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Development and Application of MEMS Devices for the Study of Liquid 3HE

Permanent Link: http://ufdc.ufl.edu/UFE0044078/00001

Material Information

Title: Development and Application of MEMS Devices for the Study of Liquid 3HE
Physical Description: 1 online resource (118 p.)
Language: english
Creator: Gonzalez, Miguel Angel
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

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

Notes

Abstract: In this work we demonstrate the use of micro-electro-mechanical oscillators for the study of quantum fluids. Using a commercial micro-machining process we designed, tested and characterized two types of oscillators which consist of a pair of parallel plates separated by well-defined gaps of 0.75 and 1.25 µm. The gap size and geometry of the devices give access to physics in the high Knudsen regime and allow the investigation of surface scattering effects in thin films of quantum fluids. A full characterization in air and at room temperature was done to quantify the effect of the so-called slide-film damping as a function of pressure from 10 mTorr to 760 Torr, ranging from a highly rarefied gas to a hydrodynamic regime. An initial test was performed in superfluid 4He, where very peculiar resonance line-shapes and hysteretic phenomena were observed below 1 K, reminiscent of quantum turbulence and vorticity effects observed in other mechanical oscillators such as tuning forks or vibrating grids. Finally, we show detailed measurements on the mechanical properties of an oscillator submerged in liquid 3He between 20 and 800 mK temperatures and at pressures, 3, 21, and 29 bar. In the Fermi liquid regime, when the bulk contribution to the damping is subtracted, these the fluid mass coupled to the oscillator as the temperature decreases below 100 mK. This phenomena is in agreement with recent experiments using torsional oscillators and provide an elegant and effective alternative to study helium physics at the micro/nano scale. Furthermore, we demonstrate the operation of a resonator in superfluid 3He. The devices show potential for use in a wide range of low temperature experiments and particularly to probe novel phenomena in quantum fluids and search for new phase transitions in reduced dimensions such as in superfluid 3He films.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Miguel Angel Gonzalez.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Lee, Yoonseok.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044078:00001

Permanent Link: http://ufdc.ufl.edu/UFE0044078/00001

Material Information

Title: Development and Application of MEMS Devices for the Study of Liquid 3HE
Physical Description: 1 online resource (118 p.)
Language: english
Creator: Gonzalez, Miguel Angel
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2012

Subjects

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

Notes

Abstract: In this work we demonstrate the use of micro-electro-mechanical oscillators for the study of quantum fluids. Using a commercial micro-machining process we designed, tested and characterized two types of oscillators which consist of a pair of parallel plates separated by well-defined gaps of 0.75 and 1.25 µm. The gap size and geometry of the devices give access to physics in the high Knudsen regime and allow the investigation of surface scattering effects in thin films of quantum fluids. A full characterization in air and at room temperature was done to quantify the effect of the so-called slide-film damping as a function of pressure from 10 mTorr to 760 Torr, ranging from a highly rarefied gas to a hydrodynamic regime. An initial test was performed in superfluid 4He, where very peculiar resonance line-shapes and hysteretic phenomena were observed below 1 K, reminiscent of quantum turbulence and vorticity effects observed in other mechanical oscillators such as tuning forks or vibrating grids. Finally, we show detailed measurements on the mechanical properties of an oscillator submerged in liquid 3He between 20 and 800 mK temperatures and at pressures, 3, 21, and 29 bar. In the Fermi liquid regime, when the bulk contribution to the damping is subtracted, these the fluid mass coupled to the oscillator as the temperature decreases below 100 mK. This phenomena is in agreement with recent experiments using torsional oscillators and provide an elegant and effective alternative to study helium physics at the micro/nano scale. Furthermore, we demonstrate the operation of a resonator in superfluid 3He. The devices show potential for use in a wide range of low temperature experiments and particularly to probe novel phenomena in quantum fluids and search for new phase transitions in reduced dimensions such as in superfluid 3He films.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by Miguel Angel Gonzalez.
Thesis: Thesis (Ph.D.)--University of Florida, 2012.
Local: Adviser: Lee, Yoonseok.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2012
System ID: UFE0044078:00001


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DEVELOPMENTANDAPPLICATIONOFMEMSDEVICESFORTHESTUDYOFLIQUID3HEByMIGUELA.GONZALEZADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOLOFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENTOFTHEREQUIREMENTSFORTHEDEGREEOFDOCTOROFPHILOSOPHYUNIVERSITYOFFLORIDA2012

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c2012MiguelA.Gonzalez 2

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Paramama,papa,mihermanitaymihermosaAnna 3

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ACKNOWLEDGMENTS Iwouldliketothankmyadvisor,Dr.YoonseokLee,forhistrustandconstantencouragement.Also,themembersofourlab,pastandpresent,whohaveprovidedvaluableknowledgeandhelpthroughthetimeofmydegree.Dr.Ho-BunChanandhisgroupfortheirenormoushelpondevelopingthedevicesusedinourexperiments.Allmembersfromthemachineshop,electronicsshop,cryogenicsservices,andothertechnicalserviceswhoinonewayoranothercontributedtothesuccessfulconclusionoftheexperiments.Finally,myfamilyandAnna,whoseunconditionalloveallowedmetoreachnewheights. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 7 LISTOFFIGURES ..................................... 8 ABSTRACT ......................................... 11 CHAPTER 1INTRODUCTION ................................... 13 2LIQUIDHELIUM ................................... 16 2.1Superuid4He ................................. 16 2.1.1PhaseDiagram ............................. 16 2.1.2SuperuidityandtheTwo-FluidModel ................ 16 2.1.3ExcitationSpectruminHe-II ...................... 18 2.2NormalLiquid3He ............................... 19 2.3BulkSuperuid3He .............................. 22 2.3.1Balian-Werthamer(BW)State ..................... 25 2.3.2Anderson-Brinkman-Morel(ABM)State ............... 26 2.4Superuid3HeFilms .............................. 27 2.4.1PlanarPhase .............................. 28 2.4.2InhomogeneousPhase ......................... 28 2.5TopologicalAspectsofSuperuid3He .................... 31 2.5.1TopologicalSuperuidsandSuperconductors ............ 33 2.5.2MajoranaFermioninSuperuid3He ................. 33 3MICRO-ELECTRO-MECHANICALDEVICES ................... 36 3.1Micro-Electro-MechanicalDeviceProperties ................. 36 3.1.1SpringDesign .............................. 38 3.1.2CombElectrodes ............................ 39 3.2FrequencyResponse ............................. 40 3.2.1Mechanical-ElectricalCorrespondence ................ 41 3.2.2ResonancePeak ............................ 42 3.3AirDamping ................................... 43 3.3.1CouetteFlow .............................. 44 3.3.2StokesFlow ............................... 45 3.3.3RareedFluidFlow ........................... 45 4EXPERIMENTALDETAILSANDDEVICECHARACTERIZATION ........ 48 4.1ManufacturingofMicro-Electro-MechanicalDevices ............ 49 5

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4.1.1DesignandFabrication ......................... 49 4.1.2DevicePreparation ........................... 49 4.2AtomicForceMicroscopySurfaceCharacterization ............. 51 4.3DeviceCharacterization ............................ 55 4.3.1DeviceSpecications .......................... 55 4.3.2DetectionScheme ........................... 56 4.3.3FrequencySweeps ........................... 57 4.3.4DirectCurrentBiasStudy ....................... 61 4.3.5DrivingForceComponents ...................... 62 4.4ResonanceasaFunctionofAirPressure .................. 63 5EXPERIMENTSINSUPERFLUID4HE ....................... 69 5.1Overview .................................... 69 5.2ExperimentalDetails .............................. 70 5.3Results ..................................... 73 6EXPERIMENTSINLIQUID3HE .......................... 84 6.1Overview .................................... 84 6.1.1EarlyExperimentsinLiquid3He .................... 84 6.1.2Liquid3HeandThinMetallicFilms .................. 87 6.2ExperimentalDetails .............................. 89 6.3Results ..................................... 92 6.4Summary .................................... 106 7CONCLUSION .................................... 110 REFERENCES ....................................... 112 BIOGRAPHICALSKETCH ................................ 118 6

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LISTOFTABLES Table page 4-1ThicknessofthedifferentlayersusedinPolyMUMPs ............... 48 4-2Calculateddevicepropertiesfromlayoutgeometry ................ 55 4-3ResonancefrequenciesandmeasuredQ-factorsinhighvacuum(3mTorr) 57 7

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LISTOFFIGURES Figure page 2-1Phasediagramfor4He ................................ 17 2-24Hedispersioncurve ................................. 19 2-3Superuid3Hephasediagram ........................... 23 2-4TheisotropicBphasegap .............................. 24 2-5TheanisotropicAphasegap ............................ 26 2-6Phasediagramforasuperuid3Helm ...................... 29 2-7Depletionoftheparallelandperpendicularcomponentsoftheorderparameterona3Helm ..................................... 30 2-8Possibletrajectoriesforaquasiparticleinahomogeneousandinhomogeneousphase ......................................... 32 2-9EdgestatesinQuantumHallstateandtopologicalsuperconductor ....... 34 3-1Computer-aideddesignofatypicalH1deviceshowingitsmainstructures ... 37 3-2Computer-aideddesignofaserpentinespringusedinbothH1andH2devices 37 3-3Cantileverandguidedbeams ............................ 38 3-4Anelectrodeshowingthecomb-shapedstructure ................. 39 3-5Aplateoscillatinglaterallyclosetoasurface ................... 43 3-6Velocityeldandsliplength ............................. 46 4-1Scanningelectronmicroscopeimagesofatypicaldeviceusedinthiswork .. 50 4-2TopographyofthesurfaceifthePoly0layerinatypicaldevice .......... 52 4-3Normalizedautocorrelationfunctioncalculatedfromthetopography ...... 53 4-4Histogramofheightsandautocorrelation ..................... 54 4-5Circuitdiagramforthecapacitancebridgetechniqueemployed ......... 56 4-6VibrationalmodesofdeviceH1 ........................... 58 4-7VibrationalmodesofdeviceH2 ........................... 59 4-8ResonancepeaksforanH1mdeviceattwodifferentexcitationvoltages .... 60 8

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4-9AmplitudeandresonancefrequencyasafunctionofbiasvoltagefortheshearmodeofanH1device ................................ 62 4-10Resonancepeaksforapivotmodeinfmodeand2fmode ......... 63 4-11Q-factorandshearmodefrequencyforH2andH1inair ............. 64 4-12DampingcoefcientforH1 ............................. 66 4-13Dampingcoefcientasafunctionofpressureforbothdevices,H1andH2,asafunctionofKn. ................................... 68 5-1Experimentalcellwithdeviceusedin4He ..................... 71 5-2Experimentalcellisontopofnucleardemagnetizationstageofcryostat .... 72 5-3Absorptioncurvesasafunctionoftemperaturefora200mVexcitation .... 73 5-4Afamilyofabsorptionanddispersioncurvesatdifferentexcitationvoltagesandthelowertemperaturesin4He ......................... 74 5-5Afamilyofabsorptionanddispersioncurvesatdifferentexcitationvoltagesandthehighertemperaturesin4He ......................... 75 5-6Afamilyofabsorptionanddispersioncurvesatdifferenttemperaturesshowinghystereticbehaviour ................................. 76 5-7PeakpositionandwidthofdeviceH1insuperuid4He .............. 78 5-8Q-factorandinverseQ-factorofdeviceH1insuperuid4Heasafunctionoftemperature ...................................... 79 5-9InverseQ-factorofdeviceH1insuperuid4Heasafunctionoftemperatureinlog-logscale .................................... 80 5-10Absorptioncurvesasafunctionoftemperatureforatuningforkinsuperuid4He .......................................... 81 5-11Absorptionanddispersioncurvesatdifferentvoltageexcitationsforatuningforkinsuperuid4He ................................. 81 5-12Resonancefrequencyandwidthforatuningforkinsuperuid4Heasafunctionoftemperature .................................... 82 5-13Q-factorandinverseQ-factorforatuningforkinsuperuid4Heasafunctionoftemperature .................................... 83 6-1DampingandfrequencyshiftfortheRoyalHollowayexperiment ........ 86 6-2Datafortheexperimentalvalueof!oscfortheRoyalHollowayexperiment .. 87 9

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6-3Experimentalcellfortheliquid3Heexperimentoncryostat ............ 89 6-4Experimentalcellwithdevice ............................ 90 6-5Absorptionanddispersioncurvesforadeviceinnormalliquid3Heasafunctionoftemperature .................................... 92 6-6Lengthscalesinnormalliquid3Heasafunctionoftemperature ......... 93 6-7Comparisonbetweentwodifferentruns ...................... 94 6-8Peakamplitude,width,frequencyandQ-factorforH2innormalliquid3Heasafunctionoftemperatureat3barpressure .................... 96 6-9Peakamplitude,width,frequencyandQ-factorforH2innormalliquid3Heasafunctionoftemperatureat21barpressure ................... 97 6-10Peakamplitude,width,frequencyandQ-factorforH2innormalliquid3Heasafunctionoftemperatureat29barpressure ................... 98 6-11DampingcoefcientofH2innormalliquid3Heasafunctionoftemperature .. 99 6-12DampingcoefcienttimesT2asafunctionoftemperature ........... 100 6-13Filmdampingcoefcient ............................... 101 6-14Knudsennumberinnormalliquid3Heasafunctionoftemperature ....... 102 6-15Massenhancementcalculatedfromtheresonancefrequencyshifts ...... 103 6-16Massenhancementcalculatedfromtheresonancefrequencyshiftstimesthetemperature ...................................... 104 6-17Filmandbulkcontributionstothetotalmassenhancement ........... 105 6-18Viscouspenetrationdepthcalculatedfromtheresonancefrequencyshifts ... 106 6-19Filmmassdecoupling ................................ 107 6-20Absorptionanddispersioncurvesforadeviceinsuperuid3Heasafunctionofexcitationvoltage ................................. 108 6-21Absorptionanddispersioncurvesforadeviceinsuperuid3Heasafunctionofexcitationvoltage ................................. 109 10

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AbstractofDissertationPresentedtotheGraduateSchooloftheUniversityofFloridainPartialFulllmentoftheRequirementsfortheDegreeofDoctorofPhilosophyDEVELOPMENTANDAPPLICATIONOFMEMSDEVICESFORTHESTUDYOFLIQUID3HEByMiguelA.GonzalezMay2012Chair:YoonseokLeeMajor:PhysicsInthisworkwedemonstratetheuseofmicro-electro-mechanicaloscillatorsforthestudyofquantumuids.Usingacommercialmicro-machiningprocesswedesigned,testedandcharacterizedtwotypesofoscillatorswhichconsistofapairofparallelplatesseparatedbywell-denedgapsof0.75and1.25m.ThegapsizeandgeometryofthedevicesgiveaccesstophysicsinthehighKnudsenregimeandallowtheinvestigationofsurfacescatteringeffectsinthinlmsofquantumuids.Afullcharacterizationinairandatroomtemperaturewasdonetoquantifytheeffectoftheso-calledslide-lmdampingasafunctionofpressurefrom10mTorrto760Torr,rangingfromahighlyrareedgastoahydrodynamicregime.Aninitialtestwasperformedinsuperuid4He,whereverypeculiarresonanceline-shapesandhystereticphenomenawereobservedbelow1K,reminiscentofquantumturbulenceandvorticityeffectsobservedinothermechanicaloscillatorssuchastuningforksorvibratinggrids.Finally,weshowdetailedmeasurementsonthemechanicalpropertiesofanoscillatorsubmergedinliquid3Hebetween20and800mKtemperaturesandatpressures,3,21,and29bar.IntheFermiliquidregime,whenthebulkcontributiontothedampingissubtracted,thesemeasurementsshowadistinctpeakinthecoefcientofdampingandareductionintheuidmasscoupledtotheoscillatorasthetemperaturedecreasesbelow100mK.Thisphenomenaisinagreementwithrecentexperimentsusingtorsionaloscillatorsandprovideanelegantandeffectivealternativetostudyheliumphysicsatthemicro/nano 11

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scale.Furthermore,wedemonstratetheoperationofaresonatorinsuperuid3He.Thedevicesshowpotentialforuseinawiderangeoflowtemperatureexperimentsandparticularlytoprobenovelphenomenainquantumuidsandsearchfornewphasetransitionsinreduceddimensionssuchasinsuperuid3Helms. 12

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CHAPTER1INTRODUCTIONThestudyofhelium,thesecondmostabundantelementintheuniverse,hasformorethanacenturybeenatthecoreofthedevelopmentofseveralmodernideasincondensedmatterphysics.Theracetoliquefythiselementhasbroughtaboutawealthofadvancementsintheeldofcryogenicsandconsequentlyallowedthedevelopmentoftechnologiessuchasnuclearmagneticresonanceforadvancedimaginginthemedicalandbiologicalsciences.Therearetwostableisotopesofhelium:4Heand3He.Bothatomsaresphericallysymmetricandarechemicallyinert.Theirsmallmassandweakinteractionbetweenatomsmakesbothisotopesremainliquiddowntoabsolutezerounlessanexternalpressureisapplied.Thisisadirectconsequenceofquantummechanics,whichkeepstheatomsfromsolidifyingthroughzero-pointmotion.Likewise,theirdifferentmassmakestheirboilingtemperaturesdifferent,4.21Kfor4Heand3.19Kfor3He.Whencooleddownmuchbelowtheirboilingtemperature,thetwoisotopesofheliumacquireamazingpropertiessuchassuperuidity,whichallowsthemtoowwithzeroviscosity.Theyalsoprovidetwoofthemostsimplemodelsofacondensedphasepossessingbosonic(4He)orfermionic(3He)properties.Below1K,liquid3Hebecomesanexemplarsystemofmany-bodyfermionphysics.ItdisplayspropertiesofadegenerateinteractingFermisystem,anditslowtemperaturepropertiesareexquisitelyexplainedbyLandau'sFermiliquidtheory,whichhasalsobeenverysuccessfulindescribingmanyaspectsofthebehaviourofelectronsinordinarymetals.Duetounconventionalpairinginthesuperuidstatebelow3mK,theorderparameterof3Hedisplayssymmetrieswhichbearresemblancetovarioussystemssuchasliquidcrystals,ferromagnets,antiferromagnetsandunconventionalsuperconductors[ 1 ].Thus,liquid3Heinitsnormalandsuperconductingstateprovides 13

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oneofthemostsophisticatedmodelsystemsavailableincondensedmatterphysics,anditsstudyhasalsobeenusefulinareaslikeastrophysicsorhighenergyphysics[ 2 ].Oneofthemostchallengingproblemsinmoderncondensedmatterphysicsistheunderstandingofdisorderanditseffectonvariousmany-bodysystemssuchaselectronsinmetals,superuids,andmagneticmaterials.Liquidhelium,withaverylowboilingtemperaturewhichfreezesoutanyimpurities,isoneofthemostpurestatesofmatterattainable.Thismakesitanidealcandidatetostudytheeffectsofdisorderinasystematicway.Inrecentyears,disorderintheformofporousmaterialslikeaerogelhasbeenintroducedinsuperuid3He.Deviationsfromthebulkphasediagramhavebeenobserved[ 3 4 ],andnewchiralphaseshavebeenstabilizedwheninducinganisotropicdisorderbydeformingtheaerogel[ 5 ].Anotherformofdisordercanbeintroducedbyconningtheliquidintotwo-dimensionallms.Inthiscase,thediffusescatteringofftheroughboundariesactsasasourceofdisorder.Innormalliquid3He,duetotheverylongquasiparticlemeanfreepaths,amesoscopicregimecanbeattainedformicro-metersizedslabs.Thisproblemresemblesthatofelectrontransportthroughmesoscopicmetalliclms.Ontheotherhand,insuperuid3Helms,quasiparticlescatteringofftheboundariesmayleadtothedestructionofthesuperuidstateorintroducenewphenomenathatmaystabilizeothersuperuidphases[ 6 7 ].Amongthemostwidelyemployedtechniquesinthestudyofbulkliquid3Hearemechanicaloscillatorssuchasvibratingwiresandtorsionaloscillators[ 8 9 ].Inrecentyears,micro-electro-mechanicalsystems,orMEMS,havepopulatedthetechnologicallandscapethroughtheirwidespreaduseassensorsandactuatorsinaplethoraofindustrialandconsumerproducts.Theirsmallsizeandtheavailabilityofcost-effectivestate-of-the-artfabricationprocessesmakethemanattractivetooltocreatenovellowtemperaturesensorsfortheexplorationofnewphysicsinquantumuidsatthemicro/nanoscale.Inthisthesis,acomprehensiveroadmapisshowndetailingthedevelopmentofaMEMSoscillatoranditsapplicationtothestudyofliquid3Helms.In 14

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addition,astudyofairdampingatroomtemperatureispresented,aswellasastudyinsuperuid4He. 15

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CHAPTER2LIQUIDHELIUMAsurveyofthepropertiesofdifferentquantumuidsispresentedinthischapter.Thisincludes:superuid4He,normalandsuperuid3Heinbulk,superuid3Helms,andtopologicalaspectsofsuperuid3He.Inadditiontoprovidingatheoreticalbackgroundtotheexperimentscarriedout,thiscompilationoftopicsservesasamotivationforfutureexperimentswithinthereachoftheexperimentaltechniquedevelopedinthisthesiswork. 2.1Superuid4He 2.1.1PhaseDiagramThephasediagramforliquid4Heisshowningure2-1.Therearetwoliquidphases:He-IandHe-II.Theformerisassociatedwiththenormalstateoftheliquid,whilethelatterisassociatedwithsuperuidity.Thesetwophasesareseparatedbyasecond-ordertransitionlinecalledthelambdaline,locatedatthelambdatemperature,T=2.17Katthesaturatedvaporpressure.Thenamereectstheshapeofananomalyobservedinthespecicheatasafunctionoftemperature,whichincreasesveryrapidlywhenapproachingTwhilecoolingandthendecreasesasymmetricallyaftercrossingthephaseseparationboundary.Otherthermodynamicquantitiessuchastheexpansioncoefcientalsoshowanomaliesatthelambdatransition.Butthemostintriguingfeatureisitsdrasticchangeinviscosity.In1938,measurementsbyKapitzaontheowofliquid4Hethroughnarrowchannelsshowedasuddendecrease,oftheorderof103,intheviscositybelowT.ThetermsuperuiditywascoinedbyKapitzatoreectthisanomalousbehaviour[ 11 ]. 2.1.2SuperuidityandtheTwo-FluidModelQuicklyafterthediscoveryofsuperuidity,Londonsuggestedthat,asaconsequenceofthebosonicnatureofthe4Heatom,Bose-Einsteincondensation,themacroscopicoccupationofasingle-particlegroundstate,wouldoccuratatemperatureof3K, 16

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Figure2-1. Phasediagramfor4Heshowingitstwoliquidphases:Normal(He-I)andsuperuid(He-II)[ 10 ]. andthisphenomenamightberelatedtothelambdatransition[ 12 ].Inspiredbythis,Tiszadevelopedfurthertheideaofatwo-uidmodel[ 13 ],whichregardstheliquidashavingtwomixedandinseparablecomponents:thenormal,correspondingtotheclassicalpartoftheuidwhichgivesrisetoviscosity,andthesuperuid,correspondingtotheparticlesinthecondensedstatewhichoweffortlesslywithzeroviscosity.Simultaneously,Landauproposedasimilarphenomenologicaltwo-uidmodelinwhichthesuperuidisidentiedwithparticlesinthegroundstateofaBoseliquidandthenormalpartisassociatedwithexcitationsoffthegroundstate[ 14 15 ].Thus,inatwo-uidmodel,thetotaldensityoftheliquidwillbetheadditionofthenormalandthesuperuiddensities,e.g.=n+s.Thesecomponentshavetheirownvelocitiesaswell,thenormalvelocity,vn,andthesuperuidvelocity,vs.Adirectconsequenceofthisisanewtypeofcollectiveoscillationscalledsecondsound,inwhichthenormalandsuperuidcomponentsoscillateoutofphasewhilekeepingthetotaldensityconstant. 17

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Thiseffectdoesnotproduceadensityoscillationasinregularsoundwaves,butcanphysicallybedetectedasatemperatureoscillation.Similartoconventionals-wavesuperconductors,theorderparameterofsuperuid4Heisascalarwavefunctiongivenby (r,t)= 0expi(r,t),(2)wheretheamplitude 0isrelatedtothesuperuiddensityand,thephaseofthewavefunction,isrelatedtothesuperuidvelocityas vs(r,t)=~ mr(r,t).(2)Animmediateconsequenceofthisdenitionis,rst,theirrotationalnatureofthevelocity rvs=0,(2)andsecond,thequantizationofow Ivs`=nh=m.(2) 2.1.3ExcitationSpectruminHe-IITheshapeoftheexcitationspectrumwasoriginallydeducedbyLandauandlaterstudiedinmoredetailbyFeynman[ 17 ],whosuggestedtheideaofusingneutronscatteringtomeasurethedispersioncurve.ThesetheoreticalpredictionswerelatervalidatedinasetofexperimentsbyPalevskyetal.[ 18 ].Asketchofthedispersioncurveisshowningure2-2.Thelowenergyexcitations,withwavevectorlessthan1A)]TJ /F2 7.97 Tf 6.59 0 Td[(1,correspondtolongwavelengthdensityoscillations,orphonons,andhavealineardispersion (p)=c1p,(2) 18

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Figure2-2. 4Hedispersioncurve[ 16 ] wherec1istheordinaryhydrodynamicsoundvelocity.Above1A)]TJ /F2 7.97 Tf 6.59 0 Td[(1,thedispersioncurvebecomesparabolic.Theexcitationsarecalledrotonsandhaveadispersiongivenby =+(p)]TJ /F6 11.955 Tf 11.96 0 Td[(p0)2 2,(2)wherep0andspecifythepositionoftherotonminimumanddeterminesthecurvature.TheinterplayofscatteringprocessesbetweenphononsandrotonswilldeterminetheviscosityofHe-II[ 19 ]. 2.2NormalLiquid3HeAlthoughchemicallyidenticaltothemorenaturallyabundantisotope4He,3Hebehavesdifferentlywhencooleddowntoverylowtemperatures.Thisisduetothefactthateachspeciesobeysdifferentquantumstatistics,becauseoftheirdifferenttotalnuclearspin.In3Hetheoddnumberofnucleonsgiveitatotalspinof~=2,makingitafermion,while4Hehaszerototalspin,whichmakesitaboson.Sincetwoelectronscompletelyllupthes-shell,theydonotcontributetothetotalspinoftheatom.Below 19

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theFermitemperature(1K),liquid3HebeginstodisplaycharacteristicsofaFermisystem.Foranon-interactingFermigas,themainthermodynamicpropertiescanbeextractedfromthedistributionfunction n(i)=1 e(i)]TJ /F10 7.97 Tf 6.59 0 Td[()=kBT+1,(2)whereiistheparticularenergystateiwithspinandisthechemicalpotential.ThelowtemperaturepropertiesofadegenerateFermigasaredominatedbythescatteringprocessesclosetotheFermisurface,withinathinshellofenergykBTintheFermisphere.Thus,theprobabilityoftwoparticlesscatteringintotwounoccupiedstatesgoesas(kBT)2[ 20 ].UsingtheGolden-Rule,therelaxationtime()forthetwoparticlescatteringcanbedeterminedtogoas1=T2.Thiswillbasicallydeterminethetransportpropertiesofthesystem,suchasviscosity,thermalconductivity,andselfdiffusion.Forinstance,theviscosityisrepresentedby =1=3v2F,(2)whereisthemassdensityandvFistheFermivelocity,whichisindependentoftemperature.Theviscositywillthereforegoas1=T2.Theviscosityisthereforestronglytemperaturedependentandbecomeslargeatlowtemperatures,unlike4He.Itisworthtomentionthattheviscosityinliquid3Hecanbeashighas0.3Prightabovesuperuidtransition,whichiscomparabletothatoflightmachineoil.Thenon-interactingdegenerateFermigasformalismturnsouttobeinsufcientbelow100mK,whenonetriestoaccountforthemagneticsusceptibilityandspecicheatofliquid3He.AmorerigorousformalismwasdevelopedbyLandautoincludeparticleinteractionsknownasFermiliquidtheory[ 21 23 ].Oneofthemainpremisesofthetheoryisthedescriptionofthelowenergyexcitationsofthemany-bodyfermionicsystemasquasiparticles,whichhaveaneffectivemass,m,aboutthreetimeshigherthanthebaremassofaheliumatom.Aquasiparticlecanbethoughtofasasingle 20

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3Heatomdressedbytheinteractionswiththenearby3Heatoms.Itshouldbenotedthatthenumberofquasiparticlesisexactlythesameasthenumberof3Heatoms.FermiliquidtheoryextendstheideaofaFermigasandassumesthattheenergylevelsfromthenon-interactinggascanbemappedtotheinteractingcasewhenadiabaticallyturningontheinteractions.Moreover,thedistributionfunctionforquasiparticlesisnotthesameasinequation2-7,sincethequasiparticleenergyisafunctionalofn(i).Inotherwords,theenergyofasinglequasiparticleisdependentonthedistributionoftheotherquasiparticles.Asaconsequence,thetotalenergyofthesystemisnotsimplytheadditionofthequasiparticlesingleenergystates.Thechangeintotalenergywhenaquasiparticleofmomentumpandspinisaddedtothesystemis[ 24 ] E=Xppnp,(2)wherenpaccountsforthechangeinthedistributionfunction.Thedoubleindexisusedtodenotethemoregeneralcasewherenpisthespindensitymatrixofquasiparticleswithmomentumpandthenpareitseigenvalues.Thetermpcanbeapproximatedas pF+vF(p)]TJ /F6 11.955 Tf 11.96 0 Td[(pF),(2)whenkBTismuchsmallerthantheFermienergy,F.Thequasiparticleenergyitselfismodiedwhenachangeinthedistributionnp00(r,t)isintroduced p=p+Xp0f(p,p0)n(p0)+Xp0g(p,p0)(p)(p0),(2)wheref(p,p0)andg(p,p0)arespin-independentandspin-dependentfunctionsthatcharacterizetheinteractionsbetweenquasiparticles.NeartheFermisurface,thesefunctionsareonlydependentonthescatteringanglebetweenthepairofparticlesand,therefore,canbeexpandedintermsofLegendrepolynomials,Pl(cos()),andthe 21

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interactionsaregivenby f(p,p0)=1 D(F)XlFslPl(cos()) (2) g(p,p0)=1 D(F)XlFalPl(cos()), (2) whereFslandFalarethespin-symmetricandspin-antisymmetricLandaucoefcients,andD(EF)=mpF=(2~3)isthedensityofstatesattheFermilevel.Noticethatthemassterminthedensityofstatesisgivenbym.ThisquasiparticleeffectivemasstermisdirectlyrelatedtothemostfundamentalLandaucoefcientas m m=1+1 3Fs1.(2)ThepreviousequationisexactandcanbeobtainedfromGalileaninvariance.Theeffectiveinteractionsalsogiverisetoanotherphenomenoncalledzerosound,whichisawave-likeoscillationoftheseeldsandcanbedetectedinacollisionlessregime,where!1.Thelow-ordervaluesoftheLandaucoefcientsarecloselyrelatedtovariousexperimentallyobtainablequantitiessuchasthespinsusceptibility,,andthecompressibility,N, =20D(EF)=(1+Fa0) (2) N=1 2nD(EF) 1+Fs0, (2) where0isthemagneticmomentof3Heandnistheparticledensity.Inparticular,thecompressibilitydeterminesthespeedofrstsound(1=p mnN) c21=1 3(1+Fs0)(1+1 3Fs1)v2F.(2) 2.3BulkSuperuid3HeBelowafewmillikelvin3HegoesthroughatransitionfromanormalFermiliquidphasetoasuperuidphase.Atthispoint,theFermiseabecomesunstabletoanew 22

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Figure2-3. Superuid3Hephasediagramatzeromagneticeld.Thedotinthelowerrightcornerindicatesroomtemperatureandpressure[ 10 ]. groundstateforarbitrarilysmallattractiveinteractionsbetweenpairsofparticles.ThesesocalledCooperpairscondenseintothisnewsuperuidstatewhere,asitsnamesuggests,asharpdropinviscosityoccurs.Therearetwosuperuidphasesinbulk3Heinzeromagneticeld:theA-phaseandtheB-phase.Thephasediagramof3Heisshowningure2-3.Cooperpairsineachphasehaveadifferentinternalstructure.Since3Heatomsarefermions,thewavefunctionforaCooperpairmustbeantisymmetricunderparticleexchange.Thisimpliesthattheorbitalandspinpartsofthewavefunctionmustbeevenandodd,respectively,orviceversa.Unlikeconventionalsuperconductors,3HeCooperpairsoccurinap-wavespin-tripletstate,whichcorrespondstooddorbital(`=1)andevenspin(S=1)wavefunctions.Theorderparameteristhenofacomplexstructurewhichcanbeparametrizedbyavector(p)andwrittenasalinearcombinationofthethree 23

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Figure2-4. TheisotropicBphasegapisshowningrey.EfandkfdenotetheFermienergyandwavevectorrespectively. possiblespinstates:j""i,j##i,j"#i+j#"ias b(p)=(iy)(p)=0B@)]TJ /F4 11.955 Tf 9.3 0 Td[(x+iyzzx+iy1CA (2) =()]TJ /F4 11.955 Tf 9.3 0 Td[(x+iy)j""i+(x+iy)j##i+zfj"#i+j#"ig, (2) wherethei'sarethePaulispinmatrices.Asdened,(p)isavectorinspinspacewhichpointsinthedirectionofzerospinprojectionfortheCooperpairs,e.g.(p)S=0.Toincludetheorbitalpartofthewavefunction,itisnecessarytoexpandthe`=1sphericalharmonicsinlinearcombinationsofthemomentumprojectionoperatorssothatY11!^pxi^py p 2andY01!^pz.The(p)vectoristhen (p)=3Xi=1i^pi,(2)wheretheindexisforspinandifortheorbitalpart. 24

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2.3.1Balian-Werthamer(BW)StateEachsuperuidphasein3Heischaracterizedbyadifferentorderparametervector(p).Inthecaseof3He-Btheorderparameterisgivenby (p)=0ei^p,(2)sothat B=0)]TJ /F4 11.955 Tf 9.44 0 Td[(^px+i^py p 2j""i+0^pz(j"#i+j#"i)+0^px+i^py p 2j##i(2)whereitcanbeshownthatjBj2=20.ThisimpliesthatthegapparameterBisisotropic,asshowningure2-4.Thustheenergygapbetweengroundpairsandexcitedpairsisalsoisotropicanditisgivenby E(p)=v2f(p)]TJ /F6 11.955 Tf 11.95 0 Td[(pf)2+jj21=2.(2)TheBWstateisdegenerateifmultipliedbyaconstantphasefactorandaspinrotationrelativetopositionspaceisperformed[ 20 ].Amoregeneralformrepresentingtheorderparameterforthisfamilyofstatesis (p)=0eiR^p,(2)whereRistherotationmatrix.Now,ifthedipolarenergy, HD=)]TJ /F6 11.955 Tf 9.3 0 Td[(gD2B)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(cos+2cos2,(2)istakenintoaccount,therelativespin-orbitdegeneracyislifted.TheminimumdipoleenergyhappenswhenbothvectorsarexedattheLeggettangle104.ThisallowstheCooperpairstobeinastatewheretheirorbitalandspinvectorsarexedwithrespecttoeachother,butareallowedtorotatethroughoutthecondensate,possessingtheso-calledbrokenrelativespin-orbitsymmetry. 25

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Figure2-5. TheanisotropicAphasegapisshowningrey.EfandkfdenotetheFermienergyandwavevectorrespectively. 2.3.2Anderson-Brinkman-Morel(ABM)StateThisphaseappearsonlyathigherpressures(above21bar)oratallpressuresinthepresenceofamagneticeld.Whenstrongcouplingeffectsaretakenintoaccount,thisstatebecomesalowerenergystatethanthatoftheBphasewithinasmallsliverinthephasediagram(seegure2-3).Theorderparameterforthisphaseisgivenby A=0(^px+i^py)(j##i)-222(j""i),(2)whereonlycomponentsofequalspinprojectionarepresent,thus,itisknownasanequalspinpairingstate.Thisphasecanberepresentedby (p)=0^(^p^m+i^p^n).(2)Thetwovectors^mand^nformatriadinorbitalspacewith`,suchthat`=^m^n.Unlike3He-B,theAphasehasananisotropicgapwithnodesalongthedirectionof`(seegure2-5).Thegapparameterisgivenby jAj2=20)]TJ /F4 11.955 Tf 5.48 -9.69 Td[(1)]TJ /F4 11.955 Tf 11.96 -.16 Td[((`^p)2(2)where`^p=cosk. 26

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Theorderparameterofthisstateisinvariantbysimultaneouslymultiplyingbyaphasefactoreiandperformingarotationabout`.Inthiscase,therelativeorbital-gaugesymmetryisspontaneouslybrokenandthespindegeneracyisagainresolvedbythedipolarenergy HD=)]TJ /F6 11.955 Tf 9.3 0 Td[(gD2A(`)2,(2)whichisminimalwhenvector^isparallelto`[ 25 ]. 2.4Superuid3HeFilmsItwasthoughtforalongtimethatthephenomenonofsuperuidityinlmswasonlypossiblein4He.Itwasexpectedthatifthethicknessofthelm,d,wasoftheorderofthecoherencelength,0,thesuperuidstatewouldbedestroyedbypair-breakingfromthewalls.Nevertheless,superuidityinlmswasobservedbySachrajdaetal.[ 26 ]in1985evidencedbythephenomenonoflmcreepfromonereservoirtoanother.Severalphysicalproperties,suchastransitiontemperatureandsuperuiddensity,havebeenmeasuredinlmssincethen[ 7 27 28 ].Nevertheless,experimentsremainchallenging,andthereareonlyalimitednumberoftechniquesthatallowthestudyandproductionofheliumlmsunderultra-lowtemperatureconditions[ 29 32 ].Uptothisdate,mostexperimentsrelyonvanderWaalslmsformedonmetallicsubstrates,thusmakingitverydifculttoreliablycontrolandmeasurethepropertiesofthelm.ShortlyaftertherealizationoftheanisotropicABMphasewasrmlyestablished[ 33 ],theeffectofboundarieswithspecularlyanddiffuselyreectingsurfaceswasstudiedbyRaineretal.[ 34 ].Twofactorswereshowntostronglyfavortherealizationofananisotropicstatenearaboundary:alongerhealinglengthintheperpendiculardirectiontothesurfaceandalossincondensationenergyclosetothesurface.Morespecically,nearaspecularboundary,theperpendicularcomponentoftheorderparameter,?,wouldbesuppressed,whiletheparallelcomponent,k,wouldremain.Ontheotherhand,ifthescatteringisdiffusive,kisalsosuppressed.Theseconditions 27

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introduceastronganisotropyinthesuperuidstateinasimilarwayamagneticeldwould.Thisobservationsledtothebeliefthataboundarynearsuperuid3Hewouldtendtoalignthe`vectorperpendiculartothesurface.Therefore,itisnaturaltoexpectthatanA-likephaseismorelikelytooccurinalm,forexample,sincethereisnolossincondensationenergyinthe`direction,wheretheenergygaphasnodes.Bythesamereasons,aphasewithanisotropicgapliketheonedescribedbytheconventionalBWstatewouldnotbeexpectedtobestableunderstrongconnement. 2.4.1PlanarPhaseIntheweakcouplinglimittheAphaseisdegeneratewithanotherphasecalledtheplanar(P)phase,althoughthisphasehasneverbeenobservedtodate.ThisphaseissimilartotheBphase,anditisacombinationof(Sz=+1,m=)]TJ /F4 11.955 Tf 9.3 0 Td[(1)and(Sz=)]TJ /F4 11.955 Tf 9.3 0 Td[(1,m=+1)states.Theorderparameterforthisphaseisgivenby P=0)]TJ /F4 11.955 Tf 9.45 0 Td[(^px+i^py p 2j""i+0^px+i^py p 2j##i,(2)whereonlytheequalspinpairssurvive.ThisphasehasthesameenergygapastheAphase,thus,theyareenergeticallydegenerate. 2.4.2InhomogeneousPhaseAninhomogeneousphasemayberealizedwhendegenerategroundstatesorganizeandformstructuresthatbreaktranslationalsymmetryinthesystem[ 25 ].ThephasediagramcalculatedbyVorontsovandSaulsinreference[ 6 ]isshowningure2-6.Theestimateddiagramshowsthatatapproximately90thetransitionbecomesre-entrant(A)166(!B)166(!A)asonemovesalongthetemperatureaxis.Thissuggestsapossibleinstabilityinthesuperuidorderparameterthatmakesthesystemunabletodecidebetweencompetingstates.Thiscompetitionmayresultinaformationofdomainwallscomposedofdegeneratephasesthusbreakingtranslationalsymmetry.ThephasesconsideredinthisphasediagramaresimilartothebulkAandBphasesbut 28

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Figure2-6. Phasediagramforasuperuid3Helm.ThethicknessesatdC1anddC2denotetheboundariesofthestripephase[ 6 ]. restrictedtotheSO(2)orbitalsymmetrygroup.Makingthefollowingdenitions[ 6 ]=(x,y,z)xjxi+yjyi+zjzi, (2)jxi=1 p 2()-166(j""i+j##i),jyi=i p 2(j""i+j##i),jzi=j""+##i,theB-likephaseandtheA-likephasesarewrittenas[ 35 ]B=)]TJ /F4 11.955 Tf 5.48 -9.68 Td[(k(z)^px,k(z)^py,?(z)^pz, (2)A=(0,0,?(z)(^px+i^px)).VorontsovandSaulsconsideredauniform3Helmofthicknessdonanatomicallyroughsubstratewheretheroughnessoccursatamuchshorterlengthscalethanthecoherencelength,0.Thetopboundaryisfreeandthereisnovapororevaporation(P!0),thusmakingthescatteringofquasiparticlesspecular.Quasiparticlescatteringfrombothsurfacesleadstoasuppressionof?,butscatteringofftheroughsurfaceleadstoasuppressionoftheorderparameterintheparalleldirection,k,aswell(see 29

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Figure2-7. Depletionoftheparallel(k)andperpendicular(?)componentsoftheorderparameterona3Helmfordifferentboundaryconditions.LeftpanelB-likephase.RightpanelA-likephase[ 35 ]. gure2-7).Theformationofdomainwallscompeteswiththesurfacepairbreakingbycompensatingthechangeindirectionofthemomentumasthequasiparticlescattersoffthesurface.Thepresenceofthesedomainwallsintroducespairbreakingforquasiparticleswhichmoveparalleltothesurface,butthesetrajectoriesbecomelesslikelyasdbecomessmaller.Thecompetitionbetweenthesetwopairbreakingmechanismsiswhatgivesrisetotheinstabilitythatleadstotheinhomogeneousphase(seegure2-8).Theorderparameternowformsastripepatternwherethesizeofthedomainwallsisdirectlyrelatedtothelevelofconnement,d=0.AtaxedtemperaturebelowthecriticaltemperatureandstartingintheBphase,wecanseethatdecreasingtheseparationbetweenthetwoboundarieswillresultrstinatransitionfromBtostripephaseatdC1andtheatransitionfromstripephasetoA(P)atdC2.AtdC1thesystemgoesthoughasecondorderphasetransitionbycontinuousdepletionoftheperpendicularcomponentoftheorderparameter.Atthispointdomainwallsbeginto 30

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formandcontinuetomultiplyastheseparationisreduced.AswegetclosetodC2,thenumberofdomainwallsbeginstodissolveuntilthesystemgoesintothePphasebymeansofacontinuousphasetransition.IfthesystemgoesintoanAphase,arstorderphasetransitiontakesplace.Recently,experimentsusingaSQUID-NMRtechniqueona0.6msuperuid3HeslabobservedsignaturesoftheA-phaseasthestablephaseintheconnedatlowpressures[ 36 ].Thepressurewasswepttomakethecoherencelengthvaryfrom40nm(5.5bar)to77nm(0bar),thuschangingthelevelofconnementfrom7.80to150(seegure2-6).Atpressurescloseto0bar(intheweakcouplinglimit),anegativeshiftintheNMRsignalcomingfromtheslabwasseendowntothelowesttemperatures,indicatingadipoleunlockedA-phasewaspresent.HysteresisintheA-phasepeakpositionwasseenbetweenwarmupandcooldowncycles,consistentwitharstorderphasetransition.Furthermore,twoNMRsignaturesofunknownoriginwereobservedatp=5.5bar.Thetwopeakswereindividuallyseenduringdifferentcooldownsat0.54Tbulkc,andwerealsoseentooccursimultaneouslyinoneparticularcooldownatthesametemperature[ 37 ].Theseobservationscouldpointtotheexistenceofaninhomogeneousphaseasdiscussedabove,buttheexpectedNMRsignaturesforthisphase,neededforadirectcomparison,haven'tbeencalculatedyet. 2.5TopologicalAspectsofSuperuid3HeSincethediscoveryofthequantumHallstate[ 38 ],theuniversaldescriptionofphasetransitionsintermsofauniqueorderparameterarisingpurelyfromdimensionalandsymmetryconsiderationsbegantoshowitslimitationstodescribethisnewstateofmatter.Awholenewmethodofclassicationneededtobedevelopedbasedonthetopologicalpropertiesofthesystem[ 39 41 ].InthequantumHallstate,thequantizationoftheHallconductanceisregardedasatopologicalconstantwhichcanonlytakeintegervaluesandisindependentofthemicroscopicdetailsofthesystem.Thus,systemswithsimilarconstantscouldbegroupedtogetherinthesamecategoryand 31

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Figure2-8. Possibletrajectoriesforaquasiparticleinahomogeneousandinhomogeneousphase.Top:Trajectory1makesupforthechangeinmomentumbyintroducingadomainwall.Trajectory2inducespairbreakingfromtheorderparameterchangefromonesideofthedomainwalltotheother.Bottom:Trajectoryonechangesdirectionofmomentumthusinducingpair-breaking.Trajectory2keepsmomentumunchangedthuspreservingthesuperuidstate. thetopologicalconstantwouldtaketheplaceoftheorderparameter.In1988,VoloviksuggestedthattheAphaseofsuperuid3Heinaslabgeometrywoulddisplayaquantizedtransversesupercurrentarisingfromagradientinthechemicalpotential[ 42 ].ThistypeofHallconductancewasshowntobeatopologicalinvariantarisingfromthenon-zerowindingnumberoftheorderparameter'sphasewhencirclingaroundtheFermisurface.Recently,thepredictionandobservationofa2DtopologicalinsulatorphaseinHgTe/CdTequantumwellshasattractedenormousattentiontotheeld[ 43 ].Therobustnessofthetopologicalstateandthepossibilitytoengineeritthroughthespin-orbitcouplingmaypotentiallybringaboutsomecutting-edgeapplicationssuchastopologicallyprotectedquantumcomputing[ 44 ]. 32

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2.5.1TopologicalSuperuidsandSuperconductorsInrecenttheoreticalwork,ageneralizationofthetime-reversalinvarianttopologicalinsulatorwasmadetoincludesuperconductorsandsuperuids[ 45 46 ].Accordingtothisclassication,superconductorswith2Dchiralstatessuchaspx+ipy,wheretime-reversalsymmetryisbroken,aredirectanaloguestothequantumhallstate.Similarly,ahelicalsuperconductorinwhichspinj""iisinastatepx+ipyofangularmomentumandspinj##iinastatepx)]TJ /F6 11.955 Tf 12.69 0 Td[(ipyarerelatedtothetime-reversalinvariantquantumspinHallstate.Figure2-8,sketchedfromreference[ 45 ],summarizesthemainsimilaritiesbetweenthefourstates.Inthetime-reversalinvariantsuperconductingstatethebulkstateshaveafullgapandtheedgestatesaregivenbyhelicalMajoranastates,whileinatopologicalinsulatortheyareDiracfermions.Infact,adirectcorrespondencecanbemadebetweentheBogoliubov-deGennesHamiltonianofasuperconductorhavingafullgapandtheHamiltonianofabandinsulatorwithitscorrespondingbandgap.TheBWphaseofsuperuid3Heisaverygoodexampleofthistopologicalstateexceptthat,duetoparticle-holesymmetry,thenumberofdegreesoffreedomisreducedbyhalfcomparedtothatofatopologicalinsulator. 2.5.2MajoranaFermioninSuperuid3HeIn1928DiracformulatedhisrelativisticversionoftheSchrodingerequationforspin1/2particles (i@)]TJ /F6 11.955 Tf 11.96 0 Td[(m) =0,(2)wherearetheso-calledgamma-matricesandobeyClifford'salgebra,e.g. =+=2,(2)whereisthemetrictensor.AmodiedversionwasintroducedlaterbyMajoranainwhichhefoundaparticularsetofgamma-matricesthatwouldrequiretheelds,~ tobereal,e.g.~ =~ .Aparticleobeyingsuchequationwouldturnouttobeitsownanti-particle.Majoranasuggestedthatneutrinosmightshowexactlythese 33

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Figure2-9. Top:QuantumHall(QH)stateandchiralsuperconductor(SC)edgestatespropagatealongthesolidarrow.Time-reversalsymmetryisbroken.Bottom:QuantumspinHall(QSH)stateandhelicalsuperconductoredgestatesofonespinpropagatealongsolidarrow,oppositespinstatespropagatesalongdashedarrow.Time-reversalsymmetryisconserved. characteristics.SubsequenteffortsbythehighenergyphysicscommunityintryingtondtheMajoranafermionhaveremainedunsuccessfuluntiltoday.Recently,thetheoreticalandexperimentalinvestigationofnewtypesofquantummattersuchastopologicalinsulatorsandsuperconductorshaverevivedtheideaofndingexcitationswiththequalitiesdescribedbytheMajoranaequation.Veryrecently,somegroupshavereportedhintsatapossiblerealizationoftheMajoranafermioninsuperconductingnanowires[ 47 ],three-dimensionaltopologicalinsulators[ 48 ],andheterostructuresofatopologicalinsulatorcoupledtoasuperconductor[ 49 ].Zero-energysurfacestatesinhighTcsuperconductorsmanifestasazero-biasconductingpeakinscanningtunnelingexperiments.In3He,thedensityofstatesoftheseso-calledAndreevboundstateshasbeenprobedthroughtransverseacousticimpedance[ 50 ]andspecicheat[ 51 ]measurements.ChungandZhanghaverecentlyproposedthedetectionofMajoranasurfacestatesin3He-Bbymeansofelectronspin 34

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relaxationforelectronbubblesaddedattheliquid'ssurface,wherethespinofthesurfacestateispolarizedperpendiculartothesurfaceinanIsinglikemanner[ 52 ].Ingeneral,forspin-tripletsuperconductorsorsuperuidswhichsatisfytheBogoliubov-deGennesequation[ 53 ] Zdr0H(r,r0)(r0)=E(r0),(2)where(r0)andEarethewavefunctionandenergyofaBogoliubovquasiparticle,theBogoliuvoveldoperators)]TJ /F5 7.97 Tf 6.77 -1.8 Td[(Eand)]TJ /F15 7.97 Tf 6.78 -1.8 Td[()]TJ /F5 7.97 Tf 6.58 0 Td[(Ehavetheproperty)]TJ /F5 7.97 Tf 6.77 -1.8 Td[(E=)]TJ /F15 7.97 Tf 20.2 5.67 Td[(y)]TJ /F5 7.97 Tf 6.58 0 Td[(Eduetoparticle-holesymmetry.Forquasiparticlesinthezeroenergystate,thisleadsto)]TJ /F2 7.97 Tf 6.78 -1.79 Td[(0=)]TJ /F15 7.97 Tf 21.69 5.67 Td[(y0,whichsatisestheMajoranacondition.Recently,TsutsumiandMachidahavecalculatedthecontributionofedgemasscurrentto3HeintheA-phaseandedgespincurrentintheB-phasewhenconnedinadisk[ 54 55 ].AreductioninthemassandspincurrentsareexpectedtofollowaT2andT3respectivelywithinalowtemperaturerange.Additionally,SinaevhasmadepredictionsofNMRsignaturesofMajoranastatesina3He-Blm[ 56 ]andNagatoetal.havecalculatedtheanisotropicspinsusceptibilitycontributionbysurfaceboundstatesina3He-Blm[ 57 ]. 35

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CHAPTER3MICRO-ELECTRO-MECHANICALDEVICESMicro-electro-mechanicalsystems,orMEMS,borrowmicrofabricationtechniquesusedinthesiliconindustrytocreatemechanicallyactivepartsofmicrometersizes.Asthenamesuggests,thesecanbeactuatedelectricallyorcantransducemechanicalmotionintoelectricalsignals.TheincreasingdemandforminiaturesensorsintheautomotiveandconsumerelectronicsindustrieshavefueledthemassproductionofMEMSandmadetheirmanufacturingeconomicallyfeasible.Today,awidevarietyofcommercialprocessesareprovidedbyspecializedfoundries,andmanyfablessdesigncompaniesmakeuseoftheirservicestocreateaplethoraofsensorsandactuatorsforuseasaccelerometers,pressuresensors,magnetometers,viscometers,etc.MEMSdevicesareaninherentlymulti-physicalsystem.InthischapterIsummarizetherelevantphysicsrelatedtotheiractuationandmotioninaviscousuidsuchasair. 3.1Micro-Electro-MechanicalDevicePropertiesThetopographyofthedevicesemployedthroughoutthisworkisthatofacomb-driveactuator.ThisissimilartothoseusedinmodernMEMSaccelerometersandwasrstintroducedbyTangetal.[ 58 59 ].Thenamecomesfromtheinter-digitatedsetofcapacitorsusedforactuationanddetectionofitsmotion.Figure2-1showsanCADimageofatypicaldeviceusedinourexperiments.Thisparticulardeviceconsistsofamovablecenterplatesuspendedbyfourserpentinesprings.Theplatecanbesetinlateralmotionbyelectrostaticinteractionbetweenthecomb-likeelectrodesattachedtothesidesandthexedelectrodesanchoredtothesubstrate.If,forexample,aDCvoltagewasappliedacrossonepairofelectrodes,thecenterplatewillbepulledtowardsthexedelectrodesalongthex-direction.Thecenterplatehoversabovethesubstrateataxeddistance,thuscreatingauniformgap.Thetwodevicesusedthroughoutthisworkhavegapsizesof1.25and0.75m,referredtoasH1andH2,respectively.Othertypes 36

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Figure3-1. CADimageofatypicalH1deviceshowingitsmainstructures.Nosubstrateisshownonthispicture. Figure3-2. CADimageofaserpentinespringusedinbothH1andH2devices.Thedimensionsoftheactiveelasticpartarelabelledast=2m(height),`=120m(length),andw=3m(width). 37

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Figure3-3. Left:CantileverbeamsubjecttoaforceF.Right:GuidedbeamsubjecttoaforceF. ofH1deviceswillbeintroducedinchapter4,labeledasH1t,m,whichhavevariationsintheconnectinglinefromthexedelectrodestothebondingpads. 3.1.1SpringDesignThespring'sgeometryisshowningure3-2.Itconsistsoffoldedbeamsarrangedinaserpentineshape.Theconnectionbetweenthebeamsisassumedrigid.Asopposedtoacantilever,thisrigidpointactsasaguideandmakeseachsinglebeamdeformintoansshapewhenaforceisappliedinthex-direction(seegure3-3).Forthisreason,thistypeofbeamiscalledaguidedbeam.Whenconnectedinseries,asetofbeamsofspringconstantkiexperiencethesameforceandtotaldisplacement.ForNspringsitisrepresentedbyxtotal=x1+x2+...+xN=F k1+F k2+...+F kN=1 k1+1 k2+...+1 kNF. (3)Thespringconstantforasinglecantileverbeamisgivenby k=E 4w l3t.(3)Therefore,thespringconstantforNseriesbeamsis k=E 4w l3t N,(3) 38

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Figure3-4. Anelectrodeshowingthecomb-shapedstructure.Theoutsetshowsapairofngers,whichformaparallelplatecapacitorattheoverlapregion. whereEisYoung'smodulus,wisthewidthofasinglebeaminthexdirection,tistheout-of-planeheightand`isthelength(seegure3-2).Theguidedbeamcanbemodelledastwocantileversoflength`=2addedinseries[ 60 ].Thus,forNguidedbeamsthetotalspringconstantisgivenby k=Ew l3t N.(3) 3.1.2CombElectrodesThemovablecomb-shapedelectrodesengageinterdigitallywiththeelectrodesxedtothesubstrate.Figure2-4showshalfofthecomb-likestructureofoneoftheelectrodes.Thecombswiththengerspointinginthepositivex-directionaremovable,whiletheonespointinginthenegativex-directionarexed.Considerthepairofngersboxedingure3-4.Eachpairofngersformaparallelplatecapacitorwhosecapacitanceisgivenbythesimpleequation C(df)=A0 df,(3) 39

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whereistherelativepermittivityofthemedium,0isthepermittivityoffreespace,dfisthedistancebetweenngersalongthey-direction,andAistheoverlappingareaofthetwoplates.Whenavoltageisappliedbetweenthepairofngerstwoforcesarise,onedirectedperpendicularlyfromonengertotheother(y-direction),andatangentialcomponentalongthengers(x-direction).Theperpendicularforceiscancelledoutwhenaddinguptheeffectofallngers,butthetangentialforceinducesadisplacementinthex-directionandchangesthecapacitanceby C=tx0 df,(3)wheretistheout-of-planeheightofasinglenger.Thiscapacitancechangeislinearintheoverlappingdistance,x.Basedonthis,thetotaltangentialelectrostaticforce(Fx=1 2dC dxV2)ontheelectrodeis[ 61 ] Fe=Nt0V2 2df,(3)whereNisthetotalnumberofngeroverlapsintheelectrode. 3.2FrequencyResponseToexcitethedeviceintoaresonantstate,asinusoidalACvoltage,Vac=V0sin!t,isappliedalongwithaDCbiasvoltage,Vb,suchthatthetotalelectrostaticforceisgivenby Fe(t)=Nt0 2df(Vb+V0sin!t)2=Nt0 2dfV2b+V20 2+2VbV0sin!t)]TJ /F6 11.955 Tf 13.15 8.09 Td[(V20 2cos2!t.(3)Itcanbeseenthattheforceonthecomb-driveisdividedintothreecomponents:F0,F1andF2.Therstterm,F0=V2b+V20=2,istheconstanttermontherighthandsideofequation3-8,F1=2VbV0sin!tisinphasewiththeexcitationandexistsonlywhenaDCbiasisapplied.TheF2=V20 2cos2!tisoutofphasewiththeexcitationattwicethe 40

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appliedfrequency.Explicitly,theelectricalforceisthen Fe(t)=Nt0 dfVbV0sin!t=Vac(t),(3)where=Nt0Vb=dfisoftencalledtheelectro-mechanicaltransductionfactor. 3.2.1Mechanical-ElectricalCorrespondenceThemechanicalsystemdescribedherecanbemodelledasadampeddrivenharmonicoscillator: mdv dt(t)+v(t)+kx(t)=Fe(t),(3)whereFe(t)istheexternalforce,v(t)thevelocity,andthesetofconstants(m,,k)arethemass,dampingcoefcientandspringconstant,respectively.Inordertocouplethemechanicalsystemtotheelectricalcounterpart,theclassicalanalogywithaseriesRLCresonantcircuitisexploited[ 62 ].Theequationdictatingtheelectricalsystemis Ldi dt(t)+Ri(t)+1 Cq(t)=Vac(t).(3)TheelectricalsystemisdeterminedbytheappliedvoltageVac(t),thecurrenti(t),andtheconstants(L,R,1=C),whicharetheinductance,resistance,andtheinverseofthecapacitance,respectively.Toconnectthetwosystems,thetransductionfactorinequation3-9isusedtocoupleFe(t)andVac(t)aswellasi(t)andv(t) i(t)=d dt[C(t)Vb]=VbdC dxdx dt(t)=v(t).(3)Thecorrespondenceacrossthetwomodelscanbeestablishedandtherelationsbetweenthedifferentconstantsaregivenby L=m=2, (3) R==2,1=C=k=2. 41

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3.2.2ResonancePeakEquation3-10cannoweasilybesolvedusingtheequivalentseriesRLCcircuit.Theadmittanceoftheelectricalsystemis 1 Z=1 j!L+R+1=j!C,(3)wherej=p )]TJ /F4 11.955 Tf 9.3 0 Td[(1and!isthedrivingfrequency.Usingtheresonancecondition!20=1=LC,equation3-14canbewrittenas 1 Z=(!=m)2=R (!20)]TJ /F7 11.955 Tf 11.95 0 Td[(!2)2+(!=m)2+j(!20)]TJ /F7 11.955 Tf 11.96 0 Td[(!2)(!=m)=R (!20)]TJ /F7 11.955 Tf 11.95 0 Td[(!2)2+(!=m)2,(3)whereL=R==mwasusedfromequation3-13.Finally,thetotalcurrentcanbewrittenas i(t)=ix(!)cos!t+iy(!)sin!t,(3)whereix(!)andiy(!)aregivenby ix(!)=i0(!!)2 (!20)]TJ /F7 11.955 Tf 11.95 0 Td[(!2)2+(!!)2, (3) iy(!)=i0(!20)]TJ /F7 11.955 Tf 11.96 0 Td[(!2)(!!) (!20)]TJ /F7 11.955 Tf 11.96 0 Td[(!2)2+(!!)2.Here!==misthefullwidthathalfmaximumandi0isthepeakamplitude.Thisequationscanalsobewrittenintermsofthemechanicalconstantsonlyas ix(!)=2V0!2 (k)]TJ /F6 11.955 Tf 11.95 0 Td[(m!2)2+(!)2, (3) iy(!)=2V0!(k)]TJ /F6 11.955 Tf 11.96 0 Td[(m!2) (k)]TJ /F6 11.955 Tf 11.96 0 Td[(m!2)2+(!)2.Thepreviousexpressionsaresimilartothoseusedforpiezoelectrictuningforkoscillatorswiththetransductionfactorsubstitutedbyafactorcalledtheforkconstant,whichistheproductofaprong'sgeometricalfactorandthepiezoelectricconstant[ 63 ]. 42

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Figure3-5. Aplate(lightgrey)oscillatinglaterallyalongthex-directionwithvelocityvclosetoasubstrate(darkgrey).Itishelddownbyaspringateachend. 3.3AirDampingAnoscillatingobjectinaviscousuidisatextbookprobleminuidmechanics[ 64 ].However,inthepresenceofanotherobject,theoweldcanchangedrastically.Thus,theproblemofaplateoscillatingincloseproximitytoasecondplatebecomesnon-trivial.Itwasrealizedintheearlystagesofminiaturizationofmechanicalstructures,thatthedampingeffectduetothesurroundingairwasseverelyincreasedunderthesecircumstances[ 65 ].Incontrastwithlargemacroscopicstructureswherevolumeforcesdominate,micro-mechanicalstructuresaremoreaffectedbysurfaceforces.Thus,theeffectofthelmofairtrappedbetweentheplatesbecomesmoreimportantatthesescalesthanthedragforce,whenthethicknessofthelmsismuchlessthanone-thirdthewidthoftheplate[ 66 ].Inthelaterallymovingcomb-driveoscillator,theso-calledslidelmdampingdominatesthedissipativeprocessofthesystem.Thiseffecthasbeenstudiedindepth,anditiscrucialintheMEMSindustry[ 67 69 ].TheNavier-Stokesequationdescribesthemostgeneralcaseforasteadyowofanincompressibleuid: @v @t+(vr)v=F)-222(rp+r2v,(3)wherevisthevelocityeld,istheuiddensity,Fistheexternalforce,pistheuidpressure,andisthedynamicviscosity.Considertheplateingure3-5movinginthex-directiononly.Thereisnoexternalpressureofforces,thepreviousequationis 43

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reducedto @vx @t+vx@vx @x= @2vx @z2.(3)Furthermore,ifthedimensionsoftheplatesatisfylpd,a0,wherelpisthecharacteristiclengthoftheplate,disthesizeofthegapanda0istheamplitudeofoscillation,thentheplatecanbeconsideredinniteandthepreviousequationcanfurthersimpliedas @vx @t= @2vx @z2(3)Thisequationisaone-dimensionaldiffusionequation,anditcanbesolvedundertwodifferentowregimes:theCouetteandtheStokesowregimes. 3.3.1CouetteFlowWhenthegapsizeismuchsmallerthantheviscouspenetrationdepth,e.g.d,where=p 2=!,equation3-21canbeapproximatedto @2v @z2=0.(3)Thepreviousequationhasasimplesolution,whichshowsalinearvelocityelddistributionasafunctionofz v(z)=vp1)]TJ /F6 11.955 Tf 13.53 8.09 Td[(z d,(3)wherevpisthevelocityoftheoscillatingplateandanon-slipboundaryconditionisassumed.Fromthisequationtheviscousforceperunitareacanbecalculatedfromtheviscousstresstensor[ 64 ],=@v=@z,andit'sfoundtobe F=Ap dvp,(3)whereAprepresentstheareaoftheoscillatingplateincontactwiththeuid.Thedampingforceontheplateislinearinvelocity,F=v,andhasadampingcoefcient=Ap=d. 44

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3.3.2StokesFlowWhentheconditiondisnotsatised,noapproximationcanbemadeonequation2-10.Inthiscase,thefullsolutionwithnon-slipboundaryconditionsisgivenby[ 69 ] v(z)=vpsinh(qz) sinh(qd),(3)whereq=p i!=,with==.Theviscousforcecanbecalculatedagainbytakingthederivativeofthevelocityproleandacomplexdampingadmittanceisobtained =Apq tanh(qd).(3)Therealpartofthecomplexadmittancegivestheactualdampingcoefcient Re()=Ap sinh(2d=)+sin(2d=) cosh(2d=))]TJ /F4 11.955 Tf 11.96 0 Td[(cos(2d=)(3)Theimaginarypartofthecomplexadmittanceisrelatedtothefrequencyshiftduetomassloadingfromtheformationofaboundarylayerofliquidontheoscillatingplate,anditisgivenby Im()=Ap sinh(2d=))]TJ /F4 11.955 Tf 11.96 0 Td[(sin(2d=) cosh(2d=))]TJ /F4 11.955 Tf 11.96 0 Td[(cos(2d=)(3) 3.3.3RareedFluidFlowAnoscillatingobjectexperiencesdampingduetoitsinteractionwiththesurroundinguid.Ingeneraltheuidcanbeconsideredtobecontinuouswhenthemeanfreepath,,islargecomparedtothecharacteristicsizeoftheobject,e.g.d.Inthiscase,theboundaryconditionfortheuidvelocityattheinterfacewiththeoscillatingplateneedstobecorrectedtoincludetheslipeffectduetothedecouplingoftheuid.Thiseffect,arisingfromthemolecularnatureoftheuid,canbecomeevidentforapairofmicrometersizedparallelplatesevenclosetoatmosphericpressure,anditisthereforeanimportantparameterinMEMSdesign.ThecrossoverfromtherareedtothecontinuousregimecanbetrackedbytheKnudsennumber,Kn==d,whichhasavalue 45

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Figure3-6. Aplate(lightgrey)oscillatinglaterallyalongthex-directionwithvelocityvclosetoasubstrate(darkgrey).Thevelocityeldoftheuidisshowninbluearrows.Theredarrowisthevelocityattheboundaryand0isthesliplength. aroundoneatthetransitionpoint.Ontheotherhand,thechangeinthevelocityprolewithinthegapasafunctionofz,e.g.slopeofthevxvszplot,shouldbeproportionaltothevelocityattheplateboundary[ 70 ].Theconstantofproportionalityiscalledthesliplength,0,anditsvalueisgivenbytheinterceptofthelinetangenttothevelocityprolewiththezaxis.Ifthelineinterceptsatz=0,thenthesliplengthiszeroandtheuidisnotrareed.Conversely,ifthelineinterceptsatz<0,thesliplengthisniteanddescribesthesizeoftheboundarylayerclosetothesurfacewithinwhichtheuidslips.Toarstapproximation,0,andtheeffectonthelateralmotionoftheparallelplatesystemistoeffectivelyincreasethegapdistancebythisamount,e.g.2.IntheCouettemodel,thedampingcoefcientisthenapproximatedby =Ap d+2,(3)or =Ap d 1+2Kn,(3)wheretheterminparenthesisrepresentsaneffectiveviscosity.IntheStokesmodel,theboundaryconditionsusedtosolve3-21arechangedtoincludethesliplengthattheboundaries.Inthiscase,Veijolaetal.calculatedthe 46

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dampingcoefcienttobe[ 69 ]=Ap sinh(2d=)+sin(2d=)+k1 cosh(2d=))]TJ /F4 11.955 Tf 11.95 0 Td[(cos(2d=)+k2, (3)k1=4(1+2)cosh(2d=)+(1)]TJ /F7 11.955 Tf 11.96 0 Td[(2)cos(2d=)+62[sinh(2d=))]TJ /F4 11.955 Tf 11.95 0 Td[(sin(2d=)],k2=4(1+22)sinh(2d=)+(1)]TJ /F4 11.955 Tf 11.96 0 Td[(22)sin(2d=)+42(2+2)cosh(2d=)+(2)]TJ /F7 11.955 Tf 11.95 0 Td[(2)cos(2d=),where==. 47

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CHAPTER4EXPERIMENTALDETAILSANDDEVICECHARACTERIZATIONThischapterpresentsacomprehensivecharacterizationofMEMSoscillatorsfabricatedusinganexternalcommercialprocess.Wedesignedtwotypesofdevices,shearandout-of-planemotionoscillators.Forthepresentresearchonlythelateralmovingoscillators(H1andH2)wereusedduetolowerviscousdamping.Section4.1describesthefabricationprocessusedandthepreparationprocedurewhichincludesrelease,cleaning,andwirebonding.Theroughnessofthesurfacesisanimportantparameterwhichhasaninuenceontheinteractionofthedevicewithitssurroundingmedium.Wehavecarriedoutacharacterizationofthesurfaceusingatomicforcemicroscopy(AFM)measurements.Theresultsfromtheseinvestigationsareshowninsection4-2.Generalpropertiesofthedevicesandthedetectionschemearedescribedinsection4-3,alongwithastudyofthefrequencyresponseandtheeffectofaDCbiasvoltage.Itsresonancepropertieswerestudiedatroomtemperaturethroughawiderangeofpressuresfrom10mTorrto1atm.Thechangeinthequalityfactorwasmonitoredandthedampingcoefcientwasextractedandfoundtobeinverygoodagreementwiththeoreticalmodelsfortheso-calledslidelmdamping.Theresultsofthisstudyarediscussedinsection4-4. Table4-1. ThicknessofthedifferentlayersusedinPolyMUMPs LayernameMaterialThickness(m) NitrideSiliconnitride0.6Poly0Polysilicon0.5FirstOxidePSG2Poly1Polysilicon2SecondOxidePSG0.75Poly2Polysilicon1.5MetalGold0.75 48

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4.1ManufacturingofMicro-Electro-MechanicalDevices 4.1.1DesignandFabricationThedeviceswerefabricatedbyanexternalfoundrycalledMEMSCAP.Amulti-usersurfacemicromachiningprocesscalledPolyMUMPsofferedbythecompanywasused.Structurescreatedwiththisprocessconsistofthefollowingcomponents:threepolysiliconlayers(Poly0,Poly1,Poly2)usedasthestructuralmaterial,onesiliconnitridelayer(Nitride)usedforelectricalisolationfromthesubstrate,twophosphosilicateglass(PSG)layers(FirstandSecondOxide)usedassacriciallayers,andametallayer(METAL)usedforelectricalconnectivity.Thenamesinparenthesisindicatethenameofthemaskusedinthelithographicprocesstocreatethepatternforthecorrespondinglayer.Thethicknessofthelayersisxedandcanbefoundintable4-1.Inadditiontothebasicmasksmentioned,othermaskscanbeusedtomakeholes,dimples,anchorsandinter-layerconnections.ThelayoutforeachlithographiclevelwasdesignedusingaspecializedCADsoftwarecalledL-edit.Thedesignswerethensenttothecompanyforfabrication.Theprocessiscarriedouton100mmwafersheavilydopedwithphosphorus.Thelayersaredepositedusinglowpressurechemicalvapordeposition(LPCVD)andpatternedbyacombinationofphotolithographyandreactiveionetching(RIE).Inaddition,PSGisusedintheprocesstodopethedifferentPolylayerswithphosphorusbyannealingthewaferat1050Cinargonduringdifferentstepsoftheprocess.Afteraddingandpatterningallthenecessarylayerstocreatethestructures,theresultingchipsaresentbacktothecostumer.Moredetailscanbefoundinthedesignhandbookatthecompany'swebsite[ 71 ]. 4.1.2DevicePreparationThediereceivedfromthefoundrywasdicedintosmallersquarechipswithasidelengthofapproximately2.5mmbyusingadiamondsaw.Thedevicesatthispointwerestillcoveredbyaphotoresistlayerandthesacriciallayerwasstillpresent,rendering 49

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Figure4-1. SEMpicturesofatypicaldeviceusedinthiswork.Themainstructuresareshownsuchas:centerplate,electrode,spring,andcombngers. themmechanicallyinactive.Afterdicing,thedevicesarereadytobereleased.ThisinvolvestheremovalofthesacricialPSGlayerstofreethemechanicallyactiveparts.Astandardprocesssuggestedbythefoundryisfollowed.First,thedicedchipsweresoakedinacetonefor20-30minutestoremovethephotoresist.Afterthis,theywereputinisopropanolforseveralminutesandthenrinsedwithdeionized(DI)water.Thechipswerethenintroducedintoabathof49%hydrouoric(HF)acidforabout7-8minutes.TheHFetchesawaythePSGlayersatarateofabout40mevery5minutes.Dependingonthegeometryandsizeofthedevicestheetchingtimecanbeadjusted.AftertheHFbaththechipsarerinsedinDIwaterfor10minutesandthencleanedagainwithisopropanol.Themechanicalstructuresinthedevicearenowfreetomoveandiftheyareallowedtodrybyexposuretotheenvironment,thesmallplatescomprisingthe 50

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devicesmightbepulledtowardseachotherbycapillaryactionandsticktogether.Thisphenomenoniscalledstictionandit'sawellknownproblemintheMEMSeld.ToavoidthisproblemthechipsweredriedusingaCO2criticalpointdryer.Afterdrying,thechipswerepackagedinacustomdesigned20-pinsocketfabricatedbyIronwoodElectronics,wheretheywerexedwithconductingsilverepoxy.Aftertheepoxywascured,electricalconnectionsweremadefromthebondingpadsonthechiptothepadsonthepackagebyusingaWest-Bondballwedgewirebonderwith0.025mmthickgoldwire.Scanningelectronmicroscope(SEM)imagesofatypicalreleaseddeviceareshowningure4-1. 4.2AtomicForceMicroscopySurfaceCharacterizationSurfacepropertiesareanessentialparametertotheunderstandingoftheinteractionbetweentheuidandtheoscillatingobject.Itis,therefore,imperativetounderstandalltherelevantlength-scalesrelatedtothetopographyofthepolycrystallineplatesinourdevices.Forthispurpose,wehavecarriedoutatomicforcemicroscopy(AFM)measurementsonasamplechipcontainingtheMEMSresonators.Thechipwassprayedwithnitrogengastoblowoffthetopplateofallthedevicespresentonthechip.ANanoscopeIIIAScanningProbeMicroscopewasusedtoscanthesurfaceoftheexposedPoly0surfacesthatconstitutethebottomplateofatypicaldevice.Twelvedifferentspotswereprobedwithscansof1010,22,and0.40.4m2ondifferentpartsofthesamedeviceandonotherdevicescontainedinthesamechip.TheAFMtipwassetintappingmodewitharesonancefrequencyof69.16kHz.Dataofscansizesof1010and22m2areshownongure4-2.AhistogramofheightsshowsasymmetricGaussiandistributionaboutthemeanplane(seegure4-4).ThewidthoftheGaussianis20.83nm,thus,theaverageheightwithrespecttothemeanplaneis10.42nm.Toinvestigatethecorrelationsbetweengrains,anormalizedautocorrelationfunctionwascalculatedfromthetwosurfacetopographies.Asshowningure4-3,thefunctionissharplypeakedatthecenter.Fromthe22m2autocorrelation,aone-dimensionalautocorrelationwascalculatedbyaveraging 51

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Figure4-2. TopographyofthesurfaceifthePoly0layerinatypicaldevice. 52

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Figure4-3. NormalizedautocorrelationfunctioncalculatedfromthetopographyofthePoly0surfaceinatypicaldevice 53

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Figure4-4. Top:Histogramofheightsfromthetopographyofthesurfaceofadevice.Bottom:1Dnormalizedautocorrelationfunctionaveragedfromdifferentspotsonthedevice. 54

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Table4-2. Calculateddevicepropertiesfromlayoutgeometry DeviceGap,dArea,AMass,mSpring,ksCapacitance,c0(m)(mm)(10)]TJ /F2 7.97 Tf 6.59 0 Td[(10kg)(4N/m)(10)]TJ /F2 7.97 Tf 6.58 0 Td[(3pF) H1t,m1.251961963.452.478.41H10.751781782.772.478.41 allthevalueswithintheperimeterofacircleorradiusrcenteredattheoriginofgure4-3,e.g.thepositionofthepeak'smaximum.Theresultingautocorrelationfunctionisshowningure4-4.TheautocorrelationgoesthroughzerotoanegativevalueatpointP1,wherethedisplacementawayfromthecenterpeak,r,is0.137m.Thisdistanceprovidesameasureofthegrain'swidth.Thisvalueis10timeshigherthantheheight,whichmightindicatethatthegrainshaveanoblatespheroidshape.Thesecondpoint,P2,goesthroughzeroagaintoapositivevalue.Usingthispointthecenterofthenextclosestgraincanbedeterminedtobe,inaverage,atadistanceof0.728m.Insummary,fromtheheighthistogram,thegraincanbeestimatedtohaveaheightof10.42nm,andfromtheautocorrelationtheestimatedwidthis0.137m.Furthermore,theaverageinter-graindistancecanbeestimatedtobe0.728mfromtheautocorrelationfunction. 4.3DeviceCharacterization 4.3.1DeviceSpecicationsSomeimportantdevicespecicationsforbothtypesofdevices,H1(1.25mgap)andH2(0.75mgap),areshownintable4-2.Theseinclude:thegapdistanced,theareaofthemainplateA,themassm,thespringconstantks,andthecapacitancec0.Thecapacitanceandthespringconstantwerecalculatedfromequationsfoundinchapter3.Thespringconstantinequation3-3isstronglydependentonthewidth,w,andtheheight,t,ofthespring(seegure3-2).ThelithographicmasksusedintheprocessbyMEMSCAPhaveapixelationof0.25m(seereference[ 71 ]),soifanerrorashighas0.5mperlateraldimensionisassumed,theerrorinthespringconstant 55

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Figure4-5. Circuitdiagramforthecapacitancebridgetechniqueemployed.Anexternalratiotransformerisusedasatunableinductivevoltagedivideralongwiththetwocapacitorsformedbythetwosetsofelectrodesatthesidesofthedevice. valuecanbeashighas38%.Themassofthedevicewascalculateddirectlyfromthegeometryofthedevices.Inthiscasetheerrorinthelateraldimensionsisnegligible,andtheonlyrelevantsourceoferroristhethicknessofthePolylayers.Thisisdeterminedbyhowthelayersaregrown,anditisaccuratewithin10nmasreportedbyfoundry.Thismakestheestimatedmasserrortobelessthan1%.Theresonancefrequenciesarecalculatedfromthevalueofthemassandthespringconstant(f=1 2p ks=m). 4.3.2DetectionSchemeThetopplateiselectrostaticallycoupledtothexedelectrodesatthesidesthroughcomb-likestructuredterminals(seegure3-1).Thisallowsthedevicetobeactuatedinshearmotion,paralleltothesubstrate.Thetwosideelectrodesformapairofseriescapacitorswiththecenterplate,thusweimplementadifferentialcapacitancemeasurementusinganexternalratiotransformer(TEGAM1011A).Acircuitdiagramisprovidedingure4-5.Ahighfrequencysignal,Vhf=150kHz,isprovidedbytheinternaloscillatorofaSignalRecovery7124lock-inamplierwhichisalsousedfordetection.Alowfrequency(Agilent33220A),Vlf,isaddedtoVhfthroughahomemade 56

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Table4-3. ResonancefrequenciesandmeasuredQ-factorsinhighvacuum(3mTorr) Devicefc(calculated)fs(simulated)fm(measured)Q)]TJ /F6 11.955 Tf 11.96 0 Td[(factor(measured)(Hz)(Hz)(Hz) H129133.324755.423063.16424.3H1t29133.324755.422006.223188.8H1m29133.324755.421516.4105472.6H232513.228438.326691.65858.6 summingamplier,andissweptthroughtheresonance.Asthedevicegoesthroughresonancetheoff-balancesignalispickedupbyanAmptekA250chargesensitiveamplier.Thissignalislaterdemodulatedbytwolock-inampliersconnectedinseries.Therstlock-inamplier(SignalRecovery7124)demodulatesthehighfrequencysignalandthesecondlock-in(SignalRecovery5210)isusedtodetecttheresonancepeak.Thefollowinglistprovidesafewclaricationsaboutthecircuitsetupupgradesintroducedduringthecourseoftheworkdescribedinthisthesiswork. Thesetupdescribedabovewasimplementedfortheairdampingmeasurementsshowninsection4.4ofthischapteraswellasfortheexperimentsinsuperuid4Heinchapter5. Itwasnoticedthatthesignal-to-noiseratiowasdegradedwhenadevicewassetuponthecryostatinlabB137.Forthe3Heexperimentsdescribedinchapter6,thisproblemwasresolvedbyreplacingthehomemadesummingamplierbypassivecomponents(MiniCircuitsZFSC-2-6-S+).Thesignal-to-noiseratiowasimproved10fold. Duringthecourseofthe3HeexperimentstheSignalRecovery7124lock-inamplierwasreplacedbyanEG&G7265DSPtodemodulatethehighfrequencysignal.Also,aTEGAM2725AwaveformgeneratorwasusedasthesourceforVhf. AnexternalDCbiasvoltagewasnotusedinthemeasurementsinsuperuid4He.Inthiscase,thedevicereliedontheintrinsicbiasvoltagedescribedlaterinsection4.3.5.Thetestin4Hewasdonebeforetheairdampingandthe3Hemeasurements. 4.3.3FrequencySweepsAfteradeviceiswirebondedandthecircuitissetup,theresonancepeakswerefoundthroughfrequencysweepswhilekeepingthedeviceinaclosedchamberunder 57

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Figure4-6. VibrationalmodesfordeviceH1. vacuum,atapressureP3mTorr.Aftersweepingthefrequency,tworesonancepeakswerefoundforallthedevicestested.Thesecorrespondtodifferentmodesofoscillation.Oneofthepeaksisfoundatalowerfrequency14KHzandtheotherat23KHz(seeTable3-3).Thepositionofthepeakschangesfromdevicetodeviceduetotheresolutionofthefabricationprocess(0.25m),whichintroducesvariationsinthespring'sthicknessthusshiftingthefrequency.Inordertoidentifythetypeofmodecorrespondingforeachresonancefrequency,simulationswereperformed,usingCOMSOLmultiphysics,tondtheeigenfrequenciesoftheoscillatingstructure.TheresultsforthedifferentmodesandtheircorrespondingfrequenciesforH1andH2aresummarizedingures4-6and4-7,respectively.Forthesesimulations,axedboundarycondition(nomotion)wasusedattheanchorpointofthespring(seegure3-2).Usingtheinformationfromthesimulations,thetwomodesfoundfromthefrequencysweeps 58

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Figure4-7. VibrationalmodesfordeviceH2. wereidentiedasthepivotmodeabouty-axisandtheshearmodeforthelowandhighfrequencypeaks,respectively.Theresultsoffrequencysweeps,simulations,andanalyticalestimationsoftheresonancefrequenciesaresummarizedinTable4-3fortheshearmode.ThreedifferentH1deviceswerestudied,indexedas:H1,H1t,andH1m.Thegeometryoftheseisthesame,butthedesignoftheelectricalconnectionfromthebondingpadstotheelectrodesisdifferent.Fordeviceslabelledwithsubscriptt, 59

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Figure4-8. ResonancepeaksforanH1mdeviceattwodifferentexcitationvoltages.Thehigherexcitationsinduceanonlinearbehaviouronbothdevices. theconnectionsweremadethicker,whilefortheoneslabelledwithsubscriptm,theelectrodeswerecoveredwithametal(gold)layertoimproveconductivity.Resonancepeaksatdifferentvoltagesareshowningure4-8fortheshear(topgures)andthepivotmode(bottomgures).Thedetectedresonancepeakwasfoundbyscanningin1fmodeatthesecondlock-inamplier,whichmeansthedetectedsignalandthedrivingsignalareatthesamefrequency.Nonlinearpeaksareobservedatthehighervoltages(seerightguresin4-8).Inthenonlinearregime,thepeaksshowhysteresisbetweenupfrequencysweepsanddownfrequencysweeps.Whensweepingdownthefrequency,theamplitudefollowsassmoothincreasingcurvethatsaturates 60

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atacertainvalue,andthen,suddenly,theamplitudesjumpdowntoavalueclosetozero,fromwhereitcontinuestodecreasesmoothlytowardszero.Whensweepingup,theamplitudefollowsasmoothlyincreasingcurveuntil,suddenly,theamplitudejumpstoahighervaluematchingthedownsweepcurve,andthen,itcontinuestodecreasesmoothlyfollowingthisdownsweepcurve.Thecombinationofthesetwosweepsiscalledahysteresisloop,anditgivestheabsorptionpeaktheslantedshapedisplayedingure4-8.Thisbehaviouriscapturedbytheso-calledDufngoscillatormodel[ 72 ],wherethenonlineartermofx3isaddedtotheequationofmotionofthelineardrivendampedharmonicoscillator. 4.3.4DirectCurrentBiasStudyADCbiasvoltagewasappliedtooneofthesideelectrodesofadevicethroughthe2kresistorandthevoltagewaschangedfrom-10to10V.Theamplitudeandresonancefrequencyaredeterminedfromafrequencysweepandtheresonancepeakisttedtooneofthecurvesgivenbyequation3-18.Theonlydifferenceisthatwedetectcharge,notcurrent,sothecurvesgivenbytheseequationsarescaledbyafactorof!andarephaseshiftedby=2.TheresultsforanH1devicewithmetallicelectrodes(H1m)areshownfortheshearmode(topofgure4-9)andforthepivotmode(bottomofgure4-9).Theamplitudeincreaseslinearly,asexpectedfromequation3-9,untilitstartssaturatingandturningbackdown.Thisisadirectconsequenceofnon-linearitiesstartingtoaffectthepeakduetohighdrivingamplitudes.Theresonancefrequencyfollowsaparabolicbehaviour.Thefrequencyoftheshearmodedecreasesasafunctionofbias,whilethefrequencyofthepivotmodeisseentoincreaseasafunctionofbias.BecauseofthelineardependenceoftheelectrostaticforceonVb,inprinciple,iftherewasnobiasvoltage,theF1terminequation3-8woulddisappearandnopeakwouldbedetectedwhenattemptingtodetectasignalatthesamefrequencyasthedrivingforce.ItwasfoundthroughthesestudiesthatevenatVbias=0thereisstillaresonancepeak.Thismeansthatthereisanintrinsicbiasvoltagebetweenthecenter 61

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Figure4-9. Top:AmplitudeandresonancefrequencyasafunctionofbiasvoltagefortheshearmodeofanH1device.Bottom:AmplitudeandresonancefrequencyasafunctionofbiasvoltageforthepivotmodeofanH1device.Thelinesrepresentattoaparabola.Theresultingequationsforthetaredisplayedineachgraph. electrodeandthedrivingelectrode.Asofnow,wehavenotfoundthesourceofthisintrinsicvoltage.Additionally,thebaseofparaboliccurvesingure4-9showasmallshiftof)]TJ /F4 11.955 Tf 9.3 0 Td[(0.5V.Thisshiftseemstoberelatedtotheintrinsicbiasvoltageasseeninthenextsubsection. 4.3.5DrivingForceComponentsTheF2componentoftheelectrostaticforceinequation3-8isoutofphasewiththeexcitationsignal.Thelock-inamplierwassettodetectasignalattwicetheinput 62

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Figure4-10. Resonancepeaksforapivotmodewhendetectingasignalatthesamefrequencyasthedrivingvoltage(fmode)andattwicethefrequencyofthedrivingvoltage(2fmode). frequency,2fmode,andthefrequencywassweptathalfthevalueoftheoriginalpositionofthepeak.Resonancepeaksforthepivotmodeatfand2fareshowningure4-10.Usingtheamplitudeofthepeakin2fmode,itispossibletoobtainameasureoftheintrinsicbiasbytakingaratiowiththeamplitudein1fmodewithnobiasvoltageapplied.TheratiooftheamplitudesforanH2deviceisfoundtobe Af A2f=0.792(mV) 0.117(mV)=6.76.(4)Thus, 2VbV0 V20=2=6.76,(4)fromwhichwegetVb=0.425VusingV0=0.25V.Similarly,usingtheshearmodepeaksatfand2f,wefoundVb=0.482V.ThesevaluesaresimilartotheestimateoftheintrinsicbiascalculatedfromthedisplacedbaseofthefrequencyversusVbiasparaboliccurve.Similarvaluesarealsoobtainedforalldevicesondifferentchips. 4.4ResonanceasaFunctionofAirPressureTheinuenceofairdampingontheperformanceofminiaturemechanicaloscillatorsisasubjectofutmostimportancewithintheMEMScommunityfromatechnological 63

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Figure4-11. Top:Q-factorandshearmodefrequencyofdeviceH1masafunctionofpressure.Bottom:Q-factorandshearmodefrequencyofdeviceH2asafunctionofpressure standpoint.Additionally,duetothehighsurfacetovolumeratio,owatthemicro/nanoscaleshowsnewphenomenanotobservedatthemacroscopicscale[ 73 75 ].Asarstapplication,acharacterizationoftheinuenceofairdampingonourdeviceswasperformed.Theresultswerecomparedtopreviousexperimentalandtheoreticaldevelopments.Thetwodifferentdevices,H1andH2,werestudiedindividuallywhileenclosedinavacuumchamber,inwhichthepressurewascontrolledbyventingandpumpingthechamberthroughaneedlevalvefrom10mTorrto1atm.Thefrequencywasswept 64

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throughtheshearmoderesonanceafterthepressurewasstabilized.Thefrequency,width,andamplitudeofthepeakwerethenobtaineddirectlyfromthettothecurvesadaptedfromequation3-18.Thequalityfactorwasobtainedbytakingtheratiooftheresonancefrequencytothewidthofthepeak.Thefrequencyandqualityfactorareplottedingure4-11fordevicesH1(topgures)andH2(bottomgures)asafunctionofpressure.Thequalityfactorshowsatextbookbehaviourfromlowesttohighestpressure.Thecurveisessentiallyatfrom10to100mTorr,sincethedampingismainlydominatedbyintrinsicdissipation,whichispressureindependent.ThequalityfactorthenstartstodecreaseinanalmostlinearmanneruntilP100Torr.Atthispointthedampingbeginstosaturateagain,sinceinthehydrodynamicregimethedampingbecomesindependentofpressure.Theresonancefrequencystaysalmostconstantatpressuresbelow4mTorr,andthenitstartstodecreasecontinuouslyasthepressureincreases.Thisindicatesatransitionfromahighlyrareed(ballistic)regimeintoahydrodynamicregime.Thecoefcientofdampingcanbeextractedexperimentallyfromthewidthofthepeak,e.g.=m!,wheremisthemassofthedevice.Theintrinsicdampingwasobtainedforbothdevicesfromtheconstantvalueofthewidthatlowpressures.Figure4-12showstheexperimentallydetermineddampingcoefcientasafunctionofpressureforH1andH2,obtainedbysubtractingtheintrinsicdamping.ThevaluesofthedampingcomingfromdifferentpartsofthedevicewerecalculatedusingtheexpressionobtainedintheCouetteowregime,e.g. =Ap d 1+2Kn.(4)Wefollowedtherecipedescribedinreference[ 68 ]toestimatethetheoreticalvalueofthedamping.Thecontributiontothedampingcomingfromthemainplateandotherstructuressuchasthecombsandthecombjointswereincludedinthecalculation.Aneffectivegapsizewasusedtoaccountforthenitesizeofthedifferentstructuresofthe 65

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Figure4-12. Top:DampingcoefcientasafunctionofpressurefordeviceH1.Bottom:DampingcoefcientasafunctionofpressurefordeviceH1.Calculatedcurvesareshownforthedampingduetothelm(bottom),thetop,andtheuidbetweenthengers. 66

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device.Thenewgapisgivenby(seereference[ 69 ]) de=d 1+8.5d=l,(4)wherelisthedimensionoftheoscillatingstructurealongthemotionoftheplate.Fromgure4-12,theslidelmdampingcomingfromthengers(cyan)andthelayerontop(green)ofthedevicearealmosttwoordersofmagnitudesmallerthanthecontributionfromtheuidtrappedbetweenthegap(blue).Thus,thetotaldampingcoefcient(red)isdominatedalmostexclusivelybytheuidinthisregion.Theseresultsarecomparabletothoseobtainedinreference[ 68 ].Finally,thedampingcoefcient,normalizedbythegeometricalconstantAp=dinequation3-30,isplottedasafunctionoftheKnudsennumber,Kn,ingure4-13.Thecurvesforthetwodevicescollapsetothesamecurveatthecross-overandhydrodynamicregimes.Intheballisticregimethetwocurvesareparallelbutslightlydeviatedfromeachother.AtaxedKnthedampingishigherfortheH2device.Bothcurvesreachthehydrodynamicregimeatthehighestpressuresstudied,butdeviceH1(blackcurve)goesslightlydeeperintothisregime.Thisisnatural,sinceithasalargergap.Also,onthehighKnregime,theH2device(redcurve)isseentogoslightlydeeperintotheballisticregime,whichisalsoexpectedsinceithasasmallergapthanH2. 67

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Figure4-13. Dampingcoefcientasafunctionofpressureforbothdevices,H1andH2,asafunctionofKn. 68

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CHAPTER5EXPERIMENTSINSUPERFLUID4HEAsaninitialtesttovalidatetheuseofourdevicesandtheexperimentalsetup,wecarriedoutexperimentsatlowtemperatureswithadevicesubmergedinsuperuid4Heandcooledbelow1Kdownto25mK.Sincethemechanicalpropertiesofthistypeofresonatorwereunknownatlowtemperaturesatthemomentoftheexperiment,weinstalledatuningforkoscillatortomakesimultaneousmeasurementsandprovideapointofreferencetoourresults.Eventhoughitwasnotthemainscopeoftheexperimenttostudyindetailthepropertiesof4He,theresultsmotivatethepotentialuseofMEMSasausefulprobeforphenomenasuchasquantumturbulenceordetailedstudiesofphonon/rotonrelaxationprocessesintheliquid. 5.1OverviewSincetheintroductionofthetorsionaloscillator,rstusedbyAndronikashviliin1946toquantifythenormaldensitycomponentofheliumII[ 76 ],therehasbeenaconstantsearchfornovelmechanicaloscillatorstoprobetheheliumliquids.Vibratingwires[ 77 ]andmorerecentlyquartztuningforks[ 63 78 ]havebeenimplementedinheliumIIforstudiesofitshydrodynamicandnon-hydrodynamicproperties.Aspureliquid4Heiscooleddownbelowit'ssuperuidtransition(2.17K),thenormalcomponentoftheheliumIIsystemdecreasescontinuouslyassuggestedbythetwouidmodelproposedbyTizsaandLandau[ 13 15 ].Thisdecreaseinthenormaluidfractionleadstodecreaseinviscosity.Ifanoscillatingobjectisintroducedintheliquidatverylowtemperatures(closeto1K),whenthenormalcomponentbecomesverysmall,thecreationoflowenergyexcitations,rotonsandphonons,provideadampingmechanismtotheobject'smotion.ThedispersioncurveforheliumII[ 17 79 ]below1Kshowsalinearenergyspectrumwithmomentum(seegure2-2),andthusthemaincontributiontotheviscosityinthisregimewillcomefromthephonongas, 69

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whichgoesas[ 19 ] =1 3ncp,(5)wherecisthespeedofsound,pisthephononmeanfreepath,andnisthenormaluiddensity,whichisgivenby[ 80 ] n=22k4B 45~3c5T4.(5)Inaddition,duetothelongmeanfreepathofthephononexcitationsbelow1K(canbeoftheorderof1mm),itispossibletoseeatransitiontoaballisticorKnudsenregimeevenformacroscopicvibratingobjects.Asaconsequenceofthisphenomena,usingdifferentmechanicaloscillators,apeakinthedampinghasbeenreported(0.7K),followedbyaT4behaviouratlowertemperatures.[ 81 84 ].Atlowertemperatures,aT4behaviourfollows.Animportantareaofresearchinuiddynamicsisthegeneration,detectionandquanticationofturbulence.Duetoitsvanishinglysmallviscosity,therefore,allowingresearcherstoattainveryhighReynoldsnumbers,superuidheliumisverywellsuitedforthispurpose.Forexample,whenanoscillatingbodyimmersedinheliumIIisagitatedaboveaparticularcriticalvelocity,atransitionfromlaminartoturbulentowoccurs.Furthermore,whenthenormalcomponentisalmostzero,anewtypeofdissipationmechanismcalledquantumturbulenceisexpectedtooccurduetotanglingofquantizedvortexlines.Manygroupshavestudiedthisphenomenabymakinguseofdifferentoscillatorssuchasvibratingmicrospheres[ 85 86 ],grids[ 87 89 ],andtuningforks[ 90 91 ].Otherphenomena,suchasthegenerationofquantizedvorticesatroughboundaries[ 92 ]andintermittentbehaviourbetweenlaminarandturbulentow[ 86 93 ],havebeenreportedforparticularrangesonthedrivingforce. 5.2ExperimentalDetailsExperimentswerecarriedoutinanOxforddilutionrefrigeratorwithabasetemperatureof5mK.Thedevicewashousedinaspeciallydesignedlowtemperature 70

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Figure5-1. Experimentalcellwithdeviceusedin4He.AquartztuningforkwasinstalledclosetotheMEMSdeviceasanindependentprobe. experimentalcellmadeofcopper,seegures5-1and5-2.Theexperimentalcellwasnotequippedwithaproperheatexchanger.Thismeansthatitwasnotpossibletocooltheliquidbelowasaturationvalue,estimatedataround100mK,duetolackofsurfacearea.ThetopofthecellwasdesignedtomaintainrigidMMCXconnectionswiththecoaxialconnectorsinthecryostat.Thecellwaspressurizedexternallywithhighpurity4Hethroughaliquidnitrogen(LN2)coldtrapandkepttoaconstantpressureof2bar.Insidethecell,aquartztuningforkoscillatorwasinstalled,closetotheMEMSdevice,tomonitortheliquidinthecell.Bysweepingthroughtheresonanceofthefork,asuperuidtransitionwasobservedcloseto2.16K.Thetemperaturewasthensweptupanddownbyapplyingheattothemixingchamberofthedilutionrefrigerator.Ateachnewheatsettingthesystemwasallowedtoreachthermalequilibriumforafewhours.Thetemperaturewasmeasuredusingacalibratedrutheniumoxidethermometerlocatedonthemixingchamberofthedilutionunit. 71

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Figure5-2. Experimentalcellisplacedontopofnucleardemagnetizationstage(demag)andbelowthemixingchamber(M/C)ofthecryostat. Thecircuitsetupforthisexperimentisslightlydifferentfromtheoneingure4-5.Thisisbecausethistestwasconductedbeforewemadeseveralimprovements.First,abiasvoltagewasnotused,thuswereliedontheintrinsicbiasasdiscussedinsection4.4.4.Second,ahomemademulti-channelsummingamplierwasusedtoaddthehighandlowfrequencysignals.ThisactivedevicewasfoundlatertointroducenoisewhentheMEMSdevicewassetuponthecryostat,probablyduetoimproperisolationofthesummingamplier'spowersupplyfromthecircuit.Thisamplierwaslaterreplacedbypassivesignalsplitters.Moredetailsabouttheseimprovementsandotherchangesarediscussedinsection4.3.2. 72

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Figure5-3. Absorptioncurvesasafunctionoftemperaturefora200mVexcitation 5.3ResultsAresonancepeakwasfoundatlowtemperaturesbetween50mKand1Katfrequenciesaround23.1kHz.Thispeakcorrespondstotheshearmode,accordingtooursimulations.Thepivotmodewasnotfound.Thus,itisbelievedthatthismodeiscompletelydampedoutduetosqueezelmdamping.Thewidthoftheresonancedecreasedsignicantlytoabout0.2Hzatthelowesttemperature,renderingthequalityfactorsintheorderof105,muchhigherthanthoseobtainedatroomtemperatureinvacuum.Whilesubmergedintheliquid,thewidthwasfoundtobestronglydependentontemperature,seegure5-3.Itdecreasedabout100timesbetween700mKand250mK.Thepeakwasstudiedasafunctionofexcitation,Vexc,atseveraltemperatures.Ourdatabelow400mKshowseveralanomaliesinthepeak'sline-shapewhendriven 73

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Figure5-4. Afamilyofabsorption(left)anddispersion(right)curvesatdifferentexcitationvoltagesfortemperatures250,305and392mK.Allgraphsshownwereobtainedwhilesweepingupinfrequency. 74

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Figure5-5. Afamilyofabsorption(left)anddispersion(curves)atdifferentexcitationvoltagesfortemperatures510,592and700mK 75

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Figure5-6. Afamilyofabsorption(left)anddispersion(right)curvesatdifferenttemperaturesshowinghystereticbehaviour.Redcolordenotesanincreasingfrequencysweepandblueadecreasingsweep 76

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aboveacertainexcitation(seegures5-4and5-5).Forexample,at392mKandwhilesweepingthefrequencyup,thepeak'samplitudegrowslinearlywithincreasingexcitationuntil1.5Visreached.Atthisvoltagetheabsorptionamplitudesuddenlydropsdownbeforereachingamaximumamplitude,thenbeginstogrowagainfollowingasteepline,andnallydropsdownquicklyagainafterreachinganapparentmaximum.Thesejumpsgivetheabsorptioncurveadouble-peakedstructure.Thisbehaviourchangeswhensweepingdowninfrequency.Lookingatthereversescanfortheabsorptionpeakat392mKasshowningure5-6,thepeakagainpresentsajumpasthefrequencydecreases,butthenitquicklyjumpsdowntoasmoothdepletionoftheamplitudewhichtendstomatchtheupwardscan.Similareventsareseenat305and250mK,buttheonsetofthenonlineareffectswasseentooccuratlowervaluesofexcitationasthetemperaturewaslowered(seegure5-6).Furthermore,asseenfromgure5-5,thesenonlinearlines-shapeswereonlyseenbelow400mK.Thephenomenadescribedaboveisveryreminiscentofwhatwasobservedinreferences[ 86 92 93 ].Theerraticbehaviourmaythenberelatedtothegenerationofquantumturbulenceaboveacriticalvelocityandsurfacephenomenathatmaybeaffectedbythecreationofquantizedvortices.Ateachtemperaturethepeakswerettedtoextracttheresonancefrequency,width,andamplitude.Fourtotenmeasurementsweremadeateachtemperature.Asthetemperaturedecreasesthefrequencyseemstoincreaseuntilitreachesastablevalue,butaround390mKasuddendipisseen,seegures5-7and5-8.Thisbehaviourwasconrmedthroughmultiplemeasurementsusingveexcitationvoltages(0.05,0.1,0.2,0.5,1V)withupwardanddownwardscans.Theerrorbarat400mKisdeterminedbythesetenmeasurements.Tothisdate,theoriginofthisfeatureremainsunknown,butitisinterestingtonotethatthetemperaturebelowwhichitoccurs(400mK),isalsothetemperatureatwhichwebegintoseenonlinearitiesinthelines-shapes.Thequalityfactor,Q,smoothlyincreasesastemperaturedecreases.Thisisduetothephonon 77

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Figure5-7. Top:Peakposition(top)andwidth(bottom)ofdeviceH1insuperuid4Heasafunctionoftemperature. populationdecreasingasthetemperaturegoesdown.TheinverseofQ(dampingcoefcient)isdirectlyrelatedtothephonondensityasshowninequation5-2.WhenplottingtheinverseofQinalog-logscale,thedataseemtofollowaT4dependenceabove300mK(gure5-9),wherecombineddatafromtwotemperaturesweepsareshown.Interestingly,thedataalsoshowasuddenchangearound300mK,whereachangeinslopehappens.Asmentionedearlier,theabsenceofaheatexchangerlimitsourabilitytocooltheliquidbelowacertaintemperature.Weestimatethistemperature 78

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Figure5-8. Q-factor(top)andinverseQ-factor(bottom)ofdeviceH1insuperuid4Heasafunctionoftemperature tobearound100mK,fromtwomeasurements,oneatthelowesttemperaturethedilutionunitcanachieve,andanotherslightlybelow100mK,whichshowedaconstantvalueofQ-factorofabout102000.Thisestimationleadsustobelievethatthekinkmightbereectingarealchangeintheliquid,butamoredetailedstudywouldbeneededtoconrmtheseobservations.Forcomparison,thedataforthetuningforkareshowningures5-10to5-13.Aninterestingfeatureintheabsorptioncurvesofgure5-11,isthetendencyofthe 79

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Figure5-9. InverseQ-factorofdeviceH1insuperuid4Heasafunctionoftemperatureinlog-logscale.AdashedlinerepresentingaT4powerlawisshown.Thisgraphincludesdatafortwotemperaturesweepswiththerstruninredandthesecondruninblack 0.02Vexcitationpeaktoshoulderontothe0.01Vpeakatthehigherfrequencyside.AsimilarbehaviourcanbeseenintheMEMSdatawhenlookingcloselyatgure5-4attemperatures250and305mK.Ingure5-12,thefrequencycorrespondingtothepositionofthepeakisshowntoincreaseastemperaturegoesdownuntilasharpdipoccursslightlyabove300mKfollowedbyanotherdipcloseto200mK.TherstdipisverysimilartowhatwasobservedwiththeMEMSdeviceandisalsocloseintemperature,consideringtheseparationintemperaturebetweenthemeasurements(100mK)wascloseto100mK.Inaddition,anotherdipisseeninthequalityfactoraround250mK.Finally,onthebottomofgure5-13theinverseofQisplottedasafunctionoftemperatureinalogscale.AlineshowingaT4powerlawisshownforcomparison.Theplotalsoshowsasuddenchangeintendencyaround400mK,similar 80

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Figure5-10. Absorptioncurvesasafunctionoftemperatureusinga20mVexcitationforatuningforkinsuperuid4He. Figure5-11. Absorptionanddispersioncurvesatdifferentvoltageexcitationsforatuningforkinsuperuid4Heat400mK.Upanddownsweepsinfrequencyareshownfor10and20mV.Anarrowpointstoananomalouskinkinthedispersioncurveata50mVexcitation 81

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Figure5-12. Peakposition(top)andwidth(bottom)foratuningforkinsuperuid4Heasafunctionoftemperature. totheMEMS'case,butataslightlyhighertemperature.Thechangeinthepositionofthisfeatureprovidessomemoreevidencethatthisfeaturemaybereectingthepropertiesoftheliquid,anditmightnotberelatedtosaturationduetoourinabilitytocooltheliquidduetoabsenceofaproperheatexchanger.Morestudiesareneededtoverifythisobservations. 82

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Figure5-13. Q-factor(top)andinverseQ-factor(bottom)foratuningforkinsuperuid4Heasafunctionoftemperature.AdashedlinerepresentingaT4powerlawisshown. 83

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CHAPTER6EXPERIMENTSINLIQUID3HEAfterseveraltestsconductedinairandsuperuid4He,wedemonstratetheuseofaMEMSdevicetostudyliquid3Heinitsnormalstate.IntheFermiliquidregime,3Heshowsverystrongtemperaturedependentproperties.Theviscouspropertiesoftheliquidaredirectlyprobedwithourdevice.Fromourmeasurements,weareabletoextractpropertiesoftheconneduidbysubtractingthebulkbehaviour.Wecompareourresultswiththoseobtainedrecentlyfromatorsionaloscillatorexperiment[ 94 96 ].Thesimilaritiesbetweenourmeasurementsandthosefromthetorsionaloscillatordemonstratethefeasibilityofusingthesenewtypeofsensorstostudypropertiesofquantumuidsatthemicro/nanoscale.Finally,wepresentourpreliminaryresultsinthesuperuidphaseofliquid3Hewhereresonancepeakswereobtainedasafunctionofdrivevoltageattemperaturesbelow0.5mK.Theabilitytooperatethesedevicesinanultra-lowtemperatureenvironmentsuggeststhepossibilityofusingtheminthesearchfornovelphasetransitionsinsuperuid3Helmsandinthestudyofzeroenergysurfacestates,wheresignaturesoftheMajoranafermionandothertopologicalphenomenacouldbesoughtfor. 6.1Overview 6.1.1EarlyExperimentsinLiquid3HeExperimentsonthecapillaryowofliquid3Heareamongtheearliestattemptstomeasureitsbulkviscosity[ 97 98 ].Inthe1960'sand1970's,manyexperimentswereconductedinanattempttocorroboratethetemperaturedependenceofvariousuidpropertiesofliquid3He.AcousticmeasurementsperformedbyWheatleyetal.[ 99 100 ]veriedthe1=T2temperaturedependenceofthesoundattenuationcoefcientfromwhichanaccuratemeasurementoftheviscositywasmade.Shortlyafter,aseriesofexperimentsimplementingmechanicaloscillatorssuchastorsionaloscillators[ 101 ]andvibratingwires[ 102 103 ]provideddetailedmeasurementsonthetemperature 84

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dependenceoftheviscositythroughawiderangeoftemperaturesspanningfromtheclassicaluidbehaviourabove1KtomillikelvintemperaturesintheFermiliquidregime.Inparticular,theseexperimentsintendedtonddeviationsfromtheexpected1=T2behaviourduetohightemperaturecorrectionswiththeform[ 104 ] 1 T2=a)]TJ /F6 11.955 Tf 11.95 0 Td[(bT.(6)Experimentalvaluesbetween1-2P)]TJ /F2 7.97 Tf 6.59 0 Td[(1K3weremeasuredfortheconstantb,whosetheoreticalvaluewaspredictedtobe0.23P)]TJ /F2 7.97 Tf 6.59 0 Td[(1K3.Invibratingwireexperiments,itwasrealizedthatcorrectionstotheirviscositymeasurementswereneededtoaccountforthenitewidthoftheirwires[ 105 106 ].Thereasonisthat,inthenormalstate,themeanfreepath,,of3Heincreasesveryquicklywithdecreasingtemperature(1=T2)asaconsequenceofFermidegeneracy.Furthermore,belowthesuperuidtransition,thenitesizeeffectisdramaticallyampliedduetoanexponentialincreaseinwithdecreasingtemperature.Attemperaturesbelow1mKandlowpressures,themeanfreepathcanreachvalueslargerthan10mm.Earlytheoreticalcalculationsmadeattemptstoaccountfortheseeffectsbyconsideringtheslipeffectofthequasiparticlegaswhenowingincloseproximitytoaboundary[ 107 109 ].Moreexperimentswereperformedinconnedgeometriestoelucidatetheeffectofalongmeanfreepathinthepreviousmeasurementsoftheviscosity[ 110 111 ].Otherexperimentsusingtorsionaloscillatorswerecarriedoutlatertoquantifytheslipeffectoflmsofliquid3Heinitsnormalandsuperuidstate[ 112 113 ].Especially,theexperimentbyTholenandParpia[ 113 ],usingatorsionaloscillatorwithhighlypolishedand4Hecoatedsurfaces,showedananomaloustemperaturedependenceintheviscosity,whichcouldnotbeexplainedbycorrectionsofarstordersliptheoryintheeffectiveviscosity.Theyattributedtheirobservationstothehighspecularityofthesurfacesused. 85

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Figure6-1. Frequencyshift(a)anddamping(b)forthetorsionaloscillatorexperimentdonebyCaseyetal.[ 95 ].Thelmthicknessesare100nm(triangle),240nm(opencircle),and350nm(circle) 86

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Figure6-2. Datafortheexperimentalvalueof!oscfortheexperimentdonebyCaseyetal.[ 96 ].Thelinegoingthroughthedatashowsagoodttothetheory.Thebottomlinesrepresentthetheoreticallinesforengineeredsurfacescomprisedofrectangularpostswithaheighth,widthw,andperiodicityp. 6.1.2Liquid3HeandThinMetallicFilmsRecently,interestingexperimentshaverevealedananomaloustemperaturedependenceintherelaxationtimeofatorsionaloscillatorwithathinlmofliquid3Hecoatingitsinnerchamber(seeref.[ 95 ]).Theoscillatorhasaresonancefrequencyof2841HzandatypicalQ-factoroforder106.Moredetailsabouttheconstructionofthistorsionaloscillatorcanbefoundinreference[ 94 ].Thelmswereproducedbyallowinggastoowintotheinnerchamberoftheoscillatorataxedtemperatureof60mK.Thecondensationofthelmtothewallsofthechamberwasveriedbymonitoringthechangeinfrequencyoftheoscillator.Withtheirexperimentalsetup,theywereabletodeterminethefrequencyoftheoscillatorbetterthanonepartin109.Filmsofthickness100,240,and350nmwerestudied.ThelmswereheldbyvanderWaalsattractiontothesubstrateofthecellwallswithoutanybulkliquidinside.Thefrequencyshiftwasobtainedwiththeemptycellresonanceasareferencepoint.Thedataforthe 87

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Q-factorandfrequencyshiftasafunctionoftemperatureareshowningure6-1.Theresonancefrequencyseemstoincreasetowardstheemptycellvaluewithtemperaturebelow30-40mK,indicatingdecouplingofthelmfromtheoscillator.Ontheotherhand,thedissipation(1/Q)goesthroughamaximumaround20mKforthe240and350nmlms.Sincetheviscouspenetrationlengthismuchlongerthanthethicknessofthelms,amodelwhichtreatstheuidlmasarigidobjectwithitsmotioncoupledwiththeoscillatorwasused.Throughthismodeltheoscillatorrelaxationtime,osc,wasdeterminedfromtheirmeasurements.Therelaxationtimeexhibiteda1=Tbehaviourinthe20to100mKrange(seegure6-2).Theanomalous1=Tbehaviourintherelaxationtimewasattributedtothesurfaceeffectsfromtheinterferenceofinelasticquasiparticle-quasiparticlescatteringwithelasticboundaryscattering[ 96 114 ].TheirdatawerewellttedbythemodeldevelopedbyMeyerovich[ 115 116 ],whichhasbeenappliedtothestudyofsurfaceeffectsinthinquantizedmetalliclmsandin3He-4Hemixtures[ 117 ].WithintheframeworkoftheMeyerovichtheory,theproblemofathinslabof3Heboundedbyroughsurfacesisconvertedtoonewithnosurfaceroughnessbutwithbulkdisorder.Theeffectiverelaxationtimeiscalculatedfromthetheoryas 1 e(p)=1 b1+dZdp0 (2~)3W(p,p0) ((p0))]TJ /F6 11.955 Tf 11.96 0 Td[(EF)2=(2~2)+1=42b,(6)wherebisthebulkrelaxationtime,disthelmthickness,andW(p,p0)istheboundaryinducedtransitionprobabilityforquasiparticlesscatteringoffthedisorderpotential.Thetransitionprobabilityfunctionisdirectlyrelatedtothecorrelationofthesurfaceroughness,whichcanbedeterminedexperimentallyfromscanningprobemeasurements.Aconsequenceofequation6-2istheviolationofMatthiessen'srule.Thesedevelopmentsencouragetheuseof3Helmsasapotentialmodelsystemformesoscopicphysicsandthestudyofphenomenasimilartothatobservedinthinmetalliclms.Forexample,quantumsizeeffects,resultinginoscillationsintheconductivity, 88

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Figure6-3. Experimentalcellfortheliquid3Heexperimentatthebottomofanucleardemagnetizationstageofthecryostat.AplatinumNMRthermometerwasusedtomeasurethetemperaturebelow1mK. andweaklocalization,resultinginanincreasedresistivity,havebeenstudiedwithinthecontextofthinmetalliclms[ 118 120 ]. 6.2ExperimentalDetailsWeusedanH2device(0.75mgap)intheexperiment.Thedevicewashousedinsidealowtemperatureexperimentalcell(seegures6-3and6-4).Thecellisthermallyattachedtothebottomofahomemadecoppernucleardemagnetizationstageconnectedtothedilutionunitthroughanindiumheatswitch.Thecellconsists 89

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Figure6-4. Experimentalcellwithdevice ofthreeparts.Therstisthemaincellbody,whichisindirectthermalcontactwiththedemagnetizationstage.Thispartcontainsasilverheatexchangermadeofa700AJapanesesilverpowder(TokurikiChemicalResearchCo.,Ltd.)packedata45%packingfraction,providingapproximately43m2ofcontactareawiththeliquid.Acigarettepaperlterwasusedtocovertheheatexchangerandprotectthedevicefromdebriscomingfromit.Theheatexchangerisdirectlyconnectedtoa3Hemeltingpressurethermometer(MPT)throughasilverrodboltedtothedemagstage.Averydetaileddocumentationabouttheconstructionofthecryostatcanbefoundinreference[ 121 ].TheMPTwasusedtomeasurethetemperaturebetween1mKto100mK.Above100mK,acalibratedrutheniumoxidethermometerinstalledonthemixingchamberwasused.Fortemperaturemeasurementsbelow1mK,ahomemadePtpowderNMRthermometerwasused.Thisthermometerwasxedtothesecondpartofthecell,whichisaninterconnectingpiecebetweenthemaincellbodyandthecellcap.Finally,thethirdpartisacellcapwherethechipcontainingtheMEMSdeviceisplaced.Thechipwasheldbyacustommadesocketasexplainedinsection4.1.2.Astandtoholdthesocketwasbuiltinsidethecellcapbyusingaphenolicspacerasasupport.Thethreepinsof 90

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thesocketwhichoperatethedevicewereconnectedthroughthincopperwiretosealedMMCXbulkheadconnectorslocatedatthebottomofthecellcap(seegure6-3).ThetwoconnectionscorrespondingtothesideplatesoftheMEMSdevice(seegure4-5)wereconnectedtoNbTisuperconductingcoaxialwiresonthecryostat.Itwasfoundthatattachingthecenterplateconnectionofthedevice(fromwheretheunbalancedsignalfromthebridgeisdetected)tothistypeofcoaxialwireresultedindegradationofthesignal.ABeCucoaxiallinewasusedinsteadfordetection.Thecoaxialwiresareconnectedtotheinstrumentsoutside.Thecircuitsetupshowningure4-5wasusedwiththemodicationsexplainedinsection4.3.2.Theexperimentalcellwasloadedhypercriticallyat4Kwiththe3Hepressuremaintainedat3bar.Thiswasdonetoavoidpotentialdamagestothedevicefromtheactionofcapillaryforces.Temperaturewassweptfromaround20mKuptoaround800mKbyapplyingacontrolledamountofheattothemixingchamber.Theresonancewasscannedataxedtemperatureduringseveralcooldownandwarmupcycles.Thecellwaspressurizedlaterto21and29barwhilekeepingthetemperaturexedat800mK.Inordertocooldownbelowthebasetemperatureofthedilutionrefrigerationsystem(5.4mK),nucleardemagnetizationwasperformedusingan8TeslasuperconductingmagnetbuiltbyAmericanMagneticswithacoilconstantof0.10269T/A.Astartingeldof7.2Teslawasappliedandthesystemwasallowedtoprecoolforsixdays.Afterprecooling,themagnetcurrentwaslowereddownto25Aatafastrateof3mA/swiththeheatswitchopen.Thesystemwasthenallowedtoreachthermalequilibriumforaboutanhour.Next,aslowdemagwasperformeddownto5Aatarateof0.4mA/sandthento2Aatthesamerate.ThelowesttemperaturemeasurementobtainedfromthePtNMRthermometerwasestimatedat340K.Theresonancewasscannedatdifferenttemperaturesduringthewarmupprocessofthecryostatbacktothebasetemperatureofthedilutionunit.ThePt-NMRthermometersignal,providedbyaPLM-3NMRthermometer,wascalibratedbytwomeasurements:oneattheNeel 91

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Figure6-5. Absorption(top)anddispersion(bottom)curvesforadeviceinnormalliquid3Heasafunctionoftemperature. transitiontemperatureintheMPTandtheotherat10mK,wherethesignalfromthePtsampleisnegligibleandthebackgroundsignalcanbemeasured. 6.3ResultsTheresonancepeakcorrespondingtotheshearmodewasobservedat23.4kHzwithaQ-factorof43ataround800mKand3bar.Themeasuredvalueatroomtemperatureinvacuumwas23.7KHzwithaQ-factorofaround40000.Thefrequencywasscannedatvarioustemperaturesfrom20mKto800mKatthreedifferent 92

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Figure6-6. Lengthscalesinnormalliquid3Heasafunctionoftemperature.Trianglesaremeanfreepathandsquarespenetrationdepth.Redisfordataat3bar,blueisfordataat21bar,greenisfordataat29bar pressures:3,21,and29bar.Eventhoughwewereabletoseeapeakbelow20mK,thesignal-to-noiseratiowastoolowtoextractanymeaningfulinformation.Typicalresonancepeaksareshowningure6-5for3bar.Thepeaksshowastrongtemperaturedependence.Asdiscussedinchaptertwo,thisstrongtemperaturedependenceonthetransportpropertiesoftheliquid,suchastheviscosity,isasignatureofFermiliquidbehaviourinnormalliquid3He.Atthispoint,itisimperativetocompareimportantlengthscalesrelevanttothissystem.Thevaluesofthemeanfreepathandtheviscouspenetrationdepthcanbecalculatedusingtabulateddatafromreference[ 122 ].Aplotwiththevaluesofthesetwolengthscalesisshowningure6-6.Fromthisgure,onecanseethatthemeanfreepathbecomescomparabletothesizeofthegapbelow10mK.Closeto200mK,theviscouspenetrationdepthoftheuidbecomescomparabletothesizeofthegap. 93

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Figure6-7. TheQ-factor(Top)andfrequency(bottom)asafunctionoftemperatureduringtwodifferentcool-downs.NohysteresisisseeninQ,butsomehysteresisisseeninthefrequency. 94

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Theresonancepeakswerettedtoequations4-18,takingintoaccountthefactthatthechargeisbeingmeasured,notthecurrent.Fromthet,alltherelevantparameterswereextractedandplottedagainstthetemperature.Asthetemperaturewasswept,veryweakhysteresiswasobservedintheQ-factorafterconsecutivecool-downs,butmorenoticeablehysteresiswasseenintheresonancefrequencyasshowningure6-7.Similarhystereticbehaviourisroutinelyobservedintuningforkoscillators[ 63 ],disappearingonlyafterafewthermalcycles.AninterestingobservationisthatafterourexperimentswerenalizedandtheMEMSdevicewastakentoroomtemperature,theQ-factorwasseentoalmostdoublefromitsoriginalvalue,buttheresonancefrequencyremainedthesame.ThereasonforthislargechangeinQ-factorisnotunderstoodyet,somorestudiesonthisthermalizationeffectareneeded.Figures6-8,6-9,and6-10showthebehaviourofvariousttingparametersasafunctionoftemperatureforpressures3,21,and29bar.Thepeakamplitudedecreasessmoothlyasthetemperatureislowered.However,theresonancefrequencyremainsalmostconstantfrom800to200mKbutstartstodecreasequicklybelow200mK.Thisshiftinresonanceindicatesalargemassloadingeffectcomingfromtheincreasingviscosityoftheliquid(1=T2),whichisdirectlyrelatedtothepenetrationdepth(=p 2=!).Thisincreaseinmassisaneffectfromthebulkuidresidingontopoftheoscillatingplate.Anestimateofthismassisshownlaterinthissection.Thewidthofthepeakincreasessharplycloseto100mK,makingthequalityfactordecreaserapidlyaswell.Thisdecreaseimpliesarapidlyincreasingviscosityintheliquid.However,thewidthstaysalmostconstantatthehighertemperatures,400to800mK.Sincetheearlydaysofthestudyofliquid3He,thevalueofitsviscosityhasbeenmeasuredbymanygroupsthroughawiderangeoftemperatures,seeforexamplereference[ 102 ].Thevalueoftheviscosityisseentobeconstantcloseto1K(Fermitemperature).Thewidthofthepeakisdirectlyrelatedtothedampingcoefcient,=m!,wheremisthetotalmassoftheoscillator.But,asdiscussedabove,themassofthedevice 95

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Figure6-8. Top:PeakamplitudeandwidthofdeviceH2innormalliquid3Heasafunctionoftemperatureandat3barpressure.Bottom:FrequencyandQ-factorofdeviceH2innormalliquid3Heasafunctionoftemperatureandat3barpressure effectivelyincreasesduetomassloadingfromtheliquid,sothatm=mH2+mf,wheremH2isthemassofthedevice(seetable4-2)andmfisthemassdraggedbytheuid.Thus,thechangeintheresonancefrequencycanbeusedtoestimatethetotalmassoftheoscillatorasm=ks=!20,where!0isthettedfrequency(peakposition)andksisthespringconstant.Thedampingcoefcientisthencalculatedintermsofthespringconstantas =!ks !20.(6) 96

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Figure6-9. Top:Peakamplitudeandwidthofdeviceinnormalliquid3Heasafunctionoftemperatureandat21barpressure.Bottom:FrequencyandQ-factorofdeviceH2innormalliquid3Heasafunctionoftemperatureandat21barpressure Asdiscussedinsection4.3.1,thecalculatedvalueofthespringconstantfromthegeometrycanbeoffbyasmuchas40%.Furthermore,atlowtemperaturesthespringconstantisnotthesameastheoneatroomtemperature.Nevertheless,ithasbeenshownthattheintrinsicdissipationinsinglecrystalsiliconmicromechanicalresonatorsdoesnotchangemuchbelow40K[ 123 ].We,therefore,assumethatthespringconstantwouldnotchangesignicantlybelow4Kwhereweextracteditbasedonaresonancemeasurementinanemptycell.Thefrequencyatroomtemperature 97

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Figure6-10. Top:PeakamplitudeandwidthofdeviceH2innormalliquid3Heasafunctionoftemperatureandat29barpressure.Bottom:FrequencyandQ-factorofdeviceH2innormalliquid3Heasafunctionoftemperatureandat29barpressure (23681.5Hz)wasslightlylowerthanthevalueat4K(23753Hz).Thus,thespringconstantforasinglespringiscalculatedtobek4K=!24KmH2=4=1.54N/m,assumingnoadditionalmassotherthanthebareH2device.Thedevicemasswascalculatedfromitsgeometryandthedensityofthepolysiliconlayersuppliedbythefoundry(seetable4-2).Thecoefcientofdampingisplottedforthethreepressuresingure6-11inalog-logscale.Aninterestingfeaturecanbeseeninthisgraph.Above300mK 98

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Figure6-11. DampingcoefcientofdeviceH2innormalliquid3Heasafunctionoftemperature.Redisfordataat3bar,blueisfordataat21bar,greenisfordataat29bar.Thesolidblackrepresenting1=Tisshownforcomparison. thedampingishigherforahigherpressure,but,asthetemperatureislowered,thecurvesbegintomergeandthencrossbelow300mK,sothatthepressuredependencereversesinthelowtemperatureregime.Thispoint(between300and400mK)marksaclearcrossoverbetweenaclassicaluidtoaquantumuid.Inthelowtemperatureregime,astheeffectsfromtheFermiliquidbegintobecomemorepronounced,theeffectivemassbecomeslargeratlargerpressuresand,therefore,theFermivelocityandtheviscosityarereduced(=1=3v2F).Therearethreedifferentregionsidentiedfromgure6-11.Therstoneisfortemperaturesabovetheclassicaltoquantumuidcrossover,whichshowsthedampingsaturatingtoaconstantvalue.Thesecondistheregionbetween250mKand40mK,whereatrendcloseto1=Tinthedampingcoefcientisobservedcloseto40mK.ThisbehaviourisveryreminiscentoftheobservationsbyCaseyetal.[ 95 96 ]ontheirtorsionaloscillatorexperiment,where 99

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Figure6-12. DampingcoefcienttimesT2asafunctionoftemperature.Inthelowtemperaturelimitthebulkcontributionisdominant.Redisfordataat3bar,blueisfordataat21bar,greenisfordataat29bar theiroscillatorrelaxationtimewasseentogoas1=T.Thethirdregionisbelow40mK,wherethetemperaturedependencestrengthenstoa1=T2.Weinterpretthisbehaviourasaresultofgrowingdominanceofthebulkcontributionatthelowertemperatures.Toquantifythebulkcontributiontothedamping,thesamedataarereplottedasT2vsT,asshowningure6-12.Thenormalizedquantity,T2,tendstoreachaconstantvaluebelow40mK.Thisveriestheobservationsabovethatthedampingcoefcienthasa1=T2tendencybelow40mKanda1=Tabovethistemperature.Itisnotknowntothisdayiftheupturninthe29barisreal.Theerrorbarforthesedatapointsislargeandmightbelargerifthenumberofmeasurementswaslarger.Ifthistrendisreal,theupturnwouldsuggestsatendencyinthedampingcoefcientof1=T,where>2.Improvementstoourdetectionschemearebeingmadetobeabletoexplorelowertemperatureandinvestigatethenatureofthe 100

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Figure6-13. Filmdampingcoefcient.Thecurvesshowapeakindissipation.Redisfordataat3bar,blueisfordataat21bar,greenisfordataat29bar upturn.Thebulkcontributioncanbeestimatedforeachpressurefromitscorrespondingconstantvaluebelow40mK.Theexpressionsforthebulkcontributionaregivenby(inunits10)]TJ /F2 7.97 Tf 6.59 0 Td[(3Ns=mwithTinmK) bulk,3bar=12.6510)]TJ /F2 7.97 Tf 6.59 0 Td[(3 T2,bulk,21bar=8.7610)]TJ /F2 7.97 Tf 6.59 0 Td[(3 T2,bulk,29bar=7.2710)]TJ /F2 7.97 Tf 6.59 0 Td[(3 T2.(6)Itisinterestingtonotethattheratiobetweenthecoefcients12.7,8.7,and7.3arewithin3%oftheratiobetweenthecoefcientsforthebulkviscosityinnormalliquid3HegivenbyWheatley[ 122 ],e.g. bulk,3bar=1.73 T2,bulk,21bar=1.22 T2,bulk,29bar=0.99 T2,(6)wheretheviscosityisgiveninpoise,andthetemperatureinmillikelvin.Thebulkcontributionscannowbesubtractedfromthetotaldampingcoefcientforeachpressure.Theresultissupposedtobethecontributionfromtheliquidconnedbetween 101

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Figure6-14. Knudsennumberinnormalliquid3Heasafunctionoftemperature.Theblacksolidlinerepresentsthetransitionfromthehydrodynamictothecross-overregime. thetwoplates,e.g.thelmdamping.Figure6-13showstheresultforthelmdampingcoefcient.ThesecurvesshowaverysimilarbehaviourtothosefromthetorsionaloscillatorexperimentdonebyCaseyetal.[ 95 96 ],buttheirdissipationpeaksoccuratalowertemperaturebetween10and20mK(seegure6-1).Themaximuminourdampingcoefcient,bycontrast,takesplacebetween55(for29bar)and75mK(for3bar).Itshouldbenotedthattheirmaximumtakesplacewhen!osc=1,e.g.,atthecrossoverpointbetweenthehydrodynamicandthenon-hydrodynamicregime.Inourexperiments,theKnudsennumberisalwayssmall(Kn1),seegure6-14,thustheslip-effectshouldbenegligibleinourcase.Abetterunderstandingofthesurfaceroughnessmightaidintheunderstandingofthisdiscrepancy.FirststepsweretakeninthisworkbyperforminganAFMcharacterizationofthesurfaceofthesedevices(seesection4.2). 102

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Figure6-15. Massenhancementcalculatedfromtheresonancefrequencyshifts.Redisfordataat3bar,blueisfordataat21bar,greenisfordataat29bar Asdescribedabovethefrequencyshiftdatawereusedtoextracttheaddedmasswhichisdraggedbythemotionoftheoscillator.Figure6-15showsalog-logplotofthetotalmassoftheoscillator(mH2+mf)asafunctionoftemperature.Similartothedampingcoefcient,threedifferentregionscanbeidentiedfromthisplot.Therstone,above400mK,showsanalmostatvalueofthetotalmass.Thisisinagreementwithourpreviousobservations,wherethisregionisidentiedwiththeclassicalregionandFermibehaviourisnotyetnoticeable.Thesecondregion,between400and30mK,showsacontinuousincreaseinthemass.Atthethirdregion,below30mK,theincreaseofthemasschangestrendtoa1=Tbehaviour.Toverifythis,thedatawerereplottedingure6-16withtheuidmassmultipliedbytemperatureinthey-axisandthetemperatureinthex-axis.Between30and40mKthevalueofmuidTgoestoaconstantvalue,suggestingthatthisaddedmassgoesas1=T.Thiscomponentisattributedtobulkbehaviour,since,thechangeinmassmustbeproportionaltothe 103

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Figure6-16. Massenhancementcalculatedfromtheresonancefrequencyshiftstimesthetemperature.Redisfordataat3bar,blueisfordataat21bar,greenisfordataat29bar penetrationdepth,whichgoesas1=T.Thebulkcontributiontothetotalmasscanthereforebeextractedandsubtractedfromthetotalmasstoobtainthelmcontribution.Thetwocontributions,bulkandlm,areshowningure6-17forthedataat3bar.Fromthegure,thebulkcontributionbecomesdominantbelow100mKanditincreasessteadily.Ontheotherhand,thelmcontributiondecreasesquiterapidlybelow100mK.Toverifytheconsistencyofourestimateofthebulkmass,wecanestimatethevalueoftheviscouspenetrationdepth,,astheverticalcolumnofuidthatisdraggedbythemotionatthetopoftheplate.First,thearea,Atop,ofthetopofthedevicecanbecalculatedfromthegeometrytobe6.2128108m2.Next,thebulkdensity,bulkcanbecalculatedtobe89.05,109.43,and115.38kg/m3fromthetabulateddatabyWheatley 104

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Figure6-17. Filmandbulkcontributionstothetotalmassenhancement.Cross-hairsymbolsrepresentthebulkmass,emptysymbolthelmcontribution,andlledsymbolisthetotalmass. [ 122 ].Finally,thepenetrationdepthwillbegivenby =mbulk bulkAtop.(6)Figure6-18showsaplotoftheestimatedvaluesusingequation6-6andthevaluescalculatedusingtheviscosityvaluesfromWheatley,e.g.,equation6-5.Thedatafortheexperimentallyobtainedpenetrationdepthisinverygoodagreementwiththecalculatedvalues.Finally,themasscontributionofthelmisplottedingure6-19for3,21,and29bar.Theyalldecreasebelow100mK,whichsuggeststhatthemassconnedbetweentheplatesoftheoscillatorisdecouplingfromthemotionoftheoscillator.ThisissimilartotheobservedbehaviourinthetorsionaloscillatorexperimentsbyCaseyetal.[ 96 ]. 105

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Figure6-18. Viscouspenetrationdepthcalculatedfromtheresonancefrequencyshifts.Redisfordataat3bar,blueisfordataat21bar,greenisfordataat29bar.ThesolidlineisthecalculatedvaluefromWheatley[ 122 ] 6.4SummaryAlaterallymovingMEMSoscillatorwassubmergedinliquid3Heandusedtostudytheviscouspropertiesoftheliquidasafunctionoftemperatureatthreedifferentpressures:3,21,and29bar.AcrossoverfromaclassicalliquidtoaFermiliquidwasobservedbythecrossingofthedampingversustemperaturecurvescorrespondingtothedifferentpressures.Atthetemperaturesabovethecrossover,thedampingislargerwithhigherpressure,and,belowthecrossover,thedampingislargerwithlesspressure.Thedampingbelowthecrossoverregionwasseentofollowatrendwhichapproachesa1=Tbehaviourcloseto40mK.Thebulkcontributiontothedamping(1=T2)dominatedbelow40mK.Whensubtractingthebulkcontribution,thedampingcomingfromtheliquidconnedbetweentheplatesoftheoscillator(lmdamping)showedapeakindissipation,withitsmaximumbetween55and75mK.Themassloading 106

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Figure6-19. Filmmassdecoupling.Redisfordataat3bar,blueisfordataat21bar,greenisfordataat29bar effectonthedevice,whichisreectedinthedecreaseintheresonancefrequency,wasusedtoestimatethemasscoupledtothemotionwithinthelmbysubtractingthebulkcontribution,whichgoesas1=T.Thisrevealedadecouplingofthemasswithinthelmfromthemotionoftheoscillator.Thesetwophenomena,peakindampingandmassdecoupling,arequalitativelysimilartotheonesobservedbyCaseyetal.intheirtorsionaloscillatorexperiment.Finally,itisworthtomentionthatweconductedapreliminarytestofthisMEMSoscillatorwhilesubmergedinsuperuid3He.Aftercoolingintothesuperuidregime,resonancepeakswereobtainedat440and340K.Asafunctionofincreasingexcitationvoltagethepeakisseentobecomebroader.Thisisduetocooper-pairbreaking,whichreducesthesuperuidcomponentandincreasesthedamping.Similarbehaviourwasobservedat340K,butwithlesspronouncedfeatures.Figures6-20and6-21showthepeaksobtainedatthesetemperatures.Thistestdemonstratethe 107

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Figure6-20. Absorption(top)anddispersion(bottom)curvesforadeviceinsuperuid3Heasafunctionofexcitationvoltage.Thetemperatureisestimatedat340Kandthepressureis29bar feasibilityofusingtheseMEMSdevicestostudysuperuid3Hephysics.Throughimprovementstoourdetectionscheme,weexpecttoincreaseoursignal-to-noiseratioandusethisdevicetostudylmsofsuperuid3He. 108

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Figure6-21. Absorption(top)anddispersion(bottom)curvesforadeviceinsuperuid3Heasafunctionofexcitationvoltage.Thetemperatureisestimatedat440Kandthepressureis29bar 109

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CHAPTER7CONCLUSIONWehaveembarkedonthedevelopmentofanovelexperimentalschemecapableofformingawell-denedslabgeometryofliquid3Heand,atthesametime,probeitsphysicalproperties.Thedeviceswerefabricatedexternallybyexploitingastate-of-the-artmicromachiningprocesscalledPolyMUMPS.Thiseffortopensawindowforresearcherstotakeadvantageoftheexibilityandefciencyofcommercialprocessesforrapiddesignandfabricationofsensorstoexploreheliumphysicsatthemicrometerscale.Thisworkprovidesacomprehensiveroadmapforthedevelopmentofthedevicesandthestudyoftheirbasicelectrostaticandmechanicalproperties,aswellastheirinteractionwithasurroundinguid.Inaddition,wehavedemonstratedthatthesedevicescanbeusedforthestudyofquantumuidsthroughourexperimentsinsuperuid4He,normalliquid3He,andsuperuid3He.Theabilitytoprobenovelandexcitingphysicsinquantumuidswithourdeviceswasevidencedthroughthroughoutthecourseoftheseexperiments.Forexample,insuperuid4He,severalanomalousline-shapesobservedaboveacertainthresholdexcitationvoltagegiveapossibleindicationoftheirpotentialforthegenerationanddetectionofquantumturbulencephenomena.Furthermore,thesensitivityofourdevicestotheseeffectswasveriedthroughourmeasurementsusingaquartztuningforkoscillator.Bytakingadvantageofcurrentmicrofabricationtechniques,itcouldbepossibletostudytheeffectoftheboundaryinthegenerationofvorticityandturbulencebyengineeringthesurfaceoftheoscillator.Forinstance,it'spossibletofabricatemicro-gridoscillatorswithsmallvariationstoourdesigns.Ourtestsin4Heshowedthattheviscousdampingcanbeobtainedandquantiedasafunctionoftemperature.Duetothelongphononmeanfreepathatlowtemperatures,moredetailedexperimentsinsuperuid4Hemightrevealapeakinthedamping,orKnudsenpeak,asseeninotherexperimentsbelow1K. 110

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ItisdemonstratedfromourstudiesatroomtemperaturethatthecurrentphysicalmodelforthedampingoflateralMEMSresonators,e.g.slidelmdamping,providesanaccuratedescriptionofourexperimentalobservations.Anextensionofthismodelmightbeusefultoobtainquantitativeinformationofthephysicsofquantumuids.Inthecaseof3He,thedampingcoefcientwasstudiedindetailthroughawiderangeoftemperaturesfrom800mKdowntoabout20mKforthreedifferentpressures:3,21,and29bar.TheFermiliquidbehaviourwasexploredandarapidincreaseinviscosityisobservedapproaching100mK.AsaturationinthelowtemperaturelimitofT2wasobservedatthethreedifferentpressures,consistentwithbulkFermiliquidbehaviour.Theseobservations,combinedwithanappropriatemodellingoftheslidelmdampinginquantumuids,makethesetypeofdevicessuitablefortheiruseaslowtemperatureviscometersandpotentiallyasthermometers,byusingthelowtemperaturevaluesofT2asacalibrationpoint.Whenisolatingthelmeffectbysubtractingthebulkbehaviour,apeakindampingisrevealedcloseto70mK.Thisagreeswithobservedphenomenabyothergroups.Withfutureimprovementsinthedetectionschemeorthedesignofthedevice,itmightbepossibletogotoevenlowertemperaturesandexplorephysicsclosetothesuperuidtransition.Finally,resonancepeakswereobtainedbelowthesuperuidtransitionatapressureof29bar.Atthelowesttemperatureobtained,340K,theresonancepeakwasstudiedasafunctionoftheexcitationvoltage.Thewidthofthepeakisseentoincreaseasthevoltageisincreased.Thesamephenomenaisobservedat440K,wherethepeakisalsobroaderdotothehighertemperature.Withimprovementsofthenewexperimentaltechniquedevelopedwiththiswork,apotentialtostudyquantumuidsatthemicro/nanoscaleisathand.ThedetectionofnovelphasetransitionsornewquasiparticleexcitationssuchastheMajoranafermionareexcellentmotivationtocontinuethedevelopmentandembracingofnewexperimentaltechniquessuchastheonedemonstratedinthisthesis. 111

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BIOGRAPHICALSKETCH MiguelGonalezwasborninBucaramanga,Colombia.Whilestilltooyoungtoremember,hisfamilymovedtothecityofBogotawherehewenttoschool.HethendecidedtostudyphysicsatUniversidaddeLosAndesinBogotaandgraduatedin2005.Fivedaysafterhisundergraduatethesisdefense,hetravelledtotheUnitedStatestopursueaPhDinphysicsattheUniversityofFloridaandjoinedthelowtemperaturegroupofprofessorYoonseokLeein2006.Hegraduatedinthespringof2012. 118