Numerical simulation of the reciprocal space maps measured from the ion-implanted single-crystal yttrium-iron garnet films on gadolinium-gallium garnet substrate has been carried out basing on the theoretical model of the triple-crystal dynamical diffractometry of crystalline multilayer systems with inhomogeneous strain distributions and randomly distributed defects. The presence of growth defects in both film and substrate as well as radiation defects created in subsurface layer of nanometer-scale thickness after 90
keV F+ ion implantation was taken into account in the proposed model of the film system

Len, E.G.

Molodkin, V.B.

Olikhovskii, S.I.

Ostafiychuk, B.K.

Pylypiv, V.M.

Skakunova, O.S.

SocioEconomic Challenges Journal (Sumy State University)

PROCEEDINGS OF THE INTERNATIONAL CONFERENCE  
NANOMATERIALS: APPLICATIONS AND PROPERTIES 
Vol. 4 No 1, 01PCSI05(4pp) (2015) 
 
 
2304-1862/2015/4(1)01PCSI05(4) 01PCSI05-1  2015 Sumy State University 
Simulation of Reciprocal Space Maps for Thin Ion-Implanted Layers in  
Yttrium-Iron Garnet Films with Defects 
 
S.I. Olikhovskii1,*, O.S. Skakunova1, V.B. Molodkin1, E. G. Len’1, B.K. Ostafiychuk2, V.M. Pylypiv2 
 
1 G.V. Kurdyumov Institute for Metal Physics of NASU, Vernadsky Blvd. 36, 03680 Kyiv, Ukraine 
2 V. Stefanyk Precarphatian National University, Shevchenko Str. 57, 76025 Ivano-Frankivsk, Ukraine 
 
(Received 31 May 2015; published online 27 August 2015) 
 
Numerical simulation of the reciprocal space maps measured from the ion-implanted single-crystal yt-
trium-iron garnet films on gadolinium-gallium garnet substrate has been carried out basing on the theoret-
ical model of the triple-crystal dynamical diffractometry of crystalline multilayer systems with inhomoge-
neous strain distributions and randomly distributed defects. The presence of growth defects in both film 
and substrate as well as radiation defects created in subsurface layer of nanometer-scale thickness after 90 
keV F+ ion implantation was taken into account in the proposed model of the film system. 
 
Keywords: X-Ray Dynamical Diffraction Theory, Yttrium Iron Garnet, Film, Ion Implantation, Strain, 
Radiation Defects. 
 
 PACS numbers: 61.05.cc, 61.05.cp, 61.72.Dd, 
61.72.jd, 61.72.Nn, 61.72.Qq, 61.80.Jh, 61.82.Ms 
 
 
                                                   
* olikhovsky@meta.ua 
1. INTRODUCTION 
 
Modern technologies allow to grow almost perfect 
multilayered crystalline systems and purposefully to 
influence on their physical properties by the controlled 
introducing different structural defects. In particular, 
epitaxial single-crystal yttrium iron garnet (YIG) 
Y3Fe5О12 films grown on gadolinium gallium garnet 
(GGG) Gd3Ga5O12 substrate are widely used in energy-
independent magnetic memory devices, magnetic mi-
croelectronics, and integral magneto-optics, etc. (see, 
e.g., Ref. [1]). Modification of physical properties of 
their functional layers, the sizes of which often are of 
nanometer scale, is carried out, in particular, by ion 
implantation. 
X-ray diffraction methods are most widely used to 
characterize structural defects and strains in the modi-
fied crystal layers, In particular, the measurements by 
high-resolution triple-crystal diffractometer (TCD) 
were carried out to characterize inhomogeneous strain 
distributions and structure imperfections in implanted 
garnet crystals [2, 3]. These methods are highly in-
formative if the analytical formulas which give the 
adequate description of measured diffraction intensity 
distributions are available. 
The purpose of this work is demonstration of diag-
nostic possibilities offered by the theoretical method for 
interpretation of results of high-resolution X-ray dif-
fraction mapping the real film and multilayered sys-
tems with defects [4]. This method of structural diag-
nostics will be applied to the simulation of the influence 
of nanometer-sized defects and inhomogeneous strain 
distributions in the epitaxial YIG films grown on GGG 
substrate, which were implanted with 90 keV F+ ions, 
on the form of coherent and diffuse components of dif-
fraction intensity distributions on reciprocal lattice 
maps measured by TCD. 
  
 
2. COHERENT AND DIFFUSE COMPONENTS 
OF RECIPROCAL LATTICE MAPS 
 
Differential X-ray diffraction intensity distributions 
measured by TCD from crystalline multilayer systems 
with inhomogeneous strain distributions and randomly 
distributed defects can be represented as a sum of co-
herent (Icoh) and diffuse (Idiff) components.  
If Bragg angles for monochromator, analyzer, and 
crystal under investigation are nearly equal, it is possi-
ble to ignore the dispersion effects and the coherent 
component of the two-dimensional intensity distribution 
can be calculated approximately in the case of quasi-non-
dispersive geometry (m, n, m) as follows [5, 6]: 
  
     
 
  
 


     
       

1
coh 0 M cohMS
A S S
,  
1 ,
m
m
x
x
I I b dzR b z R z
R b z b
 (1) 
 
where I0 is an intensity of incident X-ray beam, RM and 
RA are the reflection coefficients of monochromator and 
analyzer crystals,   and   are the angular devia-
tions of the crystal under investigation and analyzer 
crystal, bM and bS are asymmetry parameters of mono-
chromator and sample crystals, respectively. The ex-
pression for the coherent component of the reflection 
coefficient Rcoh(Δ) for imperfect crystal has been de-
rived elsewhere earlier [7] and takes into account all 
the dynamical scattering effects including the extinc-
tion of Bragg waves due to diffuse scattering by defects.   
Particularly, the dynamical consideration is neces-
sary in the case of X-ray diffraction by the implanted 
YIG films, thickness of which is of the order of an ex-
tinction length. The coherent component of reflection 
coefficient of such inhomogeneous crystalline system 
with chaotically distributed defects can be calculated in 
so called ‘layer approximation’ by using the recurrence 
relations between coherent components of amplitude 
 S.I. OLIKHOVSKII, O.S. SKAKUNOVA, V.B. MOLODKIN ET AL. PROC. NAP 4, 01PCSI05 (2015) 
 
 
01PCSI05-2 
reflection coefficients of adjacent layers for Bragg dif-
fraction geometry [5]: 
 
 




 


2 2
1
1
( )
1
j j jj j
j
j j j
r R t r
R
r R
, (2) 
 
where jr  and jt  are amplitude reflectivity and trans-
missivity of j th layer, respectively, and  1j .  
The diffuse component of the diffraction intensity 
measured by TCD can be represented approximately by 
the following expression [5, 6]: 
 
       diff 0 iM iA diff,  I I R R r κ . (3) 
 
where MiR  and AiR  are the integrated reflectivities of 
monochromator and analyzer crystals, respectively, 
and components kx and kz of two-dimensional vector κ  
describe a deviation from the reciprocal lattice point H 
in the coherent scattering plane.  
Function diff ( )r κ  in Eq. (3) represents the diffuse 
component of the differential reflection coefficient of 
the investigated sample after integration over vertical 
divergence [7]: 
 
 

 diff ext abs diff
0
( ) ( ),
M
j j j
j
r F F rκ κ  (4) 
 
where ext
jF  is the extinction factor describing the influ-
ence of redistribution of X-ray intensity between 
transmitted and diffracted coherent waves in j th layer, 
the factor abs
jF  describes photoelectric absorption and 
absorption due to diffuse scattering in layers lying 
above j th layer,  and diff ( )
jr κ is the diffuse component of 
the differential reflection coefficient of j th layer.  
Thus, the formulas presented above provide the pos-
sibility for the effective, i.e., with minimization of re-
quired calculation time expenses, and quantitative self-
consistent analysis of coherent and diffuse components of 
reciprocal lattice maps, which are measured from crys-
talline film or multilayer systems, including those with 
inhomogeneous strain distributions, large strain gradi-
ents at layer interfaces, and randomly distributed de-
fects of different types in all the layers and substrate. 
 
3. MODEL OF THE DEFECT STRUCTURE IN 
ION-IMPLANTED YIG FILM 
 
The sample of the epitaxial YIG film with 5.33 µm 
thickness, which was grown on GGG substrate and im-
planted with F+ ions of the energy of 90 keV  at dose D = 
61013 cm2, has been chosen as a model object for simula-
tion.  
This sample was investigated earlier by using the 
high-resolution double-crystal X-ray diffractometer [8]. 
As has been established by simultaneous treatment of 
the rocking curves measured for two reflections, the 
strain profile in YIG film implanted with F+ ions has the 
shape shown in Fig. 1. This strain profile is formed due 
to the radiation defects, which are distributed inhomo-
geneously in depth and cause the corresponding «in 
average» inhomogeneous strain. The depth profile of the 
strain caused by ion implantation was calculated by 
using the determined characteristics of spherical amor-
phous clusters in implanted layer, namely, concentration 
of 51020 cm3, radius of 0.75 nm, and volume misfit 
strain at the boundary between cluster and crystal ma-
trix of 0.0375. 
 
 
 
Fig. 1 – Strain profile in YIG film implanted with F+ ions 
(E = 90 keV) at dose D = 61013 cm2 
 
 Besides, in the model of defect structure of the im-
planted epitaxial single-crystalline system of YIG film 
grown on GGG substrate the presence of two types of 
growth microdefects, namely, spherical clusters and 
circular prismatic dislocation loops was taken into 
account.  Their characteristics were supposed to be the 
same as determined in Ref. [8], namely, for spherical 
clusters in YIG film and GGG substrate we putted 
cluster radii of 10 and 8 nm, respectively, concentration 
of 1014 cm3, and volume misfit strain at the boundary 
between cluster and crystal matrix of 0.03. Similarly, 
for dislocation loops of <110>-type in YIG film we 
putted dislocation loop radius of 5 nm and concentra-
tion of 1015 cm3, whereas in GGG substrate we putted 
dislocation loop radii of 5 and 90 nm, and concentra-
tions of 1015 and 1,21012 cm3, respectively. 
The YIG/GGG film system in whole was considered 
as a multilayer system in each layer of which the strain 
was consisted of sum of average and fluctuating com-
ponents. At modeling the average strain distribution, 
the implanted layer was subdivided into 10 nm thick 
laminae, whereas the transition layer between film and 
substrate was subdivided into layers with thicknesses 
of order of few hundreds of nanometers. 
 
4. SIMULATION OF RECIPROCAL LATTICE 
MAPS FOR ION-IMPLANTED YIG FILMS 
WITH DEFECTS  
 
The results of numeral simulation of reciprocal lattice 
maps are described below for the YIG film, which was 
implanted with F+ ions of the energy of 90 keV  at dose D 
= 61013 cm2. The maps were simulated for the case of 
measurements by the high-resolution PANalytical X`Pert 
Pro MRD XL diffractometer for (444) reflection of charac-
teristic CuK1 radiation in the symmetrical Bragg diffrac-
tion geometry at the sample under investigation. As far 
as the Bragg angle for Ge (220) reflection used in mono-
chromator and analyzer crystals of the diffractometer is 
nearly equal to Bragg angles for (444) reflection of YIG 
film and GGG substrate, the influence of dispersion ef-
fects can be neglected and for the calculation of coherent 
and diffuse components of reciprocal lattice maps the 
simplified formulas (1) and (3) can be used. 
 SIMULATION OF RECIPROCAL SPACE MAPS… PROC. NAP 4, 01PCSI05 (2015) 
 
 
01PCSI05-3 
To elucidate the role of growth microdefects in YIG 
film and GGG substrate in the formation of diffraction 
patterns, we first calculated the reciprocal lattice map 
for the symmetrical 444 reflection of CuK1 radiation 
from as-grown YIG/GGG film system (Fig. 2). As is evi-
dent from comparison between the total reciprocal lattice 
map and its coherent component, the diffuse scattering 
intensity from microdefects in both YIG film and GGG 
substrate give a substantial contribution to the for-
mation of observed diffraction intensity distributions. 
 
 
 
Fig. 2 – Reciprocal lattice map (middle), its coherent compo-
nent (left), and longitudinal cross section (right) for 444 reflec-
tion of CuK1 radiation from as-grown YIG/GGG film system 
 
The reciprocal lattice map simulated for the sym-
metrical 444 reflection of CuK1 radiation from the 
YIG/GGG film system implanted with F+ ions of 90 keV 
energy at dose D = 61013 cm2 is shown in Fig. 3.  
 
  
 
Fig. 3 – Reciprocal lattice map (middle), its coherent compo-
nent (left), and longitudinal cross section (right) for 444 reflec-
tion of CuK1 radiation from YIG/GGG film system implanted 
with F+ ions (E = 90 keV) at dose D = 61013 cm2 
 
As can be seen by comparing Figs. 2 and 3, the ir-
regularly shaped diffuse scattering intensity distribu-
tion from spherical clusters in implanted YIG layer 
appears additionally to the diffuse scattering intensity 
distributions from microdefects in both YIG film and 
GGG substrate around corresponding reciprocal lattice 
points. It should be remarked that the irregular shape 
of diffuse scattering intensity distribution from spheri-
cal clusters in implanted layer is caused by the inho-
mogeneous ‘in-average’ strain distribution in this layer, 
which leads to smoothing and deformation of the dou-
ble-drop form known for isodiffuse lines from spherical 
clusters.  
It should be emphasized especially that longitudinal 
cross sections of the simulated reciprocal lattice maps 
in Figs. 2 and 3 closely coincide in form with the rock-
ing curves, which were measured by using the high-
resolution double-crystal X-ray diffractometer for the 
corresponding as-grown and implanted epitaxial YIG 
films [8]. Only the distinction consists in that the oscil-
lations at the tail of the rocking curve measured from 
implanted epitaxial YIG film are something smoothed 
as compared with longitudinal cross section of the sim-
ulated reciprocal lattice map in Fig. 3 what is explained 
by an additional contribution of diffuse scattering in-
tensity from defects in implanted YIG layer due to 
integration over vertical divergence.  
 
5. RESUME AND CONCLUSIONS 
 
On the basis of the created theoretical model of the 
triple-crystal X-ray dynamical diffractometry of the 
multilayered crystalline systems with inhomogeneous 
strain profile and randomly distributed defects, the 
numeral simulation of reciprocal lattice maps which 
are measured from the ion-implanted single-crystal 
YIG films on GGG substrate has been carried out. The 
presence of growth defects in both film and substrate as 
well as radiation defects created in subsurface layer 
was taken into account  
It is established that, as a result of superposition of 
coherent and diffuse scattering intensities from defects 
of various types in different layers with heterogeneous 
strain, the reciprocal lattice maps take the form which 
substantially differs from that characteristic for crys-
tals with homogeneous strain and one type of defects.   
Self-consistent description of coherent and diffuse 
components of dynamical X-ray scattering with account 
for instrumental factors of TCD allows the correct 
quantitative analysis of complete reciprocal lattice 
maps measured from inhomogeneous imperfect crystal 
structures with defects.  
 
 
 
  
 S.I. OLIKHOVSKII, O.S. SKAKUNOVA, V.B. MOLODKIN ET AL. PROC. NAP 4, 01PCSI05 (2015) 
 
 
01PCSI05-4 
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Simulation of Reciprocal Space Maps for Thin Ion-Implanted Layers in Yttrium-Iron Garnet Films with Defects

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Simulation of Reciprocal Space Maps for Thin Ion-Implanted Layers in Yttrium-Iron Garnet Films with Defects

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SocioEconomic Challenges Journal (Sumy State University)