EE C245 ME C218 Introduction to MEMS Design Fall 2007 Prof. Clark T.-C. Nguyen Dept. of Electrical Engineering & Computer Sciences University of California at Berkeley Berkeley, CA 94720 Lecture 13: Material Properties EE C245: Introduction to MEMS Design Lecture 13 C. Nguyen 10/9/07 1 Lecture Outline Reading: Senturia Chpt. 8 Lecture Topics: Elasticity: Nomenclature Stress Strain Poisson Ratio Material Properties Young s modulus Yield strength Quality factor On-chip Measurement of Material Properties Anisotropic Material Properties EE C245: Introduction to MEMS Design Lecture 13 C. Nguyen 10/9/07 2 1
Normal Stress (1D) If the force acts normal to a surface, then the stress is called a normal stress z σ z x z σ y σ x y x y Differential volume element EE C245: Introduction to MEMS Design Lecture 13 C. Nguyen 10/9/07 3 Strain (1D) EE C245: Introduction to MEMS Design Lecture 13 C. Nguyen 10/9/07 4 2
The Poisson Ratio EE C245: Introduction to MEMS Design Lecture 13 C. Nguyen 10/9/07 5 Shear Stress & Strain (1D) EE C245: Introduction to MEMS Design Lecture 13 C. Nguyen 10/9/07 6 3
2D and 3D Considerations 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 C245: Introduction to MEMS Design Lecture 13 C. Nguyen 10/9/07 7 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 C245: Introduction to MEMS Design Lecture 13 C. Nguyen 10/9/07 8 x ux = x 4
2D Shear Strain EE C245: Introduction to MEMS Design Lecture 13 C. Nguyen 10/9/07 9 Volume Change for a Uniaxial Stress Stresses acting on a differential volume element EE C245: Introduction to MEMS Design Lecture 13 C. Nguyen 10/9/07 10 5
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 C245: Introduction to MEMS Design Lecture 13 C. Nguyen 10/9/07 11 Important Case: Plane Stress Common 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 = 0 Get two components of in-plane stress: ε x = ( 1 E)[ σ x ν ( σ y + 0)] ε y = ( 1 E)[ σ y ν ( σ x + 0)] EE C245: Introduction to MEMS Design Lecture 13 C. Nguyen 10/9/07 12 6
Important Case: 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 = E = E 1 ν EE C245: Introduction to MEMS Design Lecture 13 C. Nguyen 10/9/07 13 Edge Region of a Tensile (σ>0) 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 C245: Introduction to MEMS Design Lecture 13 C. Nguyen 10/9/07 14 7
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 10-6 to 10-7 range Can capture the 10-6 by using dimensions of μstrain/k, where 10-6 K -1 = 1 μstrain/k In 3D, get volume thermal expansion coefficient = 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 C245: Introduction to MEMS Design Lecture 13 C. Nguyen 10/9/07 15 α T As a Function of Temperature [Madou, Fundamentals of Microfabrication, CRC Press, 1998] Cubic symmetry implies that α is independent of direction EE C245: Introduction to MEMS Design Lecture 13 C. Nguyen 10/9/07 16 8
Thin-Film Thermal Stress Thin Film (α Tf ) Silicon Substrate (α Ts = 2.8 x 10-6 K -1 ) Substrate much thicker than thin film Assume film is deposited stress-free at a temperature T d, 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 C245: Introduction to MEMS Design Lecture 13 C. Nguyen 10/9/07 17 Linear Thermal Expansion EE C245: Introduction to MEMS Design Lecture 13 C. Nguyen 10/9/07 18 9
MEMS Material Properties EE C245: Introduction to MEMS Design Lecture 13 C. Nguyen 10/9/07 19 Material Properties for MEMS (E/ρ) is acoustic velocity [Mark Spearing, MIT] EE C245: Introduction to MEMS Design Lecture 13 C. Nguyen 10/9/07 20 10
Young s Modulus Versus Density Lines of constant acoustic velocity [Ashby, Mechanics of Materials, Pergamon, 1992] EE C245: Introduction to MEMS Design Lecture 13 C. Nguyen 10/9/07 21 Yield 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-reversible Yield Strength: defined at 0.2% 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 C245: Introduction to MEMS Design Lecture 13 C. Nguyen 10/9/07 22 11
Yield Strength (cont.) Below: typical stress vs. strain curves for brittle (e.g., Si) and ductile (e.g. steel) materials Stress Tensile Strength (Si @ T=30 o C) Brittle (Si) Proportional Limit Ductile (Mild Steel) Fracture (or Si @ T>900 o C) Strain [Maluf] EE C245: Introduction to MEMS Design Lecture 13 C. Nguyen 10/9/07 23 Young s Modulus and Useful Strength EE C245: Introduction to MEMS Design Lecture 13 C. Nguyen 10/9/07 24 12
Young s Modulus Versus Strength Lines of constant maximum strain [Ashby, Mechanics of Materials, Pergamon, 1992] EE C245: Introduction to MEMS Design Lecture 13 C. Nguyen 10/9/07 25 Quality Factor (or Q) EE C245: Introduction to MEMS Design Lecture 13 C. Nguyen 10/9/07 26 13
Measure of the frequency selectivity of a tuned circuit Definition: Quality Factor (or Q) Total Energy Per Cycle fo Q = = Energy Lost Per Cycle BW Example: series LCR circuit Example: parallel LCR circuit 3dB Im Q = Re f o ( Z) ol = ω = 1 ( Z ) R ω CR o BW -3dB f Im Q = Re ( Y ) oc = ω = 1 ( Y ) G ω LG o EE C245: Introduction to MEMS Design Lecture 13 C. Nguyen 10/9/07 27 Selective 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) heavy insertion loss for resonator Q < 10,000-40 Frequency [MHz] EE C245: Introduction to MEMS Design Lecture 13 C. Nguyen 10/9/07 28 Transmission [db] 0-5 -10-15 -20-25 -30-35 Increasing Insertion Loss Tank Q 30,000 20,000 10,000 5,000 4,000 998 999 1000 1001 1002 14
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-Crossing Point ω ο =2π/T O Tighter Tighter Spectrum Spectrum EE C245: Introduction to MEMS Design Lecture 13 C. Nguyen 10/9/07 29 ω ο ω ω 15