Mathematic modeling the oxygen distribution mechanism in Si ingots during growing processes
The article provides specified mathematic modeling of oxygen distribution mechanism in Si ingots. Experimentally such model parameters as quartz melting speed for different melting zones, initial oxygen concentration in melt, influence of crucible rotation speed on melting rate. The work outlines th...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2004
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nasplib_isofts_kiev_ua-123456789-1191162025-02-09T20:58:09Z Mathematic modeling the oxygen distribution mechanism in Si ingots during growing processes Oksanich, A.P. Pritchin, S.E. Vasheruk, A.V. The article provides specified mathematic modeling of oxygen distribution mechanism in Si ingots. Experimentally such model parameters as quartz melting speed for different melting zones, initial oxygen concentration in melt, influence of crucible rotation speed on melting rate. The work outlines the results of computer modeling. The results of theoretical and experimental investigations carried make possible to predict oxygen concentration in Si ingot and define the technology parameters for growing ingots of stated concentration. 2004 Article Mathematic modeling the oxygen distribution mechanism in Si ingots during growing processes / A.P. Oksanich, S.E. Pritchin, A.V. Vasheruk // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2004. — Т. 7, № 3. — С. 236-239. — Бібліогр.: 8 назв. — англ. 1560-8034 PACS: 42.65; 42.70; 61.70 https://nasplib.isofts.kiev.ua/handle/123456789/119116 en Semiconductor Physics Quantum Electronics & Optoelectronics application/pdf Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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The article provides specified mathematic modeling of oxygen distribution mechanism in Si ingots. Experimentally such model parameters as quartz melting speed for different melting zones, initial oxygen concentration in melt, influence of crucible rotation speed on melting rate. The work outlines the results of computer modeling. The results of theoretical and experimental investigations carried make possible to predict oxygen concentration in Si ingot and define the technology parameters for growing ingots of stated concentration. |
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Oksanich, A.P. Pritchin, S.E. Vasheruk, A.V. |
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Oksanich, A.P. Pritchin, S.E. Vasheruk, A.V. Mathematic modeling the oxygen distribution mechanism in Si ingots during growing processes Semiconductor Physics Quantum Electronics & Optoelectronics |
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Oksanich, A.P. Pritchin, S.E. Vasheruk, A.V. |
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Oksanich, A.P. |
| title |
Mathematic modeling the oxygen distribution mechanism in Si ingots during growing processes |
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Mathematic modeling the oxygen distribution mechanism in Si ingots during growing processes |
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Mathematic modeling the oxygen distribution mechanism in Si ingots during growing processes |
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Mathematic modeling the oxygen distribution mechanism in Si ingots during growing processes |
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Mathematic modeling the oxygen distribution mechanism in Si ingots during growing processes |
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mathematic modeling the oxygen distribution mechanism in si ingots during growing processes |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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2004 |
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https://nasplib.isofts.kiev.ua/handle/123456789/119116 |
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Mathematic modeling the oxygen distribution mechanism in Si ingots during growing processes / A.P. Oksanich, S.E. Pritchin, A.V. Vasheruk // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2004. — Т. 7, № 3. — С. 236-239. — Бібліогр.: 8 назв. — англ. |
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Semiconductor Physics Quantum Electronics & Optoelectronics |
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AT oksanichap mathematicmodelingtheoxygendistributionmechanisminsiingotsduringgrowingprocesses AT pritchinse mathematicmodelingtheoxygendistributionmechanisminsiingotsduringgrowingprocesses AT vasherukav mathematicmodelingtheoxygendistributionmechanisminsiingotsduringgrowingprocesses |
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2025-11-30T16:57:01Z |
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Semiconductor Physics, Quantum Electronics & Optoelectronics. 2004. V. 7, N 3. P. 236-239 .
© 2004, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine236
PACS: 42.65; 42.70; 61.70
Mathematic modeling the oxygen distribution mechanism
in Si ingots during growing processes
A.P. Oksanich, S.E. Pritchin, A.V. Vasheruk
Institute of Economics and Advanced Technologies, 24/37 , Proletarska street, 36900, Kremenchuk, Ukraine
Phone/fax.: +380 (5366) 31124, E-mail oksanich@cat-ua.com
Abstract. The article provides specified mathematic modeling of oxygen distribution mecha-
nism in Si ingots. Experimentally such model parameters as quartz melting speed for differ-
ent melting zones, initial oxygen concentration in melt, influence of crucible rotation speed
on melting rate. The work outlines the results of computer modeling. The results of theoretical
and experimental investigations carried make possible to predict oxygen concentration in Si
ingot and define the technology parameters for growing ingots of stated concentration.
Keywords: Si ingots, oxygen, crucible, mathematic modeling.
Paper received: 27.08.04; accepted for publication: 21.10.04.
1. Introduction
In microelectronics and instrument-making the most com-
mon growing method of Si single-crystal ingots is that of
Czochralski one though during the process silicon is satu-
rated with hydrogen from quartz crucible. Hydrogen is
the main dopant in silicon [1], determining the thermo-
donor behavior, the support of charge life thermal stabil-
ity, microdefects formation.
The Si ingot diameter and length being increased, the
hydrogen dopant influence on their electro-physical
properties extends [2].
Obtaining Si ingots with stated oxygen distribution
along the ingot is subject to carrying out technological
process with programmed variations of technological
parameters. In this case, mathematic modeling of oxy-
gen distribution along the ingot is necessary to deter-
mine these parameters.
Existing mathematic models [3] that describe oxygen
inflow in the melt and its distribution in the ingot need
further specification as they cannot be used for obtaining
accurate modeling results to practise oxygen concentra-
tion prediction and choose technological parameters in
the process of the ingot growth with the diameter more
than 150 mm. Consequently, the problem of oxygen dis-
tribution during the growth of Si ingots with the above
mentioned diameter is the issue of the day.
2. Problem definition and solution suggested.
During melting different dopant, otherwise stated, may
occur in silicon. The most important of them is oxygen
with concentration dependent on process conditions and
the ingot diameter [4]. The main source of oxygen in the
melt during ingot growth is quartz crucible in which melted
burden is loaded. Then the ingot is grown from the melt
with oxygen in it. The crystallization process is carried
out at high temperatures; crucible melts slowly and under
convection current oxygen of the crucible transfers to melt.
The problem of oxygen inflow from the crucible in the
melt and its influence on dopant distribution in growing
crystal has been widely discussed. Formulas obtained
are sophisticated and require digital integration. Thus,
for example in work [5], the equilibrium obtained for de-
termining the oxygen concentration in the melt does not
take into account the cylindrical form of the crucible.
In this case, we can use equilibrium describing the
dopant distribution along the ingot [6]:
,
1
)1(
)2(
2
)1(
]
)2(
2
1
1[
0
0
2
0
001
0000
2
00
0
)(kv
kvR
g
kRv
kvV
g
CkRv
vV
)C(kv
vR
CkC
s
p
s
pk
s
pp
s
p
−
+−
−
+−×
×
−
−
−
−=
− π
π
(1)
A.P. Oksanich et al.: Mathematic modeling of oxygen distribution mechanism in ...
237SQO, 7(3), 2004
where k0 is the equilibrium distribution ratio equal to 0.25;
C0 � initial oxygen concentration in the melt; p � melted
silicon density equal to 2.53 g/cm3; R � crucible radius;
vp � speed of the oxygen inflow in the melt with 1 cm2
contact surface of crucible and melt; Vp � melt amount;
V0 � initial melt amount; vs � growth speed; g �part of the
melt crystallized calculated as
g =
0W
W
, (2)
where W � ingot oblong mass; W0 � melt and ingot total
mass.
Programming technological process parameters is
achieved by bringing a variable parameter to the ingot
length and radius, and equilibrium (1) should be expressed
through the ingot length L and radius Rs.
After simple equilibrium transformation (1), we have:
,
1
)1(
)2(
2
)1(
]
)2(
2
1
[
0
0
2
0
001
00
0
2
00
0
)(kv
kvR
g
kRv
kvV
g
kRv
kvV
)(kv
kvR
CkC
s
p
s
pk
s
pp
s
p
−
+−
−
+−×
×
−
−
−
−=
− π
π
(3)
The oxygen concentration in ingot is defined as
ss VCC ×= , (4)
where Vs is the amount of grown ingot part equal to
LRV ss
2π= , (5)
where Rs � ingot radius, L � ingot length.
Then, using (3) and (5) in (4) and defining k0C0 as À
we have:
( ),
1
2
)1(
)2(
2
)1(]
)2(
2
1
2
[
2
0
0
0
00
1
00
0 0
LR
)(kv
kvR
g
kRv
kvV
g
kRv
kvV
)(kv
kvR
AC
s
s
p
s
p
k
s
pp
s
p
π
π
π
−
+−
−
+
+−
×
−
−
−
−= −
(6)
concentration distribution along the ingot is determined
by equilibrium:
∫=
L
sl dLCC
0
. (7)
Then, integrating (6) we have:
∫ =
L
s
s
LRA
LdlRA
0
22
2
2
π
π ; (8)
∫ −
=
−
L
s
p
s
sp
)(kv
LkvR
dL
)(kv
LRkvR
0 0
2
0
22
0
2
0
2
121
πππ
; (9)
∫ −
=
−
L
s
s
s
ss
kR
kLR
dL
kR
LRLkR
0 0
0
342
0
2
0
2
)2(3
2
)2(
2
ν
π
ν
ππ
; (10)
)(kv
LRkvR
dL
)(kv
LRkvR
s
sp
L
s
sp
121 0
2
0
22
0 0
2
0
2
−
=
−∫
πππ
. (11)
Putting equilibrium (8) as À1, equilibrium (9) as À2,
equilibrium (10) as À3, equilibrium (11) as À4 and intro-
ducing À5 = πRs
2L, we get as a result:
( )( ) ( ) 54
1
321 )11( AgAgAAAC k
l ×−+−−−= − . (12)
It is necessary to determine speed transfer of the dopant
in the melt as well as initial oxygen concentration at the
beginning of the cylindrical ingot part growth for calcu-
lating equilibrium (12).
Oxygen inflow speed from quartz crucible in melt is
determined by speed of the crucible melting dependant
on the interaction character between melt and crucible.
Heater temperature and crucible rotation ratio are main
technological parameters affecting this process. As it is
shown in [6] this problem has not been sent analytically.
Authors used measuring method for determining cru-
cible melting speed. It conveys weighing of crucible parts
with definite dimensions after finishing of ingot growth
process. We used crucible of natural quartz glass made
by GE Quartz Europe GmbH with diameter 350 mm,
during ingots growth, diameter being 155 mm. the growth
was effected with utmost usage of the melt in the crucible.
In the growing process by means of automated con-
trol growth system [7] archiving data of ingot diameter,
length, rotation ratio, lifting speed, rotation frequency
of ingot and crucible and data discrete filing, equal to
120 sec. We define 4 contact zones crucible-melt (See
figure 1).
First zone mostly takes no part in oxygen saturation
of the melt, consequently its examining poses no practi-
cal interest.
Second zone is characterized by well defined flute of
breadth up to 2 mm. This zone is developed due to the
fact that at the process beginning flute is melted at higher
temperature and the melt, after complete flute melting,
has a higher temperature up to 1480°Ñ. It determines the
oxygen concentration at the initial stage of growth process.
Third and fourth zones determine oxygen concentra-
tion in melt during basic part ingot growth and, conse-
quently, oxygen concentration in ingot.
After ingot withdrawal rectangular wafers were cut of
the quartz, with size 10 × 100 mm, which were weighted
with accuracy to 0.001 g.
Fig. 1. Contact zones of melt and the crucible.
1 � Zone defines crucible upper surface
with no contact with melt
2 Zone of contact melt surface with
crucible
�
3 Crucible walls�
4 Crucible bottom�.
238
SQO, 7(3), 2004
A.P. Oksanich et al.: Mathematic modeling of oxygen distribution mechanism in ...
During the ingot growth process for lower melt level
compensation that takes place due to silicon transforma-
tions from liquid phase (melt) to solid (ingot) crucible
lifting is made with ratio determined by means of equilib-
rium [8]:
−×
+=
1
1
1
2
2
m
s
s
t
st
D
D
ρ
ρ
νν , (13)
where Ds � ingot diameter, Dt � crucible diameter.
Thus, the time of measured wafer in the melt is de-
fined as:
t
p
H
t
ν
= , (14)
where Í � melt level from flute beginning at crucible to
lower wafer, νt � crucible lifting speed.
The crucible melting speed is calculated as:
pp
p
p
tS
MM
⋅
−
= 0ν , (15)
where M0 � wafer sample mass cut from the crucible not
used in the growth process = 10.98 g; Mp � wafer sample
mass; Sp � wafer measured dimensions = 500 mm2.
Experiments were carried out for the fixed crucible
rotation ratio 5, 10, and 15 r/min. At every rotation ratio
three processes took place and measured results of cruci-
ble melting speed were approximated.
The results of the experiments are shown in Fig. 2.
Experiments proved that crucible melting speed varies
in different zones. Crucible melting speed for these zones
for crucible of natural quartz glass made by GE Quartz
Europe GmbH with the diameter 330 mm is 7.5 mg/(cm2⋅h)
� for the second zone; 3.2 mg/(cm2⋅h) � for the third
zone, and 4.3 mg/(cm2⋅h) � for the fourth one.
The initial oxygen concentration in the melt Ñ0 is de-
fined by the degree of crucible melt in the process of flute
melting at high temperatures. It is possible to evaluate
the initial oxygen concentration in the melt by means of
the oxygen concentration varying in the upper part of the
ingot. We carried out a research of the oxygen concen-
tration in the upper part of 128 ingots grown in identical
conditions to define the initial oxygen concentration. The
results of verifications are given in Fig. 3.
The initial oxygen concentration in melt was defined
as average meaning of the concentration in ingots with
equilibrium:
∑=
n
N
n
C
1
00
1
, (16)
where n � quantity of samples measured, N0 � oxygen
concentration in samples.
On the basis of these experiments, we established the
meaning of the initial oxygen concentration in the melt
equal to 1.14⋅1018 cm�3.
Applying LabView and using the equilibrium obtained
by the authors (12) as well as the results of experimental
data concerning establishing quartz crucible melting
speed and the initial oxygen concentration in melt, we
made computer modeling the oxygen concentration in
melt. For this purpose, a virtual instrument was worked
out. The results of modeling are given in Fig. 4.
W = 15
W = 10
W = 5t
t
t
p
2
Third zoneSecond zone Fourth zone
L , %
n
,
m
g
/c
m
h ⋅
0 10 20 30 40 50 60 70 80 90 100
0
1
2
3
4
5
6
7
8
9
10
T
Fig. 2. Dependence of quartz crucible melting speed on cruci-
ble rotation ratio: vp � quartz crucible melting speed, mg/(cm2⋅h);
Wt - crucible rotation ratio, r/min; LT � ratio melt height in cruci-
ble to crucible height, %.
Fig. 3. Initial oxygen concentration in Si melt.
1
7
�
3
Quantity of ingots in a lot, units O
x
y
g
e
n
c
o
n
c
e
n
tr
a
ti
o
n
,
1
0
c
m
×
10
10.5
11
11.5
12
12.5
13
36 14 35 14 16 6 1 6
Fig. 4. Oxygen concentration in silicon.
A.P. Oksanich et al.: Mathematic modeling of oxygen distribution mechanism in ...
239SQO, 7(3), 2004
Comparison of the modeling results with those experi-
mental data of oxygen concentration in Si ingots along
their length presented quite accurate coincidence of theo-
retical and experimental data.
3. Conclusions
The given model makes it possible to describe approxi-
mately the character of oxygen concentration verifica-
tions in the melt and correct technological process pa-
rameters on the basis of data obtained.
Model parameters were determined, namely:
� Natural quartz crucible melting speed with diam-
eter 350 mm, that makes 7.5 mg/(cm2⋅h) � for the
second zone; 3.2 mg/(cm2⋅h) � for the third zone,
and 4.3 mg/(cm2⋅h) � for the fourth one;
� Initial oxygen concentration in melt that makes
1.14⋅1018 cm�1.
The experiment to find the dependence between quartz
crucible melting speed and crucible rotation ratio has
been carried out. This dependence possesses a non-linear
character.
The results of theoretical and experimental data make
it possible to predict the oxygen concentration in Si ingot
and determine technologic process parameters for obtain-
ing ingots with a given oxygen concentration.
References
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Moscow, �Mir�, p. 176 (1984), (in Russian).
2. B.M. Turovskiy, I.L. Shenderovich, A.I. Popov, V.P. Grishin,
Peculiarities of oxugen ditribution in silicon single crystals
grown by the Czochralski method with an automated con-
trolling the diameter // Nauchnyye trudy Giredmeta, 102,
p. 33 (1980), (in Russian).
3. Lin W., Hill D.W. Oxygen Segregation in Czochralski Sili-
con Growth // J. Appl. Phys.V., 54(2) pp. 1082-1085 (1983)
4. R.G. Parr., W. Yang, Hyper Silicon crystals crucible pulled
(CZ) // Phys. Rev. Lett., 78(5), pp. 887-890 (1997).
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silica interaction with a doped silicon melt // Electronnaya
technika. Ser. Materialy, 2(175) pp. 48-50 (1983), (in Rus-
sian).
6. J.C. Mikkelsen, The Metallurgy of Oxygen in Silicon // J.
Metals, 37(5), pp. 53, (1985).
7. A.P. Oksanich, S.E. Pritchin, N.D Vdovichenko, Principles
of the control system for dislocation-free silicon single crys-
tal growing under maintaining the crystal diameter and melt
temperature // Functional materials, 8(2), pp. 377-380 (2001).
8. W.R. Runyan. Technology semiconductor silicon // McGraw-
Hill San Fransisco, p. 95 (1963).
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