Of the High Performance Gallium Arsenide (GaAs) Nanowires (NWs)
The Gallium Arsenide Wafer Problem
description
Transcript of The Gallium Arsenide Wafer Problem
DFG Research Center MATHEONMathematics for key technologiesBMS Days Berlin 18 / 02 / 2008
The Gallium Arsenide Wafer
Problem
Margarita Naldzhieva
joint work with Wolfgang Dreyer, Barbara Niethammer
Industrial Needs versus Mathematical Capabilities
Outline
The Becker-Döring model
From Becker-Döring to Fokker-Planck
Industrial problem
Quasi-stationary and longtime behaviour of solutions
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The Gallium Arsenide Wafer Problem
Single crystal gallium arsenide
Arsen concentration = 0.500082
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The Gallium Arsenide Wafer Problem
Distribution of liquid droplets
Single crystal gallium arsenide
BMS Days Berlin 18 / 02 / 2008
The Gallium Arsenide Wafer Problem
Single crystal gallium arsenide
Distribution of liquid droplets
Modeling of liquid droplets
BMS Days Berlin 18 / 02 / 2008
The Gallium Arsenide Wafer Problem
Single crystal gallium arsenide
Distribution of liquid droplets
Modeling of liquid droplets
BMS Days Berlin 18 / 02 / 2008
The Gallium Arsenide Wafer Problem
Single crystal gallium arsenide
Distribution of liquid droplets
Modeling of liquid droplets
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The Becker – Döring Process
1 n + 1n +
CnW
EnW 1
n-1n
EnW
CnW 1
1+
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The Becker – Döring Process
Variables
)(tn density of n - cluster
1 n + 1n +
CnW
EnW 1
n-1n
EnW
CnW 1
1+
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The Becker – Döring Process
Variables
)(tn density of n - cluster
1 n + 1n +
CnW
EnW 1
n-1n
EnW
CnW 1
1+
)()( 11 tWtWJ nEnn
Cnn
11
)(1
JJtk
k
2)( 1 nJJt nnn
with
Becker-Döring system
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The Becker – Döring Process
Variables
)(tn density of n - cluster
1 n + 1n +
CnW
EnW 1
n-1n
EnW
CnW 1
1+
)()( 11 tWtWJ nEnn
Cnn
11
)(1
JJtk
k
2)( 1 nJJt nnn
with
Becker-Döring system
consttnn
n
1
)(
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Different Strategies, I: Ball Carr Penrose
Lyapunov Function
)1(ln)(1
n
n
nn Q
V
Transition rates
11
1 )(
nEn
nCn
bW
taW Equilibria
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)()( 11 tWtWJ nEnn
Cnn
11
)(1
JJtk
k
2)( 1 nJJt nnn
with
Becker-Döring system
Different Strategies, II: Dreyer Duderstadt
Lyapunov Function
1
11
ln)(
kk
n
nnn
nnAA
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)()( 11 tWtWJ nEnn
Cnn
11
)(1
JJtk
k
2)( 1 nJJt nnn
with
Becker-Döring system
Different Strategies, II: Dreyer Duderstadt
Lyapunov Function
1
11
ln)(
kk
n
nnn
nnAA
n n
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Different Strategies, II: Dreyer Duderstadt
Lyapunov Function
1
11
ln)(
kk
n
nnn
nnAA
Transition rates
))((11 tW
WEn
En
Cn
Cn
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)()( 11 tWtWJ nEnn
Cnn
11
)(1
JJtk
k
2)( 1 nJJt nnn
with
Becker-Döring system
Different Strategies, II: Dreyer Duderstadt
Lyapunov Function
1
11
ln)(
kk
n
nnn
nnAA
Transition rates
))((11 tW
WEn
En
Cn
Cn
2nd law of thermodynamics
)(
)(exp
))((
1
11
1
t
tAA
t nn
nnCn
En
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Comparison of Transition Rates
Evaporation rates
EnW 1
))((1 tEn
1nb
Condensation rates
CnW
Cn
)(1 tan
2nd law of thermodynamics
)(
)(exp))((
1
111 t
tAAt n
n
nnCn
En
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)()( 11 tWtWJ nEnn
Cnn
11
)(1
JJtk
k
2)( 1 nJJt nnn
with
Lyapunov Functions
Lyapunov Functions
)1(ln)(1
n
n
nn Q
V
)ln()(
1
11
kk
n
nnn
nnAA
)(L
0dt
dL
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NNNN
Water droplets in vapour
p0T0
0 20 40 60 80 100 120 140Number of droplet atoms
1.5 107
1 107
5 108
0
T0 275 °C p011
An
n
NNNN
p0T0
0 20 40 60 80 100 120 140Number of droplet atoms
1.5 107
1 107
5 108
0
T0 275 °C p011
An
n
p0
Quasi stationary Flux
0 20 40 60 80 100time intau
0
2.51017
51017
7.51017
11018
1.251018
1.51018
Jni
s^
1mc
^3
T2 °C
S
12.8
J25(t)
nmax = 40
nmax = 50
nmax = 70
t
NNNN
Quasi stationary Flux
p0
0 20 40 60 80 100time intau
0
2.51017
51017
7.51017
11018
1.251018
1.51018
Jni
s^
1mc
^3
T2 °CS
12.8
J25(t)
nmax = 40
nmax = 50
nmax = 70
J25(t)
T0
p0
Model system for calculation
Skim of droplets with nmax + 1 atoms: zn + 1= 0
monomer density constant
max
NNNN
Quasi stationary Flux
p0
0 20 40 60 80 100time intau
0
2.51017
51017
7.51017
11018
1.251018
1.51018
Jni
s^
1mc
^3
T2 °CS
12.8
J25(t)
nmax = 40
nmax = 50
nmax = 70
0 20 40 60 80 100time intau
0
2.51017
51017
7.51017
11018
1.251018
1.51018
Jni
s^
1mc
^3
T2°C
S
12.8
t t
Model system
J25(t)
T0
p0
Model system for calculation
Skim of droplets with nmax + 1 atoms: zn + 1= 0
monomer density constant
max
Equilibrium solutions : Jn=0
Fluxes
nJ111 nnnn cbcca
Conservation of mass
1
)(1
tnn
n
Variables
)(tn
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number n-clusters
total volumenumber n-clusters
total number
)()( 11 tWtWJ nEnn
Cnn
11
)(1
JJtk
k
2)( 1 nJJt nnn
with
Becker-Döring system
Equilibrium solutions
n n
nD
nn
qN
Q
)(
Equilibrium solutions : Jn=0
1)( ,111
1
k
kkD
k
kk
kk
k
kqNq
Qk
Constraints
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Equilibrium solutions
n n
nD
nn
qN
Q
)(
Equilibrium solutions : Jn=0
1)( ,111
1
k
kkD
k
kk
kk
k
kqNq
Qk
Constraints
Convergence radii RBCP and RDD
nDD
nn
DDD
nBCP
nn
BCP
RqN
RnQ
1
1
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Equilibrium solutions
n n
nD
nn
qN
Q
)(
Equilibrium solutions : Jn=0
Constraints
1)( ,111
1
k
kkD
k
kk
kk
k
kqNq
Qk
Convergence radii RBCP and RDD
nDD
nn
DDD
DDD
BCP
RnqNN1
and1or,1
Equilibrium conditions
nDD
nn
DDD
nBCP
nn
BCP
RqN
RnQ
1
1
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nn
n
Qn1
with
Equilibrium solutions
nnn Q
Longtime behaviour of solutions I: Penrose et al.
nBCP
nn
BCP RnQ1
Equilibrium condition
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Longtime behaviour of solutions I: Penrose et al.
nn
n
Qn1
with
Equilibrium solutions
nnn Q
nBCP
nn
BCP RnQ1
Equilibrium condition
. (weak*)
(strong) )( then If 0
nBCPn
nnnBCP
BCP
RQ
Qt
Asymptotical behaviour
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• existence of metastable states (Penrose 1989)
• excess density described by the LSW equations (Penrose 1997, Niethammer 2003)
• convergence rate to equilibrium e-ct1/3 (Niethammer, Jabin 2003)
Longtime behaviour of solutions I: Penrose et al.
. (weak*)
(strong) )( then If 0
nBCPn
nnnBCP
BCP
RQ
Qt
Asymptotical behaviour
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Modified Flux
11
11exp
n
kk
nnCnn
Cnn AAJ
)exp(1
11
111
n
kknn
Cnn
Cn AA
nJ~
Simplified Dreyer/Duderstadt Model
Current state of the artMathematical results for a modified DD model!
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11
11
~~exp
~~~
n
kknn
Cnn
Cnn AAJ
11
1
1
~~)(
~
2~~
)(~
JJ
nJJ
kk
nnn
with
Modified Becker-Döring system
New time scale
t
dss0 1 )(
1
)()(
~tnn
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Simplified Dreyer/Duderstadt Model
Longtime behaviour of solutions II: mod. Dreyer
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n
nn
n
nn
n
nn
RnqRq
Rq
11
1
and1
or,1
Equilibrium condition
1
11
)( ,1
k
kkD
k
kk kqNq
Equilibrium solutions
with )( nnDn qN
n
nn
n
nn
n
nn
RnqRq
Rq
11
1
and1
or,1
Equilibrium condition
1
11
)( ,1
k
kkD
k
kk kqNq
Equilibrium solutions
with )( nnDn qN
for )(weak 0
(strong) )(
~ nn
depending on the Availability of the System:
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Longtime behaviour of solutions II: mod. Dreyer
Longtime behaviour of solutions: GaAs wafer
Distribution of liquid droplets
1for nnaAn
Assumption
n
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Longtime behaviour of solutions: GaAs wafer
Distribution of liquid droplets
1for nnaAn
Assumption
1
11
)( ,1
k
kkD
k
kk kqNq
Equilibrium solutions
with )( nnDn qN
relevant ?
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Longtime behaviour of solutions: GaAs wafer
Distribution of liquid droplets
1for nnaAn
Assumption
relevant ?
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0 20 40 60 80 100time intau
0
2.51017
51017
7.51017
11018
1.251018
1.51018
Jni
s^
1mc
^3
T2 °C
S
12.8
J25(t)
nmax = 40
nmax = 50
nmax = 70
t
J25(t)
n
n
n
nn
Cnn QQQWJ
1
1
1, 111 QQWQW nEnn
Cn)()( 11 tWtWJ n
Enn
Cnn
11
)(1
JJtk
k
2)( 1 nJJt nnn
with
Becker-Döring system
From Becker-Döring to Fokker-Planck
2for)1(
)( 1
n
nn
JJt nn
n
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n
n
n
nn
Cnn QQQWJ
1
1
1, 111 QQWQW nEnn
Cn)()( 11 tWtWJ n
Enn
Cnn
11
)(1
JJtk
k
2)( 1 nJJt nnn
with
Becker-Döring system
From Becker-Döring to Fokker-Planck
2for)1(
)( 1
n
nn
JJt nn
n
))(,(
),(
tfxQ
xtfx
,Jxtf Dxt 1in ),(
fffxQxWJ xxxC
D ),(ln)(
Continuous System (Duncan)
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DxJ
,Jxtf Dxt 1in ),(
fffxQxWJ xxxC
D ),(ln)(
Continuous System (Duncan)
From Becker-Döring to Fokker-Planck
)1,(
),(ln)(
tf
dxxtfxa
Dreyer/Duderstadt
Continuous thermodynamics
dx
tfNxq
xtfxtftfA
D ))(()(
),(ln),())((
with)()(,),())(( xa
D exqdxxtftfN
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,Jxtf Dxt 1in ),(
fffxQxWJ xxxC
D ),(ln)(
Continuous System (Duncan)
From Becker-Döring to Fokker-Planck
)1,(
),(ln)(
tf
dxxtfxa
Dreyer/Duderstadt
Continuous thermodynamics
dx
tfNxq
xtfxtftfA
D ))(()(
),(ln),())((
Lyapunov Funktion, minimal at equilibria!
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Mixed System
,MJxtf
MnJJt
dxJJJt
Dxt
nnn
D
M
kk
1in ),(
2)(
)(
1
11
1
Mixed System
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Mixed System
,MJxtf
MnJJt
dxJJJt
Dxt
nnn
D
M
kk
1in ),(
2)(
)(
1
11
1
Mixed System
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fffxQxWJ xxxC
D ),(ln)(
Mixed System
,MJxtf
MnJJt
dxJJJt
Dxt
nnn
D
M
kk
1in ),(
2)(
)(
1
11
1
Mixed System fffxQxWJ xxxC
D ),(ln)(
Thermodynamics and mass conservation
constdxxtxftn
dxtfNxq
xtfxtf
NqtfA
M
nn
D
M
n Dn
nn
),()(
))(()(
),(ln),(
)(ln))((
1
1
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Mixed System
,MJxtf
MnJJt
dxJJJt
Dxt
nnn
D
M
kk
1in ),(
2)(
)(
1
11
1
Mixed System fffxQxWJ xxxC
D ),(ln)(
)1,(
0lim and
1
1
Mf
xJJJ
M
Dx
MMxD
Boundary values
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Summary and outlook
thermodynamically consistent nucleation rates
existence of a metastable phase befor equilibrium?
Becker-Döring model for homogeneous nucleation
equilibrium solutions and asymptotical behaviour
Duncan`s PDE approximation of the discrete System?
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