ARTICLE IN PRESS
Journal of Crystal Growth 266 (2004) 109–116
Numerical analysis of continuous charge of lithium niobate in
a double-crucible Czochralski system using the accelerated
crucible rotation technique
Tomonori Kitashimaa, Lijun Liub, Kenji Kitamurac, Koichi Kakimotob,*
b
a
Graduate School of Engineering, Kyushu University, 6-1, Kasuga-Koen, Kasuga 816-8580, Japan
Research Institute for Applied Mechanics, Kyushu University, 6-1, Kasuga-Koen, Kasuga 816-8580, Japan
c
National Institute for Material Science, 1-1 Namiki, Tsukuba 305-0044, Japan
Abstract
The transport mechanism of supplied raw material in a double-crucible Czochralski system using the accelerated
crucible rotation technique (ACRT) was investigated by three-dimensional and time-dependent numerical simulation.
The calculation clarified that use of the ACRT resulted in enhancement of the mixing effect of the supplied raw
material. It is, therefore, possible to maintain the composition of the melt in an inner crucible during crystal growth by
using the ACRT. The effect of the continuous charge of the raw material on melt temperature was also investigated.
Our results showed that the effect of feeding lithium niobate granules on melt temperature was small, since the feeding
rate of the granules is small. Therefore, solidification of the melt surface due to the heat of fusion in this system is not
likely.
r 2004 Elsevier B.V. All rights reserved.
PACS: 81.10.h
Keywords: A1. Fluid flows; A1. Solidification; A2. Accelerated crucible rotation technique; A2. Czochralski method; A2. Doublecrucible technique; B1. Oxides
1. Introduction
Oxide crystal of LiNbO3 (LN) with congruent
composition contains a large number of anti-site
defects and vacancies [1]. Therefore, it is necessary
to grow stoichiometric LiNbO3 single crystals to
reduce the number of such defects in the crystals.
*Corresponding author. Tel.: +81-92-583-7741; fax: +8192-583-7743.
E-mail address: [email protected]
(K. Kakimoto).
Kitamura et al. [2] and Furukawa et al. [3]
demonstrated experimentally that the doublecrucible Czochralski method enables crystal stoichiometry to be controlled during crystal growth.
In their studies, stoichiometric LiNbO3 single
crystals were grown from Li-rich melt in the inner
crucible.
In order to maintain a fixed composition of the
melt in an inner crucible, the raw material supplied
onto the melt surface must be spread rapidly over
the whole melt. Knowledge of the transport
mechanism of the raw material supplied onto the
0022-0248/$ - see front matter r 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.jcrysgro.2004.02.036
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T. Kitashima et al. / Journal of Crystal Growth 266 (2004) 109–116
melt surface would enable estimation of the mixing
effect of the melt. The accelerated crucible rotation
technique (ACRT) was used to enhance the mixing
effect of the melt in this study. There have been
many reports on the mixing effects of the ACRT
applied to various crystal growth methods such as
the Czochralski, Bridgman and zone-melting
methods. However, the double-crucible Czochralski method was not used in any of those studies.
There has been no report on the mixing effect of
the melt in a double crucible using the ACRT.
Estimation of the likelihood of solidification of
the melt surface between inner and outer crucibles
due to supply of raw material with low temperature is also important for maintaining a fixed
composition of the melt in an inner crucible. Since
solidification on the melt surface interrupts melting of the supplied raw material, such solidification
makes it difficult to control melt composition.
There have been few studies on the effects of the
continuous charge of lithium niobate on melt
temperature.
The aim of this study was to clarify the
transport mechanism of the raw material of
lithium niobate supplied onto the melt surface
and the effect of heat for melting the raw material
on the melt temperature.
2. Modeling
Schematic diagrams of a double crucible are
shown in Figs. 1(a) and (b). The inner crucible has
three windows in three-folded symmetry, and it
was placed on the bottom of the outer crucible.
The two crucibles were, therefore, rotated at the
same rotation rates. The melt in the outer crucible
could flow into the inner crucible through the
windows. We used some physical properties of
platinum for the inner crucible. The parameters
used in this study are listed in Table 1. A part of
the nonuniform grid used in the present calculations is shown in Fig. 1(c). The black belt shows
the inner crucible, and windows are also shown in
the same figure. Fig. 2 shows the crucible rotation
rate using the ACRT as a function of time in this
study. This condition of the ACRT such as the
crucible rotation rate, modulation period is
Fig. 1. Schematics of the system: top (a), meriodional view (b)
and grid used (c).
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T. Kitashima et al. / Journal of Crystal Growth 266 (2004) 109–116
10
(2) Properties of LiNbO3 melt
Melting point (K)
Emissivity ()
Density (kg m3)
Viscosity (kg m1 s1)
Thermal conductivity (W m1 K1)
Thermal expansion coefficient (K1)
Temperature coefficient of surface tension
(N m1 K1)
Density of the melted raw material (kg m3)
Diffusion coefficient of melted raw material
(m2 s1)
Heat of fusion (J kg1)
Initial temperature of the granules (K)
(3) Properties of platinum
Specific heat (J kg1 K1)
Temperature coefficient of surface tension
(N m1 K1)
Emissivity ()
(4) Nondimensional parameters
Prandtl number ()
Reynolds number ()
Rayleigh number ()
Grashof number ()
crucible rotation rate [r.p.m.]
Table 1
Calculation parameters
(1) Processing parameters
Diameter of crystal (m)
Inside diameter of the inner crucible (m)
Inside diameter of the inner crucible (m)
Outside diameter of the inner crucible (m)
Depth of the melt (m)
Height of windows in the inner crucible (m)
Highest temperature of the outer crucible (K)
Lowest temperature of the sidewall of the
outer crucible (K)
Lowest temperature of the bottom wall of the
outer crucible (K)
Temperature of the gas (K)
0.06
0.2
0.16
0.166
0.07
0.020
1480
1465
5
0
-5
-10
480
1465
520 560
600 640
680
720 760
800
time [sec]
1300
Fig. 2. Crucible rotation rate using the ACRT as a function of
time.
1440
0.3
3.58 103
2.90 102
3.09
1.70 104
3.0 104
3.58 103
1.0 108
4.56 105
1000
1.02 103
3.0 104
0.2
2
dur uf
1 qp
ur 2 quf
2
þ n r ur 2 2
¼
;
r qr
r qf
dt
r
r
duf ur uf
1 1 qp
þ
¼
r r qf
dt
r
2 qur uf
2
þ n r uf þ 2
;
r qf r2
ð2Þ
ð3Þ
duz
1 qp
gbðT Tm Þ þ nr2 uz ;
¼
r qr
dt
ð4Þ
dT
l
¼
r2 T;
dt
rCp
ð5Þ
dðrm Þ
¼ ar2 ðrm Þ;
dt
ð6Þ
d
q
q ðuf uG Þ q
q
¼ þ ur þ
þ uz ;
dt qt
qr
qf
qz
r
13.5
468.4
7.2 107
5.4 105
actually used for the crystal growth of lithium
niobate in National Institute for Material Science
(NIMS).
The governing equations of mass, momentum,
energy and conservative of the supplied raw
material are as follows:
1q
1 quf quz
ðrur Þ þ
þ
¼ 0;
r qr
r qf
qz
111
ð1Þ
where r; f and z are common coordinates in a
cylindrical system, ur ; uf and uz are components of
melt velocity in r; f and z directions, respectively,
p and T are pressure and temperature of the melt,
respectively, b is thermal expansion coefficient, n is
kinematic viscosity, r is melt density, rm is mass
concentration of melted raw material rm ¼ rM;
M is mass fraction, Cp is specific heat, l is thermal
conductivity, g is acceleration due to gravity, a is
diffusion coefficient of the melted raw material
and uG is velocity of the grid in the f direction,
which is equal to the crucible rotation rate. The
coordination frame is fixed in space. The third
term of the right-hand side in the definition of d=dt
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112
takes account of the moving mesh that rotates at
the rotation rate of crucibles.
The following boundary conditions were imposed in this study. The density of the melted raw
material was assumed to be that of Li-rich melt,
although actual density of the raw material is the
value of stoichiometric LiNbO3. Since few studies
have been carried out on the composition dependence of properties of LiNbO3 melt [4], we
imposed the above condition for simplicity. Supply
of granules to the melt was started at 480 s after
the initiation of crucible rotation, and the feed rate
of the granules was set at a constant rate of 0.1 g/
min during crystal growth. In the actual growth
system, the supplied granules float on the melt
surface because of their low density compared to
that of the melt. Moreover, the shape of the
floating region is affected by the melt velocity on
the melt surface. It was assumed that the outside
line of the floating region was a quadratic curve [5]
defined by the maximum and minimum velocities
in the floating region, and it was assumed for
simplicity that the shape of the floating region
depends on only azimuthal velocity in the floating
region. The shape of the floating region is
expressed as follows:
Djðr; uf Þ ¼
uf ðrÞ
Dt;
r
jðr; uf Þ ¼ jmin þ Djðr; uf Þ;
ð7Þ
ð8Þ
where jmin pjðr; uf Þpjmax ; jðr; uf Þ is the expansion angle of the floating region at radial r as
shown in Fig. 1(a), jmin is 15 , jman is 70 and Dt
is the time step. m ¼ raðqM=qzÞ Af was adopted in
the floating region. Af is the area of the floating
region. It was assumed that qM=qn ¼ 0 at the
walls of crucibles, at the melt surface and at the
melt–crystal interface. The granules in the floating
region absorbed the heat of the melt. The
following boundary condition of temperature at
the floating region is used [5]:
Z Tm
qT
¼
Af l
Cp dT þ Hf m;
ð9Þ
qz
Ti
where Tm is the melting point, Ti the initial
temperature of the supplied granules, Hf the heat
of fusion of lithium niobate and m the feed rate of
the granules. The right-hand side of Eq. (9) shows
the total amount of heat absorbed by the supplied
granules. The temperature distributions of the
sidewall and bottom wall of the outer crucible
were assumed to be parabolic. The temperatures at
the top of the sidewall and at the center of crucible
bottom are the lowest, while the temperature at the
bottom of the sidewall is the highest. These values
are shown in Table 1. The melt surface was
assumed to be stress-free and cooled by radiation.
It was assumed for simplicity that the melt–crystal
interface was flat. The temperature along the
interface was fixed to the melting point. The effect
of thermo-capillary force was taken into account
in this study as follows for the boundary conditions:
m
qur
qT
;
¼ gT
qr
qz
m
quf
qT
;
¼ gT
rqf
qz
uz ¼ 0;
ð10Þ
where gT is the temperature coefficient of surface
tension. The crystal was rotated at a constant rate
of 2.8 revolutions per minute throughout the
calculation.
A finite volume method was employed for
numerical analysis of this system with threedimensional and time-dependent conditions. The
convection terms in Eqs. (1)–(5) were discretized
by a quadratic upstream interpolation for convective kinetics (QUICK) scheme reported by
Hayase et al. Bram van Leer’s third-order monotone upstream-centered scheme for conservation
laws (MUSCL) with van Albada’s limiter was used
for the convection terms in Eq. (6). Pressure and
velocity corrections were solved by the semiimplicit method for pressure-linked equations
(SIMPLE) algorithm. A set of simulations with
modulation period of 100 s had been conducted
with different grid sizes. When the grid size is finer
than 90r 86y 70z ; the results showed to be gridindependent. Therefore, this grid size was used.
The time step was set to 0.1 s in this study.
3. Results and discussion
Figs. 3(a)–(c) show the mass fraction distributions of the supplied raw material and the flow
field on the melt surface and in a vertical plane at
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113
Fig. 3. Mass fraction distributions on the melt surface (a, d), in a vertical plane (b, e) and flow fields in a vertical plane (c, f) at 764 s
(a–c) and at 780 s (d–f) after the initiation of crucible rotation. The ACRT was used.
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T. Kitashima et al. / Journal of Crystal Growth 266 (2004) 109–116
764 s after the initiation of crucible rotation when
the ACRT was used. Figs. 3(d)–(f) show the mass
fraction distributions and the flow field at 780 s
after the initiation of crucible rotation. Granules
of the raw material were supplied to the area of
circles shown by arrows in Figs. 3(a) and (d). The
supplied raw material flowed into the inner
crucible along the outside wall of the inner
crucible, then moved toward the melt–crystal
interface along the melt surface, and finally spread
throughout the melt in the inner crucible.
In order to estimate the mixing effect of the raw
material under the condition of use of the ACRT,
we compared the results with the case of rotation
of the crucibles at a constant rate. Figs. 4(a)–(c)
show the mass fraction distributions of the
supplied raw material and the flow field on the
melt surface and in a vertical plane at 780 s after
the initiation of crucible rotation when the
crucibles were rotated at a constant rate 5.0
revolutions per minute. When the crucibles were
mass concentration of supplied raw
material under the interface [g/m3]
Fig. 4. Mass fraction distributions on the melt surface (a), in a vertical plane (b) and a flow field in a vertical plane (c) at 780 s after the
initiation of crucible rotation. The crucible rotation rate was constant at 5.0 r.p.m.
0. 3
ACRT
constant rotation rate
0. 2
0. 1
0
480 520 560 600 640 680 720 760 800
time [sec]
Fig. 5. Mass concentration of the supplied raw material 1 mm
under the melt–crystal interface.
rotated at a constant rate, the raw material was
concentrated near the inside wall of the inner
crucible after flowing into the inner crucible from
the outer crucible, and it was then transferred
toward the interface with upward and downward
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115
Fig. 6. Temperature distribution on the melt surface. (a) The ACRT was used. Ti was 1000 K. m was 0.1 g/min. (b) Only the crystal
was rotated. Ti was 300 K. m was 0.1 g/min. (c) Only the crystal was rotated. Ti was 1000 K. m was 3.0 g/min.
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movements. As shown in Fig. 4(b), the mass
fraction gradient of the raw material in the inner
crucible is large in comparison with the case of
using the ACRT as shown in Figs. 3(b) and (e).
Fig. 5 shows the averaged mass concentration of
the supplied raw material 1 mm under the melt–
crystal interface as a function of time. It was found
that the supplied raw material reached the melt–
crystal interface in the case of using the ACRT 40 s
faster than that in the case of constant rotation
rate. It was also found that the gradient of the
mass concentration in the case of using the ACRT
is larger than that in the case of constant rotation
rate. These results indicate that the mixing effect of
the raw material in the case of using the ACRT is
large. Therefore, the ACRT is an effective
technique for maintaining a fixed composition of
lithium niobate melts in the crucible.
Fig. 6(a) shows the temperature distribution on
the melt surface at 764 s after the initiation of
crucible rotation when the ACRT was used. The
granules were supplied to the area of circles shown
by arrows. It was shown that the change in
temperature of the melt surface due to the heat
of fusion was small despite the supply of granules
onto the melt. Ono et al. [5] carried out a
numerical simulation of the influence of feeding
polycrystalline-silicon granules on the melt in a
double-crucible system. They noticed the possibility that solidification of the silicon melt is related
to the initial temperature of the granules and the
crucible rotation rate. The factors were considered
in the present calculation. Fig. 6(b) shows the
temperature distribution on the melt surface at
764 s when only the crystal was rotated. The initial
temperature of the granules supplied onto the melt
was set to 300 K, and the feed rate of the granules
was set to 0.1 g/min. It was shown that the
temperature on the melt surface was not greatly
reduced when the crucibles were not rotated and
the initial temperature of the granules was 300 K.
Therefore, solidification of lithium niobate melt is
not likely. This is due to the low feed rate of
lithium niobate granules during crystal growth.
Fig. 6(c) shows the temperature distribution on the
melt surface at 764 s when the feed rate of the
granules was set to 3.0 g/min. Only the crystal was
rotated. The initial temperature of the granules
was 1000 K. It was found that the temperature in
the floating region on the melt surface became
remarkably low. However, such a large amount of
granules is not suitable for the small crystallization
rate of lithium niobate. The effect of heat for
melting the granules on the melt temperature was
small when the feed rate was set to a suitable rate
of 0.1 g/min for the crystallization of lithium
niobate.
4. Conclusions
Use of the ACRT resulted in enhancement of
the mixing effect of the supplied raw material,
making it easier to maintain a fixed composition of
the melt in the inner crucible. The temperature on
the melt surface did not change noticeably despite
the supply of raw material onto the melt. Therefore, solidification of lithium niobate melt due to
the heat of fusion is not likely.
Acknowledgements
This work was supported by a Grant-in-Aid for
Scientific Research (B) 14350010 from the Ministry of Education, Science and Culture, Japan.
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