x ray reflectometry

8/10/2019 x Ray Reflectometry

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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/

x ray reflectometry

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