alloy is generally used in the form of epilayers that have to be ..... U. +. = , and solute concentration is given in terms of silicon mass fraction. ..... to its higher carrier mobility, amendable band gap energy at any value between silicon ...... was interpreted as a switch over from germanium like L -band structure to the silicon like.
A Combined Experimental and Modeling Study for the Growth of SixGe1-x Single Crystals by Liquid Phase Diffusion (LPD) by
Mehmet YILDIZ B.A.Sc., Yildiz Technical University, 1996 M.A.Sc., Istanbul Technical University, 2000
A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of
DOCTOR OF PHILOSOPHY in the Department of Mechanical Engineering
© Mehmet Yildiz, 2005 University of Victoria All rights reserved. This dissertation may not be produced in whole or in part by photocopy or other means, without the permission of the author.
ii Supervisor: Dr. S. Dost
Abstract
Si x Ge1− x alloy is an emerging semiconductor material with many important potential
applications in the electronic industry due to its adjustable physical, electronic and optical properties. It has been scrutinized for the fabrication of high-speed micro electronics (e.g., SiGe heterojunction bipolar transistors (HBT) and high electron mobility field effect
transistors)
and
thermo-photovoltaics
(e.g.,
photodetectors,
solar
cells,
thermoelectric power generators and temperature sensor). Other applications of Si x Ge1− x include tuneable neutron and x-ray monochromators and γ-ray detectors. In these applications, Si x Ge1− x alloy is generally used in the form of epilayers that have to be deposited on a lattice-matched substrate (wafer). Therefore, Si x Ge1− x bulk single crystals with a specific composition ( x ) are needed for the extraction of such wafers. LPEE (Liquid Phase Electroepitaxy) was considered as a technique of choice for the growth of single crystals. However, LPEE growth process needs a single crystal seed with the same composition as the crystal to be grown. Yet, such a seed substrate with particularly higher composition is not commercially available. In order to address this important issue in LPEE, a crystal growth technique, which is named “Liquid Phase Diffusion” (LPD), was developed and used to produce the needed seed substrate materials. This was the main motivation of the present research. This thesis presents a combined experimental and modelling study for LPD growth of compositionally graded, germanium-rich single crystals of 25 mm in diameter for use as lattice-matched seed substrates. The experimental part focuses on the design and development of a complete LPD grow system. The experimental set-up was tested by growing ten Si x Ge1− x single crystals. Grown crystals were characterized by macroscopic and microscopic examinations after chemical etching for delineation of the degree of
Abstract
iii
single crystallinity and growth striations. Compositional mapping of selected crystals were performed by using Electron Probe Micro Analysis (EPMA) as well as Energy Dispersive X-ray analysis (EDX). It was shown that the LPD technique can be successfully utilized to obtain Si x Ge1− x single crystals up to 6-8 % at.Si with uniform radial composition distribution. The modeling part presents a rational continuum mixture model developed to study transport phenomena (heat and mass transfer, fluid flow) occurring during the LPD growth of Si x Ge1− x . Based on the continuum model developed, two and three-dimensional transient numerical simulations were carried out. The numerical simulation models presented account for some important physical features of the LPD growth process of Si x Ge1− x , namely (1) a growth zone design on the thermal field, (2) the structure of the buoyancy induced convective flow and its effect on the growth and transport mechanisms, (3) the shape and evolution of the initial and progressing growth interfaces, and (4) the spatial and time variation of the crystal growth velocity. It was numerically shown that, as the name LPD implies, the growth of Si x Ge1− x by LPD is mainly a diffusion driven process except the initial stages of the growth process during which the natural convection in the solution zone is prominent and has significant effects on the composition of the grown crystal. The simulated evolution of the growth interface agrees with experimental observations. In addition, the numerical growth velocities are in good agreement with those of experiments. The numerical model developed can be used to study other crystal growth processes such as LPEE, Traveling Heater Method THM, and vertical Bridgman with slight modifications.
Supervisor: Dr. S. Dost, (Department of Mechanical Engineering)
iv
Table of Contents Abstract
ii
Table of Contents
iv
List of Figures
ix
List of Tables
xiii
Nomenclature
xiv
Acknowledgements
xxiv
Dedication
xxv
1 Introduction
1
1.1
Motivation and Goals………………………………………………………
1
1.2
Outline of the Thesis……………………………………………………….
8
2 Structural Properties of SixGe1-x Alloy System
9
2.1 Introduction………………………………………………………………...
9
2.2
Crystal Structure and Equilibrium Phase Diagram of SixGe1-x…………….
9
2.3
Band Gap Structure of Bulk SixGe1-x Alloys………………………………
16
3 Crystal Growth Methods for SixGe1-x Alloys
19
3.1 Introduction..……………………………………………………………….
19
3.2
Melt Growth Methods……………………………………………………...
19
3.2.1
Crystal Pulling………………………………………………...........
21
3.2.2
Vertical Bridgman Technique (VB) ………………………………
28
Table of Contents
v
3.2.3
Floating Zone Technique (FZ)……………………………………..
34
3.2.4
Zone Melting (ZM)………………………………………………...
36
3.2.5
Difficulties in Melt Growths……………………………………….
38
3.3 Solution Growth……………………………………………………………
39
3.4
3.5
3.3.1
Liquid Phase Epitaxy (LPE)………………………………………..
41
3.3.2
Liquid Phase Electroepitaxy (LPE)………………………………...
43
Vapour Phase Growth……………………………………………………...
45
3.4.1
Chemical Vapour Deposition (CVD)………………………………
45
3.4.2
Molecular Beam Epitaxy (MBE)…………………………………..
46
Strained and Relaxed Si1− x Ge x Layers in Si1− x Ge x / Si System……………
46
4 Applications of SixGe1-x
51
4.1 Introduction………………………………………………………………...
51
4.2 Photodetectors……………………………………………………………...
51
4.2.1 PIN Photodetector………………………………………………….
52
4.2.2 HIP Infrared Detector………………………………………………
53
Heterojunction Bipolar Transistor…………………………………………
54
4.3.1
SiGe/Si HBT for RF Applications…………………………………
55
4.3.2 High Electron Mobility Transistor…………………………………
56
4.4 Solar Cells………………………………………………………………….
57
4.5
58
4.3
Summary…………………………………………………………………...
5 A Rational Continuum Mixture Model for LPD Growth of SixGe1-x
59
5.1 Introduction………………………………………………………………...
59
5.2
The Rational Mixture Model……………………………………………….
63
5.2.1
Kinematics of Mixtures…………………………………………….
63
5.2.2
Axioms of Thermomechanics and the Field Equations……………
73
5.2.2.1
Balance of Mass………………………………………….
74
5.2.2.2
Balance of Linear Momentum……………………………
76
Table of Contents
vi
5.2.2.3
Balance of Moment of Momentum………………………
81
5.2.2.4
Balance of Energy………………………………………..
82
5.2.2.5 Entropy Inequality………………………………………..
87
Constitutive Equations……………………………………………..
91
5.2.3.1
The Liquid Phase…………………………………………
91
5.2.3.2
The Solid Phase…………………………………………..
100
5.2.3.3
Interface Conditions……………………………………...
105
Governing Equations and Boundary Conditions for the Computational Model…...………………………………………………………………..... 5.3.1 The Liquid Phase…………………………………………….……..
106
5.3.2
The Solid Phase…………………………………………………….
112
5.4
Physical Parameters of the LPD SixGe1-x System………………………….
113
5.5
Summary…………………………………………………………………...
116
5.2.3
5.3
107
6 Numerical Simulation for the Growth of SixGe1-x Single Crystals by LPD 117 6.1 Introduction………………………………………………………………...
117
6.2
Order of Magnitude Analysis for the LPD Growth System………………..
118
6.3
Discretisation Method and Solution Algorithms…………………………...
120
6.4
Finite Volume Mesh………………………………………………………..
126
6.5
Grid Modification and Interface Movement……………………………….
127
6.6
Results of Two-Dimensional Numerical Simulation………………………
131
6.6.1
Temperature Field………………………………………………….
131
6.6.2 Flow and Concentration Fields…………………………………….
135
6.6.3 Growth Velocity……………………………………………………
142
6.7
Outcomes of Three-Dimensional Numerical Simulation…………………..
146
6.8
Summary…………………………………………………………………...
152
7 The Growth of Compositionally Graded SixGe1-x Single Crystals by Liquid Phase Diffusion (LPD) 7.1 Introduction………………………………………………………………...
153 153
Table of Contents 7.2
vii
LPD Crystal Growth System……………………………………………….
154
7.3 LPD Growth Steps…………………………………………………………
163
7.3.1
Temperature Profile………………………………………………...
163
7.3.2
Preparation of Growth Charges…………………………………….
165
7.3.2.1
Cutting and Core-drilling Charge Materials……………...
166
7.3.2.2
Cleaning and Chemical Treatment……………………….
167
7.4 LPD Growth Principles………………………………………………….....
169
7.4.1
A Typical Procedure for an LPD Crystal Growth Experiment…….
172
7.5
Experimental Results and Characterization………………………………..
174
7.6
Summary………………………………………………………………….
187
8 Conclusion and Outlook
188
8.1 Conclusions………………………………………………………………...
188
8.2 Contributions……………………………………………………………….
190
8.3
190
Future Work………………………………………………………………..
192
Bibliography Appendix A
Summary of Literature Review on Crystal Growth Methods for SiGe Alloy System
204
Appendix B
Derivations of Model Equations for Rational Continuum Mixture Model
208
B.1
Introduction …………………………………………...............................
208
B.2
Transport Theorem for an Arbitrary Field……………………………….
208
B.3
Derivations of Thermomechanical Balance Laws and Associated Jumps.
212
B.3.1
Conservation of Mass…………………………………………..
212
B.3.2
Balance of Linear Momentum………………………………….
216
B.3.3
Balance of Moment of Momentum…………………………….
223
B.3.4
Balance of Energy……………………………………………...
225
B.3.4.1
231
Jump Energy Balance for the Mixture……………..
Table of Contents
viii
B.3.5
Entropy Inequality …………………………………………......
233
B.3.6
Open Form of Jump Balances………………………………….
242
ix
List of Figures Figure 2.1
a-) The sketch of the diamond unit cell unit, b-) projection of the cube face (numbers indicate displacements of atoms normal to the paper plane.)……………………………………….............................
11
Figure 2.2
Equilibrium phase diagram of Si x Ge1− x system.................................... 12
Figure 2.3
A representative equilibrium phase diagram of Si x Ge1− x system…….
Figure 2.4
Lattice parameter of
Si x Ge1− x system as a function of
13
Ge
composition: circles indicate experimentally measured lattice parameters and square symbols indicate lattice parameters calculated using Vegard’s law…………………………………………………... 15 Figure 2.5
Band gap energy versus composition x for Si x Ge1− x as determined from absorption measurements (cross) and low temperature PL spectra (filled-in black squares)……………………………………...
18
Figure 3.1
A schematic drawing of a radio frequency heated Cz furnace………. 22
Figure 3.2
Schematic drawing of a standard and detached VB technique………
33
Figure 3.3
A representative diagram for a zone melting process
37
Figure 3.4
Representative figure for the multi compartment sliding boat technique……………………………………………………………..
42
Figure 3.5
Formation of strained coherent epitaxial layer………………………. 47
Figure 3.6
Misfit dislocation formation at the interface…………………………
48
Figure 5.1
The schematic view of the LPD growth system……………………..
62
Figure 5.2
Coordinate system……………………………………………………
64
Figure 5.3
Mapping defined for a mixture of two constituents, (i.e. N = 2 )…… 70
List of Figures Figure 5.4
x
The schematics of the vertical cross section of the LPD growth system with the representative thermal profile where t is the unit tangential vector to the growth interface, while n is the unit normal to
the
growth
interface
pointing
into
the
liquid
domain…………..……………………………………………………
107
Figure 6.1
Flow diagram of the solution procedure……………………………..
125
Figure 6.2
Sample finite volume mesh used in the current simulation, a-) a typical mesh for 3-D simulation, b-) mesh for 2-D simulation……… 127
Figure 6.3
Representative vertical cross-section and top view of the 3-D computational domain………………………………………………..
129
Figure 6.4
Grid plane perpendicular to the axial direction………………………
130
Figure 6.5
Computed thermal field within entire computational domain (2-D) for varying growth times (Temperature in the labels are given in Kelvin)……………………………………………………………….
Figure 6.6
133
Computed thermal field within entire computational domain (2-D) for varying growth times (Temperature in the labels are given in Kelvin)……………………………………………………………….. 134
Figure 6.7
Flow (left column) and concentration (right column) fields for the growth time 0.5 hours. Flow field is given in terms of magnitude of the flow velocity, U = v r2 + v z2 , and solute concentration is given in terms of silicon mass fraction.………………………………………..
136
Figure 6.8
Flow and concentration fields at various hours of the growth……….
137
Figure 6.9
Flow and concentration fields at various hours of the growth……….
138
Figure 6.10
Decrease in the magnitude of flow velocity as a function of silicon mass fraction………………………………………………………....
Figure 6.11
139
Flow and concentration fields for different hours of growth where
β C = 0 ……………………………………………………………….. 141 Figure.6.12
Interface position as a function of growth time for r = 0…………….
143
Figure 6.13
Interface position as a function of growth time for r = r……………..
144
List of Figures Figure 6.14
xi
The time evolution of the growth interface for the half geometrical domain is shown on the left (the time interval between each line is three hours, and total simulated growth time is 39 hours), while the cross section of an LPD grown crystal on the right. Agreement between the experimental and the simulation results are quite good………………………………….…………………………….… 145
Figure 6.15
Computed thermal field within entire computational domain (3-D) at the growth time t=1 h. Temperature in the labels is given in Kelvin…………………………………………………………….…..
146
Figure 6.16
Flow (up) and concentration fields (down) at = 0.5 h………………..
147
Figure 6.17
Flow and concentration fields at growth time, t = 3.5 h……………..
149
Figure 6.18
Flow and concentration fields at growth time t = 5.5 h……………… 150
Figure 6.19
Flow fields at different growth time: 0.5 2.5, 3.5, 4.5 and 5.5 hours. The horizontal plane is at a distance of 14 mm from the bottom of
151
the substrate……………………………………………………..….... Figure 7.1
A view of the LPD growth set-up used in this thesis work………….. 155
Figure 7.2
Schematic illustration of LPD crystal growth platform……………...
156
Figure 7.3
The detailed drawing for the experimental LPD growth set up……...
158
Figure 7.4
Top view of the furnace……………………………………………...
159
Figure 7.5
Picture of quartz components………………………………………...
161
Figure 7.6
Gas distribution network……………………………………………..
163
Figure 7.7
Thermal profile measured along the quartz ampoule (square) and the axis of a center drilled silicon dummy load (circle), and the schematic diagram of the LPD growth system………………………. 165
Figure 7.8
A representative figure for the growth mechanism of LPD growth system………………………………………………………………... 171
Figure 7.9
Typical heating and cooling cycle for the growth experiment………. 174
Figure 7.10
Cutting configuration for the characterization of grown crytals (up), and pictures of two mm thick vertical slices for several grown single crystals (down)………………………………………………...……..
176
List of Figures
xii
Figure 7.11
Several LPD grown compositionally graded SixGe1-x single crystals.. 180
Figure 7.12
Vertical cross-section of four grown single crystals for determining the growth velocity…………………………………………………... 184
Figure 7.13
Interface displacement versus growth time for the center and edge regions of the grown crystals………………………………………...
Figure 7.14
Radial silicon distribution for LPD-5 along with corresponding axial distribution…………………………………………………………...
Figure 7.15
185 186
Silicon concentration distribution for LPD-19; radial silicon distribution at various axial steps (left), corresponding axial silicon distribution for different radial locations.……………………………
187
Figure B.1
Discontinuity surface………………………………………………..
211
Figure B.2
Representative figure for unit normal, binormal, and tangent vector..
217
xiii
List of Tables Table 5.1
SiGe phase diagram coefficients……………………………………..
111
Table 5.2
Design parameters for LPD growth system……………………….…
114
Table 5.3
Physical properties of growth charges and the quartz crucible………
115
Table 6.1
Non-dimensionless
numbers
and
their
characteristic
values
calculated for the growth of germanium-rich SiGe single crystals by LPD…………………………………………………………………..
120
Table 6.2
Solver parameters…………………………………………………….
123
Table 7.1
Dimensions of various components of the LPD growth platform.
157
Table A.1
Summary of SiGe growth by Cz technique………………………….. 204
Table A.2
Summary of SiGe growth by VB technique…………………………. 205
Table A.3
Summary of SiGe growth by FZ technique………………………….
206
Table A.4
Summary of SiGe growth by LPE technique………………………...
206
.
xiv
Nomenclature Latin Letters Symbol
Description
a(ij )
Phenomenological coefficients
BN
Material bodies
Bα
Material body of the α th constituent
b(ij )
Phenomenological coefficients
bα
Gravitational body force per unit mass, acting on the
Units
N / kg
α th constituent blα
Gravitational body force per unit mass, acting on the
N / kg
α th constituent in component form b
Gravitational body force per unit mass, acting on the
N / kg
mass averaged particle b
Binormal unit vector (refer to appendix B)
C
Mass fraction increment for solute
kg / kg
cα
Mass fraction of the α th constituent
kg / kg
c
Mass fraction of solute
kg / kg
co
Reference mass fraction of solute
kg / kg
D t kl
Dissipative part of the stress tensor
N / m2
Nomenclature DCl
Concentration gradient related diffusion coefficient for
xv m2 / s
the liquid phase D Sil
Molecular diffusion coefficient
m2 / s
DTl
Temperature gradient related diffusion coefficient for
m 2 / sK
the liquid phase (Soret coefficient) d kl
Symmetric deformation rate tensor
1/ s
d kk
Trace of symmetric deformation rate tensor
1/ s
da
Area element
m2
ds
Length element
m
dυ
Volume element
m3
E3
Euclidian space (Cartesian space)
Eα
Internal energy for the α th constituent
elmn
Permutation symbol
Fα
Deformation gradient for the α th constituent
FkKα
Deformation gradient for the α th constituent in
J
component form Fα
The resultant external force acting upon the α th
N
constituent Fl α
The resultant external force acting upon the α th
N
constituent in the component form F αβ
Other external forces acting upon the α th constituent
N
hα
Internal heat generation rate per unit mass for the α th
J / kgs
constituent
h
Internal heat generation rate per unit mass for the
J / kgs
mixture
hl
The height of liquid phase
h
Tangential unit normal
I
Identity tensor
m
Nomenclature
IK
Base vector in the reference state
ik
Base vector in the deformed state
J
Jacobian
jα
Mass flux for the α th constituent, with respect to a
xvi
kg / m 2 s
stationary coordinate frame jα K
α
k Cl
Diffusion flux for the α th constituent
kg / m 2 s
Kinetic energy of the α th constituent
J
Concentration gradient-related material coefficient for
J / sm
the liquid phase (Dufour coefficient) k Tl
Temperature gradient-related material coefficient for
J / smK
the liquid phase ( k Cl
Combined concentration gradient-related conductivity
J / sm
for the liquid phase ( k Tl
Combined temperature gradient-related conductivity
J / smK
for the liquid phase Mα
Molecular weigh of the α th constituent
kg / mol
M
Molecular weigh of the mixture
kg / mol
ˆα m
Moment of momentum supply vector
N / m2
ˆ nα m
Moment of momentum supply vector in component
N / m2
form mα
Mass of the α th constituent
N
The number of constituents within the mixture
n
Unit normal
P
Projection tensor
Pkl
Projection tensor in component form
P
Material point
Pα
A particle of the α th constituent in the reference state
p
Spatial point
kg
Nomenclature
xvii
p
Unknown pressure function
N / m2
) p
Modified pressure
N / m2
ˆα p
Momentum production rate per unit volume for the
N / m3
α th constituent ˆp lα
Momentum production rate per unit volume for the
N / m3
α th constituent in component form Qα
Heat energy supply per unit time for the α th
J/s
constituent q kα
Heat influx vector of the α th constituent in component
J / m2 s
form
qI k
Inner part of the heat influx vector in component form
J / m2 s
qk
Heat influx vector for the mixture
J / m2 s
q kα
Modified heat influx vector for the αth constituent
J / m2 s
qk
Modified heat influx vector for the mixture
J / m2 s
r
Radial coordinate direction
m
rˆ α
Mass production rate per unit volume for the α th
kg / m 3 s
constituent S ασ
Viscous portion of the interface stress tensor for the
N / m2
α th constituent S ~ S ~ ~ S
Entropy of a thermodynamic system
J/K
Entropy of a thermodynamic system per unit volume
J / m3K
Entropy of a thermodynamic system per unit mass
J / kgK
S
Surface bounding the material volume V
m2
S
Boundary of the volume V
m2
ˆs α
Mass production rate per unit area for the α th
kg / m 2 s
constituent To
Reference temperature
K
T
Temperature increment
K
Nomenclature t α(n )
xviii
Partial surface traction per unit area
N / m2
Partial stress tensor surface traction per unit area
N / m2
t klα
Partial stress tensor in component form
N / m2
t ασ (b )
Partial interface stress vector
N/m
ασ
Partial interface stress tensor
N/m
Magnitude of flow intensity
m/ s
Internal energy of a thermodynamic system per unit
J / m3
t
t
α
U ~ U
volume
~ ~ U
Internal energy of a thermodynamic system per unit
J / kg
mass U αβ
Other energies per unit time
J/s
u
Velocity of the discontinuity surface
m/ s
uk
Velocity of the discontinuity surface in component
m/ s
form
~ ~ V
Volume per unit mass (Specific volume)
m 3 / kg
V
Volume of the material body
m3
V
Volume of a body in the deformed state (spatial
m3
volume) vα
The material velocity of the material particle X α
m/ s
v kα
The material velocity of the material particle X α in
m/ s
component form
v
Mass averaged velocity of the mixture
m/ s
vk
Mass averaged velocity of the mixture in component
m/ s
form vα
Diffusion velocity for the α th constituent
m/ s
vkα
Diffusion velocity for the α th constituent in component
m/ s
form
Nomenclature
xix
v ασ
Surface velocity vector for the α th constituent
m/ s
vr
Velocity component in radial direction
m/ s
vϕ
Velocity component in circumferential direction
m/ s
vz
Velocity component in vertical direction
m/ s
Wα
Mechanical energy per unit time for the
α th
J/s
constituent Xα
Position vector of a particle of the α th constituent or
m
body X Kα
Position vector of a particle of the α th constituent or
m
body in component form X
Position vector for a mass centered particle (fictitious
m
particle) x
Position vector of the particle
Xα
in spatial
m
Xα
in spatial
m
configuration xk
Position vector of the particle configuration in component form
x′α
Material velocity of the material particle X α
m/ s
xα
Molar fraction of the α th constituent
mole / mole
x
Molar fraction of the mixture
mole / mole
z
Vertical coordinate direction
m
Greek Letters Symbol
Description
Units
βC
Solutal expansion coefficient
1 / wt .Si %
βT
Thermal expansion coefficient
1/ K
χα
Deformation function for the αth constituent
Nomenclature
χ kα
xx
Deformation function for the αth constituent in index notation
χ −1α
Inverse deformation function for the αth constituent
χ K−1α
Inverse deformation function for the αth constituent in index notation
εα
Internal energy per unit mass for the α th constituent
J / kg
εI
Inner part of the internal energy per unit mass
J / kg
ε
Internal energy per unit mass for the mixture
J / kg
εˆ α
Energy supply density to the α th constituent per unit
J / m3 s
time
φ
An arbitrary field (scalar, vector or tensor)
Γ (X α ,t ) An arbitrary material function (i.e. scalar, vector or tensor quantity)
Γ (x ,t )
An arbitrary spatial function (i.e. scalar, vector or tensor quantity)
Γ α (x , t )
An arbitrary spatial function for the αth constituent (i.e. scalar, vector or tensor quantity)
Γ′
Material time derivative of the function Γ , following the motion of the particle X α of the α th constituent
Γ&
Material time derivative of the function Γ , following the motion of the mass averaged particle X
ϕ
An arbitrary field (scalar, vector or tensor)
γl
Specific heat for the liquid phase
J / kgK
γ ασ
Surface tension for the α th constituent
N/m
γσ
Surface tension for the mixture
N/m
ηα
Entropy per unit mass for the α th constituent
J / kgK
η
Entropy per unit mass for the mixture
J / kgK
µα
Mass basis chemical potential of the αth constituent
J / kg
Nomenclature
µ
xxi
Difference in chemical potentials between solute and
J / kg
solvent (
µ
Coefficient related to the chemical potential differences
J / kg
between solute and solvent µv
Viscosity
kg / ms
µ vT
Temperature related viscosity
kg / msK
µ vC
Concentration related viscosity
kg / ms
ν
Kinematic viscosity
m2 / s
π
Thermodynamic pressure
N / m2
θα
Absolute temperature for the α th constituent
K
θ
Absolute temperature for the mixture
K
ρα
Mass density of the αth constituent
kg / m 3
ρ
Mass density of the mixture
kg / m 3
σ
Discontinuity surface
m2
Ω
Mean curvature
1/ m
ωα
Molar density of the αth constituent
mole / m 3
ω
Molar density of mixture
mole / m 3
ξ
~
An arbitrarily field per unit volume
( ) / m3
ξ
~ ~
An arbitrarily field per unit mass
( ) / kg
ψ
Specific Helmholtz free energy for the mixture
J / kg
Operators Symbol
Description
d( ) dt
Total time derivative
Nomenclature Dα ( Dt
)
xxii Material time derivative, following the motion of the particle X α of the α th constituent
D( ) Dt
Material time derivative, following the motion of the
∂( ) ∂t
Partial time derivative
mass centered particle X
( ),k
Partial spatial derivative
∇s
Surface gradient operator
∇φ
grad φ =
∇u
grad u =
∇⋅u
div u =
∇×u
curl u =
∇ 2φ
∂φ i k = φ ,k i k , gradient of a scalar field ∂x k
∂u k i l i k = u k ,l i l i k , gradient of a vector field ∂xl
∂u k = u k ,k , divergence of a vector field ∂x k ∂u l ε klm i m = u l ,k ε klm i m , curl of a vector field ∂x k
div ( gradφ ) =
∂ 2φ = φ ,kk , Laplacian operator on a ∂x k ∂x k
scalar
(u ⋅ ∇ )φ (u ⋅ ∇ )v
u ⋅ ∇φ = u k
∂φ = u ,k φ ,k , compound nabla operator ∂x k
u ⋅ ∇v = u k
∂vl i l = u k vl ,k i l , compound nabla ∂x k
operator u ⊗ v, or uv
u k vl i k i l , dyadic product of two vectors
Nomenclature
Indices and Symbols Symbol
Description
α
α th constituent, (α =1,...,N)
β
β th additional force
/
( ),( )
/
•
()
Prime indicates the material time derivative, following the motion of the particle of X α of the α th constituent Dot indicates the material time derivative, following the motion of the mass averaged particle X The jump of the enclosed quantities
( )o ,( )o
Variable evaluated at reference state
( )l , ( )l
Liquid phase
( )s ,( )s
Solid phase
( )T
Coefficients associated with temperature
( )C
Coefficients associated with solute mass fraction
xxiii
xxiv
Acknowledgements I would like to thank all the people who did not leave me alone in this four-year journey of hardship by their endless support and encouragement. First and foremost, I would like to express my deepest gratitude to my supervisor Dr. Sadik Dost for giving me the opportunity to work with him and for introducing me to a challenging and exciting research field, crystal growth. His guidance throughout the course of my study played a significant role to conclude this work. It was a privilege for me to work with such an expert in the field of crystal growth. I also would like to thank my lovely wife for her continuous support and help in spite of many lonely hours. This dissertation would not have been possible without her. I would like to thank my little son for giving me peace and comfort with his smiley face and hugs. I wish to express my thanks to Mr. Brian Lent and Dr. Yongcai Liu for sharing their expertise in the field of crystal growth. Many thanks to Dr. Hamdi Sheibani, Mrs. Sema Dost, and all my fellow graduate students for their kind helps and friendship. Most of all, I would like to thank my parents, and my brothers and sister for their unconditional support, encouragement and prays in all these years to achieve my goals. I thank GOD, the most merciful, the most compassionate, for his blessings and love. The financial support provided by BL Consulting Ltd. of Victoria, DL Crystals Inc. of Victoria, and the Microgravity Science Program of the Canadian Space Agency is gratefully acknowledged. The financial support for the LPD equipment and its operation is provided by NSERC and CFI.
Dedication
Dedicated To My Beloved Parents, Caring Wife, Lovely Son (Emir Berk), All My Brothers and Sister in recognition of their contribution to this dissertation.
CHAPTER 1 Introduction 1.1 Motivation and Goals Research on the Si x Ge1− x alloy system dates back to as early as 1954 [1]. However, an impressive body of research associated with the growth of high quality Si x Ge1− x single crystals, epitaxial techniques leading to thin Si x Ge1− x layers and development of Si x Ge1− x - based microelectronic semiconductor devices started coming into existence
approximately two decades ago. The Si x Ge1− x alloy system offers very promising material features such that the band structure and thus the effective mass and mobility of both electrons and holes are significantly affected by alloy composition, temperature and strain. Therefore, the electrical and optical properties of this material can be tailored according to the needs of device applications. Furthermore, other very important aspects of Si x Ge1− x alloy system are its ease of integration with the well-developed and longexisting silicon technology, and its low-cost production compared to III-V group semiconductor materials. The above-enumerated features make the Si x Ge1− x system a very promising candidate for a variety of micro electronic and optoelectronic device applications. For instance,
Chapter 1-Introduction •
2
Si x Ge1− x has been attempted for use as a base in Si/SiGe heterojunction bipolar
transistors (HBT) [2,3,4,5] and high electron mobility field effect transistors [6,7,8] due to its higher carrier mobility, amendable band gap energy at any value between silicon and germanium and adjustable band gap offset at the junction between Si x Ge1− x and silicon. •
Si x Ge1− x has been utilized for photodetector [9,10] and solar cell [11,12] applications
because of its enhanced sensitivity to the detection of light and high solar cell response in the infrared region of solar spectrum. The band structure of Si x Ge1− x alloys enables Auger generation processes, thereby providing solar cell efficiency of approximately 40% [13]. •
Si x Ge1− x can be employed as a substrate to fabricate exactly lattice-matched
GaAs / SiGe heterostructures for higher efficiency solar cell applications [14, 15].
•
Other applications of Si x Ge1− x include thermoelectric power generators (due to its
low thermal conductivity and high Seebeck coefficients [16,17,18]), tuneable neutron and x-ray monochromators based on the large lattice parameter variation of the Ge-Si solid solution [19-20] and high speed temperature sensor for the range of 20-400 oC based on conductivity change, and γ-ray detectors [21]. Si x Ge1− x single crystal substrates with a specific composition ( x ) are needed for the
applications mentioned above. Si x Ge1− x
single crystals for microelectronic and
optoelectronic device applications have been generally prepared in the form of thin films grown on a silicon substrate by different epitaxial growth techniques such as Molecular Beam Epitaxy (MBE) [22,23], Liquid Phase Epitaxy (LPE) [24,25], Rapid Thermal Chemical Vapour Deposition (RTCVD) [26], Chemical Vapour Deposition (CVD) [27], Ultra High Vacuum Chemical Vapour Deposition (UHV/CVD) [28,29,30]. On the other hand, when a Si x Ge1− x alloy is epitaxially deposited on a silicon substrate, the lattices of the deposited layer try to embrace in-plane lattice parameters to form a coherent interface with the underlying substrate. As a result, the alloy layer will be compressively strained since the freestanding lattice parameter of the silicon is smaller than those of Si x Ge1− x
Chapter 1-Introduction
3
alloys and pure germanium. When the thickness of the strained layer exceeds a so-called critical thickness, a high density of misfit dislocations at the interface of the Si x Ge1− x and Si and many threading dislocations, traversing the Si x Ge1− x layer, are invariably created to relieve the built-in compressive strain. Upon relief of the strain, the layer relaxes to its freestanding lattice parameter. The existence of misfit and threading dislocations severely reduces the mobility and electronic quality of the material [2]. The critical layer thickness decreases significantly with increasing germanium content. Nevertheless, most of the applications require a much thicker Si x Ge1− x layer with high germanium content than that imposed by the critical thickness limit and thus, a large number of researches are still underway to reduce the dislocation density. One extensively researched method is to form fully relaxed and compositionally graded Si x Ge1− x layers on a silicon substrate, increasing the germanium content at each step up to the composition of interest and hence less lattice mismatched interfaces will be created, resulting in lower dislocation densities confined in graded layers. Such a graded structure is known as a “virtual substrate” and is used as a fully or partially lattice matched substrate on which active Si x Ge1− x layers can be grown for device applications. Nevertheless, virtual substrates are also accompanied by other problems such as surface undulation or roughness created between layer interfaces, which are harmful to device performances as well. In addition, preparation of virtual substrates is expensive, time consuming and not an easy process. Considering these problems associated with the virtual substrates, the need for high quality and relatively defect free and compositionally uniform Si x Ge1− x substrates can easily be appreciated. Availability of such substrates will allow the deposition of thicker lattice matched thin film alloys with high germanium content, which have been highly needed for development of high performance microelectronic and optoelectronic devices. In order to achieve high quality Si x Ge1− x crystals with different germanium contents, a variety of melt crystal growth techniques, such as Czochralski [31-36], floating zone [37], Bridgman [38-40], multi component [14] and liquid encapsulated zone melting [41] and others [42], have been tried.
Chapter 1-Introduction
4
It is however very difficult to grow single crystals of uniform composition and low defect densities from a binary system of Si x Ge1− x since Si x Ge1− x has a large miscibility gap and there are significant differences in physical properties of the constituent elements, such as density, melting temperature, and lattice parameter. Because of the large miscibility gap, which gives rise to segregation coefficients far from unity, ( k o = c s / c l =5.5 for silicon), ( k o = c s / c l =0.3 for germanium) in Si x Ge1− x alloy, any small changes in the solidification rate will lead to significant composition variation. Upon the formation of a Si x Ge1− x solid solution, depletion of silicon atoms in the melt just ahead of the solid and liquid interface takes places. This is because silicon atoms are preferentially consumed by the growing crystal from the limited amount of growth melt. In the mean time, germanium atoms are rejected into the melt. This increases the germanium content and while decreases the silicon content in the melt along the growth direction. Consequently, composition of the grown crystal varies in the growth direction. Various types of defects may occur during growth. For instance, strong segregation of constituents may cause constitutional supercooling in the melt near the crystallization front, which is the main mechanism for transformation from a single crystal structure to a polycrystalline one. Other main defects in melt grown crystal can be grown-in dislocations (electrically active defects) in the order of 104-106 cm-2 depending on growth methods and concentric circles (so called striation marks) caused by the variation in composition and lattice parameters in the radial direction. Striation marks in the crystal are because of fluctuations in temperature. If the concentration gradient in a single crystal exceeds a critical limit, it may result in crack formation in the structure. Despite the fact that Cz is one of the most used growth techniques to grow single crystals of large diameter, it is difficult with this technique to obtain single crystals with uniform compositions and without striations [31-36]. Similarly, there are non-homogeneous concentration distributions along the growth direction and radial striations in Si x Ge1− x single crystals, which are grown by Bridgman [38,39] and floating zone [37]
techniques. As it stands, growing Si x Ge1− x single crystals of uniform solute composition
Chapter 1-Introduction
5
and high quality compatible with the requirements of device applications is a challenging task. The problems encountered in growth of Si x Ge1− x single crystals by either the aforementioned deposition techniques or high temperature melt growth techniques may be successfully circumvented by employing a low temperature solution growth technique such as Liquid Phase Electroepitaxy (LPEE) and Traveling Heater Method (THM). LPEE was considered as a technique of choice for the growth of Si x Ge1− x single crystals of high quality unattainable by melt growth because it offers, in principle, better controllability of defects than other techniques discussed previously. LPEE has been proven to be successful in the growth of III-V alloy semiconductor materials [43,44]. In LPEE, single crystals are grown from metallic solutions at lower growth temperatures than melt growth. Thermal gradients developed in the LPEE growth cell (liquid solution zone) are generally sufficiently low so that the amount of grown-in defects such as dislocations and point defects in the final structure can be controlled and reduced. Growth in LPEE is initiated and sustained by the passage of an electric current through a substrate-solution-source system by maintaining the growth temperature constant. The application of the electric current gives rise to two well-known mechanisms, namely Peltier cooling/heating, and electromigration. Electromigration takes places due to the momentum-exchange between electrons and atoms, leading to the transport of solute species towards to the seed crystal (substrate). Peltier cooling reduces the temperature at the crystallization front in the order of approximately 1 oC and in so doing, it results in supersaturation of the advancing growth interface and thus growth. LPEE growth process requires a Si x Ge1− x single crystal seed with the same composition as the crystal to be grown (a small composition difference within the limit of acceptable lattice mismatch is tolerable). In order to address this issue, a crystal growth technique, which is called in this work “Liquid Phase Diffusion, (LPD)”, has been utilized. Two
Chapter 1-Introduction
6
objectives were in mind in initiating the present study. One was the growth of compositionally graded bulk Si x Ge1− x single crystals from which the seed substrates with required compositions can be extracted for LPEE. At the time this work was commenced, no successful growth of bulk Si x Ge1− x single crystals of large size for whole composition ranges in the phase diagram was reported, and such crystals were not commercially available. The second objective was the development of the first stage of a hybrid growth technique that combines LPD and LPEE in a single process. In this hybrid technique in mind, a compositionally graded single crystal would be grown by LPD up to the composition of interest, and then at this stage, the LPEE process would be initiated by passing an electric current through the growth system at a uniform temperature, leading to the growth single crystal with desired uniform composition. This hybrid growth process would eliminate the adverse effects of growing crystals in two stages. Such a growth process will be developed in the near future. LPD growth technique was selected over other melt growth methods due to its simplicity and low cost of growth equipment. LPD was used by Nakajima et al. under the name of “Multicomponent Zone Melting” [14,45] for the growth of compositionally uniform germanium–rich Si x Ge1− x single crystals of 15 mm in diameter. Their crystals were reported to be single in compositionally graded region and partially polycrystalline in compositionally uniform region due to the pulling process. In LPD technique, the solvent material (Ge) is sandwiched between single crystal substrate (seed, Ge) and polycrystalline source material (feed, Si). Initially, all these three layers of materials (the growth charge) are solid. On locating the growth charge (or growth zone) in an axial thermal gradient, solid germanium in the middle totally melts to form the liquid solvent for the growth. The germanium substrate partially melts. The silicon source, on the other hand, remains in the solid state due to its higher melting point. The molten germanium dissolves the silicon source according to thermodynamic equilibrium. Dissolved silicon species are incorporated into the germanium liquid, thereby forming a binary Si x Ge1− x growth solution (mixture). Silicon species in the solution zone are transported to the
Chapter 1-Introduction
7
growth interface by combination of several driving forces. When the mixture at the growth interface is supersaturated, solidification takes place due to the presence of constitutional supercooling. Since the growth solution (SiGe binary mixture) is prepared through the dissolution of solute (Si) by the solvent (Ge), LPD technique utilized here resembles a solution growth technique. The composition range required from a Si x Ge1− x alloy system is very wide depending on the device application. Our literature survey for determining the target composition has shown that the entire compositional spectrum is feasible. Therefore, in this work, we focused on the germanium rich region of the phase diagram since the other end of the phase diagram brings some operational difficulties due to the high melting temperature of silicon. In conclusion, the objective of this work is to perform a combined experimental and theoretical study to show the applicability of LPD to produce relatively large seed crystals of desired compositions. The objective has been successfully realized by growing several Si x Ge1− x single crystals of 25 mm in diameter up to 6-8 % at. Si and by developing a rational mixture model and performing two-and three-dimensional numerical simulations to study transport phenomena in detail during the growth process. The seed materials for LPEE growth can be extracted from the composition of interest by cutting a horizontal slice from the LPD grown crystals. To the best of our knowledge, there has been no attempt to grow Si x Ge1− x single crystals in 25 mm in diameter by LPD. As well, no comprehensive two –and three-dimensional simulations of the LPD growth system have been performed. These significant contributions to the field of crystal growth make this work original.
Chapter 1-Introduction
1.2
8
Outline of the Thesis
Following a brief introduction on the subject matter, Chapter 2 presents the crystal structure, phase diagram and the band gap structure of the Si x Ge1− x alloy system. Chapter 3 introduces short reviews on most crystal growth techniques including melt, solution and deposition methods, which have been used in the production of Si x Ge1− x single crystals. It also addresses the current state of such studies, and examines their status in the growth of Si x Ge1− x alloy systems. Chapter 4 is devoted to a literature survey to find out the possible applications of Si x Ge1− x along with the compositional requirements for the application of interest. Chapter 5 presents to the development of a rational macroscopic mixture model for the LPD growth system. Chapter 6 introduces two-and threedimensional transient simulations of the LPD growth system with moving boundary by employing the model developed in this thesis. Chapter 7 addresses the design, construction and testing of the LPD growth system. It also introduces experimental results and characterization of grown crystals. Chapter 8 concludes the work presented in this thesis with a short summary followed by the contributions to the field of crystal growth and a discussion about future works. Appendix-A summarizes the results of the studies for the growth of Si x Ge1− x by various growth methods published in the literature. Appendix B includes detailed derivations of the continuum mixture model.
9
CHAPTER 2 Structural Properties of SixGe1-x Alloy System 2.1 Introduction This chapter provides relatively comprehensive information on the crystal (or the lattice structure), lattice parameters, equilibrium phase diagram, electronic band structure, thermal and physical properties of Si x Ge1− x solid solution since it forms the basis for understanding of forthcoming discussions about possible device applications and growth of Si x Ge1− x systems by different techniques, particularly by Liquid Phase Diffusion (LPD).
2.2 Crystal Structure and Equilibrium Phase Diagram of SixGe1-x Lattice structure of a crystal is formed by the periodic reoccurrence of a unit cell in threedimensional space, which is defined as the smallest building block of the crystal, so that the unit cell completely describes the lattice structure. Thus, to visualize the lattice structure of a crystal, it is adequate to examine the unit cell of the crystal.
Chapter 2 – Structural Properties of SixGe1-x Alloy System
10
Silicon and germanium are the first two elemental semiconductor materials discovered. Both belong to the IV-group of the periodic table. These two elements and their mixture (hereafter referred to as Si x Ge1− x solid solutions) crystallize with the diamond lattice structure under the atmospheric pressure. The diamond lattice structure is a generic name, which characterizes all crystals crystallizing in the same pattern as the diamond. The arrangement of atoms in the diamond unit cell is pictured in figure 2.1a. The diamond unit cell is a cubic structure with atoms at each corner and at each face of the cube akin to the well-known face-centered cubic (fcc) cell. Unlike the fcc unit cell, the diamond cell hosts four additional interior atoms represented by yellow filled-in circles in figure 2.1a. Each interior atom is located on one of the four body diagonals of the cube. All interior atoms are displaced one-quarter of the body diagonal with respect to associated corners along the corresponding diagonals. The numbers inside the circles in diamond unit cell denote the height of the atoms with reference to base as a fraction of the cell dimension, denoted by ‘ a ’. Each atom in the unit cell is covalently bonded with its four nearest neighbours forming a tetrahedral structure as illustrated by a dashed tetrahedral in the interior of the diamond unit cell. The bonds between nearest-neighbour atoms are represented by lines. An alternative and easier way of visualizing the diamond space lattice might be to consider a lattice formed by the interpenetration of two fcc unit cells. Namely, the second fcc unit cell is displaced a quarter of the body diagonal relative to the first cell along the diagonal direction. In this case, the corner and face atoms of the diamond unit cell can be viewed as if they belong to first fcc unit cell, whereas atoms totally confined within the diamond unit cell belong to the second fcc cell. As each unit cell is brought adjacent to each other in the crystal, then each cell contains one-eighth of each corner atom ( 8 *1 / 8 ) and one-half of each face atom ( 6 *1 / 2 ). Therefore, with four interior atoms, each diamond unit cell contains eight complete atoms.
Chapter 2 – Structural Properties of SixGe1-x Alloy System
11
a /2 0
1/ 2
1/ 4
3/ 4
3/ 4
0
3/ 4
a
1/ 2
1/ 2
1/ 2
1/ 2
1/ 4
1/ 4
1/ 4 0
0
0
0
0
1/ 2
0
1/ 2
3/ 4
1/ 2
0
0
Base of the cube
(a )
(b)
Figure 2.1: a-) The sketch of the diamond unit cell unit, b-) projection of the cube face (numbers indicate displacements of atoms normal to the paper plane.)
The equilibrium phase diagram for SixGe1-x system (refer to figure 2.2) consists of three distinct regions, namely the upper region describing the liquid state, the lower region describing the solid state and an intermediate region where the solid and liquid states coexist. The curves that separate intermediate region of the phase diagram from the liquid state and solid state are called the liquidus and solidus curve, respectively. The region between the liquidus and solidus curves is usually referred to as the miscibility gap. Silicon and germanium have complete miscibility in each other both in solid and liquid states over the entire composition range as can be seen from equilibrium phase diagram. Upon solidification, the mixture of silicon and germanium forms a substitutional solid solution. For a given composition in the phase diagram, the concentration of the silicon in the liquid and solid states can be determined by drawing a tie line intercepting with the liquidus and solidus line at a given temperature. It can easily be noticed from the phase diagram that the concentration of silicon in the solid is always higher than that in the
Chapter 2 – Structural Properties of SixGe1-x Alloy System
12
liquid state, which implies that silicon solidifies or crystallizes faster than germanium since the melting temperature of silicon is higher than that of germanium.
Figure 2.2: Equilibrium phase diagram of Si x Ge1− x system [46] To be more particular, solidification steps of the Si x Ge1− x system are explained by using the “alloy-L” that is indicated on the representative phase diagram given in figure 2.3. When the “alloy-L” is cooled to the temperature T1 from liquid state, the formation of a solid solution starts. The composition of the first solidified solid solution can be obtained by drawing a vertical line from the meeting point of the tie line at T1 with the solidus curve to the composition axis. In this case, the composition of the first solidified solid solution is x sSi1 .
Chapter 2 – Structural Properties of SixGe1-x Alloy System
13
Temperature
TSi
L1 Liquid
T1 T2
Liquid + Solid
T3 Solid
TGe Ge
x Sil 2
Si x Sil1 x o
Si x Sis 2 x s1
Si
Atomic percent Si → Figure 2.3: A representative equilibrium phase diagram of the Si x Ge1− x system The percentage of Si at x sSi1 composition is obviously greater than that in initial melt composition x oSi . On decreasing the temperature of the alloy further to T2 , the composition x oSi moves to xlSi1 along the liquidus line because silicon in the melt is preferentially consumed during the formation of a crystal with composition x sSi1 . Therefore, the composition of the crystal layer to solidify next changes to x sSi2 . As the solidification progresses, the germanium concentration increases in both solid and liquid at the solidification front due to the rejection of germanium atoms into the liquid, and silicon concentration decreases due to its preferential consumption, thereby resulting in continuous compositional variations along the solidification direction. The intensity of the compositional non-homogeneity, the so-called macrosegregation, in a solidified crystal depends on the location in the phase diagram where the alloy is solidified. The macrosegregation is usually characterized by a coefficient k o that is known as equilibrium segregation coefficient. “ k o ” is defined as the ratio of the concentration of the solute in the solid to that in the liquid at the solid-liquid interface, namely,
Chapter 2 – Structural Properties of SixGe1-x Alloy System
14
k o = c s / c l . It is a function of temperature. If k o =1, the alloy is known to be of a
concurrent melting temperature. For the Si x Ge1− x alloy system, k o values are far from unity. For germanium-rich alloys, where germanium is a background fluid or solvent, and silicon is the solute, k o is bigger than unity and takes values up to 5.5. Regarding the silicon-rich alloys, k o is less than one and varies between one and about 0.3. Since the segregation coefficient is further away from unity for germanium rich-alloys than for silicon rich-alloys, it is relatively easy to obtain a single crystal of a uniform composition from silicon-rich side of the phase diagram. To conclude, Si x Ge1− x is a good example for an alloy system having a strong tendency towards the formation of a composition gradient along the solidification direction due to the large miscibility gap between the solid and liquid phase and the high temperature difference in melting points of silicon and germanium. The lattice parameter of the Si x Ge1− x system at room temperature varies linearly from 0.54310 nm to 0.56575 nm [47] as the germanium content increases from x = 0 to x = 1 as indicated in figure 2.4. Figure 2.4 also indicates that experimentally measured lattice parameters represented by circles can be approximated closely by taking a linear interpolation between the lattice parameters of pure silicon and germanium. This approximation is known as Vegard’s rule [48], and mathematically expressed as
a Si1− xGex = a Si + [aGe − a Si ]x , where a Si1− xGex , a Si , aGe are lattice parameters of silicon– germanium alloy, silicon and germanium, respectively and x is the atomic fraction of
Ge . The lattice constant difference between pure silicon and germanium is 4.17 %, which can be calculated by using the equation ( [(aGe − a Si ) / aGe ]* 100 ).
Chapter 2 – Structural Properties of SixGe1-x Alloy System
15
0.57
Lattice param eter, nm
0.565 0.56 0.555 0.55 0.545 0.54 0
20
40
60
80
100
Co mposition, at. % Ge
Figure 2.4: Lattice parameter of the Si x Ge1− x system as a function of Ge composition: circles indicate experimentally measured lattice parameters [47] and square symbols indicate lattice parameters calculated using Vegard’s law.
Having introduced the lattice structure of Si x Ge1− x alloy, the focus of the discussion can be turned to the microscopic ordering of constituents within the Si x Ge1− x solid solution. No experimental observation of any ordered structure has been reported for both silicon and germanium rich bulk Si x Ge1− x crystals of various compositions and for Si0.5 Ge0.5 bulk crystals grown from the melt by vertical Bridgman technique [49]. On the other hand, existence of long range ordering was reported on Si x Ge1− x thin films grown on a silicon (001), germanium (001) and fully relaxed
Si x Ge1− x buffer layer having (100)
crystallographic orientation by Molecular Beam Epitaxy (MBE) at lower temperature than the usual growth temperature (550 oC) [49]. The ordering of silicon and germanium is limited to thin films grown on substrates of (100) orientation, and received considerable interest since it may affect the electronic band structure and the material’s adjustable band gap.
Chapter 2 – Structural Properties of SixGe1-x Alloy System
16
2.3 Band Gap Structure of Bulk SixGe1-x Alloys Silicon-germanium alloys are in the category of indirect band gap semiconductors like their constituents. Optical studies confirm that in pure silicon and germanium, the maximum in the valence band occurs at the centre of the energy (E) versus k = p / h diagram where k = 0 , whereas the minimum in the conduction band is located at
k ≠ 0 [50]. Here, k is the wave number, p is the momentum and h = h / 2π is the modified Plank’s constant. A semiconductor whose maximum in the valence band and minimum in the conduction band do not occur at the same k value is called an indirect band gap semiconductor. In silicon and germanium, the conduction band minima occur along and crystallographic directions, respectively. In the indirect band semiconductors, when an electron jumps down from the conduction band to the valence band to recombine with a hole in a semiconductor material, it must release some of its energy to embrace the energy level of the hole. According to the law of conservation of energy the energy released must be absorbed by some outside sources. This energy release process may happen in two different ways; firstly, energy can be released in the form of radiation creating photons, in which case the recombination process is called radiative, or secondly, it can be transferred directly to a lattice in the form of vibration forming a phonon. In addition, because the maximum in the valance band and minimum in the conduction band occur at different k values, the momentum of the electron has to be conserved during the transition from the conduction band into the valence band. This dictates the interaction of the electron with either phonons or impurities in the semiconductor. This indirect nature of the electron transition has significant effect on the optical properties of the semiconductor material. Therefore, indirect band gap semiconductors do not show as good optical properties as direct band gap semiconductors.
Chapter 2 – Structural Properties of SixGe1-x Alloy System
17
In 1954, by performing optical absorption investigations on Si x Ge1− x alloys for 0 < x < 1 , Johnson and Christian [1] proposed that the indirect band structure depends only on the alloy composition. They observed a sudden change in the slope of energy band gap versus composition curve. The abrupt change in the curve occurring at about 15 at % Si was interpreted as a switch over from germanium like L -band structure to the silicon like ∆ - band structure [51]. Four years later from the study of Johnson and Christian,
Braunstein et al. [50] have performed absorption measurements as a function of composition, somewhat more detailed than the one performed previously [1], in order to gather more accurate information about the band structure of the alloy system. They observed that the indirect energy band gap of the Si x Ge1− x system does not depend only on composition but also on temperature, and determined the values of band gap by fitting the absorption data at low absorption levels to the Macfarlane-Roberts expression. Their work [50] confirmed the discontinuity in the curve obtained by Johnson et al. However, while Johnson et al’s curve consists of two linear portions with different slopes, the first portion of Braunstein et al’s curve is linear in the 0-15 mole percent germanium and the second potion of the curve is quadratic in the remaining composition range as can be seen from figure 2.5. The discrepancy between earlier data [1] and their data [50] was largely attributed to the lower material purity of samples used in the study [1]. Recently, Weber and Alonso [52] measured the composition dependence of the energy band gap of Si x Ge1− x system at 4.2 K by employing photoluminescence (PL). They found that the
band gap varies with the increasing germanium content from pure silicon energy gap at 1.155 eV to the pure germanium band gap at 0.740 eV, as shown in figure 2.5. All three studies indicate that in the Si x Ge1− x system, the crossover from “the lowest lying conduction band minimum” along crystallographic direction in reduced Brillion zone, to “the lowest lying conduction band minimum” along directions takes places at about 15 atomic percent silicon.
18
Energy (eV) →
Chapter 2 – Structural Properties of SixGe1-x Alloy System
Atomic percent germanium→ Figure 2.5: Band gap energy versus composition x for Si x Ge1− x as determined from absorption measurements (cross [50]) and low temperature PL spectra (filled-in black squares [52]).
Experiments have shown that the presence of tensile or compressive strain in a Si x Ge1− x layer, induced by the lattice mismatch, gives rise to drastic changes in the band
structure of the Si x Ge1− x system by shifting and splitting the valence and conduction band edges [53,54]. In addition to the band gap reduction caused by increasing germanium content, the built-in compressive strain further reduces the energy gap of the Si x Ge1− x alloy for a coherent Si x Ge1− x layer on a silicon substrate [55, 56]. Consequently, concentration, temperature and strain dependency of the band structure of Si x Ge1− x alloys can be used to tailor the band structure according to the needs for a
particular application.
19
CHAPTER 3 Crystal Growth Methods for SixGe1-x Alloys 3.1 Introduction In this chapter, the crystal growth techniques that have been utilized to date to grow Si x Ge1− x single crystals are briefly introduced. Before proceeding with the discussion, it
must be emphasized that speaking of the superiority of one crystal growth technique over another would not be appropriate without clearly specifying the types of and expected physical and chemical properties from the grown crystal since each class of a material brings its own special difficulties. A technique that results in a successful growth for a particular semiconductor crystal may fail in the growth of another material. Briefly, there exists no such a universal technique, which guaranties the growth of all kinds of single crystal semiconductor materials. Therefore, here, each technique is introduced, in general sense, to provide the reader with some background knowledge. Drawbacks and advantages of these growth methods are also discussed for the Si x Ge1− x alloy system in details to reflect their current status in the crystal growth field.
3.2 Melt Growth Methods Melt growth techniques have been well accepted owing to allowing the production of large semiconductor crystals in relatively short time. Nevertheless, as the name implies,
Chapter 3 – Crystal Growth Methods for SixGe1-x Alloys
20
the crystal growth has to be performed above the melting temperature of the crystal of interest. Hence, the high temperature requirement brings about some experimental and operational difficulties. In addition, the grown crystal may include defects particularly resulting from the high temperature processing. Such defects may not be introduced into the structure of the material if the crystal in concern can be grown by low temperature growth processes such as solution growth techniques. Almost all classical melt growth (bulk growth) techniques (i.e., Czochralski crystal pulling, vertical Bridgman, floating zone, and zone melting) have been applied in an attempt of obtaining monocrystalline Si x Ge1− x alloys of uniform composition. The melt growth methods utilized prior to1995 pertaining to growth of the Si x Ge1− x alloy system at different composition ranges were well documented in the review by Schilz and Romanenko [21]. This review discusses the difficulties and limitations that are encountered during the growth of the bulk Si x Ge1− x single crystal from a melt, and provides a comprehensive list of the growth parameters involved. For the sake of efficiency, in this review section, we will only cover the studies that were either not included in [21] or conducted after 1995. Nearly the entire region of binary phase diagram has been covered so far. Results of these studies, on the other hand, show that the bulk growth of Si x Ge1− x single crystals of a satisfactory quality over the whole composition range have had a limited success due to the lack of large size, low defect density and high compositional uniformity. Especially, almost all attempts to grow a single crystal from the intermediate composition range in the Si-Ge phase diagram resulted in either fully polycrystalline or partially single crystalline solid solutions. As for either silicon rich or germanium rich solid solutions, they do have non-homogenous axial concentration distributions. To this end, it is evident from the literature that growing Si x Ge1− x binary alloy system is still at an early stage and needs extensive research to overcome difficulties in growing reproducible, commercially viable, larger size single crystals of a uniform composition from any region of the phase
Chapter 3 – Crystal Growth Methods for SixGe1-x Alloys
21
diagram (especially from the region where the largest miscibility gap exists). Here, addressed is the melt grow techniques employed for the growth of Si x Ge1− x solid solutions along with their advantages and shortcomings.
3.2.1 Crystal Pulling All crystal-pulling processes are originated from the technique developed in 1927, which was named after its inventor Czochralski. This technique and its many derivatives have become a foremost utilized process in the industry for the production of various classes of semiconductor materials due to being a relatively faster crystal growth process and enabling a crystal in larger size than its counterparts. The focus here is to be on only the Czochralski growth process (Cz) among the crystal pulling techniques given that it has been used for Si x Ge1− x growth. A routine step utilized for the pulling process is the following. First, the charge materials are placed in a compatible container or crucible. Then, an inert gas such as Helium or Argon is introduced into the growth environment and maintained at a certain pressure. The crucible is then heated to a temperature a few degrees above the melting temperature of the charge material by either a resistance or induction heating. The seed material is lowered to a position just above the surface of the liquid for preheating. The temperature of the melt at the free surface needs to be adjusted in such a way that the seed material be partially dissolved after being brought in contact with the liquid. If the seed material does not remain in contact with the melt, the melt temperature at the surface is too high, and hence, needs adjustment. Having kept the seed dipped in the melt for a few minutes, the crystal growth or “pulling” process is commenced by withdrawing the seed at a slow rate. The growth is sustained until reaching the desired length. The growth process can be terminated by either slowly increasing the temperature of the melt to reduce the crystal diameter gradually to zero or quickly raising the crystal so that the contact between the melt and crystal is disrupted. After terminating the growth, the temperature of the system
Chapter 3 – Crystal Growth Methods for SixGe1-x Alloys
22
is lowered to the room temperature slowly, and the grown crystal is removed from the apparatus. Figure 3.1 illustrates a schematic drawing of Radio frequency heated Cz furnace.
Figure 3.1: A schematic drawing of a radio frequency heated Cz furnace. In a standard Cz method, seeding is started with a crystal smaller in diameter than the crystal to be grown. The diameter of the crystal then can be increased to a desired size by means of a process referred to as “capping”. Capping is achieved by reducing the temperature of the liquid at the seed-liquid interface. Before performing the capping, an elegant technique called “intentional necking” can be applied to reduce the dislocation density in the grown crystal. The necking is realized by increasing the temperature of the liquid, which results in decrease in the diameter of the crystal, and then by decreasing the temperature, thereby causing diameter re-enlargement. In so doing, dislocations grow out of the crystal toward the surface rather than growing in the bulk of the crystal. Usually,
Chapter 3 – Crystal Growth Methods for SixGe1-x Alloys
23
the length of the necked region is about five or ten times that of the seed diameter. If an excessive capping is invoked to obtain a crystal significantly larger in diameter than the seed, randomly oriented crystals may nucleate at the circumference of the melt-seed interface precluding the growth of monocrystalline cap. If the focus is on obtaining a larger single crystal, it would be more realistic to initiate the growth with a larger seed so that a single crystal might be grown with nearly as large diameter as the seed. Richard et al.[57,58] used pre-grown large diameter silicon single seeds with different crystallographic orientations to grow Si1− x Ge x single crystals of large diameters for the composition range of 0 < x < 0.17 . It was reported that the grown crystals were of 63 mm in diameter, but only 50 mm central region was single crystal. Throughout the growth, a stable thermal environment is required to maintain the crystal diameter constant since thermal fluctuations result in a crystal of non-uniform diameter. The diameter variation is generally associated with twinning in the grown crystals. The diameter of the crystal can be monitored by means of either a visual access to the growth environment (using optical equipments) or a weighing technique. Any deviations from the diameter of the interest can be compensated through a temperature adjustment (either increasing or decreasing the temperature). For example, if the temperature is not lowered during the pulling, a tapered boule is resulted. Common to almost all growth techniques, the natural convection driven by density and temperature variations occurs in the melt. Convection in the melt enhances the transport of species giving rise to a faster growth rate, which is desirable, but at the same time it results in uneven concentration distributions, unstable interface formation and inclusion trapping in the solid. Since Cz is a growth technique of a free surface, an additional convection (the so-called Marangoni convection) in the melt is induced by surface tension. To alleviate adverse effects of the convection and smooth out concentration distribution in the melt, it is a common practice to keep the seed and crucible in opposite
Chapter 3 – Crystal Growth Methods for SixGe1-x Alloys
24
rotational motions at a constant rate. Depending on the size of the crystal, typical values of rotations are in the order of 2 to 30 rpm for the crystal, while 2 to 20 rpm for the crucible. Growth steps and the schematic figure for Cz provided should not let us underestimate the complexity of the equipment used in Cz and the importance of growth parameters that play significant roles in obtaining a single crystal of high quality. Here, discussing the details of the equipment used in the Cz process is beyond the scope of this section; however, it is suffice to say that Cz has higher demands on equipment than many other melt growth techniques, for instance, vertical Bridgman. Cz technique can be considered as a semi-containerless method since the crystal is not in contact with any part of the crucible except the liquid. Therefore, the grown crystal may not suffer from some of the problems usually observed in the contained growth techniques such as cracking, sticking and straining of the crystal. As for the growth parameters, a few of them that appear imperative in the context of the discussion are enumerated. In order for achieving a successful single crystal growth, the correct combination of the growth parameters has to be imposed on the growth system. These parameters are essentially thermal stability of the growth environment, crystal-pulling rate, rotation rate of the crystal and crucible, flow rate of an inert gas, seeding and crystal size grown. As can be understood from the Si x Ge1− x binary phase diagram presented in chapter 2, the Si x Ge1− x system does not possess congruent melting temperature, meaning that upon
solidification of a liquid, the solidified crystal does not have the same solute composition as the liquid. Namely, during the growth, second component in the liquid (mixture) segregates leading to a crystal of non-homogeneous concentration distribution along the growth direction. The segregation problems of the solute become more apparent if a crystal is grown from the intermediate region of the phase diagram, where the largest separation (miscibility gap) exists between the liquidus and solidus lines. The compositional variation or segregation is a serious problem for the Cz growth of
Chapter 3 – Crystal Growth Methods for SixGe1-x Alloys
25
Si x Ge1− x bulk crystals since it may lead to constitutional supercooling and in turn
polycrystalline growth if the appropriate growth rate is not selected. Typical pulling rates applied for the growth of SixGe1-x vary from 0.5 to 3 mm/hour. Non-homogeneous composition distribution in the grown crystals is less severe when the growth is performed from the region close to the end sides of the phase diagram [35, 59]. Constitutional supercooling is associated with the distribution of the solute in the liquid and takes places due to a change in composition, not in temperature. When the solubility of solute in the solid is less than that in the liquid (ko P2
Figure 3.2: Schematic drawing of a standard and detached VB technique Unlike the Cz technique, the VB technique was mainly used to grow germanium rich Si x Ge1− x solid solutions. Growth experiments reported were confined to obtain small diameter crystals ranging from 9-13 mm as can be seen from Appendix A, Table A.2. The dislocation density in the grown crystals is usually in the order of 104-105 cm-2. Another immediate observation is that the VB method has not been subjected to intense research to grow Si x Ge1− x single crystals. This might be mainly attributed to the fact that Si x Ge1− x is a profoundly segregating system upon solidification almost for the entire composition range due to its large miscibility gap. It is not possible to obtain uniform composition without replenishment of preferentially consumed species in the limited amount of liquid.
Chapter 3 – Crystal Growth Methods for SixGe1-x Alloys
34
3.2.3 Floating Zone Technique (FZ) In the floating zone technique, polycrystalline feed and single crystal seed materials are vertically standing rods clamped at their ends. The feed and seed materials are enclosed in a quartz ampoule within which an inert atmosphere is maintained. During the growth process, a relatively small molten zone (a few centimetres in length) is created between the ends of seed and feed materials by proper heating equipment, usually radio frequency induction heating. To achieve a good mixing in the liquid zone during growth, either feed and seed rods or only the feed rod can be rotated. When rotating both feed and seed rods, opposite rotation directions are used. Nucleation of a single crystal starts from the seed material and the growth proceeds by moving the molten zone over the length of the feed rod by the translation of the heater. The molten zone is suspended freely like a drop between the seed and feed rods against the gravity by the counter effect of the surface tension. The repeated zone melting can be performed over the entire length of the feed rod to achieve a substantial purification in the grown crystal. In particular, impurities with a segregation coefficient, k o 0.4) have higher carrier velocity than silicon so that they can suggest better transistor characteristics.
4.3.1 SiGe/Si HBT for RF Applications The fast development observed in wireless and telecommunication technologies necessitates achieving high receiver sensitivity, which requires low noise from the amplifiers. Low noise amplifier is a key component in the receiver-end [98] and
Chapter 4 –Applications of SixGe1-x
56
employed to amplify the incoming RF signal to by introducing a minimum amount of noise [99]. Besides, portable telecommunication equipments need to have low current consumption for improved battery lifetime. Therefore, recently, appreciable amount of research has been conducted on the SiGe HBT for RF technology in order to achieve trade off between the noise performance and current gain of the SiGe HBTs because increasing the germanium content in the base region provides high current gains, but deteriorates the noise performance of SiGe HBTs [73-78]. SiGe HBT technology exhibits very attractive high frequency noise performance due to
the SiGe/Si heterojunction structure [100,101]. It is superior to III-V technology because of not only its noise performance but also its better-cost performance [100,102]. In the following, the composition of SiGe layers in the base region of SiGe HBT for RF applications is provided. •
The peak mole fraction of germanium available in the SiGe HBT that Niu et al. [99] used for their experiments is 10 at. %, 14 at. % and 18 at.%. They reported that better noise profile was obtained at 18 at % Ge.
•
Regis et al. [100] reported that the germanium content of the base region in SiGe HBTs for RF applications should range from 20%-30% to get a better noise performance.
•
Application of a SiGe HBT containing 28 at. % Ge to an ultra noise oscillator in the 4 GHz range was investigated and it was reported SiGe HBTs yields impressive low frequency noise performance [103].
4.3.2 High Electron Mobility Transistor Si1− x Ge x epitaxial layers can be deposited on a Si substrate without any relaxation of
Si1− x Ge x layers. In this structure, silicon substrate is unstrained and the Si1− x Ge x is compressed in the growth plane. In such a totally strained structure, the band offset at the
Chapter 4 –Applications of SixGe1-x
57
interface between the Si1− x Ge x and silicon substrate lies almost entirely in the valence band. Therefore, holes can be confined in such structures, but not electrons. These two dimensional hole gas structures (2DHG) have lower mobility than two dimensional electron gas structures (2DEG) due to the alloy scattering in the Si1− x Ge x conduction band. To confine the much higher mobility electrons in a 2DEG, Si1− x Ge x / Si has to be grown in a way that silicon layers have to be under tensile strain. These structures have significant conduction band offset (a type two band alignment). Si1− x Ge x alloys with large lattice constant (high germanium content) are very useful to change band alignment at the Si / SiGe heterojuction interface. A type two-band alignment where electron can be confined in Si can be only achieved when silicon is grown on a larger lattice constant. The graded Si1− x Ge x buffers have been used as a virtual substrate on which Si layers are grown to achieve a very high mobility two-dimensional electron gas. This structure can be used to fabricate high electron mobility transistors (MODFET). Graded layers include relatively high dislocation density and increase over all thickness of the devices. Therefore, using a high quality bulk single crystal with the same composition as graded layers seems more appropriate.
4.4 Solar Cells A solar cell is a p-n junction device, which converts the photon power into the electric power and delivers this power to an external circuit. Si1− x Ge x alloys have been considered for solar cell applications because of its enhanced sensitivity in the long wave-length region compared to pure silicon. As in the case of Si1− x Ge x -based photodetectors, almost all the research conducted thus far has been focused on producing compositionally graded Si1− x Ge x layers on which active Si1− x Ge x absorbing layers can be epitaxially grown. Bremond et al. [11] have recently studied the growth of graded layers on a silicon substrate (using LPE), with the germanium contents varying from 3.5 % to
Chapter 4 –Applications of SixGe1-x
58
11.5 % for solar cells applications. In addition, Said et al. [12] studied relaxed Si1− x Ge x epitaxial layers with x ≤ 0.1 for solar cell application.
4.5 Summary A literature survey was performed to determine the target crystal composition of Si x Ge x alloy system. The literature survey has shown that the potential applications of the Si x Ge x system require a wide spectrum of compositions depending on the device
applications. Therefore, it is feasible to consider entire range of Si x Ge x for the crystal growth.
59
CHAPTER 5 A Rational Continuum Mixture Model for LPD Growth of SixGe1-x 5.1 Introduction Every physical object is formed of molecules, which in turn consists of atoms, and subatomic particles. These particles are not continuously distributed within the body of the object. As well, microscopic examinations reveal that there is presence of empty spaces (microscopic discontinuities) among the particles. In the study of external effects on the physical objects, the molecular structure of the object may or may not be taken into account depending on the assumptions established. Disciplines such as statistical and quantum mechanics handle the physical objects on microscopic scale. The study ignoring the presence of microscopic discontinuities, and treating the object as being formed by continuous distributions of particles, is referred to as macroscopic study. The classical continuum theory of materials falls into the category of macroscopic study, and is the basis for modeling most engineering processes. In the classical continuum theory, all fine details of the molecular and atomic structures and their close interactions are excluded, and the physical object is considered continuous disregarding the empty spaces between particles. It is assumed that the particles are
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
60
continuously distributed within the object. These assumptions laying the foundations of the continuum theory can be satisfactorily justified for the study of solid and liquid media considering the concept of the mean free path. The mean free path is defined as the average distance that a molecule travels between successive collisions with other molecules. The ratio of mean free path to the smallest characteristic length of physical boundary is small in a solid or liquid medium, which allows one to treat them as a continuum. Continuum mechanics is a branch of physical science dealing with the study of deformation and motion of a continuum under the influence of external effects. The thermomechanical balance laws of a continuum are (1) the balance of mass, (2) the balance of linear momentum, (3) the balance of angular momentum, (4) the conservation of energy, and (5) the second law of thermodynamics (principle of entropy). These balance laws are valid for all types of media (fluids, solids, gases) regardless of their internal constitutions. Different materials respond differently when subjected to external effects such as forces, heat inputs, and any other disturbances. For instance, solids and liquids obey the same thermomechanical balance laws, but have distinct responses to external agents. To this end, to account for these differences, additional sets of equations, the so-called constitutive equations, are required. Continuum mechanics provides a unified approach consisting of three steps to the study of the global behaviour of materials. The first step involves the thermomechanical balance laws common to all materials. The second step deals with constitutive equations responsible for clarifying differences in the nature of different materials. The third step is concerned with obtaining the governing equations for a material under consideration through combining the thermomechanical
balance
laws
with
appropriate
constitutive
equations.
The
thermomechanical balance laws can be derived by employing the basic kinematic concept of deformation, strain, and motion as well as the concept of stress. The constitutive equations can be formulated based on the combination of experimental results such as Newton’s law of viscosity, Fourier law of heat transfer and eight axioms, namely the
61
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
axioms of causality, determinism, equipresence, objectivity, material invariance, neighbourhood, memory and admissibility. In the establishment of constitutive equations, intuition plays a very important role; however, in no case, must constitutive equations violate these eight axioms. This
chapter
presents
a
rational
continuum
mixture
model
based
on
the
thermomechanical balance laws and related irreversible thermodynamics to study transport phenomena (fluid flow, heat and mass transfer) occurring during the Liquid Phase Diffusion (LPD) growth process for a binary system of Si x Ge1− x single crystal semiconductors. The current model takes into account various physical phenomena that may take place during the growth process, including the Soret effect in mass balance, the Dufour effect in energy balance. In addition, the contribution of body forces in energy balance is taken into account. Diffusion velocities of species cause the creation of kinetic energy in a multicomponent mixture. This phenomenon is incorporated into the jump condition for the energy balance. Finally, the effect of the interface curvature is included in the jump condition for the energy balance. The above listed effects may or may not be significant depending on the crystal growth process considered. For LPD process considered in this thesis, these effects are negligible. Therefore, they are not taken into account in the numerical model presented in chapter 6. The LPD technique developed and utilized in this work can be considered as a solution growth technique since SixGe1− x binary liquid solution is formed by the dissolution of solute (Si) in Ge solvent. In this technique the solvent material (Ge) is sandwiched between single crystal substrate (Ge) and poly crystalline source material (Si). These vertically stacked charge materials are held in a quartz crucible, forming the LPD growth cell as illustrated in figure 5.1. Initially, all three layers are solid materials. On locating the growth zone in an axial thermal gradient, the middle solid germanium totally melts to form the liquid solvent for the growth. The germanium substrate partially melts. The silicon source, on the other hand, remains in the solid state due to its higher melting
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
62
point. The molten germanium dissolves silicon source according to thermodynamic equilibrium. Dissolved silicon species are incorporated into germanium liquid, thereby forming a binary SixGe1− x mixture. Silicon species are transported to the growth interface between substrate and solution by the combination of several driving forces. When the mixture at the growth interface is supersaturated, solidification takes places. The growth mechanism of SixGe1− x by LPD will be discussed in coming chapters in detail. In the present mixture model, the solid phase represents the substrate, source (single component) and the growing crystal in the form of a binary system as well as the growth crucible. The solid phase is treated as a rigid and heat conducting binary mixture in which the diffusive mass transports takes place. The growth crucible is only dealt with as a heat conducting solid phase. The liquid phase, namely the solution, is treated as a mixture of two viscous heat-conducting fluids. Quartz rod
Sealing cap
≈
≈
Source (Si) Solvent (Ge) Seed (Ge)
Quartz ampoule
Quartz crucible
Insulator material 3 - Zone Furnace
Quartz pedestal
Figure 5.1: The schematic view of the LPD growth system.
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
63
5.2 The Rational Mixture Model In this section, an introduction to the kinematics of the mixture is given. Then, the governing equations and related boundary and interface conditions of the model have been derived from the fundamental principles of thermomechanics of continua. The constitutive equations of the solid phase (substrate, source, the growing crystal in the form of a binary system and the crucible) and the liquid phase (solution) are derived employing an irreversible rational thermodynamic theory along with linearization about a reference state. Special forms of the governing equations and associated interface conditions are obtained by way of systematic assumptions and then simplifications. The governing and constitutive equations are presented in the spatial form. The governing equations, interface conditions and constitutive equations are obtained for each component of the mixture. Then, summing these equations over the number of the constituents, a set of equations for the mixture is obtained. A treatment of the subject of mixtures can also be found in [104-106].
5.2.1 Kinematics of Mixtures Here, the kinematics of a mixture is formulated. The mixture is treated as a continuum composed of N material bodies, B1 ,....,B N , which might be envisaged by means of the positions they occupy in three-dimensional Euclidian space E3 (sometimes called Cartesian space). Material bodies are defined as those which always contain the same set
of particles even though their shapes and locations change during the motion. As in the theories of a continuum with a single constituent, let each material body get assigned to a fixed reference configuration (also referred to as undeformed state). The reference configuration is a geometrical region in E3 where a material body with volume V and its
surface S occupies at time t=0. Assuming that a particular material point P is a position of a particular particle Pα in the reference state, then the particle for each material body
64
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
might be identified by the position vector X α of the particle of αth body or constituent in its reference state (refer to figure 5.2).
x3
S
X3
P
V
p Motion
Xα
x i3
I3 I1
S
V
i1
X2
I2
x2 i2 Deformed state t>0
Undeformed state t=0
x1
X1
Figure 5.2: Coordinate system. Since each particle has a unique position defined by X α at time t=0 and no two distinct particles of a material body can occupy the same initial position in space, the vector X α can be regarded as an unique identifier or in a sense, name for the particle Pα . Even as time passes, the initial position of the particle will remain the same; naturally, the initial location of the particle is time independent. Therefore, as a convention, X α will be referred to as the particle of the αth constituent rather than the position filled by a particle of the αth constituent. After motion and deformation have occurred, at time t, the material body is carried to a new configuration, so are all the material points. This new configuration is often referred to as the spatial configuration (deformed state). The unique material particle located in point P is now located at the spatial point p which might be described by a new position vector x . Throughout this chapter, rectangular coordinates shall be used when writing component formulas. The mutually orthogonal unit vectors for undeformed and deformed states are
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
65
I1 , I 2 , I 3 , and i1 , i 2 , i 3 , respectively. Thus, x = x k i k , X α = X Kα I K , where the summation convention on coordinate indices is in force. Note that no summation is allowed over α unless a term is inside a summation sign. An identity known as the
Kronecker delta is defined as δ kl = i k i l (i.e. δ kl = 1 , if i k = i l and δ kl = 0 , if i k ≠ i l ). The coordinates X Kα are called material or Lagrangian, whereas the coordinates x k are referred to as spatial or Eulerian. Note that lower case Latin indices shall be used for
spatial coordinates, while upper case Latin indices shall be for material coordinates. Material points of constituents in motion experience varying spatial positions in E3, leading to the generation of successive configurations. Thus, the motion and deformation of constituents carries various material points through spatial positions. Mathematically, the motion of constituents can be described by an equation of the form
(
)
x = χ α X α ,t ;
(
xk = χ kα X Kα ,t
)
t >0
(5.1)
The left-hand side of the equation (5.1) shows the spatial position x occupied at time t > 0 by the particle Xα , the right-hand side represents the function χ α called the
deformation function for the αth constituent. The above equation finds out the position x of every particle Pα (or Xα ) of B α at any instant of time. If the attention is focused on a specific particle, the equation determines the successive position of that particle at different instants of time. For a given time, on the other hand, the equation defines the position of all particles of B α , and the totality of all these positions forms the configuration of B α at that time. Hence, for a given instant, the equation defines a mapping that carries the particle Xα of B α to the spatial position x . According to the axiom of continuity, a particle Xα of B α cannot occupy two spatial positions and no two different spatial points can be the position of the same particle. In other words, the mapping defined by equation (5.1) should be one-to-one and invertible mapping for each
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
66
time t. Thus, the mapping defined by (5.1) is of a unique solution of the form, so called
inverse motion X α = χ −1α (x ,t );
X Kα = χ K−1α (x k ,t )
t>0
(5.2)
where the function χ −1α is called the inverse deformation function. For any instant of time equation (5.2) defines particle Xα of each position x . Considering a specific point x , the equation determines all the particles Xα of B α passing through that specific
point at different instants of time. At a given time, the equation specifies all particles positioned at different points of the current configuration. The totality of all particles forms the body B α . Thus, for a given time, the equation defines a mapping which carries a point x to the particle Xα of B α . In the description of the motion given by (5.1), attention is focused on a particle, and the particle is observed as it moves. This description is called the material or Lagrangian
(
)
description, and the independent variables Xα , t are referred to as material variables. However, in the inverse motion, attention is given to a point in the space; hence what is happening at the point is the center of interest as the time passes. This description is referred to as spatial or Eulerian description and the independent variables (x, t ) are
(
)
called spatial variables. Let’s assume a function given as f X α ,t . In the material description, such a function shows how the value of the function at the particle Xα changes as the particle Xα pass through different spatial locations x . If equation (5.2) is used in this function, it follows that
f (χ −1α (x ,t ),t ) = f (x ,t )
(5.3)
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
(
67
)
The function χ α X α ,t and χ −1α (x ,t ) are inverse of each other and either equation (5.1) or equation (5.2) is the unique solution of the other. To secure the invertibility of χ α , the mappings defined in (5.1) and (5.2) are assumed to possess continuous partial derivatives with respect to their arguments and the Jacobian J defined by
J = det F α ,
(5.4)
must not be zero. Here, F α is the deformation gradient defined as Fα = dχ α (X α ,t ) or in component form as FkKα = ∂xk / ∂X Kα . The material velocity of a material particle X α for which x is the spatial position at time
t is defined by
v α = x ′α =
dx dχ α (X α , t ) ∂χ α (X α , t ) = = dt dt ∂t
Xα
=
∂x ∂t
Xα
(5.5)
where the subscript Xα followed by a bar denotes that Xα is kept fixed in the derivation. Let us consider a differentiable material function Γ (X α ,t ) (i.e. scalar, vector or tensor quantity). This function can be interpreted as a value of the function, possessed by the particle Xα for which x is the position at time t. The total time derivative of this function at time t can read as
dΓ (X α , t ) ∂Γ (X α , t ) = dt ∂t
Xα
(5.6)
The partial time derivative of this function at time t, with Xα being fixed, is considered as the time rate of change of the value of the function possessed by the particle Xα . This derivative is referred to as the material time derivative of a function, since it is taken
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
68
holding Xα constant. Physically, it represents the rate at which the function changes as a function of time, as seen by an observer (Lagrangian observer) located at the particle and moving with it. Since the total time derivative of a function is taken keeping Xα constant, the total time derivative symbol
Dα ( ) d( ) is then replaced by , and thus Dt dt
equation (5.6) can read as
Dα Γ (Xα ,t ) ∂Γ (Xα ,t ) =Γ′ = Dt ∂t
(5.7)
where the prime indicates that it is a material time derivative, following the motion of the particle Xα of the αth constituent. Let us this time assume a function in the form of Γ (x , t ) , where the total time derivative for the function, with x held fixed is written as dΓ (x ,t ) ∂Γ (x ,t ) = dt ∂t
x
(5.8)
Since x is a point in the spatial (current) configuration, then Γ (x , t ) is regarded as the value experienced by particles Xα whose position is currently x at time t. Also, the partial time derivative of the function of interest can be interpreted as the time rate of change of the value of the function, experienced by the particle whose position is currently x at time t. This derivative is called local time derivative of a spatial function, and physically it represents the rate at which the function changes observed by an observer (Eulerian observer) currently stationed at the point x .
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
69
Considering the function Γ (x ,t ) again, this function can also be expressed as
Γ (χ α (X α , t ),t ). Using the chain rule, the material time derivative of Γ (x ,t ) , following the motion of particle Xα of the αth constituent, can be defined as
( (
))
D α Γ (x ,t ) ∂Γ χ α Xα ,t ,t = Dt ∂t
Xα
=
∂Γ (x ,t ) ∂t
x
+
∂Γ (x ,t ) ∂x t ∂x ∂t
Xα
∂Γ (x ,t ) ∂Γ + v α (x ,t ) ⋅ ∇ Γ (x ,t ), or , Γ ′ = + Γ ,k v kα ∂t ∂t ∂Γ α ∂Γ α ∂Γ α α v1 + v2 + v3 Γ ,k v k = ∂x1 ∂x 2 ∂x3
(
Γ ′(x ,t ) =
)
(5.9)
For notational simplicity, fields can be written without including the independent variables (x ,t ) . Indices following comma indicate partial derivatives with respect to x k . The material time derivative of a quantity describes the rate of change of the quantity, which is observed by an observer who moves at the velocity v kα .
Assuming that all continuous bodies B1 ,...B N , in short B α , in a mixture occupy common three-dimensional physical space, then the continuum for the mixture can be considered a superposition of these N number of bodies. In what follows, in each spatial location within the mixture, there are N number of particles; i.e. for a given spatial position x at time t, x = x α , simply x = x (1) = x (2 ) = xN (refer to figure 5.3). Each constituent (species) is of a distinct mass density field ρ α (x ,t ) , briefly ρ α . Thus, the
mass density of the mixture at (x ,t ) can be written as
N
ρ = ρ (x , t ) = ∑ ρ α (x , t ) α =1
The mass fraction of the αth constituent at (x ,t ) can be introduced as
(5.10)
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System c α = c α (x ,t ) = ρ α (x , t ) / ρ (x , t )
70 (5.11)
The mass fractions of different constituents are related to each other through the formula
N
cα ∑ α
=1
(5.12)
=1
The molar density of the αth species is defined by
ω α = ω α (x , t ) = ρ α (x , t ) / M α
(5.13)
where M α is the molecular weight of the αth constituent.
x3 X3
i3
X (1) I3 I1
x2
i2
X (2 )
X2
I2
x1 X1
Figure 5.3: Mapping defined for a mixture of two constituents, (i.e. N = 2 ). The molar density for the mixture is then
N
ρ
α =1
M
ω = ω (x , t ) = ∑ ω α (x , t ) =
(5.14)
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
71
where M is the molecular weight of the mixture defined as
N
M = ∑ xα M α
(5.15)
α =1
The molar fraction of the αth species at (x ,t ) is
xα = xα (x , t ) = ω α (x , t ) / ω (x , t )
(5.16)
The molar fractions must obey the formula given as
N
xα ∑ α
=1
(5.17)
=1
It must be noted that the molar fraction denoted by x α should not be confused with spatial coordinates. The mass fraction of the αth constituent can be related to the molar
fraction of the αth species through xα M α c = M α
(5.18)
The velocity of the mixture at (x ,t ) is the mass weighted average of the constituent velocities and is defined by
v (x , t ) = v =
1 N α α 1 N α α ρ v ; v = ∑ ∑ ρ vk k ρ α =1 ρ α =1
(5.19)
For a fluid system, the mass averaged velocity vk refers to the local velocity of a fluid stream. In fluid systems, it is much more common to define the velocity of a given
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
72
constituent with respect to the mass averaged velocity of the mixture than with respect to a stationary reference frame (i.e. laboratory walls). This situation results in the definition of the diffusion velocity v α for the α constituent at (x ,t ) , which is defined as v α (x , t ) = v α (x , t ) − v (x , t ); vkα = vkα − vk
(5.20)
The material time derivative of the function Γ (x ,t ) following the motion of mass centered particle moving with a mass averaged velocity vk of the mixture is defined by DΓ (x ,t ) & ∂Γ (χ (X ,t ),t ) = Γ (x ,t ) = Γ& = Dt ∂t
X = const .
(5.21)
Applying the chain rule for partial differentiation, it follows from equation (5.21) that DΓ (x ,t ) & ∂Γ (x ,t ) = Γ (x ,t ) = Γ& = + (v(x ,t ) ⋅ d )Γ (x ,t ) Dt ∂t DΓ ∂Γ = Γ& = + Γ ,k vk Dt ∂t
(5.22)
On comparison of equation (5.9)3 and (5.21)2, an identity very often used in the formulation of mixture theory is introduced as
Γ ′ − Γ& = Γ ,k ( v kα − v k ) = Γ ,k v kα
(5.23)
There are two ways in a mixture to define the mass flux of the αth constituent. The first one is the mass flux jα defined relative to a stationary coordinate frame. The second one is the mass flux jα specified with respect to the mass averaged velocity of the mixture, and is widely referred to as the diffusion flux. The former and latter can be formulated as follows
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
jα = ρ α v α ; jkα = ρ α vkα
(
)
(
)
jα = ρ α v α − v ; jkα = ρ α vkα − vk ; jkα = ρ α vkα
73
(5.24)
It is important to underline that the diffusion flux must satisfy the condition
N
jα = 0 ∑ α =1
(5.25)
k
which can be readily proven from equation (5.10) and (5.19).
5.2.2 Axioms of Thermomechanics and Field Equations Here, we introduce the application of the thermomechanical balance laws to a mixture as well as the derivation of the field equations for the mixture. The thermomechanical balance laws for each constituent B α of the mixture consist of balance of mass, balance
of momentum, balance of moment of momentum, balance of energy, and entropy inequality (also see [104,107,108]). All five axioms consist of two parts, namely, a statement for each constituent and for the mixture as a whole. For the derivations of these thermomechanical laws for each constituent and the mixture, we start with introducing the well-known transport and Green-Gauss theorems, respectively, for a material body containing a discontinuity surface σ (t ) sweeping the volume of the body with its own velocity u k ;
D ∂ϕ ϕ dυ = ∫ + (ϕv k ),k dυ + ∫ ϕ (v k − uk ) nk da ∫ Dt V -σ ∂t V -σ σ (t )
∫σφ
V-
,k
dυ + ∫ φ nk da = σ
∫σφ n da k
S-
(5.26)
(5.27)
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
74
where ϕ and φ are an arbitrary fields, V is a fixed spatial volume enclosed by a surface S (volume boundary), dυ and da are the volume and area elements, respectively,
where V - σ and S - σ denote the volume and surface excluding points lying on the discontinuity surface σ ( t ) . The symbol
indicates the jump of the enclosed quantities
across the discontinuity surface; for instance, ϕ = ϕ + − ϕ − , where ϕ + and ϕ − are the values of ϕ on the positive and negative sides of σ ( t ) . For the derivation of equation (5.26), refer to appendix B, equation (B.10).
5.2.2.1 Balance of Mass The axiom of balance of mass comprises two parts, i.e. a statement of balance of mass for each constituent (species), and that for the entire mixture. The axiom of balance of mass for each constituent states that the time rate of the change of the mass of each species α in a multicomponent mixture is equal to the rate at which the mass of chemical species α is produced within the volume and on the discontinuity surface. If V is a fixed spatial
volume at a given time (i.e. boundaries of the volume is time independent), the integral form of this axiom for α th constituent is given by
Dα Dt
α α α ∫σρ (x ,t )dυ = ∫σrˆ (x ,t )dυ + σ∫( )ˆs (x ,t )da
V−
V−
(5.28)
t
where the quantities rˆ α (x ,t ) and ˆs α (x ,t ) are called as mass production and represent the rate of mass supplied to the α th constituent per unit volume and area, respectively from the other constituents occupying the position of x at time t . That is to say, the mass
supply terms are due to the interaction among the constituents such as chemical reactions. Using the transport theorem (5.26) for a volume containing a discontinuity surface σ (t ) derived for an arbitrary field ϕ , and setting ϕ = ρ α and vk = vkα ,equation (5.28) can be expressed in the form
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
∂ρ α + ρ α v kα ,k = rˆ α in V − σ ∂t ρ α v kα − u k nk = ˆs α on σ
(
)
(
)
75
(5.29)
A derivation of this form is given in appendix B, equation (B.17). These two equations are local forms of the balance of mass (the continuity equation) for each species α expressed in a spatial (Eulerian) form. The first equation is valid at all parts of the body excluding the points located on the singularity, while the second is valid only on the discontinuity surface and represents the jump condition across σ (t ) . If equation (5.29)1 and (5.29)2 are summed over indices α along with (5.10) and (5.19), we can obtain balance of mass for the mixture as N ∂ρ + (ρv k ),k = ∑ rˆ α in V − σ ∂t α =1
(5.30)
N
ρ (v k − u k ) n k = ∑ ˆs α on σ α =1
where
N
rˆ α ∑ α
= 0 and
=1
N
ˆs α ∑ α
= 0 because of the requirement that there be no net
=1
production of mass in the body. The alternative form of equation (5.29)1 and (5.29)2 can be introduced as
⎛ ∂cα ⎞ + cα,k vk ⎟⎟ + jkα,k = rˆα in V − σ ⎝ ∂t ⎠ ρcα (vk − uk ) + jkα nk = ˆsα on σ
ρ ⎜⎜
(5.31)
where equation (5.31)1 follows from (5.23), (5.20), (5.11), (5.30)1, and (5.24)1. Similarly, (5.31)2 can be obtained using (5.20), (5.11), and (5.24)1. For the detailed derivations, refer to appendix B, equation (B.21). Before introducing the second axiom of
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
76
thermomechanics, an identity of fundamental importance in mixture theories, used extensively in the coming sections must be introduced. Let us assume that Γ α (x , t ) is a property of the αth constituent at (x ,t ) . The mass averaged value of this quantity can be written as 1 N α α ∑ ρ Γ (x ,t ) ρ α =1
Γ (x , t ) =
(5.32)
By making use of equation (5.11), (5.23), (5.32), and (5.31)1, we can write
ρΓ& = ∑ (ρ α Γ ′α − (ρ α Γ α v kα ),k + Γ α rˆ α ) N
(5.33)
α =1
For the derivation of (5.33), refer to appendix B, equation (B.23).
5.2.2.2 Balance of Linear Momentum The axiom of balance of linear momentum states that the time rate of change of the momentum of a body is equal to resultant force acting upon the body. Similar to the balance of mass, this axiom also involves statements of the balance of linear momentum for the α th constituent and the mixture. The axiom of balance of linear momentum for the α th constituent in a fixed volume is
Dα Dt Dα Dt
∫ρ
α
∫ρ
α α
v α dυ = F α (5.34)
V
vl dυ = Fl
α
V
The resultant external force acting upon the αth constituent is written as
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System ˆ α + rˆα vα ))dυ + ∫ tα(n )da + ∑ F αβ F α = ∫ (ρ α bα + (p β =1
(5.35)
Fl = ∫ (ρ bl + ( ˆpl + rˆ vl ))dυ + ∫ tkl nk da + ∑ Fl αβ V
α
V
α α
α
α α
S
S
77
α
β =1
ˆ α is the rate of where bα is the body force per unit mass acting on the αth constituent, p momentum production and t α(n ) is the partial surface traction per unit area, defined as tα(n ) = n ⋅ tα
(5.36)
where tα is the partial stress tensor. In component form, the partial surface traction can written as tα(n ) = ns i s ⋅ tklα i k i l = tkαl nk i l , where t klα is the partial stress tensor in component
ˆ α can be described as a local or internal body force, and represents the local form. p
(
ˆ α + rˆ α v α interaction of constituents with each other. The term p
)
accounts for the
complete local interaction. The partial stress tensor describes the contact forces on the
αth constituent resulting from the contact from other constituents. The term
F αβ ∑ β =1
accounts for some other forces and interfacial effects such as electromagnetic force which may act on the body and on the discontinuity surface and surface tension acting on the discontinuity surface. If a surface effect such as surface tension is included, the last summation term can be replaced by
∫σ t ( )ds ασ b
(5.37)
∂
ασ where tασ (b ) is the interface stress vector. t (b ) is defined as a contact force per unit length
on ∂σ and is a function of position on ∂σ . ∂σ is the curved line formed by the intersection of the discontinuity surface σ with the closed surface S bounding the
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
78
volume V . ds is an infinitesimally small length element. The interface stress vector can be expressed as follows [109]
tασ (b ) = t
ασ
where t
⋅b
ασ
(5.38)
is the symmetric interface stress tensor acting on the αth constituent.
b = h × n is the binormal vector which is tangent to the discontinuity surface σ , normal
to the curve ∂σ , and outwardly directed with respected to V . h is the tangential unit normal. With the application of the divergence theorem in a plane (surface divergence theorem), we can write
∫t
ασ
⋅ bds =
∂σ
∫ (∇ s ⋅ t
ασ
)
+ 2Ω tασ ⋅ n da
(5.39)
σ
The interface stress tensor tσα can be decomposed into two parts as [109]
t
ασ
ασ
=S
where S
+ γ ασ P
ασ
(5.40)
is the viscous portion of interface stress tensor, and γ ασ is thermodynamic
surface tension due to the surface stress generated by the deformation of the interface. Ignoring the viscous effect at the interface, we can write tασ = γ ασ P
(5.41)
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
79
where P = I - n ⊗ n or in index notation, Pkl = δ kl − nk nl and I is the identity tensor. Derivation of the projection tensor and surface divergence theorem is presented in
appendix B, equation (B.26). On combining equations (5.34), (5. 35),(5.37), (5.38), (5.39), and (5.41), the integral form of the balance of linear momentum for a volume swept by a singularity yields Dα Dt
∫ρ
α α
vl dυ =
V −σ
∫ (ρ
(
α α
))
bl + ˆplα + rˆ α vlα dυ +
V −σ
∫ tkl nk da + ∫ (∇ sγ α
S −σ
ασ
)
+ 2Ωγ ασ nl da
(5.42)
σ
Surface tension balances the jump condition in the stress. Applying (5.26) to the left hand side of (5.42) with ϕ = ρ α vlα ; vk = vkα , and using the Green-Gauss theorem in (5.27) for the second integral on the right hand side, the local form of the balance of linear momentum in V - σ and on σ can be obtained in spatial form as (refer to appendix B,
equation (B.35)). Dα vlα = tklα ,k + ρ α blα + ˆplα in V - σ Dt ρ α vlα (vkα − uk ) − tklα nk = ∇ sγ ασ + 2γ ασ Ω nl
ρα
on σ
(5.43)
where nl is the unit normal to the discontinuity surface, ∇s is surface gradient operator and Ω is the mean curvature. If we assume that the surface tension is independent of the position on the interface, ∇ sγ ασ becomes zero. To consider the axiom of balance of linear momentum for the mixture, the external body force per unit mass for the mixture is defined as
b=
1
N
ρ α bα ; b ∑ ρα =1
l
=
1
N
ρ α bα ∑ ρα =1
l
(5.44)
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
80
The stress tensor and the interface stress tensor for the mixture are defined respectively as
(
)
t = ∑ t − ρ α vα ⊗ v α ; tkl = ∑ (tklα − ρ α vkα vlα ) N
α =1
α
N
t = ∑t σ
ασ
α =1
N
, tkl = ∑ tkl σ
N
α =1
(5.45)
ασ
α =1
As a consequence of the balance of linear momentum for the mixture, the momentum production is subject to the condition
(ˆpα + v α rˆα ) = 0 ∑ α N
=1
l
l
(5.46)
By employing (5.33) for Γ = vl , we can write
ρv&l = ∑ (ρ α vl′α − (ρ α vlα v kα ),k + vlα rˆ α ) N
(5.47)
α =1
Alternatively, equation (5.47) can be written as (refer to appendix B, equation (B.37))
ρv&l = ∑ (ρ α vl′α − (ρ α vkα vlα ),k + vlα rˆ α ) N
(5.48)
α =1
On summing equation (5.43)1 over α and then using (5.48), (5.44), (5.45)1 and (5.46), the local form of the balance of linear momentum for the mixture in V − σ reads as
ρv&l = tkl ,k + ρbl in V − σ ρvl (vk − uk ) − tkl nk = ∇ sγ σ + 2γ σ Ω nl on σ
(5.49)
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
81
where equation (5.49)2 follows from equation (5.45) and (5.20). For the derivation of the forms given in (5.49), refer to appendix B, equation (B.43).
5.2.2.3 Balance of Moment of Momentum This axiom also involves two statements, namely, one for constituents and one for the mixture. The axiom states that the time rate of moment of momentum of the body is equal to the resultant moment of all forces acting upon the body. The integral form of this axiom for the αth constituent in a fixed volume V with a discontinuity surface σ is introduced as
Dα Dt Dα Dt
∫ (ρ
) × xdυ = ∫ ((ρ α bα + pˆ α + rˆα vα )×x + mˆ α )dυ + ∫ tα( ) × xda α α α α α α α α α ∫ ρ v x e dυ = ∫ ((ρ b + ˆp + rˆ v )x e + mˆ )dυ + ∫ t n x e α α
v
n
V −σ
V −σ
l
m lmn
V −σ
S −σ
l
l
l
m lmn
V −σ
n
(5.50)
kl k m lmn da S −σ
ˆ α (or m ˆ nα ) is the moment of momentum supply vector. It is analogous to the term where m ˆ α in (5.35). The terms inside the volume integral on the right hand side of the equation p (5.50) represent the moment due to the local interaction of the αth constituent with the other constituents within V . The symbol elmn is called the alternating or permutation symbol. As a reminder, the cross product of two vectors can be written in component form as: vα × x = vlα xmi l × i m = elmnvlα xmi n for v α = vlα i l , and x = xm i m .
With the
application of similar procedure which has led to equation (5.43), the local form of the balance of moment of momentum in the spatial form is α ˆ nα = 0 in V-σ elmntml +m
(5.51)
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
82
Summing equation (5.51) over α and using the definition of the stress tensor for the mixture as well as neglecting second order terms in diffusion velocities, it can be shown that the stress tensor for the mixture is symmetric, namely
t ml = tlm
(5.52)
For the derivation of local forms of balance of moment of momentum for the αth constituent and mixture, refer to appendix B, equation (B.49).
5.2.2.4 Balance of Energy The axiom of balance of energy deals with the thermodynamics of continuous media. This axiom is also known as the first axiom of thermodynamics. The axiom states that the time rate of change of the sum of internal and kinetic energies of a body, considered as a closed system, equals the sum of the rate of work done upon the body by all external forces and all other energies which may enter or depart the body per unit time. This axiom includes balance of energy for the mixture and the α th species. The statement of the balance of energy for the α th species reads
(
)
Dα K α + E α = W α + ∑U αβ Dt β
(5.53)
where K α and E α are, respectively, kinetic and internal energy for the α th constituent. They are defined as follows
1 α α vl vl dυ 2 V E α = ∫ ρ α ε α dυ K α = ∫ ρα V
(5.54)
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
83
where ε α is the internal energy per unit mass for the α th constituent. W α is the mechanical energy per unit time for the α th constituent and is defined as Wα =
∫ (ρ
V
α α α
)
bl vl + ˆplα vlα dυ + ∫ tklα vlα nk da
(5.55)
S
Equation (5.55) implies that the mechanical energy per unit time for the α th constituent is composed of two parts; that is, the mechanical work by the contact and body forces per unit time. In order to include effect of the surface tension on energy balance, several assumptions have been made; (1) the discontinuity surface contains surface particles of different species, and is a material surface in which surfaces particles are confined. Therefore, the normal component of the velocity of surface particles is equal to the normal component of the velocity of the discontinuity surface. The velocity of surface particles for the α th species is referred to as surface velocity v ασ and is defined as the time rate of change of the position of a particle. It is also assumed that tangential components of the surface velocity are continuous across the discontinuity surface, which means that the discontinuity surface does not have a discontinuity line. The rate at which work is done upon the body due to the surface tension is defined as
∫v
ασ
(
)
⋅ tασ ⋅ b ds
(5.56)
∂σ
With the application of the surface divergence theorem to equation (5.56), recalling the definition of surface stress tensor from equation (5.41), we can write
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System v ⋅ γ P ⋅ bds = ∫ (∇ s ⋅ (v ∫1 44244 3
∂σ
ασ
ασ
(v
ασ
ασ
⋅γ
)
ασ
)
(
) )
⋅ γ ασ P + 2Ω vασ ⋅ γ ασ P ⋅ n da
84 (5.57)
σ
P ⋅b
Combining the RHS of equation (5.57) with equation (5.55), it follows that Wα =
∫ (ρ
α α α
)
bl vl + ˆplα vlα dυ +
V -σ
∫ tkl vl nk da + ∫ (vl α α
ασ ασ
S -σ
)
γ ,l + 2Ωγ ασ vlασ nl da
(5.58)
σ
where in the derivation of terms inside the third integration on the RHS of equation
(
)
(5.58), the term ∇ s ⋅ vασ has been neglected due to the assumption of incompressibility, refer to the Appendix B, equation B.56.
The
∑U αβ
represents other energies entering or departing the body such as heat,
β
electrical, or chemical energies per unit time. For the continuum considered here, only the thermal energy is present. Therefore,
∑β U αβ
can be replaced by Qα which is the rate of
heat energy supply to the body. Hence, ⎛ ⎞ ⎛1 ⎞ Qα = ∫ ⎜⎜ ρ α hα + rˆ α ⎜ vlα vlα + ε α ⎟ + εˆ α ⎟⎟dυ − ∫ qkα nk da ⎝2 ⎠ ⎠ V⎝ S
(5.59)
where q kα is the heat influx vector of the α th constituent through the boundary of the body S with an exterior unit normal pointing outward with respect to the body, hα is the internal heat generation rate per unit mass within the volume and εˆ α is the rate of energy supply to the α th constituent due to the local interaction of α th constituent with other species within the body.
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
85
By combining equation (5.53), (5.54), (5.55) and (5.59), the integral form of the balance of energy for the α th species in a fixed volume including a discontinuity surface is given by Dα Dt +
∫(
α⎛1 α α
∫(
σ
)
α⎞ α α α α α α α ⎜ vl vl + ε ⎟dυ = ρ bl vl + ˆpl vl dυ + tkl vl nk dυ 2 ⎝ ⎠ V-σ V-σ S-σ ⎛ α α ⎞ 1 α α ⎛ ⎞ ασ ασ ασ ασ α vl γ ,l + 2Ωγ vl nl da + ⎜⎜ ρ h + rˆ ⎜ vl vl + ε α ⎟ + εˆ α ⎟⎟dυ − qkα nk da ⎝2 ⎠ ⎠ V-σ ⎝ S-σ
∫ρ
)
∫
∫
∫
(5.60)
With the classical procedure followed for the derivations of previous balance equations, the local form of the energy balance for the α th constituent is given by (refer to appendix B, equation (B.56) for derivations) Dα ε α = t klα vlα,k − q kα,k + ρ α hα + εˆ α in V − σ ρ Dt ⎞ α⎛1 α α ρ ⎜ vl vl + ε α ⎟ v kα − u k + q kα − t klα vlα nk = γ ασ vlασ 2Ωnl on σ ⎝2 ⎠ α
(
)
(5.61)
where, in obtaining equation (5.61), it is assumed that surface tension is independent of position. To evaluate the axiom of balance of energy for the mixture, the inner part of the internal energy density is introduced as
N
ρε I = ∑ ρ α ε α α =1
The inner part of the heat flux vector for the mixture is defined by
(5.62)
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System q I k = ∑ (qkα + ρ α ε α vkα − t klα vlα ) N
86
(5.63)
α =1
whereas the heat flux for the mixture is defined by
qk = q I k +
1 N α α α α ∑ (ρ vl vl vk ) 2 α =1
(5.64)
The internal heat generation rate per unit mass for the mixture is defined as
N
ρh = ∑ ρ α h α
(5.65)
α =1
The specific internal energy for the mixture is introduced as 1 ε = εI + 2ρ
N
ραvαvα ∑ α l
=1
l
(5.66)
Surface tension for the mixture is defined as
N
γ σ vlσ = ∑ γ ασ vlασ
(5.67)
α =1
Balance of energy for the mixture requires that ⎛ ⎛ α vlα vlα ⎜⎜ε + ∑ ⎜⎜ 2 α =1 ⎝ ⎝ N
⎞ ⎞ α ⎟⎟rˆ + ˆplα vlα + εˆ α ⎟ = 0 ⎟ ⎠ ⎠
Using equation (5.62) and (5.33) we can write
(5.68)
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
ρε&I = ∑ (ρ α ε ′α − (ρ α ε α vkα ),k + ε α rˆ α ) N
87
(5.69)
α =1
Summing equation (5.61)1 and using equations (5.62-5.69) as well as tiresome mathematical manipulations, the balance of energy for the mixture is of the following form N
ρε& = − qk ,k + tkl vl ,k + ρh + ∑ ρ α blα vlα α =1
N 1 ⎛1 ⎞ ρ ⎜ vl vl + ε + ∑ cα vlα vlα ⎟(vk − uk ) + qk − tkl vl nk = γ σ vlσ 2Ωnl on σ α =1 2 ⎝2 ⎠
(5.70)
where the balance of energy on the discontinuity surface for the mixture is obtained by summing equation (5.61)2 over the number of constituents in the mixture. Refer to appendix B, equation (B.75) for the detailed derivations of equation (5.70). Note that in comparison with the balance of energy for a single component body, equation (5.70)1 has an additional term which accounts for the rate of work of the external body force density. This term is in general omitted because its contribution to internal energy change with respect to time is negligibly small considering the other terms on the right hand side of equation (5.70)1. As well, it is worth noting equation (5.70)2 is also slightly different from the jump condition for a single component body. In the above jump balance equation, diffusion velocities of species contribute to the creation of kinetic energy in the multicomponent mixture. Ignoring, however, the higher order terms in diffusion velocities will lead to a jump equation identical to the one which can be written for a single component body.
5.2.2.5 Entropy Inequality The axiom of entropy inequality is also called the second axiom of thermodynamics. This axiom states that the time rate of entropy production in a multicomponent body (mixture)
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
88
owing to the external and mutual energy transmission must not be less than zero. Each species or constituent is bestowed with specific entropy η α (entropy per unit mass) and a temperature θ α which is a positive definite. The specific entropy for the mixture at (x ,t ) is defined as
N
ρη = ∑ ρ αη α
(5.71)
α =1
The integral form of entropy inequality for the α th constituent in a fixed region V with a discontinuity surface at a given time t is postulated as Dα Dt
α α ∫ ρ η dυ ≥
V -σ
⎛ α hα qkα α α⎞ ⎜ ⎟ ˆ + r d − ρ η υ ∫ ⎜ θα ∫ θ α nk da ⎟ ⎠ V -σ ⎝ S -σ
(5.72)
where the term qkα / θ α is an entropy influx of αth constituent through the boundary of the body S with an exterior unit normal pointing outward with respect to the body. The second term on the right hand side represents the entropy generation per unit time, and the term hα / θ α is the internal entropy source for αth constituent per unit mass. By following the standard procedure, the local form of the entropy inequality and the corresponding jump condition in Eulerian description can be written as (refer to Appendix B, equation (B.106) for derivations)
ρ αη ′α + (qkα / θ α ),k − ρ α (hα / θ α ) ≥ 0 in V − σ ρ αη α (vkα − uk ) + qkα / θ α nk ≥ 0 on σ
(5.73)
To be able to write the local form of the entropy inequality for the mixture, an influx vector q kα of the αth constituent is defined as
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System q kα = qkα + ρ αθ αη α vkα
89 (5.74)
Using equation (5.33) and setting Γ α = η α , we can write
ρη& = ∑ (ρ αη ′α − (ρ αη α vkα ),k + η α rˆ α ) N
(5.75)
α =1
As a consequence of the entropy inequality for the mixture,
N
η α rˆ α = 0 ∑ α
(5.76)
=1
Summing equation (5.73) over α and employing (5.75), entropy inequality for the mixture is written as
ρη& + ∑ (q kα / θ α ),k − ∑ ρ α N
N
α =1
α =1
hα
θα
≥0
(5.77)
Considering that the mixture has a single temperature, i.e. the temperature of all constituents are the same, (i.e, θ α = θ ), we can write
N
q k = ∑ q kα
(5.78)
α =1
By employing the well-known Gibbs equation in combination with the balance of energy for the mixture, it can be shown that N 1 ⎛ ⎛ ⎞ ⎞ q k = ⎜⎜ qk − ∑ ⎜ µ α + vkα vkα ⎟ jkα ⎟⎟ 2 ⎠ ⎠ α =1 ⎝ ⎝
(5.79)
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
90
On comparison of equations (5.77-5.79), and omitting higher order terms in diffusion velocities, the entropy inequality for the mixture reads
⎛ qk
ρη& + ⎜
⎝θ
−
1
⎞
N
µ α jα ⎟ ∑ θα ⎠ =1
− ρh / θ ≥ 0 in V − σ
k
(5.80)
,k
where µ α is the mass basis chemical potential of the αth constituent. Similarly, summing equation (5.73)2 over α and after some mathematical manipulations, the entropy inequality for the mixture on the discontinuity surface reads as
1⎛
N
⎞
ρη (vk − uk ) + ⎜ qk − ∑ µ α jkα ⎟ nk = 0 on σ θ⎝ α =1 ⎠
(5.81)
For the derivation of equation (5.80) and (5.81), refer to appendix B, equation (B.111). Defining the specific Helmholtz free energy function for the mixture as ψ = ε − θη and eliminating ρh between (5.70)1, and (5.80), the entropy inequality takes the following form N ⎛N ⎞ θ ⎛ ⎞ − ρ ψ& + ηθ& − ⎜ ∑ µ α jkα ⎟ − ,k ⎜ qk − ∑ µ α jkα ⎟ + tkl vl ,k ≥ 0 α =1 ⎝ α =1 ⎠ ,k θ ⎝ ⎠
(
)
(5.82)
This form of entropy inequality will be used to eliminate the dependence on certain variables appearing in the constitutive equations.
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
91
5.2.3 Constitutive Equations In this section, the constitutive equations of the liquid and solid phases are presented using constitutive theory given in [104, 107, 110, 111].
5.2.3.1 The Liquid Phase The liquid phase is considered as a mixture of two viscous and heat conducting incompressible Newtonian fluids, and is assumed to be a dilute binary solution of two components; the solute and the solvent. In the case of a Si x Ge1− x binary system, silicon ( Si ) is the solute, and germanium ( Ge ) is the solvent. Thus, equation (5. 31)1, (5. 31)2 and (5.82) can be written as ⎛ ∂c ⎞ + c,k vk ⎟ + jk ,k = 0 ⎝ ∂t ⎠ ρc (vk − uk ) + jk nk = 0
ρ⎜
(5.83)
θ − ρ ψ& + ηθ& − (µ jk ),k − ,k (qk − µ jk ,k ) + tkl vl ,k ≥ 0
(
)
θ
(5.84)
where superscript α is dropped for the sake of notational simplicity and the following notation is embraced henceforth. c = c Si = 1 − c Ge , jk = jkSi = − jkGe and µ = µ Si − µ Ge , which is the difference of chemical potentials of the solute and the solvent. In the present model, both source terms are neglected, so ˆr Si and ˆs Si are equal to zero. According to the axioms of the constitutive theory, the fields are divided into two distinct categories; dependent and independent variables. The axiom of equipresence states that at the outset, a field appearing as an independent variable in one constitutive equation should appear likewise in all constitutive equations unless ruled out by physical laws.
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
92
Thus, all dependent variables, i.e. ψ , η , t kl , q k and jk are considered to be functions of the same independent variables, i.e. ρ −1 , θ , θ ,k , c , c ,k and d kl . Here, d kl is the symmetric deformation rate tensor, defined as 2d kl ≡ (v k ,l + vl ,k ) . The deformation rate tensor gives the time variation of the distance separating two neighboring material points. The material time derivative of the field ψ can be written as follows by applying the chain rule •
∂ψ ∂ψ & ∂ψ & ∂ψ ∂ψ ∂ψ & ψ& = −1 ρ −1 + c& + c&,k + d kl θ+ θ ,k + ∂ρ ∂θ ∂θ ,k ∂c ∂c,k ∂d kl
(5.85)
Substituting equation (5.85) into (5.84) and noting that
• −1
ρ
= d kl δ kl / ρ
and
c& = − j k ,k / ρ from equations (5.30)1 and (5.83)1, respectively, we can obtain ∂ψ ∂ψ & ∂ψ ∂ψ & ⎞ ⎛ ∂ψ c&,k − ρ d kl − ρ ( − µ ⎟c& θ ,k − ρ + η )θ& − ρ ⎜ ∂θ ∂d kl ∂c,k ∂θ ,k ⎠ ⎝ ∂c θ ⎛ ∂ψ ⎞ + ⎜⎜ − −1 δ kl + tkl ⎟⎟d kl − µ,k jk − ,k (qk − µ jk ) ≥ 0 θ ⎝ ∂ρ ⎠ −ρ
(5.86)
.
The inequality in (5.86) is linear in θ& , θ&,k , c& , c& ,k and d kl since the dependent fields i.e.
ψ , η , t kl , q k and jk are independent of these quantities. For arbitrary variations of these quantities, the entropy inequality can’t be satisfied unless the coefficients of these quantities become zero, so it follows that ∂ψ ∂ψ ∂ψ = 0, = 0, = 0, ∂θ ,k ∂c,k ∂d kl ∂ψ ∂ψ ∂ψ = −η , = µ , − −1 = π ∂θ ∂c ∂ρ −1 ψ = ψ (ρ ,θ , c ), η = η (ρ −1 ,θ ,c ), µ = µ (ρ −1 ,θ , c )
(5.87)
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
93
and the entropy inequality reads
t d kl − µ,k jk −
D kl
where
D t kl
θ ,k (qk − µjk ) ≥ 0 θ
is the dissipative part of the stress tensor defined as
(5.88)
D t kl
= πδ kl + t kl and π is
the thermodynamic pressure defined as π = −∂ψ / ∂ρ −1 .
For an incompressible fluid, the mixture density is assumed constant. Therefore, the balance of mass in equation (5.30)1 reduces to vk ,k = d kk = 0
(5.89)
On replacing π by an unknown pressure function p , all dependent fields become independent of ρ −1 . Casting the following relation
µ ,k =
∂µ ∂µ θ ,k + c,k ∂θ ∂c
(5.90)
into equation (5.88), we can write the entropy inequality as
t d kl −
D kl
If
θ ,k θ
∂µ ⎞ ⎞ ∂µ ⎛ ⎛ c,k jk ≥ 0 θ ⎟ jk ⎟ − ⎜⎜ qk − ⎜ µ + ∂θ ⎠ ⎟⎠ ∂c ⎝ ⎝
(5.91)
t , q k and jk are continuous functions of d kl , θ ,k and c ,k , then from equation
D kl
(5.91), it follows that D tkl = qk = jk = 0 when d kl = θ ,k = c ,k = 0
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
94
A process is called thermodynamically admissible if and only if it obeys the entropy in equality and possesses a non-negative temperature. Thus, it follows from above discussion that the binary mixture under consideration is thermodynamically admissible if and only if they have constitutive equations of the form. ∂ψ ∂ψ = −η , =µ ∂θ ∂c ψ = ψ (θ , c ), η = η (θ , c ), µ = µ (θ , c ) qk = qk (θ ,k , c,k , d kl ;θ , c ) jk = jk (θ ,k , c,k , d kl ;θ , c ) tkl = − pδ kl + D tkl (θ ,k , c,k , d kl ;θ , c )
(5.92)
By using equation (5.92)1-2, the material time derivative of the specific Helmholtz free energy (ψ = ε − θη ) can be written as
ε& = µ c& + θη&
(5.93)
Substituting equation (5.93)in equation (5.70)1 and using equation (5.83) and (5.89), we can write the equation for balance of energy as
ρθη& = D tkl d kl − qk ,k + µ jk ,k
(5.94)
where the heat source h is neglected since there is no heat generation in the liquid. The constitutive equations in (5.92) are still in the general form, and we can not yet combine them with the thermomechanical balance laws for the mixture of concern. Thus, for our purposes, they need further simplifications. If we assumed that changes in θ ,k , c ,k and d kl are small, the linear constitutive equations in the independent variables will be sufficient in the treatment of the binary mixture. Linear constitutive equations are obtained by writing a second degree-order polynomial for ψ and linear equations for
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
95
t , jk and q k . If we expand q k , jk and D tkl into the Taylor series about the reference
D kl
state indicated by superscript “o” we can write
qk = qko +
∂q ∂qk ∂qk ∂q ∂q T + k C + k θ ,l + k c,l + d mn ∂d ∂θ ∂c ∂θ ,l ∂c,l { { 12mn 3 { { a a 11
b11
12
b12
(5.95)
b13
qk = qko + a11T + a12C + b11T,k + b12C,l + b13d mn
jk = jko +
∂j ∂j ∂j ∂j ∂jk T + k C + k θ ,l + k c,l + k d mn ∂d ∂c,l ∂θ ,l ∂c ∂θ { { 12mn 3 { { a a 21
b21
22
b22
(5.96)
b23
jk = jko + a21T + a22C + b21T,l + b22C,l + b23d mn o D t kl = D t kl
+
∂ t ∂ t ∂ t ∂ t ∂ D tkl T + D kl C + D kl θ ,m + D kl c,m + D kl d mn ∂d θ3 ∂c m ∂θ ,m ∂c3 1∂2 12 12mn 3 123 12,3 a31
o D t kl = D t kl
a32
b31
b32
+ a31T + a32C + b31T,m + b32C,m + b33d mn
b33
(5.97)
where θ = To + T , To >> T , To > 0 and c = co + C , co >> C , co > 0 . To and co are the reference temperature and mass fraction, respectively. Here it must be re-emphasized that q k is a continuous functions of θ ,l and c,l , then from equation (5.91), it follows that qk = 0 when θ ,l = c,l = d mn = 0 , which dictates that q ko = a11 = a12 = 0 . However, to account for temperature and concentration dependence
of the heat flux vector imposed by the entropy inequality, coefficients b11 and b12 are considered to be functions of both the temperature θ and mass fraction c . In what follows, neglecting contribution of deformation gradient to heat flux included in equation (5.95), we can write qk = b11(θ ,c )T,k + b12 (θ ,c )C,k
Likewise, the linear constitutive equations for jk and D t kl are can be written as
(5.98)
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System jk = b21 (θ ,c )T,k + b22 (θ , c )C ,k
96
(5.99)
D t kl = b33 (θ , c )d kl
Since coefficients in (5.98) and (5.99) are still functions of θ and c , further linearization is required by assuming that deviation in θ and c from their corresponding reference values, namely To and co , are small. Expanding these coefficients into the Taylor series about the reference state, ∂b11 ∂θ ∂b o + 21 b21 = b21 ∂θ ∂b o b33 = b33 + 33 ∂θ o + b11 = b11
∂b11 ∂b ∂b o + 12 T ,c T + 12 T ,c C C , b12 = b12 To ,co o o ∂c ∂θ ∂c o o ∂b ∂b ∂b o + 22 T ,c T + 22 T ,c C T + 21 T ,c C , b22 = b22 To ,co o o o o ∂c ∂θ ∂c o o ∂b T + 33 T ,c C To ,co ∂c o o
To ,co
T+
(5.100)
then substituting into the corresponding equations (namely, (5.98) and (5.99)) and finally eliminating all high order terms in T and C will yield q k = − kTl T,k − k Cl C ,k j k = − ρ l DTl T,k − ρ l DCl C ,k D tkl = 2 µ v + µ vT T + µ vC C d kl
(
(5.101)
)
where − kTl = b11o − kCl = b12o o o − ρ L DTl = b21 − ρ L DCl = b22 ∂b ∂b o 2 µv = b33 µvT = 33 To ,co µvC = 33 ∂θ ∂c
(5.102) To ,co
If we expand the specific Helmholtz free energy into a Taylor series about the reference state, and keep only the terms up to second order, we can write
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
ψ =ψ o +
∂ ⎛ ∂ψ ⎞ 1 ∂ 2ψ 1 ∂ 2ψ ∂ψ ∂ψ 2 T C T C2 + + + + ⎜ ⎟ To ,co TC 2 To ,co 2 To ,co To ,co To ,co ∂ ∂ θ24 θ 24 θ c24 c 24 c ∂4 21 21 ∂4 ∂4 ∂4 ⎝ ⎠4 1 3 1 3 3 3 14 424 3 o o −η
µ
ψ = ψ o − η oT + µ o C −
−
∂η ∂θ
1 γ l 2 1 ∂µ T + 2 To 2 ∂c
To ,co
To ,co
C2 +
∂µ ∂θ
∂µ ∂c
To ,co
∂µ ∂θ
To ,co
To ,co
97
(5.103)
TC
Taking the derivative of ψ in (5.103)2 with respect to θ and c yields, respectively
η =η o +
γl To
T−
∂µ ∂θ
To ,c o
C, µ = µ o +
∂µ ∂c
To ,c o
C+
∂µ ∂θ
To ,c o
(5.104)
T
where ψ o , η o and µ o denote specific Helmholtz free energy, entropy and chemical potential difference at the reference state, respectively, and γ l is the specific heat for the ∂η ∂ 2ψ . The procedure of linearization of liquid phase, defined as γ = To = −To ∂θ To ,co ∂θ 2 To ,co a binary mixture in crystal growth can also be found in [112]. l
Now we come to the stage of combining linear constitutive relations with thermomechanical balance equations. Substituting equation (5.83), (5.101) and (5.104) into equation (5.94) and then neglecting the non linear terms in T and C and noting that
θ ,k = T,k and c,k = C ,k , we can obtain the equation for balance of energy as
(
)
( ⎞ ( ⎛ ∂θ + θ ,k v k ⎟ = kTl θ ,kk + k Cl c ,kk + 2 µ v + µ vT T + µ vC C d kl d kl ⎠ ⎝ ∂t
ρl γ l ⎜
(5.105)
where ( ( ( ( kTl = (kTl − ρ l DTl µ ), kCl = (kCl − ρ l DCl µ ),
(
⎛ ⎝
µ = ⎜ µo − To
∂µ ∂θ
To ,co
⎞ ⎟ ⎠
(5.106)
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
98
where ρ l DTl and ρ l DCl are constant. Similarly, casting equation (5.101)2 into (5.83)1 gives the equation of continuity for solute in terms of mass fraction for constant ρ l DTl and ρ l DCl . ∂c + c ,k v k = DTl θ ,kk + DCl c,kk ∂t
(5.107)
As for the equation (5.49)1 for the momentum balance, several assumptions are made. First, the well-known Boussinesq approximation is employed to include the effect of natural convection in the momentum equation due to density variation arising from the temperature and concentration gradients across the liquid phase. Secondly, the gravitational body force is the only body force term, namely, bl = − g . Boussinesq approximation assumes that the density is constant in all the field equations except the one appearing in the body force term in the momentum balance equation. To account for the density variation as a function of temperature and concentration, the density is expanded into a Taylor series about the reference state. For the Six Ge1− x liquid solution, the density of the mixture decreases as the temperature increases (positive thermal expansion), namely ρ l β T = −∂ρ / ∂θ . Silicon is the lighter component in the solution; hence with the addition of silicon to the Six Ge1− x solution, the density of the mixture decreases as well, (positive solutal expansion), ρ l β T = −∂ρ / ∂c .
ρ = ρl +
∂ρ ∂ρ T+ C To ,co To ,co θ ∂ ∂ c 123 123 − βT ρ l
− βC ρ l
ρ = ρ l (1 − βT T − β C C )
(5.108)
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
99
where βT and β C , respectively, represent the thermal and solutal expansion coefficients evaluated at the reference state. ρ l is the density of the mixture at the reference state. Substituting (5.92)5 and (5.108)2 into equation (5.49)1 under the above assumptions, we can obtain the linear form of the momentum balance as follows ) ⎛ ∂vl ⎞ + vl ,k vk ⎟ = − p,l + D tkl ,k + ρ l βT Tg + ρ l β C Cg ⎝ ∂t ⎠
ρl ⎜
(5.109)
) ) where p is the modified pressure and defined as p = p + ρ l hl g , where h l is the height
of the liquid phase. If there is a temperature gradient in a mixture, a temperature-induced diffusion takes place. This kind of diffusion is generally referred to as the thermal diffusion or Soret effect. Existence of a concentration gradient in the mixture, on the other hand, contributes to the heat flux. This phenomenon is known as the Dufour effect. The Soret effect is represent by the first term on the right hand side of equation (5.101)2, while the Dufour effect by the last term on the right hand side of equation (5.101)1. The Soret and Dufour effects may or may not be significant depending on the crystal growth process considered (see [113,114] for details). In dilute systems, the Dufour effect is generally neglected, which requires that kCl = 0 . At the moment, the Soret effect is neglected; hence DTl = 0 ( ( ( From equation (5.106), it flows that kTl = kTl and kCl = − ρ l DCl µ . If required, the inclusion of the Soret effect into the model is possible. Since the liquid phase has relatively low viscosity, concentration and temperature dependence of shear viscosity is neglected as well, which necessitates that µ vT = µ vC = 0 . Following these simplifications, the linear constitutive equations for the liquid phase can be re-written from equation (5.101) as follows
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
100
qk = −kTl T,k jk = − ρ l DCl C,k
(5.110)
t = 2 µv d kl
D kl
To make further simplifications in equation (5.105), we neglect the contribution of the ( term kCl c,kk to the energy equation because, particularly in dilute systems, the variation of the temperature field due to the concentration gradient is small. Finally, neglecting the contribution of the dissipative mechanical energy to the energy balance, we can write the energy balance equation as follows. ⎛ ∂θ ⎞ ( + θ ,k vk ⎟ = kTl θ ,kk ⎝ ∂t ⎠
ρlγ l ⎜
(5.111)
Neglecting the Soret effect in equation (5.107), the equation of mass balance for the solute (silicon) is further simplified to ∂c + c,k vk = DCl c,kk ∂t
(5.112)
Substituting equation (5.110)3 into (5.109) and using the definition of the symmetric deformation rate tensor, we can obtain the equation for the balance of momentum as ⎛ ∂vl ⎞ ) + vl ,k v k ⎟ = − p ,l + µ v vl ,kk + ρ l β T (θ − To )g + ρ l β C (c − co )g ⎝ ∂t ⎠
ρl ⎜
(5.113)
5.2.3.2 The Solid Phase In the present model, the substrate, source, and the growing crystal is referred to as the solid phase, which will be treated as a rigid ( d kl = 0 ) but heat and mass conducting
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
101
binary solid mixture. The crucible hosting the growth charge is modeled as a rigid, heat conducting solid. This model allows mass transport in the solid phase and also between the solid and liquid phases. To be consistent with the liquid phase, mass fraction of silicon is selected as the independent compositional variable. Thus, the balance of mass for solute (silicon) reads as
ρ
∂c + jk ,k = 0 ∂t
(5.114)
where, c = c Si = 1 − c Ge , jk = jkSi = − jkGe . It must be recognized that the material time derivative D( ) / Dt for a field in the solid phase is reduced to ∂ ( ) / ∂t owing to the assumption of the rigidity of the solid phase. In other words, velocity field vk vanishes, but it is preserved in the diffusion flux term to include the effect of the solid state diffusion. Besides, the mass density of the body remains unchanged due to the assumption of rigidity. Hence, the equations of mass and momentum balances will be excluded from the list of thermomechanical balance equations. Following the derivation (equation 5.84) for the liquid phase, the entropy inequality is obtained as follows
(
)
− ρ ψ& + ηθ& − (µ jk ),k −
θ ,k (qk − µjk ,k ) ≥ 0 θ
(5.115)
where µ = µ Si − µ Ge , which is the difference of chemical potentials of silicon and germanium, respectively. All dependent variables i.e. ψ , η , q k and jk , are assumed to be functions of the same independent variables, i.e. θ , θ ,k , c , and c,k . When the specific Helmholtz free energy is introduced into the entropy inequality in equation (5.115), the resulting entropy inequality becomes linear in θ& , θ&,k , c& and c&,k . Therefore, in order to
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
102
satisfy the inequality, the coefficient of these quantities must vanish; from which, we can write ∂ψ ∂ψ = 0, =0 ∂θ ,k ∂c ,k ∂ψ ∂ψ = −η , = µ , ψ = ψ (θ ,c ) ∂θ ∂c η = η (θ ,c ), µ = µ (θ ,c )
(5.116)
The resulting entropy inequality becomes
− µ ,k jk −
θ ,k (qk − µjk ) ≥ 0 θ
(5.117)
Similarly, using the definition of specific Helmholtz free energy, ψ = ψ (θ ,c ) , in equation (5.70)1 along with the equation (5.114) and neglecting the heat source, the equation for the balance of energy can be written as follows
ρθ
∂η = − q k , k + µ j k ,k ∂t
(5.118)
Upon following the similar procedure introduced in the derivation of the linear constitutive equations for liquid phase, the linear constitutive equations for the solid phase read as qk = − kTsT,k − kCs C,k jk = − ρ s DTsT,k − ρ s DCs C,k
(5.119)
where θ = To + T , To >> T , To > 0 and c = co + C , co >> C , co > 0 . To and co are the reference temperature and mass fraction for the solid phase, respectively.
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
103
Expanding the specific Helmholtz free energy in equation (5.116)5 into Taylor series around the reference state, maintaining only up to second order terms yields
ψ =ψ o +
∂ψ ∂ψ 1 ∂ 2ψ 1 ∂ 2ψ ∂ ⎛ ∂ψ ⎞ 2 + + + T C T C2 + ⎜ ⎟ To ,co TC 2 To ,co 2 To ,co To ,co To ,co ∂4 ∂4 ∂4 ∂4 21 21 c24 c 24 c ∂ ∂ θ24 θ 24 θ ⎝ ⎠4 1 3 1 3 3 3 14 424 3 −η o
µo
ψ = ψ o − η oT + µ oC −
∂η ∂θ
−
1 γ s 2 1 ∂µ T + 2 To 2 ∂c
∂µ ∂c
To ,co
To ,co
C2 +
∂µ ∂θ
To ,co
∂µ ∂θ
To ,co
To ,co
(5.120)
TC
Taking the derivative of ψ in (5.120)2 with respect to θ and c yields, respectively
η =η o +
γs To
T−
∂µ ∂θ
To ,c o
C,
µ =µo +
∂µ ∂c
To ,c o
C+
∂µ ∂θ
To ,c o
(5.121)
T
where ψ o , η o and µ o denote the specific Helmholtz free energy, entropy and chemical potential difference at the reference state for the solid phase, respectively, and γ s is the specific heat for the solid phase, defined as γ s = To
∂η ∂θ
To ,co
= −To
∂ 2ψ ∂θ 2
To ,co
.
Substituting equation (5.114), (5.119) and (5.121) into equation (5.118) and then neglecting the nonlinear terms in T and C and noting that θ ,k = T,k , θ& = T& and c,k = C ,k , we can obtain the equation for balance of energy as
ρ sγ s
( ∂θ ( s = kT θ ,kk + kCs c,kk ∂t
where
(5.122)
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System ( ∂µ ( ( ( ( ⎛ kTs = (kTs − ρ s DTs µ ), kCs = (kCs − ρ s DCs µ ), µ = ⎜ µo − To ∂θ ⎝
To ,co
⎞ ⎟ ⎠
104
(5.123)
where ρ s DTs and ρ s DCs are constant. Likewise, casting equation (5.119)2 into (5.114), gives the mass conservation equation for solute in terms of mass fraction for constant ρ s DTs and ρ s DCs ∂c = DTsθ ,kk + DCs c,kk ∂t
(5.124)
Neglecting the Soret ( DTs = 0 ) and the Dufour ( kCs = 0 ) effect in the solid phase, we can write qk = − kTsT,k jk = − ρ s DCs C,k ( ( ( kTs = kTs , kCs = − ρ s DCs µ
(5.125)
( In what follows, neglecting the contribution of the term kCs to the energy balance, we can write the equation for the energy balance for the solid phase as
ρ sγ s
∂θ = kTsθ ,kk ∂t
(5.126)
Similarly, omitting the Soret effect ( DTs = 0 ) in equation (5.124), we can write ∂c = DCs c,kk ∂t
(5.127)
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
105
5.2.3.3 Interface Conditions We now write the interface conditions for the model considered. The interface between the substrate (germanium) and the liquid solution is called the growth interface, whereas the one between the liquid zone and source material (silicon) is referred to as the dissolution interface. We begin with the interface condition corresponding to the mass balance for the overall mixture. This condition is obtained from equation (5.30)2 as
ρ l (vk − uk )nk = − ρ s uk nk
(5.128)
Likewise, using equation (5.128) in (5.49)2, the interface condition for the balance of linear momentum can be written as follows − ρ s vl u k n k + 2γ σ Ωnl = t kl n k
(5.129)
Combining (5.128) and (5.129) with (5.70)2, the interface condition for the energy balance can read as
(q
l k
)
⎛ 2γ σ Ω − q ks n k = ρ s ⎜ ∆H F + ⎜ ρl ⎝
⎞ ⎟u k n k ⎟ ⎠
(5.130)
where ∆H F is as the enthalpy of fusion (latent heat) released during the solidification and is defined by
∆H F = ε l − ε s −
vl v l 2
(5.131)
Similarly, combining equation (5.83)2 with (5.128), the interface condition related to the mass balance for silicon can be written as
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
(ρ u (c s
s
k
)
)
− c l + jkl nk = jks nk
106 (5.132)
Similarly, substituting equation (5.110)1 and (5.125) 1 into equation (5.130) gives
(k θ
s s T ,k
)
⎛ 2γ σ Ω − kTl θ ,lk n k = ρ s ⎜ ∆H F + ⎜ ρl ⎝
⎞ ⎟u k n k ⎟ ⎠
(5.133)
Likewise, making use of equation (5.110)2 , (5.125) 2 and (5.132), we can write
(ρ u (c s
k
s
)
)
− c l + ρ s DCs c,sk nk = ρ l DCl c,lk nk
(5.133)
For the derivation of the open form of the jump conditions, refer to Appendix B, equation (B.117). Corresponding boundary conditions for the growth cell can be obtained from above equations by setting uk = 0 .
5.3 Governing Equations and Boundary Conditions for the Computational Model In this section, we present the field equations and the associated interface and boundary conditions used in the simulations for the growth of germanium rich Si x Ge1− x single crystals by LPD. Three–dimensional numerical simulations were performed using a finite volume-based commercial package (CFX-4.4). Simulations were conducted for only an initial period of growth due to due to the usage of a fine mesh in the computational domain, which requires significant computational time. In order to able to simulate the entire growth process, two-dimensional simulations are also conducted. Through out the derivation of model equations, the assumptions adopted have been discussed; hence, they are not included here except those not discussed previously. The flow in the solution or liquid is assumed to be of laminar regime.
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
107
5.3.1 The Liquid Phase Figure 5.4 illustrates schematically the vertical cross section of the LPD growth system along with a representative thermal profile on the right hand side.
Figure 5.4: The schematics of the vertical cross section of the LPD growth system with the representative thermal profile where t is the unit tangential vector to the growth interface, while n is the unit normal to the growth interface pointing into the liquid domain. Three-dimensional time-dependent governing equations describing the fluid flow, heat and mass transport for the binary liquid phase are written, respectively, from equation (5.89,5.113, 5.111, and 5.112 ) in cylindrical coordinates ( r ,ϕ , z ). However, the field equations of the two-dimensional simulation model are not given separately for the sake of space. In 2-D, the growth cell is assumed axisymmetric with respect to the centerline as shown in figure 5.4, and the field variables will be independent of the azimuthal coordinate ( ϕ ). Thus, the two dimensional field equations and associated boundary
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
108
conditions can be obtained from the three-dimensional model equations by simply dropping the dependency on the azimuthal coordinate ( ϕ ) in the field variables. When it is necessary, additional boundary conditions for the two- dimensional model are introduced.
In this section, diverting from some of the notations used earlier, the
governing equations will be written with a notation commonly seen in the literature. However, new notations will be defined clearly to avoid any confusion. Mass Balance:
∂v 1 ∂ (rvr ) + 1 ϕ + ∂vz = 0 ∂z r ∂r r ∂ϕ
(5.135)
Momentum Balance:
) 2 ⎛ ∂vr ∂v v ∂v ∂v v v 1 ∂p 2 ∂v ⎞ + vr r + ϕ r + vz r − ϕ = − l +ν ⎜⎜ ∇ 2vr − r2 − 2 ϕ ⎟⎟ ∂t ∂r ∂z r ∂ϕ r r r ∂ϕ ⎠ ρ ∂r ⎝
∂vϕ ∂t
+ vr
∂vϕ ∂r
+
vϕ ∂vϕ r ∂ϕ
+ vz
∂vϕ ∂z
+
vr vϕ r
=−
) vϕ ⎛ 2 1 ∂p 2 ∂vr ⎞ ⎜ ⎟⎟ v ν + ∇ − + ϕ 2 2 ⎜ ∂ ϕ rρ l ∂r r r ⎝ ⎠
vϕ ∂v z ∂v z ∂v ∂v + vr z + + vz z = ∂t ∂r ∂z ) r ∂ϕ 1 ∂p + ν∇ 2 v z + β T ( T − To )g + β C ( c − co )g − l ρ ∂z
(5.136)
(5.137)
(5.138)
where v z , vϕ , and vr and are, respectively, velocity components in the radial ( r ), circumferential direction ( ϕ ) and vertical ( z ) directions. ρ l is the density of the background fluid (germanium), ν is the kinematic viscosity of the solution, T is the absolute temperature at a point in the liquid, c is the mass fraction of silicon and To and ( co denote reference temperature and mass fraction, respectively. p is the modified
pressure. βT and β C are the thermal and solutal expansion coefficient, respectively.
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
109
Energy Balance:
∂T ∂T vϕ ∂T ∂T kl + vr + + vz = l l ∇ 2T ∂t ∂r ∂z ρ γ r ∂ϕ where k l and γ l
(5.139)
represent thermal conductivity and specific heat capacity of the
solution, respectively.
Mass Transport:
∂c ∂c vϕ ∂c ∂c + vr + + vz = DSil ∇ 2 c ∂t ∂r r ∂ϕ ∂z
(5.140)
where DSil denotes the molecular diffusion coefficient of silicon in the solution. The Laplacian operator is defined in cylindrical coordinates as
∇2 =
1 ∂ ⎛ ∂ ⎞ 1 ∂2 ∂2 + ⎜r ⎟ + 2 r ∂r ⎝ ∂r ⎠ r ∂ϕ 2 ∂z 2
(5.141)
Boundary and Interface Conditions:
Boundary and interface conditions corresponding to governing equations (5.135 -5.140) are introduced from equations (5.128, 5.133, 5.134) as follows. At the vertical wall: The no-slip boundary condition is employed at the vertical walls of the growth cell; namely, at the interface between the surrounding crucible and the liquid, the fluid velocity equals the velocity with which the solid surface is moving. As for the boundary
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
110
condition for the mass balance equation for solute (silicon), the vertical walls are assumed to be impermeable to the species in the solution. Thus,
vr = 0 , v z = 0 , vϕ = 0 ,
∂c =0 ∂r
(5.142)
At the axis of symmetry for only two-dimensional model: At the axis of symmetry, for physical (finite) results, it is required that the radial velocity component be zero. Hence, ∂vz ∂T ∂c = 0, = 0, = 0 , vr = 0 ∂r ∂r ∂r
(5.143)
At the growth interface: The local thermodynamic equilibrium is assumed at both growth and dissolution interfaces; this implies that the concentration at the interface is determined from the SiGe binary phase diagram namely, ceq = f (Teq ). Equation 5.144 gives the liquid and solid equilibrium compositions for the germanium rich side of the phase diagram in terms of the silicon mass fraction, recalling that temperature is given in Kelvin.
(
)
l ceq = a l T 3 + b l T 2 + c l T + d l , and ceqs = T − a s / b s
The numerical values of the phase diagram coefficients are given in table 5.1.
(5.144)
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
111
Table 5.1. SiGe phase diagram coefficients al
bl
cl
dl
as
bs
2.5984*10-9
-8.7189*10-6
0.0099041
-3.822
1211.45
639.63
Since the growth rate in LPD is approximately five order of magnitude smaller than the velocities of the fluid particle, radial and axial velocity components can be reasonably set to zero at the growth interface. In addition, the diffusion of solute in the solid state is far smaller than the one in the liquid phase, so, it is prudent to neglect the solid state diffusion at the interface. Regarding the energy balance at the interface, due to the small growth rate, the latent heat dissipates without disturbing the continuity of the heat flux at the interface. Besides, since the curvature of the interface is fairly small ( Ω =80 1/m), the effect of surface tension in the energy balance is also excluded. The surface tension effect in the interface energy balance, however, becomes important in the presence of an interface with large curvature such as in the case of a dendritic growth or lateral overgrowth ( Ω = 2.5 * 105 1/m) [115-116]. By solving the mass balance equation at the interface, the evolution of the growth interface shape is computed. c = c gil = ceql at Tgi , v r = 0, vϕ , v z = 0 ∂c ∂T s ∂T l ρ s u g (c gis − c gil ) = ρ l DSil , ks − kl =0 ∂n ∂n ∂n
(5.145)
l where c gi is the equilibrium mass fraction of the solute (silicon) at the growth interface,
c gis is the equilibrium mass fraction of silicon in the solidified crystal, u g is the rate of displacement of the substrate-solution interface along the unit normal direction, also known as growth velocity. ∂c / ∂n is the normal derivative of the solute mass fraction at the interface, c gis is the equilibrium mass fraction of silicon in the solid phase at the
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
112
growth interface and k s represent thermal conductivity of the solid phase. T l and T s are the temperature of liquid and solid at the growth interface. At dissolution interface: Since the experimentally observed dissolution rate of the silicon source material is so small (approximately 0.3 mm/day), the mass balance equation at the dissolution interface is excluded. c = c dil = ceql at Tdi , v r = 0, vϕ = 0, v z = 0 s l l ∂T s ∂T k −k =0 ∂n ∂n
(5.146)
where cdil is the equilibrium mass fractions of the solute (silicon) at the dissolution interface. The initial conditions are c = c o , v r = 0 , vϕ = 0 and vz = 0 at t = 0.
5.3.2 The Solid Phase Energy Balance:
The temperature distribution in the substrate, source, solidified crystal and the ampoule walls is governed by the following energy equation: ∂T ks = s s ∇ 2T ∂t ρ γ
(5.147)
where T is the temperature at a point in the solid phase, and k s , ρ s and γ s are the thermal conductivity, density and specific heat of the solid phases.
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
113
Boundary Conditions:
The thermal boundary conditions for the quartz crucible (vertical wall, top and bottom surface) and the symmetry axis of the crucible (only for 2-D model) are given as
− ks
[
]
∂T ) ∂T = h T − T f (z ) , =0 ∂n ∂n
(5.148)
where T f (z ) is the experimentally measured ambient temperature inside the furnace ) along the quartz ampoule wall or on top and bottom surfaces (see figure 7.7) , h is the modified heat transfer coefficient, including the contribution of convective and radiation ) effect on heat loss, namely, h = h + εσ (T + T f (z ))(T 2 + T f2 (z )) . Here, ε is the thermal emissivity of the quartz ampoule, and σ is the Stefan Boltzmann constant. n designates that the gradient operation is carried out in the normal direction to the crucible walls. The modified heat transfer coefficient is approximated using preliminary experimental results (such as the measured thermal profile, solute distribution in a grown crystal, and the position of the initial growth interface). In addition, perfect thermal contact at the meltampoule, crystal -ampoule and inner-outer crucible is assumed. Therefore at these boundaries, the heat flux is continuous.
5.4 Physical Parameters of the LPD SixGe1-x System Here, we have listed physical properties of Si x Ge1− x and some input parameters used in numerical computations. Since the liquid mixture is dilute in silicon solute (maximum silicon molar fraction in liquid, x Sil = 0.04, determined by the temperature across the dissolution interface), physical properties of germanium liquid are adopted when the Si x Ge1− x physical properties are lack. The composition dependence of the specific heat
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
114
capacity of solid and liquid Si x Ge1− x is approximated by a linear change between the values of end constituents at a given temperature as
γ Si Ge x
1− x
= γ Si x + γ Ge (1 − x )
(5.149)
The composition dependence of density of growing crystal is calculated by using the same argument given in the above equation. The reported diffusion coefficient for silicon in a germanium melt is highly diverse varying between 1*10-8 to5*10-8 m2/s. We adopted the value 2.5*10-8 m2/s. Table 5.2: Design Parameters for LPD growth system Total length of growth zone
40 mm
Initial height of the sources
3 mm
Initial height of the substrate
10 mm
Crystal diameter
25 mm
Total wall thickness of quartz (quartz growth reactor plus crucible) 4 mm
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
115
Table 5.3: Physical properties of growth charges and the quartz crucible. The physical parameters were compiled from references [117-123].
Physical Parameter
Source
Substrate
Si
Ge
Growing Crystal
Liquid Solution
Si x Ge1− x
Si x Ge1− x
396.1-487
380-406
γ ( J / kg .K )
Growth Charges 967 at 1300 K [117] 396.1 at 1210 K [118] 1037 (liquid) [118] 380 (liquid) [118]
Thermal Conductivity
23.7 at 1273 K [119]
10.60 at 1210 K [120]
10.60 [120]
42.8 [119]
Mass Diffusivity
Not applicable
Not applicable
1*10-20
2.5*10-8
ρ ( kg / m 3 )
2301.6 at 1300 K [117]
5323 at 1210 K [117]
4839-5323
5633 [121]
Thermal Expansion coefficient
Not applicable
Not applicable
Not applicable
1.2*10-4 [122]
Solutal Expansion coefficient
Not applicable
Not applicable
Not applicable
0.0053 [122]
Enthalpy of fusion,
1807.9
466.5
-----------
------------
Not applicable
Not applicable
Not applicable
7.35*10-4 [122]
Specific heat
k ( W / m .K )
2
D (m / s) Mass Density
βT ( 1 / K )
β c ( 1 / atm.% Si ∆H F ( kJ / kg ) Viscosity
µ v = kg / m.s
Quartz Crucible [123] Specific heat
γ ( J / kg .K ) Thermal Conductivity
k ( W / m .K ) Mass Density
ρ ( kg / m 3 )
1200 at 1300 K 2 at 1300 K 2200 at 1300 K
Chapter 5 – A Rational Continuum Mixture Model for the LPD Growth System
116
5.9 Summary In this chapter, we have presented a detailed continuum mixture model for the growth of Si x Ge1− x single crystals by LPD from the germanium rich side of the Si x Ge1− x binary phase diagrams. The model equations were obtained by using thermomechanical balance laws and irreversible thermodynamics. The current model considers various physical phenomena that may occur in LPD growth process of Si x Ge1− x binary system. The importance of these phenomena was discussed within the context of LPD growth of Si x Ge1− x single crystals. There were two main reasons for keeping this chapter extended. First, it has a notable contribution to the thesis work by including the effect of curvature on the jump energy balance, derived by using the continuum mechanics. Second, this chapter will be very valuable in terms of offering a well-documented background for the researchers who are new to the continuum mixture theory and theoretical aspect of crystal growth processes.
117
CHAPTER 6 Numerical Simulation for the Growth of SixGe1-x by LPD 6.1 Introduction In this chapter, based on the rational continuum mixture model developed and presented in the previous chapter, two-and three-dimensional transient numerical simulations were performed for the LPD growth process of SiGe single crystal. The numerical simulation models presented here account for many important physical features of the LPD growth process of Si x Ge1− x , namely growth zone design on thermal field, the structure of the buoyancy induced convective flow and its effect on the growth and transport mechanism, the shape and evolution of initial and dynamic growth interface, and the variation of growth velocity with time. It was numerically shown that as the name LPD implies, the growth of Si x Ge1− x by LPD is mainly a diffusion driven process except the initial stages of the growth process during which the natural convection is prominent in the solution zone, and has significant effects on the composition of the grown crystal. The present model successfully simulates the evolution of the growth interface that agrees with experimental observations. In addition, experimental and numerical growth velocities are in good agreement.
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
118
6.2. Order of Magnitude Analysis for the LPD Growth System Order of magnitude (scaling) analysis is an important and swift way to shed some light on how physical and transport parameters (i.e., density, viscosity, solutal and thermal expansion coefficients, etc) contributes to the structure of field variables developed during the crystal growth process without performing lengthy simulations. It allows us to evaluate qualitatively the relative contributions of these parameters to the transport phenomena occurring during the growth. To this end, the transport equations are expressed in their non-dimensional forms. These non-dimensional equations give rise to a number of dimensionless numbers such as the thermal and solutal Grashof numbers, the Prandtl number, and the Schmidt number. The thermal and solutal Grashof numbers appear as multiplication factors in front of the dimensionless temperature and concentration, respectively, in the non-dimensionless linear momentum balance equation in the axial direction, while the Prandtl and Schmidt numbers in the non-dimensionless energy and species mass balance equations. Two important material properties which relate the density of mixture to temperature and solute concentration are the thermal and solutal expansion coefficients. The values of these two coefficients (namely, being positive or negative) are dependent on whether the density of the solvent increase or decrease with temperature and solute concentration. Densities of the liquid silicon and germanium decrease with increasing temperature [124,125]. It is reasonable to expect that a liquid SiGe mixture shows the same behaviour for the density variation as a function of temperature, which means that volume of the mixture expands on heating (positive thermal expansion). Solubility of silicon (solute) in germanium (solvent) increases with temperature and since the density of silicon less than that of germanium, the mixture density decreases with the increase in the solute concentration (positive solutal expansion). The solutal and thermal expansion coefficients evaluated at the reference state for the germanium-rich SiGe solution are given as
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
βT = −
1 ∂ρ ρ l ∂T
1 ∂ρ βc = − l ρ ∂c
119
To
(6.1) co
Dimensionless thermal and solutal Grashof numbers tell us the relative contributions of solutal and thermal gradients to the natural convection occurring during the LPD growth process. The Grashof number represents the relative importance of the buoyancy force over viscous force. Definitions of the dimensionless numbers and their characteristic values (computed using the LPD experimental set-up and SiGe transport parameters evaluated around the melting temperature of germanium) are given in Table 6.1. For an accurate scaling analysis, dimensionless numbers must be defined with respect to accurate characteristic parameters of the system. For instance, the thermal and solutal Grashof are usually defined with respect to the radial length and the thermal and solutal gradients in this direction. However, in the LPD system, this would not be reliable since radial temperature and solutal gradients vary even over a small length scale. Due to this difficulty, in this work the solutal and thermal Grashof numbers are calculated in terms of the axial concentration and temperature gradients despite the fact that they do not scale process very accurately. The computed values show that in the present system the solutal Grashof number is approximately two times greater than the thermal Grashof number, which implies that the contribution of the solutal gradient to natural convection, positively or negatively, is larger than that of thermal gradient. Other important dimensionless parameters are the Schmidt and Prandtl numbers which are defined as the ratios of momentum diffusivity to solutal diffusivity and thermal diffusivity, respectively. Small value of the Prandtl number suggests that the heat transport across the solution is mainly due to conduction. This was verified by numerical simulations which led to the similar temperature distributions computed with and with no
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
120
convection. The relatively large number of Schmidt number indicates that convection has a significant effect on the transportation of solute during the growth of SiGe crystal. Table 6.1: Non-dimensionless numbers and their characteristic values calculated for the growth of germanium-rich SiGe single crystals by LPD
Dimensionless numbers
Formulation
Thermal Grashof number
GrT =
Solutal Grashof number
Grc =
(β
glo3 ) ∆T v2
T
(β gl ) ∆c 3 o
c
v
Prandtl Number
Pr =
Schmidt Number
Sc =
2
v
α v D
Characteristic Values for LPD growth system 5.09*107 10.7*107 0.0075 7.6743
In the table, lo is the characteristic length taken as the diameter of the solution zone (25 mm), g is the magnitude of gravitational force per unit mass, v is the kinematic viscosity, and ∆c = c1 − co and ∆T = T1 − To are characteristic concentration and temperature. c1 , co and T1 , T0 are boundary values of concentration (in terms of silicon mass fraction) and temperature in Kelvin at the top and bottom of the liquid.
6.3 Discretisation Method and Solution Algorithms Two-and three-dimensional time dependent partial differential equations and related boundary and interface conditions, which describe the fluid flow, heat and mass transport, are non linear, and coupled to each other; hence, they cannot be solved analytically. Numerical techniques are the only means to solve such complex equations.
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
121
In order to solve these transport equations numerically, they have to be discretized. There are several techniques for the discretization process, such as finite volume, finite difference, and finite elements. In the solution of the governing equations together with the associated boundary and interface conditions, the finite volume method was selected because of its versatility in solving problems with moving boundaries. In the following, some aspects of the discretisation methods and solution algorithms used for the implementation of the present model are described in sufficient detail. CFX-4.4, a commercial software package developed by AE technology, is used for computations. CFX-4.4 is a finite volume based code, which is capable of solving a variety of discrete transport equations for fluid and solid phases. In the finite volume approach, the region of interest where the computation is to be performed is divided into a number of finite volumes, also called control volumes. Each partial differential transport equation is integrated over each control volume to obtain their discrete or linear forms. The discrete transport equations are defined at the center of finite volume cells. CFX-4.4 offers a number of different methods for the solution of the linearised transport equations. The methods and the parameters which control them can be set in command file (prepared by a user) using keywords and subcommands. CFX-4.4 solves linearised transport equations through iterative techniques. Iteration is used at two levels; namely, an inner iteration to solve for the spatial coupling for each variable and an outer iteration to solve for the coupling between variables. Spatial coupling of each variable is computed sequentially, keeping all other variables as fixed, which means that a discrete transport equation for a particular variable is formed for every cell centre in the computational domain and the problem is handed over to a linear equation solver. An exact solution is not required for the inner iteration because this is just one step in the non-linear outer (global) iteration. The equation solver returns the updated values of that variable for the computation of spatial coupling of the next variable. The non-linearity of the original
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
122
equations is simulated by reforming the coefficients of the discrete equations, using the most recently calculated values of the variables prior to each global iteration. The treatment of the linear momentum equation is different, to some extent, from the above description since it does not obey a transport equation due to the priori unknown pressure term. The velocity fields to be computed have to satisfy the continuity equation, which might only be achieved if the pressure field was known. Since there is no transport equation for the pressure, a method is needed to calculate the pressure field. Of several techniques (algorithms) to deal with the coupling between pressure and velocity is the SIMPLE (Semi-Implicit Method for Pressure-Linked Equations), which is used in this work. The SIMPLE algorithm is a guess-and-correct procedure for the calculation of pressure. The implementation of the SIMPLE velocity-pressure coupling algorithm is briefly such that simplified versions of the discrete momentum equations are used to derive a functional relationship between a correction to the pressure and corrections to the velocity components in each cell. Substitution of this expression into the continuity equation leads to an equation linking the pressure-correction with the continuity error in the cell. This set of simultaneous equations is passed to a linear equation solver. The solution is used both to update pressure and to correct the velocity field through the functional relationship in order to enforce mass conservation. The finite difference scheme (for advection terms), the linear equation solver, and the under relaxation must be selected properly in order to achieve the desired convergence of the solution procedure. In buoyancy-driven flows, an accurate solution of the enthalpy equation is important. Hence, setting the reduction factor to be low, under-relaxing this equation minimally and using massive under-relaxation of velocities were found to be helpful for obtaining convergence in this simulation. Table 6.2 summarizes solver parameters used in the current model. For more information about solver parameters, the interested reader is referred to CFX user manual [126].
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
123
Table 6.2: Solver parameters
Velocity -U
Finite Difference Scheme HYBRID
Full field Stone
0.3
MNSL/ MXSL/ RDFC 1/5/0.25
Velocity -V
HYBRID
Full field Stone
0.3
1/5/0.25
Velocity -W
HYBRID
Full field Stone
0.3
1/5/0.25
Pressure - P
HYBRID
Algebraic Multi-grid
0.4
1/30/0.1
Enthalpy -H
HYBRID
Algebraic Multi-grid
1
1/5/0.25
Scalar Eq.
HYBRID
Full field Stone
0.2
1/5/0.25
Transport Equation
Linear Equation Solver
Under Relaxation
For the discretisation of time derivation, a fully implicit backward time difference stepping procedure has been implemented. The time iterations are initiated to calculate time dependence of concentration, flow velocity in liquid and temperature in both liquid and solid. In so doing, it becomes possible to investigate the changes in these field variables, the substrate thickness due to the movement of growth interface. Computational process starts with defining initial guesses and boundary conditions. Then the time dependent flow field is computed. The computed flow field is used to calculate thermal and concentration profiles in the liquid zone since the energy, and species mass balance, and momentum equations are coupled to each other through the Boussinesq approximation. Since the liquid and solid phases have large differences in their thermal conductivity, the thermal field has to be updated at each time step to include the influence of increasing substrate thickness. Local thermodynamic equilibrium is assumed at the growth and dissolution interfaces; this implies that the concentration at the interface is determined from the SiGe binary phase diagram namely, ceq = f (Teq ). Hence, using the computed temperate field at the interface, interfacial concentrations are computed, thereby the concentration field in the liquid domain at each time step as the thickness of the grown crystal increases. Figure 6.1 presents the order of computation schematically.
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
124
There are different possible ways available to stop the solution process for the present time step of the transient problem such as setting a tolerance on mass residuals, maximum number of outer iterations, and CPU time limit. Mass residual is defined as the sum of the absolute values of the net mass fluxes into or out of every cell in the flow, and thus has the dimensions of mass per unit time. A tolerance on mass residuals is set by user and the convergence criteria are satisfied if it has fallen below the tolerance set. Of course, the selection of convergence criteria depends on the type of simulation carried out. In the present work, as a convergence criterion, predefined maximum number of outer iterations is used. The number of outer iterations needed may vary from a few hundreds to several thousands depending on the accuracy desired. The number of outer iteration was selected after several trial runs by judging if the problem is converged satisfactorily.
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
125
START
New time step t=t+∆t Initial guess and boundary conditions for v r ,vϕ ,v z p ,T , c
Calculate the growth velocity, advect the interface and update the grid
Outer iteration
Correction to pressure and velocity fields
Solve energy balance to obtain temperature field (T )
NO Convergence criterion satisfied?
Solve mass transport equation to obtain mass fraction of silicon ( c )
YES YES
t < tmax
NO
STOP
Figure 6.1: Flow diagram of solution procedure
Inner iteration
Solve pressure correction equation ( p )
Inner iteration
Set updated fields vr ,vϕ ,v z p ,T , c
Inner iteration
Solve discrete linear momentum equations to obtain velocity fields
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
126
6.4 Finite Volume Mesh In the foregoing, typical finite element meshes for two-and three-dimensional simulation is introduced. For two- and three-dimensional simulations, body fitted grid is used. Bodyfitted coordinates are utilized to allow the treatment of complex two-and threedimensional geometries such as evolving curved growth interface. The basic idea behind body-fitted coordinates is to use a curvilinear coordinate transformation to map the complex flow domain in physical space to a simple (i.e. rectangular) flow domain in computational space. In other words, the Cartesian coordinate system ( x , y , z ) in the physical domain is replaced by a curvilinear coordinate system (ξ ,η ,ζ ) that fits all complex geometries. The equations are then discretized with respect to the computational space coordinates. The selection of finite volume mesh is of direct effect on the accuracy of computed results. The accuracy of the numerical results is checked by refining the finite volume mesh until mesh-independent solutions are obtained in both the two- and threedimensional simulations. The mesh size giving mesh-independent results is used during the simulation. In the present work, we considered an equidistant finite volume meshes The mesh size used in two dimensional simulations, given the order of substrate, liquid domain, source and quartz ampoule, is 1500, 4000, 500, and 1200. As for the threedimensional mesh, following the order of substrate, liquid zone, source and ampoule, the mesh size is 18000, 54000, 7200, and 42240 as shown in figure 6.2.
127
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
C L Source
Source
Melt
Melt
Substrate
Substrate
z
r
(a)
Quartz ampoule
(b)
Figure 6.2: Sample finite volume mesh used in the current simulation, a-) a typical mesh for 3-D simulation, b-) mesh for 2-D simulation.
6.5 Grid Modification and Interface Movement Simulation of the growth process of SiGe by LPD is a typical moving boundary problem, also known as a Stefan problem where a particular portion of the computational domain changes as a function of time. Therefore, the shape and the location of the boundary (interface) is a priori unknown function of time, which has to be predicted with the solution of the transport equations. In these cases, two sets of boundary conditions are
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
128
required; the first one is to determine the position of the moving boundary and the second one is to complete the solution of the transport equations. For the grid movement, the governing transport equations are solved in the current mesh, and subsequently, the movement of interface (growth velocity) is calculated through solving the phase equilibrium and mass balance equations for the interface. Upon employing the computed growth velocity, the new location and shape of the interface is calculated. Subsequently, the old mesh is modified by advecting the nodes to embrace the latest shape of the moving boundary so that the meshes adapt to the deformed substrate, and solution. The mesh is updated in a way that always results in the same number of grid nodes equally distributed in the computational domain. The new interface conditions are calculated to compute thermal, concentration and flow fields in the liquid and solid domains. The above-described procedure continues for the sequential growth steps until predetermined growth time is reached. In CFX-4.4, complicated grid motions need to be set in User Fortran routine by the user as was done in this study. For a flow calculation with a moving grid, the transport equations have to be modified. There are two main changes; the transient term is different to allow for the possibly of changing volume and the advection terms include the grid velocity. These modifications are included in the present simulation by defining the problem in command file as a transient grid. To calculate the initial interface shape, all field equations are solved for several time steps until a steady state solution for temperature within the entire computational domain is achieved. Subsequently, the location of an isotherm line with the temperature equal to the melting temperature of the substrate material (germanium in this simulation) is computed. The initial flat interface between the liquid and solid domain is deformed to take the shape of the isotherm since the solid domain (germanium) cannot be present physically above this reference temperature. Computational procedure for the deformation of the substrate is as follows. The code developed for 2-and 3-D simulation
129
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
scans all the finite volume cells within liquid and solid domain to find out the cells that stores the reference temperature or above. Then among these cells, the axial coordinates of first cell centers are retrieved and stored in a two dimensional array. As can be seen from figure 6.3, it might be probable for the reference thermal isotherm to pass slightly above the centre points of the cells. In this case, the code takes the next cell as a cell including the centre point with higher temperature than the reference temperature. This case results in non- smooth interface shape. This problem is tackled by curve fitting to axial coordinates of cell centers which was stored in a two dimensional array. For curve fitting, discrete least squares polynomial of degree three has been used.
Feed Quartz ampoule
I K
Liquid binary mixture
J
Quartz
Substrate Figure 6.3: Representative vertical cross-section and top view of the 3-D computational domain.
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
130
So far, the deformed thickness of the substrate has been calculated at cell centers; however, coordinates of cell vertices are needed to move the grid. Thus, from available values of axial coordinates of cell centers, axial coordinates of cell vertices at the new interface have to be approximated. To simplify the coding, an imaginary grid is enclosed around the grid plane perpendicular to the axial direction as illustrated in figure 6.4. The vertices of the grid with dashed lines represent the center points of the finite volumes, whereas the vertices of the grid with solid lines illustrate the vertices of the finite volumes. Arithmetic averages of the neighbouring center coordinate values are assigned to the corresponding cell vertices. The center points not storing any coordinates values do not contribute to the arithmetic averaging. The axial coordinates of the cell vertices obtained as a result of arithmetic averaging are used as an initial deformed thickness of the substrate to move the grid. To prevent the formation of grid discontinuities between shared nodal points at the substrate-quartz and liquid-quartz interfaces, grid points on the quartz domain have to be moved in accordance with the new locations of shared nodal points.
i, j
i, j - 1
I, J i - 1, j
i - 1, j - 1
Figure 6.4: Grid plane perpendicular to the axial direction
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
131
The grid motion, namely the growth velocity, is computed at cell centers by solving interface mass balance equation. Then, the velocity of grid vertices is approximated by employing the above argument. Since the experimentally observed dissolution rate of the silicon source is so small (being approximately 0.031mm /hour), source material is not deformed.
6.6 Results of Two-Dimensional Simulations 6.6.1 Temperature Field Figure 6.5 and 6.6 shows typical temperature distributions in the substrate, solution, source, and quartz crucible for different hours of the growth process. The initial temperature difference in the liquid domain is about 45 Kelvin. As can be seen from the figure 6.5a, the initial growth interface deformed according to a reference isothermal line is of a concave shape. The formation concave interface could be attributed to several reasons. The large mismatch in thermal conductivities of the substrate, solution and surrounding quartz crucible bends isothermal lines in the vicinity of interface. Recalling from the simulated experimental configuration, there is an annular insulator material located inside the furnace, which encompasses the quartz crucible with 3-5 mm clearance. The insulator material spans from slightly above the substrate to the bottom of the source material. The convective heat transfer in this semi-insulated region is handled by a proper estimation for the heat transfer coefficient, including both radiative and convective heat transfers. The presence of the annular insulator material distorts the isothermal lines around the boundary between the regions with and without insulator. Furthermore, the convective heat transfer from the bottom of the crucible to surrounding furnace atmosphere is smaller than that from the periphery of the ampoule due to the
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
132
presence of hollow quartz pedestal underneath the crucible. Away from the substrate and source material, thermal lines are almost flat due to being inside the insulator covered region. Within the source, thermal lines are convex since the top of the source material is in contact with the vacuum environment; therefore, heat transfer from the surrounding to the source material through its surface is less than that from its periphery through the quartz crucible. As for the thermal field nearby the dissolution interface, it has a convex shape with large radial thermal gradients. Convex thermal profile can also be reasoned out by envisaging the difference in the thermal boundary conditions between the regions with and without insulation. Besides, there is a large mismatch in thermal conductivities of solution, source and crucible as well. Upon comparing the simulated thermal profiles of the growth zone for different time of the growth, it can easily be concluded that the flow field does not have a strong influence on the thermal field. This is an expected result since the Prandtl number for the germanium rich SixGe1-x mixture is far smaller than unity. Therefore, heat transport within the liquid domain is mainly due to conduction with very little contribution of convective flow on the temperature profile. It is interesting to note that as the growth proceeds, the growth interface is crossed by thermal isotherms, and hence a non-isothermal interface is developed. This situation gives rise to different levels of local saturations at the growth interface, and in turn varying growth velocity across the interface as will be discussed in more detail in the next section.
b-) Growth time:7.5 hours
c-) Growth time:15.5 hours
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
a-) Growth time:0.5 hours
Figure 6.5: Computed thermal field within the entire computational domain (2-D) at various growth times (Temperature in the labels is given in Kelvin). 133
b-) Growth time: 31.5 hours
c-) Growth time: 39.5 hours
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
a-) Growth time: 23.5 hours
Figure 6.6: Computed thermal field within the entire computational domain (2-D) at various growth times (Temperature in the labels is given in Kelvin). 134
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
135
6.6.2 Flow and Concentration Fields Figure 6.7-6.9 present the time evolution of the flow (left column) and solute concentration (right column) structures. The flow field is given in terms of the magnitude of the flow velocity, namely, U = v r2 + v z2 , and the concentration field in silicon mass fraction. As can be seen, the computed flow field develops only one main eye-shaped convective cell which consists of radially-stacked several small cells. This circular flow cell is located near the growth interface. Within the rest of the liquid domain, the convection is very weak, consequently is not notable numerically. The convective flow cell circulates anti-clockwise, pushing the melt upward along the axis of symmetry, and downwards along the vertical crucible wall. The fastest circulation of fluid particles is at the centre of the main cell. The observation of weak convection in the rest of the liquid domain may be explained by keeping in mind the fact that silicon is constantly added into the germanium melt through dissolution. Silicon is the lighter constituent of the SiGe liquid mixture, and in turn, the addition of silicon into the mixture decreases its density, thus causing an axial density gradient and making the mixture heavier near the growth interface. This is an interesting feature of the LPD growth of SiGe. This axial density gradient due to the addition of solute acts as a stabilizing effect on the flow structure developed by the radial temperature gradient. At the early stages of the flow, since there is not enough silicon species in the vicinity of the growth interface, this stabilizing effect of silicon cannot retard the formation of the convective cell. As the growth proceeds, more silicon species is transported towards the growth interface, thereby weakening the convective flow. Subsequently, the circular flow cell shrinks continuously and disappears completely after 5.2 hours of growth. The complete suppression of this toroidal convective cell makes the LPD growth processes of SiGe dominated by diffusion. The corresponding concentration
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
136
profiles are naturally very stable giving rise to a very stable growth. The formation of axially stabilizing density gradient was also reported for the numerical simulation of the growth of SixGe1-x by the vertical Bridgman method [127]. As can also be seen from figures 6.7-6.9, due to the mixing effect of convection, the region with strong convection has a uniform concentration distribution, while that without convection, concentration distribution resembles the one observed in a diffusion limited process.
Figure 6.7: Flow (left column) and concentration (right column) fields for the growth time of 0.5 hours. Flow field is given in terms of magnitude of the flow velocity, U = v r2 + v z2 , and the solute concentration is given in silicon mass fraction.
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
a-) Growth time: 3.5 hour
b-) Growth time: 5 hours Figure 6.8: Flow and concentration fields at various hours of growth
137
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
a-) Growth time: 5.2 hours
b-) Growth time: 6.5 hours Figure 6.9: Flow and concentration fields at various hours of growth
138
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
139
In crystal growth, uncontrolled convection is not desired because it brings about instabilities in the growth process such as inclusion trapping at the interface, local variation of growth velocity which may cause polycrystalline growth even though it enhances transport process, in turn, results in faster growth rate. Figure 6.10 shows the decrease in the magnitude of the flow velocity in the close vicinity of the growth interface as a function of silicon mass fraction.
Figure 6.10: Decrease in the magnitude of flow velocity as a function of silicon mass fraction To elucidate clearly that the formation of solutal-based axial density gradients brings about stabilizing effect in the flow structure, two-dimensional simulation has been also performed for the case where the contribution of solutal gradients on convection is neglected, i.e., we take β C = 0 in the momentum equation. (refer to figure 6.11). Since the focus was on the flow structure and concentration distribution in the liquid for β C = 0 ,
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
140
the interface shape was assumed not to be deviating from its initial shape for early several hours of the growth process. Therefore, the growth interface has been moved at a constant growth velocity, 0.9mm/h. In figure 6.11, the flow field is presented in the left column, while the concentration field is in the right. The flow has two separate toroidal convective cells, one being close to the dissolution interface, and the second adjacent to the growth interface as seen from figure 6.11. The top and bottom cells circulate in opposite directions. The top cell rotates in clockwise, while the bottom cell in the anticlockwise direction. The top flow cell moves the melt upward along the vertical quartz ampoule wall and downwards along the axis of the ampoule. The bottom cell, on the other hand, pushes the melt downward along the quartz ampoule and upward along the axis of the ampoule. Existence of two convective cells can be attributed mainly to the radial thermal gradient. Magnitude and direction of the velocity field are dependent on the steepness of the radial temperature gradient and the geometrical shape of radial thermal isotherms, respectively. In the current growth configuration, the radial thermal gradient near the dissolution interface is steeper than that near the growth interface so that the flow intensity of the top toroidal cell is greater than that of bottom cell. There is no convective cell present in the middle region (between the top and bottom convective cells) because there is practically no radial temperature gradient in this region. This numerical result suggests that the existence of a convective cell is mainly due to the presence of radial thermal gradients. Because of the presence of two convective cells in the liquid domain, the concentration profile developed exhibits a totally distinct behaviour as compared to those introduced in figures 6.7-6.9. In the present case, each convection cell creates a concentration vortex. Since the flow field is stronger at the center of both convection cells, concentration gradients at the corresponding regions are the smallest. Flow circulation pushes solute species towards the boundaries of the liquid domain, causing relatively larger solutal gradients. Concentration gradient is the largest in the area between the upper and bottom concentration vorteses for two reasons: the lack of large radial thermal gradients and the accumulation of solute caused by flow circulation.
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
141
a-) Growth time: 0.5 hours
b-) Growth time: 3.5 hours Figure 6.11: Flow and concentration fields for different hours of growth where β C = 0
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
142
6.6.3 Growth Velocity The interface displacement rate (growth velocity) is computed by solving the mass balance equation coupled with the SiGe binary phase diagram at each time step. The initial computed growth interface is geometrically concave. Figure 6.12 and 6.13 reveals both spatial and time variations of the numerical and experimental growth velocity. Spatial variation of the growth velocity can easily be observed by comparing the interface displacement values for the axis (r = 0) and periphery (r = r) of the crystal. At the onset of the growth, for several hours, the growth velocity at the periphery of the grown crystal is faster than that at the center point. This situation makes the interface become more rounded inward. As the growth progress, the center regions start growing more rapidly than the periphery, leading to a flatter interface shape. As the growth advances further, the curve of the interface changes direction and becomes convex. The reason behind a faster growth along the vertical crucible wall at the early stages of growth can be attributed to the presence of non-uniform mass flux associated with the convective flow cell which is of larger magnitude nearby the crucible wall. In subsequent hours of the growth process, thermal field develops at the interface such that it begins crossing the growth interface. As a result, the growth interface no longer follows the shape of the thermal isotherm near the center region of the crystal. This creates a non-isothermal growth interface. Across the non-isothermal interface, concentration gradient (driving force for the growth) is different. In the regions with larger concentration gradients, the growth velocity is faster. The time variation of the growth velocity can be best visualized considering mass balance at the growth interface, i.e.,
ug =
ρ l DSil ∂c 1 (c gis − c gil ) ρ s ∂n
(6.2)
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
143
where c gis and c gil are the equilibrium mass fractions of silicon in the solution and the solidified crystal at the interface, computed from the phase diagram. According to the SiGe phase diagram, the term
(c
s gi
− cgil ) increases with the increasing interface
temperature as the growth interface moves upward. Considering that during the growth, the concentration gradient along the growth direction remains nearly constant, one can conclude that the growth velocity will be inversely proportional to the term (cgis − cgil ) .
Figure.6.12: Interface position as a function of growth time for r = 0 (at the center) In what follows is that the growth velocity must decrease as the growth interface moves to the higher temperature regions. This purely mathematical definition refers to the fact that at the onset of the growth process, less amount of silicon atoms are needed to saturate the interface, and causing solidification; however, at the later stages of the growth, more silicon atoms are needed for super saturation, so that growth velocity is expected to be not as fast as before.
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
144
Figure 6.12 presents the comparison of experimentally measured and numerically computed interface displacement for different hours of the growth for r = 0 (at the center). The numerically computed average growth velocity (averaged over 29 hours of the growth) is 0.63 mm/h for r = 0, while experimentally measured one is 0.67 mm/h for r = 0. The difference between experimental and numerical growth velocity for the center line is approximately 6 percent.
Figure 6.13: Interface position as a function of growth time for r = r (at the wall) Figure 6.13 shows the experimentally and numerically obtained interface displacement for different hours of the growth for r = r (at the wall). The average growth velocity (averaged over 29 hours of the growth) obtained from numerical computation for r = r is 0.58 mm/h, while experimental result is 0.55 mm/h for r = r. In figure 6.14, the computed time evolution of the growth interface is shown on the left, and the cross section of an LPD grown crystal on the right. Comparison of circular mark
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
145
(growth striations) on the grown crystal with the computed growth interfaces shows that the numerical model developed successfully simulates the tendency of the time evolution of growth interface. In experiments, the interface starts becoming flatter after 16hours of the growth, while numerically, the flat interface is observed around 24th hour of the growth.
Figure 6.14. Time evolution of the growth interface for the half geometrical domain is shown on the left (the time interval between each line is three hours, and total simulated growth time is 39 hours), while the cross-section of an LPD grown crystal on the right. Agreement between the experimental and the simulation results are quite good.
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
146
6.7 Results of Three-Dimensional Simulation In this section results of a three-dimensional simulation are presented. Figure 6.15 shows the temperature distribution computed for the vertical section where ϕ = π at time t = 1 hour.
Figure 6.15: Computed thermal field within the entire computational domain (3-D) at the growth time t =1 h. Temperature in the scale is given in Kelvin. Computed flow and concentration fields are presented in Figure 6.16-6.19 in the vertical plane (where ϕ = π ) of the computational domain where the flow field is given in U = v r2 + v z2 + vϕ2 (magnitude of the flow velocity) and the concentration field
silicon mass fraction.
in
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
Figure 6.16: Flow (up) and concentration (down) fields at t = 0.5 h
147
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
148
As shown in figure 6.16, the flow field has two main cells bridged to each other at the vertical axis of the liquid domain. The cells circulate in the opposite directions. The cell on the right moves in clockwise pushing the fluid particles down along the crucible wall and up along the axis of symmetry, whereas the one on the left circulates in the anticlockwise direction. Due to the presence of convection, solute distribution in the mixture shows a perfect mixing. The regions deprived of convection cells show diffusion like solute distribution profiles. Two-and three-dimensional computational results show similar quantitative behaviour regarding the flow structure and solute distribution in the liquid domain. This is an expected result since there are no strong external effects such as applied magnetic field or crucible rotation. However, even in the present case, the development of three-dimensional flow and concentration structures in the liquid zone can clearly be seen in figure 6.18 and 6.19. This is particularly obvious in the structures presented in the horizontal plane (figure 6.19) at a distance of 14 mm from the bottom of the substrate. This is the main difference between 2-D and 3-D computations. As the growth progresses, and in turn, more silicon is transported towards the growth interface, as was observed in two-dimensional simulations, the convective flow cells begin losing their intensity, and finally become almost suppressed when the solute concentration reaches a critical value in the vicinity of the growth interface.
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
Figure 6.17: Flow and concentration fields at growth time t = 3.5 h
149
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
Figure 6.18: Flow and concentration fields at growth time t = 5.5 h
150
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
151
Figure 6.19: Flow fields at different growth times: 0.5, 2.5, 3.5, 4.5 and 5.5 hours; the horizontal plane is at a distance of 14 mm from the bottom of the substrate.
Chapter 6 –Numerical Simulations for the Growth of SixGe1-x by LPD
152
6.8. Summary In this chapter, we have introduced the results of two and three-dimensional transient simulations of the LPD growth process for the growth of SixGe1-x. Numerical simulation results agree with experiments. Results show that the natural convection in the liquid zone is strong at the beginning of the growth process, but gets weaker as the growth progresses. After about 5.2 hours of growth time, the convection becomes unobservable numerically, and the diffusion becomes the dominant growth mechanism. 2-D and 3-D simulation results are very similar quantitatively, however, 3-D simulations lead to the development of three dimensional structures in the flow fields as expected. The presence of convection during the early stages of the growth, indeed, is not desired because it gives rise to non-uniform and uncontrolled transport of species to the growth interface. The non uniform mass flux to the growth interfaces will most likely result in local variations in the growth velocity. Such variations across the growth interface generally reveal itself as the driving force for the formation of low angle grain boundaries, or in extreme cases, of extensive polycrystalline structure. The suppression of the convection or reducing it to a minimum, in fact, can be helpful for increasing the likelihood of the single crystalline growth. A possible way to suppress the natural convection is either to carry our experiments in a microgravity environment, or to make use of an applied magnetic field
153
CHAPTER 7 The Growth of Compositionally Graded SixGe1-x Single Crystals by Liquid Phase Diffusion (LPD) 7.1 Introduction This chapter introduces the experimental apparatus designed and constructed for the growth of germanium rich Si x Ge1− x single crystals with axially varying compositions by LPD techniques, the LPD growth steps, growth mechanisms, and experimental results. The motivation behind building an LPD growth facility at the University of Victoria, Crystal Growth Laboratory is to show the applicability of LPD growth technique to produce Si x Ge1− x single crystals that can be used by other crystal growth processes as lattice-matched substrate. Grown crystals were characterized by macroscopic and microscopic examination after chemical etching for delineation of the degree of single crystallinity and growth striations. Compositional mapping of selected crystals was performed by using Electron Probe Micro Analysis (EPMA) as well as Energy Dispersive X-Ray Analysis (EDX). It was shown that LPD technique can be successfully utilized to obtain Si x Ge1− x single crystals up to 6-8 % at.Si with uniform radial composition.
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 154
7.2 LPD Crystal Growth System For the LPD growth of Si x Ge1−x from the germanium rich side of the phase diagram, a three-zone solid tubular furnace capable of operation at temperatures up to 1250 0C is used. The furnace is mounted on a platform with brackets as shown in figure 7.1 and 7.2. The platform allows the rotation of the furnace by 180o around the axis perpendicular to the platform. Since the furnace is held off the ground level by the platform, it is accessible from its both ends, which facilitates to set up the growth experiments. At the centre of the furnace is a ceramic chamber which is winded by Fe-Cr-Al heating elements. There is a long firebrick vestibule of the same inside diameter as the ceramic chamber at each end of the furnace. The bottom of the furnace can be sealed with a sliding ceramic door if required. Temperature of each zone of the furnace is separately controllable through PID, phase angle fired, SCR (Silicon Controlled Rectifier) controllers, connected to the furnace by a set of K-type thermocouples whose junction is formed by Ni-Cr and Ni-Al alloys. SCR is also sometimes referred to as a thyristor. At each zone of the furnace, there is a pair of thermocouple connection ports. One thermocouple is used to monitor the temperature of the corresponding zone, while the other is for the safety purpose. If somehow the temperature exceeds the predefined temperature set point, the furnace automatically shuts off. All three controllers are programmable either as a master or slave controller. They have the feature of 12 segments of ramps and soaks. In our case, the controller for the middle zone was used as a master controller over the other two since the growth zone is positioned therein. The slave controllers follow the set point from the master controller. A ring-shaped insulator material is placed on the top surface of the furnace to reduce the amount of heat loss to the surroundings. Sitting on this insulator is a mechanical assembly, which suspends a quartz ampoule/tube (growth reactor) inside the furnace (refer to figure 7.3). The mechanical assembly is comprised of two main parts, namely, a
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 155
water-cooled flange and end-cap, which are made of a commercial grade aluminium alloy.
Figure 7.1: A view of the LPD growth set-up used in this thesis work.
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 156
water inlet and outlet ← → water inlet and outlet pipes
three zone furnace ↑ firebrick vestibule ← top zone ← middle zone ←
TC connection port ←
→ rotating shaft
bottom zone ← firebrick vestibule ←
↓
ceramic chamber
→ supports
locking pin ← 4 − levelling feet ←
→ base plate
Figure 7.2: Schematic illustration of the LPD crystal growth platform The flange is placed on the insulator material, and is kept stationary by four aluminium clamps fastened to the top surface of the furnace. It serves as a platform for the end cap and prevents the end cap from being in direct contact with hot furnace atmosphere.
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 157
The end-cap is used to suspend the quartz ampoule (growth reactor) inside the furnace and to seal it. It seats inside a shallow counter bore machined on the flange so that the end cap and the flange can be centred axially. The positioning of the growth reactor inside the furnace is extremely important not to disturb thermal symmetry. The end cap is fixed to the flange by several aluminium clamps. It consists of two parts, namely, male and female parts, which can be fastened to each other by a set of screws. There is an Oring sandwiched between the male and female part. For the detailed assembly of the growth set-up as well as the dimension of various components, refer to figure 7.3 and table 7.1, respectively. Table 7.1: Dimensions of various components of the LPD growth platform. Item
Furnace
Specification
Dimensions
Total length
29 inch
Total heated length
24 inch
Outer diameter
14 inch
Inner diameter of the hot zone (ceramic chamber)
5 inch
Length of the firebrick vestibule
2.5 inch
Length of the top, bottom, and middle zones,
6, 6, 12 inch
respectively Quartz crucible and
Length
80/40 mm
pedestal, respectively
Inner diameter
25/25 mm
Outer diameter
29.5/29.5 mm
Length
559 mm
Inner diameter
30 mm
Outer diameter
33 mm
Quartz reactor tube
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 158 1
6 4
2
3 5 7 8
9 10
16
11 14
12
15 13
17
1 − Quartz hook 2 − Fitting assembly 3 − Al − end cap 4 − Al − water cooled flange 5 − Brass sleeve 6 − Quartz T − rod 7 − Al − clamp 8 − Insulator 9 − Quartz sealing cap 10 − Quartz ampoule 11 − Quartz crucible 12 − Ring shaped insulator 13 − Quartz pedestal 14 − Poly − Si source 15 − Single − Ge seed 16 − Ge − solution material 17 − 3 − Zone heated furnace
Figure 7.3: The detailed drawing for the experimental LPD growth set up
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 159
Figure 7.4. Top view of the furnace As seen from figure 7.4, there is a fitting installation threaded to the end cap from top, which is composed of a positonable male run tee, union cross, two needle valves and hose adapters. The fitting installation allows insertion of the thermocouple protection tube for obtaining temperature profile and of quartz hook, (which is used to hold quartz sealing cap) into the quartz growth reactor through vertical ports of the male run tee. The vacuum tight sealing at the insertion point is achieved by using a viton O-ring, and back ferrule. The fitting assembly also enables the quartz growth reactor to be depressurized and then flushed with high purity hydrogen before the sealing process. The quartz reactor tube is inserted into the end cap through the female part, the O-ring and the male part until its open end is set against a Teflon cushion that is placed on the inner top surface of the male part. The Teflon cushion is used to prevent the occurrence of cracking of the quartz ampoule rim. The end cap is winded spirally by a cupper tube
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 160
through which cooling water runs to eliminate the risk of burning of the o-ring and the Teflon cushion. There are several circumferentially located holes on the water cooled flange to insert the ceramic thermocouple protection tube and T shaped quartz roads into the furnace. The thermocouple located outside the quartz growth reactor is utilized to monitor the temperature of the growth system since mechanical access to the inside of the quartz reactor may not be possible due to the sealing or may not be desired not to induce any contamination during the growth. The quartz roads are used to hold the annular insulator material in the air, which is located close to the bottom of the middle zone of the furnace. The insulator material of one-inch thickness is used to achieve a steep axial temperature drop at the location of interest inside the furnace. Sufficient clearance between the ceramic chamber and the insulator is provided to be able to move the insulator up and down if required. In contained single crystal growth techniques such as LPD where molten material is in direct contact with the crucible inner walls, the selection of a suitable crucible material is a very important issue. Incompatible crucible materials may bring about many experimental and operational difficulties, thereby producing undesired results such as growing polycrystalline crystal or single crystal of highly defective structure. Quartz is a material of choice for several sound reasons; it is inexpensive, easy to clean and etch. Besides, covalent bonded metals do expand on cooling. Therefore, a container which can burst under tension is advantageous not to compress the grown crystal, hence reducing the possible occurrence of microcracks. In the designed LPD experimental set-up, the crystal growth operation is carried out by using double wall quartz ampoules, namely quartz growth reactor and quartz crucible. The outer quartz ampoule (growth reactor) is used as an envelope for the inner quartz ampoule (crucible). Figure 7.5 shows a picture of quartz parts.
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 161
10
13 11
1
Figure 7.5: Picture of quartz components (see also figure 7.3) The inner quartz ampoule is used as a crucible and serves for three main purposes; first, it prevents spilling of the growth material into the furnace in case the quartz reactor tube is broken; second, it facilitates the loading process of the growth charge into the ampoule; third, it creates spaces for the deposition of potential volatile reaction products such as SiO, which might be formed due to the reaction between the silicon and quartz crucible walls as well as the residual oxidizing atmosphere. At the bottom of the crucible is a 40 mm long hollow quartz pedestal used to move the growth cell up and down inside the ampoule during the determination of temperature profile since the quartz ampoule is stationary. At the early stages of the research for temperature profiling, graphite blocks were used as a pedestal; however, it was observed that graphite burns by the residual oxygen in the growth chamber, causing to increase in the temperature of the growth zone. The crystal growth process has to be performed under well-controlled environment which can be achieved through depressurizing and then purging the growth chamber by inert or non-oxidizing gases. For the depressurization of the growth reactor, the combination of turbo molecular pump (TMP) and a rotary vane type backing pump is used. The pumping system used during the experimental work allows direct measurements of pressure level
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 162
from the intake flange of the TMP. Pressure down to the 10-3 mbar is gauged by a Priani type sensor, while the pressures lower than this value is red by using a hot cathode filament ion sensor. Gas and water networks are composed of series of element to control the gas and water flows. Water system is utilized to cool down aluminium flange and end cap holding the quartz growth reactor. Gas distribution network is used to create controlled non-oxidizing environment. The existing gas distribution system (refer to figure 7.6) is capable of handling three different gases, Hydrogen, Nitrogen, and their mixture. To purge the growth reactor tube and also create a controlled growth environment, high purity hydrogen has been employed. Hydrogen distribution platform comprises flow meter, flash arrester positioned nearby the hydrogen tube outlet, the heated hydrogen purifier (getter), and the bubbler to visualize the flow, installed close to the ventilation. To reduce the risk of contamination between the purifier and the reactor inlet, only stainless steel tubing is used. Since high temperature operation and highly explosive gas coexist during some of the growth experiments, there must be serious preventive measures taken. There are two levels of alarm defined in the UVic crystal growth lab facility, namely low and high level. Some of the reasons which trigger low level alarm are those; if the pressure of the water cooling line drops under 30 psig, and if the temperature with in the furnace exceeds preset safety limit. The factors which can provoke the high level alarm are such; if the hydrogen level within the faculty goes beyond 10%, if any flame is detected in the growth area, and if the temperature level exceeds the set acceptable level. If any of above mentioned possibilities are satisfied, the furnace and hydrogen flow are shut off.
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 163
Figure 7.6: Gas distribution network
7.3 LPD Growth Steps 7.3.1 Temperature profile A well-determined, a steep and stable temperature profile is an important element of a successful LPD growth experiment. The steep temperature gradient is required for two reasons; one being able to obtain large solute composition gradient in a grown crystal, the other is to prevent the total melting down of the pure germanium substrate, which is a main difficulty of growing Si x Ge1− x with a germanium substrate. Therefore, prior to growth trials, a great deal of temperature profile measurements was taken to determine
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 164
the best-possible location for the growth crucible within the outer quartz tube as well to find appropriate temperatures of each zone of the furnace. Initially, the growth crucible was located arbitrarily at the middle zone of the furnace. Then, temperature measurements were taken at the center of and on the outside wall of the quartz tube along the growth zone by 2-5 mm axial intervals while the quartz reactor tube is under dynamic vacuum. Through trial and error attempts, a zone temperature configuration which gives highest temperature gradient throughout the growth zone was obtained. During this period, the position of the quartz crucible was adjusted by using various lengths of pedestals. Since the obtained thermal profile was not as high as desired, an annular insulator material of 25 mm in length was positioned around the growth zone arbitrarily. Following the trial and error procedure repeatedly by changing the position of the insulator and temperature of each zone, the final location of the insulator was determined. The obtained configuration was used to run the first growth experiment; however, it did lead to slight melting of charge in the vicinity of the source material since the presence of growth charge altered the thermal field drastically. In several successive thermal profile experiments, pure germanium blocks were used as the charge material. The temperature of each zone of the furnace was increased from their old values to sensible new values gradually for several experiments until the unmelted thickness of the germanium charge became 7 to 8 mm. Several preliminary growth experiments were preformed until the set thermal configuration did not cause the melting of the germanium substrate. The determination of the thermal configuration by which successful growth experiments were conducted took almost 6 months. Figure 7.7 shows the thermal profile measured along outside wall of the quartz tube and the axis of center drilled silicon dummy load. This thermal profile with fairly small modifications when required is used for all growth experiments conducted. The temperature gradient measured within the growth pathway is around 23 oC/cm inside the crucible with center drilled silicon dummy load, and 40 o
C/cm on the outside wall of the quartz ampoule.
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 165
≈
≈
90
Source (Si) Solution Si x Ge 1− x
80 70
Axial distance in mm
↑
⇑
60
⇑
Grown crystal Si x Ge 1− x
50
Ge Seed
Insulator material
40 30 20 10 0 850
900
950
1000
1050
Temperature in oC → Figure 7.7: Thermal profile measured along the quartz ampoule (square) and the axis of a center drilled silicon dummy load (circle), and the schematic diagram of the LPD growth system
7.3.2 Preparation of Growth Charges Cleaning and chemical treatment stage of the growth charge and components is one of the most important steps in the preparation of growth experiments. This step is a deciding factor on the quality of crystals. Here, we introduce the procedure followed for cutting, surface cleaning and chemical treatment of the charge materials, namely substrate (single crystal pure germanium), solution materials (germanium chunk or block) and source (polycrystalline pure silicon). This section also discusses cleaning and chemical treatment
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 166
procedure for all quartz parts i.e., ampoule, crucible, sealing cap, pedestal, and hook as well as the procedure for cleaning of the Viton o-rings, Teflon cushion.
7.3.2.1 Cutting and Core-drilling Charge Materials Polycrystalline silicon road of 50 by 300 mm in diameter and length, respectively, and Czochralski-grown single crystal germanium boules (50 by 10 mm in diameter and length, respectively) of (111) crystallographic orientation were available in our crystal growth laboratory. The silicon and germanium rods were diced into various sizes of round plates by a low speed precision saw with a Diamond wafering blade. Diced silicon and germanium sections are core drilled by a diamond impregnated core drill tip. Germanium block for a substrate is core drilled with the core drill tip of 25.4 mm inner diameter, which is slightly larger than the inside diameter of the quartz crucible. Silicon is drilled to a size slightly smaller than the inner diameter of the quartz crucible. The concentricity of the drill tip is significantly important to obtain a core-drilled sample with a uniform diameter along its length. This is particularly essential for the germanium sample to be used as substrate. During experimental works, it was observed that a relatively large clearance between the substrate and the inside of the crucible is not tolerable. If the clearance is greater than 0.3 mm, the melt (germanium) above the substrate leaks through the gap and solidifies. Upon the solidification, the volume of the leaked liquid expands, thus causing severe burst of the quartz crucible. The volume expansion is a common characteristic of covalently bonded materials such as silicon and germanium and its alloys. Clearance less than 0.3 mm does not pose any leakage problem due to the surface tension and capillary effect. The reason why silicon is drilled to a diameter smaller than the inner diameter of the crucible is following. When the growth charge is heated to growth temperature, germanium that forms the solution zone melts collapses down. In order for sustaining the full physical contacts between the melt and silicon source, the silicon source has to move down easily as well. If the clearance
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 167
between the source silicon and crucible walls were so small, silicon might get wedged and hence, might not be able to move downward in the crucible. Such an occasion was experienced once; therefore, as can be expected, no growth of Si x Ge1− x was observed. On the other hand, this clearance should not be so large to allow the occurrence of surface tension driven convection.
7.3.2.2 Cleaning and Chemical Treatment Here is presented the procedures followed to clean and etch the quartz parts as well as the growth load. Before launching the growth experiments, it is crucial that every part contained inside the growth environment or in direct contact with growth charge be cleaned and etched carefully. This reduces the risk of contamination of the growth charge. Any contamination left behind in the growth environment burns at high temperature, hence creating oxidizing surroundings, which certainly precludes the growth. One important point which must be born in mind is that cleaned parts never be handled with bare hands. The procedure described below has been used for each growth and temperature profiling experiments. 1. Wash the quartz parts with soup and then rinse with distilled water (18 MΩ resistivity) several times. Small parts which can fit into a beaker are washed in an ultrasonic cleaner for 3-5 minutes. Before putting the small parts into water-filled glass beakers, beakers have to be washed with distilled water in the ultrasonic cleaner as well. This process removes dusts or any undesired particles from surfaces and, therefore, reduce the amount of acetone and methanol cleaning agents to be used. 2. Washed quartz parts are cleaned with acetone (CH3COCH3) cleaning agent grade for 3-5 minutes in the ultrasonic cleaners to remove humidity from the material surfaces.
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 168
3. Wash the parts with methanol (CH3OH) in the ultrasonic cleaner for 3-5 minutes to remove the acetone from the material surfaces. 4.
The mixture of HCl (36.5-38 %): HNO3 (68-70 %) with the ratio of (1:1 or 3:1) is used as a chemical agent to etch the surfaces of the quartz parts. The small parts are etched inside beakers from their both interior and exterior surfaces. The parts are etched for 6 hour durations. For short etching time, quartz parts can also be etched in diluted HF (10%) for 15 minutes.
5. When the etching process is completed, the etchant is filled into a container to dispose and then, the parts are rinsed with distilled water for a few times. To remove the residual etchant from the surfaces of the parts through the adsorption, parts are kept in water for several hours. A few hours before the experiments, parts are rinsed with methanol to remove the moisture from the surfaces and also speed up the drying process. During the drying process, the quartz ampoule should be held vertically by a platform in an open end down position. Small parts should be resting on a clean surface vertically and covered by beakers. 6. For the reusable parts such as quartz pedestal and crucible, Hydrofluoric acid HF (49 %) is used to remove reaction products deposited on the surface of these parts during the growth experiment. HF reacts with quartz and removes a thin layer from the inside surface. It important that Teflon beakers or HF-resistant beakers be used for the etching process, which requires usage of HF mixture. Since HF is a deadly agent, its usage has to be kept at minimum level, and considered as last alternative. 7. As for the cleaning of Viton o-rings and Teflon cushion, they are wiped with distilled water and then methanol. The charge materials are chemically etched in a mixture of HF (49%):HNO3 (68-70%) with the ratio of 1:3 to remove surface oxides and to fit their diameter to the quartz crucible for 5 to 8 minutes. Etching reaction is killed with methanol for both germanium
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 169
and silicon. The etched samples are kept in beakers filled with methanol until loading the charge into the growth chamber.
7.4 LPD Growth Principles During the growth process, the quartz growth reactor is held stationary inside the furnace. The quartz reactor is located inside the furnace such that the quartz crucible is placed in an appropriate temperature gradient as illustrated in figure 7.7. The quartz crucible contains three layers of vertically stacked charge materials, namely single crystal germanium as a substrate with (111) crystallographic orientation, germanium blocks or chunks to form liquid solvent, and finally polycrystalline silicon as a source material. The total height of the growth domain is 40 mm. For several initial experiments, thicker germanium substrates with the thickness of 17-20 mm were used to guaranty partial melting of the substrate. As the temperature profile was mastered, the substrate thickness was reduced down to 10-12 mm to save on single crystal substrates used. Various sizes of silicon source material were used such as 10 mm, and 1.5-3 mm. The LPD growth process is initiated by heating the furnace to the predetermined temperature profile. Then, this temperature profile is kept constant throughout the growth. The procedure is as follows. The furnace was first heated up to a temperature profile, which is 50 oC lower than the real growth temperature (at which germanium charges are still solid) and then kept at this temperature until the temperature profile is stabilized. Thereafter, it is heated to the growth temperature profile within 2.5 hours (refer to figure 7.9). When the temperature around the growth cell reaches to the growth temperature, the polycrystalline germanium is entirely molten to prepare a growth solvent for the growth. As for single crystal germanium substrate, it is partially melted until the remaining thickness is 8-10 mm. On the other hand, the silicon source material remains
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 170
solid due to its higher melting temperature. However, at the growth temperature, the SiGe binary phase diagram requires that silicon be dissolved the by germanium melt. The maximum amount of dissolved silicon, of course, depends on the temperature value across the melt-source interface. With the transportation of dissolved silicon species from the silicon source crystal into molten germanium, the SixGe1-x binary growth solution starts forming. Numerical simulation work introduced in previous chapter has showed that dissolved silicon solute atoms are transported into the growth melt and to the growth interface by mainly ordinary diffusion process due to concentration gradient with little contribution of bouncy induced convection at early stages of the growth process.
The growth mechanism of LPD technique can be best visualized considering the equilibrium phase diagram of SixGe1-x alloy system as shown in figure 2.2. A sketch of growth cell is presented in figure 7.8 along with the germanium rich section of the representative binary phase diagram of SixGe1-x. The transportation of silicon species (solute) towards the growth interface enriches the solute concentration in the vicinity of the growth interface, thereby forming a supersaturated solution ahead of the interface. Assuming that the silicon concentration of the first super saturated liquid is x1L , from the phase diagram, the corresponding liquidus (equilibrium) temperature of the supersaturated solution can be obtained as T1L . T1L is obviously higher than the temperature in the solution at the interface, illustrated by a dashed line. Hence, the
supersaturated solution must be constitutionally supercooled, and then consequently leading to solidification with the silicon composition, x1S . Thus, the growth interface moves to a new position. The silicon composition in the solid (Constitutionally SuperCooled Liquid, CSCL), x1S is higher than that in the supersaturated solution. This means that germanium concentration increases near the solidification front due to the rejection of germanium species into the liquid, and the silicon concentration decreases owing to its preferential consumption upon
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 171
the solidification. As a result, the liquidus (equilibrium) temperature of the solution at the progressing interface drops below the temperature in the liquid, giving rise to a slight melt back of the newly grown layer. This situation may cause formation of micro level striations in the grown crystals. Transportation of silicon species towards the growth interface supersaturates the solution near the interface, and leads to crystallization. This process repeat itself during the growth, and the growth of the SixGe1-x crystal is maintained by the continuous reoccurrence of constitutional supercooling induced by the supply of silicon solute species through mass transport processes.
Figure 7.8: A representative figure for the growth mechanism of LPD growth system.
Throughout the growth process, unremitting silicon depletion in solution is compensated by continuous supply of silicon species from the top- located silicon feed. As the growth progresses, growing interface moves up leading to an increase in the growth temperature, and in turn in the silicon concentration at the interface. The germanium composition in
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 172
the solution decreases as the SixGe1-x crystal grows since the growth temperature becomes continuously higher and the interface approaches the silicon source material. The above-discussed LPD growth configuration allows the growth of compositionally graded SixGe1-x single bulk crystals with an increasing silicon composition along the growth direction by cooling the melt along the liquidus curve. The growth process is terminated when the entire solution solidifies completely. The furnace is first cooled at a rate of 15 0C/h from the growth temperature to the somewhat below the melting temperature of pure germanium to prevent crack generation due to the thermal shock in the solidified crystal and the quartz crucible, and then in the furnace atmosphere down to room temperature. In most of the growth experiments, the total growth time was selected to be 96-120 hours.
7.4.1 A Typical Procedure for an LPD Crystal Growth Experiment Here, we enumerate the order followed for the set up of a typical LPD growth experiment. It is vital that the experimental operator wear necessary protective equipments corresponding to the steps of the following procedure not only to protect himself/herself but also not to cause any contamination on the growth charge and the crucible. Some of the protective items are cleanroom cap, mask, gloves, goggles, acid proofed smack. 1. Cut and core-drill the growth charge; substrate, solvent and source materials. 2. Drill a pair of holes in quartz sealing cap at locations adjacent to its rim. Holes are drilled by a diamond impregnated core of 5 mm diameter and are 180o apart one to another. Cut 30 mm long quartz rod of 6.25 mm diameter to be used as a horizontal bar for the quartz hook. 3. Prepare sufficient number of cleaned glass and HF resistant plastic beakers. 4. Clean and then etch all the quartz parts; quartz tube (reactor), quartz crucible, quartz sealing cap, quartz hook, and quartz pedestal. Leave them inside or filled
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 173
with distilled water until loading. Just before loading, rinse them with methanol and dry with Argon inert gas. 5. Clean and etch the growth charge. Until loading, keep the charge inside the methanol. During the loading, dry the charge with Argon. 6. Wipe out all the mechanical parts, namely, aluminium end cap and screws as well as Viton O-ring and Teflon cushion by methanol. 7. Place the quartz pedestal into the quartz reactor. Then, load the growth charge into the quartz crucible, following the order of substrate, solution, and source material. Put the loaded crucible into the reactor. Assemble the quartz reactor and endcap. The procedure followed from loading to start of hydrogen purging is generally completed in 4-5 minutes. This loading procedure has to be completed as quickly as possible to reduce the exposure time of the etched growth charges to oxidizing environment. 8. Connect the hydrogen and vacuum lines to end cap’s hydrogen and vacuum ports, respectively. Initiate the depressurization of the reactor tube with the backing pump. When the pressure drops down to 10-2 mbar levels, launch the turbomolecular pump. Keep vacuuming until pressure level drops down to 10-5 mbar levels. 9. Turn off vacuum outlet and turn on the hydrogen ports. Then let the hydrogen fill the reactor tube slowly at 5 psig. Purge the hydrogen through vacuuming. Repeat this procedure several times to reduce amount of oxidizing molecules in the reactor tube. 10. Adjust the position of the quartz sealing cap in the quartz rector tube by sliding the quartz hook through the male run tee. Then, heat up the quartz sealing cap until the quartz reactor tube yields on the sealing cap. 11. Load the reactor tube to the furnace and make sure that it is centered with respect to the axis of the furnace. Fix the end cap by aluminium clamps. Adjust the position of the annular insulator surrounding the reactor tube. Make water
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 174
connections to flange and endcap for cooling. Adjust the position of monitoring TC. 12. Heat the furnace to the growth temperature following the heating cycle given in figure 7.9. 13. When the growth time is over, cool down the furnace by following the cooling cycle in figure 7.9. 14. When the furnace temperature is down to the room temperature, remove the quartz reactor from the furnace and disassemble it. Take the grown crystal out of the furnace.
Top zone
1050 955 750
688
1100 o C 1007 o C
Middle zone
800
738 o C
707
675 Bottom zone
418
1h
1h 1h
1h
2 - 2.5 h
438
96 h
20 h
Figure 7.9: Typical heating and cooling cycle for the growth experiment
7.5 Experimental Results and Characterization A total of twenty growth experiments were performed. One of the main difficulties of growing SixGe1-x with a germanium substrate is to prevent the substrate from being totally
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 175
melted. Therefore, six experiments were devoted to find appropriate temperatures for each zone of the furnace since visual access to the growth zone is not possible. A total of ten compositionally graded fully and partially single SixGe1-x crystals (including the crystals grown to measure the growth velocity) were grown. Remaining four experiments were unsuccessful due to the various reasons, such as unexpected power outage, failures of furnace thermocouples, and failure of quartz components such as quartz reactor, and crucible (mainly cracks). In one occasion, the growth experiment was performed under top flowing hydrogen. Since hydrogen flow was not circulating across the reactor, it could not remove volatile SiO reaction product and confined it to the growth zone. Hence, the molten growth charge and silicon source were terribly oxidized. There was no wetting of silicon source by the liquid germanium due to the oxidization. The presence of hydrogen changed the thermal profile within the quartz ampoule so that the germanium substrate was not melted at all. The diameter and the length of grown crystal are 25 mm and between 20 and 25 mm, respectively. For compositional analysis and delineation of single crystallinity, the grown crystals were bisected along the growth axis. A 2-mm thick plate was cut from the fist half to determine axial and radial compositional distributions of silicon as seen in figure 7.10. The cut samples were polished by using SiC papers of 1200 mesh size followed by diamond suspensions of 6 µm and 1µm particle size sequentially and then were etched at room temperature in the mixture of HF (49 %): H2O2 (30 %):H2O with the ratio of 1:1:4 for 12-15 minutes to delineate the extent of single crystallinity and growth striations.
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 176
Figure 7.10: Cutting configuration for the characterization of grown crytals (up), and pictures of two mm thick vertical slices for several grown single crystals (down). The compositions of the grown single crystals were measured at various axial and radial locations by Electron Probe Microanalysis (EPMA) and Energy Dispersive X-ray Analysis (EDX) with the acceleration voltage of 20 kV and SiK 1.739 keV and GeK 9.873 KeV peaks. EDX analysis was performed at Prof.Karen Kavanagh’s laboratory, Department of Physic at Simon Fraser University. No serious sticking problem has been noticed between the walls of crucibles and grown crystals. Crystals have highly smooth and shiny appearance. However, nearly in all experiments, the inner quartz crucibles included many cracks, but held their integrity. The cracks in quartz crucible might be attributable to the expansion of the solidified solution. All crystals grown using 10 mm thick silicon source materials involve extensive
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 177
amount of cracks in the close vicinity of the dissolution interface as seen in figure 7.11. The cracks penetrate into the crystal body to a certain extent. The cracks have been generated possibly because of the mismatch in the thermal expansion coefficients between the undissolved silicon source and the Si1− x Ge x solid solution. There might be a possible contribution of large lattice mismatch (approximately 4.2%) between the silicon source and the Si1− x Ge x crystal. Since the objective of this work is to grow single crystals of high silicon composition that can be used a substrate material, the presence of such cracks is not acceptable. Therefore, elimination of the cracks or reducing them to a minimum level is of paramount importance. To address this issue, growth experiments were carried out using silicon sources (2-3 mm) thin enough to ensure that the silicon sources are entirely dissolved or little left over. It was observed that cracks emanated from the remaining silicon source and the silicon source free regions did not show any sign of cracks (refer to figure 7.11). Figure 7.11 shows pictures of vertical sections of several compositionally graded LPD grown SixGe1-x crystals with thick and thin silicon sources. In the pictures, the regions, namely the germanium substrate, the initial interface, the grown crystal, and the silicon source, are labelled. In all grown crystals, growth interface shape is initially concave with approximately 1.5-2 mm curvature depth. The steepness of the interface depends on the remaining thickness of the substrate as well. The larger the remaining thickness, the shallower the growth interface since the thicker substrate facilities heat transfer in the axial direction. The development of the initial concave shape is due to (1) large variations between the thermal conductivities of the substrate, grown crystal and the quartz crucible, and (2) significant differences in thermal boundary conditions in the regions (with and without the annular insulator). The concave interface shape implies that heat loss from the periphery of the crucible is larger than that from the bottom in the present LPD configuration. Therefore, it is possible to control the interface shape changing the heat transfer characteristics of the system.
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 178
The initial growth interfaces shapes seen in fig 7.11 for instance are formed as a result of the melting process as the furnace reaches its predetermined thermal profile. At the same time, as seen in figure 7.11, LPD-5), on the right side of the picture close to the growth interface, there are formations of multi-domain structures commencing from the initial growth interface. Such a structure is not common to all grown crystals. The presence of this irregular structure might be due to the strong natural convection induced by the radial temperature gradient in the vicinity of the growth interface. This adverse effect of convection may be minimized to a certain extent by other means such as the optimization of design parameters or the application of stationary and/or rotating magnetic fields. All the grown crystals show a high degree of single crystallinity up to the 6-8 % atm Si. This tells us that it is possible to extract SixGe1-x single crystal substrates from the region that offers the specific composition of interest. As for the side initiated cracks observed in some of the crystals, they are believed to be due to the cutting process. For example, all the grown crystals except LPD-19 and LPD-20 were vertically cut by using a saw with a 5 inch diamond wafering blade. During the cutting process of long vertical section, the saw induces vibrations which may cause propagation of micro cracks likely present on the outer surface of the grown crystals. The LPD-19 and LPD-20 grown crystal were cut utilizing a more sophisticated tape saw. As can be seen from the picture, there is no sign of any horizontally propagating cracks. One important point worthy of mentioning is that the polycrystalline growth from the periphery of the crystals is only particular to LPD-19. This might be attributed to severe vibration due to the construction work near the growth laboratory at the time of performing the growth experiment LPD-19.
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 179
LPD-5
LPD-9
LPD-10
LPD-11
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 180
LPD-19
LPD-20
Figure 7.11: Several LPD grown compositionally graded SixGe1-x single crystals In all the LPD grown crystals, there are ring-shaped marks that are referred to as “growth striations”. The occurrence of growth striations is known in crystals grown by the
techniques (e.g., Czochralski, and Vertical Bridgman) involving intentional mechanical movements such as pulling, rotation, or translation. The presence of growth striations in the crystals grown by these techniques can be attributed to the disturbance of the growth interface caused by such mechanical movements. However, striations were not expected in the LPD grown crystals since there is no mechanical movement of any kind in the present set-up, neither the growth reactor nor the crucible moves. This issue is discussed below. To calculate the rate of displacement of the growth interface (the growth velocity, u g ), and also to examine whether the growth striations can be correlated to the evolution of the growth interface, five growth experiments were devised. Each growth experiment was terminated after a certain time period by switching off the furnace so that the growth zone is quenched within the furnace atmosphere. Then, by observing the interface
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 181
between single and polycrystalline regions, the average thickness of the grown crystal is determined (see figure 7.12). Examining the striation lines in the crystals presented in fig.7.12, one can see that the striation lines follow closely the shape of the evolving growth interface. This information was extremely beneficial for our study in terms of comparing the evolution of the numerically computed growth interfaces with experiments (see figure 6.14 in previous chapter). Agreement between experimental and simulation results are quite good. As mentioned earlier there are no external reasons for the presence of growth striations in our crystals. However, one may think of three possible contributing sources: (i) the continuous change in the temperature field during growth, (ii) the convection in the liquid zone, and (iii) the constitutional supercooling near the growth interface. The continuous change in the temperature field is due to the thermal character of the growth crucible, and may contribute to the fluctuations in the concentration field. Secondly, since the convection is very strong at the beginning, although it gets weaker as the growth progresses, it may also cause continuous temperature fluctuations leading to concentration fluctuations in the solution. Finally, during the growth process, silicon is preferentially consumed at the growth front. This may, although very small, further contribute to the fluctuations in the concentration field. These fluctuations in the concentration field might consequently have led to growth striations in the grown crystals. One interesting point to note is that the spacing between striation lines is wider at the early stages of the growth (where the growth velocity is faster), and it gets narrower as the growth interface moves closer to source material (where the growth velocity is slower). As can be seen in figure 7.12, the initially concave growth interface gradually becomes flatter as the interface moves up. In the middle region, the interface is almost flat and then becomes convex as it approaches the source material. This change in the curvature of the growth interface is due to the change in the temperature field in the solution as growth
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 182
progresses. The temperature field changes mainly because of the effect of the annular ceramic insulator on the thermal profile as well as because of the increase in the thickness of the solid region. The interface displacement was measured at the centre and at the edge points of the grown crystals for a given time.
LPD-16 (Growth time: 2 hours, average growth thickness: 2mm )
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 183
LPD-15(Growth time: 7 hours, average growth thickness: 7.5 mm )
LPD-14 (Growth time: 14 hours, average growth thickness: 11.8 mm)
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 184
LPD-18 (Growth time: 28 hours, average growth thickness: 17.7 mm )
Figure 7.12: Vertical cross-section of four single crystals for determining the growth velocity Figure 7.13 presents the interface displacement versus growth time at the center (r=0) and edge points (r=r, near the wall) on the grown crystals. As can be seen, the growth velocity is not uniform along the interface, and also is not constant along the growth direction. For the growth interface to become flatter, and later convex as the crystal grows, the central region of the crystal has to grow faster than the edges. The both variations in the crystal growth velocity were explained in the previous chapter in detail by comparing experimentally and numerically obtained results.
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 185
Figure 7.13: Interface displacement versus growth time for the center and edge regions of the grown crystals Figure 7.14 and 7.15 present the silicon composition distribution along both the axial and radial directions measured by EPMA and EDX. It can be concluded from the figures that as the growth proceeds, the concentration of silicon becomes higher because of higher growth interface temperature and the fact that the growth interfaces moves to the silicon source. The variation in silicon concentration in the radial direction is due to the shape of the growth interface. As the growth interfaces gets flatter, so does the radial silicon distribution.
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 186
a-) Radial silicon distribution at 2 mm vertical steps, measured by EPMA
b-) Corresponding axial distribution at various radial steps, measured by EPMA.
c-) Radial silicon distribution at various vertical steps, measured by EDX.
d-) Corresponding axial distribution at various radial steps, measured by EDX
Figure 7.14: Radial silicon distribution for LPD-5 along with corresponding axial distribution. Figure 7.14 and 7.15 also clearly reveals the evolution of the growth interface shape during the growth process since radial concentration distribution can be considered as reverse image of the thermal interface shape. The region with homogeneous radial
Chapter 7 –The Growth of Compositionally Graded SixGe1-x Single Crystals by LPD 187
concentration distribution offers possibility of extracting wafers to be used as a substrate material (seed). Since the grown crystals are almost axisymmetric in silicon composition distribution, the EDX measurements on LPD-19 was performed for only the left half side of the vertical cross-section.
Figure 7.15: Silicon concentration distribution for LPD-19; radial silicon distribution at various axial steps (left), corresponding axial silicon distribution for different radial locations.
7.6 Summary This chapter presented the experimental part of the current research program. In the context of the present work, a complete LPD growth system was designed, built and tested. A total number of ten compositionally graded crystals of 25 mm in diameter have been grown to show that the growth procedure is repeatable. The single crystallinity up to 6-8 percent atom silicon was achieved. Solidification behaviour of germanium rich silicon germanium solid solidification was investigated by measuring the growth velocity and performing characterization.
188
CHAPTER 8 Conclusion 8.1 Conclusions In this thesis, a combined experimental and modeling study for the growth of Si x Ge1− x single crystals by Liquid Phase Diffusion (LPD) has been presented. A complete
LPD experimental set-up that allows the repeatable production of Si x Ge1− x single crystals of 25 mm in diameter for use as lattice-matched substrates was established in the Crystal Growth Laboratory, at University of Victoria. Grown crystals were characterized by macroscopic and microscopic examinations after chemical etching to discern the degree of single crystallinity and growth striations. Compositional distributions of silicon on selected crystals were determined by performing EPMA and EDX. It was shown that LPD can efficiently and successfully be used to grow Si x Ge1− x with high degree of single crystallinity and uniform radial composition distribution up to 6-8 % at.Si. Since LPD crystal growth process involves fluid flow, heat and mass transfer, and their interaction, a rational continuum mixture model was developed to study transport phenomena in detail during the growth process by using thermomechanical balance laws and irreversible thermodynamics. Based on the continuum model developed, two-and three-dimensional transient numerical simulation models were conducted. Simulation
Chapter 8-Conclusion
189
results are given for temperature distribution within the entire growth domain, and for flow and concentration fields in the liquid domain. The numerical models presented here give explanations for some important physical features of the LPD growth process of SiGe: (1) effect of growth zone design on the thermal field, (2) effect of buoyancy induced flow field on growth and transport mechanism, and (3) the variation of crystal growth velocity as a function of time and space. The initial and the progressing growth interface shapes evolve in accordance with the shape of thermal isotherms near the interfaces. Therefore, the shape of growth interface can be controlled by changing heat transfer characteristics of the system. Regarding to buoyancy induced convection on transport and growth mechanism, the numerical simulation results show that the buoyancy induced natural convection in the liquid zone is strong at the initial stages of the growth process, but it gets weaker and then becomes unobservable after 5.2 hours of growth due to the formation of a stabilizing axial density gradient. The complete suppression of natural convection makes the LPD growth of SiGe diffusion dominated. Therefore, corresponding solute distribution in the liquid is very stable, leading to a very stable growth. As for the growth velocity, it varies in both radial and axial directions. Due to the variation of growth velocity in radial direction, the shape of the growth interface changes from concave to flat and then to convex form as the growth proceeds. Regarding the growth velocity in axial direction, it is faster at early hours of growth and, it decreases nonlinearly as the growth interface gets closer to the silicon source. Numerically obtained growth velocities are in very good agreement with those of experiments. Therefore, the simulated evolution of the growth interface agrees with experimental observations. In conclusion, in the light of a comparison between experimental and numerical findings, one can conclude that the two-and three-dimensional models successfully simulate LPD growth process of SiGe.
Chapter 8-Conclusion
190
8.2 Contributions The research presented in this thesis is unique due to being a combined experimental and theoretical study. During the course of this research program, following contributions were made to the field of crystal growth. •
A fully operational LPD growth set-up was established. The growth set-up tested by growing several SiGe bulk single crystals with composition up to to 6-8 % at.Si.
•
The work presented in this dissertation is a significant step in pawing a way to develop a novel hybrid growth technique, which combines the LPD and LPEE growth process in a single process to grow SiGe single crystals with uniform composition in one attempt.
•
The numerical model presented in this dissertation is a significant contribution in terms of being the first two-and three-dimensional transient models, which study LPD growth process of SiGe in detail.
8.3 Future work •
By using a furnace with operation capability up to 1500 oC, after some slight modifications on the current experimental set-up, it is possible to grow SiGe single crystals from the silicon rich side of the binary phase diagram, which is highly feasible for device applications.
•
In this study, we have only attempted to grow SiGe single crystals. However, the experimental set-up developed has a flexibility to be modified to grow seed substrate materials for other semiconductor materials, which are not available in the market commercially such as GaInSb.
Chapter 8-Conclusion
•
191
In crystal growth process, a flat or slightly convex initial growth interface is favourable. The current LPD process gives a concave initial growth interface. It might be possible to force the growth interface to be flat or convex by changing the heat transfer characteristics of the system. This can be achieved by increasing the heat transfer rate from the bottom of the crucible through using a dummy conductive block.
•
Due to its simplicity in terms of its operation and growth components, LPD growth process appears to be good candidate to perform crystal growth experiments in space to investigate solidification behaviours of semiconductor materials under no gravity condition.
•
The current three-dimensional transient numerical model can be further developed to study the effect of stationary and rotating magnetic field to suppress the natural convection at the initial stage of the LPD growth process and also to induce a controlled mixing to have better solute distribution in the binary liquid mixture, respectively.
•
The numerical model can be easily modified to simulate other crystal growth processes such as LPEE, THM, and Vertical Bridgman.
192
Bibliography [1]
E.R. Johnson, and S.M. Christian, “Some properties of germanium- silicon alloys,” Physical Review, vol. 95, pp. 560-561,1954.
[2]
D.J. Paul, “Silicon-Germanium strained layer materials in microelectronics,” Advanced Materials, vol.11 (3), pp. 191-204,1999.
[3]
D.J. Paul, “The physics, materials and devices of silicon germanium technology,” Physics World, vol. 13(2), pp. 27-32, 2000.
[4]
D.J. Paul, “Silicon germanium heterostructures in electronics: -The present and the future,” Thin Solid Films, vol. 321, pp. 172-180,1998.
[5]
J.D. Cressler, “Re-Engineering Silicon: Si-Ge heterojunction bipolar technology. IEEE Spectrum, pp. 49-55, 1995.
[6]
Y.J. Mii, Y.H. Xie, E.A. Fitzgerald, D. Monroe, F.A. Thiel, B.E. Weir, and L.C. Feldman, “Extremely high electron mobility in Si / Si1− x Gex structures grown by molecular beam Epitaxy,” Applied Physics Letters, vol. 59, pp. 1611-1613, 1991.
[7]
M. Willander, “Field-effect transistors, infrared detectors, and resonant tunnelling devices in silicon/silicon-germanium and delta-doped silicon,” Advanced Silicon and Semiconductor Silicon-Alloy Based Materials and Devices, Institute of Physics Publication, pp. 251-282, 1994.
[8]
F. Schaffler, D. Tobben, H.-J. Herzog, G. Abtreiter, and B. Hollander, “Highelectron mobility Si / SiGe heterostructures: influence of relaxed SiGe buffer layer,” Semiconductors Science Technology, vol. 7 .pp. 260-266, 1992.
[9]
M. Jutzi, and M. Berroth, “ SiGe -based photodetectors for optical communications,” Properties of Silicon Germanium and SiGe : Carbon, edited by Erich Kasper and Klara Lyutovich, p. 342, 2000.
Bibliography
193
[10]
A. Spleet, T. Zinke, B. Schuppert, K. Petermann, and et al., “Integrated optoelectronic waveguide-detectors in SiGe for optical communications,” SPIE, vol. 2550, pp. 224-234.
[11]
G. Bremond, A. Daami, A. Laugier, W. Seifert, M. Kittler, and et al., “ SiGe thin films structures for solar cells,” Mat.Res.Soc.Symp.Proc., vol. 485, pp. 43-48, 1998.
[12]
K. Said, J. Poortmans, M. Caymax, and et al., “High quality relaxed SiGe epitaxial layers for solar cell application,” Thin Solid Films, vol. 337, pp. 85-89, 1999.
[13]
S.A. Healy, and M.A. Green, “Efficiency enhancements in crystalline silicon solar cells by alloying with germanium,” Solar Energy Materials and Solar Cells, vol. 28, pp. 273-284,1992.
[14]
K. Nakajima, S. Kodama, S. Miyashita, G. Sazaki, and S. Hiyamizu, “Growth of Ge -Rich Si x Ge1− x single crystal with uniform composition ( x = 0.02 ) on a compositionally graded crystal for use as GaAs solar cells,” Journal of Crystal Growth, vol. 205, pp. 270-276,1999.
[15]
A.E. Fitzgerald, Y.-H. Xie., D. Monroe, P.J. Silverman, M.J. Kuo, A.R. Kortan, and et al., “Relaxed Gex Si1− x structures for III-V integration with Si and high mobility two-dimensional electron gases in Si ,” Journal of Vacum Science and Technology B, vol. 10(4), pp. 1807-1819, 1992.
[16]
C.M. Bhandari, and D.M. Rowe, “Silicon-Germanium alloys as high-temperature thermoelectric materials,” Contemporary Physics, vol. 21, pp. 219-242,1980.
[17]
C.B. Vining, “A model for the high-temperature transport properties of heavily doped n-type Silicon-Germanium alloys,” Journal of Applied Physics, vol. 69, pp. 331-341, 1991.
[18]
G.A. Slack, and M.A. Hussain, “The maximum possible conversion efficiency of Silicon-Germanium thermoelectric generators,” Journal Applied Physics, vol. 70, pp. 2694-2718, 1991.
[19]
M.L. Kozhukh, I.N. Belokurova, S.B. Vahrushev, A.N. Titkov, and I.L. Shul’pina, “Study of Ge − Si solid solution crystals by X-Ray topography and neutron diffraction,” Nuclear Instruments and Methods, vol. 213, pp. 483487,1983.
Bibliography
194
[20]
R. Madar, E. Mastromatteo, A. Magerl, K.D. Liss, C. Bernard, “Low pressure chemical vapor deposition of massive Si1− x Ge x gradient crystals and applications in short wavelength diffractions,” Surface and Coatings Technology, vol. 54/55, pp.229-233, 1992.
[21]
J. Schilz and V.N. Romanenko, “Review: Bulk growth of Silicon-Germanium solid solutions,” Journal of Materials Science: Material in Electronics, vol. 6, pp. 265-279, 1995.
[22]
Y.H. Luo, J. Wan, R.L. Forrest, J.L. Liu, M.S. Goorsky and K.L. Wang, “ Highquality strain-relaxed SiGe films grown with low temperature Si buffer,” Journal of Applied Physics, vol. 89, pp. 8279-8283, 2001.
[23]
P.W.J. Hsu, A.E. Fitzgerald, Y.-H. Xie, P.J. Silverman, and M.J. Cradillo, “Surface morphology of related Gex Si1− x films,” Applied Physics Letters, vol. 61, pp. 1293-1295, 1992.
[24]
M.I. Alonso, and E. Bauser, “Growth of Si1− x Ge x on silicon by Liquid-Phase Epitaxy, Journalof Applied Physics, vol. 62, pp. 4445-4449,1987.
[25]
A.M. Sembian, I. Silier, K. Davies, A. Gutjahr, K. Lyutovich, and et al., “ Surface morphology of LPE SiGE layers grown on (100)Si substrates,” Mat. Res. Soc. Symp. Proc., vol. 485, pp. 19-24,1998.
[26]
A.E. Fitzgerald, Y.-H. Xie,.M.L. Green, and et al., “Totally relaxed Gex Si1− x layer with low threading dislocations densities grown on Si substrates,” Applied Physics Letters, vol. 59, pp. 811-813,1991.
[27]
P. Bhattacharya, J. Singh, and E. Gularl, “ Epitaxial growth and characterization of Si1− x Ge x materials and devices,” Final Technical Report, October 15, 1994October 14, 1997, Solid State Electronic Laboratory, Department of Electrical Engineering and Computer Science, The University of Michigan, December 26, 1997.
[28]
S.B. Samavedam, and E.A. Fitzgerald, “ Novel dislocation structure and surface morphology effects in relaxed Ge / Si and Ge (graded)/ Si structures,” Journal of Applied Physics, vol.81, pp. 3108-3116,1997.
[29]
K. Ismail, O.J. Chu, and B.S. Meyerson, “High hole mobility in SiGe alloys for device applications,” Applied Physics Letters, vol. 64, pp. 3124-3126.1994.
Bibliography
195
[30]
B. S. Meyerson, K.E. Ismail, D.L. Harame, F.K. LeGoues, and J.M.C. Stork., “UHVCVD Growth of Si / SiGe heterostructures and their applications,” Semiconductors Science and Technology, vol. 9, pp. 2005-2010,1994.
[31]
N. V. Abrasimov, S. N. Rossolenko, V. Alex, A. Gerhardt, and W. Schroder, “Single crystal growth of Si1− x Ge x by the Czochralski technique,” Journal of Crystal Growth, vol. 166, pp. 657-662, 1996.
[32]
N. V. Abrasimov, S. N. Rossolenko, W. Thieme, A. Gerhardt, and W. Schroder, “Czochralski growth of Si -and Ge -Rich SiGe single crystals”, Journal of Crystal Growth, vol. 174, pp.182-186, 1997.
[33]
A. Matsui, I. Yonenaga, and K. Sumino, “ Czochralski growth of bulk crystals of Ge1− x Si x alloys”, Journal of Crystal growth, vol. 183, pp. 109-116, 1998.
[34]
M. Kurten, J. Schilz, “Czochralski growth of Si x Ge1− x single crystals”, Journal of Crystal Growth, vol. 139, pp. 1-5, 1994.
[35]
I. Yonenaga, and M. Nonaka, “Czochralski growth of bulk crystals of Ge1− x Si x alloys II. Si -Rich alloys”, Journal of Crystal Growth, vol. 191, pp.393398, 1998.
[36]
I. Yonenaga, A. Matsui, S. Tozawa, K. Sumino and T. Fukuda, “Czochralski growth of Ge1− x Si x alloy crystals”, Journal of Crystal growth, vol. 154, pp. 275279, 1995.
[37]
J. Wollweber, D. Schulz, and W. Schroder, “ Si x Ge1− x single crystals grown by the RF-heated Float Zone Technique”, Journal of Crystal growth, vol. 163 , pp. 243248, 1996.
[38]
P. Dold, A. Barz, S. Recha, K. Pressel, M. Franz, K.W. Benz, “Growth and characterization of Ge1− x Si x ( x ≤ 10 at % ) single crystals. Journal of Crystal growth, vol. 192, pp. 125-135, 1998.
[39]
A. Dahlen, A. Fattah, G. Hanke, and E. Karthaus, “Bridgman and Czochralski growth of Ge-Si alloy crystals”, Crystal Research Technology, vol. 29, pp. 187198,1994.
[40]
K. Kadakuro, and Y. Takano, “Germanium-Silicon single crystal growth using an encapsulant in a silica ampoule” Journal of Crystal growth, vol. 171, pp. 5660,1997.
Bibliography
196
[41]
D. Bliss, B. Demczyk, A. Anselmo, and J. Bailey, “Silicon Germanium bulk alloy growth by Liquid Encapsulated Zone Melting”, Journal of Crystal growth, vol. 174, pp. 187-193, 1997.
[42]
T. Honda, M. Suezawa, and K. Sumino, “Growth and characterization of bulk SiGe single crystal”, Jpn Journal of Applied Physics, vol. 35, pp. 5980-5985, 1996.
[43]
H. Sheibani, S. Dost, S. Sakai, and B. Lent, “Growth of bulk single crystals under applied magnetic field by Liquid Phase Electroepitaxy”, Journal of Crystal Growth,vol. 258, pp. 283-295, 2003.
[44]
H. Sheibani, Y.C.Liu, S.Sakai, B.Lent, and S.Dost, “ The effect of applied magnetic field on the growth mechanisms of Liquid Phase Epitaxy”, International Journal of Engineering Science, vol. 41, pp. 401-415, 2003.
[45]
Y. Azuma, N. Usami, T. Ujihara, G. Sazaki, Y. Murakami, S. Miyashita, K. Fijiwara, .K. Nakajima, “Growth of SiGe bulk crystals with uniform composition by directly controlling the growth temperature at the crystal-melt interface using in situ monitoring system”, Journal of Crystal growth, vol. 224, pp. 204-211, 2001.
[46]
R. W.Olesinski and G. J. Abbaschian. Bull. Alloy Phase Diagrams, vol. 5, pp. 180-183, 1984.
[47]
J. P. Dismukes, L. Ekstrom, and R.J. Paff, “ Lattice parameter and density in Germanium-Silicon alloys. Journal of Physical Chemistry, vol. 68 , pp. 30213027, 1964.
[48]
H-J. Herzog, “Crystal structure, Lattice Parameters and Liquidus – Solidus Curve of the SiGe system”, Properties of Silicon Germanium and SiGe : Carbon, edited by Erich Kasper and Klara Lyutovich, p. 45, 2000.
[49]
M. Jager, “Ordering in SiGe Alloys”, Properties of Silicon Germanium and SiGe : Carbon, edited by Erich Kasper and Klara Lyutovich, p.50, 2000.
[50]
R. Braunstein, A. R. Moore, and F. Herman, “Intrinsic optical absorption in Ge-Si alloys”, Physical Review, vol. 109, pp. 695, 1958.
[51]
F. Herman, “Speculations on the energy band structure of Ge—Si alloys”, Physical Review, vol. 95, pp. 847-848, 1954.
[52]
J. Weber, and M. I. Alonso, “Near-Band-Gap photoluminescence of Si-Ge alloys”, Physical Review B, vol. 40, pp. 5683-5693, 1989.
Bibliography
197
[53]
C. G Van de Walle, “Strain effects on the valence band structure of SiGE. properties of silicon germanium and SiGe:Carbon”, Edited by Erich Kasper and Klara Lyutovich, p.135, 2000.
[54]
C. G Van de Walle, “ Strain effects on the conduction band structure of SiGe. properties of silicon germanium and SiGe:carbon”, Edited by Erich Kasper and Klara Lyutovich, p. 135, 2000.
[55]
R. People, “Indirect band gap of coherently strained Ge x Si1− x bulk alloys on silicon substrate”, Physical Review B, vol. 32, pp.1405, 1985.
[56]
C. Penn,.T. Fromherz and G. Bauer, “Energy gaps and band structure of SiGe and their temperature dependence”, Properties of Silicon Germanium and SiGe:Carbon, Edited by Erich Kasper and Klara Lyutovich, p. 125, 2000.
[57]
R. H. Deitch, S. H. Jones and T. G. Digges, “Growth of large diameter silicongermanium monocrystals”, Material Research Society Symposium, vol. 587, pp. O5.7.1-O5.7.5, 2000.
[58]
R. H. Deitch, S. H.Jones and T. G.Digges, “Bulk single crystal growth of Silicon-Germanium” Journal of Electronic Materials. Vol. 29 , pp.1074-1078, 2000.
[59]
I. Yonenaga, Y. Murakami, “Segregation during the seeding process in the Czochralski Growth of Ge1− x Si x alloy”, Journal of Crystal Growth, vol. 191, pp. 399-404, 1998.
[60]
W. A. Tiller, K. A. Jackson, J. W. Rutter, and B. Chalmers, “ The redistributions of solute atoms during the solidifications of metals”, Acta Metallurgica, vol. 1 pp. 428-437, 1953.
[61]
G. K. Azhdarov, T. Kucukomerlioglu, A. Varilci, A. Altunbas, A. Kobya, and P.G. Azhdarov, “Distribution of components in Ge-Si bulk single crystals grown under continuous feeding of the melt with the second Ccomponent (Si)”, Journal of Crystal Growth, vol. 226, pp. 437-442, 2001.
[62]
A. Croll, N. Salk, F.R. Szofran, S.D. Cobb, and M.P. Volz, “Wetting angles and surfaces tension of Ge1−x Si x melts on different substrate materials”, Journal of Crystal Growth, vol. 242, pp. 45-54, 2002.
[63]
N. Kaiser, A. Croll, F.R. Szofran, S.D. Cobb, and K.W. Benz, “Wetting angles and surfaces tension of Ge melts on different substrate materials”, Journal of Crystal Growth, vol. 231, pp. 448-457, 2001.
Bibliography
198
[64]
P.S. Ravishankar, “Liquid Encapsulated Bridgman (LEB) Method for directional solidification of silicon using calcium chloride”, Journal of Crystal Growth, vol. 94, pp. 62-68, 1989.
[65]
M.P. Volz, M. Schweizer, N. Kaiser, S.D. Cobb, L. Vujisic, S. Motakef, and F.R. Szofran, “Bridgman growth of detached GeSi crystals”, Journal of Crystal Growth, vol. 237,pp.1844-1848, 2002.
[66]
T.A. Campell, M. Schweizer, P. Dold, A. Croll, and K.W. Benz, “Float zone growth and characterization of Ge1−x Si x ( x ≤ 10 at.% ) single crystals”, Journal of Crystal Growth, vol. 226, pp. 231-239, 2001.
[67]
T. Sukegawa, M. Izawa, H. Katsuno, and A. Tanaka, “Growth of GeSi thick alloy layers by the Yo-Yo solute feeding method”, Journal of Crystal Growth, vol. 99, pp.274-277, 1990.
[68]
M. Kumagawa, A.F. Witt, and et. al.,“Current-controlled and dopant modulation in Liquid Phase Epitaxy”, Journal of Electrochemical Society, vol. 120, pp. 583584, 1973.
[69]
A. Abul-Fadl, and E.K. Stefanakos, “Current controlled LPE Growth of InP”, Journal of Crystal Growth, vol. 39, pp. 341-345, 1977.
[70]
D. J. Lawrance, and L. F. Eastman, “Electric current controlled growth and doping modulation in GaAs Liquid Phase Epitaxy”, Journal of Crystal Growth, vol. 30, pp. 267-275, 1975.
[71]
L. Jastrzebski, H.C. Gatos, and A.F. Witt, “Electromigration in current-controlled LPE”, Journal of Electrochemical Society, vol.123, pp. 1121-1122, 1976.
[72]
L. Jastrzebski, J. Lagowski, H.C. Gatos, and A.F. Witt, “Liquid Phase Electroepitaxy: Growth kinetics”, Journal of Applied Physics, vol. 49, pp. 59095919, 1978.
[73]
J. J. Daniele, “Peltier-induced LPE and composition stabilization of GaAlAs”, Applied Physics Letters, vol. 27, pp. 373-375, 1975.
[74]
J. J. Daniele, and A.J. Hebling, “Peltier-induced LPE and composition control of mm-thick layers of (Al,Ga)As”, Journal of Applied Physics, vol.52, pp. 43254327, 1981.
[75]
T. Bryskiewicz, C. F. Boucher, J. J. Lagowski, and H. C. Gatos, “Bulk GaAs crystal growth by Liquid Phase Electroepitaxy”, Journal of Crystal Growth, vol. 82 ,pp. 279-288, 1987.
Bibliography
199
[76]
S. Dost and Z. Qin, “ A model for Liquid Phase Electroepitaxy under an external magnetic field; I. Theory”, Journal of Crystal Growth, vol. 153, pp. 123-130, 1995.
[77]
Y. Malmejac, P. Desre, and E. Bonnier, “Contribution a letude du diagramme de phases ternaire Ge-Si-Sb”, Mem.Sci.Rev.Metallurg, vol. 69, pp. 565-577, 1972.
[78]
D. T. J. Hurle, and E.R. Pike, “Computation of ternary phase diagrams from binary data: Immiscibility in the Ge-Bi-Si System”, Journal of Materials Science, vol. 1, pp. 399-402, 1966.
[79]
G. B. Stringfellow, and P.E. Greene, “A Quasi-chemical equilibrium calculation of the Ge-Si-Sn and Ge-Si-Sb ternary phase diagrams”, Journal of Electrochemical Society, vol. 117, pp. 1075-1079, 1975.
[80]
E.A. Guggenheim, “Mixtures; the theory of the equilibrium properties of some simple classes of mixtures solutions and alloys”, Clarendon Press, Oxford, 1952.
[81]
J.W. Matthews, “Defects associated with the accommodation of misfit between crystals”, Journal of Vacuum Science and Technology, vol. 12, pp. 126-132, 1975.
[82]
J, Poortmans, S. C. Jain, J. Nijs, and V.R. Overstraeten, “Material properties of strained SiGe layers”, Advanced Silicon and Semiconducting Silicon-Alloy Based Materials and Devices, Institute of Physics Publisher, p.185, 1994
[83]
R. Hull, “Equilibrium theories of misfit dislocation networks in SiGe/Si systems”, Properties of Silicon Germanium and SiGe:Carbon, edited by Erich Kasper and Klara Lyutovich, INSPEC, p.9, 2000.
[84]
D. E. Jesson, “Strain-induced morphological evolution of SiGe thin films”, Properties of Silicon Germanium and SiGe:Carbon, edited by Erich Kasper and Klara Lyutovich, INSPEC, p.3, 2000.
[85]
A. A. Darhuber, V. Holy, J. Stangl, G. Bauer, J. Nutzel, and G. Abstreiter, “Interface roughness in strained Si/SiGe multilayers”, Mat.Res.Soc.Symp.Proc. vol. 448, p.153, 1997.
[86]
S. Y. Shiryaev, F. Jensen, and J.W. Petersen, “On the nature of cross-hatch patterns on compositionally graded Gex Si1− x alloy layers”, Applied Physics Letter, vol. 64, pp. 3305-3307, 1994.
[87]
M. Ya. Bakirov, R. S. Madatov, and Yu.M. Mustagaev, and R.B. Gasymov, “PV cells on solute-solution Ge − Si single crystals”, Applied Solar Cell Energy, vol. 22, pp. 9, 1986.
Bibliography
200
[88]
E. Kasper, and H. Presting, “Device concept for SiGe optoelectronics”, SPIE, vol.1361, pp.302-312, 1990.
[89]
H. Kimura, H. Winston, D.J. O’Connor, J.A. Henige, “ Silicon-Germanium alloys for infrared detection”, SPIE Infrared Detector Materials, vol. 285, pp.155-163, 1981.
[90]
R. Baron, M.H. Young, H. Kimura, and H. Winston, “Silicon-rich Si-Ge alloys for infrared detectors-material properties”, SPIE Infrared Detector Materials, vol. 285, pp.164-174, 1981.
[91]
T. L. Lin, T. George, E. W. Jones, A. Ksendzov, “Elemental boron-doped p + − SiGe layers grown by Molecular Beam Epitaxy for infrared detector applications”, Applied Physics Letters, vol. 60, pp. 380-382, 1992.
[92]
J. Konle, H. Presting, H. Kibbel, K. Thonke, R. Sauer, “ Enhanced performance of silicon based photodetectors using Silicon/Germanium nanostructures”, SolidState Electronics, vol. 45, pp.1927-1925, 2001.
[93]
R. Strong, D.W. Greve, R. Misra, M. Weeks, and Pellegrini, “ GeSi infrared detectors”, Thin Solid Films, vol. 294, pp.343-346, 1997.
[94]
S. Vyas, D.W. Greve, T.J. Knight, R.M. Strong, and S. Mahajan, “Growth of epitaxial Ge x Si1− x for infrared detectors by UHV/CVD”, Vacuum, vol. 46, pp.1065-1069, 1995.
[95]
J. D. Cressler, “Silicon-Germanium heterojunction bipolar transistor”, Device and Circuit Cryogenic Operation for Low Temperature Electronics. Kluwer Academic Publishers, Netherlands, pp.69-84, 2001.
[96]
H. Presting, “Challenges for SiGe -heterotechnology”, Mat. Res. Soc. Symp. Proc., vol. 379, pp. 417-432, 1995.
[97]
A. Konig, “SiGe RR-electronics: Devices, circuits, competitors, markets”, Mat. Res. Soc. Symp. Proc.,vol. 533, pp.3-17, 1998.
[98]
G. Schuppener, T. Harada and et. al., “A 23 G Hz low noise amplifier in SiGe heterojunction bipolar technology”, 2001 IEEE Radio Frequency Integrated Circuits (RFIC) Symposium: Digest of Papers, p.177, 2001.
[99]
G. Niu, J. D. Cressler, S. Zhang, A. J. Joseph and D. Harame, “Noise-gain tradeoff in RF SiGE HBTs”, 2001 Topical Meetings on Silicon Monolithic Integrated Circuits in RF systems, p.187, 2001.
Bibliography
201
[100] M. Regis, M. Borgarino, L. Bary, O. Llopis, J. Graffeuil, L. Escotte, U. Koenig and R. Plana, “Noise behavior in SiGe devices”, Solid-State Electronics, vol. 45, pp. 1891-1897 ,2001. [101] K. B. Schad, U. Erben, E. Sonmez, P. Abele and H. Schumacher, “A Ku Band SiGe low noise amplifier”, 2000 Topical Meetings on Silicon Monolithic Integrated Circuits in RF systems, p. 52, 2000. [102] G. Niu, S. Zhang, J. D. Cressler, A. J. Joseph and et.al., “Noise parameter modeling and SiGe profile design tradeoffs for RF applications”, 2000 Topical Meetings on Silicon Monolithic Integrated Circuits in RF systems, p.9, 2000. [103] B. V. Haaren, A. Gruhle, M. Regis, C. Mahner, L. Escotte and et al., “HBT SiGe technology dedicated to ultra low phase noise applications”, 1997 Workshop on High Performance Electron Devices for Microwave and Optoelectronic Applicatios, p. 291, 1997. [104] R. M. Bowen, “Theory of Mixtures”, Continuum Physics, Volume III-Mixtures and Electromagnetic Field Theories edited by A. C. Eringen, Academic Press, 1976. [105] C. Truesdell, and R. A.Toupin, Handbuch der Physik, Vol.3/1, edited by S. Flugge, Springer Verlag, Berlin, 1960. [106]
R. B. Bird, W. E. Stewart, E. N. Lightfoot, Transport Phenomena, Second Edition. John Wiley and Sons Inc., New York, 2002.
[107] A. C. Eringen, and G. Maugin, Electrodynamics of Continua, Vol.I, SpringerVerlag, New York, 1989. [108] A. C. Eringen, Mechanics of Continua, John Wiley and Sons, Inc. New York, London, 1967. [109] J. C. Slattery, Interfacial Transport Phenomena, Springer-Verlag, 1990. [110] A. C. Eringen, “A mixture theory of electromagnetism and superconductivity”, International Journal of Engineering Science, vol. 36, no.5/6, pp.525-543, 1998. [111] A. C. Eringen, “Micropolar mixture theory of porous media”, Journal of Applied Physic, vol. 94, no. 6, pp. 4184-4190, 2003. [112] S. Dost. and H. A. Erbay, “A continuum model for Liquid Phase Electroepitaxy”, International Journal of Engineering Science, vol. 33 , pp. 1385-1402, 1995.
Bibliography
202
[113] S. Dost, H. Sheibani, Y. C. Liu, and B. Lent: “On the high growth rates in Electroepitaxial growth of bulk semiconductor crystals in magnetic field”, J. Crystal Growth, (in press). [114] V. Timchenko, P. Y. P.Chen, G. de Vahl Davis, E. Leonardi, and R. Abbaschian, “A computational study of binary alloy solidification in the Mephisto experiment”, International Journal of Heat and Fluid Flow, vol. 23 , pp. 258268, 2002. [115] T. Nishinaga, Journal of Crystal Growth, vol.237-239, pp.1410-1417, 2002. [116] Y. Liu, Z. R. Zytkiewicz, S. Dost, “A model for Epitaxial Lateral Overgrowth of GaAs by Liquid Phase Electroepitaxy”, Journal of Crystal Growth, 265, pp.341350, 2004. [117] Metals Handbook, Volume 2, and 10th edition. Properties and Selection: Nonferrous Alloys and Special Purpose Materials. ASM international, The materials information society, 1990. [118] Landon-Bornsten (LB), Semiconductors: Physics of Group IV Elements and III-V Compounds, vol. 17, edited by O. Modelung. Springer-Verlag, Berlin, 1982. [119] E. Yamasue, M. Susa, H. Fukuyama, and K. Nagata, “Thermal conductivities of silicon and germanium in solid and liquid states measured by non-stationary hot wire method with silica coated probe”, Journal of Crystal Growth, vol. 234, pp. 121-131, 2002. [120] V. Palankovski, R. Schultheis, and S. Selberherr, “Simulation of power hetorojunction bipolar transistors on Gallium Arsenide”, IEEE Transactions on Electron Devices, vol. 48, No.6, 2001. [121] H. Nakanishi, K. Nakazato, S. Asaba, K. Abe, S. Maeda, “Temperature dependence of density of molten germanium measured by a newly developed archimedian technique”, Journal of Crystal Growth, vol. 191, pp. 711-717, 1998. [122] S. Yesilyurt, L. Vujisic, S. Motakef, F. R. Szofran, and M. P. Volz, “A numerical investigation of the effect of thermoelectromagnetic convection ( TEMC) on the Bridgman growth of Ge1-xSix”, Journal of Crystal Growth, vol. 207, pp. 278-291, 1999. [123] Supplier provided properties. www.gequartz.com/en/fig11.htm.
Bibliography
203
[124] H. Nakanishi, K. Nakazato, K. Abe, S. Maeda, K. Terashima, “Temperature dependence of the molten germanium and silicon measured by a newly developed archimedian technique”, Journal of Crystal Growth, vol. 203, pp. 75-79, 1999. [125] S. Kimura, K. Terashima, “ A review of measurement of thermophysical properties of silicon melt”, Journal of Crystal Growth, vol. 180, pp. 323-333, 1997. [126] CFX-4.3 Solver Manual. AEA Technology plc, Didcot. UK, 2000. [127] P. M. Adornato, R. A. Brown, “Convection and segregation in directional solidification of dilute and non-dilute alloys: Effects of ampoule and furnace design”, Journal of Crystal Growth, vol. 80, pp.155-190, 1987. [128] H.-P. Trah, “Liquid Phase Epitaxy in the ternary system Si-Ge-Bi”, Journal of Crystal Growth, vol. 102, pp. 175-182, 1990. [129] F. Banhat, and A.Gutjahr,“Stress relaxation in SiGe layers on oxide-patterned Si substrates”, Journal of Applied Physics, vol. 80, pp. 6223-228, 1996.
204
Appendix A Summary of Literature Review on Crystal Growth Methods for SiGe Alloy System In this section, we briefly summarize the experimental works reported in literatures for the growth SiGe alloy system by various growth techniques. Table A.1: Summary of SiGe growth by Cz technique
Reference/ Composition [31] Si1− xGex 0 < x < 0.1 [32] Si1− xGex 0 < x < 0.15 Ge1− x Six 0 < x < 0.3
Outcomes
[34] Si1− xGex 0.02 < x < 0.2
Single and poly-crystals of 4-6 mm in diameter and 15-25 mm in length. EPD: 102-105 per cm2. Concentration striations are observed. Axial compositional variation up to 4 at % silicon depending on the melt composition.
Single crystal up to 48 mm in diameter. High dislocation density, rotational striations and non-uniform concentration distribution along the growth.
0 < x < 0.15 ; Single crystal (48 mm in diameter). 0 < x < 0.3 ; Single crystal for up to 0.1 silicon content. Non-homogeneous concentration distribution along the growth direction.
205
Appendix A
[35] SixGe1− x 0.86 < x < 1
[36] SixGe1− x 0 < x < 0.64
[59] SixGe1− x 0 < x < 0.15
[57,58] Si1− xGex , 0 < x < 0.17
Single crystals; 27-50 mm in length, 13-27 mm in diameters. Oxygen impurities originated from the reaction between silicon and quartz crucible. Growth striations are observed. Dislocation densities in the order of 104 per cm2 . Crystals with compositions corresponding to the intermediate region of the phase are partly single crystal, whereas those from either silicon or germanium-rich side are single crystal. For some crystals, twins are observed at the interface. Crystal dimensions; 6-12 mm in diameter, and 4-60 mm in length. Increase in the diameter of the solidified crystal shows steeper concentration gradient development. Crystals with non-uniform concentration distributions are obtained. The density of grown-in dislocations is in the order of 104-105 cm-2. Single crystals are obtained from a melt of a low silicon content or pure germanium using silicon seed. At the interface, growth striations, high dislocation densities and cracks are observed due to the lattice mismatch. Steep concentration non-uniformities in the grown crystals. Crystal dimensions: 30 mm in length and diameter (non constant diameter). Single and poly crystals. Unintentional doping (varying amount of carbon, oxygen, boron, and phosphorous) due to the contamination. Most of the single crystals are obtained by using (100) seeds. 63 mm diameter crystal is grown, but 50 mm central region is only single crystal. Dislocation density in the order of 5* 104 per cm2. Non-uniform crystal diameters (mainly tapered crystals). Rotational striations and non-uniform concentration distribution.
Table A2: Summary of SiGe growth by VB technique.
Reference/ Composition
[38 ] SixGe1− x 0 < x < 0.10
Outcomes
Cracks in the middle of the crystal due to the sticking problem of the grown crystal to the quartz ampoule. Profound axial non-uniformity in silicon composition. Uniform radial concentration. Average EPD (etch pit density) in the order of 6*104-1*105 cm-2 (augmented at the seedcrystal interface, which might be due to back melting and lattice mismatch). Single crystal of 9mm in diameter and 30-40 mm in length
206
Appendix A
[40] SixGe1− x 0 < x < 0.40
[65] SixGe1− x 0 < x < 0.12
Among studied encapsulant material, only CaCl2 suppressed the erosion and devitrification of silica ampoules; however, crystallization (devitrification) of silica ampoule to cristobalite was not avoided due to the diffusion of calcium ion above 1000 oC. CaCl2 did not contaminate the melt. Good temperature stability and in turn, no striation was observed. The average dislocation density in the order of 104-105 cm-2. Non-uniform silicon distributions along the axial direction. Partially single crystal, 63 mm in length and 13 mm in diameter Single crystals of 12 mm in diameter including detached and undetached regions were grown.
Table A3: Summary of SiGe growth by FZ technique
Reference/ Composition
[37] SixGe1− x 0.78 ≤ x < 1
[66] SixGe1− x x ≤ 10
Outcomes
With germanium feeding technique, single crystallinity is maintained up to 22 at % Ge. EPD is in the order of 105-106 per cm2. Axial non uniformity in germanium composition due to the germanium feeding. Slightly convex interface shape. Pronounced striation in the grown crystals. By using pre-grown SiGe seed materials, dislocation free crystals are obtained up to 5.5 at % Ge. Single crystals were grown with the sizes; 80-100 mm in length. 13-32 mm in diameter. Axial non-uniformity in silicon composition Average EPD (etch pit density) in the range of 7*103-2*104 cm-2 (enhanced at the seedcrystal interface, 2.4*104-2.9*104 cm-2). Boron doping levels in the range of 1-2*1017 at %cm-3 (boron is incorporated into the melt from the pBN crucible during the pre- synthesizing). Lack of wall contact (due to being a containerless growth technique) reduces the defect densities in grown crystals. Single crystals with maximum growth length 44 mm, and diameter 8 mm. Table A.4: Summary of SiGe growth by LPE technique
Reference/ Composition [24] S1− x Ge x up to x = 0.75
Outcomes
Conventional tipping and sliding boat system were used. Indium was employed as a solvent material. Thickness of epitaxial layers was in the range of 20 mµ. S1− x Ge x alloys up to x = 0.75 were reported.
Appendix A
[128] S1− x Ge x x = 0.98
[129] S x Ge1− x x = 0.86
[67] Ge x Si1− x x = 0.24
[25] Ge x Si1− x up to x = 0.13
207 Conventional tipping boat system was used. Bismuth was selected to be a solvent material because it is a suitable solvent for the growth of germanium-rich epitaxial layers. The growth experiments were performed on (111) and (100) silicon substrates. The growth process was carried out by using conventional tipping boat system with indium as a solvent material. Silicon was used as a seed material. At Ge contents above 10 at. % misfit dislocations were generated at the growth interface. Critical thickness depends on the germanium content. For example, for 10 % percent germanium, the critical thickness was reported to be 0.8 mµ. A novel technique called solute feeding yo-yo LPE was used for the growth of silicon rich epitaxial layers on silicon substrate by using Tin as a solvent material. The thickness of the epitaxial layers is 800 µm . This is the thickest relaxed epitaxial layers reported for the S1− x Ge x alloy system by LPE. Ge x Si1− x alloys of different germanium content up to 13 at. % was grown by using a tipping boat system. Silicon substrates (100) were used. Indium was used as a solvent material. Layer thicknesses were around 20-30 mµ.
208
Appendix B Derivations of Model Equations for Rational Continuum Mixture Model B.1 Introduction Appendix A includes detailed derivations of the transport theorem and local forms of thermomechanical balance laws together with associated jump conditions for a volume
including a discontinuity surface. The local forms of balance laws are first derived for each constituent, and then by summing these laws over number of species, these laws are obtained for a multicomponent mixture.
B.2 Transport Theorem for an Arbitrary Field Let a material body of initial volume V enclosed by a surface S in the reference state (t=0) be transferred into volume V (t ) with its boundary S (t ) after motion takes place. Let us evaluate the material time derivative of a function integrated over the volume where both the integrand ϕdυ and integration limits are functions of time.
Appendix B D ϕ dυ Dt V ∫(t )
209
(B.2)
It is important to realize that without taking into account the movement of the boundaries, it would not simply be possible to bring the time derivative inside the integral since the shape of the volume changes in time. The only condition to bring the total time derivative inside the integral is to fix the limits of the integration. At a given instant during the motion of the continuum, the volume element dυ of V (t ) is related to the volume element dυ 0 of V by dυ = Jdυ 0
(B.2)
Thus,
D D D(ϕ J ) dυo ϕ dυ = ϕ Jdυo = ∫ ∫ ∫ Dt V (t ) Dt V Dt V D DJ ⎞ ⎛ Dϕ +ϕ ϕ dυ = ∫ ⎜ J ⎟dυ0 ∫ Dt V (t ) Dt Dt ⎠ V⎝
(B.3)
where dυ and dυ 0 are volume elements in the reference state and at time t, respectively (refer to [108] for derivation). ϕ is an arbitrary field, which might be a tensorial, vectorial or scalar quantity. By the use of formulation for the material time derivative of Jacobian DJ = Jvk ,k Dt
it follows from equation (B.3) that
(B.4)
Appendix B D ⎞ ⎛ Dϕ ⎞ ⎛ Dϕ ϕ dυ = ∫ ⎜ + ϕ v k ,k ⎟dυ + ϕ v k ,k ⎟ Jdυ 0 = ∫ ⎜ ∫ Dt V (t ) Dt Dt ⎠ ⎠ V⎝ V (t ) ⎝
210
(B.5)
By employing the expression for the material time derivative of the field ϕ , Dϕ ∂ϕ = + ϕ ,k v k , equation (B.5) becomes ∂t Dt D ⎞ ⎛ ∂ϕ ϕ dυ == ∫ ⎜ + (ϕv k ),k ⎟dυ ∫ Dt V (t ) ∂t ⎠ V (t ) ⎝
(B.6)
where v k are the components of velocity of a material particle on the boundary of the volume. By use of the Green-Gauss theorem for an arbitrary field φ , defined as
∫φ
V
,k
dυ = ∫ φ nk da
(B.7)
S
and then setting φ ,k = (ϕv k ),k , equation (B.6) becomes
D ∂ϕ ϕ dυ = ∫ dυ + ∫ ϕ vk nk da ∫ Dt V (t ) ∂t V (t ) S (t )
(B.8)
where da is a surface element, and n k is an exterior unit normal to the surface S (t ) (volume boundary). The integrand of the volume integral on the RHS gives time rate of the change of the field within the volume, whereas, the second integrant of the surface integral shows the net flux of the field ϕ due to the motion of the boundary.
Appendix B
211
The result obtained in equation (B.8) can be extended for a body of volume V (t ) , which includes a singularity, namely, discontinuity surfaces. As illustrated in figure (B.1), the volume V (t ) swept by the discontinuity surface σ ( t ) moving with an absolute velocity of u k is considered to be composed of two distinct volumes, namely, V enclosed by S
+
+
(t ) and
V
−
(t )
(t ) + σ + (t ) and S − (t ) + σ − (t ) , respectively. n
V n
n−
σ
u
S
+
σ(t )
+
σ V
n+
+
−
−
n
S
−
Figure B.1. Discontinuity surface Applying the result obtained in (B.8) to these two volumes separately, one can write
∂ϕ D ϕ dυ = ∫ dυ + ∫ ϕv k nk da + ∫ ϕ + u k nk+ da ∫ ∂t Dt V + (t ) σ + (t ) V + (t ) S + (t ) D ∂ϕ ϕ dυ = ∫ dυ + ∫ ϕv k nk da + ∫ ϕ − u k nk− da ∫ Dt V − (t ) ∂t V − (t ) S − (t ) σ − (t )
(B.9)
Summing these two equations side by side (noting that n k+ = − n k− = − n k ) and letting σ + (t ) and σ − (t ) approach σ (t ) , we can obtain
D ∂ϕ ϕ dυ = ∫ dυ + ∫ ϕv k nk da − ∫ ϕu k nk da ∫ Dt V (t )−σ (t ) ∂t V (t )−σ (t ) S (t )−σ (t ) σ (t )
(B.10)
Appendix B
212
where V (t )- σ (t ) = V
+
(t ) + V − (t )
and S (t )- σ (t ) = S
+
(t ) + S − (t )
respectively denote
the volume and surface excluding points lying on the discontinuity surface σ ( t ) . The symbol
indicates the jump of the enclosed quantities across the discontinuity surface;
for instance, ϕ = ϕ + − ϕ − , where ϕ + and ϕ − are the values of ϕ on the positive and negative sides of σ ( t ) . By employing the Green-Gauss theorem for a body including a discontinuity surface
∫φ
V -σ
,k
dυ + ∫ φ nk da = σ
∫ φ n da k
(B.11)
S -σ
in the second term on the right hand side of equation (B.10), one can write D ⎞ ⎛ ∂ϕ ϕ dυ = ∫ ⎜ + (ϕv k ),k ⎟dυ + ∫ ϕ (v k − u k ) n k da ∫ Dt V (t )−σ (t ) ∂t ⎠ V (t )−σ (t ) ⎝ σ (t )
(B.12)
Equation (B.12) is the well-known transport theorem for a body containing a discontinuity surface. The transport theorem allows one to evaluate the material time derivative of the volume integral of any quantity in the presence of a discontinuity surface within a volume. From onward, the time dependence of integration limits will not be explicitly indicated for the notational convenience.
B.3 Derivations of Thermomechanical Balance Laws and Associated Jumps B.3.1 Conservation of Mass The integral form of the conservation of mass for the α th constituent is given by
Appendix B Dα Dt
213
α α ∫ ρ (x ,t )dυ = ∫ rˆ (x ,t )dυ
V
(B.13)
V
In the presence of a discontinuity surface in the fixed spatial volume V , by using the argument in equation (B.9), equation (B.13) becomes
Dα Dt
α α α ∫σρ (x ,t )dυ = ∫σrˆ (x,t )dυ + σ∫( ˆ)s (x ,t )da
V−
V−
(B.14)
t
where it is assumed that, on the discontinuity surface, mass is supplied to the α th constituent from the other constituents. The term ˆsα (x ,t ) is called mass production and gives the rate at which the mass of the α th constituent is produced on the discontinuity surface. It must be noted that some of the terms which might be appearing in the
formulations are not defined as such rˆ α if they have been introduced in the main body of the text. Using the transport theorem given in equation (B.12) and replacing ϕ with ρ α and vk with vkα
Dα Dt
α ∫ ρ dυ =
V −σ
⎛ ∂ρ α ⎞ ⎜⎜ + (ρ α v kα ),k ⎟⎟dυ + ∫ ρ α (v kα − u k ) nk da ∫ ∂t ⎠ V −σ ⎝ σ (t )
(B.15)
It follows that
⎛ ∂ρ α ⎞ ⎜⎜ + (ρ α v kα ),k ⎟⎟dυ + ∫ ρ α (v kα − u k ) nk da = ∫ rˆ α dυ + ∫ ˆs α da ∫ ∂t ⎠ σ (t ) σ (t ) V −σ ⎝ V −σ
(
)
⎛ ∂ρ α ⎞ ⎜⎜ + (ρ α v kα ),k − rˆ α ⎟⎟dυ + ∫ ρ α (v kα − u k ) nk − ˆs α da = 0 ∫ ∂t ⎠ V −σ ⎝ σ (t )
(B.16)
Appendix B
214
The localization principle states that if the mass is conserved within the entire material body, it must also be conserved in any part of the body. This is only satisfied on the condition that the integrands must be equal to zero. Thus,
∂ρ α + ρ α vkα ,k = rˆ α ∂t ρ α vkα − u k nk = ˆs α
(
)
(
(B.17)
)
An alternative form of (B.17)1 can be derived as ∂ρ α + ρ ,αk vkα + ρ α vkα,k = rˆ α ⇒ ρ ′α + ρ α vkα,k = rˆ α ∂t4243 1 Dα ρ α = ρ ′α Dt
Γ ′ − Γ& = Γ ,k vkα ⇒ ρ ′α = ρ& α + ρ& − ρ vk ,k + (ρ vk α
α
α
α
α
)
,k
ρ ,αk vkα 12 3
(
= − ρ α v kα,k + ρ α v kα
α α
+ ρ vk ,k = rˆ
(B.18)
)
,k
α
ρ α vkα,k = ρ α (vkα − vk ),k = ρ α vkα,k − ρ α vk ,k
ρ& α − (ρ α vkα,k − ρ α vk ,k ) + (ρ α vkα ),k + ρ α vkα,k = rˆ α ρ& α + ρ α vk ,k = rˆ α − (ρ α vkα ),k
ρ α = ρcα ⇒ ρ& α = ρ&cα + ρc&α
ρ&c + ρc& + ρc vk ,k = rˆ − (ρ vk α
α
α
α
α
α
)
,k
⎞ ⎛ cα ⎜ ρ& + ρvk ,k ⎟ + ρc&α = rˆ α − (ρ α vkα ),k ⎜1 424 3⎟ =0 ⎠ ⎝ ⎞ ⎛ ⎛ ∂cα ⎞ ⎟ ⎜ ρc&α = rˆ α − ⎜ ρ α vkα ⎟ = ρ ⎜⎜ + c,αk vk ⎟⎟ + jkα,k = rˆ α 3⎟ ⎜ 1=2 ⎝ ∂t ⎠ ⎝ j kα ⎠ ,k
The derivation of an alternative form of (B.17)2 follows as
(B.19)
215
Appendix B ⎛
⎞
ρ α ⎜⎜ v{kα − u k ⎟⎟ nk = ˆs α ⇒ ρ α (vkα + vk − u k ) nk = ˆs α ⎜ =(v α +v ) ⎟ ⎝ ⎠ k
k
(B.20)
⇒ { ρ α (vk − u k ) + ρ α vkα nk = ˆs α ⇒ ρcα (vk − u k ) + jkα nk = ˆs α 123 = ρcα
= jkα
⎛ ∂cα ⎞ + cα,k vk ⎟⎟ + jkα,k = rˆα ⎝ ∂t ⎠ α ρc (vk − uk ) + jkα nk = ˆsα
ρ ⎜⎜
(B.21)
In the following, we introduce the derivation of an identity relating the material time derivative of a function Γ α (x ,t ) , following the motion of particle X α of α th species, with that of a function Γ (x ,t ) , following the motion of the mass centered particle (fictitious particle).
Γ (x ,t ) =
1
N
N
ρ α Γ α (x ,t ) = ∑ cα Γ α (x ,t ) ∑ ρα α =1
=1
N ⎛ DΓ α (x ,t ) DΓ (x ,t ) Dcα = ρ ∑ ⎜⎜ cα + Γ α (x ,t ) ρ Dt Dt Dt α =1 ⎝
ρΓ& = ∑ (ρcα Γ& α + Γ α ρc&α ) N
⎞ ⎟⎟ ⎠
α =1
Γ ′ − Γ& = Γ ,k vkα ⇒ Γ& α = Γ ′α − Γ ,kα vkα
ρΓ& = ∑ (ρcα (Γ ′α − Γ ,kα vkα ) + Γ α ρc&α ) N
α =1
ρΓ& = ∑ (ρcα (Γ ′α − Γ ,kα vkα ) + Γ α (rˆα − (ρ α vkα ),k )) N
α =1
ρΓ& = ∑ (ρ α Γ ′α − ρ α Γ ,kα vkα + Γ α rˆα − Γ α (ρ α vkα ),k ) N
α =1
ρΓ& = ∑ (ρ α Γ ′α − (ρ α Γ α vkα ),k + Γ α rˆα ) N
α =1
(B.22)
216
Appendix B
ρΓ& = ∑ (ρ α Γ ′α − (ρ α Γ α v kα ),k + Γ α rˆ α ) N
(B.23)
α =1
B.3.2 Balance of Linear Momentum Before starting the derivation of the balance of linear momentum, some mathematical operators and the surface divergence theorem need to be defined. Let x be an arbitrary vector field and n be a unit normal to a plane. To find the scalar component of x in the unit normal direction, we can write. x ⋅ n = xk i k ⋅ nl i l = xk nlδ kl = xk nk
(B.24)
If the above result (B.24) is multiplied by the unit normal vector, the vector in the direction of unit normal is obtained as
(x ⋅ n )n = xk nk nl i l
(B.25)
To find the tangential component of the vector x , indicated by p , we can write p = x - (x ⋅ n )n ⇒ pl i l = xl i l − x k nk nl i l
pl = (δ kl − nk nl )xk 14243
(B.26)
Pkl
where Pkl is projector or also known as projection tensor and when it is operated on a vector field, it gives the tangential component of the vector field. To introduce the surface divergence theorem, we will use the well-known Stokes’ theorem which is given as
217
Appendix B
∫ u ⋅ tds = ∫ (∇ × u )⋅ nda
∂σ
(B.27)
∂σ
If we represent the arbitrary vector field u in the form of a × n where a is a vector field, and n is the unit normal to the surface of the interface, we can write
)3 ⋅ h ds = ∫ (∇ × (a × n )) ⋅ nda ∫ (1a4×2n4
∂σ a ⋅(n×h )= −a ⋅b
(B.28)
σ
where b is binormal, h is tangent vector, and ∂σ is the closed curve bounding the discontinuity surface σ as illustrated in figure B.2. S
σ
n
∂σ
ds
h b
Figure B.2: Representative figure for unit normal, binormal and tangent vector. After performing following manipulations, ε
∇ × (a × n ) = (
rpm }
), p i p × (1 ak i k × nl i l ) = ε klm (ak nl ), p i p × i m = ε klm ε pmr (ak nl ), p i r ⇒ 4243 123
q k nl ε klm i m
(ak nl ), p (δ krδ lp − δ kpδ lr )i r = (ar n p ), p − (a p nr ), p i r
ε pmr i r
1 424 3
δ kr δ lp −δ kp δ lr
⎞
⎛
⎟ (ar , p n p + ar n p , p − a p , p nr − a p nr , p )i r ⋅ n ⇒ ⎜⎜ ar , p n p nr + ar n p , p nr − a p , p n{ r nr − a p nr , p nr ⎟ 123 ⎜ ⎝
⎞ ⎛ ⎟ ⎜ Ωar nr = −(∇ s ⋅ a + 2Ωa ⋅ n ) ⎜ ar , p n p nr − δ pr + ar n p , p nr ⎟ = − ar , p Ppr − 21 3 { ⎟ 14243 123 424 ⎜ 2 Ωa ⋅n − 2Ω − ∇ ⋅ a P pr s ⎠ ⎝
(
)
1
0
⎟ ⎠
(B.29)
218
Appendix B we can obtain ⎛ ⎞ ⎜ ⎟ a ⋅ bds = ⎜ ∇ s ⋅ a + 2Ωa ⋅ n ⎟da 123 ⎟ ∂σ σ ⎜ P ⋅∇a ⎝ ⎠
∫
∫
(B.30)
where Ω is mean curvature. Equation (B.30) is referred to as the surface divergence theorem. It is shown in reference [109] that the surface divergence theorem holds the same for a tensor field, denoted by A . Hence,
∫ A ⋅ bds = ∫ (∇ s ⋅ A + 2Ω A ⋅ n )da
∂σ
(B.31)
σ
Let A be expressible in the form of Pϕ where ϕ is a scalar quantity, function of spatial variables. ⎛
⎞
⎜ σ⎝
{⎟ 0 ⎠
⎜ ⎟ ∫ ϕ P ⋅ bds = ∫ ∇ s ⋅ ϕ P + 2Ωϕ P ⋅ n da
∂σ
(B.32)
1 }
where P ⋅ n =0 since (I - nn ) ⋅ n = n - n (n ⋅ n ) . The first term on the right hand side is dealt as follows
219
Appendix B
(
)
P ⋅ ∇(ϕ P ) = Pkl i k i l ⋅ (ϕPst ), p i p i s i t = Pkl (ϕPst ), p δ kpδ ls i t = Pkl ϕ , p Pst + ϕPst , p δ kpδ ls i t ⎛ ⎞ Pps Pst = δ ps − n p ns (δ st − ns nt ) = ⎜ δ pt − n p nt − n p nt + n p ns ns nt ⎟ = δ pt − n p nt = Ppt { ⎟ ⎜ 1 ⎝ ⎠
(
)
(
(
)(
)
)
Pptϕ , p i t + ϕPps Pst , p i t = Pptϕ , p i t - ϕ δ ps − n p ns ns , p nt + ns nt , p i t = Pptϕ , p i t − ϕn p , p nt { 14444 4244444 3 - (n s nt ), p n p , p nt + n p nt , p − n p n s n s , p nt − n p n s n s n t , p { 123
(B.33)
1
0
P ⋅ ∇(ϕ P ) = ∇ s (ϕ P ) = Pptϕ , p i t − ϕ n p , p nt = ∇ sϕi t + 2Ωϕnt i t 1442443 { − 2Ω
we will use this
where ∇(n ⋅ n ) = 0 since n ⋅ n = 1 . The proof follows that ∇(n ⋅ n ) = (∇n ) ⋅ n + (∇n ) ⋅ n = 2(∇n ) ⋅ n = 0 ⇒ (∇n ) ⋅ n = 0
(B.34)
( ),k nl i k i l ⋅ nmi m = nl ,k nmδlm i k = nl ,k nl i k = 0 To summarize,
∫ ϕ P ⋅ bds = ∫ (∇ sϕ + 2Ωϕn )da
∂σ
(B.35)
σ
If we replace the field ϕ in equation (B.35) by the surface tension γ ασ for the α th constituent, the integral form of the balance of linear momentum for a volume intersected by a discontinuity surface is given by Dα Dt
∫ρ
V-σ
α α
vl dυ =
∫ (ρ
α α
(
))
∫
α bl + ˆplα + rˆ α vlα dυ + t kl nk da +
V-σ
S-σ
∫ (∇ sγ
ασ
)
+ 2Ωγ ασ nl da
(B.36)
σ
Following the same argument which has resulted in equation (B.15), and then setting
ϕ = ρ α vlα and vk = vkα , one can write
220
Appendix B Dα Dt
α α ∫ ρ vl dυ =
V −σ
⎛ ∂ (ρ α vlα ) ⎞ ⎜⎜ + (ρ α vlα vkα ),k ⎟⎟dυ + ∫ ρ α vlα (vkα − uk ) nk da ∫ ∂t ⎠ V −σ ⎝ σ (t )
(B.37)
with some considerable mathematical manipulations
(
) (
⎛ ∂ ρ α vlα ⎞ ⎜⎜ + ρ α vlα vkα ,k ⎟⎟dυ = ∫ ∂t ⎠ V −σ ⎝ ⎛ ⎞ ⎜ ⎟ ⎛ ⎞ ⎜ ⎜ α ⎟ α α α ⎟ ⎛ α α α D vl ⎜ vlα ⎜ ∂ρ + ρ α vkα ⎟ + ρ α ⎛⎜ ∂vl + vlα,k vkα ⎞⎟ ⎟dυ = ˆ ⎜ + ρ v r ,k ∫ ⎜ ⎜ 1∂4 ∫⎜l ⎟⎟ ⎜ ∂t Dt t 42443 ⎟⎟ ⎠ ⎝144244 V −σ V −σ ⎝ ⎜ 3 ⎜ ⎝ ⎟ α = rˆ ⎠ Dvα ⎜ ⎟ = l Dt ⎝ ⎠ α α ⎛ α α Dα α α α D vl ⎞ ˆ ⎟⎟dυ + ∫ ρ α vlα vkα − u k nk da ⎜ ρ υ ρ = + v d v r l ∫ ⎜ l Dt Dt V ∫−σ ⎠ V −σ ⎝ σ (t )
(
)
)
(
⎞ ⎟⎟dυ ⎠
(B.38)
)
Applying the Green-Gauss theorem in (B.11) to the second term on the RHS in (B.36),
∫σt
α kl
nk da =
S-
∫σt
V-
α kl ,k
dυ +
∫t σ( )
α kl
nk da
(B.39)
t
On comparison of (B.36), (B.38)2 and (B.39), one can write D α vlα ρ = t klα ,k + ρ α blα + ˆplα Dt ρ α vlα (vkα − uk ) − tklα nk = 2γ ασ Ωnl α
(B.40)
where in obtaining (B.40)2, it is assumed that surface γ ασ tension is independent of position.
221
Appendix B
To obtain the balance of the linear momentum for the mixture as a whole, some important mathematical identities are introduced as follows. By equation (B.23) for Γ = vl , we can write
ρv&l = ∑ (ρ α vl′α − (ρ α vlα v kα ),k + vlα rˆ α ) N
(B.41)
α =1
An alternative form of (B.41) can be derived as
ρv&l = ∑ (ρ α vl′α − (ρ α vkα vlα ),k + vlα rˆ α ) N
(B.42)
α =1
after the following steps given (B.43)
(
)
vlα = vlα − vl ⇒
ρv&l = ∑ (ρ α vl′α − (ρ α vkα vlα + ρ α vkα vl ),k + vlα rˆ α ) + ∑ vl rˆ α N
N
α =1
α =1
(B.43)
1 424 3 =0
N
(
(
ρv&l = ∑ ρ α vl′α − ρ α vkα vlα α =1
)
,k
α α
+ vl rˆ
)
⎛ ⎞ ⎜ α α ⎟ − ∑ ⎜ ρ vk vl ⎟ 123 ⎟ α =1 ⎜ = j kα ⎝ ⎠3 ,k 144244 N
=0
On summing equation (B.40)1 over α and then using (B.42), the balance of linear momentum can be obtained as
222
Appendix B N Dα vlα = ∑ tklα ,k + ρ α blα + ˆplα ρ ∑ Dt α =1 α =1 N
(
α
)
∑ ρ α vl′α = ρv&l + ∑ (ρ α vkα vlα ),k − ∑ vlα rˆ α N
N
α =1
N
α =1
α =1
ρv&l + ∑ (ρ α vkα vlα ),k − ∑ vlα rˆ α = ∑ (tklα ,k + ρ α blα + ˆplα ) N
N
N
α =1
α =1
α =1
=tkl 6447 448
(B.44)
ρv&l = ∑ (tklα − ρ α vkα vlα ) + ∑ ( ˆplα + vlα rˆ α ) + ∑ ρ α blα N
α =1
N
N
=1 α =14 α4 1 4244 3 1 24 3
,k
= ρbl
=0
ρv&l = tkl ,k + ρbl + ∑ ( ˆplα + vlα rˆ α ) N
α =14 1 4244 3 =0
where body force per unit mass and stress tensor for the mixture are defined as
b=
1
N
ρ α bα ; b ∑ ρα
l
(
=1
=
1
N
ρ α bα ∑ ρα l
)
=1
t = ∑ t − ρ v ⊗ v ; tkl = ∑ (tkl − ρ vk vl N
α =1
α
α
α
α
N
α
α
α
α
)
(B.45)
α =1
Whenever two vectors are written side by side without the symbol ⊗ , it will be understood that they are multiplied dyadically, namely, ab = a ⊗ b . Defining the interface stress tensor as
N
t = ∑t σ
α =1
ασ
N
, tkl = ∑ tkl σ
we can proceed as
α =1
ασ
(B.46)
223
Appendix B
ρ α vlα (vkα − uk ) − tklα nk = 2γ ασ Ω nl , tkl = ∑ (tklα − ρ α vkα vlα ) N
N
∑ α =1
α =1
ρ α vlα (vkα − uk ) − tkl − ρ α vkα vlα nk = ∑ 2γ ασ Ω nl N
α =1
14243
ρ α v α v α =∑ ρ α (vα − v )(vα − v ) ∑ α α N
2γ σ Ω n l
N
k
=1 N
l
k
=1
k
l
l
∑ ρ α (vkα − vk )(vlα − vl ) = ∑ ρ α vkα vlα − ∑ ρ α vkα vl − ∑ ρ α vk vlα + ∑ ρ α vk vl
α =1
N
N
α =1
α =1
N
N
(B.47)
=1 =1 14243 α1 4243 α1 4243 ρv k v l ρv k v l ρv k v l 1444444244444 43 − ρv k v l
N
N
ρ α v α v α = ∑ ρ α v α v α − ρv v ∑ α α =1 N
∑ α =1
k
l
=1
k
l
k l
⇒
ρ α vlα vkα − ρ α vlα uk − tkl + ρvk vl − ρ α vlα vkα = ρvl (vk − uk ) − tkl nk = 2γ σ Ω nl
The local form of the balance of linear momentum for the mixture in V − σ , and on σ reads as
ρv&l = tkl ,k + ρbl in V − σ ρvl (vk − uk ) − tkl nk = 2γ σ Ω nl on σ
(B.48)
B.3.3 Balance of Moment of Momentum The integral form of the balance of moment of momentum for the α th constituent is introduced as
Dα Dt Dα Dt
∫ (ρ
α
vα ) × xdυ =
V-σ
∫ρ
V-σ
∫ ((ρ
V-σ
α α
vl xm elmn dυ =
α
∫ ((ρ
V-σ
ˆ α + rˆα vα ) × x + m ˆ α )dυ + bα + p
∫ t ( ) × xda α
n
S-σ
ˆ n )dυ + ∫ tkl nk xm elmn da bl + ˆpl + rˆ vl )xm elmn + m
α α
α
α α
α
α
S-σ
(B.49)
Appendix B
224
On using the transport theorem in equation (B.12) and then replacing ϕ with ρ α vlα xm and vk with vkα , we can write
elmn
Dα Dt
α α ∫ ρ vl xm dυ = elmn
V −σ
⎛ ∂ρ α vlα x m ⎞ ⎜⎜ + (ρ α vlα x m v kα ),k ⎟⎟dυ ∫ ∂t V −σ ⎝ 14444 4244444 3⎠ ρα
(B.50)
⎞ D α vlα x m α ⎛ ∂ρ α + vl x m ⎜⎜ + ρ α v kα ,k ⎟⎟ Dt ∂t ⎝14 442444 3⎠
(
)
ˆrα
elmn
Dα Dt
∫ρ
α α
vl xm dυ =
V −σ
∫
V −σ
α α ⎛ α D vl xm ⎞ ⎜⎜ elmn ρ ⎟⎟ dυ + ∫ elmn vlα xm rˆα dυ Dt ⎠ ⎝144 V −σ 42444 3 elmn ρ α x m
(B.51)
D α v lα Dα xm + ρ α elmn v lα Dt 14424Dt 4 3 =0
With the usage of the Green-Gauss theorem in (B.11), it follows that
α ∫ elmn tkl xm nk da =
S -σ
⎛ ⎞ ⎜ e t α x + e t α x ⎟dυ m ,k ∫ ⎜ lmn kl ,k m lmn kl { ⎟ V -σ δ mk ⎠ ⎝
(B.52)
Comparison of equation (B.49)2, (B.51), (B.52) and (B.40)1 suggests that α ˆ nα = 0 elmntml +m
(B.53)
Summing (B.53)over α , and using equation (B.45)2 and then neglecting the second order terms in diffusion velocities, it can be seen that the stress tensor for the mixture is symmetric, namely
t ml = tlm
(B.54)
Appendix B
where
225
N
ˆα =0. m ∑ α =1
n
In passing, note that the jump equation for the balance of moment of momentum is not included since it will not be used in the current model. If its derivation is desired to derive, it can be done so, following the same argument leading to the other jump balance equations.
B.3.4 Balance of Energy The integral form of the balance of energy for the α th constituent in a fixed volume V of discontinuity surface σ is given by Dα Dt +
∫
V −σ
∫ (vl
ασ ασ
σ
⎞ ⎠
⎛1 ⎝2
ρ α ⎜ vlα vlα + ε α ⎟dυ = γ ,l + 2Ωγ
ασ ασ
vl
)
∫ (ρ
V −σ
α α α
)
bl vl + ˆplα vlα dυ +
∫ tkl vl nk da α α
S −σ
⎛ α α ⎞ ⎛1 ⎞ ⎜⎜ ρ h + rˆ α ⎜ vlα vlα + ε α ⎟ + εˆ α ⎟⎟dυ − qkα nk da nl da + ⎝2 ⎠ ⎠ V −σ ⎝ S −σ
∫
∫
(B.55)
In flowing, we introduce the treatment terms inside each integral. The terms inside the third integral on the right hand side of equation (B.55) can be obtained as
Appendix B
226
v ⋅ γ P ⋅ bds = ∫ (∇ s ⋅ (v ∫1 44244 3 ασ
∂σ
(v
ασ
ασ
)
)
ασ
(
) )
⋅ γ ασ P + 2Ω vασ ⋅ γ ασ P ⋅ n da
σ
⋅γ ασ P ⋅b
⎛ ⎞ ⎜ ⎟ vασ ⋅ γ ασ P ⋅ bds = ⎜ ∇ s ⋅ vασ ⋅ γ ασ P + 2Ωγ ασ vασ ⋅ (P ⋅ n )⎟da 144244 3 1 2 3 ⎜ ⎟ ∂σ σ⎝ (v⋅γ P )⋅b 0 ⎠
∫
(
∫
v ⋅ γ P ⋅ bds = ∫ P ⋅ ∇(v ∫1 44244 3 ασ
∂σ
(v
(
ασ
ασ
)
)
ασ
)
⋅ γ ασ P da
(B.56)
σ
⋅γ P ⋅b
)
(
P ⋅ ∇ vασ ⋅ γ ασ P = Pkl i k i l ⋅ γvασ s Pst
)
,m i m i t
(
= Pkl γ ασ vασ s Pst
ασ ασ ασ ασ Pmt vασ vs ,m + Pmt γ ασ vασ s Pst γ ,m + Pmt Pst γ s Pst ,m = vt γ ,t
)
,m δ mk δ tl
⇒
04 6 47 8 ασ + γ ∇ s ⋅ v + 2Ωγvt nt 1 424 3 ( Incomp .)
Assuming that surface tension is independent of position ( ∇ sγ ασ = 0 ) and omitting the
(
)
contribution of the term ∇ s ⋅ vασ due to the assumption of incompressibility, the above equation simplifies to
(
)
P ⋅ ∇ vασ ⋅ γ ασ P = γ ασ vlασ 2Ωnl
(B.57)
By the same type of manipulation which produced equation (B.40)1, we can write,
⎛ α⎛1 α α ⎞ ⎞ ⎜ ∂ρ ⎜ vl vl + ε α ⎟ ⎟ ⎝2 ⎠ + ⎛⎜ ρ α ⎛ 1 vα vα + ε α ⎞vα ⎞⎟ ⎟dυ ⎜ ⎜ l l ⎟ k ∫⎜ ∂t ⎠ ⎠ ,k ⎟ ⎝ ⎝2 V −σ ⎜ ⎟ ⎝ ⎠ 1 ⎛ ⎞ + ∫ ρ α ⎜ vlα vlα + ε α ⎟(vkα − uk ) nk da ⎝2 ⎠ σ (t )
(B.58)
After some tedious mathematical manipulations, the term inside the volume integral in equation can be expressed in the following form
Appendix B
227
⎛ ⎜ α α α α ⎜ ⎛⎜ 1 vlα vlα + ε α ⎞⎟rˆ α + vlα ρ α D vl + ρ α D ε ∫⎜ 2 Dt ⎠ V −σ ⎝ 142Dt 4 3 ⎜ α α α α ˆ t + b + p ρ kl ,k l l ⎝ +
ρ ∫ σ( )
α
t
(
⎞ ⎟ ⎟dυ ⎟ ⎟ ⎠
(B.59)
)
⎛1 α α α ⎞ α ⎜ vl vl + ε ⎟ vk − u k nk da ⎠ ⎝2
With the application of the divergence theorem in (B.11), the surface integrals in equation (B.55) can be written
∫σt
∫σ(t
α α
v nk da =
kl l
S−
vα + tklα vlα,k )dυ +
α
kl ,k l
V−
α ∫ qk nk da =
S −σ
∫( ) t
α α
v nk da
kl l
σ t
α ∫ qk , k dυ +
V −σ
∫
σ (t )
(B.60)
qkα nk da
Combining equation (B.55), (B.59), and (B. 60), the local form of the energy balance in spatial form in V - σ and on σ can be represented as Dα ε α = tklα vlα,k − qkα,k + ρ α hα + εˆ α in V − σ Dt α⎛1 α α ρ ⎜ vl vl + ε α ⎞⎟(vkα − uk ) + qkα − tklα vlα nk = γ ασ vlασ 2Ωnl on σ ⎠ ⎝2
ρα
(B.61)
The derivation to obtain the balance of energy for the mixture is given below. Summing the first term (i.e. tklα vlα,k ) on the RHS of equation (B.61)1 over α and using (B.40)1, we can write
tklα (vlα,k ) = ∑ (tklα vlα,k ) + ∑ tklα vl ,k = ∑ (tklα vlα ),k − ∑ ∑ { α =1 123 α =1 α =1 α =1 α =1 N
N
N
(t
α
v l ,k + v l ,k
α α
kl v l
)
t α vα = ∑ ((t α v α ) ∑ α α N
N
kl l ,k
=1
,k
kl l
=1
,k
α
α
− t kl ,k v l
N
N
({ t ) α
kl ,k
α
α
ρ v l′ − ρ α blα − ˆp lα
− ρ α vl′α vlα + ρ α blα vlα + ˆplα vlα + tklα vl ,k )
N
vlα + ∑ tklα vl ,k α =1
(B.62)
Appendix B
228
To deal with the second term (i.e. ρ α vl′α vlα ) on the RHS of equation (B.62), the following manipulations are performed
(
)
(
vlα = vlα − vl ⇒ vl′α = vl′α − vl′
)
vl′ − v&l = vl ,k vkα ⇒ vl′ = vl ,k vkα + v&l
(B.63)
vl′α = vl′α − v&l − vl ,k vkα vl′α = vl′α + v&l + vl ,k vkα
N
∑ ρα
α =1
α
vl′α {
N
α
v l′ + v&l + v l ,k v k
N
(
vlα = ∑ ρ α vl′α vlα + ρ α v&l vlα + ρ α vl ,k vkα vlα α =1
N
N
N
ρ α v′α v α = ∑ ρ α v ′α v α + v& ∑ ρ α v α + ∑ ρ α v ∑ 123 123 α α α α =1
l
l
=1
l
l
l
/
⎛ α/ α ⎜v v ρ α vl′α vlα = ∑ ρ α ⎜ l l ∑ α =1 α =1 ⎜ 2 ⎝ N
Assuming that Γ α =
N
⎛1 ⎝2
=1
α
jl 1 424 3
vαvα = l l 2
N
l
)
v α vlα
l ,k k
=1
(B.64)
=0
⎞ N ⎟ α α α ⎟ + ∑ ρ vl ,k vk vl ⎟ α =1 ⎠
N vlα vlα , then from ρΓ = ∑ ρ α Γ α , it follows that 2 α =1
⎞ ⎠
N
⎛1 ⎝2
⎞ ⎠
1 α =1 2 N
•
ρΓ = ∑ ρ α ⎜ vlα vlα ⎟ ⇒ Γ = ∑ cα ⎜ vlα vlα ⎟ ⇒ Γ& = ∑ cα vlα vlα α =1
α =1
•
1 ⇒ ρΓ& = ρ ∑ cα vlα vlα α =1 2 N
By using the identity ρΓ& = ∑ (ρ α Γ ′α − (ρ α Γ α vkα ),k + Γ α rˆ α ) N
α =1
(B.65)
229
Appendix B •
⎛1
⎛
/
⎞
⎞
ρΓ& = ρ ∑ cα vlα vlα = ∑ ⎜ ρ α ⎜⎜ vlα vlα ⎟⎟ − (ρ α vlα vlα vkα ),k + vlα vlα rˆ α ⎟ ⎜ ⎟ 2 α =1 2 α =1 ⎝ 2 ⎝ ⎠ 2 ⎠ 1
N
N
1
• / 1 α α α N ⎛⎜ 1 α α α 1 α α α α ρ v v ρ c v ρ vl vl vk = ∑ ∑ l l l vl + ⎜ 2 2 α =1 2 α =1 ⎝
(
N
1
)
,k
−
1 α α α ⎞⎟ vl vl rˆ ⎟ 2 ⎠
(B.66)
(B.67)
Casting equation (B.67) in (B.64)3, we can obtain ⎛ 1 α •α α 1 α α α α 1 α α α ⎞⎟ N α ⎜ ′ ( ) = + − ρ v v ρ c v v ρ v v v vl vl rˆ + ∑ ρ vl ,k vkα vlα ∑ ∑ l l l l l l k ,k ⎜ ⎟ α =1 2 2 2 α =1 α =1 ⎝ ⎠ N
α
α
N
α
(B.68)
Substituting equation (B.68) into (B.62)2, it follows that
N
t α vα ∑ α
kl l ,k
=1
• ⎛ α α ⎞ ⎜ (tkl vl ),k − ρ 1 cα vlα vlα − 1 (ρ α vlα vlα vkα ),k + 1 vlα vlα rˆα − ρ α vl ,k vkα vlα ⎟ = ∑⎜ ⎟ 2 2 2 α α α α α α α =1 ⎜ ⎟ ˆ + + + ρ b v p v t v l l l l kl l ,k ⎝ ⎠ N
(B.69)
Summing equation (B.61)1 over α and then using (B.69), we can write • ⎛ α α ⎞ 1 α α α 1 α α α α 1 ⎜ tkl vl ,k − ρ c vl vl − ρ vl vl vk ,k + vlα vlα rˆ α ⎟ α D ε 2 2 2 ⎟ ∑ ρ Dt = α∑=1 ⎜ α =1 ⎜ − ρ α v v α v α + ρ α bα v α + ˆpα v α + t α v ⎟ l ,k k l l l l l kl l ,k ⎝ ⎠
α
N
α
N
(
)
N
N
N
α =1
α =1
α =1
(
)
(B.70)
− ∑ qkα,k + ∑ ρ α hα + ∑ εˆ α
The inner part of the internal energy density ε I for the mixture is introduced as
N
ρε I = ∑ ρ α ε α α =1
(B.71)
230
Appendix B Using our famous mathematical identity given in equation (B.23), we can have
ρε&I = ∑ (ρ α ε ′α − (ρ α ε α vkα ),k + ε α rˆα ) N
α =1
(B.72)
∑ ρ ε ′α = ρε&I + ∑ (ρ α ε α vkα ),k − ∑ ε α rˆα N
α
α =1
N
N
α =1
α =1
Casting the result in equation (B.72) into (B.70), one can write
ρε&I + ∑ (ρ α ε α vkα ),k − ∑ ε α rˆα N
N
α =1
α =1
• ⎞ N ⎛ α α ⎜ (tkl vl ),k − ρ 1 cα vlα vlα − 1 (ρ α vlα vlα vkα ),k + 1 vlα vlα ˆrα − ρ α vl ,k vkα vlα ⎟ = ∑⎜ ⎟ 2 2 2 α α α α α α α =1 ⎜ ⎟ ˆ b v p v t v + + + ρ l l l l kl l ,k ⎝ ⎠ N
N
N
α =1
α =1
α =1
(B.73)
− ∑ qkα,k + ∑ ρ α hα + ∑ εˆ α
After some considerable arrangement, we can get ⎛ ⎜ 1 ρ⎜ε I + 2ρ ⎜ ⎝
⎞ N ⎟ 1 α α α α ⎞ ⎛ α α α α α α ρ v v = − ⎜ qk + ρ ε vk − t kl vl + ρ vl vl vk ⎟ ∑ ∑ l l ⎟ 2 ⎠ ,k α =1 α =1 ⎝ ⎟ ⎠ • N
α
N ⎛ ⎛ vαvα + ∑ ⎜⎜ ⎜⎜ ε α + l l 2 α =1 ⎝ ⎝
α
(
α
)
(B.74)
N N ⎞ N ⎞ α ⎟⎟rˆ + ˆplα vlα + εˆ α ⎟ + ∑ t klα − ρ α vkα vlα vl ,k + ∑ ρ α hα + ∑ ρ α blα vlα ⎟ α =1 α =1 α =1 ⎠ ⎠
(
)
The inner part of the heat flux vector for the mixture is defined by q I k = ∑ (qkα + ρ α ε α vkα − t klα vlα ) N
α =1
whereas the heat flux for the mixture is defined
(B.75)
231
Appendix B
qk = q I k +
1 N α α α α ∑ (ρ vl vl vk ) 2 α =1
(B.76)
The internal heat generation rate per unit mass for the mixture is defined as
N
ρh = ∑ ρ α h α
(B.77)
α =1
The specific internal energy for the mixture is introduced as
ε = εI +
1 2ρ
N
ραvαvα ∑ α =1
l
(B.78)
l
Balance of energy for the mixture requires that ⎛ ⎛ α vlα vlα ⎜⎜ε + ∑ ⎜⎜ 2 α =1 ⎝ ⎝ N
⎞ ⎞ α ⎟⎟rˆ + ˆplα vlα + εˆ α ⎟ = 0 ⎟ ⎠ ⎠
(B.79)
Comparison of equations (B.74-B.79) implies that the balance of energy for the mixture is of the following form
N
ρε& = − qk ,k + t kl vl ,k + ρh + ∑ ρ α blα vlα α =1
B.3.4.1 Jump Energy Balance for the Mixture Summing (B.61)2 over α along with using the identities given in equation (B.81)
(B.80)
232
Appendix B
∑ (ρ N
α =1
vk vl vl ) = ∑ (ρ α (vkα + vk )(vlα + vl )(vlα + vl )) = N
α α α α
α =1
⎜ ρ α v α v α v α + 2ρ α v α v α v ∑ ⎜ α
⎞ + ρ α vkα vl vl + ρ α vk vlα vlα + 2 ρ α vlα vk vl + ρ α vk vl vl ⎟ = 1 424 3 14243 ⎟ =0 =0 ⎠
(ρ α v α v α v α + 2 ρ α v α v α v ∑ α
+ ρ α vk vlα vlα + ρ α vk vl vl )
⎛
N
k
⎝
=1
l
l
k
l
l
N
k
=1
N
l
l
k
(
t kl = ∑ t klα − ρ α vkα vlα α =1
N
∑ α =1
N
l
l
) N
t klα vlα = ∑ t klα vlα + ∑ t klα vl α =1
α =1
(ρ α vα vα vα ) = ∑ (ρ α v α v α v α + 2 ρ α v α v α v ∑ α α N
N
=1
k
l
l
k
=1
l
N
N
N
α =1
α =1
α =1
l
k
∑ qkα = qk − ∑ ρ α ε α vkα +∑ tklα vlα −
l
l
(
+ ρ α vk vlα vlα + ρ α vk vl vl
1 N α α α α ∑ ρ vl vl vk 2 α =1
)
)
(B.81)
⎞ ⎛1 ρ α ⎜ vα vα + ε α ⎟(vα − u ) = ∑ ⎠ ⎝2 α N
=1
l
l
k
k
N 1 α α α α N α α α N 1 α α α ρ v v v + ρ ε v − ρ v v u − ρ α ε α uk ∑ ∑ ∑ ∑ l l k k l l k 242443 α1 α =1 α =14 α =1 2 =1 1 4243 N
N
∑
1
α =1 2
1 2
ρ α vlα vlα uk + ρvl vl uk
ρεuk
we can have the local form of balance of energy on the discontinuity surface written in spatial form as
⎛1 ⎝2
1 α =1 2 N
⎞
ρ ⎜ vl vl + ε + ∑ cα vlα vlα ⎟(vk − uk ) + qk − tkl vl nk = γ σ vlσ 2Ωnl on σ ⎠
(B.82)
N
where surface tension for the mixture is defined as γ σ vlσ = ∑ γ ασ vlασ α =1
It is worth, in passing, to note that the jump condition for energy balance in equation (B.82) is somewhat different from the jump condition, which can be derived easily, for a
233
Appendix B
single component body. In the above equation, relative motion of particles with respect to mass averaged velocity contributes to the creation of kinetic energy in the multicomponent mixture. Ignoring, however, the higher order terms in diffusion velocities will lead to a jump equation identical to the one which can be written for a single component body.
B.3.5 Entropy Inequality In the derivation of energy balance for species and the mixture as a whole, it was assumed that at each point in the continuum model, there is a certain amount of internal energy. From the classical equilibrium thermodynamics, the internal energy of the system is function of entropy S , volume V and masses of the constituents mα . Thus, U = U (S ,V ,mα ), α = 1,2...,N
(B.83)
If the internal energy is defined per unit volume, equation (B.83) is written as
(
)
~ ~ ~ U = U S , ρ α , α = 1,2...,N
(B.84)
where the tilde symbol denotes that a quantity is expressed in per unit volume. Differentiation of the internal energy per unit volume with respect to its arguments gives ~ ~ ~ ⎛ ∂U ⎞ ~ N ⎛ ∂U ⎜ ⎟ ⎜ dU = ⎜ ~ ⎟ dS + ∑ ⎜ α α =1 ⎝ ∂ρ ⎝ ∂S ⎠ ρ α
⎞ ⎟⎟ dρ α ⎠ S~ ,ρ β ,β ≠α ,N
(B.85)
On defining temperature and mass basis chemical potential of the αth constituent, respectively, as
234
Appendix B ~ ~ ⎛ ∂U ⎛ ∂U ⎞ α θ = ⎜⎜ ~ ⎟⎟ µ = ⎜⎜ α ⎝ ∂S ⎠ ρ α ⎝ ∂ρ
⎞ ⎟⎟ ⎠ S~ ,ρ β ,β ≠α ,N
(B.86) T
T
equation (B.85) is introduced as ~ N ~ d U = θdS + ∑ µ α d ρ α
(B.87) T
T
α =1
where equation (B.87) is the well-known Gibbs equation. Defining the internal energy in equation (B.83) per unit mass (specific internal energy), we can write ~ ~ ~ ~ ~ ~ ~ ~ U = U ⎛⎜ S ,V ,cα ⎞⎟ ,α = 1,2...,N -1 ⎝ ⎠
(B.88) T
T
~ ~ ~ ~ where U has N -1 mass fractions as a independent variable, and V is specific volume and defined as ~ ~ 1 V = =
ρ
1 N
∑ρ
(B.89) T
T
α
α =1
Note that the double tilde on a quantity implies that the quantity is defined per unit mass. On differentiating equation (B.88), we can write
235
Appendix B ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ⎛⎜ ∂U ⎞⎟ ~ ⎛⎜ ∂U ⎞⎟ ~ N-1⎛⎜ ∂U ⎞⎟ dcα dU = ~ dV + ∑ dS + ~ α ~ ~ ⎜ ∂V ⎟ ~~ ⎜ ∂S ⎟ ~~ α =1 ⎜ ∂c ⎟ ~ ⎠ S~ ,V~~ ,c β ,β ≠α ,N ⎝ ⎝142⎠4 ⎝142⎠4 S ,c α c α ,V 3 3 θ
−p
(B.90) T
~ ~ ~ ~ ~ ~ ~ ~ N-1⎛⎜ ∂U ⎞⎟ dU = θdS − pdV + ∑ dcα α ⎜ ⎟ ~~ ~~ α =1 ∂c ⎝ ⎠ S ,V ,c β ,β ≠α ,N
T
where p is the thermodynamic pressure.
Equation (B.90) is another form of Gibbs equation. The Gibbs equation might be assumed to be valid locally for a system which doesn’t deviate too far from the equilibrium (quasi-equilibrium postulate). A quantity defined per unit volume is related to that defined per unit mass through the following relation ~ ~ ξ = ~ ~ V ~
ξ
(B.91) T
T
where ξ is an arbitrary quantity.
Using the relation in (B.91), equation (B.87) can be written as
~ ~ α ⎛ S~ ⎞ N ⎛ U~ ⎞ ⎟ = θd ⎜ ~ ⎟ + µ α d ⎛⎜ c~ d⎜ ~ ⎜ ~ ⎜ V~ ⎟ α∑ ⎜ V~ ⎟ ⎝V ⎝ ⎠ ⎝ ⎠ =1
⎞ ⎟⎟ ⎠
(B.92)
and after rearranging equation (B.92) and recalling the Euler’s relation, we can have
T
T
236
Appendix B Euler' s equation 644 474448 ~ ~ ~ ~ ~ N α α ⎛~ ~ N α α ⎞ dV ~ ~ ~ dU = θdS + ∑ µ dc + ⎜U − θS − ∑ µ c ⎟ ~ ~ α =1 =1 ⎝14442α4 ⎠V 4 4 3 ~
~ ~ ~ ~ ~ ~ N dU = θdS − pdV + ∑ µ α dc α
(B.93) T
~ − pV
T
α =1
Since
∑ µ α dcα = ∑ (µ α − µ N )dcα , equation (B.93)2 becomes N
N-1
α =1
α =1
P
P
~ ~ ~ ~ ~ ~ N-1 dU = θdS − PdV + ∑ (µ α − µ N )dcα 1424 3 α =1
(B.94) T
T
µ
Comparison of (B.90) with (B.94) reveals that ~ ⎛ ∂U~ ⎞ ⎜ ⎟ =µ ⎜ ∂cα ⎟ ~~ ~~ β ⎝ ⎠ S ,V ,c ,β ≠α ,N
(B.95) T
T
Taking time derivative of equation (B.93)2, we can write P
~ ~ dU =θ dt
~ ~ dρ −1 N α dcα dS −p + ∑µ dt dt dt α =1 { −
~ ~ dU =θ dt
1 dρ ρ 2 dt
P
(B.96) T
T
~ ~ dS p dρ N α dcα + 2 + ∑µ dt ρ dt α =1 dt
where θ , p and µ α are assumed to be time independent due to being evaluated around the equilibrium.
237
Appendix B
To apply the relation in equation (B.96)2 to a particle in the continuum moving with the P
P
mass averaged velocity, all total time derivative operators must be replaced with the material time derivative operators. Then, casting the mass balance for the mixture and for the αth constituent, equation (B. 96)2 can be put into the form of P
P
N ~ ρ ~ µα α ~& p ~& (rˆ − jkα,k ) U = ρS − vk ,k + ∑ θ θ θ α =1
(B.97) T
T
Let’s denote the specific internal energy and entropy with the new symbols such that ~ ~ ~ ~ U = ε I and S = η , then we can write N p ρ µα ε&I = ρη& − vk ,k + ∑ (rˆ α − jkα,k ) θ θ α =1 θ
(B.98) T
T
The local energy balance for the mixture in equation (B.74) may be written as •
1 ρ ρ h 1 N α α α α α 1 & ε = − qk ,k − ∑ c vl vl + tkl vl ,k + ρ + ∑ jl bl 2θ α =1 θ I θ θ θ θ α =1 • N N 1 1 1 ρ ρ h 1 N ε&I = − qk ,k − ∑ c&α vlα vlα − ∑ jlα vlα + tkl vl ,k + ρ + ∑ jlα blα 2θ α =1 θ θ θ α =1 θ θ θ α =1 • N N θ q ρ h 1 N 1 1 1 ⎛q ⎞ ε&I = −⎜ k ⎟ − ,k 2 k − ∑ rˆ α − jkα,k vlα vlα − ∑ jlα vlα + tkl vl ,k + ρ + ∑ jlα blα 3 θ 2θ α =1 142α 4 θ α =1 θ θ θ α =1 θ ⎝ θ ⎠ ,k N
(
)
ρc&
(B.99) T
T
Combining equation (B.98) and (B. 99) to eliminate ε&I , it follows that
ρη& −
p
θ
µα α (rˆ − jkα,k ) = θ α =1 N
v k ,k + ∑
θ q 1 ⎛q ⎞ − ⎜ k ⎟ − ,k 2 k − 2θ θ ⎝ θ ⎠ ,k
(rˆ ∑ α N
=1
α
α
)
α
α
− j k ,k vl vl −
1
θ
N
j ∑ α =1
•
α l
α
vl +
1
θ
t kl vl ,k + ρ
h
θ
+
1
θ
N
j ∑ α =1
B.100 T
α l
α
bl
T
238
Appendix B
Further arranging equation (B.100), one can write
θ ,k qk t 1 N α α h ⎛ qk ⎞ ⎟ − 2 + D kl vl ,k + ∑ jl bl + ρ θ θ θ α =1 θ ⎝ θ ⎠ ,k • 1 N 1 ⎛ ⎞ 1 N − ∑ (ˆr α − jkα,k )⎜ µ α + vlα vlα ⎟ − ∑ jlα vlα θ α =1 2 ⎝ ⎠ θ α =1
ρη& = −⎜
(B.101) T
T
By means of the following manipulation
⎡1 ⎛ 1 1 1 ⎛ ⎞ ⎡1 ⎛ ⎞⎤ ⎞⎤ jkα,k ⎜ µ α + vlα vlα ⎟ = ⎢ jkα ⎜ µ α + vlα vlα ⎟⎥ − jkα ⎢ ⎜ µ α + vlα vlα ⎟⎥ θ 2 2 2 ⎝ ⎠ ⎣θ ⎝ ⎠⎦ ,k ⎠⎦ ,k ⎣θ ⎝
1
(B.102) T
T
We may put equation (B.101) in to the form given in equation (B.103) ⎛ ⎞ ⎜ ⎟ 1⎛ 1 α α ⎞⎞⎟ α⎛ α ⎜ ρη& = − ⎜ qk − jk ⎜ µ + vl vl ⎟ ⎟ ⎜θ ⎝ 2 ⎠⎠⎟ ⎝ 44443 ⎟ ⎜ 144442 qk ⎠ ,k ⎝
⎛ θ ,k qk D t kl ⎞ h ⎜− 2 + ⎟ vl ,k + ρ θ θ ⎜ θ ⎟ • N ⎜ ⎛ ⎡1 ⎛ ⎞⎟ 1 1 ⎞⎤ + ⎜ − ∑ jkα ⎜ θ ⎢ ⎜ µ α + vlα vlα ⎟⎥ − blα + vlα ⎟ ⎟ ⎟ 2 θ α =1 ⎜⎝ ⎣θ ⎝ ⎠⎦ ,k ⎜ ⎠⎟ ⎜ 1 N α⎛ α 1 α α ⎞ ⎟ ⎜ − ∑ ˆr ⎜ µ + vl vl ⎟ ⎟ 2 θ α =1 ⎝ ⎠ ⎝14 444444442444444444 3⎠ ρ
h
θ
(B.103) T
T
where D tkl = pδ kl + tkl Careful evaluation of the above expression implies that the entropy inequality might be formulated as h ⎛ qk ⎞ ⎟ +ρ θ ⎝ θ ⎠ ,k
ρη& ≥ −⎜
(B.104) T
T
Appendix B
239
Integrating this result over the volume V , and employing equation (B.6) and the transport theorem in equation (B.8), the entropy balance fro any continuous body might be written as q D h ρηdυ ≥ ∫ ρ dυ − ∫ k nk da ∫ Dt V θ θ V S
(B.105) T
T
For the fixed spatial volume with a discontinuity surface, equation (B.105) reads
q h D ρηdυ ≥ − ∫ k nk da + ∫ ρ dυ ∫ Dt V − σ θ θ S −σ V −σ
(B.106) T
T
The first term on the right hand side of the above inequality gives the net entropy influx through the bounding surface S of the multicomponent body with an exterior unit normal n k directed outward with respect to the body. The second term indicates the supply of the entropy to the body as the result of external energy transmission. To recapitulate, the time rate of change of the total entropy of the body must be never less than the sum of the entropy supplied by the body source and the entropy influx through the surface of the body. The axiom of entropy inequality is also called the second axiom of thermodynamics. In the light of the entropy inequality postulated for the multicomponent mixture in (B.106), we will write the entropy inequality for the αth constituent in the mixture. Assume that each species or constituent is bestowed with specific entropy η α (entropy per unit mass) and a temperature θ α . The specific entropy for the mixture (x ,t ) is defined as
N
ρη = ∑ ρ αη α α =1
(B.107) T
T
Appendix B
240
The integral form of the entropy inequality for the αth constituent in a fixed region V at a given time t is postulated by
Dα Dt
qkα α α⎞ ˆ ⎟ η υ r d − + ∫ θ α nk da ⎟ θα ⎠ S-σ hα
⎛
α α α ∫ ρ η dυ ≥ ∫ ⎜⎜ ρ
⎝
V-σ
V-σ
(B.108) T
T
where the term qkα / θ α is an entropy influx of the αth constituent through the boundary of the body S with an exterior unit normal pointing outward with respect to the body. The second term on the right hand side represents the entropy generation per unit time, and the term hα / θ α is the internal entropy source for αth constituent per unit mass. Setting ϕ = ρ αη α and vk = vkα
in equation (B.12), the left hand side of the equation
(B.108) can be treated as Dα Dt
α α ∫ ρ η dυ =
V −σ
⎛ ∂ρ αη α ⎞ ⎜⎜ + ρ αη α vkα ,k ⎟⎟dυ + ∫ ρ αη α vkα − uk nk da ∫ ∂t V −σ ⎝ σ (t ) 1444 424444 3⎠
(
ρα
)
(
)
(B.109) T
T
Dαη α ⎛ ∂ρ ⎞ +η α ⎜ + ( ρv k ) ,k ⎟ Dt t ⎝1∂4 42443⎠ rˆα
Employing the divergence theorem in (B.108), the second term in the right hand side reads as
∫σ(q
S-
α k
/ θ α )nk da =
∫σ(q
V-
α k
/ θ α ),k dυ +
q ∫ σ( )
α k
/ θ α nk da
(B.110) T
T
t
On combining equation (B.108), (B.109) and (B. 110) with the usage of the localization theorem, we can write
Appendix B
241
ρ αη ′α + (qkα / θ α ),k − ρ α (hα / θ α ) ≥ 0 in V − σ
(B.111)
ρ αη α (vkα − uk ) + qkα / θ α nk on σ
T
T
Using identity in (B.23) with setting Γ α = η α , we can get
ρη& = ∑ (ρ αη ′α − (ρ αη α vkα ),k + η α rˆ α ) N
(B.112) T
T
α =1
On summing (B.111)1 over α and then substituting (B.112) in, one can write P
α
P
B
B
α
qk /θ 6444 7444 8 N hα N ρη& + ∑ qkα / θ α + ρ αη α vkα ,k − ∑ ρ α α −∑η α rˆ α ≥ 0 θ α1 α =1 α =1 =1 4 24 3 N
(
)
(B.113) T
T
0
Comparison of equation (B.104) with (B.113) indicates that q kα = qkα + ρ αθ αη α vkα
(B.114) T
T
N
where q k = ∑ q kα and we have assumed that the mixture has a single temperature i.e. the α =1
temperature of all constituents are the same (i.e., θ α = θ ). Thus,
ρη& + ∑ (q kα / θ α ),k − ∑ ρ α N
N
α =1
α =1
ρη& + (q k / θ ),k − ρh / θ ≥ 0
hα
θα
≥0
where q k is a influx vector for the mixture.
(B.115) T
T
242
Appendix B
Recall from equation (B.103) that the influx vector for the mixture was defined as N 1 ⎛ ⎛ ⎞ ⎞ q k = ⎜ qk − ∑ ⎜ µ α + v kα v kα ⎟ jkα ⎟ 2 ⎠ ⎠ α =1 ⎝ ⎝
(B.116) T
T
In the formulation of field equation for a mixture, it is a common practice to omit higher order terms in diffusion velocities; thus ⎛ qk
ρη& + ⎜
−
⎝θ
1
⎞
N
µ α jα ⎟ ∑ θα ⎠
− ρh / θ ≥ 0
k
=1
(B.117) T
T
,k
where µ α is the chemical potential of the αth constituent. Summing equation (B.111)2, we can obtain the entropy inequality on the discontinuity P
P
surface as
N
∑
α =1
⎛
⎞
⎜ v α +v ⎝ k k
⎟ ⎠
ρ αη α ⎜⎜ v{kα − uk ⎟⎟ + qkα / θ α nk = 0
N
N
α =1
α =1
∑ ρ αη α (vk − uk ) + ∑ ρ αη α vkα + qkα / θ α nk = 0 1⎛
144 42444 3
(B.118) T
T
q kα / θ α
N
⎞
ρη (vk − uk ) + ⎜ qk − ∑ µ α jkα ⎟ nk = 0 θ⎝ α =1 ⎠
B.3.6 Open Form of Jump Balances ρ (vk − uk ) nk = 0 ρ (vk − uk )nk = − ρ uk nk l
s
(B.119) T
T
243
Appendix B
ρvl (vk − uk ) − tkl nk = 2γ σ Ωnl vl ρ l (vk − uk )nk − tkl nk = 2γ σ Ωnl 14 4244 3 − ρ u k nk − tkl nk = ρ s vl uk nk
(B.120) T
T
s
+ 2γ σ Ωnl
ρ ⎛⎜
vl v l ⎞ + ε ⎟(vk − uk ) + qk − tkl vl nk = γ σ vlσ 2Ωnl ⎝ 2 ⎠ ⎛ vl vl ⎞ + ε l ⎟(vk − uk )nk + qkl nk − tkl vl nk − (− ρ sε s uk nk + qks nk ) = γ σ vlσ 2Ωnl ⎝ 2 ⎠
ρl ⎜
⎛ vl vl +εl ⎝ 2
ρl ⎜
(B.121)
⎞ vk − uk )nk + qkl nk − tkl vl nk − (− ρ sε s uk nk + qks nk ) = γ σ vlσ 2Ωnl ⎟(1 43 ⎠ 42 s −
ρ uk nk ρl
⎞ ⎛vv − ρ s ⎜ l l + ε l ⎟uk nk + qkl nk + (ρ s vl uk nk + 2γ σ Ωnl )vl + ρ sε s uk nk − qks nk = γ σ vlσ 2Ωnl ⎝ 2 ⎠ ⎛ ⎝
ρ s ⎜ vl v l −
vl v l ⎞ + ε s − ε l ⎟uk nk + qkl nk − qks nk + 2γ σ Ω (vl − vlσ )nl = 0 14243 2 ⎠ s s −
⎛ ⎜ v v 2γ σ Ω − ρ ⎜ε l − ε s − l l + 23 ρl ⎜ 14 4244 ∆H F ⎝ s
⎞ ⎟ ⎟uk nk + qkl nk − qks nk = 0 ⎟ ⎠
⎛ 2γ σ Ω qkl nk − qks nk = ρ s ⎜⎜ ∆H F + ρl ⎝
⎞ ⎟⎟uk nk ⎠
(B.122) T
where vlσ nl = ul nl is used.
T
ρ ρ ul nl = − l u k nk ρl ρ
T
T
PARTIAL COPYRIGHT LICENSE U
I hereby grant the right to lend my dissertation to users of the University of Victoria Library, and to make single copies only for such users or in response to a request from the Library of any other university, or similar institution, on its behalf or for one of its users. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by me or a member of the University designated by me. It is understood that copying or publication of this thesis for financial gain by the University of Victoria shall not be allowed without my written permission. Title of Dissertation: A COMBINED EXPERIMENTAL AND MODELING STUDY FOR THE GROWTH OF SixGe1-x SINGLE CRYSTALS BY LIQUID PHASE DIFFUSION (LPD) B
Author ___________________ MEHMET YILDIZ April 18, 2005
B
B
PB
P
