Transformations of microdefect structure in silicon crystals under the influence of weak magnetic field
Quantitative characterization of complex microdefect structures in annealed silicon crystals (1150 °С, 40 h) and their transformations after exposing for one day in a weak magnetic field (1 T) has been performed by analyzing the rocking curves, which have been measured by a high-resolution double...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
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| Zitieren: | Transformations of microdefect structure in silicon crystals under the influence of weak magnetic field / T. P. Vladimirova, Ye. M. Kyslovs`kyy, V. B. Molodkin, S. I. Olikhovskii,O. V. Koplak, E. V. Kochelab // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2011. — Т. 14, № 4. — С. 470-477. — Бібліогр.: 27 назв. — англ. |
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irk-123456789-1177992017-05-27T03:04:40Z Transformations of microdefect structure in silicon crystals under the influence of weak magnetic field Vladimirova, T.P. Kyslovs’kyy, Ye.M. Molodkin, V.B. Olikhovskii, S.I. Koplak, O.V. Kochelab, E.V. Quantitative characterization of complex microdefect structures in annealed silicon crystals (1150 °С, 40 h) and their transformations after exposing for one day in a weak magnetic field (1 T) has been performed by analyzing the rocking curves, which have been measured by a high-resolution double-crystal X-ray diffractometer. Based on the characterization results, which have been obtained by using the formulas of the dynamical theory of X-ray diffraction by imperfect crystals with randomly distributed microdefects of several types, the concentrations and average sizes of oxygen precipitates and dislocation loops after imposing the magnetic field and their dependences on time after its removing have been determined. 2011 Article Transformations of microdefect structure in silicon crystals under the influence of weak magnetic field / T. P. Vladimirova, Ye. M. Kyslovs`kyy, V. B. Molodkin, S. I. Olikhovskii,O. V. Koplak, E. V. Kochelab // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2011. — Т. 14, № 4. — С. 470-477. — Бібліогр.: 27 назв. — англ. 1560-8034 PACS 61.72.Dd http://dspace.nbuv.gov.ua/handle/123456789/117799 en Semiconductor Physics Quantum Electronics & Optoelectronics Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
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English |
| description |
Quantitative characterization of complex microdefect structures in annealed
silicon crystals (1150 °С, 40 h) and their transformations after exposing for one day in a
weak magnetic field (1 T) has been performed by analyzing the rocking curves, which
have been measured by a high-resolution double-crystal X-ray diffractometer. Based on
the characterization results, which have been obtained by using the formulas of the
dynamical theory of X-ray diffraction by imperfect crystals with randomly distributed
microdefects of several types, the concentrations and average sizes of oxygen precipitates
and dislocation loops after imposing the magnetic field and their dependences on time
after its removing have been determined. |
| format |
Article |
| author |
Vladimirova, T.P. Kyslovs’kyy, Ye.M. Molodkin, V.B. Olikhovskii, S.I. Koplak, O.V. Kochelab, E.V. |
| spellingShingle |
Vladimirova, T.P. Kyslovs’kyy, Ye.M. Molodkin, V.B. Olikhovskii, S.I. Koplak, O.V. Kochelab, E.V. Transformations of microdefect structure in silicon crystals under the influence of weak magnetic field Semiconductor Physics Quantum Electronics & Optoelectronics |
| author_facet |
Vladimirova, T.P. Kyslovs’kyy, Ye.M. Molodkin, V.B. Olikhovskii, S.I. Koplak, O.V. Kochelab, E.V. |
| author_sort |
Vladimirova, T.P. |
| title |
Transformations of microdefect structure in silicon crystals under the influence of weak magnetic field |
| title_short |
Transformations of microdefect structure in silicon crystals under the influence of weak magnetic field |
| title_full |
Transformations of microdefect structure in silicon crystals under the influence of weak magnetic field |
| title_fullStr |
Transformations of microdefect structure in silicon crystals under the influence of weak magnetic field |
| title_full_unstemmed |
Transformations of microdefect structure in silicon crystals under the influence of weak magnetic field |
| title_sort |
transformations of microdefect structure in silicon crystals under the influence of weak magnetic field |
| publisher |
Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| publishDate |
2011 |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/117799 |
| citation_txt |
Transformations of microdefect structure in silicon crystals under the influence of weak magnetic field / T. P. Vladimirova, Ye. M. Kyslovs`kyy, V. B. Molodkin, S. I. Olikhovskii,O. V. Koplak, E. V. Kochelab // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2011. — Т. 14, № 4. — С. 470-477. — Бібліогр.: 27 назв. — англ. |
| series |
Semiconductor Physics Quantum Electronics & Optoelectronics |
| work_keys_str_mv |
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2025-07-08T12:49:05Z |
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2025-07-08T12:49:05Z |
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| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 4. P. 470-477.
PACS 61.72.Dd
Transformations of microdefect structure in silicon crystals
under the influence of weak magnetic field
T.P. Vladimirova, Ye.M. Kyslovs’kyy, V.B. Molodkin, S.I. Olikhovskii, O.V. Koplak, E.V. Kochelab
G.V. Kurdyumov Institute for Metal Physics, NAS of Ukraine,
36, Vernadsky blvd., 03680 Kyiv, Ukraine
Abstract. Quantitative characterization of complex microdefect structures in annealed
silicon crystals (1150 °С, 40 h) and their transformations after exposing for one day in a
weak magnetic field (1 T) has been performed by analyzing the rocking curves, which
have been measured by a high-resolution double-crystal X-ray diffractometer. Based on
the characterization results, which have been obtained by using the formulas of the
dynamical theory of X-ray diffraction by imperfect crystals with randomly distributed
microdefects of several types, the concentrations and average sizes of oxygen precipitates
and dislocation loops after imposing the magnetic field and their dependences on time
after its removing have been determined.
Keywords: silicon, supersaturated solid solution, oxygen precipitate, dislocation loop,
magnetic field, X-ray diffuse scattering.
Manuscript received 15.05.11; revised manuscript received 19.08.11; accepted for
publication 14.09.11; published online 30.11.11.
1. Introduction
Traditionally, external magnetic fields are widely used in
semiconductor industry to control behavior of the melt
during crystal growth, in particular, the growth of silicon
single crystals with a large diameter [1]. The influence
of magnetic field on the melt, which reduces the
inhomogeneity and melt turbulence and provides the
achievement of significant improvement in uniformity of
distribution of impurities in grown crystals, is provided
due to the action of Lorentz force on moving charged
particles – electrons.
Another application of magnetic fields in modern
silicon microelectronic industry is to study the
interaction of dislocations with impurities such as
oxygen or nitrogen [2–6]. This approach is used to lock
dislocations in order to prevent their propagation under
mechanical stresses arising in technological processes
[7, 8]. The importance of these investigations is raised in
view of increasing diameters of silicon wafers used in
microelectronics and development of defect
engineering [9].
For two last decades, the giant and colossal physical
and mechanical responses caused by magnetic fields have
been discovered in many solids, from oxides of transition
metals to the intermetallic compounds and magnetic
alloys. The investigations of behavior inherent to various
materials in magnetic fields have revealed the existence of
significant interactions between spin and crystal lattice, in
particular, the manipulation by the spin degree of freedom
often leads to structural reconfigurations at the atomic
level. The observed significant magnetic responses are in
essence the manifestations of structural transitions caused
by magnetic field (see reviews [10–12]).
Colossal magneto-physical, thermal, and
mechanical phenomena can provide more rapid and
energy-saving techniques used in communication
systems, storage and processing of information, etc. In
particular, as recently shown, the spin degree of freedom
for an electron plays a key role in creation of the next
generation of electronics, the so-called spintronics [13].
At this forefront of condensed matter physics and
technology, the use of interactions between the spin and
other, e.g., charge, orbital, and lattice degrees of freedom
has both fundamental and applied importance,
particularly, in creating the quantum computers [14, 15].
The important fact, which can contribute to rapid
integration of new devices into existing systems, is their
creation on the basis of silicon crystals which are widely
used in modern microelectronics.
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
470
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 4. P. 470-477.
One of the impressing experimental observations
was that the weak magnetic fields can have a significant
impact on defect structure and some physical properties
of different non-magnetic crystals, including metal-oxide
semiconductor structures, oxygen enriched silicon
crystals, semiconductors A2B
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
B6 and A3B5B , etc. (see
reviews [10–12]). Analysis of experimental data shows
that the structural changes induced by magnetic field in
non-magnetic solids can be interpreted as the
disintegration of defect complexes. This disintegration is
accompanied by the generation of mobile point defects
which are involved in long-term processes and form new
defects [10–12].
The observed effects in one of the existing
theoretical models are ascribed to the spin-dependent
reactions including intrinsic defects and impurities,
which are similar to the well-known spin-dependent
chemical reactions in liquids and gases [16–20]. The
more general model that can describe a wider range of
effects in non-magnetic materials, which are caused by
imposing the weak magnetic field, is based on the
assumption of magnon mechanism for reactions with
defects [21, 22]. Nevertheless, theoretical description of
the influence of weak magnetic fields on structural
transformations in non-magnetic crystals is far from
being completed and, in particular, requires detailed
quantitative information on the characteristics of small
and large clusters involved in these transformations.
X-ray diffraction methods are the most sensitive
and informative tool for nondestructive diagnostics of
structural changes in crystalline materials and products
under various external influences. In recent years, these
methods have been enhanced due to the developed
theoretical basis of dynamical X-ray diffraction
characterization, which allows to reliably determine the
quantitative characteristics simultaneously for several
types of microdefects with arbitrary radii (from nano- to
micrometers) [23]. Thus, the possibility is opened for the
adequate interpretation of dynamical diffraction profiles
and diffuse scattering intensity distributions measured by
high-resolution diffractometers from crystals with a
complex defect structure.
The purpose of this paper is to determine the
changes of statistical characteristics of microdefects in
single-crystalline silicon, namely: concentrations and
average sizes of oxygen precipitates and dislocation
loops after imposing a weak magnetic field as well as
their dependences versus time after its removal, by using
the method of high-resolution double-crystal dynamical
X-ray diffractometry.
2. Basic theoretical relations
According to the statistical dynamical theory, the
diffraction profile measured by the double-crystal
diffractometer (DCD) with widely open detector window
from the crystal with defects is the sum of coherent
(Rcoh) and diffuse (Rdiff) components of the crystal
reflectivity [23]:
R(Δθ) = Rcoh(Δθ) + Rdiff (Δθ). (1)
Coherent component of the reflectivity is described
by the expression:
( ) ( ) 2
B yrRcoh =θΔ , (2)
where the coherent amplitude reflection coefficient in
the case of Bragg diffraction geometry is described by
the expression [23]:
( )
1
2221
B 1ctg1
−
⎥⎦
⎤
⎢⎣
⎡ ⎟
⎠
⎞⎜
⎝
⎛ −−+ς= yAyiyyr . (3)
In the equation (3), the following notations were
used:
Λπ= /tA , σγγλ=Λ /0 H ,
,222
HH −χχ=σ EC HH −χχ=ς / ,
1
0
−
γγ= Hb , ( )ψ−θ=γ B0 sin ,
( )ψ+θ−=γ BsinH ,
where t is the crystal thickness, Λ – extinction length, C
– polarization factor, ( HLE − )= exp – static Debye-
Waller factor, χH – Fourier component of the crystal
polarizability, b – parameter of diffraction asymmetry,
θВ – Bragg angle, ψ – angle between surface and
reflecting planes of the crystal. The normalized angular
deviation of an imperfect crystal can be written as
( ) ,/0 σα−α= by (4)
( )B2sin θθΔ−=α ,
( )[ ] ,2//00000 bχΔ+χ+χΔ+χ=α HH (5)
where Δθ is the angular deviation of the crystal from its
exact Bragg reflection position.
The dispersion corrections and HHχΔ 00χΔ in
Eq. (5) take into account additional absorption caused by
diffuse scattering (DS) from defects. These corrections
are complex functions depending on the deviation
Δθ (G = 0, H):
( ) ( ) ( ) KiP /θΔμ−θΔ=θΔχΔ GGGGGG . (6)
The absorption coefficients in Eq. (6) are
calculated as:
GGμ
( ) ( ),'d
4 2
2
∫λ
≈θΔμ qk HHHH SVC (7)
( ) ( ),00 θΔμ≈θΔμ HHb
where V is the crystal volume, λ – X-ray wavelength.
Integration in Eq. (7) is carried out over the plane
tangent to the Ewald sphere, and the correlation function
S(q) is expressed through the Fourier-components of
fluctuating part of the crystal polarizability:
( ) .Re '22' GHqGHqGG q −+−+− χδχδ=S (8)
471
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 4. P. 470-477.
Real parts of dispersion corrections (6) are of the same
order of magnitude as their imaginary parts (7), and
therefore for numerical calculations one can adopt the
approximate relation ./ KP GGGG μ−≈
The diffuse component of the crystal reflectivity in
the case of Bragg diffraction in single crystals with
randomly distributed defects, which is measured using
DCD with widely open detector window, can be
expressed through the imaginary part of dispersion
correction accounting for DS [23]:
( ) ( ) ( )
( ) ,2 0 θΔμγ
θΔμ
θΔ≈θΔ HH
dyndiff FR (9)
where is the interference factor of the order of
unity, and is the interference absorption
coefficient. If randomly distributed defects of several
types (α) with size distribution (i) are present in the
crystal, then the absorption coefficient related with DS
in Eq. (9) can be described by the expression:
dynF
)( θΔμ
HHμ
( ) ( ) ( )0
i
αi
ds0ds kk ∑∑
α
μ=μ=θΔμHH , (10)
where , K = 2π/ λ, and are
absorption coefficients due to DS from defects of α type
with size.
( B0 2sin θθΔ= Kk
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
) αi
dsμ
th-i
Similarly, if the correlation in defect positions is
absent, the exponent of static Debye-Waller factor
consists of the sum of contributions from each
population of defects {αi}:
.
i
i∑∑
α
α= HH LL (11)
Eqs (9) to (11) provide the possibility to describe
the angular distribution of the diffuse component in
measured rocking curves, which consists of
contributions from several types of defects with size
distributions. Additionally, Eq. (10) describes the
attenuation of the coherent component in rocking curves
due to DS from defects.
3. Experimental
X-ray diffraction measurements were carried out with a
characteristic CuKα1 radiation from X-ray tube of
type with a power of 0.625 kW
(25kV×25mA). The
29-BSV
high-resolution measurements of
RCs for the samples under investigation were performed
in the symmetrical Bragg diffraction geometry for Si
(111) and (333) reflections by using X-ray optical
scheme of DCD with two flat Ge monochromator
crystals in a mutually dispersive position.
One of the investigated silicon samples (FZ Si),
with sizes 3×2 cm, surface orientation (111), and the
diameter 10 cm, was cut from the center of the silicon
crystal plate which has been grown by the floating zone
method. The sample thickness after lapping and chemo-
mechanical polishing was nearly 525 μm. Measurements
by using X-ray topography and scanning electron
microscopy have not revealed any defects in this sample.
The second investigated sample (Cz Si) was cut
from the center of the silicon ingot grown by
Czochralski method with a surface orientation (111).
The ingot had the p-type conductivity with the resistivity
close to 10.5 Ohm⋅cm, the concentrations of oxygen and
carbon impurities were and less than
, respectively. The sample with sizes
10×20 mm after finishing was etched chemically on both
sides to a depth 10 μm to the thickness close to 480 μm.
This sample was subjected to heat treatment for 40 h at
1150 °С in a sealed silica ampoule in Ar atmosphere.
318cm101.1 −×
317 cm10 −
First of all, the RCs of both samples have been
measured before imposing the magnetic field. Then,
after the exposition of the samples for one day in the
constant weak magnetic field with a strength of 1 T, the
RCs have been measured after 1, 5, 8, and 22 days from
the moment of switching off the magnetic field.
4. Analysis of measurement results
All the RCs of FZ Si sample that were measured before
and after imposing the magnetic field coincide with each
other within the limits of statistical errors. This fact
indicates that the magnetic field has a little effect on the
defect structure of this sample or does not influence it at
all. This result is consistent with known literature data
on the key role of impurity point defects and their
complexes (clusters), which provides mechanisms of the
influence of magnetic fields on structural
transformations in non-magnetic crystals [10–12]. Since
the concentrations of impurities in high-purity silicon
crystals grown by the floating-zone method are very low
[8], we do not observe any effects of magnetic field on
the measured RCs, which are sensitive to a defect
structure of the crystal.
At the same time, for Cz Si sample the noticeable
differences between all the measured RCs are observed
in their central parts (Fig. 1).
Fig. 1. Experimental RCs (Cz Si 333 reflection, CuKα1
radiation) measured before imposing and after one-day
472
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 4. P. 470-477.
exposition in the weak magnetic field with subsequent ageing
in air during 1, 8, and 22 days.
This specific form of RCs requires a careful choice
of an adequate model of the defect structure of the
sample and accurate fit of the measured RCs during their
treatment. It is known that silicon crystals grown by the
Czochralski method contain usually several types of
microdefects (oxygen precipitates, stacking faults or
dislocation loops of interstitial type) with size
distributions from nano- to micrometers [2]. These
distributions evolve during thermal treatments at
elevated temperatures due to interaction of oxygen
impurity and intrinsic point defects.
In the model of microdefect structure of Cz Si
sample, which was used in the treatment of the measured
RCs, it was assumed that three types of microdefects are
present in the crystal, namely: disc-shaped and spherical
precipitates of radius RP (and thickness hP for disc-
shaped precipitates), and concentration nP, as well as
circular dislocation loops of radius RL and concentration
nL. When analyzing the measured RCs, also contribution
of thermal DS and influence of instrumental factors were
taken into account [23]. The fit quality was estimated
using an ordinary (R) and weighted (Rw) reliability
factors.
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
Fig. 2. Experimental and calculated RCs (333 reflection,
CuKα1 radiation) measured before imposing (a) and one day
after removing (b) the weak magnetic field from Cz Si crystal
exposed during one day (markers and solid lines, respectively).
The contributions of coherent scattering and DS from
dislocation loops of different sizes (1, 2, 3), and disc-
shaped (4) and spherical (5) oxygen precipitates (see Table 1)
are described by thin solid, dashed, and dotted lines,
respectively.
The results of X-ray diffraction characterization of
the annealed Cz Si sample before and after imposing a
weak magnetic field are shown in Table 1 (see also
Fig. 2). The values of defect characteristics have been
determined by fitting all RCs with ordinary and
weighted reliability factors not exceeding R = 10% and
Rw = 15%, respectively.
It should be remarked that the possibility of the
simultaneous determination of the characteristics of
three types of microdefects has been realized due to self-
consistent description of coherent and diffuse
components in the RCs. Besides, an important factor for
achieving uniqueness and increasing diagnostic accuracy
was the self-consistency of contributions of coherent and
diffuse RCs components both at the tails and in the total
reflection range. For this purpose, in particular, it was
necessary to take into account the existence of a wide
spread of dislocation loop radii from tens to hundreds of
nanometers. The reliability of characterization was
enhanced also due to the account for the presence of
antisymmetric components in DS intensity distributions
from oxygen precipitates and dislocation loops.
As can be seen from the results of X-ray dynamical
diffraction characterization (Table 1), the sufficiently
significant transformations of microdefect structure in
the Cz Si sample under the influence of the weak
magnetic field have occurred. They are exhibited in the
decrease of average sizes of oxygen precipitates (Fig. 3)
and corresponding decrease of oxygen concentration in
the disc-shaped and spherical oxygen precipitates
(Fig. 4). Simultaneously, the average radii of large
dislocation loops (Fig. 5) and concentration of silicon
atoms in them has been increased (Fig. 6). Then,
after removing the magnetic field, the defect structure
was relaxed, i.e., the microdefect characteristics have
been gradually recovered to their initial values that
existed prior to imposing the magnetic field (see Figs 3
to 6, where marker sizes correspond to error bars).
PCO
LCi
Fig. 3. Dependences of the radii of disc-shaped and spherical
oxygen precipitates (filled and empty markers, respectively) in
473
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 4. P. 470-477.
the Cz Si crystal versus time after removing the magnetic field.
Marker sizes correspond to error bars.
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
Fig. 4. Dependences of concentrations of oxygen atoms in
disc-shaped and spherical oxygen precipitates in Cz Si crystal
(filled and empty markers, respectively) versus time after
removing the magnetic field for 333 reflection.
Fig. 5. Dependences of radii of large and medium dislocation
loops (filled and empty markers, respectively) in Cz Si crystal
versus time after removing the magnetic field.
Fig. 6. Dependences of concentrations of silicon atoms in
large- and medium-size dislocation loops (filled and empty
markers, respectively) versus time after removing the magnetic
field for 333 reflection.
5. Discussion
The intrigue of the observations described above is that
the weak magnetic field with the induction of 1 T can
transfer to a paramagnetic particle only the energy of
~μВВ ~ eV10 4− (μВ is the Bohr magneton) [ 1210 − ],
which is about two orders of magnitude smaller than the
mean energy of thermal fluctuations at room temperature
~kBT ~ kB eV10 2− ( BB is the Boltzmann constant, T –
absolute temperature) and four orders smaller than the
activation energy for joining the atom to the cluster
Еаct ~ 1 eV. For this reason, it is not clear which physical
mechanism is responsible for the observed structural
changes. Despite the recently published first attempts of
the theoretical interpretation of physical phenomena,
including structural changes, in non-magnetic crystals
under the influence of weak magnetic fields [ 2216 − ],
the search for an adequate theoretical model to describe
observed phenomena remains urgent.
In particular, when developing the kinetic models
of defect clustering in non-magnetic crystals under the
influence of magnetic field [19] and its application to the
case of silicon crystals grown by the Czochralski
method, the important circumstance should be taken into
account that these crystals are very supersaturated solid
solutions of oxygen in silicon [8]. During cooling in the
growth process and after various thermal treatments, the
oxygen precipitates are formed due to decomposition of
these solutions, and simultaneously dislocation loops are
formed because of accompanying rise and
decomposition of the supersaturated solid solution of
interstitial silicon atoms [8, 9]. Very often, the
decomposition processes are not completed fully, and
the metastable defect structure is “frozen” in the crystal,
as it is observed in the case of the investigated Cz Si
sample.
One can suppose that during exposition in a
magnetic field this defect structure transits to another
metastable state due to its interaction with the spin
degree of freedom of defects. This process of the
transition is characterized by detaching oxygen atoms
from the precipitates and attaching excess silicon
interstitial atoms to existing dislocation loops, what
leads to the decrease of mean average radius of oxygen
precipitates and increase of average radius of dislocation
loops (Table 1). After removal of the magnetic field, a
gradual recovery of the metastable defect structure to its
initial state occurs during several weeks under the
influence of thermal fluctuations (Figs. 4 to 6).
To describe in details the evolution of the defect
structure in the silicon crystal grown by the Czochralski
method and, in particular, its transformation under the
influence of weak magnetic fields, a complicated system
of coupled equations of chemical reactions, differential
Fokker-Planck equations, and mass conservation laws
474
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 4. P. 470-477.
for point defects should be solved numerically
[ 2624 − ]. Nevertheless, some laws of the behavior of
such complicated physical systems can be established by
using the dimensional analysis of these equations and
investigation of the relevant characteristic constants
[25, 27].
One of important parameters that characterizes the
process of oxygen precipitation in silicon and is closely
related with the average radius of oxygen precipitates is
their critical radius [ 2624 − ]. If the radius of an oxygen
precipitate formed due to stochastic processes of
attachment of oxygen atoms exceeds some critical value,
it continues to grow, while a precipitate with a smaller
radius dissolves. When interaction of the oxygen
precipitate with interstitial silicon atoms and vacancies is
taken into account, the critical radius of the oxygen
precipitate has the form [25]:
© 2011, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
( ) p
pcr
P uVSSSTk
uV
R
δεμ−
σ
= γ− 6ln
2
vi
v
γ
iOB
,
( ) ⎟
⎠
⎞
⎜
⎝
⎛
δ+
ε+
γ+γ+=
1
11 vi xxu ,
(12)
where σ is the surface energy of the boundary between
oxygen precipitates and silicon matrix, Vp is volume of
the SiOх molecule, , , and
are supersaturations of interstitial oxygen
and silicon atoms as well as vacancies, respectively, OC ,
iC , vC and eq
O , eq
iC , eq
vC –
eq
OOO /CCS = eq
iii / CCS =
eq
vvv /CCS =
C actual and equilibrium
concentrations of interstitial oxygen and silicon atoms,
and vacancies, respectively, iγ and vγ – ratios of
amounts of emitted interstitial silicon atoms and
absorbed vacancies to the amount of oxygen atoms
attached to the precipitate, μ is shear modulus of silicon,
ε and δ are bulk and linear strains at the boundary
between precipitate and matrix, respectively.
Similarly, for dislocation loops of the interstitial
type there exists a characteristic size, namely, the critical
radius cr
LR that can be found as a solution of the
transcendental equation [25]:
( ) ( ) ( /2
12/
/ 1
γ
viB
v
−
−
=
ε−
ν−π
μ
=
+
∗
bVSSTkV
cb
bR
msf
m
RRL
L
cr
LL
)[ ] ,ln
ln R
(13)
where the constant с = 2.96, b is the module of Burgers’
vector , ν – Poisson ratio, εsf – surface energy of stacking
fault, Vm – volume of Si atom.
By using the determined average radii of oxygen
precipitates and dislocation loops (Table 1) and the
expressions for the corresponding critical radii of these
microdefects (12) and (13), one can determine the
parameters of supersaturation of point defects
475
Table 1. Characteristics of oxygen precipitates and dislocation loops in Cz Si crystal before and after imposing the
magnetic field as determined from RCs for 333 reflection. The fit quality was estimated with ordinary and weighted
reliability factors not exceeding R = 10% and Rw = 15%, respectively.
Data RL, nm nL, cm–3 17
i 10×LC , cm–3 RP, nm hP, nm nP, cm–3 17
O 10×PC , cm–3
Before
imposing
field
800
300
0.5
1.0×109
6.0×1010
4.0×1018
0.3
2.7
492.4
110
300
6.2
–
1.3×1011
8.0×107
14.3
3.9
∑ = 495.4 18.2
Day 1
2800
450
0.5
1.0×108
4.0×1010
1.5×1018
0.4
4.0
184.7
70
200
5.2
–
2.5×1011
2.0×108
9.3
2.9
∑ = 189.1 12.2
Day 5
2500
400
0.5
1.2×108
4.1×1010
1.5×1018
0.4
3.2
184.7
83
230
5.4
–
2.0×1011
8.0×107
10.9
1.8
∑ = 188.3 12.7
Day 8
2000
380
0.5
2.5×108
4.5×1010
1.5×1018
0.5
3.2
184.7
85
250
5.6
–
2.0×1011
8.0×107
11.8
2.3
∑ = 188.4 14.1
Day 22
1000
350
0.5
1.0×109
6.0×1010
1.5×1018
2.0
3.6
184.7
110
290
6.2
–
1.0×1011
8.0×107
11.0
3.6
∑ = 190.3 14.6
Table 2. Supersaturation parameters for point defects in Сz Si crystal before imposing the magnetic field and one day
after removing it ( , ).
∗γ= vSSA vi
vSSSB γγ= v
-
iO
i
Data RL, nm ln A A RP, nm ln B B
Before imposing field 800
300
0.29
0.68
1.34
1.97
110
300
1.671
1.660
5.32
5.26
Day 1 2800
450
0.10
0.48
1.11
1.61
70
200
1.681
1.664
5.37
5.28
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2011. V. 14, N 4. P. 470-477.
(interstitial oxygen and silicon atoms, and vacancies)
and their corresponding equilibrium concentrations
before and after imposing the magnetic field (see
Table 2). These values, in turn, can be used to estimate
changes in activation energies of point defects and to
analyze the driving forces of structural changes at the
imposition of a weak magnetic field on silicon crystals.
It should be also remarked that after switching off
the magnetic field transformation of the defect structure
in the investigated Cz Si sample is continued, and its
gradual and non-uniform relaxation occurs to the state
that preceded the imposition of the magnetic field (Figs
3 to 6). The growth of oxygen precipitates during this
relaxation is accompanied by decrease in the oxygen
concentration and increase in the concentration of
interstitial silicon atoms due to their emission from
growing oxygen precipitates and due to shrinkage of
large dislocation loops. Since two interacting
supersaturated solid solutions of interstitial oxygen and
silicon atoms coexist in the silicon crystal and, according
to Eq. (12), the increase of supersaturation by interstitial
silicon atoms inhibits the growth of oxygen precipitates,
whereas, according to Eq. (13), the same supersaturation
slows the shrinkage of dislocation loops. Thereof, one
can conclude that the critical radii of oxygen precipitates
and dislocation loops and the corresponding average
radii of these microdefects will drift during the
relaxation over corresponding hypersurfaces in a space
of concentrations of point defects. At the same time,
both microdefect radii and concentrations of interstitial
oxygen and silicon atoms can have wave-like time
dependences (Figs 4 to 6).
In conclusion, it should be emphasized that the
obtained results suggest the possibility of an effective
application of the dynamical high-resolution X-ray
diffraction measurements, which allow reliable
determining the quantitative characteristics
simultaneously of several types of microdefects, for the
quantitative characterization of structural changes in
non-magnetic crystals under the influence of weak
magnetic fields. In turn, information obtained in this way
can contribute to the creation of adequate models of such
structural transformations.
6. Summary and conclusions
The measurements of RCs of the silicon single crystal
annealed at 1150 °C for 40 h before superposition of a
weak magnetic field (1 T) and after exposure in this field
for one day as well as after certain time intervals have
been carried out by using high-resolution double-crystal
X-ray diffractometer. Based on characterization results,
which have been obtained using formulas of the
dynamical theory of X-ray diffraction in imperfect
crystals with randomly distributed microdefects of
several types, the change of average sizes of oxygen
precipitates and dislocation loops after imposing the
weak magnetic field has been determined as well as their
dependence on the time passed after the removal of the
magnetic field.
It has been established that during aging the silicon
crystal in a magnetic field transformation of its
metastable defect structure to another metastable state
occurs due to its interaction with the spin degree of
freedom of defects. This transformation is characterized
by removing oxygen atoms from the precipitates and
attaching excess interstitial silicon atoms to existing
dislocation loops. These processes lead to decrease of
the average radius of oxygen precipitates and increase of
the average radius of large dislocation loops. After
removal of the magnetic field, a gradual recovery of the
defect structure to its initial metastable state under the
influence of thermal fluctuations occurs for several
weeks.
The obtained results indicate promising
applications of the dynamical X-ray diffractometry for
quantitative characterization of complicated structural
transformations, which occur under the influence of
magnetic fields in non-magnetic crystals, including
silicon single crystals grown by the Czochralski method.
This work was performed with the financial
support of the National Academy of Sciences of Ukraine
(Contract No. D10/713.3.6.3 −− ).
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