GROWTH & RELAXATION OF EPITAXIAL THIN FILMS: ATOMISTIC AND CONTINUUM MODELING

1. GROWTH & RELAXATION OF EPITAXIAL THIN FILMS: ATOMISTIC AND CONTINUUM MODELING Theory & Modeling: Jim Evans, Da-Jiang Liu, Maozhi Li, Kyle Caspersen1, Maria Bartelt2 Experiment: Patricia Thiel, Cynthia Jenks, Conrad Stoldt3, Tony Layson4, Vincent Fournee5 Iowa State University $$$ NSF Grants CHE-0078956 and EEC-0085604 Mound Formation during Ag/Ag(100) Multilayer Growth @ 230K ~0.02ML/s STM vs. Atomistic Modeling (KMC) 50  50 nm2 PRL 85 (2000) 800 PRB 63 (2001) 085401 *PRB 65 (2002) 193407 1UCLA 2LLNL 3U. Colorado 4U. Tulsa 5CNRS-EMN (Nancy)

2. SUBMONOLAYER GROWTH Central Problem: description of the island distribution (sizes & spatial correlations) ANALYTIC THEORIES: rate & diffusion equations 1960’s- ATOMISTIC MODELS & KMC SIMULATION: 1980’s & 1990’s- Irreversible Formation of Square Islands: Deposition of Ag/Ag(100) @ 300 RT-STM @ ISU: 400  400 nm2 Zhang et al. Surf. Sci 406 (1998) 178 OTHER METHODS: (continuum, hybrid, geometric) 1990’s & 2000’s-

3. ANALYTIC THEORIES FOR ISLAND FORMATION RATE EQUATION THEORIES FOR DENSITIES NS OF ISLANDS OF SIZE S: (ZINSMEISTER 1960’S) Key input are CAPTURE NUMBERS describing intrinsic island-size-dependence of the capture of diffusing adatom and thus of island growth SELF-CONSISTENT MEAN-FIELD THEORY (MFT) FOR ADATOM CAPTURE: (VENABLES Phil. Mag. 1973, BALES-CHRZAN PRB 1994). Solve diffusion equation for adatom capture assuming the typical environment of an island is independent of its size FAILURE OF MFT; ALT. EXACT THEORY: (BARTELT-EVANS PRB 54 R17359 1996) Larger islands have larger “capture zones” and thus capture numbers contrasting MFT Island size distribution can be described exactly using non-MF capture numbers BEYOND-MEAN-FIELD-THEORIES: (EVANS & BARTELT in “Morphological Organization…” (World Sci. 1998); PRB 63 235408 2001; MULHERAN-ROBBIE EPL 49 617 2000; AMAR et al. PRL 14 3092 2001; PRB 64 205404 2001) Note: MR analysis of joint probability distribution (JPD) for island sizes and capture zone areas Amar et al. do not account for spatial aspects of nucleation leading to an unphysical JPD ESSENTIAL PHYSICS: SPATIAL ASPECTS OF NUCLEATION (& GROWTH): (EVANS & BARTELT PRB 66 2354xx 2002 15 Dec) Details of island nucleation near capture zone boundaries controls evolution of capture zone (and thus island size) distribution

4. ATOMISTIC MODELING AND KMC SIMULATION “TAILORED” SIMPLE MODELS: (Bartelt & Evans) POINT-ISLAND MODEL (1992): ...islands occupy a single site but carry a size label PRESCRIBED (SQUARE) ISLAND SHAPE (1993): ...aggregating atoms added at perimeter sites according to prescribed growth sequence (Archimedian spiral) …colliding islands growth as overlapping squares “TAILORED” EFFICIENT EDGE-DIFFUSION MODEL: (Caspersen et al. 2001, 2002). Atoms reaching island edges undergo rapid edge diffusion to the nearest kink site - if necessary rounding kink sites. Metal(100) homoepitaxy. “GENERIC” BOND-COUNTING-TYPE MODELS: (Vvedensky, Smilauer, Ratsch, Zangwill,… 1980’s & 1990’s) Rate (or activation energy) for adatom hopping depends on local environment according to simple bond-counting rule . . . 17 18 . . . . . . . . . . . . . . . . .    o   o o o o o

5. OTHER METHODS (Connecting-the-Length-Scales, etc) CONTINUUM TREATMENT OF DIFFUSION & AGGREGATION: (BARTELT et al. PRL 98, PRB 99) …replace atomistic simulation with analysis of BVP for approp. diffusion equn in complex geometry LEVEL-SET FORMULATION OF ISLAND EDGE EVOLUTION: (UCLA NSF-DARPA VIP Initiative, NSF-FRG PRE, PRB,...1998-) …level-set treatment of edge locations facilitates description and propagation of complex morphologies associated with coalescence; ...couple to continuum treatment of diffusion & stochastic trt. of nucleation REFINEMENTS AND HYBRID METHODS: (Schulze, Smereka, E 2002) …continuum trt. of terrace diffusion & atomistic trt. of island edge dynamics GEOMETRIC MODELING: (Li, Bartelt, Evans MRS F2002) Island growth from capture zone (CZ) areas; nucleation along CZ boundaries. Co/Ru(0001)

6. SUBMONOLAYER RELAXATION SINTERING IN TWO-DIMENSIONS ISU-STM (5050 nm2): dumbbell formation and relaxation after corner-to-corner collision of Ag islands on Ag(100) at 300K. Image times are: (a) -36 min.; (b) 10 min.; (c) 91 min.; (d) 271 min. POST-DEPOSITION COARSENING OF ISLAND DISTRIBUTIONS ...driven by minimization of the free energy “cost” of step edges ...Ostwald Ripening (OR) or Smoluchowski Ripening (SR) OR = 2D evaporation of small islands SR= island diffusion & coalescence* *DIFFUSION of LARGE 2D ISLANDS …atomistic mechanism = periphery-diffusion …size-scaling of island diffusivity *ISLAND COALESCENCE (i.e., SHAPE RELAXATION) …atomistic mech = periphery-diffusion (PD) …size-scaling of relaxation time Stoldt et al. PRL 81 (1998) 2950 Liu & Evans PRB 66 (2002) 165407

7. COARSENING: Sav~tn, self-similarity, shape-selection of NS as t OSTWALD RIPENING (n=2/3): mass-transport is diffusion-limited (for homoepitaxy) *Smaller islands (higher curvature, step energy & vapor pressure) dissolve and are incorporated into larger islands (lower vapor pressure) *Island evolution dept. on local environment (non-MF) …approx. described in terms of relative size of just NN SMOLUCHOWSKI RIPENING (n dependent on ) Rate equations effective in describing evolution of size distr. CLUSTER DIFFUSIVITY: D(L) ~ L- where L = linear size = S1/2 *Continuum Langevin equation theory for PD: =3 *Expt. and atomistic simulation for PD: typically  < 3 *Analytic theory for PD: # config=C  CC Master equn. CLUSTER COALESCENCE (SHAPE RELAXATION): RELAXATION TIME(L) ~ Ln where L = characteristic linear size *Continuum Mullins-type theory for PD: n=4 and “pinch-off” possible *Expt. and atomistic simulation: often n < 4, but pinch-off observed

8. MULTILAYER GROWTH Film height: h=h(x, t)  <h>=Ft; Roughness: W=<(h-<h>)2>1/2 ~t (kinetic roughening) 50x50 nm2(a) STM (b) KMC 25ML Ag/Ag(100) @230K ESSENTIAL PHYSICS: Step-edge barriers inhibiting downward transport at island edges produce a “growth instability” (mound formation) Mounds grow in height (roughening) and in lateral dimension (coarsening) MODELING STRATEGIES: ATOMISTIC MODELS and KMC …most studies use idealized model, …but now realistic predictive models (MULTI-) LEVEL-SET (UCLA) or PHASE-FIELD (Karma et al.) PDE for 3D CONTINUUM h(x, t) …valuable conceptual tool

9. MULTILAYER RELAXATION i.e., post-deposition smoothing of kinetically roughened films INTERLAYER TRANSPORT DURING SMOOTHING: (GIESEN & IBACH) Smoothing at 300K of 1ML Ag/Ag(100): Thiel & Evans, J. Phys. Chem. B 104 (2000) SMOOTHING OF ROUGH OR MOUNDED FILM MORPHOLOGIES: CLASSIC MULLINS THEORY ABOVE THERMAL ROUGHENING TRANSITION FUNDAMENTAL CHALLENGE TO DEVELOP THEORY BELOW TRANSITION DUE TO FACETING (theory - Rettori & Villain 1988; KMC - Ramana Murty 2000) NEW HYBRID CONTINUUM-STEP APPROACH: (Israeli & Kandel, PRL 88 2002)