MEMS技术-第四讲-电子零件原理

MEMS和微系统设计

MEMS和微系统设计

1课程内容MEMS概述及MEMS设计的概述工艺简要回顾系统设计、工艺设计及版图设计主要的机械、电子元件及其设计基础多域耦合设计：以机电耦合为例子器件性能的估计简单的其他域的元件及其简要设计要点设计实例课程内容MEMS概述及MEMS设计的概述2第4讲主要内容(3)1、弹簧设计原理及计算例子2、薄膜设计原理及计算例子3、电容设计原理及计算例子4、电阻设计原理及计算例子5、压电模型第4讲主要内容(3)1、弹簧设计原理及计算例子3电容变化静电力电容变化4图2-17电容式微传感器的基本结构

平行板电容器的电容为电容敏感原理

图2-17电容式微传感器的基本结构平行板电容器的电容为电容5

式中A为极板面积为真空介电常数为极板间介质的相对介电常数当介质为空气时，；为两极板间距离间隙变化型：改变两极板间隙面积变化型：改变形成电容的有效面积A介质变化型：改变两极间介质的介电常数

式中间隙变化型：改变两极板间隙6间隙变化型电容式微传感器利用泰勒级数展开，由麦克劳林公式可得间隙变化型电容式微传感器利用泰勒级数展开，由麦克7略除高阶无穷小项，得这时传感器的灵敏度和非线性误差分别为：略除高阶无穷小项，得这时传感器的灵敏度和非线性误差分别为：8采用差动电容结构可以大大减小传感器输出的非线性：

(2-12)

(2-13)

(2-14)

(2-15)采用差动电容结构可以大大减小传感器输出的非线 (29

在小位移情况下，外加作用和成比例关系，可见电容的倒数差及电容的差除和都与输入作用力成线性关系。

式(2-14)表明，用电容的差除和表达传感器的性能，其输出还要受到介质介电常数的影响。

式(2-15)表明电容差除和只受电容极板间隙和间隙变化的影响。目前，硅电容变送器普遍采取式(2-15)的方法来描述传感器的性能。

在小位移情况下，外加作用和成比例关系，可见电容的10其他的电容变化形式变面积电容器其他的电容变化形式变面积电容器11Aexample:calculatetoCandtheshiftofC两种电容变化形式的变化量对比

（电容原值、导线的电容值、电容变化值）Wire：L=1m,r=0.2mm,d=1mmgap=g=1Thickness=t=2fingerlength=L=100overlaplengthx=75Aexample:calculatetoCan12电容readout—位置检测和速度检测Whymodulatev(t)?Idealbuffer:cin=0电容readout—位置检测和速度检测Whymodulat13MatchedAir-GapReferenceCapacitors

MatchedAir-GapReferenceCapa14SimpleCapacitorDivider(con.)matchedair-gapreferencecapacitoroffsetsignalSimpleCapacitorDivider(con.15CapacitorDividerWithDifferentialExcitationWhymodulatev+andv-?Idealbuffer:cin=0Impedancedividerwithsuperposition:CapacitorDividerWithDiffere16ImprovedCapacitiveDivider(cont.)nooffset!distortionImprovedCapacitiveDivider(c17ThecapacitiveHalf-BridgeImpedancedividerwithsuperposition:ThecapacitiveHalf-BridgeImp18ThecapacitiveHalf–Bridge(cont.)Simplifyexpression:Nooffset，2xsignalincreaseThecapacitiveHalf–Bridge(c19ParasiticCapacitancesSurfacemicromachinedz-axisparallel-platecapacitorParasiticCapacitancesSurface20EquivalentcircuitCpp(x):nominal||platesensecapacitorCf1(x):fringecapacitance(varieswithplatedisplacement)Cf2:fringecapacitancebetweenupperplate(connectedtoanchorplane)andlowerplate…slightdependenceonxCpu:parasiticcapacitancefromupperplatetosubstrateCpl:parasiticcapacitancefromlowerplatetosubstrateEquivalentcircuitCpp(x):nom21VelocitySensingFundamentalcurrent-voltagerelationshipforatime-varyingcapacitor:Considerspecialcase:v=vp=constant…usedinhigh-qualitycapacitancemicrophonesVelocitySensingFundamentalcu22VelocitySensing(cont.)Sensecapacitor’stimevariation:Parallel-platesensecapacitorwithgapgo:Harmonicmotion:VelocitySensing(cont.)Sense23SomeNumbersSurfacemicromachinedcapacitor:

Isthisreal?…noiseinbufferampSomeNumbersSurfacemicromachi24WorldRecordCapacitivePosition-SenseResolution*AnalogDevicesADRS-150vibratoryrategyroscopeJohnGeen,SteveSherman,JohnChang,andSteveLewis,IEEEJ.Solid-StateCircuits,37,Dec.2002,1860-1866FullscaleCorillis-induceddisplacement=20ÅSensecapacitance≈1000fFMinimumdetectablecapacitancechange≈12zF=0.012aFNominalsensegap=1.6

m

Minimumdisplacement:16fm!*Surfacemicromachiningclassaudiofrequencyband

EEC245-MEC218Fall2003Lecture12WorldRecordCapacitiveFullsc25IsADLSplittingElectrons?AtV+=5V,thechargeonthesensecapacitoris:qs=c+v+=(1000fF)(5V)=5000fCNumberofelectronsatMinimumdetectablechangeinsensecharge:Minimumdetectedchangeinnumberofelectrons:IsADLSplittingElectrons?At26电容变化静电力电容变化27变间隙电容驱动器的基本理论

BasicphysicsofElectrostaticActuation

Twowaystochangetheenergy:

1.Changethechargeq2.changetheseparationxNote:weassumethattheplatesaresupportedelastically,sotheydon’tcollapse.

变间隙电容驱动器的基本理论

Basicphysicsof28Charge-ControlCase(cont.)Storedenergy:Force(attractive,internal):Voltage:Independentofthegap!constantCharge-ControlCase(cont.)Sto29ElectrostaticForce(VoltageControl)Findco-energyintermsofvoltageVariationofco-energywithrespecttogapyieldsv.s.force:Variationofco-energywithrespecttovoltageyieldschargeasexpectedElectrostaticForce(VoltageC30LinearizingtheVoltageSquare-LawPolarizethecapacitorbyapplyingaDCoffsetvoltageVPtogetherwitha(small)signalvoltageVsig(t)<<VPDCoffsetneglect(small)LinearizingtheVoltageSquare31TheDifferentialElectrostaticActuatorNetforceonsuspendedcenterelectrodeisthedifferenceTheDifferentialElectrostatic32Parallel–PlateCapacitiveNonlinearityExample:laterallydrivenspringsuspendedplate(eventuallywithbalancedelectrodes)NomenclatureConductivestructureelectrodeValueACorsignalcomponent(lowercasevariablesubscript)DCComponent(uppercasevariable:uppercasesubscript)Parallel–PlateCapacitiveNon33Parallel–PlateCapacitiveNonlinearityExample:clamped-clampedlaterallydrivenbeam

withbalancedelectrodesExpressionfor

ExpandtheTaylorSeriesfurtherConductivestructureelectrodeParallel–PlateCapacitiveNon34Parallel–plateCapacitiveNonlinearityParallel–plateCapacitiveNon35Parallel–PlateCapacitiveNonlinearityRetainingonlytermsatthedrivefrequency:Thesetwotogethermeanthatthisforceactsagainstthespringrestoringforce!AnegativespringconstantsinceitderivesfromVPwecallittheelectricalstiffness,givenby:DriveforcearisingfromtheinputexcitationvoltageatthefrequencyofthisvoltageProportionaltodisplacement900phase-shiftedfromdrive,soinphasewithdisplacementParallel–PlateCapacitiveNon36Electricalstiffness,KeTheelectricalstiffnesskebehaveslikeanyotherstiffnessItaffectsresonancefrequency:Frequencyisnowafunctionofdc-biasVp1Electricalstiffness,KeTheel37CanOneCancelKewithTwoElectrodes?Whatifwedon’tlikethedependenceoffrequencyonVP?CanwecancelKCviaadifferentialinputelectrodeconfiguration?IfwedoasimilaranalysisforFd2atElectrode2:

SubtractsfromtheFd1term,asexpectedAddtothequadraturetermKc’sadd,nomattertheelectrodeconfiguration!CanOneCancelKewithTwoE38ThecapacitiveHalf-BridgeImpedancedividerwithsuperposition:ThecapacitiveHalf-BridgeImp39ThecapacitiveHalf–Bridge(cont.)Simplifyexpression:Electrostaticforce:ThecapacitiveHalf–Bridge(c40ElectrostaticForce(Cont.)Outputvoltageisproportionaltothedisplacement(forx<<go)DCand2wtermsElectrostaticForce(Cont.)Out41ElectrostaticSpringConstantkenotedirection:springappliesforceoppositetodisplacement

BothDCand2wcomponents:usesquarewaveexcitationtoyieldconstantkeElectrostaticSpringConstant42GraphicalSolutionforPlateStability

Plotnormalizedelectrostaticandspringforcesvs.normalizeddisplacement1-(g/go)GraphicalSolutionforPlateS43SowhyareelectrostaticactuatorsimportantinMEMS,anyway?Easytomakeinmicromachiningprocesses,sinceconductorsandairgapsallthat’sneededEnergy–conserving

onlyparasiticenergylossthroughi2RlossesinconductorsandinterconnectsPull-inphenomenoncanbeexploitedtomakeahystereticactuatorsimplifiescontrolMultipleplatestructures(combs,3D)canbeusedtotailortheforce(displacementvoltage)functionScalingoftheelectrostaticforceisfavorableduetoPaschen’scurveSamestructurecanbeusedforpositionsensingSowhyareelectrostaticactua44Paschen’sCurvePaschen’sCurve45叉指驱动器的理论模型(Electrostaticcombdrive)Useofcomb-capacitivetranducersbringsmanybenefits

．Linearizesvoltage-generatedinputforces

．(Ideally)eliminatesdependenceoffrequencyondc-bias．Allowalargerangeofmotion叉指驱动器的理论模型(Electrostaticcomb46ElectrostaticForce:aFirstPassStator(fixedelectrode)Rotor(not…butmoving)Gap=g,thickness=tL=fingerlengthX=overlaplengthElectrostaticForce:aFirstP47First-PassElectrostaticForce(cont.)NeglectfringingfieldsParallel-platecapacitancebetweenstatorandrotorIndependentofx！First-PassElectrostaticForce48RelativeForceforSurfaceMicrostructuresCombdrive(x-direction)(V1=V2=VS=1V)Differential||plate(y-direction)(V1=0V,V2=1V)||platewinsbig…forsurfaceMEMSGap=g=1Thickness=t=2Fingerlength=L=100Overlaplengthx=75RelativeForceforSurfaceMic49CombDriveForce:aSecondPassEnergymustincludecapacitancebetweenthestatorandrotorandunderlyinggroundplane,whichistypicallybiasedatthestatorvoltageVs…why？CombDriveForce:aSecondPa50CombDriveForcewithGroundplaneCorrectionFingerdisplacementchangescapacitancesfromstatorandrotortothegroundplanemodifiestheelectrostaticenergy

CombDriveForcewithGroundp51CapacitanceExpressionsConsidercasewhereVr=VP=0VCsp=dependsonwhetherornotfingersareengagedCapacitanceperlengthunitCapacitanceExpressionsConside52Simulation(2DFiniteElement)20-40%reductionofFeSimulation(2DFiniteElement)53VerticalForce(Levitation)

considerVr=0Vasshown:VerticalForce(Levitation)co54LevitationForce“electricalspringconst.”constantLevitationforceaddstothemechanicalspringconstantinthezdirectionincreasestheresonantfrequencyLevitationForce“electricalsp55VerticalResonantFrequencyMustaccountforelectricalspringsinfindingMEMSresonantfrequenciesComb(x-axis)Ke=0Comb(z-axis)Ke>0ParallelplateKe<0VerticalResonantFrequencyMus56第4讲主要内容(3)1、弹簧设计原理及计算例子2、薄膜设计原理及计算例子3、电容设计原理及计算例子4、电阻设计原理及计算例子5、压电模型第4讲主要内容(3)1、弹簧设计原理及计算例子57MEMS技术-第四讲-电子零件原理58

1、金属的电阻改变：由材料几何尺寸的变化引起的；与相关

2、半导体的电阻改变：由材料受力后电阻率的变化引起，与相关；

3、半导体的灵敏度因子比金属的高得多，一般在70-170之间

59当电阻为立体结构时，有立体单元电阻的应力图当电阻为立体结构时，有立体单元电阻的应力图60(7-6)其中{ΔR}=代表与应力分量{σ}=（如图7.13）相对应的一个无限小的立方压电电阻晶体单元的电阻变化。(7-6)其中{ΔR}=61式7-6、7-7立体电阻的压阻系数(7-7)式7-6、7-7立体电阻的压阻系数(7-7)62得出：得出：63若电阻为薄膜电阻，在正交坐标系中，当坐标轴与晶轴一致时，电阻的相对变化与应力的关系为若电阻为薄膜电阻，在正交坐标系中，当坐标轴与64

表示纵向应力

为横向应力表示、垂直方向上的应力，它比和小很多，一般都略去。、、分别为、、相对应的压阻系数，为纵向压阻系数，为横向压阻系数。表示纵向应力65

当电阻处于任意晶向P时，如果有纵向应力沿此方向作用在单晶硅电阻上，则会引起纵向压阻系数，如果电阻上同时作用有和电阻方向垂直的横向应力，则会引起横向压阻系数，那么任意晶向的压阻系数为（2-6）当电阻处于任意晶向P时，如果有纵向应力66（2-7）式中，、、分别为单晶硅晶轴上的纵向压阻系数、横向压阻系数和剪切压阻系数；、、分别为电阻的纵向应力相对于晶体主轴坐标系中的方向余弦；、、分别为电阻的横向应力相对于晶体主轴系中的方向余弦。（2-7）式中，、、分别为单晶硅晶轴上的纵向67RelativeresistancechangecanbeexpressedbythelongitudinalandtransversepiezoresistivecoefficientsPiezoresistorsareoftenalignedtothewaferflatof(100)wafers,whichisinthe[110]direction.Senturia,p.473providestheresultofcoordinatetransformations:

Relativeresistancechangecan68SiliconpiezoresistivecoefficientsFunctionoftype,doping,andtemperatureLongitudinalandtransversecoefficientsin[110]directionn-type11.7-102.253.4-13.6P-type7.86.6-1.1138.1Units[-cm],10-1Pa-1valuesareatT=250Cn-type

P-typeSiliconpiezoresistivecoeffic69一般地，当晶面为(100)时，有表7-9P型压电阻在各方向的压阻系数晶面取向<x>取向<y>πLπT(100)<111><211>+0.66π44-0.33π44(100)<110><100>+0.5π44~0(100)<110><110>+0.5π44-0.5π44(100)<100><100>+0.02π440.02π44一般地，当晶面为(100)时，有表7-9P型压电阻在各70PiezoresistorPlacementBulkmicromachineddiaphragmpressuresensorPiezoresistorPlacementBulkmi71电阻变化的read-out公式?电阻变化的read-out公式?72举例计算电阻的变化导致电压的变化举例计算电阻的变化导致电压的变化73第4讲主要内容(3)1、弹簧设计原理及计算例子2、薄膜设计原理及计算例子3、电容设计原理及计算例子4、电阻设计原理及计算例子5、压电模型第4讲主要内容(3)1、弹簧设计原理及计算例子74OriginofPiezoelectricEffectSeveralviewsofanα-quartzcrystalOriginofPiezoelectricEffect75OriginofPiezoelectricEffectForr>>a,theelectricfieldatthepointPis:Thepotentialandelectricfieldappearasifthechargesarecoincidentattheircenterofgravity(pointO)OriginofPiezoelectricEffect76OriginofPiezoelectricEffectAssumetheappliedforceFcausesthelineODtorotatecounterclockwisebyasmallangleThisstrainshiftsthecenterofgravityofthethreepositiveandnegativechargestotheleftandright,respectivelyAdipolemoment,p=qr,iscreatedwhichhasanarm(r)of:p=qrqa33/2AssumingthecrystalcontainsNsuchmoleculesperunitvolume,eachsubjecttothesamestrain,thepolarization(ordipolemomentperunitvolume)is:

polarizationstrainOriginofPiezoelectricEffect77OriginofPiezoelectricEffectForsufficientlysmalldeformations,polarization(p)islinearlyrelatedtothestrain(s)by:p=gswheregisthepiezoelectricvoltagecoefficient.ConversePiezoelectricEffectWhenapiezoelectriccrystalisplacedinanelectricfield,positiveandnegativeionsarepushedinoppositedirectionsandadipoletendstorotatetoalignitselfwiththeelectricfield.TheresultingmotiongivesrisetostrainsthatisproportionaltoelectricfieldES=dEwheredisthepiezoelectricchargecoefficient.OriginofPiezoelectricEffect78AnisotropicCrystalProperties:GeneralizedStress-StrainIn

anisotropicmaterialsatensilestresscanproducebothaxialandshearstrain.Forexample,athin,x-cutrodofquartzsubjecttoatensileforcewillnotonlybecomelongerandthinner,longitudinalaxis.Sincewehave6componentsofstress(T)and6componentsofstrain(S),36constantsmustbeusedtodescribebehaviorinthegeneralcase.Crystalsymmetry(e.g.trigonal,hexagonal)greatlyreducesthenumberofindependentconstants.AnisotropicCrystalProperties79AnisotropicCrystalProperties:GeneralizedStress-StrainForsmalldeformations,stress(T)andstrain(S)arerelatedthoughthecompliancematrix(s)Conservationofenergyrequiressij=sji.Performingrotationsbasedupontrigonalsymmetryconsiderations,thecompliancematrixreducesto6independentcoefficients:Quartzhasthreefoldsymmetry,physical

propertiesrepeatevery1200.Quartzisalsosymmetricaboutthex-axisAnisotropicCrystalProperties80AnisotropicCrystalProperties:GeneralizedStress-StrainRecallthat

thestrain(S)isrelatedtotheelectric

(E)bythepiezoelectricchargecoefficientmatrix(d)Applyingthesymmetryconditionsforquartz,thepiezoelectricstrainmatrix(d)simplifiesto:AnisotropicCrystalProperties81AnistropicCrystalPropertiesElasticmodulusandcomplianceThermalconductivityE