Dislocation Dynamics for Computational Design of Thin Film Systems - - PowerPoint PPT Presentation

The University of Iowa The University of Iowa

• L. Sun

Dislocation Dynamics

for Computational Design of Thin Film Systems

Lizhi Sun

Department of Civil and Environmental Engineering and Center for Computer-Aided Design The University of Iowa Presented at NSF-DMR Materials Theory Program Overview Meeting University of Illinois at Urbana-Champaign June 19-20, 2002

The University of Iowa The University of Iowa

• L. Sun

Collaborative ITR/AP Research on

Large-scale Dislocation Dynamics Simulations for Computational Design of Thin Film Systems

PIs: N.M. Ghoniem (UCLA) Post-doc: X.L. Han (UCLA) Lizhi Sun (Univ. of Iowa) Students: E.H. Tan (U. Iowa) H.T. Liu (U. Iowa) Z.Q. Wang (UCLA) (September 2001 – August 2004)

NSF-DMR 0113172/0113555

The University of Iowa The University of Iowa

• L. Sun

Materials Modeling

in Terms of Length Scales

• Ab initio method
• Molecular dynamics method
• Dislocation dynamics method
• Finite element method

The University of Iowa The University of Iowa

• L. Sun

Project Objectives

• Investigating single and collective dislocation activities

in anisotropic materials, which determines the mechanical behavior of semiconductor thin film systems.

• Coupling dislocation dynamics with continuum mechanics

to consider the surface and interface effects.

The University of Iowa The University of Iowa

• L. Sun

Project Objectives (Cont’d)

• Applying dislocation dynamics software to investigate

a number of physical mechanisms including misfit and threading dislocation motion, annihilation, multiplication, interaction, and junction; dislocation interaction with point defects, precipitates and inclusions; thermal residual stress effect; design of buffer layers and superlattices, etc.

• Conducting large-scale simulation and optimization of

semiconductor systems to provide guidelines for engineering design of microelectronics.

The University of Iowa The University of Iowa

• L. Sun

Presentation Outline

• 1. Anisotropic Dislocation Dynamics Approach
• 2. Dislocation Interaction with Inclusions
• 3. Dislocations in Thin Films

The University of Iowa The University of Iowa

• L. Sun

Elastic Fields of Dislocation Loops in Anisotropic Materials

b

Cijkl

( ) ( )

ij ijkl lk

C σ β = x x

2

1 ( ) ( ) ( ) 8

h ji jnh klmn m lik L

d C b I dl dl r ν β π = ∈

∫ !

x m,n

2

( sin cos ) ( sin cos ) ( ) cos ( ) [ ] ( sin cos ) sin ( sin cos ) ( )

l ik ik ik lik l

m N N N I n d D D D

π

φ φ φ φ φ φ φ φ φ φ φ   − + − + = − −   − + − +  

∫

m n m n n m,n m n m n n

Mura formula:

Dislocation Dynamics in Anisotropic Materials

! f: Forces (external, internal , self force,

and friction resistant (Perierls) force, etc.)

! B: Resistive matrix (inverse mobility) ! V: Dislocation movement

( )

t i ij j i

f B V r ds δ

Γ

− =

∫

• Equation of Dislocation Motion (variational waek form)

Dislocation Dynamics in Anisotropic Materials

! Self-force (Barnett)

P

j i ij ij

n b P P t E F )] ( ) ( [ 5 . ) (

2 1

σ σ κ + − =

core

F P L J t E t E t E F + − + − = ) , ( )] / 8 ln[ )] ( " ) ( [ ) ( εκ κ κ

Stretch Line Tension Non-local

P1 P2

Dislocation Dynamics in Anisotropic Materials

Circular dislocation loop on the plane (1,1,1)

Slip direction [-1,1,0] Loop radius R

b

Dislocation Dynamics in Anisotropic Materials

x [-110] z [111]

Stress σ σ σ σ (divided by 0.5(C11-C12)b/R)

Anisotropic Ratio A=2C44/(C11-C12)

0.0 0.5 1.0 1.5 2.0 2.5 3.0

• 3
• 2
• 1

1 2 3

σ zx

x/R A=1 (isotropic) A=2.0 A=4.0 A=0.5 A=0.25

0.0 0.5 1.0 1.5 2.0 2.5 3.0

• 2.0
• 1.5
• 1.0
• 0.5

0.0 0.5 1.0 1.5 2.0

σ xy

x/R A=1 (isotropic) A=2.0 A=4.0 A=0.5 A=0.25

Dislocation Dynamics in Anisotropic Materials

[-1 1 0] [-1 -1 2]

• 500
• 250

250 500

• 500
• 400
• 300
• 200
• 100

Full calculation Local only Line tension

b

t=1 ns Stable

[-1 1 0] [-1 -1 2]

• 2000
• 1000

1000 2000

• 3000
• 2500
• 2000
• 1500
• 1000
• 500

500 1000

Full calculation Local only Line tension

t=1 ns

b

t=20 ns

Isotropic A=1 Anisotropic A=0.5

Dislocation Dynamics in Anisotropic Materials No applied load

[-1 1 0] [-1 -1 2]

• 500

500 100 200 300

Stable dipole

A=1 A=2 A=0.5

t=0.5 ns

b

a h 3 25 =

] 01 1 [ 2 1 = b Parallel plane (111) seperated

Dislocation Dynamics in Anisotropic Materials

[0 0 1] [1 0 0] [0 1 0]

A=1 A=2 A=0.5

) 111 ( ] 1 01 [ 2 1

) 1 11 ( ] 101 [ 2 1

and

The University of Iowa The University of Iowa

• L. Sun

Dislocation Interaction with Inclusions

The University of Iowa The University of Iowa

• L. Sun

b

Cijkl Cijkl

*

Dislocation Loops Interacting with Heterogeneous Inclusions

* * 1 1

[ ( ) ]

d ij ijkl ijmn ijmn mnkl kl

S C C C ε ε

− −

= − + −

Ω

=

b

Cijkl Cijkl

* ij

ε

The University of Iowa The University of Iowa

• L. Sun

Dislocation Loops Interacting with Heterogeneous Inclusions

*

( ) ( )

ij ijkl klmn mn

C G σ ε ′ = x x ( ) ( ) ( )

d ij ij ij

σ σ σ ′ = + x x x

b

Cijkl Cijkl

* ij

ε

The University of Iowa The University of Iowa

• L. Sun

Dislocation Loops Interacting with Heterogeneous Inclusions

Cijkl

σ σ σ σ σ σ

ij ij d total ij ij d total m ij d total ij d n

n

( ) ( ) ( )| ( ) ( ) ( ) x x x x x x = + ′ =

− − −

∑ ∑

The University of Iowa The University of Iowa

• L. Sun

Dislocation Loops Interacting with Heterogeneous Inclusions

b

x y z

Dislocation Loop: R=100 b Particle: r=20 b Particle center: (R, 0, 0.4R) Constants of Materials: Em=73 GPa, ν ν ν νm=0.33 Ep=485 GPa, ν ν ν νp=0.20

The University of Iowa The University of Iowa

• L. Sun

Dislocation Loops Interacting with Heterogeneous Inclusions

y

b

x z

0.0 0.3 0.6 0.9 1.2 1.5 0.00 0.03 0.06 0.09 0.12 0.15

σxx (GPa) z/R Without Particle With Particle

0.0 0.3 0.6 0.9 1.2 1.5 0.0 0.1 0.2 0.3 0.4 0.5 0.6

σxz (GPa) z/R Without Particle With Particle

The University of Iowa The University of Iowa

• L. Sun

Dislocation Loops Interacting with Heterogeneous Inclusions

y

b

x z

0.0 0.3 0.6 0.9 1.2 1.5

• 0.02
• 0.01

0.00 0.01 0.02 0.03 0.04

σyy (GPa) z/R Without Particle With Particle

0.0 0.4 0.8 1.2 0.00 0.05 0.10 0.15 0.20 0.25

σzz (GPa) z/R Without Particle With Particle

The University of Iowa The University of Iowa

• L. Sun

Dislocations in Thin Films

The University of Iowa The University of Iowa

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Elastic Fields of Dislocation Loops in the Film-Substrate Systems

( ) ( )

I m i j ijkl km l V

u b n C G dA x

∂

∂ ′ = − ∂

∫

x x , x

Cijkl

I

Cijkl

II

G km ( ) ? ′ = x , x

,

( ) ( )

ij ijkl k l

C u σ = x x

The University of Iowa The University of Iowa

• L. Sun

Elastic Fields of Dislocation Loops in the Film-Substrate Systems

Cijkl

I

Cijkl

II

Cijkl

I

Cijkl

II

Cijkl

I

Cijkl

II

= +

−σ ij

j

n

G km ( ) ′ x , x

Rongved Solution Integral Transforms

The University of Iowa The University of Iowa

• L. Sun

Elastic Fields of Dislocation Loops in the Film-Substrate Systems

X2 X1 X3 X3

'

X2

'

X1

'

b

60O

The University of Iowa The University of Iowa

• L. Sun

Elastic Fields of Dislocation Loops in the Film-Substrate Systems

• 0.4
• 0.2

0.0 0.2 0.4 0.6 0.8 1.0

• 0.2
• 0.1

0.0 0.1 0.2 0.3 0.4 0.5

Ef=85.5GPa, vf=0.31, Es=165.5GPa, vs=0.25 hf=0.1µ

µ µ µm, R=0.25µ µ µ µm, x1=0.0, x2=0.05µ µ µ µm)

σ σ σ σ33hf/b(GPa) X3/hf Half-space Infinite domian Film-substrate Two half-space

Infinite domain Half-space Two half-space Film-substrate

The University of Iowa The University of Iowa

• L. Sun

Elastic Fields of Dislocation Loops in the Film-Substrate Systems

• 0.4
• 0.2

0.0 0.2 0.4 0.6 0.8 1.0

• 0.4
• 0.3
• 0.2
• 0.1

0.0 0.1 0.2 0.3 0.4

Half-space Infinite domain Film-substrate Two half-space X3/hf σ σ σ σ32hf/b(GPa) E

f=85.5GPa, v f=0.31, E s=165.5GPa, vs=0.25

hf=0.1µ

µ µ µm, R=0.25µ µ µ µm, x1=0.0, x2=0.05µ µ µ µm)

• 0.4
• 0.2

0.0 0.2 0.4 0.6 0.8 1.0 0.00 0.05 0.10 0.15 0.20 0.25 0.30

E

f=85.5GPa, v f=0.31, E s=165.5GPa, vs=0.25

hf=0.1µ

µ µ µm, R=0.25µ µ µ µm, x1=0.0, x2=0.05µ µ µ µm)

X3/hf σ σ σ σ31hf/b(GPa) Infinite domain Half-space Film-substrate Two half-space

The University of Iowa The University of Iowa

• L. Sun

Elastic Fields of Dislocation Loops in the Film-Substrate Systems

• 0.4
• 0.2

0.0 0.2 0.4 0.6 0.8 1.0

• 0.35
• 0.30
• 0.25
• 0.20
• 0.15
• 0.10
• 0.05

0.00 0.05

Infinite domain Half-space Film-substrate Two half-space Ef=85.5GPa, vf=0.31, Es=165.5GPa, vs=0.25 hf=0.1µ

µ µ µm, R=0.25µ µ µ µm, x1=0.0, x2=0.05µ µ µ µm)

X3/hf σ σ σ σ11hf/b(GPa)

• 0.4
• 0.2

0.0 0.2 0.4 0.6 0.8 1.0

• 0.5
• 0.4
• 0.3
• 0.2
• 0.1

0.0 0.1 0.2

Infinite domain Half-space Two half-space Film-substrate σ σ σ σ22hf/b(GPa) X3/hf Ef=85.5GPa, vf=0.31, Es=165.5GPa, vs=0.25 hf=0.1µ

µ µ µm, R=0.25µ µ µ µm, x1=0.0, x2=0.05µ µ µ µm)

The University of Iowa The University of Iowa

• L. Sun

Summary

• Dislocation activities in anisotropic materials

(UCLA Team)

• Dislocation interaction with inclusions and precipitates

(Univ. of Iowa Team)

• Dislocation statics in thin film – substrate systems

(Univ. of Iowa Team)