Cet ouvrage fait partie de la bibliothèque YouScribe
Obtenez un accès à la bibliothèque pour le lire en ligne
En savoir plus

Continuum modeling, analysis and simulation of the self-assembly of thin crystalline films [Elektronische Ressource] / vorgelegt von Maciej Dominik Korzec

171 pages
83WConSkutellatinDr.uummoerdeling,M?ncanalysisandrer.simorsitzender:ulationProf.ofter:theter:self-assemblyTof11.10.2010thinhaftengenehmigtelmsuss:Dr.vhorgelegtVvonDr.Diplom-MathematikerDr.wDominikter:KRybkdergebAusspracoren2010in??Dr.d?nat.(PDissertationolen)hVVonProf.derMartinFakult?tter:IDr.Iolk-MehrmannMathematikhundPDBarbarahaftenagnerderhTPDecAndreashheneitererUnivhersit?tProf.BerlinPiotrzuraErlangungagdeswissenscakhenhe:henBerlinGradesDDoktorderolutionself-assemDerivtheationshooftracandtinorkuumones,mospacingdelsaforusedepitaxialbgrowthobservofarrathinbiggersolidquanlmsriponWdrivingsub-stratesparameteryieldsmethoCahn-Hilliardeent[102ypsimeyequationsyofpseudospfourthedorsixthtoorder.edTeningosmalldescribhede℄andtheunderstandexpressionssolutionstheandsolutionssolutionphasespaceshniquetothesesolutionssemi-forormoquasilineartumpartialextendsdierenaligntialanequations(PDEs),Stranski-Krastanotheout.devfourthelopmeneecttofulationselabWhileoratedtheoryisundredsso.
Voir plus Voir moins

83

W
Con
Skutella
tin
Dr.
uum

mo
er
deling,
M?nc
analysis

and
rer.
sim
orsitzender:
ulation
Prof.
of
ter:
the
ter:
self-assem

bly
T
of
11.10.2010
thin
haften

genehmigte
lms
uss:

Dr.
v
h
orgelegt
V
v

on
Dr.
Diplom-Mathematik

er
Dr.

w
Dominik
ter:
K
Rybk

der
geb
Aussprac
oren
2010
in

??
Dr.
d?
nat.
(P
Dissertation
olen)
h
V
V
on
Prof.
der
Martin
F

akult?t
ter:
I
Dr.
I
olk
-
Mehrmann
Mathematik
h
und
PD

Barbara
haften
agner
der
h
T
PD
ec
Andreas

h
hen
eiterer
Univ
h
ersit?t
Prof.
Berlin
Piotr
zur
a
Erlangung
ag
des
wissensc
ak
hen

he:
hen
Berlin
Grades
D
Doktor
derolution

self-assem
Deriv
the
ation
sho
of


trac

and
tin
ork
uum
ones,
mo
spacing
dels
a
for
used
epitaxial
b
gro

wth
observ
of
arra
thin
bigger
solid
quan
lms
rip
on
W

driving
sub-

strates
parameter
yields
metho
Cahn-Hilliard
een
t
[102
yp
sim
e
y
equations
y
of
pseudosp
fourth
ed
or

sixth
to
order.
ed
T
ening
o
small
describ
hed
e

and
the
understand
expressions
solutions
the
and
solutions
solution
phase
spaces
hnique
to

these
solutions
semi-
for
or
mo
quasilinear
tum
partial
extends
dieren
align
tial
an
equations

(PDEs),
Stranski-Krastano
the
out.
dev
fourth
elopmen
eect
t

of
ulations
elab
While
orated

theory

is
undreds

so
.

Existence
Higher
of
smaller
solutions
or
has
Ost
to
ords:
b
tin
e
pseudosp
sho
exp
wn
solutions,
in
y
un
h
t
related
ypical
b
high
and
order
found
Sob
parameter
olev
of
spaces,
strength.
the
e
n
e
umerics
and
has
uation
to
ws
b
them
e
in

The
to
b
deal
initial
with
n
high
A
order
for
deriv
of
ativ
has
es
ed.
for
w
the
T
time-dep
Sp
enden
b
t
surface
problems
atomic
and
that
with
of
high
gro
order
e
phase
linear
spaces
to
for
quasilinear
the
the
stationary
the

whic
and
b
metho
in
ds
on
of
metho
matc
the
hed
the
asymptotics
b
require
w
matc
h
hing

at

man
sim
y
the
orders.
the
In
e
this
erimen
w
rates
ork
densities
new
are
theory
fa
is
the
presen
in
ted
ald
for


bly
mo
dots,
dels
mo
of
e
high
metho
order.
energy
F
tial
or
existence
a
w
sixth
linear
order
[62
PDE
the
that
ump
describ
is
es
to
the
Lam

ert
of
function
a
analytical
gro
are
wing
for
surface
far-eld
in
in
2D
limit
[89
small

force
it
These
is
liv
sho
in
wn
v
that
dimensional
w
space
eak
a
solutions
tin
exist.

While
allo
for
to
a
k
related
on

hes
v
a
ectiv
plane.
e
asymptotic
Cahn-Hilliard

equation
e
pro
as
ving
input
the
the
existence
umerical
of
d.
absorbing
new
balls
del

the
brings
bly
along
quan
existence
dots
of
b
solutions
deriv
[24
It

a
estimates
ork
for
y
the
ek
sixth
and
order

mo

del
y
are

more
energy
dicult
an
to
ux
obtain
h
since

the
ulations

a
surface
v
energy
wth
leads
b
to

undesired
A
terms.
stabilit
The
analysis
problem
the
is
order
solv
PDE
ed
ws
b
destabilizing
y
of
application
anisotrop
of
,
fractional
h
op
also
erators
e
to
ed
deriv
sim
e
based
lo
a
w
ectral
er
d.
order
in
b
w
ounds
for
from

a
single
transformed
ell-shap
equation,
dots
whic
ere
h
here
are
uge
then
ys
used
h
to
of
obtain
nanoislands
higher
are
order
ulated,
b
that
ounds
ev
from
of
the

original
b
equation.

Next,
exp
new
ts.
t
ux
yp
yield
es
island
of
and
stationary
dots
solutions
absorb
are
in
found
v
b
of
y
bigger
an
resulting
extension
an
of
w
a
rip
metho
pro
d
Keyw
of
Self-assem
matc
of
hed
tum
asymptotics

where
uum
exp
deling,
onen
slop
tially

small
ectral
terms
d,
are
surface
retained.
,
By
onen
using
matc
this
asymptotics,
generalization
of
of
Ost
the
ald
ansatz
ening,
b
stabilit
y
analysis
Lange.
Con
2
ten
.
ts
.
In
.
tro
.

.
1
.
0.1
Spaces
A
.
quan
.
tum
.
of
gro
self-assem
HCCH
bled
2.2
solids
2.3
.
and
.
of
.

.
.
.
.
.
.
.
.
.
.
.
of
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
42
.
.
.
and
.
.
.
.
.
solutions
.
.
.
d
.
.
.
onen
1
.
0.2
w
Pro
.

.
pro
.

.
and
Deriv
applications
The
of
.
quan
.
tum
.
dots
.
.
.
.
.
.
.
.
.
.
systems
.
.
.
.
.
.
.
functional
.
.
.
.
.
fractions,
.
.
.
.
5
time,
0.3
.
Gro
HCCH
wth
.
t
.
yp
Stationary
es
equation
and
equation

.
prop
.
erties
A
.
.
.
.
.
.
.
.
.
asymptotics
.
.
.
.
.
3.1.3
.
data
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
HCCH
.
and
.
34
.
of
.
surface
.
.
.
.
.
.
8
.
0.4
.
Ge/Si(001)
.
quan
2.1.1
tum
.
dots
.
.
.
.
.
.
.
.
.
.
.
.
.
.
phase
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Concepts
.
.
.
.
.
.
.
.
.
.
.
Op
.
v
.
.
.
.
.
.
.
43
.
v
.
spaces,
.
useful
.
2.4
.
to
11
.
0.5
.
Con
.
ten
.
t,
.
results
53
and
and

the
of
3.1
this
the
w
.
ork
.
.
.
.
.
.
.
.
61
.
space
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3.1.2
.
matc
.
.
.
.
.
.
.
.
.
.
.
.
.
b
13
n
1
analytical
Surface
.
diusion
.
based
iii

.
tin
.
uum
.
mo
.
deling
.
17
.
1.1
.
A
.

.
ux
.
.
.
.
.
.
31
.
The
.
equation:
.
ation
.
existence
.
solutions
.
2.1
.

.
a
.
wing
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
35
.
The
.
equation
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
19
.
1.2
.
T
.
yp
36
es
Related
of
separation
surface
.
energies
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
40
.
Preliminaries:
.
from
.
analysis
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
2.3.1
.
erators,
.
eigen
.
alues
.

.
.
.
.
20
.
1.2.1
.
F
.

.
deriv
2.3.2
ativ
in
es
olving
of
dual
surface
inequalities
energy
other
form
results
ulas
46
.
Existence
.
solutions
.
the
.
equation
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
23
3
1.2.2
solutions

kink
surface
to
energy
HCCH
of
60
regular
Stationary
surfaces
to
.
HCCH
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
3.1.1
.
phase
.
metho
26
.
1.3
.
The
.
strain
.
energy
.
densit
.
y
.
for
.
Ge/Si
.
lik
.
e
.
systems
.
.
.
.
.
.
64
.
Exp
.
tial
.
hed
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
67
28
Comparison
1.3.1
et
The
een
base
umerical
state
and
.
results
.
.
.
.
.
.
.
.
.
78
.
..
3.2
.
Coarsening
.


for
ds
the
.
HCCH
.
equation
.
.
.
.
(FDMs)
.
ds
.
ectral
.
.
.
discussion
.
.
.
osition
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
149
79
.
4
.
The
4.5
QDM
.
equation:
.
Deriv
.
ation,
.
analysis
108
and
equations
sim
Finite
ulation
.
results
.
86
.
4.1
5.2
Deriv
.
ation
.
of
.
the
.
QDM

equation
.
.
.
.
PSMs
.
.
.
.
.
.
.
131
.

.

.
.
.
.
.
.
.
.
.
.
.
of
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
ev
.
p
.
114
87
metho
4.2
.
Linear
.
stabilit
.
y
.
analysis
.
.
.
.
.
.
ectral
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
5.2.1
.
tiation
.
.
.
.
.
.
.
.
.
.
.
124
.
3D
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
Summary
.
A
.
mo
.
Mathematical
.
152
.
155
94
.
4.3
.
P
.
erio
.

.
stationary
.
solutions
.
.
.
.
104
.
Eect
.
dep
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
5
.
metho
.
for
.
olution
.
on
.
erio
.
domains
97
5.1
4.4
dierence
Ev
ds
olution
.
on
.
big
.
domains
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
116
.
Pseudosp
.
metho
.
(PSMs)
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
124
.
Sp
.
dieren
.
and
.
.
101
.
4.4.1
.
Coarsening
.
of
.
t
.
w
.
o-dimensional
.
arra
.
ys
.
.
.
.
5.2.2
.
for
.
problems
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
6
.
and
102
137
4.4.2
Mathematical
Three-dimensional
Surface
self-assem
deling
bly
B
.

.
systems
.
C
.
y
.
iv
.{105}1−x x
an
F
treatmen
or
During
a

true
their
writer
w
e
In
ach
eryda
b
not
o
All
ok
ecome
should
emplo
b
of
e

a
self-assem
new
or
b
out
e
plasmas,
ginning
of
wher
with
e
in
he
are
tries

again
ater-resistan
for
other
something
to
that
to
is
the
b
surface
eyond
hnology
attainment.
founders
He
generate
should
When
always
b
try

for
on
something
Figure
that
bled
has
A
never
T
b

e
of
en
opular
done
elop
or
lengths,
that

others
the
have
h-pro
trie
equipmen
d
ossible
and
on
faile
pro
d.
mathematical
Then
of
sometimes,

with
imp
gr
It
e
that
at
ely

in
he
life.
wil
the
l
w

hanges

visible
e
eople
e
a
d.
they
Ernest
ab
Hemingw
b
a
resulting
y
inside
(1899
v
-
STM
1961)
Si
0.1
Ge
A
are
quan
reprin
tum
ermission
of
hert
self-assem
last
bled
of
solids
arious
Ma
has
yb
ery
e

true
to
scien
with
tists
v
ha

v
phones
e
Nano-coatings
a
ed
similar
of
motiv
or
ation
materials
as
of
true
Man
writers,
are
since
the
it
w
is
is
the
v

pro
hers'
w
job
ects
to
gro
try
on
for
in
something
y
that
energy
is
t
b
small
ey
added
ond
mo
attainmen
e
t.
of
Sometimes,
nano-tec
with
settles
great
ev

y
k,
The
they
of

nano-
and
mesoscopic
their
orld
results


on
hange
in
the
scale.
w
p
orld.
drink
Who
of
w

ould
ottle,
ha
do
v
think
e
out
b
ham
eliev
ers,
ed
and
in

a
the
wireless
of
w
drinking
orld
essel.
thirt
1:
y

y
a
ears
solids
1
self-assem
/Si(001)
tum
ago
quan
when
dot.
programming

tum
quan
of
0.1
ted.
mean
orien
t

prin
ted
ting
p
holes
from
in

to

pap
the
er?

Who
oration
b
nano-structures
eliev
v
ed
elds
in
engineering
ying
b
200
v
y
p
ears
and
ago?
They
These
used
ideas
dev
w
lasers
ere
short
out
a
of
e
range
pro
for
for
ev
mobile
ery
or
one


are
for
y
a
for
few

visionaries.
w
By
t
no
scratc
w
of
a
or
ma
t
jorit

y
t.
is
y
a
applications
w
p
are
and
of
understanding
the
the
existence
orld
of

the

nano-w
impro
orld,
e
of



nano-
this

ork
and
asp


Ho
epitaxial
w
wth
ev
solids
er,
the
most
are
p
fo
eople
Anisotrop
migh
of
t
surface
still
is
miss
ortan
sp
on
ots
h
where
scales.
'nano'
is
pla
to
ys
diusion
a
dels
role.
describ
Quietly
the
and
bly
unin
thin
trusiv1−x x
ensiv
resulting
w
partial
surfaces.
dieren
though
tial
already
equations
the
(PDEs)
ears,
are
are
of
of
high
situations
order
b
and
role
new
the
theory
mark
is
visionary
dev
giv
elop
prop
ed
man
to
ect
analyze
in
them
(b
on
substrate
sev
sizes
eral
oltaic
asp
ears.
ects,
explo

QDs
h

as
e
equilibrium
sizes
states,
prop
linear
er
stabilit
self-assem
y
w
or
kno
general

existence
of
of
Gen-
solutions.
on
The
m
results
on
are
small
supp
When
orted

b

y
resp
sim
nor
ulations
their
that
the
are
whic

tin
out
next
with
and
help
pro
of
F
pseudosp
is
ectral
of
metho
it
ds.
eral
One
of
of
for
the

small


but
that
mak
has
p
raised
kind
great
e
atten

tion
eryda
in
ysical

ed
t
In
y
opular
ears
54
and
ears
whic
sites,
h
or
pla
sand,
ys
sky
a
b
big
formation),
role
or
in
small
this
with
w
osited
ork
manner,
is
ed.
the
this
so-called
the
quantum
Since
dot
ordering
,
the
or
photo
also
or
nano
will
dot
tly
,
few
nanoisland
are
or
state
articial
for
atom

.
during
Quan
e
tum
displa
dots
based
(QDs)
b
are
for
v
[88
ery
quan
small
on


whose
enden
t
t
ypical
implemen
sizes
remained
range
for
b
[66].
et
shap
w
tum
een
an
1
opto
and
Com-
100

nm.
prop
This
o
scale
shap

are
the
whic
motion
alternativ
of
prop
electrons
Understanding
in

the
gro

impro
band
qualit
and
pro
holes
bly
in
from
the
life,
v
other

where
band
e
in
v
all
een

the
lab
is
eling
this

[45
h
72
a
self-organization
nanoisland
man
a
man
zero-dimensional
e

on
In
sho
Figure
2
1
ater
one
in

on
h
atterning
p
observ
yramidal
scales

termediate

w
germanium
v
and


they
gro
made
wn

on
are

to
with
a
a
of
particular
e

understanding
grid
hanisms
orien
bly
tation,

Si
and
The
the
lms.
diusion

for
Ge
the
2
on
et,
mark
as
neither
/Si(001)
LEDs,

as
is
v
made
systems
visible
lasers,
b

y
gro
scanning

tunneling
in

next
y
y
(STM).
Predictions
Anisotrop
made
y
h
leads
that
to
demand
preferred
the

y

will
during
de
gro
the
wth,
v
so
y
that
when

ys

lasers
slop
on
es
will
app
e
ear

that
the
are
et
t

ypically
urthermore
small.
tum
In
based
the
QDs

under
the
h,
v
indep

tly
scale
the
is
yp
exaggerated.
of
A
tation
nano
has
dot
a

goal
b
sev
e


The
and
and
the
es
band
quan
gap
dots
energy
relev
is
t

their
related
electronic
to
erties.
its
plicated
size.
hing
The
hniques
w
e
a
er
v
trol
e
v
length
the
of
es,
the
they
emitted
exp
ligh
e,
t
h
is
es

e
trollable
bly

erties

opular.
trol
and
o
erly
v
this
er
of
the
wth
gro
ould
wth
v
is
the
ac
y
hiev
the
ed.

This
Self-assem
prop
is
ert
wn
y
ev
mak
y
es
so
articial
y
atoms
ph
v
systems
ery
patterning
useful
b
for
observ
opto
ha
electronic
e

b

analyzed.
h
particular
as
eld
LEDs
uids
or
p
for
in
lasers.
resp
The
(e.g.

,
of
,
solar

p
erally
o
app
w
in
er
y
systems
at
with
y
higher
b

it
v
glaciers,
ersion
stalagmites

as

wn
solar
Figure

on
of
w
third
or
generation
ud,
[20
the

and


is
P
a

promising
e
idea
ed
for
large
the
(cloud
application
in
of
scales
the
oiling
nano
ater)

on

ery
of
scales
the
so

that
t

gro
e
wth
visible
of

the
es.
photo
atoms
v
dep
oltaic
on
industry
a
,
in
an
suitable
eectiv
formation
e
patterns
implemen
b
tation
observ

An
rev
of
olutionize

the
of
nano
assem

is
mark
to
et.
trol
Although
distribution
so
the
far
of
QDs
nano-structures.
do
surface
not
is
pla
onsible
y
the
a
of
ma
surface
jor∗ ∗

∗ ∗

h : Ω × I → R, (x,y,t) 7! h(x,y,t),T
2Ω ⊂R
2Ω = [0,L] I = [0,T]T
(x,y) ∈ Ω t ∈ IT
∗ ∗h : Ω × I × P → R, (x,y,t,p ,...,p ) 7! h (x,y,t,p ,...,p ), (p ,...,p )∈PT 1 k 1 k 1 k
h
a
bly
;
on
nonlinearities.
small
One
scales.
In
(c)
(e)

o

for
of
lm
the
mathematics

e
kr
will
mem
is
b

ers
endencies
Mr.
Chiara
Mark,
(d)
Daveyb
patterns,
ot,
semi-
self-assem

with
e,
mo
tr

acito
p
dd.
at
atoms,
)
its
terv
analysis
m
is
w
fundamen
function
tal.
p
The
ersit

substrate
tral
(c)
fo


t
in

this
(reprin
thesis
that
lies
New
in
e
elab
ativ
orate
t

ulation
tin
hosen
uum
(f
mo
(a),
deling

based
e
on
b
surface
formation
diusion,
;
mo
time
del
so

erall
analysis
(e)
and
and
sim
e.g.
ulation

of
more
self-assem
lms
bled
dew
patterning
of
of
The
thin-lm


ydrophobic
nano-structures.
dew
Here
p
not
Corp
only
ternational
QD
gro

wth
hines
will

b
Courtesy
e
of

quasilinear
but
the
also
their
thin
elds

to
surfaces
to
that
high
undergo
and

are
during
hes
gro
and
wth.
b
Mathematically

a
examples
big
)
task
and
lies
(b),
in
and
the
erio
analysis
y
of
b
the
assumed
prop
the
erties
oundaries.
of
A
a
Cloud
smo
(f
oth
Si,Ge
real
a
function,
in
;
al
for
that
w
v
a
ud
ector
Drying
parameters
[72]);
material
with
erties
ork
h
her

.


or
a
wth
with
lik
dep
temp
see
or
olymer
osition
etting
Ho
(d)
ev
Sydney;
these
y
tities
Univ
usually
Neto,
bined
of
a
(image
v
and
The
h

ets
nondimensionalization
lm
applied
olymer
w
A
with
oration);
units
hines
this
Business
es
In
rst
while
preferable.
2009

yrigh
of
oration,
studies
Corp
dep
where
are
Business
sho
ternational
substrate
of
hallenge
t
in
high
explicitly
Mathematically
plates
is
glass
v
een
of
w

et
prop
b

ater
as
W
y
(b)
ts
;
gro
desert

the
e
in
erature
Sand
dep
(a)
rate.
2:
w
Figure
er,
3
quan
solids
are
bled

self-assem
in
of
few
tum
ariables.
quan
mathematical
A
of
that
is
describ
to
es
ork
the
dimensionless
ev
and
olution
mak
of
the
a
notation
self-assem
During
bled
t
surface
parameter

the
similar
endencies
to
then
the
discussed.
thin

lm
lies

the
in
order
Figure
the
2
or
(c).
PDEs
Here
dene
Si(001)
functions
a
as
on
solutions.
theory
dieren
in
0.1
t
of
of
gro
has
diering
b
their
established
their
deal
el
the
detail
deriv
their
es
foundation
the

There
the
dieren
innite

domain.
for
T
deling
ypically
sim
it
will
epitaxial
is
wth,
a
in
xed
scale,
domain
lev
with
of

and
hitz
mathematical
b
[106
oundary
On
or

anx 1−x
1.
mo
[69
dels
are
describ

e
previously
atom
83
in
PDE

89
and
ts
are
a
quite
gro

w
(molecular
the

more
metho
found
ds,
a
Mon
tain
te-Carlo
tify
sim
e
ulations

[49
idea

45
but
in
the
ansatz

has

the
for
gained
in

v
uum
estigating
[16
long-time
116
b
throughout
eha
b
vior
easier,
of

big
b
arra
the
ys
example
of
ork
QDs
are
are
wth

quarter
e.
ys
Only
systems
early
for
stages

of
oth
the
surface
self-assem

bly
dened
and
surface
relativ
o
ely
on
small
y
arra
ortan
ys
A

publications
b
surfaces
e
wing
studied.
27
Mean-eld
102
mo
far
dels
will
as
in
in
mo
the
simpler
w
nonlinearities,
ork
stable.
b

y
y
Ross
suitable
et
has
al.
of
for
es
QD
mo
self-assem
Homsy
bly
pap
[86
h

for

the
yield
equations
a


since
description
The
of
quan
the
in
island
of
distribution,
allo
but
A
they
bly
do
is
not


y
t
mo
for
from
the
form
shap
A
es
oten
of
b
the
mo
dots
driving
and
Ov
the
t
la
the
y
dels
out

of
y
the
more
arra
the
ys,
eects
whic
gro
h
of
are
app
essen

tial
for
information
b
needed
the
for
of
the
26
an
42

97
impro
103
v
It
emen

t
detailed
of
e
the
w

in
prop
tin
erties.
often
Con

tin
that
uum
terms
mo
sim
dels
and
describ
small
e
dieren
the
length
ev
e
olution
to
of
terms
islands,
that

neglected.
p
een
ositions
the
and

shap
Na
es

on
to
the
(see
analyzed
therton
time
or
domain.


the
surface
y
energy
al.
is
mo
used
t
to
ed).
describ

e
to
gro
the
wing
thin


surfaces
is
that
out
ha
tury
v
of
e

preferred
dot
orien
of
tations.
[39
F
and
or
phase
mo
e
deling
[79
of
109
QD
mo
gro
QD
wth
b
an

additional
the
stress
del
description
an
is
dieren
needed.
existing
Opp
the
osite
deling
to
originates
homo
Mullins'
epitaxy
diusion
,
ula
whic

h

is
p
a
tial
one
to
solid
e
system,
to
in
del
hetero
forces
epitaxy
the
lm
diusion.
and
er
substrate
last
ha
w
v

e
resulting
dieren
mo
t
based
lattice
this
spacings.
h
The

lm
b
gro

ws
and

of
tly
imp
on
t
top
that
of
the
the
wth.
substrate,
few
so
the
that
tly
a
eared

on
strain
tin

theory
stresses
self-arranging
inside

b
e
oth
in
solids
follo
that
list

references
ete
,
with
,
the
,
surface
,
energy
,
.
,
T
,
ypically
,
linear


is
y
from
theory
and
in
more
form
discussion
of
b
the

Na
the

ork,
h
particular
y
Chapter
equations
Con
is
uum
applied.
dels
These


e
b
to
e
PDEs
solv

ed
less
n
and
umerically
making
in
ulations
terms
faster
of
more
a
Therefore
three-
quotien
dimensional
of
nite
t
elemen

t
scales

b
de
emplo
as
ed
done
iden
b
small
y
in
Zhang
expansions
et

al.
e
[117
This
,
b
118
done

in
Without
eld
optimized
uid
FEM
where

full
des
vier-Stok
and
equations
high-sp
b
eed


lubrication
hines
dels
this
for

A
h
and
again

results
for
in
more
run
t
time
er
problems
w
for
b
large-scale
M?nc
sim
et
ulations
[70
whic
where
h
dels
motiv
dieren
ates
slip-regimes
the
deriv
idea
Here
to

deriv
hes
e
applied
simplied
obtain
expressions
describing
that
gro
do
of
not
solid
rely
lms
on
an
FEM
that

pursued
T
ab
w
a
o

equations
.
are

deriv

ed
wing
and
and
analyzed
tum
in
arra
this
reminds
w

ork.
liquids
One
,
describ

es
also
the
solid

separating
of
lik
a
binary
gro
ys
wing
,
surface
,
and

the
new
other
del
the
the
ev
self-assem
olution
will
of
e
Ge/Si
tro
or
it
Si
probably
the
most
Ge
mo
el
for
lev
h
4
Ost
rip
ald
system.
ening
/Si
A
QDs.
t,
F
et
or
bx 1−x
◦> 500
{105}
allo
for
some
the
man

form
of
olution.
a
it
gro
instabilit
wing
v
surface
(see
will
t
b
imp
e
substrate
analyzed
and
on
Asaro-Tiller-Grinfeld
man
ha
y

asp
see
ects.
la
A
from
detailed
a
listing
to
of
Only
the
self-assem
results
planar
will
b
b

e
of
giv
h
en
that
in
dots

bigger
0.5,
ell
where
of
also
they
the

general
QDs

distributed
of
island
the
ts
thesis
from
is
y
explained.
for
Readers
observ
familiar
dieren
with
metho

or
epitaxy
initially
and
an
in
h
particular
the
with
stresses
the
t

sur-
of
bulk
self-assem
y
bled
pr
QDs
e
migh
pro
t
bases
w

an
v
t
dev
to
ening
jump
elds,
forw
other
ard
mo
and
al.

in
tin
,
ue
e
with
the
this
sucien

while
others
w

of
get
Sim
an
e
o

v
ork
erview
tal
o
questions
v
some
er
The
these
ndings
imp

ortan
self-assem
t
w
asp
materials
ects
the
and
for
an
.
understanding
lm
for
binations
wh
gro
y
dev
self-assem
y
bled
A

analogously

w
b
of
e
leads
v
are
ery

v

aluable.
,
Observ
energy
ations
of
during
is
the
TG)
dep

osition
often
pro
amids

ev
in
p
the
throughout
Ge/Si
(these
system,
e
whic
mo
h
explained
is
Smaller
qualitativ
in
ely
of
v
whic
ery
more
similar
h
to
are
Si
in

example
Ge
mixtures
pro
b

(for
Pro
pap
/Si,
ton
are

outlined
app
step-b
droplet
y-step.
example
These
,
will
hers
b
predict
e
out
useful
ys
when
form
they
long
will
some
b
need
e
the

b
with
kno
the
w
sim
is
ulation
with
results

ac
exp
hiev
t
ed
v
in
of
this

w
exp
ork.
orks
The
op
most
sev
imp
b
ortan
them
t
tradict
asp
jor
ects
rather
of
as
QDs
y
are
in
outlined
of
on
of
the
Ross
follo
o
wing
t
pages,

ho
is
w

ev
d
er,
QD
a
bly

F
discussion

is
and
not

oered.
an
F
monotonously
or
wing
further
lm
details
elops
the
instabilit
reader
after
is
time.
referred

to
(or
a
mist)
b
et
o
een
ok
lattices
b
lm
y
substrate
F
to
reund
that
and
released
Suresh
a
[34
heigh

is
0.2
This
Pro
y

where
pro


is
and
order
applications
the
of
stresses,
quan

tum
(A
dots
instabilit
The
[22
sizes
Small
and
umps,
shap

es
e-pyr
of
,
QDs
and
dep
olv
end
to
on
yramids
man

y
the
asp

ects
islands
of
v
fabrication,
square
in
and
del
5

tum

particular
Miller
on
are
the
in
materi-
0.3).
als
islands
and
anish
temp
fa
erature
or
used
the
in
ones
the
h
gro
elop
wth
pronounced


ham
rip
b
phenomena
er.
w
Sp
analyzed
on
related
taneous
for
arrangemen
in
t
y
of
or
nano-structures
phase

oundaries
b
solids
e
a
observ
deling
ed
er
during
Thorn
a
et
pro
[104

and
that
also
is
ear

liquid
epitaxial

gro
for
wth.
[38
It
45
is
54


out
hop
at
to
high
the
tem-
y
p
of
eratures,
arra
t
of
ypically
that
quan
after
of
tly
applications
ev
and
Since
need
applications
arra
dense
0.2
ys,
equally
others
artical
and/or
sized
C,
atoms
so
equally
that
dots,
surface
nano-industry
diusion
ould
pla
enet
ys
the
a
wledge
ma
ho
jor
the
role.
distribution
A


ulations
material
dieren
is
parameters
precipitated
mak
with

a
erimen
lo
redundan
w
and
ux
sa
rate
e
on
lot
to
w
a
and
substrate.
Results
As
early
men
erimen
tioned
w
b
left
efore,
y
if
en
the
for
same
eral
material
ears,
is
ecause
used
of
for
seemed
lm

and
others.
substrate,
ma
one
reason
talks
the
ab
obscure
out
w
homo
the
epitaxial
ossibilit
gro
to
wth,
out
while
situ
hetero
ations
epitaxy
the
is
bly
the
QDs.

since
for
t1−x x
◦690

absorb
y
ening

and
hers
ears
from
and
IBM.
dep
As
a
in
as
the
the
Ge/Si(001)
their
system
since
one
A
sees
visible.
that
term
Si
also
b

wn
er
gro
of
Ge
ab
dots
ulti-faceted
dots
size
gro
redistributed
wn
one
on
p
Si(001)
also

giv
(here
bination
at
the
tum
Ho
Quan
the
3:
and
Figure
that
C
a
and
the
a
n
ux
bigger
rate
a
of
p
5
and
monola
ear
y
the
er
gro
(ML)
v
p
oring
er
place.
min
en
ute),
b
that
The
smaller
,
h
the
umps
Ge/Si(001)
v
0.4.
anish
ell
and
QDs
that
rip
bigger
en
articial
with
atoms

dev
similar
elop
QDs.
a
in
p
Here
yramidal
observ
shap
islands
e
b
that
la
ev
h
en
so
tually
v
b
b
ecomes

a
gro
m
they
ulti-faceted
length
dome-structure.
50nm,
Reprin

t
e
Courtesy

of

In
in
ternational
).
Business
v

the
hines
with
Corp
atoms
oration,
islands

neigh
yrigh
while
t
tak
2009
Figure


6
is
dots
dot

three
gro
is
whole
Ost
e
In
ue
ternational
Throughout
Business
olution

ald
hines

Corp
h
oration.
survival
and
,
her
details

epitaxy
ork
in
ers
Ge/Si
managed
and
to
material
implemen
fabrication
t
Originally
a
w
transmission
refers
electron
eect

that
y
a
(TEM)
v
apparatus
ev
that
it
w
for
orks
lik
in
ening
a
dots

seen
ham
(c)
b
(d).
er
one
during
also
gro
e
wth,
smaller
the
are
understanding
ed
of
y
the
thin
ev
y
olution
whic
of

QDs
dots,
has
that
lead
o
to
erall
a
um

er
t
islands
view
The
[85
dots

w
Results
when
of

the
base
imp
of
ortan
out
t
the
w
yramids
ork
hange
are
shap
sho
again
wn
domes
in
m
Figure
islands
3.
app
The
as
IBM
(e)

(f
visualize
Clearly
ho
a
w
erage
a
of

nano-structures
substrate
ws
is
time

the
v
from
ered
anishing
with
are
germanium
to
and
b

QDs
atoms
further
and
osition
ho
es
w
In
the
4
resulting
particular
lm
ev
ev
t
olv

es.
smaller
In
surrounded
(a)
y
a
bigger
at
yramids
state
'eaten'.
is
'fat'
visible.
surviv
A
and
toms
tin
are
to
dep
w.
osited
the

ev
tin
this
uously
w
on
rip
to
pro
the
1
surface
whic
and
is
the

lm
of
gro
fattest
ws
is
for
More
some
on
time
hetero
in
is
v
en



Although
as
is
at

surface.
w
In
studied
(b)

the
for
Asaro-
of
Tiller-Grinfeld
1
(A
the
TG)
Ost
instabilit
ald
y
ening
sets
to
in
Gibbs-Thomson
and
driv
leads

to
app
formation
in
of
system
rather
xed
rounded
olume.

w
the
er,
pre-p
tly
yramids.
is
These
used
pass
other,
in
phenomena,
to
e
p
rip
yramidally
of
formed
nano

Un pour Un
Permettre à tous d'accéder à la lecture
Pour chaque accès à la bibliothèque, YouScribe donne un accès à une personne dans le besoin