1. Solid mechanics Define the terms
Calculate stresses in deposited thin films using the disk method
Stress
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

2. Why?
Solid mechanics...
Why?
Why?

4. A bi-layer of TiNi and SiO2. (From Wang, 2004)
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)

5. 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

6. Why? Hot arm actuator + How much does it move here? e-
Zap it with a voltage here…
How much does it move here? i + e - ω
Joule heating leads to different rates of thermal expansion, in turn causing stress and deflection.

8. Stress and strain ε = — σ = — = — Normal stress Normal strain
A = w·t t P P w w L δ δ P P
Normal stress
ε = —
Normal strain
σ = — = — L A wt
Dimensions
Typical units
Dimensions
Typical units
[L ]
[F ]
[F ] N
μ-strain = 10-6
(dimensionless) Pa
[A ]
[L ]2
m 2
[L ]

9. (Modulus of elasticity,
How are stress and strain related to each other? P σ X
fracture
F = kx
plastic (permanent) deformation X
fracture E
σ = ε E L
elastic (permanent) deformation
Young’s modulus
(Modulus of elasticity,
Elastic modulus)
brittle
ductile E δ ε P

10. Strain in one direction causes strain in other directions
Elasticity
Strain in one direction causes strain in other directions y x
εy = εx -ν
Poisson’s ratio

11. Stress generalized ΣF = 0, ΣMo = 0 τxy = τyx τyz = τzy τzx = τxz
Stress is a surface phenomenon. z σz
ΣF = 0, ΣMo = 0
τxy = τyx
τyz = τzy
τzx = τxz
τzy
τzx
τyz
τxz σy
τxy
τyx y σx
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 x

12. Strain generalized
Essentially, strain is just differential deformation.
Deforms
Δy + dΔy Δy Δx
Δx + dΔx
Break into two pieces:
dux ux
ux + dx = dy + θ2
Shear strain is strain with no volume change.
duy dx θ1
uniaxial strain
shear strain
u: displacement

13. Relation of shear stress to shear strain
Just as normal stress causes uniaxial (normal) strain, shear stress causes shear strain.
dux
τyx
τxy = γxy
τxy G
τxy θ2
duy
shear modulus
τyx θ1
Si sabes cualquiera dos de E, G, y ν, sabes el tercer.
Limits on ν:
0 < ν < 0.5
Magic Algebra Box
ν = 0.5  incompressible

14. 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
Deforms
εx = [ ] + [ ] + [ ] -ν
τxy = G γxy -ν
x normal strain due to x normal stress
x normal strain due to y normal stress
x normal strain due to z normal stress

15. Generalized Hooke’s Law
For a general 3-D deformation of an isotropic material, then
εx =
γxy =
εy =
γyz =
εz =
γzx =
Generalized Hooke’s Law

16. Special cases σ = Eε σx = σy = σz = σ = K•(ΔV/V)
Uniaxial stress/strain
σ = Eε
No shear stress, todos esfuerzos normales son iguales
σx = σy = σz = σ = K•(ΔV/V)
Biaxial stress
Stress in a plane, los dos esfuerzos normales son iguales
σx = σy = σ = [E / (1 - ν)] • ε
volume strain
bulk modulus
biaxial modulus

17. Elasticity for a crystalline silicon
The previous equations are for isotropic materials. Is crystalline silicon isotropic?
E  Cij Compliance coefficients
For crystalline silicon
C11 = 166 GPa, C12 = 64 GPa and C44 = 80 GPa

18. Te toca a ti
Assuming that elastic theory holds, choose the appropriate modulus and/or stress-strain relationship for each of the following situations.
A monkey is hanging on a rope, causing it to stretch. How do you model the deformation/stress-strain in the rope?
A water balloon is being filled with water. How do you model the deformation/stress-strain in the balloon membrane?
A nail is hammered into a piece of plywood. How do you model the deformation/stress-strain in the nail?
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? 
A thin film is deposited on a much thicker silicon wafer. How do you model the deformation/stress-strain in the thin film?
A thin film is deposited on a much thicker silicon wafer. How do you model the deformation/stress-strain in the wafer?
A thin film is deposited on a much thicker glass substrate. How do you model the deformation/stress-strain in the glass substrate?
Uniaxial stress/strain
Biaxial stress/strain
Use bulk modulus (no shear, all three normal stresses the same)
Anisotropic stress/strain (Using Cij  compliance coefficients)
Generalized Hooke’s Law. I.e., ε = (1/E)(σx – ν(σy + σz)) etc.

19. Thermal expansion coefficient
Thermal strain
Thermal Expansion
Most things expand upon heating, and shrink upon cooling.
δ(T) = αT (T-T0)
Notes:
αT ≈ constant ≠ f(T)
ε(T) ≈ ε(T0) + αT (T-T0)
Thermal strain tends to be the same in all directions even when material is otherwise anisotropic.
If no initial strain
Thermal expansion coefficient

20. 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 headache
Tension
Compression

21. 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

22. εboth = εsubstrate or εfilm ?
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.)
Initially the film is in a stress free state.
The film and substrate are then allowed to cool to room temperature, Tr
thin film deposited at Td
substrate
Since the two materials are hooked together, they both experience the same ____________ as they cool.
strain
both cooled to Tr
εboth = εsubstrate or εfilm ?
εmismmatch = αT,s(Tr - Td) - αT,f (Tr - Td)
= (αT,f - αT,s)(Td - Tr)
εsubstrate =
αT,s(Tr - Td) =
εfilm =
αT,f (Tr - Td)
+ εmismmatch

23. 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)
If αT,f > αT,s  σmismatch = (+) or (-)  Film is in ___________________.
If αT,f < αT,s  σmismatch = (+) or (-)  Film is in ___________________.
tension
compression
Thin film
Initially stress free cantilever
σmismatch > 0
σmismatch < 0
Sacrificial layer

24. Compression or tension? Compression or tension?
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 ?

25. How were these fabricated?
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)]

26. 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.

27. Biaxial modulus of the wafer strain at wafer/film interface
Measuring thin film stress
The disk method
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.
Stressed wafer (after thin film) R
Unstressed wafer (before thin film)
R = _________________________,
T = _________________________ and
t = _________________________.
radius of curvature
wafer thickness
Biaxial modulus of the wafer
strain at wafer/film interface
thin film thickness