeldsandtoohighacriticaleldforthepresenttechnology.Figure 5-6 plotsthemagnetizationofDIMPYasafunctionoftemperatureateldsHHc,aftersubtractingthecontributionoftheparamagneticimpurities.At4T,slightlyaboveHc,themagnetizationattainsaminimumattemperatureTmof0.7K.Athighereldstheminimumoccursatevenhighertemperaturesensuringthatourlow 73

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Figure5-4. MagnetizationofDIMPYasafunctionofmagneticeldatxedtemperatures.Themagneticeldwasappliedalongthecaxis.Temperaturesare,frombottomtotop:45mK,300mK,700mK,1.8K,4.3K.Eachcurvetakenattemperaturehigherthan45mKhasbeenshiftedby100emu/molfromthepreviouscurveforclarity. 74

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Figure5-5. MagneticsusceptibilityofDIMPYasafunctionofthemagneticeld.ThepeaknearHcatlowtemperaturesprovidesstrongevidencefortheexcellent1Dcharacterofthiscompound. temperaturesusceptibilitydataarefromdeepintheTLLregime,atleastatandabove4T.Thepositionoftheminimum,Tm,predictedbythefreefermiontheoryisdescribedbytheuniversalrelation[ 64 ]Tm=0.76238gB kB(H)]TJ /F6 11.955 Tf 11.96 0 Td[(Hc). (5)Figure 5-7 plotsTmfromthedataalongwiththeuniversalfreefermionbehavior(Eq. 5 ).AteldsnearHc,wherethedensityoffermionsissmall,theoryandexperimentareinexcellentagreement.Astheeldincreases,Tmfallsbelowtheuniversalrelation,aspredictedbyquantumMonteCarlo(QMC)simulations[ 64 ].Thisdownwarddeviationisunderstoodasaneffectofrepulsiveinteractionbetweenthefermions. 75

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Figure5-6. MagnetizationofDIMPYasafunctionoftemperature.Symbolsareexperimentaldata,fromwhichthemagnetizationofparamagneticimpuritieshasbeensubtracted;solidlinesarequantumMonteCarlosimulations.Fieldsare,frombottomtotop:(a)3T,4T,4.5T,5.5T;(b)7.5T,9T. 76

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Tofurthercomparetheexperimentwiththeory,ourcollaboratorsTakuyaManabeandProf.ChisaHottaperformedQMCsimulationsofthemagnetizationofanS=1=2ladderconsistingof120rungs(Figure 5-7 ).Bycomparisonwiththeexperimentwehavefoundthattheg-factorg=2.2andtheexchangeratiox=2.0givethebestagreement,consistentwiththevaluex=2.2(2)extractedfromspecic-heatmeasurements[ 2 ].Ateachtemperatureandmagneticeld,50runsof1.2107MonteCarlostepswereusedforaveraging.Boththesimulationsandtheexperimentaldatasharetwosignicantfeatures.First,themagnetizationminimaTmarefoundatsimilartemperatures,exceptat3Twhichisatorveryclosetothecriticaleld.Second,asthemagneticeldisraisedtheminimumbecomeslesspronounced. Figure5-7. Thepositionofthemagnetizationminimumasafunctionofmagneticeld.DashedlineistheuniversalbehaviorforfreefermionsgivenbyEq. 5 InadditiontoQMCsimulations,density-matrixrenormalization-group(DMRG)calculationswereperformed,bythesamecollaborators,toconrmthatthevaluesg=2.2andx=2.0foundbytheQMCsimulationsyieldthebestoverallagreementbetweenexperimentandtheory.Figure 5-8 plotsourmagnetizationdataat300mKasafunctionofthemagneticeldalongwiththeDMRGresultsforg=2.2andx=2.0. 77

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Figure5-8. Comparisonofthemagnetizationasafunctinoftheeldat300mKwithDMRGresults.TheopencirclesrepresentDMRGresultsforx=2.0andg=2.2.IntheinsertthemagnetizationcurveadaptedfromRef.[ 68 ]iscomparedwithQMCforx=2.0andx=2.2andDMRGcalculationsforx=2.0.Inbothcasesg=2.2. 78

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Theagreementisexcellent.IntheinsetofFigure 5-8 QMCandDMRGdataareplottedalongwiththemagnetizationmeasuredbyWhiteetal.[ 68 ].Againthebestoverallagreementisachievedusingg=2.2andx=2.0.However,themagnetizationcurvefromRef.[ 68 ]ismeasuredusingtheinductionmethodinapulsedmagnet(seeChapter 3 ).Asaresultthespintemperaturecanvaryasafunctionoftheeldduetothemagnetocaloriceffectonthespinsandeddy-currentheatingonanymetallicparts.Additionally,sincethepickupcoilcanneverbecompletelybalanced,systematicerrorsmayenterthenalresult.Thispossiblyexplainsthe10%disagreementinthevalueofsaturationmagnetizationbytheDMRGcalculationandthepulsedeldmagnetization. Figure5-9. Magneticspecicheat,Cm,offullydeuteratedDIMPY,plottedasCm=T. InChapter 2 weexaminedthesignicanceoftheWilsonratioRWforaTLL.DespiteitsimportancetheWilsonratiohasneverbeendeterminedexperimentallyfora1Dsystem.Withtheexcellent1DbehaviorofDIMPYrmlyestablished,weproceedtocalculateRWforthersttimeinaTLL.Forthispurpose,HongandTakanohavemeasuredthespecicheatCmofa6.6mgsinglecrystaloffullydeuteratedDIMPYasdescribedinSection 5.2 79

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At6T,7T,and8.9T,CmshowsT-linearbehavior,demonstratedbyconstantCm=T,attemperaturesbelowabout1K(Figure 5-9 ).Asharppeakappearingatapproximately340mKsignalsorderingatlowertemperatures.Longrangeorderoccurringinthissampleattemperatureslowerthan340mKisinstrongcontrasttotheabsenceoforderinginthe67%deuteratedsampleofRef.[ 2 ]attemperaturesdownto150mK.ThisresultindicatesthatfulldeuterationinDIMPYdrasticallyenhancesinterladderinteractions.DistwiceasheavyasH.Asaresult,thezero-pointmotionofDislessthanthatofH,andthusthelengthofDCbondsisshorterthanthelengthofHCbonds[ 69 ].Therefore,theC5X12Nion(whereX=DorH)thatseatsbetweenthespinladders(Figure 5-1 B)issmallerinfullydeuteratedDIMPYthanin67%deuterated.ThismakestheinterladderdistancesinfullydeuteratedDIMPYshorterthanin67%deuterated,enhancingtheinterladderinteractions.TostudyorderingateldsnearHcHongandTakanoalsomademagnetocaloriceffectmeasurements.Usingthespecic-heatpeaksandthemagnetocaloriceffect,theboundaryoftheorderedphasehasbeenmappedupto18T(Figure 5-10 ).Thephaseboundaryishighlyasymmetric,withamaximumatabout8.9T,wherethefermionvelocityisatornearmaximum[ 2 ].Cm=TneededtodetermineRWisobtainedfromthespecicheatdataat6T,7T,and8.9Tattemperaturesbelow0.75K,excludingtheregionwherelongrangeorderoccurs.AdditionalCm=T,at5Tand8T,aretakenfromthe67%deuteratedsampledataofRef.[ 2 ].Forconsistency,weusethesusceptibilitydataat300mKratherthan45mK,eventhoughtheyarenearlyidentical.RWdeterminedfromthesusceptibilityandspecicheat,assumingthatginEq. 2 ais2.2,isshowninFigure 5-11 asafunctionofthenormalizedmagnetizationm=M(NASgB))]TJ /F3 7.97 Tf 6.58 0 Td[(1.TheWilsonratiotakesvaluescloseto4,increasingwithincreasingm.ToexaminewhetherthisresultconrmstherelationRW=4K(Eq. 2 ),ManabeandHottahaveemployedtheDMRGmethodtocalculatetheTLLparameterKfor 80

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Figure5-10. PhasediagramoffullydeuteratedDIMPY.Solidcirclesarefromspecic-heatpeaks.Opencirclesarefromthemagnetocaloriceffectdatashownintheinset,wherepinkandbluelinesrepresentthesampletemperatureduringupwardanddownwardeldsweeps,respectively.Otherlinesareguidestotheeye. x=2.02.TheresultisshowninFigure 5-11 asRWbyassumingEq. 2 .Kis1atm=0.AsmisincreasedKdeviatesupwardfromtheuniversalfree-fermionvalue1characteristicbehaviorofastrong-legladder[ 70 ],incontrasttoastrong-rungladder.RWfromtheexperimentisdependentonthevalueofthegfactorthatentersEq. 2 a.Infact,ifg=2.04isusedassuggestedbysaturation-magnetizationdata[ 68 ]instead 2ThiscalculationisperformedfollowingRef.[ 70 ].Itinvolves100rungswithopenboundarycondition.Alocalmagneticeldwasimposedontheedgespinstosuppressboundaryeffects,and200basisstateswerekept.Kwasextractedfromtransversetwo-spin-two-spincorrelation. 81

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Figure5-11. DependenceoftheWilsonratioRWofDIMPYonthenormalizedmagnetizationm.Opensymbolsareexperimentaldata,assumingg=2.2inEq. 2 a;specic-heatdatafromFigure 5-9 havebeenusedforcircles,andthosefromRef.[ 2 ]fortriangles.Theerrorbarsrepresentonlythecombineduncertaintiesofthemagnetizationandheatcapacitymeasurements;theuncertaintyofthegfactorhasnotbeenincluded.SmalllledcirclesaretheTLLparameterKforx=2.0,computedbytheDMRGmethodanddisplayedasRWbyassumingEq. 2 .Thelineisaguidetotheeye. ofg=2.2chosenhereonthebasisofthecomparisonofthemagnetizationdatawiththeQMCsimulationsandDMRGcalculationsRWwillrisebyabout16%,resultinginabetteragreementwith4KfromtheDMRGcalculations.Takingthisintoaccount,weconcludethatwithinthecombineduncertaintiesofexperimentandcalculations,theresultstronglysupportstheequivalencebetweenRWandK(Eq. 2 ). 5.4ProceduresFollowedforAnalyzingtheData 5.4.1ForceMeasurementandMagnetizationCalibrationInthissectionwewilldescribetheproceduresusedinanalyzingthedata.Werststartwiththemeasurementofthemagnetizationasafunctionofthemagneticeld.Duetotorquecontributionsintherawsignal,eachmagnetizationmeasurementwasperformedtwiceundertwooppositeeldgradients.Figure 5-12 Aplotstherawsignal 82

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ofthemagnetometerasafunctionoftheeldat45mKandwitheldgradients2T/mand)]TJ /F1 11.955 Tf 9.3 0 Td[(2T/m.Whilethemagneticforcechangessigntogetherwiththeeldgradient,thetorque~T=~m~Honthesampledoesnotdependonthegradient.Therefore,theforceonthesamplecanbeobtainedbysubtractingthetwooppositeeldgradientdata.ThemagneticforceisplottedinFigure 5-12 B.Theleftaxisisintheunitsofthemeasurement,whiletherightaxisiscalibratedinnNusingtheelectrostaticcalibrationdescribedinChapter 4 .Toverifythevalidityofourprocedureofremovingthetorquecontributionsfromthesignalbysubtractingthetwooppositegradientdata,weperformedthefollowingcheck.First,weturnedofftheeldgradientgeneratedbythegradientcoiltoeliminatethemagneticforceandweperformedamagneticeldsweepat45mK.ThissweepcontainsonlythetorquecontributionsinthesignalFigure 5-13 A.Thenaxeddcelectrostaticforce(50nN)wasappliedonthemovableplateandasecondmagneticeldsweepwasperformed.Thissignalconsistedoftorqueandelectrostaticforcecontributions.Theelectrostaticforcewascomparabletothemaximummagneticforceonthesampleat9T(Figure 5-12 B).ThesubtractionofthetwocurvesofFigure 5-13 A,theonewiththeelectrostaticforcetunedoffandtheotherwithitturnedoneliminatesthemagnetictorquecontributionsandgivesasaresulttheelectrostaticforce.Figure 5-13 Bplotstheelectrostaticforceextractedwiththismethod.Asthemagneticeldwasswept,theelectrostaticforceremainedconstantwithinlessthan1%(Figure 5-13 C)ofitsvalueat0Twherethereexistsnomagnetictorque.Thisresultensuresthatsubtractingthetorquefromoursignalusingtwodatasetswithoppositeeldgradientisaccurateandtheoutputofthesubtractionisproportionaltothemagneticmomentperunitformulaofthesample.Althoughthemagneticforceismeasuredprecisely,theuncertaintyinthemassofthesampleentersthemagnetizationperunitformula(Section 4.5.2 ).Inordertoobtaintheproportionalityconstantbetweentheforcesignalandtheabsolutemagnetization,weusedthemethoddescribedinSection 4.5.2 .A30.5mgsinglecrystalofDIMPY 83

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Figure5-12. Rawsignalofthemagnetometer.(a)Rawsignalofthemagnetometerasafunctionofthemagneticeldat45mK.Thetwodifferentlinesrepresentdatatakenwithtwooppositeeldgradients.(b)ThemagneticforceonDIMPYasafunctionofthemagneticeldat45mKasextractedfromthecurvesinpanel(a).Theunitsoftherightaxiscorrespondtothecalibratedsignal.Thecalibrationwasdoneusingelectrostaticforceonthemovableplate. 84

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Figure5-13. Testusingelectrostaticforce.(a)Signalfromthedeviceasafunctionofthemagneticeldwiththeelectrostaticforceturnedoff(topcurve)andon(bottomcurve)at45mK.Thesignalwithouttheelectrostaticforcecontainsonlytorquecontributions.(b)Extractedelectrostaticforceasafunctionofthemagneticeldat45mK.(c)Thepercentagechangeoftheelectrostaticforcefromitsvalueat0T.Theforcechangesbylessthan1%asthemagnetictorqueincreaseswiththeeld. 85

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wasmeasuredinaSQUIDmagnetometerat1.8Kand4.3Kandeldsupto5T.TheoutputsoftheSQUIDwerecomparedwiththoseofthemicromechanicalmagnetometeratthesametemperatures.Then,themicromechanicalmagnetometer'sdatawerescaledtomatchtheSQUIDdata(Figure 5-14 )usingasinglescalingfactor.Usingthisscalingfactorthemassofthesinglecrystalwhosemagnetizationwasmeasuredwiththemicromechanicalmagnetometerswasfoundtobe13.8g,andthedensityofthepolysiliconplatewasfoundtobe889kg/m3. Figure5-14. ThemagnetizationperformulaunitofDIMPYat1.8Kand4.3K.SymbolsaretheSQUIDdata.Solidlinesarethescaledmicromechanicalmagnetometerdata.Thesamescalingfactorwasusedforbothtemperatures. Themagneticforceresolutionachievedusingthemicromechanicalmagnetometersinthe2T/meldgradientwas2.210)]TJ /F3 7.97 Tf 6.59 0 Td[(11N/p Hz,correspondingtoamagneticmoment 86

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resolutionof1.110)]TJ /F3 7.97 Tf 6.59 0 Td[(8emu/p Hz.ThisresolutioniscomparabletothatofcommercialSQUIDmagnetometers.However,itisnotpossibletooperateSQUIDsatthelowesttemperaturesofourexperiment. 5.4.2CorrectionsontheDataWhileparamagnetism,attractionofamaterialtoamagneticeld,isapropertyofsubstancesthathaveunpairedelectrons,diamagnetism,aweakrepulsionfromamagneticeld,isacharacteristicofallatomsinmolecules.Thediamagneticsusceptibilityisnegativeandtemperatureindependent,incontrasttoparamagneticsusceptibilitythatispositiveandtemperaturedependent.Inlaboratoryexperimentsthatstudymagneticpropertiesotherthandiamagnetism,itisimportanttoaccountforthediamagneticresponseofthesample(diamagneticcorrections).Themeasuredsusceptibility(meas)ismeas=+diam, (5)whereisthemagneticsusceptibilityofinterestanddiamisthesusceptibilityduetodiamagnetism.Thediamagneticcorrectionsarecalculatedfromliteraturesourcesthatlistthediamagnetismofwholemolecules,fragmentsofmolecules,orindividualatoms,ions,orbonds[ 71 ].Followingthechemicalformulaofthecompoundofinterestthediamagneticcorrectionsarecalculatedandsubtractedfromthedata.ThediamagneticcorrectionsforDIMPYare)]TJ /F1 11.955 Tf 9.3 0 Td[(210)]TJ /F3 7.97 Tf 6.58 0 Td[(4emu/mol,approximately1.3%ofthesusceptibilityintheTLLregime,andtheyhavealreadybeenappliedinallthepreviousmagnetizationandsusceptibilityplots.Inaddition,themeasurementsperformedintheSQUIDmagnetometerhavebeencorrectedfordiamagneticcontributionsofthegelatincapsuleandgreasethatwereusedtostabilizethesampleintheexperimentalprobe.Themagneticresponseofthecapsuleandgrease(2.7%ofthetotalsignal)wasmeasuredasafunctionoftheeldandsubtractedfromoursignal.FurthermorethemagnetizationoftheSQUIDmagnetometer 87

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wascalibratedusinga263.1mgpalladiumstandard.WethankProf.MarkMeiselforthegenerousloanofthemagnetizationstandard.Moreover,themagnetizationresponseofanemptymicromechanicalmagnetometer,identicaltotheoneusedfortheexperiment,wasmeasuredat4.2Kandeldsupto9Twitha2T/meldgradient.Themagnetometerwasnotabletodetectanymagneticforcelargerthan30pN(theresolutionofthemeasurement)onthepolysiliconplate.Weconcludethatnofurthercorrectionswererequiredtobeappliedonthedata. 5.5ConclusionsWehavemeasuredthemagnetizationofDIMPYasafunctionofthemagneticeldattemperaturesdownto45mK.OurmeasurementsfurtherconrmthatDIMPYisanideal1DsystemwithaTLLgaplessphaseaboveHcatlowtemperatures.ThemagnetizationintheTLLphaseexhibitsaminimumatTm,whosedependenceonmagneticeldisingoodagreementwithQMCsimulations.ThemaximumappearinginthelowtemperaturesusceptibilitynearHcisconsistentwiththesquarerootdivergenceexpectedforagapped1Dantiferromagnetatzerotemperature.Thelowtemperaturesusceptibilityinconjunctionwithspecic-heatmeasurementsenabledthesuccessfuldeterminationoftheWilsonratio.ComparisonoftheexperimentalWilsonratiowiththedensity-matrixrenormalization-groupcalculationsoftheTLLparameterKmakesDIMPYtherstlaboratory1DsysteminwhichtherelationRW=4Kisveriedwithinthecombineduncertaintiesofexperimentandcalculations. 88

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CHAPTER6BA3CR2O8ThischapterstudiesthemagneticpropertiesofBa3Cr2O8.Ba3Cr2O8isasystemofinteractingspindimersarrangedina3Dlattice.Inspindimercompoundsasingletgroundstateisseparatedfromexcitedtripletbandsbyanenergygap.Whenamagneticeldisapplied,thegapdecreasesandclosesatacriticaleldHc(Chapter 2 ).AteldHcaQPTtakesplaceandleadstoagaplessphase.Thisgaplessphaseisdominatedbylongrangemagneticorderandisasuperpositionofsingletsandtriplets.Thetriplets,whicharebosons,mayformaBECaboveHc.ThisbehaviorhasbeenobservedinTlCuCl3[ 72 ]andBaCuSi2O6[ 25 ].Recently,anewclassofspindimercompoundswiththegeneralformBa3M2O8,whereM=CrorMn,havebeendiscovered[ 73 74 ].ThemagneticM5+ions(withspineither1/2or1),arearrangedindimerswhichformtriangularlattices,witheachdimeronacornerofatriangle.Antiferromagneticinteractionsbetweenspinsonatriangularlatticeleadtogeometricfrustrationtheinteractionenergybetweenthespinpairscannottakeontheminimumvaluesimultaneously(Figure 6-1 ).Asaresult,thetripletsaboveHcmayformasuperlatticewhichleadstomagnetizationplateaus[ 75 76 ].Ontheotherhand,ifthefrustrationisrelievedbyastructuraltransitionasthetemperatureislowered,thestateofthesystemaboveHccanbeaBECoftriplets.ThismakesBa3Cr2O8amaterialofspecialinterest.Inaddition,thecriticaleldsofBa3Cr2O8arerelativelylowcomparedtothoseofothercompoundsanditprovidesagoodopportunitytostudyeld-inducedquantumphasetransitionsingeneral. 6.1BasicPropertiesofBa3Cr2O8Ba3Cr2O8consistsofBa2+ionsandCrO3)]TJ /F3 7.97 Tf -4.44 -7.97 Td[(4tetrahedra,thatbuildarhombohedralR3mstructure[ 77 78 ].TheCrionsexistasCr5+withspin1/2thatformdouble-layeredtriangularlattices,stackedalongthecrystallographiccaxis(Figure 6-2 ).Thelatticeparametersarea=b=5.7450(2)Aandc=21.3881(1)A[ 78 ].TheCrCrdistances, 89

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Figure6-1. Geometricfrustration.(a)Antiferromagneticspinsonatriangularlattice.Alltheinteractionscannotbesatisedsimultaneously.(b)Onewaytopartiallysatisfyalltheinteractions. correspondingtotheexchangeinteractionsshowninFigure 6-2 aredJ0=3.934(6)A,dJ1=4.599(4)A,dJ2=5.739(1)A,anddJ3=6.598(3)A[ 77 ].HigheldmagnetizationandsusceptibilitymeasurementswereinitiallyperformedonpolycrystallinesamplesbyNakajimaetal[ 74 ].ThesemeasurementssuggestedthatBa3Cr2O8isaspindimersystemwithasingletgroundstateandaspingapofonly16.1K.Thevaluesofintra-andinterdimerinteractionsobtainedwereJ0=kB=25.04KandJ0=kB=7.69Krespectively,whereJ0=3J1+6J2+6J3[ 74 ].Inlaterinelasticneutron-scatteringmeasurementsthevaluesofJ0andJ0werefoundtobeJ027.6KandJ05K[ 79 ].Thepolycrystallinemagnetizationmeasurements[ 74 ]revealedthatamagneticeldofjust12T(Hc)wasnecessarytoclosethespingapandaeldHs23Twasenoughtoreachsaturationmagnetization,incontrasttootherspindimercompoundsthatrequiredmuchhigherelds.ElasticneutronscatteringexperimentsinzeroappliedeldrevealedthatBa3Cr2O8undergoesastructuralphasetransitionat70Kthatliftsthefrustrationofthetriangularlattice[ 79 ].ThismakesBa3Cr2O8agoodcandidateforBECoftriplets.Heat-capacitymeasurementsonasinglecrystalofBa3Cr2O8,byAczeletal.,revealedalarge-anomalyforappliedeldsbetweenHc12.5TandHs23.5T 90

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Figure6-2. CrystalstructureofBa3Cr2O8.(a)ThesolidspheresaretheBacationsandthetetrahedraaretheanions.(b)NetworkoftheCr5+ionsdenotedasM5+.J0,J1,J2,J3aretherst,second,thirdandfourthnearestneighborexchangeinteractions.Reprintedwithpermissionfrom[ 74 ]. (Figure 6-3 A)[ 5 ],evidentofBECoftripletsinthisregion.AstheeldapproachesHcorHsthisanomalybecomeslesspronouncedandthemagnitudeoftheheatcapacitydropsoffsignicantly.Inaddition,althoughtheanomalyremainsclearly-likedownto13T,at23TitstartstolookmoresymmetricsuggestingthatthephasetransitionatHsbecomesrstorder(Figure 6-3 B)[ 5 ].Highmagneticeldmagnetizationmeasurementswereperformedusingmicromechanicalforcemagnetometersina35TresistivemagnetattheNHMFLfacilityinTallahassee.A3Herefrigeratorwasloadedinthemagnetandthelowesttemperatureachievedwas0.6K.Themagnetwasnotequippedwithaseparategradientcoil.Instead,thenaturaleldgradient,causedbyasmalldisplacementofthesamplefromthemagnetcenter,wasused.Theeldgradientwas20T/mat26T. 91

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Figure6-3. HeatcapacityofBa3Cr2O8.(a)Heat-capacityasafunctionoftemperatureatconstantmagneticelds.(b)Aclose-upviewofthe13and23Theat-capacitycurves.Reprintedwithpermissionfrom[ 5 ],copyright2009bytheAmericanPhysicalSociety. Achipcontaining4micromechanicalmagnetometers1wasreleasedanddriedusingtheprocessdescribedinFabrication.Then,thechipwasmountedona20pin 1Thischip(named3M)wasfabricatedbyMEMSCAPPolyMUMPsRun77.Themovableplateofthemagnetometerwhichwasusedforthisexperiment,wassupportedby4springsoneateachcorner.Thespringslookedlikeserpentineswithoneturn,andtheirdimensionswere7032m. 92

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berglasspackage(Figure 6-4 ).Goldwireswerebondedbetweenthebondingpadsofthedeviceandgoldpadsonthepackageusingawirebonder.AsmallsinglecrystalofBa3Cr2O8(mass2(1)g)wasgluedontothemovableplateofamicromechanicalmagnetometer,withaspringconstantof14N/m,usingUVglue.Thepackagewasinsertedintoasocketlocatedatthebottomofanexperimentalprobewhichwasthenloadedintothe3Herefrigerator.Thesamplewasmountedsothatitscrystallographiccaxiswasparalleltothemagneticeld. Figure6-4. The20pinberglasspackageusedformagnetizationmeasurementsattheNHMFLfacilityinTallahassee.Goldwiresarebondedbetweenthebondingpadsofthedeviceandgoldpadsonthepackage.ThecrystalsmountedonthemagnetometersarenotBa3Cr2O8. 6.2ResultsThemagnetizationofBa3Cr2O8wasmeasuredasafunctionoftheappliedmagneticelddownto0.6K(Figure 6-5 ).Ateldslowerthanthecriticaleldthegroundstateisasingletandthemagnetizationisnearlyzero.However,asthegapclosesaroundHc12.5T,themagnetizationincreasesrapidlyandalmostlinearlyup 93

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tothesaturationeldHs23.5T.AthighertemperaturesthemagnetizationfeaturesaroundthetwotransitionsarethermallybroadenedbutthesignaturesofHcandHsarestillevidentbecausethemagnitudeofthespingap16.1Kishigherthantheexperimentaltemperatures.TheseobservationsareconsistentwiththemagnetizationbehaviorofotherspindimersystemsthatundergoBECateldshigherthanHc[ 25 72 ]. Figure6-5. MagnetizationofasinglecrystalofBa3Cr2O8asafunctionofmagneticeldatconstanttemperatures.Themagneticeldwasappliedalongthecaxis.Temperaturesare,frombottomtotop:0.6K,0.9K,2K.Eachcurvetakenattemperaturehigherthan0.6Khasbeenshiftedby0.410)]TJ /F3 7.97 Tf 6.58 0 Td[(5emufromthepreviouscurveforclarity. Figure 6-6 plotsthemagnetizationofBa3Cr2O8at0.6Kwithsweepingthemagneticeldupanddown.Themagnetichysteresisobservedinassociationwiththeupper 94

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transitionatHsisadditionalevidenceforarst-orderphasetransition,assuggestedbytheheatcapacitymeasurements(Section 6.1 ) Figure6-6. HystereticbehavioratHs.MagnetizationofBa3Cr2O8at0.6Ksweepingthemagneticeldup(blueline)anddown(redline).NearHsthemagneticresponseishysteretic. UsingthemagnetizationmeasurementsthephaseboundaryofBa3Cr2O8wasdeterminedattemperaturesupto2.3K.Thetransitionpointswerefoundbylocatingtheextremainthesecondderivativeofthemagnetizationwithrespecttothemagneticeld(Figure 6-7 ),aswasdoneinpreviousstudiesofspindimermaterials[ 80 81 ].ForconsistencywithmagnetictorquemeasurementsperformedbyourcollaboratorsDr.AdamAczelandDr.LuisBalicas,thephasediagramwasextractedusingthemagnetizationdatawiththeeldsweepingup.Figure 6-8 showsthecombinedphasediagramofBa3Cr2O8acquiredbyvariousexperimentaltechniquesincludingmicromechanicalforcemagnetometers[ 5 ].InalltheexperimentsBa3Cr2O8singlecrystalswereorientedsothatthecrystallographiccaxiswasparalleltothemagneticeld.ThephaseboundaryinthevicinityofHcwasttedbyAczelusingawindowinganalysistechnique[ 5 ]toseewhetheritobeystheuniversalpowerlawT/(H)]TJ /F6 11.955 Tf 11.96 0 Td[(Hc)2=d(Section 2.5 ).Inthetemperaturerangeof333 95

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Figure6-7. CriticaleldsdeterminationforBa3Cr2O8.(a)MagnetizationofBa3Cr2O8at0.9Kwithsweepingupthemagneticeld.ThecriticaleldextractedfromthesecondderivativeofthemagnetizationarenotedasHcandHs(b)Firstderivativeofthemagnetizationwithrespecttotheeld(b)Secondderivativeofthemagnetizationwithrespecttotheeld. 96

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Figure6-8. PhasediagramofBa3Cr2O8.Solidsquaresaremagnetocaloriceffectdata,solidcirclescomefromtoquemagnetometry,solidtrianglespointinguparefromheatcapacitymeasurementsandsolidtrianglespointingdownarefrommicromechanicalforcemagnetometrymeasurementsofthisdissertation.Reprintedwithpermissionfrom[ 5 ],copyright2009bytheAmericanPhysicalSociety. mK
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powerlaw;T/(H)]TJ /F6 11.955 Tf 11.96 0 Td[(Hc)2=d,withacriticalexponent2=d=2=3inagreementwiththe3DBECuniversalityclass.ThisprovidedtherstquantitativeevidencethatthismaterialisanewrealizationofBECoftriplets.Inaddition,electron-spinresonance(ESR)experimentsrevealedthatthegfactorofBa3Cr2O8isnearlyisotropicandslightlylowerthan2:gac=1.94andgc=1.93[ 83 ]. 6.3MagneticForceandMagnetizationCalibrationInthissectionwewilldescribetheproceduresusedinanalyzingtherawdataofthemagnetometers.Duetotorquecontributionsinthesignal,eachmagnetizationmeasurementwasperformedtwice,therstwiththesamplelocated1cmaboveandthesecond1cmbelowtheeldcenter.Thenaturaleldgradientsatthesepositionsareopposite.Asaresult,themagneticforcechangessignwhilethetorqueonthesampleremainsthesame.Therefore,theforceonthesamplecanbeobtainedbysubtractingthetwooppositeeldgradientdata. Figure6-9. Rawsignalofthemagnetometerat0.6K.Bluecurveistakenattheeldcenter,black1cmabovethecenterandred1cmbelowthecenter. 98

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Duetolimitedmagnettime,wewerenotabletoperformalinear-forcecheckinthisexperimentaswedidwithDIMPY(Section 5.4.1 ).Toconrmthatournaldataareindeedproportionalonlytotheforceonthesample,weperformedacheckusingtheavailabledata.Therawsignalofthedeviceat0.6KandatthreedifferentpositionsrelativetotheeldcenterareshowninFigure 6-9 .Attheeldcenter,wheretheeldgradientiszero,thesignalcontainsonlytorquecontributions.Thissignalfallsexactlyinbetweenthetwoothercurves,whichweretaken1cmaboveandbelowthemagnetcenter,justifyingourprocedureofsubtractingthetorquesignalfromtotalsignaltoyieldtheforce.TheforcesignalthatresultsfromthesubtractionofthetwooffcentercurvesisshowninFigure 6-10 togetherwiththetorquesignalofFigure 6-9 .Evidently,theforcesignalisalmost1.7timeslargerthanthetorquesignal,meaningthatthemagnetometertorstordercanbeconsideredasaparallelplatecapacitor.Inaddition,electrostaticcalibrationofthemagnetometerensuresthatevenatthehighestmagneticeldthedeviceresponseislinear.Therefore,themaximumdisplacementofthedeviceissmallcomparedtotheinitialgapbetweentheplates,whichisinsupportoftheparallelplatehypothesis.Themagneticforceresolutionachievedusingthemicromechanicalmagnetometerwas610)]TJ /F3 7.97 Tf 6.58 0 Td[(10N/p Hz.Thevalueisrelatedtothemagneticmomentresolutionthroughtheeldgradient(Eq. 4 ).Duetothenatureoftheeldgradientinthisexperimentthegradientisnotxedbutdependsonthevalueoftheeldthemagneticmomentresolutionvariesovertheeldrange.At26Twiththenaturaleldgradientof20T/mthemagneticmomentresolutionis310)]TJ /F3 7.97 Tf 6.59 0 Td[(8emu/p Hz,whichdecreaseslinearlywithdecreasingeld. 6.4ConclusionsWehavemeasuredthemagnetizationofBa3Cr2O8asafunctionofthemagneticelddownto0.6K.OurmeasurementsrevealthatthemagneticbehaviorofthiscompoundisconsistentwithBECoftripletsateldsbetweenHcandHs.Thehysteresis 99

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Figure6-10. Forceandtorquesignalsofthemagnetometerat0.6K.Thetorquecontributionsare1.7timessmallerthantheforce. inthemagnetizationinthevicinityofHssupportsthesuggestiongivenbyheatcapacitydatathatthephasetransitionatHsisrstorder[ 5 ].UsingthemagnetizationdatainconjunctionwithotherexperimentalmethodsthephaseboundaryofBa3Cr2O8wasmappedattemperaturesupto2.3K.ThehightemperaturephaseboundarynearHcisnotconsistentwithuniversal3DBECbehaviorduetothesmallnumberofdatapointsintheuniversalregimeanddatascatter.However,recentlowtemperatureinelastic-neutronscatteringandheat-capacitymeasurementsbyadifferentgroup[ 83 ]haveconrmedthatthelowtemperaturephaseboundaryisinagreementwiththe3DBECuniversalityclass. 100

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CHAPTER7SUMMARYInthisdissertationwedevelopednewmagnetizationmeasurementdevicesthathavebeenusedsuccessfullyforphysicsandmaterialsscienceexperimentsathighmagneticeldsandlowtemperatures.InChapter 2 ,weintroducedthemagneticmomentanddescribedtheinteractionofamagneticmomentwithamagneticeld.Inaddition,weexaminedthetheoreticalbackgroundofquantummagnets.Thedispersionrelationsofthemagnons,theelementaryexcitationsinmagneticsolids,werederivedusingasemiclassicalapproximation.WealsoreviewedthepredictionsofHaldaneforintegerspinchains,andthephysicsofaspindimer.Finally,weintroducedtheconceptsofBECofmagnonsandTLL.InChapter 3 ,wedescribedtheprinciplesofoperationoftheacsusceptometer,theVSM,theSQUID,thecantilevermagnetometer,andtheFaradaybalancemagnetometer.Inadditionwereviewedtheadvantagesanddrawbacksofeachdevice.ThedevelopmentandoperationofthemicromechanicalforcemagnetometersweredescribedinChapter 4 .Themagnetometersaremadeusingasiliconsurfacemicromachiningprocess.Theirsmallsizeandsymmetricdesignmakesthemidealformagnetizationmeasurementsoftinysamplesathigheldsandlowtemperatures.Thesedevicesofferhighresolutions,inavarietyofexperimentalconditions,aswellasdesignexibilityandcostefcientfabrication.Theadvantagesofthemicromechanicalmagnetometersoverconventionalmagnetizationmeasurementdeviceswereanalyzed.InChapter 5 magnetizationmeasurementsonDIMPYwerepresented.OurmeasurementsconrmedthatDIMPYisanideal1DsystemwithaTLLgaplessphaseabovethecriticaleldHcatlowtemperatures.Inaddition,wesawthatthemagnetizationintheTLLphaseexhibitsaminimumatTm.ThedependenceofTmonthemagneticeldisingoodagreementwithQMCsimulations.Furthermore,themaximumappearinginthelowtemperaturesusceptibilitynearHcisconsistentwiththe 101

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squarerootdivergenceexpectedforagapped1Dantiferromagnetatzerotemperature.Finally,thelowtemperaturesusceptibilityinconjunctionwithspecic-heatdataenabledsuccessfuldeterminationoftheWilsonratio.ComparisonoftheexperimentalWilsonratiowithdensity-matrixrenormalization-groupcalculationsoftheTLLparameterKmakesDIMPYtherstlaboratory1DsysteminwhichtherelationRW=4Kisveriedwithinthecombineduncertaintiesofexperimentandcalculations.ThemagnetizationmeasurementsofBa3Cr2O8asafunctionofthemagneticeldwerepresentedinChapter 6 .OurmeasurementsrevealedthatthemagneticbehaviorofthiscompoundisconsistentwithBECoftripletsateldsbetweenHcandHs.ThehysteresisinthemagnetizationatthevicinityofHssupportsthesuggestiongivenbyheatcapacitydatathatthephasetransitionatHsisrstorder.UsingthemagnetizationdatainconjunctionwithotherexperimentalmethodsthephaseboundaryofBa3Cr2O8wasmappedattemperaturesupto2.3K.ThehightemperaturephaseboundarynearHcisnotconsistentwithuniversal3DBECbehaviorduetothesmallnumberofdatapointsintheuniversalregimeanddatascatter.However,recentlowtemperatureinelastic-neutronscatteringandheat-capacitymeasurementsbyadifferentgrouphaveconrmedthatthelowtemperaturephaseboundaryisinagreementwiththe3DBECuniversalityclass. 102

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APPENDIXAEXPERIMENTINPULSEDMAGNETICFIELDSWehaveusedmicromechanicalmagnetometerstomeasurethemagnetizationperformulaunitofSmFeAsOatthepulsedeldfacilityoftheNHMFLinLosAlamos.Magnetizationmeasurementsinpulsedeldsarechallengingduetotheshortdurationofthepulsesandthemechanicalnoisegeneratedduringthepulse.Themicromechanicalmagnetometersaresuitedtobeusedinsuchenvironmentsbecausetheirhighresonancefrequencies,usuallyhigherthan15kHz,yieldfastmechanicalresponseandmakethemagnetometersrobustagainstvibrationsandmechanicalnoisewhichusuallyoccuratlowerfrequencies.SmFeAsO1)]TJ /F3 7.97 Tf 6.59 0 Td[(xFyisahightemperaturesuperconductorinwhichthesuperconductivitycoexistswithlongrangeantiferromagneticorder[ 84 ].ThisremarkablecoexistencemakesthestudyofthelowtemperatureantiferromagneticstateofSmFeAsO1)]TJ /F3 7.97 Tf 6.59 0 Td[(xFyanditsundopednonsuperconductingcounterpartSmFeAsOveryinteresting.MagnetictorquemeasurementsonSmFeAsOrevealevidenceforametamagnetictransitionwhichmaycorrespondtoeitheragradualspinreorientationoradiscontinuousspincanting[ 6 ].WemountedasmallsinglecrystalofSmFeAsOontothemovableplateofamicromechanicalmagnetometer1withaspringconstantof56N/m.Thedevicewasloadedintoa3Hecryostatplacedina65Tshortpulsecapacitor-drivenmagnetwithapulsedurationofapproximately25ms(Figure A-1 ).Inaddition,itwaspositioned1.9cmabovetheeldcenterwherethepeakeldis62.6Tandtheeldgradientis270T/m.Thecrystallographiccaxisofthesamplewasparalleltothemagneticeld. 1Thisdevicebelongedtoachip(named50M)whichwasfabricatedbyMEMSCAPPolyMUMPsRun88.Themovableplateofthemagnetometerwassupportedby4springsoneateachcorner.Thespringslookedlikeserpentineswithoneturn,andtheirdimensionswere3742m. 103

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FigureA-1. Thetimeproleofthemagneticeldinashotofthe65TshortpulsemagnetatNHMFLinLosAlamos. Themagnetizationwasmeasuredbythechangeincapacitanceofthemicromechanicalmagnetometerusingastandardcapacitancebridge.Beforethepulse,thebridgeoutputwastunedtozero.Duringthepulsethecapacitanceofthedevicechangedduetothemagneticforceonthesampleandthebridgeoutputwasmeasuredusingalock-inamplierconnectedtoadigitizer.Inthisexperimenttherewerenotorquecontributionsinthesignalandtherawsignalwasdirectlyproportionaltothemagneticforceonthesample.Themagneticforcewascalibratedusingtheelectrostaticforce.Figure A-2 Aplotsthemagneticforceonthesampleasafunctionofthemagneticeldatdifferenttemperatures.Themassofthesamplewasestimatedbytheshiftoftheresonancefrequencyofthedeviceatroomtemperatureandwasfoundtobe2(1)g.Themagnetizationperformulaunitwasextractedfromthemagneticforce,keepinginmindthattheeldgradientinthisexperimentwasthenaturaleldgradientofthemagnetanddependedontheinstantaneousvalueoftheeld. 104

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FigureA-2. SmFeAsOresponseinapulsedmanget.(a)ThemagneticforceonasinglecrystalofSmFeAsOasafunctionofthemagneticeld.(b)Themagnetizationperunitformulaasafunctionofthemagneticeld.Thevalueofthesaturationmagnetizationcontainsanuncertaintyof50%duetotheuncertaintyinthemassofthesample. Figure A-2 BplotsthemagnetizationperunitformulaofSmFeAsO.Ajumpinthemagnetizationisclearlyobservedat35Tand0.6K.ThisjumpisattributedtoareorientationofthemagneticmomentsofSmat35T,inagreementwithmeasurementsperformedbytorquemagnetometry[ 6 ].Thesaturationvalueofthemagnetizationisonly0.06(3)BandissmallcomparedwiththefullSmmagneticmomentreportedintheliterature0.4-0.6B[ 85 86 ].Thereforeonlypartialspinreorientationoccursat35Tandmuchgreatereldsarerequiredtofullysuppresstheantiferromagneticorder[ 6 ]. 105

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Thisresultisinagreementwithmuonspinrotation(SR)resultsforSmFeAsO,whichsuggestacomplexmagneticstructureofthesublatticeofSmmagneticmoments[ 6 ]. 106

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APPENDIXBFINITEELEMENTANALYSISDuringthisresearch,wehavedesignedseveraldifferentmagnetometers.Thedesignsweresimulatedusingniteelementanalysissoftware(COMSOL),beforefabricationoftheactualdevices.AnewdevicethatissuitableforanisotropicsamplesisshowninFigure B-1 .Thisdeviceusesdifferentplatesformountingthesampleandfordetection.Theplateonwhichthesampleismountedisdesignedtohaveverysofttorsionalsprings.Thisdesignallowstheplatetoturnsothattheeasymagnetizationaxisofthesamplealignswiththemagneticeld.Asaresult,onlythemagneticforceonthesampleismeasuredbythedetectionplates.Figure B-2 plotsthetopplateofamagnetometersubjectedtoapointforce(2.5N)atthecenter.ThissimulationwasperformedusingCOMSOL.Springsinthisdesignlooklikeserpentineswithonlyoneturnandtheirdimensionsare7032m.Thepointforceatthecenterofthetopplateisvetimeslargerthanthelargestmagneticforceinanyofourexperiments(0.5N).Simulationresultsshowthatthedeformationatthecenterofthemovableplateisonly1.5%oftheinitialseparationbetweenthetwoplatesthedeformationatthecenterofthetopplateis0.03m,whiletheinitialgapis2mensuringthatthedevicebehavesasaparallelplatecapacitorevenwhenahugepointforceisappliedtothemovableplate. 107

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FigureB-1. Magnetometersuitableforanisotropicsamples.(a)COMSOLsimulation.Theplateonwhichthesampleismountedissubjectedtoatorque.Inthisdesignthemiddleplateturnseasilywithouttiltingthedetectionplates.(b)Scanningelectronmicrographofthedevice.Topright:thesofttorsionalspring. 108

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FigureB-2. COMSOLsimulationofthemovableplateofatypicalmagnetometersubjectedtoa2.5Npointforceatthecenter.Thedeformationatthecenteroftheplateisabout1.5%oftheinitialplateseparation(2m). 109

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APPENDIXCSAMPLEPREPARATIONPROCEDUREThefollowingproceduresareperformedonchipsthathavebeenalreadydiced.1.Pre-etchclean.(a)SoakthesampleinAcetonefor5min.TransferitintoIPAandblowdrywithN2.(b)O2dryetchusingUnaxisICPetchertoremoveorganicresidues.Theetchingparametersare:O2=60sccm,Pressure=10mT,RF1=100W,RF2=300W,Time=10min.2.Oxideetch-Release.(a)Soakthesamplein49%HFsolution.Forchipsnamed"lessholes"etchingtimeis6minand30s.Forallotherchipstheetchingtimeis5minand30s.(b)Afteretchrinsethesamplefor15mininDIwater. 110

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BIOGRAPHICALSKETCH KonstantinosNinioswasborninPatras,Greece,nearthehistoricsitewheretherstOlympicgamestookplace,AncientOlympia.HegraduatedfromtheUniversityofPatraswithabachelor'sdegreein2004.DuringhissenioryearattheUniversityofPatrashedecidedtocontinuehisstudiesandpursueaPh.D.inphysics.HecametotheUniversityofFloridainthefallof2005andhejoinedthegroupofProf.HoBunChaninthesummerof2006.Hegraduatedinthefallof2011withaPh.D.inphysics. 116





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AMERICAN PHYSICAL SOCIETY One Physics Ellipse, College Park, MD 20740 http://www.aps.org November 22, 2011 Konstantinos Ninios Department of Physics, University of Florida, P.O Box 118440, Gainesville, FL 32611 Ref # 10828 Thank you for your permission request dated on November 17 2011 W e are pleased to grant you a non exclusive, non transferable permission, English and German rights limited to print and electronic format provided you meet the criteria outlined below. Permission is for a one time use and does not include permission for future editions updates, databases, translations, or any other matters. Permission must be sought for each additional use. This permission does not include the right to modify APS material. Please print the required copyright credit line on the first page that the material appe ars: Reprinted (abstract/excerpt/figure) with permission from [FULL REFERENCE CITATION] as follows: authors names, journal title, volume number, page number and year of publication. Copyright (YEAR) by the American Physical Society. The following lang uage must appear somewhere on the website: Readers may view, browse, and/or download material for temporary copying purposes only, provided these uses are for noncommercial personal purposes. Except as provided by law, this material may not be further r eproduced, distributed, transmitted, modified, adapted, performed, displayed, published, or sold in whole or part, without p rior written permission from the American Physical Society Provide a hyperlink from the reprinted APS material ( the hyperlink may be embedded in the copyright credit line). APSs link manager technology makes it convenient and easy to provide links to individual articles in APS journals. For information, see: http://link.aps.org/ You must also o btain permission from at least one of the authors for each separate work, if you havent done so already. The authors name and address can be found on the first page of the published Article. Use of the APS material must not imply any endorsement by the American Physical Society. Permission is granted for use of the following APS material only Fig. 3(c,d), 4, Phys. Rev. B Vol. 79, 100409 (2009) F ig. 3, Phys. Rev. Lett. Vol. 105, 137207 (2010) Permission is limited to the single title speci fied or single edition of the publication as follows: A dissertation entitled "MICROMECHANICAL FORCE MAGNETOMETERS FOR MEASURING MAGNETIZATION AT HIGH MAGNETIC FIELDS AND LOW TEMPERATURES" to be published by Konstantinos Ninios If you have any questions, please refer to the Copyright FAQ at: http://publish.aps.org/copyrightFAQ.html or send an email to H assocpub@aps.org Sincerely, Eile en LaManca Publications Marketing Coordinator



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