growth and pattern formation for epitaxial surfacesatomistic modeling of strain in thin films •...

CSCAMM, 4/24/2007

Growth and Pattern Formationfor Epitaxial Surfaces

Russel Caflisch

Mathematics DepartmentDepartment of Materials Science & Engineering

UCLA

CSCAMM, 4/24/2007

Collaborators

• UCLA:Youri Bae (U Arizona)Sunmi Lee (Ntl. Inst. Math. Sci., Korea)Cameron Connell (Citigroup)Young-Ju Lee (→ Rutgers)Christian Ratsch

• Xiangtan University, ChinaShi ShuYingxiong Xiao

• HRL Labs:Mark GyureGeoff Simms

• Imperial College:Dimitri VvedenskyAlexander Schindler

• Penn State Jinchao Xu

Support from NSF and from FENA (MARCO Center) www.math.ucla.edu/~material

CSCAMM, 4/24/2007

Outline

• Strain in epitaxial systems – Leads to structure– Quantum dots and their arrays

• Atomistic strain model– Lattice statics model– Lattice mismatch

• Numerical methods– Algebraic multigrid (AMG)– Artificial boundary conditions (ABC)

• Application to nanowires and nanocrystals– Step bunching instability

• Summary

CSCAMM, 4/24/2007

Strain in Epitaxial Systems

• Lattice mismatch leads to strain– Heteroepitaxy– E.g., Ge/Si has 4% lattice mismatch

• Relief of strain energy can lead to geometric structures– Quantum dots and q dot arrays

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Quantum dots and Q Dot Arrays

Ge/Si, Mo et al. PRL 1990

Si.25Ge.75/Si, (5 μm)2

MRSEC, U Wisconsin

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Directed Self-Assembly of Quantum Dots

B. Lita et al. (Goldman group), APL 74, (1999) H. J. Kim, Z. M. Zhao, Y. H. Xie, PRB 68, (2003).

In both systems strain leads to ordering!

AlxGa1-xAs system GeSi system

•Vertical allignment of q dots in epitaxial overgrowth (left)• Control of q dot growth over mesh of buried dislocation lines (right)

CSCAMM, 4/24/2007

Outline

• Strain in epitaxial systems– Leads to structure– Quantum dots and their arrays




• Summary

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Atomistic Modeling of Strain in Thin Films• Lattice statics for discrete atomistic system,

– minimize discrete strain energy (Born & Huang, 1954)– Application to epitaxial films, e.g.

• E.g., Stewart, Pohland & Gibson (1994), Orr, Kessler, Snyder & Sander (1992),

• Idealizations – Harmonic potentials, Simple cubic lattice– General, qualitative properties

• Independent of system parameters– Computational speed enable additional physics & geometry

• 3D, alloying, surface stress• Atomistic vs. continuum

– atomistic scale required for thin layer morphology• strain at steps

– continuum scale required for efficiency• KMC requires small time steps, frequent updates of strain field

CSCAMM, 4/24/2007

Microscopic Model of Elasticitywith Harmonic Potentials

• Continuum Energy density– isotropic

– cubic symmetry

• Atomistic Energy density – Nearest neighbor springs

– Diagonal springs

– Bond bending terms

• Elastic equations ∂u E [u] = 0

2 2( )xx yyE k S S= +

2 2( 2 ) ( 2 )xx xy yy xx xy yyE S S S S S S= + + + − +l l

2xyE mS=

2 2 22( ) ( 2 )xx yy xx yy xyE S S S S Sλ μ= + + + +2 2 2( )xx yy xy xx yyE S S S S Sα β γ= + + +

CSCAMM, 4/24/2007

Strain in an Epitaxial FilmDue to Lattice Mismatch

• lattice mismatch– lattice constant in film a– lattice constant in substrate h– relative lattice mismatch ε=(a-h)/h

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Deformation of Surface due to Intrinsic Surface Stress

film misfitNo misfit in film

Surface stress included by variation of lattice constant for surface atoms

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Strain TensorStep with No Surface Stress

0 5 10 15 20 25 30 350

10

20

30

40

50

60

0 5 10 15 20 25 30 350

10

20

30

40

50

60

0 5 10 15 20 25 30 350

10

20

30

40

50

60

Sxx SxySyy

CSCAMM, 4/24/2007

Strain TensorStep with Surface Stress

No lattice mismatch

0 5 10 15 20 25 30 350

10

20

30

40

50

60

0 5 10 15 20 25 30 350

10

20

30

40

50

60

0 5 10 15 20 25 30 350

10

20

30

40

50

60

Sxx SxySyy

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Interaction of Surface Steps

• Steps of like “sign”– Lattice mismatch → step attraction– Surface stress → step repulsion

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Energy vs. Step Separation

0 10 20 30 40 50 602.156

2.1565

2.157

2.1575

2.158

2.1585

2.159

2.1595ε = 0.015, with intrinsic tension

Tot

al E

nerg

y

separation of steps. solid line for the force with only monopole, o for the lattice elastic 0 10 20 30 40 50 60

1.3289

1.329

1.3291

1.3292

1.3293

1.3294

1.3295

1.3296

1.3297ε = 0.01, with intrinsic tension

Tot

al E

nerg

y

separation of steps.

Step attraction due to lattice mismatch

Repulsion of nearby steps due to surface stress

CSCAMM, 4/24/2007

Outline





• Summary

CSCAMM, 4/24/2007

Numerical method for Discrete Strain Equations

• Algebraic multigrid with PCG• Artificial boundary conditions at top of substrate

– Exact for discrete equations

• 2D and 3D, MG and ABC combined• Russo & Smereka (JCP 2006), Lee, REC & Lee (SIAP 2006), REC, et al. (JCP

2006)

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Multigrid

CPU speed (sec) vs. lattice size for strain computation in a 2D quantum dot system

100 x 155 150 x 205 200 x 2550

10

20

30

40

50

60

CP

U ti

me

in S

econ

ds

CPU time comparisons

Super LU

ILU(0)−CG

AMG−V

AMG−CG

(a) (b)

(c) (d)

Strain energy density for 160 atom wide pyramid in 2D with trenches, for various trench depths

CSCAMM, 4/24/2007

Artificial Boundary Conditions

ΩΩ

Ω

Γ

Γ

Γ1 1

2

1

2

0 20 40 60 80 100 1202.3

2.35

2.4

2.45

2.5

2.55

NC

Tot

al E

nerg

y

Total Energies versus Thickness of Substrates 2D

2 4 6 8 10 12 142.41

2.415

2.42

2.425

2.43

2.435

2.44Total Energies versus Thickness of Substrates 3D

NC

Tot

al E

nerg

y

ABCZero BC

ABCZero BC

CSCAMM, 4/24/2007

Outline




• Application to nanowiresand nanocrystals– Step bunching instability

• Summary

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Nanowires

• Growth catalyzed by metal cluster (Au, Ti, …)• Epitaxial• Application to nano-electronics• Stability difficulties

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Ti-Nucleated Si NanowiresKamins, Li & Williams, APL 2003

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Nanowire

Gudiksen, Wang & Lieber. JPhysChem B 2001

•InP wire•20nm Au clusterat tip•Scale bar =5 nm•Oxide coating,Not present during growth•TEM

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Instability in Metal Catalyzed Growth of Nanowires

• Epitaxial structure– Tapered shape due to side

attachment• Instability at high

temperature– Tapered shape → terraced

shape– Step bunching

Kamins, Li & Williams, APL 2003

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2D Simulation of Nanowires

• 2 steps – looking for step bunching by energy minimization

• Model– Harmonic potential– Surface stress

• No step bunching in 2D 0 20 40 60 80 100 120 140 160 180 200

8.775

8.78

8.785

8.79

8.795

8.8

8.805

8.81

8.815

Tot

al E

nerg

y

separation between two steps (in atom unit)

ε = 0.05, width = 16, 12

Strain Energy vs. distance between steps

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3D Simulation of Interactionbetween Steps on Nanowires

• Homogeneous, epitaxial nanowire with surface stress

• Interactions of two steps – r = R1 for z<z1– r = R2 for z1 < z < z2– r = R3 for z2 < z – L= z2-z1 = inter-step distance– z = axial distance, r = wire radius

• Energy minimum occurs for small L– Step bunching

• Results are insensitive to parameters– Step size (R1 – R2 or R2 – R3)– Surface stress– Wire radius, shape

• Lowest value of energy E occurs for small value of separation L

– System prefers bunched steps

L

E

(R1 , R2 , R3) = (3,4,5)

5 10 15 20 25 30 35 40

20.6

20.7

20.8

20.9

21

21.1

21.2

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Nanocrystals

• Sphere or rod (diameter =10 - 50 nm)• Coated by shell (thickness = 1 – 10 nm)• Epitaxial structure• Wide range of new properties and

applications• Difficulty with instability of shell due to

strain

CSCAMM, 4/24/2007

Epitaxial Nanocrytalscore/shell=CdSe/CdS Rcore=34Å, Rshell=9Å

Peng, Schlamp, Kadavanich, Alivisatos. JACS 1997

HRTEM

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Epitaxial Nanorodscore/shell=CdS/ZnS

Manna,Sher, Li, Alivisatos. JACS 2002

•Left to rightIncreasing shell thickness•Epitaxial structure breaks down at larger shell size•HRTEM

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Simulation of Strain Fieldin Nanocrystals

• Core 0 < r < Rcore• Shell Rcore < r < Rshell• Strain model

– Harmonic potential– Equal elastic parameters– lattice mismatch– No surface stress

• Max energy density occurs at critical shell thickness• Critical shell thickness is at peak in photoluminescence• Robust results: same in 2D, variation of parameters

CSCAMM, 4/24/2007

Energy Density for 3D NanocrystalShowing Critical Shell Thickness

1 2 3 4 5 6 7

0.011

0.012

0.013

0.014

0.015

0.016

0.017

0.018

0.019

0.02

(Rc,Rs)=(8,1)Rslice = 4



Rs

Rc=8

Em

E onlyin core

E incore, shell

E incore, shell

E onlyin shell

• Plots of E for 3 values of Rs– slice containing max of E– subcritical (Rs =1), – critical (Rs =2) – supercritical (Rs=7)

• Graph of Em vs. Rs• E=energy density, Em=max(E)• (Rc, Rs) = (core, shell) thickness

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Comparison to Critical Size for Photoluminescence

Peng, Schlamp, Kadavanich, Alivisatos. JACS 1997

Peak in photoluminescence is at same shell thickness as peak in elastic energy density

CSCAMM, 4/24/2007

Outline





• Summary

CSCAMM, 4/24/2007

Summary

• Strain model– Harmonic potential– Minimal stencil– Surface stress represented by variation in lattice constant

• Numerical methods– AMG– ABC

• Nanowires– Surface stress– No step bunching in 2D– Step bunching in 3D

growth and pattern formation for epitaxial surfacesatomistic modeling of strain in thin films •...

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