Some Aspects of Surface Diffusion Martin Burger Institut für Numerische und Angewandte Mathematik, Center for Nonlinear Science CeNoS Westfälische Willhelms-Universität Münster

Some Aspects of Surface Diffusion Outline Introduction: Motivation, Applications of Surface Diffusion Strong anisotropies: Including strong anisotropies, curvature regularization, equilibria, dynamics, numerical simulation Adatom diffusion: Change from 4th order to 2nd order system, change of equilibria, numerical simulation Chemotaxis: limiting behaviour of packed cell densities Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Collaborations Frank Hausser, Christina Stöcker, Axel Voigt (CAESAR Bonn) Christian Schmeiser, Yasmin Dolak-Struss (Universität Wien) Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Introduction 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. Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Growth Mechanisms Various fundamental surface growth mechanisms can determine the dynamics, most important: Attachment / Detachment of atoms to / from surfaces Diffusion of adatoms on surfaces Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Growth Mechanisms Other effects influencing dynamics: Anisotropy Bulk diffusion of atoms (phase separation) Exchange of atoms between surface and bulk Elastic Relaxation in the bulk Surface Stresses Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Growth Mechanisms Other effects influencing dynamics: Deposition of atoms on surfaces Effects induced by electromagnetic forces (Electromigration) Some Aspects of Surface Diffusion Erlangen, February 2007

Isotropic Surface Diffusion Simple 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 (ask Harald Garcke) Some Aspects of Surface Diffusion Erlangen, February 2007

Level Set / Graph Formulation Level set function or graph parametrization u of surface determined from - (graph) (level set) @ t u = ¡ d i v ( P r · ) · = d i v ( r u Q ) P = Q ( I r u ­ ) Q = p 1 + j r u 2 Q = j r u Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Level Set Formulation We have to deal with fourth-order equation, no maximum principle No global level set formulation Efficient computations and proofs still widely open (One of the „major mathematical challenges in materials science“, Jean Taylor, AMS, 2002 / Robert Kohn, SIAM, 2002) Some Aspects of Surface Diffusion Erlangen, February 2007

Applications: Nanostructures SiGe/Si Quantum Dots Bauer et. al. 99 Some Aspects of Surface Diffusion Erlangen, February 2007

Applications: Nanostructures SiGe/Si Quantum Dots Some Aspects of Surface Diffusion Erlangen, February 2007

Applications: Nanostructures InAs/GaAs Quantum Dots Some Aspects of Surface Diffusion Erlangen, February 2007

Applications: Nano / Micro Electromigration of voids in electrical circuits Nix et. Al. 92 Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Applications: Macro Formation of Basalt Columns: Giant‘s Causeway Panska Skala (Northern Ireland) (Czech Republic) See: http://physics.peter-kohlert.de/grinfeld.htmld Some Aspects of Surface Diffusion Erlangen, February 2007

Energy The energy of the system is composed of various terms: Total Energy = (Anisotropic) Surface Energy + (Anisotropic) Elastic Energy + Compositional Energy + ..... We start with first term only Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Surface Energy Surface energy is given by Standard model for surface free energy Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Chemical Potential Chemical potential m 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 Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Surface diffusion appears in many important applications - in particular in material and nano science Growth of a surface G with velocity Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion F ... Deposition flux Ds .. Diffusion coefficient W ... Atomic volume s ... Surface density k ... Boltzmann constant T ... Temperature n ... Unit outer normal m ... Chemical potential = energy variation Some Aspects of Surface Diffusion Erlangen, February 2007

Analysis and Simulation Isotropic / Weakly Anisotropic: Existence results Elliott-Garcke 1996 Numerical simulation Bänsch-Morin-Nocchetto 2003, Deckelnick-Dziuk-Elliott 2004 Anisotropic: Studies of equilibrium structures, Gurtin 1993, Spencer 2003, Cecil-Osher 2004 Numerical simulation of asymptotic model (obtained from long-wave expansion), Golovin-Davies-Nepomnyaschy 2002 / 2003 Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Surface Energy In several situations, the surface free energy (respectively its one-homogeneous extension) is not convex. Nonconvex energies can result from different reasons: Special materials with strong anisotropy: Gjostein 1963, Cahn-Hoffmann1974 Strained Vicinal Surfaces: Shenoy-Freund 2003 Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Surface Energy Effective surface free energy of a compressively strained vicinal surface (Shenoy 2004) Some Aspects of Surface Diffusion Erlangen, February 2007

Curvature Regularization In order to regularize problem (and possibly since higher order terms become important in atomistic homogenization), curvature regularization has beeen proposed by several authors (DiCarlo-Gurtin-Podio-Guidugli 1993, Gurtin-Jabbour 2002, Tersoff, Spencer, Rastelli, Von Kähnel 2003) Some Aspects of Surface Diffusion Erlangen, February 2007

Anisotropic Surface energy Cubic anisotropy surface energy becomes non-convex for e > 1/3 Faceting of the surface Microstructure possible without curvature term Equilibria are local energy minimizers only ° ( n ) = 1 + ² P 4 j Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Chemical Potential We obtain Energy variation corresponds to fourth-order term (due to curvature variation) Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Curvature Term Derivative with matrix Some Aspects of Surface Diffusion Erlangen, February 2007

Minimizing Movement: SD SD can be obtained as the limit (t →0) of minimization subject to Some Aspects of Surface Diffusion Erlangen, February 2007

Minimizing Movement: SD Level set version: subject to Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Numerical Solution Basic idea: Semi-implicit time discretization + Splitting into two / three second-order equations + Finite element discretization in space Natural variables for splitting: Height u, Mean Curvature k, Chemical potential m (denoted u, v, w in the following) Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Time Discretization Based on variational principle, minimizing movement subject to Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Time Discretization Quadratic approximation of the convex terms in the energy, linear approximation of the non-convex terms around u(t) Rewrite local variational problem as minimization over u, v, and w With constraints defining v and w KKT condition yields indefinite linear system, Lagrangian variables are multiples of v and w Some Aspects of Surface Diffusion Erlangen, February 2007

Spatial Discretization Discretization of the variational problem in space by piecewise linear finite elements and P(u) are piecewise constant on the triangularization, all integrals needed for stiffness matrix and right-hand side can be computed exactly Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Discrete Problem After few manipulations we obtain indefinite linear system for the nodal values A stiffness matrix from diffusion coefficient 1/Q B stiffness matrix from diffusion coefficient P/Q M mass matrix for identity, C mass matrix for 1/Q Iterative solution by multigrid-precond. GMRES Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion SD e = 3.5, a = 0.02, 10t = 5 10-5 Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion SD e = 1.5, a = 0.02, 10t = 1.66 10-3 Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Faceting Graph Simulation: mb JCP 04, Level Set Simulation: mb-Hausser-Stöcker-Voigt 06 Adaptive FE grid around zero level set Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Faceting Anisotropic mean curvature flow Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Faceting of Thin Films Anisotropic Mean Curvature Anisotropic Surface Diffusion mb 04, mb-Hausser- Stöcker-Voigt-05 Some Aspects of Surface Diffusion Erlangen, February 2007

Faceting of Bulk Crystals Anisotropic surface diffusion Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Modelling Aspects Standard surface diffusion models have some strange aspects, in particular for nanostructures / epitaxy: No kinetic effects Problems with topology change (atoms can only hop on single surface, not on a second one, even for small distances) They do not correspond to the atomistic picture Some Aspects of Surface Diffusion Erlangen, February 2007

Atomistic Models on (Nano-)Surfaces Standard Description (e.g. Pimpinelli-Villain): (Free) Adatoms hop on surfaces Coupled with attachment detachment kinetics for the surface atoms on a crystal lattice Some Aspects of Surface Diffusion Erlangen, February 2007

Atomistic Models on (Nano-)Surfaces From Caflisch et. Al. 1999 Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Modelling Need two equations for two coupled processes Need diffusion equation for adatoms Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Modelling Explicit model for surface diffusion including adatoms Fried-Gurtin 2004, mb 2006 Adatom density d, chemical potential m, normal velocity V, tangential velocity v, mean curvature k, bulk density r Kinetic coefficient b, diffusion coefficient L, deposition term r Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Surface Free Energy Surface free energy y is a function of the adatom density Chemical potential is the free energy variation Surface energy: Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Modelling Relation to standard surface diffusion: convergence as the cost of free adatoms (in the surface free energy tends to infinity) Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Equilibrium Shapes Equilibrium shapes minimize the surface energy at constant mass Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Equilibrium Shapes Equilibrium films: minimum at vanishing adatom density, flat surface. Same as without adatoms. Equilibrium crystals: Wulff shape with vanishing adatom density is NEVER an equilibrium ! Isotropic equilibrium has nonzero adatom density and smaller radius than Wulff shape Some Aspects of Surface Diffusion Erlangen, February 2007

Equilibrium Crystals (Isotropic) Model free energy Parameter g measures the cost of free adatoms Some Aspects of Surface Diffusion Erlangen, February 2007

Equilibrium Crystals (Isotropic) Equilibrium radius Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Surface Energy Different regimes for surface energy: Convex for small adatom densities and shapes close to equilibrium Nonconvex for large adatom densities and shapes far away from equilibrium. The surface energy is consequently not lower semicontinuous Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Numerical Simulation Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Numerical Simulation Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Numerical Simulation Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Numerical Simulation Flat initial shape, nonhomogeneous deposition Some Aspects of Surface Diffusion Erlangen, February 2007

Numerical Simulation - Surfaces Some Aspects of Surface Diffusion Erlangen, February 2007

Numerical Simulation - Surfaces Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Chemotaxis Keller-Segel Model with small diffusion and logistic sensitivity Sensitivity function for quorum sensing derived by Painter and Hillen 2003 from microscopic model Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Chemotaxis Keller-Segel Model with small diffusion and logistic sensitivity: Plateau formation Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Chemotaxis Keller-Segel Model with small diffusion and logistic sensitivity: Plateau motion Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Chemotaxis Keller-Segel Model with small diffusion and logistic sensitivity Asymptotics at hyperbolic time-scale Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Chemotaxis Limit is a nonlinear, nonlocal conservation law: we need entropy solutions Entropy inequality Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Chemotaxis Stationary solutions These are entropy solutions iff Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Chemotaxis Asymptotics for large time by time rescaling Look for limiting solutions Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Chemotaxis Asymptotic expansion in interfacial layer (as for Cahn-Hilliard) Note: entropy condition Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Chemotaxis We obtain a surface diffusion law with diffusivity and chemical potential Corresponding energy functional D = ¡ 2 @ n S ¹ = ¡ S 2 [ ­ ] Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Chemotaxis Flow is volume conserving Flow has energy dissipation property Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Chemotaxis Stability of stationary solutions can be studied based on second (shape) variations on the energy functional Stability condition for normal perturbation Instability without entropy condition ! Otherwise high-frequency stability, possible low-frequency instability Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Chemotaxis Instability Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Chemotaxis Instability Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Chemotaxis Instability Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Chemotaxis Surface Diffusion Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Chemotaxis Surface Diffusion Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Chemotaxis Surface Diffusion, 3D Some Aspects of Surface Diffusion Erlangen, February 2007

Some Aspects of Surface Diffusion Download and Contact Papers and Talks: Anisotropy: mb, JCP 2005 mb-Hausser-Stöcker-Voigt JCP 2007 Adatoms: mb, Comm. Math. Sci. 2006 Chemotaxis: mb-DiFrancesco-DolakStruss, SIMA 2007 mb-DolakStruss-Schmeiser, Preprint, 2006 www.math.uni-muenster.de/u/burger e-mail: martin.burger@uni-muenster.de Some Aspects of Surface Diffusion Erlangen, February 2007