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Multiscale Modelling of Nanostructures on Surfaces

Multiscale Modelling of Nanostructures on Surfaces. Dimitri D. Vvedensky and Christoph A. Haselwandter Imperial College London. Outline. Multiscale Modelling: Quantum Dots Lattice Models of Epitaxial Growth Exact Langevin Equations on a Lattice Continuum Equations of Motion

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Multiscale Modelling of Nanostructures on Surfaces

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  1. Multiscale Modelling ofNanostructures on Surfaces Dimitri D. Vvedensky and Christoph A. Haselwandter Imperial College London

  2. 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

  3. Synthesis of Semiconductor Nanostructures

  4. Structure of Quantum Dots Georgsson et al. Appl. Phys. Lett.67, 2981–2983 (1995) K. Jacobi, Prog. Surf. Sci.71, 185–215 (2003)

  5. Stacks of Quantum Dots Goldman, J. Phys. D37, R163–R178 (2004)

  6. 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?

  7. Size Matters

  8. Review: Vvedensky, J. Phys: Condens. Matter16, R1537 (2004)

  9. Basic Atoms-to-Continuum Method

  10. Edwards–Wilkinson Model Edwards and Wilkinson, Proc. Roy. Soc. London Ser. A381, 17 (1982)

  11. The Wolf-Villain Model Clarke and Vvedensky, Phys. Rev. B37, 6559 (1988) Wolf and Villain, Europhys. Lett.13, 389 (1990)

  12. Coarse-Graining “Road Map” renormalization group Macroscopic equation Continuum equations (crossover, scaling, self-organization) Haselwandter and DDV (2005) Lattice Langevin equation KMC simulations exact Chua et al. Phys Rev. E (2005) equivalent analytic Master & Chapman– Kolmogorov equations Lattice rules for growth model formulation

  13. Coarse-Graining “Road Map”

  14. Renormalization Group Equations

  15. Wolf–Villain Model in 1D

  16. Wolf–Villain Model in 2D (f) (i)

  17. Analysis of Linear Equation

  18. Low-Temperature Growth of Ge(001) Bratland et al., Phys. Rev. B67, 125322 (2003) • T = 95–170 ºC • F = 0.1 ML/s • DGe = 0.6 eV • tGe ≈ hours!

  19. Model for Quantum Dot Formation Rb > Ra Rc > Ra Rd < Ra Ratsch, et al., J. Phys. I (France)6, 575 (1996)

  20. 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)

  21. 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)

  22. PDE for Quantum Dots

  23. Numerical Morphology

  24. 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?

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