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Theoretische Physik und Astrophysik

& Sonderforschungsbereich 410

Julius-Maximilians-Universitt Wrzburg

Am Hubland, D-97074 Wrzburg, Germany

Mathematics and Computing Science

Intelligent Systems

Rijksuniversiteit Groningen, Postbus 800,

NL-9718 DD Groningen, The Netherlands

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Non-equilibrium growth - Molecular Beam Epitaxy (MBE)

several levels of theoretical description

deposition and transient kineticsthermally activated processes, Arrhenius dynamics

problems and limitations

mound formation and coarsening dynamicsII) (ALE) growth of II-VI(001) systems

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control parameters:substrate/adsorbate materialsdeposition ratesubstrate temperature T

ultra high vacuumdirected deposition ofadsorbate

material onto a substrate crystal

production of, for instance, high quality

layered semiconductor devices

magnetic thin films

nano-structures: quantum dots, wires

clear-cut non-equilibrium situation

interplay: microscopic processes macroscopic properties

self-organized phenomena, e.g. dot formation

Mikrostrukturlabor, Wrzburg

oven

UHV

T

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faithful material specific description

including electronic properties

often: configuration of a few atoms/molecules,

unit cells of periodic structures,

zero temperature treatment

important tool:

description in terms of electron densities

energy/stability of surface reconstructions,

preferred arrangement of surface atoms

CdTe (001) surface reconstructions

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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 (10-6 s)

example: on a surface

atomic vibrations ( ~10-12 s)

with occasional hops to the next local minimum

dissociation of deposited

dimers at the surface,

transient mobility of arriving atoms

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stochastic dynamics, consider only significant changes of the configuration

simplifying

of

empty / occupied sites

hops from site to site

models:

exclude bulk vacancies, overhangs,

defects, stacking faults, etc.

d+1 dim. crystal represented by

integer array above d-dim. substrate lattice

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

processes, examples:

upon deposition

knockout-processes

at terrace edges

downhillfunnelling

steering

weakly bound, highly mobileintermediate states

regular lattice site

potentialenergy

distance from the surface

vac.

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waiting timeTBk

oe

rateTBk

oeR

attempt frequency , e.g.o

energy barrier , e.g. for

simplifying representation:

112

o10~

s

after incorporation: mobile at surface sites

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

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

bEb

Ea

more general:

transition states and energy barriers affect onlythe of the system

Ett

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

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

for states of type a,b,...

in absence of deposition and desorption:system approaches

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an example

ESE

E

adatoms attach toupper terraces preferentially

uphill current favors

additional

hinders inter-layer diffusion

non-equilibrium, kinetic effect:

additional barrier ES is irrelevant forequilibrium properties of the system

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

:correct treatment takes into account entropies / free energies

- temperature independent

-

disregard actual shape of the

energy landscape

o(ab) = o(ba)consistent with discretized state spaceand concept of

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

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(rejection free)

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

R1

R2

R3

Rn

... rat

pick one of the possible events

with probability pi Ri

0

1

ra

ndomnumber

update the catalogue of possible processes

and associated energy barriers and rates

R3

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e.g. exchange vs. hopping diffusion

direct / indirect experimental measurement

calculations/estimates: first principlessemi-empirical potentials

quantitative match of simulations and experiments

potentially relevant processes:

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defects, dislocations ?

hetero-epitaxial growth ?

strain and other mismatch effects ?

realistic lattices or off-lattice simulations

interaction potentials, realistic energy barriers

particularities of materials / material classes

basic questions

example: (universal?) dynamical scaling behavior

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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 forterrace / step edge diffusion

effective representation of

nucleation events

single particle picture

characteristic length of step edge diffusion

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initial

due to Schwoebel effect

merging of mounds driven by

- deposition noise

and/or - step edge diffusion

finite system size single mound

example: (associated length lsed=1 lattice const. )

16 ML 256 ML 4096 ML

selection of a

compensating particle currents

upward (Schwoebel)

downward (knockout)

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time t (film thickness)

~mound height

w =t for t< tx

wsat L for t> tx

growth exponent

roughness exponent

saturation time tx L dynamic exponent z= /

systemsizesL = 80

100125

140256512

w

/L

scaling plot, data collapse

=1 (slope selection)

z=4=1/4

relatively slowcoarsening

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for the morphology and coarsening dynamics

64ML

(lsed L)

sed drivencoarsening

128ML

(lsed 1)

noise assistedcoarsening

128ML

absence of

slope selection,

rough surface

hindered sed,

noise assisted

coarsening

128ML

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characteristic exponents:

for 1

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anisotropicbinding structure:

011

110

example system:

lattice, (001) orientation:alternating layers (square lattices) of, e.g.,

representation, four sub-lattices

observed:

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

- (2x1) Te-terminated

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properties of

maximum coverage 50 %

two competing vacancy structures: checkerboard or missing rows

simultaneous occupation

of NN sites in y-direction,

i.e. [1-10], is(extremely unfavorable)

TeCd

xempty

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

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isotropic N.N. interaction

additional Te dimerization

: coverage 3/2 observed under flux of excess Teallows for Te deposition on a perfect c(2x2) Cd surface

weakly bound, highly mobile Te-atoms 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

for elementary processes = o = 1012/s

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

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alternating pulses (1s)of Cd and Te

flux: 5ML/sdead time: 0.1s

at high temperature

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

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

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Non-equilibrium growth - Molecular Beam Epitaxy (MBE)

several levels of theoretical description

following talks: continuum descriptions, multi-scale approach,...

deposition and transient kinetics

thermally activated processes, Arrhenius dynamics

problems and limitations

mound formation and coarsening dynamics

II) (ALE) growth of II-VI(001) systems

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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