the gallium arsenide wafer problem
DESCRIPTION
The Gallium Arsenide Wafer Problem. Industrial Needs versus Mathematical Capabilities. Margarita Naldzhieva joint work with Wolfgang Dreyer, Barbara Niethammer. DFG Research Center M ATHEON Mathematics for key technologies. - PowerPoint PPT PresentationTRANSCRIPT
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
BMS Days Berlin 18 / 02 / 2008
The Gallium Arsenide Wafer Problem
Single crystal gallium arsenide
Arsen concentration = 0.500082
BMS Days Berlin 18 / 02 / 2008
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
BMS Days Berlin 18 / 02 / 2008
The Becker – Döring Process
1 n + 1n +
CnW
EnW 1
n-1n
EnW
CnW 1
1+
BMS Days Berlin 18 / 02 / 2008
The Becker – Döring Process
Variables
)(tn density of n - cluster
1 n + 1n +
CnW
EnW 1
n-1n
EnW
CnW 1
1+
BMS Days Berlin 18 / 02 / 2008
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
BMS Days Berlin 18 / 02 / 2008
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
BMS Days Berlin 18 / 02 / 2008
)()( 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
BMS Days Berlin 18 / 02 / 2008
Different Strategies, II: Dreyer Duderstadt
Lyapunov Function
1
11
ln)(
kk
n
nnn
nnAA
Transition rates
))((11 tW
WEn
En
Cn
Cn
BMS Days Berlin 18 / 02 / 2008
)()( 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
BMS Days Berlin 18 / 02 / 2008
)()( 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
BMS Days Berlin 18 / 02 / 2008
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
BMS Days Berlin 18 / 02 / 2008
nn
n
Qn1
with
Equilibrium solutions
nnn Q
Longtime behaviour of solutions I: Penrose et al.
nBCP
nn
BCP RnQ1
Equilibrium condition
BMS Days Berlin 18 / 02 / 2008
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
BMS Days Berlin 18 / 02 / 2008
• 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
BMS Days Berlin 18 / 02 / 2008
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!
Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008
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
Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008
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:
Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008
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 ?
BMS Days Berlin 18 / 02 / 2008
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
Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008
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)
Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008
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
Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008
,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!
Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008
Mixed System
,MJxtf
MnJJt
dxJJJt
Dxt
nnn
D
M
kk
1in ),(
2)(
)(
1
11
1
Mixed System
Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008
Mixed System
,MJxtf
MnJJt
dxJJJt
Dxt
nnn
D
M
kk
1in ),(
2)(
)(
1
11
1
Mixed System
Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008
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
Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008
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
Mathematics for key technologies and innovation Warsaw 21 - 22 / 02 / 2008
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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