EE 245: Introduction to MEMS Lecture 14m: Mechanics of Materials CTN 10/12/10. Copyright 2010 Regents of the University of California

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1 EE 45: Introduction to MEMS Lecture 4m: Mecanics o Materials Measure o te requency selectivity o a tuned circuit Deinition: uality Factor (or ) Total Energy Per ycle o Energy Lost Per ycle BW Example: series LR circuit Example: parallel LR circuit 3dB Im Re o ( Z ) ol ω ( Z ) R ω R o BW -3dB omb-drive Resonator in Action TN // Below: ully integrated micromecanical resonator oscillator using a MEMS-last integration approac Im Re ( Y ) o ω ( Y ) G ω LG o EE 45: Introduction to MEMS Design LecM 7. Nguyen 9/8/7 3 EE 45: Introduction to MEMS Design LecM 7. Nguyen 9/8/7 3 Selective Low-Loss Filters: Need Oscillator: Need or Hig Main Function: provide a stable output requency Diiculty: superposed noise degrades requency stability Sustaining Ampliier A v o Ideal Sinusoid: v ( ) o t Vosin π t o In resonator-based ilters: ig tank low insertion loss At rigt: a.% bandwidt, 3- res GHz (simulated) eavy insertion loss or resonator <, EE 45: Introduction to MEMS Design LecM 7. Nguyen 9/8/7 3 Transmission [db] Increasing Insertion Loss Tank 3,,, 5, 4, Frequency-Selective Tank i v o i ω ο Higer Higer ω T O Real Sinusoid: v ( ) ( ) ( ) o t Vo+ ε t sin π t + t o θ Zero-rossing Point ω ο π/t O Tigter Tigter Spectrum Spectrum EE 45: Introduction to MEMS Design LecM 7. Nguyen 9/8/7 33 ω ο ω ω opyrigt Regents o te University o aliornia

2 EE 45: Introduction to MEMS Lecture 4m: Mecanics o Materials Attaining Hig Problem: I s cannot acieve s in te tousands transistors consume too muc power to get on-cip spiral inductors s no iger tan ~ o-cip inductors s in te range o s Observation: vibrating mecanical resonances >, Example: quartz crystal resonators (e.g., in wristwatces) extremely ig s ~, or iger ( ~ 6 possible) mecanically vibrates at a distinct requency in a tickness-sear mode TN // Energy Dissipation and Resonator Material Deect Losses Bending -Beam deects + + Gas Gas Damping viscous Termoelastic Damping () Tension old Spot ompression Hot Spot + support Ancor Losses At At ig requency, tis tis is is our our big big problem! Heat Flux ( Loss) Elastic Wave Radiation (Ancor Loss) EE 45: Introduction to MEMS Design LecM 7. Nguyen 9/8/7 34 EE 45: Introduction to MEMS Design LecM 7. Nguyen 9/8/7 35 Termoelastic Damping () aracteristic Frequency Occurs wen eat moves rom compressed parts to tensioned parts eat lux energy loss ς Γ( T ) Ω( ) α TE Γ( T ) 4ρ ( p ) πk ρ + Ω o p Bending -Beam Tension old Spot ompression Hot Spot Heat Flux ( Loss) ζ termoelastic damping actor α termal expansion coeicient T beam temperature E elastic modulus ρ material density p eat capacity at const. pressure K termal conductivity beam requency beam tickness caracteristic requency EE 45: Introduction to MEMS Design LecM 7. Nguyen 9/8/7 36 πk ρ Governed by Resonator dimensions Material properties p ρ material density p eat capacity at const. pressure K termal conductivity beam tickness caracteristic requency ritical Damping Factor, ζ [rom Roszart, Hilton Head 99] Peak were is is minimized Relative Frequency, / EE 45: Introduction to MEMS Design LecM 7. Nguyen 9/8/7 37 opyrigt Regents o te University o aliornia

3 EE 45: Introduction to MEMS Lecture 4m: Mecanics o Materials uartz rystal ~3,, at at 4K 4K vs. Temperature ~5,, at at 3K 3K Mecanism or or increase wit wit decreasing temperature tougt to to be be linked linked to to less less ysteretic motion motion o o material deects less less energy loss loss per per cycle cycle Aluminum Vibrating Resonator ~5, at at 3K 3K [rom Braginsky, Systems Wit Small Dissipation] ~,5, at at 4K 4K Even Even aluminum acieves exceptional s s at at cryogenic temperatures EE 45: Introduction to MEMS Design LecM 7. Nguyen 9/8/7 38 ompound Mode (,) Ancor Wine Glass Disk Resonator Support Beams [Y.-W. Lin, Nguyen, JSS Dec. 4] R 3 μm Ancor TN // Polysilicon Wine-Glass Disk Resonator Resonator Data R 3 3 μm, μm, 3 μm μm d 8 8 nm, nm, V p p 3 V EE 45: Introduction to MEMS Design LecM 7. Nguyen 9/8/7 39 Unmatced Transmission [db] o 6.37 MHz 45, GHz,,555 Nanocrystalline Diamond Disk μmecanical Resonator Impedance-mismatced stem or reduced ancor dissipation Operated in te nd radial-contour mode ~,555 (vacuum); ~, (air) Below: μm diameter disk Polysilicon Stem (Impedance Mismatced -84 to Diamond Disk) -86 o.5 GHz,555 (vac) -88, (air) -9 Polysilicon Electrode R -9, (air) VD Diamond μmecanical Disk Ground Resonator Plane EE 45: Introduction to MEMS Design LecM 7 [Wang,. Butler, Nguyen Nguyen 9/8/7 MEMS 4] 4 Mixed Amplitude [db] Design/Perormance: Rμm, t.μm, d8å, V P 7V o.5 GHz ( nd mode),,555 Disk Resonator Loss Mecanisms (Not Dominant in Vacuum) Gas Gas Damping Hysteretic Motion o o Deect Deect (Dwared By Substrate Loss) Strain Energy Flow Disk Stem Substrate Nodal Axis Electronic arrier (Dwared By Drit Drit Loss Loss Substrate Loss) No No motion along along te te nodal nodal axis, axis, but but motion along along e- te te inite inite widt widt o o te te stem stem Stem Heigt λ/4 λ/4 elps elps reduce loss, loss, but but not not perect Substrate Loss Loss Tru Tru Ancors (Dominates) EE 45: Introduction to MEMS Design LecM 7. Nguyen 9/8/7 4 opyrigt Regents o te University o aliornia 3

4 EE 45: Introduction to MEMS Lecture 4m: Mecanics o Materials TN // Stress Measurement Via Waer urvature MEMS Material Property Test Structures EE 45: Introduction to MEMS Design LecM 7. Nguyen 9/8/7 4 ompressively stressed ilm bends a waer into a convex sape Tensile stressed ilm bends a waer into a concave sape an optically measure te delection o te waer beore and ater te ilm is deposited Determine te radius o curvature R, ten apply: σ E 6Rt Laser Scan te mirror Si-substrate Detector Mirror σ ilm stress [Pa] R E E/(-ν) biaxial elastic modulus [Pa] substrate tickness [m] t ilm tickness R substrate radius o curvature [m] EE 45: Introduction to MEMS Design LecM 7. Nguyen 9/8/7 43 θ Slope /R x x θ t MEMS Stress Test Structure More Eective Stress Diagnostic Simple Approac: use a clampedclamped beam ompressive stress causes buckling Arrays wit increasing lengt are used to determine te critical buckling load, were tickness L W Single structure measures bot compressive and tensile stress Expansion or contraction o test beam delection o pointer Vernier movement indicates type and magnitude o stress σ critical π E 3 L E Young s modulus [Pa] I (/)W 3 moment o inertia L, W, indicated in te igure Expansion ompression ontraction Tensile Limitation: Only compressive stress is measurable EE 45: Introduction to MEMS Design LecM 7. Nguyen 9/8/7 44 EE 45: Introduction to MEMS Design LecM 7. Nguyen 9/8/7 45 opyrigt Regents o te University o aliornia 4

5 EE 45: Introduction to MEMS Lecture 4m: Mecanics o Materials Measurement Using Resonators TN // Folded-Beam omb-drive Resonator ompound Mode (,) Ancor Wine Glass Disk Resonator Support Beams [Y.-W. Lin, Nguyen, JSS Dec. 4] R 3 μm Ancor Resonator Data R 3 3 μm, μm, 3 μm μm d 8 8 nm, nm, V p p 3 V EE 45: Introduction to MEMS Design LecM 7. Nguyen 9/8/7 46 Unmatced Transmission [db] o 6.37 MHz 45, Issue w/ Wine-Glass Resonator: non-standard ab process Solution: use a olded-beam comb-drive resonator o 34.5kHz 4, 34, Hz EE 45: Introduction to MEMS Design LecM 7. Nguyen 9/8/7 47 omb-drive Resonator in Action Below: ully integrated micromecanical resonator oscillator using a MEMS-last integration approac Folded-Beam omb-drive Resonator Issue w/ Wine-Glass Resonator: non-standard ab process Solution: use a olded-beam comb-drive resonator o 34.5kHz 4, 34, Hz EE 45: Introduction to MEMS Design LecM 7. Nguyen 9/8/7 48 EE 45: Introduction to MEMS Design LecM 7. Nguyen 9/8/7 49 opyrigt Regents o te University o aliornia 5

6 EE 45: Introduction to MEMS Lecture 4m: Mecanics o Materials TN // Measurement o Young s Modulus Use micromecanical resonators Young s modulus Resonance requency depends on E For a olded-beam resonator: 3 W Resonance Frequency tickness L 4E( W M L) EE 45: Introduction to MEMS Design LecM 7. Nguyen 9/8/7 5 o eq Equivalent mass Extract E rom measured requency o Measure o or several resonators wit varying dimensions Use multiple data points to remove uncertainty in some parameters Anisotropic Materials EE 45: Introduction to MEMS Design LecM 7. Nguyen 9/8/7 5 Elastic onstants in rystalline Materials Get dierent elastic constants in dierent crystallograpic directions 8 o tem in all ubic symmetries make 6 o tese terms zero, leaving o tem remaining tat need be accounted or Tus, describe stress-strain relations using a 6x6 matrix σ x σ y σ z τ yz τ zx τ xy Stresses ε x ε y ε z γ yz γ zx γ xy EE 45: Introduction to MEMS Design LecM 7. Nguyen 9/8/ Stiness oeicients Strains Stiness oeicients o Silicon Due to symmetry, only a ew o te coeicients are non-zero Wit cubic symmetry, silicon as only 3 independent components, and its stiness matrix can be written as: σ x σ y σ z τ yz τ zx τ xy 3 were ε x ε y ε z γ yz γ zx 66 γ xy EE 45: Introduction to MEMS Design LecM 7. Nguyen 9/8/ GPa 63.9 GPa GPa opyrigt Regents o te University o aliornia 6

7 EE 45: Introduction to MEMS Lecture 4m: Mecanics o Materials Young s Modulus in te () Plane Poisson Ratio in () Plane TN // [units GPa] [units GPa] EE 45: Introduction to MEMS Design LecM 7. Nguyen 9/8/7 54 EE 45: Introduction to MEMS Design LecM 7. Nguyen 9/8/7 55 Anisotropic Design Implications Young s modulus and Poisson ratio variations in anisotropic materials can pose problems in te design o certain structures E.g., disk or ring resonators, wic rely on isotropic properties in te radial directions Okay to ignore variation in RF resonators, altoug some it is probably being taken E.g., ring vibratory rate gyroscopes Mode matcing is required, were requencies along dierent axes o a ring must be te same Not okay to ignore anisotropic variations, ere Drive Axis Ring Gyroscope Wine-Glass Mode Disk Sense Axis EE 45: Introduction to MEMS Design LecM 7. Nguyen 9/8/7 56 opyrigt Regents o te University o aliornia 7