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 Renormalization Group Analysis Heteroepitaxial Systems

Synthesis of Semiconductor Nanostructures

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

Stacks of Quantum Dots Goldman, J. Phys. D 37, R163–R178 (2004)

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?

Size Matters

Review: Vvedensky, J. Phys: Condens. Matter 16, R1537 (2004)

Basic Atoms-to-Continuum Method

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

The Wolf-Villain Model Clarke and Vvedensky, Phys. Rev. B 37, 6559 (1988) Wolf and Villain, Europhys. Lett. 13, 389 (1990)

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

Coarse-Graining “Road Map”

Renormalization Group Equations

Wolf–Villain Model in 1D

Wolf–Villain Model in 2D

Analysis of Linear Equation

Low-Temperature Growth of Ge(001) Bratland et al., Phys. Rev. B 67, 125322 (2003) T = 95–170 ºC F = 0.1 ML/s DGe = 0.6 eV tGe ≈ hours!

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

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)

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)

PDE for Quantum Dots

Numerical Morphology

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?