Calculated images of dislocations in crystals on section topograms
By means of numerical solution of the Takagi equations, modeling of X-ray
 topographic images of deformation fields of the dislocation loops and dislocation of
 different types. Diffraction images created by dislocations and dislocation loops of
 different size and spatial lo...
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| Опубліковано в: : | Semiconductor Physics Quantum Electronics & Optoelectronics |
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| Дата: | 2010 |
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2010
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| Цитувати: | Calculated images of dislocations in crystals on section topograms / S.M. Novіkov, І.M. Fodchuk, D.G. Fedortsov, A.Ya. Struk // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2010. — Т. 13, № 3. — С. 268-272. — Бібліогр.: 19 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1860254609676697600 |
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| author | Novіkov, S.M. Fodchuk, І.M. Fedortsov, D.G. Struk, A.Ya. |
| author_facet | Novіkov, S.M. Fodchuk, І.M. Fedortsov, D.G. Struk, A.Ya. |
| citation_txt | Calculated images of dislocations in crystals on section topograms / S.M. Novіkov, І.M. Fodchuk, D.G. Fedortsov, A.Ya. Struk // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2010. — Т. 13, № 3. — С. 268-272. — Бібліогр.: 19 назв. — англ. |
| collection | DSpace DC |
| container_title | Semiconductor Physics Quantum Electronics & Optoelectronics |
| description | By means of numerical solution of the Takagi equations, modeling of X-ray
topographic images of deformation fields of the dislocation loops and dislocation of
different types. Diffraction images created by dislocations and dislocation loops of
different size and spatial location are complicated and versatile in their thin structure.
|
| first_indexed | 2025-12-07T18:47:44Z |
| format | Article |
| fulltext |
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 1. P. 268-272.
PACS 61.05.cp, 61.10.Nz, 61.43.Bn, 61.72.Lk, 61,85. + Р, 61.80.Cb
Calculated images of dislocations in crystals on section topograms
S.M. Novіkov, І.M. Fodchuk, D.G. Fedortsov, A.Ya. Struk
Yuriy Fedkovych Chernivtsi National University,2 Kotsyubinskiy Str, 58012 Chernivtsi, Ukraine
Abstract. By means of numerical solution of the Takagi equations, modeling of X-ray
topographic images of deformation fields of the dislocation loops and dislocation of
different types. Diffraction images created by dislocations and dislocation loops of
different size and spatial location are complicated and versatile in their thin structure.
Keywords: Х-ray diffraction, topography, dislocation, dislocation loop, numerical
simulation, image formation mechanisms.
Manuscript received 14.01.10; accepted for publication 08.07.10; published online 30.09.10.
1. Introduction
X-ray topography methods have gained wide acceptance
for study of various defects in solids [1-3]. However,
there is a problem related to interpretation of observed
defect images (inverse problem solution). Various
defects, being located in different positions with respect
to scattering plane, can create diffraction images that
practically do not differ.
© 2010, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
With the advent of a variety of methods for digital
processing of images in the experimental topograms [4],
interest was rekindled to refinement of thin structure of
topographic images of dislocations and their complexes
[5,6].
At the same time, using well-proven methods for
numerical calculation of the Takagi equations [7-9] and
achievements of elaticity theory in description of
deformation fields created by defects, one can improve
considerably the diagnostic possibilities of X-ray
topography.
In this paper, by means of numerical solution of the
Takagi equations, modeling of X-ray topographic
images of deformation fields of the dislocation loops and
edge dislocations is made depending on their spatial
location in the scattering plane with respect to diffraction
vector h
r
.
2. Basic relationships
If a crystal has inhomogeneities in the form of local or
bulk-distributed elastic deformations, then Х-ray wave
field propagating in the crystal can be described by a
system of the Takagi equations [1,7]. In the majority of
cases this system of equations in partial derivatives is
solved by numerical methods of finite differences [1,8].
For slowly varying amplitudes D0 and Dh in a crystal
that has defects with a weakly varying distortion field
(∂Uk/∂xi<<1), the combined Takagi equation is written
as a relationship [7]:
( ) ( )
( ) ( ) ( ) ( )[ ]
⎪
⎪
⎩
⎪
⎪
⎨
⎧
′−′−=
∂
′∂
′−=
∂
′∂
rDrrDCki
s
rD
rDkCi
s
rD
hh
h
h
hh
rrr
r
r
r
αχπ
χπ
20
0
0
, (1)
where χ0, χh are the Fourier components of
polarizability.
( )rvThe local misorientation function α is
proportional to projection of gradient from scalar
product of diffraction vector h
r
and displacement vector
U
r
to direction of diffracted rays propagation : hsr
( )( ) Uhsr h
rr rrv
⋅∇⋅~α , (2)
For an isotropic medium in the case of placement
of the dislocation line along the Oz axis expression for
the dislocation is given in [16]:
,][
1
1
ln][
1
21
4
)(
2 ⎟
⎟
⎠
⎞⋅′×′
−
+
⎜
⎝
⎛ +′×′
−
−
−′−
Ω
=
ρ
ρ
ρ
ν
ρ
ν
ν
π
ρ
r
rrr
rrrrr
bt
btbU
(3)
where Ω – the solid angle of the observation point,
which limits the dislocation segment is selected for
modeling, ρ
r
– radius vector, ν - Poisson's ratio.
Note that the combined equation (1) in the case of
( ( ) 0χα >rv ) is easily transformed into a kinematic one.
Areas with different signs of ( )rvα functions differently
268
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 1. P. 268-272.
deflect strongly and weakly absorbing Bloch waves [1].
From the appearance of intensity rosette R(x,y) one can
reproduce local misorientation rosette of ( )rvα defect and
determine the sign of the Burgers vector [1,10-12].
3. Simulation results
© 2010, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
3.1. Dislocation loops
Typical for silicon are the simp whose lest dislocations
Burgers vector equals b
r
=a/2 ><110 [15], i.e.: screw
dislocation; 60° dislocation; ed ocation located in
slip plane {100}; and dislocation loops. Representation
of a dislocation loop as a hexagon, whose segments are
oriented along the <110> directions (60° dislocations),
and Burgers vectors b
ge disl
r
=a/2<110> of each dislocation
segment lie in the loop plane, is apparently the most
probable model presentation of sliding dislocation loop
[13,14].
However, the simplest model of a dislocation loop
(DL) can be constructed in the form of a spherically
symmetric inclusions of different sizes (Rp) and powes
(Сv). Geometrically such loop can be represented as a
circle (Fig. 1) [13,17].
Modeling of X-ray diffraction images of sliding DL
(Fig.2) was made for μt≈1 and μt>10 (μ is photoelectric
absorption coefficient), crystallographic orientation of
entrance surface (111), (440) reflection of MoKα- and
CuKα-radiation in the diffraction geometry according to
Laue. Under diffraction conditions μt>10 the case of
anomalous transmission of X-rays is realized (Fig.2b).
Fig. 3 shows the calculation of images was
performed for prismatic loop, located at an angle to the
crystal surface (110), where b
r
=a/3[1 1 1], b
r
=a/3[ 111],
which is typical of Frank dislo [1 ].
cation loop 4
Fig. 1.. Section topograms of dislocation loops. Si, (440
Fig. 2. R(x,y) of dislocation loop at the distance of 300 μm t
ig. 3. R(x,y) of prismatic dislocation loops (D=10 μm
)
MoKα, μt=3. Сv=10-16, Rp=50 μm. а) Depth of occurrence is
1800 μm, б) arbitrary (random) distribution in the volume.
o
exit surface obtained by means of (440) MoKα-, μt=1,2 (а) and
CuKα-radiation μt=12 (b).
F ):
z=1000 μm (а,c), z=1800 μm (b,d), μt=3. b
r
=1/3 [1 1 1] (а,b)
and b
r
=1/3 [ 111] (c,d). Occurrence plane ( 111).
If the crystal thickness, where the DL is placed, is
large enough then only one weakly absorbing Bloch
wave comes to it. In strongly distorted defect region
( hr χ>>α )(r ) a weakly absorbing Bloch wave is
su erband scattering [1]. As a result, a new
wave filed is excited that is composed of two types of
Bloch waves. At point of existence of both Bloch waves
their interference takes place and as a result extinction
intensity modulations can be observed (Fig.3).
Depending on the plane of dislocation loop
bject to int
occurrence and its type, diffraction image can be
changed not only the distribution of black-white contrast
to the opposite (in Fig.3 for the (111) and ( 111) planes,
respectively), but it also become considerabl weaker or
stronger. On the whole, except for separate details, this
variety of DL diffraction images depends on the
contribution of scalar products )( bh
y
rr
of individual
dislocation loop segments to total local misorientation
function [1,17,19], whereas DL occurrence plane and its
orientation with respect to diffraction plane are "hidden"
in these products.
3.2. Inclined to the surface 72° and 60° dislocations
inclined to the entrance crystal surface In the case of
dislocations (Fig.4, 5) intensity modulations, which cau-
sed by overlapping of normal and abnormal refracted
Bloch waves are observed on topograms. For all
possible cases of placing the input and output points of
dislocations through the Bormann fan, the section
images of dislocation, which parameters are defined by
the spatial orientation of the Burgers vector with respect
to the diffraction vector, are obtained. Crucial for the
formation of a clear contrast for 72° dislocation in Fig.4
is a cross section of the Bormann fan in the direction of
incidence X-ray 0s
r
.
As the dista e fnc rom the dislocation location to the
exit surface is increased, X-ray diffraction image of
dislocation expands due to divergence of rays within
Bormann fan. As a result the interferention interaction of
newly created and existing wave fields is weakened.
y
–200 0 200 x, μm
10
0
a)
0
с)
–1
00
d)b)
y
–400 x, μm–2
00
20
0
0
0
a) b)
–3
00
y
x, μm 0 –300
30
0
0
a)
–3
00
y
x, μm0 –700
30
0
0
b)
269
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 1. P. 268-272.
© 2010, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
ion
(dark
of interbranch
scatte
, 5)
is fo
of contrast change and the formation
of di
Fig. 4. Section topograms of inclined 72° dislocations: Si,
Fig.5. Section topograms of 60° dislocations, whih are parallel
to the exit surface. Si, MoKα, (220): μt=1 (a-c), μt=6 (d-f). T e
ar to the sample surface and Burgers vector
ns that are perpendicular to
In the case of μt=1, the positive contrast reg
relatively background) around dislocations begins
where there is a noticeable interbranch scattering of
Bloch X-wave. Depending on the geometry of location
of dislocation and its orientation characteristics
interference fringes can take diferent forms. For the case
of dislocation lines inclined orientation relative to the
diffraction vector in the plane of scattering (Fig. 4),
interference pattern on appearance reminds sectional
image of the inclined stacking fault [1]. A somewhat
different situation is observed for 60° dislocation in
fig.5. Its image essentially depends on where the
dislocation is located - near the entrance, in the center or
far from the exit surface of the crystal.
Overall, the picture of display
ring essentially depends on the type of defect. In
particular, for thick crystal on fig.4c under the formation of
dislocation image dynamic part of the image dominates,
while its direct (kinematic) image is hardly noticeable.
Direct image of the sectional topographs (Fig.4
rmed by the crystal with a few microns thickness,
while the misorientation value of reflecting planes in this
area is about twice bigger than the width of the reflection
curve of a perfect crystal. This image is formed mainly
by the kinematic scattering with indirect influence of
dynamic effects.
The basic law
ffraction images in the case of projection and
section topography for thin (μt=1) and thick(μt=10)
crystals are the same [1]. However, the thin structure of
image is very different: the intensity oscillations are less
contrast.
MoKα (220), μt=1 (a,b), μt=10 (c,d). Output point of
dislocations to the exit surface is at the center (а,с) and closely
to the edge of the Borrmann fan (b,d).
h
depth of location : а) z=240, b) 360, c) 600, d) 500, e) 2000, f)
3000 μm.
3.3. Edge dislocation with a dislocation line
perpendicul
parallel to diffraction vector
The features of formation of images in section
topographs of edge dislocatio
the surface of the crystal are investigated. This
orientation of dislocations is interesting that the function
)(r
r
α does not depend on the thickness of the crystal and
the formation of X-wave diffraction image is due to
fic effects of channeling [3, 18].speci
Analysis of the calculated and experimental
topograms (Fig. 6) reveals some features of formation of
the thickness oscillations (Fig. 7) and topographic
images of elastic deformation fields in a strongly
distorted crystal regions. In general, the distribution of
intensities of Rh(x,y) in the large deformation areas
repeats a function of xyx ∂α∂ ),( (Fig. 6a,c). From the
analysis of topograms on Fig. 6, it follows that scattering
areas with large intens served in places where
changing of values of the strain is non-significant along
the diffraction vector (the central region of topogram).
ity are ob
However, in the crystal region where deformations
are changing rapidly, "specific" effects of channeling X-
waves along the reflecting planes are completely
inhibited.
On the distributions R (x,z) h (Fig. 7a-c) as distance
from the gliding plane the asymmetrical inhibition of
extinction oscillations of intensity in the base of Bormann
fan and intensity concentration in the center are occur. In
fact a quasi-periodic standing wave with maсroperiod
order of 7-10 Λσ which extends as through the waveguide
in large values of function α(x,y) is arising.
The amplitude of new intensity oscillations at the
thickness distributions in the scattering plane in the areas
of tensile strain under dislocation extra plane
dramatically increases when approaching to the line of
dislocation (Fig.7a).
x, μm –1000 x, μm –7500
0
–
50
y
0
–
30
0
y
a) d)
b) e)
с) f)
–1000 x,μm
y
0 1000
d) с)
–100 x,μm
y
0 100
a) b)
270
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 1. P. 268-272.
© 2010, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
Fig. 6. Section topograms of edge dislocations: Sі, МоКα1-
radiation: а), c) ( 022
– calcul
); b), d) (004). (а, b) – experimental
image [5], (c, d) ated image.
Fig. 7. Distributions , where indent from the line of
dislocation parallell ide plane is equal to y0 =– 1 (а, d,
e), y0=–5 (b), y0=–10 МоКα1-radiation. (
),( zxRh
y to the gl
μm (c) for 2 2 0) (а-c),
(4 4 0) (d), (004) (e).
Period of new intensity oscillations exc eds the
ex nction period (5-10 times) and non-linearly
increases with decr
e
ti Λ
easing of value
),0( yx =α =
yk
bh 1
)1(2
)21())(sin(
⋅⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
ν−
ν+
π
θ
− r
rr
. (4)
Fig. 8. Section topograms of diffraction contrast from screw
dislocation μt∼1, MoКα (220) respective diffraction sections
(a) and planes (у=10 μm) (b).
tion of the phase of newly
interfering fields in the strongly deformed crystal region
is eq
fraction image for screw dislocations, whose
axis is oriented parallel to the diffraction vector is given.
dislocations is equal to half the
This is due to increase of the effective thickness of
the crystal, where accumula
ual to 2π. The same pattern is observed in the area
of compressive strain (above the glide plane). At the
same time period and amplitude of mode oscillations are
significantly lower. On the topographs in tension areas
under the extra plane the dark (positive to the
background) contrast is formed.
Similar regularities of intensity propagation along
the dislocation line are inherent and for other reflections
(fig.7d,e).
3.4. Screw dislocation
In Fig.8 dif
Depth of occurrence of
thickness of the crystal μt=1. This case of the orientation
of dislocation is also "special", since the displacement of
the reflecting planes dur do not depend on coordinate x.
Features and mechanisms of formation of
diffraction images for described case of screw
dislocation orientation re well studied in [1,8]. a
n
hs depending on crystalographic and
was explaned by the combined
In [19] the features of the impact of acoustic strains
on formation as X-ray topographic screw dislocation
contrast in general, and the formation extinction
oscillations in particular are investigated. It is noted that
with a small acoustic deformations significant changes
in image of screw dislocations occur near the output
point of kinematic image on the surface of the crystal.
4. Conclusions
. The variety of X-ray images of dislocation loops o1
section topograp
patial locations
contribution of products ( )bh
rr
of each of the loop
segments to total local misorientation function ( )r
r
α .
x, μm –100 0
0
–5
0
50
y
а)
z
15
0
37
5
–100 0 x,μm
b) a) b)
х,μm 0 –100 100 100 –200
–1
00
0
y
10
0 d)с)
d) e)
88
x –200
а)
40
0
80
0
12
00
16
0
z
c) b)
200 200
271
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2010. V. 13, N 1. P. 268-272.
© 2010, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
rma
s
ffects of channeling and complete external
Dynamical Theory of X-Ray Diffraction.
University Press, New York, 2001.
4. , V.A. Tkal’, A.O. Okunev, Yu.A.
5.
6.
2. The features of fo tion of the diffraction
images on section and projection topograms of crew,
edge, 60° and 72° dislocations depending thon eir
location in Bormann fan in thin and thick crystal were
established.
3. In the case of a special (perpendicular to
surface) location of edge dislocations in the Bormann
fan specific e
reflection of X-waves were observed. In strongly
distorted crystal region near the dislocation line the
spatial quasi-periodic wave field with macroperiod in
several extinction distances Λσ which extends as through
the waveguide in large values of function local
misorientations α(x,y) is arising.
References
1. A. Authier,
Oxford
2. D.K. Bowen, B.K. Tanner, High Resolution X-Ray
Diffractometry and Topography. Taylor Francis,
London, 1998.
3. E.V. Suvorov, I.L. Shulpina, X-Ray Optics of Defect
Crystals // Surface No.7, pp. 3-32 (2001) (in Russian).
L.N. Danil’chuk
Drozdov, Digital processing of X-ray topographic
and polarization-optical images of single crystal
structure defects. Y. Mudryi Novgorod State
University, Velikyi Novgorod, 2004 (in Russian).
E.V. Suvorov, І.А. Smirnova, E.V. Shulakov,
Formation of the image of an edge dislocation in an
absorbing crystal // FTT 49(6), pp. 1050-1057
(2007) (in Russian).
J. Hartwig, Hierarchy of dynamical theories of x-
ray diffraction for deformed and perfect crystals //
J. Applied Physics. 34(10A), pp. A70-A77 (2001).
7. S. Takagi, Dynamical theory of difraction
applicable to crystals with any Kind of small
distortion // Acta Cryst. 15, pp.1311-1312 (1962).
8. Y. Epelboin, Simulatіon of X-ray topografs //
Material Science and Engineering 73, pp.1-43
(1985).
9. Y.Epelboin, A varying-step algorithm for the
numerical integration of Takagi-Taupin equations //
Acta Cryst. A37, pp.132-133 (1981).
10. V.L. Indenbom, V.M. Kaganer, The formation of
plane-wave X-ray images of microdefects // Phys.
status solidi A 87, pp. 253-265 (1985).
11. L.N.Danil’chuk, Рентгеновское наблюдение
полей деформаций вокруг краевых дислокаций
в монокристаллах кремния // FTT 11, pp. 3085-
3091 (1969) (in Russian).
12. L.N.Danil’chuk, Ju.А.Drosdov, А.О.Оkunev et. al.,
X-ray topography of single crystal structure defects
in semiconductors on the basis of the Borrmann
effect (review) // Factory Lab. Diagnostics of
Materials 68(11), pp. 24-33 (2002) (in Russian).
13. C. Teodosiu, Elastic Models of Crystal Defects.
Mir, Moscow, 1985 (in Russian).
14. J.P.Hirth, J. Lothe, Theory of Dislocations. Eds.:
Atomizdat, Moscow, 1972 (in Russian).
15. R.G.Rodes, Imperfections and active centres in
semiconductors. Metallurgiya, Moscow, 1968 (in
Russian).
16. R. De Wit, Some Relations for Straight Dislo-
cations // Phys. Stat. Sol. 20, pp.567-573 (1967).
17. V.B. Gevyk, S.N. Novikov, D.G. Fedortsov,
I.M. Fodchuk, Simulation of X-ray diffraction
images of dislocation loops in crystals // Met. Phys.
Adv. Tech. 27(9), pp.1237-1250 (2005) (in Russian).
18. A. Authier, F. Balibar, Création de Nouveaux
Champs d’Ondes Dus à la Présence d’un Objet
Diffractant à I’Intérieur d’un Cristal Parfait. II. Cas
d’un Defaut Isole // Acta Cryst. A26, pp.647-654
(1970).
19. S. Novіkov, І. Fodchuk, D. Fedortsov, A. Struk, X-
ray section images of dislocations and dislocation
barriers in Si // Phys.stat.sol.(a) 206(8), pp. 1820-
1824 (2009).
272
|
| id | nasplib_isofts_kiev_ua-123456789-118401 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1560-8034 |
| language | English |
| last_indexed | 2025-12-07T18:47:44Z |
| publishDate | 2010 |
| publisher | Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| record_format | dspace |
| spelling | Novіkov, S.M. Fodchuk, І.M. Fedortsov, D.G. Struk, A.Ya. 2017-05-30T06:55:45Z 2017-05-30T06:55:45Z 2010 Calculated images of dislocations in crystals on section topograms / S.M. Novіkov, І.M. Fodchuk, D.G. Fedortsov, A.Ya. Struk // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2010. — Т. 13, № 3. — С. 268-272. — Бібліогр.: 19 назв. — англ. 1560-8034 PACS 61.05.cp, 61.10.Nz, 61.43.Bn, 61.72.Lk, 61,85. + Р, 61.80.Cb https://nasplib.isofts.kiev.ua/handle/123456789/118401 By means of numerical solution of the Takagi equations, modeling of X-ray
 topographic images of deformation fields of the dislocation loops and dislocation of
 different types. Diffraction images created by dislocations and dislocation loops of
 different size and spatial location are complicated and versatile in their thin structure. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Calculated images of dislocations in crystals on section topograms Article published earlier |
| spellingShingle | Calculated images of dislocations in crystals on section topograms Novіkov, S.M. Fodchuk, І.M. Fedortsov, D.G. Struk, A.Ya. |
| title | Calculated images of dislocations in crystals on section topograms |
| title_full | Calculated images of dislocations in crystals on section topograms |
| title_fullStr | Calculated images of dislocations in crystals on section topograms |
| title_full_unstemmed | Calculated images of dislocations in crystals on section topograms |
| title_short | Calculated images of dislocations in crystals on section topograms |
| title_sort | calculated images of dislocations in crystals on section topograms |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/118401 |
| work_keys_str_mv | AT novíkovsm calculatedimagesofdislocationsincrystalsonsectiontopograms AT fodchukím calculatedimagesofdislocationsincrystalsonsectiontopograms AT fedortsovdg calculatedimagesofdislocationsincrystalsonsectiontopograms AT strukaya calculatedimagesofdislocationsincrystalsonsectiontopograms |