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Casimir Force on Nanostructured Surfaces

Permanent Link: http://ufdc.ufl.edu/UFE0042576/00001

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Title: Casimir Force on Nanostructured Surfaces Geometry and Finite Conductivity Effects
Physical Description: 1 online resource (127 p.)
Language: english
Creator: BAO,YILIANG
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: CASIMIR -- GEOMETRY -- LIFSHITZ -- MATERIAL -- MEMS -- NANOSCALE -- PAA -- PFA -- QUANTUM -- SCATTERING
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
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theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In this dissertation, we study the geometry and the finite conductivity effects on the Casimir force by measuring the interaction between a gold sphere and a heavily doped silicon plate with nano-scale, rectangular corrugations. The Casimir force is a quantum effect that strongly depends on not only the material properties but also the shape of the boundary of the interacting objects. The majority of past experiments focus on simple geometries such as plate-sphere, two parallel plates or two cylinders, where the interactions are not expected to deviate significantly from the pairwise additive approximation (PAA) and proximity force approximation (PFA). To demonstrate the strong shape dependence of the Casimir force, we use artificial strongly deformed surfaces, which consist of nano-scale, periodic rectangular trenches. We fabricated three sets of samples. One of them is an shallow trench array with a depth of 100 nm and a periodicity of 400 nm. The other two are high aspect ratio trenches with a depth of 1 um and a periodicity of 400 nm and 1 um respectively. A microelectromechanical torsional oscillator was used in our experiments to precisely measure the force. To improve the detection sensitivity, we use a dynamic approach, where the the Casimir force gradient is measured by the shifts in the resonant frequency of the oscillator. At distance between 150 nm and 500 nm, the measured force gradient shows significant deviations from the value expected from the PAA and the PFA, demonstrating that the Casimir force cannot be obtained from pairwise addition of van der Waals forces between particles. The observed deviation has a good agreement with the theoretical calculations based on scattering theory that includes the finite conductivity of the material, demonstrating the strong shape dependence of the Casimir force. Compared to the calculated values for perfectly conducting surfaces, the deviation is ~ 50% smaller, revealing the interplay between the material and the geometry effects.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by YILIANG BAO.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Chan, Ho Bun.

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Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0042576:00001

Permanent Link: http://ufdc.ufl.edu/UFE0042576/00001

Material Information

Title: Casimir Force on Nanostructured Surfaces Geometry and Finite Conductivity Effects
Physical Description: 1 online resource (127 p.)
Language: english
Creator: BAO,YILIANG
Publisher: University of Florida
Place of Publication: Gainesville, Fla.
Publication Date: 2011

Subjects

Subjects / Keywords: CASIMIR -- GEOMETRY -- LIFSHITZ -- MATERIAL -- MEMS -- NANOSCALE -- PAA -- PFA -- QUANTUM -- SCATTERING
Physics -- Dissertations, Academic -- UF
Genre: Physics thesis, Ph.D.
bibliography   ( marcgt )
theses   ( marcgt )
government publication (state, provincial, terriorial, dependent)   ( marcgt )
born-digital   ( sobekcm )
Electronic Thesis or Dissertation

Notes

Abstract: In this dissertation, we study the geometry and the finite conductivity effects on the Casimir force by measuring the interaction between a gold sphere and a heavily doped silicon plate with nano-scale, rectangular corrugations. The Casimir force is a quantum effect that strongly depends on not only the material properties but also the shape of the boundary of the interacting objects. The majority of past experiments focus on simple geometries such as plate-sphere, two parallel plates or two cylinders, where the interactions are not expected to deviate significantly from the pairwise additive approximation (PAA) and proximity force approximation (PFA). To demonstrate the strong shape dependence of the Casimir force, we use artificial strongly deformed surfaces, which consist of nano-scale, periodic rectangular trenches. We fabricated three sets of samples. One of them is an shallow trench array with a depth of 100 nm and a periodicity of 400 nm. The other two are high aspect ratio trenches with a depth of 1 um and a periodicity of 400 nm and 1 um respectively. A microelectromechanical torsional oscillator was used in our experiments to precisely measure the force. To improve the detection sensitivity, we use a dynamic approach, where the the Casimir force gradient is measured by the shifts in the resonant frequency of the oscillator. At distance between 150 nm and 500 nm, the measured force gradient shows significant deviations from the value expected from the PAA and the PFA, demonstrating that the Casimir force cannot be obtained from pairwise addition of van der Waals forces between particles. The observed deviation has a good agreement with the theoretical calculations based on scattering theory that includes the finite conductivity of the material, demonstrating the strong shape dependence of the Casimir force. Compared to the calculated values for perfectly conducting surfaces, the deviation is ~ 50% smaller, revealing the interplay between the material and the geometry effects.
General Note: In the series University of Florida Digital Collections.
General Note: Includes vita.
Bibliography: Includes bibliographical references.
Source of Description: Description based on online resource; title from PDF title page.
Source of Description: This bibliographic record is available under the Creative Commons CC0 public domain dedication. The University of Florida Libraries, as creator of this bibliographic record, has waived all rights to it worldwide under copyright law, including all related and neighboring rights, to the extent allowed by law.
Statement of Responsibility: by YILIANG BAO.
Thesis: Thesis (Ph.D.)--University of Florida, 2011.
Local: Adviser: Chan, Ho Bun.

Record Information

Source Institution: UFRGP
Rights Management: Applicable rights reserved.
Classification: lcc - LD1780 2011
System ID: UFE0042576:00001


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CASIMIRFORCEONNANOSTRUCTUREDSURFACES:GEOMETRYANDFINITE CONDUCTIVITYEFFECTS By YILIANGBAO ADISSERTATIONPRESENTEDTOTHEGRADUATESCHOOL OFTHEUNIVERSITYOFFLORIDAINPARTIALFULFILLMENT OFTHEREQUIREMENTSFORTHEDEGREEOF DOCTOROFPHILOSOPHY UNIVERSITYOFFLORIDA 2011

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c r 2011YiliangBao 2

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Tomyparents,whosacricedtoprovidemewithabetterlife 3

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ACKNOWLEDGMENTS Thisworkwouldnothavebeenpossibleifwerenotforthehelp frommanyUF graduatestudents,facultyandstaff.EspeciallyIwouldlik etoexpressmydeepest gratitudetomyPhDdegreesupervisor,Dr.H.B.Chanwhohasbe eneverlastinghelpful andhasofferedallkindsofassistance,supportandguidanc eonmyresearchaswell asonthewritingofthisthesis.Inthepastseveralyearsheh asspentahugeamountof timewithmeinlabteachingmetheessentialexperimentalsk ills. Iwouldalsoliketoshowmygratitudetomycommitteemembers ,ArthurHebard, AndrewRinzler,SergeiObukhovandHuikaiXie,fortakingtheir precioustimetohelpme fulllmyrequirementsforthePhDdegree. Ialsooweadeepamountofthankstothetechnicalstaff.Thep hysicsmachine shopstaffMarcLink,BillMalphursandEdStorchhaveconstruct edtheprecision partsformyexperiments.JayHortonhelpedmebuildaplatfo rmwhichreducesthe mechanicalnoise,theacousticnoiseandthethermaldrift. Thephysicselectronicshop staffLarryPhelpsandPeteAxsonbuilttheMulti-ChannelDAC, preamplierPCboard, multipliersandsummers.Thenanofabricationtechnicians IvanKravchenko,AlOgden andBillLewisprovidedextensivehelpinmysamplefabricati on. IwouldalsoliketoshowmygratitudetomylabcolleaguesZso ltMarcet,Konstantinos NinosandJieZoufortheirsupportiveassistancesandprodu ctivediscussions. EspeciallyIwouldliketothankCoreyStambaughforputtingup withmeashis ofce-mate,makingtheharddaysbearable. Finally,Iwouldliketoexpressmyloveandgratitudetomyfa milyfortheirsupport andencouragement.Especially,Iowemypassionategratitud eandlovetomyhusband, alsoacademicpartner,YinanYu,forhisheartfulandconsid erateassistancetobothmy lifeandcareer. 4

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TABLEOFCONTENTS page ACKNOWLEDGMENTS .................................. 4 LISTOFTABLES ...................................... 8 LISTOFFIGURES ..................................... 9 ABSTRACT ......................................... 12 CHAPTER 1INTRODUCTION ................................... 14 1.1IntroductiontotheCasimirForce ....................... 15 1.2GeometryandMaterialDependenceoftheCasimirForce ......... 18 1.3CasimirForceMeasurements ......................... 20 1.4ApproximationforNon-planarGeometries .................. 23 1.4.1ProximityForceApproximation .................... 23 1.4.2PairwiseAdditiveApproximation ................... 25 1.5ExperimentalDemonstrationoftheGeometryandMaterial Dependence 26 2SAMPLEDESIGN .................................. 28 2.1Approximation ................................. 28 2.2NonperturbativeApproachtotheCasimirForceinPlate-Tr enchStructure 30 2.3SampleDesign ................................. 32 2.3.1MaterialChosen ............................ 32 2.3.2HighAspectRatioRectangularCorrugations ............ 33 2.3.3ShallowRectangularCorrugations .................. 34 3SAMPLEFABRICATION ............................... 35 3.1Gold-CoatedSpheres ............................. 35 3.1.1BaseSphere .............................. 35 3.1.2SputterDeposition ........................... 36 3.2FabricationofSiliconSample ......................... 39 3.2.1SiliconSamplewithSiliconOxideEtchMask ............ 40 3.2.2DeepReactiveIonEtch ........................ 40 3.2.3ReactiveIonEtch ............................ 42 3.3PreparationofSiliconSurface ......................... 44 4THEEXPERIMENTALSETUP ........................... 46 4.1FabricationofMEMS .............................. 46 4.2MicroelectromechanicalTorsionalOscillator ................. 48 4.3DevicePreparation ............................... 51 4.4ExperimentalSetup .............................. 53 5

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4.5DetectionScheme ............................... 55 5SAMPLECHARACTERIZATION .......................... 61 5.1TopographiesofGoldSpheresandSiliconPlates .............. 61 5.2DimensionofSampleswithHigh-aspectRatioTrenchArrays ....... 63 5.3DimensionofSampleswithShallowTrenchArrays ............. 65 6FORCEMEASUREMENTS ............................. 69 6.1ElectrostaticForce ............................... 69 6.1.1ElectrostaticForcebetweenaSphereandaFlatSurface ...... 69 6.1.2ElectrostaticForcebetweenaSphereandaPlatewithPeri odical Trenches ................................. 70 6.1.2.1Poissonsolverusingmatlabprogram ........... 70 6.1.2.2COMSOLmultiphysics ................... 76 6.2ResidualVoltage ................................ 77 6.3CalibrationofCand z 0 ............................. 80 6.4TheCasimirForceMeasurements ...................... 83 7SPHERE-PLATESTRUCTURE:EXPERIMENTANDTHEORY ......... 86 7.1DescriptionoftheMaterial ........................... 86 7.2FiniteConductivityCorrection ......................... 89 7.3RoughnessCorrection ............................. 91 7.4ThermalCorrection ............................... 92 8DEMONSTRATINGTHEGEOMETRYDEPENDENCEOFTHECASIMIR FORCE ........................................ 94 8.1HighAspectRatioRectangularCorrugations ................ 94 8.1.1PredictionbyPFA ............................ 94 8.1.2DeviationsfromPFA .......................... 96 8.1.3TheoreticalCalculationincludingmaterialpropert ies ........ 98 8.1.3.1ScatteringtheoryapproachtotheCasimirforce ..... 98 8.1.3.2TheoreticalcalculationI ................... 99 8.1.3.3TheoreticalcalculationII .................. 100 8.1.4ComparewithTheoryincludingMaterialProperties ......... 100 8.2ShallowTrenches ................................ 101 8.2.1PFAPrediction ............................. 101 8.2.2ComparewithTheory ......................... 103 8.3DeterminationofErrors ............................ 104 8.3.1DeterminationoftheExperimentalErrors .............. 104 8.3.2DeterminationoftheTheoreticalErrors ................ 105 6

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9SUMMARY ...................................... 108 9.1SummaryoftheChapters ........................... 108 9.2FutureExperiments .............................. 110 9.3Conclusion ................................... 111 APPENDIX AFabricationProcedureforGoldCoatedSpheres ................. 112 BGeneralPrepareProcedureforSiliconSample .................. 113 CEtchingProcedureforSampleswithDeepTrenchArrays ............ 115 DEtchingProcedureforSampleswithShallowTrenchArrays ........... 117 EMatlabCodeUsedinCOMSOL ........................... 118 FMatlabCodesUsedtoCalculatetheCasimirForceUsingLifs hitzFormula ... 121 REFERENCES ....................................... 123 BIOGRAPHICALSKETCH ................................ 127 7

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LISTOFTABLES Table page 5-1Dimensionsofthehighaspectratiotrencharrays ................. 64 5-2Dimensionsoftheshallowtrencharrays ...................... 68 6-1Cand z 0 determinedfromcalibration ........................ 83 A-1Recipeparametersof O 2 plasmaand Ar plasmaforsputteringsystem ..... 112 A-2Recipeparametersofgolddepositionandtitaniumdeposi tion .......... 112 B-1 O 2 etchingrecipeparameterforUnaxisICPetcher ................ 114 C-1 O 2 etchingrecipeparametersforSTSDRIE .................... 115 C-2RecipeparametersforDRIE ............................ 115 D-1RecipeparameterforUnaxisRIEchamberclean ................. 117 D-2RecipeparameterforUnaxisRIE .......................... 117 8

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LISTOFFIGURES Figure page 1-1Boxwithtwosides( x and y )oflength L ,andthethird( z )oflength d ,inthe case d << L ..................................... 17 1-2SchematicoftheexperimentalsetupusingAFM ................. 21 1-3Schematicoftheexperimentalsetupusingmicroelectrom echanicaltorsional oscillator ....................................... 22 1-4ExampleofPFA ................................... 24 1-5Geometryofasphericalsurfaceandacorrugatedplate ............. 27 2-1Geometryofaatsurfaceandthecorrugatedsurface .............. 29 2-2ExampleofPAA ................................... 30 2-3B ¨ uscherandEmig'sprediction ........................... 32 3-1AFMimageofaceramicsphere .......................... 36 3-2AFMimageofa200 mdiameterglasssphere .................. 37 3-3AFMimageofa103 mdiameterglasssphere .................. 38 3-4AFMimageofa103 mdiametergoldcoatedglasssphere ........... 39 3-5SchemeofDRIEprocess .............................. 41 3-6SEMimageofsiliconsampleetchedbyBoschprocess ............. 41 3-7SEMimageofcrosssectionviewofsampleAandB ............... 43 3-8SchematicdiagramofanICP-RIEetchingsystem ................ 44 3-9SEMimageofcrosssectionviewofsampleC .................. 45 4-1FabricationstepsinthePolyMUMPSprocess ................... 47 4-2ExampleofallsevenlayersinthePolyMUMPSprocess ............. 48 4-3SEMimageofamicroelectromechanicaloscillator ................ 49 4-4Capacitancebetweenthemovabletopplateandoneoftheb ottomelectrodes 50 4-5Schematicoftheexperimentalsetup ........................ 53 4-6Realpictureoftheexperimentalsetup ....................... 55 4-7Diagramoftheelectroniccircuitforbasicdetectionsc heme ........... 57 9

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4-8Electricalrepresentationoftheoscillator ...................... 58 4-9Circuitdiagramforthedividers ........................... 58 4-10Diagramoftheelectroniccircuitforthedynamicmeasu rementscheme .... 60 5-1AFMimageofgoldcoatedspheresurface ..................... 61 5-2Histogramofsurfaceroughness .......................... 62 5-3SEMimagesofdeeptrenches ........................... 63 5-4Periodictrencharrayplottedas1-Darrayingrayscale .............. 64 5-5HistogramofSEMtopviewimagefordeeptrenches ............... 65 5-6Topviewcompareofdeeptrenchesandshallowtrenches ............ 66 5-7Crosssectionviewofshallowtrenches ....................... 66 5-8AFMimageofashallowtrencharray ........................ 67 5-9CrosssectionofAFMimageforshallowtrenches ................. 67 6-1Triangularmeshofdeeptrenches ......................... 71 6-2Potentialdistributionfordeeptrencheswithaperiodi cityof400nm ....... 73 6-3Potentialdistributionfordeeptrencheswithaperiodi cityof1 m ........ 73 6-4Numericalcalculatedelectrostaticforcefordeeptren ches ............ 75 6-5ConvergencetestoftheNumericalcalculation .................. 76 6-6Triangularmeshofshallowtrenches ........................ 77 6-7Potentialdistributionforshallowtrencheswithaperi odicityof400nm ..... 78 6-8Residualvoltageonsphere-plate .......................... 80 6-9Distancedependenceoftheresidualvoltage ................... 81 6-10Residualvoltageonsphere-trenches ........................ 81 6-11Electrostaticforcegradientfordeeptrenches ................... 83 6-12Electrostaticforcegradientforshallowtrenches .................. 84 7-1Calculatedpermittivityalongtheimaginaryaxisforgo ld ............. 88 7-2Calculatedpermittivityalongtheimaginaryaxisforsi licon ............ 89 7-3Casimirforcebetweenagoldsphereandasiliconplate ............. 92 10

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8-1MeasuredCasimirforcegradientfordeeptrenches ................ 96 8-2Comparisonofexperimentaldataandtheoreticalcalcul ationforperfectmetal 97 8-3Comparisonofexperimentaldataandtheexactnumericsf orsampleA .... 102 8-4Comparisonofexperimentaldataandtheexactnumericsf orsampleB .... 103 8-5Comparisonofexperimentaldataandtheexactnumericsf orsampleB .... 104 11

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AbstractofDissertationPresentedtotheGraduateSchool oftheUniversityofFloridainPartialFulllmentofthe RequirementsfortheDegreeofDoctorofPhilosophy CASIMIRFORCEONNANOSTRUCTUREDSURFACES:GEOMETRYANDFINITE CONDUCTIVITYEFFECTS By YiliangBao May2011 Chair:H.B.ChanMajor:Physics Inthisdissertation,westudythegeometryandthenitecon ductivityeffectsonthe Casimirforcebymeasuringtheinteractionbetweenagoldsp hereandaheavilydoped siliconplatewithnano-scale,rectangularcorrugations. TheCasimirforceisaquantumeffectthatstronglydependso nnotonlythematerial propertiesbutalsotheshapeoftheboundaryoftheinteract ingobjects.Themajorityof pastexperimentsfocusonsimplegeometriessuchasplate-s phere,twoparallelplates ortwocylinders,wheretheinteractionsarenotexpectedto deviatesignicantlyfrom thepairwiseadditiveapproximation(PAA)andproximityforc eapproximation(PFA). TodemonstratethestrongshapedependenceoftheCasimirfo rce,weusearticial stronglydeformedsurfaces,whichconsistofnano-scale,p eriodicrectangulartrenches. Wefabricatedthreesetsofsamples.Oneofthemisanshallow trencharraywitha depthof100nmandaperiodicityof400nm.Theothertwoarehi ghaspectratio trencheswithadepthof1 mandaperiodicityof400nmand1 mrespectively. Amicroelectromechanicaltorsionaloscillatorwasusedin ourexperimentsto preciselymeasuretheforce.Toimprovethedetectionsensi tivity,weuseadynamic approach,wherethetheCasimirforcegradientismeasuredb ytheshiftsintheresonant frequencyoftheoscillator. Atdistancebetween150nmand500nm,themeasuredforcegradi entshows signicantdeviationsfromthevalueexpectedfromthePAAan dthePFA,demonstrating 12

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thattheCasimirforcecannotbeobtainedfrompairwiseaddi tionofvanderWaals forcesbetweenparticles.Theobserveddeviationhasagood agreementwiththe theoreticalcalculationsbasedonscatteringtheorythati ncludestheniteconductivity ofthematerial,demonstratingthestrongshapedependence oftheCasimirforce. Comparedtothecalculatedvaluesforperfectlyconducting surfaces,thedeviationis 50 % smaller,revealingtheinterplaybetweenthematerialandt hegeometryeffects. 13

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CHAPTER1 INTRODUCTION TheCasimirforce,predictedbyH.G.Casimirin1948,arises fromthechangeofthe zeropointenergyoftheelectromagneticeldinthepresenc eofboundaries.Between twoperfectconductingparallelatplates,theforceisatt ractiveandisgivenby F c = 2 ~ cA 240 z 4 (1–1) where A istheareaoftheplates, c isthespeedoflight, ~ isthereducedPlanck's constantand z istheseparationbetweentheplates.Itisquantumvacuumu ctuations whichcausesinteractionbetweenneutralbodiesinvacuum. TheCasimirforceisoneof fewexampleswherechangesinthezeropointenergyisexperi mentallyobservableand isthusofgreatinterest. TheCasimirforceisrelevanttoadiverserangeofsubjectsf romthecondensed matterphysics[ 1 ],elementaryparticlephysics[ 2 ]togravitationandcosmology[ 3 ].In particular,oneofthemostimportantapplicationsisinnan otechnology[ 4 ].TheCasimir forcebecomesthedominantforcebetweentwoneutralconduc torsatasubmicron scalesinceitincreasesrapidlywithdecreaseofthedistan ce.Forexample,theCasimir effectproducesapressureofapproximately1.3mPabetween twoparallel,perfectly conductingplatesataseparationof1 mandincreasesto 10 5 Pa(1atmosphere)at 10nmseparation.TheCasimirforcehasbeenprimarilyconsi deredasapossiblecause ofstictionbetweenmoveablepartsinmicro-andnano-elect romechanicalsystems (MEMSandNEMS).Onthethehand,theCasimirforcecanalsobeput ingooduse, suchastheactuationofaMEMSorNEMSdevice.Thus,controllin gtheCasimirforce canhavesignicanteffectsonthedesign,fabricationandf unctionofMEMSandNEMS devices. Equation 1–1 isonlyvalidwhenthetwoatsurfacesaremadeofperfectcon ductors. Inreality,theCasimirforcedependsonnotonlythegeometr yoftheinteracting 14

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bodiesbutalsodielectricpropertiesoftheboundarymater ials.Inthisdissertation,we investigatethegeometryandthematerialeffectsoftheCas imirforcebymeasuringthe forcebetweenagoldsphereandasiliconplatewithanano-sc aleperiodicrectangular trencharray.Amicroelectromechanicaltorsionaloscilla torwasusedtodetecttheforce. WeuseadynamicapproachwheretheCasimirforcegradientis measuredfromthe shiftintheresonantfrequencyoftheoscillatortoimprove thedetectionsensitivity. Ourresultsareingoodagreementwiththetheoreticalcalcu lationswhichincludeboth theshapeandtheniteconductivityofthematerial.Wewere abletoexperimentally demonstratenotonlythegeometrydependenceoftheCasimir forcebutalsothe profoundinterplaybetweenthegeometryeffectsandthemat erialproperties. Thestructureofthisdissertationisasfollows.Thesample designisdescribed inChapter 2 .InChapter 3 ,themethodsofthesamplefabricationandthespecial preparationforsiliconsurfacearepresented.InChapter 4 ,themicroelectromechanical deviceusedourexperiments:amicroelectromechanicaltor sionaloscillator,is introduced.Thisisfollowedbythedescriptionoftheexper imentalsetupandthe detectionscheme.InChapter 5 ,thesamplecharacterizationmethodsareprovided.In Chapter 6 ,thedetailedforcemeasurementsarediscussed,including thecalibrationfor themeasurementsystem.InChapters 7 and 8 ,themeasurementresultsarecompared tothetheory.Chapter 9 summarizesthemainbodyofthedissertation. TherestofthischaptergivesabriefintroductiontotheCas imirforceandexplains whytheexperimentalinvestigationofthegeometryandmate rialdependenceofthe Casimirforceisaninterestingtopic. 1.1IntroductiontotheCasimirForce TheCasimirforceispurelyaquantumeffect.Intheclassica lpicture,theelectromagnetic eldcanbeequaltozerointheabsenceofchargesandcurrent swhentemperatureis loweredtoabsolutezero.Thusthereisnoelectromagnetic eldbetweenneutralbodies inthevacuum,yieldingazerointeractingforceofelectrom agneticorigin.However, 15

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thequantumtheoryhastotallychangedthenotionofthevacu um.Accordingtothe energy-timeuncertaintyrelation E t ~ 2 (1–2) particles,orstrictlyspeakingvirtualparticles,withen ergyuctuation E abovethe vacuumlevelmaybecreatedforashortamountoftimeinvacuu m.Thus,thequantum vacuumisnottrulyempty,butinsteadcontainselectromagn eticeldsandvirtual particlesthatpopintoandoutofexistences.Thecreationa ndtheannihilationofvirtual particlesresultinthevacuumuctuation.Inotherwords,e lectromagneticeldshave uctuationsandarenotzeroinvacuum. ThezeropointenergyisacrucialconcepttounderstandtheC asimirforce.Inthe quantumpicture,theenergyofanyelectromagneticmodewit hfrequency isgivenby: E n = ~ ( n + 1 2 ), (1–3) where n isanintegervalueandrepresentsthenumberofphotonsinth emode.Atthe groundstate( n =0 ),eachelectromagneticmodecontributeshalftheenergyof aphoton eventhoughthereisnophotonsinthemode.Thus,thetotalze ropointenergyisgiven by E 0 = 1 2 X j ~ j (1–4) where j labelstheallowedmodes.Infreespace,modeswithallfrequ enciesexist.Inthe presenceofconductingsurfaces,theboundaryconditionsa lterthefrequencyspectrum andthereforealterthezeropointenergydensity.Thechang einthevacuumenergy E 0 = E 0, freespace E 0, surfaces (1–5) leadstotheforceonthesurfaces. Anexample .ThesimplestcaseoftheCasimirforceisthatoftwoparalle lperfectly conductingmirrorsatzerotemperaturewithaseparation d .Weconsideraboxwithtwo 16

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Figure1-1.Aboxwithtwosides( x and y )oflength L ,andthethird( z )oflength d ,inthe case d << L sides( x and y )oflength L ,andthethird( z )oflength d ,inthecase d << L (Figure 1-1 ).Thepresenceofthecavityallowsonlydiscretemodes,wit hadensityofmodes k x = L n x k y = L n y and k z = d n z .Thus,wecanwritethezeropointenergyas E 0 =2 X j 1 2 ~ j = ~ c X n x n y n z r ( n x L ) 2 +( n y L ) 2 +( n z b ) 2 (1–6) wherethefactor2resultsfromthetwopossiblepolarizatio nsper k .Becausemodesin x and y directionsarecontinuous,thesumover n x and n y transformintointegrals.One obtainsthezeropointenergybetweentwomirrors: E 0, surfaces = L 2 ~ c 2 1 X n x =1 Z 1 0 dk x Z 1 0 dk y ( k 2 x + k 2 y + 2 d 2 n 2 ) 1 = 2 (1–7) Infreespace,modesinalldirectionsarecontinuoussowene edtointegrateoverall threedirections: E 0, freespace = L 2 d ~ c 3 Z 1 0 dk x Z 1 0 dk y Z 1 0 dk z ( k 2 x + k 2 y + k 2 z ) 1 = 2 (1–8) BothEq. 1–7 andEq. 1–8 divergeduetocontributionsfromlargemomenta.Inorder tocalculatethechangeofthezeropointenergyinpresenceo fboundaries(Eq. 1–5 ), someregularizationssuchasadampingfunctionofthefrequ encyormodernzeta 17

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functionneedtobeintroduced[ 5 ].UsingtheEuler-Maclaurinsummationformula[ 6 7 ] 1 X n =1 F ( n ) Z 1 0 dkF ( k )= 1 2 F (0) 1 12 F 0 (0)+ 1 720 F 000 (0), (1–9) andthenremovingtheregularization,weobtaintheniteva lue E = ( 2 ~ c 720 d 3 ) L 2 (1–10) Therefore,theCasimirforcecanbewrittenas F c = E 0 ( d )= ( 2 ~ c 240 d 4 ) L 2 (1–11) 1.2GeometryandMaterialDependenceoftheCasimirForce Historically,thepredictionoftheCasimirforceisrelate dtotheinvestigationof thevanderWaals(vdW)force.Infact,thetheoryoftheCasimi rforceandthevdW forcecanbeobtainedinauniedwayinthemicroscopicappro ach.Itiswellknown thatthevdWforceistheinteractionbetweenneutralatomsd uetoquantumvacuum uctuations.Inthepictureofquantumvacuumuctuations, virtualphotonswhichcarry theenergyfromplacetoplace,popinandoutofexistencefro mthevacuumstate. ThelifetimesofthevirtualphotonsaredeterminedbytheHe isenberguncertainty relation(Eq. 1–2 ).Whenthedistance R betweentwoatomsismuchsmallerthanthe characteristicabsorptionwavelength,aphotonemittedby oneatomcanpropagate betweentwoatomsduringitslifetime.Thecorrelatedoscil lationsoftheinstantaneously induceddipolemomentsofthoseatomsgiverisetothenonret ardedvdWforce,which isinverselyproportionalto R 7 .Ifweincreasethedistancebetweentwoatomstobe ontheorderoforlargerthanthecharacteristicabsorption wavelength,therelativistic retardationplaysanimportantrolebecauseofthenitevel ocityoflight.Atsucha separation,thevirtualphotonemittedbyoneatomcannotre achtheotheratomduring itslifetime.However,thequantizedelectromagneticeld atthepointswherethetwo atomsaresituatedarecorrelatedinthevacuumstate.Thus, theinducedatomicdipole 18

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momentsarecorrelatedthroughtheeld,resultinginthere tardedvdWforce.Inthe microscopicview,theretardedvdWforceisalsoreferredto astheCasimirforce. Eventhoughboththeforcesoriginatefromquantumvacuumuc tuations,the CasimirforceisratherdifferentfromthevdWforce.TheCas imirforcebetween extendedbodiescannotbesimplyobtainedfromthepairwise summationoftheretarded vdWforceoftheirconstituents.WhilethevdWforcebetweenm oleculesisalways attractive,theCasimirforcecanevenchangeitssigndepen dingontheshapeofthe boundaries.Forexample,theCasimirenergyforanidealmet alshellhasanopposite signtoparallelplateswhichindicatesthepossibilityofg eneratingrepulsiveCasimir forces[ 8 ].Anotherexampleofthegeometrydependencewasfoundonape rfect conductingrectangularcavitywithdimension a 1 a 2 a 3 ,wherethecalculatedCasimir energycanbeeitherpositiveornegativedependingonthera tioofthesizeofthesides [ 9 10 ].Motivatedbytheseresults,therehavebeenanextensivet heoreticalstudiesof theshapedependenceoftheCasimirforceforperfectconduc tors[ 2 11 12 ]. AnotherimportantcharacteristicoftheCasimirforceisits materialdependence.In Eq. 1–1 ,H.G.Casimirconsideredanidealcongurationwithtwoper fectlyconducting parallelplates.Inthisidealsituation,thesurfacesree ctallfrequenciesperfectly. However,forrealsurfaces,theyreectsomefrequencieswe llwhileothersarereected poorlydependingonthematerialproperties.Inaddition,m etalsurfacesbecome transparentforelectromagneticmodeswithfrequency !>! p ,where p istheplasma frequency.Thus,themodeswithfrequency !>! p arenotsubjecttoanyboundary conditions.Inotherwords,theimperfectreectionofther ealsurfacesyieldsadecrease ofthedifferenceintotalzeropointenergybetweeninsidea ndoutsidetheboundaries. Consequently,thestrengthoftheCasimirforceissmallert hanthepredictionforperfect metals.Thefrequencydependenceofthematerialisessenti alforcalculatingtheactual Casimirforce. 19

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Lifshitzetal.extendedtheCasimirformulatorealmateria lsinplanargeometry basedontheuctuation-dissipationtheorem[ 13 14 ].TheLifshitztheoryrelieson theknowledgeofthedielectricpermittivityofboundaryma terialsalongtheimaginary frequencyaxis.Fortwoparallelplateswithdielectricfun ction 1 and 2 immersedina mediawithdielectricfunction 3 ,theCasimirenergyperunitareabasedontheLifshitz theorycanbegivenby E c = ~ 4 2 c 2 Z 1 1 pdp Z 1 0 2 3 [ln(1 (1)31 (1)32 e x )+ln(1 (2)31 (2)32 e x )] d (1–12) where (1)3 k = s k 3 s 3 k s k 3 + s 3 k (2)3 k = s k s 3 s k + s 3 x = 2 d p 3 p c s k = r p 2 1+ k 3 Withtheknowledgeofthefrequency-dependentdielectricsu sceptibilityofthematerial, Lifshitztheorycanbeappliedtoanymaterialbodybetweenp lanarsurfaces.Ithasbeen generallyacceptedandbeensupportedbyrecentprecisemea surements. 1.3CasimirForceMeasurements Duetothelimitationofmeasurementtechniques,themeasur ementoftheCasimir forcehaslaggedbehindthetheoryfordecadessincetherst predictionbyH.B.G. Casimir.TherstattempttoobservetheCasimirforcebySpar naayin1958[ 15 ]was notconclusivedueto100 % uncertaintyinthemeasurements.Aftermanyyearsofpure theoreticalresearch,thephenomenonhasblossomedinthee xperimentaleldfromthe late90 0 s.Inaddition,anewgenerationofmoderntechniquesallowe dtheCasimirforce tobemeasuredatshortseparationdistancewithanaccuracy ofafewpercent.The effectsoftheCasimirforceonnanomachineryandthepotent ialapplicationsreignited theinterestofexperimentalscientists. ThesituationoriginallydiscussedbyCasimirinvolvedtwo parallelconducting surfaces.However,thetwo-platecongurationisseldomus edinexperiments.Theonly 20

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Figure1-2.Aschematicdiagram(nottoscale)oftheexperim entalsetupusinganAFM. Ametalizedsphereismountedonthetipofthecantilever.Al aserbeamis reectedonthetopofthecantileverintophotodiodes. recentexperimentthatmeasuredtheCasimirforceinthisco ngurationwascarriedout byG.Bressi etal [ 16 ]whichagreedtowithin15 % tothetheoreticalprediction.That isbecausetheCasimirforcemeasurementinsuchacongurat ionrequiresaccurate alignmenttoensurethetwosurfacesareparallel,whichisd ifcultinpractice.For experimentalconvenience,thesimplicationofthealignm entisachievedbyreplacing oneorbothoftheplatesbyacurvedsurfacesuchasalens,cyl inderorsphere.The majorityofprecisemodernCasimirforcemeasurementsarep erformedinasphere-plate geometry. TwomainmoderntechniqueshavebeenwidelyusedintheCasim irforce measurements.Oneofthemisatomicforcemicroscopy(AFM).As showninFig. 1-2 ,ametalizedsphereismountedonthetipoftheAFMcantilever [ 17 ].Theforce betweenthesphereandtheplateisdetectedbasedonmeasuri ngthedeectionof thecantilever.Thedeectionismeasuredbymonitoringthe intensityofthelaser beamusingphotodiodesafteritisreectedoffthetopofthe cantilever.Another techniqueinvolvesusingamicroelectromechanicaltorsio naloscillatortodetectthe interactioncapacitively(Fig. 1-3 ).ThistechniquewasrstintroducedbyH.B.Chanet al.[ 18 ].Then,F.Chenetal.adaptedalaserinterferometertothes etuptocontrolthe separation,achievingthemostprecisemeasurementssofar [ 19 ]. 21

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Figure1-3.Aschematicdiagram(nottoscale)oftheexperim entalsetupusingan microelectromechanicaltorsionaloscillator. Usingthemoderntechniques,thematerialdependenceofthe Casimirforcehas beenthoroughlyinvestigatedinasphere-plategeometry.St artingfrom1998,the rstseriesofprecisemeasurementsoftheCasimirforcewer eperformedbetweena metalizedsphereandplate.Differentmetalsurfaceshaveb eentested,suchasAl-Al [ 17 ],Au-Au[ 18 ]andfordissimilarmetalsAu-Cu[ 19 ].Thoseexperimentsdemonstrated theroleoftheniteconductivitycorrectionstotheCasimi rforce.Forexample,the Casimirforcebetweenagoldsphereandagoldplateatasepar ationof150nmcan bemorethan30 % smallerthantheforceforperfectmetal.Inaddition,theef fectofthe surfaceroughnessontheCasimirforcewasobservedexperim entally.InRef.[ 17 ],the roughnesscancontributeupto20 % totheCasimirforceatcloseseparations.Thenext seriesofmeasurementswasaimedattheinvestigationofthe Casimirforcebetween ametalizedsphereandasiliconplate.Intherststageofth eresearch,F.Chenetal. demonstratedthatthemagnitudeoftheCasimirforcecanbem odiedbyusingsilicon sampleswithdifferentcarrierdensities[ 20 21 ].Then,theymeasuredtheCasimirforce betweenagoldcoatedsphereandasiliconmembrane.Themodi cationoftheCasimir forcewasachievedbychangingthecarrierdensityofthesil iconmembraneusingalight pulse[ 22 ].Sincesiliconisthebasicmaterialinnanofabrication,co ntrollingtheCasimir forceusingsiliconsurfacescanleadtomanyapplicationsi nnanotechnology.Another 22

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importantseriesofmeasurementswasperformedinuid.Ina nearlyattempt,Munday etal.measuredtheCasimirforcebetweenagoldcoatedspher eandplateinethanol, yieldinganattractiveforcewhichisapproximately80 % smallerthantheforceforideal metalsinvacuum[ 23 ].Bychoosingthemediacarefully,Mundayetal.succeededto obtaintherepulsiveCasimirforcebetweenagoldsphereand asilicaplateimmersedin bromobenzene[ 24 ],whichistherstexperimentalobservationoftherepulsi veCasimir force. 1.4ApproximationforNon-planarGeometries Foralongtime,therehasbeenlackofexactcalculationresu ltsoftheCasimir forceincludingmaterialdependenceforgeometriesothert hantwoparallelplates. Proximityforceapproximation(PFA)andpairwiseadditiveapp roximation(PAA)are twomethodsthatarewidelyusedtoestimatetheCasimirforc ebetweenbodiesthat deviatesonlyslightlyfromplanargeometries.Bothmethods neglectthenonadditivityof theCasimirforce.Forrelativelysmoothobjects,suchassp here-plateandcylinder-plate atseparationsmuchsmallerthantheradius,theseapproxim ationsarehighlyaccurate. Fornontrivialgeometries,wherethenonadditivityoftheC asimirforceplaysanimportant role,cleardeviationscanbeobservedfromPFAandPAA.Theexpe rimentalverication ofthenonadditivityisacentralthemeofthisthesis.1.4.1ProximityForceApproximation Theproximityforceapproximationisaneffectiveapproxim ationmethodtocalculate theCasimirforcebetweenbodieswithsmoothgeometries.It hasbeenwidelyaccepted forcomparingofthemeasurementresultswiththetheoryfor thesphere-plategeometry. Generallyspeaking,theapproachofPFAistoassumethatthec urvedsurfaceofthe testbodiesismadeofinnitesimalplanarelements.Ifweas sumetwotestbodies V 1 and V 2 ,atanarbitrarypoint ( x y ) ,thecurvedsurfaceelementson V 1 and V 2 around z 1 ( x y ) and z 2 ( x y ) arereplacedbyparallelplanarelements dxdy asshowninFig. 1-4 Indoingso,wecanreplacetheunknownpressurebetweenthec urvedsurfaceelements 23

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Figure1-4.AnexamplethatillustratethePFA.Thecurvedsurfa ceelementsonthetest bodiesarereplacedbyparallelplanarelements dxdy z istheclosest distancebetweentwobodies. w istheseparationbetweensmallelements. withaknownpressure P ( x y ) betweentwoparallelplanarelements.Consequently theinteractionbetweentwobodiescanberepresentedasasu mmationoftheplanar interactionsbetweenparallelsurfaceelements[ 5 25 ] F PFA ( z )= Z s dsP ( w ), (1–13) where P istheknownpressurebetweentwoparallelplatesatasepara tionof w = z 1 ( x y ) z 2 ( x y )( z 1 > z 2 ). (1–14) Thevariable z istheclosestdistancebetweenthetwobodies.Inotherword s, z isthe smallest w Inthecongurationofasphere(witharadiusof R )andalargeplate,theCasimir forceforseparation z << R isgivenby F PFA ( z )=2 RE ( z ), (1–15) where E istheCasimirenergyperunitareabetweentwoparallelplat es. 24

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1.4.2PairwiseAdditiveApproximation pairwiseadditiveapproximation(PAA)isanothercommonlyus edapproximation. ThebasicconceptofPAAisthattheinteractionbetweentwobo diescanbeobtained fromthepairwisesummationofatwobodypotentialbetweena tomsormolecules ( U ( r ) ).UnderPAA,theCasimirenergybetweentwobodiesataseparat ion z canbe givenbythesummationoftheretardedvdWpotentialsoveral latomsoftheinteracting bodies U PAA ( z )= Z V 1 d 3 r 1 Z V 2 d 3 r 2 U ( r ), (1–16) where r = j r 1 r 2 j .However,theadditiveresultoverestimatedtheCasimiren ergysince itdoesnottakeintoaccountthenon-additiveeffects,whic hresultsfromthefactthatthe interactionbetweentwoparticlesareaffectedbytheprese nceofthethirdone.Inorder toapproximatelyaccountforthenon-additivity,anormali zationprocedureisnormally used[ 2 ].Thus, U ( r ) canbechosentousea“renormalized”retardedvanderWaals potential[ 5 26 ] U ( r )= ~ c 24 r 7 (1–17) suchthatinthecongurationoftwoperfectlyconductingpa rallelplatestheexact Casimirenergyisrecovered. ApplyingthenormalizedPAAtothecongurationofasphereand alargeplateat separations z << R ,weobtain U PAA ( z )= 3 ~ cR 720 z 2 (1–18) Thus,theforcecanbegivenby F PAA ( z )= 3 ~ cR 360 z 3 (1–19) Recenttheoreticalapproachesprovideexactanalytical[ 27 – 29 ]andnumerical[ 30 ] resultsfortheCasimirforceinsphere-plateortwocylinde rsgeometries,which demonstratethatthedeviationsoftheexactforcefromPFAar elessthan z = R .Thus,the 25

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PFAprovidesgoodaccuracywhentheseparation z ismuchsmallerthantheradiusof thesphere R 1.5ExperimentalDemonstrationoftheGeometryandMaterial Dependence AllthepreciseexperimentsoftheCasimirforcementionedab oveusedasimple geometry,suchasasphereandaplate,twoparallelplatesan dtwocylinders,to investigatethematerialdependence.Forthesesmoothgeom etriestheCasimirforce isnotexpectedtoshowsignicantdeviationsfromPFAandPAA.I notherwords,the geometrydependencecannotberevealedbytheseexperiment s. Therstattempttodemonstratethestronggeometrydepende nceoftheCasimir forcewasperformedbyRoyandMonideen,wheretheforcewasm easuredbetweena largesphereandaplatewithsmallsinusoidalcorrugations attheseparationbetween 0.1and0.9 m.AsshowninFig. 1-5 ,theamplitudeofthecorrugation a 60 nmand theperiod 1.1 m.TheobservedforceshowscleardeviationsfromPFA/PAA. However,thetheoreticalcalculationinthisgeometrywith outtheassumptionofpairwise additivityindicatesthattheexpecteddeviationsarenots ignicantenoughtoaccount fortheobserveddeviation.Instead,thelateralmovemento fthetwosurfacesmay beabletoaccountforthedeviations.Thetheoreticalpredi ctionssuggestthatthe shapedependenceoftheCasimirforcecanberevealedfromco rrugationswithsmaller periods. Duetothedifcultyinthesamplefabrication,nootherexpe rimentshavebeen reportedonthedemonstrationofthegeometrydependenceof theCasimirforcenormal tothesurface.Althoughrelevanttheoreticalresearchhass tudiedthegeometryeffect comprehensively,thereisnoconclusiveexperimentalveri cationofthegeometry dependenceoftheCasimirforce.Thus,investigatingthest rongdependenceofthe Casimirforceforrealmaterialexperimentallybecomesana ttractivetopic.Itwillnotonly demonstratethevalidityofthetheorybycomparingtheexpe rimentalresultswiththe 26

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Figure1-5.GeometryusedforcalculatingtheCasimirenerg yofasphericalsurfaceand acorrugatedplateatseparation z theoreticalpredictions,butcanalsobeappliedtotheeld ofnanotechnology,suchas nanomechanicaldevices[ 4 31 ],noncontactfriction[ 32 ]andcarbonnanotubes[ 33 ]. 27

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CHAPTER2 SAMPLEDESIGN TodemonstratethestronggeometrydependenceoftheCasimi rforce,acrucial stepintheexperimentalsetupistochooseapropergeometry conguration.Inthe wellknownconguration,twoparallelplatesorasphereand aplate,theCasimir forceisattractivewhichhasbeenobservedexperimentally .Theoreticalcalculations indicatethattheCasimirenergyforgeometriessuchasasph ericalshellcouldhave oppositesigncomparedtoparallelplatespossiblyleading torepulsiveCasimirforces. Althoughitissubstantiallyinterestingtodemonstratethe repulsiveCasimirforcefor suchgeometries,inpractice,relevantmeasurementsareex perimentallydifcultforsuch closedstructures.Ontheotherhand,theCasimirforcemeas urementsbetweentwo parallelplatesorasphereandaplatehavealreadybeenwell developed.Oneofthe mostpromisingapproachistoreplacetheatplatewithanar ticiallydeformedsurface intheparallelplateorsphere-plategeometry.Thus,thest rongshapedependenceof theCasimirforcecanbedemonstratedbyrevealingastrongd eviationfromtheusual pairwiseadditiveapproximation(PAA)orproximityforceapp roximation(PFA). FollowingthetheoreticalpredictionsfromB ¨ uscherandEmig[ 11 34 ],wechoose oneofthesurfacestobeaplatewithnano-scaleperiodicrec tangulartrenches.The othersurfaceischosentobeasphericalsurfaceforexperim entalconvenience,as discussedinSec. 1.3 .Inthischapter,IwillrstintroducetheexpectedCasimir force underPFAandPAAforageometryofrectangulartrencharray.Th en,Iwilldiscussthe theoreticalpredictionanddescribethesamplesthatwecho osetomeasure. 2.1Approximation Weconsidertheinteractionbetweenaperiodicrectangular trencharraywitha50 % dutycycleandaatparallelplate,asshowninFig. 2-1 AsdiscussedinSec. 1.4.1 ,underPFA,theinteractionbetweentwobodiescan berepresentedasasummationoftheplanarinteractionsbet weenparallelsurface 28

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Figure2-1.Geometryconsistingofaatplateandaplatewit hrectangulartrencharray. elements.Thus,forthetrencharraywitha50 % dutycycle,halfoftheatplateinteracts withthetopofthetrenchesatdistance z andtheotherhalfinteractswiththebottomof thetrenchesatdistance z + t .ThetotalforceunderPFAcanbegivenby F PFA ( z )= 1 2 F rat ( z )+ 1 2 F rat ( z + t ). (2–1) UnderPAA,theforce F PAA ona50 % dutycyclerectangularcorrugationcanbe consideredasthecontributionoftowparts:theforce F array ( z ) onthearrayandthe force F bottom onthetrenchbottom(Fig. 2-2 a). F PAA canbeobtainedinthefollowing procedure.Weconsidertheforcesbetweenaatplateandtwo seperatetrencharrays, asshowninFig. 2-2 b.Allparametersandmaterialpropertiesforthetwotrencha rrays areidenticalexceptthatarrayIIislaterallyshiftedfrom arrayIbyhalftheperiod.Thus, theforce F arrayI onarrayIand F arrayII onarrayIIarethesame.IfarrayI,arrayIIanda bottomplatearesuperimposedoneachother,asolidatsurf aceisrecoved.InthePAA picture, F rat ( z )= F arrayI ( z )+ F arrayII ( z )+ F bottom (2–2) 29

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Figure2-2.(a)Thetrenchstructurecanbeconsideredasthe superimpositionofanarray andabottomplate.(b)IfarrayI,arrayIIandabottomplatea re superimposedoneachother,asolidatsurfaceisrecoved. where F arrayI ( z )= F arrayII ( z )= F array ( z ) .Therefore,underPAA,theforceonthe corrugatedsurfacecanbegivenby F PAA ( z )= 1 2 F rat ( z )+ 1 2 F rat ( z + t ). (2–3) Generally,PFAandPAAdonotproducethesameresultsinthecal culationofthe Casimirforce.However,fortheplate-trenchsituationino urexperiments,thePAA(Eq. 2–1 )andthePFA(Eq. 2–3 )predictthesameforceregardlessoftheperiodicity and materialofthetrencharrays. 2.2NonperturbativeApproachtotheCasimirForceinPlate-T renchStructure AfterRoyandMohidden'srstattempttodemonstratethegeom etrydependenceof theCasimirforce[ 35 ],B ¨ uscherandEmigperformedaseriesoftheoreticalcalculatio nto probethestronggeometrydependenceoftheCasimirforce[ 11 12 34 ].Oursamples weredesignedbasedontheircalculationoftheCasimirinte ractionbetweenaatplate 30

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andaplatewitharectangularcorrugationforperfectmetal .Inthissection,Iwillexplain alimitingcaseofthistheoreticalprediction. B ¨ uscherandEmigcalculatetheCasimirforceusinganonpertur bativeapproach basedonpathintegralquantizationoftheelectromagnetic eld.Withthenumerical computationofthisapproach,theCasimirforcecanbecalcu latedpreciselywithoutany approximation. Letusrstconsiderperfectlyconductingtrencharraywith small .Asweknown, thequantumuctuationsoftheelectromagneticmodeswithc haracteristicwavelength comparabletotheseparation z givethemaincontributiontotheCasimirforce.With decreasing ,theeldthatpenetratesintothetrencheswillbeaffected .Intheextreme situation 0 << z ,theeldcannolongergetintothenarrowtrenches.Theforc e betweentheatsurfaceandthetrencharrayisequaltothefo rcebetweentwoparallel platesataseparation z .Theforceperunitareacanbegivenby F 0 = A = F rat = A = 2 480 1 z 4 (2–4) Next,welookattheoppositelimitationwhere isverylarge.Incaseof >> z ,the diffractionofthedominantmodesfromthetrenchescanbene glected.Thusinthelimit !1 ,theforcecanbegivenbyPFA F 1 = A = F PFA = A = 2 480 1 2 ( 1 z 4 + 1 ( z + t ) 4 ). (2–5) InFig. 2-3 ,theratiooftheforcecalculatedusingnonperturbativeap proachtothe forceexpectedfromPFAasafunctionof z = t isplotted.NotethatsincethePFAand thePAApredictthesameforcefortheplate-trenchgeometrie s,Ionlyusethenotionof PFAfortherestofthedissertation.Theforce F 0 providesanupperboundoftheCasimir forceinplate-trenchstructurewhile F 1 providesalowerbound(shownbythesolidline andthedashlineinFig. 2-3 respectively).Ataxed z = t ,theforceconvergestothe 31

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1 2 3 4 5 6 7 0.9 1 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 z / tF trench / F PFA l / t = 0.05 l / t = 2.5 l / t = 5 l / t = 15 l / t = 150 Figure2-3.TheCasimirforcecalculatedbyB ¨ uscherandEmigbetweenageometryas showninFig. 2-1 forperfectmetal.Theforceconvergestotheupperbound forsmall = t .Forlarge = t thelowerboundisapproached. lowerboundforlarge = t ,whereastheupperboundisapproachedforsmall = t which stronglydeviatesfromthePFA. 2.3SampleDesign Theplateswithperiodicrectangulartrenchesaredesigned basedonthefollowing criteria:(1)achievingstrongdeviationfromPFAthatcanbe detectedbytheexperimental setupsand(2)feasibilitytobefabricatedbasedonavailab lemicrofabricationtechniques. 2.3.1MaterialChosen Atthebeginningofmydoctoralprogram,theonlyavailableth eoreticalcalculation ontrenchstructureswasforperfectmetals.Therstchoice ofthesamplematerial wasametalsuchasgold.However,thereisnoeasymethodtofa bricatewelldened 32

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nano-scalehighaspectratiotrenchstructuresinmetal.On theotherhand,microfabrication techniqueshavebeenwelldevelopedonsiliconstructures. ElectronBeamLithography ordeepultra-violetstepperlithographyallowsustogener atepatternswithresolutions betterthan200nm.Reactiveionetching,plasmadryetching ,providesanisotropic siliconetch.Afterdecidingthatcreatingwell-denedcorr ugatedstructuresisexperimentally feasible,westartedcreatingourstructuresusingheavily dopedsiliconsubstrate. Inaddition,siliconisthebasicfabricationmaterialforn anotechnology.Itcanlead manyapplicationstocontroltheCasimirforceusingsilico nsurfacethroughtheinterplay ofthegeometryandniteconductivityeffects.2.3.2HighAspectRatioRectangularCorrugations TherstsetofsamplesthatwedesignedfortheCasimirforce measurements aredeep,rectangulartrencheswithadepthof1 m.Twoperiodicity =1 mand =400 nmarechosen,whichleadto = t =1 and = t =0.4 .Accordingtotheoretical prediction,strongdeviationfromPFAcanbeobservedforthe sehighaspectratio trenches. InthePFAview,thetotalinteractionbetweenthetrenchstru ctureandtheparallel platecanbegivenbyEq. 2–4 ,whichisasumoftwocontributions:(1)theinteraction betweenhalfoftheatsurfaceandthetopsurfaceofthetren charrayseparatedby distance z and(2)theinteractionbetweenhalfoftheatsurfaceandth ebottomofthe trencharrayatdistance z + t .Fordeeptrenches,thesecondpartisnegligiblesincethe Casimirforceatthisseparation( z + t > 1 m)istoosmalltobedetected.Therefore,the forceondeeptrenchesunderPFAishalfoftheforcebetweentw oparallelatsurfaces atseparation zF PFA = 1 2 F rat ( z ) .Consequently,thedistancedependenceoftheforce underthePFAisthesameasaatsurface. 33

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2.3.3ShallowRectangularCorrugations AfterdemonstratingthestrongdeviationfromPFAusingthehi ghaspectratio trenchesdescribedinSec. 8.1 ,anothersetofsamplesweredesignedtobeshallow trencheswithadepthof100nmandaperiodof400nm. Thedimensionsleadto = t =4 ,whichwillgiverisetoaweakerdeviationfrom PFAthanthepreviousdeeptrenches.However,sincethedepth ofthetrenchesis comparabletotheseparationbetweenthesurfaces,boththe topandbottomsurfacesof thecorrugationscontributetotheforceunderPFA. F PFA ( z )= 1 2 F rat ( z )+ 1 2 F rat ( z + t ) yieldsadistancedependencethatisdistinctfromaatsurf ace.Thus,thisstructure providesadditionalevidenceforthegeometrydependenceo ftheCasimirforce. 34

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CHAPTER3 SAMPLEFABRICATION Inthischapterthesamplesusedintheexperimentsandthefa bricationprocedures aredescribed.Oneofthemeasurementsurfaceischosentobe asphericalgold surface.Theothersurfaceisasiliconplatewithrectangul artrencharraysoraat surface.Threekindsofsiliconsampleswithcorrugatedsur facewerefabricated.Twoof them(sampleAwithaperiodicityof400nmandsampleBwithap eriodicityof1 m)are deeptrencheswithadepthofaround1 mandtheother(sampleCwithaperiodicity of400nm)isashallowtrencharraywithadepthofaround100n m.SamplesAandB arefromthesamewaferwhilesampleCisfromanotherwafer.I nadditiontothesilicon sampleswithrectangularcorrugations,twoothersamplesw ithaatsurfacearealso fabricated.OneisfromthesamewaferassampleA/Bandtheoth erisfromthesame waferassampleC. 3.1Gold-CoatedSpheres Inourexperiments,thesphericalgoldsurfaceismadefroma glassspherecoated witha5nmlayeroftitaniumfollowedbya400nmlayerofgoldu singsputtering deposition.Thegoalofthefabricationistwo-fold:(1)mai ntainingasmoothgoldsurface toensurethattheseparationbetweentwomeasuredsurfaces canbereducedtobelow 100nmand(2)providingauniformgoldcoverageofthesphere toensurethegood electricalconnection.Adetailedfabricationprocedurec anbefoundinAppendix A 3.1.1BaseSphere Thebasespherewasselectedcarefullyfromavarietyofmicr o-spheresby measuringthemorphologyviaatomicforcemicroscopy(AFM). Thediameterof thetestspheresrangesfrom100 mto300 mandthematerialsincludeceramic, glass,polystyreneandPMMA.Mostspheresareveryroughwitha variationinsurface heightexceeding300nm(Fig. 3-1 ).Sometypesofglassbeadsaresmoothlocallybut withscatteredparticles(Fig. 3-2 ).Suchsurfacemorphologywouldpreventusfrom 35

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Figure3-1.Surfacemorphologyofaceramicspheremeasuredb yAFM.Themaximum heightinsurfacevariationis358nmandthermsvalueis45.9 nm. measuringtheCasimirforceinsubmicronregions.Thesmoot hestspheresaretheglass sphereswithadiameterof103 mfromMicrospheres-Nanospheres.AsshowninFig. 3-3 ,thepeak-to-peakvariationinsurfaceheightisaround30n mandthermsvalueis 2.5nm.3.1.2SputterDeposition WeusedaKurtJ.LeskerCMS-18Multi-TargetSputterDepositio nsystemtocoat theglasssphereinourexperiments.TheSputterdepositionp rocessinvolvesthe followingsteps:(1)Aninertgas,usuallyargon,isionizedg eneratingaplasma.(2) Theionsaredirectedatatargetmaterialandsputteratomsf romthetarget.(3)The sputteredatomsaretransportedtothesubstratethroughar egionofreducedpressure. (4)Thesputteredatomscondenseonthesubstrate,forminga thinlm.Thesystem hasbothDCandradiofrequency(RF)sputterpowersupplies. Usually,electrically 36

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Figure3-2.Surfacemorphologyofa200 mdiameterglassspheremeasuredbyAFM. Themaximumheightinsurfacevariationis280nmandthermsv alueis 35.4nm. conductingmetaltargetsthatcausenoionchargingissuesa resputteredwithDC power.RFsputteringistypicallyusedfortargetmaterialw ithpoorelectricalconductivity topreventexcessivechargebuild-uponthetargetsurface. RFsputteringprovidesa slowerdepositionratecomparedtoDCsputtering. Forourpurpose,sputterdepositionhastwomainadvantages comparingtoother depositionmethods.Therstadvantageisthattheprocessp rovidesagoodcoverage ofthesphere.Becauseofthediffusivetransport(character isticofsputtering),theatoms approachthesubstrate'ssurfacefrompartiallyrandomize ddirections,producinga reasonablyuniformlmthicknessacrossatexturedsubstra te'ssurface.Inotherwords, boththetopandthesideofthespherearereasonablyuniform lycoated.Theother advantageisasmoothsurfacewillbegeneratedonthelmdue tothesmallgrainsize fromthesputterdepositionprocess. 37

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Figure3-3.Surfacemorphologyofa103 mdiameterglasssphere(fromMicrospheres Nnanospheres)measuredbyAFM.Themaximumheightinsurface variation is30nmandthermsvalueis2.5nm. Togenerateassmoothadepositedlmaspossible,weusearad iofrequency(RF) magnetronsputterdepositionandtunetheparameterssucha sthepowerandchamber pressure.Althoughsputterdepositioncantypicallyproduc erelativelysmoothlmsover ashortdistance,theprocessofsputteringalsogeneratess catteredclumpedparticles. HereRFsputteringisusedforthemetaltargetsincethedimi nutionofactiveatomscan leadtolessclumping[ 36 ],whichhelpstoproduceasmoothlmsurfaceoveralong distancerange.Thebestconditionforgoldsputterdeposit ionwerefoundtobe3mTorr usinga100WRFpowersupply. Inaddition,thesputteringsystemallowsustoperforman insitu pre-treatmentof thespheresimmediatelybeforethelmdeposition.Thepretreatment,whichimprove processconsistency,includesa20-secondoxygenplasmacl eananda5-secondargon plasmaclean.Theoxygenplasmahelpstocleanofftheorgani ccontaminationonthe 38

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Figure3-4.Surfacemorphologyofa103 mdiameterglassspherecoatedwithgold. Themaximumheightinsurfacevariationis33nmandthermsva lueis 2.4nm. surfaceandtheargonplasmaphysicallyremovessmallparti clesbybombardingthe surface. Undertheconditionsdiscussedabove,goldcoatedspheres( Fig. 3-4 )with peak-to-peakvariationsof 30nmandrms 3nmwerefabricatedfortheCasimir forcemeasurements. 3.2FabricationofSiliconSample Thesiliconplateswithtrencharrayswerefabricatedonahi ghlyp-dopedsilicon wafer.Theprocessconsistsoftwostages.Therststageinv olvescreatingaresist patternwhichwillserveastheetchmaskfortrenchesetchin g(performedatBellLabs). Thesecondstageistocreatetrenchesbytransferringthepa tternintosiliconusingdry etch(performedatUniversityofFloridaNanofabricationF acility(UFNF)).Thegeneral 39

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sampleprepareprocedureandtheetchingproceduresarelis tedinAppendix B C and D 3.2.1SiliconSamplewithSiliconOxideEtchMask Thesiliconwaferswithsiliconoxideetchmaskwereprovide dbyourcollaborators atBellLabs.Thesampleswerepreparedbyrstdepositingona layerofsiliconoxide (0.2 m)onthehighlyp-dopedblanksiliconwaferbychemicalvapo rdeposition. Then,aphotoresistpatternwasmadebylithographyusingad eepultravioletstepper. Followingthis,thepatternwastransferredfromthephotor esisttothesiliconoxideusing reactiveionetching.Afterstrippingtheremainingphotore sist,thewholewaferiscoated withalayerofphotoresistforprotection.3.2.2DeepReactiveIonEtch SampleAandBwerefabricatedusingdeepreactiveionetch(DR IE).DRIEisa highlyanisotropicplasmaetchprocess[ 37 ].ThebasicBoschprocessinDRIEcan beseparatedintotwocyclingsteps(Fig. 3-5 a)[ 38 ]: SF 6 plasmaetchingand C 4 F 8 plasmapassivation.Duringthepassivationprocess,thesi liconsurfacewascoatedwith athinteon-likepolymerlmwhichresultedfrom C 4 F 8 .Thislayerprotectstheentire substratefromchemicalattackandpreventsfurtheretchin g.Intheetchingprocess,the bottomofthefeatureswasexposedtotheuorineradicalswh ilethesidewallswerestill protectedbecausethedirectionalionsbombardthesubstra tewhichremovethebottom passivationlmatamuchhigherratethanthesidewalls.The uorineradicalsattack andetchthesiliconisotropically.Byrepeatingtheetch/pa ssivatesteps,thebottom ofthefeatureisetchedbysmallisotropicetcheswhilethes idewallsareprotected, whichresultsananisotropicetch.Thetwo-phaseprocesspr ovideshighmaskselectivity andhighanisotropicetchwhichcanbehundredsmicrondeep. Butitalsocausesthe sidewallstohaveanundulatingshape.Atypicalimageofund ulatesidewallofasilicon featurecreatedbyDRIEBoschprocessisshowninFig. 3-6 a. 40

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Figure3-5.(a)TheschemeofBoschprocess.Twocyclingsteps : SF 6 isotropicetchand C 4 F 8 passivation.Thetwo-phaseprocessresultsunwantedsidew all scallopingandundercut.(b)Theschemeofasimultaniouset ch/passivation process.Theteon-likelmisformedtoprotectthesidewal lsimultaneously duringtheetch. Figure3-6.(a)Atypicalimageofundulatesidewallofasili constructurecreatedby DRIEBoschprocess.(b)Ourrstattempttoetchasilicontren chusing Boschprocess.Thesiliconstructureis2.2 mindepthandtheamplitudeof thescallopingis200nm. 41

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Thesamplesthatweaimedtofabricatearetrencharrayswith aperiodicityof 400nmor1 mandadepthof1 monsiliconwafer.WedeterminethattheBosch processisnotsuitableforcreatingthetrencharraysbecau setheamplitudeofthe undulatedsidewallsiscomparabletothewidthofthetrench es.Figure 3-6 bshowsan preliminaryattemptusingthecycledprocesstocreatea2.2 mtrench. Tominimizetheunwantedsidewallscallopingandundercut, weuseasimultaneous etch/passivationrecipeinsteadofthecycledetch/passiv ationprocess(Fig. 3-5 b). Theplasmaisgeneratedfromamixtureof SF 6 and C 4 F 8 gas.Thepassivationgas formathinteon-likelmonthesiliconstructure.Simultan eously,thedirectionalions bombardthesubstratetopreventthepassivationbuildingu ponthebottomofthetrench structures,whichallowstheuorineradicalstoetchthesi licon.Theprocessresultsin straightsidewalls,withlittleundercutandnoscalloping .Byadjustingthebiaspowerand theratioofetchinggasandpassivationgas,wecancontrolt heslopeofthesidewalls andtuneittobenearlyvertical. Figure 3-7 showsthecrosssectionviewofsampleAwithaperiodof1 mand sampleBwithaperiodof400nm.Thesidewallsofthetrenches aresmoothand nearlyvertical.However,wenoticethatacertaindegreeof roundingthatshowsup inthebottomsectionsmakesthestructureimperfectrectan gularshape.Thedetailed characherizationwillbedescribedinSec. 5.2 3.2.3ReactiveIonEtch Forshallowtrenches(sampleC),theroundingfromDRIEproc essbecomes non-negligiblecomparedtothetrenchdepth.Tofabricatet rencharrayswithatbottom surface,wedevelopedarecipeusingareactiveionetcherwi thainductivelycoupled plasmamodule(ICP-RIE). AschematicdiagramofICP-RIEisshowninFig. 3-8 .Thesystemconsistsof tworadiofrequency(RF)sources.Oneiscoupledinductivel ytocreateaplasmaof ionizedatomsandradicalsofreactivegas.Theotherisappl iedtothelowerelectrodeto 42

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Figure3-7.Acrosssectionviewof(a)sampleAwithaperiodo f1 m,(b)sampleBwith periodof400nm. produceasubstratebiaswhichcanextractandaccelerateth eionsfromplasmatoetch thesample. Wedevelopedtherecipeusing Ar and SF 6 astheetchinggaswithoutany passivationgas.Theetchingprocesscanbeconsideredastw oparts:physicalmilling fromthebombardmentofdirectionalionsandchemicaletchf romthereactiveradicals resultedfromthe SF 6 plasma.Becausetheelectriceldacceleratesreactiveradi cals towardsthesurface,theetchingcausedbytheseradicalsis muchstrongerthanthose travelinginotherdirections.Argonplasmaisusedtoprovid eonlyionstogivepurely physicalmillingwithoutanychemicalreaction. SF 6 plasmaprovidesmainlydirectional etchingwithsomeattackofthesidewallsdependingonthebi aspower.Bytuning theratioof SF 6 / Ar andadjustingthebiasRFpoweranearlystraightsidewallca nbe achieved. Oneofthedifcultiesindevelopingtherecipeistosuppres stheetchingratetoa lowlevel.Normallytheetchingrateforsiliconusing SF 6 isacouplemicronsperminutes. Inourcase,however,arateofacouplehundredsnanometerpe rminutesisdesired. Theetchingrateneedstobereducedbyatleastafactorof10. Toachievethis,thegas 43

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Figure3-8.SchematicdiagramofanICP-RIEetchingsystem.On eRFsourceis coupledinductivelytocreateaplasma.Theotherisapplied tothelower electrodetoproduceasubstratebias. ow,chamberpressureandtheRFpowerareallminimizedtoth eextentthataplasma isjustabletobegenerated. Figure 3-9 showsthecrosssectionviewofsampleCwithaperiodof400nm .The bottomstructureofthefeaturesismuchatterusingthiset chingprocessthantheresult fromDRIE.However,suchaprocessconsumesthemasksatamuch higherratedue todirectionalionsbombardments.Theselectivityofsilic onoxidetosiliconisabout1:1, whichmakeitimpossibletoetchtrenchesthataredeepertha n200nm. 3.3PreparationofSiliconSurface Sincethesiliconsurfaceisveryreactiveinair,athinlayer ofnativeoxideispresent onthesurface.TopreparethesiliconsurfacefortheCasimi rforcemeasurements, hydrouoricacid(HF)isusedtoetchthesamples.HFcanremo vethenativeoxideon thesiliconsurface.Inaddition,HFleadstohydrogentermi nationofthesurface,which temporarilypreventoxideformationatambientpressure.F ollowingthisstep,thesilicon 44

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Figure3-9.AcrosssectionviewofsampleCwithperiodof400 nm. sampleisbakedat120 Cformorethan15minutes.Thiseliminatesresidualwater thatmightbetrappedinthetrenches.Afterthisprocessthem easurementsetupis immediatelyassembledandkeptinavacuumchamberatapress ureof 10 6 Torr.In suchanenvironment,thesiliconsurfacecanbestableformo rethanoneweek. 45

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CHAPTER4 THEEXPERIMENTALSETUP Micro-electromechanicalsystems(MEMS)aredeviceswithmov ingpartslinked toelectricalcomponentsfordetectionsandactuations.Th eadvancesinMEMS technologyhaveproducedultrasensitivetransducers,mak ingitpossibletoexplore novelinteractionswithhighsensitivitybetweensurfaces .Allforcemeasurements discussedinthisdissertationwereperformedusingMEMS.In thischapter,thegeneral fabricationofMEMSisdiscussed.Thespecicdeviceusedino urmeasurements,a microelectromechanicaltorsionaloscillator,isintrodu ced.Then,adetaildescriptionis givenwithrespecttodevicepreparation,includingdicing ,releasing,metaldeposition andpackaging.Finally,theexperimentalsetupanddetecti onschemearepresented. 4.1FabricationofMEMS TheMEMSdevicesusedinourexperimentswerefabricatedatth ecommercial foundryMEMSCAP.Thefabricationprocess,knownasPolyMUMPS,i sathree-layer polysiliconsurfacemicromachiningprocess[ 39 ]. Theprocessstartswithaheavilyn-dopedsiliconwaferonwh ichathinlayer ( 600 nm)ofsiliconnitridewhichactasanelectricalisolationl ayer,isdeposited(Fig. 4-1 a).Therstlayerofpolysilicon(POLY0)withathicknessof5 00nmisdeposited usinglowpressurechemicalvapordeposition(LPCVD)(Fig. 4-1 b).Itisthenpatterned bythefollowingsteps.First,alayerofphotoresistisspun onthewafer(Fig. 4-1 c).The photoresististhenexposedtoultravioletlightwhichissh ieldthroughtheappropriate photomask(Fig. 4-1 d).Thesectionsofphotoresistthatarenotcoveredbythema sk havetheirchemicalpropertiesalter,whichmakestheirrem ovaleithereasierormore difcultthantheunexposedregions.Thedeveloperisthenu sedtowashawaythe photoresistattheundesiredregions(Fig. 4-1 e).Next,areactiveionetch(RIE)isusedto transferthepatternfromphotoresistintopolysilicon.Fi nally,theremainingphotoresist isstrippedfromthewafer(Fig. 4-1 f).Thismethodincludingdepositing,patterning 46

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Figure4-1.Thesestepsarerepeatedforallsevenlayersint hePolyMUMPSprocess. (a)Athinlayerofsiliconnitrideisdepositedonthehighly n-dopedsilicon waferasanelectricalisolationlayer.(b)Alayerofpolysi liconisdeposited usingLPCVD.(c)Alayerofphotoresistisspunonthewafer.(d) The photoresistisexposedtoultravioletlightusingaphotoma sk.(e)The undesiredphotoresistiswashedawayandRIEisusedtoetcht he polysilicon.(f)Theremainingphotoresistisstripaway. thewaferwithphotoresist,etchingandstrippingremainin gphotoresistisrepeated forallthelayersintheprocess.FollowingPOLY0,a2.0 mphosphosilicateglass (PSG)isdepositedbyLPCVDandannealed@1050 Cfor1hour.Thislayerserves asthesacriciallayerandthedopantsource.Theannealdop esthepolysiliconwith phosphorusfromthePSGlayerandalsosignicantlyreducesth enetstressinthe polysiliconlayer. 47

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Figure4-2.(a)Acrosssectionviewofallsevenlayersinthe PolyMUMPSprocess(not toscale).Polysiliconlayersareusedatstructuremateria l.PSGlayersare sacriciallayers.Siliconnitrideservesaselectricaliso lationbetweenthe polysiliconandthesubstrate.Metalprovidesbondingsurf acesandhighly reectivemirrorsurfaces.(b)AnexampleofMEMSdevicewithm oveable componentwhichisfreeafterremovingthesacriciallayer s. TheMEMSdevicesarefabricatedlayerbylayerusingthemetho ddescribedabove. AcrosssectionviewofallsevenlayersofthePolyMUMPSproce ssisshowninFig. 4-2 a.Itincludes:(1)polysiliconwhichisusedasstructurema terial,(2)depositedoxide (PSG)usedassacriciallayer,(3)siliconnitridewhichserv esaselectricalisolation betweenthepolysiliconandthesubstrateand(4)metalwhic hprovidesforprobing, bonding,electricalroutingandhighlyreectivemirrorsu rfaces.AMEMSdevicewith moveablecomponentscanbeobtainedbyremovingthesacric iallayers(Fig. 4-2 b). 4.2MicroelectromechanicalTorsionalOscillator TheMEMSdeviceusedinourexperimentswasamicroelectromec hanicaltorsional oscillatorfabricatedusingPolyMUMPSprocess,asshowninF ig. 4-3 .Theoscillator consistsofa3.5 mthick,500 500 m 2 heavilydopedpolysiliconplate,whichis suspendedbytwotorsionalrods.Therodsareanchoredtothe siliconnitridecovered Siplatformandconnectedtoabondingpadtoprovideelectric alconnectiontothe plate.Thesizeofthespringusedintheexperimentsdiscuss edinthisdissertationis 20 m 3 m 2 m.Underneaththetopplate,therearetwoseparateelectrod eswithan 48

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Figure4-3.(a)Scanningelectronmicroscopeimageofamicro electromechanical oscillator.Thetopplate(bigsquare)issuspendedbytwoto rsionalrods whichareanchoredtothesiliconnitridesurface.Theholes onthetopplate areetchingholestomakethewetetchmoreefcient.Threebo ndingpads (smallsquare)provideelectricalconnectionstothetoppl ateandtwo separateelectrodesunderneaththetopplate.(b)Amagnie dviewofthe oscillator.Underneaththetopplate,therearetwoseparat eelectrodes. areaofapproximatelyhalfofthatofthetopplate.Agapof2 mbetweentheelectrodes andthetopplateiscreatedbythesiliconoxidesacriciall ayer. Thetwoelectrodesandtheoscillatortopplatecanbeconsid eredastwovariable capacitors.Here,Idemonstratethebasicbehaviorofthede vicebymeasurementsof thecapacitance.Whenthetopplateistiltedbyasmallangle (InsetofFig. 4-4 ),the capacitanceoftheleftsidecapacitor C 1 andtherightside C 2 canbeapproximately givenby C 1 = 0 A d b 1 (4–1) C 2 = 0 A d + b 1 (4–2) whereAistheareaoftheelectrode, istheangleofrotation, d isthexedgapdistance betweenthemovabletopplateandtheelectrodes, 0 istheelectricalpermittivityoffree spaceand b 1 isthetheeffectivemomentarmofthetopplate. 49

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0 0.5 1 1.5 2 0.55 0.65 0.75 0.85 0.95Capacitance (pF)DC voltage (V) Figure4-4.Thecapacitancebetweenthemovabletopplatean doneofthebottom electrodesplottedasafunctionoftheDCvoltageappliedto thesame electrode. WhenweapplyaDCvoltagetooneofthebottomelectrodes,them ovabletopplate tiltstowardthiselectrodeduetotheattractiveelectrost atictorque.Thisexternaltorque isbalancedbytherestoringtorqueofthetorsionalspringo ftheoscillator: T osc = k (4–3) InFig. 4-4 ,thecapacitancebetweenthetopplateandoneofthebottome lectrodeis plottedasafunctionoftheDCvoltageappliedtothesameele ctrode.Withtheincrease oftheDCvoltage,themeasuredcapacitancealsoincrease,w hichcorrespondsto thechangeoftiltedangle (thereddotsinFig. 4-4 ).Aboveacertainvoltage,the capacitancesuddenlyjumpsandthenstaysataxedvaluefor higherDCvoltages.This inherentinstabilitysituationisknownasthepull-ineffe ct[ 40 ],whichhappenswhenthe externaltorqueexceedstherestoringtorqueoftheoscilla tor.Asshownbytheblack dotsinFig. 4-4 ,thecapacitanceremainsthesameevenifthevoltageisdecr easedback 50

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down,becausethetopplateisstuckatthesnappeddownposit ion.Thepermanent stictionislikelyresultedfromtheadhesionforcesduetot heresidualelectrostatic charges. Afteranoscillatorissnappeddown,therearetwowaystofree thetopplate.One istomanuallyfreethedeviceusingaglasscapillarywithav erynetipcontrolledbya micromanipulator.Thisneedstobeperformedunderamicros cope,assuchthedevice needtobedisconnectedfromtheexperimentalsetup.Theoth ermethodistoshakethe topplatebyapplyingalargeACvoltagetothetopplateforas hortamountoftime(less than2s).Thismethodisnotalwayssuccessfulandmightdama gethetorsionalspring duetothelargecurrent. 4.3DevicePreparation TheproceduretoprepareaMEMSdeviceforexperimentsinvolv esthefollowing steps:dicing,releasing,metaldepositionandpackaging. Dicing .ThediereceivedfromMEMSCAPisa1cm 2 squareandcomposedof severalchips.Adicingsawisusedtoseparatethedieintosm allerchips.Thechip designedfortheseexperimentsisa2.5mm 2 square.Eachchipconsistsofthree oscillatordevices. Releasing .Atthisstage,thechipisprotectedbyalayerofphotoresist andthe movablepartsofthedevicesaresupportedbythesacricial PSGlayer.Thedevice needstobereleasedbyetchingawaythePSGlayer.Theprocessi sdescribedas below: Stripthephotoresistondicedchip.Firstthechipisrinsedb yDIwatertoremove smallparticlesonthesurface.Then,thechipisrinsedwith acetoneandsoakedin acetonefor5minutes.Thechipshouldbetransferedintofre shacetoneatleasttwice duringthesoakingtopreventtheaccumulationofparticles .Followingthis,thechipis transferedintoIPAandblowndryusing N 2 gas.Intheend,thechipisetchedusing O 2 plasmatoremovepossibleorganicparticles. 51

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EtchthesacricialPSGlayer.Thechipissoakedin49 % HFfor5minutesto removethesacricialPSGlayerfollowedbya10minutesDIwate rrinsingtostopthe etchingprocess. Drythereleasedchip.Ifthedeviceisdriedinair,thesurfa cetensionatthe solid-liquidinterfacepullsagainstthelayerstheliquid isattachedto,whichmight resultinthetwolayers(themovabletopplateandtheelectr odes)stickingtogether. Toavoidthis,acriticalpointdryerisused.Aftertheetchin gisstoppedbyDIwater, methanolisrstusedtowashawayallwateraroundthesample .Then,thesampleis submergedinamethanolbathlocatedinsidethechamberofth ecriticalpointdryer. Whilethechamberispressurizedandkeptat10 C,methanolisgraduallyreplacedby liquidcarbondioxide.Aftermorethansixreplacingcycles, themajorityofliquidinside thechamberisliquidcarbondioxide.Inthenextstep,thete mperatureandthepressure areslowlyincreased.Insteadofcrossingthephaseboundar y,thetransitionfromliquid togaspassesthroughthesupercriticalregion,wherethedi stinctionbetweengasand liquidceasestoapply.Thisprocesscandramaticallydecre asesthesurfacetension effectandeffectivelypreventsthetwosurfacesfromstick ingtogether. Metaldeposition .Thegoldcoatedsphereasoneofthemeasurementsurface, isattachedontheoscillatortopplate.Ourearlyattemptsb ygluingthespheredirectly onthepolysilicontopplateencounteredaproblemofpoorel ectricalconnections.This isbecausethesiliconsurfaceisveryreactiveinairsuchth atalayerofnativeoxide canbeformedonthesurfacewhichisolatesthespherefromth eheavilydopedsilicon plate.Tosolvethisproblem,werstmodiedthedesignofth edevicebyaddingone smallgoldsquareononesideoftheoscillatortopplateatad istanceof210 mfromthe rotationaxis.Then,athinlayerofgold(athicknessof50nm )isdepositedonthetop platethroughashadowmaskafterthedeviceisreleased.Wep erformedthedeposition ourselvesinsteadofaddingawholemetallayeronMEMSdesign .Thereasonisthat themetallayerprovidedbyMEMSCAPhasathicknessof2 m.Thestressgenerated 52

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Figure4-5.Schematicoftheexperimentalsetup(nottoscale )in(a)3-Dviewand(b) 2-Ddiagram. betweenathickgoldlayerandthesiliconsurfacewillbendt hemoveableplate.The smallgoldareaensurestheconnectionwithoutaffectingth eatnessofthesiliconplate. Packaging .Afterthegolddeposition,thechipisgluedtoa16-pinceram icpackage usingconductivesilverepoxy.Thedeviceisthenwirebonde dtothepackage.Toensure thedevicefunctionproperlyacapacitancemeasurementisp erformedonthedevice. 4.4ExperimentalSetup TheschematicoftheexperimentalsetupisshowninFig. 4-5 .Thesiliconsample representsoneofthesurfacesintheCasimirforcemeasurem ents,whichcanbea siliconplatewithacorrugatedstructureoraatsurface.T hesiliconsampleisgluedon analuminumholderwhichismountedonacoarsezdirectionpo sitioner.Thecoarsez positioneralsoallowsustoadjusttheorientationofthesi liconsample.Forthesilicon sampleswithcorrugatedstructures,thesampleisposition edinawaythatthetrench arrayisperpendiculartotherotationaxisofthemoveableo scillatorplate.Sincethe springconstantforthetranslationalongthetorsionalaxi sisordersofmagnitudelarger thantheorthogonaldirectionintheplaneofthesubstrate, sucharrangementscan eliminatethemotionofthemoveableplateinresponsetolat eralCasimirforces[ 41 42 ]. 53

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Twogoldcoatedspheres,eachwitharadiusRof51.5 m,arestackedandattached ontoonesideofthetopplateusingconductiveepoxyatadist anceofb=210 mfrom therotationaxis.Thedifcultyofthisstepistoapplyasma llamountofepoxywithout breakingthetorsionalspringsthatsupportthetopplate.T oachievethispurpose,we takeadvantageofthemanipulatorsonawirebondingmachine .Ashortlengthofgold wireispulledoutfromthebondingtipandservesastheepoxy applyingtool.The bondingmachinecanpreciselypositionthewirewhilethesi zeofthewireallowsusto applyonlyasmalldropofepoxy.Theexibilityofthewireen suresthepushingforce willnotbreakthespringwhenthewiretouchesthemovableto pplate.Thesphereis thenmanuallyplacedontheglueusinganetiptweezer.Weus ethetweezertopick upthesphereata45 angleandthentiltthetweezerby45 toplacethesphereonthe glue.Thereisnogoldlmdepositednearthecontactregionb etweenthebottomofthe spheresandthedoublesizetape.Byrotatingthesphere,thec onductiveepoxycan contactwiththedepositedgoldlmwhichprovideselectric alconnection. Theceramicpackagecontainingtheoscillatordeviceismou ntedonastagewith acombinationofxydirectionmanualtranslationstageandz directionclosed-loop piezoelectrictranslationstage.Themaximumextensionof theclosed-looppiezois 10 m. Duringtheassembling,thesiliconsampleisrstmovedtoth espherewithina distanceof10 musingthecoarsezpositionerunderthemicroscope.Meanwh ile thespheresarepositionedunderthecenterofthesiliconsa mplebythexymanual translationstage.Inthenextstep,thewholeassembledsta ge(Fig. 4-6 a)ismoved totheplatformandthedeviceisconnectedtotheelectrical circuits.Apreliminary electrostaticforcemeasurementisperformedtocheckthep erformanceofthedevice andtoestimatetheseparationbetweenthetwosurfaces.Sinc etheseparationbetween thetwosurfaceswillchangeduringpumpingdownofthechamb er,asafeseparation isempiricallydeterminedtobebetween3 mto7 m.Aftertheelectrostaticforce 54

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Figure4-6.(a)Thewholeassembledstagewithelectricalco nnections.(b)Theglass belljarchamber.Insidethebelljar,theassembledstageis coveredwitha electricshield. verication,thecoarsezpositionerisxedbythemodestti ghteningofthetopscrew andthestageiscoveredwithametalshield.Theseparationt ypicallydecreasesby 1 maftertighteningthescrew.Intheend,theglassbelljaris placedoverthe apparatusandthechamberisevacuatedusingavacuumpumpto apressureof 10 6 torr(Fig. 4-6 b).Afterpumpingdown,theseparationtypicallyincreasesb y 1 m. 4.5DetectionScheme Inthissectionthepreliminaryschemeoftheforcedetectio nandrelevantenhancements toincreasethemeasurementsensitivityaredescribed.The basicprincipleoftheforce detectionistomeasurethecapacitancechangewhichisprop ortionaltotheforceusing anoscillatoronthecapacitorbridge. AsshowninFig. 4-5 (b),theforceFactingbetweenthesphereandsiliconsample producesanexternaltorque = Fb ,whichisbalancedbytherestoringtorqueofthe torsionalspringoftheoscillator = Fb = k (4–4) 55

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Aswediscussedabove,theoscillator'stopplateandthebott omelectrodescanbe consideredastwovariablecapacitorswiththecapacitance givenbyEq. 4–1 and 4–2 Foralltheforcemeasurementsreportedinthisdissertatio n 2 10 5 rad,i.e., << 1, b 1 << d .Underthiscircumstance,Eq. 4–1 and 4–2 canbewrittenas C 1 = 0 A d b 1 = C 0 (1+ b 1 d ) (4–5) C 2 = 0 A d + b 1 = C 0 (1 b 1 d ), (4–6) where C 0 = 0 A d .Therefore / C = C 1 C 2 .Consequently,theforcebetweenthetwo surfacesataseparation z canbegivenby F ( z )= A C (4–7) Theelectricalcircuitstomeasurethecapacitancechangea reschematicallyshown inFig. 4-7 [ 43 ].Weapplya102kHzACvoltagetoeachbottomelectrodesasth ecarrier signal.TheACvoltages, V AC 1 and V AC 2 ,haveanequalmagnitude V ac andfrequency ,butare180 outofphase.Theelectricalrepresentationoftheoscillat orisshownin Fig. 4-8 .Whenthetopplateisat,thetwocapacitorsareidenticalwi thacapacitance C 0 .Thusthebridgecircuitisinbalanceandtheoutputcurrent iszero.Ifthereisan externalforceappliedtothetopplatethattilttheplateby asmallangle ,theinduced capacitancechangeislinearlyproportionaltotheangle underthecondition << 1 Consequently,theoutputcurrent i isalsolinearlyproportionalto ThetopplateisconnectedtoanAmptek250chargesensitivepr eamplier,which detectsthechargevariationandgeneratesanampliedvolt age.Theampliedvoltage isthensenttoaSRS830lock-inamplier,whichmeasuresthevo ltageatthefrequency ofthecarrierwave.Thelock-inamplieressentiallyperfo rmsahomodynephase detectionbetweenthereferencesignalandalocaloscillat ortoextractthermsvoltage viaphase-locking.Therefore,noiseatanyotherfrequency canberejectedbythelock-in ampliersuchthatthegeneratedvoltageiseffectivelyiso latedfromthenoisesources. 56

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Figure4-7.Thediagramoftheelectroniccircuitforbasicd etectionscheme.Ahigh frequencycarrierACvoltageisappliedtoeachofthebottom electrodes. V AC 1 and V AC 2 havethesamemagnitudeandfrequency,but180 outof phase.Thedetectedsignalisfedintoachargesensitivepre amplier. InoursetupweuseaHP3314functiongeneratortocreatetheAC voltages.An analogsplittermanufacturedbyMiniCircuitsisusedtopro duceACsignalsofequal magnitudeandfrequency,but180 outofphase.Foracertainacvoltageinput,the magnitudeofthesplitvoltageisaxedvalue.Therefore,we builtatuneabledividerto adjustthemagnitudeofthevoltage(Fig. 4-9 ).Sinceaddingadividerwillshiftthephase ofthevoltage,anotherxeddividerisaddedtotheotherarm ofthesplittertokeepthe twosignalat180 outofphase. Dynamicmeasurements .Toimprovethesensitivityoftheforcedetection,weset upadynamicmeasurementschemewhichtakesadvantageofthe highqualityfactorof themicroelectromechanicaltorsionaloscillator.Thedia gramfortheelectroniccircuitis 57

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Figure4-8.Electricalrepresentationoftheoscillator.Tw oacexcitations( V AC 1 and V AC 2 ),withanequalmagnitudeandfrequencybut180 outofphase,are appliedtoeachbottomelectrode.Whennoforceisapplied,th esymmetrical gapsgivewaytotwoidenticalcapacitances C 0 ,theoutputcurrentiszero. Whenaforceisapplied,thetopplateoftheoscillatortiltsb y .Forasmall thechangesofthecapacitancesarelinearwith .Acurrent i ,whichislinear to ,willbedetected. Figure4-9.Thecircuitdiagramofaxed-ratioamplitudedi vider(left)andavariable amplitudedivider(right).Eachoutputofthesplitteriscon nectedwithan amplitudedividerwhichadjuststheamplitudetothedesire dvaluetoreach thebridgebalanceaswellasmaintainsthephasedifference betweenthe twoarms. 58

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showninFig. 4-10 .Inadditiontotheapproachwediscussedabove,adrivingsi gnalis appliedtooneofthebottomelectrodestodrivetheoscillat oratitsresonantfrequency. Asthedeviceoscillatesinresponsetothisexcitationtheou tputsignalfromthecapacitor istheampliedvoltagewiththecarrierfrequencymodulate dbythedrivingsignal. Toextractthedrivingsignalfromthecapacitoroutput,weu setwolock-inampliers: Therstoneislockedtothecarrierfrequencysuchthatitl tersoutnoisesignalsand itsoutputisproportionaltothemechanicalresponseofthe oscillatoratthedriving frequency.Thelocaloscillatorofthesecondlock-inampli erisphase-lockedtothe mechanicalresponsesignalandisthenfedbacktokeepthedr ivingsignalxedatthe resonantfrequency.Thisentirecircuitworksasaphase-lo ckedlooptotracktheshiftof theresonantfrequencyoftheoscillator[ 44 ]. Inthisapproach,thetopplateoftheoscillatorisexciteda titsresonantfrequency. Atsmalloscillationwherenon-lineareffectcanbeneglecte d,themotionoftheoscillator isgivenby f r = f 0 (1 b 2 8 2 If 2 0 @ F @ z ), (4–8) where f r istheresonantfrequencyoftheoscillatorinthepresenceo ftheexternalforce Fand f 0 = p k = I istheintrinsicresonantfrequencyoftheoscillator( 1783 Hz).bisthe distancebetweenthesphereandtheaxisoftheoscillatoran dIisthemomentofinertia ofthetopplatetogetherwithtwospheres.Thedistance z isgivenby z = z 0 z piezo b where z 0 istheinitialgapbetweentwosurfaces, z piezo isthepiezoextensionand b is thecorrectionduetothetiltingoftheoscillatortopplate .Theforcegradientismeasured throughtheresonancefrequencychange.TheEq. 4–8 canberewrittenas f = C @ F @ z (4–9) where C = b 2 = 8 2 If 0 .TheproportionalconstantCandtheinitialgap z 0 aretwo parametersthatneedtobecalibratedforthemeasurementsy stem. 59

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Figure4-10.Thediagramoftheelectroniccircuitforadyna micmeasurementscheme. InadditiontothehighfrequencycarrierACvoltageapplied toeachofthe bottomelectrodes,adrivingsignalisappliedtooneofthee lectrodesto modulatethetopplateatitsresonantfrequency.Thedetect edsignalis ampliedbyachargesensitivepreamplierandthenfedinto therst lock-in.Therstlock-inoutputsavoltagethatisproporti onaltothetilt angleoftheoscillator,whichisltered,ampliedandlock edtotheresonant frequencyoftheoscillatorbythesecondlock-inandfedbac ktodrivethe oscillatoratitsresonantfrequency.Thephaselockloopma intainsa specicphasedifferenceof = 2 betweenthedriveandthevibrationsofthe device,ensuringthattheMEMSoscillatorisdrivenatresona nce. 60

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CHAPTER5 SAMPLECHARACTERIZATION Inthischapter,thedetailedcharacterizationofthesampl esisdescribed.First, thetopographiesofboththegoldspheresandthesiliconpla tes,obtainedusingan atomicforcemicroscopy(AFM),arepresented.Thenthemetho dstomeasuretheactual dimensionsofthetrencharraysareintroduced.Whiletheper iodofthetrencharrays isaknownparameter,determinedbythemaskdesign,theothe rparameters,suchas thefractionofsolidvolume,thedepthofthetrenchandthes idewallangle,mustbe obtainedfrommeasurementsusingSEMorAFM.Differentapproac heswereusedto characterizeforhighaspectratiotrenchesandshallowtre nchestoachievethebest results. 5.1TopographiesofGoldSpheresandSiliconPlates Wemeasuredthetopographiesofboththegoldspheresandthe siliconplates usinganAFM.ThoseAFMimagesareusedtodeterminethesurface roughnessofthe samples,whichhastobetakenintoaccountwhenwecompareth emeasurementswith theoreticalpredictions.InFig. 5-1 ,weshowatypicalAFMimageforthesurfaceofa goldcoatedspherewitha1 m 1 mscanningarea. Figure5-1.1 m 1 mAFMimageofagoldcoatedsphere. 61

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0 0.5 1 1.5 2 2.5 3 3.5 0 1000 2000 3000 4000 Number of countsSurface height (nm) (a) 0 10 20 30 40 50 60 0 100 200 300 400 Number of countsSurface height (nm) (b) Figure5-2.Histogramsofthesurfaceheightinascansizeof 10 m 10 mfor(a)at Sisurfaceand(b)goldcoatedspheresurface.Thepeaktopeak variationin thesurfaceheightoftheSisurfaceislessthan4nmandthatof theAu surfaceis55nm.ThermsvaluefortheSisurfaceis0.3nmandth atforthe Ausurfaceis3nm. Toanalyzethesurfaceroughnessofthesamples,wetakeAFMsc answitha10 m 10 mscanningareaandthenobtainahistogramofthesurfacehei ghtforeachsetof imagedata.Ingure 5-2 ,thehistogramofthesurfaceheightofoneAFMimagefor(a)a goldcoatedsphereand(b)aatsiliconplateisshown.Thest atisticalanalysisindicates thatthepeaktopeakvariationinthesurfaceheightoftheSis urfaceislessthan4nm andthatoftheAusurfaceis55nm.ThermsvaluefortheSisurfa ceis0.3nmandthat fortheAusurfaceis3nm. Inpractice,wecannotmeasurethetopographiesoftheactua lsphereorsilicon plateusedintheforcemeasurements.Toensurethattheobta inedAFMimages representthesurfaceroughnessforthesphereusedintheme asurements,we measuredmultiplespheresfromthesamebatchofsputtering deposition.Wenote 62

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Figure5-3.(a)Crosssectionofrectangulartrenchesinsil icon,withaperiodicityof 400nmandadepthof0.98 m.(b)Topviewofthestructure. thatalltheinvestigatedspheresyieldconsistentstatist icalresults.Similarexaminations werealsoperformedforsiliconplates.Againtheresultswer estatisticallyconsistent. Therefore,theaverageofmultipleAFMimagedatawereusedto representthesurface roughnessoftheactualsamples. 5.2DimensionofSampleswithHigh-aspectRatioTrenchArray s Weaimedtofabricatetrencharrayswithadutycycleat 50% .However,inpractice, thetrencheswerefabricatedatadutycycleclosetobutnote xactlyat50 % .By analyzingSEMimages,thefractionofsolidvolume p canbedeterminedforsamples withhigh-aspectratiotrencharraysusingthefollowingst eps.(1)Topviews(Fig. 5-3 b) aretakenbySEMattendifferentlocationsforeachsample.Each imagecontains morethan30periods.(2)EachimageisloadedintoMatlabwhic hreturnsaM-by-N array(MandNaredeterminedbythedimensionoftheimage).Ea chpointinthearray representsthegrayscaledataforapixelinthetopviewimag e.InFig. 5-4 ,across sectionofthestructureconsistingofa1-Darrayofthegray scaledataisplotted.The highvaluesingrayscalerepresentthetopofthetrencharra ywhilethelowvalues representthebottom.Althoughacoupletransitionpointsex istinthedata,themajority ofthegrayscaledataareclearlyseparatedsuchthatthetop andthebottomarewell dened.(3)Wechooseaxednumberofpixelswhichcontainsi ntegermultiplesof oneperiodalongasinglerowofpixels.Thegrayscaledatafo rthosepixelsareused 63

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50 100 150 200 250 300 350 400 450 500 0 50 100 150 200 250 300 GrayscalePixels Figure5-4.ThecrosssectionofSEMdatafortheperiodictrenc harrayplottedas1-D arrayingrayscale. tocreateahistogram.AsshowninFig. 5-5 ,brightpixels(largenumberinxaxis) representthetopsolidpartofthetrenchwhiledarkpixelsr epresentthebottompart. Themiddlevalue(125inFig. 5-5 )issetasthedividinglineofthetopsurfaceandthe bottomsurface.(4) p isdenedasthesumofthecountsforpixelswithgrayscaleda ta thatislargerthan125dividedbythetotalpixelcounts.The fractionofsolidvolume p is calculatedfromeachhistogramandaveragedforalltenimag es. Thetrenchdepthandtheangleofthesidewallstothetopsurf acearemeasured fromtheSEMimagesofthecrosssectionofthetrenches.Forthe sehighaspectratio trenches,anAFMtipcannotreachthebottomofthetrenches.T herefore,AFMcannot beusedtomeasurethedepthofthesedeeptrenches.Asummary ofthedimensionsfor sampleswithdeeptrenchesislistedinTable 5-1 Table5-1.Thephysicaldimensionsofthehighaspectratiot rencharrays. istheperiod ofthetrencharray, p isthefractionofsolidvolume, t isthedepthofthe trenchesand istheangleofthesidewallstothetopsurfaces. Sample ( m )pt( m ) ( C ) A10.5100.9890.3B0.40.4781.0191.0 64

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0 50 100 150 200 250 0 0.5 1 1.5 2 x 10 5 Number of countsGrayscale Figure5-5.HistogramofthegrayscaledataforaSEMtopviewim ageofdeeptrenches. Themiddlevalue(125)issetasthedividinglineofthetopsu rfaceandthe bottomsurface. Inalltheoreticalanalysis,thetrenchesareassumedtohav eaperfectlyrectangular shape.Butinpractice,thebottomsectionsshowacertaindeg reeofrounding.For suchhighaspectratiotrenches,theoverallelectrostatic forceisinsensitivetosmall variationsinthedepthofthetrenches.Infact,thecalcula tedelectrostaticforcechanges onlyby0.01 % whenthedepthofthetrenchesvariesby10 % .Thisalsoshowsthat thedeterminationofthedepthisnotascriticalasthefract ionofsolidpartpinthe theoreticalcomparision. 5.3DimensionofSampleswithShallowTrenchArrays Themethodusedfordeeptrenchestoobtainthefractionofso lidvolumepcannot beappliedtoshallowtrenchesasthecharacteristicoftheSEM imagesfortwokindsof trenchesaretotallydifferent.Ingure 5-6 ,wecomparetheSEMimageofdeeptrenches andshallowtrenchesinazoomedintopview.Fordeeptrenche s,thecontrastofthe topsurfacesandthebottomsurfacesintheimageissharp(Fi g. 5-6 b)sincethebottom 65

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Figure5-6.TheSEMimageof(a)shallowtrenchesand(b)deeptr enchesinazoomed intopview.Forshallowtrenches,thecontrastofthetopand thebottomare comparable.Inaddition,thebrightlinesontheedgesofthe shallow trenchesmaketheedgehardlydened. surfacesisfaraway.Inthatcase,thetwosurfacesarewelld enedinthegrayscale datareturnedbyMatlab.However,forshallowtrenchestheb ottomsurfacesaretoo closetothetopsurfacesresultinginlessdistincttones(F ig. 5-6 a).Inaddition,there arebrightlinesontheedgesoftrenchesintheimageofshall owtrenches,whichmay haveresultedfromedgeeffectsorthesidewallsofthetrenc hes.Theseeffectsmakeit impossibletoobtainprecisedimensionsoftheshallowtren chesfromtheSEMtopview image. Figure5-7.(a)ASEMimageand(b)aschematicofthecrosssecti onviewofthe shallowtrenches. WeusetheSEMimageofthecrosssectionviewtoobtaintheactua ldimension oftheshallowtrenches.TenSEMimagesofthecrosssectionwer etakenatdifferent 66

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Figure5-8.AnAFMimageofshallowtrencheswithascanningare aof1 m 1 m. positions.Insteadofusingarulertomeasureitmanually,w eloadtheimageusinga Matlabprogram.Asdescribedabove,thegrayscaledataforea chpixelispresented byaM-by-Narray.AslabelledinFig. 5-7 (b),wecountthepixelnumberforthetop L 1 thebottom L 2 andperiod .Sincetheperiodofthetrenchesisxedbylithographyto be =400 nm,wecanobtainthelength L 1 and L 2 byscalinginthepixelnumbersto theperiod.Thedisadvantageofthismethodisthelimitedsa mpling.Sincewecanonly cleavetheplatewithtrenchstructuresonce,thecrosssect ionviewislimitedtothis lineinsteadofrandomlycoveringthewholesample.AnAFMimag eoftheshallow 0 200 400 600 800 1000 0 50 100 150 surface height (nm)x (nm) Figure5-9.AcrosssectionoftheAFMimageshowninFig. 5-8 67

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trenchesisshowninFig. 5-8 andonecrosssectionofthisimageisshowningure 5-9 .Theatbottomregion,asshowninFig. 5-9 ,demonstratesthattheAFMtipcan reachthebottomofthetrenchesandprovidecorrectdepthme asurements.Thedepth iscalculatedfromtheaverageoftwentyAFMscanswhicharera ndomlytakenonthe sample.Tenofthesescansarewithascanningareaof1 m 1 m,theothersare 2 m 2 m.Asummaryofthedimensionsforsampleswithshallowtrenc hesarelisted inTable 5-2 Table5-2.Theactualdimensionsoftheshallowtrencharray s. istheperiodoftrench array, L 1 isthelengthofthetopsurface, L 2 isthelengthofthebottomsurface, tisthedepthofthetrenchesand istheangleofthesidewallstothetop surfaces. Sample ( m ) L 1 ( nm ) L 2 ( nm )t( nm ) ( C ) C0.4185.3199.19894.6 68

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CHAPTER6 FORCEMEASUREMENTS Inourexperiments,boththeelectrostaticforceandtheCas imirforcewere measured.Infact,theelectrostaticforceplaysanessenti alroleintheCasimirforce measurements.Throughthemeasurementsoftheelectrostat icforce,wecandetermine speciccalibrationparametersaswellastheresidualvolt age.Inthischapterwestart byintroducingtheelectrostaticforceforourgeometries. Then,theresidualvoltagein oursystem,whichresultsfromtheworkfunctiondifference betweenthetwosurfaces, isdiscussed.ThisresidualvoltagewillbeusedintheCasim irforcemeasurements toeliminatetheeffectofelectrostaticforce.Thisisfoll owedbythedescriptionofthe calibrationofourmeasurementsystem.Twoparametersneed tobedetermined.One istheproportionalityconstantbetweentheforcegradient andthemeasuredresonant frequencyshift.Theotheristheinitialseparation z 0 betweenthetwosurfaces.Finally, themeasurementsoftheCasimirforcearepresented. 6.1ElectrostaticForce 6.1.1ElectrostaticForcebetweenaSphereandaFlatSurface Theelectrostaticforcebetweenthegroundedgoldspherean dtheatplateat voltage V isgivenby[ 45 ]: F e rat =2 0 ( V V 0 ) 2 1 X n =1 [coth( ) n coth( n )] sinh( n ) (6–1) where 0 isthepermittivityofvacuum, V 0 istheresidualvoltage, =cosh 1 (1+ z = R ) and z istheseparationbetweenthesphereandtheplate.Inpracti ceanapproximate expansionissufcienttocalculatethenumericalvalue.Us ingtheperturbative expansion,wecanrearrangetheequationasfollows F e rat ( z )= 2 0 ( V V 0 ) 2 6 X i = 1 c i ( z R ) i (6–2) 69

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where C 1 =0.5 C 0 = 1.18260 C 1 =22.2375 C 2 = 571.366 C 3 =9592.45 C 4 = 90200.5 C 5 =383084 and C 6 = 300357 .Comparedtothecomplete sphere-plateexpression,therelativeerrorintroducedby usingEq. 6–2 is 4.7 10 5 and 1.5 10 5 atseparationof1.5 mand5 mrespectively(assume R =200 m)[ 46 ]. 6.1.2ElectrostaticForcebetweenaSphereandaPlatewithPeri odicalTrenches Thereisnoanalyticexpressionfortheelectrostaticforce betweenasphereand aplatewithcorrugatedstructures.Thuswecalculatetheel ectrostaticforceinsucha congurationbynumericalmethods. Foracertainstructurewithinnitelengthononedimension ,thestructurecan berepresenteda2-Dmodel.Sinceinourcasethelengthofthet renchescanbe consideredinnitecomparedtothewidth,wefollowtheappr oximationtonumerically calculatethestructureinaperiodical2-Dmodel. Tocalculatetheelectrostaticforce,westartwithevaluat ingtheelectricpotential distributionbetweenaatsurfaceandaplatewithperiodic altrencharrayinwhichthe boundaryconditionsontheatsurfaceandthetrenchesarea lreadyknown.Thenthe potentialenergyperunitarea E plate trench canbederivedfromthecalculatedpotential distribution.Finally,theelectrostaticforcebetweenas phereandaplatewithtrenches canbegivenbytheproximityforcetheorem(PFA). Twodifferentsolvershavebeenusedfornumericalcalculat ion.Duringtheearlypart ofmydoctoralprogram,whenwewereaimingtomeasuretheCas imirforceusinghigh aspectratiotrenchsamples,weusedafreeprogramthatsolv espoissonequationby meansofFEMinMatlablanguage.Later,wepurchasedacommerc ialprogramcalled COMSOLMultiphysics.Twosolversprovideconsistentresult sontheelectrostaticforce calculationforourstructures.6.1.2.1Poissonsolverusingmatlabprogram ThepoissonsolverthatweusedisapackageofMatlabcodewhi chappliesthe niteelementmethodtosolveaformofPoisson'sequationov eranarbitrarytriangulated 70

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z V A A ¢ B B ¢ C C ¢ Figure6-1.Ageometryconsistingofaatsurfaceandaperio dicaltrenchsurfaceintwo periods.Theareabetweenthemarenonuniformlydividedint otriangular mesh. region.TheMatlabcodewasrunningonthecomputerclustero ftheUniversityofFlorida HighPerformanceComputingCenter.HereIwilldescribethe numericalapproachin detail. Denethegeometryandboundarycondition .AsshowninFig. 6-1 ,therst stepistodeneageometryconsistingofaatsurfaceandape riodicaltrenchsurface intwoperiods.Theboundaryconditionscanbedescribedasf ollows:theatsurfaceis grounded,thetrenchsurfaceisatpotentialV.Periodicalb oundaryconditionisapplied toAA 0 ,CC 0 whichisachievedbyiteration.Theinitialboundarycondit ionissetsothat thepotentialoftheleftsideAA 0 andtherightsideCC 0 canbeconsideredlinear,which meansthemagnitudeoftheelectriceldonthesetwosidesar egivenbyV/z.Asthe boundaryconditionsarealldetermined,theelectriceldo nanarbitrarypointwithinthe structurecanbeevaluatedbythePoissonequation.Thenthe boundaryconditionson 71

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lineAA 0 andCC 0 canbereplacedbyobtainedelectriceldonthelineBB 0 duetothe periodicity.Followingthisway,aniterationisusedtoobt ainthenumericalresultofthe periodicalboundaryconditions. TriangulateMesh .Todiscretizethedomainofinterestintoelements,wedivi de thedomainintoatriangulatedmesh.Althoughitisdesirable tohaveasmanytriangles aspossible,inpracticewearelimitedbythememorycapacit yofthecomputer.To achievethebestresultwithanitenumberoftriangles,weu seanonuniformmesh distribution:Thedistributionissparseonthebottomofth etrencheswhilethetriangles areconcentratedaroundtheareaswheretheforcegradienti ssensitivetothegeometry (Fig. 6-1 ). Obtaintheelectricpotentialdistribution .Theelectricpotentialdistributioncanbe obtainedbynumericallysolvingthePoissonequation r 2 U ( x y )=0, (6–3) withtheboundaryconditionsdiscussedabove.ThePoissons olver,writtenbyJohn Burk,solvesthePoissonequationinatriangulatedregionin theplaneusingFEM.The codeusescontinuouspiecewiselinearbasisfunctionsontr iangles.Figure 6-2 and 6-3 demonstratethepotentialdistributionforthetrencharra ywithaperiodicityof400nm and1 m,respectively. Electricpotentialenergy .Inthisparagraphwewillevaluatetheelectricpotential energybetweenthetwosurfacesstartingfromthe2-Ddistri butionoftheelectrical potential ( x y ) .Thenegativegradientofthepotentialfunctionyieldsthe electriceld E : E = r ( x y ) = @ @ x ^ e x @ @ y ^ e y (6–4) 72

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0 0.4 0.8 0 0.5 1 1.5 x ( m m) y ( m m) 0 0.2 0.4 0.6 0.8 1 Figure6-2.Thepotentialdistributionforthetrencharray withaperiodicityof400nmand adepthof1 m.Theappliedvoltageonthetrenchsurfaceis1Vandtheat plateisgrounded. 0 0.5 1 1.5 2 0 0.5 1 1.5 x ( m m) y ( m m) 0 0.2 0.4 0.6 0.8 1 Figure6-3.Thepotentialdistributionforthetrencharray withaperiodicityof1 manda depthof1 m.Theappliedvoltageonthetrenchsurfaceis1Vandtheat plateisgrounded. 73

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Sinceforeachpoint ( x y ) intheareathereisacorrespondingpotentialvalue given byamatrixinMatlab,thepartialderivative @ @ x and @ @ y canberepresentedbythematrix elements: @ @ x ( y )= ( x m y ) ( x n y ) x m x n @ @ y ( x )= ( x y m ) ( x y n ) y m y n (6–5) Thereforetheelectriceld E asa2-Ddistributionfunction E ( x y ) canbenumerically obtained.Tocalculatethetotalelectricpotentialenergy W inthisarea,weusethe formula W = 1 2 0 ZZ E ( x y ) 2 dxdy (6–6) wheretheintegralrangeistheentireareabetweenthetwosu rfaces.Fordiscretevalues of E ( x y ) theintegralisreducedtothesummation: W = 1 2 0 N X k =1 ( E 2 x + E 2 y ). (6–7) CalculateelectricstaticforceusingPFA .Thecalculationsdiscussedaboveare forthegeometriesbetweenaatsurfaceandatrencharrayst ructure.Intheactualforce measurements,theatsurfaceismodiedintoasphericalsu rface.Thus,asdescribed inSec. 1.4.1 weusethePFA[ 47 48 ]torelatethesphere-planeandtheplane-plane geometriesaccordingto F sphere trench =2 RE plate trench (6–8) where E plate trench istheelectricpotentialenergyperunitareabetweenaats urface andatrencharraystructure.The E plate trench isobtainedfrom E plate trench = W = (2 ) whereWthetotalelectricpotentialenergybetweenaatsur faceanda2periodtrench structureand istheperiodofthetrencharray.Thecalculatedelectrosta ticforcesare presentedinFig. 6-4 74

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0 0.2 0.4 0.6 2 4 6 8 10 12 14 16 x 10 -9 F ele (N)z ( m m) Figure6-4.Thecalculatedelectrostaticforcebetweenasp hereandaplatewithaat surface(blueline),sampleA(redline),sampeB(blackline ).Thediameterof thesphereis103 m.Theappliedvoltageontheplateis1Vandthesphere isgrounded. ConvergenceTest .Intheprocedureofgeneratingtriangulatemesh,wedivide the areabetweenthetwosurfacesinto N triangles.Toensurethat N issufcientlarge,we checktheconvergenceofthenumericalcalculation.Ataxed separation( z =150 nm), wecalculatetheelectrostaticforceforatrencharraywith aperiodicityof400nmand adepthof1 musingdifferenttrianglenumbers(Fig. 6-5 ).When N increasesfrom 5000to10,000,thecalculatedforcechangesby 1 % .When N increasesfrom10,000 to20,000,thecalculatedforceonlychangesbylessthan0.1 % .Thisconvergencetest isrepeatedatmultipleseparationsandfordifferentstruc turesprovidingconsistance results.Baseonthistest,wechosetouse N > 10,000 forournumericalcalculation. 75

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0 0.5 1 1.5 2 x 10 4 8.5 8.55 8.6 8.65 8.7 8.75 8.8 x 10 -9 F ele (N)Triangle Numbers Figure6-5.Thenumericalcalculatedelectrostaticforceb etweenasphereandaplate withtrenchstructuresplottedasafunctionofdividedtria nglenumbers.The periodicityofthetrencharrayis400nmandthedepthis1 m.The separationbetweenthesphereandthetrenchsurfaceis150n m.Whenthe trianglenumber N increasesfrom10,000to20,000,thecalculatedforceonly changesbylessthan0.1 % 6.1.2.2COMSOLmultiphysics COMSOLMultiphysicsisasimulationsoftwareformodelingan dsolvingphysics problemsbasedonpartialdifferentialequations(PDEs).Iti sapackageofnumerical solversthatsolvethePDEsusingtheniteelementanalysisto getherwithadaptive meshinganderrorcontrol. ThebasicprocessofCOMSOLincludesthestepsofbuildingmod elgeometries, creatingameshfortheniteelements,specifyingthephysi cs,solvingtheproblemand postprocessingthesolutions.Althoughtheprocessandbasi cconceptsaresimilartothe numericalapproachdiscussedinSec. 6.1.2.1 ,COMSOLisamorepowerfulsoftware.It providesaexibleuserinterference,quickset-upofmodel andefcientsolving. 76

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z Figure6-6.Ageometryconsistingofaatsurfaceandashall owperiodicaltrench surfaceinoneperiodataseparationof z =200 nm.Theareabetweenthem arenonuniformlydividedintotriangularmeshbyCOMSOL.The numberof trianglesischosentobelargerthan10,000intheactualcal culation. OurlabpurchasedCOMSOLMultiphysicsinthelaterpartofmyd octoralprogram. Themodelsfordeeptrenches(sampleAandB)weresetupandthe electrostaticforce wascalculated.Consistentresultsareobtainedcomparedt othecalculationdiscussed abovewhichdemonstratethevalidityofbothmethods.Then, COMSOLwasappliedto sampleC(shallowtrencheswithaslightlytrapezoidalshap e)toobtainthenumerical valueoftheelectrostaticforce. InFig. 6-6 ,atriangularmeshgeneratedbyCOMSOLisplotted.Thenon-un iformity ofthetriangledistributionisdifferentwiththatfordeep trenches.Sincethedepthof thetrenchesaresmallerthantheseparationsofthemeasure dforce,thedensityof thetrianglesinsidethetrenchiscomparabletothatintheg apbetweentheatsurface andthetrench.Thesharpcornersofthetrenchrequiresane rtriangledivision.The calculatedpotentialdistributionisshowninFig. 6-7 .Intheend,theelectrostaticforce canbecalculatedusingthesamepost-processdescribedabo ve. 6.2ResidualVoltage Theresidualvoltage V 0 isthecontactpotentialbetweentwogroundedsurfaces resultingfromtheworkfunctiondifferencebetweenthetwo surfaces.Sinceitcan 77

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0 100 200 300 400 0 100 200 300 y (nm) x (nm) 0 0.2 0.4 0.6 0.8 1 Figure6-7.Thepotentialdistributionforthetrencharray withaperiodicityof400nmand adepthof100nm.Theappliedvoltageonthetrenchsurfaceis 1Vandthe atplateisgrounded. generateanattractiveelectrostaticforcebetweentwogro undedsurfaces,itisimportant todeterminetheresidualvoltagebetweenoursamplessotha twecanisolatethe electrostaticforcefromtheCasimirforceoreliminatethe electrostaticforceeffectsinthe Casimirforcemeasurements. Theworkfunctionistheminimumenergyneededtomoveanelec tronfromthe Fermilevelintovacuum.Ifwechoosetheenergyofafreeelec tron(withnokinetic energy)inthevacuumasthezeroenergy,theworkfunctionca nbewrittenas W work = E F (6–9) where E F istheFermienergy.Forrealsurfaces,therearepotentialp atchesonthe surfacesthatcanbecausedbymultiplefactorsincludingst rainsinthesurfaceor contaminationinthesurface.Thesepatchescreateasurfac edipolelayer,which generateapotentialdifferenceacrossthesurface W S .Theactualworkfunctionisgiven 78

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by W work = E F + W S (6–10) Inotherwords,thelocalchangesinsurfacecrystallinestr ucturegiverisetovarying workfunctions,andhencecausethesurfacepotentialvaria tion(patcheffect). Therehavebeenexperimentsstudyingthepatchpotential[ 49 ].Ideally,itis desirabletomeasuretherelevantpatchpotential insitu sothatwecouldcharacterize theelectrostaticforcegeneratedbypatchpotential.Ther earetechniquessuchasKelvin electrometersandprobesthatmightbeabletooffersufcie ntresolutionandsensitivity forthisapplication.Inourexperiments,weuseasimplemet hoddescribedbelow. Thetotalforcebetweenagroundedsphereandaatplatewith voltageVis F total = F 0 + F e rat (6–11) where F 0 isthevoltageindependentoffsetcorrespondingtotheCasi mirpartand F e rat istheelectrostaticforcedenedbyEq. 6–2 .Usingtheparabolicdependence ofthetotalforceontheappliedvoltageV,wedeterminether esidualvoltage V 0 by ndingthevoltagebetweenthetwosurfacesthatminimizest heforce.Inourdynamic measurements,theforcegradientismeasuredthroughthere sonantfrequencyshiftof themicro-mechanicaloscillator.Atseparation z ,theresonantfrequencyoftheoscillator ismeasuredasafunctionoftheappliedvoltageonthesilico nplate.Then,weusea ttingproceduretoobtainthevoltageatwhichthefrequenc yachievesthemaximumin theparabola.Atthispointofsystemcalibration,weonlykno wthepiezoextensionand theexactvalueof z isuncertain.Apreliminarytestisusuallyperformedtoest imatethe valueof z .Intheactualmeasurements,theparabolaisrepeatedatmul tiple z ranging from2 mto100nm.FourttedparabolasareplottedinFig. 6-8 withthepurpleline correspondingto 100nmandtheredlinecorrespondingto 400nm.Aplotof V 0 as afunctionofseparation z isplottedinFig. 6-9 V 0 isfoundtochangebylessthan2mV for z rangingfrom100nmto1 m.Duetothedecreaseinthesignal-to-noiseratio,the 79

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-0.8 -0.6 -0.4 -0.2 1778 1779 1780 1781 1782 1783 1784 1785 Resonant frequency (Hz)Voltage on silicon plate (V) Figure6-8.Thefrequencysignalplottedasafunctionofthe appliedvoltageataxed separationdistance z forsiliconatplate.Thecirclesarethemeasured frequencyandthelinesarethettingdata.Thepurple,blac k,blueandred curvescorrespondtothedatasetwithaseparationofapprox imate100nm, 150nm,200nmand400nm,respectively. randomerrorincreaseswiththeincreasingoftheseparatio n.Wecalculatetheresidual voltagebyaveragingthemeasured V 0 attheseparationbetween100nmand1 m.In thissetofdata,theresidualvoltage V 0 isdeterminedtobe-0.499V. Thesameapproachisusedtoobtaintheresidualvoltagefort hesampleswith trenchstructures.InFig. 6-10 ,weplottheresidualvoltageforsampleA,BandC. 6.3CalibrationofCand z 0 AsdiscussedinSec. 4.5 ,thegradientoftheforce F 0 ( z ) betweenthesurfaces dependslinearlyontheshiftsintheresonantfrequencyoft heoscillator f = CF 0 ( z ), (6–12) where z isgivenby z = z 0 z piezo b 80

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0 0.5 1 1.5 -0.5 -0.499 -0.498 -0.497 -0.496 -0.495 z ( m m)residual voltage (V) Figure6-9.Theresidualsphere-platevoltage V 0 plottedasafunctionoftheseparation distance z V 0 isfoundtochangebylessthan2mVfor z rangingfrom 100nmto1 m. -0.8 -0.6 -0.4 -0.2 1779 1780 1781 1782 1783 Resonant frequency (Hz)Voltage on sample A (V) (a) -0.8 -0.6 -0.4 -0.2 1780 1781 1782 1783 Resonant frequency (Hz)Voltage on sample B (V) (b) -0.8 -0.6 -0.4 -0.2 1781 1782 1783 1784 Resonant frequency (Hz)Voltage on sample C (V) (c) Figure6-10.Thefrequencysignalplottedasafunctionofth eappliedvoltageataxed separationdistance z for(a)sampleA,(b)sampleBand(c)sampleC.The circlesarethemeasuredfrequencyandthelinesarethetti ngdata. 81

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b isthedistancechangeduetotherotationofthetopplate,wh ichneedtobe takenintoaccountinthecalculationofseparation z .Inordertocalculate b ,the followingstepsaretaken.Weslowlyextendthepiezountilt hegoldsphereandthe siliconplatecomeintocontact.Aswefurtherextendthepiez o,thedistancechange b ,whichequalstothepiezoextensionchange,isaknownvalue .Weextendthe piezoforacouplestepsatastepsizeof2.5nmandrecordthec orrespondingchange incapacitancesignaloftheoscillator.Thus,thechangein thecapacitancesignalis calibratedwithrespecttothedistancechangewhichiscorr espondingtotherotationof thetopplate. Calibrationof z 0 andCisperformedusingelectrostaticforcemeasurements byapplyingadcvoltageVtothesiliconsamplewhilethegold sphereiselectrically grounded.Inonecalibrationprocedure,sixsetsofelectro staticforcemeasurements weretaken.Thevoltageswerechosentobe V 0 +245 mV, V 0 +283 mVand V 0 +300 mV. Theforceforeachvoltageismeasuredtwicetocheckthecons istency.Onlyvoltages largerthan V 0 areusedtoavoiddepletingthesurfaceofthep-dopedsilico nwithcharge carriers. Thetotalmeasuredforcegradientconsistsoftwoparts:the Casimirforceandthe electrostaticforce.Aftersubtractingthecontributionof theCasimirforcegradienttothe frequencyshift,themeasuredfrequencyisttedbythecalc ulatedelectrostaticforce gradientusingatwoparameterleastsquaret. InFig. 6-11 ,thesolidcirclesrepresentthemeasuredelectrostaticfo rcegradient, withtheproportionalityconstantCand z 0 determinedfromttingtoEq. 6–2 (solidline). ThethecalibratedCand z 0 forsixelectrostaticmeasurementsarelistedinTable 6-1 Inthissetofcalibration, C isdeterminedtobe555 3mN 1 s 1 anduncertainties inthedistance z isfoundtobe 0.4nmfromttingtotheelectrostaticforceforsix differentmeasurements.Forsampleswithtrenchstructure s,thenumericalcalculated 82

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0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 10 3F e ¢ (pN m m -1 )z ( m m) z V Figure6-11.Measuredgradientoftheelectrostaticforcea t V = V 0 +0.3 V ontheat siliconsurface(solidcircles)andsampleB(hollowsquare s).Thesolidline isatusingEq. 6–2 .Thedashedlineisatusingtheforcegradient obtainedfromnumericalcalculation(Poissonsolverusing Matlab).Inset:A twoperiodtrenchstructurewithaperiodicityof400nmanda depthof 1 m.Thespacebetweenthecorrugatedstructureandaatsurfa ceis dividedintotriangularmeshtosolvethePoissonequationi n2D ( z =150 nm). electrostaticforcevaluesareusedtottheexperimentald ata,asshownbythedash linesinFig. 6-11 andFig. 6-12 Table6-1.Cand z 0 obtainedfromthettingofthegradientoftheelectrostati cforce.Six setsofforcegradientaremeasuredwiththreedifferentvol tages. 123456 C(m N 1 s 1 )555550559554557552 Z 0 ( m)0.17270.17190.17260.17210.17230.1716 6.4TheCasimirForceMeasurements TheCasimirforcegradientbetweenthegoldsphereandthesi liconsample(ator corrugated)ismeasuredbygroundingthesphereandapplyin gtheresidualvoltage V 0 tothesiliconsample.Bydoingso,theeffectoftheelectrost aticforcecanbeeffectively eliminatedintheCasimirforcemeasurements. 83

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0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 10 3F e ¢ (pN m m -1 )z ( m m) z V Figure6-12.Measuredgradientoftheelectrostaticforcea t V = V 0 +0.3 V ontheat siliconsurface(solidcircles)andsampleC(hollowsquare s).Thesolidline isatusingEq. 6–2 .Thedashedlineisatusingtheforcegradient obtainedfromnumericalcalculation(COMSOLMultiphysics) .Inset:Atwo periodtrenchstructurewithaperiodicityof400nmandadep thof100nm. Thespacebetweenthecorrugatedstructureandaatsurface isdivided intotriangularmeshtosolvethePoissonequationin2D( z =150 nm). Asweknow,theCasimirforcechangesrapidlywhenthetwosurf acesaregetting close.Thus,itisveryimportanttopreciselydeterminethe distancebetweentwo surfaceswhentheirseparationissmall.Giventhattheunkn owncalibrationparameters aremainlydeterminedbytheelectrostaticforcemeasureme ntsatclosedistances,both measurementsforsmallseparationsneedtobenishedwithi narelativelyshortamount oftimeinordertominimizeanypotentialsystemdrift.This isachievedbythefollowing approach.Theelectrostaticforceismeasuredwhenthesphe reisbroughtclosetothe siliconsample.Whentheelectrostaticforcemeasurementis nished,thesphereiskept atthesmallestmeasurementseparation( 100nm).Thenthevoltageappliedonthe siliconsampleisimmediatelyswitchedtotheresidualvolt ageandtheCasimirforceis measuredfromsmalltolargeseparation. 84

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Thereisonemoreparameterthatneedstobechosencarefully ,namelythe amplitudeofthedrivingsignal.AsdescribedinChapter 4 ,weuseadynamicapproach tomeasuretheCasimirforcegradient.Adrivingsignalisap pliedtooneofthebottom electrodestomodulatethetopplateatitsresonantfrequen cy z ( t )= A z cos( r t ), (6–13) where r istheresonantfrequencyoftheoscillator.Forappropriat elysmall A z ,the motionoftheoscillatorcanbegivenbyEq. 4–8 .However,iftheoscillationamplitudeis large,thenon-lineareffectoftheoscillatorduetotheCas imirforceproducesadditional amplitude-dependentfrequencyshifts,asdemonstratedin Ref.[ 31 ].Thus A z needsto besufcientlysmallsuchthatthenon-linearitiescanbene glected.Ontheotherhand, A z shouldbeaslargeaspossibletoprovideareliablesignal. A z ischosenbasedon preliminarymeasurements.Forelectrostaticforcemeasur ements, A z is 1.5nmatthe separationsfrom150nmto500nm.FortheCasimirforcemeasu rements, A z is 1.5nm attheseparationsbelow170nm, 6nmattheseparationsfrom170nmto270nmand 15nmattheseparationsabove270nm. Thedataofthemeasurementswillbepresentedinthefollowi ngchapters. 85

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CHAPTER7 SPHERE-PLATESTRUCTURE:EXPERIMENTANDTHEORY TheCasimirforce,intheoriginalcongurationdiscussedb yCasimir,isthe interactionbetweenidealmetalplatesatzerotemperature .Incaseofidealmetal, allfrequenciescanbereectedperfectly.Forrealsurface s,however,therearemany nonidealities,suchasniteconductivity,surfaceroughn essandnitetemperature. Inthischapter,IwilldiscussthecorrectionstotheCasimi rforceduetothenonideal surfaces.Themeasurementresultsforsphere-platestruct urewillbecomparedwith theoreticalpredictionsincludingrealsurfacecorrectio ns. 7.1DescriptionoftheMaterial Toaccountfortheniteconductivityofmaterials,thediel ectricpermittivitiesalong theimaginaryaxis ( i ) areusedinthetheoreticalcalculationoftheCasimirforce However,thereisoftennosimpleformofthepermittivityas afunctionofthefrequency. Differentapproacheshavebeenusedtoobtain ( i ) Oneofthecommonapproachestoestimatetheopticalpropert iesofmetalsis theplasmamodel.Accordingtothisapproach,thedielectric functioncanbegivenby [ 50 51 ] ( )=1 2 p 2 (7–1) where p istheplasmafrequency.Forimaginaryfrequency = i ( i )=1+ 2 p 2 (7–2) Theplasmamodelisadescriptionofthehighfrequencyoptic alproperties.Itis unphysicalformetalduetothedivergenceof 1 = 2 forsmallfrequencies.For metal,a 1 =! behaviorhasalreadybeenclearlyshowninthelowfrequency “tail”of theexperimentaldata[ 52 ]. 86

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Anotherapproachisbasedonthetabulatedopticaldata.Inth isapproach, ( i ) is foundthroughtheKramers-Kronigrelation[ 53 54 ] ( i )= 2 Z 1 0 00 ( ) 2 + 2 d +1, (7–3) where 00 isthecomplexpermittivitygivenby 0 + i 00 = n 2 k 2 +2 ink (7–4) n istherealpartofthecomplexindexand k istheimaginarypart.Bothofthemare tabulatedasafunctionoffrequencyinseveralreferences[ 52 ].Sincethetabulated opticaldataareavailableonlyforparticularregions,the complexpermittivityneedstobe extrapolatedforlowerfrequencies.Onecommonmethodisto usetheDrudemodel,in whichthedielectricfunctionisgivenby ( )=1 2 p ( + i r ) (7–5) where r istherelaxationparameter.FromEq. 7–5 wecanobtain 00 = 2 p r ( 2 + r 2 ) (7–6) Thedivergence 1 =! oftheDrudemodelprovidesaproperlowfrequencydescripti on formetals.However,aspointedoutbybysometheoreticalan alysis,ifweuse ( i ) obtainedfromtheextrapolationintheLifshitztheory,itc anleadtoaviolationofthe Nernstheattheoremforperfectcrystallattices[ 55 ]possiblyresultinginthecalculated Casimirforcebeingincontradictionwiththeexperiments. Itisstillacontroversyon whethertheplasmamodelortheDrudemodelismoreappropria tefordescribingthe opticalpropertiesofmetalsinCasimirforcecalculations Forgold,theplasmafrequency p g is9eVandtherelaxationrate r is35meV [ 56 ].InFig. 7-1 ,thedielectricpermittivitiesalongtheimaginaryaxis g ( i ) ,whichare obtainedfromtheapproachesdiscussedabove,areplotteda safunctionoffrequency. 87

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10 14 10 16 10 18 10 0 10 2 10 4 10 6 e g (i w )w (rad s -1 ) Figure7-1.Thepermittivityalongtheimaginaryaxisforgo ldcalculatedusingthe plasmamodel(reddashline)andthetabulateddatatogether withDrude model(blacksolidline). Forintrinsicsiliconwitharesistivity 0 =1000n cm,thetabulatedopticaldataare availablefor !> 0.00496 eV[ 52 ].Sincethecomplexpermittivityisavailableforsufcient lowfrequenciesnoexptraplationisneed.Thus,Eq. 7–3 isusedtoobtain si i ( i ) ,which isshownbytheblacksolidlineinFig. 7-2 However,inourexperimentsthesiliconplatesareheavilyp -dopedwitharesistivity muchlowerthantheoneusedinthetables.Thetabulatedopti caldataareadaptedfor ourheavilydopedsiliconbyaddingtheimaginarypartofthe Drudedielectricfunctionto thatobtainedfromtables si d ( i )= si i ( i )+ 2 p si ( + r si ) (7–7) 88

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10 14 10 16 10 18 10 0 10 1 10 2 e si (i w )w (rad s -1 ) Figure7-2.Thepermittivityalongtheimaginaryaxiscalcu latedusingEq. 7–3 for intrinsicsilicon(blacksolidline)andheavilydopedsili con(reddashline). where p si istheplasmafrequencyforsiliconand r si istherelaxationparameter.The valueof p si and r si canbegivenby p si = e p n p 0 m (7–8) r si = 0 2 p si (7–9) where n isthecarrierdensity, 0 isthepermittivityofvacuum, istheresistivityand m =0.34 m e istheelectroneffectivemassforp-dopedsilicon.Usingaf our-probe technique,theresistivityofoursiliconsampleisdetermi nedtobe =0.028n cm,which leadstoacarrierdensity n =2 10 18 cm 3 .InFig. 7-2 thedependenceof si d ( i ) on frequencyisshownbythereddashline. 7.2FiniteConductivityCorrection Inourexperiments,theCasimirforcegradient F 0 c = @ F c =@ z ismeasured betweenagoldspherewitharadiusRandasiliconplate.Since R >> z ,weuse 89

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PFA(asdescribedinSec. 1.4.1 )torelatetheinteractionforthesphere-planeandthe plane-planegeometriesaccordingto F 0 c s p =2 RP c p p ,where P c p p istheforceper unitareabetweentwoparallelinniteplates.Tocompareth emeasurementresultswith thetheory,werstcalculatetheCasimirforceincludingn iteconductivitycorrections usingLifshitzformula P c p p ( z )= ~ 2 2 c 3 R 1 0 3 d R 1 1 p 2 h ( s 1 + p )( s 2 + p ) ( s 1 p )( s 2 p ) e 2 p z = c 1 i 1 + h ( s 1 + 1 p )( s 2 + 2 p ) ( s 1 1 p )( s 2 2 p ) e 2 p z = c 1 io 1 dp (7–10) where = i isthecomplexfrequency, j ( i ) isthedielectricfunctionofmaterialand S j = p j 1+ p 2 Withthedetermined j ( i ) ,theCasimirforcecanbecalculatedbynumerically integratingEq. 7–10 [ 54 ].Thebasicstepsaredescribedasbelow.InEq. 7–10 ,we makeasubstitution x =2 p z = c (7–11) First,ataxedseparation z theintegrationisdonebyintegratingover x whilexing p Therangeandthestepsizeforthe x integrationisdeterminedbytherangeandstep sizeof j ( i ) .Thisstepyieldsanintegralwhosevaluedependson p .Then,Eq. 7–10 isintegratedover p ataadjustedstepsize dp p = 100 toobtaintheCasimirforceat separation z .Finally,theintegrationisrepeatedasafunctionof z Inthetheoreticalcalculationpresentedinthischapter, j ( i ) iscalculatedfrom tabulateddataasdescribedinSec. 7.1 .Thisiscommonlyusedinrecentexperimentsto comparethemeasurementdatawiththetheory.Notethatitis stillcontroversialamong researchersonwhethertheplasmamodelorthetabulateddat aapproachismore appropriatefordescribingtheopticalpropertiesofgold, wecomparetheCasimirforce forbothapproaches.Thedifferencebetweenthecalculated forcesis2 % at150nm separationanddecreasesto0.3 % at300nmseparation.Suchdifferencesaretoosmall toberesolvedinoursetup. 90

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7.3RoughnessCorrection TheroughnesscorrectioncanplayanimportantroleintheCa simirforcecalculation. Ithasbeenreportedthattheroughnesscorrectioncancontr ibute20 % ofthemeasured forceattheshortestseparationinthemeasurements[ 19 ].However,whenspecial preparationproceduresareperformedtodecreasethesurfa ceroughness,thecorrection canbereducedtolessthan1 % evenattheshortestseparation[ 20 ].Thus,acareful analysisoftheroughnesscorrectionisnecessaryforcompa risionoftheorywith experiments. Theinvestigationofthetopographyofthesiliconplateand thegoldcoatingonthe spherehasbeendiscussedinSec. 5.1 .Toestimatetheroughnesscorrection,werst calculatethezeroroughnesslevel H 0 denedas X i ( H 0 h i ) v i =0, (7–12) where h i istheroughnessheightand v i isthefractionofthesurfaceareawithheights h i h < h i +1 .Notethattheseparationbetweentwosurfacesismeasuredf romthezero roughnesslevelinourforcemeasurements. Weestimatetheroughnesscorrectionbyaveragingthediffe rentpossibleseparation distancesonthetwosurfacesresultingfromthesurfacerou ghness P ( z i )= X k X l v k v l P c p p ( z i + H (1) 0 + H (2) 0 h k h l ), (7–13) where P c p p istheCasimirforceincludingniteconductivitycorrecti on.Asdiscussedin Section 5.1 ,goldcoatedsurfacedominatestheroughnesscorrection.W esimplifythe calculationto P ( z i )= X l v l P c p p ( z i + H ( Au ) 0 h l ). (7–14) Thecalculationshowsthattheroughnesscorrectionismode rateinourmeasurements intheseparationsrangedfrom150nmto500nm.Whenthesepara tionincreasesfrom 150nmto500nm,thecorrectionstotheCasimirforceduetoro ughnessdecreasefrom 91

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0.2 0.3 0.4 0.5 0 50 100 150 200 250 300 F c,flat ¢ (pN m m -1 )z ( m m) Figure7-3.Dots:measuredCasimirforcegradientbetweena goldsphereandaat siliconsurface.Solidline:thetheoreticalCasimirforceg radientincludingthe niteconductivityandsurfaceroughnesscorrections. 0.73 % to0.07 % .Thusthemaximumcorrectionduetoroughnessisabout 0.73% forour sample. InFig. 7-3 ,thetheoreticalCasimirforceincludingtheniteconduct ivityandsurface roughnesscorrectionsisplottedasthesolidline. 7.4ThermalCorrection Allthecomputationdiscussedabovearedoneforzerotempera ture.However,the Casimirforcemeasurementswereperformedatroomtemperat ure.Inordertoprovidea comprehensivecomparisonbetweentheoryandexperiments, itisnecessarytoestimate thethermalcorrection. TocalculatethethermalCasimirforcebetweentwoparallel platesataseparation z inthermalequilibrium,Eq. 7–10 ismodiedbyreplacingtheintegrationincontinuous 92

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withasummationoverthediscreteMatsubarafrequencies l l = 2 k B T ~ 10 X l =0 (7–15) where k B istheBoltzmannconstantandprimereferstotheadditionofm ultiple 1 = 2 in theterm l =0 Recently,therehasbeenextensivediscussiononthermalco rrectionstotheCasimir force.Differentapproacheshavebeenusedtocalculatethe temperatureeffectonthe Casimirforcebetweenrealmaterialwithniteconductivit y.Theseincludeusingthe DrudemodelalongtheimaginaryaxisintheLifshitzformula [ 57 58 ]andusingthe free-electronplasmadielectricfunctionintotheLifshit zformula[ 50 51 ],and(c)usethe surfaceimpedanceboundaryconditiontodescribethetherm alCasimirforce.However, theresultsproducedbytheseapproachesarenotconsistent witheachother,dueto thedifferentwaysthatthezeroMatsubarafrequencyofthee lectromagneticeldwould contributetotheCasimirforce.Forexample,usingDrudemo del,unexpectedlarge temperaturecorrectionsresultatsmallseparations. Generallyspeaking,theinuenceofthethermalelductua tionsontheCasimir forceisimportantforseparationsontheorderof T = ~ c k B T (7–16) Atroomtemperature T =300 K ,then T 7 m.Thereforethetemperaturecorrection shouldbenegligibleatsmallseparations.Sincethemeasure mentrangeforour experimentsarefrom150nmto500nm,thethermalcorrection sarelessthan1 % inthisregion,whichissmallerthanthemeasurementuncert aintyinoursetup. 93

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CHAPTER8 DEMONSTRATINGTHEGEOMETRYDEPENDENCEOFTHECASIMIRFORCE Inmydoctoralresearch,theCasimirforcegradientismeasu redonplateswith periodicrectangulartrenchesinthreedifferentdimensio ns.Signicantdeviations fromPFA/PAAareobserved,demonstratingthestronggeometryd ependenceofthe Casimirforce.Themeasureddeviationsareingoodagreemen twiththetheoretical calculationsthattakeintoaccounttheniteconductivity ofthematerials.Ourresults demonstratetheinterplaybetweengeometryeffectsandthe materialeffects.Inthis chapter,theexperimentalresultsarepresented.Thenadet ailedcomparisonbetween theexperimentaldataandthetheoreticalcalculationsisp rovided. 8.1HighAspectRatioRectangularCorrugations Inourrstseriesofexperiments,theCasimirforcewasinve stigatedusingsilicon plateswithhighaspectratiorectangularcorrugationswit hadepth 1 m.Twosets ofsamplesaremeasured,sampleAwithaperiodicityof1 mandsampleBwitha periodicityof400nm.Theexperimentresultsinthissectio ncanbefoundinarticle MeasurementoftheCasimirForcebetweenaGoldSphereandaSil iconSurfacewith NanoscaleTrenchArrays ,H.B.Chan etal. ,Phys.Rev.Lett.101,030401(2008). 8.1.1PredictionbyPFA TodemonstratethestrongdependenceoftheCasimirforce,w eevaluatethe deviationofthemeasuredCasimirforcefromtheprediction ofPFA.Asdiscussedin Chapter 2 ,underPFAtheinteractioninplate-trenchgeometrycanbegi venby F PFA ( z )= 1 2 F rat ( z )+ 1 2 F rat ( z + t ). (8–1) F rat istheforcebetweentwoparallelplates,where z ismeasuredfromthetoppartof thetrencharrayand t isthedepthofthetrenches.Fordeeptrenches,thecontribu tion fromthebottompartisnegligiblesincetheCasimirforceat thisseparation( z + t > 1 m) 94

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istoosmalltobedetected.Therefore,equation 8–1 issimpliedinto F PFA ( z )= 1 2 F rat ( z ). (8–2) Consideringthefractionofsolidvolumeofthetrencharray is p insteadof50 % ,the interactionunderPFAcanbegivenby F PFA ( z )= pF rat ( z ). (8–3) Anothercommonlyusedapproximationisthepairwiseadditiv eapproximation(PAA) whichcalculatestheinteractionbypairwiseadditionofth evdWforce.Asdiscussedin Sec. 2.1 ,thePFAandthePAApredictthesameforcefortheplate-trench situationin ourexperiments.Thus,weonlywritethePFAfortheremaining chapter. Inourexperiments,theCasimirforcegradientismeasuredi nthesphere-trench geometry.ThisisconnectedtotheCasimirforceinplate-tr enchgeometryinthe followingapproach.Accordingtotheproximityforcetheore m[ 48 ] F sphere trench ( z )=2 RE plate trench ( z ), (8–4) where E plate trench ( z ) istheCasimirenergyperunitareaforaplate-trenchgeomet ry. DifferentiatingEq. 8–4 withrespecttoz,weobtain F 0 sphere trench ( z )= 2 RP plate trench ( z ), (8–5) where P plate trench ( z ) istheCasimirforceperunitareaforaplate-trenchgeometr y.Thus, wehave F 0 PFA ( z )= pF 0 rat ( z ). (8–6) InFig. 8-1 ,themeasuredCasimirforcegradientsareplottedforsampl eAandB togetherwiththePFAprediction. 95

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0.2 0.3 0.4 0.5 0 50 100 150 F c,B ¢ (pN m m -1 )z ( m m) (b) 0.2 0.3 0.4 0.5 0 50 100 150 F c,A ¢ (pN m m -1 )z ( m m) (a) Figure8-1.MeasuredCasimirforcegradientbetweenthesam egoldsphereand (a)sampleA, F 0 c A ( =1 m)and(c)sampleB, F 0 c B ( =400nm).thelines representtheforcegradientsexpectedfromPFA( F 0 c rat ). 8.1.2DeviationsfromPFA ToanalyzethedeviationfromPFA,weconsidertheratiooftheC asimirforcein trenchgeometrytothepredictionunderPFA = F 0 c trench = pF 0 c rat (8–7) Theratios A = F 0 c trench = p A F 0 c rat and B = F 0 c trench = p B F 0 c rat areplottedinFig. 8-2 Notethat equalsoneifPFAisvalid.ForsampleAwith = t =0.94 ,themeasured forcegradientdeviatesfromPFAby 10 % .Thedeviationincreasesto 20 % for sampleBwith = t =0.41 .BothsamplesshowcleardeviationfromPFAprediction fortheseparationbetween150nmand250nm,demonstratingt hestronggeometry dependenceoftheCasimirforce.Theuncertaintyincreases considerablyatlarger separationsbecauseofthedecreaseoftheforcegradient.T heobserveddeviations occurbecausetheCasimirforceisassociatedwithconnede lectromagneticmodes withwavelengthcomparabletotheseparationbetweenthein teractingobjects,which areaffectedbythepresenceoftrenches.Inthelimitation << z ,thesemodescan nolongerpenetrateintothetrenches,renderingtheCasimi rforceonthecorrugated surfaceequaltoaatone,whichleadtodeviationsfromPFAby afactorof2. 96

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0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.8 1 1.2 1.4 rz ( m m) Figure8-2.RatiooftheCasimirforceintrenchgeometrytot hepredictionunderPFA = F 0 c trench = pF 0 c rat forsampleAwith = t =0.94 (bluesolidcircles)andB with = t =0.41 (redhollowsquares).Theoreticalpredictionsforperfect conductingsurfacesareplottedasasolidline( = t =1 )andadashline ( = t =0.5 ). Theexperimentalresultsarecomparedtothecalculationsb yB ¨ uscherandEmig. Intheirtheoreticalcalculation,theCasimirforceiscalc ulatedonthetrenchstructures with p =0.5 forperfectconductor.AsshowninFig. 8-2 ,thesolidlineandthedashline arethecalculationontrenchstructurewith = t =1 and = t =0.5 respectively.Note thegeometryusedinthetheoreticalcalculationisslightl ydifferentfromoursample. Nevertheless,thequalitativetrendscanbeobtainedfromt hecomparison.First,the deviationfromPFAinsampleBislargerthaninsampleA,whichi sconsistentwith thetheoreticalpredictionthatthedeviationfromPFAisstr ongerwhen = t decreases. Second,eventhoughupto 20% deviationisobserved,themeasureddeviationsforboth samplesaresmallerthanthetheoreticalpredictionforape rfectconductorby 50% Suchdiscrepancyofmeasurementfrompredictionbasedonper fectmetalarisesdue totheinterplaybetweengeometryeffectsandniteconduct ivity,asdiscussedinthe followingsections. 97

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8.1.3TheoreticalCalculationincludingmaterialpropert ies Thediscrepancybetweenourmeasureddeviationandthatofp redictionsforperfect conductorsstimulatetheinterestsoftheorists.Twogroup scalculatedtheCasimir forceforourgeometriesbasedonscatteringtheorywhichta keintoaccountthenite conductivityofthematerials.8.1.3.1ScatteringtheoryapproachtotheCasimirforce ThescatteringtheoryapproachtotheCasimirforceisapowe rfulmethodthat cancalculatetheCasimirforceforarbitrarygeometriesin cludingtheniteconductivity correctionandthenitetemperaturecorrection[ 59 60 ].Inthissection,thebasic conceptofthescatteringtheoryisbrieysummarized. Basedonthescatteringtheory,theinuenceofagroupofobje ctsontheelectromagnetic eldisdescribedbytheirscatteringproperties.Inthefra meworkofthisapproach,there aretwokeycomponents:(1)thescatteringamplitudeofeach objectconsideringthe objectasisolatedscattererand(2)thetranslationmatric eswhichdependonlyonthe separationsandorientationsoftheobjects.Atzerotempera ture,theinteractionfree energybetweentwobodiesisgivenby E ( z )= ~ c 2 Z 1 0 dk Tr log[1 R 1 X 12 ( z ) R 2 X 21 ( z )], (8–8) where R i isthescatteringoperator, X 12 ( z ) and X 21 ( z ) arethetranslationoperators. Equation 8–8 ingeneralcanrepresentextremelycomplicatedsituations .However, itcanbesimpliedforcertainsystems.Inparticular,fort hefollowinggeometry:aplate withaperiodic,rectangulartrencharrayandaparallelpla te,aspecicbasecanbe chosentosimplifyEq. 8–8 basedonthesymmetrypropertiesofperiodicsystem.Thus, theCasimirforceperunitareacanbegivenby F ( z )= ~ ZZZ Tr ((1 M 1 @ z ( M ) d 2 k ? d ), (8–9) 98

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where k ? gatherthecomponentsofthewavevectorintheplaneoftheob jects, isthe imaginaryfrequency. M isdenedas M = R 1 ( ) e z R 2 ( ) e z (8–10) where = p 2 = c 2 + k 2 ? InordertocalculatetheCasimirforceatnon-zerotemperat ure,theintegral ~ c 2 R 1 0 dk isreplacedbythesummationoverMatsubara(imaginary)fre quencies 1 P 0l .Equation 8–8 isthenmodiedinto E ( z )= 1 10 X l Tr log[1 R 1 X 12 ( z ) R 2 X 21 ( z )], (8–11) 8.1.3.2TheoreticalcalculationI Davids etal. calculatedthenitetemperatureCasimirforcebetweenasi liconplate withdeeptrenchesandaparallelgoldsurfacebasedonthesc atteringtheory.Intheir approach,Eq. 8–11 isnumericallycalculatedat T =300 Kusingamodalexpansion. Thebasicconceptofthemodalexpansionisaplanewaveexpan sionoftheeldsand aFourierdecompositionofthepermittivityofthestructur e.Inordertoprovideaprecise comparisonbetweenthetheoreticalcalculationandtheexp erimentresults,theoptical propertiesofthesamplesusedinourexperimentshavebeeni ncludedinthecalculation code.ThegoldsurfaceismodeledbytheDrudemodel g ( i )=1+ 2 p g ( + r g ) (8–12) where p g =1.27524 10 16 rad/sand r g =6.59631 10 13 rad/s.Theintrinsicsilicon permittivityismodeledbytheDrude-Lorentzmodel si ( i )= 1 +( 0 1 ) 2 0 2 + 2 0 (8–13) 99

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where 0 =11.87 1 =1.035 and 0 =6.6 10 15 rad/s.Toadaptittothepropertyof p-dopedsilicon,aDrudebackgroundisaddedtotheintrinsi cpart si d ( i )= si ( i )+ 2 p si ( + r si ) (8–14) where p si =3.6151 10 14 rad/sand r si =7.868 10 13 rad/s.Note,these materialsmodelsandparametersarecitedfrom[ 61 ],whichisslightlydifferentfrom theparametersthatweusedtocalculatetheCasimirforcebe tweenagoldsphereanda siliconplate.8.1.3.3TheoreticalcalculationII R.Gu erout etal. calculatedthezerotemperatureCasimirforceinplate-tre nch geometriesbasedonscatteringtheoryforboththedeepands hallowtrenches.In thisapproach,theCasimirforceperunitareabetweentwore ectingobjectscanbe givenbyEq. 8–9 (Sec. 8.1.3 ).Thedescriptionofthematerialsisdoneviatheuseof thedielectricfunctionsevaluatedatimaginaryfrequenci es ( i ) whichenterintothe descriptionofthereectionoperators.Thedielectricfun ctionsaremodeledusingEq. 7–3 and 7–4 withtabulatedopticaldataasdescribedinSec. 7.1 .Forgold,optical dataareextrapolatedatlowfrequenciesbytheDrudemodelEq 7–6 ,withtheplasma frequency p g =9 eVandtherelaxationrate r g =35 meV.Thep-dopedsiliconis modeledbyaddingaDrudebackgroundtothedielectricfunct ionoftheintrinsicsilicon (Eq. 7–7 ),with p si =1.36 10 14 rad/sand r si =4.75 10 13 rad/s. 8.1.4ComparewithTheoryincludingMaterialProperties Inthisstep,amoreprecisecomparisonisperformedbetween theexperimental resultsandtheexactcalculationsincludingniteconduct ivitycorrections.InFig. 8-3 and 8-4 ,thesolidlinesrepresenttheratiooftheexactnumericsto thetheoreticalPFA predictioncalculatedbyDavids etal. andthedashlinesrepresentthecalculationsby R.Gu erout etal. .Theexactnumericsarecalculatedingeometrieswhicharei dentical tooursamplesbasedonscatteringtheoryincludingniteco nductivitycorrection.The 100

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F 0 c rat inthetheoreticalpredictioniscalculatedusingtheLifsh itzformulawiththesame opticaldataasthecalculationinthetrenchstructure.The hollowsquareswitherror barsaretheratioofmeasuredforcegradientandthe“experi mental”PFAprediction, inwhich F 0 c rat isobtainedfromaseparatemeasurementinsphere-planegeo metry. Theatsurfaceandthetrencharraysamplesarefromthesame waferandfollowthe samepreparationproceduresothatwecanexpectthesameopt icalpropertiesfor them.Inaddition,bothmeasurementsintrenchgeometryand planegeometryused thesameexperimentalsetup(samemicromechenicaloscilla torandgoldsphere)to providethesamemeasurementenvironment.Giventheuncert aintyinthenumerical calculationandtheexperimentalmeasurements,theexperi mentaldataagreeswellwith thetheoreticalcalculation.Themeasuredforcegradienta ttheclosestmeasurement separationdeviatesfromPFAby 10% forsampleAand 20% forsampleB. 8.2ShallowTrenches AfterobservingthestrongdeviationsfromPFAinhighaspectr atiotrenches,we designedandmeasuredanothersetoftrenchstructurewitha depthof98nmand aperiodicityof400nm.Forthisgeometry,boththetopandbo ttomsurfacesofthe corrugationscontributetothePFAprediction.Theexperime ntandthetheoretical calculationresultsinthissectioncanbefoundin TheCasimirforceonasurfacewith shallownanoscalecorrugations:Geometryandniteconduc tivityeffects ,Y.Bao etal. Phys.Rev.Lett.Adetailedtheoreticalapproachwaspresent edin CasimirInteractionof DielectricGratings ,AstridLambrechtandValeryN.Marachevsky,Phys.Rev.Lett. 101, 160403(2008).8.2.1PFAPrediction Thedepthoftheshallowtrenches( t =97.8 nm)issmallerthanthetypical separationbetweenthetwointeractingbodies,sothatthec ontributionofthebottom parttothePFApredictionisnotnegligible.Inaddition,the trencheshaveaslightly trapezoidalshapeinsteadofperfectrectangular.Aschema ticaschematicofthecross 101

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0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0.8 0.9 1 1.1 1.2 1.3 r Az ( m m) Figure8-3.Theratio A oftheexactCasimirforcegradientintrenchgeometrytothe PFApredictionforsampleA.Thehollowsquaresarethemeasure dforce gradientdividedbythe“experimental”PFA.The“experimenta l”PFArefersto themeasuredCasimirforceinsphere-planegeometrymultip liedby p A .The solidanddashlinesarethetheoreticalcalculationinthee xactgeometry includingniteconductivitycorrectionstothetheoretic alPFAcalculatedbyP. S.Davids etal. fortemperatureof300KandR.Gu erout etal. forzero temperaturerespectively.ThetheoreticalPFAiscalculate dusetheLifshitz formula. sectionoftheshallowtrenchesisshowninFig. 5-7 ,with l 1 isthetoplengthinone period, l 2 isthebottomlength, istheperiodicityand t isthedepth.Toaccountforthe contributionofboththesidewallsandthebottomparts,the interactionunderPFAcan begivenby F PFA = 1 Z 0 F rat ( z ( x )) dx = p 1 F rat ( z )+ p 2 F rat ( z + t )+2 Z p 3 0 F rat ( z t x = p 3 ) dx (8–15) where p 1 = l 1 = p 2 = l 2 = and p 3 =(1 p 1 p 2 ) = 2 .For“theoreticalPFA”, F rat is calculatedusingtheLifshitzformulawiththesameoptical parametersasthetrench 102

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0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 1 1.2 1.4 r Bz ( m m) Figure8-4.Theratio B oftheexactCasimirforcegradientintrenchgeometrytothe PFApredictionforsampleB.Thehollowsquaresarethemeasur edforce gradientdividedbythe“experimental”PFA.The“experimenta l”PFArefersto themeasuredCasimirforceinsphere-planegeometrymultip liedby p B .The solidanddashlinesarethetheoreticalcalculationinthee xactgeometry includingniteconductivitycorrectionstothetheoretic alPFAcalculatedbyP. S.Davids etal. fortemperatureof300KandR.Gu erout etal. forzero temperaturerespectively.ThetheoreticalPFAiscalculate dusetheLifshitz formula. geometrycalculation.For“experimentalPFA”, F rat ismeasuredfromaatsurface madeofthesamematerialasthesamplewiththeshallowtrenc harray. 8.2.2ComparewithTheory ToanalyzethedeviationfromPFA,theratio = F 0 c trench = F 0 c PFA iscalculated.In Fig. 8-5 ,thehollowsquareswitherrorbarsarethemeasuredCasimir forceinshallow trenchgeometrydividedbythe“experimentalPFA”.Themeasur edforcegradientclearly deviatesthePFApredictionbyupto 15% .ThesolidlineistheexactCasimirforce calculatedforshallowtrenchgeometrydividedbythe“theo reticalPFA”.Thetheoretical calculation,includingmaterialproperties,yieldsagood agreementwithmeasurements. Ifwereplacethetwosurfaceswithperfectconductingsurfa ce,asrepresentedbythe dashlineinFig. 8-5 ,thedeviationofthecalculatedforceexceedsthemeasured value 103

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0.2 0.3 0.4 0.5 0.8 0.9 1 1.1 1.2 1.3 r Cz ( m m) Figure8-5.Theratio C oftheexactCasimirforcegradientintrenchgeometrytothe PFApredictionforsampleC,wherethePFApredictioniscalcul atedusing Eq. 8–15 .Thehollowsquaresarethemeasuredforcegradientdivided by the“experimental”PFA.Thesolidlineisthetheoreticalcalc ulationinthe exactgeometrytothetheoreticalPFAatzerotemperature.Th edashedline isalinearinterpolationbetweenthetwotheoreticalvalue s(solidcircles) assumingperfectconductors. byafactorof2.Thisresultdemonstratethenon-trivialint erplaybetweenthematerial dependenceandthegeometrydependenceoftheCasimirforce 8.3DeterminationofErrors Toestimatethetheoreticalprecisionandtheexperimental precision,theerrors originatedfromdifferentsourcesareanalyzedinthissect ion. 8.3.1DeterminationoftheExperimentalErrors Randomerrors .Inthemeasurements,therandomerrorsarisemainlyfrom thermomechanicaluctuationsoftheoscillator.Attheclos estseparation z =150 nm, withtheaverageof10measurements,therandomerror R =3.5 pN/ mandthe relativerandomerror R =2% .Attheseparation z =300 nm,withtheaverageof 104

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50measurements,therandomerror R =0.3 pN/ mandtherelativerandomerror R =2% Systematicerrors .Therearethreemainsourcesofsystematicerror: R = R R (8–16) C = C C (8–17) z 0 =3 z 0 z 0 (8–18) Inourexperiments,thevalueof R wastakenfromthemanufacturespecication insteadofactualmeasurements.Therecanbeupto10 % errorresultingfromthe uncertaintyofthedeterminationofR.However,inourcompa risonofexperimentand theory,theCasimirforcehasbeennormalizedbythePFApredi ction.For“experimental PFA”,theforcegradientbetweenasphereandaplaneismeasure dusingthesame sphereasthemeasurementsoftrenchstructures.Thus,ther adius R isruledoutfrom thedivisionandnolongercontributestotheerror.Fromthe electrostaticcalibration, C 0.6% .Attheclosestdistance z =150 nm, z 0 1% 8.3.2DeterminationoftheTheoreticalErrors Numericalerrors .InthecomputationoftheCasimirforce,numericalerrorsc an originatefrommultiplesources.CitedfromRef.[ 61 ],anexampleofnumericalerrorsin thedeeptrenchcalculationispresented.Therearethreema inerrorsources.Oneis theterminationoftheMatsubarafrequencysummation,whic hismainlydeterminedby theminimumseparationbetweentwotestbodies.InDavids'c alculation,36Matsubara frequencieswereusedyieldingaconvergenceofbetterthan 10 4 attheseparationof 100nm.Anothererrorsourceisthetruncationofthediscrete spatialfrequencyspectrum (diffractionorders)resultinginnitedimensionalreec tionmatrices.Increasingthe diffractionorder N willimprovethemodalapproach,butalsoincreasethecompu tational 105

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cost.Using N =5 ,theconvergenceisat 1% accuracy.Thelasterrorsourceis thenumericalintegrationoverthecontinouswavevectorin therstBrilliounzone,the relativeerrorofwhichislessthan3 % forourexperimentalseparationrange.Thus,the estimatedtotalerrorislessthan 3% fortheseparationrangeinourexperiments. Errorfromthedescriptionofthematerial .Toincludetheniteconductivity correctionsintotheCasimirforcecalculation,thedielec tricpermittivitiesalongthe imaginaryaxis ( i ) areusedtodescribethematerials.Sofar,theactualoptical propertieshaveneverbeenmeasuredineachindividualexpe rimentforsufcientlarge rangeoffrequencies.The ( i ) isobtainedfromdifferentmodels,suchasplasmamodel [ 62 ],DrudeModel[ 63 ]andgeneralizedDrudeorplasmamodel[ 64 ],resultinginan errorinthetheoreticalcomputations.Baseonourcalculati onbetweenagoldsphere andap-dopedsiliconplate(Sec. 7.2 ),thedifferenceresultingfromusingplasmamodel ortabulateddataforgoldis2 % ,0.3 % and0.2 % at150,300and500nmrespectively. Thus,weestimatetheupperboundoftheerrorsresultingfro mthemodeloftheoptical propertiestobe2 % Errorfromproximityforcetheorem .Inthetheoreticalcalculationsofthetrench geometry,theCasimirforcegradientinsphere-trenchgeom etryiscalculatedfromthe Casimirforceplate-trenchgeometryusingPFA F 0 sphere trench =2 RF plane trench .The approximationisvalidat R >> z ,where R istheradiusofthesphereand z isthe separationbetweentotestbodies.Therehavebeenanumbero ftheoreticalcalculations tostudytheCasimirforcebetweenasphereandaplanebeyond PFA[ 62 65 ].To specifytheaccuracyofPFA,weconsiderthecorrectionfactor p s = F p s F PFA p s (8–19) Forsmallvaluesof(L/R),theexpressionof p s isassumedtobeoftheform[ 65 – 67 ] p s =1 z R + O ( z 2 R 2 ), (8–20) 106

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where = 5 2 1 3 .Thus,weconsidertheupperlimitoferrorintroducedbyPFAi s z = R Inourexperiments,theradius R =51.5 mand z rangesfrom150nmto500nmsothat theerrorislessthan1 % Errorfromthesurfaceroughness .Inthecomparisonofexperimentaldataand theoryfortrenchstructures,wedidnottakeintoaccountth econtributionofthesurface roughnesscorrection.Weestimatetheerrorintroducedbyt hesurfaceroughnessbased ontheroughnesscorrectiontotheCasimirforcebetweenasp hereandaatsurface. FromthecalculationinSec. 7.3 ,theerrorshouldbelessthan 1% inthemeasurement range. 107

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CHAPTER9 SUMMARY InthisdissertationtheCasimirforceismeasuredbetweena goldsphericalsurface andasiliconnanostructuredsurface,allowingfortheinve stigationofthegeometryand theniteconductivityeffectsoftheCasimirforce. 9.1SummaryoftheChapters InChapter 1 ,weintroducedtheCasimirforceanddevelopmentsintheCas imir forcemeasurements.TwoimportantpropertiesoftheCasimi rforceareitsstrongshape dependenceandmaterialdependence.Whilethematerialdepe ndencehasbeenwidely studiedinsmoothgeometries,fewattemptshavebeenmaketo demonstratetheshape dependenceexperimentally.Themainfocusofthisdisserta tionwastoexperimentally investigatethestronggeometrydependenceoftheCasimirf orceforrealmaterials. InChapter 2 ,weproposedtheexperimentalapproachofreplacingtheat plate withanarticiallydeformedsurfaceforthewelldeveloped sphere-platesetuptoreveal thestronggeometrydependence.Thedeformedsurfaceisint roducedasaplate withanarrayofnano-scaleperiodicrectangulartrenches. Thedimensionsofthe trenchesarechosenbasedonB ¨ uscherandEmigstheoreticalpredictionstoexhibit signicantdeviationfromthepairwiseadditiveapproxima tionandtheproximityforce approximation.Trencheswithtwodepthswereselected:hig haspectratiotrencheswith adepthof1 mandshallowtrencheswithadepthof100nm. Thedetailedsamplefabricationprocedurewasdiscussedin Chapter 3 .The sphericalgoldsurfaceismadefroma100 mdiameterglassspherecoatedwitha 5nmlayeroftitaniumfollowedbya400nmlayerofgoldusingR Fsputteringdeposition, whichprovidedthesmoothestgoldsurfacethatwecouldachi eve.Thesiliconsamples withtrencharrayswerefabricatedonahighlydopedsilicon wafercoveredwitharesist patternandwereprovidedbyBellLabs.Twodifferentdryetch approacheswereused. Thedeepreactiveionetchwithasimultaneousetch/passiva tionrecipewasusedto 108

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createhighaspectratiotrenches.Areactiveionetcherwit hainductivelycoupled plasmamodulewasusedtocreateshallowtrenches.Nopassiv ationgaswasused intheetchingprocessofshallowtrenchesinordertominimi zetheroundingissueat thebottomofthetrenches.Notethatthesiliconsurfaceisv eryreactiveinair,aquick hydrouoricacidetchingisnecessarytoremovethenativeo xideonthesiliconsurface andleadtohydrogenterminationofthesurface. InChapter 4 werstintroducedMEMS,inparticular,themicroelectromec hanical torsionaloscillatorusedinourmeasurementswhichisfabr icatedusingthePolyMUMPS processbyMEMSCAP.Theoscillatorwasshowntobeabletoprovid esensitiveforce detectionbasedoncapacitivemeasurements.Then,theexpe rimentalsetupwas presented.Thisisfollowedbyadescriptionofthedetectio nscheme.Byutilizingthe electricalcircuitasdescribed,theforcegradientcanbem easuredfromtheshiftinthe resonantfrequencyoftheoscillator. InordertocalculatetheelectrostaticforceandtheCasimi rforceonthetrench arrays,itiscrucialtoobtainaccuratesampledimensions. Adetailedsamplecharacterization waspresentedinChapter 5 .AFMandSEMaretwomaintechniquethathavebeen used.Differentapproacheswereusedtocharacterizethehi ghaspectratiotrenchesand shallowtrenchestoachievethebestresults. InChapter 6 boththeelectrostaticforceandtheCasimirforcemeasurem ents werediscussed.Theelectrostaticforcemeasurementswere essentialintheCasimir forcemeasurements.Thisisbecauseweusetheelectrostati cforcetocalibratethe measurementsystem.Threeparameterscanbeobtainedfromt heelectrostaticforce measurements:theresidualvoltageresultingfromthework functiondifferencebetween theinteractingsurfaces,theproportionalityconstant C betweentheforcegradientand themeasuredresonantfrequencyshift,andtheinitialsepa ration z 0 betweenthetwo surfaces.Whiletheelectrostaticforcebetweenasphereand aplatecanbecalculated fromtheanalyticexpression,anumericalapproachwasneed edtocalculatetheforce 109

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betweenthesphereandthetrencharrays.Followingthis,th edetailoftheCasimirforce measurementswerepresented. InChapter 7 themeasuredCasimirforcegradientbetweenagoldspherean da siliconplatewascomparedwiththetheoreticalcalculatio nincludingthecorrectionsdue torealsurfaces.Inourcalculation,theniteconductivit ywastakenintoaccountusing theLifshitzformula.Theroughnesscorrectionwereestima tedbyaveragingtheCasimir forceatdifferentpossibleseparationdistancesonthetwo surfacesresultingfromthe surfaceroughness,yieldingacorrectionof0.73 % at150nmseperation.Inaddition, sinceourmeasurementswereperfomedatroomtemperature,t hecorrectiondueto nitetemperaturewasalsodiscussed,whichislessthan1 % inourmeasurementrange. InChapter 8 wediscussedthecomparisonofthemeasurementsandthetheo ry forsphere-gratinggeometry.Themeasuredforcegradiento ntrencharraysdeviates fromthePFAandPAAby 10 % 20 % and 10 % forsampleA,BandCrespectively. Theobserveddeviationagreeswiththetheoreticalcalcula tionsbasedonscattering theoryincludingtheniteconductivitycorrection,butsm allerthanthepredictionfor perfectmetals.Theerrorsoriginatedfromdifferentsourc esbothinmeasurementsand intheoreticalcalculationswerealsoanalyzedinthischap ter. 9.2FutureExperiments Theresultsofthisdissertationlaidthefoundationforfut ureexperimentsto investigatesampleswithdifferentstructuresandmateria lstoprovideacomplete understandingofthegeometryandthematerialdependence. Inparticular,metamaterials, whicharearticialmicrostructuredmaterialswithdesign edproperties,havebeen theoreticallyproposedasagoodcandidatetoprovidesigni cantmodulationofthe amplitudeandeventhesignoftheCasimirforce.Itisofgrea tinteresttogenerate repulsiveCasimirforcessincesuchforcecanbeusefulinMEM S,suchaspreventing stictionbetweenmoveablecomponents.Ithasbeenshowntha taperfectlens(madeof left-handmaterialwith = = 1 overabroadrangeoffrequencies)sandwiched 110

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betweentwomirrorscanleadtoarepulsiveCasimirforcebet weenthemirrors [ 68 ].Inaddition,Rosa etal. discussedthepursuitofrepulsiveCasimirforceviathe congurationofametallichalfspaceandametallicmetamat erial[ 69 ].Therehavebeen anumberoftheoreticresearchesontheCasimirforceinvolv ingmetamaterials[ 70 71 ]. However,Casimirforceexperimentsonthesestructuresare stillchallenginginpractice. LifshitzformulafortheCasimirforceneedswelldenedper mittivitiesandpermeabilities. Forthetheoreticalworkonmetamaterials, and areobtainedusingeffective-medium approximation.Thisrequirestheeldwavelengths,whichc ontributessignicantlytothe force,tobemuchlargerthanthestructuredimensionofthem etamaterial.Currently,the Casimirforcecanbeaccuratelymeasuredattheseparationl essthan500nm,which meansthatthemetamaterialunitcellneedstobemuchsmalle rsothatthetheories predictingrepulsiveforcesarevalid.Fabricatingsuchst ructuresatlengthsscaleof < 100 nmfortheCasimirforcemeasurementsisachallenginggoal. 9.3Conclusion Inconclusion,wemeasuredtheCasimirforcegradientbetwe enagoldspherical surfaceandaheavilydopedsiliconplatewithanarrayofnan o-scale,rectangular trenchesusingaMEMS-basedtechnique.Signicantdeviations ofupto20 % fromthe PFAandthePAAwereobserved.Recentlytheoreticaladvancesa llowtheCasimir forcetobeaccuratelycalculatedonnanostructuredsurfac esforrealmaterials. Thegoodagreementoftheexperimentalresultsandthetheor eticalcalculations demonstratethestronggeometrydependenceoftheCasimirf orce.Inaddition,our resultsdemonstratethecontrollingoftheCasimirforceth roughtheinterplaybetween thegeometryandtheniteconductivityeffect,whichcanle adtoavarietyofapplications inthenanotechnology. 111

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APPENDIXA FABRICATIONPROCEDUREFORGOLDCOATEDSPHERES 1. Placeapieceofremovabledoublesidetapeonaglasssideandp ourasmall amountofspheresonthetape. 2. Loadthesampleintothesputteringchamber(KurtJ.LeskerC MS-18Multi-Target SputterDepositionsystem)andperformthecleaningprocess .Theprocess includesa5secargonplasmacleanfollowedbya20secoxygen plasmaclean. TherecipeisshowninTable A-1 TableA-1.Therecipeparametersof O 2 plasmaand Ar plasmaforsputtering system.Itisusedtocleantheglassspheresbeforethedepos ition. processGasPowerpressuretime Ar clean Ar 100W10mT5s O 2 clean O 2 200W10mT20s 3. Deposition.Inthesamechamber,withoutremovingthesampl efromvacuum, deposita5nmlayeroftitaniumfollowedbya400nmlayerofgo ldonthespheres. TherecipeislistedinTable A-2 .Thesampleholdercontinuouslyrotatesduringthe depositiontogiveauniformcoverageonthesideofthespher es. TableA-2.Therecipeparametersofgolddepositionandtitan iumdeposition. processTargetPowerpressuretime TidepositionTiatgun2200WDC5mT50s AudepositionAuatgun1100WRF3mT4000s 4. Unloadsample. 112

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APPENDIXB GENERALPREPAREPROCEDUREFORSILICONSAMPLE Thefollowingproceduresareperformedonthesiliconwafer swithsiliconoxideetch maskprovidedbyourcollaboratorsatBellLabs. 1. Pre-etchclean (a) CleanthesampleinAcetonetostripawaythephotoresist.Soak thesample inacetonefor5minutes,thentransferitintoIPAandblowdr ywith N 2 (b) Piranhacleantoremoveorganiccompoundsfromsubstrates.T hesample issoakedinamixturesolutionof3:1concentratedsulfuric acidto30 % hydrogenperoxideat110 Cfor15minutes.Thenthesampleisrinsedusing DIwaterfor15minutesandnallyblowndryusing N 2 (c) Afteraquicketchusing1:1000HFtoDIfor30s,thesampleisri nsedusing DIwaterfor15minutesandblowndryusing N 2 .Itisthenbakedat120 C for15minutes.Thequicketchistoremovethethinoxidelaye rthatbuildson siliconsubstratewithoutdamagingtheoxidemask.Thebaki ngistoeliminate theresidualwateronthesubstrate. 2. Etching (a) DRIE(DeepReactiveIonEtchsystemfromSTS)isusedforhighasp ectratio trenchesetch. (b) RIE(UnaxisICP-RIEetcher)isusedforshallowtrenchetch. 3. Post-etch (a) SEM(ScanningElectronMicroscope)andAFM(AtomicForceMicrosco pe) areusedtoobtainnecessarydatalesforsampleproleanal ysis. (b) Thesampleiscoatedwithphotoresist1813.Thesampleisspu nat500 rpmfor5stodispensephotoresistand3500rpmfor45sforcoa ting.The sampleisthenbakedonahotplateat115 Cfor1minutes.Thepurposeof thephotoresististoprotectthesamplesurfaceandprepare itfordicing.This stepisusuallytakenafterSEMandAFMimagesaretakentoavoida dditional samplecleaningsteps. (c) Sampleisdicedinto0.7mmby0.7mmpiecesfortheforcemeasur ements. (d) Stripthephotoresistondicedsamples.Samplesarerstrinse dusingDI water.AfterbeingsprayedbyAcetone,thesamplesaresoakedi nacetone for5minutes.Afterallthephotoresisthavebeenremoved,sa mplesare transferedintoIPAanddriedby N 2 113

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(e) An O 2 dryetchusingUnaxisICPetcherisperformedtoremovepossi ble organiccompounds.TheetchingrecipeisshowninTable B-1 TableB-1.The O 2 etchingrecipeparameterforUnaxisICPetcher.Itisusedto removeorganicresidualcompoundsonthesample. O 2 PressureRF1RF2Time 60sccm10mT100W300W3min (f) Rightbeforetheforcemeasurements,thedicedsampleissoa kedin1:100 HFtoDIfor2minutestogetridofallpossiblenativeoxideon theSisample. Afteretching,thesampleisrinsedusingDIwaterfor15minut esandblown dryusing N 2 (g) GluethesampleontoanAluminumholderusingsilverepoxyand bakeitat 120 Cfor30minutes.Thepurposeofbakingistocurethesilverep oxyand eliminatethepossiblewaterresidualintrencharray. (h) Loadsample,alignandpumpdownthechamber.Sincesiliconsu rfaceisvery reactiveinair,thetimebetweenHFetchandpumpingdownisu suallylimited to3hours. 114

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APPENDIXC ETCHINGPROCEDUREFORSAMPLESWITHDEEPTRENCHARRAYS ThesampleswithdeeptrencharraysareetchedbyDRIE(deepr eactiveionetch systemfromSTS)usingacontinuousetchrecipe.Theadvantage sofusingDRIEare thatthepassivategasprotectsthesidewallwhichprovides adeepandstraightsidewall andthatetchinginthesystemconsumesverylittleoxidemas kwhichwillensurethe patternprole.Thedisadvantageisthatthebottomofthetr enchesareroundeddueto passivation. 1. Gluethesampleontoa4inchesSiwafer(carrierwafer).TheDR IEetcherin UFNFonlyaccept4incheswaferwhilethesampleweusedisabo ut1cm square.CoverthecarrierwaferwithKaptontapeandonlyleav eaopeningfor thesample.Gluethesampleusingaheatconductivepastetoe nsurethehelium owunderneaththecarrierwaferwillcoolthesampleduring theetch.Makesure nogluepastewillbeexposedtoetchinggasduringtheetch. 2. LoadablankwaferintotheDRIEetcher.Performaoxygenetch tocleanthe chamber,asshowninTable C-1 .Then,runetchrecipeontheblankwafertoverify thattheetcherfunctionsproperly. TableC-1.The O 2 etchingrecipeparametersforSTSDRIE.Itisusedtocleanthe chamberbeforerealsampleetch. O 2 PressureRF1RF2Time 60sccm40mT15W700W3min 3. Unloadtheblankwaferandloadthewaferwithsample.Runthe etchrecipeas showninTable C-2 TableC-2.TherecipeparametersforDRIE.Ititusedtocreate deeptrencheson siliconwafer. Sampleperiod C 4 F 8 SF 6 RF1RF2PressureTime sampelA500nm90sccm45sccm30W700W10mT140s sampleB200nm90sccm45sccm30W700W10mT160s 4. Removethesamplefromthecarrierwafer.Spraythesamplewit hAcetoneand soakthesampleinAcetonefor5minutestoremovethebackside glue.After Acetone,transfersampletoIPAandthenblowndryusing N 2 gas. 5. O 2 cleantoremovepossiblepolymerlayerproducedfromthepas sivationduring DRIEetchusingUnaxisICP(Table B-1 ). 115

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6. Removeoxidemaskusing1:10HFtoDIfor4minutes. 116

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APPENDIXD ETCHINGPROCEDUREFORSAMPLESWITHSHALLOWTRENCHARRAYS ThesampleswithshallowtrencharraysareetchedbyRIEetch er(RIEetchsystem fromUnaxis).TheroundingbottomproblemfromDRIEbecomes non-negligiblefor shallowtrenches.TherecipeusedforRIEmainlyisphysical millingfromions.Sucha recipeprovidesaatbottom,butconsumetheetchingmaskmu chfaster. 1. Loadacleanwaferandrunachambercleanrecipe(Table D-1 ). O 2 gasisusedto etchallpossibleorganiccompoundswhile Ar gasisusedtoprovidesomephysical millingonresidualparticles. TableD-1.TherecipeparameterforUnaxisRIEchamberclean O 2 gasisusedto etchpossibleorganiccompoundsand Ar gasisusedtoprovidesome physicalmillingonresidualparticles. O 2 Ar PressureRF1RF2Time 60sccm5sccm10mT100W300W15min 2. Runetchingrecipeonblankwaferfortesting. 3. Unloadblankwaferandreloaditwithsample.Thesampleisju stsetontheblank waferwithoutanyglue. 4. Runetchingrecipeonthesample(Table D-2 ). TableD-2.TherecipeparameterforUnaxisRIE.Itisusedtocr eateshallow trenchesonsiliconwafer. Sampleperiod SF 6 Ar PressureRF1RF2Time sampleC200nm3sccm6sccm1mT200W600W22s 5. Removeoxidemaskusing1:10HFtoDIfor4minutes. 117

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APPENDIXE MATLABCODEUSEDINCOMSOL Thefollowingcodecalculatetheelectrostaticenergyperu nitareaonshallowtrench structure(sampleC)inCOMSOLusingmatlablanguage. %COMSOLMultiphysicsModelM file %GeneratedbyCOMSOL3.5a(COMSOL3.5.0.603,$Date:2008/1 2/0317:02:19 $ ) clearall n=0; fori=2 E 6:50e 9:2e 6 flclearfem %COMSOLversion clearvrsn vrsn.name='COMSOL3.5'; vrsn.ext='a'; vrsn.major=0; vrsn.build=603; vrsn.rcs='$Name: $ '; vrsn.date='$Date:2008/12/0317:02:19 $ '; fem.version=vrsn; %Geometry a=i; g1=rect2(400e 9,a,'base','corner','pos', f '0','0' g ,'rot','0'); g2=rect2(199.26e 9,98e 9,... 'base','corner','pos', f '100.42e 9',' 98e 9' g ,'rot','0'); g3=geomcoerce('solid', f g1,g2 g ); gg=geomedit(g3); gg f 4 g =beziercurve2([1.0042 E 7,9.266 E 8],[ 9.8 E 8,0],[1,1]); g4=geomedit(g3,gg); gg=geomedit(g4); gg f 8 g =beziercurve2([2.9958 E 7,3.0734 E 7],[ 9.8 E 8,0],[1,1]); g5=geomedit(g4,gg); %geomplot(g5) %Geometryobjects clears s.objs= f g5 g ; s.name= f 'CO1' g ; s.tags= f 'g5' g ; fem.draw=struct('s',s); fem.geom=geomcsg(fem); %( Default valuesarenotincluded) %Initializemesh fem.mesh=meshinit(fem,'hmax',0.04e 7) %% Refine mesh %fem.mesh=meshrefine(fem,... %'mcase',0,... %'rmethod','regular'); 118

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% %% Refine mesh %fem.mesh=meshrefine(fem,... %'mcase',0,... %'rmethod','regular'); % %% Refine mesh %fem.mesh=meshrefine(fem,... %'mcase',0,... %'rmethod','regular'); %figure,meshplot(fem),axisequal %Applicationmode1 %setboundarycondition clearappl appl.mode.class='Electrostatics'; appl.assignsuffix=' es'; clearbnd bnd.V0= f 0,0,0,1 g ; bnd.name= f 'plate','','gap','trench' g ; bnd.type= f 'V0','cont','nD0','V' g ; bnd.ind=[3,4,1,4,2,4,2,4,2,4,3]; appl.bnd=bnd; fem.appl f 1 g =appl; fem.frame= f 'ref' g ; fem.border=1; clearunits; units.basesystem='SI'; fem.units=units; %ODESettings clearode clearunits; units.basesystem='SI'; ode.units=units; fem.ode=ode; %Multiphysics fem=multiphysics(fem); %Extendmesh fem.xmesh=meshextend(fem); % Solve problem fem.sol=femstatic(fem,... 'solcomp', f 'V' g ,... 'outcomp', f 'V' g ,... 'blocksize','auto'); % Save currentfemstructureforrestartpurposes fem0=fem; % Plot solution 119

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%postplot(fem,... %'tridata', f 'V','cont','internal','unit','V' g ,... %'trimap','Rainbow',... %'title','Surface:Electricpotential[V]',... %'axis',[ 1.2371541501976275 E 7,5.237154150197628 E 7, 1.5 E 7,2.0 E 7]); % Integrate I1=postint(fem,'We es',... 'unit',' N ',... 'recover','off',... 'dl',[1,2]) n=n+1; Energy(n)=I1; g(n)=a; makef output='energy d100w200.txt' output unit=fopen(makef output,'at'); fprintf(output unit,'%e%e n n',I1,a); fclose(output unit); end dat=[Energy;g]; 120

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APPENDIXF MATLABCODESUSEDTOCALCULATETHECASIMIRFORCEUSINGLIFSHITZ FORMULA ThefollowingmatlabcodescalculatethetheCasimirForceU singLifshitzFormula. functionmake f two(astart,astep,astop) %astart=60e 9; %astep=10e 9; %astop=60e 9; hbar=1.055e 34; c=3e8; foraa=astart:astep:astop f=hbar/(2 piˆ2 cˆ3) int p two(aa); fp=fopen('si f1.dat','a'); fprintf(fp,'%e n t%e n n',aa,f); fclose(fp); end function[output]=int p two(a) uplimit=2000; p=1 total=0; while(p < uplimit) dp=p/100; total=total+int x two(a,p) pˆ2 dp; p=p+dp end output=total; function[output]=int x two(a,p) load('au ei.dat'); xx1=au ei(:,1); yy1=au ei(:,2); clearau ei; load('si ei.dat'); xx2=si ei(:,1); yy2=si ei(:,2); clearsi ei; 121

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c=3e8; step=mean(diff(xx1)); [m,n]=size(xx1); total=0; fornn=1:m y1=yy1(nn); x1=xx1(nn); y2=yy2(nn); x2=xx2(nn); s1=sqrt(y1 1+pˆ2); s2=sqrt(y2 1+pˆ2); term1=((s1+p) (s2+p))/((s1 p) (s2 p)) exp((2 p a/c) x1) 1; term2=((s1+y1 p) (s2+y2 p))/((s1 y1 p) (s2 y2 p)) exp((2 p a/c) x integrand=x1ˆ3 (1/(term1)+1/(term2)); total=total+integrand step; end output=total; 122

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BIOGRAPHICALSKETCH YiliangBaowasborninJilininthenortheastofChina,butmov edtoShanghai, whenshewasakid.Shespentmostofherchildhoodandadolesce nceinShanghai andgraduatedfromFudanUniversitywithBachelor'sdegreei nsciencein2003.Inher senioryear,shedecidedtoapplyforgraduateschooltopurs ueherPhDdegreeaswell astoexpandhercareerofscienticresearch.Shewasaccepte dbytheDepartmentof PhysicsattheUniversityofFloridawithteachingassistant shipin2003.Inthesummer of2004,shejoinedHoBunChansresearchgroupandbeganworki ngwithhimon theCasimirforcemeasurementsusingmicroelectromechani caldevices.Thiswork demonstratedthestrongshapeandmaterialdependenceofth eCasimirforce.After sevenyearsofresearch,shegraduatedfromtheUniversityo fFloridawithaDoctoreof Philosophyinphysics. 127