Off-lattice Kinetic Monte Carlo simulations of strained hetero-epitaxial growth

1. MFO Mini-Workshop on Multiscale Modeling in Epitaxial Growth, Oberwolfach 2004 Off-lattice Kinetic Monte Carlo simulations of strained hetero-epitaxial growth Florian Much, Christian Vey, Martin Ahr, Wolfgang Kinzel 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 {~much} Mathematics and Computing Science Intelligent Systems Rijksuniversiteit Groningen, Postbus 800, NL-9700 AV Groningen, The Netherlands biehl@cs.rug.nl

2. Outline Hetero-epitaxial crystal growth - mismatchedadsorbate/substrate lattice - model: simple pair interactions, 1+1 dim. growth - off-lattice KMC method Formation of dislocations - characteristic layer thickness - relaxation of adsorbate lattice constant Stranski-Krastanov growth - self-assembled islands, SK-transition - kinetic / stationary wetting layer - mismatch-controlled island properties 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

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, frequent case: substrate and adsorbate with identical crystal structure, but Aadsorbate lattice constants mismatch Ssubstrate strain relief

5. Ball and spring KMC models, e.g. [Madhukar, 1983] continuous variation of particle distances, but within preserved (substrate) lattice topology, excludes defects, dislocations activation energy for diffusion jumps: 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. continuous space Monte Carlo based on empirical pair-potentials, rates determined by energies of the binding states e.g. [Plotz, Hingerl, Sitter, 1992], [Kew, Wilby, Vvedensky, 1994] 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, A. Schindler (PhD thesis Duisburg, 1999)

7. A simple lattice mismatched system continuous particle positions, without pre-defined lattice 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 possible simplifications: 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 with respect to particles in a 3 o neighborhood of latest event 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. · interactions US=UA=UAS  diffusion barrier E  1 eV for =0 · lattice mismatch -15%   +11% · deposition rate Rd = 1 ML / s · substrate temperature T = 450 K · 6 ... 11 layers of substrate particles, bottom layer immobile · system sizes L=100, ..., 800 (# of particles per substrate layer) large misfits: dislocations at the substrate/adsorbate interface  = 10 %  = 6 % (grey level: deviation from A,S , light: compression) Simulation of dislocations dislokationen

12. sudden appearance of dislocations at a film thickness hc experimental results (semiconductors): misfit-dependence hc = a* ||-3/2 solid lines: <0:a*=0.15 >0:a*=0.05 adsorbate under compression, earlier dislocations hc vs. || =- 4 % Critical film thickness • small misfits: • initial growth of adsorbate coherent with the substrate

13. Pseudomorphic growth up to film thickness  -3/2 • enlarged vertical lattice constant in the adsorbate - Relaxation of the lattice constant above dislocations = 0.31% vertical lattice spacing ZnSe / GaAs, in situ x-ray diffraction KMC re-scaled film thickness [F. Much, C. Vey] [A. Bader, J. Geurts, R. Neder] SFB-410, Würzburg, in preparation qualitatively the same: 6-12-, m-n-, Morsepotential Critical film thickness

14. experimental results for various II-VI semiconductors -3/2 Matthews, Blakeslee

15. potential route for the fabrication of self-assembled quantum dots desired properties: (applications) - dislocation free - narrow size distribution - well-defined shape - spatial ordering PbSe on PbTe(111) hetero-epitaxy G. Springholz et al., Linz/Austria Stranski-Krastanov growth experimental observation ( Ge/Si, InAs/GeAs, PbSe/PbTe, CdSe/ZnSe, PTCDA/Ag) deposition of a few ML adsorbate material with lattice mismatch, typically 0 % <  7 % • - initial adsorbate wetting layer of characteristic thickness • sudden transition from 2d to 3d islands (SK-transition) • separated 3d islands upon a (reduced) persisting wetting layer

16. 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  favors wetting layer formation US > UAS > UA small misfit, e.g.= 4% deposition of a few ML  dislocation free growth Stranski-Krastanov growth S-K growth observed in very different materials hope: fundamental mechanism can be identified by investigation of very simple model systems

17. - 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 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

18. PTCDA / Ag ? [Chkoda et al., Chem Phys. Lett. 371, 2004] Nature of the SK-transition  • thermodynamic instability ? Island size ~ -2 • triggered by segregation and/or intermixing effects ? • e.g. InAs/GaAs[Cullis et al.] [Heyn et al.] reduced effective misfit concentration and strain gradient • kinetic effects, strain induced diffusion properties ?

19. substrate E [eV] WL (1)(2) UAS Adsorbate adatom diffusion on the surface slow on the substrate fast on the wetting layer - qualitatively as, e.g., for Ge on Si [B. Voigtländer et al.] • stabilizing effect: favors existence of a wetting layer • LJ-potential: no further decrease for more than 3 WL, • limited (stationary) wetting layer thickness

20. energy barrier (hops to the left) diffusion bias towards the center stabilizes existing islands (on relaxed ads.) 5 3 island height 1 (on 1 WL) position above island base Adsorbate adatom diffusion on the surface single adatom on a (partially) relaxed island on top of 1 WL base: 24 particles, height h ML

21. Rd=3.5 ML/s, T=500 K hw*= • = o (  – hw*)  •  1.5, hw* 2.1 ML  experiment: InAs on GeAs simulations: [ Leonard et al., Phys. Rev. B 50 (1994) 11687 ] Rd=3.5 ML/s, T=500 K = 4 % Determination of the kinetic wetting layer thickness analogous to experiment: end of layer-by-layer roughness oscillations or: (3rd and 4th layer) island density  vs. coverage  fit: = o(  – hw*)   [ML]

22. hw* [ML] Rup US = 1 eV UA = 0.74 eV  = 4 % island formation triggered by significant rate Rup for upward moves at the 2d-3d transition hw* = ho ( Rd / Rup )   0.2 Fit (500K): [ J. Johansson, W. Seifert, J. Cryst. Growth 234 (2002) 132 ] Kinetic wetting layer thickness hw* grows with - increasing flux - decreasing temperature T= 480 K T= 500 K

23. saturation behavior: island properties depend only on  density  base length b distance d for large enough deposition rate Rd become constant and T-independent d T=500 K T=480 K 70 b T=500 K 50 T=480 K  = 4 % 30 b  T=480 K 0.03 T=500 K 0.02 0.01 Characterization of islands

24. saturation behavior: island properties depend only on  density  base length b distance d for large enough deposition rate Rd become constant and T-independent # of islands b T=500 K T=480 K  = 4 % Rd= 7 ML/s T = 500 K b b  -1 length scale -1 introduced by S A Characterization of islands

25. Summary Method off-lattice Kinetic Monte Carlo application: simple model of hetero-epitaxy Dislocations characteristic length -1, critical layer thickness -3/2 Stranski-Krastanov growth strain induced formation of mounds, kinetic / stationary wetting layer large deposition rates: misfit controlled islanddensity, size b -1 SK-transition: slow diffusion on the substrate significant rate for upward jumps fast diffusion on the wetting layer diff. bias towards island centers

26. Stranski-Krastanov growth: - island formation mechanism for  <0 ? universality (Morse, mn-Potentials) interaction potentials, lattices • spatial distribution of islands material specific (e.g. RGL-Potentials) simulations • long time behavior, e.g. annealing / ripening after deposition • kinetic vs. equilibrium dots, e.g. b -2forRd0 ? Outlook 2+1 dimensional growth Growth modes - Volmer-Weber growth for ? UAS < UA - Layer-by-layer growth for small misfit ?