EE C245 – ME C218 Introduction to MEMS Design Fall...

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EE C245: Introduction to MEMS Design Lecture 14 C. Nguyen 10/11/07 1

EE C245 – ME C218Introduction to MEMS Design

Fall 2007

Prof. Clark T.-C. Nguyen

Dept. of Electrical Engineering & Computer SciencesUniversity of California at Berkeley

Berkeley, CA 94720

Lecture 14: Beam Bending


Announcements

•Move midterm date to Thursday, Nov. 1 (1 week later than original tentative date)

•Website issues

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Lecture Outline

• Reading: Senturia Chpts. 8, 9• Lecture Topics:

Material PropertiesYoung’s modulusYield strengthQuality factor

On-chip Measurement of Material PropertiesAnisotropic Material PropertiesBeam Bending


Quality Factor (or Q)

3


Lr hWr

VPvi

Clamped-Clamped Beam μResonator

Smaller mass higher freq. range and lower series Rx

Smaller mass higher freq. range and lower series Rx(e.g., mr = 10-13 kg)(e.g., mr = 10-13 kg)

Young’s Modulus

DensityMass

Stiffness

ωωο

ivoi

203.121

rr

ro L

hEmkf

ρπ==

Frequency:

Q ~10,000

vi

Resonator Beam

Electrode

VP

C(t)

dtdCVi Po =

Note: If VP = 0V device off

Note: If VP = 0V device off

io


Quality Factor (or Q)•Measure of the frequency selectivity of a tuned circuit

• Definition:

• Example: series LCR circuit

• Example: parallel LCR circuit

dB3CyclePer Lost EnergyCyclePer Energy Total

BWfQ o==

( )( ) CRR

LZZQ

o

o

ωω 1

ReIm

===

( )( ) LGG

CYYQ

o

o

ωω 1

ReIm

===

BW-3dB

fo f

4


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

-35

-30

-25

-20

-15

-10

-5

0

998 999 1000 1001 1002

Frequency [MHz]

Tran

smis

sion

[dB

]

Tank Q30,00020,00010,0005,0004,000

Increasing Insertion

Loss


•Main Function: provide a stable output frequency• Difficulty: superposed noise degrades frequency stability

ωωο

ivoi

A

Frequency-SelectiveTank

SustainingAmplifier

ov

Ideal Sinusoid: ( ) ⎟⎠⎞⎜

⎝⎛= tofVotov π2sin

Real Sinusoid: ( ) ( ) ( )⎟⎠⎞⎜

⎝⎛⎟

⎠⎞⎜

⎝⎛ ++= ttoftVotov θπε 2sin

ωωο

ωωο=2π/TO

TO

Zero-Crossing Point

Tighter SpectrumTighter Spectrum

Oscillator: Need for High Q

Higher QHigher Q

5


• Problem: IC’s cannot achieve Q’s in the thousandstransistors consume too much power to get Qon-chip spiral inductors Q’s no higher than ~10off-chip inductors Q’s in the range of 100’s

•Observation: vibrating mechanical resonances Q > 1,000• Example: quartz crystal resonators (e.g., in wristwatches)

extremely high Q’s ~ 10,000 or higher (Q ~ 106 possible)mechanically vibrates at a distinct frequency in a thickness-shear mode

Attaining High Q


Energy Dissipation and Resonator Q

supportviscousTEDdefects QQQQQ11111

+++=

Anchor LossesAnchor Losses

Material Defect Losses

Material Defect Losses Gas DampingGas Damping

Thermoelastic Damping (TED)Thermoelastic Damping (TED)

Bending CC-Beam

CompressionHot Spot

Tension Cold Spot Heat Flux(TED Loss)

Elastic Wave Radiation(Anchor Loss)

At high frequency, this is our big problem!

At high frequency, this is our big problem!

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Thermoelastic Damping (TED)

•Occurs when heat moves from compressed parts to tensioned parts → heat flux = energy loss

QfT

21)()( =ΩΓ=ς

pCTETρ

α4

)(2

=Γ

⎥⎦

⎤⎢⎣

⎡

+=Ω 222)(

fffff

TED

TEDo

22 hCKf

pTED ρ

π=

Bending CC-Beam

CompressionHot Spot

Tension Cold Spot Heat Flux(TED Loss)

ζ = thermoelastic damping factorα = thermal expansion coefficientT = beam temperatureE = elastic modulusρ = material densityCp = heat capacity at const. pressureK = thermal conductivityf = beam frequencyh = beam thicknessfTED = characteristic TED frequency

h


TED Characteristic Frequency

• Governed byResonator dimensionsMaterial properties

22 hCKf

pTED ρ

π=

ρ = material densityCp = heat capacity at const. pressureK = thermal conductivityh = beam thicknessfTED = characteristic TED frequency

Critical D

amping

Fac

tor,

ζ

Relative Frequency, f/fTED

Q

[from Roszhart, Hilton Head 1990]

Peak where Q is minimizedPeak where Q is minimized

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Q vs. Temperature

Quartz Crystal Aluminum Vibrating Resonator

Q ~5,000,000 at 30K

Q ~5,000,000 at 30K

Q ~300,000,000 at 4KQ ~300,000,000 at 4K

Q ~500,000 at 30K

Q ~500,000 at 30K

Q ~1,250,000 at 4KQ ~1,250,000 at 4K

Even aluminum achieves exceptional Q’s at

cryogenic temperatures

Even aluminum achieves exceptional Q’s at

cryogenic temperatures

Mechanism for Q increase with decreasing temperature thought to be linked to less hysteretic motion of material defects

less energy loss per cycle

Mechanism for Q increase with decreasing temperature thought to be linked to less hysteretic motion of material defects

less energy loss per cycle

[from Braginsky, Systems With

Small Dissipation]


Output

Input

Input

Output

Support Beams

Wine Glass Disk Resonator

R = 32 μm

Anchor

Anchor

Resonator DataR = 32 μm, h = 3 μmd = 80 nm, Vp = 3 V


-100

-80

-60

-40

61.325 61.375 61.425

fo = 61.37 MHzQ = 145,780

Frequency [MHz]

Unm

atch

ed T

rans

mis

sion

[dB

]

Polysilicon Wine-Glass Disk Resonator

[Y.-W. Lin, Nguyen, JSSC Dec. 04]

Compound Mode (2,1)

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-100-98-96-94-92-90-88-86-84

1507.4 1507.6 1507.8 1508 1508.2

1.51-GHz, Q=11,555 Nanocrystalline Diamond Disk μMechanical Resonator

• Impedance-mismatched stem for reduced anchor dissipation

•Operated in the 2nd radial-contour mode•Q ~11,555 (vacuum); Q ~10,100 (air)• Below: 20 μm diameter disk

PolysiliconElectrode R

Polysilicon Stem(Impedance Mismatched

to Diamond Disk)

GroundPlane

CVD DiamondμMechanical Disk

Resonator Frequency [MHz]

Mix

ed A

mpl

itude

[dB

]

Design/Performance:R=10μm, t=2.2μm, d=800Å, VP=7Vfo=1.51 GHz (2nd mode), Q=11,555

fo = 1.51 GHzQ = 11,555 (vac)Q = 10,100 (air)

[Wang, Butler, Nguyen MEMS’04]

Q = 10,100 (air)


Disk Resonator Loss Mechanisms

Disk Stem

Nodal Axis

Strain Energy Flow No motion along

the nodal axis,but motion along the finite width

of the stem

No motion along the nodal axis,

but motion along the finite width

of the stem

Stem Heightλ/4 helps

reduce loss, but not perfect

λ/4 helps reduce loss,

but not perfect

Substrate Loss Thru Anchors

Substrate Loss Thru Anchors

Gas Damping

Gas Damping

Substrate

Hysteretic Motion of

Defect

Hysteretic Motion of

Defect

e-

Electronic Carrier Drift Loss

Electronic Carrier Drift Loss

(Not Dominant in Vacuum)

(Dwarfed By Substrate Loss)

(Dwarfed By Substrate Loss)

(Dominates)

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MEMS Material Property Test Structures


Stress Measurement Via Wafer Curvature

• Compressively stressed film →bends a wafer into a convex shape

• Tensile stressed film → bends a wafer into a concave shape

• Can optically measure the deflection of the wafer before and after the film is deposited

• Determine the radius of curvature R, then apply:

RthE

6

2′=σ

σ = film stress [Pa]E′ = E/(1-ν) = biaxial elastic modulus [Pa]h = substrate thickness [m]t = film thicknessR = substrate radius of curvature [m]

Si-substrate

Laser

Detector

MirrorxScan the

mirror θ

θ

xSlope = 1/R

h

t

R

10


MEMS Stress Test Structure

• Simple Approach: use a clamped-clamped beam

Compressive stress causes bucklingArrays with increasing length are used to determine the critical buckling load, where

Limitation: Only compressive stress is measurable

2

22

3 LEh

criticalπσ −=

E = Young’s modulus [Pa]I = (1/12)Wh3 = moment of inertiaL, W, h indicated in the figure

L

W

h = thickness


More Effective Stress Diagnostic

• Single structure measures both compressive and tensile stress

• Expansion or contraction of test beam → deflection of pointer

• Vernier movement indicates type and magnitude of stress

Expansion → Compression

Contraction → Tensile

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Output

Input

Input

Output

Support Beams

Wine Glass Disk Resonator

R = 32 μm

Anchor

Anchor



-100

-80

-60

-40

61.325 61.375 61.425

fo = 61.37 MHzQ = 145,780

Frequency [MHz]

Unm

atch

ed T

rans

mis

sion

[dB

]

Q Measurement Using Resonators

[Y.-W. Lin, Nguyen, JSSC Dec. 04]

Compound Mode (2,1)


Folded-Beam Comb-Drive Resonator

• Issue w/ Wine-Glass Resonator: non-standard fab process• Solution: use a folded-beam comb-drive resonator

8.3 Hz

3.8500,342

=Q

fo=342.5kHzQ=41,000

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Comb-Drive Resonator in Action

• Below: fully integrated micromechanical resonator oscillator using a MEMS-last integration approach


Folded-Beam Comb-Drive Resonator

• Issue w/ Wine-Glass Resonator: non-standard fab process• Solution: use a folded-beam comb-drive resonator

8.3 Hz

3.8500,342

=Q

fo=342.5kHzQ=41,000

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Measurement of Young’s Modulus

• Use micromechanical resonatorsResonance frequency depends on EFor a folded-beam resonator: 213)(4

⎥⎥⎦

⎤

⎢⎢⎣

⎡=

eqo M

LWEhfResonance Frequency =

LW

h = thickness

Young’s modulus

Equivalent mass

• Extract E from measured frequency fo

•Measure fo for several resonators with varying dimensions

• Use multiple data points to remove uncertainty in some parameters


Anisotropic Materials

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Elastic Constants in Crystalline Materials

• Get different elastic constants in different crystallographic directions → 81 of them in all

Cubic symmetries make 60 of these terms zero, leaving 21 of them remaining that need be accounted for

• Thus, describe stress-strain relations using a 6x6 matrix

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

xy

zx

yz

z

y

x

xy

zx

yz

z

y

x

CCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCCC

γγγεεε

τττσσσ

665646362616

565545352515

464544342414

363534332313

262524232212

161514131211

Stiffness CoefficientsStresses Strains


Stiffness Coefficients of Silicon

• Due to symmetry, only a few of the 21 coefficients are non-zero

•With cubic symmetry, silicon has only 3 independent components, and its stiffness matrix can be written as:

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

=

⎥⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

xy

zx

yz

z

y

x

xy

zx

yz

z

y

x

CC

CCCCCCCCCC

γγγεεε

τττσσσ

66

55

44

332313

232212

131211

000000000000000000000000

whereC11 = 165.7 GPaC12 = 63.9 GPaC44 = 79.6 GPa

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Young’s Modulus in the (001) Plane

[units = 100 GPa]

[units

= 1

00 G

Pa]


Poisson Ratio in (001) Plane

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Anisotropic Design Implications

• Young’s modulus and Poisson ratio variations in anisotropic materials can pose problems in the design of certain structures

• E.g., disk or ring resonators, which rely on isotropic properties in the radial directions

Okay to ignore variation in RF resonators, although some Q hit is probably being taken

• E.g., ring vibratory rate gyroscopes

Mode matching is required, where frequencies along different axes of a ring must be the sameNot okay to ignore anisotropic variations, here Ring Gyroscope

Drive AxisDrive Axis

Sense

Axis

Sense

Axis

Wine-Glass Mode Disk


Bending of Beams

17


Beams: The Springs of Most MEMS

• Springs and suspensions very common in MEMSCoils are popular in the macro-world; but not easy to make in the micro-worldBeams: simpler to fabricate and analyze; become “stronger” on the micro-scale → use beams for MEMS

Comb-Driven Folded Beam Actuator


Bending a Cantilever Beam

•Objective: Find relation between tip deflection y(x=L) and applied load F

• Assumptions:1. Tip deflection is small compared with beam length2. Plane sections (normal to beam’s axis) remain plane and

normal during bending, i.e., “pure bending”3. Shear stresses are negligible

F

x′ L

x

Clamped end condition:At x=0:

z=0dz/dx = 0

Free end condition

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Reaction Forces and Moments


Sign Conventions for Moments & Shear Forces

(+) moment leads to deformation with a (+) radius of curvature

(i.e., upwards)

(-) moment leads to deformation with a (-) radius of curvature (i.e., downwards)

(+) shear forces produce clockwise

rotation

(-) shear forces produce counter-clockwise rotation

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Beam Segment in Pure Bending

Small section of a beam bent in response to a tranverse load R


Beam Segment in Pure Bending (cont.)

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