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

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

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 vacuumdirected deposition of adsorbatematerial(s) onto a substrate crystal

oven

UHV

T

Hetero-epitaxy

lattice constants A adsorbate

S substratemismatch

S

SA

σσσ

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

A

strain relief

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: E = Ebond - Estrain

bond counting

elasticenergy

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) 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 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

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

possible simplifications: 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 with respect to particles in a 3 o neighborhood of latest event

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 ... 11 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

hc 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 hc

experimental results (semiconductors):

misfit-dependence hc = a* ||-3/2

re-scaled film thickness

vert

ical

latt

ice

spac

ing

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 .hlnh c

1c const

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:

US > UAS > UA favors wetting layer 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 %

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

concentrationand 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 UAS

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)

1

35

isla

nd

heig

ht

(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 ( – hw* )

simulations:

Rd=3.5 ML/s, T=500 K

= o ( – hw* )

1.5, hw* 2.1 ML

Rd=3.5 ML/s, T=500 K

= 4 %

[ Leonard et al., Phys. Rev. B 50 (1994) 11687 ]

experiment: InAs on GeAs

hw*=

[ML]

Kinetic wetting layer thickness

hw* grows with

- increasing flux

- decreasing temperature

US = 1 eV

UA = 0.74 eV

= 4 %

hw

*

[ML]

T= 480 KT= 500 K

hw* = ho ( Rd / Rup )

0.2

Fit (500K):

Rup

island formation triggered by

significant rate Rup 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 bdistance d

become constant and T-independent

for large enough deposition rate Rd

T=500 K

T=480 K

= 4 %

b

b

dT=500 KT=480 K

0.0

10

.03

T=500 K

T=480 K 0

.02

30

50

70

Rd= 7 ML/s

T = 500 K #

of

isl

ands

Characterization of islands

saturation behavior: island properties depend only on

density base length bdistance d

become constant and T-independent

for large enough deposition rate Rd

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 Rd0 ?

Growth modes

- Volmer-Weber growth for

?

UAS < UA

- Layer-by-layer growth for small misfit ?