Task 1. Diffractometry of polycrystalline thin film
1.1. Task
Experiment of polycrystalline thin film diffraction, phase analysis, Rietveld refinement
1.2. Theory
The experiment is performed with monochromatic x-ray beam. Let us assume the polycrystalline grains
are small, thus the irradiated volume contains huge number of grains. The diffraction condition is satisfied
only for small number of grains
2d sin θ = nλ , (1.1)
where angle of incidence equals Bragg angle θ, d is inter-planar distance, n diffraction order and λ wavelength
of x-ray radiation.
The intensity of particular diffraction peak is given by structure factor, depending of structure of crystallographic
unit cell
F(h) =
N
i=1
fi(h)e−ih·ri
, (1.2)
where summation runs over N atoms in the unit cell, h is diffraction vector, fi(h) is atomic form factor of
i-th atom in the unit cell and ri its position. Several high symmetry types of unit cells have certain selection
rules of forbidden reflection where structure factor equals zero.
For cubic crystals following selection rules for allowed reflections apply:
simple cubic lattice – all reflections are allowed,
face centered cubic – only those are allowed for which Laue indices have same parity,
body centered cubic – allowed diffractions have h+k+l even,
diamond lattice – all Laue indices odd; or all Laue indices are even and their sum is divisible by 4.
1.3. Experimental setup
Angular distribution of diffracted intensity will be measured in Bragg-Brentano setup and grazingincidence
diffraction setup for thin films.
1.3..1 Bragg-Brentano method
The principle of Bragg-Brentano is shown in figure 1.1. Let us assume the x-ray source S is ideal points
source and both source and detector D are moving along circle with the goniometer radius kBB. The sample
is positioned in the center of the goniometer circle kBB. If the sample was circular with radius kv defined
by source S, detector D and goniometer center V, the beam from the point source scattered at any sample
point with scattering angle of 2θ would meet at the same detector point D. In reality we use flat samples; the
focusing condition is still satisfied as far as the sample is small with respect to the radius of focusing circle.
In reality we will measure with setup shown in figure 1.2. The divergence of the primary beam is limited
by a divergence slit. The detection is performed with linear position sensitive detector. We are using copper
x-ray tube with dominant Kα1 line with wavelength of 1,540601 Å and Kα2 line 1,544430 Å. Kβ line is
suppressed using nickel filter in the scattered beam. Soller slits are used to define the divergence in the
direction perpendicular to the diffraction plane.
First we have to align the diffractometer and sample. We cane use automatic macros which align the
beam to the center of the goniometer and also sample surface to the center of goniometer. Measurement is
the performed for the selected rage of scattering angles, typically 15–120◦. Angular step size in the selected
setup can be set to ∆2θ = 0.01◦. Better angular resolution cannot be achieved because of detector pixel size.
The date are saved in the text format with column corresponding to 2θ angle and the second column to
scattered intensity.
1
Obrázek 1.1. Principle of Bragg–Brentano setup for powder diffraction. X-ray is originating in a point source
S and is detected by a point detector D.
1.4. Evaluation of diffraction peak positions
The lattice parameters can be determined from peak positions only. If the lattice is cubic you can simply
determine lattice parameter by following procedure
1. Determine angular positions of diffraction peaks in 2θ. At high angles the profiles are split, why?
2. Sort the positions and try to find corresponding allowed Laue indices N = h2 + k2 + l2 for translation
type. Well corresponding type should give same lattice parameter given by formula (1.1) for any
diffraction order.
3. determine average lattice parameter. Every diffraction line has different uncertainty of lattice parameter
determination; higher orders leads to a more precise determination. We suggest to use weighted average
¯a =
n
i=1 wiai
n
i=1 wi
, (1.3)
where the weight is inversely proportional to a standard uncertainty of each point σi as wi = 1/σ2
i .
The standard deviation of weighted average is then
σ2
¯a =
1
(n − 1) n
i=1 wi
n
i=1
wi(ai − ¯a)2
, (1.4)
where n is number of measured diffraction lines.
Obrázek 1.2. Bragg–Brentano setup with linear position sensitive
detector for powder diffraction measurements.
20 30 40 50 60 70 80 90
0
0.5
1
1.5
2
2.5
3
3.5
4
x 10
4
2θ (deg)
Intenzita(#pulzu)
80 81 82 83 84 85
Obrázek 1.3. Example of diffractogramm
measured using Bragg–
Brentano setup. Inset shows detail of
80–85◦.
2
Fast evaluation can be done using computer programs and database of powder diffraction data (PDB –
powder diffraction database, paid license). Free available database is COD (crystallography open database)
at webpage: http://www.crystallography.net/cod/
1.5. Width of diffraction peaks
The diffraction line width is given by several factors; the basic ones are the following
• crystallites size
• inhomogeneous strain
• instrumental resolution
• wavelength spread
First we will discuss the crystallites size broadening. The width of the diffraction peak in reciprocal space is
inversely proportional to crystallite size, or more precisely size of coherently diffracting domain ∆Q ≈ 2π
D .
The scattering vector is proportional to sin of Bragg angle
Q =
4π
λ
sin θ. (1.5)
Differentiating preceding formula we obtain formula for angular width of diffraction lines
∆Q = 2
2π
λ
cos θ∆θ ≈
2π
D
, ∆2θ ≈
λ
D cos θ
. (1.6)
More precise derivation of Scherrer formula can be find in literature, see for example [2]
2w(2θ,rad) =
0.94λ
D cos θ
, (1.7)
where 2w is full width at half maximum of diffraction peak measured in 2θ angle and D is mean crystallite
size in the direction perpendicular to diffracting crystallographic planes, i.e. in the direction of the scattering
vector. One can express similar formula using integral peak width β, defined as ration of integral peak
intensity and its maximum intensity. Then we will obtain Stokes-Wilson formula
β(2θ,rad) =
λ
L cos θ
, (1.8)
where L is volume averaged crystallite size in the scattering vector direction. The quantities D and L equals
for a case of monodisperse particles – all crystallites have the same size. For polydisperse particles those
quantities differ according size distribution.
The second contribution to diffraction profile width is an inhomogeneous strain. Let us assume the local
inter-planar distance d + ∆d differs from the average inter-planar distance d by a small difference ∆d. The
corresponding Bragg angle would by also different by a value ∆θ which we can derive differentiating Bragg
equation
∆θ = −
∆d
d
tan θ, (1.9)
where ε = ∆d
d deformation in the diffraction vector direction. The local deformation has statistical distribution
width η and then the diffraction peak would have width
β(2θ,rad) = 4η tan θ. (1.10)
We denote the strain inhomogeneity leads to peak widening; homogeneous strain leads to shift of the diffraction
peak.
Total width is convolution of both components. If both profiles are Lorentzian the total width is sum of
both components
β(2θ,rad) =
λ
L cos θ
+ 4η tan θ. (1.11)
3
Since both contribution has different Bragg angle dependence we can decouple their values. Simple analysis
is based on formula multiplied by a factor of cos θ
β(2θ,rad) cos θ =
λ
L
+ 4η sin θ, (1.12)
which is used to construct Williamson-Hall plot. We plot peak width multiplied by cos θ as a function of
sin θ. The graph should show linear dependence whose slope is proportional to strain and absolute value to
crystallite size. The Williamson-Hall analysis is very simplified and in present time better methods are used.
However, this analysis can still give simple and fast quantitative description of the sample.
The real profiles are widened also by an experimental resolution and non monochromatic radiation. The
instrumental resolution is in the first approximation angle independent; the derivation of wavelength spread
to a peak width is left to the reader.
Real samples could exhibit very different behavior than previous simplified analysis. Among other effects
we would mention crystallite shape and elastic anisotropy. Shape of real crystallites are often elongated in
certain crystallographic direction (or could have flat shape). The crystallite size would than strongly vary
with scattering vector direction and the peak width would not be monotone function of Bragg angle. Similar
effect could have elastic anisotropy; the deformation could vary strongly with crystallographic direction.
Those effects could lead to a significant deviation of the formula (1.12).
This analysis is currently not very often used. Nowadays, the most popular methods are fitting of the
full diffraction profile and not only width of the peaks is taken into account.
1.6. Experimental setup
The experiment will be performed using X-ray diffractometer Rigaku SmartLab. The components will be
x-ray tube with copper anode – voltage 40 kV, current 30 mA, two Soller slits with divergence ∼ 5 ◦, linear
position sensitive detector D/Tex Ultra or 2D detector Hypix 3000 .
1.7. Questions
1. Why do we observe line splitting at high diffraction angles (figure 1.3)?
4
Task 2. Epitaxial thin film diffractometry, reciprocal space mapping
2.1. Deformation and stress in epitaxial layers
In the following text we will suppose cubic material with (001) surface orientation. Cubic materials have
three independent elastic parameters C11, C12 a C44 [4]. Also the substrate is assumed to be much thicker
than the layer and therefore not deformed. Lattice parameter of substrate is as and lattice parameter of the
layer in undeformed state is al. The lattice mismatch is defined as
ǫl =
al − as
as
. (2.1)
In the general case layer is deformed so its lattice parameters are a and a⊥, in the surface plane and
perpendicular to surface, respectively. We can define in-plane deformation ǫ and out-of-plane deformation
ǫ =
a − al
al
, ǫ⊥ =
a⊥ − al
al
. (2.2)
There are possible limiting cases
• Pseudomorph layer – in-plane lattice parameter of layer equals substrate lattice parameter a = as;
layer is deformed in such a way that it follows substrate. Usually for very thin films or very small
mismatch. In-plane deformation equals to negative mismatch value.
ǫ = −ǫl. (2.3)
• Relaxed layer – layer lattice parameters equal the bulk lattice parameter of the layer material a =
a⊥ = al. The layer is not deformed at all; this case occurs for thick films with large lattice mismatch.
The deformation is followed with existence of misfit dislocations. Deformation of the layer equals zero
ǫ = ǫ⊥ = 0. (2.4)
• General case – anything between the above mentioned states. We can define degree of relaxation as
R =
a − as
al − as
, (2.5)
which equals 0 for pseudomorphic layer and 1 for relaxed layer.
• Layer lattice is tilted by an angle φ with respect to substrate lattice – figure 2.2 right panel. This case
can be combined with the above mentioned cases.
Elastic strain components σ can be evaluated using Hook’s law
σ =
E
(1 + ν)(1 − 2ν)
ε + νε⊥ , σ⊥ =
E
(1 + ν)(1 − 2ν)
(1 − ν)ε⊥ + 2νε , (2.6)
where E is Young’s modulus and ν Poisson’s ratio. Layer surface is free of force, therefore the out of plane
stress equals zero
σ⊥ = 0. (2.7)
Using the preceding formula one can obtain the following expression of the stress free layer lattice para-
meter
al =
(1 − νl)a⊥ + 2νla
1 + νl
. (2.8)
where layer lattice parameter is determined using in-plane a and out-of-plane lattice parameters a⊥ measured
using x-ray diffraction. The in-plane stress component than equals
σ =
El
1 − νl
ǫ . (2.9)
For cubic lattice and surface orientation (001) following formulas hold for Young’s modulus and Poisson’s
ratio
ν001 =
C12
C11 + C12
, E001 =
(C11 + 2C12)(C11 − C12)
C11 + C12
. (2.10)
and for (111) orientation
ν111 =
1
2
C11 + 2C12 − 2C44
C11 + 2C12 + 2C44
, E111 = 3
(C11 + 2C12)C44
C11 + 2C12 + C44
. (2.11)
5
2.2. Reciprocal space mapping
This section describes basics of reciprocal space mapping using x-ray diffraction and lattice parameters
determination. General experiment scheme is plotted in figure 2.1. Incident beam is described with a wave
Qx
Qz
fK
Q
wafer
2θ
ω
Ki
2θ
ω
detector
crystal
analyzer
sample
crystal
monochromator
mirror
parabolic
x−ray tube
Obrázek 2.1. Left panel: general schematics of x-ray scattering. Right: experimental setup of high-resolution
x-ray diffraction with crystal monochromator and analyzer.
vector Ki of a length K = 2π/λ, where λ is wavelength of x-ray radiation. Diffracted beam has wavevector
Kf and we define scattering vector Q = Kf −Ki. Diffraction maxima are observed if the scattering vector Q
Q Q
Q
x
zQQzz
x
Qx
φ
φ
φ
Obrázek 2.2. Top: position of layer (red) and substrate (black) diffraction maxima in a symmetric (points
close to Qz axis) and asymmetric diffraction (right with respect to Qz axis). Bottom: schematic arrangement
of layer and substrate lattices. Left: relaxed epitaxial layer. Center: pseudomorphic epitaxial layer. Right:
relaxed and tilted epitaxial layer.
coincides with any reciprocal lattice point. Relative positions of substrate and layer reciprocal lattice points
are plotted in figure 2.2 for three examples of relaxed and pseudomorphic layer. In usual definition the axis
Qz denotes surface normal.
At symmetric diffraction the diffraction vector Q is perpendicular to the surface. If the layer lattice is not
tilted with respect to the substrate one can easily determine out-of-plane lattice parameter of both substrate
and layer
Qz,l =
2π
a⊥
l, and Qz,s =
2π
as
l, (2.12)
6
where l is diffraction order. For tilted lattice the layer diffraction maximum does not lie at Qz axis; the tilt
angle φ can be easily determined from the diffraction maximum position according figure 2.2. The in-plane
lattice parameter has to be determined using asymmetric diffraction
Qz,l =
2π
a⊥
l, and Qx,l =
2π
a
h2 + k2, (2.13)
where these formulas hold for cubic lattice with surface orientation (001) and asymmetric diffraction with
Laue indices hkl.
In a case of tilted layer one has to correct the position by a tilt angle φ. Therefore both symmetric
and asymmetric diffractions are necessary to be measured; tilt angle can be determined using symmetric
diffraction and asymmetric one is needed to determine both in-plane and out-of-plane components of lattice
parameter. Using known values of Poisson’s ratio we can determine the unstrained layer lattice parameter
and strain components and relaxation degree using formulas (2.8), (2.2), and (2.5).
2.3. Experimental setup
We will use copper x-ray tube with crystal monochromator and analyzer. The measurement is performed
as mapping in two angles ω and 2θ. The experiment is very sensitive to a precise alignment, it is necessary
to align properly angle of incidence, tilt angle and azimuthal rotation of the sample.
2.4. Procedure
1. Align sample to measure map of one symmetric and one asymmetric diffractions.
2. Measure intensity distribution in the vicinity of aligned reciprocal lattice points.
3. Determine lattice parameters, tilt angle, strain and relaxation degree of epitaxial layer.
2.5. Questions
1. How to correct diffraction position of tilted layer?
2. How would you align the sample?
7
Task 3. X-ray reflectivity
3.1. Task formulation
Measurement of x-ray reflectivity on a thin film sample. Determination of layer thickness and roughness
of individual interfaces fitting the reflectivity profile.
3.2. Theory
Refraction index od x-ray radiation n = 1 − δ + iβ is very close and smaller than one. According the
Fresnel formulae the reflectivity is measurable only for grazing incidence angles. The angle of incidence θ
is therefore usually measured with respect to the sample surface in contrary to the common terminology in
viible range. Since the material has refraction index smaller than air we can bserve total external reflection
at critical angle θC (in radians)
θC = arccos n ≈
√
2δ. (3.1)
Values of δ can be found for example at webpage: http://x-server.gmca.aps.anl.gov/x0h.html [6].
Let us assume sample with thickness t on a substrate, sse figure 3.1. Fresnel coefficients of reflectivity of
layer surface and layer-subtsrate interface in a small angle limit can evaluated as
r1 =
θ − θl
θ + θl
a r2 =
θl − θs
θl + θs
, (3.2)
where direction of beam propagation in layer follows Snell’s law θv =
√
θ2 − 2δv and in substrate θs =√
θ2 − 2δs.
The amplitude of reflected wave is
R(θ) =
r1 + r2e−iφ
1 + r1r2e−iφ
, (3.3)
where φ = (4π/λ)θlt is a phase retardation of wave in the layer.
Measured intensity I(θ) is proportional to |R(θ)|2.
From the previous formulas follows the reflectivity has oscillatory behavior with a pseudo-period of
θlt = t
√
θ2 − 2δl. The angular position of m-th maximum θ(m) is then
(θ(m))2 − θ2
Cl =
λ
2t
(m − m0) . (3.4)
One can see the maxima are not strictly equidistant, since they are shifted by a angle of total external
reflection θCl. For the practical evaluation it is easier to introduce the angle of first observable maximum of
the order m0. In many cases the first interference maximum is allways observable. The we can obtain simple
linear dependence of the quantity on the left side of formule (3.4) with a slope λ/2t. In the previous formula
we have used small angle limit of gonimetric functions and the angles have to be allways used in radians.
The approximate value of interference period for rough thickness determination is 180/π ·
√
(λ/2t)2 + 2δl.
3.3. Experimental setup
X-ray diffractometer equipped with a copper x-ray tube, slits, computer controlled gonimeter and a
detector. First the adjustment of the instrument is performed; the zero position of a detector arm angle is
adjusted to a direct beam. The next step is sample alignment; sample position and angle of incidence is
aligned. The measurement output is a reflectivity curve I(θ) of a reflected intensity as a function of angle of
incidence.
layer
substrate
Obrázek 3.1. Schematic drawing of wavevectors in layer/substrate system.
8
3.4. Data evaluation
Perform simple evaluation of layer thickness using approximative formula (3.4). Also the data will be
fitted using specialized software – either the commercial software Rigaku or freeware software GenX [7]. The
fitting of full curve will provide us with values of surface and interface roughness and density of individual
layers. Compare the obtained resuts with ones obtained using approximative formula.
10
−6
10
−5
10−4
10−3
10
−2
10−1
1
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
layer
substrate
layer
substrate
layer
substrate
layer
substrate
angle of incidence [deg]
reflectivity
Obrázek 3.2. Reflectivity simulation of 20 nm thick tungsten film on saphire substrate. The interface and
surface are smooth or hav roughness of 0.5 nm. Four cases are simulated: (a) both interfaces smooth, (b)
surface smooth and interface rough (c) interface smooth and surface rough, and (d) both rough.
3.5. Questions
1. How will affect the measurement finite sizes of the sample and beam at low angles of incidence? Guess the
angle when whole beam would irradiate sample surface for realistic parameters (beam size 0.2 mm, sample
size 10 mm).
2. How would look reflectivity of semiinfinite sample with an oxide or adsorbed layer at surface? Would affect
a single monolayer the experimental data?
3. What is the roughness effect? Try to identify the curves in figure 3.2?
4. The divergence of beam defined using a parabolic mirror is ∼ 0.03◦ and the dynamical range allows us to
measure up maximal angle of incidince θmax ∼ 5◦. Guess minimal and maximal layer thickness available for
x-ray reflectivity experiment.
9
Task 4. Small angle x-ray scattering
4.1. Task formulation
Measure and analyze small angle scattering of nanoparticles.
4.2. Theory
Small angle scattering is measured at low scattering angle up to 10◦. The crystalline ordering usually
does not play any role since the lowest diffraction peaks begin to appear at bigger scattering angles. The
scattered amplitude is proportional to a Fourier transform of the electron density
E(Q) = drρ(r)e−iQ·r
. (4.1)
And the scattered intensity is proportional to a square of its absolute value
I(Q) = drρ(r)e−iQ·r
2
. (4.2)
Assuming constant density of the particles we can express the density distribution using shape function and
constant electron density
ρ(r) = ρ0Ω(r). (4.3)
In many cases the particles are randomly oriented in a dilute solution. As long as the particles are far from
each other we can neglect correlation effects of their positions. Then the intensity can be expressed as an
orientational average
I(Q) = 4π
Dmax
0
p(r)
sin Qr
Qr
dr, p(r) =
r2
2π2
∞
0
QI(Q)
sin Qr
Qr
dQ, (4.4)
where p(r) is distance distribution function.
The distance distribution function of intensity distribution has to be calculated for a given particle shape
and size model including size dispersion. We can use several of available software packages. List of them is
available online http://smallangle.org/content/software.
4.3. Limiting cases – simple quantitative analysis
Several quantities can be evaluated without fitting, and even without any particle model. At low angle
limit the intensity can be expressed using Guinier’s formula
I(Q) = I0e−Q2R2
g/3
, (4.5)
where Rg is particle gyration radius. Guinier’s plot is logarithm of intensity plotted as a function of square
of scattering vector – the intensity dependence in this region is linear. We note the zero angle scattered
intensity I(0) cannot be directly measured (because of primary beam) and Giunier fit is one of possibilities
to determine its value. Gyration radius is defined using distance distribution function
R2
g =
∞
0 r2p(r)dr
∞
0 p(r)dr
. (4.6)
The high angle limit has a power dependence according Porod’s law
I(Q) = CQ−4
, (4.7)
where C is a scale parameter. This law is valid for particles with sharp interface. If the particle surface is
diffuse or has fractal behavior the exponent changes from 4 to a general exponent n = 6 − d, where d fractal
dimension of the particle surface. Last of the simple evaluation methods is Porod’s invariant
P = I(Q)Q2
dQ (4.8)
Combining Porod’s invariant P and zero angle scattered intensity I(0) we can get particle volume Vp
Vp =
2π2I(0)
P
. (4.9)
10
x−ray source
beamstop
detector
sample
pinholes
Obrázek 4.1. SAXS setup in transmission geometry with two pinholes.
4.4. Experimental setup
The experiment will be performed in setup with a small beam and two-dimensional detector. The incident
beam has to be very narrow and collimated in both directions. The beam will be collimated using two pinholes;
one positioned close to the x-ray tube and the other one as close as possible to the sample.
The beam divergence is limited by the pinholes diameter, the smallest ones are 0.1 mm and 0.05 mm. The
beam divergence limits the smallest possible scattering vector accessible for experiment. For smaller angles
the signal is covered by primary beam. By the properties of Fourier transform it translate to the upper limit
of particle size. Typical values are in the order of lowest scattering angle 2θ ≈ 0.1◦, corresponding scattering
vector Qmin ≈ 0.007 Å−1 and largest particle size D = π
Qmin
≈ 50 nm.
For SAXS in solution the scattered intensity has rotational symmetry. The data of two-dimensional
detector will be then azimuthally averaged to increase signal to noise ration and the line profile will be fitted
using specialized software.
Literature
[1] U. Pietsch, V. Holý, T. Baumbach, High-Resolution X-Ray Scattering From Thin Films and Multilayers,
Springer, Berlin, 1999 and 2004.
[2] B. E. Warren, X-ray diffraction, Dover Publications, New York 1990.
[3] FullProf homepage: http://www.ill.eu/sites/fullprof/
[4] Lattice parameters of semiconductors: v O. Madelung, Semiconductors: Data Handbook, Springer 2004,
or http://www.ioffe.ru/SVA/NSM/Semicond/index.html.
[5] Crystallography open database: http://www.crystallography.net/.
[6] Sergey Stepanov’s x-ray server: http://x-server.gmca.aps.anl.gov/x0h.html.
[7] Genx homepage: http://genx.sourceforge.net/.
11