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

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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**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**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**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**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**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**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**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**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.**R(ab) = 0exp[ / (kBT) ]**R(ba) = 0exp[ ( Ea-Eb+ ) / (kBT) ] more general: Ea a Eb b t Et R (ab) exp[ - Ea / (kBT) ] = R (ba) 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**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**deep (local) minima, infrequent events**exclude, e.g., double or multiple jumps: t b a ? o(ab) = o(ba) 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**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, ...**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)**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**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****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**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**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= / **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**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**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**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**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**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**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,...**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