1 Model Hierarchies for Surface Diffusion Martin Burger Johannes Kepler University Linz SFB Numerical-Symbolic-Geometric Scientific Computing Radon Institute for Computational & Applied Mathematics

Surface Diffusion Wien, Feb Outline Introduction Modelling Stages: Atomistic and continuum Small Slopes: Coherent coarse-graining of BCF Joint work with Axel Voigt

Surface Diffusion Wien, Feb Surface diffusion processes appear in various materials science applications, in particular in the (self-assembled) growth of nanostructures Schematic description: particles are deposited on a surface and become adsorbed (adatoms). They diffuse around the surface and can be bound to the surface. Vice versa, unbinding and desorption happens. Introduction

Surface Diffusion Wien, Feb Various fundamental surface growth mechanisms can determine the dynamics, most important: - Attachment / Detachment of atoms to / from surfaces / steps - Diffusion of adatoms on surfaces / along steps, over steps Growth Mechanisms

Surface Diffusion Wien, Feb From Caflisch et. Al Atomistic Models on (Nano-)Surfaces

Surface Diffusion Wien, Feb Other effects influencing dynamics: - Anisotropy - Bulk diffusion of atoms (phase separation) - Elastic Relaxation in the bulk - Surface Stresses - Effects induced by electromagnetic forces Growth Mechanisms

Surface Diffusion Wien, Feb Applications: Nanostructures SiGe/Si Quantum Dots Bauer et. al. 99

Surface Diffusion Wien, Feb Applications: Nanostructures SiGe/Si Quantum Dots

Surface Diffusion Wien, Feb Applications: Nano / Micro Electromigration of voids in electrical circuits Nix et. Al. 92

Surface Diffusion Wien, Feb Applications: Nano / Micro Butterfly shape transition in Ni-based superalloys Colin et. Al. 98

Surface Diffusion Wien, Feb Applications: Macro Formation of Basalt Columns: Giant‘s Causeway Panska Skala (Northern Ireland) (Czech Republic) See:

Surface Diffusion Wien, Feb Standard Description (e.g. Pimpinelli-Villain): - (Free) Atoms hop on surfaces - Coupled with attachment-detachment kinetics for the surface atoms on a crystal lattice - Hopping and binding parameters obtained from quantum energy calculations Atomistic Models on (Nano-)Surfaces

Surface Diffusion Wien, Feb Atomistic simulations (DFT -> MD -> KMC) limited to small / medium scale systems Continuum models for surfaces easy to couple with large scale models Need for Continuum Models

Surface Diffusion Wien, Feb Continuum Surface Diffusion Simple continuum model for surface diffusion in the isotropic case: Normal motion of the surface by minus surface Laplacian of mean curvature Can be derived as limit of Cahn-Hilliard model with degenerate diffusivity Physical conditions for validity difficult to verify

Surface Diffusion Wien, Feb Continuum Surface Diffusion Simple continuum model for surface diffusion in the isotropic case: Normal motion of the surface by minus surface Laplacian of mean curvature Can be derived as limit of Cahn-Hilliard model with degenerate diffusivity Physical conditions for validity difficult to verify

Surface Diffusion Wien, Feb Growth of a surface  with velocity F... Deposition flux, D s.. Diffusion coefficient ... Atomic volume, ... Surface density k... Boltzmann constant, T... Temperature n... Unit outer normal, ... chemical potential Surface Diffusion

Wien, Feb Chemical potential  is the change of energy when adding / removing single atoms In a continuum model, the chemical potential can be represented as a surface gradient of the energy (obtained as the variation of total energy with respect to the surface) For surfaces represented by a graph, the chemical potential is the functional derivative of the energy Chemical Potential

Surface Diffusion Wien, Feb Surface energy is given by Standard model for anisotropic surface free energy Surface Energy

Surface Diffusion Wien, Feb Faceting of Thin Films Anisotropic Surface Diffusion mb-Hausser-Stöcker-Voigt-05

Surface Diffusion Wien, Feb Faceting of Crystals Anisotropic surface diffusion

Surface Diffusion Wien, Feb Parameters (anisotropy, diffusion coefficients,..) not known at continuum level Relation to atomistic models not obvious Several effects not included in standard continuum models: Ehrlich-Schwoebel barriers, nucleation, adatom diffusion, step interaction.. Disadvantages of Continuum Models

Surface Diffusion Wien, Feb Large distance between steps in z-direction Diffusion of adatoms mainly in (x,y)-plane Introduce intermediate model step: continuous in (x,y)-direction, discrete in z-direction Small Slope Approximations

Surface Diffusion Wien, Feb Step Interaction Models To understand continuum limit, start with simple 1D models Steps are described by their position X i and their sign s i (+1 for up or -1 for down) Height of a step equals atomic distance a Step height function

Surface Diffusion Wien, Feb Step Interaction Models Energy models for step interaction, e.g. nearest neighbour only Scaling of height to maximal value 1, relative scale  between x and z, monotone steps

Surface Diffusion Wien, Feb Step Interaction Models Simplest dynamics by direct step interaction Dissipative evolution for X

Surface Diffusion Wien, Feb Continuum Limit Introduce piecewise linear function w N on [0,1] with values X k at z=k/N Energy Evolution

Surface Diffusion Wien, Feb Continuum Height Function Function w is inverse of height function u Continuum equation by change of variables Transport equation in the limit, gradient flow in the Wasserstein metric of probability measures (u equals distribution function)

Surface Diffusion Wien, Feb Continuum Height Function Transport equation in the limit, gradient flow in the Wasserstein metric of probability measures (u equals distribution function) Rigorous convergence to continuum: standard numerical analysis problem Max / Min of the height function do not change (obvious for discrete, maximum principle for continuum). Large flat areas remain flat

Surface Diffusion Wien, Feb Non-monotone Step Trains Treatment with inverse function not possible Models can still be formulated as metric gradient flow on manifolds of measures Manifold defined by structure of the initial value (number of hills and valleys)

Surface Diffusion Wien, Feb BCF Models In practice, more interesting class are BCF- type models (Burton-Cabrera-Frank 54) Micro-scale simulations by level set methods etc (Caflisch et. al ) Simplest BCF-model

Surface Diffusion Wien, Feb Chemical Potential Chemical potential is the difference between adatom density and equilibrium density From equilibrium boundary conditions for adatoms From adatom diffusion equation (stationary)

Surface Diffusion Wien, Feb Continuum Limit Two additional spatial derivatives lead to formal 4-th order limit (Pimpinelli-Villain 97, Krug 2004, Krug-Tonchev-Stoyanov-Pimpinelli 2005) 4-th order equations destroy various properties of the microscale model (flat regions stay never flat, global max / min not conserved..) Is this formal limit correct ?

Surface Diffusion Wien, Feb Continuum Limit Formal 4-th order limit

Surface Diffusion Wien, Feb Gradient Flow Formulation Reformulate BCF-model as dissipative flow Analogous as above, we only need to change metric  appropriate projection operator

Surface Diffusion Wien, Feb Gradient Flow Structure Time-discrete formulation Minimization over manifold for suitable deformation T

Surface Diffusion Wien, Feb Continuum Limit Manifold constraint for continuous time for a velocity V Modified continuum equations

Surface Diffusion Wien, Feb Continuum Limit 4th order vs. modified 4th order

Surface Diffusion Wien, Feb Explicit model for surface diffusion including adatoms Fried-Gurtin 2004, mb 2006 Adatom density , chemical potential , normal velocity V, tangential velocity v, mean curvature , bulk density  Kinetic coefficient b, diffusion coefficient L, deposition term r Example: adatoms

Surface Diffusion Wien, Feb Surface free energy  is a function of the adatom density Chemical potential is the free energy variation Surface energy: Surface Free Energy

Surface Diffusion Wien, Feb Numerical Simulation - Surfaces

Surface Diffusion Wien, Feb Outlook Limiting procedure analogous for more complicated and realistic BCF-models, various effects incorporated in continuum. Direct relation of parameters to BCF models Relation of parameters from BCF to atomistic models Possibility for multiscale schemes: continuum simulation of surface evolution, local atomistic computations of parameters

Surface Diffusion Wien, Feb Download and Contact Papers and Talks: