EE C245 ME C218 Introduction to MEMS Design Fall 2012

Transcription

1 TN 1/2/12 EE 245 ME 218 Introduction to MEMS Design Fall 212 Prof. lark T.-. Nguyen Dept. of Electrical Engineering & omputer Sciences University of alifornia at Berkeley Berkeley, A 9472 Lecture EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 1 Outline Reading: Senturia, hpt. 8 Lecture Topics: Stress, strain, etc., for isotropic materials Thin films: thermal stress, residual stress, and stress gradients Internal dissipation MEMS material properties and performance metrics EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 2 opyright 212 Regents of the University of alifornia 1

2 TN 1/2/12 Vertical Stress Gradients Variation of residual stress in the direction of film growth an warp released structures in z-direction EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 3 Elasticity EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 4 opyright 212 Regents of the University of alifornia 2

3 TN 1/2/12 Normal Stress (1D) If the force acts normal to a surface, then the stress is called a normal stress z σ z x Δz σ y σ x y Δx Δy Differential volume element EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 5 Strain (1D) EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 6 opyright 212 Regents of the University of alifornia 3

4 TN 1/2/12 The Poisson Ratio EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 7 Shear Stress & Strain (1D) EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 8 opyright 212 Regents of the University of alifornia 4

5 TN 1/2/12 2D and 3D onsiderations Important assumption: the differential volume element is in static equilibrium no net forces or torques (i.e., rotational movements) Every σ must have an equal σ in the opposite direction on the other side of the element For no net torque, the shear forces on different faces must also be matched as follows: Stresses acting on a differential volume element τ xy = τ yx τ xz = τ zx τ yz = τ zy EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 9 2D Strain In general, motion consists of rigid-body displacement (motion of the center of mass) rigid-body rotation (rotation about the center of mass) Deformation relative to displacement and rotation Area element experiences both displacement and deformation Must work with displacement vectors Differential definition ux x of axial strain: ε x = ( + Δx) u ( x) Δx EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 1 x ux = x opyright 212 Regents of the University of alifornia 5

6 TN 1/2/12 2D Shear Strain EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 11 Volume hange for a Uniaxial Stress Stresses acting on a differential volume element EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 12 opyright 212 Regents of the University of alifornia 6

7 TN 1/2/12 Isotropic Elasticity in 3D Isotropic = same in all directions The complete stress-strain relations for an isotropic elastic solid in 3D: (i.e., a generalized Hooke s Law) [ σ ν ( σ σ )] ε + x = 1 x y z E y σ y ν ( σ z σ x ) = 1 [ E ] = 1 [ σ ν ( σ σ )] ε + ε z z x + E y γ 1 xy = G τ γ 1 yz = G τ γ 1 zx = G τ Basically, add in off-axis strains from normal stresses in other directions xy yz zx EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 13 Important ase: Plane Stress ommon case: very thin film coating a thin, relatively rigid substrate (e.g., a silicon wafer) At regions more than 3 thicknesses from edges, the top surface is stress-free σ z = Get two components of in-plane stress: ε x = ( 1 E)[ σ x ν ( σ y + )] ε y = ( 1 E)[ σ y ν ( σ x + )] EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 14 opyright 212 Regents of the University of alifornia 7

8 TN 1/2/12 Important ase: Plane Stress (cont.) Symmetry in the xy-plane σ x = σ y = σ Thus, the in-plane strain components are: ε x = ε y = ε where σ σ ε x = (1 E)[ σ νσ ] = = [ E (1 ν )] E and where Biaxial Modulus = Δ E = E 1 ν EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 15 Edge Region of a Tensile (σ>) Film Net non-zero inplane force (that we just analyzed) At free edge, in-plane force must be zero Film must be bent back, here There s no Poisson contraction, so the film is slightly thicker, here Discontinuity of stress at the attached corner stress concentration Peel forces that can peel the film off the surface EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 16 opyright 212 Regents of the University of alifornia 8

9 TN 1/2/12 Linear Thermal Expansion As temperature increases, most solids expand in volume Definition: linear thermal expansion coefficient Linear thermal expansion coefficient Remarks: α T values tend to be in the 1-6 to 1-7 range an capture the 1-6 by using dimensions of μstrain/k, where 1-6 K -1 = 1 μstrain/k In 3D, get volume thermal expansion coefficient = Δ d x ε α T = [Kelvin -1 ] dt ΔV V = 3α TΔT For moderate temperature excursions, α T can be treated as a constant of the material, but in actuality, it is a function of temperature EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 17 α T As a Function of Temperature [Madou, Fundamentals of Microfabrication, R Press, 1998] ubic symmetry implies that α is independent of direction EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 18 opyright 212 Regents of the University of alifornia 9

10 TN 1/2/12 Thin-Film Thermal Stress Thin Film (α Tf ) Silicon Substrate (α Ts = 2.8 x 1-6 K -1 ) Substrate much thicker than thin film Assume film is deposited stress-free at a temperature T r, then the whole thing is cooled to room temperature T r Substrate much thicker than thin film substrate dictates the amount of contraction for both it and the thin film EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 19 Linear Thermal Expansion EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 2 opyright 212 Regents of the University of alifornia 1

11 TN 1/2/12 MEMS Material Properties EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 21 Material Properties for MEMS (E/ρ) is acoustic velocity [Mark Spearing, MIT] EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 22 opyright 212 Regents of the University of alifornia 11

12 TN 1/2/12 Young s Modulus Versus Density Lines of constant acoustic velocity [Ashby, Mechanics of Materials, Pergamon, 1992] EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 23 Yield Strength Definition: the stress at which a material experiences significant plastic deformation (defined at.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-reversible Yield Strength: defined at.2% offset pt. Elastic Limit: stress at which permanent deformation begins Proportionality Limit: point at which curve goes nonlinear True Elastic Limit: lowest stress at which dislocations move EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 24 opyright 212 Regents of the University of alifornia 12

13 TN 1/2/12 Yield Strength (cont.) Below: typical stress vs. strain curves for brittle (e.g., Si) and ductile (e.g. steel) materials Stress Tensile Strength T=3 o ) Brittle (Si) Proportional Limit Ductile (Mild Steel) Fracture (or T>9 o ) Strain [Maluf] EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 25 Young s Modulus and Useful Strength EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 26 opyright 212 Regents of the University of alifornia 13

14 TN 1/2/12 Young s Modulus Versus Strength Lines of constant maximum strain [Ashby, Mechanics of Materials, Pergamon, 1992] EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 27 Quality Factor (or Q) EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 28 opyright 212 Regents of the University of alifornia 14

15 TN 1/2/12 f W r Frequency: Stiffness o = v i 1 2π Mass lamped-lamped Beam μresonator Resonator Beam Electrode vi kr m r L r Young s Modulus = 1.3 Density E h ρ L 2 r (e.g., m r = r kg) kg) EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 29 V P V P h i o (t) i v o i Q ~1, ω ο V d dt Smaller mass higher freq. range and and lower series R x x i o = P ω Note: If If V P = P V V device off off Measure of the frequency selectivity of a tuned circuit Definition: Quality Factor (or Q) Total Energy Per ycle fo Q = = Energy Lost Per ycle BW Example: series LR circuit Example: parallel LR circuit 3dB Im Q = Re f o ( Z) ol = ω = 1 ( Z ) R ω R o BW -3dB f Im Q = Re ( Y ) o = ω = 1 ( Y ) G ω LG o EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 3 opyright 212 Regents of the University of alifornia 15

16 TN 1/2/12 Selective Low-Loss Filters: Need Q In resonator-based filters: high tank Q low insertion loss At right: a.1% bandwidth, 3- res 1 GHz (simulated) heavy insertion loss for resonator Q < 1, -4 Frequency [MHz] EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 31 Transmission [db] Increasing Insertion Loss Tank Q 3, 2, 1, 5, 4, Main Function: provide a stable output frequency Difficulty: superposed noise degrades frequency stability Sustaining Amplifier A Oscillator: Need for High Q v o Ideal Sinusoid: v ( ) o t = Vosin 2πf t o Frequency-Selective Tank i v o i ω ο Higher Higher Q ω T O Real Sinusoid: v ( ) ( ) ( ) o t = Vo+ ε t sin 2πf t + t o θ Zero-rossing Point ω ο =2π/T O Tighter Tighter Spectrum Spectrum EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 32 ω ο ω ω opyright 212 Regents of the University of alifornia 16

17 TN 1/2/12 Attaining High Q Problem: I s cannot achieve Q s in the thousands transistors consume too much power to get Q on-chip spiral inductors Q s no higher than ~1 off-chip inductors Q s in the range of 1 s Observation: vibrating mechanical resonances Q > 1, Example: quartz crystal resonators (e.g., in wristwatches) extremely high Q s ~ 1, or higher (Q ~ 1 6 possible) mechanically vibrates at a distinct frequency in a thickness-shear mode EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 33 Energy Dissipation and Resonator Q Material Defect Losses 1 1 = Q Q defects 1 + Q TED 1 + Q Gas Gas Damping viscous 1 + Q support Bending -Beam Thermoelastic Damping (TED) Tension old Spot ompression Hot Spot Anchor Losses At At high frequency, this this is is our our big big problem! Heat Flux (TED Loss) Elastic Wave Radiation (Anchor Loss) EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 34 opyright 212 Regents of the University of alifornia 17

18 TN 1/2/12 Thermoelastic Damping (TED) Occurs when heat moves from compressed parts to tensioned parts heat flux = energy loss ς = Γ( T ) Ω( f ) = 2 α TE Γ( T ) = 4ρ ( f p f ) = 2 2 f πk = 2ρ h 1 2Q f + f Ω o TED 2 TED f TED p 2 Bending -Beam h Tension old Spot ompression Hot Spot Heat Flux (TED Loss) ζ = thermoelastic damping factor α = thermal expansion coefficient T = beam temperature E = elastic modulus ρ = material density p = heat capacity at const. pressure K = thermal conductivity f = beam frequency h = beam thickness f TED = characteristic TED frequency EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 35 TED haracteristic Frequency f TED πk = 2ρ h Governed by Resonator dimensions Material properties p 2 ρ = material density p = heat capacity at const. pressure K = thermal conductivity h = beam thickness f TED = characteristic TED frequency ritical Damping Factor, ζ Peak where Q is is minimized Q [from Roszhart, Hilton Head 199] Relative Frequency, f/f TED EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 36 opyright 212 Regents of the University of alifornia 18

19 TN 1/2/12 Q vs. Temperature Quartz rystal Aluminum Vibrating Resonator Q ~5, at at 3K 3K Q ~3,, at at 4K 4K Q ~5,, at at 3K 3K Mechanism for for Q increase with with decreasing temperature thought to to be be linked linked to to less less hysteretic motion motion of of material defects less less energy energy loss loss per per cycle cycle [from Braginsky, Systems With Small Dissipation] Q ~1,25, at at 4K 4K Even Even aluminum achieves exceptional Q s Q s at at cryogenic temperatures EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 37 Polysilicon Wine-Glass Disk Resonator ompound Mode (2,1) Anchor Output Input Wine Glass Disk Resonator Support Beams Input [Y.-W. Lin, Nguyen, JSS Dec. 4] R = 32 μm Anchor Output Resonator Data R = μm, μm, h = 3 μm μm d = 8 8 nm, nm, V p = p 3 V EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 38 Unmatched Transmission [db] f o = MHz Q = 145, Frequency [MHz] opyright 212 Regents of the University of alifornia 19

20 TN 1/2/ GHz, Q=11,555 Nanocrystalline Diamond Disk μmechanical Resonator Impedance-mismatched stem for reduced anchor dissipation Operated in the 2 nd radial-contour mode Q ~11,555 (vacuum); Q ~1,1 (air) Below: 2 μm diameter disk Polysilicon Electrode Polysilicon Stem (Impedance Mismatched to Diamond Disk) VD Diamond μmechanical Disk Resonator R Ground Plane EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 39 Mixed Amplitude [db] Design/Performance: R=1μm, t=2.2μm, d=8å, V P =7V f o =1.51 GHz (2 nd mode), Q=11, Frequency [MHz] f o = 1.51 GHz Q = 11,555 (vac) Q = 1,1 (air) Q = 1,1 (air) [Wang, Butler, Nguyen MEMS 4] Disk Resonator Loss Mechanisms (Not Dominant in Vacuum) Gas Gas Damping Strain Energy Flow Nodal Axis Electronic arrier arrier Drift Drift Loss Loss e- (Dwarfed By Substrate Loss) No No motion along along the the nodal nodal axis, axis, but but motion along along the the finite finite width width of of the the stem stem Hysteretic Motion Motion of of Defect Defect (Dwarfed By Substrate Loss) Disk Stem Substrate Stem Height λ/4 λ/4 helps helps reduce reduce loss, loss, but but not not perfect perfect Substrate Loss Loss Thru Thru Anchors (Dominates) EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 4 opyright 212 Regents of the University of alifornia 2

21 TN 1/2/12 MEMS Material Property Test Structures EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 41 Stress Measurement Via Wafer urvature ompressively stressed film bends a wafer into a convex shape Tensile stressed film bends a wafer into a concave shape an optically measure the deflection of the wafer before and after the film is deposited Determine the radius of curvature R, then apply: θ Laser Scan the mirror h Si-substrate Slope = 1/R x x Detector Mirror θ t σ = E h 6Rt 2 σ = film stress [Pa] R E = E/(1-ν) = biaxial elastic modulus [Pa] h = substrate thickness [m] t = film thickness R = substrate radius of curvature [m] EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 42 opyright 212 Regents of the University of alifornia 21

22 TN 1/2/12 MEMS Stress Test Structure Simple Approach: use a clampedclamped beam ompressive stress causes buckling Arrays with increasing length are used to determine the critical buckling load, where h = thickness L W σ critical 2 π Eh = 2 3 L 2 E = Young s modulus [Pa] I = (1/12)Wh 3 = moment of inertia L, W, h indicated in the figure Limitation: Only compressive stress is measurable EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 43 More Effective Stress Diagnostic Single structure measures both compressive and tensile stress Expansion or contraction of test beam deflection of pointer Vernier movement indicates type and magnitude of stress Expansion ompression ontraction Tensile EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 44 opyright 212 Regents of the University of alifornia 22

23 TN 1/2/12 Q Measurement Using Resonators ompound Mode (2,1) Anchor Output Input Wine Glass Disk Resonator Support Beams Input [Y.-W. Lin, Nguyen, JSS Dec. 4] R = 32 μm Anchor Output Resonator Data R = μm, μm, h = 3 μm μm d = 8 8 nm, nm, V p = p 3 V EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 45 Unmatched Transmission [db] f o = MHz Q = 145, Frequency [MHz] Folded-Beam omb-drive Resonator Issue w/ Wine-Glass Resonator: non-standard fab process Solution: use a folded-beam comb-drive resonator f o =342.5kHz Q=41, 342,5 Q = Hz EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 46 opyright 212 Regents of the University of alifornia 23

24 TN 1/2/12 omb-drive Resonator in Action Below: fully integrated micromechanical resonator oscillator using a MEMS-last integration approach EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 47 Folded-Beam omb-drive Resonator Issue w/ Wine-Glass Resonator: non-standard fab process Solution: use a folded-beam comb-drive resonator f o =342.5kHz Q=41, 342,5 Q = Hz EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 48 opyright 212 Regents of the University of alifornia 24

25 TN 1/2/12 Measurement of Young s Modulus Use micromechanical resonators Young s modulus Resonance frequency depends on E For a folded-beam resonator: Resonance Frequency = h = thickness W L 4Eh( W = M L) EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 49 f o eq Equivalent mass Extract E from measured frequency f o Measure f o for several resonators with varying dimensions Use multiple data points to remove uncertainty in some parameters Anisotropic Materials EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 5 opyright 212 Regents of the University of alifornia 25

26 TN 1/2/12 Elastic onstants in rystalline Materials Get different elastic constants in different crystallographic directions 81 of them in all ubic symmetries make 6 of these terms zero, leaving 21 of them remaining that need be accounted for Thus, describe stress-strain relations using a 6x6 matrix σ x σ y σ z = τ yz τ zx τ xy ε x ε y ε z γ yz γ zx γ xy Stresses Stiffness oefficients Strains EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 51 Stiffness oefficients of Silicon Due to symmetry, only a few of the 21 coefficients are non-zero With cubic symmetry, silicon has only 3 independent components, and its stiffness matrix can be written as: σ x σ y σ z = τ yz τ zx τ xy where ε x ε y ε z γ yz γ zx γ xy EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/ = GPa 12 = 63.9 GPa 44 = 79.6 GPa opyright 212 Regents of the University of alifornia 26

27 TN 1/2/12 Young s Modulus in the (1) Plane [units = 1 GPa] [units = 1 GPa] EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 53 Poisson Ratio in (1) Plane EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 54 opyright 212 Regents of the University of alifornia 27

28 TN 1/2/12 Anisotropic Design Implications Young s modulus and Poisson ratio variations in anisotropic materials can pose problems in the design of certain structures E.g., disk or ring resonators, which rely on isotropic properties in the radial directions Okay to ignore variation in RF resonators, although some Q hit is probably being taken E.g., ring vibratory rate gyroscopes Mode matching is required, where frequencies along different axes of a ring must be the same Not okay to ignore anisotropic variations, here Drive Axis Ring Gyroscope Wine-Glass Mode Disk Sense Axis EE 245: Introduction to MEMS Design LecM 7. Nguyen 9/28/7 55 opyright 212 Regents of the University of alifornia 28