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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
CSCAMM, 4/24/2007
Quantum dots and Q Dot Arrays
Ge/Si, Mo et al. PRL 1990
Si.25Ge.75/Si, (5 μm)2
MRSEC, U Wisconsin
CSCAMM, 4/24/2007
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
• 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
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
CSCAMM, 4/24/2007
Deformation of Surface due to Intrinsic Surface Stress
film misfitNo misfit in film
Surface stress included by variation of lattice constant for surface atoms
CSCAMM, 4/24/2007
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
CSCAMM, 4/24/2007
Interaction of Surface Steps
• Steps of like “sign”– Lattice mismatch → step attraction– Surface stress → step repulsion
CSCAMM, 4/24/2007
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
• 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
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)
CSCAMM, 4/24/2007
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
• 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 nanowiresand nanocrystals– Step bunching instability
• Summary
CSCAMM, 4/24/2007
Nanowires
• Growth catalyzed by metal cluster (Au, Ti, …)• Epitaxial• Application to nano-electronics• Stability difficulties
CSCAMM, 4/24/2007
Ti-Nucleated Si NanowiresKamins, Li & Williams, APL 2003
CSCAMM, 4/24/2007
Nanowire
Gudiksen, Wang & Lieber. JPhysChem B 2001
•InP wire•20nm Au clusterat tip•Scale bar =5 nm•Oxide coating,Not present during growth•TEM
CSCAMM, 4/24/2007
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
CSCAMM, 4/24/2007
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
CSCAMM, 4/24/2007
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
CSCAMM, 4/24/2007
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
CSCAMM, 4/24/2007
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
CSCAMM, 4/24/2007
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
(Rc,Rs)=(8,2)Rslice = 5
(Rc,Rs)=(8,7)Rslice = 10
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
CSCAMM, 4/24/2007
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
• 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
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
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