K.J. Hemker

MEMS 487

Class 04, Feb. 13, 2003

K.J. Hemker

Materials Come As:

!Amorphous – Glasses, polymers, some metal alloys– Processing can result in amorphous

structures!Crystalline

– Single crystals– Textured crystals– Polycrystalline

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Cubic Crystals

Body Centered CubicBCC

Face Centered CubicFCC

SiliconInterlaced FCC

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BCC Atomic Planes: {001}

4-fold

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BCC Atomic Planes: {011}

2-fold

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BCC Atomic Planes: {111}

3-fold

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FCC Atomic Planes: {001}

4-fold

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FCC Atomic Planes: {011}

2-fold

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FCC Atomic Planes: {111}

3-fold

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Atomic Planes in Si: {001}

4-fold

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Atomic Planes in Si: {011}

2-fold

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Atomic Planes in Si: {111}

3-fold

K.J. Hemker

Useful crystallography web sites

EMS On Line at http://cimesg1.epfl.ch/CIOL/ems.html

http://www.geo.ucalgary.ca/~tmenard/crystal/crystal.htmlnote: Si is the same structure as diamond, gold is FCC and Fe is BCC

K.J. Hemker

Crystallographic directions :

[100][100]

[001][001]

[010][010]

<100> cube edges

<011> face diagonals

<111> cube diagonals

[110][110]

[101][101]

[011][011][111][111]

K.J. Hemker

Silicon wafers:

Primaryflat

<110>±3°

[010]

[001]

{100} [001]

[00-1]

[010][0-10]

[01-1]

[011]

[0-11]

[0-1-1]

45°

45°

45°

45°

Primary

flat

<110>±3°

[00-1]

[010

{100}

K.J. Hemker

Isotropic Elasticityσ = E ε

τ = G γ

p = -K ∆

ν = εyy / εxx

σyy= λ εxx

All directions the sameAll directions the same

polycrystallinepolycrystalline

σyy

εxx

Lamé Coefficient (λ)

5 constantsonly

2 independent

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Anisotropic elasticity (stiffness)

σi = Cij εj

σxx C11 C12 C13 C14 C15 C16 εxx

σyy C21 C22 C23 C24 C25 C26 εyy

σzz = C31 C32 C33 C34 C35 C36 εzz

σyz C41 C42 C43 C44 C45 C46 εyz

σxz C51 C52 C53 C54 C55 C56 εxz

σxy C61 C62 C63 C64 C65 C66 εxy

K.J. Hemker

Complianceεi = Sij σj

εxx S11 S12 S13 S14 S15 S16 σxx

εyy S21 S22 S23 S24 S25 S26 σyy

εzz = S31 S32 S33 S34 S35 S36 σzz

εyz S41 S42 S43 S44 S45 S46 σyz

εxz S51 S52 S53 S54 S55 S56 σxz

εxy S61 S62 S63 S64 S65 S66 σxy

K.J. Hemker

Cij and Sij for Cubic Crystals

C11 C12 C12

C12 C11 C12

Cij = C12 C12 C11

C44

C44

C44

0

0

C11, C12, C44

K.J. Hemker

1-Dimensional LoadingE[100], E[110], and E[111]

E[100] = 1s11

E[110] = 2[s11 + s12 + s44/2]

E[111] = 3[s11 + 2s12 + s44]

K.J. Hemker

Relations :

c11 = s11 + s12(s11 - s12) (s11 + 2s12)

c12 = -s12(s11 - s12) (s11 + 2s12)

c44 = 1s44

s11 = c11 + c12(c11 - c12) (c11 + 2c12)

s12 = -c12(c11 - c12) (c11 + 2c12)

s44 = 1c44

K.J. Hemker

Example of Anisotropic Elasticity:

!To be addressed in homework problem.– Assume that for Si

• C11 = 166 GPa• C12 = 64 GPa• C44 = 80 GPa

– Calculate Young’s modulus (E) along <100>, <110> and <111>

K.J. Hemker

Elasticity of Textured Films:

1. Consider a textured thin film[001]

2. Calculate E from Sij's

3. Estimate Voight and Reuss bounds for E

EReuss = 1

ΣΣΣΣ (((( ViΕΕΕΕi

))))i=1

1000

((((000011111111))))αααα

αααα'

ββββββββ'

EVoight = ΣΣΣΣ ViEii=1

1000

K.J. Hemker

LIGA-Ni texture model

50

100

150

200

250

300

350

400

450

0 50 100 150 200

(011)(111)(001)

You

ng's

mod

ulus

(G

Pa)

Angle (degree)

<011>, 232 GPa

<001>, 136 GPa

<111>, 303 GPa

232 GPa

E(001) = 177 GPa

Emeasured = 180 GPa

• <001> out-of-plane • No in-plane texture

JJOHNS HOPKINSOHNS HOPKINSEENNGGII NNEEEERRII NNGG

Isotropic E

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Headlines about elasticity:

! [Cij]’s will not change at micro-scale.

!What out for texture effects.

!Single-crystalline materials require anisotropic elasticity.

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Mechanical Strength:

!Ductile Materials (Metals, hi T Si)– Deform plastically or yield

!Brittle Materials (Ceramics, Glass, RT Si)– Fracture

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Yield Strength:

!Dislocations lead to plastic deformation

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Atomic description of plasticity:

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Yield Strength:

!Dislocations lead to plastic deformation!Slow down dislocations = strengthening

– lattice– solute atoms– precipitates– grain boundaries

Note: MEMS is a young field many people still use pure materials.

K.J. Hemker

Fracture Strength:

σf

1 2 3 4 5

K.J. Hemker

Stress Magnification:Stress Magnification:

2c

2h

σ

σ

σ

σο

σmax

Inglis 1913

σmax = σ (1+2c / h)for an ellipse: ρ = h2 / cσmax = 2σ (c / ρ)1/2

limρ−>0 (ρ) = ao

σmax = 2σ (10-2 / 10−10)1/2

σmax = 20,000 σ !!!!!

ρ

K.J. Hemker

Basis for Fracture MechanicsBasis for Fracture Mechanics

Geometry and loading

“Stress Intensity Factor”

Material Parameter

“Fracture Toughness”

σ (πc )1/2 = Kc

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Key to Fracture MechanicsKey to Fracture Mechanics

1. Determine Kc- measure on specimens of known geometry

2. Calculate K- from current geometry and loading

3. Compare K with Kc- K < Kc is OK- K > Kc will fracture

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Typical Values for KICTypical Values for KIC

Silicon

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Comparison of structural members: beams

Micro Macro

polysilicon

steel

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Comparison of structural members: turbines

Micro Macro

GE 90 Jet Engine :

GE 90 Jet Engine :

Ni superalloys

silicon -> SiC

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Comparison of structural members: gears

Micro Macro

Steel,Ti

polysilcion

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Design views:

!MEMS view:– “Silicon a wonderful structural material”

!Macro view:– Silicon too brittle and too expensive

Who is right ?

K.J. Hemker

When will Si fracture ?

!Macro world defects ~ 1 mm

σf = KIc/(πc)1/2 = 0.9 MPa m1/2 / (π 10-3 m)1/2 = 16 MPa

!MEMS world defects ~1 µm

σf = KIc/(πc)1/2 = 0.9 MPa m1/2 / (π 10-6 m)1/2

= 508 MPa

K.J. Hemker

How much will Si cost ?Total Cost = Cmaterial + Cmanufacture + Cdisposal

!Macro worldCsteel = $0.20/lb x tons + $

CSi = $4.00/lb x tons + $$$

!MEMS worldCsteel = $0.20/lb x grams + $$$$

CSi = $4.00/lb x tons + $$$

K.J. Hemker

Farmer’s view on polysilcion:

!Cost:– Tolerable, should get better, key is batch

processing.

!Mechanical performance:– Ok for low T applications where components

are “over-sized” and flaws are controlled.