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

BMS Days Berlin 18 / 02 / 2008

The Gallium Arsenide Wafer Problem

Single crystal gallium arsenide

Arsen concentration = 0.500082



Distribution of liquid droplets






Modeling of liquid droplets


The Becker – Döring Process

1 n + 1n +

CnW

EnW 1

n-1n

EnW

CnW 1

1+



Variables

)(tn density of n - cluster

1 n + 1n +

CnW

EnW 1

n-1n

EnW

CnW 1

1+



Variables


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



Variables


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


consttnn

n

1

)(


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


)()( 11 tWtWJ nEnn

Cnn

11

)(1

JJtk

k

2)( 1 nJJt nnn

with


Different Strategies, II: Dreyer Duderstadt

Lyapunov Function

1

11

ln)(

kk

n

nnn

nnAA


)()( 11 tWtWJ nEnn

Cnn

11

)(1

JJtk

k

2)( 1 nJJt nnn

with



Lyapunov Function

1

11

ln)(

kk

n

nnn

nnAA

n n



Lyapunov Function

1

11

ln)(

kk

n

nnn

nnAA

Transition rates

))((11 tW

WEn

En

Cn

Cn


)()( 11 tWtWJ nEnn

Cnn

11

)(1

JJtk

k

2)( 1 nJJt nnn

with



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


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


)()( 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


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


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


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


number n-clusters

total volumenumber n-clusters

total number

)()( 11 tWtWJ nEnn

Cnn

11

)(1

JJtk

k

2)( 1 nJJt nnn

with


Equilibrium solutions

n n

nD

nn

qN

Q

)(


1)( ,111

1

k

kkD

k

kk

kk

k

kqNq

Qk

Constraints



n n

nD

nn

qN

Q

)(


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



n n

nD

nn

qN

Q

)(


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


nn

n

Qn1

with


nnn Q

Longtime behaviour of solutions I: Penrose et al.

nBCP

nn

BCP RnQ1

Equilibrium condition



nn

n

Qn1

with


nnn Q

nBCP

nn

BCP RnQ1


. (weak*)

(strong) )( then If 0

nBCPn

nnnBCP

BCP

RQ

Qt

Asymptotical behaviour


• 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)


. (weak*)

(strong) )( then If 0

nBCPn

nnnBCP

BCP

RQ

Qt

Asymptotical behaviour


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


Simplified Dreyer/Duderstadt Model

Longtime behaviour of solutions II: mod. Dreyer


n

nn

n

nn

n

nn

RnqRq

Rq

11

1

and1

or,1


1

11

)( ,1

k

kkD

k

kk kqNq


with )( nnDn qN

n

nn

n

nn

n

nn

RnqRq

Rq

11

1

and1

or,1


1

11

)( ,1

k

kkD

k

kk kqNq


with )( nnDn qN

for )(weak 0

(strong) )(

~ nn

depending on the Availability of the System:


Longtime behaviour of solutions II: mod. Dreyer

Longtime behaviour of solutions: GaAs wafer


1for nnaAn

Assumption

n




1for nnaAn

Assumption

1

11

)( ,1

k

kkD

k

kk kqNq


with )( nnDn qN

relevant ?




1for nnaAn

Assumption

relevant ?


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



2for)1(

)( 1

n

nn

JJt nn

n


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



2for)1(

)( 1

n

nn

JJt nn

n

))(,(

),(

tfxQ

xtfx

,Jxtf Dxt 1in ),(

fffxQxWJ xxxC

D ),(ln)(

Continuous System (Duncan)


DxJ

,Jxtf Dxt 1in ),(

fffxQxWJ xxxC

D ),(ln)(



)1,(

),(ln)(

tf

dxxtfxa

Dreyer/Duderstadt

Continuous thermodynamics

dx

tfNxq

xtfxtftfA

D ))(()(

),(ln),())((

with)()(,),())(( xa

D exqdxxtftfN


,Jxtf Dxt 1in ),(

fffxQxWJ xxxC

D ),(ln)(



)1,(

),(ln)(

tf

dxxtfxa

Dreyer/Duderstadt

Continuous thermodynamics

dx

tfNxq

xtfxtftfA

D ))(()(

),(ln),())((

Lyapunov Funktion, minimal at equilibria!


Mixed System

,MJxtf

MnJJt

dxJJJt

Dxt

nnn

D

M

kk

1in ),(

2)(

)(

1

11

1

Mixed System


Mixed System

,MJxtf

MnJJt

dxJJJt

Dxt

nnn

D

M

kk

1in ),(

2)(

)(

1

11

1

Mixed System


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


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


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?


The Gallium Arsenide Wafer Problem

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