Off-lattice KMCsimulations of hetero-epitaxial growth: the formation of nano-structured surface alloys

1. Michael Biehl Mathematics and Computing Science Rijksuniversiteit Groningen biehl@cs.rug.nl www.cs.rug.nl/~biehl Off-lattice KMCsimulations of hetero-epitaxial growth: the formation of nano-structured surface alloys Florian Much, Thorsten Volkmann, Sebastian Weber, Markus Walther Theoretische Physik und Astrophysik & Sonderforschungsbereich 410 Julius-Maximilians-Universität Würzburg http://theorie.physik.uni-wuerzburg.de/~volkmann ~much, ~biehl Miroslav Kotrla Institute for Theoretical Physics Academy of Sciences, Prague

2. Outline Hetero-epitaxial crystal growth - mismatchedadsorbate/substrate lattice - model: simple pair interactions - off-lattice KMC method Formation of dislocations - misfit dislocations and strain relaxation 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

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

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

5. continuous space Monte Carlo based on empirical pair-potentials, rates determined by energies e.g. [Plotz, Hingerl, Sitter, 1992], [Kew, Wilby, Vvedensky, 1994] Ball and spring KMC models, e.g. [Madhukar, 1983] activation energy for diffusion jumps: preserved lattice topology + elastic interactions E =  Ebond - Estrain elastic energy bond counting Lattice gas + elasticity theory: SOS lattice gas : binding energies, barriers continuum theory: elastic energy for given configurations e.g.: monolayer islands [Meixner, Schöll, Shchukin, Bimberg, PRL 87 (2001) 236101] Modelling/simulation of mismatch effects Molecular Dynamics limited system sizes / time scales, e.g. [Dong et al., 1998]

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

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

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

9. e.g. hopping diffusion binding energy Eb (minimum) transition state energy Et (saddle) 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!) diffusion barrier E =Et - Eb Schwoebel barrierEs important simplifications: neglect concerted moves, exchange processes cut off potential at 3 o frozen crystal approximation

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

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

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

13. Simple off-lattice model: 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  favors WL formation US > UAS > UA 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

14. - kinetic WL hw* 2 ML - stationary WL hw 1 ML mean distance from neighbor atoms growth: deposition + WL particles splitting of larger structures self-assembled quantum dots mean base length1/ S A Stranski-Krastanov growth US= 1 eV, UA= 0.74 eV Rd= 7 ML/s T = 500 K dislocation free multilayer islands  = 4 % aspect ratio 2:1

15. 750 nm 50 nm co-deposition of A andB dendritic growth, ramified islands, nm-scale stripe sub-structure [ R.W. Hwang, PRL 76 (1996) 4757 ] 175 nm 600 nm Nano-structured surface alloys (bulk) immiscible metal adsorbates A andB, e.g. Coand Ag form surface alloys on appropriate substrates e.g. Ru (0001) with intermediate lattice constante.g.Co +6% Ag -5% deposition of only A orB compact island growth, (characteristic size for >0)

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

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

18. =4.5% =5.5% UAB /UAA = 0.6 0.8 0.9 1.0 UAB /UAA=0.6 misfit small strip widths favors small UAB large domains segr. along <01> equilibrium MC simulations: completely filled monolayer, non-local particle exchange dynamics (LJ) stripes in <11> directions:

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

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

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

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

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

24. Summary Method off-lattice Kinetic Monte Carlo application: simple model of hetero-epitaxy Dislocations formation of misfit-dislocations, critical film thickness Stranski-Krastanov growth strain induced island formation, kinetic/stationary wetting layer 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), 157-173

25. 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,UAS 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 . . .