Coarse-Grained Theory of Surface Nanostructure Formation Dimitri D. Vvedensky The Blackett Laboratory, Imperial College London Christoph A. Haselwandter Department of Applied Physics, Caltech

Imperial College London

Outline Systems with many scales The coarse-graining road map Renormalization-group trajectories Transient effects and crossover Experimental realizations Extension to the submonolayer regime

Outline Systems with many scales The coarse-graining road map Renormalization-group trajectories Transient effects and crossover Experimental realizations Extension to the submonolayer regime

Multiscale Physics: Turbulence C. H. K. Williamson, Cornell (Source: Free gliding of delta-wing in water Fluorescent dye illuminated by laser Vortices in near field Turbulence in far field Both panels have same scale Energy cascades from large to small scales

Multiscale Physics: Fracture F. F. Abraham, D. Brodbeck, R. A. Rafey, and W. E. Rudge, Phys. Rev. Lett. 73, 272 (1994). 2D simulation of atom system Bond-breaking at crack tip Dislocation emission blunts crack tip Feed-back between atomic and continuum modes

Multiscale Physics: Critical Phenomena 2D Ising Model J. D. Noh, Chungham National University, Korea

Multiscale “Physics”: Medical Implants B. Kasemo, Surface Science 500, 656 (2002) Time Scale ns µm ms

Outline Systems with many scales The coarse-graining road map Renormalization-group trajectories Transient effects and crossover Experimental realizations Extension to the submonolayer regime

formulation Master & Chapman- Kolmogorov equations Lattice Langevin equation Hierarchy of equations KMC simulations Lattice model Macroscopic equation continuum variables renormalization-group (crossover, scaling, self-organization) analytic stable fixed point Chua et al., PRE 72, (2005), C. A. H. & D. D. V., PRE 76, (2007) Direct analysis/solution C. A. H. & D. D. V. PRL, EPL, PRE (2007, 2008) Coarse-Graining Road Map Continuum equation

Lattice-to-Continuum Method “Atomistic” Continuum Equation

Compare atomistic equation directly to computer simulations? Extract qualitative multiscale surface features via RG analysis… Continuum Equation for Random Deposition/Diffusion

The Multiscale Paradigm M. Scheffler, FHI–Berlin

Outline The coarse-graining road map Renormalization-group trajectories Transient effects and crossover Experimental realizations Extension to the submonolayer regime

Renormalization Group Transformations Real Space Momentum Space Original SystemCoarse GrainingRescaling

Renormalization-Group Equations Points along RG trajectory constitute a hierarchy of equations. RG “weeds out” terms that become irrelevant as the scale is increased, and absorbs their contributions into other terms.

Outline The coarse-graining road map Renormalization-group trajectories Transient effects and crossover Experimental realizations Extension to the submonolayer regime

Stable & Unstable Fixed Points

Initial Conditions & Crossover

Outline The coarse-graining road map Renormalization-group trajectories Transient effects and crossover Experimental realizations Extension to the submonolayer regime

Regimes of Growth D/F>>1. Typical MBE conditions. Initially, conserved Mullins-Herring. Submonolayer regime. D/F ≈ 1. Diffusion noise diminished in importance. Initially, Mullins-Herring. Al on silicone oil (Fang et al., Thin Solid Films 517, 3408 (2009)). D/F<<1. Growth dominated by shot noise.

Growth on Patterned Substrates. 1. H.-C. Kan et al., Phys. Rev. Lett. 92, (2004). KPZ cVLDS VLDS Moun d

Growth on Patterned Substrates. 2. H.-C. Kan et al., Phys. Rev. Lett. 92, (2004). ExperimentKPZcVLDS

Analysis from Initial Conditions

Outline The coarse-graining road map Renormalization-group trajectories Transient effects and crossover Experimental realizations Extension to the submonolayer regime

The Submonolayer Regime

Summary Continuum formulation that retains connection to atomistic processes Unifies a wide range of experimental scenarios Extension to submonolayer regime Large-scale morphologies on patterned substrates

Ongoing and Future Work Coarse-graining magnetohydrodynamics Coarse-grained molecular dynamics