Dynamical Theory of Triple-Crystal X-ray Diffractometry and Characterization of Microdefects and Strains in Imperfect Single Crystals
A short review of basic principles and limitations in obtaining the analytical expressions for the coherent and diffuse scattering intensities measured by the triple-crystal diffractometer (TCD) are presented. Explicit analytical expressions are given for both the diffuse components of TCD profiles...
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Інститут металофізики ім. Г.В. Курдюмова НАН України
2016
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| Цитувати: | Dynamical Theory of Triple-Crystal X-ray Diffractometry and Characterization of Microdefects and Strains in Imperfect Single Crystals / V. B. Molodkin, S. I. Olikhovskii, E. G. Len, Ye. M. Kyslovskyy, O. V. Reshetnyk, T. P. Vladimirova, B. V. Sheludchenko, E. S. Skakunova, V. V. Lizunov, E. V. Kochelab, I. M. Fodchuk, and V. P. Klad’ko // Металлофизика и новейшие технологии. — 2016. — Т. 38, № 1. — С. 99-139. — Бібліогр.: 48 назв. — англ. |
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irk-123456789-1124662017-01-23T03:02:32Z Dynamical Theory of Triple-Crystal X-ray Diffractometry and Characterization of Microdefects and Strains in Imperfect Single Crystals Molodkin, V.B. Olikhovskii, S.I. Len, E.G. Kyslovskyy, Ye.M. Reshetnyk, O.V. Vladimirova, T.P. Sheludchenko, B.V. Skakunova, E.S. Lizunov, V.V. Kochelab, E.V. Fodchuk, I.M. Klad’ko, V.P. Дефекты кристаллической решётки A short review of basic principles and limitations in obtaining the analytical expressions for the coherent and diffuse scattering intensities measured by the triple-crystal diffractometer (TCD) are presented. Explicit analytical expressions are given for both the diffuse components of TCD profiles and the reciprocal-lattice maps measured within the Bragg diffraction geometry from crystals containing microdefects of several types. These formulas are derived by using the generalized dynamical theory of X-ray scattering by imperfect crystals with randomly distributed microdefects. Some examples demonstrating possibilities of the developed theory for quantitative characterization of structural imperfections in real single crystals are represented. In particular, characteristics of the complicated microdefect structures fabricated in various silicon crystals grown by Czochralski technique and floating-zone melting method are determined by analytical treatment of the measured TCD profiles, rocking curves, and reciprocal space maps. В роботі представлено короткий огляд основних принципів, що використовуються при одержанні аналітичних виразів для когерентної та дифузної інтенсивностей розсіяння, виміряних трикристальним дифрактометром (ТКД). Одержано точні аналітичні вирази для дифузних компонент як ТКД-профілів, так і мап оберненого простору, виміряних у геометрії дифракції за Бреґґом для кристалів, які містять мікродефекти декількох типів. Ці формули одержано при використанні узагальненої динамічної теорії розсіяння Рентґенових променів неідеальними кристалами з випадково розподіленими мікродефектами. Представлено деякі приклади, які демонструють можливості розробленої теорії для кількісної характеризації недосконалостей структури в реальних монокристалах. Зокрема, шляхом аналітичного обробляння виміряних ТКД-профілів, кривих відбивання і мап оберненого простору визначено характеристики складних структур мікродефектів, створених у кристалах силіцію методами Чохральського і зонного топлення. В работе представлен краткий обзор основных принципов, используемых при получении аналитических выражений для когерентной и диффузной интенсивностей рассеяния, измеряемых трёхкристальным дифрактометром (ТКД). Получены точные аналитические выражения для диффузных компонент как ТКД-профилей, так и карт обратного пространства, измеренных в геометрии дифракции по Брэггу для кристаллов, содержащих микродефекты нескольких типов. Эти формулы получены при использовании обобщённой динамической теории рассеяния рентгеновских лучей неидеальными кристаллами со случайно распределёнными микродефектами. Представлены некоторые примеры, демонстрирующие возможности разработанной теории для количественной характеризации несовершенств структуры в реальных монокристаллах. В частности, путём аналитической обработки измеренных ТКД-профилей, кривых отражения и карт обратного пространства определены характеристики сложных структур микродефектов, созданных в кристаллах кремния методами Чохральского и зонной плавки. 2016 Article Dynamical Theory of Triple-Crystal X-ray Diffractometry and Characterization of Microdefects and Strains in Imperfect Single Crystals / V. B. Molodkin, S. I. Olikhovskii, E. G. Len, Ye. M. Kyslovskyy, O. V. Reshetnyk, T. P. Vladimirova, B. V. Sheludchenko, E. S. Skakunova, V. V. Lizunov, E. V. Kochelab, I. M. Fodchuk, and V. P. Klad’ko // Металлофизика и новейшие технологии. — 2016. — Т. 38, № 1. — С. 99-139. — Бібліогр.: 48 назв. — англ. 1024-1809 PACS: 61.05.cc, 61.05.cp, 61.72.Dd, 61.72.J-, 68.35.Gy, 81.40.Ef, 81.40.Wx DOI: 10.15407/mfint.38.01.0099 http://dspace.nbuv.gov.ua/handle/123456789/112466 en Металлофизика и новейшие технологии Інститут металофізики ім. Г.В. Курдюмова НАН України |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| topic |
Дефекты кристаллической решётки Дефекты кристаллической решётки |
| spellingShingle |
Дефекты кристаллической решётки Дефекты кристаллической решётки Molodkin, V.B. Olikhovskii, S.I. Len, E.G. Kyslovskyy, Ye.M. Reshetnyk, O.V. Vladimirova, T.P. Sheludchenko, B.V. Skakunova, E.S. Lizunov, V.V. Kochelab, E.V. Fodchuk, I.M. Klad’ko, V.P. Dynamical Theory of Triple-Crystal X-ray Diffractometry and Characterization of Microdefects and Strains in Imperfect Single Crystals Металлофизика и новейшие технологии |
| description |
A short review of basic principles and limitations in obtaining the analytical expressions for the coherent and diffuse scattering intensities measured by the triple-crystal diffractometer (TCD) are presented. Explicit analytical expressions are given for both the diffuse components of TCD profiles and the reciprocal-lattice maps measured within the Bragg diffraction geometry from crystals containing microdefects of several types. These formulas are derived by using the generalized dynamical theory of X-ray scattering by imperfect crystals with randomly distributed microdefects. Some examples demonstrating possibilities of the developed theory for quantitative characterization of structural imperfections in real single crystals are represented. In particular, characteristics of the complicated microdefect structures fabricated in various silicon crystals grown by Czochralski technique and floating-zone melting method are determined by analytical treatment of the measured TCD profiles, rocking curves, and reciprocal space maps. |
| format |
Article |
| author |
Molodkin, V.B. Olikhovskii, S.I. Len, E.G. Kyslovskyy, Ye.M. Reshetnyk, O.V. Vladimirova, T.P. Sheludchenko, B.V. Skakunova, E.S. Lizunov, V.V. Kochelab, E.V. Fodchuk, I.M. Klad’ko, V.P. |
| author_facet |
Molodkin, V.B. Olikhovskii, S.I. Len, E.G. Kyslovskyy, Ye.M. Reshetnyk, O.V. Vladimirova, T.P. Sheludchenko, B.V. Skakunova, E.S. Lizunov, V.V. Kochelab, E.V. Fodchuk, I.M. Klad’ko, V.P. |
| author_sort |
Molodkin, V.B. |
| title |
Dynamical Theory of Triple-Crystal X-ray Diffractometry and Characterization of Microdefects and Strains in Imperfect Single Crystals |
| title_short |
Dynamical Theory of Triple-Crystal X-ray Diffractometry and Characterization of Microdefects and Strains in Imperfect Single Crystals |
| title_full |
Dynamical Theory of Triple-Crystal X-ray Diffractometry and Characterization of Microdefects and Strains in Imperfect Single Crystals |
| title_fullStr |
Dynamical Theory of Triple-Crystal X-ray Diffractometry and Characterization of Microdefects and Strains in Imperfect Single Crystals |
| title_full_unstemmed |
Dynamical Theory of Triple-Crystal X-ray Diffractometry and Characterization of Microdefects and Strains in Imperfect Single Crystals |
| title_sort |
dynamical theory of triple-crystal x-ray diffractometry and characterization of microdefects and strains in imperfect single crystals |
| publisher |
Інститут металофізики ім. Г.В. Курдюмова НАН України |
| publishDate |
2016 |
| topic_facet |
Дефекты кристаллической решётки |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/112466 |
| citation_txt |
Dynamical Theory of Triple-Crystal X-ray Diffractometry and Characterization of Microdefects and Strains in Imperfect Single Crystals / V. B. Molodkin, S. I. Olikhovskii, E. G. Len, Ye. M. Kyslovskyy, O. V. Reshetnyk, T. P. Vladimirova, B. V. Sheludchenko, E. S. Skakunova, V. V. Lizunov, E. V. Kochelab, I. M. Fodchuk, and V. P. Klad’ko // Металлофизика и новейшие технологии. — 2016. — Т. 38, № 1. — С. 99-139. — Бібліогр.: 48 назв. — англ. |
| series |
Металлофизика и новейшие технологии |
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99
ВЗАИМОДЕЙСТВИЯ ИЗЛУЧЕНИЯ И ЧАСТИЦ
С КОНДЕНСИРОВАННЫМ ВЕЩЕСТВОМ
PACS numbers:61.05.cc, 61.05.cp,61.72.Dd,61.72.J-,68.35.Gy,81.40.Ef, 81.40.Wx
Dynamical Theory of Triple-Crystal X-Ray Diffractometry and
Characterization of Microdefects and Strains in Imperfect
Single Crystals
V. B. Molodkin, S. I. Olikhovskii, E. G. Len, Ye. M. Kyslovskyy,
O. V. Reshetnyk, T. P. Vladimirova, B. V. Sheludchenko, O. S. Skakunova,
V. V. Lizunov, E. V. Kochelab, I. M. Fodchuk*, and V. P. Klad’ko**
G. V. Kurdyumov Institute for Metal Physics, N.A.S. of Ukraine,
36 Academician Vernadsky Blvd.,
UA-03680 Kyiv, Ukraine
*Yuriy Fedkovych Chernivtsi National University,
2 Kotsyubyns’kyy Ave.,
58012 Chernivtsi, Ukraine
**V. Ye. Lashkaryov Institute of Semiconductor Physics, N.A.S. of Ukraine,
41 Nauky Ave.,
03028 Kyiv, Ukraine
A short review of basic principles and limitations in obtaining the analytical
expressions for the coherent and diffuse scattering intensities measured by the
triple-crystal diffractometer (TCD) are presented. Explicit analytical expres-
sions are given for both the diffuse components of TCD profiles and the recipro-
cal-lattice maps measured within the Bragg diffraction geometry from crystals
containing microdefects of several types. These formulas are derived by using
the generalized dynamical theory of X-ray scattering by imperfect crystals with
randomly distributed microdefects. Some examples demonstrating possibilities
of the developed theory for quantitative characterization of structural imper-
fections in real single crystals are represented. In particular, characteristics of
the complicated microdefect structures fabricated in various silicon crystals
Correspondence author: Stepan Iosypovych Olikhovskii
E-mail: olikhovsky@meta.ua
Please cite this article as: V. B. Molodkin, S. I. Olikhovskii, E. G. Len,
Ye. M. Kyslovskyy, O. V. Reshetnyk, T. P. Vladimirova, B. V. Sheludchenko,
O. S. Skakunova, V. V. Lizunov, E. V. Kochelab, I. M. Fodchuk, and V. P. Klad’ko,
Dynamical Theory of Triple-Crystal X-ray Diffractometry and Characterization of
Microdefects and Strains in Imperfect Single Crystals, Metallofiz. Noveishie Tekhnol.,
38, No. 1: 99—139 (2016), DOI: 10.15407/mfint.38.01.0099.
Металлофиз. новейшие технол. / Metallofiz. Noveishie Tekhnol.
2016, т. 38, № 1, сс. 99—139 / DOI: 10.15407/mfint.38.01.0099
Оттиски доступны непосредственно от издателя
Фотокопирование разрешено только
в соответствии с лицензией
2016 ИМФ (Институт металлофизики
им. Г. В. Курдюмова НАН Украины)
Напечатано в Украине.
100 V. B. MOLODKIN, S. I. OLIKHOVSKII, E. G. LEN et al.
grown by Czochralski technique and floating-zone melting method are deter-
mined by analytical treatment of the measured TCD profiles, rocking curves,
and reciprocal space maps.
Key words: dynamical scattering, triple-crystal diffractometer, double-crystal
diffractometer, microdefects.
В роботі представлено короткий огляд основних принципів, що використо-
вуються при одержанні аналітичних виразів для когерентної та дифузної
інтенсивностей розсіяння, виміряних трикристальним дифрактометром
(ТКД). Одержано точні аналітичні вирази для дифузних компонент як ТКД-
профілів, так і мап оберненого простору, виміряних у геометрії дифракції за
Бреґґом для кристалів, які містять мікродефекти декількох типів. Ці фор-
мули одержано при використанні узагальненої динамічної теорії розсіяння
Рентґенових променів неідеальними кристалами з випадково розподілени-
ми мікродефектами. Представлено деякі приклади, які демонструють мож-
ливості розробленої теорії для кількісної характеризації недосконалостей
структури в реальних монокристалах. Зокрема, шляхом аналітичного обро-
бляння виміряних ТКД-профілів, кривих відбивання і мап оберненого прос-
тору визначено характеристики складних структур мікродефектів, створе-
них у кристалах силіцію методами Чохральського і зонного топлення.
Ключові слова: динамічне розсіяння, трикристальний дифрактометр, двок-
ристальний дифрактометр, мікродефекти.
В работе представлен краткий обзор основных принципов, используемых
при получении аналитических выражений для когерентной и диффузной
интенсивностей рассеяния, измеряемых трёхкристальным дифрактомет-
ром (ТКД). Получены точные аналитические выражения для диффузных
компонент как ТКД-профилей, так и карт обратного пространства, изме-
ренных в геометрии дифракции по Брэггу для кристаллов, содержащих
микродефекты нескольких типов. Эти формулы получены при использова-
нии обобщённой динамической теории рассеяния рентгеновских лучей не-
идеальными кристаллами со случайно распределёнными микродефектами.
Представлены некоторые примеры, демонстрирующие возможности разра-
ботанной теории для количественной характеризации несовершенств
структуры в реальных монокристаллах. В частности, путём аналитической
обработки измеренных ТКД-профилей, кривых отражения и карт обратного
пространства определены характеристики сложных структур микродефек-
тов, созданных в кристаллах кремния методами Чохральского и зонной
плавки.
Ключевые слова: динамическое рассеяние, трёхкристальный дифракто-
метр, двухкристальный дифрактометр, микродефекты.
(Received November 10, 2015)
1. INTRODUCTION
Modern growth technologies provide producing nearly perfect single
DYNAMICAL THEORY OF TRIPLE-CRYSTAL X-RAY DIFFRACTOMETRY 101
crystals and crystalline devices. The existing level of growth tech-
niques including ‘defect engineering’ allows for purposeful introduc-
ing defects in a crystal lattice to satisfy needs for a wide variety of
their mechanical, optical, electric, magnetic properties, etc. The non-
destructive control of defect structures in such crystal systems is ex-
clusively important to control growth processes. The most informative
among different physical characterization methods are the X-ray dif-
fraction techniques; in particular, those based on measurements of
diffraction intensity distributions in the reciprocal lattice space by
means of the triple-crystal diffractometer (TCD).
The triple-crystal X-ray diffractometry provides the possibilities of
very complete and precise characterization of both structural defects
in crystal bulk and strains in disturbed surface layers [1—12] as well as
chemical compositions and strain distributions in thin film structures,
multilayer systems, superlattices, etc. [13, 14]. However, at the analy-
sis of TCD measurements, the consideration is usually restricted to the
investigation of the diffuse scattering intensity distributions only in
those regions where the coherent component of the diffraction intensi-
ty can be neglected, or this component can be subtracted by using the
equations for perfect crystals. Similarly, the attenuation of coherent
scattering intensity due to diffuse scattering from defects is as rule
neglected. Such kind approaches can lead to systematic errors when
determining the characteristics of crystal structure imperfections.
Strongly speaking, the more rigorous treatment of the diffraction
patterns measured by TCD requires the joint consideration of the su-
perimposed coherent and diffuse scattering intensity distributions.
This implies also the account for dynamical effects in diffuse scatter-
ing intensity and diffuse scattering effects in coherent intensity dis-
tributions. Such self-consistent approach in the triple-crystal X-ray
diffractometry [15—19], which is necessary for the reliable quantita-
tive characterization of defects and strains in crystals, was conse-
quently developed on the base of the generalized dynamical theory of
X-ray scattering by imperfect crystals with randomly distributed mi-
crodefects [20—22].
The purpose of this article is reviewing basic principles of the gener-
alized dynamical theory of X-ray diffraction by crystals with chaoti-
cally distributed microdefects and the description of corresponding
analytical expressions for diffraction profiles and reciprocal lattice
maps measured by TCD. Also, the application of these theoretical re-
sults on some examples demonstrating possibilities of the developed
theory for quantitative characterization of structural imperfections in
real single crystals will be represented.
Below, the basic principles of the generalized dynamical theory of X-
ray scattering by imperfect single crystals with randomly distributed
microdefects will shortly be stated and the analytical expressions for
102 V. B. MOLODKIN, S. I. OLIKHOVSKII, E. G. LEN et al.
the differential dynamical coherent and diffuse scattering amplitudes
in such crystals will be given in Sect. 2. Various methods of triple-
crystal X-ray diffractometry, which are based on this theory, such as
reciprocal space mapping, differential-integral method, and separa-
tion of coherent and diffuse components of integrated diffraction in-
tensity will be described in Sect. 3. Examples of the quantitative char-
acterization of microdefect structures in various silicon crystals will
be given in Sect. 4.
2. DIFFERENTIAL REFLECTIVITY OF SINGLE CRYSTALS WITH
MICRODEFECTS
2.1. Basic Assumptions of the Generalized Statistical Dynamical
Theory
The polarizability (r) of a crystal with random distribution of defects
is a non-periodic function which can be represented as Fourier integral
over the whole momentum space. Then, from Maxwell’s equations af-
ter the Fourier transformation at || 1, the set of basic equations of
dynamical theory for the imperfect crystal in momentum space can be
derived [21]:
2 2 χ( ) ,K k k G q k G q
G q
D k k D 0 (1)
where K 2/ is the modulus of the wave vector K of the incident
plane wave, is the X-ray wavelength, Dq and q are Fourier compo-
nents of electric induction and crystal polarizability, respectively, G
runs over all reciprocal lattice vectors, and q and k run over N discrete
values in first Brillouin zone (N is the number of crystal lattice
points). This set consists of an infinite number of linear algebraic
equations for an infinite number of unknown amplitudes of coherent
and diffusely scattered plane waves within the crystal.
The set of basic equations (1) is valid only for the homogeneous ran-
dom distribution of microdefects. Such distribution provides the
equivalence of averaging over crystal lattice points and averaging over
defect ensemble as well as the translation invariance of the ensemble-
averaged polarizability:
0
( ) ( ) exp( ),s i G
G
r R r Gr (2)
where corner brackets denote the ensemble averaging,
0
s sR R are
the average lattice vectors (s runs over N lattice points), and G are the
corresponding 2 times reciprocal lattice vectors. Fourier component
of polarizability in Eq. (2) is equal to
0 E G G
, if the changes of struc-
DYNAMICAL THEORY OF TRIPLE-CRYSTAL X-RAY DIFFRACTOMETRY 103
tural factors of unit cells due to static distortions can be neglected.
Here,
0G is the Fourier component of the perfect crystal polarizability
and the static Krivoglaz—Debye—Waller factor is defined as
exp( ) exp( ( )) ,sE L i G Gu R where the static displacement field
0( )
s s s
u R R R can be described at the small concentration of random-
ly distributed defects c 1 by the superposition of static displace-
ments of matrix atoms U(r), which are caused by a single defect [23]:
( ) ( ) ( ).
s t s t
t
c c u R U R R (3)
The random occupation numbers ct are equal to 1, if the lattice point
with radius-vector
0
t
R is occupied by the centre of a microdefect, and
0, if not. The averaging is performed just over these random numbers
for which .tc c
Thus, the total polarizability of the crystal with randomly distribut-
ed microdefects can be represented as the sum of average and fluctuat-
ing (random) parts, i.e.
( ) ( ) ( ), r r r (4)
where ( ) r is described by Eq. (2). Similarly, the wave field in the
crystal can be subdivided into average and fluctuating parts:
( ) ( ) ( ), D r D r D r (5)
which correspond to coherent and diffusely scattered waves, respec-
tively:
( ) e e ,
i i 0K r Gr
G
G
D r D (6)
( )
( ) e e ,
i i
0K r G q r
G q
G q 0
D r D (7)
where the constant wave vector K0 of the transmitted coherent plane
wave in crystal has been introduced for the sake of convenience in sub-
sequent consideration.
In the two-beam case of diffraction, only two so-called strong Bragg
(coherent) waves are excited in the crystal and, consequently, in Eqs.
(1) as well as in Eqs. (6) and (7) only two amplitudes of coherent waves
(DG) and corresponding 2N amplitudes of diffusely scattered waves
(DGq) with G 0 and H should be retained, whereas the rest of ampli-
tudes of (weak) Bragg and diffusely scattered waves can be neglected.
The infinite equations set (1) can be represented then for each of two (
and ) polarization states as two equation sets, namely, one for two
amplitudes of coherent transmitted (D0) and diffracted (DH) plane
104 V. B. MOLODKIN, S. I. OLIKHOVSKII, E. G. LEN et al.
waves with wave vectors k K0 and k KH K0 H, respectively, and
another one for 2N amplitudes of transmitted (Dq) and diffracted
(DHq) diffuse plane waves with wave vectors k K0q K0 q and
k KHq KH q, respectively.
These sets of coupled equations can be decoupled and solved by using
the perturbation theory. [21, 22] if inequalities hold for the dispersion
corrections to the wave vectors of coherent and diffusely scattered
waves, respectively, which describe the attenuation of these waves due
to diffuse scattering:
( ) , ( , ) ,E E GG H GG H
(8)
where dispersion corrections are defined as quadratic combinations of
Fourier components of the fluctuating part of crystal polarizability,
and are the angular deviations of the wave vectors of coherent and
diffusely scattered waves from their exact Bragg conditions, G and
G 0 or H.
The dispersion corrections to the wave vectors of coherent and dif-
fusely scattered waves describe the dynamical effects of influence of
the elastic diffuse scattering from defects through virtual ( , )P P
GG GG
and energy-conserved ( , )
GG GG scattering channels on the angular
redistribution and attenuation of the coherent and diffusely scattered
intensity, respectively:
( ) ( ) ( ) / ,P i K
GG GG GG
(9)
( , ) ( , ) ( , ) / ,P i K
GG GG GG
(10)
where and 1, 2 numerate coherent and diffuse wave fields, respec-
tively, and the quantities ( )P P
GG GG and ( )
GG GG are of the same or-
der of magnitude.
In the derivation of compact analytical expressions for the diffuse
scattering intensity, the inequalities were taken into account:
,
q q q q H q H q H (11)
where
G H (G 0, H) are fluctuating Fourier components of crystal
polarizability. These inequalities are valid for typical microdefects at
sufficiently small q H.
2.2. Differential Reflectivity of Imperfect Single Crystals
The differential intensity of X-ray scattering by the imperfect single
crystal containing homogeneously distributed microdefects can be rep-
resented as the sum of coherent or Bragg (RB) and diffuse (RD) compo-
DYNAMICAL THEORY OF TRIPLE-CRYSTAL X-RAY DIFFRACTOMETRY 105
nents (see Refs. [20—22]):
RS(k) RB(k) RD(k), (12)
where k K K H is the deviation of the wave vector K of the scat-
tered plane wave from the reciprocal lattice point H, which corre-
sponds to the reciprocal lattice vector H, and K is the wave vector of
the incident plane wave.
The coherent component of the crystal reflectivity has the form
B coh
( ) ( , ) ( ) ( ),R R b k (13)
2
coh
( ) ( ) ,R r (14)
where (x) is Dirac’s -function, and are the angular deviations
of the wave vectors of incident and scattered plane waves in the hori-
zontal plane from the directions satisfying exactly the Bragg condi-
tion,
1
| |b 0 H is the parameter of diffraction asymmetry,
0 sin(B ) and H sin(B ) are direction cosines relatively to
the inner normal n to the entrance crystal surface for wave vectors K
and K, respectively, is the angle between surface and reflecting
planes of the crystal, В is the Bragg angle, and are the angular de-
viations of the wave vectors K and K from the horizontal plane, re-
spectively.
2.2.1. Coherent Component of Reflectivity of Imperfect Crystal
According to the generalized dynamical theory of the X-ray scattering
in imperfect single crystals [21, 22], the amplitude reflection coeffi-
cient in Eq. (14) can be written for each polarization state ( and ) in
the form:
1
2 2
( ) 1ctg( 1) ,r y i y A y
(15)
where the notation was used:
/ , / ,A t 0 H (16)
1 2( )( ) , ( )( ) ,CE CE CE CE
H H0 H 0H H H0 H 0H
with t and being the crystal thickness and extinction length, respec-
tively, and C is the polarization factor equal to 1 or cos(2B) for - and
-polarization states, respectively. The normalized angular deviation
of the crystal is defined as
0 B
( ) / , sin(2 ),y b (17)
106 V. B. MOLODKIN, S. I. OLIKHOVSKII, E. G. LEN et al.
0 0 0
2 ( ) / .b
HH 00
In the limiting case of the thick crystal, i.e., when 0t 1, where 0
is the linear coefficient of photoelectric absorption, Eq. (15) can be
simplified to the form:
1/2 2( ) ( 1),r y s y (18)
where s sgn(yr), yr Re(y).
It should be remarked that the dispersion corrections to the wave
vectors of coherent and diffusely scattered waves, which appeared in
Eq. (15) to (18) and describe the attenuation of these waves due to dif-
fuse scattering, are connected immediately with the coefficient of ab-
sorption due to diffuse scattering, ds():
ds
( ) ( ), ( ) ( ),b HH 00 HH (19)
1
( ) ( ), ( ) ( ).P K P bP HH HH 00 HH
In turn, this coefficient is related to the correlation function S(q)
describing the differential distribution of diffusely scattered waves:
2
ds 2
( ) d ( ),
4
C V
S
k q (20)
where the integration is performed over the plane tangent to Ewald
sphere near the reciprocal lattice point considered. The correlation
function in Eq. (20) is defined as follows:
( ) Re ,S q H q Hq (21)
where q k iin, and i is the limiting value of an interference absorp-
tion coefficient at and .
It should also be noted that the non-diagonal dispersion corrections
to the wave vectors of coherent waves can be neglected for typical mi-
crodefects with radii significantly smaller than extinction length:
( ) ( ) 0, ( ) ( ) 0.P P 0H H0 0H H0
(22)
The relations for the dispersion corrections to the wave vectors of
diffusely scattered waves are similar to those described above in Eqs.
(19) to (22) for the dispersion corrections to the wave vectors of coher-
ent waves.
Thus, the complex dispersion corrections due to diffuse scattering
and static Krivoglaz—Debye—Waller factor do account for diffuse scat-
tering effects in the coherent component of the crystal reflectivity and
DYNAMICAL THEORY OF TRIPLE-CRYSTAL X-RAY DIFFRACTOMETRY 107
establish its relation to defect characteristics.
2.2.2. Influence of Distorted Thin Surface Layer
Disturbed or distorted surface layers often are present in crystals with
strains caused by artificial modification [3, 5, 23] or natural relaxa-
tion, particularly, due to ‘mirror image forces’ from point defects [24,
25] and microdefects [20, 26]. The coherent component of amplitude
reflectivity of an imperfect crystal with defects and disturbed surface
layer can be found from the reduced Takagi—Taupin equation for the
case of an arbitrary (but not fluctuating) one-dimensional strain field
[18]:
2
2 1,
dX
i X X
dZ
(23)
1/2
0
( ) ( ) / , ( )/ ,X z b D D Z z d H
(24)
with the boundary condition X(d) X0, where d is the layer thickness
and z 0 corresponds to the crystal surface. The equation (23) should
be modified by renormalizing variables and replacing diffraction pa-
rameters for perfect crystal by those for imperfect crystal with defects
to account for the presence of randomly distributed Coulomb-type de-
fects in the layer and substrate, namely:
S S S S B
, / , sin(2 ),y y y b (25)
2 2
B
( cos sin ) tan ( ) sin cos sgn(1 ),b (26)
where and are normal and parallel strain components, respective-
ly.
The corresponding boundary condition has the form:
1/2
0
( ) ( ),X d X r (27)
where the amplitude reflectivity of substrate is described by Eq. (15).
For the sufficiently thin disturbed surface layer whose thickness is
significantly smaller as compared with extinction length, i.e.,
d Re(), the dynamical effects of coherent scattering can be neglect-
ed and corresponding quadratic term in the Takagi—Taupin equation
(15) can be dropped:
2 1.
dX
i X
dZ
(28)
Then, the coherent component of amplitude reflectivity of the imper-
108 V. B. MOLODKIN, S. I. OLIKHOVSKII, E. G. LEN et al.
fect crystal with defects and disturbed surface layer with constant
strain can be found in kinematical approximation:
2
2
0
1
( ) .
2
i Z
i Z e
X z X e
(29)
It is obvious that representation of the strain in disturbed layer by
one-step profile is rather a rough model. More realistic is the model
presented by the multi-step strain profile which corresponds to so-
called ‘layer approximation’. For the multilayer structure with con-
stant average strain i
in each layer of thickness ti, the solution of Eq.
(28) can be easily found by iteration. The iteration procedure is per-
formed by using the solution (29) for the first layer adjacent to the
substrate as the boundary condition for equation in the second layer,
etc.
As result, we obtain the coherent component of reflectivity of the
imperfect crystal with defects and inhomogeneous disturbed surface
layer:
2 0 int kin
coh coh coh coh
(0) ,R X R R R (30)
20
coh 0 0
exp( 2Im ),R X (31)
int *
coh kin 0 0
2 Re exp( ) ,R X X i (32)
2kin
coh kin
,R X (33)
kin
1
1 exp(2 )
e ,
2
j
N
i j j
j j
i T
X
where the notation was used (N is the number of sublayers in the dis-
turbed layer):
1
2 , 0, 1, 0, / ,
N
j i i N i i
i j
T j N T t
(34)
i
S S S S B
, / , sin(2 ),
i i i
i iy y y b (35)
2
B
( cos sin ) tan ( ) sin cos sgn(1 ).
i i i i
i
b (36)
It should be emphasized that Eqs. (30) to (36) for the coherent com-
ponent of reflectivity of the imperfect crystal with defects and inho-
mogeneous disturbed surface layer provide the possibility for account-
ing the influence of not only randomly distributed microdefects in
crystal bulk but also subsurface strains caused by artificial modifica-
tion or natural relaxation, particularly, due to ‘mirror image forces’
from point defects.
DYNAMICAL THEORY OF TRIPLE-CRYSTAL X-RAY DIFFRACTOMETRY 109
2.2.3. Diffuse Component of Imperfect Crystal Reflectivity
The diffuse component of the dynamical differential reflection coeffi-
cient of imperfect crystal with randomly distributed microdefects has
the form [22]:
2
D 2
0 0
,
( ) ,
f
R
S E
H
K K
k (37)
where ( , )f
H
K K is the diffuse scattering amplitude, and S is the illu-
minated area of the entrance crystal surface, E0 is the amplitudes of
incident plane wave in vacuum. The diffuse scattering amplitude in
Eq. (37) is formed by contributions from partial diffuse scattering am-
plitudes ( , )F
HG K K describing the scattering of coherent waves ( )D
G K
into diffuse ones:
( , ) ( ) ( , ),f D F
H G HG
G
K K K K K (38)
where 1, 2 and G 0, H.
In the limiting case of the thick crystal, the expression (37) can be
reduced to the form:
2
2
2
D dyn
0
1
( ) ,
4
CVK
R F
S
H q
k (39)
where V is the crystal volume, the factor Fdyn describes the modulation
of diffuse scattering intensity caused by the dynamical interference of
strong Bragg waves, Hq is the Fourier component of the fluctuation
part of the crystal polarizability, and the interference absorption coef-
ficient i in the complex momentum transfer q k iin describes the
extinction effect for diffusely scattered waves.
The diffuse component of the differential crystal reflectivity can be
written after averaging over random distribution of defects as follows:
1 2
D
( ) ( ),R MK F k q (40)
2 2 2
0 0 0 c
1
i i
( ) / , ( ) /4,
( ) 1 exp( 2 ) 2 ,
M cm C E p t t m v H
p t t t
H
where c is the concentration of randomly distributed microdefects per
one lattice site, vc is the unit cell volume, and t is the crystal thickness.
Function F(q) in Eq. (40) for momentum transfers in Huang scatter-
ing region, i.e., at k km 1/Reff, where Reff is an effective radius of
defect [22], can approximately be written in the form:
110 V. B. MOLODKIN, S. I. OLIKHOVSKII, E. G. LEN et al.
2
0H 2
1 2 34 2
1 1
( ) ( ) ,F F H B B HB
H q
q q
qq q
(41)
where the effective radii of defects are defined as
1/2
eff C
( )R E HA and
1/2
eff L
( )R ER H b for clusters and dislocation loops, respectively, AC
is the cluster strength, b is the Burgers vector, RL is the dislocation
loop radius, and H0 H/H is a unit vector. Constants В1 and В2 are de-
fined for spherical clusters as follow:
2 3 1
1 2 C c C C
0, (4 / ) , , (1 )(1 ) /3,B B A v A R (42)
where is the strain at cluster boundary, RC is the cluster radius, is
the Poisson ratio. For circular dislocation loops, the definitions are fol-
lowing:
2 2 2 2
1 L c 2 1
4 1
( / ) , , (3 6 1) /(1 ) .
15 4
B R v B B b (43)
The constant B3 in Eq. (41) can be put equal to
1/2
3
2 /B L B c H
, where
for clusters B B2 and for dislocation loops B B1.
For the Stockes—Wilson scattering region (k km), the function
F(q) should be rewritten as follows [22]:
S W H 2 2 2 2
m i i
( ) ( ) ( )( )/( ),F F F k k q q q (44)
where the antisymmetric term with B3 coefficient from Eq. (41) should
be dropped.
The expressions (40) to (44) for the diffuse component of reflectivity
are valid at arbitrary radii of microdefects (up to the extinction length
and larger) and in the whole angular range including the total reflec-
tion range due to the account for multiple diffuse scattering processes.
Also, these expressions can be used with some precautions to describe
contributions from thermal diffuse scattering and diffuse scattering
from point defects (see, e.g., Refs. [27, 28]).
3. METHODS OF TRIPLE-CRYSTAL X-RAY DIFFRACTOMETRY
3.1. Reciprocal Space Mapping
3.1.1. Coherent Component of Reciprocal Space Map
The X-ray diffraction intensity from imperfect crystal with defects,
which is registered by TCD detector, is the sum of coherent and diffuse
components; both reflected from third crystal-analyser and integrated
DYNAMICAL THEORY OF TRIPLE-CRYSTAL X-RAY DIFFRACTOMETRY 111
over horizontal and vertical divergence [15—19]:
coh diff
( , ) ( , ) ( , ).I I I (45)
This intensity depends only on two angular variables, namely, the an-
gular deviations of investigated crystal () and analyser-crystal ()
from their exact reflecting positions (Fig. 1).
In the case of quasi-non-dispersive geometry (m, n, m) of the opti-
cal scheme used conventionally, the coherent component of the intensi-
ty measured by TCD can be represented in the form [18]:
1 1 1
coh 0 M M S coh S A
( , ) { [ ( ) ]} [ ( )] ( ),I I dxR b b x R b x R x
(46)
where I0 is the incident X-ray intensity, RM and RA are reflection coef-
ficients of flat or grooved monochromator and analyser crystals, bM
and bS are asymmetry parameters of monochromator and investigated
crystal, respectively. If structural defects are present in monochroma-
tor and analyser crystals, their reflection coefficients also consist of
coherent and diffuse components similarly to that of the investigated
crystal:
M,A M,A
M,A coh diff
( ) ( ) ( ),R x R x R x (47)
Fig. 1. Scheme applied for mapping the coherent and diffuse scattering inten-
sity distributions in reciprocal space by using TCD measurements at Bragg
diffraction geometry. The notation: X–X-ray tube, M–monochromator, S–
investigated crystal, A–analyser, D–detector, K0 and KH–wave vectors of
incident and diffracted waves, respectively, H–reciprocal lattice vector,
and –angular deviations of S and A crystals from their exact reflecting
positions, respectively, q and q –orts of the oblique coordinate system [29].
112 V. B. MOLODKIN, S. I. OLIKHOVSKII, E. G. LEN et al.
where diffuse components of reflection coefficients RM and RA are de-
scribed by expressions like Eq. (40). When the multiple reflections are
used in monochromator and analyser crystals (Fig. 2), the diffuse com-
ponents of monochromator and analyser reflection coefficients are
strongly suppressed and can be neglected. However, in the case of flat
monochromator and analyser crystals these diffuse components can
substantially modify observed diffraction patterns [15—17, 30, 31].
3.1.2. Diffuse Component of Reciprocal Space Map
The diffuse component of the diffraction intensity measured by TCD
can be represented in the form [19, 32]:
diff 0 M diff A
( , ) ( ) ( ) ( ),I I dxR x dx r R x
(48)
where the function rdiff represents the diffuse component of differen-
tial reflection coefficient (40) integrated over a vertical divergence:
1
diff D
( ) ( ).
y
r K dk R k (49)
Here, k kyey, components kx and kz of the vector kxex + kzez lie in
the coherent scattering plane (K, H), ey is a normal to this plane, ez n,
and components kx and ky lie in the crystal surface.
The components kx and kz are connected by simple relationships with
angular deviations of wave vectors of incident (x) and scattered (x)
plane waves from their exact Bragg directions in the scattering plane.
In the case of the asymmetric Bragg diffraction geometry these rela-
tionships have the form:
Fig. 2. Scheme used in the typical TCD with the multiple reflections in
grooved monochromator (М1, М2) and analyser (A) crystals at Bragg diffrac-
tion geometry on the investigated crystal (S) [32].
DYNAMICAL THEORY OF TRIPLE-CRYSTAL X-RAY DIFFRACTOMETRY 113
B B
B B
( ) sin( ) 2 ( ) sin cos ,
( ) cos( ) 2 ( ) sin sin .
x
z
k K x x K x
k K x x K x
(50)
If the half-width of the function rdiff is much larger as compared
with half-widths of reflection coefficients of monochromator and ana-
lyser crystals, the functions RM and RA in Eq. (48) can be replaced by -
functions:
diff 0 iM iA diff
( , ) ( , ),x zI I R R r k k (51)
where RiM and RiA are integrated reflectivities of monochromator and
analyser crystals, and components kx and kz are described by expres-
sions:
B B
B B
sin( ) 2 sin cos ,
cos( ) 2 sin sin .
x
z
k K K
k K K
(52)
Now, after integrating in Eq. (49) over vertical divergence, the dif-
fuse component of the reciprocal space map for imperfect crystals with
randomly distributed microdefects can be represented in the Huang
scattering region ( km) as follows:
diff H S W a
( ) [ ( ) ( ) ( )],r M A A A (53)
2 2
m
H 1 2
2 2 2 2
i i
( ) (2 ) arctan ,
kK
A B B a
20
2 2
i
a
H
, (54)
m m mm
S W 1 2
2 2
i
2 2 2 2 2 22 2
i
22 2 2 2 2 2 2 2
i i i i
3
( )
4
arctan ,
2 2
K
A B B a
k k kk K
B a
(55)
2 2 2 2 2 22 2
m i m mm i
a 3 2 22 2 2 2 2 2
im i m m i
( ) ln 2 1 .
k k kkK
A B
H k k k
(56)
Similarly, in the Stokes—Wilson scattering region ( km), we obtain:
diff S W
( ) ( ),r MB (57)
2
m
S W 1 22 2 3 2
i
3
( ) .
22( )
Kk
B B a B
(58)
114 V. B. MOLODKIN, S. I. OLIKHOVSKII, E. G. LEN et al.
The interference absorption coefficient describes the extinction ef-
fect for diffusely scattered waves and is given by the relationship:
0
i i i
0
1
( , ) [ ( ) ( )],
2 2
b E
r z r z
g
(59)
where the notation was used:
2 2
i
( ) ( ) /2,r z u v u
2 2 2 2 2( ) 1, 2( ),u z g E v zgE p
B B
r r
sin(2 ) ( ) sin(2 )
, ,z b z b
C C
H H
i0 ir i r i
2
r i0r
Re( ) Re( ) Im( ) Im( )1
, , .
2
b b
g p
C
HH H H H
H H
The interference absorption coefficient i plays in Eqs. (53) to (58) a
role of the cut-off parameter and removes the non-physical divergence
at 0. This coefficient also allows for the correct description of the
diffuse component contribution to TCD diffraction profiles in the re-
gion of coherent peak, where this contribution is rapidly decreased ap-
proximately by an order of magnitude.
If several types of microdefects are present within the crystal simul-
taneously, without mutual correlation, the Eqs. (53) and (57) should be
simply replaced by the sum of corresponding expressions for each type
of defects.
The above consideration allows to describe the complex diffraction
patterns measured by TCD from single crystals with complicated de-
fect structures and, particularly, to explain the asymmetrical behav-
iour of coherent peaks on TCD profiles due to taking into account the
influence of microdefects and disturbed layers in monochromator, ana-
lyser, and sample crystals.
3.2. Differential-Integral Method
The diffraction profiles measured by TCD from the investigated crys-
tal, which contains homogeneously distributed microdefects, as func-
tions of deviation angle of the analyser crystal at fixed deviation
angle of the investigated sample are usually consisting of three
peaks. Namely, there are observed the main peak at M S
(1 ) ,b
pseudo-peak at S
and diffuse peak at
2
D B
2 sin . A form
of the pseudo-peak is determined by the reflection coefficient of the
monochromator, whereas the forms of main and diffuse peaks are de-
DYNAMICAL THEORY OF TRIPLE-CRYSTAL X-RAY DIFFRACTOMETRY 115
termined by the coherent and diffuse components of the reflection co-
efficient of the investigated sample, respectively. These peaks are well
separated at sufficiently large deviations on the sample, which are
significantly greater than the width W of its total reflection range.
The integrated intensities of coherent (main) and diffuse peaks de-
pend only on the angular deviation of the investigated crystal and
are connected unambiguously with the coherent and diffuse compo-
nents of the reflection coefficient of this crystal, the sum of which is
measured by the double-crystal diffractometer (DCD) with widely open
detector window. These connections can be established easily by inte-
grating the expressions (46) and (48) over in the vicinity of points
M
and D
. If the condition W is fulfilled and the peaks are
well separated, the corresponding integrals are factorized simply [16]:
M 0 iM iA coh
( ) ( ),I I R R R (60)
D 0 iM iA diff
( ) ( ).I I R R R (61)
In Eqs. (60) and (61), the next notation was used:
2
coh
( ) ( ) ,R r (62)
diff diff D
'
( ) ( ) ( ) ( )d ,
K K
R d r R
K
k (63)
where d
K is a solid angle in K direction, and Rcoh() and Rdiff() de-
scribe the coherent and diffuse components of rocking curve measured
by DCD [22], respectively. The value of the error made at such factori-
zation is of the order of
2 2/| | .W
The ratio of the integrated intensities of coherent and diffuse peaks
D M
( ) ( ) / ( ),Q I I as can be seen from Eqs. (46) and (48), will not
depend on the instrumental factors of TCD. After accounting for the
summation over X-ray polarizations, this ratio takes the form:
diff diff
diff diff
( ) ( )
( ) ,
( ) ( )
R R
Q
R R
(64)
where
M A
B B
cos(2 ) cos(2 ) ,
M
B
and
A
B
are the Bragg angles of mon-
ochromator and analyser, respectively. When obtaining the formula
(64), it was supposed that the approximate relationships hold
M
iM B iM
cos(2 )R R and
A
iA B iA
cos(2 )R R .
The diffuse component of the reflection coefficient of the crystal
with homogeneously distributed defects after the integration over
output angles can be represented as follows [22, 28]:
ds
diff dyn
0
( )
( ) ( ) ,
2 ( )
R F
(65)
116 V. B. MOLODKIN, S. I. OLIKHOVSKII, E. G. LEN et al.
where the dependence of the interference absorption coefficient i on
exit angles was neglected because its influence is smoothed due to
integration, and thus i in Eq. (59) can be replaced by its limiting value
at W :
0
i i
0
1
( ) 1 ( ) / .
2 2
b
Er z g
(66)
The absorption coefficient due to diffuse scattering from microde-
fects in Eq. (65) is described by the expression:
2 2
ds 0 0 0
( ) ( ),k cC E m J k (67)
H 0 H SW 0 H 0 0 m
0
SW 0 0 m
( ) ( ) ( ), at ,
( )
( ), at ;
J k J k J k k k
J k
J k k k
(68)
here, 0 B
sin(2 ).k K The angular dependence of the ‘integral’ diffuse
scattering intensity in the Huang scattering region is described by sym-
metric (JH) and antisymmetric ( H
J ) components in Eq. (68) as follows:
2 2
2 2m
H 0 2 3 0 42 2 2 2 2 2
0 m 0
1 1
( ) ln ( ) ,
k
J k b b k b
k k k
(69)
2 2
3 0 4
H SW 0 2 2 2
m
1
( ) ,
2
b k b
J k b
k
(70)
2 22 2
3 0 4m
SW 0 22 2 2 2
0 0
1
( ) ,
2
b k bk
J k b
k k
(71)
2 2 2 2
H 0 1 m 0
( ) ( ).J k b k k (72)
The coefficients ( 1,4)
i
b i in Eqs. (69)—(72) are connected with char-
acteristics of microdefects by relations:
2 2
1 2 1 2 B
4
, ( cos )/2,
L B
b b B B
cH
H (73)
2 2 2 2
3 2 B B 4 2 B
((cos )/2 sin ), ((cos )/2 cos ).b B b B
It should be remarked that the absorption coefficient due to diffuse
scattering in the above equations is the sum of corresponding coeffi-
cients in the case of several randomly distributed defect types. Similar
superposition law is supposed to be valid for the exponent of static
Krivoglaz—Debye—Waller factor as well. The last is also immediately
connected with defect characteristics by following relationships for
DYNAMICAL THEORY OF TRIPLE-CRYSTAL X-RAY DIFFRACTOMETRY 117
dislocation loops and spherical clusters, respectively [24, 33]:
3 3/2
L L
1
( )
2
L n R Hb
H
, (74)
C C
C C
4
0
3
0
0.525 , 1.9,
, 1.9,
n v n
L
n v n
H
(75)
C
C
3
0
c
4 R
R , ,
3
H n
v
where nL cL/vc and nC cC/vc are number densities of dislocation loops
and spherical clusters, respectively.
The above-written formulas establish analytical relations between
coherent and diffuse components of the reflection coefficients meas-
ured by high-resolution double- and triple-crystal diffractometers
from imperfect crystals in Bragg diffraction geometry, on the one
hand, and statistical characteristics of defects, on the other hand.
Thus, a closed set of the analytical relationships is formed which can
provide the self-consistent dynamical description and quantitative
analysis of the rocking curves and diffraction profiles measured by the
high-resolution double- and triple-crystal diffractometers, respectively,
from imperfect crystals containing randomly distributed defects of sev-
eral types to determine more reliably their structural characteristics.
3.3. Separation of Coherent and Diffuse Components of Integrated
Diffraction Intensity
The investigation of crystal imperfections by using measurements of
the integrated X-ray diffraction intensities has a number of advantages
such as rapidity, high sensitivity, and simple data treatment. The mul-
ticrystal arrangements provide the unique possibility to separate the
coherent and diffuse components of the integrated diffraction intensity
from imperfect crystals containing randomly distributed defects and
thus to enhance the sensitivity to defect characteristics due to their di-
rect connections with the integrated diffuse component [34, 35].
In particular, the experimental TCD setup has been proposed for di-
rect measurements of the integrated diffuse component at Laue dif-
fraction geometry on the investigated crystal [36]. In this setup (Fig.
3, a), the mutual disorientation of monochromator and sample crystals
has been used to screen the X-rays exciting the coherent component of
scattering intensity in the sample under investigation. In another pro-
posed TCD setup (Fig. 3, b), the thin analyser crystal set in the exact
Bragg reflection geometry was used as screen for the coherent compo-
nent of diffraction intensity from the sample whereas the diffuse com-
118 V. B. MOLODKIN, S. I. OLIKHOVSKII, E. G. LEN et al.
ponent was transmitted to the detector [37].
4. CHARACTERIZATION OF DEFECTS AND STRAINS IN SINGLE
CRYSTALS BY USING TCD
4.1. Defect Structure of Annealed Czochralski-Grown Silicon Crystal
4.1.1. Annealing at High Temperatures
A) Anneal at 1100C during 40 min and 5.5 h. The two investigated
samples of 1 mm thickness with (100) surface orientation were cut
from Czochralski-grown silicon (Cz-Si) ingot of n-type conductivity
(10 cm) with initial oxygen concentration about 1018
cm
3. The sam-
ples were annealed in wet oxygen atmosphere at 1100C during 40 min
and 5.5 h. As known, the enhanced oxygen precipitation is occurred in
the silicon samples at such conditions of the thermal treatment.
Measurements of diffraction profiles (see Fig. 4) were carried out by
using the non-dispersive TCD scheme with flat monochromator and
a
b
Fig. 3. TCD schemes used to measure separately the coherent (a) and diffuse
(b) components of total integrated reflecting power at Laue X-ray diffraction
geometry on the investigated crystal (S) [36, 37].
DYNAMICAL THEORY OF TRIPLE-CRYSTAL X-RAY DIFFRACTOMETRY 119
analyser silicon crystals (see Fig. 1) and the samples were set in the
symmetrical Bragg diffraction geometry [15]. The measured profiles
showed the high and narrow diffuse peaks located at 0 and evi-
denced for the existence of strong diffuse scattering from large oxygen
precipitates. Moreover, the analysis of these profiles led to the necessi-
ty for accounting the influence of the loose precipitate interface creat-
ed by emitted silicon interstitials and like that described recently in
the experimental investigation of oxygen precipitation in highly bo-
ron-doped Cz-Si [38].
In the case of clusters with loose interface, the diffuse component of
the differential crystal reflectivity (40) can be modified according to
the model proposed by Krivoglaz as follows [15, 24 ]:
2
D
( ) ( ) ( ) ,R F sk q k (76)
where the Fourier component of the ellipsoidal cluster form-factor is
described by the expression:
2 2
0 c
, ,
1
( ) exp ,
2
i i
i x y z
s n v k R
k (77)
3/2
0 c
(2 ) / ,
x y z
n R R R v
and Ri are the lengths of ellipsoid semi-principal axes. After integrat-
ing Eq. (76) over vertical divergence, the diffuse component of the dif-
fraction profiles for imperfect crystals with randomly distributed el-
lipsoidal microdefects having the loose interface can be approximately
described by the following expression:
2 2 2 2
diff diff
( , ) ( , ) exp( ),
x z x x z z
I r k k k R k R (78)
Fig. 4. Measured and calculated diffraction profiles (dashed and solid lines,
respectively) measured by TCD from Cz-Si samples annealed at 1100C during
40 min (a, b) and 5.5 h (c) for Si(004) reflection of MoK-radiation [15].
120 V. B. MOLODKIN, S. I. OLIKHOVSKII, E. G. LEN et al.
where rdiff(kx, kz) is given by Eq. (53).
The fit of the measured diffraction profiles was performed to
achieve coincidence of heights and positions of main, pseudo-, and dif-
fuse peaks at measured and calculated curves by varying oxygen pre-
cipitate radii, concentration, and the static Krivoglaz—Debye—Waller
factor. The treatment of the diffraction profiles measured from the
sample annealed during 40 min (Fig. 4, a and b) has given the following
values: Rx Ry 13 nm and Rz 650 nm, with small differences in
c(AC/vc)
2
1.410
6
and E 0.7 for the profile shown in Fig. 4, a and
c(AC/vc)
2
2.210
6
and E 0.6 for the profile shown in Fig. 4, b. The
treatment of the diffraction profiles measured from the sample an-
Fig. 5. Measured (left) and calculated (right) reciprocal space maps for the Cz-Si
sample annealed at 1150C during 50 h, Si(111) reflection, CuK-radiation [39].
Fig. 6. Measured and calculated diffraction profile (markers and solid line,
respectively) from Cz-Si sample annealed at 1150C during 50 h, Si(111) re-
flection, CuK-radiation [39].
DYNAMICAL THEORY OF TRIPLE-CRYSTAL X-RAY DIFFRACTOMETRY 121
nealed during 5.5 h (Fig. 4, c) has given the following values: Rx Ry
260 nm, Rz 1300 nm, c(AC/vc)
2
2.510
6, and E 0.4. The remained
visible discrepancies between measured and calculated diffraction pro-
files can be caused by the existing spread of oxygen precipitate radii
and ignoring instrumental factors.
B) Anneal at 1150C during 50 h. The investigated sample was cut
from Cz-Si ingot and annealed in argon atmosphere at 1150C during
50 h. The measurements of reciprocal space maps (Fig. 5) were per-
formed by using the diffractometer PANanalytical X`Pert Pro MRD
XL at symmetrical Si(111) reflection on the sample in Bragg diffrac-
tion geometry of characteristic CuK1-radiation (see Fig. 2).
Additionally, the measurements of diffraction profiles (Fig. 6) were
performed in the non-dispersive scheme of the home-made TCD with a
higher resolution at symmetrical Si(111) reflections in Bragg diffrac-
tion geometry of characteristic CuK1-radiation on all three flat crys-
tals [39]. To describe defect structure in the investigated Cz-Si sample,
the model was used supposing the presence of two microdefect types
(see, e.g., Ref. [40]), namely, the spherical amorphous oxygen precipi-
tates SiO2 (radius RP, thickness hP, and number density nP) and circular
dislocation loops with Burgers vector b <110>/2 (radius RL and num-
ber density nL).
As can be seen from Table 1, the characterization results obtained
from the reciprocal space map coincide sufficiently well for predomi-
nant type of defects (large dislocation loops) with those obtained from
the set of diffraction profiles measured by the home-made TCD one of
which is shown in Fig. 6. Some discrepancy in the concentration can be
explained by the presence of smaller dislocation loops and oxygen pre-
cipitates, which also give contributions to the diffraction pattern. The
detailed quantitative characterization of small oxygen precipitates and
dislocation loops was possible due to additional using the set of diffrac-
tion profiles measured by the home-made TCD (see also Ref. [41]).
It should be emphasized that the exclusively important factor for
the reliable determination of characteristics of these defects was the
use of the generalized dynamical theory, which permits determining
simultaneously a set of defect characteristics due to the self-consistent
TABLE 1. Radii and number densities of spherical oxygen precipitates and
circular dislocation loops in Cz-Si sample annealed at 1150C during 50 h [39].
Data type RP, nm nP, cm
3 RL, m nL, cm
3
Profile 10 1 (1.0 0.1)1013
0.05 0.005
0.3 0.03
5.0 0.3
(8.0 2)1011
(9.0 3)109
(4 0.5)107
Map — — 4.3 0.6 (1.9 1)108
122 V. B. MOLODKIN, S. I. OLIKHOVSKII, E. G. LEN et al.
consideration of both diffuse and coherent components of diffraction
patterns. The availability of the explicit analytical expressions for dif-
fuse scattering intensity, which take the anisotropy of displacement
fields around dislocation loops with discrete orientations into account
[39], was also important.
C) Anneal at 1160C during 50 h. The investigated Si single crystal was
cut from Cz-Si ingot with p-type conductivity ( 10 cm) and initial
oxygen concentration about 1.11018
cm
3. The sample of 478 m thick-
ness was annealed in argon atmosphere at 1160C during 50 h with sub-
sequent heat hardening in air. The measurements of diffraction pro-
files (Fig. 7) were performed in the non-dispersive TCD scheme with
symmetrical reflections Si(111) in Bragg diffraction geometry of char-
acteristic CuK1-radiation on all three flat crystals (see Fig. 1) [16].
The set of TCD profiles was measured at fixed angular positions of
the sample under investigation versus angular deviation of the analys-
er crystal. As can be seen form an example in Fig. 7, the presence of
diffuse scattering from defects in monochromator crystal leads to a
rapid increase or decrease of the pseudo-peak height in comparison
with that of main peak in dependence on the sign of the sample devia-
tion. Such asymmetrical behaviour can be explained only by account-
ing for the contribution of the antisymmetric part of diffuse scatter-
ing intensity from microdefects in monochromator crystal [18].
The angular dependence of the ratio of diffuse to coherent integrated
peak intensities calculated from the set of measured TCD profiles is
shown in Fig. 8. The treatment of this dependence was performed for the
defect model of spherical amorphous new phase SiO2 particles by using
Eq. (64). The fit shows a good agreement between experimental observa-
tions and theoretical calculations at following values of oxygen precipi-
tate characteristics: RC (0.94 0.10) m and nC (1.2 0.1)106
cm
3.
D) Anneal at 1180C during 6 h. The silicon single crystal was cut from
the Cz-Si ingot perpendicularly to (111) growth axis. The sample of
about 500 m thickness was annealed at 1180C during 6 h. Measure-
ments of thickness dependences of the separated coherent and diffuse
components of the total integrated reflecting power (Fig. 9) were car-
ried out by the inclination method at Laue diffraction geometry on the
investigated crystal in both TCD (Fig. 3) and DCD optical schemes with
symmetrical Si(220) reflections of characteristic MoK1-radiation on
all three crystals [36, 37].
In the optical TCD scheme used to separate coherent component of
total integrated reflecting power (Fig. 3, a), the asymmetrically cut Si
analyser crystal was chosen to strongly decrease the receiving aperture
for the X-ray beam diffracted from the sample.
In the optical TCD scheme used to separate diffuse component of to-
tal integrated reflecting power (Fig. 3, b), the thin Si analyser crystal
was set in the exact reflecting Bragg diffraction geometry to reflect
DYNAMICAL THEORY OF TRIPLE-CRYSTAL X-RAY DIFFRACTOMETRY 123
the coherent component and to provide the transmission with low at-
tenuation for X-ray beam diffracted from the sample outside the total
reflection range.
The fit of the measured thickness dependence of the total integrated
reflecting power evidenced its low sensitivity to defect characteristics
of the investigated nearly perfect silicon crystal. Namely, the follow-
ing values of the absorption coefficient due to diffuse scattering, nor-
malized to the photoelectric absorption coefficient,
0
ds 0
/ 0.3 0.5 ,
and the exponent of static Krivoglaz—Debye—Waller factor L220
a b
Fig. 7. Diffraction profiles measured by TCD for the symmetrical Si(111) re-
flection of CuK1-radiation from the Cz-Si sample annealed at 1160C during
50 h [16] at the negative (a) and positive (b) angles of sample deviation.
Fig. 8. Angular dependence of the ratio of diffuse to coherent integrated peak
intensities calculated from measured TCD profiles (points–experiment, solid
line–theory) [16].
124 V. B. MOLODKIN, S. I. OLIKHOVSKII, E. G. LEN et al.
0.005 0.001 were found. However, the additional fit of the meas-
ured thickness dependence of the separated coherent component of to-
tal integrated reflecting power (see Figs. 3, a and 9, a) at fixed value of
L220 0.005 allowed the determination of the absorption coefficient
due to diffuse scattering with sufficiently high precision, namely,
0
ds 0
/ 0.23 0.02 .
The thickness dependences of the total integrated reflecting power
and its diffuse component separated by using the optical TCD scheme
shown in Fig. 3, b are represented in Fig. 9, b. The fit of the separated
diffuse component at fixed value of L220 0.005 allowed determining
the value of absorption coefficient due to diffuse scattering,
0
ds 0
/ 0.25 0.02 , which is very close to that determined by the
previous method.
Thus, both the optical TCD schemes used to separate measured co-
herent and diffuse components of the total integrated reflecting power
allow for increasing, in comparison with the conventional inclination
a b
Fig. 9. Thickness dependences of the separated coherent and diffuse compo-
nents of total integrated reflecting power measured by the inclination method
using optical TCD schemes shown: (a)–in Fig. 3, a and (b)–in Fig. 3, b corre-
sponding to Bragg and Laue diffraction geometries for the investigated sili-
con crystal, respectively (markers–experiment; solid, dot, dashed, and dot-
dashed lines–theory) [36, 37].
DYNAMICAL THEORY OF TRIPLE-CRYSTAL X-RAY DIFFRACTOMETRY 125
method, the precision of determination of the second important defect
characteristics, namely, the absorption coefficient due to diffuse scat-
tering, additionally to static Krivoglaz—Debye—Waller factor.
4.1.2. Annealing at Low Temperature
The investigated Cz-Si crystal of 480 m thickness has been cut from
the central part of 10 cm wafer with (111) surface orientation perpen-
dicular to growth axis. The sample of р-type conductivity with the re-
sistance of 10.5 сm, which contained 1.11018
сm
3
and less than
1.01017
сm
3
of oxygen and carbon atoms, respectively, was annealed
at 750C during 50 h.
Measurements of TCD profiles (Fig. 10, a) were carried out in Bragg
diffraction geometry near reciprocal lattice point (111) in the non-
dispersive scheme (n, n, n) by automated home-made TCD with flat
monochromator and analyser crystals (see Fig. 1). Also, the rocking
curves were measured in the mode of DCD with widely open detector
window [42].
In the model used to describe defect structure in the investigated Cz-
Si sample, the presence of two types of microdefects was supposed,
namely, the disk-shaped amorphous oxygen precipitates SiO2 (radius
RP, thickness hP, and number density nP) and chaotically oriented cir-
cular dislocation loops of interstitial type with Burgers vector
b <110>/2 (radius RL and number density nL) (see, e.g., [20]). The
contribution from thermal diffuse scattering was simulated by the dif-
fuse scattering from spherical clusters with the radius equal to the co-
valent one of silicon atom RP 1.17 Å and concentration nP
51022
сm
3, i.e., concentration of silicon atoms. The value of strain at
a b
Fig. 10. Fitted TCD profile (a) and rocking curve (b) for Si(111) reflection of
CuK1-radiation from Cz-Si sample annealed at 750C during 50 h [42].
126 V. B. MOLODKIN, S. I. OLIKHOVSKII, E. G. LEN et al.
such ‘cluster’ boundary 0.25 has been found by fitting and corre-
sponds well with the mean-square deviation of vibrating Si atoms. The
defect characteristics given in Table 2 have been determined by the
self-consistent fitting the measured rocking curve and TCD profiles.
The successful treatment of diffraction profiles measured by TCD
and DCD was possible due to their different sensitivities to microde-
fects of different types and sizes. In particular, the contributions of
diffuse scattering intensity from small microdefects, point defects,
and thermal vibrations to TCD profiles are significantly smaller in
magnitude as compared to those from large microdefects. On the other
hand, these contributions are comparable on DCD profiles due to the
additional integration of diffuse scattering intensity over vertical di-
vergence. More important, however, was the use of the common theo-
retical approach to the description of diffraction profiles measured by
TCD and DCD, namely, the generalized dynamical theory of X-ray
scattering in real single crystals [10—17]. Thus, the most complete
characterization of the defect structure in the investigated sample can
be achieved after combined measurements by TCD and high-resolution
DCD with joint treatment of experimental data by using this theory.
TABLE 2. Characteristics of disk-shaped oxygen precipitates and circular dis-
location loops in Cz-Si crystal annealed at 750C during 50 h [42].
RP, nm hP, nm nP, cm
3 RL, m nL, cm
3
10 1 5.0 0.5 (3.0 0.3)1013 0.0242
0.045 0.003
0.25 0.03
5.0 0.3
(5.0 1.5)1011
(5.8 1.6)109
(4 1)107
a b
Fig. 11. Measured and calculated TCD diffraction profiles (markers and solid
line, respectively) versus analyser deviation angle () at fixed values of de-
viation angle () of the investigated Cz-Si sample annealed at 750C during
50 h, Si(111) reflection, CuK-radiation [19].
DYNAMICAL THEORY OF TRIPLE-CRYSTAL X-RAY DIFFRACTOMETRY 127
In Ref. [42], such approach has allowed to determine not only the
characteristics of small and large microdefects in the crystal bulk but
also the strain in subsurface layer, which is caused by the so called
‘mirror image forces’ from all the defects in crystal bulk. This strain
was described by the exponential law
0
0
( ) exp( / )z z t , where z is
the depth, and the parameters have been determined to be equal
t0 7 nm and
0 410 .
It is of interest to compare the determined defect characteristics
with those obtained for the same sample two years ago in a similar re-
search [19], where the measurements of TCD profiles (Fig. 11) were
carried out also in Bragg diffraction geometry by the home-made TCD.
The model of defect structure of the investigated Cz-Si sample, which
was used to fit the measured TCD diffraction profiles and rocking
curves, was different only in that the presence of the spherical instead
of disk-shaped amorphous oxygen precipitates SiO2 was supposed. The
defect characteristics given in Table 3 have been determined by the
self-consistent fitting the measured rocking curves and TCD profiles.
As can be seen by comparing the defect characterization data from
Tables 2 and 3, the average volume of oxygen precipitates remained
unchanged within the fit error after two-year conservation at the room
temperature. Moreover, the characteristics of large dislocation loops
are constant. At the same time, the small dislocation loops have been
partially dissolved after two-year ageing by emitting nearly half of
their silicon interstitials.
4.2. Defect Structure of the Silicon Crystal Grown by Floating-Zone
Method
The investigated sample has been cut from the central part of the oxy-
gen-free silicon single-crystalline plate grown by the floating-zone
method (FZ-Si). The conventional X-ray topography and scanning elec-
tron microscopy have not revealed any defects in the sample.
The model of defect structure in the investigated FZ-Si sample,
which was chosen to describe the measured diffraction profiles [17],
was based on results of direct observations of FZ-Si crystals by the
TABLE 3. Characteristics of spherical oxygen precipitates and circular dislo-
cation loops in Cz-Si crystal annealed at 750C during 50 h [19].
RP, nm nP, cm
3 RL, m nL, cm
3
0.01 11013
0.05
0.3
5.0
81011
9109
4107
128 V. B. MOLODKIN, S. I. OLIKHOVSKII, E. G. LEN et al.
transmission electron microscopy [43—46]. This model includes three
types of defects, namely, circular dislocation loops of interstitial type
with the Burgers vector <110>/2, disc-shaped precipitates of un-
known nature, and point defects. The later ones give a small contribu-
tion to rocking curve tails in the measured range, being comparable
to that of thermal diffuse scattering, but affect appreciably the dif-
fraction pattern through the corresponding significant addition to the
exponent of the static Krivoglaz—Debye—Waller factor.
Monochromator and analyser crystals used in TCD were identical to
the investigated sample, i.e., they have been cut from the central part
of the same FZ-Si plate. TCD measurements were carried out in non-
dispersive scheme (n, n, n) with symmetric (111) reflections in Bragg
diffraction geometry on all three crystals by using CuK1-radiation
from the conventional X-ray tube (see Fig. 1). The TCD diffraction
profiles (Fig. 12) were measured at fixed sample deviation by rotat-
ing analyser crystal () with using the step motor. The very low
heights of diffuse peaks as compared to coherent ones indicate imme-
diately the high perfection of the crystal under investigation.
The rocking curves for symmetrical Bragg (111) and (333) reflec-
tions of CuK1-radiation were measured by the high-resolution DCD
with widely open detector window (Fig. 13). The dispersive (m, m, n)
scheme was used in DCD with one-fold symmetrical Ge(333) reflections
on both monochromator crystals.
The fit of diffuse peaks on TCD profiles (Fig. 12) and DCD rocking
curves (Fig. 13) was performed supposing the same defect model. It
should be emphasized also that the separate measurement of diffuse
a b
Fig. 12. Diffuse peaks on the diffraction profiles measured by TCD at sample
deviations 49 (a) and 73 arc sec (b), FZ-Si(111) reflection of CuK1-
radiation. The calculated diffraction profile is constituted of coherent and
diffuse components (thick solid, thin solid, and dash-dotted lines, respective-
ly), the last one is the sum of diffuse scattering intensities from dislocation
loops and precipitates (dashed and dotted lines, respectively) [17].
DYNAMICAL THEORY OF TRIPLE-CRYSTAL X-RAY DIFFRACTOMETRY 129
scattering intensity by TCD has allowed the determination of the in-
terstitial nature of the precipitates, which give the noticeable anti-
symmetric part to the observed diffuse scattering intensity. It was im-
portant also to verify the obtained results, which follow from the anal-
ysis of the diffuse scattering intensity superimposed in DCD meas-
urements with the coherent scattering intensity, by the more direct
observation of diffuse scattering intensity profiles measured by TCD
separately from coherent ones.
The determined characteristics of microdefects (radii RL and RP as
well as number densities nL and nP of dislocation loops and disc-shaped
precipitates, respectively) are listed in Table 4. The values of these
characteristics, which have been obtained from different diffraction
experiments, are sufficiently close one to other. The sizes of both type
microdefects are in surprisingly good correspondence with those found
in the direct observations of another FZ-Si crystal [43]. The discrepan-
cies of number densities and size values can originate from the small-
ness of crystal volumes probed in the transmission electron microscopy
and, obviously, from different growth conditions.
The characteristics of microdefects in the investigated FZ-Si sample
have been repeatedly determined from the diffraction profiles (Fig. 14)
and reciprocal space map (Fig. 15) measured by TCD five years later
(see Table 5). As can be seen from Tables 4 and 5, the characteristics of
dislocation loops in the FZ-Si sample have been only little changed af-
ter five year conservation at the room temperature. However, the
measurement and analysis of the reciprocal space map enables for the
refinement of their average increased radius. At the same time, the
average number of silicon interstitials in dislocation loops remained
a b
Fig. 13. Rocking curves measured for FZ-Si 111 (a) and 333 (b) reflections,
CuK1-radiation. The calculated total rocking curves and their coherent com-
ponents are shown by thick and thin solid lines, respectively, and diffuse scat-
tering intensity contributions from dislocation loops, precipitates, and point
defects are shown by dashed, dotted, and dash-dotted lines, respectively [17].
130 V. B. MOLODKIN, S. I. OLIKHOVSKII, E. G. LEN et al.
nearly unchanged due to the decreased concentration of these disloca-
tion loops.
4.3. Defect Structure of the Cz-Si Crystal after High-Energy Electron
Irradiation
The investigated Cz-Si samples of (111) surface orientation and thick-
nesses t 4.26 mm have been irradiated with doses 1.8 and 3.6 kGy
(Nos. 1A and 1B, respectively) of high-energy electrons (18 MeV). The
as-grown sample No. 1 was used as etalon for the comparison with the
irradiated samples.
Reciprocal space maps of the samples under investigation were
measured by PANalytical X`Pert Pro MRD XL diffractometer (see Fig.
2) for the symmetrical Si(333) reflection of characteristic CuK1-
radiation [47, 48]. Additionally, the rocking curves for Si(333) reflec-
tion were measured in the -scanning mode by using the TCD with re-
moved analyser crystal.
The thorough comparative analysis of the measured reciprocal space
maps (Fig. 16), their cross-sections (Fig. 17), and rocking curves (Fig.
18) was performed to achieve their mutual consistence on the base of a
common set of defect characteristics. For this sake, the appropriate
model of the defect structure was chosen which is typical for both as
grown and annealed Cz-Si samples and consists of two types of random-
ly distributed microdefects, namely, oxygen precipitates and intersti-
tial dislocation loops [40].
The fit of the measured diffraction profiles and reciprocal space
maps was carried out at neglecting the contributions from the thermal
diffuse scattering and diffuse scattering from point defects as being
TABLE 4. Microdefect characteristics determined from the diffraction pro-
files measured by DCD and TCD for FZ-Si sample [17].
Method hkl RL, m nL, cm
3 RP, nm nP, cm
3
DCD
111
333
1.0
0.8
2.9108
1.8108
50
50
1.81011
3.21011
TCD 111 0.7 4.5108 50 1.01011
TABLE 5. Characteristics of circular dislocation loops in FZ-Si sample ob-
tained by fitting total TCD profile and reciprocal space map [39].
Data type hkl RL, m nL, cm
3
Profile 111 1.15 0.1 (3.5 1)108
Map 111 2.1 0.5 (7.3 3)107
DYNAMICAL THEORY OF TRIPLE-CRYSTAL X-RAY DIFFRACTOMETRY 131
relatively small in the considered reciprocal space region [28].
In the analysis first of all, the fact was established that it was impossi-
ble to describe any of the experimental reciprocal space maps having
nearly oval shapes by using only spherical oxygen precipitates because
the typical shape of the diffuse scattering intensity distribution for these
defects is the ‘double droplet’. Therefore, the attempts were undertaken
to fit jointly the diffraction profiles and rocking curves supposing the
simultaneous presence of both types of microdefects, namely, spherical
oxygen precipitates and circular interstitial dislocation loops. However,
the account for the presence of both type defects with single sizes does
Fig. 14. Measured and calculated TCD diffraction profile (markers and solid
line, respectively) versus analyser deviation angle at fixed deviation angle
of the investigated FZ-Si sample 55 arc sec, Si(111) reflection, CuK1-
radiation [39].
Fig. 15. Measured (left) and calculated (right) reciprocal space maps for the
FZ-Si sample, Si(111) reflection, CuK-radiation [39].
132 V. B. MOLODKIN, S. I. OLIKHOVSKII, E. G. LEN et al.
not provide sufficiently good fits of all the diffraction patterns.
a b
c d
e f
Fig. 16. Measured (a, c, e) and calculated (b, d, f) reciprocal space maps of the
as grown (No. 1–a, b) and electron-irradiated (No. 1A–c, d, and No. 1B–e,
f) Cz-Si samples; (333) reflection, CuK1-radiation. The intensities of neigh-
bouring equal intensity lines differ by a factor of 101/3, the quantities kx and kz
are given in units of the reciprocal lattice parameter d110
4 [48].
DYNAMICAL THEORY OF TRIPLE-CRYSTAL X-RAY DIFFRACTOMETRY 133
To obtain the acceptable fit, the assumption on the presence of dislo-
a B
c D
e f
Fig. 17. Transverse (a, c, e) and longitudinal (b, d, f) cross-sections of the
measured reciprocal space maps from Fig. 11 and their calculated profiles
(markers and thick solid line, respectively), the last consist of coherent and
diffuse components with contributions from small and large dislocation loops,
and oxygen precipitates (thin solid, dashed, dotted, and dot-dashed lines, re-
spectively) [48].
134 V. B. MOLODKIN, S. I. OLIKHOVSKII, E. G. LEN et al.
cation loops with two different sizes, namely, large and small ones, was
necessary to be made (see the defect characteristics in Table 6).
Thus, the simultaneous description of the rocking curves and recip-
rocal space maps measured from both as-grown and irradiated Cz-Si
samples is possible only if contributions from defects with at least
three effective radii are considered. On the other hand, the appropriate
shape of the equal intensity lines on all the maps can only be ensured by
the predominant contribution of diffuse scattering from dislocation
loops. The analysis of these maps demonstrated that the shape of equal
intensity lines is typical for the case of circular dislocation loops with 〈111〉 orientations of Burgers vectors [39]. The radii of these (large)
dislocation loops RL were estimated using the measured reciprocal
space maps and their cross sections (Figs. 16 and 17). For the irradiat-
ed samples, they turned out to be two fold less than for the as-grown
a b
c
Fig. 18. Experimental and theoretical rocking curves (markers and thick lines)
for the as-grown (a) and electron-irradiated Cz-Si samples Nos. 1A (b) and 1B
(c); (333) reflection of CuK1-radiation. Other lines are explained in caption of
Fig. 17. Central parts of the rocking curves are shown in the insets [48].
DYNAMICAL THEORY OF TRIPLE-CRYSTAL X-RAY DIFFRACTOMETRY 135
crystal. The concentration nL of these loops in the irradiated samples
proved to be significantly (by an order of magnitude and more) larger
as compared to the concentration in the as-grown sample (see Table 6).
The estimate for radii of large spherical oxygen precipitates
RP 1 m in all samples was obtained simultaneously with that for
large loops. Their concentration nP in the irradiated samples signifi-
cantly increased as compared with as-grown one.
Final values of the characteristics of large dislocation loops and ox-
ygen precipitates were obtained after introducing small dislocation
loops into the combined treatment. These loops with radii of several
nanometres give significant contributions at far tails of the rocking
curves for all studied samples but their contributions are negligible in
the TCD diffraction profiles (compare Figs. 17 and 18). As can be seen
from the Table 6, the concentrations of small dislocation loops in the
irradiated samples are approximately two fold smaller than in the as-
grown crystal.
The obtained results have been interpreted as follows. Under irradi-
ation with high energy electrons the defect structure in the bulk of the
silicon crystal is transformed due to the influence of two factors,
namely, the creation of radiation defects, first of all, vacancies and in-
terstitial silicon atoms, and the increase of crystal temperature (to ap-
proximately 1000C at the chosen irradiation parameters) which en-
hances diffusion processes. As can be seen from the results of diagnos-
tics (Table 6), both these factors have significant influence on the de-
composition processes of supersaturated solid solution of oxygen in
silicon and results in considerable changes of the growth defect charac-
teristics in both irradiated silicon samples as compared to the as-grown
crystal.
Thus, the obtained results demonstrate the possibility of increasing
the unambiguity of the quantitative characterization of complex de-
fect structures in imperfect single crystals owing to the joint treat-
ment of reciprocal space maps and rocking curves.
TABLE 6. Characteristics of dislocation loops (radius RL and concentration
nL) and oxygen precipitates (RP and nP) in the as-grown (No. 1) and electron-
irradiated (Nos. 1A and 1B) Cz-Si samples [48].
Sample No. RL, m nL, cm
3 RP, m nP, cm
3
1
0.002
0.15
1.151016
3.31011
1.0 7.5106
1A
0.003
0.065
5.01015
1.01013
1.0 1.25107
1B
0.003
0.07
5.01015
3.51012
1.0 9.0106
136 V. B. MOLODKIN, S. I. OLIKHOVSKII, E. G. LEN et al.
5. RESUME AND CONCLUSIONS
The basic principles of the dynamical theory of X-ray diffraction by
crystals with chaotically distributed microdefects are described. Based
on this theory, the analytical expressions for the coherent and diffuse
scattering intensities measured by TCD from crystals containing mi-
crodefects of several types have been derived. Peculiarities of TCD
measurements, which are connected with the presence of additional
contributions in reflection coefficients of all the crystals of X-ray opti-
cal scheme due to both strained subsurface layers and diffuse scatter-
ing from defects in these crystals, have been taken into account as well.
The developed theoretical model for the description of total TCD
profiles and reciprocal lattice maps has allowed for performing the
quantitative characterization of complicated defect structures in im-
perfect single crystals at qualitatively new level of uniqueness and re-
liability. Besides, the analysis of total diffraction profiles measured by
TCD with flat monochromator and analyser crystals has provided the
possibility of reliable establishment of the role of diffuse scattering
from the monochromator crystal in the formation of coherent peaks on
TCD profiles. In particular, the account for the influence of these fac-
tors has allowed for giving the correct explanation of the asymmetrical
behaviour of main and pseudo-peaks on TCD profiles.
It is necessary to emphasize the especially important role of the joint
processing reciprocal space maps and rocking curves for reliable quan-
titative determination of the microdefect characteristics with a wide
spread in their effective sizes. The analysis of reciprocal space maps is
the simplest way to unambiguously establish the dominant type of mi-
crodefects that make the main contribution to the measured differen-
tial diffuse scattering intensity distributions and to determine the ap-
proximate characteristics of these defects. At the same time, owing to
integration of the diffuse scattering intensity over the Ewald sphere,
rocking curves are more sensitive to small size defects as compared to
reciprocal space maps in which the contribution from small defects is
almost negligible.
The developed self-consistent dynamical theoretical model for the
description of total TCD diffraction profiles, reciprocal space maps,
and rocking curves has allowed for achieving the qualitatively new lev-
el of uniqueness and reliability of defect characterization by using
their joint treatment. In particular, the quantitative characterization
of the complicated microdefect structures created in various silicon
crystals grown by the Czochralski and floating-zone methods have been
determined by analytical treatment of the measured TCD profiles,
rocking curves, and reciprocal space maps.
This research was supported by Project No. 28/15-Н, N.A.S. of
Ukraine.
DYNAMICAL THEORY OF TRIPLE-CRYSTAL X-RAY DIFFRACTOMETRY 137
REFERENCES
1. P. Eisenberger, N. G. Alexanropoulos, and P. M. Platzmann, Phys. Rev. Lett.,
28, No. 23: 1519 (1972).
2. B. C. Larson and W. G. Schmatz, Phys. Rev. B, 10, No. 6: 2307 (1974).
3. A. Iida and K. Kohra, phys. status solidi (a), 51, No. 2: 533 (1979).
4. A. Iida, phys. status solidi (a), 54, No. 2: 701 (1979).
5. A. M. Afanas’ev, M. V. Koval’chuk, E. F. Lobanovich, R. M. Imamov,
P. A. Aleksandrov, and M. K. Melkonyan, Kristallografiya, 26, Iss. 1: 28
(1981) (in Russian).
6. E. K. Kov’ev, V. V. Ratnikov, and L. M. Sorokin, Fizika Tverdogo Tela, 23,
No. 6: 1626 (1981) (in Russian).
7. P. Zaumseil and U. Winter, phys. status solidi (a), 70, No. 2: 497 (1982).
8. P. Zaumseil and U. Winter, phys. status solidi (a), 73, No. 2: 455 (1982).
9. А. А. Lomov, P. Zaumseil, and U. Winter, Acta Crystallogr. A, 41, No. 3: 223
(1985).
10. R. N. Kyutt, A. A. Sitnikova, and L. M. Sorokin, Fizika Tverdogo Tela, 27,
No. 3: 673 (1985) (in Russian).
11. V. Holý and J. Kubĕna, phys. status solidi (b), 170, Iss. 1: 9 (1992).
12. L. A. Charniy, A. N. Morozov, V. T. Bublik, K. D. Scherbachev,
I. V. Stepantsova, and V. M. Kaganer, J. Cryst. Growth, 118: 163 (1992).
13. V. Holý, U. Pietch, and T. Baumbach, High-Resolution X-Ray Scattering from
Thin Films and Multilayers (Berlin—Heideiberg: Springer: 1998).
14. P. F. Fewster, X-Ray Scattering from Semiconductors (London: Imperial
College Press: 2000).
15. V. B. Molodkin, S. I. Olikhovskii, M. E. Osinovskii, V. V. Kochelab,
A. Yu. Kazimirov, M. V. Koval’chuk, and F. N. Chukhovskiy, Metallofizika, 6,
No. 3: 7 (1984) (in Russian).
16. V. B. Molodkin, V. V. Nemoshkalenko, S. I. Olikhovskii, E. N. Kislovskii,
O. V. Reshetnyk, T. P. Vladimirova, V. P. Krivitsky, V. F. Machulin,
I. V. Prokopenko, G. E. Ice, and B. C. Larson, Metallofiz. Noveishie Tekhnol.,
20, No. 11: 29 (1998).
17. V. B. Molodkin, M. Ando, E. N. Kislovskii, S. I. Olikhovskii, T. P. Vladimirova,
O. V. Reshetnyk, E. G. Len, E. A. Evgrafova, and E. V. Pervak, Metallofiz.
Noveishie Tekhnol., 24, No. 4: 541 (2002).
18. S. I. Olikhovskii, V. B. Molodkin, E. M. Kislovskii, O. V. Reshetnyk,
T. P. Vladimirova, E. G. Len, G. E. Ice, R. I. Barabash, R. Köhler, and
D. O. Grigoriev, Metallofiz. Noveishie Tekhnol., 27, No. 7: 949 (2005).
19. S. I. Olikhovskii, V. B. Molodkin, E. M. Kislovskii, O. V. Reshetnyk,
T. P. Vladimirova, G. E. Ice, R. I. Barabash, R. Köhler, and D. O. Grigoriev,
Metallofiz. Noveishie Tekhnol., 27, No. 9: 1251 (2005).
20. V. B. Molodkin, S. I. Olikhovskii, and M. E. Osinovskii, Metallofizika, 5, No. 1:
3 (1983) (in Russian).
21. V. B. Molodkin, S. I. Olikhovskii, E. N. Kislovskii, E. G. Len, and E. V. Pervak,
phys. status solidi (b), 227: 429 (2001).
22. S. I. Olikhovskii, V. B. Molodkin, E. N. Kislovskii, E. G. Len, and E. V. Pervak,
phys. status solidi (b), 231: 199 (2002).
23. V. S. Wang and R. J. Matyi, J. Appl. Phys., 72, Iss. 11: 5158 (1992).
24. M. A. Krivoglaz, X-Ray and Neutron Scattering in Nonideal Crystals (Berlin:
138 V. B. MOLODKIN, S. I. OLIKHOVSKII, E. G. LEN et al.
Springer: 1996).
25. P. H. Dederichs, Phys. Rev. B, 4: 1041 (1971).
26. R. Bloch, D. Bahr, J. Olde, L. Brügemann, and W. Press, Phys. Rev. B, 42: 5093
(1990).
27. T. P. Vladimirova, R. F. Seredenko, V. B. Molodkin, S. I. Olikhovskii, and
E. N. Kislovskii, Metallofiz. Noveishie Tekhnol., 29, No. 6: 711 (2007)
(in Ukrainian).
28. V. B. Molodkin, S. I. Olikhovskii, E. N. Kislovskii, T. P. Vladimirova,
E. S. Skakunova, R. F. Seredenko, and B. V. Sheludchenko, Phys. Rev. B, 78,
No. 22: 224109 (2008).
29. V. P. Klad’ko, L. I. Datsenko, J. Bak-Misiuk, V. F. Machulin, I. V. Prokopenko,
V. B. Molodkin, S. I. Olikhovskii, and Z. V. Maksimenko, J. Phys. D: Applied
Physics, 34, No. 10А: A87 (2001).
30. E. N. Kislovskii, O. V. Reshetnyk, T. P. Vladimirova, V. B. Molodkin,
S. I. Olikhovskii, B. V. Sheludchenko, R. F. Seredenko, and E. S. Skakunova,
Metallofiz. Noveishie Tekhnol., 29, No. 5: 701 (2007) (in Ukrainian).
31. A. P. Shpak, V. B. Molodkin, S. I. Olikhovskii, Ye. M. Kislovskii,
O. V. Reshetnyk, T. P. Vladimirova, E. G. Len, A. I. Nizkova, V. M. Venger,
and S. V. Dmitriev, phys. status solidi (b), 204, No. 8: 2651 (2007).
32. V. B. Molodkin, S. I. Olikhovskii, E. S. Skakunova, E. G. Len, E. N. Kislovskii,
O. V. Reshetnyk, T. P. Vladimirova, V. V. Lizunov, L. N. Skapa,
S. V. Lizunova, E. V. Fuzik, N. G. Tolmachev, B. K. Ostafiychuk,
V. M. Pylypiv, and O. Z. Garpul’. Metallofiz. Noveishie Tekhnol., 37, No. 8:
1017 (2015).
33. E. S. Skakunova, Metallofiz. Noveishie Tekhnol., 37, No. 4: 555 (2015).
34. J. Chikawa, Y. Asaeda, and I. Fujimoto, J. Appl. Phys., 41: 1922 (1970).
35. V. Holý, and J. Kubĕna, phys. status solidi (b), 141, No. 1: 35 (1987).
36. V. V. Nemoshkalenko, V. B. Molodkin, E. N. Kislovskii, S. I. Olikhovskii,
T. A. Grishchenko, M. T. Kogut, and E. V. Pervak, Metallofiz. Noveishie
Tekhnol., 22, No. 2: 42 (2000) (in Russian).
37. V. G. Bar’yahtar, V. V. Nemoshkalenko, V. B. Molodkin, A. P. Shpak,
D. Chikava, K. Kohra, R. N. Kyutt, S. I. Olikhovskii, A. I. Nizkova,
T. A. Grishchenko, E. N. Kislovskii, M. T. Kogut, and M. V. Koval’chuk,
Metallofizika, 15, No. 12: 72 (1993) (in Russian).
38. C. Bergmann, A. Gröschel, J. Will, and A. Magerl, J. Appl. Phys., 118: 015707
(2015).
39. V. B. Molodkin, S. I. Olikhovskii, E. G. Len, E. N. Kislovskii, V. P. Kladko,
O. V. Reshetnyk, T. P. Vladimirova, and B. V. Sheludchenko, phys. status solidi
(a), 206, No. 8: 1761 (2009).
40. A. Borgesi, B. Pivac, A. Sasella, and A. Stella, J. Appl. Phys., 77: 4169 (1995).
41. V. B. Molodkin, S. I. Olikhovskii, Ye. M. Kyslovskyy, E. G. Len,
O. V. Reshetnyk, T. P. Vladimirova, V. V. Lizunov, and S. V. Lizunova,
Semiconductor Physics, Quantum Electronics and Optoelectronics, 13, No. 4:
353 (2010).
42. A. P. Shpak, V. B. Molodkin, S. I. Olikhovskii, Ye. M. Kyslovskyy,
O. V. Reshetnyk, T. P. Vladimirova, E. G. Len, A. I. Nizkova, V. M. Venger,
and S. V. Dmitriev, phys. status solidi (a), 204, No. 8: 2651 (2007).
43. P. M. Petroff and A. J. R. de Kock, J. Cryst. Growth, 30: 117 (1975).
44. H. Föll, V. Gösele, and B. O. Kolbesen, J. Cryst. Growth, 52, No. 2: 907 (1981).
45. P. J. Roksnoer and M. M. B. van der Boom, J. Cryst. Growth, 53, No. 3: 563
DYNAMICAL THEORY OF TRIPLE-CRYSTAL X-RAY DIFFRACTOMETRY 139
(1981).
46. P. J. Roksnoer, J. Cryst. Growth, 68, No. 2: 596 (1984).
47. V. B. Molodkin, S. I. Olikhovskii, Ye. M. Kyslovskyy, T. P. Vladimirova,
E. V. Kochelab, O. V. Reshetnyk, V. V. Dovganyuk, І. М. Fodchuk,
T. V. Lytvynchuk, V. P. Klad’ko, and Z. Świątek, phys. status solidi (a), 208,
No. 11: 2552 (2011).
48. E. N. Kislovskii, V. B. Molodkin, S. I. Olikhovskii, E. G. Len,
B. V. Sheludchenko, S. V. Lizunova, T. P. Vladimirova, E. V. Kochelab,
O. V. Reshetnik, V. V. Dovganyuk, I. M. Fodchuk, T. V. Litvinchuk, and
V. P. Klad’ko, J. Surface Investigation. X-Ray, Synchrotron and Neutron
Techniques, 7, No. 3: 523 (2013).
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/HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. Stvoreni PDF dokumenti mogu se otvoriti Acrobat i Adobe Reader 5.0 i kasnijim verzijama.)
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/NLD (Gebruik deze instellingen om Adobe PDF-documenten te maken die zijn geoptimaliseerd voor prepress-afdrukken van hoge kwaliteit. De gemaakte PDF-documenten kunnen worden geopend met Acrobat en Adobe Reader 5.0 en hoger.)
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/UKR <FEFF04120438043a043e0440043804410442043e043204430439044204350020044604560020043f043004400430043c043504420440043800200434043b044f0020044104420432043e04400435043d043d044f00200434043e043a0443043c0435043d044204560432002000410064006f006200650020005000440046002c0020044f043a04560020043d04300439043a04400430044904350020043f045604340445043e0434044f0442044c00200434043b044f0020043204380441043e043a043e044f043a04560441043d043e0433043e0020043f0435044004350434043404400443043a043e0432043e0433043e0020043404400443043a0443002e00200020042104420432043e04400435043d045600200434043e043a0443043c0435043d0442043800200050004400460020043c043e0436043d04300020043204560434043a0440043804420438002004430020004100630072006f006200610074002004420430002000410064006f00620065002000520065006100640065007200200035002e0030002004300431043e0020043f04560437043d04560448043e04570020043204350440044104560457002e>
/ENU (Use these settings to create Adobe PDF documents best suited for high-quality prepress printing. Created PDF documents can be opened with Acrobat and Adobe Reader 5.0 and later.)
>>
/Namespace [
(Adobe)
(Common)
(1.0)
]
/OtherNamespaces [
<<
/AsReaderSpreads false
/CropImagesToFrames true
/ErrorControl /WarnAndContinue
/FlattenerIgnoreSpreadOverrides false
/IncludeGuidesGrids false
/IncludeNonPrinting false
/IncludeSlug false
/Namespace [
(Adobe)
(InDesign)
(4.0)
]
/OmitPlacedBitmaps false
/OmitPlacedEPS false
/OmitPlacedPDF false
/SimulateOverprint /Legacy
>>
<<
/AddBleedMarks false
/AddColorBars false
/AddCropMarks false
/AddPageInfo false
/AddRegMarks false
/ConvertColors /ConvertToCMYK
/DestinationProfileName ()
/DestinationProfileSelector /DocumentCMYK
/Downsample16BitImages true
/FlattenerPreset <<
/PresetSelector /MediumResolution
>>
/FormElements false
/GenerateStructure false
/IncludeBookmarks false
/IncludeHyperlinks false
/IncludeInteractive false
/IncludeLayers false
/IncludeProfiles false
/MultimediaHandling /UseObjectSettings
/Namespace [
(Adobe)
(CreativeSuite)
(2.0)
]
/PDFXOutputIntentProfileSelector /DocumentCMYK
/PreserveEditing true
/UntaggedCMYKHandling /LeaveUntagged
/UntaggedRGBHandling /UseDocumentProfile
/UseDocumentBleed false
>>
]
>> setdistillerparams
<<
/HWResolution [2400 2400]
/PageSize [612.000 792.000]
>> setpagedevice
|