Dynamics of Surface Pattern Evolution in Thin Films Rui Huang Center for Mechanics of Solids, Structures and Materials Department of Aerospace Engineering.

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

Yang, Liu and Lagally, 2004.

Granados and Garcia, 2003.

Tabe et al., 2002 & 2003.

Wrinkle Patterns

Bowden et al., 1998 & 1999 .

Muller-Wiegand et al., 2002.

Stafford et al., 2004.

Cahill 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: 2

Ec (~300nm)

Chemical potential on surface: EU

Surface evolution: 2 MJvn

Linear analysis: tAA exp0 cm 3

4

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

WE UU

Surface chemical potential:

sMJ

WEs UUh

M

t

h

2

21

Surface flux:

Equation of surface evolution:

Nonlinear terms arise from wavy surface as the boundary condition for the stress field and from the wetting effect.

Stress Analysis

0jijn

x

h

hn

21

123

1

1

hn

Boundary condition on the surface:

)2()1()0(

ijijijij (In the order of )h

Zeroth-order: 0

)0( 0)0(

3 j 2

00

)0(1

f

f

E EUU

First-order: ,0

)1(

3 h 0)1(

33

x

uU E

)1(

0

)1(

hCiku ˆˆ0

)1(

(B.C.)

Linear Evolution Equation

hx

uM

t

h 2)1(

0

2

hkkE

Mkt

h

s

ˆ2ˆ2

2

02

Fourier transform 2

02 sE

L

8

0

43

16

M

Es

Length scale:

Time scale:

Critical wavelength:2

0

2

sc

EL

Fastest growing wavelength:2

03

4

3

4

s

cm

E

Nonlinear Stresses

Second-order: ,

)1()2(

3 h 2

0

)2(

33 h

x

uhUU

f

f

E

)2(

0

)1()1(2

0

)2(

2

1

1

1

(B.C.)

)2(

3

)2( ˆˆjiji Cu

x

uhUh

x

uM

t

h

f

f)2(

0

)1()1(2

0

2)1(

0

2

2

1

)1(2

3

Nonlinear evolution equation:

Spectral Method

hfhkkE

Mkt

h

s

ˆˆ2ˆ2

2

02

Fourier transform of the nonlinear equation:

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

tkP

thfhh

nnn

)(1

ˆˆˆ

)()()1(

Semi-implicit integration:

1D Simulations

Consideration of nonlinear stress leads to unstable evolution and formation of deep grooves.

0 5 10 15 20-0.05

0

0.05

0.1

0.15

0.2

x/L

h/L

0 5 10 15 20-1.5

-1

-0.5

0

0.5

1

x/Lh

/L

Linear equation Nonlinear equation

2D Simulations

Downward blow-up instability: nanopits?

t = 50

t = 85

t = 0 t = 20

Effect of Wetting

Linear evolution equation with wetting:

hh fsfs arctan

1

2

1

Transition of surface energy (Spencer, 1999):

)(1)( 222 hh

hU fSW

Sf

hh

hhx

uM

t

hfs

3

0

2

0

)1(

0

2 2

hh

khkE

Mkt

hfs

s

ˆ22ˆ3

0

2

0

2

02

Linear Analysis

0 0.2 0.4 0.6 0.8 1 1.2

-0.05

0

0.05

0.1

Wave Number, kL

Gro

wth

Rat

e,

h

0/h

c = 0.9

1

1.1

1.5

2

h0/h

c > 5

Critical film thickness:

3/1

2

f

c LLh

nm 1.0~ N/m, 1.0~

N/m, 1~ GPa, 150~ GPa, 1~0

sE

nm 5~

nm 75~

ch

L

Thick films: no effect;

Very thin films: stabilized.

Typical values:

0 5 10 15 200

0.05

0.1

0.15

0.2

x/L

h/L

t = 0

t = 200

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

x/L

h/L

t = 2350

t = 2250

t = 200~2000

1D Simulations

0 5 10 15 20-1

0

1

2

3

x/L

h/L

t=2350t=2358t=2359

Stable growth

Coarsening

Blow-up instability

2D Simulations

Upward blow-up instability: nano whiskers?

t = 0 t = 50 t = 200

t = 259 t = 260

Nonlinear Stress + Wetting: 1D Simulation

0 5 10 15 200

0.05

0.1

0.15

0.2

x/L

h/L

t=0

t=200

0 5 10 15 200

0.1

0.2

0.3

0.4

x/L

h/L

t = 200~5000t = 5700t = 6000~30000

Stable growth

Coarsening

No blow-up instability!

t = 0

t = 1000t = 500

t = 200

t = 250

t = 50

Nonlinear Stress + Wetting: 2D

Part I: Summary

• Nonlinear stress field leads to downward blowup instability.

• Wetting effect leads to upward blowup instability.• Combination of nonlinear stress and wetting stabilizes

the evolution.

Nonlinear analysis of surface diffusion-controlled pattern evolution in strained epitaxial films:

Part II:

Compression-Induced Wrinkle Patterns

(Lee at al., 2004)

Freestanding film: Euler bucklingCritical load:

22

112

L

hc

Other equilibrium states: energetically unfavorable

• Buckling relaxes compressive stress

• Bending energy favors long wavelength

On elastic substrates• Deformation of the substrate

disfavors wrinkling of long wavelengths and competes with bending to select an intermediate wavelength

Elastic substrate

Wrinkling: short wavelength, on soft substrates, no delamination

Buckling: long wavelength, on hard substrates, with delamination

Critical Condition for Wrinkling

3/2

21

3

4

1

f

scf E

E

0 0.002 0.004 0.006 0.008 0.010

0.005

0.01

0.015

0.02

0.025

Stiffness Ratio, Es/E

f

Co

mp

ress

ive

Str

ain

, -

wrinkling

flat film

Thick substrate (hs >> hf):

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.

Equilibrium Wrinkle Wavelength

Thick substrate (hs >> hf):

3/1

32

s

f

f E

Eh

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.

0 0.002 0.004 0.006 0.008 0.010

20

40

60

80

100

Stiffness Ratio, Es/E

f

Wrin

kle

Wav

elen

gth,

/

hf

Measure wavelength to determine film stiffness

Equilibrium Wrinkle Amplitude

Thick substrate (hs >> hf):

2/1

21 1

c

fhA

0 2 4 6 8 100

0.5

1

1.5

2

2.5

3

Compressive Strain, /c

Wrin

kle

Am

plitu

de, A

/hf

Measure amplitude to determine film stress/strain.

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 SubstratesCross-linked polymers

Compressive Strain

Wrinkle Amplitude

0

Evolution of wrinkles:

(I) Viscous to Rubbery

(II) Glassy to Rubbery

Rubbery State

R

Glassy State

G

Wrinkling Kinetics I: GR

Fastest mode

m 0

GrowthRate

Wrinkles of intermediate wavelengths grow exponentially;

The fastest growing mode dominates the initial growth.

1

f

m

h

For hs >> hf :

The kinetically selected wavelength is independent of substrate!

stAtA exp)( 0

Wrinkling Kinetics II: G

Instantaneous wrinkle at the glassy state:

2/1

0 1

G

fhA

3/1

0 32

G

f

f E

Eh

Kinetic growth at the initial stage:

1)exp()( 0 tBAtA

Long-term evolution: 3/1

32

R

f

f E

Eh

0

2/1

1

R

fhA

0A

Evolution Equations

wqh

t

w

p

p

p

p

p

p

)1(2

21

u

x

Nh

t

u

p

p

p

p

mmmmm hEhN )1()0(

x

w

x

w

x

u

x

u

2

1

2

1

x

w

x

N

xx

wN

xxxx

whEq mm

243

12

t = 0

t = 1104

Numerical Simulation

0 200 400 600-0.1

0

0.1

x/hf

w/h

f

0 200 400 600-0.1

0

0.1

x/hf

w/h

f

0 200 400 600-2

0

2

x/hf

w/h

f

0 200 400 600-2

0

2

x/hf

w/h

f0 50 100

W avelength, L/hf

0 50 100W avelength, L/h

f

0 50 100W avelength, L/h

f

0 50 100W avelength, L/h

f

t = 1105

t = 1107

Growing wavelengths

Coarsening

Equilibrium wavelength

Evolution of Wrinkle Wavelength

0 2 4 6 8 10

x 104

20

30

40

50

Normalized time, t/

Wa

vele

ng

th,

L/h f

/E

f=0.0001

/E

f=0.00001

Lm

= 26.9hf

104

105

106

107

20

30

40

50

60

70

Normalized time, t/

Wa

vele

ng

th,

L/h f

/E

f=0.0001

/E

f=0.00001

Lm

= 26.9hf

Leq

= 33.7hf

Leq

= 60.0hf

Initial stage: kinetically selected wavelengths

Intermediate stage: coarsening of wavelength

Final stage: equilibrium wavelength at the rubbery state

0 2 4 6 8 10

x 104

0.01

0.1

1

Normalized time, t/

RM

S

/E

f=0.0001

/E

f=0.00001

104

105

106

107

0

0.5

1

1.5

Normalized time, t/

RM

S

/E

f=0.0001

/E

f=0.00001

Aeq

= 0.619hf

Aeq

= 1.63hf

Evolution of Wrinkle Amplitude

Initial stage: exponential growth

Intermediate stage: slow growth

Final stage: saturating

t = 0 t = 104 t = 105

t = 107t = 106

2D Wrinkle Patterns I

t = 0 t = 105

t = 2X107

t = 106

t = 5X106

2D Wrinkle Patterns II

t = 107

t = 5X105

t = 106

t = 104

2D Wrinkle Patterns IIIt = 0

t = 0 t = 104 t = 105

t = 106 t = 107

On a Patterned Substrate

Circular Perturbationt = 0 t = 104 t = 105

t = 5105 t = 106 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=3X107t=1X107t=1X106t=1X105

t=8X104t=1X104 t=3X104 t=5X104

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 !