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

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 L. Sun The University of Iowa The University of Iowa

NSF-DMR 0113172/0113555 Collaborative ITR/AP Research on Large-scale Dislocation Dynamics Simulations for Computational Design of Thin Film Systems (September 2001 – August 2004) 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) L. Sun The University of Iowa The University of Iowa

Materials Modeling in Terms of Length Scales • Ab initio method • Molecular dynamics method • Dislocation dynamics method • Finite element method L. Sun The University of Iowa The University of Iowa

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. L. Sun The University of Iowa The University of Iowa

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. L. Sun The University of Iowa The University of Iowa

Presentation Outline 1. Anisotropic Dislocation Dynamics Approach 2. Dislocation Interaction with Inclusions 3. Dislocations in Thin Films L. Sun The University of Iowa The University of Iowa

Elastic Fields of Dislocation Loops in Anisotropic Materials Mura formula: b σ = β ( ) x C ( ) x ij ijkl lk ν 1 d ∫ β = ∈ ! ( ) ( h ) ( ) x C b I m,n dl π ji jnh klmn m lik 2 8 dl r C ijkl L π  − φ + φ − φ + φ  2 φ ( sin cos ) ( sin cos ) ( ) m N m n cos N m n N n ∫ = − − φ  l ik ik ik  ( ) [ ] I m,n n d − φ + φ φ − φ + φ lik l  ( sin cos ) sin ( sin cos ) ( )  D m n D m n D n 0 L. Sun The University of Iowa The University of Iowa

Dislocation Dynamics in Anisotropic Materials • Equation of Dislocation Motion (variational waek form) ( ) ∫ − δ = t 0 f B V r ds i ij j i Γ ! f : Forces (external, internal , self force, and friction resistant (Perierls) force, etc.) ! B : Resistive matrix (inverse mobility) ! V : Dislocation movement

Dislocation Dynamics in Anisotropic Materials ! Self-force (Barnett) = κ − κ + εκ − + ( ) [ ( ) " ( )] ln[ 8 / )] ( , ) F E t E t E t J L P F core Line Tension Stretch Non-local = κ − σ + σ ( ) 0 . 5 [ ( ) ( )] F E t P P b n P 1 1 2 ij ij i j P P 2

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 z [111] Anisotropic Ratio A=2C 44 /(C 11 -C 12 ) x [-110] A=1 (isotropic) 2.0 A=2.0 A=1 (isotropic) 3 1.5 A=4.0 A=2.0 2 A=0.5 A=4.0 1.0 A=0.25 A=0.5 0.5 1 A=0.25 0.0 σ xy 0 σ zx -0.5 -1 -1.0 -2 -1.5 -3 -2.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 0.0 0.5 1.0 1.5 2.0 2.5 3.0 x/R x/R Stress σ σ (divided by 0.5(C 11 -C 12 ) b/R) σ σ

Dislocation Dynamics in Anisotropic Materials 0 1000 Full calculation Full calculation Local only Local only Line tension 500 Line tension -100 t=1 ns 0 t=1 ns -200 -500 [-1 -1 2] [-1 -1 2] -1000 -300 Stable t=20 ns -1500 -400 -2000 b -2500 b -500 -3000 -500 -250 0 250 500 -2000 -1000 0 1000 2000 [-1 1 0] [-1 1 0] Anisotropic A=0.5 Isotropic A=1

Dislocation Dynamics in Anisotropic Materials 300 A=1 A=2 1 = A=0.5 [ 1 01 ] b t=0.5 ns 2 200 Parallel plane (111) seperated b [-1 -1 2] Stable dipole = 25 3 h a 100 No applied load 0 -500 0 500 [-1 1 0]

Dislocation Dynamics in Anisotropic Materials A=1 A=2 A=0.5 [0 0 1] [1 0 0] [0 1 0] 1 1 [ 101 ] ( 11 1 ) [ 01 1 ] ( 111 ) and 2 2

Dislocation Interaction with Inclusions L. Sun The University of Iowa The University of Iowa

Dislocation Loops Interacting with Heterogeneous Inclusions b b = * C ijkl C ijkl ε * Ω ij C ijkl C ijkl ε = − + − − − ε * * 1 1 d [ ( ) ] S C C C ij ijkl ijmn ijmn mnkl kl L. Sun The University of Iowa The University of Iowa

Dislocation Loops Interacting with Heterogeneous Inclusions b σ ′ σ = σ + d ( ) ( ) ( ) x x x ij ij ij C ijkl ′ σ = ε * ( ) ( ) x C G x ij ijkl klmn mn ε * ij C ijkl L. Sun The University of Iowa The University of Iowa

Dislocation Loops Interacting with Heterogeneous Inclusions − − σ = σ + ∑ σ σ ′ d total d total ( ) ( ) ( )| ( ) x x x x ij ij ij ij m − σ = ∑ σ d total d ( ) ( ) x x n ij ij n C ijkl L. Sun The University of Iowa The University of Iowa

Dislocation Loops Interacting with Heterogeneous Inclusions z Dislocation Loop: R=100 b Particle: r=20 b b y Particle center: (R, 0, 0.4R) x Constants of Materials: E m =73 GPa, ν ν ν ν m =0.33 E p =485 GPa, ν ν ν ν p =0.20 L. Sun The University of Iowa The University of Iowa

Dislocation Loops Interacting with Heterogeneous Inclusions z 0.15 0.6 Without Particle Without Particle 0.5 0.12 With Particle With Particle b y 0.4 0.09 σ xx (GPa) σ xz (GPa) 0.3 0.06 0.2 x 0.03 0.1 0.00 0.0 0.0 0.3 0.6 0.9 1.2 1.5 0.0 0.3 0.6 0.9 1.2 1.5 z/R z/R L. Sun The University of Iowa The University of Iowa

Dislocation Loops Interacting with Heterogeneous Inclusions z 0.04 0.25 Without Particle Without Particle 0.03 With Particle With Particle 0.20 0.02 0.15 σ yy (GPa) b y σ zz (GPa) 0.01 0.10 0.00 0.05 -0.01 x -0.02 0.00 0.0 0.3 0.6 0.9 1.2 1.5 0.0 0.4 0.8 1.2 z/R z/R L. Sun The University of Iowa The University of Iowa

Dislocations in Thin Films L. Sun The University of Iowa The University of Iowa

Elastic Fields of Dislocation Loops in the Film-Substrate Systems σ = I ( ) ( ) x x C ijkl C u , ij ijkl k l II C ijkl ∂ ∫ ′ = − I ( x ) ( x , x ) u b n C G dA ∂ m i j ijkl km x ∂ l V ′ = G km ( ) ? x , x L. Sun The University of Iowa The University of Iowa

Elastic Fields of Dislocation Loops in the Film-Substrate Systems − σ ij n j I I I C ijkl C ijkl C ijkl + = II II II C ijkl C ijkl C ijkl Rongved Integral ′ G km ( ) x , x Solution Transforms L. Sun The University of Iowa The University of Iowa

Elastic Fields of Dislocation Loops in the Film-Substrate Systems X 3 ' X 3 ' X 1 ' X 2 b X 2 60 O X 1 L. Sun The University of Iowa The University of Iowa

Elastic Fields of Dislocation Loops in the Film-Substrate Systems E f =85.5GPa, v f =0.31, E s =165.5GPa, v s =0.25 h f =0.1 µ µ µ m, R=0.25 µ µ µ m, x 1 =0.0, x 2 =0.05 µ µ µ µ µ µ m ) 0.5 Half-space 0.4 Infinite domian Film-substrate Half-space Film-substrate Two half-space 0.3 σ 33 h f /b(GPa) 0.2 0.1 σ σ σ 0.0 -0.1 Infinite domain -0.2 Two half-space -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 X 3 /h f L. Sun The University of Iowa The University of Iowa

Elastic Fields of Dislocation Loops in the Film-Substrate Systems E f =85.5GPa, v f =0.31, E s =165.5GPa, v s =0.25 E f =85.5GPa, v f =0.31, E s =165.5GPa, v s =0.25 h f =0.1 µ µ µ µ m, R=0.25 µ µ µ m, x 1 =0.0, x 2 =0.05 µ µ µ µ µ m ) h f =0.1 µ µ µ µ m, R=0.25 µ µ µ µ m, x 1 =0.0, x 2 =0.05 µ µ µ µ m ) 0.4 0.30 Infinite domain 0.3 0.25 Half-space Film-substrate 0.2 Two half-space 0.20 0.1 σ 32 h f /b(GPa) σ 31 h f /b(GPa) 0.0 0.15 -0.1 σ σ σ 0.10 σ σ σ Half-space -0.2 Infinite domain 0.05 Film-substrate -0.3 Two half-space -0.4 0.00 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 X 3 /h f X 3 /h f L. Sun The University of Iowa The University of Iowa