Off-lattice Kinetic Monte Carlo simulations of strained hetero-epitaxial growth Theoretische Physik und Astrophysik & Sonderforschungsbereich 410 Julius-Maximilians-Universität Würzburg Am Hubland, D Würzburg, Germany {~much} Mathematics and Computing Science Intelligent Systems Rijksuniversiteit Groningen, Postbus 800, NL-9700 AV Groningen, The Netherlands Michael Biehl Florian Much, Christian Vey, Martin Ahr, Wolfgang Kinzel MFO Mini-Workshop on Multiscale Modeling in Epitaxial Growth, Oberwolfach 2004

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

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

Hetero-epitaxy lattice constants  A adsorbate  S substrate mismatch different materials involved in the growth process, frequent 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 relief 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:  E =  E bond -  E strain bond counting elastic energy continuous variation of particle distances, but within preserved (substrate) lattice topology, excludes defects, dislocations 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 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 [D. Wolf, 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 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 possible simplifications: 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 with respect to particles in a 3  o neighborhood of latest event

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 · layers of substrate particles, bottom layer immobile  = 6 %  = 10 % large misfits: dislocations at the substrate/adsorbate interface (grey level: deviation from  A,S, light: compression )

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

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

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

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 % PbSe on PbTe(111) hetero-epitaxy G. Springholz et al., Linz/Austria potential route for the fabrication of self-assembled quantum dots desired properties: (  applications) - dislocation free - narrow size distribution - well-defined shape - spatial ordering - 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

Stranski-Krastanov growth S-K growth observed in very different materials hope: fundamental mechanism can be identified by investigation of very simple model systems 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 wetting layer 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 % self-assembled quantum dots dislocation free multilayer islands

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 ? PTCDA / Ag ? [Chkoda et al., Chem Phys. Lett. 371, 2004]

Adsorbate adatom diffusion on the surface slow on the substrate fast on the wetting layer U AS  E [eV] substrate WL (1) (2) - 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

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

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 (  – h w * )  simulations: R d =3.5 ML/s, T=500 K  =  o (  – h w * )    1.5, h w *  2.1 ML  R d =3.5 ML/s, T=500 K  = 4 % [ Leonard et al., Phys. Rev. B 50 (1994) ] experiment: InAs on GeAs hw*=hw*=  [ML]

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

Characterization of islands saturation behavior: island properties depend only on  density  base length b distance d become constant and T-independent for large enough deposition rate R d T=500 K T=480 K  = 4 % b b d T=500 K T=480 K  T=500 K T=480 K

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

Summary Method off-lattice Kinetic Monte Carlo 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 island density, 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 application: simple model of hetero-epitaxy

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