1. (Some recent results on) Coarse graining of step edge kinetic models Dionisios Margetis MIT, Department of Mathematics Joint work with : Russel E. Caflisch, Axel Voigt IPAM, Lake Arrowhead Meeting December 14, 2005

2. General perspective In traditional settings (classical elasticity, fluids) ``truth’’ is identified with continuum (PDEs etc), by phenomenology. Continuum eqs  (discrete) numerical schemes To compute: Many cases of materials modeling : ``truth’’ is ``discrete’’. Discrete scheme  Continuum Objective: Understand detail transmission across scales; interfaces. Predict large-scale mechanical behavior. Motivation (why continuum?):  Numerical schemes Need : Rich family of examples

3. Macroscale [AFM, Si(001)] 15  m [Blakely,Tanaka, Japan J. Electron Microscopy (1999) ] [STM image of Si(001) steps; B. S. Swartzentruber’s website, Sandia Lab] 25 nm terrace step Nanoscale [STM, Si(001)] void cluster kink Prototypical case: Crystal surface evolution ~1 atom; height a Diffusion: adatom density  i Bc’s Steps as smooth curves (top view): vivi xixi x i+1 … … PDE(s) etc for height z, especially for processes out of equilibrium? z [Burton, Cabrera, Frank, 1951: BCF theory]

4. Some continuum theories for steps PDEs in 1+1 dims from microscopic step motion. [Nozieres, 1987; Rettori, Villain, 1988; Ozdemir, Zangwill, 1990] Free-boundary problems for crystal surfaces. [Spohn, 1993] Variational principle for surface dynamics. [Shenoy, Freund, 2002; Shenoy et. al., 2004] Continuum theories from various microscopic models. [E., Yip, 2001; Margetis, Kohn, submitted] Feature: Evolution near equilibrium, in close correspondence to BCF theory

5. Review: Relaxation (to flatness) near equilibrium; surface diffusion f+f+ f-f- From step velocity law Normal adatom flux From bc’s at step edges mobility tensor chem. potential From definition of  and step energetics;    ~  : step curvature 4 th -order, highly nonlinear; stiff PDE for z Lyapunov function; H -1 grad. flow ``Out-of-equilibrium’’ theory? Near equilibrium

6. Step edge (discrete) kinetic model k r : right kinks k l : left kinks  step edge adatoms  terrace adatoms Densities: w v Velocities: v: step edge (normal to edge) w: kinks (along edge) x Equations of motion for densities: k=k r +k l y  v k r -k l =-tan  + - a along edge [Caflisch et al., 1999; Balykov, Voigt 2005]

7. Constitutive laws Express fluxes and velocities in terms of densities by mean-field theory : edge adatoms to kinks [Caflisch et al., 1999; Balykov, Voigt, 2005] w : kink velocity (along step edge)  : step curvature Polynomial functions; linear with f + + - (Top view of step edge) Edge adatoms to fixed kink site w1w1 Upper-terrace adatoms to kink w2w2 Lower-terrace adatoms to kink w3w3

8. Constitutive laws (cont.): Bc’s at step edges Top view of step edge : From nucleation of kink pairs via attach.-detach. at edge atom From step edge adatoms Directly from kinks From annihilation of kink pairs via attach.-detach. at edge atom [Caflisch et al., 1999; Balykov, Voigt, 2005] Related to Ehrlich-Schwoebel barrier

9. Continuum limit Assumption: Step edges ``closely aligned’’; conveniently, cos  fast variable slow variable Other conditions: Terrace width, d i << length of step density variation f i (s,t)  J ter. (x,t),  i (s,t)   (x,t) etc d i (top view)

10. Continuum equations (formal results) ~ ~ Source: edge adatoms to kinks From step velocity Bc's at step edges: Convec. for kinks: Compare to Compare to: Mean field:  linear in 

11. How is the near-equilibrium continuum approached? Our knowledge is incomplete (e.g., effects of initial data). Attempt via asymptotics: ~ ~

12. Take-Home Messages Continuum theory for out-of-equilibrium step edge kinetics. Variables: Surface height, edge adatom and kink densities. Coupled PDEs. Promising direction: Criteria for near-equilibrium evolution. Issues to be resolved: Energy/variational framework? Evolution of facets w/ microsc. detail. Numerical treatment. Inclusion of elasticity; other effects. Assumptions for continuum limit