Transcript Document

Dynamics of Surface Pattern Evolution in Thin Films
Rui Huang
Center for Mechanics of Solids, Structures and Materials
Department of Aerospace Engineering and Engineering Mechanics
The University of Texas at Austin
Self-Assembled Surface Patterns
Granados and Garcia, 2003.
Tabe et al., 2002 & 2003.
Yang, Liu and Lagally, 2004.
Wrinkle Patterns
Bowden et al., 1998 & 1999 .
Stafford et al., 2004.
Cahill et al., 2002.
Muller-Wiegand et al., 2002.
Part I:
Surface Diffusion-Controlled Patterns
Surface Instability of Stressed Solid

Asaro and Tiller (1972);
Grinfeld (1986);
Srolovitz (1989)……
Competition between surface energy and
strain energy leads to a critical wavelength:
Chemical potential on surface:
Surface evolution:
Linear analysis:
E 
c  2

(~300nm)
  U E  
vn    J  M2 
A  A0 expt 
4
m  c
3
Nonlinear analysis: develop crack-like grooves or cusps.
Instability of Epitaxial Films
Spencer, Voorhees and Davis (1991);
Freund and Jonsdottir (1993);
Gao (1993)……
The film is stressed due to lattice mismatch between the film and the substrate
(e.g., Ge on Si).
Stress relaxation leads to formation of dislocations and/or surface roughening.
Linear analysis: similar to that of stressed solids
Nonlinear analysis: self-assembly of quantum dots
How to control the size and order of quantum dots?
The Base Model
Surface chemical potential:
  U E    UW
Surface flux:
J  M s 
Equation of surface evolution:
h
M
2
U E    U W 


s
2
t
1  h
Nonlinear terms arise from wavy surface as the boundary condition
for the stress field and from the wetting effect.
Stress Analysis
Boundary condition on the surface:
n  
 ij n j  0
h
2
1  h x
1
 ij   ij(0)   ij(1)   ij( 2)  
Zeroth-order:     0
( 0)
(1)

First-order: (B.C.) 3   0 h,

1
n3 
1  h
2
(In the order of h )
( 0)
3j
0
U
( 0)
E
 U0 
 33(1)  0
uˆ(1)  ik  C  0 hˆ
U
(1)
E
1  f
Ef
 02
u(1)
0
x
Linear Evolution Equation
 u(1)

h
2
2
 M  0
  h 
t
x


Fourier transform
 2

hˆ
2 2 0
2 ˆ
 Mk 
k  k h
t
 Es

Critical wavelength:
Length scale:
Es
L
2 02
Time scale:
 3 Es4

16M 08
Es
c  2L  2
0
Fastest growing wavelength:
4Es
4
m  c 
3
3 02
Nonlinear Stresses
Second-order: (B.C.)  3(2)   (1) h, 

(2)
33
  0 h
2
uˆi( 2)  Cij ˆ 3( 2j )
U E( 2)
1  f
u( 2)
1 (1) (1)

U 0 h       0
1  f
2
x
2
Nonlinear evolution equation:
(1)
( 2)


3



u

u
h
1
2
f
2
2
(1) (1)



 M   0
  h 
U 0 h        0


t

x
2
(
1


)
2

x

f



Spectral Method
Fourier transform of the nonlinear equation:

2


2

hˆ
2
2 ˆ
0
 Mk 
k  k h  f hˆ
t
 Es

Semi-implicit integration:
hˆ ( n1)
 
hˆ ( n )  f hˆ ( n ) t

1  P(k )t
Numerical simulations:
Calculate spatial differentiation in the Fourier space
Calculate nonlinear terms in the physical space
Communicate between physical and Fourier spaces via FFT and its inverse
1D Simulations
Linear equation
Nonlinear equation
1
0.2
0.5
h/L
h/L
0.15
0.1
0
-0.5
0.05
-1
0
-0.05
0
5
10
x/L
15
20
-1.5
0
5
10
15
x/L
Consideration of nonlinear stress leads to unstable evolution and
formation of deep grooves.
20
2D Simulations
t=0
t = 20
t = 85
Downward blow-up
instability: nanopits?
t = 50
Effect of Wetting
Transition of surface energy (Spencer, 1999):
1
1
h
 h    s   f    s   f arctan 
2

 
U W ( h)  
S  f

2
2

(


h
)
1  h
2
Linear evolution equation with wetting:
(1)



u
h
2
2
2

 M  0
  h0  h  3  s   f h 
t
x
h0


 2
ˆ
hˆ
2
2 2 0
2
 Mk 
k   h0 k  3  s   f h
t
h0
 Es

 f  S
Linear Analysis
Thick films: no effect;
0.1
h /h > 5
Very thin films: stabilized.
Critical film thickness:
 
hc  2 L
 L
f

1/ 3




Growth Rate, 
0
c
2
0.05
1.5
1.1
0
1
h /h = 0.9
0
-0.05
0
0.2
c
0.4
0.6
0.8
Wave Number, kL
1
Typical values:
 0 ~ 1 GPa, Es ~ 150GPa,  ~ 1 N/m,
 ~ 0.1 N/m, ~ 0.1 nm
L ~ 75 nm
hc ~ 5 nm
1.2
1D Simulations
0.2
Stable growth
t = 200
0.1
Coarsening
t=0
0.5
0.05
0
0
t = 2350
0.4
5
0.3
10
x/L
0.2
h/L
t = 2250
15
20
t = 200~2000
3
t=2350
t=2358
t=2359
0.1
0
0
2
5
h/L
h/L
0.15
Blow-up instability
x/L
10
15
20
1
0
-1
0
5
x/L
10
15
20
2D Simulations
t=0
t = 259
t = 50
t = 200
t = 260
Upward blow-up instability: nano whiskers?
Nonlinear Stress + Wetting: 1D Simulation
0.2
t=200
h/L
0.15
Stable growth
0.1
t=0
0.05
0
0
5
x/L
10
15
20
0.4
t = 200~5000
t = 5700
t = 6000~30000
h/L
0.3
Coarsening
No blow-up instability!
0.2
0.1
0
0
5
10
x/L
15
20
Nonlinear Stress + Wetting: 2D
t=0
t = 50
t = 200
t = 250
t = 500
t = 1000
Part I: Summary
Nonlinear analysis of surface diffusion-controlled
pattern evolution in strained epitaxial films:
• Nonlinear stress field leads to downward blowup
instability.
• Wetting effect leads to upward blowup instability.
• Combination of nonlinear stress and wetting stabilizes
the evolution.
Part II:
Compression-Induced Wrinkle Patterns
(Lee at al., 2004)
Freestanding film: Euler buckling
Critical load:

h
c 
 
121    L 
2
• Buckling relaxes compressive stress
• Bending energy favors long wavelength
Other equilibrium states: energetically unfavorable
2
On elastic substrates
Elastic substrate
• Deformation of the substrate
disfavors wrinkling of long
wavelengths and competes
with bending to select an
intermediate wavelength
Wrinkling: short wavelength, on soft substrates, no delamination
Buckling: long wavelength, on hard substrates, with delamination
Critical Condition for Wrinkling
Compressive Strain, - 
0.025
Thick substrate (hs >> hf):
0.02
0.015
wrinkling
1  3Es 
1   f  2   c   
4  E f 
0.01
flat film
0.005
0
0
0.002
0.004
0.006
0.008
Stiffness Ratio, E /E
s
0.01
f
The critical strain decreases as the substrate stiffness decreases.
In general, the critical strain depends on the thickness ratio and
Poisson’s ratios too.
In addition, the interface must be well bonded.
2/3
Equilibrium Wrinkle Wavelength
100
Wrinkle Wavelength,  /h f
Thick substrate (hs >> hf):
80
1/ 3
 Ef 

  2h f 

3
E
 s
60
40
Measure wavelength to
determine film stiffness
20
0
0
0.002
0.004
0.006
0.008
Stiffness Ratio, E /E
s
0.01
f
The wrinkle wavelength is independent of compressive strain.
The wavelength increases as the substrate stiffness decreases.
In general, the wavelength depends on thickness ratio and Poisson’s
ratios too.
Equilibrium Wrinkle Amplitude
Wrinkle Amplitude, A/h f
3
Thick substrate (hs >> hf):
2.5
1/ 2
 1  2 
A  h f 
 1
 c

2
1.5
1
Measure amplitude to
determine film stress/strain.
0.5
0
0
2
4
6
Compressive Strain, /
8
10
c
The wrinkle amplitude increases as the compressive strain increases.
For large deformation, however, nonlinear elastic behavior must be
considered.
Equilibrium Wrinkle Patterns
In an elastic system, the equilibrium state minimizes the total strain energy.
However, it is extremely difficult to find such a state for large film areas.
More practically, one compares the energy of several possible patterns to
determine the preferred pattern.
How does the pattern emerge?
How to control wrinkle patterns?
Wrinkling on Viscoelastic Substrates
Cross-linked polymers
Wrinkle
Amplitude
Rubbery State
Glassy State
0
R
G
Evolution of wrinkles:
(I) Viscous to Rubbery
(II) Glassy to Rubbery
Compressive
Strain
Wrinkling Kinetics I:  R     G
Growth
Rate
Fastest mode
Wrinkles of intermediate
wavelengths grow exponentially;
The fastest growing mode
dominates the initial growth.
0
m
A(t )  A0 expst 

For hs >> hf :
m 
h f
1    
The kinetically selected wavelength is independent of substrate!
Wrinkling Kinetics II:     G
Instantaneous wrinkle at the glassy state:
1/ 3
 Ef 

0  2h f 

3
E
 G
1/ 2
  
A0  h f 
 1
 G

Kinetic growth at the initial stage:
A(t )  A0  Bexp(t )  1
Long-term evolution:
1/ 3
0
 Ef 

  2h f 

3
E
 R
1/ 2
A0
  
A  h f 
 1
 R

Evolution Equations
p
w 1  2 p hp

q
w
t 2(1   p )  p
p
hp N
p
u


u
t
 p x
p
N w
Emhm3
4w
2w
q
 N

12 x x x x
x x
x x
N   (0) hm  Emhm (1 m )  m  
 
1  u u 



2  x  x
 1 w w

 2 x x



Numerical Simulation
t=0
w/h
f
0.1
0
-0.1
0
200
400
600
0
x/hf
0.1
Growing wavelengths
f
w/h
t=
1104
0
-0.1
0
200
0
w/h
f
200
400
600
x/hf
0
50
100
Wavelength, L/hf
Equilibrium wavelength
f
w/h
50
100
Wavelength, L/hf
Coarsening
2
t=
600
0
-2
0
1107
400
x/hf
2
t = 1105
50
100
Wavelength, L/hf
0
-2
0
200
400
600
0
50
100
Wavelength, L/h
Evolution of Wrinkle Wavelength
70
50
/Ef=0.0001
/Ef=0.00001
Wavelength, L/h
f
Wavelength, L/h
f
/Ef=0.0001
40
30
Lm = 26.9hf
20
0
2
4
6
Normalized time, t/
8
10
x 10
4
60
Leq = 60.0hf
/Ef=0.00001
50
40
Leq = 33.7hf
30
Lm = 26.9hf
20 4
10
10
5
10
6
Normalized time, t/
Initial stage: kinetically selected wavelengths
Intermediate stage: coarsening of wavelength
Final stage: equilibrium wavelength at the rubbery state
10
7
Evolution of Wrinkle Amplitude
1.5
1
/Ef=0.0001
Aeq = 1.63hf
1
0.1
RMS
RMS
/Ef=0.00001
Aeq = 0.619hf
0.5
/Ef=0.0001
0.01
/Ef=0.00001
0
2
4
6
Normalized time, t/
8
10
x 10
4
0 4
10
10
5
6
Normalized time, t/
Initial stage: exponential growth
Intermediate stage: slow growth
Final stage: saturating
10
10
7
2D Wrinkle Patterns I
t=0
t = 106
t = 104
t = 107
t = 105
2D Wrinkle Patterns II
t=0
t = 5X106
t = 105
t = 2X107
t = 106
2D Wrinkle Patterns III
t=0
t = 106
t = 104
t = 107
t = 5X105
On a Patterned Substrate
t=0
t = 104
t = 106
t = 107
t = 105
Circular Perturbation
t=0
t = 5105
t = 104
t = 106
t = 105
t = 107
Evolution of Wrinkle Patterns
• Symmetry breaking in isotropic system:
– from spherical caps to elongated ridges
– from labyrinth to herringbone.
• Symmetry breaking due to anisotropic strain
– from labyrinth to parallel stripes
• Controlling the wrinkle patterns
– On patterned substrates
– By introducing initial defects
Large-Cell Simulation
t=1X104
t=1X105
t=3X104
t=1X106
t=5X104
t=1X107
t=8X104
t=3X107
Acknowledgments
• Co-workers: Se Hyuk Im, Yaoyu Pang, Hai Liu,
S.K. Banerjee, H.H. Lee, C.M. Stafford
• Funding: NSF, ATP, Texas AMRC
Thank you !