Mechanics of Micro Structures Chang Liu Micro Actuators, Sensors, Systems Group University of Illinois at Urbana-Champaign 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 MASS UIUC

Miller indices A popular method of designating crystal planes (hkm) and orientations <hkm> Identify the axial intercepts Take reciprocal Clear fractions (not taking lowest integers) Enclose the number with ( ) : no comma <hkm> designate the direction normal to the plane (hkm) (100), (110), (111) MASS UIUC

Stress and Strain Definition of Stress and Strain The normal stress (Pa, N/m2) The strain Poisson’s ratio MASS UIUC

E: Modulus of Elasticity, Young’s Modulus Hooke’s Law E: Modulus of Elasticity, Young’s Modulus The shear stress The shear strain The shear modulus of elasticity The relationship MASS UIUC

General Relation Between Tensile Stress and Strain 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 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; 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.) MASS UIUC

General Stress-Strain Relations C: stiffness matrix S: compliance matrix For many materials of interest to MEMS, the stiffness can be simplified MASS UIUC

Flexural Beam Bending Types of Beams; Fig. 3.15 Possible Boundary Conditions MASS UIUC

Longitudinal Strain Under Pure Bending Pure Bending; The moment is constant throughout the beam MASS UIUC

Deflection of Beams Appendix B MASS UIUC

Finding the Spring Constant MASS UIUC

Calculate spring constant MASS UIUC

Vertical Translational Plates MASS UIUC

Torsional Deflections Pure Torsion; Every cross section of the bar is identical MASS UIUC

Intrinsic Stress Many thin film materials experience internal stress even when they are under room temperature and zero external loading conditions In many cases related to MEMS structures, the intrinsic stress results from the temperature difference during deposition and use MASS UIUC

Intrinsic Stress The flatness of the membrane is guaranteed when the membrane material is under tensile stress MASS UIUC

Intrinsic Stress There are three strategies for minimizing undesirable intrinsic bending Use materials that inherently have zero or very low intrinsic stress For materials whose intrinsic stress depends on material processing parameters, fine tune the stress by calibrating and controlling deposition conditions Use multiple-layered structures to compensate for stress-induced bending 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 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 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 fixed- guided beams Four fixed-guided beams MASS UIUC

Examples 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 MASS UIUC

A Clamped-Clamped Beam Fixed-guided Fixed-guided MASS UIUC

A Clamped-Free Beam MASS UIUC

One-end Supported, “Clamped-Free” Beams 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. 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. MASS UIUC

Modulus of Elasticity Names Definition Young’s modulus Elastic modulus Definition Values of E for various materials can be found in notes, text books, MEMS clearing house, etc. MASS UIUC

Large Displacement vs. Small Displacement end displacement less than 10-20 times the thickness. Used somewhat loosely because of the difficulty to invoke large-deformation analysis. Large deformation needs finite element computer-aided 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) I= for rectangular cross section Maximum angular displacement Maximum vertical displacement under F is Therefore, the equivalent force constant is Formula for 1st order resonant frequency where is the beam weight per unit length. MASS UIUC

Zig-Zag Beams Saves chip real-estate 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 MASS UIUC

An Example 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. MASS UIUC

Resonant frequency of typical spring-mass system Self-mass or concentrated mass being m The resonant frequency is 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. 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 MASS UIUC

Electrostatic Sensors and Actuators Chang Liu MASS UIUC

Outline Basic Principles Applications examples capacitance formula capacitance configuration Applications examples sensors actuators Analysis of electrostatic actuator second order effect - “pull in” effect Application examples and detailed analysis MASS UIUC

Basic Principles Sensing Actuation Two major configurations 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) Interdigitated finger configuration Parallel plate configuration MASS UIUC

Examples Parallel Plate Capacitor Comb Drive Capacitor MASS UIUC

Parallel Plate Capacitor Fringe electric field (ignored in first order analysis) 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. MASS UIUC

Fabrication Methods Flip and bond Surface micromachining Wafer bonding 3D assembly Flip and bond Movable vertical plate MASS UIUC

Forces of Capacitor Actuators Stored energy Force is derivative of energy with respect to pertinent dimensional variable Plug in the expression for capacitor We arrive at the expression for force 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. 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 MASS UIUC

Analysis of Electrostatic Actuator What happens to a parallel plate capacitor when the applied voltage is gradually increased? MASS UIUC

An Equivalent Electromechanical Model x If top plate moves down- ward, x<0. 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, At equilibrium, the two forces, electrical force and mechanical restoring force, must be equal. Less the plate would move under Newton’s first law. Gravity is generally ignored. 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 MASS UIUC

Electrical And Mechanical Forces V3 V2 V3>V2>V1 Equilibrium: |electric force|=|mechanical force| V1 X0 Km X0+x1 fixed X0+x2 X0+x3 MASS UIUC

Force Balance Equation at Given Applied Voltage V 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 spacing (x0+x). 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. 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” MASS UIUC

fixed A threshold point Equilibrium: VPI Equilibrium: |electric force|=|mechanical force| X=-x0/3 X0 Km Positive feedback -snap, pull in fixed 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. When voltage is below the pull-in voltage, the magnitude of Ke and Km are not equal at equilibrium. MASS UIUC

Review of Equations Related To Parallel Plate The electrostatic force is The electric force constant is 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. This is on top of the condition that the magnitude of Fm and Fe are equal. Force balance yields Eq.(*) Plug in expression of V2 into the expression for Ke, we get This yield the position for the pull-in condition, x=-x0/3. Irrespective of the magnitude of km. 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 At pull in, C=1.5 Co Thus, 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%. 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. MASS UIUC

Counteracting Pull-in Effect: Variable Gap Capacitor Existing Tunable Capacitor Tuning range: 88% (with parasitic capacitance) Counter capacitor plate Suspension spring d0 Actuation electrode Capacitor plate Actuation electrode NEW DESIGN Variable Gap Variable Capacitor Suspension spring Counter capacitor plate <(1/3)d0 d0 Actuation electrode 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. MASS UIUC

Step 1: Find mechanical force constants Calculate force constant of one beam first use model of left end guided, right end fixed. Under force F, the max deflection is The force constant is therefore 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 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 Find the pull-in voltage value When the applied voltage is 0.4 volt, the beam has been pulled-in. The displacement is therefore 2 mm. 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 or How to solve it? MASS UIUC

Solving Third Order Equation ... To solve Apply Use the following definition The only real solution is 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. 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) MASS UIUC

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BU Adaptive Micro Mirrors MASS UIUC

Optical Micro Switches Texas Instrument DLP Torsional parallel plate capacitor support Two stable positions (+/- 10 degrees with respect to rest) All aluminum structure No process steps entails temperature above 300-350 oC. MASS UIUC

“Digital Light” Mirror Pixels Mirrors are on 17 m center-to-center spacing Gaps are 1.0 m nominal Mirror transit time is <20 s from state to state Tilt Angles are minute at ±10 degrees Four mirrors equal the width of a human hair MASS UIUC

Digital Micromirror Device (DMD) -10 deg Mirror +10 deg Hinge CMOS Substrate Yoke MASS UIUC

Perspective View of Lateral Comb Drive 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: 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. N=4 in above diagram. 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. 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. MASS UIUC

Lateral comb drive banks Sandia Electrostatically driven gears - translating linear motion into continuous rotary motion Lateral comb drive banks Mechanical springs Gear train Optical shutter http://www.mdl.sandia.gov/micromachine/images11.html MASS UIUC

Sandia Gears Mechanical springs Position limiter Use five layer polysilicon to increase the thickness t in lateral comb drive actuators. Mechanical springs Position limiter MASS UIUC

More Sophisticated Micro Gears 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. MASS UIUC

Three Phase Motor Operation Principle MASS UIUC

Starting Position -> Apply voltage to group A electrodes MASS UIUC

Motor tooth aligned to A -> Apply voltage to Group C electrodes MASS UIUC

Motor tooth aligned to C -> Apply voltage to Group B electrodes MASS UIUC

Motor tooth aligned to B -> Apply voltage to Group A electrodes MASS UIUC

Motor tooth aligned to A -> Apply voltage to Group C electrodes MASS UIUC

Example of High Aspect Ratio Structures MASS UIUC

Some variations Large angle Long distance Low voltage Linear movement MASS UIUC

1x4 Optical Switch John Grade and Hal Jerman, “A large deflection electrostatic actuator for optical switching applications”, IEEE S&A Workshop, 2000, p. 97. MASS UIUC

Torsional mechanical spring Actuators that Use Fringe Field - Micro Mirrors with Large Displacement Angle Torsional mechanical spring R. Conant, “A flat high freq scanning micromirror”, IEEE Sen &Act Workshop, Hilton Head Island, 2000. MASS UIUC

Curled Hinge Comb Drives MASS UIUC

Other Parallel Plate Capacitor - Scratch Drive Actuator Mechanism for realizing continuous long range movement. Scratch drive invented by H. Fujita of Tokyo University. The motor shown above was made by U. of Colorado, Victor Bright. MASS UIUC