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Experimental Studies of Low Dimensional Quantum Antiferromagnets

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Title:
Experimental Studies of Low Dimensional Quantum Antiferromagnets
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Aoyama, Christopher P
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[Gainesville, Fla.]
Florida
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University of Florida
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english
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1 online resource (96 p.)

Thesis/Dissertation Information

Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Physics
Committee Chair:
TAKANO,YASUMASA
Committee Co-Chair:
ANDRAKA,BOHDAN
Committee Members:
BISWAS,AMLAN
DETWEILER,STEVEN L
CHRISTOU,GEORGE
Graduation Date:
5/2/2015

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Subjects / Keywords:
Capacitance ( jstor )
Electric fields ( jstor )
Electrodes ( jstor )
Magnetic fields ( jstor )
Magnetism ( jstor )
Magnetization ( jstor )
Magnetometers ( jstor )
Magnets ( jstor )
Specific heat ( jstor )
Torque ( jstor )
Physics -- Dissertations, Academic -- UF
antiferromagnets
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bibliography ( marcgt )
theses ( marcgt )
government publication (state, provincial, terriorial, dependent) ( marcgt )
born-digital ( sobekcm )
Electronic Thesis or Dissertation
Physics thesis, Ph.D.

Notes

Abstract:
Quantum magnets in low dimensions lead to some of the most interesting physics in condensed matter physics. The study of these systems requires measurements at low temperatures, below $100$\,mK, and high magnetic fields up to $20$\,T. Magnetization measurements in this parameter space are often limited in one way or another. Advancements in broadening the capabilities of these measurements can only open new doors for future research. This dissertation features the development of a new magnetization instrument, a force magnetometer that surpasses current devices in resolution and its ability to operate in preferable experimental conditions. Using the new force magnetometer we have measured the magnetization of a spin-$\frac{1}{2}$ linear-chain Heisenberg antiferromagnet, (CuPzN) and coupled two-leg spin-$\frac{1}{2}$ ladder (DCLB) at temperatures down to $20$\,mK. Studying low-dimensional antiferromagnets with small spin quantum numbers is important because quantum fluctuations are significant, leading to novel effects that are of fundamental and technical interest. Exactly solvable models of these systems are rare, yet they occupy a special place in physics. These simple models offer the foundations for understanding more complex, realistic models which require approximations to study. One of the simplest systems is the Heisenberg spin-$\frac{1}{2}$ antiferromagnetic linear chain, which an exact solution for the system has been found. [Cu(C$_{4}$H$_{4}$N$_{2}$)(NO$_{3}$)$_{2}$] (CuPzN) is an example of such a system. Accurate magnetization and specific heat measurements of this material allow us to find the two parameters that completely characterize the Tomonaga-Luttinger Liquid(TLL) phase, in which low energy magnetic excitation are spinons, whose fermionic character dictates the thermodynamic properties. The optimal way to obtain the parameters to characterize all the dynamic properties of the system is to obtain the Wilson Ratio. Using Bethe-ansatz calculations and these accurate measurements we compare a real material, CuPzN to the solved model for a Heisenberg spin-$\frac{1}{2}$ linear chain for the first time. Another important system is the coupled two-leg spin-$\frac{1}{2}$ ladder. (Dimethylammonium) ($3,5-$dimethylpyridinium)CuBr$_{4}$ (DCLB) is an example of such a system that has a 3-dimensional long-range magnetic order that coexists with a spin energy gap. Magnetization measurements provide part or the measurements used to construct the phase diagram that displays a spin-flop transition, the saturation field, and an Ising anisotropy of the system. The ratio $J_{int}/J_{leg}$ of the interladder exchange, $J_{int}$, to the leg exchange, $J_{leg}$ indicates that DLCB is very close to a quantum critical point at which the LRO vanishes. These characteristics demonstrate agreement with the two-leg spin-$\frac{1}{2}$ ladder model predictions. Spin triangles in single-molecule magnets have been a system of interest due to quantum effects such as spin electric couplings, frustrated magnetism and multiferroic behavior. Through specific heat, magnetocaloric effect, magnetic torque, dielectric, and magnetization measurements we have revealed multiferroic behavior in Tris[4-(1-oxyl-3-oxide-4,4,5,5-tetramethylimidazolin-2-yl)phenyl]amine (TNN$\bullet$CH$_3$CN), a molecular spin triangle system. Never had this behavior been observed in such a system, although theory had predicted it. The data from these various measurements were used to construct the phase diagram up to the saturation field. \end{abstract} ( en )
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In the series University of Florida Digital Collections.
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Includes vita.
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Includes bibliographical references.
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Description based on online resource; title from PDF title page.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Thesis:
Thesis (Ph.D.)--University of Florida, 2015.
Local:
Adviser: TAKANO,YASUMASA.
Local:
Co-adviser: ANDRAKA,BOHDAN.
Electronic Access:
RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2017-05-31
Statement of Responsibility:
by Christopher P Aoyama.

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Embargo Date:
5/31/2017
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LD1780 2015 ( lcc )

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EXPERIMENTALSTUDIESOFLOWDIMENSIONALQUANTUMANTIFERROMAGNETSByCHRISTOPHERP.AOYAMAADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2015

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c2015ChristopherP.Aoyama

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Idedicatethistomygodmother.

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ACKNOWLEDGMENTS First,IwouldliketothankmyPh.D.adviserDr.YasumasaTakano.Hisunendingpatienceandguidancehasledmethroughamyriadofchallengingandrewardingexperiences.Hehasbeenconsistentlyadeepwellofinformationandwisdom,whichhasimpactedwhoIamandhowIwork.WithouthismotivationandguidanceIcouldnothaveaccomplishedanyofthis.Iwouldalsoliketothankmysupervisorycommitteemembers,ProfessorsAmlanBiswas,StevenDetweiler,BohdanAndrakaandGeorgeChristou,fortheirinsightsandencouragement.Alongthispath,IhavehadtheprivilegeofworkingwithmanytalentedanddedicatedpeoplewhomIwouldliketoshowappreciation.SpecialthankstoTaoHong,EunSangChoi,MinseongLee,HaidongZhou,ScottHannahsandNatFortunefortheircollaborationandworkattheNationalHighMagneticFieldLaboratory(NHMFL)runningexperiments.ThanksarealsoduetoJu-HyunPark,GloverJonesandTimMurphy,whohavebeenatremendouspartofourtimeattheNHMFLMillikelvinLaboratory.Theyalwayskeptexperimentsrunningsmoothlyandproductively.Totheguysinthemachineshop,EdStorchandMarcLink,whospenttheirtimeandpatienceonmeandmyproject,Ioweadebtofgratitude.Thankyoutoallcollaboratorsabroadwhoseexpertiseandhardworkcomplementedsowellwhatwehaveaccomplished.TheyareToshiroSakakibara,YoheiKono,YasuyukiShimura,ChisaHotta,YukoHosokoshi,KosukeTakada,SeitaroIisaka,AyakaHigashiguchi,KosukeTakada,HironoriYamaguchi,NaokiAmaya,andFirasAwwadi.Thankyoualsotoourdomesticcollaboratorswhoprovidedsamplesforourresearch,ChrisLandee,MarkTurnbull,andShenLiQiu.IwouldalsoliketothankDr.RobertDeSerioandCharlesParksforthepleasureofteachingundergraduatelabsalongsidethem.Toallthoseinthedepartment,theuniversity,andtheNHMFL,thesupportandservicesyouhaveprovidedhavebeeninvaluable.Finally,tomyfamily,friends,Heather,andCaptain,yourlove,support,andunderstandingareappreciatedwithallmyheart.IapologizetothoseIhaveneglectedforthesepastyears,butyour 4

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patiencehasbeenvaluedaboveallelse.WithouttheknowledgeofyourencouragementanddevotionIwouldnothavecompletedthisquest. 5

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 13 2THEORETICALBACKGROUND .......................... 15 2.1PropertiesofMagneticMoments ....................... 15 2.2MagnonExcitations .............................. 16 2.2.1FerromagneticMagnons ........................ 16 2.2.2AntiferromagneticMagnons ...................... 17 2.3Spin-1 2LinearChains .............................. 18 2.3.1Spinons ................................. 18 2.3.2Tomonaga-LuttingerLiquid ....................... 18 2.4SpinLadders .................................. 23 2.5SpinTriangles .................................. 24 3EXPERIMENTALTECHNIQUES .......................... 32 3.1Calorimetry ................................... 32 3.1.1RelaxationCalorimetry ......................... 32 3.1.2MagnetocaloricEffect ......................... 36 3.2MagneticTorque ................................ 37 3.3MagnetizationMeasurements ......................... 38 3.3.1SQUIDMagnetometry ......................... 38 3.3.2ForceMagnetometry .......................... 39 3.4DielectricMeasurements ........................... 40 4DEVELOPMENTOFLOW-TEMPERATUREFORCEMAGNETOMETERS ... 42 4.1ComparisonwithOtherMethods ....................... 42 4.1.1VSM ................................... 42 4.1.2SQUIDMagnetometers ........................ 43 4.1.3CapacitiveCantileverMagnetometers ................ 43 4.1.4InductionMethod ............................ 44 4.1.5ForceMagnetometers ......................... 44 4.2DesignofForceMagnetometers ....................... 45 4.3FabricationofaPrototype ........................... 47 4.4ReducingtheBackgroundMagnetization .................. 49 4.5Field-proleMeasurements .......................... 50 6

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4.6MagnetometerCalibration ........................... 52 5COPPERPYRAZINEDINITRATE ......................... 54 5.1PreviousWorkonCuPzN ........................... 54 5.2Experimental .................................. 57 5.2.1SQUIDMagnetometry ......................... 57 5.2.2Calorimetry ............................... 58 5.2.3ForceMagnetometry .......................... 58 5.3Results ..................................... 58 5.3.1SpecicHeat .............................. 60 5.3.2SpinonVelocity ............................. 63 5.3.3Magnetization .............................. 64 5.3.4WilsonRatio ............................... 66 5.4Discussion ................................... 66 6DLCB ......................................... 68 6.1PreviousWorkonDLCB ............................ 68 6.2Experimental .................................. 69 6.2.1Calorimetry ............................... 70 6.2.2MagnetizationMeasurements ..................... 70 6.2.3MagneticTorque ............................ 71 6.3Results ..................................... 71 6.4Discussion ................................... 76 7TNNCH3CN ..................................... 78 7.1Experimental .................................. 80 7.1.1SpecicHeat .............................. 80 7.1.2MagnetizationMeasurements ..................... 80 7.1.3DielectricMeasurements ........................ 81 7.2Results ..................................... 81 7.3Discussion ................................... 87 8CONCLUSIONS ................................... 90 REFERENCES ....................................... 92 BIOGRAPHICALSKETCH ................................ 96 7

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LISTOFFIGURES Figure page 2-1Spinwave ....................................... 17 2-2Dispersionrelationformagnonexcitations ..................... 17 2-3Dispersionrelationforspinonexcitations ...................... 19 2-4SpinonvelocityandTLLparameterofthespin-1 2linear-chainHAFasafunctionoftheappliedeld .................................. 20 2-5Specicheatofaspin-1 2linear-chainHAFatzeroeld .............. 21 2-6Specicheatofspin-1 2linear-chainHAFfordifferentmagneticelds ...... 22 2-7Magneticsusceptibilityofspin-1 2linearchainHAFfordifferentmagneticelds 22 2-8Coupledspinladders ................................ 23 2-9Energygapofaspin-1 2ladder ............................ 24 2-10TLLparameterKforspin-1 2Heisenbergladders .................. 25 2-11Spintrianglewithequalinteractions ........................ 26 2-12Energyofthespin-1 2spintrianglewiththemagneticmomentofasinglespintriangleasafunctionofeldandmagnetizationasafunctionofeldandaspintriangleinteractingwithneighbors ....................... 27 2-13Magneticstatewithnonzeroelectricpolarization ................. 28 2-14Groundstateswithnonzeroelectriccurrent .................... 29 2-15Triangularlatticeofspintriangles .......................... 30 2-16Multiferroicphasediagramforaspin-trianglesystem ............... 31 3-1Schematicofthecalorimeterusedforspecic-heatandmagnetocaloric-effectmeasurements .................................... 33 3-2Temperaturerelaxationcurvesforrelaxationcalorimetry ............. 35 3-3Capacitivecantilevertorquemagnetometer .................... 37 3-4Silver-paintelectrodeonasinglecrystalofTNNCH3CN ............. 41 4-1Crosssectionalviewofforcemagnetometer .................... 45 4-2Ourgroup'smagnetometer ............................. 47 4-3Prototypetestforresponseofcapacitancetochangeinforce .......... 48 8

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4-4Comparisonofmaterialsconsideredfortheconstructionofthesuspendedstructureofthemagnetometers ........................... 50 4-5Magnetizationmeasurementsmadeforbackgroundsubtraction ......... 51 4-6FieldprolemeasuredbyaHallsensorovertherangeofmotionforthesampleposition ........................................ 52 4-7Resultsofmagnetizationmeasurementsmadeonnickeldisksforcalibration 53 5-1CrystalstructureofCuPzN ............................. 55 5-2SpecicheatofhydrogenousCuPzNmeasuredbyHammaretal. ....... 56 5-3MagnetizationofCuPzN ............................... 56 5-4MagneticinelasticneutronscatteringintensityofCuPzN ............. 57 5-5SQUID-magnetizationmeasurementofCuPzN .................. 59 5-6SpecicheatofCuPzNwithphononcontribution ................. 60 5-7MagneticentropyofCuPzNasafunctionoftemperature ............. 61 5-8C/TvsTofCuPzN ................................. 62 5-9MagneticspecicheatofCuPzNplottedasC/TvsT .............. 63 5-10MagnetizationanddifferentialsusceptibilityofCuPzN .............. 64 5-11SpinonvelocityinCuPzNasafunctionoftheappliedeld ............ 65 5-12TemperaturedependenceofM/HofCuPzN .................... 66 5-13WilsonratioofCuPzN ................................ 67 6-1LadderchainsthatcompriseDLCB ......................... 69 6-2SpecicheatasafunctionoftemperatureofDLCB ................ 72 6-3SpecicheatmeasurementswithrespecttomagneticeldofDLCB ...... 72 6-4DLCBmagnetocaloric-effectdata .......................... 73 6-5MagnetizationfordifferentcrystalorientationsofDLCB ............. 74 6-6Magnetictorqueasafunctionofthemagneticeld ................ 75 6-7SaturationeldofDLCBdeterminedbytorquemeasurements ......... 76 6-8PhasediagramsforDLCB .............................. 77 7-1StructureoftheTNNmolecule. ........................... 79 9

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7-2TNNCH3CNcrystalstructure ............................ 79 7-3DielectricresponseofTNNCH3CNasafunctionoftemperature ........ 82 7-4MagnetizationofTNNCH3CN ............................ 83 7-5MagnetizationovereldofTNNCH3CNasafunctionoftemperature ...... 84 7-6SpecicheatofTNNCH3CN ............................ 84 7-7ChangeindielectricresponseasafunctionoftemperatureforTNNCH3CN .. 85 7-8ChangeindielectricresponseasafunctionofmagneticeldforTNNCH3CN 86 7-9PhasediagramofTNNCH3CN ........................... 87 7-10MagneticentropyofTNNCH3CN .......................... 89 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyEXPERIMENTALSTUDIESOFLOWDIMENSIONALQUANTUMANTIFERROMAGNETSByChristopherP.AoyamaMay2015Chair:YasumasaTakanoMajor:PhysicsQuantummagnetsinlowdimensionsleadtosomeofthemostinterestingphysicsincondensedmatterphysics.Thestudyofthesesystemsrequiresmeasurementsatlowtemperatures,oftenbelow100mK,andhighmagneticeldsupto20T.Magnetizationmeasurementsinthisparameterspaceareoftenlimitedinonewayoranother.Broadeningthecapabilitiesofthesemeasurementscanonlyopennewdoorsforfutureresearch.Thisdissertationfeaturesthedevelopmentofnewmagnetization-measuringinstruments,forcemagnetometersthatsurpassconventionalmagnetometersintheirabilitytooperateatlowertemperaturesandinhighermagneticelds.Usingthenewforcemagnetometers,wehavemeasuredthemagnetizationofaspin-1 2linear-chainHeisenbergantiferromagnet,andasystemofcoupledspin-1 2laddersattemperaturesdownto25mKand0.23K,respectively.Exactlysolvablemodelsoflow-dimensionalquantummagnetsarerare,yettheyoccupyaspecialplaceinphysics.Thesemodelsofferthefoundationsforunderstandingmodelsthatrequireapproximationstostudy.Onesuchmodelisthespin-1 2linear-chainHeisenbergantiferromagnet,ofwhichCu(C4H4N2)(NO3)2,hereafterCuPzN,isanexample.Accuratespecic-heatmeasurementsofthismaterial,inconjunctionwithaccuratemagnetizationmeasurements,allowustodeterminethetwoparametersthatcompletelycharacterizetheTomonaga-Luttingerliquidphase,inwhichthefermionic 11

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characterofspinonsdictatesthethermodynamicproperties.OurresultsareinexcellentagreementwithexactcalculationsbasedontheBetheansatz.Anotherimportantsystemisanarrayofcoupledspin-1 2ladders.(Dimethylammonium)(3,5)]TJ /F1 11.955 Tf 9.3 0 Td[(dimethylpyridinium)CuBr4,DLCBforshort,isanexampleofsuchasystem.Inthismaterial,athree-dimensionallong-rangemagneticordercoexistswithaspinenergygap.Magnetization,magnetic-torque,specic-heat,andmagnetocaloric-effectmeasurementswereusedtoconstructthephasediagram,whichdisplaysaspin-optransition,saturation,andanIsinganisotropyofthesystem.TheratioJ0/Jkoftheinterladderexchange,J0,tothelegexchange,Jk,indicatesthatDLCBisveryclosetoaquantumcriticalpointatwhichthelong-rangeordervanishes.Spintriangleshavebeenasystemofinterestduetoquantumeffects,geometricfrustration,andmultiferroicbehavior.Throughspecic-heat,magnetocaloric-effect,magnetic-torque,magnetization,anddielectricmeasurements,wehaverevealedmultiferroicbehaviorinthespintrianglesystemTNNCH3CN,whichTNNstandsfortris[4-(1-oxyl-3-oxide-4,4,5,5-tetramethylimidazolin-2-yl)phenyl]amine.Neverhadthisbehaviorbeenobservedinsuchasystem,althoughtheoryhadpredictedit.Thedatafromthemeasurementswereusedtoconstructthephasediagramuptothesaturationeld. 12

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CHAPTER1INTRODUCTIONLow-dimensionalantiferromagnetswithsmallspinquantumnumbersareattractingmuchinterest,becausequantumuctuationsaresignicantinthesesystems,leadingtonoveleffectsthatareoffundamental,andsometimestechnical,importance.Inlowdimensionspronounceductuationsarise,becauseindividualspinshaveonlyafewneighbors.Forcemagnetometryatlowtemperatureandinhighmagneticeld,inconjunctionwithspecic-heatmeasurements,willimproveourunderstandingofsuchquantummagnets.Forthisreason,Ihavedevelopednewforcemagnetometersforhigh-sensitivitymeasurementsatlowtemperatures,downto25mK,ineldsupto20Tinasuperconductingmagnetordownto0.4Kandupto31Tinaresistivemagnet.Theresultsofthiswork,incollaborationwithToshiroSakakibaraattheInstituteforSolidStatePhysicsoftheUniversityofTokyo,composeapartofthisdissertation.Althoughtheystillrequireimprovements,thesemagnetometerswillallowustoperformaccuratemagnetizationmeasurementsonlow-dimensionalantiferromagnetsthatpreviouslycouldnotbedoneinthiscountryinthetemperatureandeldregionsofourinterest.Inaddition,thisdissertationdescribesthestudyofthreelow-dimensionalantiferromagnets.TherstisCuPzN,aone-dimensional(1D)spin-1 2Heisenbergantiferromagnet.Supplementedbymagnetizationmeasurements,specic-heatmeasurementsonthisquantummagnetareexpectedtoleadtobetterunderstandingofquantumuctuationsin1Dquantumsystems.Unfortunately,thesensitivityofourmagnetometerswasinsufcientforthisstudy,whichrequiredprecisionaswellasaccuracy,andthemagnetizationmeasurementswereperformedinsteadbyYoheiKonoinSakakibara'slaboratory.ThesecondisDLCB,aspin-1 2laddercompound.Spinladderspossessanenergygapformagnonssometimescalledtriplonsiftherunginteractionisantiferromagnetic,asinthecaseofDLCB.Asaresult,suchspinladdersdonotorder 13

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exceptathighmagneticelds,strongenoughtodestroythegap.However,DLCBordersevenatzeroeld,becausethestrengthoftheinterladderinteractionexceeds,albeitslightly,thecriticalvalueforlong-rangeordering.Despitetheordering,magnonsaregappedbecauseofaweakIsinganisotropy,whichalsohelpstheordering.ThethirdmaterialisTNNCH3CN,inwhichthemolecularspintrianglesTNNformatriangularlattice.Recenttheorypredictsthatsuchasystemexhibitsmultiferroicbehavior.Unlikebetterknownspintriangles,TNNallowsustotestthisprediction,becausethespinsinthismoleculearecarriedbyorganicradicals,not3dions.Consequently,Jahn-Tellerdistortionisabsent,keepingthestrengthsoftheexchangeinteractionsequalbetweenthespins.Wehaveconductedspecic-heatanddielectricmeasurementsonTNNCH3CNandrevealeditsmultiferroicbehavior.BysupplementingtheresultsofthesemeasurementswiththemagnetizationdataobtainedinSakakibara'slaboratoryandwithmagnetic-torqueandmagnetocaloric-effectdatatakenearlierbyourgroup,wehaveconstructedaphasediagramofthesystemuptothesaturationeld. 14

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CHAPTER2THEORETICALBACKGROUND 2.1PropertiesofMagneticMomentsThefundamentalconstituentinmagnetismisthemagneticmoment.ThesimplestconceptualizationofthisistheclassicalpictureofacurrentloopwithcurrentiandanareaA.Themagneticmomentisthendenedas ~m=i~A,(2)wherevector~Aisnormaltotheplaneofthecurrentloop.Inamagneticeldthemagneticmomentexperiencesatorque ~=~m~H,(2)where~Histheeld.Asaresultofthistorque,themagneticmomenttendstoalignitselfwiththeeld.Associatedwiththetorqueistheenergy E=)]TJ /F8 11.955 Tf 10.44 .5 Td[(~m~H.(2)Microscopically,themagneticmomentofanionisconnectedtoitsspinthroughtherelation ~m=gB~S,(2)wheregisthegfactorwhichhasavaluecloseto2,BistheBohrmagnetonand~Sisthespin.Sincethespinisquantized,themagneticmomentisalsoquantized.Asaresult,theenergygivenbyEq. 2 isdiscrete.ThissplittingofanenergylevelcalledZeemansplittingplaysacrucialroleinthebehaviorofdifferentmagneticmaterialsinmagneticelds.Inadditiontothetorquethemagneticmomentexperiencesaforce ~F=(~mr)~H(2)inthepresenceofaeldgradient. 15

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2.2MagnonExcitationsInasolid,atnon-zerotemperatures,ionsoratomsvibratearoundequilibriumpositions.Thesevibrationsarequantizedasphonons.Ifthesolidismagneticallyordered,thereareadditionalexcitationsduetothespinscalledspinwaves.Thespinwaveisaperiodicprecessionofspinswithacharacteristicwavelength,asillustratedinFig. 2-1 .Spinwavesarequantizedasmagnons,whicharespin-1bosons.Therearefundamentaldifferencesbetweenmagnonsinferromagnetsandantiferromagnets,aswewilldescribehere.WewillfocusonsolidsforwhichtheinteractionsbetweenspinscanbeexpressedbytheHeisenbergHamiltonian H=Xi>jJ~Si~Sj,(2)whereSiandSjarespinsinneighboringsitesiandj,andJisthestrengthoftheinteraction.J<0indicatesthattheinteractionbetweenspinsisferromagnetic,whereasJ>0meansthatitisantiferromagnetic. 2.2.1FerromagneticMagnonsWerstconsideraone-dimensionalferromagnet,inwhichtheinteractionbetweenneighboringspinsfavorsthemtoaligninthesamedirection.Atzerotemperature,theferromagnetispolarizedperfectlyinadirectionspeciedatrandom.Thetotalmagnetizationofthesystemis M=NgBS,(2)whereNisthenumberofspins.Atnonzerotemperatures,magnonsareexcited.Theenergyofamagnonisgivenby =~!=2jJjS(1)]TJ /F5 11.955 Tf 11.96 0 Td[(coska),(2)where!isthefrequency,kisthewavevector,andaisthelatticeconstant.Excitationofonemagnonchangesthetotalspinoftheferromagnetby1,therebydecreasingthemagnetizationbygB.AsshowninFig. 2-2 (left),thedispersionisquadraticatk=0. 16

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Figure2-1. Sideview(top)andtopview(bottom)ofaspin-waveexcitationinaferromagnet.TakenfromRef.[ 1 ]. Figure2-2. Dispersionrelationforferromagneticmagnons(left)andantiferromagneticmagnons(right). 2.2.2AntiferromagneticMagnonsAHeisenbergantiferromagnetdoesnotorderinonedimension.Buthereweassumethatitdoesorderbyformingtwosublatticeswithalternatingspins.Thedispersionofmagnonsinsuchanorderedstateisgivenby =~!=2JSjsinkaj,(2)whichislinearatk=0and=a,asopposedtothequadraticbehaviorofferromagneticmagnons,asshowninFig. 2-2 (right).Botharegapless.However,onlytheantiferromagneticmagnonsareNambu-Goldstonebosons.MagnonsareknowntohaveS=1Sz=)]TJ /F5 11.955 Tf 9.3 0 Td[(1forferromagnets,butS=1Sz=1forantiferromagnets.Equation 2 isbasedonthe 17

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assumptionthatthespinsorderinanantiferromagnet.Averydifferentpicturearisesinquantummagnets,wheremagneticorderingisabsent.Welookataspin-1 2linear-chainHeisenbergantiferromagnet,thesubjectofChapter5. 2.3Spin-1 2LinearChainsOneofthesimplest1Dquantumantiferromagnetsisalinearchainofspin-1 2ionsantiferromagneticallycoupledbytheHeisenbergexchangeinteraction,orthespin-1 2one-dimensionalHeisenbergantiferromagnet(1DHAF).Thissystemisparticularlyinterestingbecauseitspropertiesevolvewithmagneticeldinanontrivialmannerthatcanbecalculatedexactly. 2.3.1SpinonsThespin-1 21DHAFdoesnotorderevenatzerotemperature,nordoesany1DHAFforthatmatter.Asaresult,itdoesnothavemagnons.Insteadexcitationsareknownasspinons,whichhavespin1 2andarethereforefermions.ThedispersionrelationofspinonswasderivedbydesCloizeauxandPearson[ 2 ]fromEq. 2 byusingtheBetheansatz[ 3 ]andisgivenby =~!= 2Jjsinkaj.(2)ThisequationrepresentsthelowerboldlineinFig. 2-3 .Thedispersionhasasinusoidalshape,qualitativelysimilartothatofantiferromagneticmagnons.However,foragivenk,spinonshaveafactorof/2largerenergythanantiferromagneticmagnons.Spinonscanonlybecreatedordestroyedinpairsbecauseoftheconservationofangularmomentum.Asaresult,theyformacontinuum,asshowninFig. 2-3 .Theexcitationspectrashowninthisgure,includingthespinoncontinuum,havebeenconrmedbyinelasticneutron-scatteringexperiments. 2.3.2Tomonaga-LuttingerLiquidAlargenumberofone-dimensionalsystemsbecomeaTomonaga-Luttingerliquid(TLL)atlowtemperature.TLLarecompletelycharacterizedbytwoparameters,thevelocityvoflow-energyexcitationsandtheTLLparameter,K[ 4 5 ].Oneofthem,v, 18

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Figure2-3. Dispersionrelationforspinonexcitationsinaspin-1 2linear-chainHAF.Theshadedregionisthespinoncontinuum. appearsinthespecicheat,whichisgivenby C= 3RkBT v,(2)whereRisthemolargasconstant,kBtheBoltzmannconstant,andTistemperature.BothvandKappearinthemagneticsusceptibilityas =NA (gB)2K v,(2)whereNAisAvagadro'snumber[ 6 ].NotethatthespecicheatislinearinTandthemagneticsusceptibilityistemperatureindependent.Figure 2-4 (left)showsvasafunctionofeld,calculatedfromtheBetheansatzintegralequations[ 7 ].AmongthetwoparametersthatcompletelycharacterizeaTLL,vandK,thelatterisofparticularimportancesinceitgovernsalldynamicproperties,includingspincorrelationfunctions.Figure 2-4 (right)showsKasafunctionofmagnetic 19

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Figure2-4. (Left)Spinonvelocity,v,and(right)TLLparameter,K,ofthespin-1 2linear-chainHAFasafunctionoftheappliedeld,calculatedfromtheBetheansatz.Thelatticeconstantahasbeentakentobe1.AdoptedfromRef.[ 7 ]. eld,againcalculatedfromtheBetheansatz[ 7 ].KincreasesmonotonicallywithH.K=1 2atH=0,andK=1atthesaturationeld,wheregBH/J=2.Figure 2-10 inthenextsectionshowsKofspin-1 2Heisenbergladders.Herethehorizontalaxisisthemagnetizationinsteadoftheeld,butthemagnetizationisamonotonicallyincreasingfunctionofH.ThustheelddependenceofKisdramaticallydifferentfromthatofthespin-1 2linear-chainHAF,showninFig. 2-4 .Asthisexampleillustrates,KofeachTLLsystemexhibitsauniqueelddependence.TheTLLparameterKhasneverbeenexperimentallydeterminedexceptinonespin-1 2Heisenbergladdermaterial[ 8 10 ].Inparticularnodeterminationhasbeenmadeinaspin-1 2linear-chainHAF.ThebestwaytoobtainthisparameteristomeasuretheWilsonratio[ 11 ],adimensionlessratioofthetemperature-independentmagneticsusceptibilitytothecoefcientoftheT-linearspecicheatC, RW=4 3kB gB2 C=T.(2) 20

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Figure2-5. (Top)Specicheatofaspin-1 2linear-chainHAFatH=0,accordingtoquantumtransfer-matrixcalculations.(Bottom)C/TslowlybecomesconstantinthelimitofT=0.Reprintedgurewithpermissionfrom[ 12 ].Copyright(2000)bytheAmericanPhysicalSociety. AscanbeimmediatelyfoundbysubstitutingEqs. 2 and 2 intoEq. 2 RW=4K.(2)DeterminingtheTLLparameterKthusrequiresaccuratespecic-heatandmagnetic-susceptibilitymeasurements[ 8 ].Figures 2-5 and 2-6 showthespecicheatofaspin-1 2linear-chainHAFcalculatedbythequantumtransfer-matrixmethod[ 12 13 ].ThespecicheatbecomessufcientlyclosetoEq. 2 onlyatlowtemperatures.Therefore,experimentsmustbeconducted 21

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Figure2-6. Specicheat,c(T),ofaspin-1 2linear-chainHAFfordifferentmagneticelds,h=2gBH/J.(a)and(b)areforeldsbelowthesaturationeld,wherealinear-Tdependenceexistsatlowtemperatures.Panel(c)isforeldsabovethesaturationeld,whereaTLLdoesnotexist.ThespecicheatisintheunitsofR.Thehorizontalaxisis2kBT/J.Reprintedgurewithpermissionfrom[ 13 ].Copyright(1998)bytheAmericanPhysicalSociety. Figure2-7. Magneticsusceptibility,,ofaspin-1 2linearchainHAFfordifferentmagneticelds.isintheunitsofNAgB,andh=2gBH/J.h=4isthesaturationeld.Thehorizontalaxisis2kBT/J.Reprintedgurewithpermissionfrom[ 13 ].Copyright(1998)bytheAmericanPhysicalSociety. atquitelowtemperatures,belowabout5%ofJ/kBevenatlowelds.InFig. 2-7 themagneticsusceptibilityofaspin-1 2linear-chainHAFisshownasafunctionoftemperature,againcalculatedbythequantumtransfer-matrixmethod[ 13 ].IntheTLLregime,thesusceptibilityissufcientlyclosetoEq. 2 onlyatlowtemperatures,belowabout5%ofJ/kBevenatlowelds.InChapter5wedeterminetheWilsonratioforthespin-1 2linear-chaincompoundCuPzN. 22

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Figure2-8. Coupledspinladders.Jkistheinteractionalongtheleg,J?istheinteractionacrosstherung,andJ0istheinterladderinteraction.Reprintedgurewithpermissionfrom[ 14 ].Copyright(1999)bytheAmericanPhysicalSociety. 2.4SpinLaddersSpinladdersaredenedbytwointeractions,theinteractionalongthechains(Jk),alsoknownasthelegsoftheladder,andtherunginteractionbetweenthechains(J?),asshowninFig. 2-8 .Therangeofladdersystemsspansfromparallellinearchains,inwhichJ?iszero,tospindimersinthelimitofzeroJk.Inthisdissertation,wefocusonladdersformedbyS=1 2spins.Atzeroeld,thesystemisinanS=0Sz=0state.Thisgroundstateisamacroscopicspinsinglet,whichmaybeapproximatedasashort-rangeresonating-valence-bondstate[ 15 ].Thespinexcitationsaregappedmagnons,sometimescalledtriplons,withS=1,Sz=1and0.Theenergygapasafunctionoftheratiox=Jk/J?isshowninFig. 2-9 .Belowthecriticaleld,Hc==gB,thesystemremainsintheS=0Sz=0state.Abovethecriticaleld,atwhichthemagnongapvanishes,thegroundstatebecomesmagneticandgraduallyevolveswiththemagneticeld,TLLbehaviorappearsatlowtemperatures.Thespinexcitationsinthisregionarespinons,whenJ?>>Jk.Nosimple 23

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Figure2-9. Energygapofaspin-1 2ladderasafunctionoftheratiox=Jk/J?.JrunginthelabelfortheverticalaxisisJ?.Reprintedgurewithpermissionfrom[ 16 ].Copyright(2010)bytheAmericanPhysicalSociety. descriptionexistsofspinexcitationsoutsidethislimit.Figure 2-10 showsthebehavioroftheTLLparameterwithrespecttomagnetizationastheratioj=J?/Jkchanges.Inarealmaterial,interactionwithneighboringladders(J0)mustalsobetakenintoaccount.AsJ0increases,thesystemstartstoresembleasquarelatticeofspins,andthegroundstatebecomesmagneticallyordered.Forexample,forx=1,thequantumcriticalpointbetweenthesingletphaseandtheNeel-orderedphaseexistsatJ0/Jk0.3[ 18 ].Intheorderedphase,theTLLbehavioriscompletelygoneandNambu-Goldstonebosonsappear.Inchapter6westudyDLCB,aspin-1 2laddercompoundinwhichJ0slightlyexceedsthecriticalvalue. 2.5SpinTrianglesSpintriangleshavebeenasystemofinterestduetoeffectssuchasspin-electriccoupling[ 19 21 ],geometricfrustration[ 22 ],andmultiferroicordering[ 23 24 ].OfparticularimportanceisatriangleofS=1 2spinswithequalantiferromagneticinteractionsbetweenthem,showninFig. 2-11 ,sincethequantumeffectsaremostpronouncedin 24

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Figure2-10. TLLparameterKforspin-1 2HeisenbergladdersasafunctionofnormalizedmagnetizationMcalculatedbythedensity-matrixrenormalization-grouptechniqueforasystemconsistingof200spins.j=1/x=J?/Jkistheratiooftherungexchangetothelegexchange.Reprintedgurewithpermissionfrom[ 17 ].Copyright(2001)bytheAmericanPhysicalSociety. thiscase.Inzeromagneticeld,thissystemhasonlytwoenergylevels,whosespinquantumnumbersareS=3 2orS=1 2,separatedby3 2J.Thegroundstate,withS=1 2,isfourfolddegenerateasshowninthetoppanelofFig. 2-12 .Inamagneticeld,thestatesplitsintotwotwofolddegeneratelevels,ofwhichthestatewithS=1 2andSz=1 2remainsthegroundstate.Asaresult,themagneticmomentjumpsformzerotogB=2immediatelyasshowninthemiddlepanel.HereSzisthequantumnumberofthecomponent,ofthetotalspin,paralleltotheeld.TheS=3 2state,whichisalsofourfolddegenerateatzeroeld,completelysplitsintofourinamagneticeld.Astheeldincreases,thelowest-energystatewithSz=3 2takesoverthe 25

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Figure2-11. SpintrianglewithequalinteractionsJbetweenthespinslocatedatthevertices. S=1 2Sz=1 2doubletatH=3J=2gB.Thisleadstoasecond,naljumpofthemagneticmoment,fromgB=2to3gB=2.WhenaninteractionJ0betweenspintrianglesisintroduced,thetwojumpsbecomeslopes,asshowninthebottompanelofFig. 2-12 .Therangeoftheeldoverwhichthemagnetizationchanges,H,isproportionaltoJ0.J0alsocausesthesystemtoorderinthetwoeldregionswherethemagnetizationchanges.Intheintermediateregion,wherethemagnetizationisconstant,nospontaneousmagneticorderingoccurs.EffectssuchasformationofspontaneouselectricdipolesorcurrentloopsinthespintrianglecannotbeunderstoodintermsoftheHeisenbergHamiltonian.TheminimalHamiltonianrequiredisthehalf-lledHubbardmodel, H=)]TJ /F10 11.955 Tf 9.3 11.35 Td[(Xijt(cyicj+cyjci)+U 2Xi(ni)]TJ /F5 11.955 Tf 11.96 0 Td[(1)2,(2)wheretisthehoppingconstant,iandjaretheelectronsites,isthespinstate,cyandcarethecreationandannihilationoperatorsofanelectron,Uistheonsiterepulsion,andnisthenumberoperator.Ifspintrianglesformalattice,andelectronsareallowed 26

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Figure2-12. (Top)Energyofthespin-1 2spintriangle.TheS=3 2andS=1 2stateshaveadifferenceofenergy3 2J.ThereddotshowstheeldatwhichtheS=3 2Sz=3 2statebecomesthegroundstate.(Middle)Magneticmomentofasinglespintriangleasafunctionofeld.(Bottom)Magneticmomentofaspintriangleinteractingwithneighbors. 27

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Figure2-13. Magneticstatewithnonzeroelectricpolarization.Twospinsinsidetheovalformasingletandtheunpairedspinmaybeupordown.Theredarrowindicatestheelectricdipolemoment.AdoptedfromRef.[ 23 ]. tohopbetweenneighboringspintriangles,thesystemmayloweritsenergybyvaryingtheelectrondensityfromsitetosite.Thiswillleadtotheappearanceofaspontaneouselectricdipolemomentineachspintriangle.Thedipolemomentisparalleltotheplaneofthetriangle,withthexandycomponentsgivenby Px=4p 3ea(t U)3[~S1(~S2+~S3))]TJ /F5 11.955 Tf 11.96 0 Td[(2~S2~S3](2)and Py=12ea(t U)3~S1(~S2)]TJ /F8 11.955 Tf 12.13 3.16 Td[(~S3),(2)whereaisthedistancebetweensites,andeistheelectroncharge.Noticethatthedipolemomentisassociatedwiththespinconguration.Asanexample,letusconsideroneofthedegenerategroundstatesofthespintriangle,illustratedinFig. 2-13 .AccordingtoEqs. 2 and 2 ,thisstatehasanelectricdipolemomentalongthexdirection.Therefore,ifsuchagroundstateisselectedbytheinteractionsbetweentheneighboringspintriangles,thenanelectricdipolemomentwillspontaneouslyappearineachspintriangle,asindicatedinthe 28

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Figure2-14. Groundstateswithnonzeroelectriccurrent.Thecirculararrowsindicatethedirectionsofthecurrents.Reprintedgurewithpermissionfrom[ 23 ].Copyright(2008)bytheAmericanPhysicalSociety. gure.Themicroscopicmechanismforthisappearanceisthattheformationofthespinsingletattractsanelectronfromtheremainingvertex.AnotherpossibleeffectduetointeractionsbetweenspintrianglesistheappearanceoforbitalcurrentsasshowninFig. 2-14 .TheorypredictsthatthisdoesnotoccurifthespintrianglesformatriangularlatticeasinTNNCH3CN,thesubjectofChapter7.Multiferroicsarematerialsinwhichcouplingexistsbetweenprimaryorderparameters.Ofparticularinterestarethosemultiferroicsinwhichoneorderparameterismagnetic,theotherelectric.Ifspintrianglesformatwo-dimensionalorthree-dimensionallattice,inwhichtheneighboringtrianglesinteract,multiferroicorderingmayoccur.Figure 2-15 showsatriangularlatticeformedbyspintriangles.Themodelcontainsstackedlayersinwhichspintriangleslieaboveeachother,withintralayer(J0)andinterlayer(J00)interactions.KamiyaandBatista[ 24 ],whohaveconsideredthismodel,predictamultiferroicphasediagramshowninFig. 2-16 .ThenumberofmultiferroicallyorderedphasesandtheirnaturedependsontheratioofthetwoJ's.ForTNNCH3CN,0
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Figure2-15. Triangularlatticeofspintriangles[ 24 ].Thebluetrianglesrepresentthespintrianglesseparatedevenlythroughoutthelattice.Thearrowslabeledu1,2,3aretheprimitivevectorsofthestackoftriangularlatticelayers,wherease1,2,3arethevectorsbetweennearestneighborspintrianglesinthesamelayer.Thehoppingamplitudest,t0,andt00ofelectronsaredenedintheinsetontheright.Reprintedgurewithpermissionfrom[ 24 ].Copyright(2012)bytheAmericanPhysicalSociety. showstwostepsappearinginthemagnetization,whereasonlyonestepappearsintheelectricpolarization.Sofar,thetheoryhasonlyconsideredthemagnetic-eldregioninwhichthespinofeachtriangleis1 2.ThisregioncorrespondstotheregionofM/gBupto1 2inthebottompanelofFig. 2-12 .MultiferroicallyorderedphasesarealsoexpectedintheeldregionwhereM/gBrisesfrom1 2to3 2,buttheorystillneedstobeworkedout.InChapter7,wewillstudyTNNCH3CN,inwhichspintrianglesformatriangularlattice. 30

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Figure2-16. Multiferroicphasediagramforaspin-trianglesystem,predictedbyKamiyaandBatista.Reprintedgurewithpermissionfrom[ 24 ].Copyright(2012)bytheAmericanPhysicalSociety. 31

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CHAPTER3EXPERIMENTALTECHNIQUESSpecic-heat,magnetocaloric-effect,magnetic-torque,andmagnetizationmeasurementsarevaluabletoolsforstudyingmagneticmaterials.Theseprobesgiveusinsighttothevariousmagneticphasesofthematerials.IntheTomonaga-Luttingerliquidphaseofaone-dimensionalHeisenbergantiferromagnet,wehavemeasuredspecicheatandmagneticsusceptibilitytoextractthespinonvelocityandtheWilsonratio.Inacoupledarrayofspinladders,alloftheprobeswereusedtorevealananisotropicphasediagraminvolvingaspin-optransition.Inatriangularlatticecomprisingspintriangles,wehaveuseddielectricmeasurementsinadditiontoallthetoolstoidentifymultiferroicbehaviorandtoconstructaphasediagram.Themajorityoftheresultspresentedinthisdissertationwereobtainedinthetop-loadingdilutionrefrigeratorwitha20Tsuperconductingmagnet[ 25 ]intheMillikelvinLaboratoryoftheNationalHighMagneticFieldLaboratoryinTallahassee.Thissetupallowedustoworkintheparameterspaceofinterest.Iwilldescribeinthischaptertheprinciplesandpracticesofthemeasurementtechniques. 3.1Calorimetry 3.1.1RelaxationCalorimetrySpontaneousorderingisafundamentalphenomenonstudiedincondensedmatterphysics.Iftheorderingiscontinuous,thenthespecicheatexhibitsasharppeak,providingaclearsignatureoftheorderingtransition.Oneofthebestmethodsformeasuringthespecicheatistherelaxationcalorimetry.Ofteneld-inducedlong-rangeorderingoccursbelow0.3K,atypicalbasetemperatureofa3Herefrigerator.Toreachsuchtemperaturesrequiresadilutionrefrigerator.Ourcalorimeter,showninFig. 3-1 ,ishousedinsidea24.6mm-diametervacuum-sealedbrasscan[ 26 ],whichthermallyinsulatesthecalorimeterfromtheliquidheliuminthemixingchamberofthedilution 32

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Figure3-1. Schematicofthecalorimeterusedforspecic-heatandmagnetocaloric-effectmeasurements.Thecalorimeterassembly(left)isattachedtothebottomofthetop-loadingprobeandinsertedintothemixingchamberofthedilutionrefrigerator.Detailsofthecalorimeterandthesampleplatformareshownattoprightandbottomright.Reprintedfrom[ 27 ].Copyright(2003)withpermissionfromElsevier. refrigerator.Thecanalsoshieldsthesamplefromthermalradiationandthecalorimeterfromeddy-currentheating.Thesample,whichtypicallyweighsafewmilligrams,isgluedwithaverythinlayerofWakeeldcompoundontoa6.4mmdiameter,0.13mmthicksapphirediskplatform.TheplatformhasacarbonresistancethermometerattachedandathinlmofTi-Cralloyevaporatedasaheater.Theplatformissuspended,bytheleadsforthethermometerandheater,froma19mmdiametersilverringconnectedtoa6.3mmthicksilverblock, 33

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whichtogetheractasthethermalreservoir.Theelectricalleadsforthethermometerandheaterserveassupportsaswellasaweakthermallinkfortheplatform.Silverisusedfortheblockandringbecauseitsnuclearspecicheatisthesmallestofallmetals,thusminimizingtheequilibrationtime.TheblockissuspendedfromthelidofthebrasscanbyaVespelSP-22rrod,whichprovidesthermalinsulation.ThecanisevacuatedatroomtemperaturethroughaCuNicapillary,connectedtoastainless-steeltubethatrunsthelengthoftheprobe,andisthentestedforleaks.Insidethecan,therearetwelveleadsthatattachtotworowsofgold-platedpins,takenfromaSIPconnectorandgluedwithStycast2850FTrepoxyintwogroovesonthelidofthebrasscan.Theyareconnectedtoleadsthatrunthroughtheprobe.Thesearetested,thentheprobeisloweredintothemixingchamberandcooled.Ontheblockisaheaterthatcontrolstheambienttemperature,whichismeasuredbyacarbonresistancethermometeralsoontheblock.TheheatercurrentisprovidedbyaDCcurrentsource,andthethermometerresistanceismeasuredbyanotherDCcurrentsourceandadigitalmultimeter.Oncethetemperatureisset,atemperaturedifferenceisestablishedacrosstheweaklinkbytheplatformheater.MeasuringaspecicheatentailsturningofftheplatformheateranddetectingtheensuingrelaxationofthetemperaturebyaPAR124Alock-inamplierasanulldetectorofaWheatstonebridgewithadecaderesistorinonearm.Thebridgeisbalancedbymatchingtheresistanceofthedecaderesistorwiththatoftheplatformthermometer.Theanalogoutputofthelock-inamplierisrecordedbyadataacquisitionboardinstalledonacomputer.Thebridgeisdrivenbytheinternaloscillatorofthelock-inamplieratafrequencyof441Hz.Theexcitationlevelismonitoredandadjustedtoensurethattheplatformthermometerisnotoverheated.Arelaxationcurveoftheplatformthermometerat140mKisshowninFig. 3-2 .Thiscurveisanaverageofmultiplecurvestakenatthesametemperature.Theaveragedcurveisttedtoanexponentialformtoobtaintherelaxationtimeconstant,,which 34

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Figure3-2. (Left)Temperaturerelaxationofa1.59mgsampleofCuPzNin12.5Tat140mK.(Right)Temperaturerelaxationofa0.37mgsampleofTNNCH3CNin0.5Tat250mKwhena2effectispresent.Inbothpanels,theverticalblacklineindicatesthetimeatwhichtheheaterwasturnedoff.Theredlineshowsatofthedatatoanexponentialform.Thesedatahavebeendigitallyprocessedtoremoveperiodicnoise,andmultiplecurveshavebeenaveraged,20forCuPzNand6forTNNCH3CNtoreducerandomnoise. yieldsthetotalheatcapacityCtotalofthesampleandaddendathroughtherelation Ctotal=,(3)whereisthethermalconductanceoftheweaklinkbetweenthesampleplatformandthethermalreservoir.Heremustbethatattheaverageoftheinitialandnaltemperaturesoftheplatform.Sincetheplatformthermometerresistancechangesfromruntorun,itmustbecalibratedateacheldagainsttheblockthermometer.FromCtotal,theaddendaheatcapacitywhichhasbeenmeasuredseparatelyissubtracted,yieldingtheheatcapacityofthesample.Theresistanceoftheblockthermometerdependsontemperatureandthemagneticeld.Therefore,thecalorimeterhasbeencalibratedatseveraleldsbyusingsamplesofsilver,platinum,andindium[ 1 ].Intheeldrangeofthepresentexperiment,calibrationuncertaintiesareabout1.0%at0.8Kandabout1.8%at0.1K.Theprincipleofthecalibrationmethodisasfollows.Wechoosetwoofthethree 35

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materialswhosespecicheatshavequitedifferenttemperaturedependencesatagivenmagneticeld.Wethenmeasure,atagiventemperature,therelaxationtimesofthesamplesofthosetwomaterialsofknownmassesinourcalorimeter.Theratioofthetworelaxationtimesyieldsthetemperatureuniquely,sincethephononandelectronspecicheatsofthesematerialsareaccuratelyknownandtheirnuclear-spinspecicheatandnuclearquadrupolarspecicheat,presentinindium,canbecalculated[ 28 ].Oncethetemperatureisfound,weknowtheheatcapacityofeithersample.Inturn,thisheatcapacityinconjunctionwiththerelaxationtimeofthesamesamplegivesthethermalconductanceoftheweaklink.Atlowtemperatures,simpleexponentialrelaxationoftheplatformtemperatureisnotalwaysobserved.Occasionallythetemperatureinsteaddropsrapidlyrst,thendecaysmoreslowly,asshownintherightpanelofFig. 3-2 .Thisrapidinitialdrop,theso-called2effect,arisesfromeitherapoorthermalcontactbetweenthesampleandtheplatformorapoorthermalconductanceofthesample.Ifthe2effectispresent,onlythepartofthedataaftertheinitialdropisttedtoanexponential,asshownintherightpanel.Thesampleheatcapacityinthiscaseisgivenby Csample=a11(1)]TJ /F9 7.97 Tf 13.15 5.11 Td[(Cadd 1)2 1)]TJ /F9 7.97 Tf 13.15 5.11 Td[(a1Cadd 1,(3)wherea1and1aretheweightandthetimeconstantfromthet,andCaddistheaddendaheatcapacity. 3.1.2MagnetocaloricEffectAlthoughspecicheatmeasurementsarereliablewaystodetectthepresenceofphasetransitions,theyaretimeconsuming.Moreover,thespecic-heatpeakbecomessmallandwideasthetransitiontemperatureapproacheszero.Magnetocaloric-effectmeasurementsprovideaquickandconvenientwaytosupplementspecic-heatmeasurementsoverawidetemperaturerange.Theymaybeusedpriortospecic-heatmeasurementstosurveyphaseboundaries,ortheymaybeusedtollintemperature 36

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andeldregionswherespecic-heatmeasurementsfallshort.Thismethodusesthesameequipmentastherelaxationcalorimetry.Themagnetocaloriceffectisachangeinthetemperatureofamagneticmaterialduetoachangeofamagneticeld.Ifthemagneticeldiscontinuouslychangedattherate_H,theeffectisgivenby T=)]TJ /F4 11.955 Tf 10.49 8.09 Td[(T [(@M @T)H+CH Td(T) dH]_H,(3)whereTisthetemperaturedifferencebetweenthesampleplatformandthereservoir,andCHisthemagneticheatcapacityofthesample[ 29 ]. 3.2MagneticTorqueMagnetictorquemeasurementsareanothertoolthatallowsidenticationofmagneticphasetransitions.Themeritofthesemeasurementslieintheirefciency.Althoughtorquemeasurementsarequickandmaygiveageneralideaofwhetherthereisatransitionornot,theyareoftennotpreciseenoughtodeterminetheexacteldatwhichthetransitionoccurs.Moreover,magnetization,andconsequentlysusceptibility,cannotbeaccuratelyextractedfromthemagnetictorquedata. Figure3-3. Schematicofthecapacitivecantilevertorquemagnetometer.(a)Thesampleisgluedto(b)aBeCucantileverusingGEvarnishorWakeeldcompound.Thesampleplatformisheldparallelto(c)axedelectrodeattachedto(d)thebaseplate.Theseparationofthetwoelectrodesismaintainedby(e)aspacer.Whentheeldisnotalignedwiththemagnetizationofthematerial,atorqueactsonthesampleandchangestheseparationandcapacitancebetweenthetwoelectrodes.TakenfromRef.[ 1 ]. 37

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TorquemeasurementsrequireacapacitivecantilevermagnetometerdevelopedattheDC-FieldFacilityoftheNHMFL,illustratedinFig. 3-3 .Aberyllium-coppercantileverisheldinplaceaboveaxedelectrode.Thetwoareseparatedbyaspacer.Usingacantileverofanappropriatethicknessisimportantinthesemeasurementsbecauseitdeterminesthesensitivityofthemeasurements.Foratypicalsampleweighingafewmilligrams,0.001inchisusuallygoodchoice.Whenthesampleislargerorhighlymagnetic,alargerthicknessmayhavetobeselectedtopreventthecantileverfromtouchingthexedelectrodeandceasingtowork.ThesampleisattachedtothecantileverwithathinlayerofGEvarnishorWakeeldcompound,dependingonthereactivityofthesamplewiththeadhesive.Thesampleispositionedneartheendofthecantilevertomaximizethesensitivity.Themagnetictorque~exertedonthesamplebythemagneticeldisgivenby ~=~M~H,(3)where~Misthemagnetizationofthesample,and~Histheappliedeld.ThisisjustamacroscopicversionofEq. 2 .Thetorquebendsthecantileverwhosedeectionchangesthecapacitancebetweenitandthexedelectrode.Forasmalldeection,thechangeisproportionaltothetorque.ThecapacitanceismeasuredwithanAndeen-HagerlingAH2700Aultra-precisioncapacitancebridge,operatedat5kHzwitha15Vrmsexcitation.AtypicalmeasuredcapacitanceattheMillikelvinLaboratoryoftheNHMFLis1to2pF. 3.3MagnetizationMeasurements 3.3.1SQUIDMagnetometrySQUIDmagnetometersarecurrentlythemostsensitivemethodofmeasuringmagnetization.BecauseoftheSQUID'ssensitivitytouctuationsinthemagneticeld,thesensormustbeprotectedbyasuperconductingshield,whichexpelstheexternalmagneticeld.TheSQUIDiscoupledtoasuperconductingpickupcoil,which 38

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comprisesthreesectionsinasecond-derivativegradiometerconguration.Asthesamplemovesfromonesectionofthecoiltoanother,counterwoundsection,thetotaluxinthepickupcoilchanges.TheSQUIDdetectsthischange.TheresolutionsofthemagneticmomentforcommercialSQUIDmagnetometersaretypicallybetterthan10)]TJ /F3 7.97 Tf 6.59 0 Td[(8emu.Inthepresentwork,acommercialSQUIDmagnetometerofourdepartment(QuantumDesignMPMS)wasusedforthestudyofCuPzN.Thebasetemperatureofthemagnetometeris2Kandthemagneticeldislimitedto6T.Eventhoughthemagnetometeryieldsverypreciseresults,careisnecessarynearthebasetemperaturetomakesurethattheeldissweptslowlyenoughinsmallsteps,toachievetemperaturestability.Whenmeasurementsaremadeasafunctionofeld,itisalsoimportantthattheledisstablebeforeeachmeasurementismade. 3.3.2ForceMagnetometryForcemagnetometryinthisworkwasperformedbyusingoneofthethreehomemademagnetometerswhosedesignandconstructionaredescribedinChapter4.Themagnetometersweredesignedtoaccuratelymeasurethemagnetizationofasample,typicallyweighingontheorderof1mg.Thesampleisattachedtoa7.62mmdiametersampleplatformwithaverythinlayerofGEvarnishorWakeeldcompound.Theplatformissuspendedbyphosphor-bronzewiresandiselectricallyinsulatedfromanelectrode,whichisattachedtotheplatformandformsacapacitorwithastationaryelectrode.Topreventthelossofthesampleintheeventthatthesamplefallsoff,themagnetometerswerecoveredwithaNylonmesh.Themagnetometersweredesignedtotinsidethe25.0mmdiametersamplespaceofthemixingchamberofthedilutionrefrigerator,whichprovidestemperaturesbetweenabout25mKmuchlowerthanthebasetemperatureofheliumrefrigeratorsandabout1.5K. 39

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Athermometerismountedonthemagnetometerandaheaterisattachedtotheadapter.Themagnetometerisattachedtotheendofatop-loadingprobethatallowsustoaccuratelyadjustthepositionofthesamplewithrespecttothemagnet.Measurementsaremade,withandwithoutthesample,attheeldcenterandoneortwooff-centerpositions.Fromthedatatakenwiththesample,thedatatakenwithoutitatthesamepositionissubtractedtoremovethemagnetometerbackground.Afterthesubtraction,theeld-centerdataissubtractedfromtheoff-centerdatatoremovethecontributionofthemagnetictorque.Attheoff-centerpositions,theeldgradient,whichisproportionaltotheeld,is5.68T/mwhentheeldisat5T.Thesamplemagnetizationisdetectedasachangeinthecapacitancebetweenthetwoelectrodesofthemagnetometer.Thischangeisproportionaltothemagnetizationandtheeldgradient,ascanbeseeninEq. 2 .ThecapacitancewasmeasuredbyanAH2700Acapacitancebridge.Thefrequencyofthebridgeistypicallysetto5kHz.Theexcitationvoltageis15Vrmsandthesignalisaveragedover1s.Wehavebuiltthreemagnetometers,aswillbedescribedinChapter4.Theresolutionsofthersttwomodelswere1.310)]TJ /F3 7.97 Tf 6.58 0 Td[(4emuand1.010)]TJ /F3 7.97 Tf 6.59 0 Td[(5emuat15T.Thethirdmagnetometersuffersfromnoiseproblems,aswillbedescribedinSection4.2. 3.4DielectricMeasurementsDielectricmeasurementsinvolvemeasuringthechangeincapacitancebetweentwoelectrodesattachedtothetwooppositesurfacesofthesample.Dependingonthesample,theelectrodesmaybesputteredgoldorplatinum,orsilverpaintappliedtothesample,asshowninFig. 3-4 .Thesampleiscutsuchthatthefacesonwhichtheelectrodesareformedareperpendicularorparalleltothecrystallographicaxisorplaneofinterest.Theelectrodeandsamplearegluedtotherotatingplatformofatop-loadingprobeofthedilutionrefrigeratorbyGEvarnish.Silverpaintisusedifthematerialissensitivetothermalstress.Theprobeisthenloweredintothemixingchamberoftherefrigerator.ThecapacitanceismeasuredbyusinganAH2700Acapacitancebridge. 40

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ThetemperatureismeasuredbyusingaLakeShoremodel370ACresistancebridge.Themixing-chamberheaterisusedtocontrolthetemperatureoftheexperiment. Figure3-4. Silver-paintelectrodeonasinglecrystalofTNNCH3CN.Anotherelectrode,ontheoppositesurface,ishiddenfromviewbythecrystal.PicturetakenbyEunSangChoi. 41

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CHAPTER4DEVELOPMENTOFLOW-TEMPERATUREFORCEMAGNETOMETERSSinceprecisemeasurementsofmagnetizationwerenecessaryforthestudiesofCuPzNandDLCBforacombinationoftemperaturesandmagneticeldsthatisunattainablewithexistingmagnetometers,developmentofaninstrumentthatovercamethelimitationsofthosemagnetometersbecameanimportantpartofmyproject.Wesucceededinbuilding,incollaborationwithToshiroSakakibaraoftheUniversityofTokyo,amagnetometeradequateforthestudyofDLCB.Theinstrumentwasinsufcient,however,fortheexperimentsonCuPzN.Forthisreason,themagnetizationmeasurementsonthismaterialwereperformedbyYoheiKonoinSakakibara'slaboratory,wherethemagnetizationofTNNCH3CNwasmeasuredearlierbyYasuyukiShimuraandothers. 4.1ComparisonwithOtherMethodsTherearefourtypesofmagnetometerscommonlyusedformeasurementsatlowtemperatures:vibratingsamplemagnetometers(VSM),SQUIDmagnetometers,capacitivecantilevermagnetometers,andinductionmagnetometers.Herewediscussthemeritsanddisadvantagesofeachmethod,followedbyadiscussionofforcemagnetometers.Adetaileddescriptionisgiveninthenextsectionoftheforce-magnetometerdesign,fabrication,andtests. 4.1.1VSMVibratingsamplemagnetometersarebasedonFaraday'slaw.Asampleismovedbackandforth,withatypicalfrequencyofabout85Hz,throughapairofdetectioncoilsinauniformmagneticeld.Theoscillatingmagneticuxcausedbythesamplemotioninducesanemfinthecoils,whichisdetectedandconvertedtothesamplemagnetization.However,themotionofthesamplealsoproducesheat.Asaresult,thesemagnetometersonlyworkattemperaturesdowntoabout0.3Katbest,moreoftenonlytoabout1.5K.Forinstance,themagnetizationofCuPzNshowninFig. 5-3 42

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inChapter5wasmeasuredbytheJohnsHopkinsgroupusingaVSMattheNHMFLinTallahassee.Thelowesttemperatureachievedinthatmeasurement,1.82K,istoohighfordeterminationoftheWilsonratioofCuPzN.Presently,themaximumeldforaVSMis35T,abovewhichthedrivingmechanismforthesampleceasestowork. 4.1.2SQUIDMagnetometersSQUIDmagnetometersdetectadifferenceinmagneticux,asthesampleismovedbetweentwosectionsofasuperconductingpickupcoil,asdescribedinSection3.3.1.Thesearethemostsensitivemagnetometers,sincetheSQUIDisthemostsensitivedevicecapableofmeasuringanextremelysmallmagneticux.Despitetheirsensitivity,SQUIDmagnetometersarelimitedtotemperaturesdowntoonly2Kandmagneticeldsuptoabout9T.Thelimitationintemperaturearisesfromtothemotionofthesamplecausingheating.Theeldrangeisrestricted,becausetheuxtransformerbetweenthepickupcoilsandtheSQUIDmustbeshieldedbyasuperconductingtube,whichdoesnotworkwhentheeldishigh.SQUIDmagnetometersaregreatforpreliminarystudies,butforthepresentworkthetemperatureandeldrangeswereinsufcient. 4.1.3CapacitiveCantileverMagnetometersTheprincipleanddesignofthecapacitivecantilevermagnetometerhavebeenexplainedinSection3.2.Formagnetizationmeasurements,asopposedtomagnetic-torquemeasurementsdescribedinthatsection,thesamplemustbealignedsuchtomakeitsmagnetizationparalleltotheeld.Themagnetometerisdeliberatelyplacedawayfromtheeldcentertotakeadvantageofthebuilt-ineldgradientofthemagnet.Bendingofthecantilever,causedbytheforceaccordingtheEq. 2 ,isdetectedasachangeinthecapacitancebetweenthecantileverandthexedelectrode.Butmagnetictorqueonthesample,whichalsobendsthecantilever,isoftenverydifculttoeliminate,makingitpracticallyimpossibletodeterminetheforce,thusthemagnetization,withaccuracy. 43

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4.1.4InductionMethodInductionmagnetometersusearapidlychangingmagneticeldappliedtoastationarysampleanddetectthechangeinthemagneticuxduetothemagnetizationofthesample.Thismethodrequiresapulsedmagnet,whichcausesthespintemperatureofthesampletochangewildlyduringmeasurements.However,measurementscanbemadeuptoabout100T,muchhigherthanthemaximumeldsofallothermethods. 4.1.5ForceMagnetometersTothispointwehavereviewedthecommonlyusedmagnetometersandhavehighlightedthelimitationsofeachmethod.Theselimitationsexcludeallmethodsasoptionsforaccuratemagnetizationmeasurementsattemperaturesbelowabout1.5K.Theforcemagnetometerswehavedevelopedallowhigh-resolutionmeasurementsatlowertemperatures.Aforcemagnetometermeasurestheforceexertedonasamplebyaspatiallyvaryingmagneticeld.Thesampleismountedonaplatformattachedtoarodsuspendedbytwosetsofphosphor-bronzewires.Ontheotherendoftherod,electricallyisolatedfromit,isanelectrodewhichformsacapacitorwithastationaryelectrode[ 30 ].Thesuspendedstructuremustbemadeofametalwithassmallamagneticsusceptibilityaspossibleorametalizedpieceofplasticwithnomagneticimpurities.Themagneticeldandthegradientarestatic.ThemagneticeldmagnetizesthesampleandtheeldgradientexertsaforceonitaccordingtoEq. 2 .Thisforceactsonthesampleplatform,movingthesuspendedstructure,therebychangingthespacing,hencethecapacitance,betweenthetwoelectrodes.ThesuperconductingmagnetwehaveusedattheMillikelvinLaboratoryoftheNHMFLlacksaseparategradientcoil,forcingustousethebuilt-ingradientoftheeldatanoff-centerposition.Next,wewilldescribetheforcemagnetometersthatwehavedeveloped. 44

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4.2DesignofForceMagnetometersOurmagnetometersfollowthedesignofToshiroSakakibara,ourcollaborator,showninFig. 4-1 [ 30 ].Thelowerpairofsuspensionwirescounterthetorqueonthesamplecausedbymagneticanisotropiesandanon-uniformdemagnetizingeld.Themagnetometerhasaresolutionof10)]TJ /F3 7.97 Tf 6.58 0 Td[(4emu.Usedinvacuum,thisdesignrequiresathermallinkfromthemixingchamberofthedilutionrefrigeratortothesampleplatform.Bycontrast,oursaredirectlyimmersedinliquidhelium,eitherinsidethemixingchamberofthedilutionrefrigeratororinsidetheliquid-3Hepotofa3Herefrigerator. Figure4-1. CrosssectionalviewofaforcemagnetometerdevelopedbySakakibaraetal.Copyright1994TheJapanSocietyofAppliedPhysics[ 30 ]. ThethreemagnetometersbuiltinourlaboratoryaresimilartoaprototypeshowninFig. 4-2 .Inallversionsofthemagnetometer,thelowerelectrodescrewsintothebottomring,allowingtheusertoadjusttheseparationbetweenthetwoelectrodestoincreaseordecreasesensitivity.Theprototypeandtherstmagnetometerhadthebottomringttedwithapolycarbonateringpiecetoinsulatethelowerelectrodeasshownintheleftpanel.Thesecondandthirdversionshadtheelectrodeepoxiedintoathreadedberylliumcopperpiecetoeliminatethepolycarbonatering.Theepoxy 45

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servedasinsulationaswellasanadhesive.Thischangewasmadeinanattempttoreducethetemperature-dependentbackgroundobservedintherstmagnetometer.Oneexplanationwehadforthetemperaturedependencewasthatthelowerelectrodemovedduetoanomalousthermalcontractionofthepolycarbonate[ 31 ].Anotherexplanationwasthatthepolycarbonatemayaffectthefringeelectriceldofthecapacitor,therebycontributingtothecapacitance.Thiswasaconcern,becauseamorphousmaterialsingeneralhaveanomaloustemperature-dependentdielectricconstantsatlowtemperatures[ 32 ].Thesewerealldismissedwhenthesametemperature-dependentbackgroundwasstillseeninthesecondmagnetometer.Ourrsttwomagnetometersenablehigh-sensitivitymagnetizationmeasurementsattemperaturesdownto25mKineldsupto20T.Thethird,smallermodelwasdevelopedformeasurementsattemperaturesdownto0.4Kandeldsupto31Tinaresistivemagnetequippedwithaseparategradientcoil.Thismagnetometerwastestedina3HerefrigeratorattheDC-FieldFacilityoftheNHMFLinTallahassee.VibrationoftherefrigeratorcausedbythecoolingwaterofthemagnetwastoohighfortheAndeen-HagerlingAH2700Aultra-precisioncapacitancebridgetofunction.InsteadweusedaGenRadmanualcapacitancebridgeinconjunctionwithadouble-lock-intechnique.Thegradientcoilwasoperatedat1Hzor10Hz,andthecapacitancebridgeat1kHz.Theoutputoftherstlock-inamplier,usedasthenulldetectorofthebridge,wasfedtothesecondlock-inamplier,whichwasslavedtothegradient-coilpowersupply.Themagnetizationofa30.62mgnickeldiskcouldbedetected,butthenoisewasaslargeas10%ofthesignal.Improvementofthevibrationisolationofthecryostatisseriouslyneededtousethismagnetometerforaccuratemeasurements.Thismagnetometercaninprinciplebeusedineldsupto45T,thehigheststaticeldavailableatpresent,andattemperaturesdownto25mK,thebasetemperatureoftop-loadingdilutionrefrigerators. 46

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Figure4-2. (Left)Sectionsideviewoftheprototypemagnetometer.(Right)Imageoftheprototype,showingthesampleplatformandapairofcrossedphosphor-bronzewiressuspendingit.Thediameteroftheprototypeis0.77inandtheheightis1.13in. 4.3FabricationofaPrototypeFirst,afull-scaleprototypeforthemagnetometerswasconstructed,withallthemetallicpartsmadeofbrassexceptthephosphor-bronzesuspensionwires.Theprototype,showninFig. 4-2 ,wastestedatroomtemperaturetodeterminewhetheritssensitivitywouldbesufcientfortherangeofmagneticforcesatypicalantiferromagneticsamplewouldexperienceina20Tsuperconductingmagnet,withatypicaleldgradient.ThemagneticforceexertedonasampleinaspatiallyvaryingmagneticeldisgivenbyEq. 2 .Forinstance,inaeldgradientof10T/mat14T,a10mgsampleofCuPzNwouldexperienceaforceof2.610)]TJ /F3 7.97 Tf 6.59 0 Td[(3N.Thetestswereconductedbyusingtestmasses,rangingfrom25mgto5.2g,placedonthesampleplatform.ThecapacitancewasmeasuredbyanAndeen-HagerlingAH2700Aultra-precisioncapacitancebridgewitharesolutionof510)]TJ /F3 7.97 Tf 6.59 0 Td[(7pF.TheresultsareshowninFig. 4-3 fortwogapsizesbetweentheelectrodes.ThegapsizeswereestimatedfromthecapacitanceC0without 47

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Figure4-3. Changeinthemagnetometercapacitancecausedbytheweightoftestmassesforasmallgap(left)andalargegap(right)betweentheelectrodes. atestmass,accordingto =0A C0,(4)whereisthegapsize,0isthepermittivityoffreespace,andAistheareaoftheelectrode.Thesmallgapwasfoundtobe0.62mmandthelargegap1.6mm.Withthesmallgap,thetestresultdemonstratesalinearrelationbetweenthecapacitanceandforceandexcellentsensitivityofthemagnetometer.A2.610)]TJ /F3 7.97 Tf 6.58 0 Td[(3Nforceonthesample,correspondingtotheweightofa0.26gmass,wouldresultinacapacitancechangeof0.026pF,nearlyveordersofmagnitudelargerthantheresolutionofthecapacitancebridge.Withthelargergap,themagnetometerhasanexcellentlinearityuptoatleast0.06N.Thefabricationoftheforcemagnetometersinvolvedmultiplestepsanddifferenttechniques,especiallysincethedesignchangedoverthreegenerations.Allofthepartsweredesignedbyourgroupandproducedbythemachineshopofourdepartment.Thebodyofthemagnetometerwasmadeofbrass.Thesampleplatformandelectrodewererstmadeofasilver-antimonyalloyandthenacopper-titaniumalloy. 48

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4.4ReducingtheBackgroundMagnetizationBecausetheplatform,oneoftheelectrodes,andtherodconnectingthemaresuspendedandexperienceaforceexertedbytheeldgradient,thechoiceofthematerialforthemisimportant.Useofapure,non-magneticmetalorplasticwillreducethebackgroundsignal.Amongmetals,silverisidealbecauseofnegligiblenuclear-spinheatcapacityandnegligiblenuclearmagnetization.However,non-magneticimpuritiesmustbeaddedtothemetaltoeliminatethedeHaas-vanAlpheneffect,whichwillresultinabackgroundmagnetizationthatoscillatesasafunctionofmagneticeld.Forourrstmagnetometer,wechose99%silverwith1wt%antimonybyweight,acompositionthathasworkedwellinotherapplicationsinourlaboratory.Figure 4-4 showsthatthesilver-antimonyalloy,hereafterAgSb,isslightlydiamagnetic,asexpected,withnodetectablecontributionfromparamagneticorferromagneticimpurities.Amongplastics,wetestedpolycarbonate,whichcanbeeasilymachinedandcanwithstandlowtemperatures.Asshowninthegure,however,themagnetizationofpolycarbonateshowsnonlinearcontributionfromparamagneticimpuritiessuperposedonthelineardiamagneticmagnetizationoftheplastic.Thematerialshouldnotbeusedforthesuspendedstructureofthemagnetometer,sincetheparamagneticimpuritieswillgiveabackgroundthatistemperaturedependent.TherstmagnetometerwasconstructedwithAgSbforthesuspendedstructure.AlthoughthediamagneticmagnetizationofAgSbwassmall,itstillturnedouttogivetoolargeabackgroundforourintendedapplications.Thesecondmagnetometerwasbuiltwiththesuspendedstructuremadeofcopperwith3wt%titanium,hereafterCuTi,whosemagnetizationissignicantlysmallerthanthatofAgSb[ 33 ].ThebackgroundgreatlyimprovedwithCuTicomparedtoAgSb,asshowninFig. 4-5 (left).Butthetimeconstantbecamelargeattemperaturesbelowabout0.2K,manifestingasalargehysteresisloop. 49

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Figure4-4. Comparisonofmaterialsconsideredfortheconstructionofthesuspendedstructureofthemagnetometers.MeasurementsweremadeinaSQUIDmagnetometerat2.0K.Brass(151.7mg,49.0mm3)andTi(290.77mg)areincludedforinformation.Polycarbonate(57.4mg,67.5mm3)containsmagneticimpuritiesthatwouldshowasabackground.Themagnetizationsofthesilver-antimonyalloy(190.6mg,18.0mm3),Ag3Mg(160.09mg),AgMg(61.78mg),andthecopper-titaniumalloy(577.39mg)shownegligiblecontributionofmagneticimpurities.TheAg3MgandAgMgsampleswerestoichiometricalloysprovidedbyShenLiQiuofFloridaAtlanticUniversity. 4.5Field-proleMeasurementsAforcemagnetometerrequiresaeldgradient,butthesuperconductingmagnetintheMillikelvinLaboratoryoftheNHMFLlacksagradientcoil.Toobtainaeldgradient,thesampleispositionedatanoff-centerpositioninthemagnet.Subsequently,measurementsarerepeatedattheeldcenter,andthedatasubtractedfromtheoff-centerdatatoremovetheparasitictorquecontribution. 50

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Figure4-5. (Left)ComparisonofthebackgroundoftherstmagnetometerutilizingAgSbforthesuspendedstructurewiththatofthesecondmagnetometerutilizingCuTi.ThebackgroundoftherstmagnetometerwasinferredfromthenickelcalibrationdatashowninFig. 4-7 ,whereasthatofthesecondmagnetometerwasdirectlymeasuredwithoutasample.(Right)Thehysteresisofthesecondmagnetometerat20mKindicatesalargetimeconstantarisingfromthenuclearheatcapacityofCu.Theeldwassweptat0.5T/minfortherstmagnetometer.Forthesecondmagnetometer,theratewas0.5T/minat0.23Kand0.2T/minat20mK. Todeterminetheeldgradientandtopinpointtheeldcenter,weemployedaToshibaTHS-118Hallsensortomeasuretheeldprole,whichisshowninFig. 4-6 .Thesensorwasmountedonasmall,squarecircuitboard,0.3inchalongthediagonal,whichinturnwasmountedonthesampleplatformofthemagnetometer.TheheightsofthecircuitboardandtheHallsensorweremeasuredtoaccountforthedifferencebetweenthesensorpositionandthesampleposition.TheHallresistancewasmeasuredslightlyabove1.6Kat5TbyusingaLakeShoreCryotronicsmodel370ACresistancebridge.Themeasurementsweremadeatevery0.025inchawayfromtheeldcenterandevery0.005inchwithin0.025inchoftheeldcenter.Thedataweretted 51

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Figure4-6. FieldprolemeasuredbyaHallsensor.Thepositionisrelativetothehighestpositionallowedbytherotarylinearfeedthroughtowhichthemagnetometerwasattached.Theredlinerepresentsthetofthedatafromwhichtheeldcenterandtheeldgradientwerefound. toaquadraticfunctionofposition,andthederivativeofthetyieldedtheeldgradientasalinearfunctionofposition.Themaximumintheresistanceindicatedtheeldcenter. 4.6MagnetometerCalibrationCalibrationofthemagnetometerisnecessarytoconvertameasuredchangeinthecapacitancetomagnetization.Thecalibrationwasmadebymakingmeasurementsonapieceof99.96%purenickelfoil,sincethemagnetizationofnickelisaccuratelyknown[ 34 ].Thenickelpieceusedforthecalibrationoftherstmagnetometer,withthesuspendedstructuremadeofAgSb,wasabout6.56mmindiameter,weighing30.62mg.Forthecalibrationofthesecondmagnetometer,withthesuspendedstructuremadeofCuTi,thenickelpiecewasabout6.68mmindiameter,weighing31.71mg.Thecapacitanceofthemagnetometerisinverselyproportionaltotheelectrodegapsize(seeEq. 4 ),andthedisplacementofthesuspendedelectrodeisproportionaltothemagneticforceexertedonthesample,whichisproportionaltotheeldgradient(Eq. 2 ),whichinturnisproportionaltotheeld.Therefore,(1/C-1/C0)/Hisproportionalto 52

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thesamplemagnetization,whereC0isthecapacitanceatH=0.ThisquantityisplottedagainstHinFig. 4-7 .Themagnetizationsaturatedatabout1T,andabovethatelditremainedconstant.Theeldwassweptatarateof0.5T/min.Bothmagnetometersshowahysteresisatleastupto5Tatnearlyallpositions.At0.540inch,therstmagnetometerexhibitsanonconstant(1/C-1/C0)/H,indicatingexcessivebackgroundmagnetizationofAgSb,whichisdiamagnetic.Thecalibrationconstantsobtainedfromthedataare2.12106emupFTfortherstmagnetometeratthe1.960-inchpositionand5.21105emupFTforthesecondmagnetometeratthe1.487-inchposition. Figure4-7. Resultsofmagnetizationmeasurementsmadeonnickeldisksforcalibration.(Left)Datafortherstmagnetometer,withthesuspendedstructuremadeofAgSb,attemperaturesbetween20mKand30mK.(Right)Dataforthesecondmagnetometer,withthesuspendedstructuremadeofCuTi,at0.23K.Inbothpanels,thepositionsindicatedarethoseoftherotarylinearfeedthroughtowhichthemagnetometerswereattached.Theeld-centerpositiondiffersbetweenthetwopanels,becausethetwomagnetometerswereattacheddifferentlytothefeedthrough. 53

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CHAPTER5COPPERPYRAZINEDINITRATEExactlysolvablemodelsarerare,yettheyoccupyaspecialplaceinphysics.Amongthem,oneofthesimplestisthespin-1 2linear-chainHeisenbergantiferromagnet.PropertiesofthismodelhavebeendescribedinSection2.3.Copperpyrazinedinitrate,Cu(C4H4N2)(NO3)2,knownasCuPzN,isoneofthefewmaterialsthatarewelldescribedbythismodel.Thiscoordinationcompoundconsistsofspin-1 2Cu2+ionslinkedbypyrazineringstoformchainsalongthecrystallographicaaxis[ 35 ],asshowninFig. 5-1 .Thechainsarefarenoughapartsothattheinteractionsbetweenspinsbelongingtodifferentchainsismuchsmallerthantheinteractionsbetweenspinsalongthechain.Asaresult,thespinsorderonlyat107mK,fromwhichtheinterchainexchangeJ0hasbeenestimatedtobelessthan4.410)]TJ /F3 7.97 Tf 6.58 0 Td[(3timestheintrachainJofabout10.3Kto10.6K[ 36 ].Thesaturationeldofthematerialis13.97T[ 37 ].Twoothermaterialsarealsowelldescribedbythespin-1 2linear-chainantiferromagneticHeisenbergmodel,DEOCC-TCNQF4[ 38 ]andSr2CuO3[ 39 ].DEOCC-TCNQF4hasthesmallestjJ0j/J=2.010)]TJ /F3 7.97 Tf 6.59 0 Td[(5knowntodate.ThetwoproblemswiththismaterialarethattherearenosinglecrystalsavailableandthatJis100K,withthesaturationeldestimatedtobe160T,muchhigherthancanbeproducednondestructivelybythepresentmagnettechnology[ 38 ].Sr2CuO3hasajJ0j/J=9.310)]TJ /F3 7.97 Tf 6.59 0 Td[(4andsinglecrystalscanbeproduced.Unfortunately,Jis2200K,withacorrespondingsaturationeldof3300T[ 40 ].CuPzNisthebestoptionbecauseoftheavailabilityofsinglecrystalsandtherelativelylowsaturationeld,whichcanbereachedinasuperconductingmagnet. 5.1PreviousWorkonCuPzNSpecicheatofCuPzNwasmeasuredbytheJohnsHopkinsgroup[ 42 ],atmagneticeldsupto9T,byemployinganundeuteratedsample.Theresults,showninFig. 5-2 ,indicatethatT-linearbehaviorispresentatlowtemperatures,indicativeof 54

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Figure5-1. CrystalstructureofCuPzN.Thestructureisorthorhombic,withspacegroupPmnaandroom-temperaturelatticeconstantsarea=6.712A,b=5.142A,andc=11.732A[ 35 ].Thecrystalismadeofcarbon(black),hydrogen(grey),nitrogen(green),oxygen(red),andcopper(gold).TheunitcellcontainstwoneighboringchainsAandB.Cu2+ionsarelinkedbypyrazinerings,C4H4N2,alongthecrystallographicaaxis,formingalinearspinchain.Theionsmakeconnectionsalsowithnitrates,NO3.Reprintedfrom[ 41 ].Copyright(2010)withpermissionfromJohnWileyandSons. aTLLasdescribedinSection2-3.ThespinonvelocityextractedfromthedatawillbeshownlaterinFig. 5-11 withourdata.TheJohnsHopkinsgrouphasalsomeasuredmagnetizationupto30Tat4.2Kand1.82K[ 42 ].AsshowninFig. 5-3 ,the1.82Kdataagreewellwithexactdiagonalizationofspin-1 2linearchainscontainingupto16spins.However,thetemperatureistoohightocomparewiththeexacttheory[ 43 ]atzerotemperature.Thedispersionoflow-energymagneticexcitationshasbeenmeasured,alsobytheJohnsHopkinsgroup,at0.25K[ 44 ].AsshowninFig. 5-4 ,thedataagreewell,withintheexperimentalresolution,withthetheoreticalspinondispersion.ThegfactorshavebeendeterminedbyESR;thevaluesforthethreeprincipaldirectionsarega=2.052,gb=2.275,andgc=2.063[ 45 ]. 55

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Figure5-2. SpecicheatofhydrogenousCuPzNmeasuredbyHammaretal.,shownasCoverTagainstT2.Reprintedgurewithpermissionfrom[ 42 ].Copyright(1999)bytheAmericanPhysicalSociety. Figure5-3. MagnetizationofCuPzNnormalizedtothesaturationmagnetizationasafunctionofeld,measuredbyHammaretal.at1.82Kand4.2K[ 42 ].Dash-dotlineistheexactcalculationforaspin-1 2linear-chainHeisenbergantiferromagnetatzerotemperature,basedontheBetheansatz[ 43 ].Reprintedgurewithpermissionfrom[ 42 ].Copyright(1999)bytheAmericanPhysicalSociety. 56

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Figure5-4. (a)MagneticinelasticneutronscatteringintensityofCuPzNatT=0.25KandH=0.Blacklinesdenotetheupperandlowerboundsofthetwo-spinoncontinuum.Inthehorizontal-axislabel,~qisthewavevector,calledkinChapter2,andthelatticeconstantahasbeentakentobe1.Reprintedgurewithpermissionfrom[ 44 ].Copyright(2003)bytheAmericanPhysicalSociety. 5.2ExperimentalSinglecrystalsofCuPzN,providedbyMarkTurnbullandChristopherLandeeofClarkUniversity,weregrownbyslowevaporationofamixtureofdeuteratedpyrazinewithaheavy-watersolutionofcoppernitrate[ 35 ].Thedeuterationwastominimizethecontributionofnuclearspinstothespecicheat,allowingaccuratedeterminationofthenon-nuclearmagneticcontributionatlowtemperaturesinhighmagneticelds. 5.2.1SQUIDMagnetometryMagnetizationmeasurementsweremadeona3.59mgsinglecrystalat2.0Kinmagneticeldsupto5TinaSQUIDmagnetometerofourdepartment.Thepurposeofthesemeasurementswastoidentifythecrystallographicbaxis.ThisistheaxisalongwhichwewishedtoapplyHforallspecicheatmeasurements,sincethegfactoristhelargestalongthisdirection,providingthelowestsaturationeldandthemoststable 57

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congurationduringthemeasurements.Additionalmagnetizationmeasurementsweremadeat4.2KtocalibratetheforcemagnetometerusedbyYoheiKonoinSakakibara'slaboratoryagainsttheSQUIDmagnetometer,whichinturnwascalibratedbymeasuringthemagnetizationofapalladiumstandard. 5.2.2CalorimetrySpecicheatwasmeasuredintheMillikelvinLaboratoryoftheNHMFLonasinglecrystalofa1.59mgmass,byusingtherelaxationmethod[ 27 ].Themeasurementsweremadeatzeroeld,3T,6T,9.5T,11T,12.5T,and13.1T,attemperaturesbetween0.1Kand7.5K,withthebaxisparalleltotheeld.Inaddition,measurementsweremadeinaphysicalpropertymeasurementsystem(PPMS),acommercialautomatedsystem,byHaidongZhouattheNHMFL.Thesemeasurementsweremadeona3.53mgsinglecrystalattemperaturesbetween2.3Kand200Katzeroeld,3T,6T,and8.5T. 5.2.3ForceMagnetometryThesamplewasa3.59mgcrystal,differentfromthesamplefortheSQUIDmagnetometry.ThemeasurementsweremadebyYoheiKonoinSakakibara'slaboratoryattheUniversityofTokyoonacapacitiveforcemagnetometerineldsupto14.7Tappliedalongthebaxis.A3He-4Hedilutionrefrigeratoranda3Herefrigeratorwereemployedforthemeasurementsinthetemperatureranges0.08KT2Kand0.3T15K,respectively.ThemagnetometerwascalibratedagainsttheSQUID-magnetometerdataat4.2K.MeasurementswerealsomadewithoursecondmagnetometerdescribedinChapter4,intheMillikelvinLaboratoryoftheNHMFLatabout25mK,thebasetemperatureofthedilutionrefrigerator,inmagneticeldsupto18T.However,thequalityofthedatawasinsufcientforourpurpose. 5.3ResultsItwasknown,priortoourstudy,thatCuPzNcrystalsalwaysgrowwithlongedgesparalleltotheaaxis,butitwasnotknownwhetherthebaxisisparallelorperpendiculartotheatfaces[ 46 ].TondthedirectionofthebaxiswithoutusingX 58

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Figure5-5. MagnetizationofCuPzNat2.0KusingaSQUIDmagnetometer.Themagnetizationwasmeasuredwiththecrystalinat-face-up(blue)andedge-up(red)congurations.Opencirclesindicatetheat-face-updatamultipliedbytheratioofthegfactorsgc=gb,withtheelddividedbythesameratio. rays,wemeasuredthemagnetizationofasampleontheSQUIDmagnetometerat2.0Kfortwocongurations,asshowninFig. 5-5 .Intheat-face-upconguration(blue),themagneticeldwasperpendiculartotheatfaces.Intheedge-upconguration(red),theeldwasparalleltothesefacesandperpendiculartothelongedges.Theat-face-upmagnetizationislargerthantheedge-upmagnetization,indicatingthatthebaxisisperpendiculartotheatfacesofthesampleandthecaxisisperpendiculartotheatfacesandthelongedges.Indeed,theat-face-updatamultipliedbytheratioofthegfactors,gc/gb(opencircles),yieldsaclosematchtotheedge-updata,conrmingthisconclusion. 59

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Figure5-6. SpecicheatofCuPzNattemperaturesupto30K.Thelinerepresentsthephononcontribution. 5.3.1SpecicHeatFigure 5-6 showsthespecicheatofCuPzNwiththebaxisparalleltothemagneticeld.ComparisonoftheresultwiththeoryrequiresthatthephononandnuclearcontributionsbesubtractedtoobtainthemagneticspecicheatC.Atzeroeld,thespecicheatofnuclearspinsisabsent,andthemagneticspecicheatdoesnotextendtoashighatemperatureasinamagneticeld.Forthesereasons,theT3contributionofphononswasdeterminedatzeroeldandsubsequentlysubtractedfromthedatatakenatallelds.ThedeterminationwasmadesuchthattheentropycomputedbyintegratingC/TintermsofTbestapproachesthevalueRln2astemperatureincreases.AsshowninFig. 5-7 ,wend1675J/molK4tobethebestvalueforthecoefcientforthe 60

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Figure5-7. MagneticentropyofCuPzNasafunctionoftemperatureatzeroeldfordifferentchoicesofthecoefcientbfortheT3phononterm.Rln2forthehigh-temperaturelimitisindicatedbythehorizontalline. phononspecicheatwhereas2425J/molK4chosenbyHammaretal.(purple)[ 42 ]isclearlyanoverestimate.ThedeuterationofthesamplereducesthenuclearspecicheatCn,whichisproportionaltoH2/T2,toabout14%ofthatofahydrogenoussample.Althoughthisisasmalleramount,itstillhastobesubtractedfromthedata.Todoso,werstexamineC/Tafterthesubtractionofthephononcontribution,plottedinFig. 5-8 ,andidentifyconstantregionsthatrepresenttheTLLregime.Sucharegionexistsinthedataupto11T.Then,wetthedataatandbelowthoseregionssimultaneouslytoaH2/T2termplusaelddependentconstantterm.Wendthat Cn=1.910)]TJ /F3 7.97 Tf 6.58 0 Td[(5H2 T2(5) 61

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Figure5-8. C/TvsTofCuPzNbelow1Kforeldsupto13.1Tafterthesubtractionofthephononcontribution.Solidlinesrepresentthenuclearcontributionforthecorrespondingcolors,withconstantsaddedtorepresentthetotalspecicheatatlowtemperatures. besttsthenuclearcontributionupto11T,asshowninthegure.Thisnuclearcontributionisthensubtractedfromthespecicheatatallelds.Themagneticspecicheat,afterthesubtractionofthephononandnuclearcontributions,isshowninFig. 5-9 asC/TasafunctionofT.Atzeroeld,thedataexhibitabroadpeakatabout3.5K.Attemperaturesbelow1K,aconstantC/TindicatestheTLLregime.Astheeldisincreased,thebroadpeakshiftstolowertemperatures,compressingtheTLLregime.Asshown,theoreticallines,calculatedbyChisaHotta[ 47 ]byusingthequantumtransfer-matrixmethod[ 12 13 ],areingoodagreementwiththeexperiment.Tocomparethetheorywiththedata,wehaveusedJ=10.81K,whichisobtainedfromthesaturationeldHsat=13.97Tandgb=2.305determinedfromthe 62

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Figure5-9. MagneticspecicheatofCuPzNplottedasC/TvsTforconstantmagneticelds.Therightpanelisanexpandedviewofthetemperatureregionbelow2K.Solidlinesdenotetheoreticalresults. magnetizationdataat80mKshowninFig. 5-10 .Theslightupturninthezero-elddataattemperaturesbelowabout0.3Kmaybeaprecursoroftheorderingat107mKoracontributionofnuclearquadrupolemoments. 5.3.2SpinonVelocityInaTLL,theT-linearspecicheatarisesfromthelineardispersionoflowenergyexcitations,whichinaspin-1 2linear-chainHeisenbergantiferromagnetarefermionicspinons.Theslopeofthedispersion,thespinonvelocity,isrelatedtothespecicheatthroughEq. 2 .Ateldsupto11T,thespinonvelocitywasextractedfromthedatausingthisrelation.At12.5Tand13.1T,wherenoregimesofconstantC/Texist,C/Tforthedeterminationofvswastakenatthelocalminimaattemperaturesbelowthatofthebroadpeak.TheresultsareshowninFig. 5-11 ,demonstratingexcellentagreementwiththetheoreticalpredictionbasedontheBetheansatz[ 7 ].OurresultsarealsoingoodagreementwiththedataoftheJohnsHopkinsgroupintheregionofoverlap,upto9T[ 42 ].Thisisthersttimethevelocityofanylow-energymagneticexcitations,letalone 63

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Figure5-10. MagnetizationM(bluecircles)anddifferentialsusceptibility(greencircles)ofCuPzNat80mKasafunctionofthemagneticeld,appliedparalleltothebaxis.ThemeasurementsweremadebyYoheiKonoinSakakibara'slaboratory[ 37 ].BestttotheexactsolutionoftheBetheequationsforthespin-1 2linear-chainHAFatzerotemperature[ 43 ]isshownbytheredline. spinons,hasbeendeterminedina1Dantiferromagnetinsuchawideeldrange,uptoaeldclosetothesaturationeld. 5.3.3MagnetizationThetemperaturedependenceofthemagnetizationisshowninFig. 5-12 ,wheremagnetizationMhasbeendividedbythemagneticeldH.Themeasurementsweremadeateldswherethespecicheatwasmeasuredexceptthat1Twaschoseninlieuofzeroeld,atwhichmagnetizationcannotbemeasured.AlsoshownaretheoreticalcurvescalculatedbyChisaHotta[ 47 ],whohasusedthequantumtransfer-matrixmethod[ 12 13 ].WehaveagainusedJ=10.81Kandgb=2.305tocomparethetheorywiththeexperiment.Thedataagreewellwiththetheory.First,thereisabroadpeakatabout7Kinzeroeld.Thepeakbecomessharperandmovestolowertemperaturesas 64

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Figure5-11. SpinonvelocityinCuPzNasafunctionoftheappliedeld.Circlesareourresultsobtainedfromthespecic-heatdatashowninFig. 5-9 .DiamondsarethedataoftheJohnsHopkinsgroup[ 42 ].LinerepresentstheorycalculatedfromtheBetheansatz[ 7 ],asshowninFig. 2-4 theeldincreases.Second,themagnetizationateldsupto11Tbecomesconstantasthetemperatureapproacheszero,consistentwiththebehaviorofaTLL.Magnetizationmeasuredasafunctionofeldupto14.7Tat80mKshowscontinuousincreaseinmagnetizationfromzeroelduptothesaturationeldof13.97T,asshowninFig. 5-10 .Thegraduallyincreasingslopeofthemagnetizationwithincreasingeldischaracteristicofalowdimensionalantiferromagnet.ThedataareinexcellentagreementwiththetheoreticalcalculationsfromtheBetheansatz[ 43 ],unlikethepreviousdatatakenat1.82KandshowinFig. 5-3 .Abovethesaturationeldaconstantmagnetizationindicatestheeld-inducedferromagneticstate. 65

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Figure5-12. TemperaturedependenceofM/HofCuPzNateldsupto13.1T.Linesaretheoreticalresults.ThedataweretakenbyYoheiKonoinSakakibara'slaboratory[ 47 ]. 5.3.4WilsonRatioToobtaintheWilsonratiodenedbyEq. 2 ,themagnetizationdataat80mKwasdifferentiatedwithrespecttothemagneticeldtondthemagneticsusceptibility,asshowninFig. 5-10 .TheC/Tvalues,alsoneededfortheWilsonratio,weretakenfromthedatashowninFig. 5-9 .OurvaluesoftheWilsonratioarecomparedinFig. 5-13 with4K,whereKistheTLLparametercalculatedfromanexactsolutionoftheBethe-ansatzequations[ 7 ]andshowninFig 2-4 .Atallelds,theexperimentyieldsaWilsonratiowithin4.7%ofthetheoreticalvalue.Thezero-elddatapointisshowninredtoindicatethatitisexpectedtobeoffduetologarithmicbehaviorofthesusceptibility[ 48 ]. 5.4DiscussionWehavemeasuredthespecicheatofdeuteratedCuPzNasafunctionoftemperatureinmagneticeldsupto13.1T,closetothesaturationeldof13.97T.Formagneticeldsupto11T,thecharacteristicsofaTomonaga-Luttingerliquidwas 66

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Figure5-13. WilsonratioofCuPzNasafunctionofthemagneticeld.LineisanexactsolutionoftheBethe-ansatzequationsfortheTLLparameterK[ 7 ],showninFig. 2-4 ,multipliedby4.Thedatapointatzeroeldisshowninredtoindicatethatweexpectittobeoffduetologarithmicbehaviorofthesusceptibility[ 48 ]. observedatlowtemperatures.Agreementswiththeoryareexcellentovertheentireeldrange.AlsoinexcellentagreementwiththeoryarethespinonvelocityextractedfromthespecicheatandtheWilsonratioobtainedfromthespecicheatanddifferentialmagneticsusceptibility.TheseresultsdemonstratethatCuPzNisanearlyidealspin-1 2linear-chainHeisenbergantiferromagnet. 67

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CHAPTER6DLCBAspinladderconsistsoftwoparallelspinchainsthatarecoupled.Quantumspinladderscomprisingspin-1 2arerichinphysicalpropertiesandhavereceivedmuchattention.Thesesystemshaveanitegapintheexcitationspectrum,asdescribedinSection2.4.Atzeroeld,inelasticneutron-scatteringexperiments[ 49 53 ]haveshownone-magnonandtwo-magnonexcitations,conrmingthetheory[ 54 56 ].Inamagneticeld[ 14 57 ]thesesystemsexhibitquantum-criticalbehaviorssuchasBose-Einsteincondensationofmagnons[ 58 59 ],magneticBoseglass[ 60 ],andTomonaga-Luttingerliquidphases[ 8 16 61 63 ].Inacoupledarrayofladders,thesubjectofthepresentstudy,aquantumphasetransitionfromthespin-liquidstatetoamagneticallyorderedstateisexpectedtooccurwhenthestrengthoftheinterladdercouplingisincreased[ 18 64 65 ].C9H18N2CuBr4,alsoknownasDLCB,isacoupledspin-1 2laddersystem,inwhichCu2+ionscarryspin1 2.ThecrystalstructureofDLCBisshowninFig. 6-1 .Thestructureistriclinic,withspacegroupP1,andthelatticeconstantsat85Karea=7.459A,b=8.270A,c=13.720A,=107.41,=90.21,and=91.37[ 66 ]. 6.1PreviousWorkonDLCBFrommagnetizationdataat1Tattemperaturesbetween2Kand300K,thelegexchangeandrungexchangehavebeendeterminedtobeJk=7.95KandJ?=4.07K[ 66 ].Magnondispersionhasbeenmeasuredbyaninelasticneutronscatteringexperiment,yieldingJk=7.00.2K,J?=7.41.0K,andJint=2.20.2K[ 67 ].Thesevalues,especiallyJ?,aredifferentfromthevaluesfromthemagnetizationdata,whichwasanalyzedontheassumptionthatJint=0[ 66 ].TheratioofJinttoJkis0.320.35,slightlyexceedingthecriticalvalueofabout0.30predictedforaladderwithJ?=Jk[ 18 ].Indeedthespinsorderinthismaterialatabout2.0K[ 67 ].However,thelong-rangemagneticordercoexistswithamagnonenergygapof3.50.2K,whicharisesfroma 68

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Figure6-1. (Top)IllustrationofthespinladderinDLCB.(Bottom)HydrogenbondinginDLCB.ReddottedlinesrepresentBrBrcontacts,andblueandorangedottedlinesindicatehydrogenbondsfromtheDMA+cationsandthe35DMP+cations,respectively.Theparallelepipedinlightgreyindicatestheunitcell;thered,green,andblueedgesareparalleltothea,b,andcaxes,respectively.Reprintedgurewithpermissionfrom[ 66 ].Copyright(2008)bytheAmericanPhysicalSociety. weakIsinganisotropy.Theinteraction-anisotropyparameterwasfoundtobe=0.93accordingtoinelasticneutronscatteringmeasurements[ 67 ].Hereisdenedby H=X,hi,jiJ[SziSzj+(SxiSxj+SyiSyj)],(6)whereisk,?,orint. 6.2ExperimentalSinglecrystalsoffullydeuteratedDLCBweregrownbyFirasAwwadioftheUniversityofJordanbyslowevaporationofaheavy-watersolution,asdescribedinRef.[ 66 ].Thepurposeofthedeuterationwastoreducethecontributionofnuclearspinstothespecicheat.Thecrystalstructurewasdeterminedat4Konthefour-cycle 69

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diffractometerHB3AattheHighFluxIsotopeReactor,OakRidgeNationalLaboratorybyTaoHong[ 67 ].TheseresultsconrmedthestructurereportedearlierinRef.[ 66 ]. 6.2.1CalorimetryPreliminaryspecic-heatmeasurementsweremadedownto0.5KbyutilizingaQuantumDesignPPMSbyHaidongZhou.Specicheatmeasurementsattemperaturesbetween0.1Kand2.4Kweremadebyusingtherelaxationmethod[ 27 ]intheMillikelvinLaboratoryoftheNHMFLinmagneticeldsupto15.7T.Themeasurementswereperformedon7.90mgand5.53mgsamples.Fortherstsample,theeldwasorientedperpendiculartotheabplaneofthecrystal(H?ab).Forthesecondsample,theelddirectionwasparalleltotheabplaneandperpendiculartothebaxis(HkabandH?b).Magnetocaloric-effectmeasurementswerealsomadenearthesaturationeldattemperaturesbelow0.5Kforbothelddirections. 6.2.2MagnetizationMeasurementsMagnetizationmeasurementsweremadeat0.23Kona78.24mgcrystalonoursecondforcemagnetometeremployingaCuTisuspendedstructure.Thecrystalwaspositionedintwoorientations.Intherstconguration,theeldwasperpendiculartotheabplane;inthesecond,theelddirectionwasabout30awayfromtheaaxistowardthe)]TJ /F4 11.955 Tf 9.3 0 Td[(bdirection.Themagnetometerwaspositioned0.425inchbelowtheeldcentertoprovideaeldgradient,whichwas5.68T/mat5T,thenattheeldcenter.Theeldwassweptat0.05T/minupto4Tand0.1T/minfrom4Tto18T.Fromeachdata,abackgroundmeasuredatthesamepositionwithoutthesamplewassubtracted.Subsequently,theeld-centerdatawassubtractedfromtheoff-centerdatatoremovethetorquecontribution.TheeldcenterwaslocatedusingaHallsensormountedonthemagnetometerasdescribedinSection4.5.1.CalibrationagainstanickelstandardwasperformedasdescribedinSection4.5.2. 70

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6.2.3MagneticTorqueMagnetictorquemeasurementsweremadebyusingberyllium-coppercantileversat80mKona5.65mgsample.Firstthesamplewasplacedona0.001-inchthickcantileversuchthatbyrotatingthetorquemagnetometerinsitu,elddirectionsfromH?abtoHkabandH?bcanbestudied.Subsequently,thesamplewasrepositionedonthecantilevertoinvestigateelddirectionsfromH?abtoHkab.However,thiscantileverturnedouttobetoosensitiveforsomeelddirectionsinthisrange,touchingthexedelectrodeatabout12.7T.A0.002-inchthickcantileverwasthenusedtodeterminethesaturationeldforelddirectionsfromH?abtoHkab.Foreachmeasurement,theeldwassweptat0.3T/min. 6.3ResultsFigure 6-2 showsspecicheatasafunctionoftemperatureforH?abandforHkabandH?b.Ateacheld,asharppeakindicatesatransitiontotheorderedphase.Fittingthezero-elddatabelow0.9Ktotheformulaforthespecicheatofaone-dimensionalS=1 2gappedHeisenbergantiferromagnet[ 68 ], Cm(T)/( kBT)3 2e)]TJ /F15 5.978 Tf 12.48 3.25 Td[( kBT,(6)givesanenergygapof=3.40.2K[ 67 ].InanS=1 2spin-laddersystem,thepresenceofagapusuallymeansthatthegroundstateisquantumdisordered.InthecaseofDLCB,whichorders,thegapiscausedbytheweakIsinganisotropy[ 67 ].Measurementswerealsomadeasafunctionofeld,whileholdingthetemperatureconstant,asshowninFig. 6-3 .Thesemeasurementsallowedustodetectthetransitionwherethetransitiontemperaturechangesrapidlywitheld,nearthesaturationeldandatthespin-optransition.Themeasurementsweremadeattemperaturesrangingfrom218mKupto1.34K.Todetectthetransitionateldsnearthesaturationeld,wealsomademagnetocaloric-effectmeasurements.Figure 6-4 showstheresultsforHkabandH?b.Ateach 71

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Figure6-2. Specicheatasafunctionoftemperatureatseveraleldsfromzeroupto15.7T.(a)WhenH?ab,notransitionoccursabove2.0K.(b)WhenHkabandH?b,thetransitiontemperatureexceeds2.0Kateldsrangingfrom3Tto8T. Figure6-3. Specicheatversuseld.Themeasurementswerefocusedaroundthepeakatthespin-optransitiontopreciselyidentifythetransitioneld.Themagneticeldwasperpendiculartotheabplane. 72

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Figure6-4. Magnetocaloric-effectdatanearthesaturationeldforHkab,H?b.Thetransitionoccurswherethecurvesforanupwardeldsweep(red)crossesthecurveforadownwardeldsweep(blue).Thechangeintemperaturehasbeenexpandedbyafactoroften. temperature,dataweretakenwhilesweepingtheeldupwardthendownwardatarateof0.05T/min.Thetransitioneldisdeterminedbythecrossingoftheupward-sweepcurvewiththedownward-sweepcurve.Below0.2Kthismethodunderestimatesthetransitioneldbecauseofthemagnetocaloriceffectofthenuclearspins.Magnetizationcurvesat0.23KareshowninFig. 6-5 forH?ab,andfortheeldmakinganangleofabout30withtheaaxis.WhenH?ab,themagnetizationisnearlyzerouptoabout1.5T,andatransitionisseenatabout2.37Twherethemagnetizationincreasesrapidly,indicatingaspin-optransition,asexpectedfromtheweakIsinganisotropyfoundbyinelasticneutronscattering[ 67 ].Thevalueofthesaturation 73

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Figure6-5. Magnetizationfor(a)H?aband(b)aelddirectionabout30awayfromtheaaxis.Themeasurementsweremadeat0.23K.Therapidrisenearthesaturationeldindicateslowdimensionality.Aspin-optransitionisobservedatabout2.37TforH?ab. magnetization,Msat,isusedtondthegfactorvia Msat=NAgSB,(6)whereS=1 2forDLCB.Wegetg=1.88forH?aband1.79fortheotherdirection.Themagnongap=3.480.23Kfoundbytheinelasticneutronscatteringpredictsaspin-optransitioneld/gB=2.80.2T,whichislargerthantheactualeldbyabout20%,suggestingthatourgfactorhasbeenunderestimated,possiblyasaresultofasystematicerrorinthecalibrationofthemagnetometer.ThesaturationeldHsatforH?abisgivenby (2Jn+J?+Jint)1+ 2=gBHsat.(6)Thisequation,inconjunctionwiththeexchangeconstantsandfrominelasticneutronscatteringgivesHsat=18.00.9T.Thisisconsiderablylargerthan16.34Tdeterminedfromthemagnetizationcurve,againsuggestingthatourgfactorisanunderestimate. 74

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Themagnetic-torquedataateldsupto3.5TareshowninFig. 6-6 forelddirectionsbetweenH?abandHkabH?b.ForH?ab,asharptransitionisobservedatabout2.42Tinagreementwiththemagnetizationdata.Thetransitionbecomessubstantiallyroundedwhentheeldistiltedby10towardstheHkabH?bdirectionanddisappearsatananglebetween15and20. Figure6-6. Magnetictorqueasafunctionofthestrengthanddirectionofthemagneticeld.Theangleisbetweenthedirectionoftheeldandthedirectionperpendiculartotheabplane.At90,theeldisparalleltotheabplaneandperpendiculartothebaxis.Inpanel(a),yellowstandsforzerotorque,andvioletthestrongesttorque.Atransitionisclearlyobservedatabout2.42T.Measurementsweremadeat80mKona0.001-inchthickBeCucantilever. Figure 6-7 showsthesaturationelddeterminedbythetorquemeasurementsasafunctionoftheelddirection.ThevalueforH?abisingoodagreementwiththatfromthemagnetizationcurve.Thesaturationeldisnearlyconstantforelddirectionsperpendiculartotheaaxis,suggestingthatthegfactorisnearlyconstantforthesedirections.Phasediagramswereconstructedfromthespecic-heat,magnetocaloric-effect,magnetization,andmagnetic-torquedata,asshowninFig. 6-8 .Theygiveclear 75

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Figure6-7. Saturationeld,determinedbythetorquemeasurementsat80mK,asafunctionoftheelddirection.istheanglebetweenthedirectionoftheeldandthedirectionperpendiculartotheabplane.Forcircles,whichrepresentdatatakenwitha0.001-inchthickcantilever,theeldisperpendiculartothebaxiswithintheabplaneat90.Forsquares,whicharefordatatakenwitha0.002-inchthickcantilever,theeldisparalleltothebaxisat90. evidencefortheanisotropyofthesystem.Thespin-optransitionoccurringintheH?abdirectionisconsistentwiththeresultofinelasticneutronscattering,whichshowsthatspinsintheorderedstateatzeroeldarealignedparalleltothisdirection[ 67 ].Thespin-optransitionoccursatabout2.42Tinthelimitofzerotemperature,movingslightlytohighereldswithincreasingtemperature.At1.49K,itreaches3.0T.WhenHkabandH?b,thespin-optransitionisabsent. 6.4DiscussionWehavemadespecic-heat,magnetocaloric-effect,magnetization,andmagnetic-torquemeasurementsondeuteratedDLCBtodeterminethecriticaleldsandtheexchange-anisotropyparameter.Aspin-optransitionisfoundwhenthemagnetic 76

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Figure6-8. Phasediagramsfor(a)H?aband(b)HkabH?b.Solidcirclesarefromspecicheatvstemperature,squaresfromspecicheatvseld,opencirclesfromthemagnetocaloriceffect,trianglesfromthemagnetization,anddiamondsfromthemagnetictorque. eldisinthedirectionperpendiculartotheabplaneofthecrystal,indicatingtheaxisoftheweakIsinganisotropyalongthisdirection.Thisagreeswiththespindirectionsdeterminedpreviouslybyneutronscattering.Finally,fromthespecicheatatzeroeldwend=3.40.2K,inexcellentagreementwith=3.50.2Kdirectlydeterminedbyinelasticneutronscattering[ 67 ]. 77

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CHAPTER7TNNCH3CNSpintrianglesarerichwithquantumeffectsthatareinteresting,asmentionedinSection2.5.However,therehasbeenachallengeinndinganidealspintriangle.Forexample,thewellknownspin-trianglesystemNa9[Cu3Na3(H2O)9(-AsW9O33)2]26H2O,theso-calledfCu3g[ 70 71 ]suffersfromJahn-TellerdistortionwhichmakestheexchangeinteractionbetweenoneCu-Cupairstrongerthanthosebetweentheotherpairs.ThisbreakingofthreefoldsymmetrysplitstheStotal=1 2groundstatesintotwoenergylevelsaliftingofdegeneracydetrimentaltothenovelmultiferroiceffectsdescribedinSection2.5.Enterorganicchemistry.Tris[4-(1-oxyl-3-oxide-4,4,5,5-tetramethyl-imidazolin-2-yl)phenyl]amine,TNNforshort,isamoleculecontainingthreenitronylnitroxideradicals,eachofwhichcarryinganS=1 2spin[ 72 ].AsshowninFig. 7-1 ,theradicalsarelinkedtoanitrogencorebyphenylgroups,thusformingaspintriangle.Sincethespinsarecarriedbyorganicradicalsinsteadoftransition-elementions,thereisnoJahn-Tellereffect,leavingtheexchangeinteractionsbetweenthepairsofspinsequal.Owingtotwistingofthearmsattachedtothenitrogencore,therearetwostereoisomersofTNNright-handedandleft-handedenuntiomers.Todate,TNNhasbeengrownonlyintomicrocrystals,toosmallfordetailedstudiesofitsmagnetism[ 72 ].However,itcanbegrownintomillimeter-sizecrystalswithacetonitryl[ 73 ].Inthismolecularcrystal,TNNCH3CN,right-handedisomersandleft-handedonesformalternatinglayersparalleltothecrystallographicabplane.Withineachlayer,theTNNmoleculesformatriangularlattice,asshowninFig. 7-2 .ThismakesTNNCH3CNanexcellentmaterialinwhichtolookforthenovelmultiferroicphasespredictedbyKamiyaandBatista[ 24 ]. 78

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Figure7-1. StructureoftheTNNmolecule.Bluespheresindicatenitrogen,redspheresareoxygen,andsmallgreyspheresarecarbon.Hydrogenatomsarenotshown.Boxesindicatenitronylnitroxideradicals,whichcarryS=1 2spins.TakenfromRef.[ 69 ]. Figure7-2. TNNCH3CNcrystalstructure,takenfromRef.[ 69 ].Thestructureistrigonal,withspacegroupR3c,andthelatticeconstantsat25Karea=b=15.048Aandc=29.916A[ 74 ].Left-handed(blue)andright-handed(red)enuntiomersofTNNformalternatingtriangular-latticelayersparalleltotheabplane.Theunitcellcontainssixsuchlayers. 79

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7.1ExperimentalTNNwassynthesizedthroughawet-laboratoryproceduresimilartotheonedescribedinRef.[ 72 ].Themoleculeswerethendissolvedindichloromethaneandplacedinanopenvial,whichinturnwasplacedinalargersealedvialcontainingacetonitryl.TheacetonitrylvaporinthesealedvialwasabsorbedbytheTNNsolution,outofwhichTNNCH3CNcrystalsgrew[ 73 ].ThesynthesisofTNNandthegrowthofTNNCH3CNcrystalswereperformedbyKosukeTakada,AyakaHigashiguchiandNaokiAmayainYukoHosokoshi'slaboratoryatOsakaPrefectureUniversity.Singlecrystalswereusedinallmeasurements,whichweremadeattheMillikelvinLaboratoryoftheNHMFLexceptformagnetizationmeasurementsandsomespecic-heatmeasurements. 7.1.1SpecicHeatSpecicheatmeasurementsweremadeintheMillikelvinLaboratoryoftheNHMFLonasinglecrystalofTNNCH3CNwithamassof0.37mg.At12T,11T,10T,5T,andzeroeld,themeasurementsweremadeasafunctionoftemperaturefromabout0.1Kuptoabout0.7Ktolocateatransition.Furthermeasurementsweremadeatseveralmagneticeldsrangingfrom0.75Tto8.5Tattemperaturesbetween0.2Kand0.6K.Inaddition,measurementsbetweenweremadeona1.39mgsampleatzeroeld,0.75T,and5TbyusingaQuantumDesignPPMSbyKosukeTakadainHosokoshi'slaboratory. 7.1.2MagnetizationMeasurementsMagnetizationmeasurementsusingaforcemagnetometer[ 30 ]wereperformedbyYasuyukiShimura,KosukeTakada,andHironoriYamaguchiinSakakibara'slaboratoryonasinglecrystalofTNNCH3CNwithamassof1.30mg.Themeasurementsweremadeat0.13Kinmagneticeldsupto14.7T.Theeldwasappliedperpendiculartothelarge(012)facesofthecrystal.Furthermeasurementsweremadeasafunctionoftemperatureat0.1T,5T,and10T.ThemagnetometerwascalibratedagainstaSQUIDmagnetometerat4.2Kineldsupto5T. 80

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7.1.3DielectricMeasurementsTosearchfortheorderingofinducedelectricdipolemomentspredictedbytheory[ 24 ],wecarriedoutdielectricmeasurementsintheMillikelvinLaboratoryoftheNHMFLasfunctionsofmagneticeldandtemperature.Twosinglecrystalswereused:sample1,anuncutcrystalweighing3.48mg,andsample2,weighing0.44mgwithtwoparallelsurfacescutalongtheabplane.Onsample1,goldwassputteredonlarge(012)faces,byEunSangChoioftheNHMFLinTallahassee,toformelectrodes.Sample2wascutbyMinseongLeeinChoi'sgroup,andsilverepoxywasappliedonthecutsurfacesbyChoitoformelectrodes.Consequently,theacelectriceldforthemeasurementswereperpendiculartothe(012)planeforsample1andparalleltothecaxisforsample2.Forbothsamples,thecapacitancebetweenthetwoelectrodesweremeasuredwithanAndeen-HagerlingAH2700Aultra-precisioncapacitancebridgeat5kHz.At5Tand0.5K,measurementsonsample1werealsomadeat100Hz,1kHz,and20kHz,andnofrequencydependencewasfound,asshowninFig. 7-3 .Thesampleswereplacedonarotatingtop-loadingprobeofthedilutionrefrigerator.Forsample1,themagneticeldwasappliedeitherparalleltothecaxis(Hkc)orparalleltothe[120]direction,perpendiculartothecaxis.Forsample2,themagneticelddirectionwaseitherHkcorperpendiculartothecaxis(H?c). 7.2ResultsThemagnetizationat0.13K,showninFig. 7-4 ,revealstheevolutionofthegroundstateasafunctionofthemagneticeld.Theone-thirdplateaurangingfromabout1.3Tto8.5Tindicatesthetwofold-degenerateS=1 2Sz=+1 2state,asexplainedinSection2.5.Infact,thisdegeneracyisliftedaswillbediscussedinSection7.4.Saturationoccursatabout11T.Theregionsinwhichthemagnetizationchanges,fromH=0toabout1.3Tandfromabout8.5Ttoabout11T,areduetointermolecularinteractions,asalsoexplainedinSection2.5,andarethereforeexpectedtobeantiferromagneticallyordered. 81

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Figure7-3. Dielectricresponseofsample1asafunctionoftemperatureat5T,measuredatfourfrequencies.Themagneticeldwasappliedparalleltothecaxis. Figure 7-5 showsthemagnetizationasafunctionoftemperatureat0.1T,5T,and10T.At0.1Tand10T,themagnetizationexhibitsananomalycharacteristicofanantiferromagnetictransitionatabout0.25Kandabout0.35K,respectively.Asexpectednomagnetictransitionoccursat5Tatleastdowntoabout80mK.AsshowninFig. 7-6 ,thespecicheatatH=0and10Texhibitsapeakat0.25Kand0.35K,respectively,inagreementwiththemagnetizationdata.However,thespecicheatalsoshowsabroadpeakat5Tat0.36K.Thisisunexpectedsincenotransitionisseenatthesameeldinthemagnetizationdata.Infact,apeakappearsinspecicheatatalleldsupto10T,attemperaturesbetweenabout0.26Kand0.37K.Dielectricmeasurementsgiveanimportantcomplementaryinformationaboutthetransitionsseenintheaforementionedmeasurements.TheresultsareshowninFig. 82

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Figure7-4. MagnetizationofTNNCH3CNat0.13K,showingthreemagnetictransitions,at1.3T,8.5T,and11T.TakenfromRef.[ 69 ] 7-7 .Peaksareobservedatintermediateeldsbetweenabout1.3Tandabout8.5T,signifyingtransitions.Transitionsathigherandlowereldsaresigniedbyshoulder-likefeatures.Thedirectionofthemagneticeld,whetherparalleltothecaxisorparalleltothe[120]directionperpendiculartothecaxis,haslittleeffectonthetemperaturesatwhichthepeaksandshoulder-likefeaturesoccur.Whentheacelectriceldisappliedparalleltothecaxis,theoveralldielectricresponsedecreasesbyafactorofabout6,insupportofthetheoreticalprediction[ 24 ]thattheelectricdipolesliewithintheabplane.Theeld-dependentdielectricdata,presentedinFig. 7-8 ,showtransitionsaspeaksandbreaksinslope.Uptovetransitionsmaybeseenforanygivenorientationofthemagneticeld,includingaatteningofthecurveatabout11T,thesaturationeld.Abovethiseldthedielectricresponseshowslittletemperaturedependence,withnofeatures,ascanbeseeninFig. 7-7 .Again,measurementsmadewiththeacelectriceldalongthecaxisresultinasubstantialdecreaseindielectricresponse,conrmingthattheelectricdipoleslieprimarily,ifnotcompletely,withintheabplane. 83

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Figure7-5. MagnetizationovereldasafunctionoftemperatureofTNNCH3CNat0.1T(left),5Tand10T(right)takeninSakakibara'slaboratory.Anantiferromagnetictransitionisseenatabout0.25Kat0.1Tandabout0.35Kat10T.TheleftpanelhasbeentakenfromRef.[ 69 ],therightpaneladoptedfromthesamesource. Figure7-6. (a)SpecicheatofTNNCH3CNatzeroeld,5T,and10T,allshowingpeaksindicativeofaphasetransition.(b)Apeakappearsinspecicheatattemperaturesbetweenabout0.26Kandabout0.37Katalleldsupto10T. 84

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Figure7-7. Changeincapacitance,C,duetothedielectricresponseofTNNCH3CN,asafunctionoftemperature.(Topleft)Sample1,withthedirectionoftheacelectriceldE?(012)andthemagneticeldHkc.(Topright)Sample1,withE?(012)andHk[120].(Bottomleft)Sample2,withEkcandHkc.(Bottomright)Samedatashownonthesamescaleasthesample1dataare,forcomparison.Inallpanels,Ciswithrespecttothecapacitanceatthelowesttemperatureat11.5T. 85

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Figure7-8. Changeincapacitance,C,asafunctionofmagneticeld.(Topleft)Sample1,withE?(012)andHkc.(Topright)Sample1,withE?(012)andHk[120].(Bottomleft)Sample2,withEkcandHkc.(Bottomright)Sample2,withEkcandH?c.Inallpanels,Ciswithrespecttothecapacitanceatthelowesttemperatureat11.5T.Themagneticeldwassweptatarateof0.2T/min. Todeterminemagneticphaseboundariesattemperaturebelowabout0.22K,therebycompletingthephasediagram,magnetictorquemeasurementsweremadebyusingacapacitivecantileverwiththeAH2700Acapacitancebridgesetat5kHz,withanexcitationvoltageof15Vrms.A0.21mgsamplewasplacedona0.0005-inchthickBeCucantilever.Fieldsweepsweremadeat0.5T/minfromzeroto12Tatvarioustemperaturesbetween17mKand475mK.ThesemeasurementsweremadebyKosukeTakada,whileheworkedinourgroupasanexchangestudent.Additionaldatapointsforthemagneticphaseboundarieswereobtainedbyperformingmagnetocaloric-effectmeasurementsbysweepingtheeldatarateof 86

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Figure7-9. PhasediagramofTNNCH3CNfromspecic-heatpeaks(triangles),dielectriccapacitancepeaksforeldsweeps(circles)andtemperaturesweeps(pentagons),magnetocaloric-effectmeasurements(squares),magnetizationmeasurements(diamonds),andtorque(invertedtriangles)measurements. 0.1T/min,fromzeroeldto2T,thenfrom8Tat13T.Thesemeasurementsweremadeona1.99mgsamplebySeitaroIisaka,anotherexchangestudentfromOsakaPrefectureUniversity[ 75 ].AlltransitionsdetectedbythevemethodswerecombinedtoconstructthephasediagramshowninFig. 7-9 7.3DiscussionWehavepresentedresultsfrommultipleexperimentsindicatingmultiferroicbehaviorinTNNCH3CN,wherespintrianglesformatriangularlattice.Thisistherstexampleinwhichspintriangleswiththreefoldsymmetryexhibitsuchbehavior.Thephasediagramcontainsveorderedregions.Magnetically,thesecanbegroupedintothreeregions:anantiferromagneticallyorderedlow-eldregionupto1.25T, 87

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anintermediateregionextendingfrom1.25Tto8.49TinwhichTNNmoleculesarenaivelyexpectedtobeinthetwofold-degenerateS=1 2Sz=1 2state,andanantiferromagneticallyorderedhigh-eldregionfrom8.49Tto11.28T,thesaturationeld.Theveorderedregionscanalsobegroupedintoanothersetofthreeregions,intermsofelectricdipolemoments,withsomeoverlapwiththerstset.Thelow-eldregionupto0.6Tisantiferroelectricallyordered,theregionfrom0.6Tto9.7Tisferroelectricallyorderedandthehigh-eldregionfrom9.7Tto11.28Tisantiferroelectricallyordered.Inallregions,thedielectricresponsetoanacelectriceldappliedparalleltotheabplaneisveryweak,supportingthetheoreticalpredictionthattheinducedelectricdipolemomentslieparalleltotheplane.Asthetemperatureisloweredintheeldrangefrom1.25Tto8.49T,thetransitiontotheferroelectricallyorderedregionoccursatabout0.35Kwithoutmagneticordering.However,thespinentropyisnearlycompletelyremovedasthetemperatureapproacheszero,asshowninFig. 7-10 indicatingthatthetwofolddegeneracyofthethreespinsofeachTNNmoleculeisremovedastheinducedelectricdipolemomentisformed. 88

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Figure7-10. MagneticentropyofTNNCH3CNatzeroeldand5Tobtainedfromthespecicheat. 89

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CHAPTER8CONCLUSIONSInthiswork,propertiesofthreequantummagnetsCuPzN,DLCB,andTNNCH3CNwerestudiedinhighmagneticeldsandlowtemperaturesbyemployingvariousexperimentaltechniques.Amongthosetechniques,forcemagnetometryrequiredadevelopmentofnewmagnetometersincollaborationwithToshiroSakakibaraoftheUniversityofTokyo.ThemagnetometerswereusedintheMillikelvinLaboratoryoftheNHMFLattemperaturesdownto20mKandinmagneticeldsupto18T.Tomyknowledge,thisistheonlysuccessfuldevelopmentofhigh-accuracywire-suspensionforcemagnetometersinthiscountry,althoughIamawareofefforts,pastandpresent,byotherresearchers.Themagnetometershaveadvantagesoveralternativeinstrumentsthroughtheirabilitytoaccuratelymeasuremagnetizationatlowertemperatures.CuPzNisaprototypicalspin-1 2linear-chainHeisenbergantiferromagnet.Inthestudyofthismaterial,adeuteratedcrystalwasusedforspecic-heatmeasurementsinmagneticeldsupto14T,closetothesaturationeldof13.97T,inconjunctionwithmagnetizationmeasurementsperformedbyYoheiKonoinSakakibaraslaboratory.Formagneticeldsupto11T,thecharacteristicsofaTomonaga-Luttingerliquidwereobservedatlowtemperatures.ThespinonvelocityextractedfromthedataandtheWilsonratio,obtainedfromthemandthemagnetizationdata,areinexcellentagreementwiththeory.TheseresultsdemonstratethatCuPzNisanearlyidealspin-1 2linear-chainHeisenbergantiferromagnet.Specic-heat,magnetocaloric-effect,magnetic-torque,andmagnetizationmeasurementsweremadeondeuteratedDLCB,amaterialcomprisingacoupledarrayofspin-1 2ladders,todetermineitsanisotropicphasediagram.Aspin-optransitionwasobservedwhenthemagneticeldisinthedirectionperpendiculartotheabplaneofthecrystal,indicatingthepresenceofaweakIsinganisotropyalongthisdirection. 90

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ThisndingisinagreementwiththespindirectionsdeterminedpreviouslybyinelasticneutronscatteringintheNeel-orderedphaseatzeroeld.TNNCH3CNisafullyorganicmaterialcomposedoftriangular-latticelayersofS=1 2spintriangles.TNN,themainingredientofthismaterial,istherstexampleofaspintrianglewithretainedthreefoldsymmetry,owingtotheabsenceofJahn-Tellerdistortion.TheresultingdegeneracyofthegroundstatesisresponsibleforthenovelmultiferroicorderingwehavefoundinTNNCH3CN.Specic-heat,magnetocaloric-effect,magnetic-torque,magnetizationanddielectricmeasurementswereusedtodeterminethemultiferroicphasediagram,whichcontainsveorderedregions.Therstregion,fromH=0to0.6T,isorderedantiferromagneticallyandantiferroelectrically.Thesecondregion,from0.6Tto1.25Tisantiferromagneticallybutferroelectricallyordered.Thethirdregion,from1.25Tto8.49Tisalsoferroelectricallyordered,butthetransitiontothisregionatabout0.35Kdoesnotinvolveorderingofspins.Thefourthregion,from8.49Tto9.7T,isorderedantiferromagneticallyandferroelectrically,analogoustothesecondregion.Thefthregion,fromabout9.7Ttothe11.28Tsaturationeld,isorderedantiferromagneticallyandantiferroelectrically,analogoustotherstregion. 91

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BIOGRAPHICALSKETCH ChristopherPedroAoyamawasborninProvidence,RI.HegraduatedfromWakeForestUniversityinWinston-Salem,NCinMay2002.Asanundergraduatehemajoredinmathematicsandphysics.Understandingthepaucityofstudentswillingtoundertaketheeducationinscienceandmathematics,hereturnedtoRhodeIslandtoeducateandmotivatestudentsasahighschoolteacher.HetaughtmathatCharlesE.SheaSeniorHighSchoolinPawtucket,RIforayear,thenphysicsatJohnstonHighSchoolinJohnston,RIforthenextsevenyears.In2010hedecidedtoreturntoschoolasagraduatestudenttopursuehisdoctorateattheUniversityofFlorida.Thefollowingyear,in2011,hejoinedDr.YasumasaTakano'sgrouptoconductexperimentalresearchincondensedmatterphysics.Hegraduatedinthespringof2015. 96