Department of Engineering Physics and Mathematics
Helsinki University of Technology
FIN-02015 HUT, Finland
Modeling of the Sublimation Growth
of
Silicon Carbide Crystals
To my parents
Peter Råback
Dissertation for the degree of Doctor of Technology
to be presented with due permission for public examination and debate
in Auditorium E at Helsinki University of Technology (Espoo, Finland)
on the 30th of June 1999, at 12 o’clock noon.
CSC Research Reports R01/99
Center for Scientific Computing, Espoo 1999
ISSN 0787–7498
ISBN 952–9821–54–9
Picaset Oy 1999
Finally, I thank my family, friends, relatives and all the others that shared
my life and kept me good company over the years of this thesis project.
Maybe you didn’t speed up the process but you certainly made the time
much more worth while.
Preface
Espoo, June 1999
This thesis is a treatise on the modeling of the seeded sublimation growth
of silicon carbide crystals. The research work was carried out at the Center
for Scientific Computing (CSC) between late 1994 and early 1999. The work
was done in close collaboration with the experimental crystal growth group
in Linköping University.
I would first like to thank my thesis supervisor Professor Risto Nieminen
and my thesis advisor Dr. Jari Järvinen for constant support and guidance
during this work. Without their encouragement this thesis might never have
been realized.
The enabling force behind this research project was Dr. Asko Vehanen
from Okmetic Ltd. For this I wish to express my gratitude. His true dedication to silicon carbide inspired also me to do my best.
I am grateful to Professor Rolf Hernberg from Tampere University of
Technology and Professor Timo Tiihonen from University of Jyväskylä for
their careful reading of the manuscript and for their constructive criticism.
I thank the experimental team at Linköping for a pleasant co-operation
and co-authoring, particularly Dr. Rositza Yakimova, Dr. Marko Tuominen,
Alexander Ellison, Mikael Syväjärvi and Professor Erik Janzén. Their experience was invaluable guidance for the modeling work.
I am indebted to my colleagues Ville Savolainen, Juha Ruokolainen,
Dr. Juha Fagerholm and Tommi Nyrönen for fruitful discussions and useful
comments about this work, and to the computer gurus and science wizards
who have helped me to solve numerous problems. The smoothly running
computers offered by CSC has been an important asset in pursuing the
research.
I have enjoyed very much the human atmosphere at CSC. I thank the
whole staff for contributing to this and especially the management for gardening the good spirit.
The financial support provided by Okmetic Ltd and the Swedish Foundation for Strategic Research is greatfully acknowledged.
iii
Peter Råback
iv
6.6 Feedback Mechanisms . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.7 Coupled Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
Contents
1 Introduction
2 General Information on SiC
2.1 Historical Background
2.2 Crystalline Structure .
2.3 Physical Properties . . .
2.4 Defects in SiC Crystals
2.5 Information Resources
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3 Growth Techniques for SiC Crystals
3.1 Growth from Melt . . . . . . . . .
3.2 Chemical Vapor Deposition . . .
3.3 Lely Growth . . . . . . . . . . . . .
3.4 Seeded Sublimation Growth . . .
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4 SiC Growth Modeling
14
4.1 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 This Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5 Physical Models
5.1 Equilibrium Chemistry . . . . . . .
5.2 Mass Transport of Reactive Gases
5.3 Adsorption and Desorption . . . .
5.4 Temperature Distribution . . . . .
5.5 Induction Heating . . . . . . . . . .
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17
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6 Numerical Models
6.1 Equilibrium Chemistry . . .
6.2 Temperature Distribution .
6.3 Diffusion of Reactive Gases
6.4 Induction Heating . . . . . .
6.5 Virtual Crystal Growth . . .
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7 Verification and Evaluation
7.1 Equilibrium Chemistry . . .
7.2 Diffusion of Reactive Gases
7.3 Temperature Distribution .
7.4 Induction Heating . . . . . .
7.5 Feedback Mechanisms . . .
7.6 Coupled Model . . . . . . . .
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8 Simulation Results
8.1 Equilibrium Chemistry . . .
8.2 Diffusion of Reactive Gases
8.3 Temperature Distribution .
8.4 Coupled Model . . . . . . . .
8.5 Virtual Crystal Growth . . .
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9 Synthesis of Simulation and Practice
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9.1 A Practical Model for the Growth Rate . . . . . . . . . . . . . . . . 97
9.2 Optimization of the Crystal Growth Process . . . . . . . . . . . . 104
10 Conclusions
107
10.1 About this Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
10.2 About Sublimation Growth of SiC Crystals . . . . . . . . . . . . . 109
A Transport Properties
111
B Villars-Cruise-Smith Method
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C Weakly Coupled Induction Heating
114
vi
been published in international journals and conferences [1, 2, 3].
The thesis starts with an overview of SiC and its properties in Chapter 2.
The different methods for growing monocrystalline SiC are presented in
Chapter 3 with an emphasis on the seeded sublimation method. Modeling
aspects of the SiC growth are treated on a general level in Chapter 4. The
physical models used in this work are introduced in Chapter 5. Numerical
solution of these models including some other computational techniques
are presented in Chapter 6. Chapter 7 is dedicated to the verification and
evaluation of the simulation code. Most of the results from the calculations
can be found in Chapter 8. The objective of Chapter 9 is to show how
the simulations and practical experiments might be better linked together.
Chapter 10 concludes the work with some final remarks. In addition, three
Appendices contain selected special details of the modeling work.
Chapter 1
Introduction
Crystalline materials are essential for modern electrical engineering. Apart
from silicon there are other crystalline semiconductors that are very attractive for certain applications. One of these is silicon carbide (SiC). It has
many remarkable properties which make it a very promising semiconductor
material.
Some of the potential applications of silicon carbide are in high-temperature,
-frequency, and -power electronic devices. Others make use of the wide
bandgap: UV radiation detectors and even blue-light lasers. Light emitting
diodes (LEDs) have already been in commercial production for some years.
Also some other electronic devices may become commercial in the near
future.
The large-scale manufacturing of electronic devices requires a continuous production of good-quality wafers. In silicon carbide growth there are
still some basic problems to be resolved that limit the commercial utilization of the material. These problems are related to crystal size and both
macroscopic and microscopic defects.
This work deals with the modeling of the seeded sublimation growth
which is the most promising technique for growing silicon carbide crystals.
The growth is studied from a macroscopic point of view. This involves equations of change that describe how mass, energy and electric currents are
transported in the growth apparatus. These are partial differential equations that are solved with the finite element method, FEM. Also chemical
reactions are considered, which leads to the minimization of Gibbs free
energy.
The physical models arising from the growth process are solved numerically using simulation tools that were developed mostly for this particular
case by the author. This thesis is an extensive description of the modeling
effort and its results. The work has given rise to several articles which have
1
2
Chapter 2
General Information on SiC
2.1
Historical Background
Figure 2.1: Tetragonal bonding of a carbon atom with the four nearest
silicon neighbours. The distances a and C–Si are about 3.08 Å and 1.89 Å
respectively [6].
to manufacture using conventional silicon technology. The potential of the
power electronics industry is much larger and its interest has grown only in
recent years as the progress of silicon devices have began to stagger. This
also largely explains the rapidly increasing research on SiC growth.
Silicon carbide is the only stable compound in the Si–C equilibrium system
at atmospheric pressure. SiC was first observed in 1824 by Jöns Berzelius.
The properties and potential of the material were, of course, not understood
at the time. The growth of polycrystalline SiC with an electric smelting
furnace was introduced by Eugene Acheson around 1885. He was also the
first to recognize it as a silicide of carbon and gave it the chemical formula
SiC. The only occurrence of SiC in nature is found in meteorites. Therefore,
SiC cannot be mined but must be manufactured with elaborate furnace
techniques.
In its polycrystalline forms, silicon carbide has long been a well proven
material in high-temperature, high-strength and abrasion resistant applications. Silicon carbide as a semiconductor is a more recent discovery. In
1955, Jan Antony Lely proposed a new method for growing high quality
crystals which still bears his name. From this point on, the interest in SiC
as an electronic material slowly began to gather momentum; the first SiC
conference was held in Boston in 1958. During the 60’s and 70’s SiC was
mainly studied in the former Soviet Union.
Year 1978 saw a major step in the development of SiC, the use of a
seeded sublimation growth technique also known as the modified Lely technique. This breakthrough led to the possibility for true bulk crystal preparation. The first blue LED was fabricated already in 1979 and in 1987, Cree
Research Inc., the first commercial supplier of SiC substrates, was founded.
Today, there are a few companies and many active research groups in the
field, but SiC industry still remains a small business. In the beginning, the
SiC industry was concentrated around the blue LEDs that are impossible
Different crystal forms of the same chemical composition are called polymorphs. Polymorphism commonly refers to a three-dimensional change
affected by either a complete alteration of the crystal structure or a slight
shift in bond angles. Polytypism is a special type of polymorphism which
occurs in certain close-packed structures. In this phenomenon, two dimensions of the basic repeating unit cell remain constant for each crystal
structure (polytype), while the third dimension is a variable integral multiple of a common unit perpendicular to the planes having the highest density (closest packing) of atoms. Silicon carbide is very rich in polytypism,
as more than 170 different one-dimensional ordering sequences have been
determined. For a theoretical treatment of SiC polytypes, see [4, 5].
The fundamental structural unit in all SiC polytypes is a covalently
bonded tetrahedron of four C atoms with a single Si atom at the center.
Each C atom is likewise surrounded by four Si atoms as shown in Figure 2.1.
The tetrahedra are linked through their corners. In a common polytype notation the letters C, H and R are used to represent cubic, hexagonal and
rhombohedral structures, respectively, and numerals are used to represent
the number of closest-packed layers in the repeating sequence. The most
common polytypes are 3C and 6H; 4H, 15R and 2H have also been identified, but are much more rare. All other polytypes are combinations of these
basic sequences. The only cubic polytype (3C) is also referred as β-SiC. The
3
4
2.2
Crystalline Structure
Table 2.1: Properties of silicon carbide compared to other some semiconductor materials.
Property
Bandgap (eV)
Max. Temperature (C)
Melting point (C)
Physical stability
Electron Mobility (cm2 /Vs)
Hole Mobility (cm2 /Vs)
Breakdown voltage (106 V/cm)
Thermal cond. (W/cmC)
Sat. velocity (107 cm/s)
Dielectric constant
Figure 2.2: Stacking sequence of double layers of the four most common SiC
polytypes [6].
most common polytypes are presented in Figure 2.2.
Up to date quite little is known about the kinetics and thermodynamics
of polytype formation, growth and stability, and also the mechanism that
produces the periodic sequences. Literature indicates that β-SiC may be
both the initial polytype that forms at virtually all temperatures and the
necessary precursor to the occurrence of other polytypes. This polytype is
particularly interesting from the standpoint of electronics.
3C–SiC
2.2
873
1800
excellent
1000
40
4.0
5.0
2.5
9.7
6H–SiC
2.9
1240
1800
excellent
600
40
4.0
5.0
2.0
10.0
Si
1.1
300
1420
good
1400
600
0.3
1.5
1.0
11.8
GaAs
1.4
460
1240
fair
8500
400
0.4
0.5
2.0
12.8
Diamond
5.5
1100
?
very good
2200
1600
10
20
2.7
5.5
in outer space. Some of the properties of silicon carbide compared to some
other semiconductors are listed in Table 2.1. It may be noticed that silicon
is inferior to SiC in many respects. Diamond would be the ultimate semiconductor for power electronics, but problems related to its use appear to
be even larger than in the case of SiC. There are also some other potential
wide-bandgap semiconductors that compete with SiC, for example, gallium
and aluminum nitride.
2.4
Defects in SiC Crystals
SiC belongs to a class of materials commonly referred to as wide-bandgap
semiconductors. This means that the energy gap between the valence and
conduction band is significantly larger than in silicon. It implies, for example, that it is less probable that thermally excited electrons would jump over
the gap. Therefore SiC devices are less sensitive to high temperatures and
should be able to operate at temperatures exceeding 500 ◦ C. The thermal
conductivity of SiC is larger than that of copper. Thus the heat generated
by the devices is efficiently removed. Also such properties as high electric
field strength and high saturation drift velocity are important for the device
technology. Consequently devices can be made smaller and more efficient.
SiC is a very hard material. This has resulted in a wide variety of applications already at the polycrystalline era. To make it a physicist’s dream, SiC
is also chemically inert and extremely radiation hard. It may thus be used
in the most hostile environments, for example, near nuclear reactors and
As any real crystal, also the SiC crystals suffer from defects [7, 8, 9]. The
most severe defect is called micropipe. It is like a small wormhole with a
diameter of the order of micrometers in the crystal structure. Most views on
micropipe formation are based around the Frank theory that a micropipe is
the hollow core of a screw dislocation with a huge Burger’s vector [10]. Since
the formation of micropipes is not yet fully understood, totally micropipefree material has not yet been grown. Usually the density of micropipes
ranges from a few to some hundreds per cm2 . The largest micropipe free
areas reported have been around a few cm2 .
Another serious problem is the polytype control of the growing material.
While the polytype 6H is the easiest to grow, 4H would be favored by the
power electronics industry. Difficulties in growing evenly doped 4H still
remain severe. On the other hand, if the problem of polytype control is
efficiently solved, the various polytypes with different physical properties
will enable more versatile applications.
In silicon industry there are strict specifications on all defect types. In
the field of SiC, the problems with micropipes and polytypes dominate to
5
6
2.3
Physical Properties
such a degree that the research of dislocations, vacancies and impurities
still remains an academic activity. When the material develops, these issues
must be targeted and the growth must be further optimized to minimize
these defects as well. At the present situation the main interest, however,
lies in more fundamental problems.
2.5
Information Resources
For further information, good generic introductions to the field of silicon
carbide are given in References [11, 12, 13, 14]. Articles on different aspects on SiC technology are published in many journals as may be seen in
the list of references of this work. Conference publications provide good
overall visions of the field. The publications from International Conference
on Silicon Carbide, III-Nitrides and Related Materials – Stockholm 1997 are
available in Materials Science Forum Vols. 264–268 (1998). The papers from
2nd European Conference on Silicon Carbide and Related Materials – Montpellier 1998 are available in Materials Science and Engineering Vols. 61–62
(1999). Also Physica status solidi (b) Vol. 202, No. 1 (Part I) and Physica
status solidi (a) Vol. 162, No. 1 (Part II) from the year 1997 are dedicated to
SiC. Some recently published doctoral theses provide nice overviews of the
subject [6, 15, 16]. The progress in the field is so rapid that the researcher
should follow carefully the latest developments.
Chapter 3
Growth Techniques for SiC
Crystals
There are several methods for growing monocrystalline silicon carbide. Unfortunately none of these have proven to be very successful and major
challenges still lie for efficient crystal growth [10, 14, 17, 18, 19].
3.1
Growth from Melt
Most commercially utilized single crystal semiconductor boules are grown
from a melt or solution, but this is not a feasible option for SiC growth.
SiC does not have any liquid phase in normal engineering conditions. Calculations have indicated that stoichiometric melting is possible only under
pressures exceeding 105 bar at temperatures higher than 3200 ◦ C. Even if
the solubility of carbon in silicon melt ranges from 0.01% to 19% in the
temperature interval from 1412 to 2830 ◦ C, at high temperatures the evaporation of silicon makes the growth unstable. The solubility of carbon can
be increased by adding certain metals to the melt (e.g., praseodymium, terbium, scandium). This would, in principle, enable the use of crystal pulling
techniques, such as Czochralski growth. Unfortunately there is no crucible
material available that would be stable with these melts. It is also speculated that the solubility of the added metals in the growing crystal is too
high to be acceptable in semiconductor materials [10, 20].
In spite of all the problems, SiC was grown from melt at 2200 ◦ C and 150
bar in a recent study. The crucible was made of graphite and it also acted as
the carbon source. A 1.4-inch crystal was demonstrated [21]. The technology is very expensive and might never be economically feasible. However,
7
8
growing from a solution would avoid many of the problems related to the
growth techniques from gas phase.
SiC source
SiC crystal
crucible
porous graphite
3.2
Chemical Vapor Deposition
A well established method for growing thin crystalline layers directly from
gas phase is chemical vapor deposition (CVD) [14]. In the process a mixture
of gases is injected to the growth chamber. The higher the temperature the
larger is the probability that the initial bonds will crack and the radicals will
attach to the surface thus leading to epitaxial growth. When temperature
is increased the probability of sticking increases but also the etch rate
from the surface increases. The growth rate is therefore determined by the
desorption of the reaction products and by the etch rate of the surface, and
by the diffusion dominated mass transport of the source molecules.
Generally, the growth rates in CVD are too low to allow boule production,
usually tens of micrometers an hour [22]. By increasing the temperature
the growth rate increases, but at the same time problems related to the
controlling of the growth become more severe, and problems such as homogeneous nucleation in the gas phase may occur. These problems might
be overcome by a very careful control of the thermal and thermodynamic
conditions. This technology is not yet available, even though research on
high temperature CVD (HTCVD) is under way [23, 24].
Figure 3.1: The Lely method for growth of SiC crystals
method does not require any seed crystals it is the method which all the
other SiC crystals originate from. The resulting crystals have low defect
and micropipe densities.
3.4
Seeded Sublimation Growth
The Lely growth method is used even nowadays to grow the crystals of the
highest quality. A schematic Lely geometry is presented in Figure 3.1. In
CVD the growth is driven by the initial gas composition, whereas in the Lely
method the growth is due to temperature gradients within the system. The
system is close to chemical equilibrium and the partial pressures of the SiC
forming species are higher where also the temperature is higher. This leads
to a pressure gradient that results in mass transport from the hot parts of
the crucible to the cooler parts of the crucible [25, 26].
In the Lely growth the temperature distribution is such that in the cylindrical crucible the temperature minimum is at the origin. Therefore the
gases travel towards the origin. The porous graphite holding the source
provides nucleation centers for infinitely small seed crystals. They will
eventually grow larger and usually obtain an energetically favorable hexagonal form. Unfortunately, the Lely grown crystals are limited and random
in size, in average they are about the size of a nail. Because the Lely growth
The seeded sublimation growth, also known as physical vapor transport
(PVT), is the method of the present study. It is historically referred to as
the modified Lely method. The geometry was initially quite similar to the
Lely geometries but the difference is the use of a seed crystal which results
in a more controlled nucleation.
The seeded sublimation process is nowadays the standard method for
growing bulk monocrystalline silicon carbide [10, 18, 19, 27, 28, 29, 30, 31].
In the process polycrystalline SiC at the source sublimes at a high temperature, (1800–2600 ◦ C) and low pressure. The resulting gases travel through
natural transport mechanisms to the cooler seed crystal where crystallization due to supersaturation takes place. The seed crystal is usually situated
at the top of the crucible in order to prevent contamination by falling particles.
There are different flavors of sublimation growth. The most important
factor that can be varied, is the crucible design and the temperature distribution related to it. The crystals tend to grow along isotherms and therefore
the shape of the temperature distribution must be carefully designed. Many
different designs have been tested over the years. Two different designs are
presented in Figure 3.2. One way to classify them is to look at the placement
of the source material.
The most common design is to put the source at the bottom of the
crucible so that the surface of the source is facing the growing surface
9
10
3.3
Lely Growth
SiC seed
SiC seed
SiC source
SiC source
SiC crystal
SiC crystal
crucible
crucible
porous graphite
insulation
induction coil
Figure 3.2: Two versions on the seeded sublimation growth geometry, the
original modified Lely on the left and the modern version on the right.
[32, 33, 34]. This minimizes the source-to-seed distance, but in large systems it makes the uniform heating of the source material difficult. A partial
solution to this problem is to set the source material in a ring-like configuration near the crucible walls. The vapors are transported through porous
graphite walls. This design enables improved temperature control of the
source and cooling of the seed. The design has shown its potential in growing large crystals [25, 29, 35, 36, 37]. The drawback is a lower growth rate
due to larger source-to-seed distance, contamination by the porous graphite
and possible problems in controlling the shape of the growing crystal. Due
to these reasons this design seems to be decreasing in popularity.
The so called sublimation sandwich method was proposed initially in
1970 to grow thin epitaxial layers of monocrystalline SiC. The design is
partially open and the environment may be used to control the gas-phase
stoichiometry [38, 39, 40, 41, 42]. The method has a high growth rate, more
parameters to control the system, but it has not yet been shown that it
can be used to grow large boules. The high growth rate is mainly achieved
with a small source-to-seed distance and a large heat flux enabled by a
small amount of source material. These facts make the growth of large
boules quite difficult. The quality of the crystals grown with this method
is, however, quite promising.
The crucibles are usually heated up by electromagnetic induction as depicted in Figure 3.3. Typically, the frequency is in the range of 10 to 100
kHz. Resistive heating might give improved temperature control, but it
requires more engineering efforts and is more expensive. Also the resistive elements may be quickly worn out at the high temperatures required.
The sublimation growth systems are controlled during the growth process
mainly by changing the pressure of the inert gas (usually argon) and coil position [33, 43, 44]. The chemical composition of the system may be affected
by the selection of the source and crucible material. Since the selection
of crucible materials is quite limited at the high temperatures required,
carbon materials are usually employed. At least two kinds of graphite are
used: dense graphite that is a good conductor of heat and electricity as
well, and porous graphite that has significantly smaller conductivities in
both respects. As chemically inert crucibles have some advantages, tantalum has recently been proposed as a crucible material [40]. The material
choices may have a decisive effect on the temperature and impurity control
of the crystal growth process.
The seed crystals for sublimation growth are produced with the Lely
method or taken from previous sublimation growths. Unfortunately, the
good properties of the original Lely-grown crystals are difficult to maintain
and therefore the resulting crystals may have considerably higher defect
and micropipe densities. The sublimation growth is an evolutionary process. By selecting the seed among the best wafers it may gradually be
possible to get rid of micropipes. The size of the crystal may be increased
only gradually by radial growth. This is very different from the growth of
silicon where the quality and size of seed crystals are not that critical.
The processing of SiC crystals is a much more difficult task than the processing of crystalline silicon. The grown boules must be cut and polished
before device manufacturing. Even though technological problems still exist in these fields, they will probably not hinder the advance of SiC industry.
The availability of high quality bulk material is the real bottleneck.
SiC wafers are already commercially available. The market leader has
11
12
Figure 3.3: A schematic model of an induction heated seeded sublimation
geometry.
introduced a 3” wafer but only 2” wafers are commercially available. There
are also some other producers of SiC wafers that provide crystals up to two
inches [10, 45].
Although the seeded sublimation growth is the most promising method
for producing SiC crystals, and has been known for more than twenty years,
there are still some major difficulties involved in the process. The polytype
formation and growth shape are poorly controlled and the doping is nonuniform. There are also severe defects, such as micropipes and dislocations
in the grown crystals. Many of the problems of the sublimation method are
inherent but the technological limits may though be pushed further.
Chapter 4
SiC Growth Modeling
To overcome the problems of SiC crystal growth, modeling can be a valuable
tool. There is no way to control the behavior of individual particles and
therefore the growth process can only be influenced through macroscopic
mechanisms, such as thermal environment, mass transport and chemical
composition. These mechanisms are highly interdependent. A useful model
should be able to predict quantitatively how these phenomena influence the
crystal growth process. The purpose of modeling should not be the model
itself, but to provide tools for the development of the process. The results
of modeling are usually twofold. Firstly, it gives an insight on what is
actually happening in the system, and secondly, it helps in evaluating new
systems without the need of actually building them.
Often the simulations give more information than can be used. It is
not easy to filter the essential data until it is fully understood what really
affects the process. Therefore, before a thorough understanding is reached
it is also not all that clear what physical phenomena should be included in
the model. As the knowledge increases it is possible to target new areas of
interest. Also during this work, the model has been developing in the same
manner with time. In the end, the work is presented as though it had been
the only path to follow. This is of course an illusion. There are several ways
to approach a problem and this represents only one possible vision.
The multitude of physical phenomena in the sublimation growth process
is so overwhelming that even the most ambitious researcher can model
the process only partially. In deciding what phenomena to include in the
model one should evaluate the importance of each feature and also the
labor involved in its implementation. Even though the complete model
has been independently implemented the work has been largely influenced
by the approaches adopted in some previous contributions in the field of
crystal growth modeling.
13
14
4.1
Previous Work
The seeded sublimation growth process has been modeled for almost as
long as the method has been known. In the beginning, the models were
simplified which made analytical solution possible. With the advance of
numerical methods and computer resources also the models have become
more sophisticated. The modeling of sublimation growth has by no means
been in the cutting edge in applying the newest methods. The methods
have usually been first developed in other fields and only later applied to
the problem of sublimation growth. Most of the simulation papers related
to the simulation of sublimation growth are therefore quite sparse in computational details.
The sublimation growth process is driven by supersaturation due to the
temperature difference between the source and the seed. This driving force
may be estimated by equilibrium thermodynamics. It is the more relevant
the closer the system is to equilibrium. Sublimation growth of SiC is close to
equilibrium conditions due to the high temperatures and slow mass transport rates. Equilibrium calculations may also be used to identify the most
important chemical species in the system. The equilibrium chemistry related to SiC growth has been analyzed in several papers [39, 46, 47, 48]. In
most papers only the three most important species (Si, Si2 C, SiC2 ) have been
considered. Reactive process gases quickly increase the number of important species [49, 50, 51]. The equilibrium calculations can unfortunately
only give trends, i.e., tell whether the growth is possible or not, but it does
not tell anything about the growth rate. The lack of accurate kinetic data at
high temperatures has been the bottleneck in applying kinetic calculations
to the sublimation growth process. Such treatise have been written for the
CVD growth, see for example References [52, 53]. In the kinetic calculations
also the reaction pathways must be given. In equilibrium thermodynamics,
the reaction pathways are meaningless because the given minimum is a
global one, and therefore independent of the reaction history.
Experiments have shown that in sublimation method the growth is mainly
controlled by the temperature distribution. Therefore, determination of the
temperature distribution is a natural first step in the modeling of the real
crucible. In the energy equation at least conduction and radiation need to
be taken into account. The temperature distribution calculations may be
used in the design of different furnace geometries. The possible correlation with growth speed, growth shape, or other growth phenomena may
then be derived heuristically by comparing the temperature distribution to
observations.
In order to target such questions as growth rate and shape of the grown
crystal, mass transport needs to be taken into account. Generally this means
15
solving a coupled system of Navier-Stokes equations, multispecies diffusion and kinetic chemistry. The kinetic nature of chemical reactions is
often omitted assuming local chemical equilibrium (LCE). This assumption
is valid if the typical time scale of reactions is much shorter than the time
scale of mass transport. Such approaches have been considered in papers
[54, 55]. Also generic commercial CFD codes may be used to model the
temperature distribution and mass transport. For example, vendors like
CFDRC and Cape Simulations have put special effort in modeling the sublimation process. If the pressure of the ambient gas is high enough the mass
transport is dominated by diffusion and convection may be neglected. This
approach has been used in [48].
If the source-to-seed distance is small the adsorption and desorption
rates at the surfaces limit the growth rate. Therefore these phenomena have
been mostly studied in relation to the sublimation sandwich method [38, 39,
42]. The growth method enables the control of the gas-phase environment
and thus the influence of the external gas-phase stoichiometry has also
been considered.
The sublimation growth systems are usually heated up by induction. Induction heating has been modeled by solving Maxwell’s equations in terms
of a vector potential. This approach has been used in [48], for example.
In recent years some effort has also been put in the modeling of the
material properties. The thermal conductivity of monocrystalline SiC is
discussed in [56] and the heat transfer through the porous source powder
is treated in [57]. Such work is important in improving the accuracy of the
thermal models.
4.2
This Work
This work considers the models of equilibrium chemistry, global heat transfer, mass transport by diffusion, adsorption-desorption and induction heating. The main problem in the work is to couple all these individual phenomena. To do this successfully a three-level strategy with three independent
meshes is introduced. A method that couples the adsorption-desorption
equations with diffusion-limited mass transport of reactive gases in a generic
manner is developed in this work. Also some heuristic methods for utilizing the complete model even better are introduced. These include feedback
for process control and the concept of virtual crystal growth. Finally an analytical model for the growth rate in a simplified geometry is presented. All
these models are also applied to a number of different cases to illuminate
the underlying phenomena.
16
The free energy of a chemical system consisting of several chemical
~ = (n1 , n2 , . . . , nN ),
species is also a function of the molar amounts n
~
G = G(T , P , n),
where N is the number of species. The equation gives rise to a corresponding equation for the complete differential of the function,
Chapter 5
δG = −SδT + V δP +
Physical Models
5.1
Equilibrium Chemistry
The process of sublimation growth is not limited to a liquid-solid phase
transition as in the silicon growth. Also chemical reactions are involved in
the process. A chemical system is made up of a number of chemical species.
Species may react to build up other species as long as the amount of basic
elements is conserved. The maximum extent of these reactions depends
on the thermodynamics of the system. The equilibrium composition may
be calculated by either minimizing the Gibbs free energy or by solving a
set of nonlinear equations which describe the equilibrium constants of all
independent chemical reactions [58]. The first approach was used in this
work. The following basic material can be found in most books of physical
chemistry, such as [59].
5.1.1
(5.2)
Free Energy
A thermodynamical system can be described with appropriate potential
functions and the corresponding thermodynamic variables [60, 61]. In equilibrium thermodynamics the most suitable potential is the Gibbs function,
which is often also referred to as free energy. The independent variables of
the Gibbs function that define the state of the system are pressure P , and
temperature T . The second law of thermodynamics states that any natural
process must be such that the Gibbs function decreases,
δGT ,P ≤ 0,
(5.1)
N
X
µi δni .
(5.3)
i=1
where S is the entropy of the system and µi is the chemical potential of
species i. At equilibrium the differential should vanish.
Equilibrium thermodynamics defines the direction of the reactions but
it does not say anything about the rate of reactions. Even an energetically
favorable reaction may be very improbable if the activation energy, E a associated with the reaction is large. An experimental observation is that the
probability of a reaction is often proportional to e −Ea /RT , where R is the
gas constant. The same result known as the Arrhenius law is obtained by
studying the random collision of Boltzmann-distributed particles. Since the
probability increases at high temperatures the system will reach the state of
equilibrium in less time. In sublimation growth it is customary to assume
that the time scale of chemical reactions is short compared to the other
time scales in the system. This is partially justified by the high temperatures involved. However, it may also be viewed as a pragmatic choice since
the parameters describing the rate of reactions are poorly known for many
of the possible reactions. Solution of the rate equations might also result
in a stiff system of differential equations that are often hard to solve. When
equilibrium thermodynamics is used to evaluate the growth rate the model
gives the upper limit of the real kinetic system.
5.1.2
Chemical Potential
In order to minimize the Gibbs function, the chemical potentials of the compounds must be known. For the condensed species the chemical potential
depends only on temperature; for the ideal gases the chemical potentials
may be expressed as
µi (T , P , pi ) = µi◦ (T ) + R T log(pi ).
(5.4)
or in other words, that the system moves towards equilibrium. The equality
is true at the state of equilibrium.
Here µi◦ is the standard chemical potential and pi is the partial pressure of
species i,
ni
P = xi P ,
(5.5)
pi =
nt
17
18
P
where xi is the corresponding molar fraction and nt =
ni is the total
material amount [62]. A more general expression accounts for non-ideal
species in a non-ideal solution,
~ = µi◦ (T , P ) + R T log ai (T , P , x),
~
µi (T , P , x)
(5.6)
where ai is the activity. Thus for ideal gases, the activity is assumed to be
directly proportional to the partial pressure of the species.
Sometimes the data for chemical potentials is directly available, but
often it must be derived from some other data. In literature the thermodynamic properties are often expressed as standard free energies of formation
∆f Gi◦ of the chemical species. The energy offset may be freely selected in
minimization problems. Therefore the standard chemical potentials may
be identified with the free energies of formation,
µi◦ = ∆f Gi◦ .
(5.7)
For a single chemical compound the free energy of formation can also be
determined from entropy and enthalpy data by [50]
∆f Gi◦ = ∆f Hi◦ − T ∆f Si◦ .
A change in the species balance can be expressed by a stoichiometric vector
~. The components of the stoichiometric vector are simply the multipliers
ν
in the corresponding chemical reaction. On the right hand side of the
reaction, the multipliers are taken as positive and on the left hand side as
negative. It can be shown that there are L = N −M such linearly independent
vectors. Any allowed state can be expressed by
~=n
~0 +
n
Chemical Stoichiometry
When the requirement of mass conservation is applied to chemical phenomena the concept chemical stoichiometry is introduced. The constraints
following from the conservation of mass may be expressed directly in terms
of conservation equations or indirectly in terms of chemical reactions that
always conserve the amounts of basic elements.
Let us assume that there are N different species distinguishable by their
molecular formula and M basic elements. Then for each basic element,
there is a direct form of the conservation law,
N
X
i=1
aki ni = bk ,
k = 1, 2, . . . , M,
(5.9)
where aki is the subscript of the k:th element in the molecular formula
of species i and bk is the fixed amount of the k:th element in the system
[62, 63]. Equation (5.9) can also be written in vector-matrix form:
~
~ = b.
An
19
(5.10)
(5.11)
AV = 0.
(5.12)
When minimizing the Gibbs free energy the material conservation may
be taken into account directly with Equation (5.9) or indirectly employing
Equation (5.11).
5.2
5.1.3
~j ,
ξj ν
j=1
where the quantities ξj are a set of real parameters that describe the extents of reactions. Equation (5.11) is therefore a general solution for Equation (5.10). The stoichiometric vectors may represent the reactions taking
place in the system, but they may also be selected otherwise. In general,
~ = 0 is a stoichiometric vecany nonzero vector satisfying the condition Aν
tor. A set of linearly independent stoichiometric vectors form a complete
~L ), which satisfies the matrix equation
stoichiometric matrix V = (~
ν1 , . . . , ν
(5.8)
One should be careful when applying thermodynamic data from different sources. The definition of the thermodynamic data may vary. For
example, the reference temperature may be the absolute zero or room temperature.
L
X
Mass Transport of Reactive Gases
Inside the crucible the chemical species travel through natural transport
mechanisms, convection and diffusion. Convection arises from the pressure difference due to chemical reactions and diffusion arises from the
concentration or temperature gradients. The species density may generally
be obtained from
∂nk
~ k − ∇ · J~k + Sk ,
= −∇ · vn
∂t
k = 1, . . . , N,
(5.13)
where nk is the molar density and Sk is the production rate due to chemical
reactions. J~k is the diffusion flux due to concentration and temperature
gradients in the fluid [48, 64],
J~k = −
X
l
Dkl ∇nl − DkT
∇T
,
T
k = 1, . . . , N,
(5.14)
~ T contain the diffusion coefficients. As thermal diffusion is
where D and D
a second order phenomenon it is usually significantly weaker than concentration diffusion.
20
~ may be calculated from the compressible NavierThe local velocity v
Stokes equations. The continuity equation is
∂ρ
~
= −∇ · (ρ v),
∂t
(5.15)
where ρ is the mass density. The momentum equation reads
~
∂v
~ · ∇v
~ = ∇ · τ − ∇P + ρ g,
~
+ ρv
∂t
(5.16)
where the stress tensor is
h
T i 2
~ I.
~ + ∇v
~
− µ ∇·v
τ = µ ∇v
3
(5.17)
ρ
The temperature may be solved from the energy equation arising from the
requirement of energy conservation. If energy is transported via convection
and conduction the energy equation yields
cV
∂(ρT )
~
= −cV ∇ · ρ vT
+ ∇ · (κ · ∇T ) + h,
∂t
(5.18)
where T is the temperature, cV is the specific heat capacity at constant volume, and κ is the thermal conductivity and h is the local heat generation in
unit volume that may include contributions due to chemical reactions or radiation within a semitransparent medium. An equation of state is required
to close the system. It defines the relation between temperature, pressure
and density. Usually ideal gases are assumed and thus the equation of state
is
N
P X
ρ=
x k Mk ,
(5.19)
RT k=1
where Mk is the molecular mass. All in all, the general set of equations includes 5 + N unknown field variables: temperature, pressure, three velocity
components, and N concentrations.
The importance of the different transport mechanisms depends on the
growth conditions. The lower the pressure of the inert gas, the more prominent is the convection due to pressure gradients (Stefan flow). The practical
experiments related to this work were performed at conditions where diffusion should dominate. In a general model convection should, however,
be taken into account. This is the main shortcoming of the model.
Neglecting convection also means that the model cannot be used without modifications to simulate the CVD growth process where forced and
natural convection play a decisive role. On the other hand the CVD process avoids the nasty coupling that occurs in sublimation growth when the
crystal growth that is limited by mass transport is also the driving force of
21
mass transport. Usually in CVD growth these phenomena may be decoupled
which means that the Navier-Stokes equations may be solved neglecting the
chemical reactions. The composition of the gas phase is taken into account
explicitly when calculating the density and transport properties. In sublimation the driving force of convection would be the pressure gradient
caused by the chemical reactions.
Neglecting convection, the stationary form of the diffusion equation
yields
∇ · J~k = Sk , k = 1, . . . , N.
(5.20)
The chemical composition of the gas phase is in local chemical equilibrium.
The assumption is justified because the time scale of chemical reactions is
much shorter than the time scale of mass transport. This implies that as
the species are transported they immediately retain the state of equilibrium.
The transport properties of the multi-component gas may be calculated with
the formulas presented in Appendix A.
5.3
Adsorption and Desorption
The crystal growth process takes place at the gas-surface interface. The rate
by which the molecules move with from the gas phase to the solid phase is
determined by the adsorption and desorption of individual molecules. The
theory involved may be found in books on surface science, such as [65].
In sublimation growth the surface processes become increasingly important when the pressure and source-to-seed distance are decreased as in the
sublimation sandwich method [38, 42].
The impingement rate of the molecules is obtained from the kinetic gas
theory and yields
pk
p
.
(5.21)
2π Mk RT
The molecules may be reflected from the surface or be adsorbed to the
surface. A true model should include reactions between the surface and
gas-phase molecules. Such a model is, however, not available and therefore
the reactions are modeled by a simple probability factor. The probability
of adsorption is called sticking coefficient γk . The sticking coefficient may
involve an activation factor of the Boltzmann type, e −Ea /RT . In sublimation
growth of SiC the sticking coefficients are usually assumed to be unity since
the temperatures are high and more accurate estimates are not available.
eq,s
In equilibrium at the surface (pk = pk ) the rate of desorption is equal
to the rate of adsorption. The adsorption and desorption processes may be
assumed to be independent. The net flux of molecules is the difference of
22
desorption and adsorption rates given by
eq,s
pk
pk −
.
Rk = γ k p
2π Mk RT
The energy equation may have fixed (Dirichlet) or flux (Neumann) boundary conditions. Often the flux boundary conditions may be written as
(5.22)
The same equation may also be expressed using the molar densities n k ,
s
RT
eq,s
Rk = γ k
(nk − nk ).
(5.23)
2π Mk
The short distance at the surface at which the partial pressures change is
called Knudsen layer.
The equilibrium partial pressures over SiC crystal are not defined in a
unique way. The equilibrium pressures depend on the stoichiometry of the
gas phase. The stoichiometry may be defined from the requirement that
~ must be equal to the one of the growing
the stoichiometry of the flux R
surface. In the growth of SiC crystals the stoichiometry of the flux must be
such that equal amounts of carbon and silicon are adsorbed by the growing
crystal.
After the species have been adsorbed to the surface they may easily
diffuse along the surface. For small supersaturation values the crystal tends
to grow layer by layer and the add-atoms are attached to kinks which give
the energy minimum. There is some evidence that SiC would grow by spiral
growth associated with a screw dislocation at the crystal surface. If the
supersaturation is increased over some critical value island growth may
occur. The models in this work are fully macroscopic and the crystal is
treated as isotropic material and surface diffusion is neglected. Therefore
we cannot consider different polytypes and growth modes.
5.4
Temperature Distribution
− (κ · ∇T ) · e~n = q(~
r , T ), = hT (~
r , T )(T − Text ),
(5.25)
where e~n is a normal of the surface, q is the heat flux, hT is the heat transfer
coefficient, and Text is the external temperature. In particular, heat transfer
by radiation is of the form
4
,
(5.26)
q(~
r , T ) = σSB T 4 − Text
where σSB is the Stefan-Bolzmann constant, is the emissivity of the surface, and Text is some external temperature. If this is linearized by factorization to the form given by Equation (5.25) the heat transfer coefficient
becomes
2
(5.27)
hT (~
r , T ) = σSB T 2 + Text
(T + Text ) .
A special case of radiation is a boundary condition for a surface that sees itself at least partially. This makes the local external temperature dependent
on the temperature of the boundary itself. Therefore the external temperature cannot be known a priori and must be calculated from the following
equation,
Z
1
4
Text
(~
r 0 )G(~
(~
r) =
r 0 , r~)T 4 (~
r 0 )dSr~0 ,
(5.28)
(~
r)
where G(~
r 0 , r~) is the Gebhart factor that represents the fraction of energy
leaving at r~0 being absorbed at r~ [66]. The Gebhart factors are solutions of
the integral equation
Z
r , r~0 )(1 − (~
r 0 ))G(~
r 0 , r~)dSr~0 = F (~
r 0 , r~)(~
r ),
(5.29)
G(~
r 0 , r~) − F (~
where the function F is the view factor between two points. It depends only
on the geometry of the system and is defined as
The temperature distribution of the crucible is solved from the energy equation. At the high temperatures radiation dominates the energy transport
and convection may be neglected. The system changes gradually with time
due to the crystal growth. The changes are, however, so slow that the system is assumed to be in a thermally stable state, and the energy equation
becomes stationary. Equation (5.18) thus simplifies to
F (~
r , r~) =
cos β(~
r , r~0 ) cos β0 (~
r 0 , r~)
χ(~
r , r~0 ),
2
π r~ − r~0 (5.30)
Now h is the local heat generation in unit volume which may result from
inductive of resistive heating. For electromagnetic induction the equations
for heat generation will be presented in Section 5.5.
where β and β0 are the sharp angles formed by the surface normals and the
line connecting the points points r~ and r~0 . Function χ is one if the points
r~ and r~0 see each other and zero if there is an obstacle between the points
[67, 68, 69].
When a system is built of several parts the thermal contact is seldom
perfect. This creates difficulties since the gaps between the different parts
are usually too small to be taken into account in creating the computational
geometry. A way to tackle this problem is to allow the temperature to be
23
24
−∇ · (κ · ∇T ) = h.
(5.24)
discontinuous over the gaps. The boundary conditions must be set on both
sides of the gap. For linear heat transfer the boundary condition at the
sides are expressed by a dual boundary condition
(
q(~
r,T)
= hT (~
r , T )(T − T 0 )
,
(5.31)
0
0
r , T ) = hT (~
r , T 0 )(T 0 − T )
q (~
In the case of radiation it is not necessary to calculate the view factors since
it is obvious that at the interface the walls see the other side of the gap in a
full angle. The heat transfer coefficient may now be analytically calculated
to be
0
hT =
(5.32)
σSB T + T 0 T 2 + T 02 .
+ 0 − 0
5.5
Induction Heating
At high temperatures the most practical method to heat up the crucible is
by electromagnetic induction. The induction coil generates an alternating
current that flows through the crucible. The Ohmic resistance encountered
by this current dissipates energy, thereby directly heating the crucible via
internal heat generation.
In induction heating it may usually be assumed that all the materials
~ and D
~ = µH
~ = εE,
~ where µ
have linear, isotropic electrical properties B
is the magnetic permeability and ε is the electric permittivity, both being
independent of the values of the field variables. This simplifies Maxwell’s
equations to
1
~ = ρc ,
(5.33)
∇·E
ε
~ = 0,
∇·B
(5.34)
~
∂B
~=−
,
∇×E
∂t
~ = µ~
∇×B
 + εµ
~
∂E
,
∂t
(5.35)
(5.36)
where ρc is the charge density and ~ is the current density vector [70].
~ for the magnetic field B
~
It is convenient to define a vector potential A
so that
~
~ = ∇ × A,
B
(5.37)
Using this expression Equation (5.35) yields
!
~
∂A
~+
=0
∇× E
∂t
25
(5.38)
A vector having a zero curl may be written as a gradient of a scalar and
therefore
~
∂A
~ = −∇Φ −
E
.
(5.39)
∂t
The above equations give the electric and magnetic fields in terms of a
~ and a scalar potential Φ. Substituting these expressions
vector potential A
~
into Equation (5.36) gives a wave equation for the vector potential A:
"
#
~
~ + µε ∂ ∇Φ + ∂ A = µ~
.
(5.40)
∇×∇×A
∂t
∂t
~ = ∇∇ · A
~ − ∇2 A
~ gives
Using the general vector identity ∇ × ∇ × A
~ + εµ
−∇2 A
~
∂2A
~ + εµ∇ ∂Φ = µ~
.
+ ∇∇ · A
∂t 2
∂t
(5.41)
~ may still be defined as most conThe divergence of the vector potential A
venient. The usual choice is the so-called Lorentz condition,
~ + εµ
∇·A
∂Φ
= 0.
∂t
(5.42)
Using this condition for the vector potential the equation is considerably
simplyfied:
2 ~
~ − εµ ∂ A = −µ~
∇2 A
.
(5.43)
∂t 2
A similar equation may be derived for the scalar potential,
∇2 Φ − εµ
∂2Φ
1
= − ρc .
∂t 2
ε
(5.44)
The Maxwell’s equations presented above have so far been quite general.
When the equations are applied to induction heating we may make some
additional assumptions. The first one is to assume that there are no free
charges. This simplifies the problem so that there is no scalar potential.
The source of the vector potential is the current density. There are two
kinds of currents in the system heated by induction. The driving force is the
current in the induction coil generated by an external current supply. Its
magnitude is well known. Using this given current the mean current density
in the coil may be calculated. In reality, the current distribution is far from
uniform but if the coil is relatively thin this has a negligible effect on the
fields at large distances. In conductors, currents are caused by electric
fields through Ohm’s law that states that current is directly proportional to
electric field. The electric field, on the other hand, may be obtained from
Equation (5.39) as the negative time derivative of the vector potential. The
26
current density for a sinusoidally driven system may therefore be written
as
~
∂A
(5.45)
~ = −σ
+ ~0 eıωt ,
∂t
where ~0 is the peak current amplitude applied to the system, σ is the
electric conductivity, and ı is the imaginary unit. The angular velocity of
the induction current ω is related to the frequency f of the current source
by
ω = 2π f .
(5.46)
The time-dependence of the current density causes also the solutions of
~=A
~ 0 eıωt ,
the field equations to vary with the same frequency. Writing A
and applying it to Equations (5.43) and (5.45) gives
~ 0 − εω2 µ A
~ 0 − ı ωσ µ A
~ 0 = −µ~
∇2 A
0 .
(5.47)
In free space εµ = 1/c 2 and the permeability is by definition equal to
4π 10−7 Ns2 /C2 . The second term in Equation (5.47) may be neglected if
L c/ω, where L is the characteristic length of the induction heating geometry. The condition is true for radio frequency induction heating and
thus the equation becomes
~ 0 − ı ωσ µ A
~ 0 = −µ~
∇2 A
0 ,
(5.48)
The instantaneous energy dissipation rate in the conductors is given by:
~=σ
−~
·E
~ ∂A
~
∂A
·
.
∂t
∂t
(5.49)
The relaxation time for the dissipation is much shorter than the time scale
for heat conduction. Thus the heat generation h(r~) can be represented by
the time-averaged quantity
h(~
r) = σ
2 +
*
∂A
1
~ 2
~
= σ ω 2 A
0 .
∂t 2
(5.50)
The Joule heating h now gives the source term in the energy equation. The
total heating power is obtained by integrating the local heat generation
over the total volume. The model presented here is basically the same as in
References [66, 71, 72].
27
Chapter 6
Numerical Models
The solution of the physical models presented in the previous chapter requires some elaborate computational techniques. The outline of the numerical and computational techniques is presented here. For further information about the methods the reader is encouraged to consult the given
references.
The description of equilibrium chemistry leads to a minimization problem with some additional constraints. It could basically be solved with
standard methods for minimization problems. However, the sometimes
very low partial pressures may make numerical derivation quite inaccurate. Therefore general optimization methods may fail and some special
methods for the case must be used [62].
Most of the physical models are expressed as partial differential equations. The finite element method (FEM) is a standard method for discretization of differential equations in complex geometries [73, 74, 75]. It was also
the method of choice in this work. There are different flavors of FEM, depending on what kind of elements are used and how the values of functions
are calculated inside the elements. In this work, the standard Galerkin formulation with linear elements is always used. The computational meshes
used are structured. Unstructured grids made of triangles would have certain advantages. Unfortunately the special requirements of the simulation
program eliminated the use of commercial grid generators, and therefore
also limited the use of sophisticated modern triangular grids. The advanced
features in the FEM calculations deal with the implementation of the specific
physical models and the multi-physics features of the code.
28
6.1
Equilibrium Chemistry
Applying Equations (5.3) and (5.11), Equation (6.3) simplifies to
There are several methods that can be used to solve the state of equilibrium.
A good method is fast and converges even with a poor initial guess. It seems
that no single method fulfills these two different requirements for all cases.
Therefore several different methods for the solution of the equilibrium
equations were implemented and tested. One is used to generate a good
initial guess while the others are useful in saving computing time later on.
Historically the problem of equilibrium chemistry of SiC growth system was often solved in terms of equilibrium constants. The number of
unknowns were then reduced by sometimes quite laborious manual manipulations [49, 50, 76]. Now, when the computational resources are more
abundant and chemical systems may be much larger such methods are no
longer needed. More generic methods are easier to apply to new systems
and are also quite fast to calculate [77, 78, 79, 80]. The methods used in
this work are mostly based on the ones presented in [62]. In addition some
heuristic methods were used to minimize the computational time and to
improve the rate of convergence.
In all the methods the problem of equilibrium chemistry is solved by
minimizing the Gibbs free energy. There are two main formulations for
minimizing the free energy. In the stoichiometric formulation the closed
system constraints are treated by means of stoichiometric equations. In the
nonstoichiometric formulation the closed system constraints are forced by
Lagrangian multipliers.
6.1.1
Stoichiometric Formulation
In the stoichiometric formulation, the molar amounts are expressed using
the extents of reactions as in Equation (5.11). Therefore the free energy
may be written as
~
G = G(T , P , ξ).
(6.1)
At equilibrium the free energy is at the minimum and hence its derivatives
at fixed temperature and pressure should vanish
!
∂G
= 0.
(6.2)
∂ ξ~
T ,P
Using the chain rule we may write the left side as:
∂G
∂ξj
!
T ,P
=
N
X
i=1
∂ni
∂ξj
!
∂G
∂ni
29
T ,P
,
j = 1, 2, . . . , L.
(6.3)
∂G
∂ξj
!
T ,P
=
N
X
νij µi ,
i=1
j = 1, 2, . . . , L.
(6.4)
The usual way to solve a set of nonlinear equations is the NewtonRaphson method [81]. As it is a second order method, if it converges it
does usually so quite fast. The Newton-Raphson method applied to Equation (6.2) suggests the following extents of reactions:
ξ(m+1) = −
∂2G
∂ ξ~2
!−1
~ (m)
n
∂G
∂ ξ~
!
.
(6.5)
~ (m)
n
Applying Equations (6.5) and (5.11) the Hessian matrix H = ∂ 2 G/∂ ξ~2 yields
Hij =
N X
N
X
∂2G
∂µk
=
νki νlj
,
∂ξi ∂ξj
∂nl
k=1 l=1
i, j = 1, 2, . . . , L,
(6.6)
The derivatives of the chemical potentials might in principle be calculated
numerically. However, this tends to be difficult since the molar fractions
may be very different in magnitude. Instead, analytical derivatives are used.
Here we have to make a distinction between gases and condensed species,
and between isometric (constant volume) and isobaric (constant pressure)
conditions. Differentiating Equation (5.4) and applying (5.5) gives
δ

1
ij

gas at constant P
 R T ni − nt
∂µi (T , P , pi ) 
δij
,
(6.7)
=
gas
at constant V
R
T

ni
∂nj


0,
condensed species
where δij is the Kronecker delta and nt is the total amount of species in the
gas phase [62]. For example, in the isobaric case with no condensed species
the Hessian yields,
Hij =
N
N X
X
δkl
∂2G
1
,
=RT
νki νlj
−
∂ξi ∂ξj
nk
nt
k=1 l=1
The equation may be rewritten as


N
X
νki νkj
ν̄i ν̄j 

,
Hij = R T
−
nk
nt
k=1
where
ν̄i =
N
X
k=1
30
νki .
i, j = 1, 2, . . . , L,
i, j = 1, 2, . . . , L,
(6.8)
(6.9)
(6.10)
The Hessian matrix can be inverted numerically. If the number of stoichiometric vectors (or reactions) is large, most of the time may be spent in that
process. For small systems that does, however, not impose any problems.
In order to increase the convergence speed and radius, the suggested
change in the species balance is checked before applying it to the system.
In order to do that a parameter w is used. It describes how much of the
suggested change ξ~ is accepted.
~ (m+1) = n
~ (m) + w (m+1)
n
L
X
j=1
(m+1)
~j ξj
ν
(6.11)
In principle it would be optimal to define the step size w so that the square
~ would be minimized. However, it is not wise to use too
norm of ∂G/∂ ξ,
much computational resources for the accurate determination of w. Starting with value w = 1 and going gradually down any value that results into
a norm that is significantly less than the original is accepted. This way also
the pathological case of negative amounts is avoided.
Very small partial pressures tend to make the Hessian matrix singular
and thus the iteration might fail. Therefore species are taken into consideration only if their molar fraction exceed some given limit, e.g., 10 −20 . The
iteration is repeated until the square norm of the left side is less than a
given cutoff value, e.g., 10−10 kJ/mol, or until the vector ξ~ vanishes.
The stoichiometric matrix can be formed from a set of known chemical reactions. This has the disadvantage that the reactions must be predetermined, which might not always be easy. Secondly, the stoichiometric
matrix arising directly from the reactions is not usually the computationally
most favorable. Instead, a method where the stoichiometric matrix V can
be deduced from the formula matrix A may be used.
The procedure is similar to Gauss-Jordan reduction [62]. Using elementary row operations the formula matrix A is set to unit matrix form:
!
IM Z
A∗ =
,
(6.12)
0
0
Where I M is M × M identity matrix. A complete stoichiometric matrix is
formed from A∗ by appending a L × L identity matrix below −Z; thus
!
−Z
V =
.
(6.13)
IL
There is also a faster, but not as well converging, method to solve the
equations arising from the stoichiometric formulation, the Villars-CruiseSmith method [62]. It makes use of the diagonal part of the stoichiometric
matrix. The method is presented in Appendix B.
31
6.1.2
Nonstoichiometric Formulation
In the nonstoichiometric formulation the material constrains are directly
taken into account by using Lagrange multipliers. Now we must minimize


M
N
X
X
~ = G(T , P , n)
~ λ)
~ +
L(n,
λk  bk −
aki ni  ,
(6.14)
k=1
i=1
~ is a vector of M unknown Lagrange multipliers. At minimum the
where λ
partial derivatives of Equation (6.14) must vanish:
and
∂L
∂ni
= µi −
∂L
∂λk
= bk −
M
X
aki λk = 0,
(6.15)
N
X
aki ni = 0.
(6.16)
k=1
i=1
These provide us with (N + M) equations necessary for solving the system.
~(m) ) gives:
~ (m) , λ
Linearization of Equation (6.15) around (n


(m)
N
M
M
X
X
X
∂µ
(m)
(m)
(m)
 i  δn(m) +
aki δλk = µi −
aki λk , i = 1, 2, . . . , N,
j
(m)
∂n
j=1
k=1
k=1
j
(6.17)
where
(m)
(m)
(6.18)
δλk = λk − λk ,
and
(m)
δnj
(m)
= nj − nj
.
(6.19)
~ and n
~ (m) are related through the element-abundance conThe quantities n
straints arising from Equation (5.9):
N
X
(m)
j=1
δnj
where
(m)
bk
=
(m)
= bk − bk
N
X
j=1
(m)
akj nj
,
,
k = 1, 2, . . . , M,
k = 1, 2, . . . , M.
(6.20)
(6.21)
For ideal systems the number of equations can be reduced from M + N to
M + 1. Substituting Equation (5.4) into (6.17) and solving for δn j yields
(m)
δnj
=
(m)
nj

(m)
M
X
µj
a
λ
kj
k
,
u +
−
RT
RT
k=1

32
j = 1, 2, . . . , N,
(6.22)
where the additional variable u is defined by
u=
PN
(m)
j=1 δnj
(m)
nt
(m)
=
δnt
(m)
nt
.
(6.23)
i=1
Substituting Equation (6.22) into (6.20) gives:


M
N
1 X X
(m) 
(m)
aik ajk nk
λi + b j u =
RT i=1 k=1
N
1 X
(m) (m)
(m)
aj knk µk + bj − bj ,
RT k=1
j = 1, 2, . . . , M.
(6.24)
The additional equation for closing the system is obtained by summing
Equation (6.22) over j to give
M
N
1 X (m)
1 X (m)
b i λi − n t u =
n
µk .
RT i=1
RT k=1 k
where U is a conveniently selected test function space. The unknown
temperature distribution is expressed as a linear combination of the basis functions ϕi ,
I
X
T (~
r) =
ti ϕi (~
r ),
(6.28)
(6.25)
where the values ti are the unknown temperatures at selected points and I
is the number of nodes.
The real-life geometries are seldom rectangular. Therefore also the elements used for discretization should be nonrectangular. It is nowadays
customary to use isoparametric elements to represent the geometry and
use local coordinates in assembling the matrix equations. In this work, all
the geometries are two-dimensional and only bilinear isoparametric basis
functions are used. In the axisymmetric case the global coordinates r and
z are expressed with the help of normalized coordinates ξ ∈ [−1, 1] and
η ∈ [−1, 1]. The mapping
r (ξ, η)
The algorithm consists of solving the set of M + 1 linear Equations (6.24)
~ (m) . The suggested
and (6.25), and using Equation (6.22) to determine δn
correction is checked and the iteration is terminated as in the stoichiometric
iteration scheme.
6.2
Temperature Distribution
The finite element method is used to solve the energy equation. This equation will be considered first since the finite element method is most conveniently introduced for this case. The discretization of the differential
equation starts with writing the corresponding variational formulation, a
so-called weak form of the equation. In the axisymmetric case equation
(5.24) yields
∂T
∂
∂T
1 ∂
−
κz
−
κr T
= h,
(6.26)
∂z
∂z
r ∂r
∂r
where the thermal conductivity tensor is assumed to be diagonal. Integrating over an axisymmetric volume Ω with boundary ∂Ω and applying Green’s
theorem, Equations (6.26) and (5.25) give
Z
Z ∂v ∂T
∂v ∂T
r dr dz +
κz
+ κr
vq r dl
∂z ∂z
∂r ∂r
Ω
∂Ω
Z
vh r dr dz. ∀v ∈ U,
(6.27)
=
Ω
33
z(ξ, η)
=
=
4
X
fi (ξ, η) ri ,
i=1
4
X
fi (ξ, η) zi .
(6.29)
i=1
transforms the local point to the global coordinate system. The bilinear
isoparametric element is defined by the following basis functions:
f1 (ξ, η)
=
f2 (ξ, η)
=
f3 (ξ, η)
=
f4 (ξ, η)
=
1
(1 − ξ)(1 − η),
4
1
(1 + ξ)(1 − η),
4
1
(1 + ξ)(1 + η),
4
1
(1 − ξ)(1 + η).
4
(6.30)
The basis functions above are local basis functions associated with a given
element. They are related to the global basis functions by
ϕi (~
r) =


 fj (ξ, η),


0,
if r~ is within an element whose
local node j is the global node i,
otherwise.
(6.31)
In the Galerkin formulation the basis functions are also used as test
functions. Inserting the basis functions gives a set of equations for the
34
unknown temperatures:
Z
I
X
∂Ω
"Z
!
#
∂ϕj ∂ϕi
∂ϕj ∂ϕi
+ κr
r dr dz ti +
∂z
∂z
∂r ∂r
Ω
i=1
Z
ϕj q(~
r , t1 , t2 , . . . , tN )r dl =
ϕj hr dr dz, j = 1, . . . , I.
κz
Ω
(6.32)
The integration is performed numerically using Gaussian quadratures. The
procedure results in a matrix equation that may be solved applying standard methods of linear algebra. The type of heat exchange q determines
the nature of equation (6.32). If it is linear with temperature also the total
equation is at least pseudo-linear. There might still be temperature dependent parameters, such as the thermal conductivity κ. If the heat exchange
is due to radiation, the equation can still be solved as if it were linear. This
requires the use of Equation (5.27).
Linear boundary conditions can be split into two,
Z
∂Ω
ϕj qi r dl =
I Z
X
i=1
∂Ω
ϕi ϕj hT r dr dz ti −
Z
∂Ω
ϕj hT Text r dr dz. (6.33)
Here, the first term is linear with respect to T and the second term is
constant, or can at least be handled as constant. Truly nonlinear boundary
conditions may lead to problems in convergence when treated as linear.
This is the case for a boundary that is coupled to itself through the Gebhart
factors. The discrete counterpart of Equation (5.26) is
4
,
(6.34)
qi = σSB i Ti4 − Text,i
where
4
Text,i
=
Is
1 X
k Ak Gki Tk4 ,
Ai i k=1
(6.35)
where Ai is the area of an surface element i and Gji the discrete Gebhart
factor and Is the number of surface elements. The Gebhart factors Gji are
solutions of
Is
X
Fjk (1 − k )Gki = Fji i ,
(6.36)
Gji −
We have basically two choices how to solve the nonlinear equation. The
simple one is to linearize radiation according to the formula (5.27). If, however, radiation is the dominating heat-transfer mechanism this may lead
to very poor convergence. In this case Newton-Raphson type of iteration
can be used. In the linearized method the external temperature is calculated from the previous iteration step (explicit method), whereas in NewtonRaphson iteration it is also regarded as an unknown (implicit method). If
the contribution of radiation is treated simply as a source term it results
in severe convergence problems for well-isolated systems even though for
most systems it works perfectly well.
6.2.1
In the linear iteration the solution satisfies the equation
A(t)t = f (t),
Aji =
Z Z
Si
Sj
cos βi (~
ri , r~j ) cos βj (~
ri , r~j ) χ r~i , r~j dSi dSj .
2
ri − r~j π ~
35
Ω
κz
∂ϕj ∂ϕi
∂ϕj ∂ϕi
+ κr
∂z ∂z
∂r ∂r
fj =
Z
Ω
!
ϕj hr dr dz, +
r dr dz +
Z
∂Ω
Z
∂Ω
ϕj ϕi hT r dl
ϕj hT Text r dl.
(6.39)
(6.40)
The iteration scheme is
t (m+1) = A−1 (t (m) )f (t (m) ),
6.2.2
(6.41)
Newton-Raphson Iteration
In the Newton-Raphson iteration we must find vector t that satisfies the
nonlinear equation
g(t) = A(t)t + b(t) − f = 0,
(6.42)
where
!
∂ϕj ∂ϕi
∂ϕj ∂ϕi
+ κr
r dr dz,
∂z ∂z
∂r ∂r
Ω


Z
Is
1 X
4
4

bj =
ϕj σSB i ti −
k Ak Gki tk r dl
Ai i k=1
∂Ω
Aji =
(6.37)
The difference between linear and nonlinear models is important in selecting the best method for solving the equations.
Z
and
where Fij is the view factor between two surface elements,
1
Ai
(6.38)
where
k=1
Fij =
Linear Iteration
and
Z
κz
fj =
Z
Ω
ϕj hr dr dz.
36
(6.43)
(6.44)
(6.45)
The iteration scheme is
t
(m+1)
=t
(m)
−J
−1
where
J =A+
and
∂bj
=
∂ti
Z
∂Ω
4ϕj σSB i
ti3
(t
(m)
)g(t
(m)
),
(6.46)
vector may be represented as a linear combination of the stoichiometric
vectors,
M
X
~i ,
S~ =
ci ν
(6.52)
(6.47)
where
i=1
∂b
,
∂t
Aj j
−
Gji tj3 r dl.
Ai i
(6.48)
The implicit Newton-Raphson type of iteration may lead to a very different type of matrix equation compared to the linear iteration. The elements
within a closure that see each other form a full block in the matrix, while
linear iteration leads to a sparse matrix with a banded structure.
6.3
Diffusion of Reactive Gases
The mass transport inside the crucible involves diffusion and homogeneous
chemical reactions. In addition the boundary conditions introduce adsorption-desorption processes and heterogeneous chemical reactions.
The diffusion equation defines the species flux and thus the equilibrium
chemistry should somehow give the production rates — not absolute molar
~ that satisfies the diffusion
quantities. Suppose that we have a candidate n
equation.
~
∇ · J = S.
(6.49)
The system should also be in gas-phase equilibrium. We introduce operator
~ s1 , . . .) that minimizes the Gibbs free energy at constant volEquilibrium(n,
ume in the presence of condensed species {s1 ,. . .}. Gas-phase equilibrium is
obtained from
~ eq = Equilibrium n
~ .
n
(6.50)
When the system is driven to equilibrium the residuals of the diffusion
equation obviously result from the chemical reactions and thereby give the
suggested source terms,
(6.51)
S~0 = ∇ · J eq ,
ci =
(S~0 −
Pi−1
~j )
j=1 cj ν
~i · ν
~i
ν
~i
·ν
.
(6.53)
The projection operator will be shortened by
S~ = P g (S~0 )
(6.54)
and it returns a vector if the argument is a vector, and a matrix if the
argument is a matrix. As the projection operation is independent of the
space coordinates it may be taken inside the divergence operator,
S~ = ∇ · P g (J eq ).
(6.55)
The projection operator certifies that the solution of Equation (6.49) is
unique by defining the direction of the solution in species space. Otherwise the solution would depend on the initial guess that is used.
From a physical point of view the boundary conditions for the diffusion
equation are determined by the adsorption-desorption flux
~ =β· n
~−n
~ eq,s ,
J · e~n = R
where the contact with condensed species is marked with s and
s
RT
βkl = δkl γk
.
2π Mk
(6.56)
(6.57)
~ eq,s are defined
The equilibrium partial pressures at the reactive surface n
in a unique way only if the number of condensed species is the same as the
number of elements. This is assumed to be the case on the subliming surface which is experimentally known to be graphitized and thus equilibrium
with condensed SiC and carbon may be assumed,
~ eq,s = Equilibrium n
~ s , SiC, C .
n
(6.58)
where the superscript refers to the species flux calculated using the equilibrium values for partial pressures. The source terms arising from chemical
reactions should conserve the amounts of basic elements. This is not, however, an inherent property of the suggested source terms. Therefore they
must be corrected using an appropriate projection [82]. The projection
must be such that the source vector S~ conserves the amount of basic elements, or in other words, is orthogonal to the formula matrix A. Any such
~ s is some initial gas-phase composition over the reactive surface.
where n
On the growing surface the two conditions for determining the adsorption-desorption flux are obtained when the condensing species is predefined. This a priori assumption may later be checked in equilibrium calcu~ eq,s . It must be in equilibrium with
lations. The problem is to determine n
the condensing species, and on the other hand, it must result to a species
37
38
~ that has the same stoichiometry as the condensing species. In the
flux R
case of SiC this means that
~ s , SiC
~ eq,s = Equilibrium n
n
and
~ ·a
~ ·a
~Si − R
~C = 0.
R
(6.59)
dk =

 0,

eq,s
nk −nk
aSi,k −aC,k ,
if aSi,k − aC,k = 0
otherwise
(6.63)
is a vector that depending on the multiplier brings either silicon or carbon
to the reactive surface and
ζ=
~ −a
~
~C · R
~Si · R
a
~ +a
~
~Si · R
~C · R
a
(6.64)
is the disbalance between the two element fluxes. The initial guess for the
concentrations is obtained from
~ eq,s(0) = Equilibrium n,
~ SiC .
n
~ 0 = e~n · J eq ,
R
(6.60)
Equations (6.56), (6.59) and (6.60) cannot be solved analytically and are thus
enforced by solving alternatingly the following two equations
~−n
~ eq,s(n) ,
(6.61)
R(n) = β · n
~ eq,s(n+1) = Equilibrium n
~ eq,s(n) + ζ (n) d~(n) , SiC ,
n
(6.62)
where
stoichiometric correction to the flux has only a minor effect on the solution.
Thus in the diffusion-limited case, the FEM iteration would converge very
slowly. In order to have good convergence the stoichiometric correction in
the diffusion-limited case should be applied directly to the diffusion flux.
The suggested diffusion flux on the reacting boundary is
(6.65)
(6.66)
where e~n is a normal of the surface. Also now the stoichiometry of the
flux must be the same as that of the growing substance. Therefore the
suggested flux must be corrected with an appropriate projection operator
for heterogeneous chemistry P s which corresponds to P g in homogeneous
chemistry. Thus the boundary source term yields
~ = e~n · P s (J eq ).
R
(6.67)
This boundary condition does not add any new physics to the system. It is
just another way of managing the stoichiometry of the surface flux.
So we have obtained two different ways for fixing the stoichiometry of
the species flux. In order to obtain good convergence, the rate limiting
step should also define which of the two formulations to use. Ultimately,
in mass transport limited growth diffusion flux would be controlled, and
in surface reaction limited growth the adsorption-desorption flux. We do
not, however, have the luxury of knowing the rate limiting step in advance.
The conditions have the convenient property that they do not affect a true
solution at all. Therefore both boundary conditions may be freely applied
in any order. This is done by uniting Equations (6.56) and (6.67) by
~ = g1 e~n · P s (J eq ) + g2 β · (n
~ −n
~ eq,s ),
R
(6.68)
The presented iterative procedure is a heuristic one, but it has physical basis because the element that is transported more efficiently will gather at
the surface and eventually decrease the mass transport of that particular
element. When the disbalance ζ vanishes no correction to the concentrations is applied. Once the iteration has converged the correct boundary
~ is obtained and n
~ eq,s is in equilibrium with the condensing species.
flux R
In practical calculations less than ten iterations were required when the
convergence criteria was ζ < 10−6 . It should be noted that this iteration is
performed for each side node individually as a part of applying boundary
conditions for the FEM calculation that requires iteration in itself.
The boundary condition resulting from the adsorption-desorption flux
results to good convergence only when the growth is limited by the surface
processes. This may be the case if the source-to-seed distance is small and
the pressure of the inert gas is low.
When the growth rate is limited by the species diffusion the partial
pressures over the Knudsen layer change very little and the above presented
In this scheme the adsorption-desorption flux is set every other iteration,
and otherwise the direction of the diffusion flow (in species space) is corrected using the stoichiometric flux condition. The scheme converges well
for most cases but if the source-to-seed distance is so small that diffusion
does not limit the growth rate a better convergence may be obtained with
the adsorption-desorption condition alone. In the diffusion limited cases
the adsorption-desorption condition is required in order to set equilibrium
with condensed SiC at the boundaries.
39
40
where g1 and g2 are logical functions that depend on the iteration step
m. The functions should be selected so that neither of the two conditions
dominates. A simple way to choose is
g1
g2
=
=
m mod 2,
(m + 1) mod 2.
(6.69)
Now we are ready to introduce the finite element formulation of the
diffusion equation. For simplicity let us assume axisymmetric geometry
and neglect diffusion due to temperature gradients. Now the diffusion
equation yields




∂ X
∂nl  1 ∂ X
∂nl 
Dkl
r Dkl
(6.70)
−
−
= Sk , k = 1, . . . , N.
∂z
∂z
r ∂r
∂r
l
l
We apply the Galerkin method in a standard manner. The variational formulation of Equation (6.70) is


Z
X
∂nl
∂nl 
∂v X
 ∂v
r dr dz +
+
Dkl
Dkl
∂z l
∂z
∂r l
∂r
Ω
Z
Z
vRk r dl =
vSk r dr dz, ∀v ∈ U.
(6.71)
∂Ω
Ω
The molar fractions are written as a linear combination of the basis functions by
I
X
nk (~
r) =
wk,i ϕi (~
r ),
(6.72)
The source terms at the right hand side are evaluated using the values given
by the thermodynamic calculations. It should be noted that the source
terms of the gas phase give also a contribution to the surface reactions. In
Equation (6.74) this is seen by the double projection.
The solution of the mass transport requires an iterative solution. To provide a good initial guess, the concentrations are first calculated assuming
a global equilibrium with the solid sources. Thereafter the finite element
problem and equilibrium chemistry problem are solved consecutively. The
source terms in the finite element calculations are evaluated using updated
values from the equilibrium calculations. When convergence is reached the
solution is at chemical equilibrium everywhere and satisfies the diffusion
equation.
It is possible that the assumed boundary conditions are not the true
ones. In the Si–C system the condensed species may include one or two
species from the set C, Si or SiC. The practical experiments suggest the most
probable condensed species. As the system deviates from equilibrium during the iteration, the condensed species may not be reliably defined. When
convergence is reached the initial assumption is checked. If it proves to be
wrong the calculation may be continued with updated boundary conditions.
i=1
where values wk,i are the unknown molar fractions at the nodal positions.
The Galerkin method gives a set of equations for the unknowns,
I X
N
X
i=1 l=1
"Z
Z
Ω
∂Ω
!
#
∂ϕj ∂ϕi
∂ϕj ∂ϕi
+
r dr dz wl,i +
∂z ∂z
∂r ∂r
Z
ϕj Rk r dl =
ϕj Sk r dr dz, j = 1, . . . , I.
Dkl
Ω
(6.73)
Applying (6.55) and (6.68) to Equation (6.73) yields
"Z
I X
N
X
i=1 l=1
+
=
+
+
I
X
Ω
g2 βkk
i=1
I X
N
X
i=1 l=1
"Z
Ω
∂Ω
Dkl
g1
g2 βkk
Z
i=1 l=1
i=1
Z
Z
N
I X
X
I
X
Dkl
∂Ω
∂Ω
∂ϕj ∂ϕi
∂ϕj ∂ϕi
+
∂z ∂z
∂r ∂r
ϕj ϕi r dl wk,i
∂ϕj ∂ϕi
∂ϕj ∂ϕi
+
∂z ∂z
∂r ∂r
Dkl ϕj en,z
ϕj ϕi r dl
!
!
#
r dr dz wl,i
r dr dz P
∂ϕi
∂ϕi
+ en,r
∂z
∂r
eq,s
wk,i ,
41
#
g
eq
r dl P s (wl,i − P g (wl,i ))
j = 1, . . . , I.
Induction Heating
Generally, Equation (5.48) for the magnetic vector potential is a complex
vector equation with six field variables. Luckily, the induction heating geometries are often axisymmetric which simplifies the problem a great deal.
The external current density is usually caused by a coil that forms a helix
around the object to be heated. Neglecting the helicity of the coil the external current density may be written in cylindrical coordinates as ~0 = 0 e~θ .
~ 0 = Ar e~r +
The unknown vector potential has three components, A
Aθ e~θ + Az e~z . At infinity all the components should vanish. Thus, the
components in the absence of any sources are zero everywhere. This leaves
us only the equation for Aθ which in cylindrical coordinates yields
1 ∂
r ∂r
eq
(wl,i )
eq
6.4
(6.74)
r
∂Aθ
∂r
−
Aθ
∂ 2 Aθ
+
− ı ωσ µAθ = −µθ .
r2
∂z 2
(6.75)
Dividing the azimuthal component of the vector potential into in-phase
and out-of-phase parts, Aθ (r , z) = C(r , z) + ı S(r , z), and inserting this
into Equation (6.75) we obtain a coupled pair of equations for C and S
1 ∂
r ∂r
r
∂C
∂r
−
C
∂2C
+
+ µσ ωS = −µ0
r2
∂z 2
42
(6.76)
1 ∂
∂2S
∂S
S
− µσ ωC = 0
(6.77)
r
− 2 +
r ∂r
∂r
r
∂z 2
Infinitely far away from the coil the components of the vector potential
should vanish:
C = S = 0, as (r , z) → ∞.
(6.78)
The average heat generation h(r , z) is now obtained using Formula (5.50):
h(r , z) =
h
i
1
σ ω2 C 2 (r , z) + S 2 (r , z) , .
2
(6.79)
Equations (6.76) and (6.77) are linear and elliptic differential equations
that are easily solved with the finite element method. Integrating (6.76)
over volume Ω and applying Green’s theorem gives
Z
Z ∂C ∂v
∂C ∂v
v
+
r dr dz +
C dr dz +
∂r ∂r
r
Ω ∂z ∂z
Ω
Z
Z
µωσ (~
r )Sv r dr dz, =
µ0 (~
r )v r dr dz ∀v ∈ U.
(6.80)
Ω
Ω
The in-phase and out-of-phase amplitudes of the vector potential are both
linear combinations of the basis functions:
C(~
r) =
I
X
ci ϕi (~
r)
S(~
r) =
i=1
I
X
si ϕi (~
r ).
(6.81)
i=1
Substituting the formulae (6.81) into Equation (6.80) and using the basis
functions also as test functions gives:
"Z
!
#
I
X
ϕi ϕj
∂ϕi ∂ϕj
∂ϕi ∂ϕj
r+
r+
dr dz ci +
∂z ∂z
∂r ∂r
r
Ω
i=1
I
X
i=1
Z
Ω
µωσ (~
r )ϕi ϕj r dr dz si =
Z
Ω
µ0 (~
r )ϕj r dr dz.
(6.82)
Likewise, for the out-of-phase component:
!
#
"Z
I
X
ϕi ϕj
∂ϕi ∂ϕj
∂ϕi ∂ϕj
dr dz si −
r+
r+
∂z ∂z
∂r ∂r
r
Ω
i=1
I Z
X
i=1
Ω
µωσ (~
r )ϕi ϕj r dr dz ci = 0.
h
i
1
σ ω2 ci2 + si2 .
2
43
P = 2π
I Z
X
i=1
Ω
ϕi h r dr dz .
(6.85)
If the heating power is to be set to a given value P0 the heat generation may
be multiplied by a factor P0 /P.
The components of the vector potentials should vanish at infinity. In
practice it is of course not possible to extend the calculations to an infinite
domain. A feasible approach is to increase the domain until a further
increase has no significant effect on the results. Another alternative would
be to use a special kind of elements that extend to infinity. This was not,
however, considered necessary as the computational time consumed was
not critical.
6.5
Virtual Crystal Growth
A problem in modeling crystal growth phenomena is that the system changes
with time, but the modeling approach is static. One way to tackle the problem is to gradually change the geometry to account for the growth. When
the iteration has converged the growth rate V on each reacting surface may
be evaluated from the species flux by
V =
~ ·M
~
R
.
ρ
(6.86)
The growth direction is perpendicular to the surface. The local coordinate
displacement at the reaction fronts is now
~0 = V ∆t e~n ,
U
(6.87)
where ρ is the density of the growing crystal, ∆t is a virtual time step, and
e~n is a normal of the surface. We solve the other coordinate displacements
from
~ = 0.
∇2 U
(6.88)
(6.83)
Equations (6.82) and (6.83) are valid for any j = 1, . . . , I and therefore form
2I linear equations that may be used to calculate values c i and si . The local
heat generation hi at node i is now obtained from Equation (6.79) to be
hi =
The total heating power P in the system is obtained from the integral
(6.84)
The natural boundary condition for this formula would be a Dirichlet condition. It turns out, however, that on the corners there may be conflicting
boundary conditions for the displacement. Also, the displacements of the
nodes must be fixed only in the direction of the surface normal. Therefore,
a flux condition is used,
~ = p e~n · (U
~−U
~0 ),
−~
en · ∇ U
44
(6.89)
where the weight (p ≈ 106 ) is chosen so that the boundary conditions are
satisfied tightly. The equation for the displacements is solved with the finite
element method and the displacements are added to the old coordinates.
All the equations may be solved also with the new coordinates when using isoparametric elements. Unfortunately, the elements may be corrupted
step by step, and finally isoparametric mapping is not possible. Also some
accuracy is lost when the elements are being deformed. Therefore this
procedure is only feasible if the growth surface progresses smoothly. Otherwise more sophisticated methods for generating the new mesh should be
used.
6.6
Feedback Mechanisms
In practise the sublimation growth process is controlled by means of various
feedback mechanisms. For instance, there is a pyrometer that measures the
temperature at the crucible top or bottom. Usually the aim is to keep the
temperature constant throughout the growth process. Therefore the power
input of the induction heating is altered so that the temperature measured
by the pyrometer remains constant. The usability of the simulation tool is
enhanced by implementing feedback mechanisms similar to the ones of the
real system. Also for optimization purposes the feedback mechanisms are
useful.
The state of the system may be described by a number of state functions which depend on a number of parameters. Because the goal is not to
generate a general feedback mechanism we assume that the state functions
may be set independently of each other.
In a simple feedback we set the state function to a predefined value.
Since a unique solution is preferred, the state function only depends on
one parameter, y = f (x) . The solution for f (x) = y0 is obtained by
applying the secant method, where the derivative of the object function is
calculated numerically,
x (m+1) = x (m) −
x (m) − x (m−1) (m)
y
− y0 .
(m)
(m−1)
y
−y
(6.90)
The first step is given manually. If the stepsize becomes too small the
derivative is not updated. The method requires a monotonic object function. Also it requires that the iteration has converged well enough to provide
reliable values for the state function. Some relaxation may be applied to
improve the convergence of the feedback.
The accuracy of the feedback is highly boosted if we have some a priori
knowledge of the functional dependencies. Consider that we want to set
45
the temperature at some point by altering the heating power. We know that
the temperature rises monotonically with the heating power. The power
loss due to conduction is proportional to the first power and the loss due
to radiation to the fourth power of temperature. Around a given operation
point the heating power P may therefore be approximated by
P = a(T − T0 )b ,
(6.91)
where a and b are free parameters. For pure conduction b obtains the
value 1. As the heating power increases the influence of radiation becomes
more and more prominent and thus b approaches the value 4. Because
the working temperature of the growth system is usually quite the same
we may estimate b using a typical geometry. Then the value a may be
estimated from Equation (6.91). If the preferred temperature is T 0 then the
corresponding heating power is suggested by
P0 =
T 0 − T0
T − T0
b
P.
(6.92)
This formula makes the feedback very robust. In practical calculations the
value b = 2.6 was used very successfully for typical sublimation growth
systems.
If the different state functions are relatively independent several feedback mechanisms may be applied simultaneously. For example, the absolute temperature depends mostly on heating efficiency while the temperature gradient depends on the coil position. Therefore, the absolute
temperature and temperature gradient may be set simultaneously.
6.7
Coupled Model
In the previous pages models for the individual physical processes occurring in the sublimation growth process have been presented. Obviously
the individual physical phenomena form a coupled system. The temperature distribution depends on the induction heating, while the induction
heating depends on the temperature via the temperature dependent electric conductivities. The chemical equilibrium depends on the temperature
distribution via the temperature dependent chemical potentials. Also the
temperature distribution may depend on the chemistry by the heat release
associated with chemical reactions. Therefore, the whole model must be
treated as a coupled system. This is a multi-physics problem that requires
special attention. We will shortly introduce the multi-physics features and
review the physical models related to it [2].
46
The second scope arises from the global temperature distribution solved
from the energy equation,
Numerical model for the
sublimation growth process
with
Maxwell’s equations
and
Energy equation
Diffusion
equation
−∇ · (κ · ∇T ) = h. in Ω2 ,
(6.95)
4
q = − (κ · ∇T ) · e~n = σSB T 4 − Text
on Γ2R ,
(6.96)
T = Text on Γ2D .
(6.97)
The temperature is solved iteratively with formulas (6.42)-(6.47). Feedback
for temperature control described by Equation (6.92) may be applied by
normalizing the heat generation h. The induction heating problem in the
first scope is resolved after every few thermal iterations. The temperature
iteration is terminated when the maximum relative change of temperature
and heat generation fall under given limits, e.g 10−5 and 10−3 .
The third scope of interest is the reactive gas inside the closure,
Equilibrium
chemistry
Figure 6.1: A schematic model of the solution procedure
∇ · J = ∇ · P g (J eq ) in Ω3 ,
(6.98)
~
J = −D · ∇n
(6.99)
~ = J · e~n = β · (n
~ −n
~ eq,s ) on Γ3 .
R
(6.100)
where
There are several requirements arising from the coupling of physical
models. The individual models tend to have different domains. For example, the induction heating problem should in principle be solved in an
infinitely large domain, whereas the diffusion of reactive gases needs to be
solved only inside the crucible. These different models also require very
different meshing. Hence, a natural solution is to use different meshes for
different physical models and to map the results between the meshes.
The complete model of the sublimation process is presented in Figure 6.1. The system is treated on three different levels each having different meshes and computational domains. All the physical phenomena may
be coupled. The field variables must be interpolated between the different
meshes until the solution is consistent in each of them.
The first scope is that of induction heating. It provides the local heat
generation for the temperature distribution calculations. It is obtained from
the equation for the vector potential
∇2 Aθ − ı ωσ µAθ = −µ0 in Ω1 ,
(6.93)
Aθ = 0 on Γ1 .
(6.94)
with
These are numerically solved with Equations (6.82) and (6.83) and the local
Joule heating is obtained from (6.84).
47
and
~ eq
Here n is obtained by minimizing the Gibbs function in the gas phase
~ eq,s is obtained by minimizing the
and at constant volume, while and n
Gibbs function in the gas-solid interface at constant pressure. This is most
often done with the stoichiometric iteration scheme given by Equations
(6.5)–(6.11). For the growing surface the flux is controlled by fixing the
~ eq,s by Equations (6.62)–(6.65). The solution of the mass
stoichiometry of n
transport is obtained by an iterative algorithm given by Equation (6.74).
The iteration is terminated when the maximum disbalance and maximum
change in molar fractions are less than the predefined limits. In the FEM
calculations the heat of crystallization is omitted and the temperature is
not recalculated after the mass transport calculations.
When the calculations of the coupled model have converged we obtain
the mass flux at the surfaces which gives the growth shape and growth rate.
These are quantities that may also be truly tested by experiments. If the
virtual crystal growth model is applied the coordinate displacements are
calculated from Equations (6.87)–(6.89) and thereafter the calculations are
continued from the beginning.
The coupling of the different equations is assumed to be weak. This
means that the equations are solved one by one and in each equation only
48
the variable under solution is treated implicitly. This assumption seems
to be well motivated since no convergence problems have occurred in the
actual simulations.
The multi-physics characteristics of the crystal growth process require
that the variables are mapped between different meshes. This results in an
inverse problem: We have to find the local coordinates ξ and η that give
the global coordinates r and z. In a general case this leads to a difficult
task. We should first find the right element in which the global point is and
then solve a fourth order equation arising from Equations (6.29). However,
if all the meshes are rectangular in the beginning the mapping is easy to
calculate. Then Equations (6.29) lead to simple first-order equations
2r − r1 − r2
,
r2 − r 1
2z − z2 − z3
η=
.
z3 − z 2
Chapter 7
Verification and Evaluation
ξ=
(6.101)
then the original mapping is conserved.
In the simulation program the mesh generator is fully integrated to the
solver. Therefore, we always have the topology of the mesh at our disposal.
Non-rectangular forms are created by moving straight lines defining the
natural boundaries to lines defined by analytical functions. Between these
natural boundaries a linear transformation is applied. When the different
meshes are transformed by the same linear operations the local coordinates required in mapping remain unaltered. Thus the weights used in the
mapping may be calculated using the original rectangular meshes. Also the
virtual crystal growth model transforms the meshes in a linear manner and
the mapping is unaltered.
To make the geometry generation as quick as possible the same geometry is used to create all the three meshes. The unnecessary parts of the
geometry may be neglected in mesh generation and thereby elements are
created only where they are needed. The mesh generator also creates a predefined number of elements while still fulfilling the constraints on relative
mesh density and local mesh refinement. The mapping of variables from
mesh to mesh is automatic. Always the newest results of a given variable
are mapped from mesh to mesh.
The implementation of the introduced physical models was a major part
of this work. The total amount of code generated was around 30000 lines.
The sparse matrix libraries are the only major component that has not been
developed within this work. For general sparse matrices the shareware
library SPARSE written by Kenneth S. Kundert and Alberto SangiovanniVincentelli of University of California was used. For special band matrix
structures the LAPACK routines were applied. Also a piece of code written
by Juha Katajamäki has been used to compute the view factors [67]. The
postprosessing is carried out with separate software packages, mainly with
Matlab and ELMERPOST, the latter developed by Juha Ruokolainen. The
preprocessor is integrated to the solver and was also written within the
work.
A correct solution for an equation is, in principle, obtained if consistency
is proven and convergence is reached. Convergence was reached for all the
results presented in this work. However, consistency needs to be proven
in some way. Therefore, the simulation codes should always be verified.
The easiest and most accurate way to do this is to compare the results of
the code against analytical solutions. This is a feasible method only for
simple equations in simple geometries. The results may also be compared
against well established codes that have been extensively tested over the
years. Unfortunately, the full model could not be verified in this way since
the software includes some features that are not available in commercial
packages and analytical solutions are not available.
When convergence and consistency have been proven the exact solution
is basically obtained by increasing the accuracy of the calculations. In finite
elements this means increasing the density of the computational mesh.
When the size of an individual element approaches zero so should the
discretization error. In practice this is of course not possible. A feasible
49
50
This kind of mapping is very quick to compute. If we perform the same
linear operations on all the meshes this simple mapping will still be valid in
the transformed coordinates. This may easily be seen by studying Equations
(6.29). If the operator O transforming the coordinates is such that


4
4
X
X
O(r ) = O 
f i ri  =
fi O(ri ),
(6.102)
i=1
i=1
7.1
Equilibrium Chemistry
The stoichiometric and nonstoichiometric models for determining the state
of equilibrium of a chemical system were implemented. In defining the
chemical equilibrium the minimization algorithm itself includes many possibilities to error. However, the free energy changes in chemical reactions
are very easy to calculate: it is just a problem of simple addition and subtraction. If the change in energy for each reaction is zero and the amount
of basic elements is conserved the solution is a true one.
The models were applied to the Si–C and Si–C–H gas-phase systems
in constant pressure. The first system included 18 different species and
the second one 68 species. The thermodynamic data was obtained from
References [83, 84, 85].
Both methods give the same results with a very good accuracy. If the
iteration converges, a residual of 10−10 kJ/mol is easily reached with both
methods. This is demonstrated in Figure 7.1 showing the residuals as a
function of iteration step. The first case is calculated with the stoichiometric method. Then temperature is raised with 20 ◦ C and both methods are
applied to find the new state of equilibrium. It may be observed that the
first state is difficult to find but in the vicinity of the solution the iteration
converges very rapidly with both methods. They also conserve the amount
of basic elements very accurately. However, the stoichiometric formulation
51
5
10
0
10
∆G (kJ/mol)
approach is to increase the resolution of the calculations until a further
increase does not affect the results significantly.
Sometimes it is useful to calculate data purely for debugging purposes.
For example, for an elliptic differential equation the solution may be evaluated by comparing the flux through the boundaries to the source over
the volume. They should be closely equal if the solution is accurate. This
provides a good way of checking the results. It does not actually matter
how the solution of an equation is reached as far as the residuals vanish.
Moreover, the residuals are often much easier to calculate.
The complete model is so complicated that errors are always possible
even after careful debugging. However, the individual models were tested
quite extensively and have been applied to a number of cases. Also, the results of the real-world experiments were in harmony with the simulations
even though they can hardly verify the code. Many documented phenomena could be explained by simulations and no experiments violated the
modeling results, or vice versa.
In the following sections it is explained how the separate parts of the
software were verified and evaluated.
−5
10
∆T=20 C
−10
10
−15
10
0
5
10
15
m
20
25
30
Figure 7.1: Gibbs energy change of the reactions as a function of iteration
step with the stoichiometric (solid line) and nonstoichiometric method (dashed
line).
has a significantly larger radius of convergence and therefore it is preferred
when determining the first state of equilibrium. The nonstoichiometric
formulation diverges easily if the chemical potentials change considerably.
This happens if, for example, the temperature step between consecutive
stages is large. As the number of iterations required depends greatly on
the quality of the initial guess, the first state usually takes the longest time
to compute.
The advantage of the nonstoichiometric formulation is the shorter calculation time. This is due to the smaller size of the matrix that has to be
inverted. In the stoichiometric formulation the size is N − M whereas the
nonstoichiometric formulation leads to a matrix of size M + 1 (ideal species
are assumed). Therefore, the difference in speed increases rapidly with the
number of species. This is demonstrated by the results in Table 7.1 that
display the total CPU time used for determining 200 states of equilibria at
different temperatures.
The computational cost does not follow any simple power law that may
be derived from the matrix inversion since a large part of the time is used
for other tasks. Also some species are ignored as their concentration falls
under a predetermined fraction, e.g. 10−20 . It may, however, be clearly
seen that the relative advantage of the nonstoichiometric method grows
with problem size.
52
Table 7.1: Comparison of time consumption of the stoichiometric and nonstoichiometric formulations for equilibrium calculations.
stoichiometric
2.36 s
98.29 s
nonstoichiometric
0.441 s
7.36 s
ratio
5.35
12.5
−2
10
err
species
18
68
0
10
−4
10
In production calculations the two methods were used adaptively so that
the stoichiometric algorithm was used to solve the initial case and the nonstoichiometric formulation took over thereafter. If the nonstoichiometric
algorithm failed to converge the stoichiometric algorithm was used again. If
the number of species was small (less than ten) the stoichiometric algorithm
was used throughout the calculations.
Since the residuals of the calculations are very small the numerical errors are negligible. Consequently, the accuracy of the solution is highly
dependent on the quality of the input data, i.e., chemical potentials.
7.2
Diffusion of Reactive Gases
−6
10
−8
10
5
10
15
20
m
25
30
35
40
Figure 7.2: Residuals in the diffusion of reactive gases. The solid line represents the maximum change suggested by the equilibrium calculations, the
dashed line the maximum change by the diffusion equation and the dash-dot
line the disbalance of the growing species.
The multispecies diffusion was verified without chemical sources by solving a number of Laplace equations simultaneously. This proves that the
diffusion equation is correctly implemented. The equilibrium chemistry
solver was already separately verified. The source terms arising from the
chemical reactions were checked by certifying that the total amount of basic
elements remained unaltered.
In order to verify that the solution of the coupled problem is accurate
several residuals were checked: the change of partial pressure in the equilibrium calculations tells how close the solution is to true equilibrium. The
diffusion equation suggests a change in the partial pressures that should
vanish if the solution is a true one. And finally the disbalance of the silicon
and carbon elements growing at the surface tells how well the stoichiometric constraint of the growth is satisfied. Figure 7.2 shows how the residuals
change with the iteration in a simple one-dimensional case. The figure
shows that the solution converges nicely with iteration even though the
final solution oscillates considerably. In the calculations the flux was also
quite accurately conserved, i.e., the same amount of elements that sublimes
at the source also grows at the seed.
In some cases accurate results for the diffusion of reactive gases is hard
to achieve. This is particularly the case if the temperature variations at
the source are large. Then the source tends to be violently active at the
The solution of the energy equation was verified in quite a complicated geometry arising from Czochralski growth of silicon crystals. The comparison
was done against the commercial FIDAP code [86] and against another inhouse code [87]. Also a sublimation type of geometry was verified against
results by two international research groups [88]. Both cases showed accurate results with a very good convergence. It was also checked that the
power input was closely the same as the heat transfer through the external
boundaries.
From the mathematical point of view the most interesting feature of the
energy equation is the nonlinearity. It may be caused by variable thermal
conductivity or by radiation. The temperature dependence of thermal conductivity caused no convergence problems whatsoever, and neither did the
linearized radiation to external temperature. However, radiation within a
53
54
hot spots and small residuals are hard to obtain. The reason may be that
the chemical reactions are not linear with temperature. Therefore the finite
element method in which the partial pressures are presented with piecewise
linear functions fails to represent the true solution accurately.
7.3
Temperature Distribution
0
10
1
0.01
−5
<|∆T|/T>
10
100
0.01
−10
10
1
−15
10
100
0
Figure 7.3: Computational mesh with 709 elements and 777 nodes
closure is a more critical problem. The Newton-Raphson iteration proved
to be quite necessary particularly for well-isolated systems. Therefore a
special test case was set up to investigate this phenomenon. It helps in deciding when the computationally more expensive Newton-Raphson iteration
should be used.
The test case consists of an axisymmetric closure surrounded by an insulation. The inner diameter and inner height are both 10 cm, as well as
the thickness of the insulation. No heat source is included. At all the surfaces diffuse gray radiation with unity emissivity is assumed. The ambient
temperature is set to be quite high, 3000 K, while the initial guess is a constant temperature distribution of 1000 K. This provides with a good test
case since it is easily concluded that the correct solution of this problem
is a constant temperature distribution of 3000 K. The computational mesh
used in the calculations is shown in Figure 7.3.
The thermal conductivity of the insulation was varied in order to investigate how it affects the rate of convergence of the two different models.
The values used for the thermal conductivity, 0.01, 1.0 and 100.0 W/mK,
cover all typical materials. The mean relative change as a function of iteration step is presented in Figure 7.4. As may be seen the error of the
Newton-Raphson iteration decreases rapidly and settles then down to a
value that characterizes the maximum numerical accuracy of the solution.
The speed of convergence does not depend on the thermal conductivity.
55
10
20
30
40
50
m
60
70
80
90
100
Figure 7.4: Mean relative change as a function of iteration step for linear
(solid line) and Newton-Raphson iteration (dashed line) with different values
of thermal conductivity (0.01, 1.0 and 100 W/mK).
Accuracy with five digits is obtained in about ten iterations. The situation
is quite different for the linear iteration. Now the speed of convergence depends very much on the thermal conductivity. When thermal conductivity
is small the error seems to be small but it is just a sign that the equation
is so stiff that the solution hardly changes. This is illustrated in Figure 7.5
that shows the minimum temperature as a function of iteration step. For
well-isolated systems the system converges extremely slowly and therefore
linear iteration fails desperately.
Why not then always use Newton-Raphson iteration which seems to be
superior to the linearized approach? The answer is given by Figure 7.6.
Linearized iteration leads to a matrix equation with a nice banded structure.
In Newton-Raphson iteration the surface elements that see each other are
coupled, which results in a partially full matrix structure. The sparse matrix
solvers are usually designed to solve banded matrices. Algorithms able to
solve general sparse matrices are much more difficult to implement and
they are bound to be considerably slower.
7.4
Induction Heating
The vector potential is solved from an elliptic differential equation with
one matrix inversion. Convergence problems are not possible in this sim56
ple model. The model was compared against an analytical model assuming
zero electric conductivity for the crucible. The analytical model is presented in Appendix C. The general case was also verified against previously
published results [71] and against the results of two other research groups
in a simple sublimation geometry [88]. The total heating efficiency showed
good agreement and also the distribution of the Joule heating was similar
in appearance.
100
3000
2800
2600
2400
1
Tmin (K)
2200
7.5
2000
Feedback Mechanisms
1800
1600
1400
1200
0.01
1000
0
10
20
30
40
50
m
60
70
80
90
100
Figure 7.5: Minimum temperature in the linear iteration as a function of
iteration step with different values of thermal conductivity (0.01, 1.0 and
100 W/mK).
The feedback mechanisms need to be robust but they should not slow down
the convergence too much. These requirements are partially contradictory.
To ensure convergence, relaxation is used, and this slows down the rate
convergence.
Figure 7.7 shows how the total heating power changes with iteration
when the feedback for temperature control is applied to a typical sublimation geometry. Two cases are presented, one where the initial heating
power is too low, and another where it is too high. The final heating power
is very accurately the same even though the first case results to a faster
convergence. The corresponding temperatures at the control point are presented in Figure 7.8. The desired temperature at the measurement point
was 2300 ◦ C and it was achieved with the accuracy of five numbers in both
cases.
7.6
0
0
100
100
200
200
300
300
400
400
500
500
600
600
0
200
400
600
0
200
400
600
Figure 7.6: Matrix form for linear (left) and Newton-Raphson (right) iteration
57
Coupled Model
The full model consists of individual partial differential equations that are
weakly coupled to each other. Because the coupling is weak it is sufficient
to verify all the individual models separately and to check the mapping of
results between the different computational meshes. The variable mapping
was verified by mapping the results repeatedly between the computational
meshes and certifying that the results remained the same.
A practical problem of the coupled model is to find an iteration scheme
that converges quickly and securely. We chose a heuristic approach where
the number of iterations was just roughly tuned to achieve good convergence. For example, if the electric conduction is temperature dependent
the heat generation is recalculated every n:th iteration for temperature, n
being in the order of 5. Figure 7.9 shows the relative maximum error as a
function of iteration step for two different values of n calculated in a typical
58
10
0
10
9
Max|∆f/f|
8
7
P (kW)
6
−5
10
5
−10
10
4
3
2
1
5
5
10
15
20
25
m
30
35
40
45
10
15
20
25
m
8000
Tp (C)
5000
4000
3000
2000
15
20
25
m
30
35
40
45
50
Figure 7.8: Feedback temperature as a function of iteration step for initial
heating powers of 1.0 kW (solid line) and 10.0 kW (dashed line).
59
45
50
sublimation geometry. With a smaller n the convergence is faster but the
computational cost is somewhat higher.
The calculation of the coupled model is terminated when the relative
error in all the individual equations falls under the required accuracy. This
certifies that the solution of the coupled model is consistent.
6000
10
40
Figure 7.9: Maximum relative errors as a function of iteration step. The solid
line represents the error of temperature when n = 7 and the dashed line the
case n = 4. The corresponding symbols for heat generation are ∗ and o.
7000
5
35
50
Figure 7.7: Heating power as a function of iteration step when the temperature feedback is applied. The initial heating powers are 1.0 kW (solid line)
and 10.0 kW (dashed line).
1000
30
60
0
10
Si2C
SiC2
−2
10
Chapter 8
−4
C
i
p (bar)
10
−6
10
Simulation Results
C3
Si
Si2
C5
Si3
−8
10
Si4 C2
C4
−10
Si–C System
The thermodynamic calculations were applied to the Si–C system. The standard chemical potentials were obtained from tabulations of thermodynamic
data [83, 84]. Functional approximations were used if available, otherwise
power-law expansions were derived from the tabulated values. If data was
not available up to the temperature of 3000 ◦ C, care was taken to make
sure that the extrapolation was realistic. The same data was used also in
the other thermodynamic calculations. The number of calculated states
was around one hundred for each line graph and around 100 × 100 for each
contour plot.
Figure 8.1 shows the equilibrium pressures of Si–C system at constant
pressure. The gaseous species considered were C, C2 , C3 , C4 , Si, Si2 , Si3 ,
SiC, SiC2 , and Si2 C, and the condensed species were SiC and carbon. The
total pressure is highly dependent on temperature. Of individual species
gaseous silicon dominates at temperatures below 2400 ◦ C, while SiC2 and
Si2 C have larger partial pressures at higher temperatures. Since the system
with liquid silicon is energetically less favorable it only occurs when there
is excess of silicon and no carbon source. The partial pressures are now
higher as shown in Figure 8.2.
In both systems there is more silicon than carbon in the gas phase. This
is demonstrated in Figure 8.3 that shows the [Si]/[C] ratios in gas phase. If
the condensed species include liquid silicon the balance is even more on
the side of silicon.
Let us take a closer look at the thermodynamics of SiC formation. The
61
1400
1600
2000
2200
2400
2600
2800
T (C)
Figure 8.1: Equilibrium partial pressures for various species in Si–C system
assuming condensed SiC and carbon. The dashed line is the total pressure.
0
10
Si2C
−2
10
SiC2
−4
10
C
Si
Si2 Si3 Si4
i
8.1.1
10
Equilibrium Chemistry
p (bar)
8.1
SiC
1800
−6
10
C4
−8
C3
10
C2
−10
10
1400
1600
SiC
1800
2000
2200
C5
2400
2600
2800
T (C)
Figure 8.2: Equilibrium partial pressures in Si–C system assuming condensed
SiC and silicon.
62
2
10
0
−0.2
−0.4
Growth
−0.6
1
10
−0.8
log10(P)
[Si]/[C]
SiC−Si
SiC−C
−1
Etching
SiC+Si
−1.2
SiC+C
−1.4
SiC
−1.6
0
10
1400
1600
1800
2000
2200
2400
2600
2800
−1.8
T (C)
−0.8
−0.6
−0.4
−0.2
0
0.2
log10([Si]/[C])
0.4
0.6
0.8
1
Figure 8.3: [Si]/[C] ratios in SiC–C (solid line) and SiC–Si (dashed line) systems.
63
Figure 8.4: Condensed species in equilibrium at 2400 ◦ C.
0
−0.2
1
−0.4
−0.6
0
−0.8
log10(P)
initial gas-phase composition may be defined by the total pressure and
by the ratio [Si]/[C]. If this system is driven to equilibrium at constant
volume in the presence of a SiC seed, what are the condensed species?
Figure 8.4 illustrates the situation at 2400 ◦ C. There is a certain region
where the only condensed species is solid SiC. For larger values of [Si]/[C]
liquid silicon is introduced, and for smaller values solid carbon occurs. If
the initial pressures are lower than the equilibrium pressures the condensed
species sublime. Thus, in order to grow solid SiC without any additional
condensed species both the pressure and stoichiometry of the initial gasphase composition need to be controlled.
Unfortunately we cannot be sure that all of the available species are used
to form crystalline SiC. If the formation of condensed silicon or carbon is
even momentarily energetically possible it might occur. Therefore, we may
apply more stringent conditions for the gas-phase composition. Figure 8.5
shows the supersaturation of the condensed species at 2400 ◦ C. Supersaturation is here defined as the ratio of some particular partial pressure to
eq
the corresponding equilibrium partial pressure, pk /pk . An ideal area to
operate is the triangle in which the supersaturation of SiC is above unity
and the supersaturation of silicon and carbon is under unity. This ensures
that the only species that may condense is solid SiC.
There are several ways to introduce supersaturation. In CVD the supersaturation is due to the initial gas composition and the process is driven by
an external gas supply. In sublimation growth the supersaturation is caused
−1
0
−1.2
−1.6
−1.8
−1
−2
−1.4
−1
1
−1
0
−0.8
−0.6
−0.4
−0.2
0
0.2
log10([Si]/[C])
0.4
0.6
0.8
1
Figure 8.5: Supersaturation of SiC (solid line), silicon (dashed line) and carbon
(dash-dot line) at 2400 ◦ C.
64
2
300
1.5
250
0.1
1
∆T (C)
200
0.25
0.75
150
0.5
0.5
100
0.75
0.25
50
0
1800
1900
2000
2100
2200
2300
T (C)
2400
2500
2600
2700
2800
Figure 8.6: Supersaturations of silicon (solid line) and carbon (dashed line)
resulting from a temperature drop at constant volume.
by a temperature difference within the system. As may be seen in Figure 8.1
the equilibrium partial pressures decrease with decreasing temperature. So
it is possible to cause supersaturation by transporting gases from a higher
temperature to a lower temperature. The partial pressures over the active surface depend on how the gases are transported. If mass transport
is dominated by convection the gas phase composition remains unaltered.
Unfortunately in sublimation the mass transport is mainly due to diffusion
and therefore the transport of different species varies and the supersaturation is also affected. However, we may argue that because diffusion is
driven by supersaturation, it cannot fully cancel the initial supersaturation
because that would also stop the species diffusion. So even though we
omit the accurate modeling of mass transport, we should be able to predict
trends with thermodynamic calculations.
We study an ideal case where the initial gas phase composition is defined by equilibrium with solid SiC and carbon. This initial gas mixture is
then moved to a lower temperature without any change in the stoichiometry. This corresponds to mass transport by pure convection. The system
is then driven to equilibrium with condensed SiC at constant volume. Constant volume conditions are valid if the carrier gas dominates the total
pressure. Thereafter the supersaturation values of silicon and carbon are
calculated. The results are shown in Figure 8.6. It may be noticed that
the supersaturation of carbon is everywhere less than unity whereas the
supersaturation of silicon is larger than unity when the temperature drop
65
is around 200 ◦ C. This means that the temperature difference between the
seed and the source should not exceed this value if the formation of liquid
silicon is to be avoided.
These equilibrium calculations do not tell anything about the polytype
formation of SiC crystals. Some studies propose that the gas phase composition would have a decisive effect on the polytype formation [89], whereas
it is also known that the polytype and the polarity of the seed crystal are
at least equally important [90]. Recent work also suggest that the quality
of crystals is favored by near-equilibrium growth conditions, which means
moderate temperature gradients [90, 91]. High growth rate seems to give
SiC ingots with inferior quality whereas high temperatures would reduce
the amount of micropipes [92]. However, the calculations suggest that the
initial state given by equilibrium with solid SiC and carbon is at least in
some way optimal for crystal growth from vapor phase. There cannot be
any more carbon in the initial phase if it is a state of equilibrium, and
on the other hand, if there is any less carbon the limitations on the maximum temperature drop will be even more stringent. Thus, the principle
of sublimation growth that supersaturation is introduced by a temperature
difference is well motivated. It results in a robust manner in an ideal gasphase stoichiometry. If the supersaturation is caused by other means, for
example by external gaseous sources, a very delicate control is required to
avoid the problems that sublimation growth avoids by nature.
8.1.2
Si–C–H System
The thermodynamic calculations of the Si–C system show that in equilibrium there is excess silicon in the gas phase which means that the mass
transport of carbon may limit the growth rate. It may also lead to the
condensation of liquid silicon. Could these problems be at least partially
resolved by growing the crystal in presence of hydrogen? Hydrogen will
actively react with carbon and thereby increase the amount of carbon in the
gas phase.
In literature the thermodynamics of Si–C–H system have been mostly
studied from the viewpoint of chemical vapor deposition (CVD), e.g. References [51, 52, 53, 93]. Sublimation growth is, however, quite a different
system: in CVD growth the supersaturation is due to the initial composition of the gases, while in sublimation growth the supersaturation results
from the temperature difference between the source and the seed. More
relevant to the sublimation growth is the study in [50] where, for example,
the effect of hydrogen to the gas-phase stoichiometry of the Si–C–H system
is calculated. Here the treatment is repeated with some additional species
and extended to the calculation of supersaturation [1].
66
0
2
10
10
0.0
1
10
−2
10
0.00001
H2
0
H
10
0.0001
−4
−1
C2H2
−6
SiH
10
SiC=CH
SiCH
[Si]/[C]
pi (bar)
10
C2H
SiCH2
SiH2
10
0.001
−2
10
0.01
−3
10
−8
10
0.1
CH4
−4
10
1.0
−10
10
1400
−5
1600
1800
2000
2200
2400
2600
2800
T (C)
Figure 8.7: Equilibrium partial pressures of some additional species in the
Si–C–H system with 10 mbar of initial hydrogen. The dashed line is the total
pressure.
The thermodynamic calculations were applied to the Si–C–H system.
In addition to the species in the Si–C system, the following compounds
were included in the calculations: H, H2 , CH, CH2 , CH3 , CH4 , C2 H, C2 H2 ,
C2 H3 , C2 H4 , C2 H5 , C2 H6 , C3 H8 , SiH, SiH2 , SiH3 , SiH4 and Si2 H6 , and the 34
organosilicon compounds from Reference [94].
Figure 8.7 shows the equilibrium pressures of the most important additional species of the Si–C–H system following from an initial state in which
hydrogen has a partial pressure of 10 mbar. Hydrogen forms compounds
with both silicon and carbon, while the partial pressures of species present
also in Si–C system remain the same. Contrary to silicon and carbon, hydrogen does not have a condensed phase and therefore all hydrogen must
remain in gas phase. This means that the total pressure is greatly affected
by the initial amount of hydrogen.
The presence of hydrogen affects the silicon to carbon ratio in the gas
phase because hydrogen is more likely to build up compounds with carbon
than with silicon. The more there is hydrogen the more the balance moves
towards carbon. Figure 8.8 shows the [Si]/[C] ratios with temperature for
several initial hydrogen pressures. As temperature increases the effect of
hydrogen diminishes as the vapor pressure of silicon and carbon species
increase.
A silicon-to-carbon ratio close to one would be favorable to efficient
mass transport of the elements in the growth process. At each temperature
67
10
1200
1400
1600
1800
2000
T (C)
2200
2400
2600
2800
Figure 8.8: [Si]/[C] ratios in the Si–C–H system for various initial hydrogen
pressures in the interval 0 . . . 1 bar.
there exists only one level of initial hydrogen pressure that leads to this
state. In order to find this unknown pressure we solve the inverse problem:
find such a hydrogen pressure that the [Si]/[C] ratio is the desired one. This
is done iteratively by applying the secant method. The pressure values for
three different [Si]/[C] ratios are presented in Figure 8.9. At most an initial
pressure of about 0.2 bar is required for an even stoichiometry. For the
value [Si]/[C]=1.1 no hydrogen is needed at the highest temperatures.
The driving force of the growth process is the supersaturation in the
gas phase. To evaluate how the addition of hydrogen affects the supersaturation we let the temperature drop, drive the system to equilibrium with
solid SiC, and calculate the resulting supersaturation values of silicon and
carbon. This means that we assume mass transport where silicon and carbon are equally efficiently transported. Supersaturation values greater than
one indicate that an additional condensed species would be formed at the
growing surface.
Figure 8.10 shows the supersaturation of silicon and carbon over condensed SiC resulting from a temperature drop. The initial pressure of hydrogen (H2 ) was 1.0 bar. The picture shows that hydrogen has a dramatic
effect on the supersaturation. In the pure Si–C system silicon condensed
if the temperature drop was large enough. Now there is excess carbon in
the system which may lead to the formation of solid carbon. The formation of liquid silicon seems no longer favored. The amount of hydrogen
68
300
1.5
4
0
10
[Si]/[C]=0.9
[Si]/[C]=1
1.5
10.75
250
1.25
2
1
−1
200
[Si]/[C]=1.1
p
H2
∆T (C)
10
0.75
0.25
150
−2
0.5
0.25
10
0.5
100
50
−3
10
0.95
0.05
1.05
0
1400
1600
1800
2000
2200
2400
2600
2800
T (C)
−4
10
1400
1600
1800
2000
2200
2400
2600
2800
°
T ( C)
Figure 8.9: Initial hydrogen pressure (bar) in the Si–C–H system leading to
values 0.9, 1.0 and 1.1 for the [Si]/[C] ratio.
300
250
200
0.75
∆T (C)
0.95
0.25
0.95
1.5
150
1
0.05
10.5
100
0.75
50
1.05
0
1400
1600
1800
2000
2200
2400
2600
2800
T (C)
Figure 8.10: Supersaturation of Si (solid line) and C (dashed line) over condensed SiC resulting from a temperature drop. The initial state is obtained
from equilibrium with condensed SiC and C and 1.0 bar of initial H 2 .
69
Figure 8.11: Supersaturation of Si (solid line) and C (dashed line) over condensed SiC resulting from a temperature drop. The initial state is obtained
from equilibrium with condensed SiC and C and 10 mbar of initial H 2 .
in the gas phase has a major effect. This is shown by Figure 8.11 where
the pressure of initial hydrogen was only 10 mbar. Now the area where the
supersaturation of carbon exceeds unity has moved to lower temperatures.
At higher temperatures the amount of hydrogen is too small to change the
stoichiometry significantly.
It is obvious that the amount of hydrogen should depend closely on the
temperature. As shown previously for each temperature there exists a level
of hydrogen that leads to a [Si]/[C] ratio of unity. Would this ratio also
lead to acceptable supersaturation values? We first find the hydrogen pressure leading to the preferred [Si]/[C] value in Si–C–H equilibrium. We then
assume ideal convective mass transport and calculate the supersaturation
values due to a temperature drop. The results are presented in Figure 8.12.
As may be seen the supersaturation of silicon is everywhere below unity.
Adding hydrogen to gas phase decreases the supersaturation of silicon.
However, the supersaturation of carbon increases at the same time. For
the line [Si]/[C]= 0.9 where the amount of hydrogen is the largest the supersaturation of carbon also exceeds unity. Therefore, too large hydrogen
pressures may lead to the carbonization of the growing SiC crystal. There is
however quite a large window of hydrogen pressures where these problems
do not occur.
It should be noted that the analysis above is only valid if the gas-phase
stoichiometry is the one defined by equilibrium with condensed SiC and
70
1
1.2
0.9
1
0.8
0.7
[Si]/[C]=1
0.8
H mass fraction
Supersaturation
[Si]/[C]=0.9
0.6
0.4
[Si]/[C]=1.1
0.2
[Si]/[C]=1.1
[Si]/[C]=1
1600
1800
0.5
0.4
0.3
[Si]/[C]=0.9
[Si]/[C]=1
0.2
[Si]/[C]=0.9
0
1400
0.6
2000
2200
0.1
2400
2600
0
1400
2800
T (°C)
Figure 8.12: Supersaturation of silicon (solid line) and carbon (dashed line)
over condensed SiC resulting from a temperature drop of 200 ◦ C. In the
initial state the H2 pressure is tuned to give the preferred [Si]/[C] ratio (0.9,
1.0 and 1.1).
carbon at the higher temperature. The results should, however, give some
hints on how the supersaturation is affected by the addition of hydrogen
even though the dominating mass transport mechanism would be diffusion.
One drawback in adding hydrogen to the system is that the increased
pressure decreases the mobility of the species and may therefore ruin the
advantages of improved stoichiometry. One measure of this effect is the
mass fraction of hydrogen in the gas phase. Higher fraction means decreased efficiency. Figure 8.13 shows the mass fraction of hydrogen as
a function of temperature for three different [Si]/[C] ratios. At low temperatures hydrogen is the dominating species. However, at typical growth
temperatures the mass fraction of hydrogen is around 10 to 20 %.
In conclusion, it would seem that hydrogen atmosphere is favorable to
the sublimation growth of SiC. The silicon to carbon ratio of the system
moves towards unity. The supersaturation of silicon decreases while that
of carbon increases. When [Si]/[C] is close to unity the formation of extra
condensed species is not thermodynamically favorable. If the amount of
hydrogen is further increased the formation of condensed carbon becomes
possible.
The presence of hydrogen also increases the adsorption and desorption
rates of silicon and carbon. Therefore, it is not quite clear what the outcome
71
[Si]/[C]=1.1
1600
1800
2000
2200
2400
2600
2800
T (C)
Figure 8.13: Hydrogen mass fraction with three different [Si]/[C] ratios.
will be even though the simple analysis suggests that hydrogen seems to
have a positive influence on the growth process. A foreseeable problem is
that the graphite crucible may be quickly worn out due to high etch rate.
In addition to hydrogen, also other active process gases and their influence
might be analyzed similarly, for example nitrogen [49] or chlorine [93, 95].
8.2
Diffusion of Reactive Gases
The thermodynamic calculations do not involve any spatial considerations
essential for the crystal growth process. Both the geometry and the temperature distribution will affect the growth rate and the shape of the growing
crystal as well as the way the powder is consumed. These questions may
be targeted by solving the diffusion equation of reactive gases.
The diffusion equation may also be solved analytically in some special
cases. This requires that the diffusion coefficients and chemical properties
of the crucible are constant, and that the shape of the crucible enables
analytical solutions, see for example References [96, 97, 98, 99]. Analytical
solutions may increase the understanding but they unfortunately fail to
model the real system which is usually far from ideal.
We solve the diffusion equation coupled with equilibrium chemistry for
some simple cases where the temperature of the source and seed are predefined. The seed is slightly colder than the source which generates a flux
from the source to the seed. This flux defines the growth rate and the shape
72
0.9
0.8
0.7
V (mm/h)
0.6
Tc=const
0.5
0.4
<T>=const
0.3
Figure 8.14: Mass flux resulting from an infinitely large source (at the bottom)
and seed (at the top).
0.1
0
of the growth front. In the calculations we assumed equilibrium with condensed SiC and carbon at the source and equilibrium with condensed SiC at
the seed. In the gas phase only the three most important species were considered; SiC2 , Si2 C and Si. Up to 3000 elements were used in the calculations
and around twenty iterations were required for good convergence.
8.2.1
Basic Dependencies of the Growth Rate
T =const
s
0.2
0
10
20
30
40
50
∆ T (C)
60
70
80
90
100
Figure 8.15: Growth rate as a function of temperature difference with p Ar =
1 bar and L = 1 cm. In the solid line the mean temperature is kept the at
constant value of 2500 ◦ C, in the dashed line at the temperature of the source
and in the dash-dot line at the temperature of the seed.
We first consider an ideal case where the temperatures of the source and
seed are both constant. The source and seed are horizontal and either infinitely large or restricted by an inert vertical wall. This makes the problem
essentially one-dimensional; thus the crystal grows evenly and the source
is evenly consumed as shown in Figure 8.14. The basic dependencies of the
growth rate are efficiently revealed by this simple geometry.
The growth is driven by the temperature gradient. How does it affect the
growth process? We assume that either the source or seed is at constant
temperature and change the temperature gradient at the interval 0–100 ◦ C.
The source-to-seed distance was set to 1 cm and the partial pressure of
inert argon was 1 bar. The results are presented in Figure 8.15. As might be
expected the growth rate increases linearly with the temperature gradient
for small temperature differences. For temperature differences exceeding
20 ◦ C the second-order effect of the temperature difference is seen. The
“central difference” gives a rather linear dependence for the whole interval. In practice the temperature difference is very seldom over 20 ◦ C and
therefore the growth rate may be assumed to be linearly dependent on the
temperature gradient.
The growth rate as a function of temperature is another basic depen-
Figure 8.16: Growth rate as a function of temperature with argon pressures
1.0, 0.01 and 0.0001 bar with L = 1 cm and ∆T = 1 ◦ C.
73
74
2
10
0
10
V (mm/h)
0.0001 bar
−2
10
0.01 bar
−4
10
1.0 bar
−6
10
1800
1900
2000
2100
2200
2300
T (C)
2400
2500
2600
2700
2800
dence. The temperature difference was chosen to be 1 ◦ C. The real growth
rate may be obtained by multiplying with the real temperature difference.
The results are presented in Figure 8.16. The growth rate increases monotonically with temperature. However, when the temperature increases it becomes more difficult to create a temperature gradient between the source
and seed since thermal radiation is proportional to the fourth power of
temperature. Diffusion limited growth rate at a given temperature is inversely proportional to the total pressure. As the pressure of the inert gas
decreases the assumption of diffusion limited growth is violated, consequently the linear dependence is no longer valid.
1
10
0.0001 bar
0.01 bar
0
V (mm/h)
10
0.1 bar
−1
10
1.0 bar
−2
10
Figure 8.18: Mass flux resulting from a smaller seed. In the upper case the
source at the bottom is large and in the lower case it is of the same size with
the seed at the top.
−3
10
−4
10
−3
10
−2
10
L (cm)
−1
10
0
10
In practice one-dimensional mass transport is difficult to obtain. The symmetry may be broken either by geometry or by temperature distribution.
The size of the seed is often limited by practical or economic reasons.
Therefore it may not be possible to have a seed that would cover the whole
top part of the cavity. This will break down the one-dimensional mass
transport. The growth at the seed edges will now be favored as may be seen
in Figure 8.18. This is an intrinsic property of the Laplace equation. The
same phenomena explains why porridge cools the most at the edge of the
plate, and why induction heats efficiently the crucible corners. Even if we
reduce the size of the source to the same radius the edge effect remains.
Now the same effect is also seen at the source.
Note that in the calculations the walls are assumed to be inert which is
an extreme case. In reality this is not quite true because the graphite may
act as a carbon source or amorphous SiC may start to grow on the walls.
Also the surface diffusion and the prevailing growth mechanism may have
a damping effect on the growth shape. Thus, the growth shape is not so
pronounced as suggested by the calculations.
The source and the seed need not be horizontal. Large boules have been
grown in geometries where the source is placed vertically even though the
seed is horizontal. We performed similar calculations also for this case
75
76
Figure 8.17: Growth rate as a function of source-to-seed distance at argon
pressures 1.0, 0.1, 0.01 and 0.001 bar with T = 2500 ◦ C and ∆T = 1 ◦ C.
In the diffusion limited region the growth rate is inversely proportional
to the source-to-seed distance. When the distance is further decreased
the growth rate becomes limited by adsorption-desorption and the dependence vanishes. The transition from diffusion limited region to adsorptiondesorption limited regime depends on the pressure. At smaller pressures
the transition distance is larger as seen in Figure 8.17.
8.2.2
Growth Shape
Figure 8.19: Mass flux in three different cases with varying source height (R,
1.5R and 1.8R). In each case the seed is at the top and the source is on the
right, and both are at constant temperature.
Figure 8.20: Mass flux in three different cases with varying source height
(2R, R and R/2). In each case the seed with a constant temperature is at the
top and the source with linear temperature distribution is on the right.
varying the source height. The smaller the distance from the source to the
seed the more the growth is favored at the seed edge. The results for three
different cases are shown in Figure 8.19. Also now the edge effect is shown
in that the source is mostly consumed close to the edge.
So far we have assumed that the temperature over the source and the
seed is constant. This assumption may be freely violated. If the source
is vertical the one-dimensional character may be nearly reproduced with
a temperature distribution that depends linearly on the axial coordinate.
This is depicted in Figure 8.20. Now the source is most active at the bottom
and the seed grows more evenly. However, if the height of the source is
reduced growth near the edges is again favored.
The active surfaces of the seed and the source do not need to be planar.
There may be some practical aspects that limit the shape of the source and
particularly of the seed. Let us have a look at some basic forms of the seed
assuming a horizontal source at constant temperature. Figure 8.21 shows
the mass flux for a cylindrical seed and Figure 8.22 for a spherical seed.
As may be seen the growth rate over the seed surface varies with position.
Therefore the shape of the growing crystal will change with time. If the seed
and source are at constant temperature only forms with some symmetry
properties will maintain the shape of the crystal. The axisymmetric case
with one-dimensional temperature distribution has this property as shown
in Figure 8.14. Also the spherical symmetry with T = T (r ) maintains the
shape of the crystal even though it may not be practical to implement. This
is demonstrated by Figure 8.23.
Purely axial temperature gradients are in practice difficult to create. In
some cases it is also preferable that the shape of the growth front is not
maintained. For example, if the seed is too small at the beginning a convex
growth front is preferred to increase the crystal diameter [100]. Figure 8.24
presents the flux arising from simple linear temperature distributions for
the seed. The growth rate is faster where the temperature is lower. If the
temperature is lower at the center the growth front will eventually turn
convex and correspondingly, if the temperature is lower at the edge the
growth front will become concave.
The calculations show that the ideal one-dimensional growth may be
broken by the temperature distribution and by the geometry of the source
and seed. Could these two phenomena be used simultaneously to cancel
each other? For example, if the seed is smaller than the top of the cavity
growth on the edges is favored. Nevertheless, if at the same time the
temperature profile at the seed is tuned carefully it may be possible to
obtain even growth after all.
77
78
Figure 8.21: Mass flux at the active surfaces with a cylindrical seed. On the
left the temperature over the seed is constant and on the right it depends
linearly on z.
Figure 8.24: Mass flux at different analytical temperature distributions. The
temperature at the source is 2700 ◦ C and the temperature at the seed is
2690 ± 200(r /cm − 0.01) ◦ C, where the plus sign is valid for the upper case
and minus for the lower case.
Figure 8.22: Mass flux at the active surfaces with a spherical seed. On the
left the temperature over the seed is constant and on the right it depends
linearly on z.
Figure 8.23: Mass flux in spherical symmetry with the seed and source at
constant temperature.
79
Consider a simple case where the source is at constant temperature and
the seed is considerably smaller than the source. If the seed is at constant
temperature the maximum growth rate is achieved at the edge of the seed.
The growth rate decays closely exponentially as we approach the origin.
This suggests that if we want to find a temperature profile that would lead
to an even growth rate an exponential dependence on the radius might be
a good candidate. We therefore look for seed temperature of the form
!
ebr /r0 − 1
.
(8.1)
Tc (r ) = Ts + ∆T0 1 + a b
e −1
The parameters a and b should be nearly independent of the geometry.
Some manual iterations suggest that almost an even growth profile is obtained by choosing a = 0.4 and b = 3.0. The original and transformed
fluxes are shown in Figure 8.25. Even though the growth front may be controlled with a delicate tuning of the temperature profile it may be difficult
to maintain a complicated temperature profile as the growth proceeds. This
80
is a strong argument for simple, one-dimensional designs.
Figure 8.26: Schematic presentation of the axisymmetric geometry used in
the calculations.
8.3
Figure 8.25: Mass flux at different analytical temperature distributions. At
the upper case the temperature difference is constant and in the lower case it
is obtained from ∆T0 (1 + a (ebr /r0 − 1)/(eb − 1)), where a = 0.4 and b = 3.0.
The equation for multispecies diffusion is a vector equation but the final
outcome of the calculations is a scalar quantity — the local growth rate, or
mass flux. If the temperature differences within the system are so small
that the growth rate is linearly dependent on temperature differences it
turns out that the mass flux is very closely proportional to the heat flux.
This is natural since the diffusion equation is of the same form as the
conduction dominated energy equation and the boundary conditions are
directly related to temperature. The chemistry has actually no independent
role, everything may ideally be reduced to temperature distribution and
geometry. Therefore, if one wants to optimize the shape of the mass flux
it may be enough just to solve the temperature equation with appropriate
boundary conditions.
81
Temperature Distribution
So far the sublimation growth of SiC may seem to be quite simple. It involves
only the problem of placing a hot source to face a slightly colder seed within
a closed cavity. The growth is controlled by the temperature gradient and
by the choice of the carrier gas and its pressure. Unfortunately, there
are some intrinsic problems that complicate the design and building of
growth furnaces. These problems may be efficiently studied by the use of
simulations.
Let us look at a typical sublimation growth geometry presented in Figure 8.26. A crucible made of dense graphite forms a semiclosed cavity. At
the top of the cavity there is a SiC seed and at the bottom of the cavity there
is a SiC source. The crucible is surrounded by insulation made of porous
graphite. For temperature measurement two openings are provided for pyrometers that determine the temperature by the radiation emitted from the
crucible surface. The whole system is surrounded by an induction coil that
may be moved in the axial direction. The inner diameter of the crucible is
5.0 cm and the outer diameter of the insulation is 15.0 cm. At the top of
the crucible the thickness of the insulation is 3.0 cm and at the bottom 10.0
cm, correspondingly. The coil is 10.0 cm in length, 16.0 cm in diameter and
its center is aligned with the inner bottom of the crucible.
82
Table 8.1: Material parameters used in the example simulations.
material
source
seed
crucible
insulation
0.8
0.8
0.9
0.9
κ (W/mK)
1.0
451700T −1.29
125868/(T+615.9)
2.16E-9 T 2 - 1.43E-5 T + 0.367
σ (1/Ωm)
100.0
10000.0
6.67E4/(1+T/2500)
245.7/(1+T/2500)
The material parameters used in the example calculations are presented
in Table 8.1. They are typical, yet not necessarily accurate values for the
various parameters. The properties of silicon carbide source and seed are
discussed in papers [56, 57].
In solving the induction heating problem it was assumed that the vector
potential vanishes at infinity which in this case meant a moderate distance
of 30 cm. An additional increase in this distance did not affect the results
significantly. In the energy equation the temperature was assumed to be
fixed to room temperature at the bottom of the insulation, and on other
directions black body radiation to room temperature was assumed. Inside
the crucible the heat transfer included conduction in the gas phase and
radiation between the crucible walls. Also the heat transfer in the pyrometer
holes was modeled assuming wall-to-wall radiation.
In solving the vector potential around 10000 bilinear elements were used
and for the temperature around 6000 elements. The iteration converged
without any problems. Around 20 iterations for the temperature and 3 iterations for the vector potential were required when the termination criteria
was a relative error of 10−5 . The overall computational time on a powerfull
workstation was a couple of minutes.
Figure 8.27 shows typical results from the vector potential calculations.
In the picture the components of the vector potential have been normalized so that their maximum values are equal. In reality the out-of-phase
component is significantly smaller in magnitude. The vector potential is
mapped to the scope used in thermal calculations as shown in Figure 8.29.
The vector potential is then used to calculate the local heat generation due
to Ohmic losses. The losses are concentrated on the crucible walls. In
the thermal calculations the heat generation is normalized so that the total
heating power agrees with the desired value. The picture shows also the
resulting temperature distribution.
The temperature distribution may be slightly affected by altering the
parameters of the induction heating. In the simulation of Figure 8.29 the
frequency was increased from 50 kHz to 200 kHz and the coil was lowered
83
Figure 8.27: Components of the vector potential. In-phase component on the
right and out-of-phase component on the left.
Figure 8.28: a) Components of the vector potential mapped to the scope used
in the thermal simulations. b) Heat generation by induction on the left and
the resulting temperature distribution on the right.
84
3200
3000
2800
T (C)
2600
2400
2200
2000
1800
1600
1400
1000
Figure 8.29: a) Components of the vector potential with increased frequency
and lower coil position. b) Heat generation by induction and the temperature
distribution.
by 1.5 cm. As a result the heating is concentrated more clearly on the
surface and on the crucible corner. Ideally we would like to heat also the
inner parts of the crucible and therefore lower frequencies are generally
favored. They also couple in a lesser extent to the porous graphite used for
thermal isolation.
Figures such as 8.27, 8.28 and 8.29 may give insight in the governing
phenomena but they do not provide the quantitative information required
in thermal design. Ideally, we could reduce the properties of a crucible to
just a few figures of merit that could be used in the evaluation process.
Figure 8.30 shows temperature at given points as a function of heating
power. As might be expected the temperatures increase monotonically with
heating power. The system is set up so that heating is strongest under the
source. The energy must escape somewhere and the natural way is through
the weakest isolation which results to a heat flux from bottom to top. Heat
flux is directly related to temperature and therefore the source is hotter
than the seed.
Unfortunately the source material is typically porous SiC powder that
has a very poor thermal conductivity. Theoretical calculations have shown
that the conductivity of ideal porous SiC is less than 1 W/mK [57]. This
means that it is difficult to apply a large heat flux through the source and
85
1500
2000
2500
P (W)
3000
3500
4000
Figure 8.30: Temperatures at lower pyrometer hole (solid line), upper pyrometer hole (dashed line), source bottom (dotted line), source surface (dash-dot
line) as a function of heating power.
that it will automatically result to large temperature differences within the
source. The thicker the source the more prominent this problem is. Closely
the same heat flux is also transferred from the surface of the source to
the seed. Now the temperature difference is much smaller since radiation
transports energy very efficiently at high temperatures. The temperature
difference between the source and the seed is presented in Figure 8.31. The
same picture also shows the temperature difference with a reduced amount
of powder. The heat flux is now larger and therefore the temperature
difference between the source and the seed is also larger.
8.3.1
Feedback for Temperature Control
When the input power is known the calculated temperatures may be compared to experimental results. Unfortunately, there are some internal losses
in the induction heating system and also the thermal conductivities at high
temperatures are badly known. When we want to know the temperatures
inside the crucible as accurately as possible we use the measured temperatures to calibrate the simulation. This is done by using a feedback loop to
set the heating power so that the temperature estimated by the simulations
agrees with the value measured by the pyrometer. A similar feedback is
also used in the practical experiments in order to keep the system at a fixed
temperature during the growth process.
86
2800
12
2700
−30 mm
11
2600
−10 mm
T (C)
∆T (C)
10
9
8
2500
+10 mm
2400
7
+30 mm
2300
6
2200
−30
5
1000
1500
2000
2500
P (W)
3000
3500
−20
−10
0
10
z (mm)
20
30
40
50
4000
Figure 8.31: Temperature difference between the source and the seed as a
function of heating power with 20 mm (solid line) and 10 mm (dashed line)
of source powder.
Figure 8.33: Temperature at the axis and inner crucible walls (dashed line)
for coil positions −30, −10, +10 and +30 mm when the temperature measured by the upper pyrometer is kept constant.
Figure 8.32: Temperature at the axis and inner crucible walls (dashed line)
for coil positions −30, −10, +10 and +30 mm when the temperature measured by the lower pyrometer is kept constant.
Figure 8.32 shows the temperature at the axis and wall temperature
projections at different coil positions, −30, −10, +10 and +30 mm. The
zero position is aligned with the bottom of the crucible cavity. By moving
the coil it is possible to influence the temperature gradient. It is important
that the sign of the temperature gradient can be changed by coil movement
because the quality of the crystal may benefit if the seed is etched in the
start of the process [43].
The temperature measured by the lower pyrometer may be quite different from the temperature of the seed. Feedback using the upper pyrometer
for temperature control might be more useful as is seen in Figure 8.33. The
temperature of the seed correlates now much better with the temperature
used for feedback. The holes used for temperature control create a cold
finger in the temperature distribution. This may result in a convex growth
front which may be desired particularly if seed enlargement is required.
The pyrometer holes should therefore be carefully designed for optimal
crystal-growth shape.
The Figures 8.32 and 8.33 also reveal a problem that is an intrinsic property of the simple geometry under discussion. The temperature gradient
on the side wall is significantly larger than the temperature gradient at the
axis. This is an unfortunate result of the poor thermal conductivity of the
source material when compared to the walls made out of dense graphite.
87
88
2600
+30 mm
2550
2500
2450
+10 mm
T (C)
2400
2350
−10 mm
2300
2250
−30 mm
2200
2150
2100
−30
−20
−10
0
10
z (mm)
20
30
40
50
Active surface
2600
−30 mm
2550
SiC source
T (C)
2500
A
−10 mm
Heating element
2450
+10 mm
2400
R
2350
+30 mm
2300
R=25 mm
2250
−30
−20
−10
0
10
z (mm)
20
30
40
50
Figure 8.34: Temperature at the axis and inner crucible walls (dashed line)
for different coil positions with a solid SiC source.
Figure 8.35: Simple heating geometry when heating from aside.
Let us pay some more attention to the problem of heating the source evenly.
Ideally, we would only heat up the source itself but it is difficult with the
means that are available. As the RF-field couples quite poorly with the
source, induction it is not a feasible method for direct heating. It seems
also impossible to heat the source directly at the active surface since it
should preferably face the colder seed. Therefore, the only practical heating
method is to heat the source via the hot walls. Unfortunately, the thermal
conduction of porous SiC is so bad that when we heat the powder up we
are bound to introduce large temperature gradients over the powder. The
warmest spots will always be situated at the hot crucible walls. It is also
difficult to control the temperature at the active surface. The larger the
amount of powder the more difficult are the problems related to heating.
The heating of the source material may be the most severe limitation of
sublimation growth.
Let us assume that we want to have constant temperature and heat flux
over the active surface. If the source is cylindrical in shape we may heat
it from the side or from the bottom. Heating from the bottom is a trivial
case since constant temperature at the bottom will also result to constant
temperature at the active surface. Heating from aside is a more involved
matter.
Consider a simple geometry heated from the side. It is presented in
Figure 8.35. The problem is to heat a cylindrical source so that the temperature at the active surface is uniform. The heating element of length
R = 25 mm is set at a distance A lower than the source surface. At the
active surface we assume radiation to external temperature of 2500 ◦ C. At
the heating element constant power generation is set. We should select A
so that both the radial temperature gradient at the surface and the temperature difference within the source would be reasonably small. These two
requirements are contradictory.
Figure 8.36 shows the temperature at the interface with different values
of A/R. When the value A/R increases the temperature distribution at the
surface becomes more settled, but the maximum temperature difference
increases. This is shown in Figure 8.37. It seems that ratio A/R close to
unity might be quite optimal.
The calculations suggest that if we require even temperature at the
source surface, heating from the side should be avoided unless the thickness of the source is significantly larger than the radius of the source. Even
89
90
This feature in the temperature distribution means that the source material
is mostly consumed near the walls whereas even consumption would be favored. The problem might be solved by increasing the thermal conductivity
of the source material. This might be achieved by using solid SiC source as
is done in Figure 8.34. Now the temperature gradient at the axis and at the
side wall are nearly equal and thus the source is consumed more evenly.
Also the temperature difference between the source and the seed is now
larger since the heat flux going through the solid source is larger than the
heat flux going through the porous source.
8.3.2
Heating the Source Evenly
2.8
2.6
∆ T (C)
2.4
2.2
1.0
0.8
0.6
2
1.8
2.0
0.4
1.6
0.2
1.4
0
0.005
0.01
0.015
0.02
0.025
r/R
Figure 8.36: Temperature difference at the interface with different values
of A/R.
650
600
550
500
∆ Ts (C)
450
400
350
300
250
200
150
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
A/R
Figure 8.37: Maximum temperature difference within the source as a function of A/R.
91
then heating from aside must be performed very accurately in order to
control the temperature at the active surface. This might not be possible
with the technical means that are at disposal at such high temperatures.
Also, maintaining constant heating during the growth process may require
some active parts. On the other hand, heating from aside may eventually
enable continuous growth techniques for long crystals. So far it seems that
the SiC industry is pursuing crystal area at the expense of crystal length.
Geometries favoring axial growth have, however, been reported for other
materials, see for example Reference [101].
The considerations above are largely true also for the cooling of the
growing crystal. However, since the crystalline SiC has a thermal conductivity of one or two orders of magnitude greater than the porous SiC powder
this problem is not the limiting factor in sublimation growth. If the difficulties in the finite size of the powder are overcome, the cooling of the growing
crystal may require further attention.
8.4
Coupled Model
In the coupled model the Joule heating, temperature distribution and mass
transport are calculated as a weakly coupled system using three different
meshes. The mass flux resulting from Figure 8.28 is presented in Figure 8.38. The hot graphite wall results to a maximum at powder temperature close to the wall. This further results in a violent etch rate at this
corner. In reality, this would settle in time because the shape of the active surface would change as would the chemical properties of the surface.
These phenomena would lead to negative feedback and eventually the powder would be more evenly consumed. Unfortunately, as the computational
model does not take into account these phenomena in the following calculations the porous SiC source is replaced with a solid SiC source in this test
geometry. The conclusions should still be reasonably valid.
Figure 8.39 shows the mass flux at the reactive surfaces when solid
SiC is used as source. Now the flux is much more settled. The effect of
temperature may still be seen in that the maximum of etch and growth
rates are close to the walls. In Figure 8.40 the size of the seed is decreased
and now the edge effect is again clearly seen. The edge effect is not so strong
if the seed is thick and may also grow radially. This is is demonstrated by
Figure 8.41.
As shown previously, the shape of the growing surface may be controlled
by manipulating the temperature distribution. Radial growth is obtained by
favoring growth at the center by cooling the seed more at the center. We
let the hole for pyrometer extend 15 mm to the dense graphite lid. This
92
Figure 8.38: Mass flux at the reactive surfaces with a porous SiC source.
Figure 8.40: Mass flux at the reactive surfaces with a solid SiC source and a
smaller seed.
Figure 8.39: Mass flux at the reactive surfaces with a solid SiC source.
Figure 8.41: Mass flux at the reactive surfaces with a smaller seed with
thickness of 2 mm.
93
94
b)
a)
3.5
3.5
3
3
2.5
2.5
z (cm)
2
original boundary
2
1.5
final boundary
1.5
0.5
0
0
−0.5
−0.5
−1
Figure 8.42: Mass flux at the reactive surfaces with an extended hole for the
pyrometer.
should cool the center of the lid thereby increasing the growth there. The
results are shown in Figure 8.42 and when compared to Figure 8.39 they
indeed do confirm this experimentally observed fact.
The results of the coupled model show that the model can nicely predict the behavior of the seed. The source is a more challenging material
to model since it is not easily treated by considering only the surface phenomena. Also mass transport in porous media, graphitization and other
complicated issues are involved. However, in designing the crucibles it
does not necessarily matter that the materials are treated in an ideal way.
The accuracy of the model may be lost but the optimal geometry may be
quite the same for the ideal and for the non-ideal system.
8.5
Virtual Crystal Growth
The virtual crystal growth model was applied to a few simple cases where
temperature distributions were set by analytical functions. Three most
important gaseous species (SiC2 , Si2 C and Si) and two solid species (SiC
and C) were considered in the equilibrium chemistry. The surface shape
was determined by ten virtual growth steps. Figure 8.43 illustrates the
different cases. The shape of the growing crystal is determined by the shape
of the temperature distribution. Most powder is consumed around the
temperature maximum while most growth occurs around the temperature
minimum.
The results are hardly surprising. Unfortunately, the virtual crystal
growth model was not well suited for most cases of practical interest. The
poor conductivity of the powder caused violent sublimation near the cru95
T=2300−10z−r
1
T=2300−10z+r
1
0.5
0
1
2
r (cm)
−1
3
0
1
c)
3
d)
3.5
3.5
3
3
2.5
2.5
2
2
1.5
1.5
T=2300−10z−0.5r(z−1.5)
1
T=2300−10z−0.5r(r−3.0)
1
0.5
0.5
0
0
−0.5
−0.5
−1
2
r (cm)
0
1
2
r (cm)
−1
3
0
1
2
r (cm)
3
Figure 8.43: Boundary shapes resulting from different analytical temperature distributions. a) T = 2300 − 10z + r , b) T = 2300 − 10z − r , c)
T = 2300 − 10z − 0.5r (z − 1.5), d) T = 2300 − 10z − 0.5r (r − 3)
cible walls which quickly led to badly deformed elements. Also if the seed
was smaller than the crucible top, the elements were ruined at the edges of
the seed. Therefore the virtual crystal growth model could only be used for
closely one-dimensional cases. Further work is required for the model to be
generally applicable to all cases. The computational mesh may need to be
regenerated when the growth has ruined the original elements so that accurate solution of the equations is no longer possible. If the seed or source
is morphologically unstable the virtual crystal growth may be very tricky to
implement.
96
Chapter 9
Synthesis of Simulation and
Practice
All too often simulations are used just as a demonstration tool. It is just
like looking out of the window and observing that it is raining without the
possibility to influence the weather. This is hardly enough. The information obtained from simulations should be used as efficiently as possible to
achieve the goals, in this case the growth of large silicon carbide boules of
high quality. It is not enough just to consider a few cases. We should be
able to map the whole parameter space with the simulations. Ideally the information obtained from simulations would be seamlessly integrated with
the practical research. The practical experiments and simulations would
just be two different ways of looking at the same thing.
We should carefully consider how to use the simulations most efficiently.
In this chapter two steps in that direction are taken. The first case presents
an analytical model for the growth rate that nicely retains the parametric
dependencies of the original system [3]. It may be of practical value when
experiments are designed and analyzed. In the second approach we consider how the information from simulations could be used in a systematic
manner to boost up the optimization of the crystal growth process, or any
similar process in general.
9.1
A Practical Model for the Growth Rate
The growth rate is, perhaps, the most important parameter in a crystal
growth process. It is closely related to the material quality and process
yield. In sublimation growth of SiC, it is very difficult to control the growth
97
rate actively. It can not be measured during the process, but must be determined afterwards. In order to achieve reproducible growth results, a
theoretical model that estimates the growth rate is of great practical use. A
good model also clarifies the growth rate dependence on specific parameters with a faster feedback than obtained by experimental means.
A problem in modeling is that complicated models tend to hide physical
dependencies while simple models often fail to model the real systems in
detail. Here we have tried to make a compromise. A simple analytical model
for the growth rate is set up and the real temperature profile and mass
transport are incorporated into it with the help of numerical simulations.
Growth rate models have been presented also by other authors [39, 47, 102,
103, 104, 105]. We follow their footsteps on free molecular transport and
diffusion. In this model these phenomena are coupled and also the latent
heat is taken into account.
An ideal sublimation growth process is determined by crucible geometry
and temperature distribution. In a one-dimensional geometry presented in
Figure 9.1, the obvious geometrical parameters are the source-to-seed distance L and the thicknesses of the source and seed, Ls and Lc , respectively.
The temperature distribution is then defined by two temperatures, or one
temperature and a heat flux. In this work the variables are T , the mean of
the source and seed temperatures (Ts and Tc ), and q, the applied heat flux.
These parameters cannot be directly measured but must be expressed with
the help of known variables. Typically, we know temperature at least in one
point, measured by a pyrometer, and the total heating efficiency. Since T
and q are monotonic functions of these it is possible to derive functional dependencies between the variables. For this purpose numerical simulations
of the global heat transfer may be used.
The temperature difference between the source and the seed is the driving force of the sublimation process. If the heat flux is fixed then the heat
of crystallization ∆H will decrease the original temperature difference ∆T 0 .
The reduced temperature difference ∆T is obtained from
!
V ∆H
∆T = ∆T0 1 −
,
(9.1)
q
where V is the so far unknown growth rate.
The species flux J~ in the sublimation process is mainly due to the gra~
dients of the molar densities n,
J~ = −D ·
~
dn
,
dz
(9.2)
where D is the transport matrix. The growth rate is obtained by multiplying
~ and dividing it by the density
the molar flux by the vector of molar masses M
98
where σSB is the Stefan-Boltzmann constant and is emissivity. In the
expression we have used the knowledge that ∆T0 T . The same heat flux
must also go through the source material, thus
q = κs
∆Ts
,
Ls
(9.8)
where κs is thermal conductivity and ∆Ts the temperature difference within
the source. Applying Equations (9.7) and (9.8) to Equation (9.5) we get
V0 = Ṽ
Figure 9.1: Idealized one dimensional geometry and temperature profile.
ρ of the crystal. By assuming that the gradient of molar densities results
from a uniform temperature gradient between the source and the seed, we
may write
~
~ ∆T
dn
∆T
M
~ T)
·D·
= Ṽ (n,
,
(9.3)
V =
ρ
dT L
L
where we have defined a function Ṽ that describes the growth rate at unity
temperature gradient. It includes the chemistry and transport properties of
the gas mixture. Applying Equation (9.1) to Equation (9.3) the growth rate
may be expressed as a harmonic sum,
1
1
1
=
+
,
V
V0
V1
where
Ṽ ∆T0
V0 =
L
(9.4)
(9.5)
is defined by mass transport and
V1 =
q
∆H
(9.6)
is defined by energy transport. The growth rate V is always smaller than
either of V0 or V1 ; usually it is limited by V0 . Only if the mass transport is
very efficient does the heat of crystallization become relevant.
At high temperatures the dominating heat transfer mechanism is radiation and the initial temperature difference may therefore be estimated by
∆T0 =
q
s + c − s c
,
4σSB T 3
s c
99
(9.7)
s + c − s c
κs
∆Ts
,
Ls L 4σSB T 3
s c
(9.9)
This formula manifests the limitations on growth rate resulting from a
poorly conducting thick source. If we try to increase the dimensions of
the system the growth rate with a fixed temperature difference is bound
to decrease. A similar formula may be derived for the seed but then the
constraints for the growth rate are not so stringent since the seed is usually
a better conductor than the source. These conclusions are valid only if the
temperature gradient is nearly axial. In more complicated geometries the
situation may be somewhat different.
The formulae above may also be used to roughly estimate the condition
for growth stability. Imagine a case where a step occurs at the seed. In
order to have stable growth, the part of the seed that lags behind must
catch up with the step. This requires that the temperature gradient inside
the seed is larger than the gradient between the seed and source. This leads
to a condition for the thermal conductivity of the seed κc . Correspondingly
for the source the condition for stable sublimation is that the temperature
gradient within the source is smaller than the one between the source and
seed. Combining these requirements we obtain
κc < 4σSB T 3 L
s c
< κs .
s + c − s c
(9.10)
This is a very rough estimate but it shows that in this simple geometry stable
growth and stable sublimation are not simultaneously possible because
always κs ≤ κc . This is bound to limit the maximum size of the the source
or that of the grown crystal. Usually the condition for stable sublimation
is much more difficult to meet which may be seen in the corruption of the
source material.
So far we have not discussed the mass transport that is described by
equation
~
~
M
dn
Ṽ =
·D·
.
(9.11)
ρ
dT
The dominating transport mechanism depends highly on process parameters. In the presence of an inert gas diffusion is often the dominating mass
100
mm/h
Ṽ = A exp(−E/RT + b)
,
K/cm
∆H = (a0 + a1 x + a2 x 2 )
(9.13)
2
W/cm
,
mm/h
(9.14)
where x = 10000K/T . In the case of free molecular transport we obtain
Af = γL/cm, Ef = 623.9 kJ/mol, bf = 30.77, a0 = 0.0441, a1 = 0.331,
and a2 = −0.0306. The corresponding values for diffusion dominated
mass transport are Ad = (T /K)/(Pa/P ), Ed = 625.7 kJ/mol, bd = 25.56,
a0 = 0.186, a1 = 0.298, and a2 = −0.0290.
In reality, mass transport is seldom accurately described by either of
the extreme cases. To account for both cases we assume that the individual
phenomena resist the mass transport in parallel and are therefore coupled
like the growth rates in Equation (9.8). We neglect the small deviations in
101
2
10
0
V (mm/h / K/cm)
where δij is the Kronecker delta, R is the gas constant, and γ is the effective
sticking coefficient.
At low pressures and large source-to-seed distances the dominating
mass transport mechanism may be convection due to pressure gradients,
i.e., the Stefan flow. In this work convection is not taken into account. Convection is nonlinear in nature and it is therefore not conveniently expressed
by the linear formalism adopted in this model.
~ depend on the species flux and vice versa. The
The molar densities n
stoichiometry of the flux must be the same as that of the growing crystal.
Also the heat of crystallization depends on the chemical composition at the
~
growing surface. Therefore J~ and dn/dT
are solved iteratively, and Ṽ and
∆H may be determined when convergence is reached.
We solved the two extreme mass transport models for the sublimation
growth of SiC considering 20 species from the Si–C system [83] at the interval 1800-3000 K. In the calculations we assumed SiC–C equilibrium at
the source and growth of stoichiometric SiC at the seed. The results are
presented in Figures 9.2 and 9.3. They are compactly expressed by the
following analytical functions:
4
10
10
−2
10
−4
10
−6
10
−8
10
1800
2000
2200
2400
T (K)
2600
2800
3000
Figure 9.2: Growth rate at unity temperature gradient in diffusion limited
growth (solid line) and free molecular transport (dashed line).
0.96
0.94
0.92
∆ H (W/cm2 / mm/h)
transport mechanism and D is the familiar diffusion matrix which may be
estimated from the kinetic gas theory [47, 64].
Another extreme case of mass transport occurs if the mean free path of
molecules is large compared to the distance L. This may be the case in the
sublimation sandwich geometry at low pressure [39, 102, 103]. The mass
transport may then be expressed similarly with
s
RT
,
(9.12)
Dij = δij γL
2π Mi
0.9
0.88
0.86
0.84
0.82
0.8
0.78
1800
2000
2200
2400
T (K)
2600
2800
3000
Figure 9.3: Heat of crystallization for diffusion-limited growth (solid line) and
free molecular transport (dashed line).
102
the temperature dependence and unite them by
Growth rate as a function of argon pressure
−1
10
V0 = ∆T0
exp(−Ef /RT + bf )
mm/Kh,
L/l + 1/γ
(9.15)
−2
10
where the parameter l is defined by the ratio of the two mass transport
mechanisms,
T /K
l=
d.
(9.16)
P /Pa
103
V (mm/h)
It describes the distance at which transition from free molecular transport
to diffusion limited growth occurs. Parameter d = 49.5 µm results from calculations in argon atmosphere. Other inert gases would result in different
values. The total pressure P includes the argon pressure and pressure of
the Si–C species. In the case of SiC–C equilibrium the latter may be obtained
from exp(−Ep /RT + bp ) Pa, where Ep = 570.1 kJ/mol and bp = 33.14.
The model is applied by first defining T and either q or ∆T0 . Then the
growth rate is obtained from Equations (9.4), (9.6) and (9.14)–(9.16). We emphasize that the model is rather simplified and needs further justification
from the practice of SiC crystal growth.
The major assumption in the model is that it interpolates the growth rate
between the two extreme cases. Therefore we made a comparison to the
sandwich growth experiments by Vodakov et al. [103]. In the calculations
we assumed that γ = 1 and fitted the heat flux q to the results since
we did not have accurate knowledge about the temperature distribution.
The results are shown in Figure 9.4. Even though the fit is not exact the
model quite nicely predicts the transition from free molecular transport to
diffusion limited mass transport. For larger source-to-seed clearances the
agreement is not that good which is probably due to the increasing effect
of the Stefan flow.
We also applied the model to our own experiments in closely one dimensional geometry [106]. First we made simulations of the true axisymmetric
geometry and derived the heat flux q and temperature T as a function of the
temperature measured by the pyrometer. We then calculated the growth
rate as a function of temperature. The overall growth rate estimated by
the model was too high with a factor close to two whereas the temperature
dependence of the growth rate was accurately reproduced, as may be seen
in Figure 9.5.
The model estimates quite well the parametric dependencies of the
growth rate and also the absolute values are within reasonable limits. It
also separates clearly the geometry, idealized temperature distribution and
the mechanism of mass transport. The parametric values included in the
model are affected by approximations and somewhat inaccurate input data.
Therefore, in practice, it may be a good idea to fit some of parameters, e.g.
−3
10
−4
10
−5
10
−6
10
−2
10
0
10
2
10
P (Pa)
4
10
6
10
Figure 9.4: Calculated growth rates compared to experimental results [103].
The four curves represent growth rates at source-to-seed clearances 10, 150,
500 and 3000 µm. The corresponding experimental results are presented by
symbols ×, 4, + and o.
bf and d, to experimental values. The model is best applicable to growth
in relatively high pressures or small seed-to-source distances. To account
for all process conditions, also convection should be taken into account.
9.2
Optimization of the Crystal Growth Process
The ultimate goal of the simulation is to create a robust, accurate model
that can be used for optimization of the growth process. A proposed ideal
optimization scheme is presented in Figure 9.6.
In the scheme the geometry and the process have been separated. The
geometry includes the crucible assembly, thermal isolation, induction heating and the material choices. The process includes, for example, heating
power and pressure as a function of time, and different attachment techniques for the seed. In order to make the development of the crystal growth
process as efficient as possible, the different optimization tools should be
used in a systematic manner. The geometry is related mostly to macroscopic phenomena that may be investigated by simulation. The process,
on the other hand, has a larger effect on the microscopic phenomena, such
as micropipe density, crystal mosaicity and purity. Therefore simulation
104
1
10
Design
objectives
Optimization of the Sublimation
Crystal Growth Process
0
Characterization
experiment planning
V (mm/h)
10
Simulation
geometry
optimization
Optimal crystal
growth process
process
optimization
Experiments
−1
10
model verification
falsification
−2
10
2000
2050
2100
2150
2200
T (K)
2250
2300
2350
New ideas
falsification
2400
Figure 9.5: Calculated growth rate as a function of temperature compared
to experimental results (argon pressure 100 Pa, L = 1 mm). The dashed line
represents calculations without the latent heat.
Figure 9.6: Schematic view of the proposed optimization process
should be the main tool for geometry optimization and experiments should
be the main tool for process optimization. Simulation may be verified with
experiments and experiments may be planned using the insight given by
simulations. In addition, there are of course new ideas that should be
tested against simulations and experiments.
When making a modification in the system there should be a way to
evaluate whether it is an improvement or not. In experiments the decisive
method for doing this is the characterization of the grown crystals. Of
course it is not always clear how different aspects should be evaluated and
weighted. This is particularly true for simulation. When a virtual change
in the geometry is made there should be a way to evaluate its effect: Is the
new geometry better or worse? For doing this we need design objectives.
For a given design objective it is possible to generate a cost function.
As an example let us consider the objective: “create a purely axial temperature gradient inside the crucible”. An appropriate cost function might
now be
P
|dT /dr |
,
(9.17)
Q= P
|dT /dz|
where the sum is calculated over the elements inside the crucible. It is easily
seen that the minimum value of Q is zero which is obtained only if the radial
temperature gradient is zero over all elements. The parameters describing
~ = (x1 , x2 , . . . , xN ), should result from the solution of the
the geometry, x
105
problem
~
Minimize Q(x).
(9.18)
The optimization procedure might be automated particularly if the number
of parameters is large. In our cases there were not, however, that many
parameters that may be changed. Therefore we simply used a manual
optimization procedure where the parameters were roughly optimized one
by one until convergence was reached. In this kind of optimization it is
essential that the mesh generation is parameterized in order to minimize
the extra work. As this feature is included in the simulation software,
the modification and calculation of each case took only a few minutes. This
included the solution of induction heating coupled with global heat transfer.
It should be noted that the geometry optimization is not that straightforward. It is difficult to decide what the design objectives should be.
Different problems in the growth process may require different solutions.
Neither is the optimum ever a global one. The parameter space is always
partially fixed when the general structure of the geometry is defined. Only
within this geometry can the solution be searched for. Also the possible
material choices limit the freedom of design. Therefore the geometry optimization requires an intuitive understanding of the growth process boosted
with some intelligent guesses.
106
Chapter 10
Conclusions
10.1
About this Work
In this work a comprehensive model for the simulation of sublimation
growth of SiC has been developed. The model includes the most important physical phenomena. It has been verified in simple cases and applied
to more complicated cases of practical interest.
The simulation program includes some features that are not yet available in any commercial software. The integrated mesh generator with the
multi-physics features makes the solution of complicated systems efficient.
They enable the simultaneous solution of different equations in different
domains. The model also links mass transport to equilibrium chemistry
and adsorption-desorption at the surface in a generic manner. To our
knowledge this has not been previously demonstrated. The virtual crystal growth is a new concept in the SiC modeling community. However, it
should be further developed before it is generally applicable. The suitability
of the code for crystal-growth modeling is enhanced by the tailored feedback mechanisms that make it possible to control the simulations with the
same parameters as the real growth process.
The results of this work have increased the understanding of the growth
process. The thermodynamics of SiC formation was studied carefully in different conditions. We have treated a larger number of chemical species than
has been traditionally customary. This does not, however, affect the results
to a significant degree. The growth rate and growth shape were addressed
in a systematic manner which increased the intuitive understanding and
helped in the design decisions. The results obtained using the feedback
routines enable a more precise process control. The simulation program
has also been used in geometry optimization. For example, the goal of
107
purely axial temperature gradient inside the crucible was closely achieved
with the help of simulations. A lot of work is still needed in further optimization of the growth process.
The model could be further developed to account for some additional
physical phenomena. The incorporation of Stefan flow would be the next
natural step. It is required in accurate growth rate models when the pressure of the inert gas is low compared to the pressure of reactive species.
During this work the Streamline Upwind Petrov-Galerkin (SUPG) method
was implemented and applied for some flow problems. However, it turned
out that this strategy was poorly suited for pressure driven flows occurring
in sublimation growth. Since in the SUPG method the basis functions for
pressure are of the zeroth degree pressure gradients at the reactive boundaries are difficult to apply. The stabilized finite element method might be a
way to resolve this problem.
The accurate modeling of the source material would be a major challenge. A porous source is not consumed evenly and it changes both its
chemical and physical properties during the growth process. Silicon may
diffuse through the porous media and result to a graphitized source that
is no longer isometric. Small channels that are created in the direction of
the mass transport ruin the initially homogeneous structure of the source
material. Also the properties of the porous graphite used for thermal isolation change both with time and process conditions. With time the isolation
properties gradually degrade. Fortunately, this effect is not significant during one growth.
In this work only diffuse gray radiation between the walls has been
taken into account. This is also a simplification since in reality radiation
is a function of direction and wavelength. Radiation occurs also within the
seed and the source, and the gas phase acts as a participating medium.
The equilibrium chemistry is also a simplification that might be replaced
by considering the reaction kinetics. This would, however, require accurate
data for the kinetic reactions.
All these additional features would, in principle, make the model more
accurate. Still the easiest way to improve the accuracy would be to gather
more accurate input data by experimental measurements. The basic phenomena may quite well be demonstrated with the existing tool. It is not
all that clear that adding a multitude of new phenomena would help in the
crucible design. In the end the only ways to influence the growth is via
temperature distribution, crucible geometry and material choices. Extending the model to the additional details that are beyond our control would
increase understanding but might not be directly usable for optimization
purposes.
All the models presented in this work have been macroscopic in nature.
108
The true crystal growth process is, however, microscopic in nature. Today
there are methods for calculating systems that may include a few hundred
atoms using ab initio calculations that stem directly from quantum mechanics. Many of the problems related to crystal growth are unfortunately
in the mesoscopic scale that has too many atoms for first-principles calculations and too few atoms for statistical treatment. We can only make
well educated guesses on the dependencies between the crystal quality and
macroscopic growth conditions. For example, it is speculated that the creation of micropipes is related to the stresses in the growing crystal. Thus
the questions of micropipe minimization and polytype control may be only
indirectly addressed by simulations. There is still a vast playground requiring experimental studies.
10.2
About Sublimation Growth of SiC Crystals
The simulations efficiently reveal the problems of sublimation growth. From
a macroscopic point of view they are mostly related to crystal size. Both the
mass transport and crystal growth require a heat flux that is increasingly
difficult to apply as the size of the source and size of the crystal increase.
A source of porous SiC limits the growth rate more than the crystal. This
is due to the very poor thermal conductivity of porous SiC. The problems
related to temperature distribution may partially be solved by geometry optimization. However, there will always be a trade-off between growth rate
and crystal size. Therefore — one dimension, the radius or the length — is
always severely limited.
In sublimation growth the supersaturation is ideally only due to a temperature difference between the source and the seed. However, because the
surface energy of porous SiC is slightly smaller than that of crystalline SiC
the partial pressures over porous SiC are also higher than over crystalline
SiC, even at the same temperature. This may lead to nucleation anywhere
within the system, not only at the cold spots. In order to control the growth
efficiently, the system should be set up so that the major cause for supersaturation is the temperature difference between the source and the seed.
This argument together with the bad thermal conductivity of porous SiC
strongly suggests that the source should rather be solid SiC.
Ideally the crystal growth process would be one-dimensional. The onedimensional symmetry is easily broken by the geometry or the temperature
distribution. Seed enlargement may also be required and it is achieved
by carefully designed convex isotherms. In Czochralski crystal growth the
shape of the growth front is easily retained because the crystal growth is
not limited by mass transport. Sublimation, on the other hand, may be
109
unstable because variations in the geometry may give rise to a positive
feedback in the growth and etch rates. Therefore, the temperature control
calls for special attention in the sublimation growth process.
The growth of SiC crystals is an evolutionary process. Within one growth
run only limited increase in the wafer size and quality is possible. The size
and quality will therefore change gradually and the technological limits are
achieved only after a lengthy process. This requires a careful selection of
seed crystals and maybe also a number of crucible geometries gradually
increasing in size. But more than anything, it requires repetitious experiments and lots of patience.
The progress of silicon technology has in many devices began to stagger.
The simple laws of physics set limits to what silicon technology can achieve.
If further advances are required the power electronics industry has to resort
to new materials and SiC is one of the most promising options. The future
will show whether SiC will remain a small niche business in optoelectronics
or fulfill all its potential and result in a significant semiconductor industry.
Here simulations will certainly play a key role.
110
where Mi is the molar mass of the ith species.
Using the definitions above the diffusion matrix may be calculated from
q
3
3 2π kB T 3 /Mij
Dij =
(A.6)
16 π P σ 2 Ω(1,1)
ij ij
Appendix A
where P is the total pressure. Thermal diffusion is a second-order effect
important only when there are large differences between the atomic masses
of the constituent species.
The thermal conductivity κ of a mixture of neutral gases may be calculated from
X
κi
,
(A.7)
κ=
P
1.065
√
j≠i nj φij
i 1 + 2 2n
Transport Properties
i
where
In order to calculate the mass transport in a multicomponent gas we need to
know the transport properties of the mixture. The properties for pure gases
are usually obtained from measurements. For mixtures this is not feasible
since it is impossible to measure all the possible combinations. Therefore
the properties of multicomponent gases are calculated from other available
data. One approach is to use kinetic gas theory assuming a Lennard-Jones
type of potential between the molecules [64, 107]. The potential is of the
form
" 6 #
σ
σ 12
−
.
(A.1)
φLJ (r ) = 4ε
r
r
where σi is the collision diameter and ε the maximum attraction. This
kind of potential is a major simplification. In reality, the potential is
three-dimensional. To account for the non-spherical shape of the colliding molecules, marked with i and j, the concept collision integral, Ω ij , is
used. The collision integral is a function of the reduced temperature,
∗
Tij
=
where
kB T
,
εij
p
εij = εi εj .
φij =
h
1 + (κi0 /κj0 )1/2 (Mi /Mj )1/4
[1 + Mi /Mj ]1/2
κi = Ei κi0 ,
Ei = 0.115 + 0.354
Mi Mj
,
Mi + M j
(A.5)
and
Mij =
111
cP ,i
,
kB
(A.10)
where cP ,i is the specific heat at constant pressure and
κi0
=
8.332 × 103
(2,2)
σi2 Ωii
(A.3)
(A.4)
(A.8)
(A.9)
where
(A.2)
1
(σi + σj ).
2
.
Here κi0 is the thermal conductivity of the ith species when the internal
degrees of freedom are neglected. The thermal conductivities κi may be
calculated from κi0 by the formula
and kB is the Boltzmann constant. The values of the collision integral that
are of the order of unity may be retrieved from lookup tables. Also the
following definitions are applied,
σij =
i2
112
s
T
.
Mi
(A.11)
Appendix B
Appendix C
Villars-Cruise-Smith Method
Weakly Coupled Induction
Heating
The stoichiometric matrix V can be chosen with considerable freedom. We
may use this freedom to choose V so that the Hessian matrix is easily
inverted. Expressing again the minor species as a linear combination of the
major species the Hessian matrix simplifies to


M
X
δij
ν̄i ν̄j 
νki νkj
+
−
, i, j = 1, 2, . . . , L,
(B.1)
Hij = R T 
pj+M k=1 nk
nt
The approximation by Villars, Cruise and Smith assumes that the Hessian
matrix is nearly diagonal [62]. This leads to an expression for the Hessian
matrix that is easily inverted to
−1
Hij
−1

M
2
X
νki
ν̄i2
1  1
 δij .
=
+
−
R T ni+M k=1 nk
nt
(B.2)
Combining Equations (6.5) and (B.2) gives an iteration scheme for the extents of reactions.

−1
(m)
M
2
X
∆Gj
ν̄i2
νki
1
(m+1)


, j = 1, 2, . . . , L.
(B.3)
=−
+
−
δξj
ni+M k=1 nk
nt
RT
The VCS method is very rapid, but it does not converge as well as the true
Newton-Raphson, because the approximated Hessian matrix may not always
give changes towards the minimum. The step-size parameter w should also
be used with this method.
If the electric conductivity is low in all the materials inside the induction coil
the problem of induction heating is further simplified. Then the coupling
of the magnetic field is so weak that it may be neglected. The condition for
p
this is that the skin depth, defined as 2/µσ ω is larger than the typical
dimensions of the absorbing media. Now, the electric conductivity can be
neglected in calculating the vector potential. The values of electric conductivities are used only in the final step to calculate local Joule heating.
Neglecting the electric conductivities Equation (6.75) the out-of-phase
part of the vector potential vanishes and the equation becomes purely real.
The equation for the vector potential now yields
(
−µ0 in the coil,
∂Aθ
Aθ
∂ 2 Aθ
1 ∂
r
− 2 +
=
(C.1)
0
elsewhere.
r ∂r
∂r
r
∂z 2
This equation gives some hope for an analytical solution, particularly if the
form of the current density is very simple. In the following subsections
some trials for an analytic solution are presented.
A Coil of Infinite Length
Let’s assume a narrow coil of infinite length. This is a valid assumption
if the coil is significantly longer than the object under heating. Now the
current density may be expressed as
0 = n I δ(r − a),
113
114
(C.2)
where n is the density of spirals, I is the coil current, and a is the radius
of the coil. The vector potential is independent of axial coordinate and
therefore
∂Aθ
Aθ
1 ∂
r
− 2 = µnIδ(r − a).
(C.3)
r ∂r
∂r
r
The homogeneous equation has an analytic solution, which is of the form
Aθ = c1 r + c2 /r . Fitting this piecewise to the boundary conditions and
demanding continuity at r = a we have
( 1
r ≤ a,
2 µnIr ,
(C.4)
Aθ =
1
2
µnIa
/r
,
r > a.
2
Because the object is usually inside the coil we get a simple formula for the
heat generation
1
h(r ) = σ (ωµnIr )2 .
(C.5)
8
This formula nicely shows the parametric dependencies of a weakly coupled induction heating. The heating is directly proportional to the electric
conductivity and to the square of the driving current. In an ideal geometry
the heating is also proportional to the square of the radius, thus it is not
possible to heat any objects at the origin.
A Coil of Finite Length
Usually the coil that drives the heating is not infinitely long compared to
the object under heating. It may even be that an uneven temperature distribution is desired in which case the position of the coil is an important
parameter. Now the current density may be expressed as
(
n I δ(r − a), zmin ≤ z ≤ zmax ,
∂ 2 Aθ 1 ∂Aθ Aθ ∂ 2 Aθ
+
=
(C.6)
− 2 +
0,
elsewhere.
∂r 2
r ∂r
r
∂z 2
This equation is very difficult, if not impossible, to solve analytically. Therefore we take another approach.
It may be proven that the vector equation
~ = −µ~
∇2 A

(C.7)
has the solution [70]
~ r2 ) = µ
A(~
4π
Z
V
~(~
r1 )
dV1 .
|~
r2 − r~1 |
(C.8)
This formula may be applied to the case of a coil of finite length. Now in
cartesian coordinates
r~1 = a cos θ e~x + a sin θ e~y ,
115
(C.9)
Thus
r~2 = r e~x + z e~z .
(C.10)
p
|~
r2 − r~1 | = z 2 + a2 + r 2 − 2ar cos θ.
(C.11)
dS~ = a (− sin θ e~x + cos θ e~y )dθ dz.
(C.12)
Because the coil is assumed to be very thin the volume integral may be
~ where
replaced by a surface integral, ~ dV → In dS,
Applying these to formula (C.6) gives
~ r2 ) = µnIa
A(~
4π
Z zmax Z 2π
zmin
0
− sin θ e~x + cos θ e~y
√
dθ dz.
z 2 + a2 + r 2 − 2ar cos θ
(C.13)
It is easily seen that in this formula only the e~y -component is nonzero. In
cylindrical coordinates this corresponds to the e~θ -component. Thus
Aθ =
µnIa
4π
Z zmax Z 2π
zmin
0
cos θ
√
dθ dz.
z 2 + a2 + r 2 − 2ar cos θ
(C.14)
For convenience we separate the part arising from the finite length of the
coil and write
1
(C.15)
Aθ = µnIr U,
2
where
Z zmax Z 2π
cos θ
a
√
U(~
r) =
dθ dz.
(C.16)
2π r zmin 0
z 2 + a2 + r 2 − 2ar cos θ
As shown previously for a coil of infinite length, the formula U( r~) = 1 is
valid. This provides us with a limiting value that U(r~) must approach as
the length of the coil increases. U cannot be calculated analytically.
Even if we integrate numerically over θ we may still perform the integration over z analytically. Using the formula [108]
Z
p
x
dx
√
= arsinh + c1 = log(x + x 2 + a2 ) + c2
(C.17)
2
2
a
x +a
we get
U=
a
2π r
Z 2π zmax
0
zmin
p
cos θ log z + z 2 + a2 + r 2 − 2ar cos θ dθ. (C.18)
To minimize the interval for numerical integration we take use of the symmetry properties of the cosine function:
!
√
Z
a π /2 zmax
z + z 2 + a2 + r 2 − 2ar cos θ
√
dθ. (C.19)
U=
cos θ log
πr 0
z + z 2 + a2 + r 2 + 2ar cos θ
zmin
116
This expression may be evaluated numerically,
U=
a
πr
Z π /2
0
f (θ) dθ,
(C.20)
where
f (θ) =
zmax
zmin
cos θ log
!
√
z + z 2 + a2 + r 2 − 2ar cos θ
√
.
z + z 2 + a2 + r 2 + 2ar cos θ
(C.21)
Note that function f (θ) is singular if r = a and z = 0. The integral may be
calculated using Simpson’s rule, for example:


m−1
m−1
X
X
a h
f0 + 4
U=
f2i+1 + 2
f2i + f2m  ,
(C.22)
πr 6
i=0
i=1
where h = π /4m, fi = f (ih) and m ∈ Z+ .
The local heat generation is obtained from Equation (5.50)
h(~
r) =
1
σ (ωµnIr )2 |U|2 .
8
(C.23)
A good test for the numerical calculations is to check that U approaches
unity as the length of the coil increases.
List of Symbols
This is a fairly comprehensive list of the mathematical symbols used in
this thesis. I have tried to follow the traditional conventions when possible
while also trying to avoid the risk of misinterpretation. Some symbols that
are used only in one occasion are not listed in this list. They are, however,
defined in the text.
A
~
A
~
B
C, S
D
~
D
Vector potential of the magnetic field
Magnetic flux density
Components of Aθ
Diffusion or transport matrix
Electric displacement
E
~
E
Energy
F
View factor
G
Gibbs free energy or Gebhart factor
Electric field
∆f G ◦
Standard free energy of formation
H
~
H
Enthalpy
H
Magnetic field
Hessian matrix of Gibbs function
I
Number of nodes
I
J~
Identity matrix
L
L
117
Formula matrix of the chemical species
Species flux
Number of stoichiometric vectors or some distance
Lagrange function
M
~
M
Number of basic elements
N
Number of chemical species
P
Total pressure
Vector of molar masses
118
P
Pg
Total heating power
δij
Projection operator in gas-phase chemistry
Surface emissivity
Kronecker delta, 1 if i = j, 0 otherwise
Ps
Projection operator in surface chemistry
ε
Electric permittivity
R
~
R
Gas constant or some radius
Thermal conductivity
Species flux at the surface
κ
~
λ
Entropy
µ
Magnetic permeability or kinematic viscosity
Gas phase chemistry source vector
~
µ
Chemical potentials of the chemical species
T
~
U
Temperature
~
ν
Stoichiometric vector
Coordinate displacement
ω
Angular velocity of sinusoidal current
V
Growth rate
Ω
Integration domain
V
Stoichiometric matrix consisting of stoichiometric vectors
Γ
Integration boundary
~
a
~
b
Formula vector of a chemical species
Φ
Electric potential
Vector of amounts of basic elements
ϕi
Global basis function of a finite element
cV
Specific heat at constant volume
ρ
Mass density
e~r , e~θ , e~z
Unit vectors in cylindrical coordinates
ρc
Charge density
e~x , e~y , e~z
Unit vectors in cartesian coordinates
σ
Electric conductivity
e~n
Normal of surface
σSB
Stefan-Boltzmann constant
f
Frequency of sinusoidal current
τ
Stress tensor
fi
Local basis function in a finite element
h
Joule heating
θ
ξ~
Extents of reactions
ξ, η
Normalized coordinates of a finite element
S
S~
hT
Heat transfer coefficient
√
−1
ı
Imaginary unit,

Current density
m
Iteration loop
~
n
Vector of molar densities
~
p
Vector of partial pressures
q
Heat flux
r
Radial coordinate
t
Time
v
Test function in variational formulation
~
v
Velocity
~
x
Vector of mole fractions
z
Axial coordinate
β
Viewing angle of an element
γ
Sticking coefficient
χ
Shading factor
119
Vector of Lagrange multipliers
Azimuthal coordinate
120
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