Lattice gas models and Kinetic Monte Carlo simulations of epitaxial crystal growth Theoretische Physik und Astrophysik & Sonderforschungsbereich 410 Julius-Maximilians-Universität.

Lattice gas models and Kinetic Monte Carlo simulations of epitaxial crystal growth

Theoretische Physik und Astrophysik

& Sonderforschungsbereich 410

Julius-Maximilians-Universität Würzburg

Am Hubland, D-97074 Würzburg, Germany

http://theorie.physik.uni-wuerzburg.de/~biehl

Mathematics and Computing Science

Intelligent Systems

Rijksuniversiteit Groningen, Postbus 800,

NL-9718 DD Groningen, The Netherlands

[email protected]

Michael Biehl Michael Biehl

Outline

• Motivation Non-equilibrium growth - Molecular Beam Epitaxy (MBE)

• Theory / modeling / simulation several levels of theoretical description

• Summary

• Lattice gas and Solid-On-Solid (SOS) models

• Kinetic Monte Carlo simulations deposition and transient kinetics thermally activated processes, Arrhenius dynamics problems and limitations

• Example applications I) unstable growth, mound formation and coarsening dynamics II) Atomic Layer Epitaxy (ALE) growth of II-VI(001) systems

Molecular Beam Epitaxy ( MBE )

control parameters: substrate/adsorbate materialsdeposition rate substrate temperature T

ultra high vacuumdirected deposition of adsorbatematerial 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. dot formation

Mikrostrukturlabor, Würzburg

oven

UHV

T

Theory / modelling of (growing) surfaces

Quantum Mechanics

faithful material specific descriptionincluding electronic propertiesoften: configuration of a few atoms/molecules, unit cells of periodic structures, zero temperature treatment

important tool: Density Functional Theory (DFT)

description in terms of electron densities

typical problem:

energy/stability of surface reconstructions,

preferred arrangement of surface atoms

CdTe (001) surface reconstructions

Molecular Dynamics

numerical integration of equations of motion + thermal fluctuations

effective interactions, e.g. classical short range pair potentials

(QM treatment: e.g. Car Parinello method )

microscopic dynamics of particles

limited system size and real time ( 10-6 s )

example: diffusion on a surfaceatomic vibrations ( ~10-12 s )with occasional hops to the next local minimum

typical problem:

dissociation of deposited

dimers at the surface,

transient mobility of arriving atoms

Kinetic Monte Carlo (KMC) simulations

stochastic dynamics, consider only significant changes of the configuration

simplifying lattice gas models: pre-defined lattice of empty / occupied siteshops from site to site

Solid-On-Solid (SOS) models:exclude bulk vacancies, overhangs,defects, stacking faults, etc.

d+1 dim. crystal represented by integer array above d-dim. substrate lattice

Deposition of particles, e.g. with flux F = 1 atom / site / s (incoming momentum, attraction to the surface...)

incorporation processes, examples:

Transient effects upon deposition

knockout-processes

at terrace edges

downhill funnelling

steering

weakly bound, highly mobile intermediate states

regular lattice sitepote

nti

al en

erg

ydistance from the surface

vac.

Arrhenius law: waiting time TBk

Δ

o eτ τ

rate TBk

Δ

o e R

attempt frequency , e.g. o

energy barrier , e.g. for hopping diffusion

thermally activated processes, simplifying representation:

112o 10~ s

after incorporation: mobile adatoms at surface sites

R (ab) = 0 exp[ / (kBT) ]

R (ba) = 0 exp[ ( Ea-Eb+ ) /

(kBT) ]a

b Eb

Ea

more general:

transition states and energy barriers affect „only“

the non-equilibrium dynamics of the system

Et

t

R (ab) exp[ - Ea / (kBT) ] = R (ba) exp[ - Eb / (kBT) ]

detailed balance condition stationary P(s) exp[- Es / (kBT) ]

for states of type a,b,...

in absence of deposition and desorption:

system approaches thermal equilibrium

an example: Ehrlich-Schwoebel instability

ES

E

E

diffusion bias: adatoms attach to

upper terraces preferentially

uphill current favors mound formation

additional Schwoebel barrier

hinders inter-layer diffusion

non-equilibrium, kinetic effect:

additional barrier ES is irrelevant for equilibrium properties of the system

implicitly used simplifications and assumptions

deep (local) minima, infrequent eventsexclude, e.g., double or multiple jumps:

transition state theory: correct treatment takes into account entropies / free energies

constant prefactor (attempt frequency) - temperature independent - state independent disregard actual shape of the energy landscape a

b

t

o(ab) = o(ba) ?

consistent with discretized state spaceand concept of detailed balance

frozen crystal : e.g. single, mobile particle in a static environment, neglectconcerted rearrangements of the entire crystal / neighborhood

desorption

islands

diffusion

nucleation

deposition

downward diffusion

edge diffusion

some microscopic processes on the growing surface

+more: incorporation, knockout attachment to edges / islands detachment processes, ...

Kinetic Monte Carlo Simulation (rejection free)

· perform the selected event

(evaluate physical real time step)

· perform the selected event

(evaluate physical real time step)

· initial configuration of the (SOS) system

· catalogue of all relevant processes i=1,2,...n

and corresponding Arrhenius rates

· initial configuration of the (SOS) system

· catalogue of all relevant processes i=1,2,...n

and corresponding Arrhenius rates

R1

R2

R3

Rn

... rate

s ...

· pick one of the possible events randomly

with probability pi Ri

· pick one of the possible events randomly

with probability pi Ri

0

1

random

num

ber

· update the catalogue of possible processes

and associated energy barriers and rates

· update the catalogue of possible processes

and associated energy barriers and rates

R3

exchange processes / concerted moves

e.g. exchange vs. hopping diffusion

dimer and island mobility

material specific input ?

direct / indirect experimental measurement

calculations/estimates: first principles semi-empirical potentials

quantitative match of simulations and experiments

complete catalogue of events ?

potentially relevant processes:

Problems and limitations

lattice gas / SOS description:

defects, dislocations ?

hetero-epitaxial growth ?

strain and other mismatch effects ?

material specific simulations

realistic lattices or off-lattice simulations

interaction potentials, realistic energy barriers

particularities of materials / material classes

Applications:abstract models, further simplifications

basic questions

example: (universal?) dynamical scaling behavior

I) Unstable growth: slope selection and coarsening

model features / simplifications

SOS lattice (e.g. simple cubic) neglect overhangs, defects

knock-out process upon deposition momentum of incoming particles

irreversible attachment

immobile islands

forbidden downward diffusion

high barriers (large enough flux)

limited diffusion length for

terrace / step edge diffusion

effective representation of

nucleation events

single particle picture

lsed : characteristic length of step edge diffusion

• initial mound formation

due to Schwoebel effect

• coarsening process

merging of mounds driven by

- deposition noise

and/or - step edge diffusion

• saturation state

finite system size single mound

example: slow step edge diffusion (associated length lsed=1 lattice const. )

16 ML 256 ML 4096 ML

• selection of a stable slope:

compensating particle currents

upward (Schwoebel)

downward (knockout)

dynamic scaling behavior time t <h> (film thickness)

surface width ~mound height

w =t for t< tx

wsat L for t> tx

growth exponent

roughness exponent

saturation time tx Lz dynamic exponent z= /

systemsizesL = 80 100 125 140 256 512

w /

Lscaling plot, data collapse

=1 (slope selection)

z=4=1/4

relatively slow coarsening

The role of step edge diffusion (sed)

for the morphology and coarsening dynamics

64ML

fast sed

(lsed L)

1.00

0.45

sed driven

coarsening 128ML

slow sed

(lsed 1)

1.00

0.25

noise assisted

coarsening

128ML

absent sed

0.70 < 1

0.20

absence of

slope selection,

rough surface

additional

corner barrier

hindered sed,

noise assisted

coarsening

128ML

significant step edge diffusion

characteristic exponents:

= 1, 1/3, z 3

for 1 << lsed << L

lsed

universality: observed in various types of lattices

simple cubic (001), body centered cubic (001)

simple hexagonal (001), hcp (001)

contradicts continuum model predictions:

0.24 for cubic lattices

1/3 for all other lattices

Siegert, 1998Moldovan, Golubovic, 2000

Ahr, Kinne, Schinzer

anisotropicbinding structure:

][ 011

][110

x

y

example system: II-VI (001) semiconductor surfaces

· zincblende lattice, (001) orientation:

alternating layers (square lattices) of, e.g., Cd / Te

SOS representation, four sub-lattices

· surface reconstructions observed:

- c(2x2), (2x1) vacancy structures Cd-terminated

- (2x1) dimerization Te-terminated

II) Competing surface reconstructions in non-equilibrium

CdTe (001)

properties of Cd-terminated surfaces

maximum coverage 50 % two competing vacancy structures: checkerboard or missing rows

simultaneous occupation

of NN sites in y-direction,

i.e. [1-10], is forbidden

(extremely unfavorable)

TeCd

x empty

electron counting rule, DFT

[Neureiter et al., 2000]

small difference in surface energies

favors checkerboard c(2x2)-order at low temperatures

e.g. DFT: E 0.008 eV per site in CdTe [Gundel, private comm.] 0.03 eV ZnSe

Te at the surface

isotropic N.N. interaction

additional Te

dimerization

motivation: coverage 3/2 observed under flux of excess Te

allows for Te deposition on a perfect c(2x2) Cd surface

beyond SOS

weakly bound, highly mobile Te-atoms ( Te* ) on the surface, e.g.

at a Cd-site (Te-trimers)

bound to a single Cd (neutralizes repulsion)temporary position

time consuming explicit treatment / mean field like Te* reservoir

Kinetic Monte Carlo simulations

Arrhenius rates for elementary processes = o e –/ (kT) o = 1012/s

choice of parameters: qualitative features, plausibility arguments semi-quantitative comparison,prospective first principle results

Atomic Layer Epitaxy (ALE)

alternating pulses (1s) of Cd and Te flux: 5ML/s dead time: 0.1s

Cd Te Cd Te

reconstructions self-limitation of the growth rate at high temperature

experiment [Faschinger, Sitter] simulation [M. Ahr, T. Volkmann]

overcome at lower T due to presence of highly mobile, weakly bound Te* :

Summary

• Motivation Non-equilibrium growth - Molecular Beam Epitaxy (MBE)

• Theory / modeling / simulation several levels of theoretical description following talks: continuum descriptions, multi-scale approach,...

• Lattice gas and Solid-On-Solid (SOS) models

• Kinetic Monte Carlo simulations deposition and transient kinetics thermally activated processes, Arrhenius dynamics problems and limitations

• Example applications I) unstable growth, mound formation and coarsening dynamics II) Atomic Layer Epitaxy (ALE) growth of II-VI(001) systems

Outlook (Wednesday)

application of KMC method in off-lattice modelstreatment of

- hetero-epitaxy, mismatched lattices

- formation of dislocations

- strain-induced island growth

- surface alloys of immiscible materials