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

Outline The problem: surface nanostructures The coarse-graining road map Renormalization-group trajectories Transient effects and crossover Experimental realizations Summary

Outline The problem: surface nanostructures The coarse-graining road map Renormalization-group trajectories Transient effects and crossover Experimental realizations Summary

Fundamental Processes during Epitaxy Ratsch and Venables, J. Vac. Sci. Technol. A 21 S96 (2003) (a)deposition (b)diffusion (c)nucleation (d)attachment (e)detachment (f)edge diffusion (g)downward hops (h)second-layer nucleation (i)break-up

Growth of SiGe Quantum Dots Ge quantum dots on Si (100) 1600 Å  1600  Å Continuous scanning by STM Courtesy Bert Voigtländer, KFA Jülich

Self-Limiting Growth of QDs Kapon et al., Physica E 25, 288 (2004)

Outline The problem: surface nanostructures The coarse-graining road map Renormalization-group trajectories Transient effects and crossover Experimental realizations Summary

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

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

Outline The problem: surface nanostructures The coarse-graining road map Renormalization-group trajectories Transient effects and crossover Experimental realizations Summary

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.

Stable & Unstable Fixed Points

Outline The problem: surface nanostructures The coarse-graining road map Renormalization-group trajectories Transient effects and crossover Experimental realizations Summary

Initial Conditions & Crossover

Outline The problem: surface nanostructures The coarse-graining road map Renormalization-group trajectories Transient effects and crossover Experimental realizations Summary

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

Kardar–Parisi–Zhang (KPZ) equation Some Growth Equations Villain–Lai–Das Sarma (VLDS) equation conservative Villain–Lai–Das Sarma (cVLDS) equation

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

Analysis from Initial Conditions

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