Simulation of X-Ray Diffraction Spectra for AlN/GaN Multiple Quantum Well Structures on AlN(0001) with Interface Roughness and Variation of Vertical Layers Thickness
A detailed XRD analysis of AlN/GaN multiple quantum well (MQW) structures grown on AlN(0001) substrates is proposed. The effect of roughness on the 2θ-ω scans measured in Bragg diffraction for symmetrical reflections is investigated together with the effect of depth variation of the well and barrier...
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Інститут металофізики ім. Г.В. Курдюмова НАН України
2018
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irk-123456789-1469532019-02-13T01:23:47Z Simulation of X-Ray Diffraction Spectra for AlN/GaN Multiple Quantum Well Structures on AlN(0001) with Interface Roughness and Variation of Vertical Layers Thickness Liubchenko, O.I. Kladko, V.P. Строение и свойства наноразмерных и мезоскопических материалов A detailed XRD analysis of AlN/GaN multiple quantum well (MQW) structures grown on AlN(0001) substrates is proposed. The effect of roughness on the 2θ-ω scans measured in Bragg diffraction for symmetrical reflections is investigated together with the effect of depth variation of the well and barrier thickness. As shown, the magnitude of depth variation of the well and barrier thickness results in an asymmetrical broadening of the satellite peaks of the 2θ-ω scans. Roughness causes their symmetrical expansion that allows separating the influence of both effects. Several reasons of asymmetrical broadening of satellite peaks are considered: variation of the thickness period, variation of the average lattice parameter inherent to the period, which depends on the thickness ratio of the layers in the period, and their combination. The efficiency of the described method is illustrated in detail by numerical simulations. В работе предложен детальный рентгенодифракционный анализ структур с множественными квантовыми ямами (МКЯ) AlN/GaN, выращенными на подложках AlN(0001). Было исследовано влияние шероховатости и вариации толщины слоёв квантовых ям и барьеров на 2θ-ω-сканы, полученные в геометрии отражения по Брэггу для симметричных рефлексов. Показано, что наличие вариации толщины слоёв AlN и GaN по глубине приводит к появлению асимметрии сателлитных пиков МКЯ на 2θ-ω-сканах. Наличие шероховатости приводит к симметричному расширению сателлитных пиков, что позволяет разделить влияние этих эффектов. Рассмотрено несколько причин асимметричного расширения сателлитных пиков: изменение толщины периода, изменение среднего параметра решётки периода, который зависит от соотношения толщин слоёв в периоде, и их комбинации. Эффективность разработанного метода показана с помощью численного моделирования. В роботі проведено детальну аналізу структур із множинними квантовими ямами (МКЯ) AlN/GaN, вирощених на підкладинках AlN(0001). Було досліджено вплив шерсткости та зміни товщини шарів структури з МКЯ по глибині на 2θ-ω-скани, виміряні в Бреґґовій геометрії дифракції для симетричних рефлексів. Показано, що зміна товщини квантових ям і бар’єрів по глибині приводить до асиметричного розширення сателітних піків МКЯ на 2θ-ω-сканах. Шерсткість спричинює симетричне розширення піків, що уможливлює розрізнити вплив цих ефектів. Розглянуто кілька причин асиметричного розширення сателітних піків: зміну товщини періоду, зміну середнього параметра ґратниці періоду, який залежить від співвідношення товщин шарів періоду, та їх комбінації. Ефективність розробленої методи показано шляхом числового моделювання рентґенівських спектрів. 2018 Article Simulation of X-Ray Diffraction Spectra for AlN/GaN Multiple Quantum Well Structures on AlN(0001) with Interface Roughness and Variation of Vertical Layers Thickness / O.I. Liubchenko, V.P. Kladko // Металлофизика и новейшие технологии. — 2018. — Т. 40, № 6. — С. 759-776. — Бібліогр.: 38 назв. — англ. 1024-1809 PACS: 61.05.cc, 61.05.cp, 68.35.Ct, 68.65.Ac, 68.65.Cd, 68.65.Fg, 81.05.Ea DOI: https://doi.org/10.15407/mfint.40.06.0759 http://dspace.nbuv.gov.ua/handle/123456789/146953 en Металлофизика и новейшие технологии Інститут металофізики ім. Г.В. Курдюмова НАН України |
| institution |
Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| language |
English |
| topic |
Строение и свойства наноразмерных и мезоскопических материалов Строение и свойства наноразмерных и мезоскопических материалов |
| spellingShingle |
Строение и свойства наноразмерных и мезоскопических материалов Строение и свойства наноразмерных и мезоскопических материалов Liubchenko, O.I. Kladko, V.P. Simulation of X-Ray Diffraction Spectra for AlN/GaN Multiple Quantum Well Structures on AlN(0001) with Interface Roughness and Variation of Vertical Layers Thickness Металлофизика и новейшие технологии |
| description |
A detailed XRD analysis of AlN/GaN multiple quantum well (MQW) structures grown on AlN(0001) substrates is proposed. The effect of roughness on the 2θ-ω scans measured in Bragg diffraction for symmetrical reflections is investigated together with the effect of depth variation of the well and barrier thickness. As shown, the magnitude of depth variation of the well and barrier thickness results in an asymmetrical broadening of the satellite peaks of the 2θ-ω scans. Roughness causes their symmetrical expansion that allows separating the influence of both effects. Several reasons of asymmetrical broadening of satellite peaks are considered: variation of the thickness period, variation of the average lattice parameter inherent to the period, which depends on the thickness ratio of the layers in the period, and their combination. The efficiency of the described method is illustrated in detail by numerical simulations. |
| format |
Article |
| author |
Liubchenko, O.I. Kladko, V.P. |
| author_facet |
Liubchenko, O.I. Kladko, V.P. |
| author_sort |
Liubchenko, O.I. |
| title |
Simulation of X-Ray Diffraction Spectra for AlN/GaN Multiple Quantum Well Structures on AlN(0001) with Interface Roughness and Variation of Vertical Layers Thickness |
| title_short |
Simulation of X-Ray Diffraction Spectra for AlN/GaN Multiple Quantum Well Structures on AlN(0001) with Interface Roughness and Variation of Vertical Layers Thickness |
| title_full |
Simulation of X-Ray Diffraction Spectra for AlN/GaN Multiple Quantum Well Structures on AlN(0001) with Interface Roughness and Variation of Vertical Layers Thickness |
| title_fullStr |
Simulation of X-Ray Diffraction Spectra for AlN/GaN Multiple Quantum Well Structures on AlN(0001) with Interface Roughness and Variation of Vertical Layers Thickness |
| title_full_unstemmed |
Simulation of X-Ray Diffraction Spectra for AlN/GaN Multiple Quantum Well Structures on AlN(0001) with Interface Roughness and Variation of Vertical Layers Thickness |
| title_sort |
simulation of x-ray diffraction spectra for aln/gan multiple quantum well structures on aln(0001) with interface roughness and variation of vertical layers thickness |
| publisher |
Інститут металофізики ім. Г.В. Курдюмова НАН України |
| publishDate |
2018 |
| topic_facet |
Строение и свойства наноразмерных и мезоскопических материалов |
| url |
http://dspace.nbuv.gov.ua/handle/123456789/146953 |
| citation_txt |
Simulation of X-Ray Diffraction Spectra for AlN/GaN Multiple Quantum Well Structures on AlN(0001) with Interface Roughness and Variation of Vertical Layers Thickness / O.I. Liubchenko, V.P. Kladko // Металлофизика и новейшие технологии. — 2018. — Т. 40, № 6. — С. 759-776. — Бібліогр.: 38 назв. — англ. |
| series |
Металлофизика и новейшие технологии |
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2025-07-11T01:01:20Z |
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| fulltext |
759
PACS numbers: 61.05.cc, 61.05.cp, 68.35.Ct, 68.65.Ac, 68.65.Cd, 68.65.Fg, 81.05.Ea
Simulation of X-Ray Diffraction Spectra for AlN/GaN Multiple
Quantum Well Structures on AlN(0001) with Interface
Roughness and Variation of Vertical Layers Thickness
O. I. Liubchenko and V. P. Kladko
V. E. Lashkaryov Institute of Semiconductor Physics, N.A.S. of Ukraine,
41 Nauky Ave.,
UA-03028 Kyiv, Ukraine
A detailed XRD analysis of AlN/GaN multiple quantum well (MQW) struc-
tures grown on AlN(0001) substrates is proposed. The effect of roughness on
the 2 scans measured in Bragg diffraction for symmetrical reflections is
investigated together with the effect of depth variation of the well and barri-
er thickness. As shown, the magnitude of depth variation of the well and bar-
rier thickness results in an asymmetrical broadening of the satellite peaks of
the 2 scans. Roughness causes their symmetrical expansion that allows
separating the influence of both effects. Several reasons of asymmetrical
broadening of satellite peaks are considered: variation of the thickness peri-
od, variation of the average lattice parameter inherent to the period, which
depends on the thickness ratio of the layers in the period, and their combina-
tion. The efficiency of the described method is illustrated in detail by numer-
ical simulations.
Key words: dynamical X-ray diffraction, multiple quantum well structure,
thickness variation with depth, computer simulation, superlattice.
В роботі проведено детальну аналізу структур із множинними квантови-
ми ямами (МКЯ) AlN/GaN, вирощених на підкладинках AlN(0001). Було
досліджено вплив шерсткости та зміни товщини шарів структури з МКЯ
по глибині на 2 -скани, виміряні в Бреґґовій геометрії дифракції для
симетричних рефлексів. Показано, що зміна товщини квантових ям і
Corresponding author: Oleksii Ihorovich Liubchenko
E-mail: lubchenco.a@gmail.com
Citation: O. I. Liubchenko and V. P. Kladko, Simulation of X-Ray Diffraction Spectra
for AlN/GaN Multiple Quantum Well Structures on AlN(0001) with Interface
Roughness and Variation of Vertical Layers Thickness, Metallofiz. Noveishie Tekhnol.,
40, No. 6: 759–776 (2018), DOI: 10.15407/mfint.40.06.0759.
Ìåòàëëîôèç. íîâåéøèå òåõíîë. / Metallofiz. Noveishie Tekhnol.
2018, т. 40, № 6, сс. 759–776 / DOI: 10.15407/mfint.40.06.0759
Îòòèñêè äîñòóïíû íåïîñðåäñòâåííî îò èçäàòåëÿ
Ôîòîêîïèðîâàíèå ðàçðåøåíî òîëüêî
â ñîîòâåòñòâèè ñ ëèöåíçèåé
2018 ÈÌÔ (Èíñòèòóò ìåòàëëîôèçèêè
èì. Ã. Â. Êóðäþìîâà ÍÀÍ Óêðàèíû)
Íàïå÷àòàíî â Óêðàèíå.
https://doi.org/10.15407/mfint.40.06.0759
https://doi.org/10.15407/mfint.40.06.0759
760 O. I. LIUBCHENKO and V. P. KLADKO
бар’єрів по глибині приводить до асиметричного розширення сателітних
піків МКЯ на 2 -сканах. Шерсткість спричинює симетричне розши-
рення піків, що уможливлює розрізнити вплив цих ефектів. Розглянуто
кілька причин асиметричного розширення сателітних піків: зміну тов-
щини періоду, зміну середнього параметра ґратниці періоду, який зале-
жить від співвідношення товщин шарів періоду, та їх комбінації. Ефек-
тивність розробленої методи показано шляхом числового моделювання
рентґенівських спектрів.
Ключові слова: динамічна дифракція X-променів, множинні квантові
ями, зміна товщини по глибині, комп’ютерне моделювання, надґратниця.
В работе предложен детальный рентгенодифракционный анализ структур
с множественными квантовыми ямами (МКЯ) AlN/GaN, выращенными
на подложках AlN(0001). Было исследовано влияние шероховатости и
вариации толщины слоёв квантовых ям и барьеров на 2 -сканы, полу-
ченные в геометрии отражения по Брэггу для симметричных рефлексов.
Показано, что наличие вариации толщины слоёв AlN и GaN по глубине
приводит к появлению асимметрии сателлитных пиков МКЯ на 2 -
сканах. Наличие шероховатости приводит к симметричному расширению
сателлитных пиков, что позволяет разделить влияние этих эффектов.
Рассмотрено несколько причин асимметричного расширения сателлит-
ных пиков: изменение толщины периода, изменение среднего параметра
решётки периода, который зависит от соотношения толщин слоёв в пери-
оде, и их комбинации. Эффективность разработанного метода показана с
помощью численного моделирования.
Ключевые слова: динамическая дифракция рентгеновских лучей, мно-
жественные квантовые ямы, изменение толщины по глубине, компью-
терное моделирование, сверхрешётка.
(Received April 6, 2018)
1. INTRODUCTION
Multiple quantum well (MQW) structures and superlattices based on
III-nitrides are extensively used in light emitting diodes with high lu-
minescent efficiency (LEDs) and laser diodes for various applications
in photonic devices operating in a wide optical range [1–6]. Specifical-
ly, AlN/GaN MQW structures are used for LEDs emitting in the visi-
ble-ultraviolet spectral range [1, 7] and solid-state THz optoelectronics
[8, 9]. Investigation and development of these materials and devices
requires advanced materials’ characterization techniques examining
structural, optical, and electronic properties. Optical characteristics
of devices based on MQW structures essentially depend on structural
parameters such as the thickness of barrier/quantum well layers [2, 3,
6, 10]. Any variation in the well or barrier width with the depth or in
plane during growth can therefore modify the electronic characteris-
SIMULATION OF X-RAY DIFFRACTION SPECTRA FOR AlN/GaN STRUCTURES 761
tics. In addition, because of the small thickness of recent QW (few na-
nometres), variations in the thickness of only one monolayer are suffi-
cient to lead to large changes in the interband and intersubband transi-
tions energies [11–13].
Therefore, a comprehensive structural characterization of the MQW
structures is needed to optimize the growth process and device perfor-
mance. The laboratory high-resolution X-ray diffraction (HRXRD) is a
rapid, non-destructive and low-cost technique widely used for charac-
terization of III-nitrides [14–22]. In the theoretical description of X-
ray diffraction in MQW structures, thin and ion-implanted layers,
both dynamical and kinematical approaches were used [21, 23]. The
main approaches to model X-ray diffraction patterns from MQW
structures are layer-by-layer simulation [24] and the period-by-period
simulation [21]. For both approaches, it is often assumed that all peri-
ods in MQWs are equal. However, only the layer-by-layer modelling
approach can also be used to study vertical structure inhomogeneities.
The analysis of HRXRD data is often based on the Takagi–Taupin
equations [25–27] or their derivatives [23, 28]. The Takagi–Taupin
equations are based on the dynamical theory of X-ray diffraction,
which assumes that the structures are crystallographically perfect (or
almost perfect) with very small fluctuations of lattice displacements
and suits well for high-quality epitaxial structures study.
In recent experimental studies of MQW structures, especially struc-
tures based on III-nitrides, an asymmetry of the peaks and a much
larger peak broadening in X-ray diffraction spectra are observed in
comparison with theoretically calculated spectra based on convention-
al diffraction models [3, 29]. Several different features cause these ef-
fects: variation of the in-plane lattice parameter [30], dislocations [18,
31], roughness [19, 32, 33], interface grading, changes in the thick-
ness of layers from the substrate to the surface and on the area of the
sample.
Previous studies were focused on the investigation of the effect of
interface grading and lateral thickness variation [32], MQW structure
lateral lattice parameter variation [19, 32], but the effect of the verti-
cal variation of the layers thickness is poorly explored from the view-
point of modern methods of HRXRD [33] and requires further investi-
gations. Commercial programs allow simulating spectra by using con-
ventional models of the diffraction, taking into account the effect of
interface grading, while the simulation of X-ray spectra, which con-
sider the effects of lattice parameter variation or variation in the layer
thickness, is often complicated or impossible.
Therefore, in this paper, the effect of thickness variation with depth
and interface roughness from MQW structures on the 2 scans in
the symmetrical Bragg case is investigated using dynamical diffrac-
tion simulation.
762 O. I. LIUBCHENKO and V. P. KLADKO
2. EXPERIMENTAL
The samples investigated in this paper were grown by plasma-assisted
MBE on a template that consists of a 600 nm AlN layer on an Al2O3
(0001) substrate. The GaN/AlN MQW structures were grown at the
substrate temperature of about 690 C. First, a 100 nm undoped GaN
buffer layer was grown on the template, followed by the deposition of a
10 and 20 periods MQW structure, samples S10 and S20, respectively.
Finally, the samples were capped by 10 nm of GaN. The nominal thick-
ness of the AlN barriers and GaN QW layers was 3 nm and 5 nm, re-
spectively.
High-resolution X-ray diffraction (HRXRD) was performed using a
PANalytical X’Pert Pro MRD XL diffractometer (delivering CuK 1
radiation with 0.154056 nm) equipped with the fourfold (220) Ge
monochromator and threefold (022) Ge analyser in front of a propor-
tional point detector.
3. THEORY
The XRD spectra for the multiple quantum well structures were simu-
lated using the dynamical theory of X-ray diffraction described by the
Takagi–Taupin equation [23, 25–28]. A complete description of the
dynamical theory used in this paper can be found in Ref. [28]. The solu-
tion of the Takagi–Taupin equation is used to calculate the X-ray com-
plex scattering amplitude X:
2
2 1.
idX
X X
dT
(1)
T is the normalized coordinate perpendicular to the crystal surface:
0
H H
H
Ct
T , (2)
and is the deviation parameter:
0 0
1
1
,
( )
2
H H
b
bC
(3)
where C is the polarization factor, which is equal to unity for polari-
zation and cos(2 ) for polarization, 0 and H are electric susceptibili-
ties, which are related to the structure factor of the crystal unit cell, 0
and H are the direction cosines of the incident and reflected beams
with respect to the crystal-surface normal, b is the asymmetry factor,
SIMULATION OF X-RAY DIFFRACTION SPECTRA FOR AlN/GaN STRUCTURES 763
t—the layer thickness, —the X-rays’ wavelength, and —the devia-
tion parameter.
The conventional deviation parameter was shown to fail at the angu-
lar distances far from the Bragg position, thus an improved function
for introduced by Zaus [28] was used:
2
2
3
sin(2 ) cos(2 ) sin1
( ) ( ) .
sin( ) 2 sin( ) sin
( )
( )
B B B
B B B
(4)
Here, B is the Bragg angle of the layer, —the inclination of the layer
lattice planes relative to the surface, B—the deviation from
the exact Bragg position, —the angle between the reflecting planes
and the incident wave.
The direction cosines 0 and H in the case of the improved deviation
parameter are given as follow [28]:
0
sin( ), sin( ),
B H H (5)
where the angle of exit H of the diffracted wave in vacuum is given
in Ref. [28]:
cos( ) cos( ) 2sin( ) sin( ).
H B
(6)
Assuming that the layers are laterally uniform and all variations occur
along the axis z normal to the sample surface, the amplitude ratio Xi at the
top of the layer is related to the amplitude ratio Xi 1 at the bottom by [28]:
2 1 2
1 2
1 ,
i
S S
X
S S
(7)
2 2
1,2 1
( 1) exp( 1).
i
S X iT (8)
The reflected amplitude from multilayered structure was calculated
starting from the bottom layer and assuming that below the bottom
layer the complex scattering amplitude is equal to zero; or using com-
plex scattering amplitude from the substrate and adding layers one-by-
one, using the recursion formula Eq. (7), until reaching the surface of
the uppermost layer. Finally, Ref. [28] gives the reflectivity:
2
,H
N
H
R X (9)
where XN is the complex scattering amplitude of the upper layer.
The main advantage of this diffraction model is its simplicity and,
consequently, a high computing speed. The disadvantage of the model
764 O. I. LIUBCHENKO and V. P. KLADKO
is a large number of simplifications, which lead to deterioration in the
quality of physical description of the diffraction phenomenon. For
even more accelerated computations, we developed a program for XRD
spectra simulation by using the C programming language and
NVIDIA CUDA parallel computing technology.
Within the framework of the above discussion, the MQW structure was
considered as a stack of N layers. Investigating the effect of thickness
variation was carried out by substitution of the thickness value in Eq. (2)
in the form t t0 t, where t0 is a given constant thickness for each layer
in the MQW period (averaged constant value), and t is the additive part
of the thickness, which can vary by arbitrary law from substrate to sur-
face of the investigated MQW structure (depending on the structure
depth). We used the exponential form of t, which is given by
1 2
exp( ),t K K N (10)
where N is the period number (the periods are counted from the sub-
strate to the top of the layer stack), K1 and K2 are parameters that spec-
ify variation of the thickness.
In Equation (4), the angles and B are incidence and Bragg angles
for the respective layer. If the Bragg plane of the substrate is chosen as
a common plane of reference, then for these layers, Ref. [28] gives the
deviation B:
,
Bs
(11)
where Bs is the Bragg angle for the substrate, and the change of the
Bragg angle is [28, 34]
||
2 2
||
cos sin tan( ) cos( ) sin( ),( ) ( ) Bs
d d d d
d d d d
(12)
where
,||
d
d
are the in- and out-of-plane mismatches of interplanar
spacing between the layer and the substrate.
4. RESULTS AND DISCUSSION
Theoretical simulations of symmetrical 2 scans were performed for
the tested sample consisting of a 20 period AlN/GaN MQW structure
(5 nm /5 nm) (layers A and B) on the AlN/GaN (600 nm/100 nm) buffer
grown on an Al2O3(0001) substrate.
SIMULATION OF X-RAY DIFFRACTION SPECTRA FOR AlN/GaN STRUCTURES 765
To investigate the influence of different kinds of thickness varia-
tions theoretically, we assume here that the GaN buffer layer and the
GaN layers in the MQW are fully strained to the AlN buffer layer and
that the AlN layers are fully relaxed. Figure 1 shows a sketch of the
MQW structure used for the 2 -scan simulation. The lattice param-
eters of bulk GaN and AlN were taken from Ref. [35]. For strained GaN
layers, the lattice parameter c was recalculated from a according to
Refs. [16, 35]. The electric susceptibilities were taken from Stepanov’s
X-ray server [36]. The instrumental function is approximated by the
Gaussian profile with the full width at half maximum (FWHM) 12
arcsec that corresponds to FWHM of the rocking curve of analyser
crystal.
4.1. Influence of Thickness Variation
Various cases of the thickness variation t for the barrier (layer A) and
QW (layer B) layers in the MQW structure were investigated according
to Eq. (10).
First, using Eq. (10), we explore the variation of the MQW period
thickness with increasing distance from the substrate (to the sample
surface). For this purpose, we simultaneously change the thickness of
both layers A and B. The thickness ratio between the layers A and B
does not change with depth; whereas, the addition to the thickness of
both layers t have the same sign and magnitude. As the average lat-
tice parameter of the period (Eq. (13)) depends on the lattice parame-
ters of both layers and their thickness ratio [16, 18, 37], it does not
change with depth:
A A B B
A B
,
av
c t c t
c
t t
(13)
where cA,B and tA,B are the lattice parameters and thickness of layers A
and B in the MQW structures.
Figure 2 shows the simulated 2 scans of the symmetrical (0002)
reflection (a), and the thickness profile of each period used for the
simulation (b). Two cases were considered: 1) a thickness change with
larger magnitude on the side of the substrate with a positive value of
t; 2) a thickness change with larger magnitude on the surface side
Fig. 1. Schematic sketch of MQW structure.
766 O. I. LIUBCHENKO and V. P. KLADKO
with a negative value of t (Fig. 2, b). It was observed that, when the
thickness of the periods in the MQW structure varies, the shape of one
satellite peak (here, the zero-order peak) does not change. Subsequent-
ly, we will call this peak as unchanged shape satellite. For other satel-
lite peaks, an asymmetry of the peak shape is observed, which depends
on the sign and magnitude of t.
First, we discuss the influence of a positive value of t. In this case
for the satellite peaks lying at lower angles than the unchanged shape
satellite (indicated on the spectra as 1, 2, 3), the asymmetry is more
pronounced on the higher-angle side of each peak. Satellite peaks at
larger angles than the unchanged shape satellite (1, 2, 3) have a clearly
visible anisotropy on their low angle shoulder. If the value of t is neg-
Fig. 2. The 2 scans of (0002) reflection for different cases of period thick-
ness variation (1 and 2) and calculated using the conventional model (a);
depth profiles of variation in the period thickness (b).
SIMULATION OF X-RAY DIFFRACTION SPECTRA FOR AlN/GaN STRUCTURES 767
ative, the asymmetric broadening of the peak shape is on the opposite
shoulder than in case of a positive t. In other words, for positive t,
the asymmetry of the satellite peaks arises from the side of the un-
changed shape satellite, and where the addition t is negative, asym-
metry emerges on peak shoulders pointing away from the unchanged
shape satellite.
The asymmetry effect is more pronounced for distant satellites rela-
tive to the unchanged shape one. In addition, there is a small angular
displacement of the satellite peaks, accompanied by an increase of the
peak width and a decrease of the intensity on direction of the asymmet-
ric broadening. At the same time, satellites, which were not observed
due to attenuation conditions [18], like the missing satellite of 3rd-
order in Fig. 2, a, may become clearly visible. It should be noted that
the XRD spectra weakly vary in dependence of the period change direc-
tion (from the substrate or from the surface). The differences are bet-
ter visible for low intense satellite peaks that are more distant from the
unchanged shape satellites. The main differences between positive and
negative t are manifested in high-frequency intensity oscillations,
which are poorly visible for this test sample due to the superposition
with high-frequency thickness oscillations from the buffer layers.
However, determining the side of a greater thickness change may be
possible for spectra with a clearly visible thin satellite structure. In
this case, the reason for the asymmetry is the variation of the period
thickness according to a certain law. The average lattice parameter of
the period, which depends on the layers thickness ratio, does not
change.
Different changes of the diffraction pattern can be observed (Fig. 3)
when the thickness of the layers A and B varies with the opposite sign
but the same magnitude of t (in antiphase). In this case, the thickness
of all GaN/AlN periods is constant, and according to Eq. (13), the av-
erage lattice parameter of the period changes with depth in dependence
on the thickness ratio between the layers A and B. Figure 3 shows the
2 scans of the symmetrical (0002), (0004), (0006) reflections (a–
c—black), which correspond to the thickness variation shown in Fig. 3,
d and are compared with the 2 scans (grey) of GaN/AlN layers of
constant thickness through the whole sample. In the case of thickness
variations as described above, the unchanged shape satellite vanishes,
and all reflections and satellite peaks of this MQW structure have
asymmetry. The peak asymmetry depends on the sign and magnitude
of t, and for the thickness profiles in Fig. 3, d, the peaks are broad-
ened asymmetrical toward the higher angles. When the thickness vari-
ation t of the AlN and GaN layers has opposite signs, the satellites
peaks are broadened toward the lower angles on the 2 axis (not
shown). From the spectra of different symmetrical reflections in Figs.
3, a–c, it is clear that with increasing diffraction order the satellite
768 O. I. LIUBCHENKO and V. P. KLADKO
peak asymmetry also increases. Other features of the satellites asym-
metry behave as in the previous case. In this case, the reason for the
asymmetry of all satellites is the change of the average lattice parame-
ter of each period, which is caused by simultaneous change in the
thickness of layers.
Next, we consider the interstitial case when the thickness of only
one layer in the period (A or B) changes. This results in change of the
period thickness and a change of the average lattice constant of each
period with increasing distance from the buffer layer. In addition, the
magnitude of the thickness change per period is two-times lower than
in the case when the thickness changes simultaneously for both layers.
Obviously, the change in the average lattice parameter of the period is
also smaller. The mutual influence of changes in the thickness and av-
erage lattice parameter leads to a MQW structure satellite’s asym-
metry, which depends on the sign of t and the law of its change. Fur-
thermore, the unchanged shape satellite has a slightly different angu-
lar position than in the previous case of period’s thickness change.
When the thickness of the GaN layers in the MQW structure changes,
the angular position of the unchanged shape satellite is close to the re-
flection of a single GaN layer with the same lattice parameters (and
state of relaxation) as in the GaN layers in the MQW structure. For
Fig. 3. The 2 scans of (0002) (a), (0004) (b) and (0006) (c) reflections corre-
sponding to depth profiles for AlN and GaN layers thickness (d) calculated
with thickness variation (black) and using the conventional model (grey).
SIMULATION OF X-RAY DIFFRACTION SPECTRA FOR AlN/GaN STRUCTURES 769
different-order reflections, the peak position of the GaN buffer layer is
close to the angular position of the unchanged shape satellite, whose
order also depends on the reflection order. For the case of thickness
variation of GaN layers in the MQW considered in this section, the un-
changed shape satellite has 1st-order for (0002) reflection, 2nd-order
for (0004) reflection, and 3rd-order for (0006) reflection. Similarly,
when only the thickness of AlN layers in MQW varies, the angular po-
sition of the unchanged shape satellite is close to the peak position of a
AlN buffer layer whose lattice parameters is the same as in the AlN
layers in the MQW. Applying this to our test structure, this means
that, when the thickness of GaN layers changes with depth, the angu-
lar position of the unchanged shape satellite is close to the angular po-
sition of the GaN buffer layer of the investigated symmetrical reflec-
tions. It should be noted that this arrangement of the unchanged shape
satellite peaks relative to the peak of the buffer layers is preserved on-
ly for fully strained or relaxed the AlN (GaN) MQW layers and AlN
(GaN) buffer layer, respectively. Other features of the satellites
asymmetry are the same as in the previous case but less pronounced
because the addition t has a lower magnitude. Thus, the mutual influ-
ence of period thickness variation and the change of the average lattice
parameter of the period lead to the appearance of satellites with un-
changing shape and with displaced angular positions relative to the
zero-order satellite and asymmetry of other satellites.
Our simulations have shown that, when the thickness of layers in
the MQW structure changes with depth, the thickness ratio of quan-
tum well/barrier layers and the average lattice parameter over the pe-
riods in the MQW structure vary as well. This leads to different asym-
metry of the satellite peak. Therefore, thickness variation in the MQW
structure layers is one of the reasons for asymmetry of the satellite
peaks in 2 scans.
4.2. Influence of Random Thickness Fluctuation
During the growth of MQW structures, interface roughness can lead to
local fluctuations in the thickness of each layer. To consider the inter-
face roughness, the following model similar to that described in Ref.
[32] was developed.
We split the sample into a large number of crystallites with random
thickness values of t for each layer in each crystallite. The resulting
spectrum was obtained as the averaged sum of the reflected intensities
from the crystallites. We considered the case of normal distribution of
random values of t. The generation of random numbers was done us-
ing the mt19937 random Number generator from the free library in
C . Figure 4 shows the effect of roughness on the 2 scans of our
MQW test structure. The increase of roughness leads to a broadening
770 O. I. LIUBCHENKO and V. P. KLADKO
of the satellite peaks with simultaneous reducing the amplitude of the
thickness oscillations. The amplitude is even more reduced for distant
satellites as compared with central satellites ( 1, 0, 1-orders). Figure 5
shows the effect of thickness fluctuation on the spectra of various
symmetrical reflections. The decrease of the oscillation amplitude is
also observed with the increase of the reflection order.
Therefore, roughness can be one of the reasons for a significant ex-
pansion of satellites in the 2 scan of the highest order reflections.
4.3. Comparing the Theory and Experimental Data
The theoretical framework described in the previous section has been
applied for the investigation of GaN/AlN MQW structures grown on
the AlN/GaN buffer layer. In contrast to the theoretical calculations
demonstrated in the previous sections, the relaxation state of real
MQW should be considered for accurate simulation.
Therefore, the in-plane lattice parameters of the MQW (aSL) and AlN
buffer layer (a) of samples S10 and S20 were determined from the re-
ciprocal space map (RSM) of the asymmetric (1233 ) reflection [16].
For both samples, the measured parameters are aSL 0.31283 nm and
Fig. 4. The 2 scans of (0002) reflection for different magnitudes of thick-
ness fluctuation.
SIMULATION OF X-RAY DIFFRACTION SPECTRA FOR AlN/GaN STRUCTURES 771
a 0.31007 nm. Consequently, the AlN buffer layer is slightly com-
pressed along the a-axis. The MQW lattice parameter aSL is closer to
that of bulk AlN (a 0.31113 nm) than to the a-parameter of GaN
(0.31893 nm). It indicates that GaN layers are in a compression state
(mismatch 1.9%), which has much larger magnitude than the tensile
stress of AlN layers. The out-of-plane lattice parameter c was recalcu-
lated from a according to Refs. [16, 35].
The 2 scans of (0002) reflection (samples S10 and S20) are shown
in Figs. 6, a, c, respectively. The figures show also the best fits using
the developed model considering a variation in the thickness and the
conventional model with a nominal thickness of the layers. The depth
profiles of thickness variation for both layers in the MQW structures
of the samples S10 and S20 are shown in Figs. 6, b and d. Since the peak
from the GaN buffer layer is almost invisible, it is not taken into ac-
count for the simulation. The simulation using the conventional model
showed a significant asymmetric broadening of satellite peaks, reveal-
ing thickness values of 4 nm and 5.5 nm for AlN and GaN layers, re-
spectively. Simulation of the spectra by using the developed model
showed significant variation of the thickness. The roughness deter-
mined by simulation was 1–2 atomic layers ( 0.3 nm to 0.5 nm). In the
investigated spectra, roughness mainly influences the intensity of the
Fig. 5. The 2 scans of (0002) (a), (0004) (b) and (0006) (c) reflections with
the same thickness fluctuation 1 nm (black) and 2 scans obtained using the
conventional model (grey).
772 O. I. LIUBCHENKO and V. P. KLADKO
2nd-order satellite.
The minimization methods and fitting procedure are described in
details in our previous work [38] for the case of implanted layers. We
used the Hooke–Jeeves direct search algorithm for fitting the spectra.
The error function is given by
exp calc
1
,Err
N
k
j j
j
I I (14)
where
exp
j
I and
calc
j
I are experimental and calculated intensity values
over all experimental points, k is the power index (we use k 0.1–0.8),
by which one can set the part of the spectrum that should be fitted bet-
ter. At lower values of k, the ‘tails’ of the spectrum are better fitted,
while the larger k promotes a better fit of the peaks.
The average relative error calculated for all points of the experi-
mental and fitted spectra (Eq. (15)) was 30–40% that is caused by the
noise, background, and high-frequency oscillation through the thick-
ness of buffer layers in the spectrum:
exp cal
rel exp
1
1
Err .
c
N
j j
j j
I I
N I
(15)
Fig. 6. Comparison of the measured (1) and simulated 2 scans of S20 (a)
and S10 (c) samples using the developed (2) and conventional (3) models. The
figures b and d show the depth profiles of thickness variation in the GaN QW
and AlN barrier layers for the samples S20 and S10, respectively.
SIMULATION OF X-RAY DIFFRACTION SPECTRA FOR AlN/GaN STRUCTURES 773
4.4. Discussion
In general, it can be difficult to separate the influence of different
MQW features (such as lateral thickness variation, lattice parameter
variation, dislocations) on the XRD spectra. The peak width and
shapes are commonly interpreted to be caused by a single dominant
feature. Because of the integrity of XRD methods, spectra from MQW
structures are sensitive to variations of the averaged lattice parame-
ter, strains and any kind of thickness variations on significant area of
the sample. As our examples showed, there is a significant asymmetry
of satellites in the investigated samples. As shown above, thickness
variation is the reason for this. Dislocations and lateral thickness vari-
ations cause only symmetrical broadening of satellites, like to that
caused by roughness. Simulating the experimental spectra by using
conventional model demonstrates the significant asymmetric peak
broadening, especially in the sample S20. For real structures, many
different features (listed above) are present simultaneously, but, in
the investigated samples S10 and S20, the asymmetrical peak broaden-
ing is predominant.
The modelling of 2 scans including thickness variation and the
roughness shows a good coincidence with the experiment data, but oth-
er effects may exist, which can cause the peaks asymmetry or both the
symmetrical and asymmetrical broadening. Simultaneous considera-
tion of these effects together with considered above thickness varia-
tion and roughness goes beyond a scope of this work. The resulting
depth profiles of the thickness can be explained by interpretation of
the spectra, if taking into account only effects of thickness variation
and roughness. It explains the significant values of thickness variation
obtained using simulation.
It is also important to choose the type of t function. The fitting
procedure can be carried out for any form of t, but it is necessary to
take into account the physical possibility of existence of this thickness
variation. It is better to perform studies by using additional depth pro-
filing methods such as secondary ion mass spectrometry or transmis-
sion electron microscopy, and then to use these data in XRD spectrum
simulation. This approach can help to separate the influence of various
effects on the spectrum and to investigate better their nature.
5. SUMMARY AND CONCLUSIONS
In this paper, a detailed X-ray diffraction analysis of the influence of
layer thickness variation and thickness fluctuations for MQW struc-
tures was carried out. It was ascertained that variation of the thick-
ness is one of the reasons for an asymmetry of satellite peaks. Varia-
tion of the period thickness with a constant ratio of A and B layer
774 O. I. LIUBCHENKO and V. P. KLADKO
thickness leads to existence of the unchanged shape satellite and a dif-
ferent degree of the asymmetry for other satellites. If there is varia-
tion of thickness in both A and B layers in MQW structure, and here-
with the period is kept constant, there is the asymmetry of all satel-
lites. In the general case, there are both the asymmetry and the un-
changed-shape satellite with angular position shifted relatively to the
previous one. The size and side of the satellite asymmetry depends on
the magnitude and sign of the layer thickness change. This asymmetry
is clearer pronounced in distant-order satellites, and it increases for
higher-order reflections. A random thickness fluctuation leads to
smoothness and broadening of the satellites. These effects are also
more pronounced for high-order satellites and significantly increase in
higher-order reflections. Simulating of experimental spectra showed
the necessity of using other independent methods for separating the
investigated effects from others such as lateral thickness variation,
lattice parameter variation, etc. Further studies should be performed
to separate the listed above effects using other analytical methods in
combination with XRD-methods.
ACKNOWLEDGMENTS
This work was funded by the Program of Fundamental Research of the
National Academy of Sciences of Ukraine ‘Fundamental Problems of
New Nanomaterials and Nanotechnologies’. Furthermore, the authors
thank PhD Hryhorii V. Stanchu and PhD Sergiy B. Kryvyi for useful
discussions, and acknowledge Y. Polishchuk for critical reading of the
manuscript.
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