Lattice gas models and Kinetic Monte Carlo simulations of epitaxial crystal growth Theoretische Physik und Astrophysik & Sonderforschungsbereich 410 Julius-Maximilians-Universität Würzburg Am Hubland, D Würzburg, Germany Mathematics and Computing Science Intelligent Systems Rijksuniversiteit Groningen, Postbus 800, NL-9718 DD Groningen, The Netherlands Michael Biehl

Outline Motivation Non-equilibrium growth - Molecular Beam Epitaxy (MBE) Theory / modeling / simulation several levels of theoretical description Summary 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

Molecular Beam Epitaxy ( MBE ) control parameters: substrate/adsorbate materials deposition rate substrate temperature T ultra high vacuum directed deposition of adsorbate material onto a substrate crystal 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 Mikrostrukturlabor, Würzburg oven UHV T

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 typical problem: energy/stability of surface reconstructions, preferred arrangement of surface atoms CdTe (001) surface reconstructions

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 (  s ) example: diffusion on a surface atomic vibrations ( ~ s ) with occasional hops to the next local minimum typical problem: dissociation of deposited dimers at the surface, transient mobility of arriving atoms

Kinetic Monte Carlo (KMC) simulations stochastic dynamics, consider only significant changes of the configuration simplifying lattice gas models: pre-defined lattice of empty / occupied sites hops from site to site 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

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

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

R (a  b) =  0 exp[  / (k B T) ] R (b  a) =  0 exp[ ( E a -E b +  ) / (k B T) ] a b EbEb EaEa  more general: transition states and energy barriers  affect „only“ the non-equilibrium dynamics of the system EtEt t R (a  b) exp[ - E a / (k B T) ] = R (b  a) exp[ - E b / (k B T) ] detailed balance condition  stationary P(s)  exp[- E s / (k B T) ] for states of type a,b,... in absence of deposition and desorption: system approaches thermal equilibrium

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

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

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

Kinetic Monte Carlo Simulation (rejection free) · perform the selected event (evaluate physical real time step) · perform the selected event (evaluate physical real time step) · initial configuration of the (SOS) system · catalogue of all relevant processes i=1,2,...n and corresponding Arrhenius rates · initial configuration of the (SOS) system · catalogue of all relevant processes i=1,2,...n and corresponding Arrhenius rates R1R1 R2R2 R3R3 RnRn... rates... · pick one of the possible events randomly with probability p i  R i · pick one of the possible events randomly with probability p i  R i 0 1 random number · update the catalogue of possible processes and associated energy barriers and rates · update the catalogue of possible processes and associated energy barriers and rates R3R3

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

lattice gas / SOS description: 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 Applications: abstract models, further simplifications basic questions example: (universal?) dynamical scaling behavior

I) Unstable growth: slope selection and coarsening model features / simplifications 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)  limited diffusion length for terrace / step edge diffusion effective representation of nucleation events single particle picture  l sed : characteristic length of step edge diffusion

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

dynamic scaling behavior time t  (film thickness) surface width ~mound height w = t  for t< t x w sat  L  for t> t x growth exponent roughness exponent saturation time t x  L z dynamic exponent z=  /  system sizes L = w / L scaling plot, data collapse  =1 (slope selection) z=4  =1/4 relatively slow coarsening

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

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

anisotropic binding structure: x y 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 · surface reconstructions observed: - c(2x2), (2x1) vacancy structures Cd-terminated - (2x1) dimerization Te-terminated II) Competing surface reconstructions in non-equilibrium

CdTe (001) properties of Cd-terminated surfaces maximum coverage 50 % two competing vacancy structures: checkerboard or missing rows simultaneous occupation of NN sites in y-direction, i.e. [1-10], is forbidden (extremely unfavorable) Te Cd x empty 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  eV per site in CdTe [Gundel, private comm.] 0.03 eV ZnSe

Te at the surface isotropic N.N. interaction additional Te dimerization motivation: coverage 3/2 observed under flux of excess Te allows for Te deposition on a perfect c(2x2) Cd surface 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 time consuming explicit treatment / mean field like Te * reservoir Kinetic Monte Carlo simulations Arrhenius rates for elementary processes = o e –  / (kT) o = /s choice of parameters: qualitative features, plausibility arguments semi-quantitative comparison, prospective first principle results

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

Summary Motivation Non-equilibrium growth - Molecular Beam Epitaxy (MBE) Theory / modeling / simulation several levels of theoretical description following talks: continuum descriptions, multi-scale approach,... 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

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