Off-lattice KMCsimulations of hetero-epitaxial growth: the formation of nano-structured surface alloys Mathematics and Computing Science Rijksuniversiteit Groningen Michael Biehl Theoretische Physik und Astrophysik & Sonderforschungsbereich 410 Julius-Maximilians-Universität Würzburg ~much, ~biehl Florian Much, Thorsten Volkmann, Sebastian Weber, Markus Walther Institute for Theoretical Physics Academy of Sciences, Prague Miroslav Kotrla

Hetero-epitaxial crystal growth - mismatched adsorbate/substrate lattice - model: simple pair interactions - off-lattice KMC method Stranski-Krastanov growth - self-assembled islands, SK-transition Nano-structured surface alloys - ternary material system: metals A/B on substrate S - equilibrium formation of stripes - growth: kinetic segregation and/or strain effect ? Summary and outlook Outline Formation of dislocations - misfit dislocations and strain relaxation

Molecular Beam Epitaxy ( MBE ) control parameters: deposition rate substrate temperature T ultra high vacuum directed deposition of adsorbate material(s) 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. mound formation · development of mathematical models, numerical methods, and simulation techniques oven UHV T

Hetero-epitaxy lattice constants  A adsorbate  S substrate mismatch different materials involved in the growth process, simplest case: substrate and adsorbate with identical crystal structure, but initial coherent growth undisturbed adsorbate enforced in first layers far from the substrate  dislocations, lattice defects SS AA strain relaxation: island and mound formation hindered layered growth self-assembled 3d structures AA SS and/or

Modelling/simulation of mismatch effects Ball and spring KMC models, e.g. [Madhukar, 1983] activation energy for diffusion jumps: preserved lattice topology + elastic interactions  E =  E bond -  E strain bond counting elastic energy e.g.: monolayer islands [Meixner, Schöll, Shchukin, Bimberg, PRL 87 (2001) ] SOS lattice gas : binding energies, barriers continuum theory: elastic energy for given configurations Lattice gas + elasticity theory: Molecular Dynamics limited system sizes / time scales, e.g. [Dong et al., 1998] continuous space Monte Carlo based on empirical pair-potentials, rates determined by energies e.g. [Plotz, Hingerl, Sitter, 1992], [Kew, Wilby, Vvedensky, 1994]

off-lattice Kinetic Monte Carlo evaluation of energy barriers in each given configuration D. Wolf and M. Schroeder (1999), A. Schindler (PhD thesis Duisburg, 1999) e.g. effects of (mechanical) strain in epitaxial growth, diffusion barriers, formation of dislocations

A simple lattice mismatched system continuous particle positions, without pre-defined lattice simplest case: (1+1)-dimensional growth equilibrium distance   o „short range“: U ij  0 for r ij > 3  o substrate-substrate U S,  S adsorbate-adsorbate substrate- adsorbate, e.g. U A,  A lattice mismatch  qualitative features of hetero-epitaxy, investigation of strain effects example: Lennard-Jones system

KMC simulations of the LJ-system - deposition of adsorbate particles with rate R d [ML/s] - diffusion of mobile atoms with Arrhenius rate simplification: for all diffusion events - preparation of (here: one-dimensional) substrate with fixed bottom layer

Evaluation of activation energies by Molecular Statics virtual moves of a particle, e.g. along x minimization of potential energy w.r.t. all other coordinates (including all other particles!) e.g. hopping diffusion binding energy E b (minimum) transition state energy E t (saddle) diffusion barrier  E = E t - E b Schwoebel barrier E s important simplifications: neglect concerted moves, exchange processes cut off potential at 3  o frozen crystal approximation

KMC simulations of the LJ-system - deposition of adsorbate particles with rate R d [ML/s] - diffusion of mobile atoms with Arrhenius rate simplification: for all diffusion events - preparation of (here: one-dimensional) substrate with fixed bottom layer - avoid accumulation of artificial strain energy (inaccuracies, frozen crystal) by (local) minimization of total potential energy all particles after each microscopic event (global) w.r.t. particles in a 3  o neighborhood of latest event (local)

Simulation of dislocations dislokationen · deposition rate R d = 1 ML / s · substrate temperature T = 450 K · lattice mismatch -15%    +11% · system sizes L=100,..., 800 (# of particles per substrate layer) · interactions U S =U A =U AS  diffusion barrier  E  1 eV for  =0  = 6 %  = 10 % large misfits: dislocations at the substrate/adsorbate interface (grey level: deviation from  A,S, light: compression )

- Relaxation of the vertical lattice spacing: KMC qualitatively the same: 6-12-, m-n-, Morsepotential [F. Much, C. Vey, M. Walther] vertical lattice spacing ZnSe / GaAs, in situ x-ray diffraction  = 0.31% [A. Bader, J. Geurts, R. Neder] SFB-410, Würzburg, in preparation small misfits: - initial pseudomorphic growth of adsorbate coherent with the substrate - sudden appearance of dislocations at a film thickness h c (KMC & experiment) misfit-dependence h c ≈ a * |  | -3/2

Stranski-Krastanov growth experimental observation ( Ge/Si, InAs/GeAs, PbSe/PbTe, CdSe/ZnSe, PTCDA/Ag) with lattice mismatch, typically 0 % <   7 % - initial adsorbate wetting layer (WL) of characteristic thickness - sudden transition from 2d to 3d islands (SK-transition) - separated 3d islands upon a (reduced) persisting WL L J pair potential, 1+1 spatial dimensions modification: Schwoebel barrier removed by hand single out strain as the cause of island formation small misfit, e.g.  = 4% deposition of a few ML  dislocation free growth Simple off-lattice model: U S > U AS > U A  favors WL formation

Stranski-Krastanov growth aspect ratio 2:1 - kinetic WL h w *  2 ML growth: deposition + WL particles splitting of larger structures - stationary WL h w  1 ML U S = 1 eV, U A = 0.74 eV R d = 7 ML/s T = 500 K AA SS mean distance from neighbor atoms  = 4 % dislocation free multilayer islands self-assembled quantum dots mean base length 1/ 

(bulk) immiscible metal adsorbates A and B, e.g. Co and Ag form surface alloys on appropriate substrates e.g. Ru (0001) with intermediate lattice constant e.g.  Co  +6%  Ag  -5% deposition of only A or B compact island growth, (characteristic size for  >0 ) Nano-structured surface alloys 50 nm 175 nm 600 nm 750 nm co-deposition of A and B dendritic growth, ramified islands, nm-scale stripe sub-structure [ R.W. Hwang, PRL 76 (1996) 4757 ]

possible mechanisms: strain-induced arrangement of adatoms  >0  0 side view smaller atoms fill gaps between larger ones  zero effective mismatch equilibrium configuration ?purely kinetic effect ? segregation due to different binding energies top view extra barrier  step edge diffusion:  =  = 0

Off-lattice simulations - substrate (6*100*100), adsorbate A/B in the sub-monolayer regime - interaction strength U AB ≤ U AA ( U AA =U BB ) example: U AB = 0.6 U AA (numerical values such that A-diffusion barrier is 0.37eV ) - ternary material system, symmetric misfits:  A,B = ±  - modulated Morse (LJ, m-n,...) potential favors simple cubic geometry [ M. Schroeder, P. Smilauer, D. E. Wolf, Surf. Sci. 375 (1997) 129 ] - random deposition of A/B with conc.  A =  B, total flux: 0.01 ML/s - diffusion only within the layer (no inter-layer transport)

equilibrium MC simulations: completely filled monolayer, non-local particle exchange dynamics (LJ) U AB /U AA =  = 4.5%  = 5.5% stripes in directions: misfit small strip widths favors small U AB large domains U AB /U AA =0.6 segr. along

non-equilibrium KMC simulations: deposition of material A only growth of compact islands, characteristic  -dependent size color-coded distance to in-plane NN (LJ)

 A,B = 0 but (only) different binding energies  kinetic segregation, smooth shapes, complete separation for long times co-deposition of (LJ) materials A / B U AB < U AA, U BB  A,B =±4 % and  persisting stripe structure larger particles (B) form backbone smaller particles (A) fill in the gaps meandering, ramified island edge U AB < U AA, U BB binding energies + strain effects

influence of (Morse) potential steepness a and misfit  both mechanisms are needed to reproduce experimental observations qualtitatively !

quantitative measure of the ramification: # of perimeter particles  = √total # in island  vs. misfit  vs. substrate temperature ( not a low T effect! )

attempt: a lattice gas formulation off-lattice Molecular Statics  set of barriers for a catalogue of events example: diffusion along an island edge energy barriers (←)  =0 off-lattice lattice gas  =5% off-lattice lattice gas non-local strain effects (elastic interactions through substrate) barriers cannot be determined from small neighborhoods

Summary Method off-lattice Kinetic Monte Carlo Dislocations formation of misfit-dislocations, critical film thickness Stranski-Krastanov growth strain induced island formation, kinetic/stationary wetting layer application: simple model of hetero-epitaxy 2D alloys ternary system, monolayer adsorbate with non-trivial composition profile island growth: ramified contour, nano-scale stripe substructure Interplay of strain relaxation and chemically induced diffusion barriers T. Volkmann, F. Much, M. Biehl, M. Kotrla, Surf. Sci. 586 (2005),

Outlook Model (2+1)-dimensional growth, realistic interaction potentials exchange diffusion processes, interdiffusion, concerted moves,... Dislocations relaxation above misfit dislocations diffusion properties on surfaces with buried dislocations Island and mound formation Stranski-Krastanov vs. Volmer-Weber growth phase diagram for variation of , T, U AS 2D alloys asymmetric situations: misfits, concentrations realistic lattices, e.g. fcc(111) substrate more realistic interaction potentials (metals) anisotropic substrates, formation of aligned stripes several layers of adsorbate, interlayer diffusion processes...