Solid mechanics
 Define the terms
 Calculate stresses in deposited thin films using the disk
 Stress
method
 Deformation
 Strain
 Thermal strain
 Thermal expansion coefficient
 Appropriately relate various types of stress to the correct
corresponding strain using elastic theory
 Give qualitative descriptions of how intrinsic stress can
form within thin films
 Calculate biaxial stress resulting from thermal mismatch
in the deposition of thin films
Why?
Solid mechanics...
Why?
Why?
Why?
Why is this thing bent?
Thermal actuator produced by Southwest Research Institute
And these?
A bi-layer of TiNi and SiO2. (From Wang, 2004)
Why?
Membrane is piezoresistive;
i.e., the electrical resistance
changes with deformation.
Adapted from MEMS: A Practical Guide to Design, Analysis, and Applications,
Ed. Jan G. Korvink and Oliver Paul, Springer, 2006
A simple piezoelectric actuator design: An applied voltage causes
stress in the piezoelectric thin film stress causing the membrane to
bend
Why?
Hot arm actuator
+
e
-
Joule heating leads to different
rates of thermal expansion, in
turn causing stress and
deflection.
Zap it with a voltage here…
i
+
e
-
How much does it move
here?
ω
Stress and strain
A = w·t
t
P
P
w
w
δ
L
P
P
σ=—=—
A

[A ]

ε=—
Normal stress
L
wt
Typical units
Dimensions
[F ]
δ
[F ]
[L
]2
N
m
2

Pa
Normal strain
Dimensions

[L ]
(dimensionless)
[L ]
Typical units
μ-strain = 10-6
Elasticity
How are stress and strain related to each other?
P
X fracture
F = kx
σ
plastic (permanent)
deformation
E
σ= E ε
L
elastic (permanent)
deformation
brittle
ductile
E
δ
P
X fracture
ε
Young’s modulus
(Modulus of elasticity,
Elastic modulus)
Elasticity
Strain in one direction causes strain in other directions
y
x
εy = -ν εx
Poisson’s ratio
Stress generalized
Stress is a surface phenomenon.
z
σz
ΣF = 0, ΣMo = 0
τzy
τzx
τxz
σy
τxy
σx
x
τxy = τyx
τyz = τzy
τzx = τxz
τyz
τyx
y
σ : normal stress
Force is normal to surface
σx  stress normal to x-surface
τ : shear stress
Force is parallel to surface
τxy  stress on x-surface in y-direction
Strain generalized
Essentially, strain is just differential deformation.
Δy
Δy + dΔy
Deforms
Δx
Δx + dΔx
Break into two pieces:
ux
=
dux
ux + dx
+
dy
θ2
θ1
dx
uniaxial strain
duy
Shear strain is strain with
no volume change.
shear strain
u: displacement
du x
x 
dx
 xy
 du x du y 
  1   2 
 

dx 
 dy
Relation of shear stress to shear strain
Just as normal stress causes
uniaxial (normal) strain, shear
stress causes shear strain.
τxy = G γxy
dux
τyx
τxy
τxy
θ2
τyx
duy
θ1
shear modulus
G
E
2(1  )
• Si sabes cualquiera dos de E,
G, y ν, sabes el tercer.
• Limits on ν:
Magic Algebra Box
0 < ν < 0.5
ν = 0.5 
incompressible
Generalized stress-strain relations
The previous stress/strain relations hold for
either pure uniaxial stress or pure shear
stress. Most real deformations, however, are
complicated combinations of both, and
these relations do not hold
εx = [
x
E
x normal
strain due to x
normal stress
] + [ -ν
y
E
] + [ -ν
x normal
strain due to y
normal stress
Deforms
z
E
]
x normal
strain due to z
normal stress
τxy = G γxy
Generalized Hooke’s Law
For a general 3-D deformation of an isotropic material, then




1
 yz
γyz =
G


1
 zx
γzx =
G
1
 x   y   z 
εx =
E
1
 y   z   x 
εy =
E
1
 z   x   y 
εz =
E
γxy =
Generalized Hooke’s Law
1
 xy
G
Special cases
• Uniaxial stress/strain
σ = Eε
• No shear stress, todos esfuerzos normales son iguales
volume strain
σx = σy = σz = σ = K•(ΔV/V)
bulk modulus
• Biaxial stress
Stress in a plane, los dos esfuerzos normales son iguales
σx = σy = σ = [E / (1 - ν)] • ε
biaxial modulus
Elasticity for a crystalline silicon
The previous equations are for isotropic materials. Is crystalline silicon isotropic?
E  Cij Compliance coefficients
 xx   C11 C12
  
C11
21
22
 yy   C12
 zz  C12
C12
31
32
 
 xyxy   C041 C042
 xzxz   C0 C0
52
   51
 yzyz   C061 C062
C12
13
C12
23
C11
33
C043
C053
C063
0x15
C014  C
 

C024  C0y25
0z35
C034  C
 
C44 C0xy45

C054 Cxz55
44
  
C064 C0yz65
C016    x 
 

C026   y 
C036    z 
 
C046   xy 
C056   xz 
  
C66
44 
  yz 
For crystalline silicon
C11 = 166 GPa, C12 = 64 GPa and C44 = 80 GPa
Te toca a ti
Assuming that elastic theory holds, choose the appropriate modulus and/or stress-strain relationship for each of the following
situations.
1. A monkey is hanging on a rope, causing it to stretch. How do you model the deformation/stress-strain in the rope?
Uniaxial stress/strain
2. A water balloon is being filled with water. How do you model the deformation/stress-strain in the balloon membrane?
Biaxial stress/strain
3. A nail is hammered into a piece of plywood. How do you model the deformation/stress-strain in the nail?
Uniaxial stress/strain
4. A microparticle is suspended in a liquid for use in a microfluidic application, causing it to compress slightly. How do you
model the deformation/stress-strain in the microparticle?
Use bulk modulus (no shear, all three normal stresses the same)
5. A thin film is deposited on a much thicker silicon wafer. How do you model the deformation/stress-strain in the thin film?
Biaxial stress/strain
6. A thin film is deposited on a much thicker silicon wafer. How do you model the deformation/stress-strain in the wafer?
Anisotropic stress/strain (Using Cij  compliance coefficients)
7. A thin film is deposited on a much thicker glass substrate. How do you model the deformation/stress-strain in the glass
substrate?
Generalized Hooke’s Law. I.e., ε = (1/E)(σx – ν(σy + σz)) etc.
Thermal strain
Thermal Expansion
Most things expand upon heating, and shrink upon cooling.
δ(T) = αT (T-T0)
Notes:
d T
T 
dT
Thermal expansion coefficient
• αT ≈ constant ≠ f(T)
If no initial strain
ε(T) ≈ ε(T0) + αT (T-T0)
• Thermal strain tends to be the same in all
directions even when material is otherwise
anisotropic.
Solid mechanics of thin films
Adhesion
Ways to help ensure adhesion of deposited
thin films:
• Ensure cleanliness
• Increase surface roughness
• Include an oxide-forming element in
between a metal deposited on oxide
Stress in thin films
positive (+)
Negative (-)
Tension
Compression
Tension headache
Stress in thin films
Two types of stress
Intrinsic stress
Extrinsic stress
Also known as growth stresses, these develop during as the
film is being formed.
These stresses result from externally imposed factors.
Thermal stress is a good example.
Doping
Sputtering
Microvoids
Gas entrapment
Polymer shrinkage
Thermal stress in thin films
Consider a thin film deposited on a substrate at a deposition
temperature, Td. (Both the film and the substrate are initially at
Td.)
thin film deposited at Td
Initially the film is in a stress free state.
The film and substrate are then allowed to cool to room
temperature, Tr
Since the two materials are hooked together, they both experience
strain as they cool.
the same ____________
substrate
both cooled to Tr
εboth = εsubstrate or εfilm ?
εsubstrate = αT,s(Tr - Td)
=
εfilm = αT,f (Tr - Td) + εmismmatch
εmismmatch = αT,s(Tr - Td) - αT,f (Tr - Td)
= (αT,f - αT,s)(Td - Tr)
Thermal stress in thin films
How would you relate σmismatch to εmismatch?
Biaxial stress/strain
σmismatch = [E / (1-ν)]·εmismatch
= [E / (1-ν)]·(αT,f - αT,s)(Td - Tr)
tension
 If αT,f > αT,s  σmismatch = (+) or (-)  Film is in ___________________.
compression
 If αT,f < αT,s  σmismatch = (+) or (-)  Film is in ___________________.
Thin film
Initially
stress free
cantilever
Sacrificial
layer
σmismatch > 0
σmismatch < 0
Stress in thin films
Compression or tension?
Compression or tension?
(a)
(b)
(a) Stress in SiO2/Al cantilevers
(b) Stress in SiO2/Ti cantilevers
[From Fang and Lo, (2000)]
αT,Al >, <, = αT,SiO2 ?
αT,Ti >, <, = αT,SiO2 ?
Stress in thin films
How were these fabricated?
(a)
(b)
(a) Stress in SiO2/Al cantilevers
(b) Stress in SiO2/Ti cantilevers
[From Fang and Lo, (2000)]
Te toca a ti
Show that the biaxial modulus is given by
E/(1 – ν)
Pistas:
• Remember what the assumptions for “biaxial” are.
• In thin films you can always find one set of x-y axes for which there is only σ and no τ.
Measuring thin film stress
The disk method
Stressed wafer (after thin film)
Unstressed wafer
(before thin film)
R
radius of curvature ,
R = _________________________
wafer thickness
T = _________________________
and
thin film thickness
t = _________________________.
Assumptions:
 The film thickness is uniform and small compared to the
wafer thickness.
 The stress in the thin film is biaxial and uniform across it’s
thickness.
 Ths stress in the wafer is equi-biaxial (biaxial at any location in
the thickness).
 The wafer is unbowed before the addition of the thin film.
 Wafer properties are isotropic in the direction normal to the
film.
 The wafer isn’t rigidly attached to anything when the
deflection measurement is made.
2
E
T


1  6Rt
Biaxial modulus
of the wafer
strain at wafer/film
interface