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Lattice gas models and Kinetic Monte Carlo simulations of epitaxial crystal growth
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Lattice gas models and Kinetic Monte Carlo simulations of epitaxial crystal growth

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  1. Lattice gas models and Kinetic Monte Carlo simulations of epitaxial crystal growth Michael Biehl Michael Biehl Theoretische Physik und Astrophysik & Sonderforschungsbereich 410 Julius-Maximilians-Universität Würzburg Am Hubland, D-97074 Würzburg, Germany http://theorie.physik.uni-wuerzburg.de/~biehl Mathematics and Computing Science Intelligent Systems Rijksuniversiteit Groningen, Postbus 800, NL-9718 DD Groningen, The Netherlands biehl@cs.rug.nl

  2. Lattice gas and Solid-On-Solid (SOS) models • Kinetic Monte Carlo simulations • deposition and transient kinetics • thermally activated processes, Arrhenius dynamics • problems and limitations • Example applications • I) unstable growth, mound formation and coarsening dynamics • II) Atomic Layer Epitaxy (ALE) growth of II-VI(001) systems Outline • Motivation • Non-equilibrium growth - Molecular Beam Epitaxy (MBE) • Theory / modeling / simulation • several levels of theoretical description • Summary

  3. UHV T oven Mikrostrukturlabor, Würzburg Molecular Beam Epitaxy ( MBE ) ultra high vacuum directed deposition of adsorbate material onto a substrate crystal control parameters: substrate/adsorbate materials deposition rate substrate temperature T production of, for instance, high quality ·layered semiconductor devices · magnetic thin films · nano-structures: quantum dots, wires theoretical challenge · clear-cut non-equilibrium situation ·interplay: microscopic processes  macroscopic properties · self-organized phenomena, e.g. dot formation

  4. typical problem: energy/stability of surface reconstructions, preferred arrangement of surface atoms CdTe (001) surface reconstructions Theory / modelling of (growing) surfaces Quantum Mechanics faithful material specific description including electronic properties often: configuration of a few atoms/molecules, unit cells of periodic structures, zero temperature treatment important tool: Density Functional Theory (DFT) description in terms of electron densities

  5. Molecular Dynamics numerical integration of equations of motion + thermal fluctuations effective interactions, e.g. classical short range pair potentials (QM treatment: e.g. Car Parinello method ) microscopic dynamics of particles limited system size and real time (10-6 s) typical problem: dissociation of deposited dimers at the surface, transient mobility of arriving atoms example: diffusion on a surface atomic vibrations ( ~10-12 s) with occasional hops to the next local minimum

  6. Kinetic Monte Carlo (KMC) simulations stochastic dynamics, consider only significant changes of the configuration Solid-On-Solid (SOS) models: exclude bulk vacancies, overhangs, defects, stacking faults, etc. d+1 dim. crystal represented by integer array above d-dim. substrate lattice simplifying lattice gas models: pre-defined lattice of empty / occupied sites hops from site to site

  7. knockout-processes at terrace edges Transient effects upon deposition vac. Deposition of particles, e.g. with flux F = 1 atom / site / s (incoming momentum, attraction to the surface...) incorporation processes, examples: potential energy regular lattice site steering distance from the surface downhill funnelling weakly bound, highly mobile intermediate states

  8. after incorporation: mobile adatoms at surface sites thermally activated processes, simplifying representation: Arrhenius law: waiting time rate energy barrier , e.g. for hopping diffusion  attempt frequency , e.g.

  9. R(ab) = 0exp[  / (kBT) ] R(ba) = 0exp[ ( Ea-Eb+  ) / (kBT) ] more general:  Ea a Eb b t Et R (ab) exp[ - Ea / (kBT) ] = R (ba) exp[ - Eb / (kBT) ] detailed balance condition  stationary P(s)  exp[- Es / (kBT) ] for states of type a,b,... in absence of deposition and desorption: system approaches thermal equilibrium transition states and energy barriers affect „only“ the non-equilibriumdynamics of the system

  10. ES E diffusion bias: adatoms attach to upper terraces preferentially E uphill current favors mound formation an example: Ehrlich-Schwoebel instability additional Schwoebel barrier hinders inter-layer diffusion non-equilibrium, kinetic effect: additional barrier ESis irrelevant for equilibrium properties of the system

  11. deep (local) minima, infrequent events exclude, e.g., double or multiple jumps: t b a ? o(ab) = o(ba) consistent with discretized state space and concept of detailed balance  implicitly used simplifications and assumptions frozen crystal : e.g. single, mobile particle in a static environment, neglect concerted rearrangements of the entire crystal / neighborhood constant prefactor (attempt frequency) - temperature independent - state independent disregard actual shape of the energy landscape transition state theory: correct treatment takes into account entropies / free energies

  12. deposition desorption downward diffusion nucleation islands diffusion edge diffusion some microscopic processes on the growing surface +more: incorporation, knockout attachment to edges / islands detachment processes, ...

  13. 0 · initial configuration of the (SOS) system · catalogue of all relevant processes i=1,2,...n and corresponding Arrhenius rates R1 R2 R3 R3 · pick one of the possible events randomly with probability pi Ri random number · perform the selected event (evaluate physical real time step) ... rates ... · update the catalogue of possible processes and associated energy barriers and rates Rn 1 Kinetic Monte Carlo Simulation (rejection free)

  14. complete catalogue of events ? potentially relevant processes: dimer and island mobility exchange processes / concerted moves e.g. exchange vs. hopping diffusion Problems and limitations material specific input ? direct / indirect experimental measurement calculations/estimates: first principles semi-empirical potentials quantitative match of simulations and experiments

  15. Applications: lattice gas / SOS description: abstract models, further simplifications basic questions example: (universal?) dynamical scaling behavior defects, dislocations ? hetero-epitaxial growth ? strain and other mismatch effects ? material specific simulations realistic lattices or off-lattice simulations interaction potentials, realistic energy barriers particularities of materials / material classes

  16. SOS lattice (e.g. simple cubic) neglect overhangs, defects  knock-out process upon deposition momentum of incoming particles irreversible attachment immobile islands forbidden downward diffusion  high barriers (large enough flux) effective representation of nucleation events single particle picture limited diffusion length for terrace / step edge diffusion  I) Unstable growth: slope selection and coarsening model features / simplifications lsed :characteristic length of step edge diffusion

  17. 256 ML 16 ML 4096 ML • selection of a stable slope: • compensating particle currents • upward (Schwoebel) • downward (knockout) example: slow step edge diffusion (associated length lsed=1 lattice const. ) • coarsening process • merging of mounds driven by • - deposition noise • and/or - step edge diffusion • initial mound formation • due to Schwoebel effect • saturation state • finite system size  single mound

  18. system sizes L = 80 100 125 140 256 512 w / L scaling plot, data collapse =1(slope selection) z=4 =1/4 relatively slow coarsening dynamic scaling behavior time t <h> (film thickness) t  for t< tx growth exponent surface width ~mound height w = roughness exponent wsat  L for t> tx saturation timetx Lzdynamic exponent z= / 

  19. fast sed (lsed  L)  1.00  0.45 sed driven coarsening slow sed (lsed 1)  1.00  0.25 noise assisted coarsening 64ML 128ML additional corner barrier hindered sed, noise assisted coarsening absent sed  0.70 < 1   0.20 absence of slope selection, rough surface 128ML 128ML The role of step edge diffusion (sed) for the morphology and coarsening dynamics

  20. significant step edge diffusion  characteristic exponents:  = 1,   1/3, z  3 for 1 << lsed << L lsed universality:observed in various types of lattices simple cubic (001), body centered cubic (001) simple hexagonal (001), hcp (001) contradicts continuum model predictions:   0.24for cubic lattices   1/3 for all other lattices Ahr, Kinne, Schinzer Siegert, 1998 Moldovan, Golubovic, 2000

  21. example system: II-VI (001) semiconductor surfaces · zincblende lattice, (001)orientation: alternating layers (square lattices) of, e.g., Cd / Te SOS representation, four sub-lattices anisotropic binding structure: y x II) Competing surface reconstructions in non-equilibrium · surface reconstructions observed: - c(2x2), (2x1)vacancy structures Cd-terminated - (2x1)dimerization Te-terminated

  22. simultaneous occupation of NN sites in y-direction, i.e. [1-10], isforbidden (extremely unfavorable) xempty Te Cd electron counting rule, DFT [Neureiter et al., 2000] small difference in surface energies favors checkerboard c(2x2)-order at low temperatures e.g. DFT: E 0.008 eV per site in CdTe[Gundel, private comm.] 0.03 eV ZnSe CdTe (001) properties of Cd-terminated surfaces maximum coverage 50 % two competing vacancy structures: checkerboard or missing rows

  23. beyond SOS weakly bound, highly mobile Te-atoms ( Te* ) on the surface, e.g. at a Cd-site (Te-trimers) bound to a single Cd (neutralizes repulsion) temporary position motivation:coverage 3/2 observed under flux of excess Te allows for Te deposition on a perfect c(2x2) Cd surface time consuming explicit treatment / mean field like Te* reservoir Kinetic Monte Carlo simulations Arrhenius rates for elementary processes o = 1012/s  = o e –/ (kT) choice of parameters: qualitative features, plausibility arguments semi-quantitative comparison, prospective first principle results Te at the surface isotropic N.N. interaction additional Te dimerization

  24. overcome at lower T due to presence of highly mobile, weakly bound Te* : experiment [Faschinger, Sitter] simulation [M. Ahr, T. Volkmann] Atomic Layer Epitaxy (ALE) alternating pulses (1s) of Cd and Te flux: 5ML/s dead time: 0.1s Cd Te Cd Te reconstructions  self-limitation of the growth rate at high temperature

  25. Lattice gas and Solid-On-Solid (SOS) models • Kinetic Monte Carlo simulations • deposition and transient kinetics • thermally activated processes, Arrhenius dynamics • problems and limitations • Example applications • I) unstable growth, mound formation and coarsening dynamics • II) Atomic Layer Epitaxy (ALE) growth of II-VI(001) systems Summary • Motivation • Non-equilibrium growth - Molecular Beam Epitaxy (MBE) • Theory / modeling / simulation • several levels of theoretical description • following talks: continuum descriptions, multi-scale approach,...

  26. Outlook (Wednesday) application of KMC method in off-lattice models treatment of - hetero-epitaxy, mismatched lattices - formation of dislocations - strain-induced island growth - surface alloys of immiscible materials