Detecting Classical and Quantum Fluctuations with Microelectromechanical Systems

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Title:
Detecting Classical and Quantum Fluctuations with Microelectromechanical Systems
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1 online resource (128 p.)
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english
Creator:
Zou, Jie
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University of Florida
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Gainesville, Fla.
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Degree:
Doctorate ( Ph.D.)
Degree Grantor:
University of Florida
Degree Disciplines:
Physics
Committee Chair:
Lee, Yoonseok
Committee Co-Chair:
Chan, Ho Bun
Committee Members:
Biswas, Amlan
Maslov, Dmitrii
Xie, Huikai

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Subjects / Keywords:
casimir -- fluctuations -- mems -- noise -- on-chip
Physics -- Dissertations, Academic -- UF
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Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
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Abstract:
Fluctuations are ubiquitous. It plays an important role in all physical and biological systems, including microelectromechanical systems (MEMS). Due to the relatively small size, MEMS devices are susceptible to different kinds of fluctuations. The fluctuations, originating from an external system or the MEMS device itself, limit the sensitivity of the MEMS device and therefore warrant careful investigations. On the other hand, the unprecedented sensitivity of the MEMS device can be exploited as a probe to study the classical and quantum fluctuations. In this dissertation, we employ the MEMS device to detect a classical non-Gaussian noise that comes from external fluctuations and a quantum electrodynamical force (the Casimir force) that arises from quantum fluctuations. For the non-Gaussian noise, we utilize a driven nonlinear micromechanical resonator and study noise-induced switching out of a metastable vibrational states. Gaussian noise and Poisson modulated pulses induce switching with qualitatively different behaviors. For the Gaussian noise, the measured logarithm of the switching rate W, or the switching exponent Q, is proportional to the reciprocal noise intensity 1/D. In contrast, in Poisson pulse induced switching, the switching exponent is proportional to the logarithm of the noise intensity, for fixed pulse area. We also observe that the scaling behaviors exhibit different scaling exponents. Casimir force studies have been an active field since the first modern demonstration of the Casimir force in the nineties. But the Casimir effect within a single chip has not been demonstrated yet, due to the necessity of a bulky off-chip positioner. In the second part of this dissertation, the bulky positioner is replaced by an on-chip actuator. A doubly clamped beam is integrated with a comb drive actuator on a single silicon chip. The beam performs as a force gradient sensor to measure the Casimir force between the beam and a nearby electrode. Our successful demonstration of the Casimir force on a single chip represents one step towards the important goal of harnessing the Casimir force on a single chip. Besides, it opens novel opportunities of tailoring Casimir forces by the geometry effect.
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In the series University of Florida Digital Collections.
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Includes vita.
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This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility:
by Jie Zou.
Thesis:
Thesis (Ph.D.)--University of Florida, 2012.
Local:
Adviser: Lee, Yoonseok.
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Co-adviser: Chan, Ho Bun.
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RESTRICTED TO UF STUDENTS, STAFF, FACULTY, AND ON-CAMPUS USE UNTIL 2014-05-31

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Sixyearsisasignicantlengthoftimeinoneperson'slife.Thisdissertationisasummaryoftheacademicsideofthesixyears'life.Forsure,thisdissertationwillnotexistwithoutthehelpfrommydearcolleaguesandfriends.Iwouldliketoacknowledgethemforhelpingmeachievethis.First,Ithankmysupervisor-Dr.HoBunChan.Heintroducedmetotheeldofmicro-(nano-)electromechanicalsystems(MEMS).WhenIjoinedhisgroup,mostessentialsetupsandbasictransductionschemesfortheMEMSdeviceswerealreadyinplace.HoBunalwayssupportsmetotrynewideas.MostinstrumentsandmaterialsIaskedforinordertoexplorenewideaswerenallypurchased,thoughalotofmyideasdidnotworkoutasIthought.Moreover,HoBunisalwayswillingtorollupthesleevestoworkwithusinthelab.Wexedmanyissuestogether,likethegroundloopsandspuriousnoise.Hisleadershipandguidanceareinvaluable.EverytimewhenIgotlostinthedetailsofanexperiment,heistheonewhoclariestheprioriesandformulatestheresultsasagoodstory.Ialsomustthankhimforhiswillingnesstointroducemetomanyotherkeyresearchersintherelatedeld.Iwouldliketoacknowledgemysupervisorycommitteemembers.IthankDr.Leeforfriendlytakingtheresponsibilityofbeingthechairandintroducingmetotheattractivelowtemperaturephysics.IamgratefultoDr.Maslov,Dr.Xie,Dr.HebardandDr.Biswasforthetimeandadvices.Dr.Hebardwasverygenerousforallowingmetousetheevaporatorinhislab.Iamindebtedtomylabmates:Dr.BaoYiliang,Dr.CoreyStambaugh,SanalBuvaev,Dr.Konstantinos(Kostas)Ninios,Dr.ZsoltMarcet,TangLuandSunFengpei.YiliangledtheprojectofmeasuringtheCasimirforcebetweenthesiliconnanotrenchesandagoldsphere,inwhichIplayedasideroleasthefabricationguy.IlearntfromheralotabouttheCasimirstudies,fromthefabricationtechniquestothenumericalcalculationoftheLifshitztheory.Shealsogavememanyacademicandnon-academic 4

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4 )mucheasier.ZsoltMarcetcollaboratedwithmeintheprojectofon-chipCasimirforcedemonstration(Chap. 5 ).Thisprojectdenitelycannotbecompletedwithouthim.Heworkswithcreativityanddiscipline.Ilearnedmanylessonsfromhim,especiallyhowtodrawbeautifulgures.TangLujoinedChan'sgroupintheHongKongUniv.ofScienceandTechnology(HKUST).HerworkinCOMSOLsimulationofthecombdriveisanindispensablepartoftheon-chipCasimirforcedemonstration.IbelieveherthoroughnessandcuriositywillleadtothesuccessintheongoingCasimirforceproject.FengpeiisanothertalentedgraduatestudentinChan'sHKUSTlab.Wehavebeenworkingtogetheronthefrequencynoisedetectionproject.Heisaquicklearnerandalwayswillingtotakechallenges.Thankstohishardwork,thisprojectstartstogeneratepromisingresults.Additionally,IthankLuandFengpeiforrevivingmyinterestinplayingbadminton.IamalsogratefultomylabmateandofcemateKostasNiniosforsharingalotoffunnystoriesandteachingmeaboutthegreatGreekscientists.Kostasalsotaughtmehowtoxthewirebonderandplaywithtransistors.Moreover,hebuiltseveralinstruments,includingacrucialprobe,whichIandothercolleaguesusedheavily.CoreyStambaughwasthebigbrotherinourlab.(Andyes.Heiswatchingyou!)Hestandsasanexampleofahardworkingandintelligentscientist.WeworkedtogetheronthefabricationinmyrstyearinChan'sgroup.Afterhegraduated,hecontinuedtohauntourgroup,eitherviaskypeorduringtheMarchmeeting.ImustthankhishospitalityandadvicesduringmyvisittoNIST.Atlast,IalsoneedtomentionKostasandZsoltweregreatroommateswhenwemovedtoHongKongandtriedtosurvivethere.Wehadalotoffuntogether. 5

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page ACKNOWLEDGMENTS .................................. 4 LISTOFFIGURES ..................................... 11 ABSTRACT ......................................... 14 CHAPTER 1INTRODUCTION ................................... 16 1.1PlentyofFluctuationsattheBottom ..................... 16 1.2MicroelectromechanicalSystemsasFluctuationDetectors ......... 20 1.2.1NonlinearOscillators,ThresholdDetectorsandNon-GaussianNoise .................................. 22 1.2.2High-QMechanicalOscillators,ForceGradientSensorsandCasimirForces .................................. 27 1.3StructureoftheDissertation .......................... 29 2FABRICATIONANDBASICBEAMTHEORY ................... 30 2.1Fabrication ................................... 30 2.1.1SiliconNitrideDevices ......................... 31 2.1.2SOIDevices ............................... 34 2.1.3PolyMUMPSRDevices ........................ 36 2.2BasicBeamTheory .............................. 36 3APPARATUSANDTRANSDUCTIONSCHEMES ................. 44 3.1Apparatus .................................... 44 3.1.1TheSampleCarrierandStage .................... 44 3.1.2TheProbeandCryostat ........................ 44 3.1.3TestingintheAmbientCondition ................... 47 3.2TransductionSchemesoftheDoublyClampedBeam ........... 48 3.2.1CapacitiveTransduction ........................ 48 3.2.1.1CapacitiveActuation ..................... 48 3.2.1.2CapacitiveDetection ..................... 50 3.2.2MagnetomotiveTransduction ..................... 52 3.2.2.1Transmission ......................... 52 3.2.2.2Reection ........................... 55 3.2.2.3DampingduetotheMagneticField ............ 57 3.2.3HybridTransduction .......................... 58 4DETECTINGNON-GAUSSIANNOISEBYNONLINEARMICROMECHANICALOSCILLATORS .................................... 59 8

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.................................... 59 4.2DeviceDesignandExperimentSetup .................... 62 4.3Theory ...................................... 67 4.4DataAnalysisandDiscussion ......................... 71 4.5ConclusionandFutureResearch ....................... 78 5DEMONSTRATINGTHECASIMIRFORCEONAMICROMECHANICALCHIP 81 5.1Motivation .................................... 81 5.2DeviceDesign ................................. 85 5.3SampleCharacterization ............................ 87 5.3.1TheConductivityandtheCarrierDensity ............... 87 5.3.2DimensionsoftheStructures ..................... 87 5.3.3GeometriesoftheInteractingStructures ............... 88 5.3.4RoughnessoftheInteractingSurfaces ................ 90 5.4TheoreticalCalculationsoftheCasimirForce ................ 90 5.4.1LifshitzTheory ............................. 90 5.4.2AccountfortheGeometry ....................... 92 5.4.3Corrections ............................... 93 5.4.3.1Roughness .......................... 94 5.4.3.2FiniteTemperatureEffects ................. 94 5.5CalibrationandMeasurements ........................ 94 5.5.1MeasuringtheMechanicalResponse ................. 94 5.5.2OperationoftheCombDrive ..................... 96 5.5.3CalibrationbyApplyingElectrostaticForces ............. 96 5.5.4MeasurementoftheCasimirforce .................. 100 5.6Double,Triple,Quadruple,...Checks ..................... 102 5.6.1StrayElectrostaticForce ........................ 102 5.6.2LevitationfromtheCombDrive .................... 103 5.6.3Parallelism ............................... 104 5.7ConclusionandFutureDirections ....................... 105 6SUMMARY ...................................... 107 APPENDIX AFABRICATIONDETAIL ............................... 109 A.1FabricationofSiliconNitrideDevices ..................... 109 A.2FabricationofSOIDevices .......................... 113 BNOTATIONSINTHEROTATINGFRAME ..................... 116 CREDUCTIONTO1DIMENSIONINTHEVICINITYOFBIFURCATIONPOINTS 117 DTHEMOSTPROBABLESWITCHINGPATH ................... 119 EPERMITTIVITYATIMAGINARYFREQUENCIES ................. 122 9

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....................................... 124 BIOGRAPHICALSKETCH ................................ 128 10

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Figure page 1-1TheschematicfortheeldsbetweentwoCasimirplates. ............ 18 1-2Aschematicofadoublyclampedbeam. ..................... 21 1-3Thenonlinearmechanicalresponseofthebeamunderstrongperiodicdrivingforce. ......................................... 23 1-4Schematicforthepotentialnearametastablestate. ............... 24 1-5ThePoissondistributionswithasmallorlargemean. .............. 25 1-6TheratiobetweenthevaluesofPoissonandGaussiandistributionswiththesamethemean=100. ............................. 26 1-7SchematicsoftheswitchinginthepresenceofGaussianandPoissonnoise. 27 1-8Themechanicalresponseofaunderdampeddrivenoscillator. ......... 28 2-1Finiteelementmethod(FEM)ofadoublyclampedbeam. ............ 31 2-2AscanningelectronmicrographofaSiNnanobeam. .............. 32 2-3ThefabricationowofaSiNnanobeam. ..................... 33 2-4ThefabricationowofSOIdevices. ........................ 34 2-5AfalsecolorSEMimageofasuspendedspringmadeinaSOIwafer. ..... 35 2-6FEM-simulateddisplacementfortheshapeofthethreelowestmodes. .... 40 2-7ThemechanicalresponsesofthebeamusedinChapter 4 underdifferentperiodicdrivingforce. ..................................... 43 3-1Thetailoftheprobe.Insetontherightdisplaysaceramicchipcarrier. .... 45 3-2Theside-viewpictureofthetopoftheprobe. ................... 46 3-3Thetop-viewpictureofthetopoftheprobe. ................... 46 3-4Acompacttestplatformforoperatingthedeviceintheambientcondition. ... 47 3-5Theschematicofcapacitivetransduction. ..................... 49 3-6Theschematicofbridgecapacitivetransduction. ................. 51 3-7Abasicmagnetomotivetransductionscheme(transmissometer). ........ 52 3-8Theequivalentcircuitforthemagnetomotivetransmissionscheme. ...... 53 11

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................. 56 4-1Themechanicalresponseofthebeamundervariousperiodicdrivingforce. .. 61 4-2Schematicoftheexperimentsetup. ........................ 63 4-3SchematicofthegenerationofPoissonRFpulses. ............... 65 4-4ThetimeseriesofPoissonRFpulses. ....................... 66 4-5Translationinthephasespace. .......................... 71 4-6Thedirectionoftranslationsandthephaseofthepulse. ............ 72 4-7UnipolarswitchinginducedbyPoissonRFpulses. ................ 73 4-8TransitionsinthepresenceofPoissonRFpulsesandthehistogram. ..... 74 4-9TherelationsbetweentransitionrateandthenoiseintensityaredistinctforGaussianandPoissonnoisecases. ........................ 75 4-10ScalingforGaussiannoise. ............................. 77 4-11Scalingforthefrequencydetuningfromthebifurcationfrequency. ....... 77 4-12ScalingforthereducedworkofasinglePoissonpulse~g. ............ 78 5-1Schematicofthedevicewithelectricalconnections. ............... 85 5-2Scanningelectronmicrographofthedevice. ................... 86 5-3Thetop-viewscanningelectronmicrographoftheinteractingobjects. ..... 88 5-4ThecrosssectionalSEMimageoftheinteractingobjects. ........... 89 5-5Themechanicalresponseofthedoublyclampedbeam. ............. 95 5-6Measuredfrequencyshift!RasafunctionoftheelectrodevoltageVe. ... 97 5-7ThemeasuredresidualvoltageV0asafunctionofthedistance. ........ 98 5-8AtypicalttothemeasuredelectrostaticforcegradientatVe=V0+100mV. 99 5-9ThemeasuredCasimirforce. ............................ 100 5-10TheimageofthecombdriveinsampleD. ..................... 102 5-11TheratiosoftheCasimirforcegradientsgivenbytheBEMcalculationandthePFAasafunctionofthedistance. ....................... 105 A-1Step1ofthefabricationofSiNbeams. ...................... 109 A-2Step2ofthefabricationofSiNbeams. ...................... 109 12

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...................... 110 A-4Step4ofthefabricationofSiNbeams. ...................... 111 A-5Step5ofthefabricationofSiNbeams. ...................... 112 A-6Step6ofthefabricationofSiNbeams. ...................... 112 A-7Step1ofthefabricationofSOIdevices. ...................... 114 A-8Step2ofthefabricationofSOIdevices. ...................... 114 A-9Step3ofthefabricationofSOIdevices. ...................... 115 A-10Step4ofthefabricationofSOIdevices. ...................... 115 D-1Themostprobableswitchingpaththatisnumericallycalculated. ........ 119 D-2Thecomparisonofthecalculatedswitchingexponentsfromdifferentmethods. ............................................. 121 13

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1 ],inwhichheenvisagedthecomingeraofnanotechnology.Feynmandiscussedaboutthepossibilitiesoftheassemblyofnanoscalemachines,theatomicmanipulationofnewmaterials,ultrahigh(atomic)resolutionmicroscopes,andtheirexcitingapplications.AlotofFeynman'senvisionshavebecomeareality.Microscale,nanoscaleorevenatomicscalesystemsarecommonlyinvestigatedinthelaboratories,whilesomeofthesemicro-ornano-systemshavealreadyfoundtheircommercialapplicationsindailylives.Microelectromechanicalsystems(MEMS),rangingfromthemicroscalecantileversinatomicforcemicroscopes(AFM)tocomb-drivebasedaccelerometersincars,standasgoodexamples.Obviously,therearenewissuesandnewchallengesatthebottom.InFeynman'stalk,severalchallengesarediscussed,forexample,thedifcultyinheatdissipationatthenanoscale.However,maybelimitedbythelengthofthetalk,hedidnotmentiononething:therearealsoplentyofuctuationsatthebottom.Fluctuationiseverywhere.Forexample,forsystemsatnitetemperature,thereexistthermaluctuations.Evenatabsolutezerotemperatureinpurevoidvacuum(bywhichImeantherearenorealparticles),thecreationandannihilationofvirtualparticlesgiverisetoquantumvacuumuctuations.Fluctuationsoftenbecomemoresignicantinsmallersystems.Letustakethermaluctuationsofaharmonicoscillatorasanillustration.Accordingtotheequipartitiontheoremmv2f=2=3kBT=2wheremisthemassoftheoscillator,vfistheuctuatingvelocity,kBistheBoltzmannconstantandTisthetemperature.Atthesametemperature,whenthemassoftheoscillatorisreduced,thermaluctuationsleadtoalargeructuatingvelocityoftheoscillator.Thisphenomenonhasdirectimplications 16

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2 4 ].Themechanicaloscillatorconsistsofbillionsofbillionsofatoms,butiscooledclosetoitsmotionalgroundstate,i.e.kBT<~!.Inthiscase,quantumuctuationsdeterminethebehaviorofthemechanicaloscillator. 17

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5 6 ].ThisquantumforceistheCasmirforce[ 7 ],whichwillbeourtopicinChapter 5 Figure1-1. Theschematicfortheeldsbetweentwoelectricallyneutral,parallelplatesmadeoutofaperfectconductor.Redwavesrepresentthestandingwavesinthezdirection,asaresultoftheboundaryconditionsimposedbytheplates. LetustotakeashortdigressiontoappreciateCasimir'sderivationoftheCasimirforcefromquantumvacuumuctuations.InH.B.G.Casimir'spioneeringworkonthevacuumuctuation,hepointedoutthatthequantumvacuumenergy,likemanyotherphysicalquantities,dependsonitsboundaryconditions.Hestudiedaspeciccasefortwoinniteparallel,electricallyneutralplatesmademadeoutofaperfectconductor 18

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1-1 ).ThecorrespondingboundaryconditionsarethatE,B=0intheperfectmetal.AlsothetangentialcomponentsofEandthenormalcomponentofBmustvanishatthesurfacesoftheplates.Hereforsimplicity,wederivetheCasimirforceformulaforscalareldsina3Dspace.Thex,yandzdirectionsaredenedinFig. 1-1 .Itmightbeinsightfultoviewthetwoplatesasacavity.Thewavevectorinthezdirectionmustbequantizedduetotheboundaryconditions.Onlythemodeswithcertainwavelengthswillresonatewithinthecavity.Thesefrequenciesare d3,(1)whereAisthearea,~isthereducedPlanckconstant,cisthespeedoflight,anddisthedistancebetweenparallelplates.Theregularizedvacuumenergyturnsouttobenite.Moreastonishingly,itisafunctionofthedistanced.Consequently,theCasimirpressure(forceperarea)arises: d4,(1)wheretheminussignmeansanattractiveforce.TheCasimirpressureforEMeldsisjusttwiceaslargetheoneforscalarelds: d4,.(1)AsshowninEq. 1 ,theCasimirforcerisessharplyatsmalldistances.Itisbelievedtobeoneofthereasonsforstiction,whichreferstothephenomenonthattwomovablemechanicalobjectssticktogetheruponcontact.Stictiondirectlyleadstomalfunctionsof 19

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8 ].Itoffersimportantinformationaboutthequantizedchargeofthechargecarrierinthecircuit.Similarly,forparticlesadsorbinganddesorbingfromthesurfaceofamicro-(nano-)mechanicalstructure,themassshotnoise[ 9 ]dependsonthemassofthemasscarrier.Arobustdetectorofthemassshotnoisemayenablenovelmassspectroscopy[ 10 ].Therefore,uctuationisawindowtounderlyingphysicsoftheuctuatingsystem.Tosummarize,micro-(nano-)scalesystemsaresusceptible,orevenvulnerabletouctuations.Agoodunderstandingofclassicalandquantumuctuationsisoffundamentalinterestandisimportantforthestate-of-artnanotechnology.Ontheotherhand,uctuationscanbehelpfulratherthanharmful.Adetectorofuctuationsprovidesagoodprobeofthesystemunderinvestigation,includingmicro-(nano-)systems. 20

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11 12 ],attometerdisplacementsensors[ 13 ],atomicmasssensors[ 9 10 14 ],ultrasensitiveelectrometers[ 15 ],andsinglespindetectors[ 11 ].Inthisthesis,micromechanicaldevicesareemployedtoinvestigateclassicalandquantumuctuations.ItisworthnoticingthatMEMSdevices,likecantileversortorsionaloscillators,havealreadyproventobeanexcellenttoolinstudiesofmassuctuations[ 9 ]andtheCasimirforce[ 6 16 17 ].Furthermore,thefabricationtechniquesoftheMEMSdevice,borrowedfromtheintegratedcircuitsindustry,areveryrobustandpowerful.Thereisplentyofroomtomodifythedesignfordifferentapplications.Itcanbedesignedasanoff-boardsensortodetecttheexternaluctuationsfromthedeviceundertest.Moreover,inprinciple,theMEMSsensorcanbeintegratedontothesamechiptodetectanon-chipuctuationsourceandworkwithotheron-chipfunctionalcomponent.Thisplatformprovidesuniqueopportunitiestoinvestigatetheuctuationinanovelwayandpossiblywithimprovedsensitivity.Inthisdissertation,usingMEMSdevices,wedemonstrateanoveldetectionmethodforthenon-Gaussiannoisethatcomesfromexternalclassicaluctuations,andtheon-chipmeasurementoftheCasimirforcethatarisesfromquantumvacuumuctuationsinelectromagneticelds.Adoublyclampedbeam(showninFig. 1-2 )actsasaforce(orforcegradient)sensorandplaysacriticalroleinbothprojects. Figure1-2. Aschematicofadoublyclampedbeam.Thebeaminthemiddleisclampedonbothends.Thelengthlismuchlargerthanthewidthwandthicknesstofthebeam.Weusuallydenethedimensionalongtheoscillatingdirectionasthewidth.Thebeamisusuallydrivenbyasinusoidalforcenearitsresonantfrequencyandactsasthedetectorofuctuations. 21

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1 andthemotionisnowdescribedbyaDufngoscillatormodel[ 18 ]: 2.2 .Figure 1-3 showsthatthemechanicalresponsearoundtheresonancebendstowardshigherfrequenciesbecauseofspring-hardening.Ahysteresisloopemergesandtwostablestateswithdifferentamplitudescoexistwithinafrequencyrange(fromabout300rad/sto500rad/sinFig. 1-3 ).Thepresenceofnoiseenablesthesystemtoswitchbetweenthestablestates.Ontheboundariesofthatfrequencyrange,onestablestatemergeswithonesaddlepoint(bluedashcurve)andtheotherstablestatebecomestheonlystablestate.Thisisthesaddle-nodebifurcation.Closetothebifurcationpoint,wecantreatthesystemasanoverdampedparticlemovinginapotentialasdepictedinFig. 1-4 [ 19 20 ].Weconsiderthestablestateclosetothesaddlepointisthemetastablestatethatisrepresentedbythelocalpotentialminimum 22

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Normalizedoscillationamplitudeasafunctionofthedrivingfrequency!D.BlackdotsaremeasureddataforthebeamweusedinChapter 4 .TheytwellwiththeDufngmodel(lines).Thenonlinearmechanicalresponseofthebeamunderstrongperiodicdrivingforce.Twostatescoexistbetweenthebifurcationfrequencies.Theamplitudescorrespondingtounstablesaddlepointsarecalculatedanddisplayedasthebluedashline. ontheleftofFig. 1-4 .TheotherstablestateisfarawayfromthesaddlepointontherightsidesoitisnotshowninFig. 1-4 .Startingatthemetastablestate,inthepresenceofanoisyforceexcitation,theoscillatorcanswitchoutofthismetastablestatewhenalargeoutburstofnoiseoccurs.Otherwiseitwillstayatthemetastablestate.Whenswitchingtotheotherstablestate,theoscillatingamplitudechangestoaverydifferentvalueandthisservesasthesignaltoidentifythelargeoutburstofnoise.Inthisway,thenonlinearmicromechanicaloscillatorperformsasathresholddetector.Athresholddetectorisusefulinnon-Gaussiannoisedetection.Usingshotnoiseasanexample,wewillshowrstwhythenon-Gaussiannoiseisdifculttodetectandthendiscusstheadvantageofathresholddetector.Sincewestudiedelectricalnoiseinourexperiment,welimitourdiscussioninthissectiontoelectricalnoise.Yetinprincipletheapproachdescribedhereworksforgeneralnon-Gaussiannoise. 23

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Schematicforthepotentialnearametastablestate.Startinginthelocalminimumontheleft,thesystemwillkeepbeingtrappedgiventheexternalnoiseislowerthanthethreshold.Butwiththeoutburstofthenoiselargerthanthethreshold,thesystemcouldovercomethebarrierandescapethelocalminimum.Thisswitchingwouldbethesignalfortheover-thresholdoutburstofnoise. Fluctuationsinanelectricsignalcanbeviewedasnoise.Althoughdifferenttypesofnoiseoriginatesfromdifferentmechanismsintheuctuatingsystems,insimpletreatments,itisassumedthattheelectricsignaluctuatesaboutameanvalueinaGaussianway[ 8 ].SuchapproximationfollowsfromthecentrallimittheoremandworkssowellthatinfactdetectingadeviationfromtheGaussiandistributionisachallengingtaskinmostcases.LetusillustratethisideawithshotnoisewhichfollowsPoissonstatistics.ForatunneljunctionbiasedwithavoltageV,thechargecarrierssingleelectronstunnelthroughthejunctionataveryfastrate.WeapproximatethetunnelingcurrentasasummationofdeltafunctionI=ePn(ttn)wheretnisthetimeatwhichthetunnelingoccursande=1.61019coulombsisthechargeofasingleelectron.TunnelingisaPoissonprocess;thereforeforanitemeasuringwindow,thenumberofthetunnelingelectronskhappeninginthatintervalfollowsaPoissondistributioncharacteristicwithonlyoneparameter: 24

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ThepinkbarshowsthePoissondistribution,whilethebluecurvedisplaysthecorrespondingGaussiandistributionwiththesamemeanandvariance.Inpanel(a),withameanatonly2thesetwodistributionaredistinct.Butinpanel(b)withameanaslargeas100,theGaussiancurvebecomesalmostidenticaltothePoissonbargraph. Themeanvalueofkisandsoisthevariance.WecanseethatrepresentstheaveragerateatwhichthetunnelingeventsoccuranditcanalsobeassociatedwiththeaveragecurrenthIibyhIi=e.Whenislarge,thePoissondistributionapproachestheGaussiandistribution.Figure 1-5 comparesthePoissondistributiontothecorrespondingGaussiandistributionwiththesamemeanandvariance.Panel(a)showsthattheyaredistinctfromeachotherwhenthemeanissmall(only2).However,whenthemeanbecomes100,itisverydifculttodistinguishthePoissondistributionfromtheGaussiandistribution,asshowedinFig. 1-5 (b).Thethresholddetectorapproachhasrecentlyattractedmuchattention[ 8 21 30 ].ThemainideaistofocusonthetailofthedistributionwherethedeviationfromtheGaussiandistributionisrelativelylarger.AsshowninFig. 1-6 ,thePoissonandGaussiandistributionsarealmostidenticalwhenkiswithintwostandarddeviationofthemeanvalue.Theeffortsspentonaccumulatingthedistributionneark=aremostlywasted,asitisverydifculttodistinguishnon-GaussiannoisefromaGaussianoneinthatparameterrange.Itismucheasiertodosoatk>+2ork<2. 25

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WereplotFig. 1-5 (b)astheratiobetweenthevaluesofPoissonandGaussiandistributionswiththesamemean=100.Thedeviationoftheratiofrom1onlybecomeslargewhenkdeviatesfromthemeanbymorethan2standarddeviations(=10). However,theeventsatthetailhaveexponentiallylowprobabilityofhappening.Bydirectlycumulatingthehistogramforthedistribution,weareusuallyunabletocollectenoughinformationaboutthedetailofthetail.Therefore,weneedathresholddetectortodiscriminatetherareevents.ThethresholddetectoronlygeneratesasignalwhenthevariablehIibecomeslargerthanthethresholdIth.SettingIthonthetailofthePoissonianorGaussiandistributions,wewillcollectinformationofthetailpart,andtherebywecandistinguishthemmoreefcientlyandaccurately.Bycouplingtheelectricalnoisetoadrivennonlinearmicromechanicaloscillator,weutilizethenonlinearoscillatorasathresholddetectortoinvestigatethenon-Gaussiancharacteristicofthenoise.ThethresholdIthcanbetunedbychangingthedistancetothebifurcationfrequency.Moreimportantly,westudyaspecictypeofnon-Gaussiannoise,whichismodulatedPoissonpulses.EachpulseoccursrandomlyandfollowsthePoissonstatistics.Moreover,eachpulseismodulatedatthedrivingfrequencyoftheoscillatorwithaxedphase.Asaresult,thePoissonpulseshaveverydifferenteffects 26

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(a)TheGaussiannoisedisturbsthesysteminametastablestatearoundthephasespace.Itisnotdirectional.ThenoiseneedstobestrongenoughtoassistthesystemtoclimbovertheenergybarrierU,inordertocauseswitching.(b)Incontrast,PoissonmodulatedpulsestranslatethesystembyacertaindistanceP1=P2=P3inthegeneralizedcoordinate.Switchingoccurswhenthesystemisdisplacedtotherightofthepotentialmaximum(thesaddlepoint). ontheoscillatorthantheGaussiannoisedo.WhiletheGaussiannoiseexcitesthesystemaroundthemetastablestateinallthedirectionsinthephasespace[Fig. 1-7 (a)],thePoissonpulsestranslatethesysteminonedirectionbyacertaindistanceinthegeneralizedcoordinate[Fig. 1-7 (b)].ItismainlybecausethemodulatedpulsehasaxedphasewhiletheGaussiannoisedoesnot.Thesetwodifferentmechanismsleadtoqualitativelydifferentswitchingbehaviorsforthenonlinearmechanicaloscillator.Thesebehaviorsareobservedinourexperimentsandingoodagreementwiththetheoreticalpredictions.MoredetailwillbegiveninChapter 4 1 )is/1=(!20!2D+j!0!D=Q)where!DisthefrequencyofthedrivingforceandQ=!0=2isthequalityfactor.Intheunderdampedlinearregime,asFig. 1-8 (a)displays,themechanicalresponseofthedoublyclampedbeamisanarrowpeakneartheresonantfrequency!0.Infact,itisthesquarerootofaLorentzianfunction.Thewidthofthepeak 27

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1-8 (b)showstheXquadratureofthemotion,whichisinphasewiththedrivingforce.ThevaluefortheXquadraturedecreasesfastaround!0.Theslopenear!0isalmostaconstantandproportionaltoQ.Wewillsoonseeitispossibletousethissteepslopetodetecttheforcegradient. Figure1-8. (a)Thenormalizedoscillationamplitudevs.thedrivingfrequencyforaunderdampeddrivenlinearoscillatorwhose=100and!0=2106rad/s.(b)TheXquadratureoftheoscillationvs.thedrivingfrequency. High-Qmechanicaloscillatorshavebeendemonstratedinmanyapplications.Inthisdissertation,weuseitasaforcegradientsensoranditshighqualityfactorisessentialforCasimirforcemeasurement.ItiseasytoextendEq. 1 tocalculatetheparametriceffectofadistance-dependentforce(Eq. 2 ).TheforcegradientF0resultsinashiftintheresonantfrequency!0/F0.Drivingtheoscillatorataxedfrequencynear!0,wemeasurethechangeintheXquadraturesignaltodetectthesmallshift!0.Therefore,thesensitivityofF0isdirectlyrelatedtotheslopeoftheXquadraturesignalwhichisdeterminedbythequalityfactorQ.EmployingadoublyclampedbeamwithQ105,weareabletomeasuretheweaksignaloftheCasimirforcegradientthatoriginatesfromquantumvacuumuctuations.TheCasimirforceisexpectedtoplaymoreandmoreimportantrolesinthefurtherminiaturizeddevices.ExploitationoftheCasimirforceonthechipscaleisoneoftheimportantgoalsinCasimirforcestudies.However,anon-chipdemonstrationofthe 28

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5 ].Thisisbecauseintheconventionalscheme,abulkypositionerisnecessaryinordertocontrolthedistancebetweentheinteractingobjects.Thenecessityofmanualalignmentlimitsthegeometryoftheobjects,duetothedifcultyinaligningtwomacroscopicobjectsatsubmicronseparations.Inmostexperiments,atleastoneoftheinteractingobjectsisspherical,inordertoovercomethealignmentissue.InChapter 5 ,wetakeadvantageoftheintegrabilityandthehighqualityfactorofaMEMSforcegradientsensor.Thishigh-Qoscillator(adoublyclampedbeam)isfabricatedonthesamechipwithacombdriveactuatorthatreplacesthebulkypositioner.ThebeamactsasasensitiveforcegradientsensortomeasuretheCasimirforcethatarisesfromthequantumuctuationsoftheEMelds.Theinteractingsurfacesonthechipareautomaticallyalignedafterthefabrication.InChapter 5 ,wesuccessfullydemonstratetheCasimirforcebetweentwoon-chipcomponentsthebeamandanearbyelectrode.ItmayleadtonewapplicationsoftheCasimirforceandopenupnovelopportunitiestoengineertheCasimirforceusingnanofabrication. 2 rstdescribesthefabricationofdifferenttypesofMEMSdevicesthatareusedinourexperiments.Itisfollowedbyasectionexplainingthebasictheoryofvibrationsinadoublyclampedbeam,whichisessentialforthedesignandoperationsofourdetectors.InChapter 3 theapparatusandtransductionschemesarediscussed.Chapter 4 talksaboutthenoveldetectionmethodfortheelectricalnon-Gaussiannoiseusingamicromechanicalresonator.InChapter 5 ,wedemonstratetheCasimirforceinasinglemicromechanicalchip.Finally,weconcludethisdissertationbyasummary. 29

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1-2 and 2-1 .Itoffershighsensitivitytoforce[ 31 ]andcharge[ 15 ]withitshighresonantfrequency,highqualityfactorandsmallmass.Insomecases,morecomplicatedstructures,forexample,combdrivesasdescribedinChapter 5 ,areintegratedwiththebeam.However,ingeneral,thesedevicessharesimilarfabricationprocesseswhichwillbediscussedtogetherinSec. 2.1 .Throughoutthedissertation,weoperatethedoublyclampedbeamdynamically,i.e.,anacforceatfrequencyclosetothemechanicalresonantfrequencyisappliedontothebeam.ThefundamentalmodeofadoublyclampedbeamisdisplayedinFig. 2-1 .InSec. 2.2 ,wedescribeabasicbeamtheorywhichpredictsthebasicparameters,suchastheresonantfrequencyandnonlinearitycoefcient,usingthedimensionsandmaterialpropertiesofthebeam.Thistheoreticaltoolisgreatlyhelpfulforourdesignandoperationofthebeam. 4 werefabricatedthroughacommercialfoundryMUMPSCAPR.Thisprocessisathree-layerpolysiliconsurfacemachiningprocessPolyMUMPsR. 30

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Finiteelementmethod(FEM)simulationofthefundamentalin-planemodeofadoublyclampedbeam.Theoriginalpositionofthebeamisshownasthe3Dblackbox.Thecoloredbeamdisplaysthenormalizeddisplacementintheoscillatingdirection.Thecolorrepresentsdifferentdisplacement,changingfromblue(theminimumdisplacement)tored(themaximumdisplacement).FEMsimulationisperformedonCOMSOL. 2-2 .Westartwithasiliconwafer.Thedopinglevelischosenbasedonhowconductivewewantittobe.Ifweplantoexcitetheout-of-planemodebyapplyingacvoltagetothesubstrate,thesiliconwafershouldbeveryconductive(<0.05cm)inordertoreducetheresistanceinseriesandtherebytheRCtime.Ifweplantominimizetheparasiticcapacitancetothegroundonthechip,itisbettertomakethesubstrateinsulatingatthetemperatureofoperation.Undopedorlightlydopedwaferswithconductivity 31

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AscanningelectronmicrographofaSiNnanobeamof50mlengthandabout700nmwidth.Thebeamissuspendedabovethesiliconsubstrateafteranisotropicetchingofthesilicon.TheSiNlmis200nmthick.Itispartlytransparentunderscanningelectronmicroscopes(andunderopticalmicroscopestoo).Theundercuttingcanbeclearlyseeninthisimageontheedgesoftheunreleasedpattern.Twosidegatescanprovidecapacitivecouplingtothenanobeam.Theimageistakenafterthebeamismetalizedbyalayerofgold. 2-3 ),thefabricationowfollows(Fig. 2-3 ):(a)DepositalayerofSiN(blue). 32

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ThefabricationowofaSiNnanobeam.Thegrey,blue,red,andyellowcolorsrepresentthesiliconsubstrate,theSiNlm,thephotoresistlayer,andthemetallayer,respectively. (b)Deneapatternofphotoresist(red)astheetchmaskusingphotolithography.(c)Etchthroughthenitridelayerbyreactiveionetching.Duringthisprocess,asmallamountofsiliconisalsoetched.Thethicknessofphotoresistisreducedduetotheetching.(d)Thebeamisnowreleasedbytheisotropicetchingofsilicon.(e)Removethephotoresistlayerbywetordryetching.(f)Evaporatealayerofgoldorotherconductingmaterial(yellow)forelectricalaccess.Figure 2-3 showsthecrosssectionalcartoonofthedeviceaftereachstep.MoretechnicaldetailsoftheSiNfabricationowarediscussedinAppendix A .SiNhasbeensuggestedtobeoneofthebestmaterialstoproducemechanicalresonatorswithextremelyhighqualityfactor[ 32 34 ].Howeverthequalityfactorsofour 33

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5 ). Figure2-4. ThefabricationowofSOIdevices.Thegrey,yellow,redanddarkredcolorsrepresentthesiliconsubstrate(orsocalledsiliconhandlelayer),theburiedoxidelayer,thesilicondevicelayerandtheAlmasklayer. ThefabricationofSOIdevicesissimilartoSiNdevices.Figure 2-4 displaythecrosssectionalcartoonoftheSOIdeviceaftereachstepofthefabrication. 34

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Figure2-5. AfalsecolorSEMimageofasuspendedspringmadeinaSOIwafer.Thepicturewastakenfromatiltedangle.RedcolorstandsforthedevicelayeroftheSOIwafer.Yellowcolormarksthesiliconoxide.Wecanseethattheserpentinespringissuspendedandthereisnooxidebeneathit.Butthepartsdesignedtobeimmobileareanchoredbytheremainingoxide. Figure 2-5 showsanexampleofthereleaseddevice.MoredetailsofthisfabricationprocesscanbefoundinAppendix A .Finally,weevaporate5nm-thickchromiumand150nm-thickgoldonlyontothebondingpadsandpartoftheleadsusingashadowmaskforelectricalcontacts.UnlikethecaseoftheSiNbeam,weusuallydonotevaporatemetalontothebeamorothersuspendedstructuresasthehighlydopedsiliconisconductiveenoughfortheelectricalcontactsandthemetalwouldleadtodeteriorationinthequalityfactoroftheresonator.Moreover,inthecaseoflargesuspendedstructures,thestressmismatchbetweenthemetallmandthedevicelayerwillcauseanundesiredcurvatureofthestructure. 35

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2.1.1 ,isrepeatedforeachdepositedlayertodenethepatternsoneachlayer,exceptfortheSiNlayer,whichinprinciplecanbeetchedandpatternedtoobutrequiresspecialcare.IntheSOIcasediscussedinSec. 2.1.2 ,wehavetocarefullydesignthedimensionofsuspendedandanchoredstructuresbecausetheimmobilepartsneedtobeanchoredbyremainingoxide.Incontrast,inPolyMUMPSRdevices,theabilitytodeneapatternofthesacriciallayerenablesustoanchorthesiliconstructuredirectlyonanothersiliconorSiNstructure.Asaresult,thetimingoftheHFreleaseforPolyMUMPsRdevicesisnolongerstringent.Itisalsopossibletodesignlessorevennoneetchingholesforalargeareaofsuspendedstructure.Moreover,whilethefabricationusingSOIwaferslimitsustoonlyonelayerofsilicon,amulti-layersprocessnaturallyopenspossibilitiestomorecomplicatedstructures.Ontheotherhand,thedisadvantagesarealsoobvious.Asastandardmulti-userprocess,thethicknessandmaterialsofeachlayerarexed.Thephotolithographyproducesaresolutionof2mandanalignmenterrorto12m,farworsethanwhatisachievedbydeepUVorelectronbeamlithography.Lastly,thereisnocontroloverthesidewallsoftheetchingprole. 36

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2-1 isinthedirectionoftheyaxis.Thedimensionofthebeamalongtheyaxisisdenedasthewidthw.Euler-Bernoullibeamtheory1describesasmalldeectionofalongbeam(thelengthlw,t)underaforcetransversetotheneutralaxisofthebeam.ThekeyassumptioninEuler-Bernoullibeamtheoryisthattheplanesofcrosssectionsremainperpendiculartotheneutralaxisduringthedeection.Thisassumptionfailsinthepresenceofashearloadorinthecaseoflargedeections.Yetitholdsinmostcasesinthisdissertation.Thedirectionperpendiculartothesubstratewillbethezdirection.ThestaticEuler-Bernoullibeamequationisgivenby 35 ]givesafullderivationsimilartowhatispresentedhere.AmoresuccinctderivationcanbefoundinYurkeetal.'sPRApaper[ 36 ]ortheAppendixA.4ofLahaye'sdissertation[ 37 ]. 37

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2 ),wecansolveforthedeectionofthedoublyclampedbeamunderarbitraryforceloaddistribution.Animportantcaseistheuniformloadff(x).Thedeection EI(l2x2 38

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2 ,then-thnormalmodeun(x)satises 2-6 .WeplugEqs. 2 and 2 intoEq. 2 .Thenbothsidesaremultipliedbyun(x)andintegratedoverthelengthofthebeam.Thesolutionforthetimevaryingpartistheequationofadrivendampedharmonicoscillator: 39

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FEM-simulateddisplacementfortheshapeofthethreelowestmodesofthedoublyclampedbeam.Fromtoptobottom,panels(a),(b)and(c)correspondtonl=4.73004,7.8532,10.9956,respectively.Theblackboxindicatestheoriginalpositionofthebeam.Colorsfrombluetoredrepresentthedisplacementfromtheminimum(includingnegativevalues)tomaximum.FEMsimulationisperformedonCOMSOL. issimpletoshowthat 40

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5 .Thehighqualityfactorandthesmalleffectivemassmeprovideahighsensitivitytotheforcegradient.Itisalwaysgoodtoknowwheretolookfortheresonancewhenwearetuningthefunctiongeneratortoexcitethenormalmodes.Oftentheresonantfrequencyisrequiredtobewithinacertainrange,eitherduetothebandwidthrestrictionofourapparatusorotherexperimentalconsiderations.Thus,!0istherstquantitytoconsiderwhenstartinganewdesign.Then-thresonantfrequency!ncanbewellpredictedbythisfollowingformula: 2 .Ifthebeamismadebyasinglematerial,thefundamentalresonantfrequency l2s 2 ,!n=!0=(n=0)2.Sotheratiosbetweentherst5harmoniceigenfrequenciesandthefrequencyofthefundamentalmodeare!15=!0=2.757,5.404,8.933,13.345,18.637.Interestingly,forsingle-materialorcompositebeams,thetheoryinthepreviousparagraphgivesusthesamedependenceof!0onthedimensions:!0/w=l2and!0doesnotvarywiththethicknesst.Siliconnitride,metallayersandsometimesthedevicelayerofaSOIwaferhavesignicantbuilt-instresses.SiNisusuallytensile-stressedwhilethedevicelayeriscompressive-stressed.Metallayersmayhavetensileorcompressivestressdependingontheannealingprocess.Ifthestressisnotrelievedafterfabrication(anditisusuallythecaseforadoublyclampedbeam),thestresscontributestotheresonant 41

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32 ]: l2s 2 : 18 ]modeledthenonlinearityasanextratensionduetothelengtheningandobtained=71E=(l4).2Cleland[ 35 ]lateremployedaLangrangianapproachandfound0.722nl2!2n.Somehow,inYurke'sresult,doesnotrelatetowandt,whileCleland's/w.Theybothpredictthat/1=l4.Figure 2-7 illustratesthefrequencyresponseofadrivenPolyMUMPSRresonatorthatisusedinChapter 5 .AlltheexperimentalpointsarettedwellbytheDufngmodel.Whenthedrivingforceissmall,ourdoublyclampedbeambehaveslikeasimpledampedharmonicoscillator[Fig. 2-7 (a)].ALorentzianttingaccordingtoEq. 2 isadequate,thoughttingtoDufngmodelwouldgeneratethesameresult.However,whilesufcientlylargeperiodicexcitationisapplied,thebeamstartstoexhibitnonlinearbehavior.AsFigures 2-7 (b)and(c)show,themechanicalresponsearoundtheresonancebendstowardshigherfrequencies,owingtoapositivecubicnonlinearity. 18 ]containsatypo.ItshouldbeL3. 42

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Themechanicalresponsesofthebeamunderdifferentperiodicdrivingforce.Fromlefttoright,thedrivingamplitudeincreases.Theblackdotsaretheexperimentaldataandthecolorlinesaretswiththetheoreticalexpressions.Wenormalizealltheresponsestotheheightofthecriticalresponse.(a)Linearresponseundersmallexcitation.(b)Criticalresponse.Ahysteresisloopisabouttoemerge.(c)Forthedrivingforcelargerthanthecriticalforce,theoscillatorpossessestwostableoscillatingstatewithdifferentamplitudeswithinacertainfrequencyrange.Theunstablesaddlepointisalsocalculatedanddisplayedasthebluedashcurve. Whenthedrivingforceislargerthanthecriticalexcitation,ahysteresisloopemergesandtwostablestateswithdifferentoscillationamplitudescoexistwithinafrequencyrange[Fig. 2-7 (c)].Ontheboundariesofthatfrequencyrangeonestablestatemergeswithonesaddlepoint(bluedashcurve)andtheotherstablestatebecomestheonlystablestate.Thisisthesaddle-nodebifurcation.Experimentally,thecriticalamplitudecorrespondingtotheonsetofthehysteresisloopcanbedeterminedaccurately.Theoretically,thecriticalamplitudeforthefundamentalmodeofadoublyclampedbeamispredictedtobewp 38 ].Thisprovidesonewaytocalibratetheamplitude. 43

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3.1.1TheSampleCarrierandStageAsamplereadytobeloadedisrstgluedtoacommercial32-pinchipcarrier(theinsetofFig. 3-1 )byeitherfastdryingsilverpaintorsilverepoxy.Afterwirebonding,thecarrierismountedonthecarriersocketthatsitsonaprintedcircuitboard(PCB)designedbytheelectronicshopofUniv.ofFlorida(Fig. 3-1 ).UnfortunatelytheleadsonPCBarenot50transmissionlines.Twenty-fourleadsaresolderedtotwistedpairsofaribboncableforlowfrequencyordcuse,whilefourleadsaresolderedtothecenterpinsofsemi-rigidcoaxesforhighfrequencyuse.Thelastfourleadsconnecttotheoutergroundshieldsofthecoaxes.Groundisprovidedbythescrewswhichconnecttothecopperextender,whichissupposedtobethegroundoftheprobe. 3-1 ).Acopperextenderisxedontheangebyscrews.ThePCBsitsontheextender.Thusintheabsenceofexchangegasorliquid,theheatfromthesampleneedstotransferthroughthegoldbondingwire(sincethesiliconsubstrateisnotagoodthermalconductor),thecarrier,theleadsonthePCB,thecopperextenderandnallytheange.Inpractice,ifthedeviceintheprobeisinvacuum,afterloadingtheprobeintoliquidheliumdewar,itusuallytakes12to18hours 44

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Thephotoofthetailoftheprobe,whichconsistsofaprintedcircuitboard(PCB),achipcarrier,acarriersocketandcopperparts.Thecoaxesandwiresruninsidethetubingtoreachthetopoftheprobe(showninFig. 3-2 .Insetontherightdisplaysa32-pinceramicchipcarrier. fortheresonantfrequencyofthedevicetobestable.AbrasscapcoversthetailoftheprobeandaheliumtightsealingcanbeachievedbyanindiumO-ring.Thetoppartoftheprobe(Figs. 3-2 and 3-3 )hasa4-wayS.S.cross(NW25).Thetopporthas7coaxeshermeticSMAfeedthroughconnectorsonaNW50blank,4ofwhichareusedforthesemi-rigidcoaxesthatgoesallthewaydowntothePCBasdisplayedinFig. 3-1 .Theother3SMAconnectorsaresavedforfutureuse.TheportwithaFisher24-pinconnectorprovideselectricalaccesstotheribboncablefordcorlowfrequencyuse.Theprobeisusuallypumpedbyapumpsystemwithaturbomolecularpump.Afterpumpingovernight,theprobecanreach105Torr.Thenweclosethevalveandloadtheprobeintoadewar.Afterthat,thecryogenperformscryo-pumping.Sometimeswealsoputcharcoal(charcoalparticlesorspecialcharcoalcloth)insidetheprobeasanabsorbantforbettervacuum.Inthatcase,heatingofthecharcoalduringtheordinarypumpingisrecommended. 45

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Theside-viewpictureofthetopoftheprobe. Figure3-3. Thetop-viewpictureofthetopoftheprobe. Theprobetsintoacryostat(OxfordinstrumentsR)withasuperconductingmagnet.Thesuperconductingmagnetprovidesupto9teslamagneticeldalongitsneutralaxis.Thehomogeneityisspeciedas0.1%over10mm.Itisalsoabletogenerateamagneticeldgradientbutthisfunctionwasnotusedinourexperiment. 46

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Figure3-4. Acompacttestplatformforoperatingthedeviceintheambientcondition.Ontherightisthechipholderwherethe32-pinchipcarrierismounted.Ontheleft,BNCconnectorsprovideuswithelectricaccesstoallthepins. 47

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39 ]forarecentreview).Herewefocusontwotechniques:capacitiveandmagnetomotivetransduction. 3-5 .BysupplyingapotentialdifferenceVdc1+Vacbetweenthem,weintroduceanin-planeforceonthebeam.Theforceisgivenby(@C=@d)(Vdc1+Vac)2=2=(@C=@d)(V2dc1+2Vdc1Vac+V2ac)=2wheredisthedistancebetweenthebeamandsidegate.Thersttermisaconstantdcforcewhichleadstoastaticdisplacementofthebeam.Itonlychangestheequilibriumpositionofthebeamandisoflittleinterest.WeapplyVdc2=Vdc1ontheoppositesidegatetobalanceoutthedcforce.ThesecondtermgeneratesanacforceatthesamefrequencyasVac.ThethirdtermprovidesanacforceattwicethefrequencyofVac,butitisregardedasahigherordertermandignoredwhenVacVdc1.Usingthe 48

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Theschematicofcapacitivetransduction.Shadedpartsrepresentthefabricatedstructures:thesidegatesandthebeaminourcase.Theupperpartofthecircuitprovidesactuationwhilethelowerpartofthecircuitdetectsmotionofthebeam. inniteparallelplatesapproximation:(@C=@d)=r0lt=d2,itiscleartoseethattheacelectrostaticforceisproportionaltoVdc1Vaclt=d2.IfV2acisreallyundesired,alternativelyonecanapplyonlyp 49

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3-5 displaysonepossibleschemewherethebiasdcvoltageandachargesensitiveamplierareconnectedtothesidegate.Itisinprinciplealsopossibletosupplythebiasvoltageandconnecttheampliertothebeam.Usingtheinniteparallelplatesapproximation,oneobtains: d0+x(t)=r0lt d01x(t) 50

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3-6 .Thisschemenegatesthecapacitancebackgroundandonlymeasuresthedynamicpartofthecapacitance.The180degreeout-of-phasesignalsVac1andVac2areusuallyprovidedbya180splitter.Ifthecapacitancesonthetwobranchesarenotequal,attenuatorsandphaseshiftersmaybeaddedinonebranchtomaketheoutputofthebridgeclosetonull.Thisschemenegatestheparasiticcapacitanceandonlymeasuresthedynamicpartofthecapacitance.Asmallimbalanceinthebridgeleadstoanelectricalbackgroundthatisindependentofmotionofthebeam.Tofurthereliminatetheelectricalbackground,heterodynemethodscanbeutilizedbysimplyaddingacarriersignalinthisscheme[ 40 ].Yetnoticethebridgeandheterodynemethodsdoesnotautomaticallysolvethebandwidthlimitation. Figure3-6. Theschematicofbridgecapacitivetransduction.Shadedpartsrepresentthesidegatesandthebeam.Vac1andVac2are180degreeoutofphase.Thiseliminatesthecapacitancebackground. 51

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41 42 ]employstheLorentzforceandtheelectromotiveforce(EMF)toactuateanddetectmotionofthebeam,respectively.Thescalingdownofthedevicedoesnotdirectlyaffectitsefciency.Itisnotwidelyusedincommercializedproductsowingtothestringentrequirementofhighmagneticelds,butitisacommontechniqueinlaboratorieswhereasuperconductingmagnetorapermanentmagnetwithsufcientlyhigheldisavailable. Schematicforabasicmagnetomotivetransduction(transmissometer).Anaccurrentpassesthroughthedoublyclampedbeam.AnEMFinducedbythemotionofthebeamwillbedetectedbytheamplierontheotherendofthebeam.Vdc1,dc2canprovideadditionalelectrostaticcoupling. Figure 3-7 showsonetypicalmagnetomotivetransductionscheme(transmissometer).Thedoublyclampedbeamisbiasedbyanacvoltageononeendandtheotherendofthebeamisconnectedtothevirtualgroundofacurrentamplieroutsidetheprobe.Toachievethebestsensitivityandbandwidth,thetransmissionlinesarerequiredtobe50coaxesandthecircuitshouldbe50matchedeverywhereexceptatthedeviceundermeasurement,whichisourdoublyclampedbeam.Inthepresenceofamagneticeldperpendiculartothebeam,theaccurrentthroughthebeamgeneratesaLorentz 52

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2.2 thatthemotionofthedoublyclampedbeamcanbemodeledasadampedharmonicoscillatorwhentheexcitationforceissmall.RearrangingEq. 2 ,wedenetheresponsefunction=1=(!2R!2D+j!R!D=Q)where!Ristheresonantfrequencyand!Disthedrivingfrequency.Theamplitudeofthenormalizeddisplacementx=F=mewhereFistheexcitationforceandmeistheeffectivemass.Accordingtothetheory[ 18 ],theEMFVemf=Blj!Dxwhereisaconstantrelatedtothemodeshape,Bisthemagneticeld,listhelengthoftheoscillatingbeam,!Disthedrivingfrequencyandxistheoscillatingamplitude.Sincexcanbefurtherwrittenasx=F=me=lBIm=mewhereImisthecurrentthroughthebeam, Figure3-8. Theequivalentcircuitforthemagnetomotivetransmissionscheme.TheexcitedresonatorinthisschemecanbeexpressedasaparallelcombinationofaresistorRm,aninductorLmandacapacitorCm.Inaddition,Ristheintrinsicresistanceofthebeam. 53

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43 ]. 3 ,itiseasytogetCm=me=l2B2,Lm=l2B2=!2RmeandRm=l2B2Q=!Rme.ThefullcircuitofourschemecanbeexpressedasthetotalelectricimpedanceoftheelectriccircuitZ0inserieswiththeelectricallyequivalentmotion-inducedimpedance,orso-calledmechanicalimpedanceZm,asshowninFig. 3-8 .Z0shouldincludetheintrinsicresistanceofthebeamandotherimpedancefromthelinesorexternalcircuits.Theconvenienceoftheelectricallyequivalentimpedancewillbeclearattheendofthissectionanditbecomesevenmoreusefulwhenwediscussaboutthereectionschemeinthenextsection.Nowweturntotheissueabouthowtoextracttheparametersoftheresonatorfromourmeasurement.Intheabsenceofthemagneticeldoratafrequencyawayfromthemechanicalresonance,wemeasurerstthebackgroundcurrentIbg=Vin=Z0.Whenthedrivingvoltagesweepsthroughthemechanicalresonanceinthepresenceofthemagneticeld,themeasuredcurrentImcanbedescribedas 3 .Vemf/,ifVemfVinandImcanbeapproximatedasaconstant. 54

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3 intoEq. 3 andsolvingfor,wendthat 3 ,weknowZm/.Intheequivalentcircuit(Fig. 3-8 ),Im=Vin=(Z0+Zm).ItissimpletosolveforZm,whichturnsouttobeVin=ImVin=Ibg.Finally,Zm=Z0(IbgIm)=Imandwereachthesameconclusion/(IbgIm)=Im. 3-9 shows,thedrivingacsignalVacdoesnotdirectlytravelintothedetectioncircuit(theamplier).Ifthebeam'simpedanceZLisperfectly50,thereisnoreected 55

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Figure3-9. Theschematicofmagnetomotivereectometer.Thepotentialsonthesidegatesarexedatconstantdcvoltages.ThedrivingacsignalVaccomesintotheCoupledportofadirectionalcoupler.ThisoutgoingwavetravelsfromtheInputportintoacoaxwhichisterminatedbythedoublyclampedbeam.Noticeitisbettertogroundtheotherendofthebeamonthechip,notoutsideoftheprobe.ThereectedwavepassesthroughthedirectionalcouplerandentersfromtheOutputportintothe50amplier. 56

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3-8 .ThusbymeasuringR,onecanextractthemechanicalimpedanceZm.InthelimitofZmZ0+Zc,R=Zm+Z0Zc 44 ].Inthecaseofmagnetomotivetransduction,thecurrentIinducedbytheEMFVemf=BlvwillgenerateaforceFv=(B2l2@I=@V)v,whichaccordingtoLenz'slawisalwaysoppositetothevelocityofthebeam.Therefore,itisadampingforce.Substituting@I=@V=1=Z0,dampingisproportionaltoB2l2=Z0.ThetotalqualityfactorQrelatestotheintrinsicqualityfactorQMandtheeddyeffectqualityfactorby1=Q=1=QM+1=Qeddy,whereQeddy=

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4 ,whereweapplynoiseforcethroughtheelectrostaticcouplingbutdetectthebeam'smotionbythemagnetomotivetransmissionscheme. 58

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45 ],andthermaluctuationsgiverisetoJohnsonnoise[ 46 ].Thenon-Gaussiannoisecharacteristics,likethehigherordercumulants,revealvaluableinformationthatcannotbededucedfromthemeanandthevariance.However,asaresultofthecentrallimittheorem,inmostcases,theGaussiannoiseapproximatesallthedifferenttypesofnoisetoanexcellentextentsothatthedetectionthenon-Gaussiancharacteristicsishighlychallenging.ThewideapplicabilityofGaussianapproximationmightbegoodnewsasitsimpliesnoiseanalysisinsomecases.Ontheotherhand,itpreventsusfromutilizingthenoiseasapowerfultooltoprobethenoise-generatingmechanism[ 47 ]andextractothervaluableinformation,sincedifferenttypesofnoiseallapproachGaussianuctuations.Forexample,atunneljunctioncanhavethesameaveragecurrentandvarianceasabiasedresistorwiththermalnoisedoes.Ifwemeasuretheircurrentdistributions,theybothlooklikeaGaussiandistribution.WhilethelatteristrulyaGaussiandistribution,theformeractuallyobeysaPoissonstatisticsthatunderliesthetunnelingmechanismoftheindependenttransportationofdiscretecharges.Onestraightforwardsolutionistomeasurethenon-GaussiandistributionwithveryhighaccuracyorchooseproperparametersthatmaximizeitsdeviationfromtheGaussianapproximation.Letustakethetunnelingjunctionasanexample.ThenumberofthetunnelingelectronskhappeninginameasurementdurationfollowsaPoissondistributioncharacteristic.Supposethatthemeantunnelingrateisandthemeasurementdurationis.Whenislarge,thePoissondistributionisasymptotictoaGaussianone,asshowninFig. 1-5 .Onewayistolowerthetunnelingcurrent,sothatthetunnelingrateisverylow.Severalexperimentsinquantumdots[ 48 49 ]have 59

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50 ]exploitedafast12-bitanalog-to-digitalconvertorwith5nssamplingtimetoprobethevoltageuctuationresultedfromthetunnelingcurrent.AsymmetryoftherecordeddistributionwasingoodagreementwiththePoissonstatistics.Thisapproachrequireshighaccuracy,sophisticatedltering,andwidebandwidth.Alternatively,wecanusenonlineardevicesasthresholddetectorstoinvestigatethetailoftheprobabilitydistributionwherethedeviationfromaGaussiandistributionispronounced[ 8 21 30 ].Sofar,therehasbeengreatprogressboththeoretically[ 8 21 26 ]andexperimentally[ 27 30 ]toexploitJosephsonjunctionsasthresholddetectorstostudythePoissonnoisefromatunneljunction.AsmalldeviationfromtheGaussianapproximationhasbeensuccessfullyextractedintheexperimentsandcomparedwiththetheory.Inthischapter,weutilizedrivennonlinearmicromechanicaloscillatorsasdetectorstoinvestigatetheswitchinginducedbyPoissonradiofrquency(RF)shortpulses[ 51 ].Ouroscillatorisamicroscaledouble-clampedbeamdrivenbyasinusoidalforce.AsshowninFig. 4-1 ,subjecttosmalldrivingforces,ourmicromechanicaloscillatorrespondsasanunderdampedlinearharmonicoscillator.Foralargeroscillationamplitude,nonlinearbehavioremergesgradually.Thecriticaldrivingamplitudehccorrespondstothelargestdrivingforcethatkeepsthesystemmonostableatalldrivenfrequencies.Whenthedrivingforceexceedsthecriticaldrivingamplitudehc,thesystemisbistableinthefrequencyrange(!1,!2),where!1,2aretwobifurcationfrequenciesmarkedbythearrowsinFig. 4-1 .Closetothebifurcationfrequency,onestablestatebecomesmetastable.Noisecanenablethesystemtoswitchoutofthemetastablestate. 60

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Measuredoscillationamplitude(circles)asafunctionofdrivingfrequency.Thets(lines)uselinearharmonicoscillatororDufngoscillatormodelstoextracttheparametersinthesemodels.Thecurvescorrespondtothedrivingamplitudesof0.2hc,hc,and4hc,frombottomtotop.Acisthecriticalamplitudethatcorrespondstotheonsetofhysteresis. Whenswitchinghappens,theoscillatingamplitudejumpstoanewvalueandthisactsasasignalofthelargeoutburstofnoise.WedeneaquantityQbythelogarithmoftheswitchingrateW.Q=ClogW,whereCisaprefactorthatisindependentofthenoise.ForGaussiannoise,theswitchingrateWobeysKramers'law[ 52 ],W/exp(R=DG)whereRistheactivationenergy,andDGistheintensityofGaussiannoise.ItfollowsthatQ/1=DG.However,forPoissonRFpulses,owingtothedifferentswitchingmechanism,weobservealogarithmicdependenceofQonthemeanrate[ 53 ].ThemeanrateofPoissonRFpulsesisproportionaltotheintensityofthenoiseD.Therefore,thisrelationcannotbereducedtothecorrespondingoneinthecaseofwhiteGaussiannoise.Furthermore,westudythescalingbehaviorofswitchingunderPoissonRFpulses.Whileapproachingthebifurcationpoint,Qshowsuniversalscaling[ 19 54 57 ].WhenthenoiseiswhiteandGaussian,Qexhibitsapower-lawdependenceonthedistancetothebifurcationpoint(fornowwecanunderstandastheslopeattheoriginpointin 61

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1-4 ),Q/withacriticalexponent=3=2forthesaddle-nodebifurcation.Thishasbeenconrmedbymicromechanicaloscillators[ 55 57 ]andRF-drivenJosephsonjunctions[ 58 ].Recently,itispredictedthatthescalingbehaviorchangesfromasimple3=2powerlawtoa1=2powerlawwithadditionallogarithmicdependenceforPoissonnoise[ 53 ].WeconrmthisqualitativedifferencebetweenthecasesofpureGaussianandPoissonnoiseinourexperiment.Apartfromnoisedetection,micro(nano)-mechanicaloscillatorscoupledwithnon-Gaussiannoisefromatunneljunction[ 31 ],quantumpointcontact[ 59 ],orevaporatedmass[ 9 10 ]havebeenrealizedinrecentexperiments.Itisimportanttounderstandtheeffectsofnon-Gaussiannoiseonthenonlinearmicro(nano)-mechanicaloscillators.Sincethescalingbehaviorispredictedtobeagenericfeatureindependentofsystems,thisexperimentalsoenhancesourunderstandingoftheperformanceofgeneralnonlineardevicesnearthebifurcationpointinthepresenceofnon-Gaussiannoise.Thestructureofthischapterisasfollows:InSec. 2.1 wewoulddescribetheessentialfabricationanddesignconsiderations.Wewouldalsoexplainthespecicexperimentsetup.WewouldderivethetheoryinSec. 4.3 .ThedataanalysiswouldbeaddressedinSec. 4.4 .Lastly,wewouldconcludewiththediscussionaboutfutureresearchinSec. 4.5 4-2 ).Itisfabricatedthroughathree-layerpolysiliconsurfacemachiningprocessPolyMUMPsRthroughacommercialfoundry,asbrieyintroducedinSec. 2.1.3 .ThedetailsofthedesignguidelinesandrulescanbedownloadedfromMEMSCAP'swebsite.Hereweonlybrieydescribeourdesignconsideration. 62

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Schematicoftheexperimentsetup.InthemiddleistheSEMpictureofthedouble-clampedpolysiliconbeam(lwt=100m1.2m1.5m).Thebeamissuspendedintheair2.75maboveapolysilicongroundingshield.ThewholedeviceisinavacuumprobeheldinliquidHe.Asuperconductingmagnetprovidesmagneticeldperpendiculartothebeam.Weactuatethebeambypassinganalternatingcurrentthroughit.Thenoiseisfeededinbyanelectrodenearbythebeam.TheotherelectrodeisheldatVdc.Thesignalisampliedandreadbyalock-inamplier. AsthePolyMUMPsisastandardmulti-userprocess,thethickness,stressandmaterialofthebeamaremoreorlessxed.Weonlyneedtochoosethewidthandlengthofthebeam,aswellasthegapbetweenthebeamandtheelectrodes.First,wedecidetominimizethegaptomaximizethecapacitivecoupling,whichwouldgeneratelargerforcefortheactuationandbettersignalforthedetection.ThePolyMUMPsguaranteesaminimalfeatureof2m,sowechoose2mforthegap.Forthebeam'swidthandlength,themainconcerniswhatresonantfrequencyrangeweareinterestedin.Wecanestimatethebeam'sfrequencyofthefundamental 63

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2 .Werewriteithere: l2.(4)whereEistheYoung'smodulusofthematerial,isthedensity,wisthewidthofthebeamandlisthelengthofthebeam.Aswewouldliketoworkover1MHz,westartswithl=100mandw=1.2m.Wewouldliketodesigntheresonantfrequencyoftheout-of-planemodefarawayfromthefrequencyofthein-planemode,inordertoavoidfurthercomplicatingthesystemowingtothemodemixing.FromEq. 4 ,weseethattheratiobetweentheresonatingfrequenciesoftheout-of-planeandin-planemodesisexactlytheaspectratiot=w=1.5=1.2.Forresonatorswithhighqualityfactors,thelinewidthofeachresonancewillbeprettynarrow,andthisratioissufcienttoseparatetheresonanceswellapart.UsingE=180GPaand=2330kg=m3,Eq. 4 predictstheresonantfrequencyofthefundamentalin-planemodetobe1.09MHz.Theexperimentalmeasurementgivesf01.135MHz.Ourestimationappearsadequateforthedesignpurpose.WeexciteanddetectthemotionofthebeambythemagnetomotivetransmissiontechniqueasdiscussedingreatdetailinSec. 3.2.2 .Thedeviceisinvacuumandheldat4.2K.Asuperconductingmagnetprovidesmagneticeldperpendiculartothebeam.WeapplyadrivingvoltageVaccos!Dttooneendofthebeam.TheotherendofthebeamisconnectedtothevirtualgroundofacurrentpreamplierAmptek250locatedatroomtemperature.ThusweapplytothebeamperiodicLorentzforceperlength,BI=BVaccos!Dt=RbeamwherethemagneticeldB=5Tthroughouttheexperiment.AnRFlock-inamplier(SR844)isusedtomeasurethein-phaseXandout-of-phaseYquadraturesoftheoutputcurrentatthedrivingfrequency!D.Aroundtheresonantfrequency,thebeamstartstooscillateandthemotionperpendiculartothemagneticeldgeneratestheelectromotiveforce(EMF).AccordingtoLenz'slaw,theEMFisintheoppositedirectionoftheactuatingvoltage.Thusthechangeinthecurrentindicatesthe 64

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SchematicofthegenerationofPoissonRFpulses.Thenoisesourcetriggersthepulsegenerator.ThepulsesobeysPoissonstatisticsbecausethetriggeringisaPoissonprocess.ThepulsesfurthercontrolaswitchconnectedtoaRFsignal.WhentheswitchisON,theRFsignalisinjectedintothedevice.WhenitisOFF,noRFsignalistransmitted.TherebywegeneratePoissonpulsesofRFsignals. amplitudeofthebeam'soscillation.ThemeasuredmechanicalresponsesareshowninFig. 4-1 .ThetsusingDufngmodelextracttheparametersofthemodel.Thelock-inamplierprovidesasamplingratesof512Sa/sandatotalmemoryof16,000datapoints.Whenfasterandreal-timedatacollectingisneeded,weemploytwodigitalmultimeters(Agilent34410A)totakedatafromtheanalogoutputsofthelock-inamplier.Thetwodigitalmultimetersaretriggeredbythesamesquarewavefromanotherfunctiongenerator.Inthisway,twomultimetersaresynchronizedandthesamplingrateisthefrequencyofthetriggeringsquarewave.Noticedestructionreadingisrequiredtocollectcontinuousreal-timedata.Thisschemeworksupto5kSa/s,whichislimitedbythemultimeter(Agilent34410A).SincethesmallesttimeconstantfortheSR844lock-inamplieris100s,apursuitofsamplingrate>10kSa/sisnotverymeaningful.Nowwedescribehowtoapplynoisetothedevice.Intheparameterrangeofoperation,thermalmotionofthebeamdoesnotleadtoswitchings.Electricalnoise 65

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4-3 displaysaschematicexplainingthegenerationofPoissonRFpulses.ThePoissonnoisehereconsistsofmodulatedPoissonRFpulses(Fig. 4-4 )triggeredbyacontinuousnoisesource.Thepulsegeneratorgeneratesapulsewhentheinputvoltageexceedsathresholdvaluethatisxedinthegenerator.Poissonstatisticsoriginatesfromthenatureoftriggering.Theonlyconcernisthatthefunctiongeneratorwouldnotbetriggeredifthelastpulsehasnotnished.Inourcase,thetotallengthofthepulsewaveis600s(400shighvoltageand200szerovoltage).Throughoutourexperimentthemeanintervalbetweenpulsesislargerthan20ms;therefore,thecasethattwotriggeringeventshappenwithin600sisveryrareandthePoissoniannatureofthenoiseisrobust.Themeanrateofthepulsesiscontrolledbychangingthestrengthofthetriggeringnoise.AcombinationofVdc+Vnoiseisappliedtooneelectrodefornoisecoupling,whiletheotherelectrodeisheldatVdc.Thustheelectrostaticforcegeneratedbythenoiseis(Vdc+Vnoise)2=2CwhereCisthecapacitancebetweenthebeamandtheelectrode. 2.2 .Giventheassumptionthatthedampingandthedrivingforcedonotchangetheshapeofmodes,wecanseparatethevariablesandreducethetwodimensiondynamicstoonedimensiondynamics.Thenthedouble-clampedbeamiswellmodeledasaDufngoscillatoroftheform[ 19 ] 66

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(a)ThetraceontheoscilloscopeofPoissonRFpulses.Eachverticallineisinfacta400slongpulse,whichconsistsof400cyclesofRFmodulation.(b)Thecloselookofone400slongpulse. whereqisthenormalizeddisplacementofthebeam,isthedampingcoefcient,!0istheintrinsicfrequency,isthecubicnonlinearityparameter,histheamplitudeoftheexternaldrivingforce,!Disthedrivingfrequency,andf(t)isthenoiseforcewhichinthepresentstudycanbePoissonnoisefP(t),GaussiannoisefG(t)oracombinationofboth.ThewhiteGaussiannoiseischaracterizedbyitscorrelatorhfG(t)fG(t0)i=2D(tt0).ThePoissonnoiseisfP(t)=gexp(i!Dt+i)PnH(ttn)H(tn+tlt)wheregistheamplitudeoftheRFsignal,tlisthedurationofonepulse,andH(x)istheHeavisidestepfunctionwithH(x)=0forx<0,H(x)=1forx0.TheexpfactorrepresentsthemodulationoftheRFsignal.Thetimetnforthen-thpulsetooccurmustobeyaPoissonstatisticswithameanvalue.Whentlismuchsmallerthantherelaxationtimetr=1=,H(ttn)H(tn+tlt)canbeapproximatedbyadeltafunctiontl(ttn).ItisconvenienttoconverttotherotatingframewherethegeneralizedcoordinatesPandQaredirectlyrelatedtothein-phaseXandout-of-phaseYreadingsfromourlock-inamplier. 67

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whereCres=p 4-1 (c)shows,werecordtwobranchesofthestablestateswithdifferentamplitudeswithincertainfrequencyrange.Theexperimentaldataagreeswellwiththepredictedstablestatesatdifferentfrequencies.Thebluedashlinecorrespondstothesaddlepoint.Itmergeswithastablestateatthebifurcationfrequencies.Outsidethatparameterrange,onlyoneattractorexists.Inthepresenceofsmallnoise,thesystemwoulductuatearoundthestablestatesintheQPphasespace.Moreover,itcanbeproved(refertoAppendix C forthedetail)thatclosetothebifurcationpoint,thedynamicsfurtherreducestoarstorderone-dimensionproblemanalogtothesituationthataoverdampedparticlemovesin1-DpotentialoftheshapeinFig. 1-4 .Weareinterestedinthedistancebetweenthestateandthebifurcationpointinareduced~Pdimension(refertoFig. 1-7 .Thexaxiscorrespondstothe~P).WedeneitasfP.ItsdynamicsisaLangevinequationoftheform 68

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1-4 .SolvingU0=0,weobtainthepositionsofthesaddlepointandtheattractor,fPs,a=(~)1=2.Thequantitativetheoryofnoise-inducedswitchingiscarriedoutinthegeneralizedFokker-Planckapproach.Thedetailscanbefoundinthereference[ 53 ].Iwouldsimplystatetheconclusionshere whereisanumericalconstantand~gisthereducedworkofsinglepulse,/theproductoftheamplitudeofthepulsegandthedurationofthepulsetl.Following[ 53 ],thescalingbehaviorsintheGaussianorPoissoncasecanbeintuitivelyunderstoodinthefollowingqualitativepictures.TheintuitivepictureoftheswitchinginducedbythewhiteGaussiannoiseisthatthestateclimbsupthepotentialUwiththenoisefGcompensatingthedeterministicforceU0.RecallingthecartoonFig. 1-7 (a)mightbehelpful.ThusweneednoiseoutburstwithdurationtherelaxationtimeetrandintensityU0~.Theprobabilityofthenoiserealizationis/exp[Rdtf2G(t)=4~D]where~D/D.TheprobabilityofswitchingWGetrcanbeestimatedbypluggingfGU0intothepreviousequationandintegratingoveretr.Thenwehave 69

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4 ,thepulsegeneratesthetranslationofourstatefP=~g.Giventhat~gissmall,asinglepulsecannottranslatethestateoverthesaddlepointfPsandafterthepulsethestatewouldrelaxbacktotheattractorfPa.Thusonlywhenn0jfPsfPaj=j~gjpulsesoccurwithintherelaxationtimeetr,canthestatemoveoverthesaddlepointfPsandswitchtotheotherstablestate.TheintuitivecartoonFig. 1-7 (b)describesthisprocess.FromPoissonstatistics,wecancalculatetheprobabilityforn0eventsduringetr,asProb(n0,etr)=(etr)n0exp(etr)=n0!whereisthemeanrateforPoissonRFpulses.Thisprobabilityistheprobabilitytoswitchduringtimeetr,whichisWPetrwiththeswitchingrateWPforthePoisson-noisecase.Sinceweassumethetranslationinducedbyonepulseissmall,thenumberofpulsesweneedn01.WiththeaidofStirling'sformula,weestimatetheswitchingbarrier: 70

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Figure4-5. (a)Translationsduetodifferencepulses.Intheabsenceofthepulses,thesystemstaysinthestablestate.Thepulseforallthreecurveshereis200slong.Inblackandmagentacurves,thetranslationistowardsthesamedirectionsincethephasesofthepulsesarethesame.YetthetranslationamplitudescalesastheamplitudeoftheRFsignalofthepulses.Thetranslationamplitudeinthebluecurveisthesameasinthemagentacurvebuttowardstheoppositedirection.(b)IfwekeepthesametheworkoftheRFsignal,i.e.theproductoftheamplitudeoftheRFsignal~gandthelengthofthepulsetl,thetranslationisalmostthesame.Theinitialstatesaredifferentbecauseofanelectriccrosstalkinourdevice. 4 ).Therebyweobtainthecubicnonlinearityparameterandthecriticaldrivingamplitudehc.Thettedcriticaldrivingamplitudeagreesreasonablywellwiththelinearextrapolationofthetteddrivingamplitudeinthelinearcases.Furthermore,usingtheseparameters,wecomparethepredictedresponse 71

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Thedirectionoftranslationschangeswhilewechangethephaseofthemodulationofthepulses.Thepulseischosentobenotlargeenoughtocauseswitching.Thesystemrelaxesbacktothestablestateafterthepulse. formorenonlinearresponseswiththeexperimentaldata.Figure 4-1 displaysgoodagreement.WerstinvestigatetheeffectofasingleRFpulse.Thesystemisfarfrombothbifurcationpoints.AsdescribedinSec. 4.3 ,thesingleshortRFpulsetranslatesthesystembyacertainvalueinthe(Q,P)phasespace.TheQandPquadraturescanbedirectlymeasuredastheXandYoutputofourlock-inamplier.Withthexeddrivingfrequency!Danddrivingamplitudeh,thevalueofthetranslationisproportionaltotheamplitudeoftheRFsignalgandthedurationofthepulsetl.InFig. 4-5 (a),weseethatthetranslationscaleswiththeamplitudeoftheRFsignalandreverseswhenthephaseoftheRFsignalisreversed.Moreover,Figure 4-5 (b)showsthatifwekeepstheproductoftheamplitudeoftheRFsignalgandthedurationofthepulsetlthesame,thetranslationisalmostthesame.Finally,weshowthedirectionsofthetranslationschangewhilewealterthephasedifferencebetweenthemodulatedpulsesandtheperiodic 72

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UnipolarswitchinginducedbyPoissonRFpulses.Thesystemstateisclosetobothbifurcationpointsandcanbemodeledasoverdampedparticlemovementaroundbothstablestates.GivenacertainphaseofPoissonRFpulses,switchingcanonlyhappeninonedirection. drivingforceinFig. 4-6 .Afterthepulse,thesystemevolvesaccordingtotheequationsofmotioninthephasespace 4 .Sincethepulsewechooseisnotlargeenoughtoleadtoswitch,aftertheinitialsharpjump,thesystemrelaxesbacktotheattractor.Nextwestudytheunipolarswitchinginthevicinityofthebifurcationpoint.Wedrivetheoscillatorwiththeamplitudeslightlyhigherthanthecriticaldrivingamplitude.Therebythehysteresisloopopensuptoonly3Hz.Weplaceoursysteminthemiddleofthehysteresisloop.Thesystemstateisclosetotwobifurcationpointsandcanbemodeledasanoverdampedparticleuctuatingaroundbothstablestates.IfweapplyPoissonpulseswiththesamephaseasthedrivingforce,i.e.thephasedifference=0,thesystemcanonlyswitchinonedirection,say,fromthestablestateAtoB(Fig. 4-7 ).Ifweipthephaseto=,theallowedswitchingisintheotherdirection,fromthestablestateBtoA.Afterinducingtheswitching,evenwiththecontinuousapplicationofPoissonpulses,anotherswitchingwouldnothappen. 73

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Figure4-8. (a)Transitionsfromthelow-amplitudestatetothehigh-amplitudestateinthepresenceofPoissonRFpulses.Theoscillatorstaysinthelow-amplitudestateforvarioustimedurations.Afteritswitchestothehigh-amplitudestate,weresettheoscillatortothelow-amplitudestate.Theamplitudeduringresettingisnotshownhere.(b)Histogramfortheresidencetimeinthelow-amplitudestateatacertainnoiseintensityandfrequencydetuning.Thetextractsthemeanresidencetime. AswelearnattheendofSec. 4.3 ,closeenoughtothebifurcationpoint,GaussiannoisecouldactivatetheswitchingmoreeffectivelyevenwhenitissmallerthantheintensityofPoissonnoise.InourexperimentforPoissonRFpulses,thepresenceofGaussiannoiseisinevitablebecauseofnitetemperatureandJohnsonnoisefromresistorsinroomtemperature.ThoughtheinjectedPoissonpulsesismuchlargerthantheGaussiannoise,weemployaproceduretoprovideexperimentalevidencetomakessurethatswitchingsareinfactduetoPoissonRFpulses.Inthefollowingexperiments,wealwayssetthephasedifferenceforthePoissonRFpulsestobe0,i.e.inphasewiththedrivingfrequency.AfterthesystemswitchesfromstateAtostateB,wetunethephasedifferencetobeandverifythatswitchinginthesamedirection(fromAtoB)doesnotoccursinthedurationthatis100timeslonger.Thisprocedureguaranteesthat 74

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Figure4-9. TherelationsbetweentransitionrateandthenoiseintensityaredistinctforGaussianandPoissonnoisecases.(a)WeplotlogWasafunctionofreciprocalfortheintensityofGaussiannoiseD.(b)ForPoissonRFpulses,theintensityofnoiseisproportionaltothemeanrateofpulses.WendthatlogWdependslinearlyonlog1=.InsetplotsthesamedatabutaslogWversus1=. Wemeasuretheswitchingratenearthebifurcationpointatthelowerbifurcationfrequency!1=7133771rad/s.Theoscillatorisdrivenbyasinusoidalforceabout4timesaslargeasthecriticalforceamplitude,sothatalargehysteresisloopisestablished(Fig. 4-1 ).Figure 4-8 (a)displaystypicalswitchingsatthefrequency!1+!closetothebifurcationfrequency!1.Theoscillatorresidesinthelow-amplitudestateforvariousdurationsbeforeswitchingtothehigh-amplitudestate.Aftertheswitching,weturnoffthenoiseandresettheoscillatortothelow-amplitudestatebythefollowingprocedure.Inabsenceofthenoise,westartfromthedrivingfrequency!D>!2outsidethehysteresisloopandsweepthefrequencydownslowlyto!1+!inordertomaintaintheoscillatoratthebranchoftheloweramplitude.Throughoutourexperiment,!issmallcomparedwiththefrequencyrangeofthehysteresisloop.Thenweturnonthenoiseandbegintomonitortheamplitudeoftheoscillator.ThisprocedureisnecessarysinceQforswitchingfromthehigh-amplitudestateismuchlargerandthe 75

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4-8 (b)].TheexponentialrelationbetweenthenumberoftransitionsandresidencetimeinfersthattheswitchingsarePoissonprocess.ItisworthnotingthatthisPoissonstatisticsinthehistogramhasnothingtodowiththenatureofthenoise.Inourexperiment,thehistograminthecaseofswitchingsinducedbyGaussiannoisealsoobeysthePoissonstatistics.TheexponentialttinginFig. 4-8 (b)extractsthemeanresidencetimemandtherebyweobtaintheswitchingrateW1=m.WeexaminethedependenceofQontheintensityofnoiseD.ThemeasurablequantityistheswitchingrateW=Cexp(Q)whereCisaprefactorthatisindependentofnoise.ThusQGisonlydifferentfromlogWbyaconstant.Figure. 4-9 (a)displaysthelinearrelationbetweenQGand1=DGforthecaseofwhiteGaussiannoise,inagreementwithEq. 4a .Incontrast,forPoissonRFpulses,ifwetuneonlythemeanrateofpulsesandxthereducedworkofsinglepulse~g,logWvarieslinearlywithlog1=[Fig. 4-9 (b)].SincethenoiseintensityDP/,itfollowsthatQPisproportionaltothelogarithmofreciprocalforthenoiseintensitylog(1=DP).ThisobservationisconsistentwithEq. 4b andrevealsthequalitativelydifferentmechanismofswitchinginducedbythePoissonpulses.Inordertostudythecriticalscaling,werepeattheaboveprocedurefordifferentdrivingfrequency!D=!1+!.Wevary~toexaminethedependenceon!,since~/!.WeplottheslopesinFigs. 4-10 (a)and 4-11 (a)asafunctionof!onalog-logscale(noticethatforthecaseofPoissonpulses,wekeepthereducedworkofasinglepulse~gxed).AsFigures 4-10 (b)and 4-11 (b)display,thepower-lawttingyieldscriticalexponents1.390.17forwhiteGaussiannoiseand0.620.07forPoissonnoise,bothinagreementwiththetheoreticalpredictionsof3=2(Gaussian)and1=2(Poisson)[Eqs. 4a and 4b ].ItisimportanttopointoutthatinthecaseofPoissonpulses,theslopesinFig. 4-11 arepredictedtobe2~1=2=~gandthelogarithmicfactorof 76

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ScalingforGaussiannoise.(a)WeplotlogWvs1=Dfordifferentfrequencydistancetothebifurcationpoint!andperformlinearttingforeachofthem.Fromlefttoright,!=2=1.5Hz,1Hz,0.7Hz,0.5Hz,and0.3Hz.(b)Logarithmofslopesfrompanel(a)asafunctionoflogarithmof!,yieldingacriticalexponent1.390.16. Figure4-11. Scalingforthefrequencydetuningfromthebifurcationfrequency!.(a)FixingthereducedworkofasinglePoissonpulses~g,WeplotlogWversuslog1=fordifferent!andextracttheslopesforeachofthem.(b)Logarithmofslopesfrompanel(a)asafunctionoflogarithmof!,yieldingacriticalexponent0.620.07.Theerrorbarwasunderestimated. 77

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58 ]andothermicro(nano)-mechanicaloscillators[ 57 60 ].Toourknowledge,ourexperimentprovidestherstevidenceofthecriticalexponentof1=2forthecaseofPoissonnoise.Weexpectthisbehaviortobeagenericpropertyindependentofsystems. Figure4-12. ScalingforthereducedworkofasinglePoissonpulse~g.(a)Fixingthefrequencydetuningfromthebifurcationfrequency!,WeplotlogWversuslog1=fordifferent~gandextracttheslopesforeachofthem.(b)Logarithmofslopesfrompanel(a)asafunctionoflogarithmof1=g,yieldingacriticalexponent1.030.07. Finallyweinvestigatetheeffectsofthereducedworkofasinglepulse~gbymaintainingthefrequencydetuningfromthebifurcationfrequency!.AsFig. 4-12 shows,werepeatthelinearttingofQPversusthelogarithmofreciprocalforthemeanrateofpulses.Theslopesareagainextractedandttedversus1=~gtoapowerlaw.Themeasuredcriticalexponentis1.050.07,ingoodagreementwiththetheoreticalpredictionof1. 78

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32 ].Thestresswouldleadtouctuationsintheresonantfrequencyofthebeam,whichcouldactivateswitchingbetweenstablestatesinthenonlinearregime.Bystudyingtheswitchingswecouldobtaincharacteristicsofthenoise.LikeotherdetectionschemeswediscussinSec. 4.1 ,oursensitivityisalsolimitedbytheinevitablethermalnoise.Weareexploringthepossibilityofdetecting 79

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60 61 ],whichdependsonlyonthenon-Gaussiannoise.Inthefrequencyrangewheretwostablestatescoexist,thekineticphasetransitionoccursatthefrequencywhereQABfromthestateAtoBisunequaltoQBAfromthestateBtoA.SinceinthecaseofwhiteGaussiannoiseQ/1=D,QABandQBAremainsequalfordifferentnoiseintensityD.However,non-Gaussiannoisemightfavoronestate.Thetwoswitchingrateswouldbecomeunequalinthepresenceofnon-Gaussiannoise.Consequently,thefrequencyofkineticphasetransitionbecomesanindicatoronlysensitivetonon-Gaussiannoise,largelyunaffectedbytheGaussianbackground.Wearealsoworkingonimprovingthecouplingbetweenthenon-Gaussiannoiseandourdetectingdevice.Thestrongerthecouplingis,themoreevidenttheeffectofnon-Gaussiannoise.Wearefabricatingatunnelingjunctionnexttothemechanicaloscillator,attemptingtomaximizethecoupling.Moreover,thisschemeispredictedtoenhancethenon-Gaussiannatureofthenoise[ 62 ].Besidesthat,awealthofinterestingphysics,suchasquantumbackaction[ 31 ],canbeexploredonthisdevice. 80

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81

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1 ,theCasimirforcebetweentwoelectricallyneutral,parallelplatesmadeinperfectmetalis 17 ].Atsmallseparations,theattractiveCasimirforcemaybecomelargeenoughtomakethemovablemechanicalcomponentstucktoothercomponents.Thisisaphenomenoncalledstiction,whichisoneofthemajorreasonsforthefailureofaMEMSdevice.Strongerspringsattachedtothemovablecomponentcanbedesignedinordertobalancetheattractiveforcebytherestoringforce,butthesensitivityandfunctionalityaresacriced.Moreover,theCasimirforceincreasessosteeplywhenthedistancereduces.Itmightbedifculttobalance 82

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63 ].InSec. 5.4.1 ,wewillshowaderivationoftheLifshitzformulaatzerotemperatureinawaythatthevacuumenergyismoreexplicit.TheLifshitzformulapredictsthattheCasimirforcebetweenparallelrealmetalplatesissmallerthantheCasimirforcebetweenidealmetalplates.TheCasimirforcebetweenparalleldielectric(forinstance,silicon)platesisagainsmallerthantheonebetweenmetalplates.Furthermore,itwaspointedoutthatifthestructure'sthicknesswascomparabletotheskindepthofthematerial,theCasimirforcewillbefurtherattenuated[ 64 65 ].Therefore,theinitialcalculationbasedonidealmetaloverestimatesthemagnitudeoftheCasimirforce.YetthesharpincreaseofCasimirforcesatnanoscaledistanceremainstrue.ThelimitationtotheminiaturizationimposedbyCasimirforcesstillexistsbutmayhappenatasmallerdistanced.Thevacuumenergyinrealmaterialsdependsstronglyontheopticalpropertiesandtheshapeoftheinteractingstructures.Then,canweengineertheopticalpropertiesandtheshapetotailortheCasimirforce?Theanswerisyes.Theopticalpropertiesinsemiconductorrelyonthecarriers.IthasbeendemonstratedthattheCasimirforcebetweenagoldsphereandasiliconplatedecreaseswhenthecarrierdensityinthesiliconplategoesdown[ 66 ].Inalaterexperimentusingalasertotunethecarrierdensityinsiliconwasalsoclaimedtoberealized[ 67 ].AbreakthroughinrecentyearswasthegenerationofrepulsiveCasimirforcesbyinsertingliquidwiththeappropriatedielectricfunctionintothegap[ 68 ].ThismightturntheCasimirforceintoatooltoghtagainststictioncausedbyotherattractiveforcesinMEMS.SuchrepulsiveCasimirforcescouldpotentiallyenablefriction-freeoperationofmechanicalcomponentsifthey 83

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69 74 ].Experimentsinvolvingnanostructuredsurfaceshavedemonstratedthenon-pairwiseadditivenatureoftheCasimirforce[ 75 77 ].Usingmorecomplexgeometries,theCasimirforcehasbeenpredictedtoberepulsiveinvacuumoverspecicdistanceranges[ 78 79 ].RepulsiveCasimirforcesinavacuumenvironmentwillopenupmanypossibilitiesdiscussedinthelastparagraphandavoidthedisadvantagesbroughtbytheinsertedliquid.Tosummarize,theCasimirforcewilllikelyplayamajorroleinthefutureMEMSdevices,andbythecorrectmodelingandengineering,wecanmakethebestuseoutoftheCasimirforce.EngineeringthegeometriesoropticalpropertiesisapowerfulwaytomanipulateCasimirforces.ItispossibletorelieveoreveneliminatetheharmfuleffectsduetoCasimirforcesbysuchcarefulengineering.FightingstictionisonlypartofabroadclassofpotentialapplicationsoftheCasimirforce.HarnessingtheCasimirforcetoactuateaMEMSdevice[ 17 ]andtunethenonlinearityoftheMEMSdevice[ 80 ]hasbeendemonstrated.Ithasalsobeenproposedtoactuatenanoscalemachinesinanon-contactwaybythelateralCasimirforce[ 81 ].ManychallengesareinthewayofharnessingandengineeringtheCasimirforceonasinglemicromechanicalchip.Surprisingly,althoughithasbeenabout15yearssincetherstmodernmeasurementofCasimirforces,anon-chipdemonstrationoftheCasimirforcestillremainselusive.ItisbecauseallexperimentalobservationsoftheCasimirforcesofarrequirepositioninganexternalobjectclosetotheforcesensor.Bulkymicropositionersandpiezoelectricactuatorsarenecessaryinorder 84

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Schematicoftheexperimentsetup.Themovableelectrodeandcombdrivearemarkedbythebluecolor,whiletheanchoredpartsaremarkedbydarkgrey.Theforcesensingbeam(red)isalsofreetooscillate. 85

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Scanningelectronmicrographofthedevice.Thewhiterectangularboxhighlightstheinteractingobjects,betweenwhichtheCasimirforceismeasured. Figure 5-1 showsasimpliedschematicoftheMEMSdevice.ThescanningelectronmicrographofthedeviceisdisplayedinFig. 5-2 .Thex,yandzaxisesaredenedinFig. 5-2 .ThedetailsofthefabricationprocessisdescribedinSec. 2.1.2 ,andwillnotberepeatedinthischapter.BoththeforcesensingelementandtheactuatoraremadefromthedevicelayerofaSOIwafer,whichissinglecrystallinesilicon.Itensuresthatthethicknessesoftheinteractingstructuresarealmostidentical.Italsoenforcesthattheirpositionsinthezdirectionarethesame,atleastatthebeginningoftheexperiment.Adoublyclampedsiliconbeamactsastheelementforforcesensing,asshowninthetoppartsofFigs. 5-1 and 5-2 .ThewhiterectangularboxshowninFig. 5-2 highlightstheinteractingobjects,betweenwhichtheCasimirforceismeasured.ByapplyingVcombtothexedcombrelativetothemovablecomb,anin-planeforceFcombisgeneratedandtherebythecombdrivemovesintheydirectionuntiltherestoringforcefromthespringsbalancesoutFcomb.ItisimportanttonotethatFcombhasnoeffectontheforcesensingbeam,asdiscussedinSec. 5.6.1 .Itonlyservestosetthedistancedbetweenthemovableelectrodeandtheforcesensingbeam.TheelectrostaticforcebetweenthebeamandelectrodewillbecontrolledbythepotentialdifferencebetweenthebeamandelectrodeVe,whichisindependentofVcomb. 86

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3.2.2 ),wemeasurethemechanicalresonanceofthefundamentalin-planemodeofthebeam.Theresonantfrequencyisabout1.1MHz,comparabletothepredictionfromEq. 2 .TheforcegradientF0(d)isdetectedbyfollowingtheshift!Roftheresonantfrequency!R(Eq. 2 inSec. 2.2 ).Inthisway,theCasimirforcegradientismeasuredandcomparedtothetheoreticalresult.ThediscussionoftheoperationandmeasurementofthedevicewillcontinueinSec. 5.5 .Nextwedescribethesamplecharacterizationandthetheoreticalcalculations. 5.3.1TheConductivityandtheCarrierDensityBoththedevicelayerandthehandlewaferarep-dopedwithboron.SincetheCasimirforcechangeswiththecarrierdensityandconductivityinsilicon,itisnecessarytomeasurethemaccuratelyat4Kenvironment.Theresistivityandcarrierconcentrationofthedevicelayeraremeasuredtobe0.011cmand7.01018cm3respectivelyat4KbythevanderPauwmethod.Forthehandlewafer,theresistivityismeasuredtobe21.5ohmcmatroomtemperatureandbecomeseffectivelyinsulatingat4K. 5-3 )weretakenwithgreatcare.WeemployedaRaith150e-beamwriterforitshighresolutionandusedthecontinuousaveragingmodetotakepicturesinordertoavoidthesystematicerrorfromdrifting.Themovablecombsandtheelectrodeareconnectedtogether,supportedbyfourserpentinespringswiththeirotherendsanchoredtothesubstrate.Eachcomb 87

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Thetop-viewscanningelectronmicrographofpartoftheinteractingobjects:anapproachingelectrodeandthedoublyclampedbeam. ngermeasuresabout8.3m2.0m2.65m.Thexedandmovablesetsofcombngersareinterdigitated,withalateraldistanceofabout1.0mbetweenadjacentelements.Eachserpentinespringistwofoldedwiththeturningat80mfromtheanchor.Thewidthandthicknessofthespringare0.78mand2.65mrespectively.Thecross-sectionalwidthischosentobesmallerthanthethicknesstominimizemotionintheout-of-planedirection.Thedoublyclampedsiliconbeammeasures100mlong,1.42mwide,and2.65mthick.Theelectrodeisabout2.80mwide.Thesewidthswerecalculatedfromthegeometriesmeasuredinthenextsection. 88

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ThecrosssectionalSEMimageoftheinteractingobjects:anapproachingelectrodeandtheforcesensingbeamonatester(sampleB). themwithoutdamagingthedevice.Sincethegeometriesofourinteractingsurfacesrelyontheetchingprocess,theymightbecomplicated.Itisnecessarytocharacterizethem.Tosolvethisproblem,wefabricatedatester(sampleB)fromthesameSOIwafertogetherwithsampleAinwhichtheCasimirforcewasmeasured.SampleBcontainsonlypatternsofabeamandanelectrode,whosenominalwidthsareidenticaltothoseinsampleA.Thelengthofthestructuresisincreasedto1.5or3mm,inordertomakecleavingeasier.Figure 5-4 showsacrosssectionalscanningelectronmicrographofthesidewallproleofsampleB.Wenoticeaslightlyconvexslopenearthetopofthecrosssection.Itisbelievedtoresultfromtheconsumptionoftheperipheryoftheetchmaskduringthedryetchingprocess.Therestofthesidewalldisplaysaconcaveproleatanangleof88tothesubstrate.Themeasuredgeometries,asshowninFig. 5-4 ,areusedinallourcalculations,includingtheniteelementmodeling(FEM)oftheelectrostaticforce,theproximityforceapproximation(PFA)andtheboundaryelementmethod(BEM)calculationoftheCasimirforce. 89

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5-3 ,weperformsoftwareanalysistolocatetheedgesandcalculatethermsroughness[ 6 ]tobe12nm. 5-4 arebynomeanscomplicated.ButitischallengingenoughforthetheoreticalcalculationofCasimirforces.Anefcientmethodforsuchcalculationswasnotdevelopeduntil2009[ 73 ].ThissectionwillstartwiththeLifshitzformulaforinnitedielectricplates[ 63 ]usingthepermittivity(i)alongtheimaginaryfrequencyaxis.Thenwediscussthecommonly-usedproximityforceapproximation(PFA)[ 69 ].NextwegiveabriefdescriptionoftheefcientBEMnumericalcalculation[ 73 ].Lastlyweendthissectionwithadiscussionofthecorrections. 82 ]byBordagetal.Consideravacuumgapofwidthdbetweentwoparalleldielectricsemispaces.1Theinterfacesarelocatedatz=d=2.AmonochromaticEMeldmustsatisfy: 90

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5 hascompleteorthogonalsolutionswiththetransversemagnetic(TM)andtransverseelectric(TE)polarizations.SolvingEq. 5 withthecontinuityboundaryconditionsofclassicalelectrodynamics,wendtheso-calledmode-generatingfunctionsfortheTMandTEwavesrespectively: 2=~A 2iZi1i1!dln!TM,TEk?,n+ZC+!dln!TM,TEk?,n,(5)whereC+representsacounterclockwisepathwithinniteradiusintherighthalfofthecomplexplane.Thesecondtermturnsouttobeinniteandindependentofthevacuumgapd.AswedidinCasimir'sanalysisforparallelidealmetalplates,wedenearegularizedenergyperareawheretheinnityissubtractedout:E(d)=A=E0(d)=AE0(1)=A.The 91

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E 5-4 .TocalculatetheCasimirforcesfornon-planarstructures,awidelyusedtechniqueistheproximityforceapproximation(PFA)[ 69 ].Essentially,itdividesthenon-planarsurfaceintoinnitesimalareaandcalculatestheCasimirpressureoneachpairofthefacingsurfacesusingtheLifshitzformulafortheinniteparallelplates.ThentheCasimirpressureisintegratedoverthesurfacetoobtainthetotalforce.Thisapproximationworkswellwhenthelocalcurvatureissmall.WewillcomparethePFAresulttotheboundaryelementmethod(BEM)numericaloneattheendofthischapter.ThemethodforcomputingCasimirforcesbetweencompactdielectricobjectsinarbitrarygeometrieswasrstdevelopedbyEmigetal.[ 70 ]in2007.ThisbeautifulworkshiftedthefocusfromtheuctuationsoftheeldsinthevacuumgaptotheuctuatingcurrentsJandchargedensitiesintheinteractingobjects,asrstsuggestedby 92

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83 ].TheCasimirenergyisapathintegraloftheenergyofalltheallowedcongurationsofJandwiththeircorrespondingactions.Emigetal.wereabletocalculateCasimirforcesbetweenspheroidalobjects,providingalmostanalyticalresults.Inprinciple,thismethodworksforarbitrarygeometry.Butthedirectapplicationofthismethodtoalittlemorecomplicatedgeometryisprohibitedbytheintimidatingcomputationaleffortsthatarerequired.ThecomputationalapproachhasalsobeenfruitfulintheinvestigationofthegeometryeffectintheCasimirforce[ 74 ].In2009,Reidetal.developedaBEMapproach[ 73 ]thatdiscretizestheinteractingsurfacesintotrianglesandexpressestheCasimir-energypathintegralbyaclassicalBEMinteractionmatrix.ThedetailsoftheBEMnumericalcalculationarebeyondthisthesis.Wewillonlydiscusshowitisappliedtoourgeometry.First,theeffectofthenitelengthofthebeamandelectrodeisomitted.Thebeamsareassumedtobeinnitelylong.Thegeometryofinterestreducestothe2-dimensionalcrosssectionshownbyFig. 5-4 .Thetotalforcegradientisgivenbytheproductofthecalculatedforcegradientperlengthandthereallength100m.Secondly,thefrequecny-dependentpermittivity(i)iscalculatedusingEq.( E )forthedopedsilicon(thebeamandelectrode)andtheintrinsicsilicon(substrate).Thirdly,itisfoundthatadicretizationofthesurfaceswith3200totalconnectingpointsandtruncationofthesubstrateto1mthickareenoughtoreach1%accuracy. 93

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84 ]: 5.3.4 ,currentlywedonothaveagoodwaytodirectlymeasuretheroughnessofthesidewalls.Usingtheedgeroughness12nmasexplainedinSec. 5.3.4 ,weareabletocalculatethecorrection3%atd300nm,whichisasmallcorrection. 85 ].Ford=1m,Te=1148K,whiletheexperimentwasperformedatT=4K.SincewecandetecttheCasimirforceonlyatdistancessmallerthand=1m,thenitetemperatureeffectinourexperimentisnegligible. 5.5.1MeasuringtheMechanicalResponseThedeviceisinvacuumatliquidheliumtemperature,inthepresenceof5Tmagneticeldperpendiculartothesubstrate.Usingthemagnetomotivetransmissionscheme(Sec. 3.2.2 ),weactuateanddetectthemotionofthefundamentalin-planemodeofthebeam,asthismodeistheonemostsensitivetothein-planeforcegradientalongtheydirection.Figures 5-5 (a)and(b)displaythemeasuredvibratingamplitude 94

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(a)Themeasuredoscillatingamplitudeofthedoublyclampedbeamasafunctionofthedrivingfrequency.(b)ThemeasuredXquadrature. ofthebeamandthecorrespondingX(in-phase)quadraturewhenthermsamplitudeoftheexcitingvoltageVac=5.7V.Theabsoluteamplitudeiscalibratedfromthemeasurementofthecriticalresponse,asdescribedattheendofSec. 2.2 [ 38 ].ThequalityfactorQisttedtobe2.09105.Theresonantfrequency!Ris7.26185106rad=s.TheresonanceinFig. 5-5 ismeasuredwhenVeandVcombarebothheldat0V.Itisfoundthatexcepttheshiftoftheresonantfrequency!R,neitherthequalityfactorQnorthevibratingamplitudeAshowsameasurablechangeduringtheoperationofthecombdriveandtheapplicationofdifferentbiasvoltagesVe.ItisinaccordancewiththemodelinSec. 2.2 .Forahigh-Qresonator,theshiftoftheresonantfrequency!RisproportionaltotheforcegradientF0(d) !R=KF0(d),(5)whereKisaproportionalityconstant.Weemploythefollowingmethodtomeasure!R.FixingtheamplitudeandfrequencyoftheexcitingacvoltageVac,giventhatthefrequencyoftheacvoltageiscloseto!R,smallchangesinthemeasuredXquadrature[Fig. 5-5 (b)]oftheoscillationareproportionalto!R.Essentially,wetakeadvantageofthelargeslopeoftheXquadraturetoamplify!R.Oneissueofthismethodisthatthelinearrelation 95

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96

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Measuredfrequencyshift!RasafunctionoftheelectrodevoltageVe.EachparabolarepresentsthemeasurementsmadeatonecertainVcomb,whichcorrespondstod=1.403m,1.065m,865nm,643nm,450nmor349nm,fromtoptobottom. thebeamandthemovableelectrodeisemployedtogenerateanelectrostaticforceFe/(VeV0)2,whereV0istheresidualvoltage.Themeasuredfrequencyshift!RdisplaystheexpectedparabolicdependenceontheelectrodevoltageVe,asFig. 5-6 shows.TheresidualvoltageV0isdeterminedtobethepotentialdifferenceVewhere!Rreachesthemaximum.Figure 5-7 depictsthemeasurementresultforV0.PreviouslytheresidualvoltageV0isbelievedtooriginatefromthedifferentworkfunctionsintheinteractingsurfaces.Inourexperiment,bothinteractingstructuresaremadeofsingle-crystallinesilicononthesamewafer.Thetwosurfacesaresupposedtohavethesameworkfunctionandtheresidualvoltageisexpectedtobezero.YetthemeasuredV0is10to25mV.Wesuspectthenon-zeroresultismainlyduetothesoldermaterialsatdifferenttemperatures.V0shiftsby15mVoverthefullrangeofthedisplacement,comparabletoseveralpreviousconventionalexperiments.Itisunclear 97

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Figure5-7. ThemeasuredresidualvoltageV0asafunctionofthedistance.Noticethattheresidualvoltagedoesnotdriftmuchatdistanced<0.8mandaroundtheinitialdistanced0.Itdriftabout15mVoverthedisplacementrangefromd0toabout300nm. Ourdistancecalibrationisslightlydifferentfromconventionalcalibrationprocedures.Inconventionalones,thechangeofthedistancedisusuallyknown,buttheinitialdistanced0betweentwointeractingsurfacesisanunknownthatneedstobedeterminedfromthetting.Here,ourinitialdistanced0iscarefullymeasuredtobe1.920.015mbythetop-viewSEMpictures(Sec. 5.3.2 )whiledneedstobedetermined.The 98

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2kkdCcomb Figure5-8. Fitthemeasuredelectrostaticforcegradient(circles)atVe=V0+100mVtotheFEMgeneratedtheoreticalvalues(line). ApplyingawiderangeofVcomb,theforcegradientF0earemeasuredwiththreedifferentvoltagesVe=V0+100mV,V0+125mV,orV0+150mV.TheproportionalityconstantsKinEq. 5 andinEq. 5 arethettingparameters.Figure 5-8 displaysatypicaltofthemeasuredfrequencyshiftsatVe=V0+100mVtotheFEMcalculated 99

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Figure5-9. ThemeasuredCasimirforcebetweenthebeamandmovableelectrodeasafunctionofthedistanced.Greyandblackdotscorrespondtotwosetsofindependentmeasurements.TheredlinerepresentstheBEMcalculationusingthemeasuredgeometrywithoutanyttingparameter.ThepurplelinerepresentsthesummationofthetheoreticalBEMcalculationandaresidualelectrostaticforcegradient. Figure 5-10 displaysthemeasuredresult(dots)comparedwiththeBEMtheoreticalresult(redline)thatiscalculatedwithoutanyttingparameter.TheBEMcalculationisbasedonthe2Dcross-sectionalgeometryandtakesintoaccounttheniteconductivity 100

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5.4 ).Clearly,theplotshowsthattheCasimirforcebecomesthedominantinteractionatnanoscaledistances.ThemeasurementresultisingoodaccordancewiththeBEMcalculation.Yettheagreementisnotexact.Onepossiblereasonisuncertaintiesof15nmintheinitialdistanced0andthegeometriesofthebeamandelectrode.Moreover,theroughnesseffectisnotincludedintheBEMcalculationshowninFig. 5-10 .However,asstatedinSec. 5.4.3 ,theroughnesscorrectionisestimatedtobe3%ofthecalculatedCasimirforceattheclosetmeasureddistanceusingtheestimatedrmsroughness12nm.Hereweanalyzeanotherpossiblecontributionfromresidualelectrostaticforcesowingtopotentialpatches.Thepatcheffectreferstothevaryingoructuatingsurfacepotentialsatdifferentlocationsonthesurfaces.FollowingamodelusedinKimetal.[ 86 ],thesmallrandompatchescontributeanelectrostaticforceequivalenttotheelectrostaticforcewithVrmspotentialdifferencebetweentheinteractingsurfaces.Forthepatcheswithsizeslargerthanthedimensionoftheinteractingobjects,thevaryingpotentialonthelargepatchesgeneratesashiftinV0(d)overthedistance.TheforceduetothelargepatchesisequivalenttoanelectrostaticforcewithV0(d)V1potentialdifference,whereV1istheresidualvoltageatinnitedistance.ThetotalforcewillbethesummationoftheCasimirforceandtheresidualelectrostaticforcethatisproportionalto[(V0(d)V1)2+V2rms].InKimetal.'sanalysis[ 86 ],VrmsandV1arebothttingparameters.Inouranalysis,sincetheresidualvoltageremainslargelyconstantclosetotheinitialdistance,wechoosetouseV(d0)asV1andtonlyVrms.TheagreementbetweenthemeasurementandthetheoryisgreatlyimprovedafteraddingtheresidualelectrostaticforcegradientwithV1=11.3mVandVrms=16.7mV.ThepurplelineinFig. 5-10 showsexcellentagreementwiththemeasurements.Nonetheless,weshouldnotjumptotheconclusionthatthediscrepancyisonlyduetothepatcheffect.Otherpossibilitieslikeuncertaintyingeometryandroughnesscannotberuledout. 101

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Figure5-10. TheimageofthecombdriveinsampleD.ComparedtoFig. 5-2 ,thecombdriveinsampleDcontainsnoetchingholes.Thereforebothcombsareanchoredtothesubstrateandtherebyimmobile,regardlessoftheappliedpotentialdifferenceacrossthecombsVcomb. 102

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87 ].Thedifferenceoftheelectricelddistributionaboveandbelowthecombresultsinalevitatingforceinthezdirection.Thislevitatingforceenablesmanyapplications,forexample,theparametricexcitationofatorsionalMEMSdevice[ 88 ].Yetitisaharmfuleffectforourexperiment,asthebeamandelectrodewillnotbealignedinthezdirection.However,thelevitationwasminimizedbytwofactors.First,thesubstrateinourcaseisineffectinsulatingwhenweoperatethedeviceandmakethemeasurementat4K.Unlikeaconductingsubstrate,theelectriceldcanpenetrateintoourdielectricsubstrateandthesymmetrybreakingislesssevere.Secondly,thespringssupportingthemovablestructurearedesignedtohaveahighaspectratio.Thethicknessofthespring2.65misabout3.4timeslargerthanitswidth0.78m,which 103

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Figure5-11. TheratiosoftheCasimirforcegradientsgivenbytheBEMcalculationandthePFAasafunctionofthedistance.Twogeometriesareconsidered:thebeamandelectrodewithsubstrate(red)andwithoutsubstrate(blue). Evenforourcurrentgeometry,thegeometryeffectisinterestingandseveralimportantpredictionshavenotbeentestedinthecurrentexperiment.Figure 5-11 showstheratiooftheBEMcalculatedCasimirforcegradientandthePFAone.Redandblue 105

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89 ]and/orfurtherminiaturizedstructures,wemightbeabletodemonstratethenon-trivialgeometryeffectthatisbeyondthePFA.AnotherinterestingeffectisthattheCasimirforcebetweenthebeamandelectrodeinthehorizontaldirectiondependsonthepresenceofthesubstrate,whichisparalleltothedirectionoftheforce.Thiscanberegardedasamacroscopicthirdbodyeffect.Removingthesubstrate,weincreasetheCasimirforcegradientby14%atd=6m.Duetosimilarlimitations,thiseffectcannotbeobservedincurrentscheme.Butitispossibletoinvestigateitusingadvancedmeasurementtechniquesand/orsmallerinteractingobjectswithnanoscalecrosssectionalgeometries. 106

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1 introducestheeffectsofuctuationsonamicro-ornano-system.WethenintroduceMEMStechnologyandthedoublyclampedbeamthatisinvolvedinbothprojects.Itprovidesuniqueopportunityfoructuationstudies.Thekeyideasofthetwoprojectsarealsobrieyexplained.Chapters 2 and 3 focusontheinstrumentaleffortsandtherelatedtheories.Westartwithadiscussionoffabricationanddesignprinciples,whichisfollowedbythedetailedEuler-Bernoullibeamtheory.Apparatusandthetransductionschemesarealsopresented.Understandingtheelectricalcircuitryisimportantinthedeviceoperationsanddataanalysis.InChapter 4 ,adoublyclampedbeamisdrivenatlargeexcitationssothattwostablestatescoexist.PoissonorGaussiannoiseenablestheoscillatingsystemtoswitchoutofametastablevibrationalstate.WeshowthatPoissonRFpulsesleadtoaqualitativelydifferentswitchingbehaviorfromtheoneinducedbyGaussiannoise.Experimentalresultsareingoodaccordancewiththetheory.InChapter 5 ,wearemotivatedtoengineertheCasimirforcebycomplicatedmicrostructuresandweachieveademonstrationoftheCasimirforcebetweentwoon-chiplithographicallydenedcomponents.Wereplacetheoff-boardbulkypositionerbyamicroscalecombdriveactuatorandintegratethedoublyclampedbeamonthesamechipwiththeactuator.ThemeasuredCasimirforceiscomparedwithatheoretical 107

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Step1ofthefabricationofSiNbeams.Itisacrosssectionalschematic.Thisgureandallthefollowingonesarenottoscale.WebeginwithabaresiliconwaferanddepositalayerofSiN. AlayerofSiNisdepositedbychemicalvapordeposition(CVD),asshowninFig. A-1 .Thenitridelayercanbehighlyorlowstressed,dependingontheCVDprocess.Inthefabricationofsamplesinvolvedinthisthesis,weboughthighlydopedsiliconwafersdepositedwith200nm-thicklayerofhighlystressedstoichiometricSiNfromUniversityWafers.RecentlywearealsoabletodepositSiNonsiliconwafersbyourselves. FigureA-2. Step2ofthefabricationofSiNbeams.Weuselithographytodeneapatternofphotoresistastheetchmask. Thenweperformaphotolithographytodenethedesiredpattern(Fig. A-2 ).TypicallyweemploythedeepUVprojection(stepper)lithographyforitssubmicronresolution(minimum400nm)andmassproduction(20identicalsegmentsexposedonthesamewaferwithin3minutes).ThesysteminHKUSTisanASMLStepper5000 109

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FigureA-3. Step3ofthefabricationofSiNbeams.Thenitridelayerisetchedthroughbyreactiveionetchingandasmallamountofsiliconisalsoetched.Thethicknessofphotoresistisreducedduetotheetching. 110

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A-3 FigureA-4. Step4ofthefabricationofSiNbeams.Thebeamisnowreleasedbytheisotropicetchingofsilicon. Thenweneedtoetchisotropicallythesiliconsubstrateinordertoreleaseourmovablestructure(Fig. A-4 ),whichusuallyisadoublyclampedbeamwithsubmicronwidth.TheSTSICPDRIEetcherisemployedtoperformtheisotropicallysiliconetchingwhilethephotoresistprotectstheSiNpattern.TherecipeisISOEtch.ItusesSF6gastoreactivelyetchthesiliconandnodissipativegas(C4F8)isinvolved.Highpressure,highgasowrateandhighRFgeneratingpowerareutilizedtoenhancetheisotropyoftheetching.OneminuteofISOSietchetchingetchesdownabout1mofsiliconandundercutsabout0.5mintheopenarea.TheundercuttingisclearlyobservableunderopticalmicroscopesasthereleasedSiNmembranebecomespartlytransparentandshowsgrayordarkcolor,distinctfromitsoriginalbluecolorwhenitcoversonthesilicon(thebluecolorisfor200nmthickSiNlm.SiNlmswithotherthicknesseswill 111

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FigureA-5. Step5ofthefabricationofSiNbeams.Thephotoresistlayeriseliminated. Finally,weneedtogetridofthephotoresist(Fig. A-4 ).Thephotoresistisstrippedawaybyoxygenplasmafor30minutesorlongerinBransonICP2000photoresistasher.Thedevice,asuspendedSiNbeam,isreadyforoptotransductionscheme.Yetsincewechoosetoemployelectricaltransductioninourexperiments,anelectricaccesstothesuspendedbeamisrequired. FigureA-6. Step6(thelaststep)ofthefabricationofSiNbeams.Alayerofgoldorotherconductingmaterialisevaporated. 112

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A-6 ).Evaporationishighlydirectionalsoitwillnotshortbetweendifferentleadsandgrounds.Sputteringthinmetallayersalsoworks,buttoothicksputteredmetalmaterialswouldprobablyshorteverythingasitisaundirectionaldeposition.Wealsoneedgoldbondingpadsforwirebondingtogetaccesstotheconductiveleadsonthechip.Alltheelectricalleadsandbondingpadsarepatternedinthephotolithography(step2inFig. A-2 ).Thebondingpadsrequireathickergoldlayer(>100nm).Ifthethicknessofgoldlayeronthebeamisdesignedtobethickerthan100nmtoo,wecanevaporatethemetallayersalltogethertothewholesample.Otherwise,weuseashadowmasktoprotectthebeamsandevaporatethickermetallayersonotherplaces.Thenweremovethemaskandevaporatethewantedthinnermetallayerstothewholesample.Thankstotheundercutting,theelectricleadswillnotbeshortedtothesubstrate.Alltheinvolvedetchingaredry.Thenisheddeviceafterstep(f)isreadytouseafterbeingcleavedintopropersizes.Ifdicingispreferred,itisbettertodosoafterSiNetching(step3),asnopartsweresuspendedthen.Nevertheless,itwasreportedthatsuspendeddoublyclampedbeamswererobusttophotoresistspinning[ 42 ],thusdicingasthenalstepshouldalsowork,thoughitnecessitatescriticalpointdryingatthelaststep. A-7 shows,aSOIwaferconsistsofthreelayers:thedevicelayer(silicon),theburiedsiliconoxidelayerandthehandlelayer(silicon).Ourdevicewillbemadeinthedevicelayer.Thewaferiscleavedintosmallerrectangularfragments.A350nmlayerofresist(950MPMMAA4)isspunat1500rpm,andastandardelectronbeamlithographywith 113

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Step1ofthefabricationofSOIdevices.Fromtoptobottom,theSOIwaferconsistsofthreelayers:thesilicondevicelayer(red),theburiedsiliconoxidelayer(yellow)andthesiliconhandlelayer(gray). FigureA-8. Step2ofthefabricationofSOIdevices.Anetchmaskisdenedbythee-beamlithography. proximitycorrectionisperformedtodenethewholepatternatonetime.Analuminumlmofthickness90nmisevaporatedontothesamplewiththeaidoflift-off,asdisplayedinFig. A-8 .Usingthealuminumlayerasanetchmask,theexposedsiliconisremovedbydeepreactiveionetching(DRIE).Acontinuousetchandpassivaterecipeischosenandcarefullytunedtoproducesmoothsidewallswithnoscallops(Fig. A-9 ).Sincetheactivegas(SF6)doesnotreactwiththesilicondioxide,theetchstopsattheburiedoxidelayer.Afterwards,anoxygenplasmaetch(usually5minsoftherecipeO2CleanbySTSDRIE)removesthehydrocarbongeneratedduringDRIE.Theselectivityoftheetchmaskissogoodthatwecannotmeasurehowmuchitisconsumedduringtheetchprocess.Thechipisthenplacedin49%hydrouoricacid(HF)toremovetheetchmaskandtheexposedburiedoxidetoreleasethestructuresthataredesignedtobemovable.Thedurationoftheisotropicwetetchistimedtoundercutthesiliconinthedevicelayer 114

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Step3ofthefabricationofSOIdevices.Thedevicelayerisetchedthroughbyadirectionaldeepreactiveetchingprocess.Theetchstopsattheburiedoxidelayer. FigureA-10. Step4ofthefabricationofSOIdevices.Atimedwetetchundercutsthedevicelayerandreleasesthestructuresthataredesignedtobemovable. byacontrollabledistance,inadditiontoremovingtheexposedoxidelayer(Fig. A-10 ).Forexample,inChapter 5 ,thedurationwasabout2.5minutesandtheundercuttingwasabout2.7m.Asaresult,structuresmadeinthedevicelayerwithwidthlessthan5maresuspendedafterthewetetch,whileotherlargerpartsarestillanchoredtothesubstratethroughtheremainingoxide.Figure 2-5 showsanexampleofasuspendedspring.Theimageistakenfromatiltedanglesothattheremainingoxide(yellow)canbeclearlyseen.Oneofthespring'sendsisanchoredthroughtheremainingoxidewhiletheserpentinespringisfreetomove. 115

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1 )[ 18 20 ].Onpage78oftheclassictextbookMechanicsbyLandauandLifshitz(andinmanyothertextbooks),thesolutionofadrivendampedharmonicoscillatorisexpressedastherealpartofq(t)=Bexp(i!Dt)whereB=F=m(!20!2D+2i!D).InChapter 4 ,wetransformthemotionintotherotatingframe: whereobviouslyuB=2.Wealsoemployedaslightlydifferentexpressionintherotatingframe: whereQ=ReuandP=Imu.AninterestingobservationhereisthatiftheLock-inamplierrecordthesignalproportationaltotheoscillator'samplitude,theXandYquadratureswouldacturallycorrespondtoQandP,respectively.Itfollowsthatu/XiY.Howeverifwearemeasuringthevelocityoftheoscillator,itrecoversu/X+iY.Noticethatmanyreferencesuseq(t)=Cres(Qcos!Dt+Psin!Dt)insteadof( Ba ).InthesecasesP=Imu. 116

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reducestotworstorderequationsforthenewvariablesP(t)andQ(t): where Cresf(t)exp(i!Dt). Wecanunderstand1asreduceddampingandasreduceddrivingforce.Bysettingf(t)=0andsolvingtheequationsK(KQ,KP)=0,wecancalculatethestablestatesoftheoscillator.Forgiven,theoscillatorhastwostablestateswithdifferentamplitudesandoneunstablestationarystateknownasthesaddlepointintherange(1)B()<<(2)B(),whileoutsidethatrangeonlyonestablestateexists. 27[1+92(132)3=2].(C)At(1)Bor(2)B,thestablestatewithlargerorsmalleramplitudes,respectively,mergeswiththesaddlestate,andtheotherstablestatecontinuestoexist.Thisisthesocalledsaddle-nodebifurcation.WealsosolvethecorrespondingQandPvalues 117

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19 20 ].ByexpandingKQ,PinQ=QQB,P=PPBandthedistancetothebifurcationpoint=B,thedynamicequationsforKQ,PtotheleadingterminQandPare whereaB=(2YB1)andb=p 1-4 .ItisaLangevinequationoftheform 4 118

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90 ].AsAppendix C explains,theswitchingdynamicsofthesystemcrossoversfrom2Dtoeffectively1Dwhenthedrivingfrequencyapproachesbifurcationfrequencies.Itisdifculttodrawadistinctlinetodistinguish1Ddynamicsfrom2D.AnumericalcalculationoftheMPSPisofgreathelpbecauseitvisualizesthemostlikelypathofthesysteminthephasespace.Aquasi-straightpathindicates1Ddynamics,while2DdynamicsshowsupasspiralsintheMPSP. FigureD-1. (a)Themostprobableswitchingpaththatisnumericallycalculatedwithtypicalparameters:!=9.4rads1,gP=3.5scaledunit,and=11.5Hz.(b)Thezoom-inofthepathneartheattractor.Thespiralingbehaviorismanifestincontrasttothequasi-straightlineinpanel(a). ThegeneralizedFokker-Planckequationfortheprobabilitydensityofthesysteminthepresenceofthenoiseis[ 53 90 ] Ca and Cb ,and~garethemeanrateandreducedworkofPoissonpulses.Wedene(q)=exp[s(q)].Makingsanalogoustotheactionofanauxiliarysystem,thegeneralizedmomentumisp=@qs.Intheweaknoisecondition,theswitchingprobabilityissmalland 119

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D ,thestationarysolutionoftheFokker-PlanckequationcanbemappedtotheproblemofHamiltoniandynamicsofanauxiliarysystemwithgeneralizedcoordinatesandgeneralizedmomentumbysolvingH(q,p)=0.TheexpressionforH(q,p)is D ,wecannumericallycalculatetheMPSPinthephasespaceandtheswitchingexponentsQ0P.Figure D-1 showsatypicalMPSPthatstartsfromtheattractor(thebluedot)andendsatthesaddlepoint(thebluestar).Themainpartofthepathisclearlyaquasi-straightlineanditconrmsthattheswitchingdynamicsisinthequasi-1Dregimeforourparameterrange.Azoom-ingure[Fig. D-1 (b)]displaysaverysmallspiralneartheattractorthatisthesignatureof2Ddynamics.Yetowingtoitssmallscale[noticethescaleofFig. D-1 (b)],suchasmallpartoftheMPSPcontributesmarginallytotheswitchingexponent.Theutilizationoftheasymptotictheoryfor1Ddynamicsisthereforejustied.Thisconclusionisreinforcedbythegoodagreementbetweentheexperimentalresult(blackdots)andthecalculationfromtheasymptotictheoryfor1Ddynamics(theblackline),showninFig. D-2 .ThecalculationusingtheasymptotictheoryincludestheprefactorCthatisalsogivenbytheasymptotictheoryinordertodirectlycomparewiththeexperiment.Unfortunately,currentlywearenotabletocalculateCusingthe 120

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Theexperimentalresult(dots)log(W=)agreeswiththeasymptotictheory(theblackline)foratypicalparameterchoice:!=9.4rads1andgP=3.5scaledunit.ThenumericalcalculatedQP(redcircles)areplottedontherightaxis.Therightaxisisshiftedby1butthescaleisthesameastheleftone.Inthisway,wendthatthenumericalresultpredictstheslopeandthedeviationmaybeexplainedbytheunknownC. numericalmethod.Asaresult,wecannotdirectlycomparetheabsolutevalueofnumericallycalculatedQ0PwithlogW=intheothertworesults.WeplotQ0P(Eq. D )inashiftedrightaxiswiththesamescale.ThecomparisonwiththeotherresultsshowsthattheslopeofQ0Pvs.log=isingoodaccordancewiththeresultsfromtheasymptotictheoryandthemeasurement. 121

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91 ]: 5 ,wemeasuretheCasimirforcebetweensiliconsurfaces.Althoughthetabulatedmeasurementsareavailable,theDrudemodelisstilluseful.First,tocalculatelongrange(i.e.dseveralmicrons)Casimirforces,thetabulatedvaluedoesnotcoverenoughlongwavelength.Inourcalculations,(i)

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66 ]andtherelaxationtime=0!p,whereeistheelectroncharge,nisthecarrierdensity,0isthevacuumpermittivity,m=0.26meistheeffectiveelectronmass,andistheresistivity.Hence,knowingtheresistivityandthecarrierdensitynisenoughtogeneratethecorresponding(i)fromthetabulatedcomplexrefractionindexfortheintrinsicsilicon.Althoughwewillnotdiscusstheplasmamodel,itisworthpointingoutthatsetting=0inEqs. E E and E willgeneratethecorrespondingEqs.intheplasmamodel. 123

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JieZouwasbornandraisedinNanping,Fujian,China,whichisconsideredasasmalltowninChinawithitspopulationofabouthalfamillion.In2002,hewent1000milesuptoattendBeijingUniversity,whereheenjoyedafewinterestingprojectsincomputationalphysics.HeearnedachancetobeanexchangestudentandlaterworkedasajuniorresearchassistantfortwomonthsinPhysicsDept.oftheHongKongUniv.ofScienceandTechnology(HKUST).HecametoUniv.ofFloridatopursuitaPh.D.degreeinphysicsin2006.HejoinedDr.HoBunChan'sgroupandworkedondetectingnon-GaussiannoiseandtheCasimirforcebymicromechanicalresonators.WhenDr.ChanmovedhislabtoHKUSTin2010,hecamewithDr.ChanandspentoneyearinHongKong.In2012,hegraduatedfromtheUniversityofFloridawithaPh.D.degreeinphysicsandcontinuedtopursuithisscienticcareer. 128