Transcript maws2 5438
Multiscale Modelling of Nanostructures on Surfaces Dimitri D. Vvedensky and Christoph A. Haselwandter Imperial College London Quic kTime™ and a TIFF (Unc ompres sed) dec ompres sor are needed to see this pic ture. 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 K. Jacobi, Prog. Surf. Sci. 71, 185–215 (2003) Georgsson et al. Appl. Phys. Lett. 67, 2981–2983 (1995) 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” Continuum equations renormalization group (crossover, scaling, self-organization) Macroscopic equation Haselwandter and DDV (2005) Lattice Langevin equation exact KMC simulations Chua et al. Phys Rev. E (2005) Master & Chapman– Kolmogorov equations equivalent analytic formulation Lattice rules for growth model Coarse-Graining “Road Map” Renormalization Group Equations Wolf–Villain Model in 1D Wolf–Villain Model in 2D (f) (i) 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?