Film in tension
Film in compression
Hutchinson, Suo
Advances in Applied Mechanics
29, 64-192 (1992).
pdf file: www.deas.harvard.edu/suo
1
Variety of Thin Film Fracture Patterns
Cook, Suo, MRS Bulletin, 27, 45 (2002)
Courtesy J. Sturn (a), (c); Q. Ma (b); M.D. Thouless (d)
2
A channel crack
T. Tsui (http://www.imechanica.org/node/248)
3
The origin of stress in a film
• Deposition process (intrinsic stress)
• Mismatch due to lattice constant
• Mismatch in the coefficient of thermal
expansion
• Bending
W.D. Nix, Mechanical Properties of Thin Films
(http://imechanica.org/node/530)
4
Stress in a thin film due to mismatch in
the coefficients of thermal expansion
•The film is very thin compared to the substrate (h << H).
•The substrate is nearly stress free.
•The film is in a state of equal biaxial stress, σ.
•At Tref, the film is stress free.
•The film deforms elastically as the temperature changes
ε s = α s (T − Tref )
ε f = α f (T − Tref
(
1 −ν )σ
)+
f
Ef
As the temperature changes, the substrate and the film remains bonded
εs = ε f
σ=
Ef
(1 −ν
f
() α
f
− α s )(Tref − T )
When αf > αs, upon cooling, the film is under tensile stress, σ > 0.
5
Stress in film due to bending
1
=
R
M
Es
3
BH
12 1 −ν s2
(
)
H /2
ε=
R
σ=
σ=
Ef
1 −ν
2
f
(
Silicon substrate
M = applied moment
B = width of the beam
H = thickness of the substrate
R = radius of curvature
ε = strain in the film
σ = stress in the film
ε
)
6 E f 1 −ν s2 M
(
)
Es 1 −ν 2f BH 2
6
Measure residual stress using wafer curvature
Line tension
Bending moment
f = ∫ σdz
M = fBH / 2
Radius of curvature
1 1 −ν s 12 M
=
3
R
Es BH
Stoney’s formula
⎛ Es ⎞ H 2
⎟⎟
f = ⎜⎜
⎝ 1 −ν s ⎠ 6 R
7
Channel crack: Initiation vs. steady propagation
Channel growth direction
A
a
B
h
G
σ 2h / E f
su
t
n
ia
l
p
com
te
a
r
t
bs
stiff substrate
A. Initiation Ginitiation = Yσ 2 a / E f
1
B. Steady propagation Gss = Zσ 2 h / E f
When the substrate is compliant, steady state is attained at
a large a/h and a large energy release rate.
a/h
8
Steady-state energy release rate
of a channel crack
Gss
Reduction in elastic energy for the crack
to advance a unit distance
U = SEupstream - SEdownstream
h
U = ∫ G ps (b )
0
h
Steady-state energy release rate
G ps
b
h
hGss = U
1
Gss = ∫ G ps (b )db
h0
Example: film and substrate have similar elastic modulus
1 −ν 2
(1.1215)2 πbσ 2
G ps (b ) =
E
(
1 −ν )
=2
hσ
2
Gss
E
2
9
Steady-state energy release rate of a channel crack
Reduction in elastic energy for the crack
to advance a unit distance
U = SEupstream - SEdownstream
Gss
h
1
U = ∫ δ (z )σ ( z )dz
20
δ ( z ) Crack opening downstream
σ (z ) Stress upstream
hGss = U
h
1
Gss =
δ ( z )σ ( z )dz
∫
2h 0
10
Driving force for steady propagation
2
Dimensionless energy release rate, (GSSE1*)/(σ0h)
40
Gss = Z
σ h
2
Ef
35
Beuth [21]
XFEM
β =α /4
30
25
20
15
10
5
0
-1
-0.5
0
Elastic mismatch, α
0.5
1
E f − Es
E f + Es
An advantage of being nano: thin films can sustain large stains.
e.g., 7 nm silica film can sustain strains of ~5%.
11
Channel crack in patterned structure
X.H. Liu (http://www.imechanica.org/node/379)
12
Cracking in low k dielectrics
Also see a thread of discussion at http://www.imechanica.org/node/165
13
Crack driving force, G
Generalized coordinates: D = displacement, A = crack area
Elastic energy
U ( D, A)
D
dU = FdD − GdA
∂U ( D, A)
G=−
∂A
A
Work supply = elastic energy + excess
In steady state, the excess is proportional to crack area, ΓA
FdD = dU + ΓdA
G=Γ
14
Fracture mechanics as a division of labor
George Rankin Irwin (1907-1998)
Crack driving force, G.
Crack resistance, Γ
G=Γ
Solve an elasticity BVP
to calculate G of a structure
G = Yσ 2 a / E
Determine
load σ
Run a test to
measure Γ of a material
Determine
crack length a
15
Small structures: a new economy
• Calculating G is prohibitively expensive
•3D architecture
•uncertain flaws
•uncertain residual stress field
•uncertain material properties
• Massive testing is cheap.
16
A new division of labor: run test to measure G!
Moisture-assisted crack growth: the V-G function
Weiderhorn, J. Am. Ceram. Soc. 59, 407 (1967)
H2O
silica in moisture
G>Γ
Γ100%
Si2O
H2O
Si
oxygen
The load opens crack and strains bonds
H2O transports to crack tip and assists in breaking bonds
17
A method to measure G
V-G function is specific to material and environment, and
remains the same when the material is integrated into a structure.
Use a simple structure to measure V-G function.
In a complex structure, an observed V gives a reading of G.
…in a way analogous to measuring temperature
Thermometer
G-meter
18
Measure the V-G function
•Scratch and bend
•Channel crack
Silicon substrate
G = Zσ 2 h / E
Crack Velocity, V(m/s)
10-5
10-6
10-7
10-9
1.3
OSG
Organosilicate glass
Room temperature
30% relative humidity
10-8
1.4
1.5
1.6
1.7
Crack Driving Force, G (J/m2)
Si substrate
1.8
19
Read G
Known V-G function
10
-5
P
Crack Velocity, V(m/s)
M
10
K
-6
NO
L
J
I
10
G
-7
E
C
10
-8
1.7
A
H
F
D
B
1.9
2.1
2.3
2.5
2
Crack Driving Force, G(J/m )
G=
σ 2 Zh ⎛
⎛ S1 ⎞
⎛ S2 ⎞
⎛ S1 + S 2 ⎞ ⎞
tanh
tanh
tanh
+
⎜
⎜
⎟
⎜
⎟−
⎜
⎟⎟
E ⎝
⎝ 2 Zh ⎠
⎝ 2 Zh ⎠
⎝ 2 Zh ⎠ ⎠
Xia, Hutchinson, J. Mech. Phys. Solids 48, 1107 (2000).
20
Measure crack driving force due to
residual stress field, GR
At GR, the crack velocity is too low to be observed.
•Bend structure, observe crack velocity, and read G.
•Extrapolate the data to obtain GR
G = K2 / E
K = K R + BF
K
KR
F
To be immortal, GR < Γ
21