Multiscale Modelling of Nanostructures on Surfaces

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Multiscale Modelling ofNanostructures on Surfaces

• Dimitri D. Vvedensky and Christoph A.
  Haselwandter
• Imperial College London

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Outline

• Multiscale Modelling Quantum Dots
• Lattice Models of Epitaxial Growth
• Exact Langevin Equations on a Lattice
• Continuum Equations of Motion
• Renormalization Group Analysis
• Heteroepitaxial Systems

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Synthesis of Semiconductor Nanostructures
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Structure of Quantum Dots
Georgsson et al. Appl. Phys. Lett. 67,
29812983 (1995)
K. Jacobi, Prog. Surf. Sci. 71,
185215 (2003)
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Stacks of Quantum Dots
Goldman, J. Phys. D 37, R163R178 (2004)
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Theories of Quantum Dot Formation

• Quantum mechanics
• Accurate, but computationally expensive
• Molecular dynamics
• Requires accurate potentials, long simulation
  times
• Statistical mechanics and kinetic theory
• Fast, easy to implement, but need parameters
• Partial differential equations
• Large length and long time scales relation to
  atomic processes?

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Size Matters
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Review Vvedensky, J. Phys Condens. Matter 16,
R1537 (2004)
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Basic Atoms-to-Continuum Method
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EdwardsWilkinson Model
Edwards and Wilkinson, Proc. Roy. Soc. London
Ser. A 381, 17 (1982)
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The Wolf-Villain Model
Clarke and Vvedensky, Phys. Rev. B 37, 6559 (1988)
Wolf and Villain, Europhys. Lett. 13, 389 (1990)
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Coarse-Graining Road Map
renormalization group
Macroscopic equation
Continuum equations
(crossover, scaling, self-organization)
Haselwandter and DDV (2005)
KMC simulations
Lattice Langevin equation
exact
Chua et al. Phys Rev. E (2005)
equivalent analytic
Master Chapman Kolmogorov equations
Lattice rules for growth model
formulation
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Coarse-Graining Road Map
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Renormalization Group Equations
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WolfVillain Model in 1D
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WolfVillain Model in 2D
(f)
(i)
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Analysis of Linear Equation
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Low-Temperature Growth of Ge(001)
Bratland et al., Phys. Rev. B 67, 125322 (2003)

• T 95170 ºC
• F 0.1 ML/s
• DGe 0.6 eV
• tGe hours!

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Model for Quantum Dot Formation
Rb gt Ra Rc gt Ra Rd lt Ra
Ratsch, et al., J. Phys. I (France) 6, 575 (1996)
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KMC Simulations of Quantum Dots

• KMC simulations with
• Random deposition
• Nearest-neighbor hopping
• Detachment barriers calculated
• from Frenkel-Kontorova
• model

Ratsch, et al., J. Phys. I (France) 6, 575 (1996)
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Basic Lattice Model for Quantum Dots

• Random deposition
• Nearest-neighbor hopping
• Total barrier to hopping ED ES nEN
• ES from substrate, EN from each nearest
  neighbor, n 0, 1, 2, 3, or 4
• Detachment barrier a function of height only EN
  EN(h)

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PDE for Quantum Dots
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Numerical Morphology
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Summary, Conclusions, Future Work

• Systematic lattice-to-continuum concurrent
  multiscale method
• Ge(001) mechanism responsible for smooth growth
  early during growth leads to instability at later
  times
• Application to simple model of quantum dot
  formation
• Applications to other models (Poster Christoph
  Haselwandter)
• Submonolayer growth
• Systematic treatment of heteroepitaxy
• More realistic lattice models?