1. Multiscale Modelling of Nanostructures 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. D 37, 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. Matter 16, R1537 (2004)

9. Basic Atoms-to-Continuum Method

10. Edwards–Wilkinson Model
Edwards and Wilkinson, Proc. Roy. Soc. London Ser. A 381, 17 (1982)

11. The Wolf-Villain Model
Clarke and Vvedensky, Phys. Rev. B 37, 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

17. Analysis of Linear Equation

18. Low-Temperature Growth of Ge(001)
Bratland et al., Phys. Rev. B 67, (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?