Modelling vacancy microvoid formation in dislocation-free silicon single crystals
An alternative mathematical model of vacancy microvoid formation in dislocation-free silicon single crystals was represented. The analysis of conditions of microvoid nucleation inside the bulk of crystals during cooling after their growth was carried out. The possibility of formation of a quasi-stat...
Збережено в:
| Дата: | 2006 |
|---|---|
| Автори: | , , , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2006
|
| Назва видання: | Semiconductor Physics Quantum Electronics & Optoelectronics |
| Онлайн доступ: | https://nasplib.isofts.kiev.ua/handle/123456789/121641 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Цитувати: | Modelling vacancy microvoid formation in dislocation-free silicon single crystals / V.I. Talanin, I.E. Talanin, S.A. Koryagin, M.Yu. Semikina // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2006. — Т. 9, № 4. — С. 77-81. — Бібліогр.: 12 назв. — англ. |
Репозитарії
Digital Library of Periodicals of National Academy of Sciences of Ukraine| id |
nasplib_isofts_kiev_ua-123456789-121641 |
|---|---|
| record_format |
dspace |
| spelling |
nasplib_isofts_kiev_ua-123456789-1216412025-02-23T19:27:38Z Modelling vacancy microvoid formation in dislocation-free silicon single crystals Talanin, V.I. Talanin, I.E. Koryagin, S.A. Semikina, M.Yu. An alternative mathematical model of vacancy microvoid formation in dislocation-free silicon single crystals was represented. The analysis of conditions of microvoid nucleation inside the bulk of crystals during cooling after their growth was carried out. The possibility of formation of a quasi-stationary microvoid profile in large-scale crystals within the temperature range 1130…1070 °С has been shown. This scientific work was made by the budgetary funds of Ministry of Education and Science of Ukraine as the grant of the President of Ukraine. 2006 Article Modelling vacancy microvoid formation in dislocation-free silicon single crystals / V.I. Talanin, I.E. Talanin, S.A. Koryagin, M.Yu. Semikina // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2006. — Т. 9, № 4. — С. 77-81. — Бібліогр.: 12 назв. — англ. 1560-8034 PACS 61.72Bb, 61.72.Jj, 61.72.Yx https://nasplib.isofts.kiev.ua/handle/123456789/121641 en Semiconductor Physics Quantum Electronics & Optoelectronics application/pdf Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| collection |
DSpace DC |
| language |
English |
| description |
An alternative mathematical model of vacancy microvoid formation in dislocation-free silicon single crystals was represented. The analysis of conditions of microvoid nucleation inside the bulk of crystals during cooling after their growth was carried out. The possibility of formation of a quasi-stationary microvoid profile in large-scale crystals within the temperature range 1130…1070 °С has been shown. |
| format |
Article |
| author |
Talanin, V.I. Talanin, I.E. Koryagin, S.A. Semikina, M.Yu. |
| spellingShingle |
Talanin, V.I. Talanin, I.E. Koryagin, S.A. Semikina, M.Yu. Modelling vacancy microvoid formation in dislocation-free silicon single crystals Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Talanin, V.I. Talanin, I.E. Koryagin, S.A. Semikina, M.Yu. |
| author_sort |
Talanin, V.I. |
| title |
Modelling vacancy microvoid formation in dislocation-free silicon single crystals |
| title_short |
Modelling vacancy microvoid formation in dislocation-free silicon single crystals |
| title_full |
Modelling vacancy microvoid formation in dislocation-free silicon single crystals |
| title_fullStr |
Modelling vacancy microvoid formation in dislocation-free silicon single crystals |
| title_full_unstemmed |
Modelling vacancy microvoid formation in dislocation-free silicon single crystals |
| title_sort |
modelling vacancy microvoid formation in dislocation-free silicon single crystals |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2006 |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/121641 |
| citation_txt |
Modelling vacancy microvoid formation in dislocation-free silicon single crystals / V.I. Talanin, I.E. Talanin, S.A. Koryagin, M.Yu. Semikina // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2006. — Т. 9, № 4. — С. 77-81. — Бібліогр.: 12 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
AT talaninvi modellingvacancymicrovoidformationindislocationfreesiliconsinglecrystals AT talaninie modellingvacancymicrovoidformationindislocationfreesiliconsinglecrystals AT koryaginsa modellingvacancymicrovoidformationindislocationfreesiliconsinglecrystals AT semikinamyu modellingvacancymicrovoidformationindislocationfreesiliconsinglecrystals |
| first_indexed |
2025-11-24T16:06:52Z |
| last_indexed |
2025-11-24T16:06:52Z |
| _version_ |
1849688496818618368 |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 4. P. 77-81.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
77
PACS 61.72Bb, 61.72.Jj, 61.72.Yx
Modelling vacancy microvoid formation
in dislocation-free silicon single crystals
V.I. Talanin, I.E. Talanin, S.A. Koryagin, M.Yu. Semikina
University of State & Municipal Government
70B, Zhukovskii str., 69002 Zaporozhye, Kyiv, Ukraine
Abstract. An alternative mathematical model of vacancy microvoid formation in
dislocation-free silicon single crystals was represented. The analysis of conditions of
microvoid nucleation inside the bulk of crystals during cooling after their growth was
carried out. The possibility of formation of a quasi-stationary microvoid profile in large-
scale crystals within the temperature range 1130…1070 °С has been shown.
Keywords: point defect, grown-in microdefect, vacancy microvoid, CZ-Si, FZ-Si.
Manuscript received 10.10.06; accepted for publication 23.10.06.
1. Introduction
In dislocation-free silicon single crystals, the formed
various point defect agglomerates (i.e. grown-in micro-
defects) sufficiently influence on both electrophysical,
mechanical properties of crystals and characteristics of
integrated chips. It is necessary to develop a theoretical
model of grown-in microdefect formation that is
adequate to experimental results of a research of a
dislocation-free silicon single crystal microdefect
structure. This is an actual task, it allows to solve the
problem of controlling the point defect ensemble.
At the moment, the model of point defect dynamics
is generally accepted [1, 2]. This model describes
convection, recombination, diffusion of intrinsic point
defects and, for all its modifications, the Voronkov
model of grown-in microdefect formation [3] is basic.
The heart of the Voronkov model is a recognition of a
critical role of "fast recombination" process of intrinsic
point defects close to a crystallization front. Depending
on temperature growth conditions, the model of point
defect dynamics assumes the formation of interstitial
dislocation loops and vacancy microvoids during crystal
cooling in the narrow temperature interval below
1200 °С in various regions of a crystal. Furthermore, it is
supposed that the formation of all types of microdefects
is of homogeneous nature, and the process of interaction
of intrinsic point defects with impurity atoms is ignored
[1]. In the model of point defect dynamics, it is
implicitly supposed that, depending on the ratio V/G (V
is the growth rate of the crystal, G – axial temperature
gradient), the defects that were formed within the tem-
perature range 1420…1200 °С are either small vacancy
microvoids or small interstitial dislocation loops [3].
As opposed to the model of point defect dynamics,
basing on a great body of experimental researches, we
have developed the qualitative heterogeneous me-
chanism of formation and transformation of grown-in
microdefects [4, 5]. It was shown that the defect
formation process was controlled by diffusion of point
defects in the temperature gradient field. It was caused
by availability of an entropy recombinational barrier that
blocks the recombination of intrinsic point defects at
high temperatures in the course of crystal growth [4].
Therefore, dissociation of the oversaturated solid
solution of point defects goes in parallel in two direc-
tions: vacancy and interstitial. At the temperatures close
to the crystallization front, the dissociation of
oversaturated solid solutions of impurities takes place,
and in the course of cooling the crystal (Т < 1200 °С) the
dissociation of those of intrinsic point defects (vacancies
and self-interstitials) occurs. The absence of a theoretical
(mathematical) model for defect formation within
the temperature ranges both 1420…1200 °С and
1200…900 °С is currently disadvantage of the hetero-
geneous mechanism.
In this research, developed was the model of
vacancy microvoid formation as a result of condensation
of non-equilibrium vacancies during cooling of growing
monocrystals at Т < 1200 °С. The aim of this paper is
the analysis of conditions of microvoid initiation taking
into account diffusion of non-equilibrium vacancies deep
into the crystal and onto its surface.
2. The statement of problem
Vacancy microvoids are formed in large-scale crystals
(100 mm and more). The growth of such crystals is
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 4. P. 77-81.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
78
characterized by large growth rates, small curvature of
crystallization front and axial temperature gradient [6].
The temperature interval of formation and growth of
microvoids is ~1130…1070 °С, the interval of oxide
film growth on microvoid walls is ~ 1070…900 °С [6-
8]. From these results, we believe that, in the crystal
layer with a thickness δ that corresponds to the
conditions of cooling the crystal during its growth from
1130 to 1070 °С (Fig. 1), an excess of non-equilibrium
vacancies n0 are produced (Fig. 2).
Binding of oxygen atoms into grown-in
microdefects ((I+V)-microdefects) with lowering the
temperature down to the values Т < 1200 °С can be the
reason of initiation of these non-equilibrium vacancies.
For these temperature conditions at the first stage, there
are both the process of dissociation of oversaturated
solution of vacancies and simultaneous process of their
diffusion. After the majority of vacancies will be used
for microvoid formation, and the concentration of non-
equilibrium vacancies becomes rather small, the
diffusion process of the remained non-equilibrium
vacancies onto the surface and into the bulk of the
crystal is accompanied by a coalescence process of
microvoids. After completion of this second stage, all
the excess vacancies pass onto the crystal surface.
At the first stage, according to [9] the system can
be described by the equations
NDrn
x
nD
t
n π42
2
−
∂
∂
=
∂
∂ , (1)
D
N
n
t
r
L
22
=
∂
∂ , (2)
where n is the concentration of non-equilibrium
vacancies, D is the diffusivity of vacancies, r is the
radius of microvoids, NL is the concentration of
vacancies in the microvoid or the reciprocal volume per
one vacancy; N is the microvoids concentration.
Fig. 1. The scheme of non-equilibrium vacancy formation
during crystal cooling in the crystal layer with a thickness δ
(d is the crystal diameter).
In the right side of Eq. (1), the first member
describes diffusion of non-equilibrium vacancies from
the A-surface (where the A-surface is the interface
between the volume of crystal formed at 1420 < T <
1200 °С and the crystal layer with a thickness δ) into the
depth of a sample, and the second one excludes the part
of vacancies that was used for microvoid formation from
a vacancy diffusion flux.
According to Figs 1, 2, the boundary and initial
conditions look like:
.,0,
at,0
at,
,0 00
0
00 ∞==
⎪⎩
⎪
⎨
⎧
>
≤
== ==== tttx dr
x
xn
nn
δ
δ
The condition 00 ==xn assumes that, on the
surface (the bulk of crystal formed at 1420 < T < 1200 °С
plays the role of this surface), very fast absorption of
excess vacancies takes place. It follows that the
characteristic time of an absorption of excess
concentration of vacancies at х = 0 is much less than all
the other characteristic times. The condition ∞==0td
assumes that in the course of solving the problem we do
not take into account a drain of vacancies onto a lateral
face of the crystal.
3. Mathematical model
Let's consider a solution of the problem for the first stage
as is shown in [10]. Let's substitute (2) into (1), integrate
with respect to time and enter new variables according to
relationships:
qrtx λντμξ === 2,, , (3)
where
.)π38(,,
)π38(
1
2
24
2
2
0
6 −−=== N
DNn
NL μλ
μ
νμ
Fig. 2. Profile of vacancies at the initial instant (n0 –
concentration of excess non-equilibrium vacancies; neq –
equilibrium concentration of vacancies).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 4. P. 77-81.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
79
Then the equations (1), (2) take the dimensionless form:
)(2/3
2
2
ξ
ξτ
fqqq
+−
∂
∂
=
∂
∂ , (4)
τ∂
∂
=
qnn 0 , (5)
where
μ
δ
ξ
ξ
ξ =Δ
⎪⎩
⎪
⎨
⎧
Δ>
Δ≤
= ,
at,0
at,1
)(f (6)
with the boundary and initial conditions
0,0,0 00 === ∞=== ξξτ qqq .
The function )(ξf can be found by substituting
the initial conditions into Eq. (4).
Duration of the first stage τI depends on the time of
outgoing the majority of excess vacancies into
microvoids, which is determined from Eq. (4) with
elimination of the diffusion member 2
2
ξ∂
∂ q that is
approximately equal to unity. The restriction τI ≤ τS = Δ2
is the necessary condition of microvoid formation in the
layer with a thickness δ. In accord with this condition,
the time of microvoid formation should be no more than
the time of vacancy going onto a surface [10].
The characteristic times of the first stage of the
process in the dimensional form look like:
D
t
2
II
μντ == , (7)
D
t SS
2δντ == . (8)
After the completion of the first stage )0/( ≈∂∂ τq ,
the obtained microvoid profile can be determined from
the equation
0)(2/3
2
2
=+−
∂
∂ ξ
ξ
fqq (9)
with the boundary conditions .0,00 == ∞→= ξξ qq
The value Δ is the parameter that determines the shape
of the structure voids. This value is meaningful of the
ratio of the characteristic time of going the vacancies
onto a surface to the characteristic one of microvoid
formation. Solutions of Eq. (7) for three ranges of
varying the coordinate ξ look like [10]:
∫ =
⎟
⎠
⎞
⎜
⎝
⎛ +−
≤≤
Δ
q
qqq
dq
0 2/5
max
5
22
,0 ξξξ , (10)
∫ =
⎟
⎠
⎞
⎜
⎝
⎛ +−
+Δ≤≤
Δ
max
2/5
maxmax
5
22
,
q
q qqq
dq ξξξξ ,
(11)
,
)(
400, 4
0ξξ
ξ
+
=∞<≤Δ q (12)
where
∫ Δ−⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
⎟
⎠
⎞
⎜
⎝
⎛ +−
=
Δ
Δ
max
0
4/1
0
2/5
max ,400,
5
22
q
q
qqq
dq ξξ
max, qqΔ are determined by the equations
,0
5
2
max
2/5
max =+− Δqqq
∫
Δ
−Δ=
⎟
⎠
⎞
⎜
⎝
⎛ +− Δ
max
.
5
22
max
2/5
q
q qqq
dq ξ
In the limiting cases, we have
,2.62.2,
2
1 2/1
2
max
Δ
+Δ≈
Δ
≈≈<<Δ Δ ξqq
.5.2,11 2/1max +Δ≈≈>>Δ ξq
The duration of the second stage of this process
(microvoid coalescence) can be evaluated using the
expression of the characteristic coalescence time [10]:
,
eq
2
3
II nD
kTrt
Ω
=
σ
(13)
where σ is the surface tension at the interface crystal-
vacuum.
If in a crystal the process of impurity diffusion
takes place, then in this case when the impurity has a
chance to pass a distance δ in a time timp that is smaller
than tII, it will decorate the vacancy profile. Therefore,
the condition of impurity profile formation is the ratio:
,II
imp
2
imp t
D
t ≤=
δ (14)
where impD is the diffusivity of impurity.
4. Experimental
The values of parameters necessary for calculations are
as follows: 322
323
cm105
cm102
11 −
−
⋅=
⋅
=
Ω
=LN
[11]; 35 cm10 −=N [8]; 2cm/erg650=G [11];
)/917.1exp(449.188 kTD −= ; eV/K106153.8 5−⋅=k ;
);/54.2exp(17.00 kTD −= )/9.3exp(1011639.1 27 kTn −⋅=
[11]; cm10nm100 5−==r ; )/1.3exp(9.1 kTDc −= .
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 4. P. 77-81.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
80
Fig. 3. Dependence of a microvoid radius squared on the
distance up to the initial boundary of the temperature interval
for microvoid formation (dimensionless, Δ = 9.767).
The temperature profile along the length of the
growing CZ-Si and FZ-Si crystals is described by the
equation [2]: ,111
2 Gx
TTT mm
+= where Т = 1685 К is the
temperature at the crystallization front, G is the axial
temperature gradient. Using the standard for majority of
CZ-Si crystals value G = 2.5 К/mm and vacancy
microvoid formation temperature range 1130…1070 °С
[6-8], we obtained δ ≈ 3.75 cm. Then μ = 0.384 cm and
Δ = 9.767 >> 1. For this value of Δ, from Eqs (10)-(12)
we determined qmax = 0.99659, εmax = 5.284. The
calculation results are shown in Fig. 3.
For the temperature range of vacancy microvoid
formation, we evaluated the characteristic times of the
first and second stages of the process and concluded that
the ratio III ttt S <<<< holds true.
Hence, the unambiguous conclusion follows that,
during the crystal growth in a case of creation of excess
vacancy concentration in the temperature range of
cooling the crystal of 1130…1070 °С, the quasi-
stationary microvoid profile is formed. The conditions of
vacancy microvoid decoration by background impurities
of carbon and oxygen (14) in the temperature interval
1070…900 °С [6, 8] are well valid everywhere over the
indicated range.
In a statement of the problem, we did not take into
account a drain of vacancies onto a lateral face of
crystal. The availability of a lateral face limits conditions
of vacancy microvoid formation in the cross-section of
the crystal to a minimum request dmin ≈ 2δ. Therefore,
under a certain temperature growth conditions in
commercial single crystals, the vacancy microvoids
should be formed since the crystal diameter is more than
80 mm.
5. Discussion
The experimental results that were carried out recently
by us using the transmission electron microscopy of the
FZ-Si and CZ-Si crystals grown in various temperature
conditions have shown that the primary oxygen-vacancy
and carbon-interstitial agglomerates formed at impurity
centres close to a crystallization front are the basis of
defect formation process [4, 5]. The main reason of this
process is the availability of a recombinational barrier
for intrinsic point defects at high temperatures [12].
Using the impurity atoms to form precipitates results
that, depending on the growth conditions when the
crystal cooling temperature is lower than 1200 °С,
created are the conditions for initiation of a non-
equilibrium concentration of vacancies and self-
interstitials of silicon. Therefore, the generation of
secondary defects (agglomerates of intrinsic point
defects), with which the growth of a new phase is
accompanied, is the key feature of dissociation of
oversaturated solid solution of point defects.
These experimental results have allowed us to enter
a new physical classification of grown-in microdefects,
which is based on two types of interaction: the "impurity
– intrinsic point defect" (primary grown-in microdefects)
and "intrinsic point defect - intrinsic point defect"
(secondary grown-in microdefects). Such consideration
allows us to enter two defect subsystems (primary and
secondary microdefects) and, for each of them, to
develop mathematical models that are based on
consideration of the phenomena of dissociation of
oversaturated solid solutions of point defects within the
framework of the dissociative diffusion model.
In this work, as opposite to the Voronkov model,
offered was the model of secondary grown-in
microdefect formation (vacancy microvoids) that:
- explains the reason of formation of excess non-
equilibrium vacancies in the observed temperature
range;
- explains the reason of absence of vacancy
microvoids in small-scale crystals;
- enters the mathematical means that shows the good
coordination of the theoretical calculations with the
experimental results;
- gives understanding of physics of defect formation
process in dislocation-free silicon single crystals;
- is well agreed with the available experimental
results.
The theoretical calculations and experimental
researches show that the vacancy microvoid formation is
caused by initiation of non-equilibrium concentrations of
vacancies when cooling the crystal as a result of binding
the impurity atoms (in particular, oxygen) in primary
grown-in microdefects. In so doing, the primary grown-
in microdefects (impurity precipitates) are the points of
nucleation of vacancy microvoids what is confirmed
experimentally. The short-term heat treatment (30 min at
Т = 1100 °С) causes the sharp decrease of sizes and
change of microvoid profile which are accompanied by
creation of precipitates of significant sizes [8]. It testifies
that the vacancies deposit on the oxygen-vacancy and
carbon-interstitial agglomerates that are formed close to
the crystallization front and grow in the course of crystal
cooling. The experimental fact of availability of oxygen
in one microvoids and carbon in another microvoids is
the confirmation of this contention [6, 7].
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2006. V. 9, N 4. P. 77-81.
© 2006, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
81
6. Conclusion
The performed mathematical calculations show that the
vacancy microvoid formation in dislocation-free silicon
single crystals can be described within the framework of
the dissociative diffusion model. In difference from the
model of point defect dynamics, the vacancy microvoid
formation in the course of crystal cooling within the
temperature range 1130…1070 °С correlates with the
experimental results on formation of a microdefect
structure when cooling within the temperature interval
1420…1130 °С. Our consideration of the problem of
vacancy microvoid formation confirms that the process
of point defect diffusion in the field of a temperature
gradient provides the basis for the defect formation in
dislocation-free silicon monocrystals during their
growth.
Acknowledgements
This scientific work was made by the budgetary funds of
Ministry of Education and Science of Ukraine as the
grant of the President of Ukraine.
References
1. T. Sinno, R.A. Brown, W. von Ammon and
E. Dornberger, Point defects dynamics and the
oxidation-induced stacking-faults in Czochralski-
grown silicon crystals // J. Electrochem. Soc. 145
(1), p. 302-318 (1998).
2. M.S. Kulkarni, V. Voronkov and R. Falster,
Quantification of defect dynamics in unsteady-state
and steady-state Czochralski growth of mono-
crystalline silicon // J. Electrochem. Soc. 151 (5),
p. G663-G669 (2004).
3. V.V. Voronkov, Mechanism of swirl defects
formation in silicon // J. Cryst. Growth 59 (3), p.
625-642 (1982).
4. V.I. Talanin and I.E. Talanin, Mechanism of
formation and physical classification of the grown
in microdefects in semiconductor silicon // Defect
& Diffusion Forum 230-232 (1), p. 177-198 (2004).
5. V.I. Talanin and I.E. Talanin, Formation of grown-
in microdefects in dislocation-free silicon
monocrystals, In: New research on semiconductors,
Ed. T.B. Elliot, p. 35-59. Nova Sci. Publ., New
York, 2006.
6. T. Ueki, M. Itsumi, T. Takeda, K. Yoshida,
A. Takaoka and S. Nakajima, Shrinkage of grown-
in defects in Czochralski silicon during thermal
annealing in vacuum // Jpn J. Appl. Phys. 37 (7),
p. L771-L773 (1998).
7. Y. Yanase, H. Nishihata, T. Ochiai, H. Tsuya,
Atomic force microscope observation of the change
in shape and subsequent disappearance of “crystal-
originated particles” after hydrogen-atmosphere
thermal annealing // Jpn J. Appl. Phys. 37 (1), p. 1-
4 (1998).
8. M. Itsumi, Octahedral void defects in Czochralski
silicon // J. Cryst. Growth 237-239 (3), p. 1773-
1778 (2002).
9. V.I. Fistul, V.I. Petrovskii, N.S. Rytova and
P.M. Grinshtein, Formation of nearsurface impurity
profile // Fizika i Technika Poluprovodnikov 13
(11), p. 1402-1410 (1979) (in Russian).
10. V.I. Fistul and M.I. Sinder // Fizika i Technika
Poluprovodnikov 15 (6), p. 1182-1186 (1981) (in
Russian).
11. M. Akatsuka, M. Okui, S. Umeno and K. Sueoka,
Calculation of size distribution of void defects in
CZ silicon // J. Electrochem. Soc. 150 (9), p. G587-
G590 (2003).
12. V.I. Talanin and I.E. Talanin, Recombination
parameters of point defects in dislocation-free
silicon single crystals // Functional Materials 13
(1), p. 1-5 (2006).
|