Off-lattice KMCsimulations of hetero-epitaxial growth:

the formation of nano-structured surface alloys

Mathematics and Computing Science

Rijksuniversiteit Groningenbiehl@cs.rug.nl 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. . .