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Electrostatic Sensors and Actuators Chang Liu Chang Liu MASS UIUC Single crystal silicon and wafers • To use Si as a substrate material, it should be pure Si in a single crystal form – The Czochralski (CZ) method: A seed crystal is attached at the tip of a puller, which slowly pulls up to form a larger crystal – 100 mm (4 in) diameter x 500 mm thick – 150 mm (6 in) diameter x 750 mm thick – 200 mm (8 in) diameter x 1000 mm thick Chang Liu MASS UIUC Miller indices • A popular method of designating crystal planes (hkm) and orientations <hkm> – – – – • <hkm> designate the direction normal to the plane (hkm) – Chang Liu Identify the axial intercepts Take reciprocal Clear fractions (not taking lowest integers) Enclose the number with ( ) : no comma (100), (110), (111) MASS UIUC Stress and Strain • Definition of Stress and Strain – The normal stress (Pa, N/m2) F A – The strain L L0 L L0 L0 – Poisson’s ratio Chang Liu y z x x MASS UIUC Hooke’s Law E E: Modulus of Elasticity, Young’s Modulus The shear stress The shear strain F A X L The shear modulus of elasticity G The relationship G Chang Liu E 21 MASS UIUC General Relation Between Tensile Stress and Strain Chang Liu MASS UIUC • The behavior of brittle materials (Si) and soft rubber used extensively in MEMS • A material is strong if it has high yield strength or ultimate strength. Si is even stronger than stainless steel • Ductility is a measure of the degree of plastic deformation that has been sustained at the point of fracture • Toughness is a mechanical measure of the material’s ability to absorb energy up to fracture (strength + ductility) • Resilience is the capacity of a material to absorb energy when it is deformed elastically, then to have this energy recovered upon unloading Chang Liu MASS UIUC Mechanical Properties of Si and Related Thin Films • 거시적인 실험데이터는 평균적인 처리로 대개 많은 변이가 없는데 미시적인 실험은 어렵고 또 박막의 조건 (공정조건, Growth 조건 등), 표면상태, 열처리 과정 때문에 일관적이지 않음 • The fracture strength is size dependent; it is 23-28 times larger than that of a millimeter-scale sample Hall Petch equation; y 0 Kd 1/ 2 • For single crystal silicon, Young’s modulus is a function of the crystal orientaiton • For plysilicon thin films, it depends on the process condition (differ from Lab. to Lab.) Chang Liu MASS UIUC General Stress-Strain Relations xx , yy , zz T1 , T2 , T3 yz , xz , xy T4 , T5 , T6 T1 C11 T C 2 21 T3 C31 T4 C41 T5 C51 T6 C61 C12 C13 C22 C23 C32 C33 C42 C43 C52 C53 C62 C63 C14 C24 C34 C44 C54 C64 C15 C25 C35 C45 C55 C65 C16 1 C26 2 C36 3 C46 4 C56 5 C66 6 T C C: stiffness matrix ST S: compliance matrix For many materials of interest to MEMS, the stiffness can be simplified Chang Liu yz , xz , xy T4 , T5 , T6 CSi,100 0 0 1.66 0.64 0.64 0 0.64 1.66 0.64 0 0 0 0.64 0.64 1.66 0 0 0 11 10 Pa 0 0 0 0 . 8 0 0 0 0 0 0 0.8 0 0 0 0 0 0.8 0 MASS UIUC Flexural Beam Bending • Types of Beams; Fig. 3.15 • Possible Boundary Conditions Chang Liu MASS UIUC Longitudinal Strain Under Pure Bending Pure Bending; The moment is constant throughout the beam My EI max Chang Liu Mt 2 EI MASS UIUC Deflection of Beams d2y EI 2 M ( x) dx max Fl 2 Fl 3 , d max 2 EI 3EI d max Fl 3 12 EI d max Fl 3 192EI max Chang Liu Appendix B Ml Ml 2 , d max EI 2 EI MASS UIUC Finding the Spring Constant Chang Liu MASS UIUC Outline • Basic Principles – capacitance formula – capacitance configuration • Applications examples – sensors – actuators • Analysis of electrostatic actuator – second order effect - “pull in” effect • Application examples and detailed analysis Chang Liu MASS UIUC Mechanics of Micro Structures Chang Liu Micro Actuators, Sensors, Systems Group University of Illinois at Urbana-Champaign Chang Liu MASS UIUC Mechanical Variables of Concern • Force constant – flexibility of a given device • Mechanical resonant frequency – response speed of device – Hooke’s law applied to DC driving • Importance of resonant freq. – Limits the actuation speed – lower energy consumption at Fr Fmechanical Felectric Km Fmechanical K m x Chang Liu MASS UIUC Types of Electrical-Mechanical Analysis • Given dimensions and materials of electrostatic structure, find – force constant of the suspension – structure displacement prior to pull-in – value of pull-in voltage • Given the range of desired applied voltage and the desired displacement, find – dimensions of a structure – layout of a structure – materials of a structure • Given the desired mechanical parameters including force constants and resonant frequency, find – – – – Chang Liu dimensions materials layout design quasistatic displacement MASS UIUC Analysis of Mechanical Force Constants • • • Concentrate on cantilever beam (micro spring boards) Three types of most relevant boundary conditions – free: max. degrees of freedom – fixed: rotation and translation both restricted – guided: rotation restricted. Beams with various combination of boundary conditions – fixed-free, one-end-fixed beam – fixed-fixed beam – fixed-guided beam Fixed-free Two fixedguided beams Four fixed-guided beams Chang Liu MASS UIUC Examples Chang Liu MASS UIUC Boundary Conditions • Six degrees of freedom: three axis translation, three axis rotation • Fixed B.C. – no translation, no rotation • Free B.C. – capable of translation AND rotation • Guided B.C. – capable of translation BUT NOT rotation Chang Liu MASS UIUC A Clamped-Clamped Beam Fixed-guided Fixed-guided Chang Liu MASS UIUC A Clamped-Free Beam Chang Liu MASS UIUC One-end Supported, “Clamped-Free” Beams Chang Liu MASS UIUC Fixed-Free Beam by Sacrificial Etching • Right anchor is fixed because its rotation is completely restricted. • Left anchor is free because it can translate as well as rotate. • Consider the beam only moves in 2D plane (paper plane). No out-of-plane translation or rotation is encountered. Chang Liu MASS UIUC Force Constants for Fixed-Free Beams • Dimensions – length, width, thickness – unit in mm. • Materials – Young’s modulus, E – Unit in Pa, or N/m2. Chang Liu MASS UIUC Modulus of Elasticity • Names – Young’s modulus – Elastic modulus • Definition F x E A x L L • Values of E for various materials can be found in notes, text books, MEMS clearing house, etc. Chang Liu MASS UIUC Large Displacement vs. Small Displacement • Small displacement – end displacement less than 1020 times the thickness. – Used somewhat loosely because of the difficulty to invoke large-deformation analysis. Chang Liu • Large deformation – needs finite element computeraided simulation to solve precisely. – In limited cases exact analytical solutions can be found. MASS UIUC Force Constants for Fixed-Free Beams • Moment of inertia I (unit: m4) 3 – I= wt for rectangular crosssection 12 • Maximum angular displacement Fl 2 2 EI Fl 3 3EI • Maximum vertical displacement under F is • Therefore, the equivalent force constant is F 3EI Ewt 3 km 3 Fl 3 l 4l 3 3EI • Formula for 1st order resonant frequency – where Chang Liu is the beam weight per unit length. 3.52 EIg 2 l 4 MASS UIUC Zig-Zag Beams • Used to pack more “L” into a given footprint area on chip to reduce the spring constant without sacrificing large chip space. Saves chip real-estate Chang Liu MASS UIUC An Example Chang Liu MASS UIUC Order of Resonance • 1st order: one node where the gradient of the beam shape is zero; – also called fundamental mode. – With lowest resonance frequency. • 2nd order: 2 nodes where the gradient of the beam shape is zero; • 3nd order: 3 nodes. • Frequency increases as the order number goes up. Chang Liu MASS UIUC Resonant frequency of typical spring-mass system • Self-mass or concentrated mass being m • The resonant frequency is 1 2 k m • Check consistency of units. • High force constant (stiff spring) leads to high resonant frequency. • Low mass (low inertia) leads to high resonant frequency. • To satisfy both high K and high resonant frequency, m must be low. Chang Liu MASS UIUC Quality Factor • If the distance between two half-power points is df, and the resonance frequency if fr, then – Q=fr/df • Q=total energy stored in system/energy loss per unit cycle • Source of mechanical energy loss – crystal domain friction – direct coupling of energy to surroundings – distrubance and friction with surrounding air • example: squeezed film damping between two parallel plate capacitors • requirement for holes: (1) to reduce squeezed film damping; (2) facilitate sacrificial layer etching (to be discussed later in detail). • Source of electrical energy loss – resistance ohmic heating – electrical radiation Chang Liu MASS UIUC Basic Principles • Sensing – capacitance between moving and fixed plates change as • distance and position is changed • media is replaced • Actuation – electrostatic force (attraction) between moving and fixed plates as • a voltage is applied between them. • Two major configurations – parallel plate capacitor (out of plane) – interdigitated fingers - IDT (in plane) A Interdigitated finger configuration d Parallel plate configuration Chang Liu MASS UIUC Examples • Parallel Plate Capacitor • Comb Drive Capacitor Chang Liu MASS UIUC Parallel Plate Capacitor A Fringe electric field (ignored in first order analysis) C d Q V E Q / A C Q A Q d d A – Equations without considering fringe electric field. – A note on fringe electric field: The fringe field is frequently ignored in first-order analysis. It is nonetheless important. Its effect can be captured accurately in finite element simulation tools. Chang Liu MASS UIUC Fabrication Methods • Surface micromachining • Wafer bonding • 3D assembly Flip and bond Movable vertical plate Chang Liu MASS UIUC Forces of Capacitor Actuators • Stored energy 1 1 Q2 2 E CV 2 2 C • Force is derivative of energy with F E 1 C V 2 d 2 d respect to pertinent dimensional variable • Plug in the expression for capacitor C Q A Q d A • We arrive at the expression for force Chang Liu d E 1 A 2 1 CV 2 F V d 2 d2 2 d MASS UIUC Relative Merits of Capacitor Actuators Pros • Nearly universal sensing and actuation; no need for special materials. • Low power. Actuation driven by voltage, not current. • High speed. Use charging and discharging, therefore realizing full mechanical response speed. Chang Liu Cons • Force and distance inversely scaled - to obtain larger force, the distance must be small. • In some applications, vulnerable to particles as the spacing is small - needs packaging. • Vulnerable to sticking phenomenon due to molecular forces. • Occasionally, sacrificial release. Efficient and clean removal of sacrificial materials. MASS UIUC Capacitive Accelerometer • Proof mass area 1x0.6 mm2, and 5 mm thick. • Net capacitance 150fF • External IC signal processing circuits Chang Liu MASS UIUC Deformable Mirrors for Adaptive Optics • 2 mm surface normal stroke • for a 300 mm square mirror, the displacement is 1.5 micron at approximately 120 V applied voltage • T. Bifano, R. Mali, Boston University (http://www.bu.edu/mfg/faculty/homepages/bifano.html) Chang Liu MASS UIUC Chang Liu MASS UIUC Chang Liu MASS UIUC BU Adaptive Micro Mirrors Chang Liu MASS UIUC Optical Micro Switches • Texas Instrument DLP Chang Liu • Torsional parallel plate capacitor support • Two stable positions (+/10 degrees with respect to rest) • All aluminum structure • No process steps entails temperature above 300350 oC. MASS UIUC “Digital Light” Mirror Pixels Mirrors are on 17 mm center-to-center spacing Gaps are 1.0 mm nominal Mirror transit time is <20 ms from state to state Tilt Angles are minute at ±10 degrees Four mirrors equal the width of a human hair Chang Liu MASS UIUC Digital Micromirror Device (DMD) Mirror -10 deg Mirror +10 deg Hinge Yoke Chang Liu CMOS Substrate MASS UIUC Perspective View of Lateral Comb Drive Chang Liu MASS UIUC Lateral Comb Drive Actuators • Total capacitance is proportional to the overlap length and depth of the fingers, and inversely proportional to the distance. • Pros: 2 0t ( x x0 ) Ctot N [ cp ] d F x0 N 0 t 2 V d N=4 in above diagram. Chang Liu – Frequently used in actuators for its relatively long achievable driving distance. • Cons – force output is a function of finger thickness. The thicker the fingers, the large force it will be. – Relatively large footprint. MASS UIUC Transverse Comb Drive Devices • Direction of finger movement is orthogonal to the direction of fingers. • Pros: Frequently used for sensing for the sensitivity and ease of fabrication • Cons: not used as actuator because of the physical limit of distance. Csl N ( Csr N ( Chang Liu 0lt Cf ) x0 x 0lt x0 x Cf ) MASS UIUC Devices Based on Transverse Comb Drive • Analog Device ADXL accelerometer • A movable mass supported by cantilever beams move in response to acceleration in one specific direction. • Relevant to device performance – sidewall vertical profile – off-axis movement compensation – temperature sensitivity. • * p 234-236. Chang Liu MASS UIUC Sandia Electrostatically driven gears - translating linear motion into continuous rotary motion Lateral comb drive banks Mechanical springs Gear train Optical shutter Chang Liu • http://www.mdl.sandia.gov/mic romachine/images11.html MASS UIUC Sandia Gears • Use five layer polysilicon to increase the thickness t in lateral comb drive actuators. Mechanical springs Position limiter Chang Liu MASS UIUC More Sophisticated Micro Gears Chang Liu MASS UIUC Actuators that Use Fringe Electric Field - Rotary Motor • Three phase electrostatic actuator. • Arrows indicate electric field and electrostatic force. The tangential components cause the motor to rotate. Chang Liu MASS UIUC Three Phase Motor Operation Principle Chang Liu MASS UIUC Starting Position -> Apply voltage to group A electrodes Chang Liu MASS UIUC Motor tooth aligned to A -> Apply voltage to Group C electrodes Chang Liu MASS UIUC Motor tooth aligned to C -> Apply voltage to Group B electrodes Chang Liu MASS UIUC Motor tooth aligned to B -> Apply voltage to Group A electrodes Chang Liu MASS UIUC Motor tooth aligned to A -> Apply voltage to Group C electrodes Chang Liu MASS UIUC Example of High Aspect Ratio Structures Chang Liu MASS UIUC Some variations • • • • Chang Liu Large angle Long distance Low voltage Linear movement MASS UIUC 1x4 Optical Switch • Chang Liu John Grade and Hal Jerman, “A large deflection electrostatic actuator for optical switching applications”, IEEE S&A Workshop, 2000, p. 97. MASS UIUC Actuators that Use Fringe Field - Micro Mirrors with Large Displacement Angle Torsional mechanical spring Chang Liu R. Conant, “A flat high freq scanning micromirror”, IEEE Sen &Act Workshop, Hilton Head Island, 2000. MASS UIUC Curled Hinge Comb Drives Chang Liu MASS UIUC Other Parallel Plate Capacitor - Scratch Drive Actuator • Mechanism for realizing continuous long range movement. Chang Liu Scratch drive invented by H. Fujita of Tokyo University. The motor shown above was made by U. of Colorado, Victor Bright. MASS UIUC Analysis of Electrostatic Actuator What happens to a parallel plate capacitor when the applied voltage is gradually increased? Chang Liu MASS UIUC An Equivalent Electromechanical Model Fmechanical x If top plate moves downward, x<0. Felectric Km Note: direction definition of variables • This diagram depicts a parallel plate capacitor at equilibrium position. The mechanical restoring spring with spring constant Km (unit: N/m) is associated with the suspension of the top plate. • According to Hooke’s law, Fmechanical Km x • At equilibrium, the two forces, electrical force and mechanical restoring force, must be equal. Less the plate would move under Newton’s first law. Chang Liu Gravity is generally ignored. MASS UIUC Mechanical Spring • Cantilever beams with various boundary conditions • Torsional bars with various boundary conditions Chang Liu MASS UIUC Electrical And Mechanical Forces If the right-hand plate moves closer to the fixed one, the magnitude of mechanical force increases linearly. Equilibrium: |electric force|=|mechanical force| If a constant voltage, V1, is applied in between two plates, the electric force changes as a function of distance. The closer the two plates, the large the force. X0 x Equilibrium position Km fixed Chang Liu MASS UIUC Electrical And Mechanical Forces V3 V2 Equilibrium: |electric force|=|mechanical force| V3>V2>V1 V1 X0 Km X0+x1 fixed X0+x2 X0+x3 Chang Liu MASS UIUC Force Balance Equation at Given Applied Voltage V km x • The linear curve represents the magnitude of mechanical restoring force as a function of x. • Each curve in the family represents magnitude of electric force as a function of V increases spacing (x0+x). AV 2 2 x x0 2 km Chang Liu • Note that x<0. The origin of x=0 is the dashed line. MASS UIUC Determining Equilibrium Position Graphically • At each specific applied voltage, the equilibrium position can be determined by the intersection of the linear line and the curved line. • For certain cases, two equilibrium positions are possible. However, as the plate moves from top to bottom, the first equilibrium position is typically assumed. • Note that one curve intersects the linear line only at one point. • As voltage increases, the curve would have no equilibrium position. • This transition voltage is called pull-in voltage. • The fact that at certain voltage, no equilibrium position can be found, is called pull-in effect. Chang Liu MASS UIUC Pull-In Effect • As the voltage bias increases from zero across a pair of parallel plates, the distance between such plates would decrease until they reach 2/3 of the original spacing, at which point the two plates would be suddenly snapped into contact. • This behavior is called the pull-in effect. – A.k.a. “snap in” Chang Liu MASS UIUC A threshold point VPI Equilibrium: |electric force|=|mechanical force| X=-x0/3 X0 Km fixed Chang Liu Positive feedback -snap, pull in MASS UIUC Mathematical Determination of Pull-in Voltage Step 1 - Defining Electrical Force Constant • Let’s define the tangent of the electric force term. It is called electrical force constant, Ke. F CV 2 ke ke 2 x d • When voltage is below the pull-in voltage, the magnitude of Ke and Km are not equal at equilibrium. Chang Liu MASS UIUC Review of Equations Related To Parallel Plate • The electrostatic force is E 1 A 2 1 CV 2 F V 2 d 2d 2 d • The electric force constant is 1 A 2 A V 2 V2 K e (2) 3 V C 2 2 2 d d d d Chang Liu MASS UIUC Mathematical Determination of Pull-in Voltage Step 2 - Pull-in Condition • At the pull-in voltage, there is only one intersection between |Fe| and |Fm| curves. • At the intersection, the gradient are the same, I.e. the two curves intersect with same tangent. Ke K m • This is on top of the condition that the magnitude of Fm and Fe are equal. 2 2 k x ( x x ) 2km x( x x0 ) m 0 – Force balance yields V A C 2 2 CV – Plug in expression of V2 into the expression for Ke, ke 2 d • we get 2k m x CV 2 ke ( x x0 ) 2 ( x xo ) – This yield the position for the pull-in condition, x=-x0/3. Irrespective of the magnitude of Km. Chang Liu MASS UIUC Mathematical Determination of Pull-in Voltage Step 3 - Pull-in Voltage Calculation • Plug in the position of pull-in into Eq. * on previous page, we get the voltage at pull-in as 2 4 x V p2 0 k m 9C • At pull in, C=1.5 Co • Thus, Vp Chang Liu 2 x0 km . 3 1.5C0 MASS UIUC Implications of Pull-in Effect • For electrostatic actuator, it is impossible to control the displacement through the full gap. Only 1/3 of gap distance can be moved reliably. • Electrostatic micro mirros – reduced range of reliable position tuning • Electrostatic tunable capacitor – reduced range of tuning and reduced tuning range – Tuning distance less than 1/3, tuning capacitance less than 50%. Chang Liu MASS UIUC Counteracting Pull-In Effect Leveraged Bending for Full Gap Positioning • E. Hung, S. Senturia, “Leveraged bending for full gap positioning with electrostatic actuation”, Sensors and Actuators Workshop, Hilton Head Island, p. 83, 2000. Chang Liu MASS UIUC Counteracting Pull-in Effect: Variable Gap Capacitor Existing Tunable Capacitor Counter capacitor plate Suspension spring Tuning range: 88% (with parasitic capacitance) d0 Actuation electrode Capacitor plate Actuation electrode NEW DESIGN Variable Gap Variable Capacitor Suspension spring d0 Actuation electrode Chang Liu <(1/3)d0 Counter capacitor plate Capacitor plate Actuation electrode MASS UIUC Example • A parallel plate capacitor suspended by two fixed-fixed cantilever beams, each with length, width and thickness denoted l, w and t, respectively. The material is made of polysilicon, with a Young’s modulus of 120GPa. • L=400 mm, w=10 mm, and t=1 mm. • The gap x0 between two plates is 2 mm. • The area is 400 mm by 400 mm. • Calculate the amount of vertical displacement when a voltage of 0.4 volts is applied. Chang Liu MASS UIUC Step 1: Find mechanical force constants • Calculate force constant of one beam first – use model of left end guided, right end fixed. 3 – Under force F, the max deflection is d Fl 12 EI – The force constant is therefore F 12EI Ewt 3 120 109 10 106 (1106 )3 Km 3 3 0.01875N / m d l l (400 106 )3 – This is a relatively “soft” spring. – Note the spring constant is stiffer than fixed-free beams. • Total force constant encountered by the parallel plate is K m 0.0375 N / m Chang Liu MASS UIUC Step 2: Find out the Pull-in Voltage • Find out pull-in voltage and compare with the applied voltage. • First, find the static capacitance value Co 8.85 1012 ( F / m) (400 106 ) 2 C0 7.083 1013 F 6 2 10 • Find the pull-in voltage value 2 x0 km 2 2 106 0.0375 Vp 0.25(volts) 13 3 1.5C0 3 1.5 7.083 10 • When the applied voltage is 0.4 volt, the beam has been pulledin. The displacement is therefore 2 mm. Chang Liu MASS UIUC What if the applied voltage is 0.2 V? • Not sufficient to pull-in • Deformation can be solved by solving the following equation 2km x( x x0 ) 2 2km x( x x0 ) V A C 2 • or v 2A x 2 x0 x x x 0 2k m 3 2 2 0 x 3 4 10 6 x 2 4 10 12 x 7.552 10 19 0 • How to solve it? Chang Liu MASS UIUC Solving Third Order Equation ... • To solve x 3 ax 2 bx c 0 • Apply y x a / 3 • Use the following definition a2 a ab p b, q 2( )3 c 3 3 3 3 p q Q 3 2 A3 • The only real solution is • 2 q q Q,B 3 Q 2 2 y A B a x A B 3 Chang Liu MASS UIUC Calculator … A Simple Way Out. • Use HP calculator, – x1=-2.45x10-7 mm – x2=-1.2x10-6 mm – x3=-2.5x10-6 mm • Accept the first answer because the other two are out side of pull-in range. • If V=0.248 Volts, the displacement is -0.54 mm. Chang Liu MASS UIUC
