Chang LiuMASSUIUC

Mechanics of Micro Structures

Chang LiuMicro Actuators, Sensors, Systems Group

University of Illinois at Urbana-Champaign

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• 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 m thick– 150 mm (6 in) diameter x 750 m thick– 200 mm (8 in) diameter x 1000 m thick

Single crystal silicon and wafers

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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)

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Stress and Strain

• Definition of Stress and Strain– The normal stress (Pa, N/m2)

– The strain

– Poisson’s ratio

A

F

00

0

L

L

L

LL

x

z

x

y

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Hooke’s Law E

A

F

L

X

E: Modulus of Elasticity, Young’s Modulus

The shear stress

The shear strain

The shear modulus of elasticity

The relationship

12

EG

G

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General Relation Between Tensile Stress and Strain

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• 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

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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.)

2/10

Kdy

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General Stress-Strain Relations

654 ,,,, TTTxyxzyz

654 ,,,, TTTxyxzyz

6

5

4

3

2

1

666564636261

565554535251

464544434241

363534333231

262524232221

161514131211

6

5

4

3

2

1

CCCCCC

CCCCCC

CCCCCC

CCCCCC

CCCCCC

CCCCCC

T

T

T

T

T

T

321 ,,,, TTTzzyyxx

CT

TSC: stiffness matrix

S: compliance matrix

PaCSi11

100, 10

8.000000

08.00000

008.0000

00066.164.064.0

00064.066.164.0

00064.064.066.1

For many materials of interest to MEMS, the stiffness can be simplified

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Flexural Beam Bending

• Types of Beams; Fig. 3.15• Possible Boundary Conditions

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Longitudinal Strain Under Pure Bending

EI

My

EI

Mt

2max

Pure Bending; The moment is constant throughout the beam

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Deflection of Beams

)(2

2

xMdx

ydEI

EI

Fld

EI

Fl

3,

2

3

max

2

max

EI

Mld

EI

Ml

2,

2

maxmax

EI

Fld

192

3

max

EI

Fld

12

3

max

Appendix B

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Finding the Spring Constant

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Calculate spring constant

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Vertical Translational Plates

3

3

3

12

l

Ewt

l

EIk

3

3

3

12

l

Ewt

l

EIk

3

3

4)(l

Ewtkb

3

3

2)(l

Ewtka

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Torsional Deflections

• Pure Torsion; Every cross section of the bar is identical

J

Tr0max

402

1rJ

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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

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Intrinsic Stress

The flatness of the membrane is guaranteed when the membrane material is under tensile stress

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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

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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

Felectric

Fmechanical

Km

xKF mmechanical

• Importance of resonant freq.– Limits the actuation speed– lower energy consumption at Fr

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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

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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

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Examples

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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

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A Clamped-Clamped Beam

Fixed-guided

Fixed-guided

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A Clamped-Free Beam

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One-end Supported, “Clamped-Free” Beams

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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.

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Force Constants for Fixed-Free Beams• Dimensions

– length, width, thickness– unit in m.

• Materials– Young’s modulus, E– Unit in Pa, or N/m2.

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Modulus of Elasticity

• Names– Young’s modulus– Elastic modulus

• Definition

• Values of E for various materials can be found in notes, text books, MEMS clearing house, etc.

LLAF

Ex

x

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Large Displacement vs. Small Displacement• 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.

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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.

EI

Fl

2

212

3wt

EI

Fl

3

3

3

3

33 4

3

3l

Ewt

l

EI

EIFl

Fkm

42

52.3

l

EIg

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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 chipreal-estate

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An Example

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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.

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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.

m

k

21

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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

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Electrostatic Sensors and Actuators

Chang Liu

Chang LiuMASSUIUC

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

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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)

dA

Parallel plate configuration

Interdigitated finger configuration

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Examples

• Parallel Plate Capacitor• Comb Drive Capacitor

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Parallel Plate Capacitor

– 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.

dA

V

QC

AQE /

d

A

dAQQ

C

Fringe electric field(ignored in first orderanalysis)

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Fabrication Methods

• Surface micromachining• Wafer bonding• 3D assembly

Flip andbond

Movablevertical plate

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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

CQ

CVU2

2

21

21

2

21

VdC

dU

F

d

A

dAQQ

C

dCV

Vd

AdU

F2

22 2

121

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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.

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Capacitive Accelerometer

• Proof mass area 1x0.6 mm2, and 5 m thick.

• Net capacitance 150fF• External IC signal processing

circuits

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Analysis of Electrostatic Actuator

What happens to a parallel plate capacitor when the applied voltage is gradually increased?

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An Equivalent Electromechanical Model

• 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.

Felectric

Fmechanical

Km

xKF mmechanical

x

Note: directiondefinition of variables

If top platemoves down-ward, x<0.

Gravity is generally ignored.

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Electrical And Mechanical Forces

fixed

Km

X0

Equilibrium:|electric force|=|mechanical force|

If the right-hand plate movescloser to the fixed one, the magnitudeof mechanical force increases linearly.

If a constant voltage, V1, is applied in between two plates, the electric forcechanges as a function of distance. Thecloser the two plates, the large the force.

Equilibriumposition

x

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Electrical And Mechanical Forces

fixed

Km

X0

Equilibrium:|electric force|=|mechanical force|

V3

V2

V1

V3>V2>V1

X0+x3

X0+x2

X0+x1

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Force Balance Equation at Given Applied Voltage V

20

2

2 xx

AVxkm

km

V increases

• 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.

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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.

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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”

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A threshold point

fixed

Km

X0

Equilibrium:|electric force|=|mechanical force|

VPI

X=-x0/3

Positivefeedback-snap, pull in

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Mathematical Determination of Pull-in VoltageStep 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.

x

Fke

2

2

d

CVke

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Review of Equations Related To Parallel Plate

• The electrostatic force is

• The electric force constant is

d

CVV

d

A

d

EF

22

2 2

1

2

1

2

2

2

22

3)2(

2

1

d

VC

d

V

d

AV

d

AKe

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Mathematical Determination of Pull-in VoltageStep 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.

me kk

C

xxxk

A

xxxkV mm )(2)(2 0

202

)(

2

)( 20

2

o

me xx

xk

xx

CVk

2

2

d

CVke

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Mathematical Determination of Pull-in VoltageStep 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,

mp kC

xV

9

4 202

.5.13

2

0

0

C

kxV m

p

dA)3/2(

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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%.

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Counteracting Pull-In EffectLeveraged 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.

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Counteracting Pull-in Effect: Variable Gap CapacitorExisting Tunable Capacitor

Capacitor plate

Actuationelectrode

Actuationelectrode

Suspensionspring

Counter capacitor plate

NEW DESIGN

Capacitor plate

Actuationelectrode

Actuationelectrode

Suspensionspring

Counter capacitor

plate

Variable Gap Variable Capacitor

d0

d0

<(1/3)d0

Tuning range: 88% (with parasitic capacitance)

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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 m, w=10 m, and t=1 m.

• The gap x0 between two plates is 2 m.

• The area is 400 m by 400 m. • Calculate the amount of vertical

displacement when a voltage of 0.4 volts is applied.

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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

EI

Fld

12

3

mNl

Ewt

l

EI

d

FKm /01875.0

)10400(

)101(1010101201236

3669

3

3

3

mNKm /0375.0

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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 m.

)(25.010083.75.1

0375.0

3

1022

5.13

213

6

0

0 voltsC

kxV m

p

FmF

C 136

2612

0 10083.7102

)10400()/(1085.8

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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?

C

xxxk

A

xxxkV mm )(2)(2 0

202

010552.7104104

02

2

1912263

220

20

3

xxx

k

Avxxxxx

m

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Solving Third Order Equation ...

• To solve

• Apply• Use the following definition

• The only real solution is•

023 cbxaxx

3/axy

33

23

32

2,

2

23

3)

3(2,

3

Qq

BQq

A

qpQ

caba

qba

p

3

aBAx

BAy

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Calculator … A Simple Way Out.

• Use HP calculator, – x1=-2.45x10-7 m– x2=-1.2x10-6 m– x3=-2.5x10-6 m

• 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 m.

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Deformable Mirrors for Adaptive Optics

• 2 m surface normal stroke • for a 300 m 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)

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BU Adaptive Micro Mirrors

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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.

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“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

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Mirror-10 deg

Mirror+10 deg

Hinge

YokeCMOS

Substrate

Digital Micromirror Device (DMD)

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Perspective View of Lateral Comb Drive

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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.

])(2

[ 00ptot c

d

xxtNC

200

Vd

tNF

x

N=4 in above diagram.

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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.

)(

)(

0

0

0

0

fsr

fsl

Cxx

ltNC

Cxx

ltNC

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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.

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Sandia Electrostatically driven gears- translating linear motion into continuous rotary motion

• http://www.mdl.sandia.gov/micromachine/images11.html

Lateral comb drive banks

Gear train

Optical shutter

Mechanicalsprings

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Sandia Gears • Use five layer polysilicon to increase the thickness t in lateral comb drive actuators.

Positionlimiter

Mechanical springs

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More Sophisticated Micro Gears

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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.

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Three Phase Motor Operation Principle

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Starting Position -> Apply voltage to group A electrodes

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

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

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

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

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Example of High Aspect Ratio Structures

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Some variations

• Large angle • Long distance• Low voltage• Linear movement

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1x4 Optical Switch

• John Grade and Hal Jerman, “A large deflection electrostatic actuator for optical switching applications”, IEEE S&A Workshop, 2000, p. 97.

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Actuators that Use Fringe Field - Micro Mirrorswith Large Displacement Angle

R. Conant, “A flat high freq scanning micromirror”, IEEE Sen &ActWorkshop, Hilton Head Island, 2000.

Torsional mechanical spring

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Curled Hinge Comb Drives

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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.