Investigating the effects of smoothness of interfaces on stability of probing nano-scale thin films by Neutron Reflectometry
Most of the reflectometry methods which are used for determining the phase of complex reflection coefficient such as Reference Method and Variation of Surroundings medium are based on solving the Schrödinger equation using a discontinuous and step-like scattering optical potential. However, during t...
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Jahromi, S.S. Masoudi, S.F. 2017-06-11T07:39:38Z 2017-06-11T07:39:38Z 2012 Investigating the effects of smoothness of interfaces on stability of probing nano-scale thin films by Neutron Reflectometry / S.S. Jahromi, S.F. Masoudi // Condensed Matter Physics. — 2012. — Т. 15, № 1. — С. 13604: 1-10. — Бібліогр.: 12 назв. — англ. 1607-324X PACS: 68.35.-p, 63.22.Np DOI:10.5488/CMP.15.13604 arXiv:1204.5825 https://nasplib.isofts.kiev.ua/handle/123456789/120155 Most of the reflectometry methods which are used for determining the phase of complex reflection coefficient such as Reference Method and Variation of Surroundings medium are based on solving the Schrödinger equation using a discontinuous and step-like scattering optical potential. However, during the deposition process for making a real sample the two adjacent layers are mixed together and the interface would not be discontinuous and sharp. The smearing of adjacent layers at the interface (smoothness of interface), would affect the the reflectivity, phase of reflection coefficient and reconstruction of the scattering length density (SLD) of the sample. In this paper, we have investigated the stability of Reference Method in the presence of smooth interfaces. The smoothness of interfaces is considered by using a continuous function scattering potential. We have also proposed a method to achieve the most reliable output result while retrieving the SLD of the sample. Бiльшiсть методiв рефлектометрiї, що використовуються для визначення фази комплексного коефiцiєнту вiдбивання такi як еталонний метод i змiна прилеглого середовища грунтуються на розв’язку рiвняння Шредiнгера з використанням розривного i сходинкоподiбного оптичного потенцiалу розсiювання. Проте, пiд час процесу напорошування для пiдготовки реального зразка два сусiднi шари змiшуються i мiжфазова границя може не бути розривною i чiткою. Розмивання сусiднiх шарiв при мiжфазовiй границi (гладкiсть iнтерфейсу), може мати вплив на вiдбивання, фазу коефiцiєнта вiдбивання i перебудову густини довжини розсiювання зразка. В цiй статтi ми дослiдили стiйкiсть еталонного методу у присутностi гладких мiжфазових границь. Гладкiсть мiжфазових границь розглядається, використовуючи неперервну функцiю потенцiалу розсiювання. Ми також запропонували метод для отримання найбiльш надiйного результату, який вiдновлює густину довжини розсiювання зразка. en Інститут фізики конденсованих систем НАН України Condensed Matter Physics Investigating the effects of smoothness of interfaces on stability of probing nano-scale thin films by Neutron Reflectometry ослiдження впливу гладкостi мiжфазових границь на стiйкiсть зондуючих наномасштабних тонких плiвок за допомогою нейтронної рефлектометрiї Article published earlier |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine |
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DSpace DC |
| title |
Investigating the effects of smoothness of interfaces on stability of probing nano-scale thin films by Neutron Reflectometry |
| spellingShingle |
Investigating the effects of smoothness of interfaces on stability of probing nano-scale thin films by Neutron Reflectometry Jahromi, S.S. Masoudi, S.F. |
| title_short |
Investigating the effects of smoothness of interfaces on stability of probing nano-scale thin films by Neutron Reflectometry |
| title_full |
Investigating the effects of smoothness of interfaces on stability of probing nano-scale thin films by Neutron Reflectometry |
| title_fullStr |
Investigating the effects of smoothness of interfaces on stability of probing nano-scale thin films by Neutron Reflectometry |
| title_full_unstemmed |
Investigating the effects of smoothness of interfaces on stability of probing nano-scale thin films by Neutron Reflectometry |
| title_sort |
investigating the effects of smoothness of interfaces on stability of probing nano-scale thin films by neutron reflectometry |
| author |
Jahromi, S.S. Masoudi, S.F. |
| author_facet |
Jahromi, S.S. Masoudi, S.F. |
| publishDate |
2012 |
| language |
English |
| container_title |
Condensed Matter Physics |
| publisher |
Інститут фізики конденсованих систем НАН України |
| format |
Article |
| title_alt |
ослiдження впливу гладкостi мiжфазових границь на стiйкiсть зондуючих наномасштабних тонких плiвок за допомогою нейтронної рефлектометрiї |
| description |
Most of the reflectometry methods which are used for determining the phase of complex reflection coefficient such as Reference Method and Variation of Surroundings medium are based on solving the Schrödinger equation using a discontinuous and step-like scattering optical potential. However, during the deposition process for making a real sample the two adjacent layers are mixed together and the interface would not be discontinuous and sharp. The smearing of adjacent layers at the interface (smoothness of interface), would affect the the reflectivity, phase of reflection coefficient and reconstruction of the scattering length density (SLD) of the sample. In this paper, we have investigated the stability of Reference Method in the presence of smooth interfaces. The smoothness of interfaces is considered by using a continuous function scattering potential. We have also proposed a method to achieve the most reliable output result while retrieving the SLD of the sample.
Бiльшiсть методiв рефлектометрiї, що використовуються для визначення фази комплексного коефiцiєнту вiдбивання такi як еталонний метод i змiна прилеглого середовища грунтуються на розв’язку рiвняння Шредiнгера з використанням розривного i сходинкоподiбного оптичного потенцiалу розсiювання.
Проте, пiд час процесу напорошування для пiдготовки реального зразка два сусiднi шари змiшуються i мiжфазова границя може не бути розривною i чiткою. Розмивання сусiднiх шарiв при мiжфазовiй границi (гладкiсть iнтерфейсу), може мати вплив на вiдбивання, фазу коефiцiєнта вiдбивання i перебудову густини довжини розсiювання зразка. В цiй статтi ми дослiдили стiйкiсть еталонного методу у присутностi гладких мiжфазових границь. Гладкiсть мiжфазових границь розглядається, використовуючи неперервну функцiю потенцiалу розсiювання. Ми також запропонували метод для отримання найбiльш надiйного результату, який вiдновлює густину довжини розсiювання зразка.
|
| issn |
1607-324X |
| url |
https://nasplib.isofts.kiev.ua/handle/123456789/120155 |
| citation_txt |
Investigating the effects of smoothness of interfaces on stability of probing nano-scale thin films by Neutron Reflectometry / S.S. Jahromi, S.F. Masoudi // Condensed Matter Physics. — 2012. — Т. 15, № 1. — С. 13604: 1-10. — Бібліогр.: 12 назв. — англ. |
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Condensed Matter Physics, 2012, Vol. 15, No 1, 13604: 1–10
DOI: 10.5488/CMP.15.13604
http://www.icmp.lviv.ua/journal
Investigating the effects of smoothness of interfaces
on stability of probing nano-scale thin films
by neutron reflectometry
S.S. Jahromi∗, S.F. Masoudi
Department of Physics, K.N. Toosi University of Technology, P.O. Box 15875-4416, Tehran, Iran
Received August 5, 2011, in final form February 7, 2012
Most of the reflectometry methods which are used for determining the phase of complex reflection coefficient
such as reference method and Variation of Surroundings medium are based on solving the Schrödinger equa-
tion using a discontinuous and step-like scattering optical potential. However, during the deposition process
for making a real sample the two adjacent layers are mixed together and the interface would not be discon-
tinuous and sharp. The smearing of adjacent layers at the interface (smoothness of interface), would affect the
reflectivity, phase of reflection coefficient and reconstruction of the scattering length density (SLD) of the sam-
ple. In this paper, we have investigated the stability of reference method in the presence of smooth interfaces.
The smoothness of interfaces is considered by using a continuous function scattering potential. We have also
proposed a method to achieve the most reliable output result while retrieving the SLD of the sample.
Key words: neutron reflectometry, thin films, smooth potential, reference method
PACS: 68.35.-p, 63.22.Np
1. Introduction
In the past decades, neutron reflectometry has been developed as an atomic-scale probe with appli-
cations to the study of surface structure ranging from liquid surfaces to the solid thin films. The type and
thickness of the nanostructure materials can be determined from measuring the number of neutrons,
reflected elastically and specularly from the unknown sample [1–3].
Neutron reflectometry problems are generally divided into two groups; “Direct problems” and “In-
verse problems”. The goal of the direct problems is to solve the Schrödinger equation for a distinct sample
so that the neutron wave function is determined; On the other hand, the application of inverse problems
is to extract the information of the interacting potential by using the complex reflection coefficient [1].
Measuring the intensity of reflected neutrons in terms of the perpendicular component of neutron
wave vector to the surface, q , provides us with useful information about the scattering length density of
the sample along its depth. As with any scattering technique in which only the intensities are measured,
the loss of the phase of reflection would result in unresolvable ambiguities on retrieving the SLD from
measurement. Without the knowledge of the phase of reflection, more than one SLD can be found for the
same reflectivity data. By knowing the phase of the reflection, a unique result would be obtained for the
depth profile of the unknown sample [4, 5].
In the recent years, several methods have beenworked out for determining the complex reflection co-
efficient such as: Dwell time method, variation of surroundings medium and the reference layer method
which seems to be the most practical [1].
All of these methods are worked out for an ideal sample in which the interface of two adjacent layers
is discontinuous and sharp. In this case, the reflection coefficient is determined by the solution of one
dimensional Schrödinger equation for a step-like optical potential. However, as we know from a real
∗E-mail: s.jahromi@dena.kntu.ac.ir
© S.S. Jahromi, S.F. Masoudi, 2012 13604-1
http://dx.doi.org/10.5488/CMP.15.13604
http://www.icmp.lviv.ua/journal
S.S. Jahromi, S.F. Masoudi
(a) (b)
Figure 1. Smearing of the SLD of two adjacent layers at the interface. The SLD of layer have been mixed
together during the deposition process. (a) Smearing of layers in atomic scale. (b) Continuous variation
of SLD at the interface in one dimension.
sample, there is some smearing at the boundaries. This generally happens during the deposition of a
top layer which is miscible with the bottom material (figure 1). This process is strongly temperature
dependent. In this case, the interface is defined as a thin layer acrosswhich, the SLD of the sample varies
smoothly around the mean position of the interface [6].
In most of the simulation methods, the reflectivity measurement and phase determination process
is performed for an ideal sample. Considering a continuous and smooth varying optical potential at the
interface, it would affect the retrieved reflection coefficient. In the present work, we have studied the ef-
fects of potential smoothness on retrieving the depth profile of the sample and the phase of the reflection
by considering a continuous potential at boundaries, using reference layers method. As retrieving the
SLD from the real and imaginary part of the reflection coefficient is very sensitive, we are going to inves-
tigate the way one can find the characteristics of an unknown sample in the presence of the smoothness
effects.
2. Theory and method
Interaction of neutron with nuclei is demonstrated by the neutron optical potential v(z), which is a
function of the scattering length density of the sample, v(z) = 2πħ2ρ(z)/4π where ρ(z) is the scattering
length density (SLD) profile as a function of the coordinate z; normal to the sample surface [2, 7]. As
the thickness of the sample is of the order of nanometer, multiple scattering is neglected in neutron
specular reflectometry and the neutron specular reflection is accurately described by a one-dimensional
Schrödingerwave equation {
∂2
z +
[
q2
0 −4πρ(z)
]}
ψ(q, z) = 0, (1)
where ψ(q, z) is the neutron wave function and q0 is the z-component of the incident neutron wave
vector in vacuum.
Specular reflection is rigorously invertible, however, if both the modulus and the phase of the com-
plex reflection coefficient, r (q0), were known, a unique result for scattering length density, ρ(z) would
be determined [5].
An exact relation for the reflection coefficient r (q0) can be derived from the transfer matrix which
is a 2×2 matrix with the elements A(q0),B(q0),C (q0), and D(q0) which are determined by solving the
equation (1), for a film with SLD ρ(z) [3, 7].
Correspondingly, for a sample with arbitrary surroundings, the solution of equation (1) in terms of
the transfer matrix elements is represented by:
(
1
ih
)
teihqL =
(
A B
C D
)(
1+ r
i f (1− r )
)
, (2)
where t(q) is the transmission coefficient and f and h are the refractive index of the fronting and backing
medium having SLD values of ρ f and ρh respectively. For non-vacuum surroundings, n =
(
1−4π
ρn
q2
0
)1/2
where n stands for f and h. Correspondingly, for vacuum fronting or backing n(q) = f = h = 1 [1].
The transfer matrix has a unit determinant and is unimodular;
A(q0)D(q0)−B(q0)C (q0)= 1. (3)
13604-2
Neutron reflectometry in the presence of smooth inter-layer interfaces
As the other property of the transfer matrix, reversing the potential, ρ(z) −→ ρ(L − z), would cause the
interchange of the diagonal elements A and D without having any effect on the off-diagonal elements, B
and C [1].
By solving the equation (1) for the reflection coefficient, r (q),we have:
r =
(
f 2h2B2 + f 2D2
)
−
(
h2 A2 +C 2
)
−2i
(
f h2 AB + f CD
)
(
f 2h2B2 + f 2D2
)
+
(
h2 A2 +C 2
)
+2 f h
. (4)
The reflectivity R(q) = |r (q)|2 can be represented in terms of the transfer matrix elements by introducing
a new quantity Σ(q):
Σ(q)= 2
1+R f h
1−R f h
=
(
f 2h2B2 + f 2D2
)
+
(
h2 A2 +C 2
)
(5)
where R is the amplitude of the complex reflection coefficient and the superscript f h, denotes a sam-
ple with fronting and backing medium with refractive index f and h, respectively. By introducing the
following new quantities
α f h =
(
f −1h A2 + f −1h−1C 2
)
,
β f h =
(
f hB2 + f h−1D2
)
,
γ f h = h AB +h−1CD. (6)
Equations (4) and (5), are represented as
r f h =
β f h −α f h −2iγ f h
β f h +α f h +2
, (7)
Σ(q) =β f h +α f h . (8)
These two equations are the two key relations in reference method. Equation (2) shows that below the
critical wave number qc = 4πMax
(
̺ f ,̺h
)1/2
, the method is not applicable since the information of r (q)
is not available. However, the missing data below the qc could be accurately retrieved by extrapola-
tion [1, 7]. In section 4 we will introduce an algebraic method for determining the missing value below
the critical q .
2.1. Reference layer method
Here we explain how one can find the complex reflection coefficient of any unknown non-absorptive
layer by using some reference layers. Suppose a known layer has been mediated between the fronting
medium and the unknown layer. In this case, the transfer matrix of the whole sample (containing the
known and unknown layers) can be expressed by the multiplication of the transfer matrix of each one as
follows: (
A B
C D
)
=
(
a b
c d
)(
w x
y z
)
, (9)
where (a, . . . ,d) refer to the matrix elements of the unknown part and (w, . . . , z) present the matrix ele-
ments of the known part. Consequently we have
Σ=
(
h2a2 +c2
)(
w2 + f 2x2
)
+
(
h2b2 +d2
)(
y2 + f 2z2
)
+2
(
h2ab +cd
) (
w y + f 2xz
)
. (10)
Alternatively, the above formula can be denoted in compact notation in terms of three unknown real-
valued parameters as
Σ(q) = 2
1+R f h
1−R f h
= h2β̃
f f
K
αhh
U + f 2α̃
f f
K
βhh
U +2 f hγ̃
f f
K
γhh
U , (11)
where the subscripts K and U refer to the known and unknown parts of the sample and the tilde rep-
resents the parameters of the sample with mirror reversed potential (A ←→ D). f f and hh denote the
same fronting and backing media having the SLD values of ρ f and ρh , respectively.
13604-3
S.S. Jahromi, S.F. Masoudi
In order to determine the reflection coefficient, r (q), we consider three measurements of reflectiv-
ity for the three known reference layers. Using equation (6) and (11) for three different reference lay-
ers (K1,K2,K3), three equations will be obtained. By solving these equations in matrix algebra as shown
in equation (12), the parameters (αhh
U
,βhh
U
,γhh
U
) are determined which represent the r hh (q) for the un-
known film in contact with a uniform identical surroundings in front and back with ρh .
αhh
U
βhh
U
γhh
U
= M
Σ1
Σ2
Σ3
, M =
h2β̃
f f
K1
f 2α̃
f f
K1
2 f hγ̃
f f
K1
h2β̃
f f
K2
f 2α̃
f f
K2
2 f hγ̃
f f
K2
h2β̃
f f
K3
f 2α̃
f f
K3
2 f hγ̃
f f
K3
−1
. (12)
For q > qc this gives us the reflection coefficient
r hh
U =
βhh
U
−αhh
U
−2iγhh
U
βhh
U
+αhh
U
+2
, (13)
which is equivalent to the reflectivity of a “free film” defined by ρequiv(z) = ρU(z)−ρh. By inverting r hh
U
and shifting by a known constant value, the SLD of the free film can be retrieved [1, 8].
2.2. Determining the phase below the critical wave number
There are several ways for retrieving the missing data below the critical wave number such as ex-
trapolation or using polarized neutron and a magnetic substrate. Here we introduce an algebraic method
to retrieve the data below qc.
The transfer matrix for a layer having constant SLD in terms of wave number is represented by:
T =
(
cos nq0L 1
n sin nq0L
−n sin nq0L cosnq0L
)
, (14)
where n =
√
1−4π̺/q2
0 , L is the thickness and ̺ is the SLD of the sample.
Introducing several new parameters, ̺eq = ̺−̺h , qeq =
√
q2
0 −4π̺h and neq =
√
1−4π̺eq/q2
eq, for
T we have
T =
(
cos neqqeqL 1
hneq
sin neqqeqL
−hneq sin neqqeqL cosneqqeqL
)
. (15)
By inserting the elements of transfer matrix in equation (13) and using equation equation (7), we
have:
r hh (̺)
∣∣∣
q0=
p
4π̺h
= r 11(̺−̺h)
∣∣∣
q j =
√
q2
0−4π̺h
. (16)
Equation (16) shows that knowing r hh (̺) for q0 Ê
√
4π̺h , the reflection coefficient of a free sample
having SLD value of ̺eq = ̺−̺h , r 11(̺−̺h), is known for specific wave number q j =
√
q2
0 −4π̺h for
q j Ê 0.
2.3. Smooth variation of interface
As we mentioned in section 2.1, there is some smearing at boundaries in real samples and the SLD
varies smoothly and continuously from one layer to another. The smooth variation of SLD at boundaries
can be expressed by Error function as follows [9]:
̺(z)= ̺1(z)+ ̺2(z)−̺1(z)
2
[
1+erf
(
z −∆
p
2σ
)]
, (17)
where σ is the smoothness factor. In other word, σ is the thickness of the smoothly varying area across
the interface. ∆ is also the mean position of the interface or the turning point of the Error function. Since
the reference layer method is based on discontinuity of the interfacial potential, it is obvious that the
effect of smoothness at boundaries would affect the reconstruction procedure. In the next section we
have investigated these effects numerically.
13604-4
Neutron reflectometry in the presence of smooth inter-layer interfaces
Figure 2. A known reference layer over the Cu-Ni sample with smooth potential at boundaries. The red
lines demonstrate the ideal sample with non-smooth boundaries.
3. Numerical example
As a realistic example to study the stability of the reference method by using a smooth potential, we
consider a sample with 20 nm thick Copper on 30 nm Nickel having a constant SLD of 6.52×10−4 nm−2
and 9.4×10−4 nm−2 as the unknown layer; and three separate reference layers with 20 nm Au, 15 nm
Cr and 10 nm Co having a constant SLD value of 4.46,3.03 and 2.23 (×10−4 nm−2), respectively. Figure 2
shows this configuration for the gold layer as reference. Figure 3 also shows the reflectivity of the sample
of figure 2 for three difference smoothness factors. The smearing of potential at boundaries is taken into
account by using σ = 0, 0.2 and 0.5 nm. The effects of smoothness on output results are clear in the
figure, particularly at large wave numbers, while the data for small wave numbers truly correspond with
none-smooth data.
Reflectivity of the unknown part of the sample with smooth varying SLD at the interface with σ= 5 Å
Figure 3. (Color online) Reflectivity from the sample of figure 2 for three cases: σ= 0, 0.2 and 0.5 nm.
13604-5
S.S. Jahromi, S.F. Masoudi
is illustrated in figure 4 (upper curve). Similar to figure 3, the effects of smoothness are more obvious
at large wave numbers. However, consideration of smoothness at interface would cause some abrupt
noises at certain values of wave numbers such as: q ≃ 0.3 and 0.6 nm−1. The noises also appear for the
real and imaginary data of reflection coefficient (figure 5 (a), (b)). These noises are due to the abrupt
changes in some of the elements of the M matrix, equation (12), in the reference method. The intensity
and distribution of these noises along the different ranges of wave numbers are extremely relevant to
the smoothness factor, thickness of the reference layers and the range of neutron wave numbers. In
order to better understand the origin of these noises, we have plotted the M(3,3) element of the M matrix
in figure 4 (lower curve). The result shows that M(3,3) has some noises in q values, exactly as for the
reflectivity data. Such severe noises in output results would cause a complete loss of reflection coefficient
data in those q values. As we mentioned in section 2, the data of reflection coefficient in the whole range
of wave numbers, even bellow the critical wave number, are needed to retrieve the SLD of the sample.
Unfortunately, the noises are distributed over different values of wave numbers ranging from small to
large q ’s. This deficiency makes the data useless in retrieving the SLD of the sample. Here we are going
to propose a method for removing or reducing these noises.
Figure 4. Reflectivity of the unknown sample of figure 2 which is reconstructed by using three reference
layers: 20 nm gold, 15 nm Cr and 10 nm Co (Circled curve is for σ= 0.5 nm and solid line is for the ideal
sample, σ= 0). M(3,3) element of the M matrix, equation (12), is plotted for better understanding of the
origin of reflectivity noises.
The stability of themethod to a great extent depends on the M matrix which is a function of the known
part of the sample. The best result for the reflection coefficient of the unknown part of the sample comes
out of the equation (12) in the case where the behavior of the M matrix is smooth. Our investigation
shows that not every arbitrary reference layer can lead to a smooth result for the M matrix. Since we
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Neutron reflectometry in the presence of smooth inter-layer interfaces
can control the choice of the reference layers, we can run a simulation to find the best layers for which
the M matrix behaves smoothly and without noise. By choosing the best reference layer, we can retrieve
reliable results for the unknown part of the sample. Those reference layers which are well behaved can
be used for experimental implementation of the reflectometry experiments. As we mentioned before,
the noise strongly depends on the thickness of the reference layers, and the choice of a proper thickness
for the reference films would enhance the accuracy of the output results. In our example, the numerical
calculations show that the best result with the smallest noise level would be obtained for 15, 10 and 20 nm
thicknesses for Au, Co and Cr, respectively. The purified curves of real and imaginary parts of reflection
coefficient in this case are shown in figures 6 (a), (b). As it is clear in the figure, we have much better
results with less noise in comparison with figure 5.
(a)
(b)
Figure 5. The reflection coefficient of the unknown part of the sample of figure 2 for two different σ= 0,
5 Å by three reference layers as; 20 nm Au, 15 nm Cr and 10 nm Co. (a) Real and (b), imaginary parts of
the reflection coefficient.
Thereupon, the choice of an appropriate thickness for the reference layers would completely remove
the noise or displace it to large wave numbers, where it would cause no ambiguity in retrieving the SLD
profile of the sample. The rest of the noises can be removed by extrapolation. By using these corrected
data as input for some useful codes such as the one which is developed by Sacks [10–12] based on the
Gel’fand-Levitan integral equation [1–3], the scattering length density of the sample is retrieved. The data
of reflection coefficient up to q = 0.6 are sufficient to retrieve the SLD by Sacks code and we can neglect
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S.S. Jahromi, S.F. Masoudi
the data of large wave numbers. Hence, after purification of data, any remaining noise in the range of
large wave numbers can be neglected.
The SLD profile corresponding to r hh (q) was retrieved by using the real and imaginary parts of the
reflection coefficient at the presence of the smoothness. As it is shown in figure 7, the method of reference
layers is stable at the presence of smoothness and the SLD corresponding to r hh(q) is truly reconstructed.
The circled curve depicts the retrieved SLD profile of the sample. The retrieved thicknesses corresponds
well with the thickness of the sample.
(a)
(b)
Figure 6. The reflection coefficient of the unknown part of the sample of figure 2 for σ= 0, 5 Å. The three
reference layers are: 15 nm Au, 20 nm Cr and 10 nm Co. Due to the proper selection of reference layers,
the noise is removed in comparison with the result of figure 4. (a) Real and (b), imaginary parts of the
reflection coefficient.
Consideration of smoothness of interfaces makes the output results more corresponding to experi-
mental data. In order to verify the efficiency of the method, the reflectivity curves were plotted in figure 8
for a sample from [9], Al2O3(6.7 nm)/FeCo(20 nm)/GaAS, for either of smooth (green solid line) and non-
smooth boundaries (red dashed line), in contrast to the experimental reflectivity curve. As it is shown
in the figure, the reflectivity curve in the presence of smoothness of interfaces better corresponds with
the experimental reflectivity curve while the simulated reflectivity for an ideal sample does not abso-
lutely correspond with the experimental curve, particularly at large wave numbers. The results verify
that consideration of the smoothness of interfaces is of highest importance.
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Neutron reflectometry in the presence of smooth inter-layer interfaces
Figure 7. The reconstructed SLD of the unknown sample shown in figure 2, surrounded by identical
medium on both sides (̺hh ), obtained by inversion of the extracted values of the reflection coefficient
in themomentum range 0 É q É 0.6 nm−1. The non-smooth profile is shown by thin solid line. The circled
curve represents the SLD of the sample in the presence of smoothness.
4. Conclusion
In most of the simulation methods of neutron reflectometry, the SLD of adjacent layers is supposed
to be discontinuous and sharp. Since in real samples there is observed some smearing at boundaries,
consideration of smoothness would effect the output results and the retrieved data.
In this paper by considering a certain type of Error function which denotes the smooth variation of
SLD across the interface and implementing it into the reference method, the effects of smoothness on
determining the phase and retrieving the SLD profile of the sample were investigated. Due to consider-
ation of smoothness of interfaces, several abrupt noises existed in the output, such as reflectivity, real
and imaginary parts of reflection amplitude. It was shown that these noises are due to the abrupt change
in the elements of M matrix of equation (12). We proposed to choose a proper thicknesses for the refer-
ence layers, which would completely remove the noise or shifted it to the large wave numbers (q > 0.6),
Figure 8. (Color online) Experimental reflectivity curve in contrast to the simulated reflectivity at the pres-
ence of smoothness of boundaries and the ideal sample. The green solid line demonstrates the reflectivity
at the presence of smoothness and better corresponds with the experimental curve.
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S.S. Jahromi, S.F. Masoudi
where the data of reflectivity are not needed in order to retrieve the SLD and we can easily neglect them.
Consequently, the SLD of the sample was retrieved by using the purified data of the reflection coefficient.
In order to make sure that the method is highly reliable, we also compared the simulated reflectivity
with the experimental curve for a sample. Astoundingly, the results show that consideration of smooth
variation of SLD at the interface would make the measured reflectivity to better correspond with the
experimental data. The results also certify that the reference method is stable in the presence of smooth
interfaces.
References
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2. Zhou X.-L., Chen S.-H., Phys. Rep., 1995, 257, 223; doi:10.1016/0370-1573(94)00110-O.
3. Penfold J., Thomas R.K., J. Phys.: Condens. Matter, 1990, 2, 1369; doi:10.1088/0953-8984/2/6/001.
4. Kasper J., Leeb H., Lipperheide R., Phys. Rev. Lett., 1998, 80, 2614; doi:10.1103/PhysRevLett.80.2614.
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7. Majkrzak C.F., Berk N.F., Physica B, 2003, 336, 27; doi:10.1016/S0921-4526(03)00266-7.
8. Majkrzak C.F., Berk N.F., Physica B, 1996, 221, 520; doi:10.1016/0921-4526(95)00974-4.
9. Fitzsimmons M.R., Majkrzak C.F., Application of polarized neutron reflectometry to studies of artificially struc-
tured magnetic materials. In: Modern Techniques for Characterizing Magnetic Materials, edited by Yimei Zhu,
Springer, New York, 2005; doi:10.1007/0-387-23395-4_3.
10. Klibanov M.V., Sacks P.E., J. Comput. Phys., 1994, 112, 273; doi:10.1006/jcph.1994.1099.
11. Chadan K., Sabatier P.C., Inverse Problem in Quantum Scattering Theory, 2nd edn, Springer, New York, 1989.
12. Aktosun T., Sacks P., Inverse Prob., 1998, 14, 211; doi:10.1088/0266-5611/14/2/001;
Sacks P., Aktosun T., SIAM J. Appl. Math., 2000, 60, 1340; doi:10.1137/S0036139999355588;
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Дослiдження впливу гладкостi мiжфазових границь
на стiйкiсть зондуючих наномасштабних тонких плiвок
за допомогою нейтронної рефлектометрiї
С.С. Джаромi, С.Ф. Масудi
Фiзичний факультет, Технологiчний унiверситет iм. K.Н. Тусi, а/с 15875-4416, Тегеран, Iран
Бiльшiсть методiв рефлектометрiї, що використовуються для визначення фази комплексного коефiцiєн-
ту вiдбивання такi як еталонний метод i змiна прилеглого середовища грунтуються на розв’язку рiвня-
ння Шредiнгера з використанням розривного i сходинкоподiбного оптичного потенцiалу розсiювання.
Проте, пiд час процесу напорошування для пiдготовки реального зразка два сусiднi шари змiшуються i
мiжфазова границя може не бути розривною i чiткою. Розмивання сусiднiх шарiв при мiжфазовiй гра-
ницi (гладкiсть iнтерфейсу), може мати вплив на вiдбивання, фазу коефiцiєнта вiдбивання i перебудову
густини довжини розсiювання зразка. В цiй статтi ми дослiдили стiйкiсть еталонного методу у присутностi
гладких мiжфазових границь. Гладкiсть мiжфазових границь розглядається, використовуючи неперерв-
ну функцiю потенцiалу розсiювання. Ми також запропонували метод для отримання найбiльш надiйного
результату, який вiдновлює густину довжини розсiювання зразка.
Ключовi слова: нейтронна рефлектометрiя, тонкi плiвки, гладкий потенцiал, еталонний метод
13604-10
http://dx.doi.org/10.1021/la0341254
http://dx.doi.org/10.1016/0370-1573(94)00110-O
http://dx.doi.org/10.1088/0953-8984/2/6/001
http://dx.doi.org/10.1103/PhysRevLett.80.2614
http://dx.doi.org/10.1016/j.physb.2004.10.038
http://dx.doi.org/10.1007/s00339-009-5515-5
http://dx.doi.org/10.1016/S0921-4526(03)00266-7
http://dx.doi.org/10.1016/0921-4526(95)00974-4
http://dx.doi.org/10.1007/0-387-23395-4_3
http://dx.doi.org/10.1006/jcph.1994.1099
http://dx.doi.org/10.1088/0266-5611/14/2/001
http://dx.doi.org/10.1137/S0036139999355588
http://dx.doi.org/10.1088/0266-5611/16/3/317
Introduction
Theory and method
Reference layer method
Determining the phase below the critical wave number
Smooth variation of interface
Numerical example
Conclusion
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