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Institute Of Physics
X-ray Reflectrometry
Supervisor
Andreas Biermanns
( Kamran Ali,Tahir kalim,Muhammad saqib)
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TABLE OF CONTENTS
1. Abstract.....3
IntroductionScope and Outline of the X-ray Reflectivity Technique3
Theoretical Introduction...4
2. ExperimentalDetails...............................................................14
Setup and procedure.14
Evalution of the measured curve.16
3. Simulation..........................26
4. Conclusion..27
5.References28
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1. X-ray Reflectivity Technique:-
Abstract:-
X-ray has become an invaluable tool to probe the structure of matter. It has revolutionized our life
by helping scientists to unravel such mysteries as the structure of DNA and in more recent times, the
structure of proteins. The main reason for this unprecedented success is that the X-ray wavelength,
which determines the smallest distance one can study with such a probe, is comparable to the inter-
atomic dimension. For research in the fields of physics, chemistry, biology and materials science,
photons ranging from radio frequencies to hard "- rays provide some of the most important tools for
scientists. Since X-rays discovered by W.C. Roentgen [1845-1923] in 1895 , Subsequent
experiments, conducted by Max von Laue and his students, showed that X-rays are of the same
nature as visible light but with shorter wavelength. In this work, X-rays are used to investigate the
properties of thin film .because most properties Of thin film is thickness dependent. Furthermore
this technique determined the density and Roughness of films and also the multilayers with high
precision.XRR is non-destructive and Non-contact technique for thickness determination between 2-
200nm with precision of about 1-3A.
Figure 1:-schmatic diagram of x-ray reflectivity
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Theoretical Introduction:-
Fundamental of specular x-ray reflectivity
Index of Refraction:-
When Roentgen discovered X-rays in 1895, he immediately searched for a means to focus
them . On the basis of slight refraction in prisms he stated that the index of refraction of X-
rays in materials cannot be much more than n = 1.05 if it differed at all from unity. Twenty
years later, Einstein proposed that the refractive index for X-rays was n = 1-Where =10-6,
allowing for total external reflection at grazing incidence angles. The reflection index n of a
medium for electromagnetic radiation depends on the frequency . The refraction index in x-ray
range of any material is slightly smaller than unity. Usually the refractive index, n, of the materials
with respect to the x-ray is written as:-
1n i
Where, considers the dispersion and the absorption. The parameters are related to the absorption
coefficient, , and electron density e.
2
0
2
er
4
u
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Where r0 is the classical electron radius. e the electron density is the x-ray wavelength. The
imaginary component of the refractive index stems from x-ray absorption. u is the linear absorption
coefficient of the material. These formulas from equation 2.14 to 2.16 refer to the literature.
Snells law, well-known from the optics, which is used also for x-rays, relates the incident grazing
angle i to the refracted angle f in the matter, as shown in equation.
cos cosi fn
As the index of refraction is smaller than unity for X-rays, the phenomenon of total external reflection
occurs for incident angles i smaller than the critical angle, c . The critical angle for total external
reflection is obtained as
Fig 2:- shows Schmatic view of X-ray reflection and refraction
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Ideal Surface: Fresnel Reflectivity:-
For homogeneous media with idealized smooth surfaces, from equation (1) together with the
components of k perpendicular to the interface, we can derive the Fresnel equations as
ri
Er
E
2t
i
Et
E
r and t are referred to amplitude reflectivity and transmittivity. The corresponding intensity
reflectivity and transmittivity are the absolute square of the r and t.Since and are small, the cosines can be expanded as
This gives that is a complex number for a given incidence angle . So can be
decomposed as
The transmitted wave falls off with increasing depth into the material. And the intensity falls
off with a 1/e penetration depth given by
In the reciprocal space, the wave vector transfers are
Q Z = 2K sin
Then the Fresnel reflectivity is written as:
22 2
22 2
2
2 2
2 2
cZ Z
F Z
cZ Z
Q Q Q i k R Q
Q Q Q i k
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The representation eliminates all setupspecific factors of a measurement.
The Fresnel reflectivity can be replaced by
Figure 3. The reflectivity R(q), the transmittivity T(q) and the penetration depth for different values of the
parameter b =/2.Small value of b correspond to low absorption.
On the surface of realistic samples, there is no abrupt change in the density (or refractive
index) takes place. The surface is always rough on atomic level (order of magnitude nm). This
cause an intensity reduction of the reflected wave and additional diffuse scattered intensity.
By using the laterally averaged electron density;
, ,
e ez x y z dxdy
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we describe the transition of the electron density from medium 1 to medium 2 as
2 11 . , ,,z f z e ee e
For ideally smooth surface, the function f(z) is a step function so the change of electron
density takes place abruptly. For a real, rough surface one usually uses a normalized Gaussian
distribution function for the gradient of the electron density perpendicular to the surface:
In which the is the root mean square roughness and 2is the routmeansquare deviation ofthe surface height z(x,y) from its mean value.
Fig.4 A rough surface with the mean height zjhas fluctuations z(x,y) around this value.
The reflectivity of a tough surface however differs from the ideal Fresnel reflectivity only
noticeable for angles above c. The ratio between the reflectivity of the rough surface
R(Q)and the Fresnel reflectivity RF(Q) is:
0
2iQz
F
R Q dfe dz
dzR Q
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Using the normalized Gaussian distribution model for function f, one gets:
2 2Q
FR Q R Q e
X-Ray Reflection from a thin Slab:-
The reflectivity from a slab of finite thickness is considered. As shown in fig.1.3.1, the finite
slab of thickness sits on an infinitely thick slab 2. There is now an infinite series of possible
reflections sums up the total reflectivity rslab.
In details:
1. The wave with wave vector k and glancing angle is partly reflected at interface 0 to 1,
amplitude r01.2. The wave is transmitted from 0 to 1, amplitude t01, and then reflected at interface 1 to 2,amplitude r12, followed by transmission at interface 1 to 0, amplitude t10, this wave is now
phase shifted due to the way through the layer and back for a phase factor;
3. The wave after another set of reflections at the upper and lower surface of the layer, with
amplitude , is now phase shifted for a phase factor
.
The total amplitude reflectivity is therefore:
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Considering the Fresnel equations, with
And we get :
Insert these relations to the equation of rslab, one obtains:
The intensity reflectivity given by |is plotted in fig. 1.3.1. Due to the phase factor with a periodof 2/, it displays oscillations known as Kiessig oscillations. The peaks in the oscillationscorrespond to the waves scattering in phase, and the dips to them scattering out of phase.
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Wave vector transfer Q
Figure. 5 Kiessig oscillations from a thin layer of tungsten
In case of n2
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X-Ray Reflection from multilayer:-
We can extend the result we got before to multilayer structures consisting of N layers on an
infinitely thick substrate. Each layer j has a refraction index:
And the thickness .
Figure 6 Composition of the multilayer system
From the solution of Maxwell equations at the boundaries it follows that x component of the
wave vector is conserved in all layers. The z component of the wave vector within the layer jis
The wave vector transfer in the layer j is,
We first neglect multiple reflections and calculate the reflectivity at interface between the
layer j and j+1
so for the lowest layer it is written as
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Next, we consider the multiple reflections. The case for the layer N on an infinite substrate is
exactly the case of a thin slab we have discussed before with replacing the medium 0 by the
layer N1. So according to rslab we can write
The notation r represents reflectivity including multiple reflections in comparison with r for
no multiple reflections. With rN1,N we can now write rN2,N1 according to the same equation
for rslab. If we do this again and again, we can get the reflectivity at the top of the multilayer
system.
Considering the roughness of the individual interfaces j on the specular reflected intensity as
we did before, we replace the Fresnel coefficient by a new set of coefficients
With is the root mean square roughness of the interface j.
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2-Experimental Set-up for X-ray Reflectivity Techniques:-
The XRD measurement were carried out using a 2 circle diffractometer at home lab siegen.
The wave length of the incident beam X-ray was 1.54A.which correspond to the Cu K line.Following are the essentials parts of the Diffractometer.
X-ray Tube: the source of X Rays
Incident-beam optics: condition the X-ray beam before it hits the sample
Thegoniometer: the platform that holds and moves the sample, optics, detector, and/tube
Thesample & sample holder
Receiving-side optics: condition the X-ray beam after it has encountered the sample
Detector: count the number of X Rays scattered by the sample
Filter is used in order to prevent the saturation
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Omega-Scan: an X-ray tool for the characterization of thin film properties:-
-scan is done to align the sample so that the incidence angle equals the exit angle.
The specimen is turned by 360 around a certain axis, e.g., the surface normal. The X-ray
source and the detector have to be adjusted depending on the crystal type to get a sufficient
number of reflections per turn. The angular positions of these reflections are used to evaluate
the orientation of the crystal lattice (completely described by three angles) with relation to the
rotation axis. In order to relate the lattice orientation exactly to the surface of a crystal, the
direction of the surface is checked by a laser beam. Also other relevant crystal reference faces
or directions can be measured by optical tools in a similar way. This measuring technique
enables to determine the orientation of arbitrary single crystals in any orientation range with
high precision. Usually, a measuring time of some seconds (during one or a few turns of the
specimen) is sufficient to get reproducibility in the range of a few arc seconds. A special
application of the Omega-Scan Method is the precision lattice-parameter determination,
especially of cubic crystal.
Advantages of the Omega-Scan Method compared with other X-ray diffraction
techniques are:
Stable and relative simple arrangement
(X-ray tube and detector in fixed positions, only one measuring circle, no
monochromator).
All the data necessary for the complete orientation determination are measured by one
turn.
High precision at low measuring time.
From the Omega-Scan diagram are calculated (e.g.): the inclination angle between onecrystallographic axis (here: the a axis) and the reference plane as well as the direction of the
inclination in this plan
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Evaluation of the measured curve:-
Our characterization of the sample was periodic Carbon and tungsten thin film deposited over
Silicon substrate. The angular range of the whole measurement ran from 0 to 17.998for
2and 0 to 8.999 for . The measurement was taken in two parts as the following table
shows.
S.NO 2(degree) (degree) Filter
Limit
(Initial-final)Step size
Limit
(Initial- Final)Step size
1 0.000 ; 3.200 0.004 0.000 ; 1.600 0.002 YES
2 3.030;17.998 0.008 1.515 ; 8.999 0.004 NO
Table 1 shows the measurement condition
In the first part, the smallest angular step is used to get a high resolution. A filter is used to
reduce the over high intensity for small angles. While for larger angles the filter is not needed
and the counting time per measured point should be increased for better statistics.
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The measurements for these two graphs in logarithmic scale are shown in following figures
0,0 0,5 1,0 1,5 2,0 2,5 3,0 3,5
1
10
100
1000
10000
Intensity
2 Theta degree
B
Fig.8.Reflectivity scan for angular range 0 to 3.2 in log scale
2 4 6 8 10 12 14 16 18 20
1
10
100
1000
10000
100000
Inten
sity
2 Theta
B
Fig.9 Reflectivity scan for angular range 3.030 to17.998 in log scale
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Critical Angle:-
Since the refractive index, n < 1 , the X-rays are refracted away from the normal to the surface
when they enter the matter (the Snells law). Consequently, there exists a critical angle of the
incidence, c, below which total reflection of X-rays occurs. From the reflectivity curve one
can determine the critical angle. It is the value of 2 Theta where the reflecting intensity Icdecreased to 50% of the maximum intensity Imax .
The precise determination of the critical angle of total external reflection is necessary for
electron density analysis.
From the first part of the reflectivity curve, we measured the critical angle
-0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5
-2000
0
2000
4000
6000
8000
10000
12000
14000
16000
18000
20000
22000
24000
Intensity
2 Theta(degree)
B
X=0.748(critical angle)
Y=10069
Fig:-11 determination of the critical angle
The critical angle c= (0.748 0.01) /2=0.374= (6.52x10-3 rad)
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Electron Density:-
The electron density el can be determined using the relation:
With wave length =1.54A and the classical electron radius rel = 2. 82x10-15 m.
The critical angle in radian unit is c =6.52x10-3 rad
Hence, el (electron density)= 19.96 x1023 cm-3for multilayer.
The precision of our density determinations can be estimated as:
So we have:
We get the electron density for multilayer with precision:
el (electron density)= (19.96 0.5) x1023 cm-3
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Determination of Layer Thickness:-
If we have a special structure, for example C/W Si substrate, the wave can be trapped
by total external reflection between the Si substrate and the Si top layer and travel parallel to
the surface in the C/W layer: We have made a wave guide for x-rays. The wave guide will
trap modes, if the thickness of the layer matches, n=1,2,3,... - then we have the nodes of the
trapped wave on the top- and bottom interface of the wave guide. The excitation of such a
waveguide mode will show up as a sharp dip in the reflected intensity - the dip will be the
sharper and shallower, the less damping or losses the guided mode experiences.. Five
resonances can be observed. If a periodic structure of layers with different refractive indices is
prepared, we will get "Bragg reflection" at the corresponding angle. The width of this Bragg
peak depends on how many layers contribute to the scattering, i.e. the penetration depth. The
angular width of the Bragg peak is directly related to the energy-acceptance at fixed angle by
Bragg's law. Thus a W:C multilayer accepts a bandwidth of 2-3% at 10 keV
0 2 4 6 8 10 12 14 16 18 20
1E-8
1E-7
1E-6
1E-5
1E-4
1E-3
0.01
0.1
1
Intensity
2 Theta degree
1
2
3
4 5
Fig 12 The total layer thickness and the period of the multilayer.
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Peak(m) 1 2 3 4 5
im(degree) 1.433 2.810 4.157 5.551 6.928
Table:- Satellite maxima and its order numbers as (2 )/ 2
0 5 10 15 20 25
0.000
0.002
0.004
0.006
0.008
0.010
0.012
0.014
0.016
one double layer
Linear
m2
Equation y = a + b*x
Weight No Weighting
Residual Sum ofSquares
4.4417618E-9
Adj. R-Square 0.9999534
Value Standard Error
B Int ercept 5. 8057753E- 5 2.7841174E-5
B Slope 5.827047E-4 1.9896707E-6
All angles are converted from degree
to radian in Y-axis and determined their square valuesand
the above slope of fitting line gives the multilayer thickness according to equation below
Y = A + B * X where c2=a , b= (
)
2
Parameter Value Error
A 5.8057753E-5 2.7841174E-5B 5.827047E-4 1.9696707E-6
2D
b
The wave length of the incident x-ray was 1.54 Awhich correspond to the Cu K
line 1.54Aand the slope give a
T= (31.89 0.05) A
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The total layer thickness and the period of the multilayer (fig below)
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5
1E-5
1E-4
1E-3
0.01
0.1
1
(Intensity)
2in Degree
reflectivity spectrum
Fig 13:-shows the interference fringes
M M2 2 2/2=im im2(rad2)
2 4 0.9471 0.4735 6.824x10-5
3 9 1.023 0.5115 7.946x10-5
4 16 1.116 0.558 9.475x10-5
5 25 1.210 0.605 1.1138x10-4
6 36 1.313 0.6565 1.3115x10-4
7 49 1.435 0.7175 1.566x10-4
8 64 1.548 0.774 1.823x10-49 81 1.660 0.83 2.096x10-4
10 100 1.731 0.8955 2.440x10-4
11 121 1.914 0.957 2.787x10-4
12 144 2.036 1.018 3.153x10-4
13 169 2.167 1.083 3.572x10-4
14 196 2.289 1.144 3.986x10-4
15 225 2.411 1.205 4.422x10-4
16 256 2.552 1.276 4.954x10-4
17 289 2.683 13.415 5.476x10-4
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0 100 200 300
0.0000
0.0001
0.0002
0.0003
0.0004
0.0005
0.0006
B
Linear Fit of B
M2
Equation y = a + b*x
Weight No Weighting
Residual Sum ofSquares
6.2741321E-11
Adj. R-Square 0.9998123
Value Standard Error
B Inte rc ept 5.7572785E -5 8.8202755E -7
B Slope 1.5167611E-6 5.3658121E-9
rad)
2
The slope gives a T= 626.3 0.06A
This means that our film consist of N=626.3/31.8=19.6~20 periods.
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3. Simulation:-
The measured reflectivity is simulated with the module IMD of the XOP program. The
programme computes the reflectivity of an arbitrary layered system.the measured data has
been divided by two to have in theta degree.The multilayers system consist of 20 reguler c/w
film deposited on silicon substrate
All of the film parameters are modified with respect to the bulk are to be entered in to the
program.
Number of Periods N=20
Carbon: =2.859 g/cm3
Tungsten: = 19.258g/cm3
Silicon: =2.330 g/cm3
Multilayer thickness: 15. 94 A
Fig.14 fitting of the entire reflectivity curve
The red color of the reflectivity curve correspond to the data in the lab and the green refer to
Simulated data over the whole range of scanned angles. we can see that the simulated and the
measured peaks do fit up exactly to a certain degrees.
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The simulated model below takes a closer look to our spectrum. We can see clearly that thesets of peaks are fitted exactly in first and second set of peaks. And we have a model to
determine the values for roughness using IMD programme.
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4. Conclusion: -
A glancing, but varying, incident angle, combined with a matching detector anglecollects the X rays reflected from the samples surface and Interference fringes. In the reflected
signal we used to measure the total thickness and multilayer Thickness, density and roughness
of the thin film through this technique which is called X-ray reflectivity and our aim was to
use this technique in this experiment and this was successful with precession of 10%. The
simulation of the reflectivity curve is in good agreement with measured curve as we can see
the positions of peaks and the intensity.
The critical angle c= (0.748 0.01) /2=0.374= (6.52x10-3 rad)
The average el (electron density)= (19.96 0.5) x1023 cm-3
For the thickness of the multilayer and the thin film thickness was determined
TDL= (31.89 0.05)A
TML= (626.3 00.6)A
The film consist of N=626.3/31.8=19.6~20 periods
From Simulation we get the roughness values which are as
Density:-
(c) = 2.859 g/cm3
(W) = 19.258 g/cm3
Thickness:-
Z(C) = 15.95Ao
Z (W) = 15.94Ao
Roughness:-
( Vacuum/C ) = interface 2.31Ao
( W/C ) = interface 3.00Ao
( C/ W ) = interface 4.30Ao
(W/ Si ) = interface 2.52Ao
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5. References
[1] http: // www. esrf. eu/ computing/ scientific/ xop2. 1/
[2] Pietsch, Ulrich ; Holy, Vaclav ; Baumbach, Tilo: High-Resolution X-Ray Scattering:
From Thin Films to Lateral Nanostructures (Advanced Texts in Physics). Springer, 2004
[3] Blodgett, K.B. ; Langmuir, I.: Built-Up Films of Barium stearate and their Optical
Properties. In: Phys. Rev. 51 (1937), S. 964(from manual)
[4] Als-Nielsen, Jens ; Mc Morrow, Des: Elements of Modern X-ray Physics. John Wiley
and Sons, 2001(from manual)
[5]www.wikipedia.com
http://www.wikipedia.com/http://www.wikipedia.com/http://www.wikipedia.com/http://www.wikipedia.com/