Modeling of X-ray rocking curves for layers after two-stage ion-implantation
In this work, we consider the approach for simulation of X-ray rocking curves inherent to InSb(111) crystals implanted with Be⁺ ions with various energies and doses. The method is based on the semi-kinematical theory of X-ray diffraction in the case of Bragg geometry. A fitting procedure that relies...
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Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України
2017
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| Cite this: | Modeling of X-ray rocking curves for layers after two-stage ion-implantation / O.I. Liubchenko, V.P. Kladko, O.Yo. Gudymenko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2017. — Т. 20, № 3. — С. 355-361. — Бібліогр.: 30 назв. — англ. |
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| author | Liubchenko, O.I. Kladko, V.P. Gudymenko, O.Yo. |
| author_facet | Liubchenko, O.I. Kladko, V.P. Gudymenko, O.Yo. |
| citation_txt | Modeling of X-ray rocking curves for layers after two-stage ion-implantation / O.I. Liubchenko, V.P. Kladko, O.Yo. Gudymenko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2017. — Т. 20, № 3. — С. 355-361. — Бібліогр.: 30 назв. — англ. |
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| description | In this work, we consider the approach for simulation of X-ray rocking curves inherent to InSb(111) crystals implanted with Be⁺ ions with various energies and doses. The method is based on the semi-kinematical theory of X-ray diffraction in the case of Bragg geometry. A fitting procedure that relies on the Hooke–Jeeves direct search algorithm was developed to determine the depth profiles of strain and structural disorders in the ion-modified layers. The thickness and maximum value of strain of ion-modified InSb(111) layers were determined. For implantation energies 66 and 80 keV, doses 25 and 50 µC, the thickness of the strained layer is about 500 nm with the maximum value of strain close to 0.1%. Additionally, an amorphous layer with significant thickness was found in the implantation region.
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Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 3. P. 355-361.
doi: https://doi.org/10.15407/spqeo20.03.355
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
355
PACS 61.05
Modeling of X-ray rocking curves for layers
after two-stage ion-implantation
O.I. Liubchenko, V.P. Kladko, O.Yo. Gudymenko
V. Lashkaryov Institute of Semiconductor Physics, NAS of Ukraine,
41, prospect Nauky, 03028 Kyiv, Ukraine;
e-mail: lubchenco.a@gmail.com
Abstract. In this work, we consider the approach for simulation of X-ray rocking curves
inherent to InSb(111) crystals implanted with Be+ ions with various energies and doses.
The method is based on the semi-kinematical theory of X-ray diffraction in the case of
Bragg geometry. A fitting procedure that relies on the Hooke–Jeeves direct search
algorithm was developed to determine the depth profiles of strain and structural disorders
in the ion-modified layers. The thickness and maximum value of strain of ion-modified
InSb(111) layers were determined. For implantation energies 66 and 80 keV, doses 25
and 50 µC, the thickness of the strained layer is about 500 nm with the maximum value
of strain close to 0.1%. Additionally, an amorphous layer with significant thickness was
found in the implantation region.
Keywords: X-ray rocking curve modeling, two-stage implantation, Hooke-Jeeves direct
search, strain distribution.
Manuscript received 18.07.17; revised version received 04.08.17; accepted for
publication 06.09.17; published online 09.10.17.
1. Introduction
Output characteristics of nano- and microelectronic
devices strongly depend on technological processes
during their fabrication. Therefore, it is important to
investigate the influence of each technological step on
the device characteristics. Accurate investigation of the
clue technologies such as ion implantation, diffusion and
passivation, allows to control the electro-physical
parameters of the devices [1–5].
Ion implantation is a widely used technique for
fabrication of variety of nano- and microelectronic
devices [1, 2, 6]. This method allows to introduce
impurities with preset concentration profiles without use
of elevated temperatures, and is mostly used for p- or n-
type doping [3–5]. Investigation of its influence on
structural changes in the post-implanted material will
allow to improve the output characteristics of these
devices, which is of great importance.
X-ray diffraction (XRD) is a widely used technique
for nondestructive structural characterization of
semiconductor materials. For ion-modified layers, the
structural changes can be qualitatively investigated by
analyzing the X-ray diffraction curves in the Bragg
geometry. In addition, simulation of X-ray diffraction
curves can be used for the depth profiles of strain
determination [7–9]. However, for the ion-modified
structures, simulation of XRD spectra is a difficult task
with ambiguous solutions [10–12]. The uncertainty
arises due to interference of X-rays on heterogeneities of
the structure, in particular, in amorphous layer [13, 14].
The kinematical [13–18] and generalized
dynamical theory [21, 22] of X-ray diffraction were used
in combination with the dynamical approach in case of
the Takagi–Topens approximation [11, 12, 19, 20], for
the ω/2Θ scans simulation for ion implanted layers. The
kinematical and the dynamical approaches are used for
thin implanted layers and high-quality bulk material,
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 3. P. 355-361.
doi: https://doi.org/10.15407/spqeo20.03.355
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
356
respectively. To perform this simulation, the ion-
modified layer was separated by several sublayers with
mean values of strain, defects and clusters concentration.
The parameters of each layer were changed, until a good
fit between the calculated and measured XRD spectra
was achieved.
Many authors [10–17, 19, 20] simulate only the
coherent part of the scattered intensity, which gives
information about the strains and composition. The
diffuse scattering sufficiently complicates the model and
is not considered here. In this work, the strain
distribution in ion implanted InSb(111) crystals were
investigated by means of X-ray ω/2Θ scans simulation.
2. Experimental
Two-stage implantation with Be+ ions was performed for
two series of InSb (111) samples with the same
parameters, to check the reproducibility of the
experiment. The implantation energies (E) and doses (D)
were E = 66 keV and D = 25 µC for the first stage of
implantation, and Е = 80 keV, D = 50 µC for the second
stage. The implantation was carried out through a thin
SiO2 mask layer to avoid the effect of substrate (surface)
destruction. More high-energy implantation was carried
out to create p-n junction in a single InSb crystal, and the
second serves as a heteromaper for defects.
The structural parameters of as-implanted samples
were examined with high-resolution X-ray diffraction
using the PANalytical X’Pert Pro MRD XL
diffractometer. The ω/2Θ scans of symmetric (111) and
(333) reflections were measured. The X-ray reflectivity
measurements were performed additionally to estimate
the thickness (~90 nm) of the SiO2 layer.
3. The model
We use the kinematical and dynamical theories of X-ray
diffraction [13–15, 23, 24] to simulate the X-ray ω/2Θ
scans for the Be+ implanted InSb layer and substrate,
respectively. The kinematical theory is feasible for the
ion-implanted layer because of its small thickness and
high structural disorder. Proper simulation of the
intensity scattered from the high quality InSb substrate
should be performed with consideration of the diffuse
component of X-ray scattering. However, to keep the
model simple we will not consider the diffuse part in our
simulations. The general expressions of the semi-
kinematical theory of X-ray diffraction are given below.
The reflectivity of the whole structure is
proportional to the reflection amplitude of the substrate
(A0) and layer (AL)
2
0 LAiAR ⋅+= , (1)
12
0 −−= yyA , (2)
( )( ) ( )i
i
ii
n
i
iL i
fy
ufy
aA φ⋅−⋅
−
⋅−
⋅=∑
=
exp
sin
1
, (3)
where iφ is the phase, y – angular deviation, fi , ui and ai
are the normalized strain, thickness and absorption
multiplier in the i-th layer.
( ) ( ) kk
i
k
iii ufyufy ⋅−⋅+⋅−= ∑
−
=
1
1
2φ , (4)
( )
HH
y
γ
γ
χ
02sin Θ⋅ΔΘ
= , (5)
( )
HH
HHi
zzif
γ
γ
χ
γγγ
ε 00 ⋅−
⋅−= , (6)
0γγλ
χπ
⋅⋅
⋅⋅
=
H
Hi
i
t
u , (7)
⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⋅
⋅⋅
+
⋅−= ∑
+=
N
ij
j
H
H
i ta
10
0
2
exp
γγ
γγ
μ (8)
where γ0, γH are direction cosines, ΔΘ is angular
deviation of the investigated crystal from its exact
substrate Bragg position, ti, i
zzε are thickness and strain
of the i-th layer, Θ0 is the Bragg angle of substrate, χH –
Fourier component of the perfect crystal polarizability,
λ = 0.1546 nm – X-ray wavelength, µ – absorption
coefficient.
The strain distribution in the implanted layers is
described with a two-sided Gaussian function [11–13,
15]. It is determined by only four parameters in
contradistinction to B-spline basis functions [18, 24, 25].
In our case of two-stage implantation, the strain
distribution in the InSb:Be+ layers is described by a sum
of two asymmetric Gaussians:
⎪
⎪
⎪
⎩
⎪⎪
⎪
⎨
⎧
>⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⋅
−
−⋅
<<⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⋅
−
−⋅
+
+
⎪
⎪
⎪
⎩
⎪⎪
⎪
⎨
⎧
>⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⋅
−
−⋅
<<⎟
⎟
⎠
⎞
⎜
⎜
⎝
⎛
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
⋅
−
−⋅
=
Δ
2
2
22
2
2
2
2
21
2
2
1
2
12
1
1
1
2
11
1
1
,
2
exp
0,
2
exp
,
2
exp
0,
2
exp
ρ
σ
ρ
τ
ρ
σ
ρ
τ
ρ
σ
ρ
τ
ρ
σ
ρ
τ
x
x
x
x
x
x
x
x
d
d
(9)
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 3. P. 355-361.
doi: https://doi.org/10.15407/spqeo20.03.355
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
357
where τi and ρi are the maximum value of strain and its
depth, respectively, and σi1, σi2 – full widths at half
maximum of the i-th Gaussian function.
Principles of the program developed for the ω/2Θ
scans simulation are reviewed below. The program is
written using the C++ programming language
(https://gcc.gnu.org/) and parallel computing platform
CUDA (https://developer.nvidia.com/cuda-toolkit). The
input data are the parameters of 2 asymmetric Gaussians,
the experimental ω/2Θ scan and the characteristics of
material. The simulation procedure is as follows:
(1) reading the input data; (2) calculating strain
distribution and intensity of scattering by the structure;
(3) convolution of the calculated ω/2Θ scan with the
instrumental function (instrumental function is
approximated by the Gaussian profile with full width at
half maximum (FWHM) 12 arc sec that is responsible
for the FWHM of an analyzer’s rocking curve);
(4) additional convolution is carried out for rough search
of optimal parameters [11]; (5) determination of the
mean error between experimental and calculated data by
using the equations (10), (11); (6) performing the fitting,
which consists of multiple repeating the steps 2 to 5 with
using the minimization algorithm; (7) after the minimum
error is found, the program saves and shows the profiles
of strain distribution, calculated data, and finishes its
work.
The Nelder–Mead Algorithm [26] and Hooke–
Jeeves Direct Search (“pattern search”) [27] were
applied for error minimization in our simulation. Both
methods are unconditional, and function minimization
is performed by setting the starting point to search for a
minimum (or maximum). Also, these algorithms have
no required derivatives of the error function; only the
value of the function is required. Finding the derivative
in our case would be a complicate problem. Also, we
note that these methods can find only the local
minimum. To find the global minimum, one must set
different starting points, or as in our case, use addi-
tional convolution.
The Nelder–Mead method, or the amoeba method,
is the commonly applied numerical method used to find
the minimum or maximum of an objective function in a
multidimensional space. The method uses the concept of
a simplex, which is a special polytope of n+1 vertices in
n dimensions. Based on the values calculated in the
vertices of the simplex, the search for the minimum
value is performed using reflection, compression and
stretching operations. When the distance between the
vertices of the simplex decrease less than to a certain
value, there is an exit from the procedure (checking for
convergence). In detail, the principle of the algorithm is
described in [26, 28]. The fitting of X-ray ω/2Θ scans is
performed to fulfill the condition of convergence or to
exceed the number of iterations. We used a ready-made
algorithm for applied statistics AS 047 [29].
Hooke–Jeeves Direct Search, also known as
pattern search, is a family of numerical optimization
methods. It consists of two stages: the exploratory
move and the pattern move. At the first stage, the
starting point 1 and the steps for each coordinate
should be defined. Then, the values of all coordinates
are fixed except for the first one, and values of the
function in points at the step distance from the initial
value are found. Next, the transition to the point with
the smallest value of the function is performed. If the
value at the starting point is less than the values for
both directions of the step, then the step in this
coordinate decreases. This procedure is performed for
all coordinates to a given minimum. In this way, we
obtain a new point with the smallest value of the
function in the neighborhood (denote its 2). At the
second stage, the point 3 is set aside in the direction
from 1 to 2 at the same distance. Then, at point 3, the
exploratory move is performed without decreasing
the step. If the point 4 another than 3 is given, then
there is a redefinition of the points: 2→1, 4→2; else
there is a redefinition of the point 2→1 and repeat an
exploratory move. The minimum search is performed
until the shift step in all coordinates will not be less
than the given value or to exceed the number of
iterations. In our work, we used the modified Hooke–
Jeeves algorithm [30].
The Nelder–Mead algorithm works faster, but
when some precision of the fit is reached, this method
“stagnates” and, despite the built-in convergence check,
the exit from the procedure is carried out on the
condition of the iterations’ number. To improve the
accuracy of the solution, one needs to set more
iterations, which is time consuming. Hooke–Jeeves
direct search works slower, but fitting quality is better
than in the previous method. On average, the number of
attempts to minimize errors by using the Hook–Jeeves
method was smaller. In general, the behavior of both
algorithms at a rough search minimum is similar. When
one set different starting points far from the global
minimum, there are often local minima, which points to
a complex profile of errors. Therefore, Hooke–Jeeves
algorithm for fitting of X-ray diffraction spectra was
more applicable. This may be caused by the complicated
hyper surface profile of the error function, depending on
the many parameters or the fact that the optimization
parameters are interrelated (e.g., σ1, σ2 – half-widths of
the asymmetric Gaussian, which give a similar
contribution to the diffraction pattern).
It was ascertained that applying the Hooke–Jeeves
method, the best fit was achieved when using the mean
error function given by the expression:
∑
=
−
=
N
i
exp
i
exp
i
theor
i
I
II
N
Err
1
1 , (10)
where N is the number of experimental points on the X-
ray spectra; theor
iI and exp
iI are the calculated and
experimental intensities, respectively.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 3. P. 355-361.
doi: https://doi.org/10.15407/spqeo20.03.355
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
358
For the Nelder–Mead algorithm, the best fitting
was achieved when using the following mean error
function:
∑
=
−=
N
i
exp
i
theor
i IIErr
1
. (11)
The intensity scattered from the layer was
calculated using Eq. (3). Herewith, the layer was
separated by n sublayers. Investigation of the effect of
the number of sublayers on the diffraction pattern
showed that with the number of sublayers higher than
50, there were no significant changes in the calculated
spectrum. However, for our fitting procedure the number
of sublayers has been increased up to 200. The depth
profile of strain in the ion-implanted InSb layer was
calculated using Eq. (9). It is important to calculate it by
choosing the point where one can ignore the strain (i.e.,
the boundary between the layer and the substrate), which
affects the X-ray spectrum near the substrate peak. We
assume that the strain of about 1% of the maximum
value has a minor effect on the X-ray spectrum and can
be neglected in simulation. This decrease of strain can be
observed at a distance of about 4.3σ1i from the maximum
value.
4. Results and discussion
Simulation of the whole spectra has shown violation of
the peaks intensity ratio between the layer and substrate.
Without convolution with instrumental function, this
difference is about one order of magnitude. When using
convolution, it depends on the width of the window of
convolution. Most likely, this is because we used the
semi-kinematical approach without consideration of the
static Debye–Waller (DW) factor. Therefore, an
additional fitting parameter was introduced in the model,
which multiplies the intensity reflected from the
substrate. With a decrease of this factor, the intensity
fluctuations in the interference pattern from the layer
become better visible. The fact that the value of the
intensity adjustment coefficient has a roughly identical
value for the whole layer can indicate a homogeneous
amorphization of this layer. Also, the fitting procedure
works far better in the case of fitting the part of spectrum
that corresponds to the layer, so we did not adjust the
part of the spectrum containing the substrate peak at an
exact Bragg angle. When the fitting of the whole
spectrum was conducted, the best result was achieved
using the error function (Eq. (11)), but there are
oscillations of intensity and poor coincidence of the
“tails” between the calculated and experimental ω/2Θ
scans. So, we used the error formula (Eq. (10)) and
simulated only the part of spectrum that corresponds to
the implanted layer.
For the sample 1, we simulated the ω/2Θ scans of
(111) (Fig. 1a) and (333) reflections (Fig. 1b). This
simulation was carried out using the above described
algorithm. The top SiO2 mask layer was not taken into
account in this simulation for the following reasons:
after the ion implantation, this layer can be considered as
amorphous, which contributes to the diffuse component
of the X-ray spectra. However, we do not consider the
diffuse component in our model, and calculate only the
coherent part of the X-ray scattering. The obtained depth
profiles of strain have been compared in Fig. 1c, and
some differences in the profiles obtained from
simulation of ω/2Θ scans for (111) and (333) reflections
are observed. It should be noted that we expect more
accurate determination of the depth profile of strain
when simulating the ω/2Θ scan of the high-order (333)
reflection as compared with the ω/2Θ scan of (111)
reflection. First, the lower order reflections are less
strain sensitive. Secondly, the ω/2Θ scans of these
reflections do not have well-defined features, and
thirdly, the peak of the layer is close to the peak of
substrate. All of this increases the error of the strain
profiles determination from the ω/2Θ scans fitting for
lower-order reflections.
Fig. 2a shows the depth profiles of strain in the
sample 2 determined by simulating the ω/2Θ scans of
both (111) and (333) reflections. For each reflection, the
strain profiles were obtained by specifying different
starting points in the fitting algorithm to achieve the
minimum fitting error (Eq. (10)). Noticeable differences
between the strain profiles obtained from simulation of
the (111) and (333) reflections can be observed, too.
Fig. 2b compares the depth profiles of strain for
both samples. Despite the same designed conditions of
the ion implantation, there are some differences between
the obtained profiles of strain distribution. The strain
profiles obtained from the less sensitive reflection (111)
almost coincide for both samples. The deformation
profiles obtained from the reflection (333) show a
greater thickness of the strained layer for the sample 1,
indicating a greater energy of implantation. There are
also some differences in the strain distribution near the
surface. This indicates that the real parameters of the
implantation process are not fully repeatable for the
samples 1 and 2.
For each ω/2Θ scan, the average fitting error do not
exceeds 10%. Finishing the fitting algorithm was carried
out, when the step δ = 10–8 was reached. The difference
in the depth profiles of strain obtained by simulating the
ω/2Θ scan of (111) and (333) reflections is explained by
the ambiguity of modeling these structures. For example,
different shapes of the strain profile can lead to similar
changes of the X-ray spectra. By taking into account the
absorption (attenuation) of X-rays can simplify the task
to determine strain profiles. However, it requires the use
of more advanced models that are more complicated and
require longer computational time.
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 3. P. 355-361.
doi: https://doi.org/10.15407/spqeo20.03.355
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
359
Fig. 1. The measured (black squares) and simulated (red circles) ω/2Θ scans of (111) (a) and (333) (b) reflections for the
sample 1 (c) with the depth profiles of strain obtained from this simulation.
Fig. 2. The depth profiles of strain in the sample 2 obtained from simulation of the ω/2Θ scans for (111) (red squares) and
(333) (black circles) reflections (a) and the depth profiles of strain for the samples 1 and 2 obtained from this simulation of
ω/2Θ scans for (111) and (333) reflections (b).
Semiconductor Physics, Quantum Electronics & Optoelectronics, 2017. V. 20, N 3. P. 355-361.
doi: https://doi.org/10.15407/spqeo20.03.355
© 2017, V. Lashkaryov Institute of Semiconductor Physics, National Academy of Sciences of Ukraine
360
5. Conclusions
In this work, we developed the method to simulate the
X-ray ω/2Θ scans by using the semi-kinematical theory
of X-ray diffraction in the Bragg geometry for two-stage
ion-implanted layers. The fitting method is developed on
the basis of the Hooke–Jeeves Direct Search algorithm
for determining the depth profiles of strain in the
implanted layers. The as-implanted layer thickness of
InSb was established to be about 500 nm, and the
maximum value of strain was ~0.1%. The presence of
amorphization in the strained layer was also ascertained.
The obtained strain profiles of 2 samples show
reproducibility of the implantation process, although
slight difference can be observed.
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| id | nasplib_isofts_kiev_ua-123456789-214945 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1560-8034 |
| language | English |
| last_indexed | 2026-03-21T19:36:26Z |
| publishDate | 2017 |
| publisher | Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України |
| record_format | dspace |
| spelling | Liubchenko, O.I. Kladko, V.P. Gudymenko, O.Yo. 2026-03-05T12:01:35Z 2017 Modeling of X-ray rocking curves for layers after two-stage ion-implantation / O.I. Liubchenko, V.P. Kladko, O.Yo. Gudymenko // Semiconductor Physics Quantum Electronics & Optoelectronics. — 2017. — Т. 20, № 3. — С. 355-361. — Бібліогр.: 30 назв. — англ. 1560-8034 PACS: 61.05 https://nasplib.isofts.kiev.ua/handle/123456789/214945 https://doi.org/10.15407/spqeo20.03.355 In this work, we consider the approach for simulation of X-ray rocking curves inherent to InSb(111) crystals implanted with Be⁺ ions with various energies and doses. The method is based on the semi-kinematical theory of X-ray diffraction in the case of Bragg geometry. A fitting procedure that relies on the Hooke–Jeeves direct search algorithm was developed to determine the depth profiles of strain and structural disorders in the ion-modified layers. The thickness and maximum value of strain of ion-modified InSb(111) layers were determined. For implantation energies 66 and 80 keV, doses 25 and 50 µC, the thickness of the strained layer is about 500 nm with the maximum value of strain close to 0.1%. Additionally, an amorphous layer with significant thickness was found in the implantation region. en Інститут фізики напівпровідників імені В.Є. Лашкарьова НАН України Semiconductor Physics Quantum Electronics & Optoelectronics Modeling of X-ray rocking curves for layers after two-stage ion-implantation Article published earlier |
| spellingShingle | Modeling of X-ray rocking curves for layers after two-stage ion-implantation Liubchenko, O.I. Kladko, V.P. Gudymenko, O.Yo. |
| title | Modeling of X-ray rocking curves for layers after two-stage ion-implantation |
| title_full | Modeling of X-ray rocking curves for layers after two-stage ion-implantation |
| title_fullStr | Modeling of X-ray rocking curves for layers after two-stage ion-implantation |
| title_full_unstemmed | Modeling of X-ray rocking curves for layers after two-stage ion-implantation |
| title_short | Modeling of X-ray rocking curves for layers after two-stage ion-implantation |
| title_sort | modeling of x-ray rocking curves for layers after two-stage ion-implantation |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/214945 |
| work_keys_str_mv | AT liubchenkooi modelingofxrayrockingcurvesforlayersaftertwostageionimplantation AT kladkovp modelingofxrayrockingcurvesforlayersaftertwostageionimplantation AT gudymenkooyo modelingofxrayrockingcurvesforlayersaftertwostageionimplantation |