1EE 245: Introduction to MEMSLecture 13: Mechanics of Materials IICTN 10/6/09Copyright © 2009 Regents of the University of CaliforniaEE C245: Introduction to MEMS Design LecM 7 C. Nguyen 9/28/07 21MEMS Material PropertiesEE C245: Introduction to MEMS Design LecM 7 C. Nguyen 9/28/07 22Material Properties for MEMS[Mark Spearing, MIT]√(E/ρ) is acoustic velocityEE C245: Introduction to MEMS Design LecM 7 C. Nguyen 9/28/07 23Young’s Modulus Versus Density[Ashby, Mechanics of Materials, Pergamon, 1992]Lines of constant acoustic velocityEE C245: Introduction to MEMS Design LecM 7 C. Nguyen 9/28/07 24Yield Strength• Definition: the stress at which a material experiences significant plastic deformation (defined at 0.2% offset pt.)• Below the yield point: material deforms elastically → returns to its original shape when the applied stress is removed• Beyond the yield point: some fraction of the deformation is permanent and non-reversibleTrue Elastic Limit: lowest stress at which dislocations moveProportionality Limit: point at which curve goes nonlinearElastic Limit: stress at which permanent deformation beginsYield Strength: defined at 0.2% offset pt.2EE 245: Introduction to MEMSLecture 13: Mechanics of Materials IICTN 10/6/09Copyright © 2009 Regents of the University of CaliforniaEE C245: Introduction to MEMS Design LecM 7 C. Nguyen 9/28/07 25Yield Strength (cont.)• Below: typical stress vs. strain curves for brittle (e.g., Si) and ductile (e.g. steel) materialsStressTensile StrengthFractureDuctile (Mild Steel)Brittle (Si)Proportional LimitStrain[Maluf](Si @ T=30oC)(or Si @ T>900oC)EE C245: Introduction to MEMS Design LecM 7 C. Nguyen 9/28/07 26Young’s Modulus and Useful StrengthEE C245: Introduction to MEMS Design LecM 7 C. Nguyen 9/28/07 27Young’s Modulus Versus Strength[Ashby, Mechanics of Materials, Pergamon, 1992]Lines of constant maximum strainEE C245: Introduction to MEMS Design LecM 7 C. Nguyen 9/28/07 28Quality Factor (or Q)3EE 245: Introduction to MEMSLecture 13: Mechanics of Materials IICTN 10/6/09Copyright © 2009 Regents of the University of CaliforniaEE C245: Introduction to MEMS Design LecM 7 C. Nguyen 9/28/07 29LrhWrVPviClamped-Clamped Beam μResonatorSmaller mass Ö higher freq. range and lower series RxSmaller mass Ö higher freq. range and lower series Rx(e.g., mr= 10-13kg)(e.g., mr= 10-13kg)Young’s ModulusDensityMassStiffnessωωοivoi203.121rrroLhEmkfρπ==Frequency:Q ~10,000viResonator BeamElectrodeVPC(t)dtdCViPo=Note: If VP = 0V Ö device offNote: If VP = 0V Ö device offioEE C245: Introduction to MEMS Design LecM 7 C. Nguyen 9/28/07 30Quality Factor (or Q)• Measure of the frequency selectivity of a tuned circuit• Definition:• Example: series LCR circuit• Example: parallel LCR circuitdB3CyclePer Lost EnergyCyclePer EnergyTotalBWfQo==()()CRRLZZQooωω1ReIm===()()LGGCYYQooωω1ReIm===ÖÖBW-3dBfofEE C245: Introduction to MEMS Design LecM 7 C. Nguyen 9/28/07 31Selective Low-Loss Filters: Need Q• In resonator-based filters: high tank Q ⇔ low insertion loss• At right: a 0.1% bandwidth, 3-res filter @ 1 GHz (simulated)ª heavy insertion loss for resonator Q < 10,000-40-35-30-25-20-15-10-50998 999 1000 1001 1002Frequency [MHz]Transmission [dB]Tank Q30,00020,00010,0005,0004,000Increasing Insertion LossEE C245: Introduction to MEMS Design LecM 7 C. Nguyen 9/28/07 32• Main Function: provide a stable output frequency• Difficulty: superposed noise degrades frequency stabilityωωοivoiAFrequency-SelectiveTankSustainingAmplifierovIdeal Sinusoid:()⎟⎠⎞⎜⎝⎛=tofVotovπ2sinReal Sinusoid:()()()⎟⎠⎞⎜⎝⎛⎟⎠⎞⎜⎝⎛++=ttoftVotovθπε2sinωωοωωο=2π/TOTOZero-Crossing PointTighter SpectrumTighter SpectrumOscillator: Need for High QHigher QHigher Q4EE 245: Introduction to MEMSLecture 13: Mechanics of Materials IICTN 10/6/09Copyright © 2009 Regents of the University of CaliforniaEE C245: Introduction to MEMS Design LecM 7 C. Nguyen 9/28/07 33• Problem: IC’s cannot achieve Q’s in the thousandsª transistors Ö consume too much power to get Qª on-chip spiral inductors Ö Q’s no higher than ~10ª off-chip inductors Ö Q’s in the range of 100’s• Observation: vibrating mechanical resonances Ö Q > 1,000• Example: quartz crystal resonators (e.g., in wristwatches)ª extremely high Q’s ~ 10,000 or higher (Q ~ 106possible)ª mechanically vibrates at a distinct frequency in a thickness-shear modeAttaining High QEE C245: Introduction to MEMS Design LecM 7 C. Nguyen 9/28/07 34Energy Dissipation and Resonator QsupportviscousTEDdefectsQQQQQ11111+++=Anchor LossesAnchor LossesMaterial Defect LossesMaterial Defect LossesGas DampingGas DampingThermoelastic Damping (TED)Thermoelastic Damping (TED)Bending CC-BeamCompressionÖ Hot SpotTension Ö Cold SpotHeat Flux(TED Loss)Elastic Wave Radiation(Anchor Loss)At high frequency, this is our big problem!At high frequency, this is our big problem!EE C245: Introduction to MEMS Design LecM 7 C. Nguyen 9/28/07 35Thermoelastic Damping (TED)• Occurs when heat moves from compressed parts to tensioned parts → heat flux = energy lossQfT21)()( =ΩΓ=ςpCTETρα4)(2=Γ⎥⎦⎤⎢⎣⎡+=Ω222)(fffffTEDTEDo22 hCKfpTEDρπ=Bending CC-BeamCompressionÖ Hot SpotTension Ö Cold SpotHeat Flux(TED Loss)ζ = thermoelastic damping factorα = thermal expansion coefficientT = beam temperatureE = elastic modulusρ = material densityCp= heat capacity at const. pressureK = thermal conductivityf = beam frequencyh = beam thicknessfTED= characteristic TED frequencyhEE C245: Introduction to MEMS Design LecM 7 C. Nguyen 9/28/07 36TED Characteristic Frequency• Governed byª Resonator dimensionsª Material properties22 hCKfpTEDρπ=ρ = material densityCp= heat capacity at const. pressureK = thermal conductivityh = beam thicknessfTED= characteristic TED frequencyCritical Damping Factor, ζRelative Frequency, f/fTEDQ[from Roszhart, Hilton Head 1990]Peak where Q is minimizedPeak where Q is minimized5EE 245: Introduction to MEMSLecture 13: Mechanics of Materials IICTN 10/6/09Copyright © 2009 Regents of the University of CaliforniaEE C245: Introduction to MEMS Design LecM 7 C. Nguyen 9/28/07 37Q vs. TemperatureQuartz Crystal Aluminum Vibrating ResonatorQ ~5,000,000 at 30KQ ~5,000,000 at 30KQ ~300,000,000 at 4KQ ~300,000,000 at 4KQ ~500,000 at 30KQ ~500,000 at 30KQ ~1,250,000 at 4KQ ~1,250,000 at 4KEven aluminum achieves exceptional Q’s at cryogenic temperaturesEven aluminum achieves exceptional
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