Off-lattice KMCsimulations of hetero-epitaxial growth:
the formation of nano-structured surface alloys
Mathematics and Computing Science
Rijksuniversiteit [email protected] www.cs.rug.nl/~biehl
Michael Biehl
Theoretische Physik und Astrophysik
& Sonderforschungsbereich 410
Julius-Maximilians-Universität Würzburghttp://theorie.physik.uni-wuerzburg.de/~volkmann
~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 vacuumdirected deposition of adsorbatematerial(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 substratemismatch
S
SA
σσσ
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
A
strain relaxation:
island and mound formationhindered layered growthself-assembled 3d structures
A
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 = Ebond - Estrain
bond counting
elasticenergy
e.g.: monolayer islands [Meixner, Schöll, Shchukin, Bimberg, PRL 87 (2001) 236101]
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 latticesimplest case: (1+1)-dimensional growth
6
ij
o
12
ij
ooooij r
σrσ
U4σUU ,
equilibrium distance o
„short range“: Uij 0 for rij > 3 o
substrate-substrate US, S
adsorbate-adsorbate
substrate- adsorbate, e.g. 2σσσUUU ASASASAS ,
UA, A
lattice mismatch SSA σσσ
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 Rd [ML/s]
- diffusion of mobile atoms with Arrhenius rate TBk
ΔE
oi
i
e R
simplification: for all diffusion events -112
o s10
- 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 Eb (minimum)
transition state energy Et (saddle)
diffusion barrier E = Et - Eb Schwoebel barrier Es
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 Rd [ML/s]
- diffusion of mobile atoms with Arrhenius rate TBk
ΔE
oi
i
e R
simplification: for all diffusion events -112
o s10
- 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)
n
1ijij
n
1itot UE
Simulation of dislocations
dislokationendislokationen
· deposition rate Rd = 1 ML / s · substrate temperature T = 450 K
· lattice mismatch -15% +11%
· system sizes L=100, ..., 800 (# of particles per substrate layer)
· interactions US=UA=UAS 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]
vert
ical
latt
ice
spac
ing
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 hc (KMC & experiment)misfit-dependence hc ≈ 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:
US > UAS > UA favors WL formation
Stranski-Krastanov growth
aspect ratio 2:1
- kinetic WL hw* 2 ML
growth: deposition + WL particles
splitting of larger structures
- stationary WL hw 1 ML
US= 1 eV, UA= 0.74 eV
Rd= 7 ML/s T = 500 K
AS
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 >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 UAB ≤ UAA ( UAA =UBB )
example: UAB = 0.6 UAA (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
),f(2eeUU r)(σar)(σaoij
oo [ 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)
UAB /UAA = 0.6 0.8 0.9 1.0
=4.5%
=5.5%
stripes in <11> directions:
misfit small strip widths favorssmall UAB large domains
UAB /UAA=0.6
segr.
alo
ng <
01
>
non-equilibrium KMC simulations: deposition of material A only
growth of compact islands, characteristic -dependent size
color-codeddistance toin-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
UAB < UAA, UBB
A,B =±4 % and
persisting stripe structure
larger particles (B) form backbone
smaller particles (A) fill in the gaps
meandering, ramified island edge
UAB < UAA, UBB
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 alongan island edge
energy
barriers (←)
=0off-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
Dislocationsformation of misfit-dislocations, critical film thickness
Stranski-Krastanov growth
strain induced island formation, kinetic/stationary wetting layer
application: simple model of hetero-epitaxy
2D alloysternary system, monolayer adsorbate with non-trivial composition profileisland growth: ramified contour, nano-scale stripe substructure
Interplay of strain relaxation and chemically induced diffusion barriersT. Volkmann, F. Much, M. Biehl, M. Kotrla, Surf. Sci. 586 (2005), 157-173
OutlookModel
(2+1)-dimensional growth, realistic interaction potentials
exchange diffusion processes, interdiffusion, concerted moves, . . .
Dislocationsrelaxation above misfit dislocationsdiffusion properties on surfaces with buried dislocations
Island and mound formation
Stranski-Krastanov vs. Volmer-Weber growth
phase diagram for variation of , T, UAS
2D alloysasymmetric situations: misfits, concentrationsrealistic lattices, e.g. fcc(111) substrate more realistic interaction potentials (metals)anisotropic substrates, formation of aligned stripesseveral layers of adsorbate, interlayer diffusion processes. . .