Lecture 4 - The University of Texas at Arlington
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Introduction to Microeletromechanical Systems (MEMS) Lecture 4 Topics • Fundamental MEMS Processes and Devices Surface Micromachined Polysilicon Comb Drives o Mechanics ¾ Stress and Strain ¾ Cantilevers ¾ Resonance o Electrostatics ¾ Parallel Plate Capacitor ¾ Pull-In Voltage ¾ Comb Drives Texas Christian University Department of Engineering Ed Kolesar MEMS Overview Introduction & Background History & Markets Methodology Devices & Structures Processes & Foundries Micromachining: lithography, deposition, etching, … Texas Christian University Department of Engineering Ed Kolesar 11 Texas Christian University Department of Engineering Ed Kolesar Fundamental MEMS Processes and Devices • Example: surface micromachined polysilicon comb drives Mechanics for MEMS - Stress and strain - Cantilever beams - Resonance Electrostatics for MEMS - Parallel-plate capacitors - Pull-in voltage - Comb drives Texas Christian University Department of Engineering Ed Kolesar 22 Electrostatic Comb Drives Principle: interlacing comb fingers create large capacitor area; electrostatically actuated suspended microstructures (Tang, Nguyen and Howe, 1989) Static (Fixed) Comb Features: • Linear relationship between capacitance and displacement • Higher surface area / capacitance than parallel plate capacitor • Electrostatic actuation: low power (no DC current) Texas Christian University Spring Suspensions Released (Moving) Comb Department of Engineering Ed Kolesar Electrostatic Comb Drives Comb drives combine mechanical and electrostatic issues: • Elasticity • Stress and strain • Resonance (natural frequency) • Capacitance • Electrostatic forces • Electrostatic work and energy Tang, Nguyen and Howe JMEMS 1989. Texas Christian University Department of Engineering Ed Kolesar 33 Axial Stress And Strain Stress: force applied to surface σ = F/A σ Si Al measured in N/m2 or Pa compressive or tensile Strain: ratio of deformation to length ε = ∆l / l wood measured in %, ppm, or microstrain ε ∆l l A Texas Christian University F Young’s Modulus: E = σ /ε Hooke’s Law: K = F/∆l = E A/l Department of Engineering Ed Kolesar Shear Stress And Strain Shear Stress: force applied parallel to surface τ = F/A measured in N/m2 or Pa Shear Strain: ratio of deformation to length γ = ∆l / l A F ∆l Shear Modulus: G=τ/γ l Texas Christian University Department of Engineering Ed Kolesar 44 Poisson’s Ratio Tensile stress in x direction results in compressive stress in y and z direction (object becomes longer and thinner) Poisson’s Ratio: ν = - εy / εx = - transverse strain / longitudinal strain Metals: ν ≈ 0.3 Rubbers: ν ≈ 0.5 ν≈0 Cork: Texas Christian University Department of Engineering Ed Kolesar Cantilever Beams Axial Strain: ε(y) = y/ρ ρ radius of curvature Axial Stress: σ(y) = E ε(y) L t w y x Assume that x axis lies in center of beam Axial Force: dF = σ(y) w dy Bending Moment: M = 1/12 t3 w E / ρ =EI/ρ I = 1/12 t3 w (area moment of inertia) Texas Christian University Department of Engineering Ed Kolesar 55 Cantilever Beams F(x) = F0 M(x) = M0 + F(L-x) L M0 y F0 x Assume that we apply a force F0 and a moment M0 on a beam with length L Texas Christian University For M0 = 0 y(x) = F / (6EI) (3 Lx2 - x3) y(L) = FL3 / 3EI Spring Constant, K = F/y = 3EI/L3 = Et3w / 4L3 Department of Engineering Ed Kolesar Cantilever Beams Point Load Distributed Load Cantilever y(x) = F/(6EI) (3 Lx2-x3) σmax = FLt / 2I y(x) = ρx2/(24EI) (6L2 - 4Lx + x2) σmax = ρL2t / 4I Bridge y(x) = Fx/(48EI) (3 Lx-4x2) for L/2 ≥ x ≥ 0 σmax = FLt / 8I y(x) = ρx2/(24EI) (L - x)2 σmax = ρL2t / 12I L length of beam, t thickness of beam, w width of beam I = wt3/12 bending moment of inertia Texas Christian University Department of Engineering Ed Kolesar 66 SCS Beam Example: E = 100 GPa K = Ea3b / 4L3 = 0.4 N/m = 0.4 µN/µm L=100µm t=2µm w=2µm y How much does beam bend in a 1g gravity field? m = ρ V (assume mass at end of beam) = 2.3 gram/cm3 400 µm3 ≈ 10-12 kg ∆y ≈ 2.5 10-11 m = 0.25 Å detectable! x Texas Christian University Department of Engineering Ed Kolesar Resonators L = 100µm t = 2µm w = 2µm y mx’’ + bx’ + Kx = F (Newton dynamics with damping and springs) For b = 0: Κ 1 Et 3 w = m 2 mL3 f0 = ω / 2π ≈ 100 kHz ω= x Notice: if t = 1µm fy = f0 / 2 fz = f0 Texas Christian University Department of Engineering Ed Kolesar 77 Electrostatic Forces A d Parallel Plate Capacitor: Capacitance: C = Q/V = ε0 εr A / d ε0 ≈ 8.854 10-12 F/m dielectric constant of free space εr dielectric permittivity Stored energy: W = ½ C V2 = ½ Q2 / C V Texas Christian University Electrostatic force between plates: F = ½ C/d V2 Department of Engineering Ed Kolesar Electrostatic Actuation x Positioning of capacitor plate: Fel = ½ ε0 εr A V2 / x2 FS = K (x - d0) d0 : distance at rest (no applied voltage) Stable equilibrium when Fel = -FS F Fel(V) Kd0 V -FS d0 Texas Christian University Department of Engineering x Ed Kolesar 88 Pull-In Point x The higher V, the closer the plate is pulled in. Fel → ∞ when d → 0. What is the closest stable distance xmin? Fel and -FS must be tangential: -ε0 εr A/x3 V2 = -K , so V2 = K x3 / ε0 εr A V Texas Christian University Substitute into Fel = -FS to get xmin = 2/3 d0 can control x only from 2/3 d0 to d0 Department of Engineering Ed Kolesar Electrostatic Comb Drive Capacitance is approximately: C = ε0 εr A/d = 2n ε0 εr lh/d Change in capacitance when moving by ∆x: ∆l ∆C = ε0 εr ∆A / d = 2n ε0 εr ∆l h/d d Electrostatic force: w Fel = ½ V2 dC/dx = n ε0 εr h/d V2 Note: Fel independent of ∆l over wide range (fringing field), and quadratic in V. Texas Christian University Department of Engineering V Ed Kolesar 99 Electrostatic Accelerometer Example: use MEMS comb structures as accelerometer h = 100 µm n = 100 d = 1 µm Spring Constant: K = 1 N/m Proof Mass: m = 0.1 mg Acceleration: a = 0.1 g ∆x = 0.1 µm ∆C = 17.7 fF Texas Christian University Department of Engineering Ed Kolesar Accelerometers SANDIA’s IMEMS Process http://www.sandia.gov/mems/ micromachine/pix/techinfo/cmos.gif Three-axis accelerometer micrograph with labeling of functional units as reported by Lemkin et al, Proceedings ISSCC 1997. Texas Christian University Texas Christian University Department of Engineering Ed Kolesar 1010 Gyroscopes F. Ayazi and K. Najafi, “Design and fabrication of a high-performance polysilicon vibrating ring gyroscope,” in Proc. IEEE Micro Electro Mechanical Systems Workshop (MEMS 1998), Heidelberg, Germany, February 1998, pp. 621–626. Texas Christian University Department of Engineering Ed Kolesar MEMS Gyroscope (B. Clark, R. Horowitz and R. T. Howe, 1996) Texas Christian University Department of Engineering Ed Kolesar 1111 Comb Drive Design Combs Suspensions Linear Cantilever / Bridge Rotational Crab Leg Texas Christian University Department of Engineering Ed Kolesar Comb Drive Failure Modes Comb drives require low stiffness in x direction but high stiffness in y, z direction as well as rotations. Note: comb fingers are in unstable equilibrium with respect to the y direction. Good Poor Poor y x Texas Christian University Department of Engineering Ed Kolesar 1212 Comb Drive Fabrication Surface micromachining with 1 released polysilicon layer Tang, Nguyen and Howe (UC Berkeley) This process formed basis for many subsequent MEMS designs Figure: Tang, Nguyen and Howe, 1989. Texas Christian University Department of Engineering Ed Kolesar Electrostatic Actuators Ideas • • • • • • • • • • Comb drive Rotors Scratch drive T-drive Parallelogram Zipper DMD (torsional mirrors) Impact actuator Microengine Inchworm motors (see actin and myosin) Texas Christian University Department of Engineering Issues • Force, F • Range, s • Frequency, 1/t P = F s/t • Linearity • Efficiency Ed Kolesar 1313 Translation Rotation Acceleration, velocity, distance Angular acc., ang. vel., angle α = ω& = φ&& a = v& = &x& Force, momentum Torque, angular momentum Kinetic energy Kinetic energy F p= mv = Ft T = r×F L = r × p = Iω = Tt E = 12 Iω 2 E = 12 mv2 Dynamics (spring,damper,mass) Dynamics (moment of inertia) F = Kx + bx& + m&x& T = Κφ + βφ& + Iφ&& Oscillation (assume b=0 ) Oscillation (assume β=0 ) f = 1 2π f = K m Texas Christian University 1 2π Κ I Department of Engineering Ed Kolesar Coriolis Force Force generated when rotating a rotating system Underlying principle: conservation of angular momentum. Torque: r r r r T = ∇L = ∇( Iω ) = I∇ω Time-dependent angular velocity: r sin Ωt ω (t ) = ω 0 cos Ωt 0 Gradient: r −cos Ωt ∇ω (t ) = ω 0 Ω sin Ωt 0 r T = I∇ω = Iω 0 Ω Texas Christian University Coriolis Force : Fc = T / r = Iω0Ω / r ring : I = mr 2 , Fc = mv0Ω disk : I = 12 mr 2 , Fc = 12 mv0Ω Department of Engineering Ed Kolesar 1414