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A PHYSICAL MODEL FOR EXPERIMENTAL X-RAY DIFFRACTION MEASUREMENTS OF TiO2 THIN
FILMS GROWN ON SrTiO3 SUBSTRATE VIA PULSED LASER DEPOSITION
PI PENG
SUPERVISORS: PROF. ANDRIVO RUSYDIDr. LUIS RODRIGUEZDr. ONG BIN LEONG
DEPARTMENT OF PHYSICS
NATIONAL UNIVERSITY OF SINGAPORE
2018
ACKNOWLEDGEMENTS
Firstly, I would like to express my deepest gratitude to my supervisor Prof.
Andrivo for allowing me learning from this subject, and for his understand and patient
guidance that helped me through all the difficulties.
I would also like to offer my special thanks to my co-supervisors: Dr. Rodriguez
and Dr. Leong, for their kind support in both the project and this report, and for the
knowledge and professional opinion they have taught me regarding this project.
Also I want to thank the principal research fellow of the XDD beamline, Dr. Yang
and the deputy director, Prof. Mark of the Singapore Synchrotron Light Source
(SSLS), for providing me the facilities for all of the experiments conducted for the
research.
ABSTRACT
This work explores a new method of XRD analysis through simulation. A layered
model is built in IMD software for TiO2 thin film grown epitaxially on STO (001)
substrate. The model is used to fit three TiO2 samples with different Ta-doping: pure
TiO2, 0.5% Ta-doping and 1% Ta-doping. Samples were deposited using via PLD and
characterized by X-ray diffractometry and glazing incidence X-ray reflectometry at
SSLS. The main parameters such as lattice constant, grain size, density, and thickness
of thin films are obtained by fitting experimental data with the model and compared
with the values obtained by traditional X-ray analysis.
1 INTRODUCTION
Titanium dioxide (TiO2) has experienced an increasing demand from end-user
industries due to its rich physical and optical properties. Among the latter, its high
absorption of ultraviolet (UV) radiation renders it suitable material for applications
such as optical coatings1, wire grid polarizers2, UV blocking pigment3 or photocatalyst
in dye-sensitized solar cell4. The UV absorption of thin films of TiO2 depends on
several parameters including the phase5 (anatase or rutile, being brookite unstable)
and doping6. The latter property is particularly interesting, as its intrinsic high UV
absorption can be tuned and further enhanced for desired energy range via doping
with materials such as Tantalum7; this novel tunability opens a new horizon for new
devices whose performance is nowadays chiefly limited by the optical properties of
TiO2 films.
The analysis of the phase and doping level of the grown TiO2 samples is thus
crucial in understanding the effect of each of these factors in the absorption spectra.
X-ray diffraction (XRD) analysis is among the most used technique for this particular
analysis; besides, additional information about grain size and roughness, that
ultimately drives the scattering loses of thin films and hence its optical quality, can be
also obtained from XRD. XRD analysis is mainly applied by comparing the
experimental diffraction pattern with database for line identification. This procedure
usually reveals the phase and crystalline orientations in the samples. Information
about grain size and doping level is usually also obtained by comparison on the angle
position, intensity and full width at half maximum (FWHM) of the respective peaks.
The measurement, however, relies on the completeness of available databases, but for
TiO2 with various level of Ta doping there is scarce literature6 for reference and hence
the method mentioned above reaches its limitation. Among common interpretation
problems, for the determination of the grain size, the traditional XRD analysis
requires accurate evaluation of the Scherrer’s constant K, an empirical, material-
dependent parameter, which is usually affected by large error bars8. Overall,
traditional XRD analysis has a key limitation, which is the inaccuracy in
determination of certain parameters.
In this work, a new model for the simulation of XRD diffraction patterns is
proposed and tested with experimental data; this model describes the unit-cell in terms
of an multilayer where the total electronic density of the atoms within the unit cell is
disjoined and distributed in different layers with a given criteria.. This model is
expected to enhance the understanding of experimental data and the quality of
analysis, and will be a fundamental tool for the team within the same research group
which, working in parallel to us, is nowadays analyzing the absorption properties of
Ta-doped TiO2 thin films.
In this work, a set of TiO2/SrTiO3 (001) samples were grown via pulsed laser
deposition (PLD) under various conditions and then XRD data were obtained. To
analyze the XRD data, a new model is presented and compared to experimental data.
Section 2 introduces the experimental setup for thin film growth by PLD and XRD
characterization of the samples. Section 3 briefly describes the theory regarding XRD
simulation and the fundamentals of the model. Section 4 discusses the results derived
from the data analysis through simulation. Finally, section 5 concludes on the results
and gives some schemes for further improvements.
2 MATERIALS & EXPERIMENTAL SETUP
2.1 Anatase and STO
Titanium dioxide has three allotropes (or phases), which are anatase, rutile and
brookite. All of them have a tetragonal configuration. The one studied in this work is
anatase, and this phase was selected because the absorption tunability by Ta-doping
was discovered for TiO2 films with anatase phase (Ref. 7). TiO2 films are grown on
SrTiO3 (STO) (100) substrates, which has a cubic configuration. The unit cell
configurations for all TiO2 phases and STO substrate are shown in Figs.1 and 2.
Fig. 1 The three allotropes of TiO2: (a) anatase (b) brookite and (c) rutile. They are all tetragonal crystals.
Fig. 2 SrTiO3 (STO). It is a cubic crystal
2.2 Pulsed laser deposition
The pulsed laser deposition (PLD) method uses a high energy pulsed laser beam
which ablates a target of the desired composition. Each laser pulse vaporizes a small
amount of the material from the target, and it is deposited onto a heated substrate to
form a thin film. The deposition process usually takes place in an ultra-high vacuum
(UHV) or in the presence of a background gas.
Fig. 3 A typical setup of a Pulsed Laser Deposition system.
A
B
C
The samples presented in this work are all deposited with the new state-of-the-art
PLD system at Singapore Synchrotron Light Source (SSLS). The system is a
component of Soft X-rays Ultraviolet (SUV) beamline; pictures of the system are
shown in Fig. 4. The PLD chamber, which works under UHV conditions (base
pressure is 5x10-9 Torr), allows the loading of samples via the load-lock chamber. The
deposition system can hold up to six targets with 1-inch diameter or three targets with
2-inches diameter. Targets are ablated by a pulsed solid-state laser operating at 266
nm, with pulse frequencies up to 10Hz and laser power up to 100 mJ/cm2. Substrates
Fig. 4 Photos of outside and inside of PLD system.
Fig. 5 The diffraction pattern obtained from RHEED system. (a) The RHEED pattern obtained from the bare STO substrate, before the deposition process. (b) The RHEED pattern obtained after the deposition. Streaks on (a) manifest a flat surface with small crystalline domains, whereas satellite streaks indicate the presence of a stepped surface. After the deposition of TiO 2 film, some spots can be hinted in the streaks shown in (b), which is a fingerprint of 3D islands growth.
(a) (b)
up to 10 mm x 10 mm in size can be positioned in the chamber by means of a
manipulator with XYZ linear-drives and can be rotated with desired rotational speed
parallel to target surface. The substrate can be heated up to 1100 ᵒC with a 140W laser
heater, with substrate temperature controlled and monitored by an optical pyrometer.
Up to three different operational gases (O2, N2 and Ar) at different pressures (up to
200mTorr) can be used at the same time in the chamber as the system integrates three
mass flow controllers.
The PLD system houses a Reflection High-Energy Electron Diffraction (RHEED)
system which can be operated at high pressures using a differential pumping system.
The RHEED system allows in-situ growth monitoring of the PLD process, enabling
selective and controlled growth of thin films in terms of monolayers. In addition to
this, information about the quality and morphology of both substrate and thin film
surfaces can be obtained by RHEED9, as displayed in Fig. 5.
The three sample for this project are all titanium dioxide (TiO2) thin film grown on
strontium titanate (SrTiO3). The deposition parameters and doping level of each
sample are shown in the table below.
Sample Doping level (%)
Temp (°C)
Oxygen Pressure (mTorr)
Deposition rate (pulses per sec)
Time (mins)
PTS0217 1 600 100 2 25
PTS0318 0.5 600 10 2 28
PSS0218 0 950 200 1 40
2.3 XRD measurements
2.3.1 XDD beamlines and experimental setup
The X-ray Diffractometry and Development (XDD) beamline10 is a user-oriented
facility at SSLS. XDD beamline and its experimental end-station are designed for a
general purpose of diffractometry, fluorescence detection and absorption
spectroscopy, and offers an invaluable service to industry and research institutions in
Singapore and the region. The main applications for the beamline are:
High-resolution diffraction (HR-XRD): to determine precisely structure
parameter, strain, composition, thickness, surface/interface roughness and
texture/stress analysis.
Grazing-incidence X-ray diffraction (GI-XRD), X-ray reflectometry (XRR)
and diffuse scattering: to determine surface and interface structure /ordering, surface
phase identification/transition, layer thickness and density.
In a typical θ-2θ configuration, a beam of monochromatic light is diffracted by the
sample from the original direction of propagation. To do this, the sample is mounted
horizontally on a sample holder, with its surface parallel with the incident beam. As
Fig. 6 XDD beamline setup and their relevant configuration parameters.
measurement starts, the sample holder slowly rotates to form angle θ between the thin
film surface and the incident beam, diffracting the beam light away from its original
direction. In the meantime, the detector rotates to an angle of 2θ to measure the
diffracted radiation. The angle of the detector is always 2 times the angle of the
sample. The measured intensities are plotted against 2θ for analysis. For highly
ordered crystal, diffraction intensities in most directions are merely zero, due to the
destructive interference among scattered light rays from different atoms. For some
specific angle, however, the light rays constructively interfere and form an intense
diffracted beam whose fingerprint in the experimental 2θ plot is a peak. Fig. 7 shows
a picture of XDD beamline end-station, and a sketch of the θ-2θ configuration.
Measurements in θ-2θ configuration were performed in θ range between 10° and 65°,
with a step of 0.02°. Incident wavelength was set to 0.15406 nm by means of a Si
monocrhomator. Typical spectral resolution ∆E/E for 0.5 nm slits is better than 10-3.
Glancing incidence x-ray reflectometry (XRR) measurements are complementary
to XRD applied to obtain sample information including density and thickness. With a
similar setup and incident beam configuration than for XRD measurements but in
angles close to glancing incidence, the diffracted intensity was recorded as a function
of the incident angle. XRR measurements were performed from 0.2° to 2° in θ with a
step of 5x10-4 degrees. Sample PSS0218 was only measured up to 0.75°.
2.3.2 Theory
The angle positions of diffraction peaks are related to the d-spacing, by Bragg
Law:
sin θ= nλ2d , (2.1)
Fig. 7 The XRD diffractometer at Singapore Synchrotron Light Source. Below, configuration for θ-2θ measurements
where θ is the incident angle, λ is the wavelength, n is an integer which gives the
Bragg order, and d is the d-spacing which in general is given by
1d2=
h2+k2+ l2
a2 , (2.2)
for cubic crystals, and
1d2=
h2+k2
a2 + l2
c2 , (2.3)
for tetragonal11 crystals.
The numbers h, k, l above, usually written in parenthesis as (hkl), are the Miller
indices and denote a set of planes which are parallel and equidistant, one of which
passes through the origin, and another intercepts the axis at (a/h, b/k, c/l )
coordinates. Miller indices are also used to show the orientation of the sample by
denoting the planes parallel to the sample surface. For example, the sample denoted
with TiO2 (001) has its c axis perpendicular to the sample surface and the other two
axes parallel to the surface. The d-spacing in this case is just c. Fig. 8 shows another
example of planes and the respective Miller index.
Fig. 8 In this example, a plane intersects with the three axes at a/5, b/3 and c/2 respectively, and therefore the corresponding Miller index is (532). Any other plane that makes intersections of coordinates as multiple times large as this plane may also denoted by (532).
In general, the nth-order reflection from (hkl) planes is equivalent to and
considered as the first-order reflection from (nh nk nl) planes. Therefore, STO (n00) is
used to refer to peaks corresponding to the nth-order reflection of STO (100) sample.
Therefore, the equation for STO (100) peak reads
sin θ= λ2a (2.4)
and for anatase (004) peak it is
sin θ=2 λc (2.5)
For example, in a 2θ scan, the anatase (004) peak appears at θ=18.7°. Therefore the
calculated value of lattice constant is:
c=2 ×0.15418 nmsin 18.7 °
≅ 0.9618 nm.
It is easy to calculate the angle position of the Bragg peak from the lattice
parameters, d-spacing, or Miller indices. It is, however, not straightforward to know
to which Miller indices a peak belongs. The procedure to find the Miller index of a
peak is called line identification. The d-spacing can be obtained from the angle
position using Eq. (2.1), and subsequently, d is related to the Miller indices by Eq.
(2.2) or (2.3) (for cubic or tetragonal crystals, respectively). But in general, there exist
more than one set of integer solutions for h, k, and l to the aforementioned equations.
One way to reduce the number of solutions is to exclude those which are called
forbidden peaks. Consider the relative intensity equation for of powder diffraction11:
I=|F|2 p ( 1+cos2θsin 2θ cosθ ) (2.6)
In the above equation, F is the structural factor, p is the multiplicity factor and that
in the parenthesis is the Lorentz-polarization (L-p) factor. F is related to the Miller
indices and takes the explicit form of:
Fhkl=∑1
N
f n e2 πi(hun+k vn+l wn ) (2.7)
fn is the atomic form factor of the nth atom which depends on the electronic density
of atoms, angle and wavelength of the light. (un , vn , wn ¿ are the relative coordinates of
the nth atom in the unit cell, in terms of the primitive translation vectors. N is the total
number of atoms in that cell.
While the multiplicity factor and L-p factor is always positive, the structural factor
can be 0, which gives 0 intensity. For anatase, as retrieved from Springer Materials
the standardized atomic coordinates in the unit cell are12:
Ti: (0, 0.25, 3/8), Symmetry: -4m2
O: (0, 0.25, 0.167), Symmetry: 2mm
Thus the final form of structural factor is
Fhkl=f Ti{2cos [ π (h+0.5 k+0.208l ) ]+2cos [π (0.5 k+0.75 l ) ]}+ f O {2 cos [ π (h+0.5 k−0.208 l ) ]+2cos [π (0.5 k+0.334 l ) ]+2cos [ π (h+0.5 k+0.624 l ) ]+2cos [π (1.5 k+0.834 l ) ] }
(2.8)
The above expressions equals to 0 when l=0 and either h or k is odd. For example,
no peaks with Miller indices like (110) or (100) would appear on the anatase XRD
pattern. Therefore, we can exclude those with 0 intensity or very low intensity and by
this identify Miller indices of the peaks observed in the XRD pattern in most of the
cases.
Another standard procedure to find the Miller indices of peaks in a well-known
material, such as TiO2, is through databases, where the indices of planes in Bragg
configuration (for a given wavelength) are tabulated as a function of the angle θ (or 2θ
or both) where the experimental peak is expected, along with the relative peak
intensity. However, doped materials such as Ta-doped TiO2 experience a shift in d-
spacing due to atom substitution, (Ref. 7) and hence this change in d produces a shift
in the angle θ where the peak is expected. This shift makes the use of databases
sometimes unadvisable.
3 MODEL FOR XRD MEASUREMENTS
3.1 Simulation: the model and parameters
When light rays are scattered by two different atoms in the crystal, there is a phase
difference due to the path difference, and it varies with the incident angle. However, if
the two scattering centers are on the same plane there would be no path difference.
“The same plane” here refers to those planes parallel to the surface of the thin film.
Light rays scattered from the atoms of the same plane are always in phase, and
therefore the interatomic distance along this plane will not affect the total intensity
and its angle distribution. Instead, the distance between these planes, the d-spacing, is
the one that needs to be modeled and studied.
Fig. 9 a, b and c are parallel monochromatic rays which are originally in phase before their interaction with the samples. Rays a and b are scattered by atoms located in different layers and have path difference of 2dsinθ after scattering. Meanwhile, b and c are scattered by different atoms in the same layer, and therefore have no path difference along their propagation.
Therefore a model to simulate the crystal diffraction of any material needs only to
take in account the d-spacing between planes parallel to the sample surface, and the
component of the lattice constant normal to sample surface. The materials studied in
this project are TiO2 (anatase) thin films grown on SrTiO3 (001) substrates. Hence,
TiO2 unit cells are modelled following this procedure:
1- All d-spacing distances related to thin films peaks are calculated from
experimental data using Eq. (2.1). There is small room for confusion between thin
film and substrate peaks, as the latter are usually very intense and narrow due to the
fact that STO substrate is an oriented single-crystal.
2- Looking at the lattice parameters of both substrate and thin film, the orientation
of crystals in TiO2 thin film can be guesstimated as the one that minimizes the lattice
mismatch between film and substrate. For instance, for TiO2 Anatase (tetragonal, a =
3.78 Å, c = 9.51 Å) on STO (cubic, a = 3.905 Å), a probable orientation is TiO2
(00X), giving a lattice mismatch of 3.2%. This step is later verified once the model is
completed, and corrected if necessary.
3- Now, the normal component to sample surface of the lattice constants (c, for
TiO2 if we assume (00X) the most probable orientation) is divided at least in m
bilayers, where
m=normal component of lattice constantd (2.9)
Each mth bilayer is also divided in two sublayers, one with a given electronic
density, and the second is vacuum. The first sublayer models the averaged electronic
density of the atoms within the unit cell, whereas the vacuum sublayer models the
fraction of volume of the unit cell where the probability of having a small electron
density is high, in a simplified and classical point of view of atomic distribution in
structured crystals. Fig. 10 shows how this layer division procedure is applied to both
TiO2 and STO. This bilayer system is a simple way to model the electronic density of
atoms within the unit cell, simulating the d-spacing distances and generating the
necessary electronic density (or refractive index) contrast between layers. One
important constraint for the density distribution of layers is that the average density of
a layered unit cell must be the same as the bulk density for a given phase.
4- A given family of Bragg peaks (e.g. STO (001), STO (002), STO (003) and
STO (004), requires one single model of the unit cell using the largest d-spacing (in
this case, the one given by STO (001)). High-order peaks will naturally appear at high
angles as the Bragg law (Eq. (2.1)) is satisfied for n>1.
Due to the small role played by the atomic bonds in short wavelengths, the optical
properties of a material can be approximated by the sum of the responses of the
electrons of all the atoms to the x-ray radiation, as if the atoms were independent of
each other. From the classical point of view, each electron bound in the atom has a
specific resonance frequency; the interaction of the incident beam with an atom of
Fig. 10 (a) Anatase. (b) STO. On the left side are the unit cell structure of the crystals. On the right side is the model needed for calculations. This model ignores the in-cell position and converts the unit cell structure into layers that possess parameters including thickness, density and atom components only.
multiple electrons is obtained by the sum of the interaction on all the resonance
frequencies of the electrons within the atom extended to all the atoms that integrate
the material. Hence, the model presented in this work assumes that as long as the size
and the average electronic density of the unit cell are kept constant and similar to
tabulated values, the internal distribution of electron density can be customized under
the criteria described above.
To simulate the diffraction pattern, the Fresnel equations for reflection and
transition are applied13. For s-polarization:
r⊥=n icosθ i−nt cos θt
nicos θi+nt cosθ t(3.1)
t⊥=2ni cosθ i
ni cosθi+nt cosθ t(3.2)
For p-polarization:
r∥=nt cosθ i−ni cosθt
ni cosθt+nt cosθ i(3.3)
t ∥=2 ni cosθi
ni cosθ t+nt cosθ i(3.4)
Where Ni and Nt, are the complex refractive indices of the incident and transmitted
medium, respectively, and θi and θt are angles between the perpendicular of the
sample surface and the incident and transmitted beams, respectively. The refractive
index of any material in a given wavelength can be easily derived from their atomic
from factors and density.
For multiple-layer system, the above Fresnel equation is recursively applied to
obtain the total reflected and transmitted coefficients:
r j=r ( j+1) j+r j ( j−1)e
2 i β j
1+r j ( j−1)r ( j+1) je2 i β j
, t j=t j ( j−1)t ( j+1 ) j e
i β j
1+r j( j−1)r ( j+1 ) j e2 i β j
, (3.5), (3.6)
With β j=2 πλ ( N jd jcos θ j ), where d is the distance between layers, and the angle θ j is
recursively calculated through the well-known Snell law, N i senθ i=N t senθ t. Fig. 11
shows a sketch with the reflexion and transmission coefficients in a multilayer.
Finally, the reflected intensity of a multilayer of M layers is obtained by
R=|r M|2 (3.7)
All these equations are included in the IMD software, which is applied here to
carry out these calculations14.
3.2 IMD: the tool for calculation
Fig. 11 Diagram of the coefficients of reflection and transmission in amplitude in a multilayer of M layers.
IMD is a free software designed by David Windt for the calculation of optical
functions of an arbitrary multilayer system. Specular optical, such as reflectance, are
computed in IMD using an algorithm based on recursive application of the Fresnel
equations, similar to the one described in the previous subsection.
The main interface of IMD is shown in the Fig. 11. To build a model for the
simulation of present samples, these steps are to be followed:
1. Under the structure built section, add two multilayers, one for TiO2 and other
for STO. Add 2m sublayers in each multilayer, including vacuum layers
following the configuration proposed in Fig. 10 for both TiO2 and STO.
2. In each of the sublayer, input the density, material (with its related
stoichiometry) and thickness; density is not needed for vacuum layers. The
software includes an extensive material database with the atomic form factors
of all elemental atoms, and the atomic form factor of compounds at the relevant
wavelength are also calculated by software.
3. Set the number of repetition for each multilayer. This number indicates the
Fig. 12 IMD interface. Each of the column on the left side introduces where to add in the parameters for the simulation. By adding grazing angle in the “independent variables”, it is able to calculate the angle dependent reflectance.
repetition of the unit cells within the crystal/grain.
4. Input under independent variables the values such as wavelength (0.15418 nm)
of the incident light, instrumental angular resolution (around 0.005°), angular
range (θ from 10° to 65°) and angle step (0.01°) , and polarization degree (is
estimated to be close to 1).
5. Select under dependent variables the optical function to be calculated: in this
case, specular reflectance.
6. Calculate. Optical functions can be computed as a function of any independent
variable; in this case, the wavelength is fixed and the incident angle is selected
as variable
In the following section, this model is used to extract the relevant parameters from
the samples under study.
4 RESULTS AND DISCUSSION
4.1 Experimental measurements and model
Three TiO2 films deposited on STO (001) substrates were grown via PLD with
three different Ta-doping level in TiO2.
XRD plots of intensity against 2θ obtained from the experiments and from model
built for each sample are shown in Fig. 12. The following general parameters were
applied to all models: wavelength of the light was set to 0.15418 nm, the actual value
used in the XRD experiments; the instrumental angle resolution was 0.005°; 2θ
angular range spanned from 18° to 130°, and polarization degree was +1. Then, for
each particular sample, the parameters listed in the table below were used:
TiO2 SrTiO3
Sample N c (nm) ρ (g/cm3) N c (nm) ρ (g/cm3)
PTS0217 3 0.93623 3.78
1000 0.39075 4.81PTS0318 12 0.97623 3.78
PSS0218 120 0.96413 3.78
In the following subsections, the comparison between the model and experimental
data will be discussed in depth. Before that, several conclusions can be drawn from
direct inspection:
1. Only anatase allotropes of TiO2 are seen in all the three samples, since all
the peaks are matched with the model that includes anatase only.
Fig. 12 The experimental XRD patterns for the three samples along with their respective models.
2. The grain size of sample PTS0217 deducted from the plot is significantly
small, whereas it is larger for samples PTS0318 and PSS0218. It will be
seen later that this is related to TiO2 film thickness and the relative size
between the latter and film thickness.
3. The d-spacing of anatase (004), and thus lattice parameter c, of all samples
does not decrease with the doping level. Apparently, sample PSS0218 is
outside the trend, however, in subsection 4.3 will be demonstrated that the
origin of that bizarre behavior is that the strain induced in film by the
deposition at 950 ᵒC has a shifting effect in the opposite direction.
4. Though the model shows a good agreement with the first and second order
of STO Bragg peaks, high-order STO peaks in model present different
bandwidth and peak intensity in comparison with experimental values.
4.2 Line identification and Phase identification
From the Bragg law presented in Eq. (2.1), the d-spacing for a peak is
calculated using its respective angular position. As shown in Appendix 1, the four
peaks located at angle positions 2θ=22.82°, 46.53°, 72.61° and 104.26° correspond to
(001), (002), (003) and (004) STO peaks. The first thin film peak at 2θ=36.7ᵒ is
undoubtedly assigned to anatase (004). The unknown peak displayed in Fig. 12 at
2θ=79.76ᵒ in samples PTS0318 and PSS0218, although it might be confused with
rutile (212), is assigned to anatase (008). The reason is, as shown in Fig. 12, from the
adjustment of model to anatase (004), a high order Bragg peak appears in the same 2θ
position as the unknown peak. Hence, it is anatase (008). This example highlights the
capabilities of the model: whereas with the traditional line identification the unknown
peak could have been assigned to rutile (212), with the utilization of present model,
once anatase (004) has been modelled, a higher Bragg order appears and matches the
peak at 79.76ᵒ.
Fig. 12 shows only anatase (004) and (008). According to the simulation, the
reason why the other anatase peaks are missing is that peaks such as (002) and (003)
are very weak, and hence covered up by noise; anatase (001) is of medium intensity,
however it locates at around 2θ=4.64° which is out of the experimental range. With
the absence of any peaks of other allotropes such as rutile, the TiO2 in the three
samples shows only anatase phase.
The presence of anatase (004) and (008) confirms the assumption made when the
model was built: due to the particular STO and anatase lattice parameters, the TiO2
film grows with a preferential orientation that minimizes the lattice mismatch with the
substrate, and therefore the c axis of TiO2 films are perpendicular to substrate surface.
This effect has also been previously reported in the literature15.
4.3 Lattice parameter c and Ta-doping effects
This section describes the process of obtaining lattice constant c of the anatase in
each of the sample through simulation methods, and how the Ta-doping of samples
PTS0217 and PTS0318 affects c. The steps for PSS0218 sample are shown below as
an example.
Fig. 13 displays how the peaks for anatase of the sample PSS0218 slightly shifts to
smaller angle position; note that c=0.95143 nm is the tabulated c-lattice parameter
value. The simulation was run with c variable, increased by steps of 0.002 nm. The
results showed that for c=0.96143 nm, the simulated anatase peak is the closest to the
experimental. Sample PSS0218 is undoped, hence it would have been expected no
shift in TiO2 d-spacing and thus a c lattice parameter value close to c=0.9514 nm. This
change in c lattice parameter and hence, the shift of the peak is attributed to the strain
produced by the high substrate temperature (950 ᵒC) of substrate during deposition.
The tabulated a lattice parameters for STO and TiO2 are 0.3905 nm and 0.3780 nm,
respectively, yielding a lattice mismatch of 3.2% of an unstrained film. If volume of
the unit cell is assumed to be constant, therefore an increase in c from 0.951 nm to
0.961 nm implies a decrease in a lattice parameter from 0.378 to 0.376 and hence,
increase in lattice mismatch between TiO2 and STO from 3.2% to 3.7%. Fig. 16
displays the Ta-doping effects on the lattice constant c.
In Figs. 14 and 15 c is determined also for PTS0217 and PTS0318. The figures
shown above include anatase (004) only, since by matching this peak position, anatase
(008) peak will be automatically matched.
4.4 Grain size
Fig. 12 shows a noticeable difference among anatase peaks, , in terms of intensity
and bandwidth. This is due to the different grain size within the thin film. The grain
size gives a rough measurement on the structural morphology and degree of order
inside a crystal. With smaller grain size, there are more crystallites that may take
various orientation, and result in broadening the angle of diffraction and decreasing
the intensity in an XRD pattern. Similar effect can be seen with very thin TiO2 films,
where the amount of crystalline planes satisfying Bragg condition is limited. For a
single crystal thin film, the measurements for grain size reveals the thin film
thickness.
In this subsection, the grain size for TiO2 in each of the sample is calculated using
two different ways and compared: first one is by Scherrer equation, and the second
Fig. 16 The Ta-doping effect on the lattice constants c of the three samples.
one is by simulation with the model described in section 3. The Scherrer equation
takes the form:
T= KλB cosθ (4.1)
T stands for the grain size, K is the Scherrer constant that can treated as equal to
0.9, λ is incident wavelength that equals to 0.15418 nm and B (in rad) is the FWHM
of the peak.
Fig. 17 The simulation pattern for anatase (004) peak for N=2,3 and 4, compared with the experimental data.
In order to obtain the grain size from the simulation, the procedure is to find the
value of repetition number N which gives the best matching to both peak intensity and
FWHM.
The results for simulation using different repetition number are plotted in Figs. 17-
19. For PTS0217 sample, N=3 gives the best matching. For the other two samples,
Fig. 18 The simulated patterns of PTS0318 for anatase (004) peak using different repetition number N.
Fig. 19 The simulated patterns of PSS0218 for anatase (004) peak for different value of N.
there is one additional difficulty as the FWHM value obtained from the model can be
distorted by the inclusion of exclusion from FWHM calculation of side-lobes, which
are resolved in the simulation but unresolved in the experimental data; this effect
mostly arise from the angular dispersion of the beam as it is not fully collimated.
Nonetheless, one can tell a range of N value. Grain sizes are shown in the table below;
for sample PTS0217, N equals to 3, which means the grain size determined for it is
3xc, 3 multiplied by its lattice constant c. It is therefore around 3 nm. For the rest two
samples, N=12 yields grain size is 11.5 nm, and N=120 is 115 nm. Compared to
results using Scherrer equation, these values are slightly smaller; one possible
explanation for the disagreement between the values obtained with the model and
from Scherrer’s analysis is that the real value of the empirical the Scherrer’s constant
K could be smaller than 0.9 for TiO2.
PTS0217 PTS0318 PSS0218
Scherrer Simulation Scherrer Simulation Scherrer Simulation7.95 nm 2.88 nm 24.32 nm 11.5 nm 111.2 nm 115.4 nm
4.5 XRR measurements
In the XRR measurements, the intensity of the reflected beam as a function of the
small grazing angles (0.1ᵒ to 2ᵒ) is recorded and studied. The XRR experimental data
can be fitted with the same individual model that was presented in subsection 4.1, but
with the exception that for XRR, thin film thicknesses do not need to be sub-layered,
and the density is averaged among the whole film. By simulating the XRR pattern,
the information about the sample thickness, density, stoichiometry, and roughness can
be obtained. Experimental data along with the model for each sample is displayed in
Fig. 20. The following table shows the main parameters determined from the fitting:
Sample TiO2 Stoichiometry Thin film layer thickness (nm) Density (g/cm3)
PTS0217 Ta:1; Ti:99; O:200 4.1 3.72
PTS0318 Ta:1; Ti:199; O:400 51.1 3.81
PSS0218 Ti:1; O:2 117 4.07
The stoichiometry obtained from the model matches with the expected value for
each sample, which is not surprising as PLD usually transfers the stoichiometry from
target to thin film. Regarding film thickness, the value obtained from PTS0217
invalidates the grain size of 7.95 nm obtained by Scherrer’s equation for the same
sample in previous subsection: obviously, the grain size cannot be larger than the
whole film thickness. This result highlights the shortcomings of the Scherrer method
to obtain grain sizes. As for PSS0218, the film thickness is similar to the grain size
values obtained from both model and Scherrer’s equation that means that sample
PSS0218 is a single crystal. This result is not unexpected, as the deposition
temperature of 950 ᵒC yields a value of Tm/Ts=0.51, where Tm is the melting
temperature of the material and Ts is the substrate temperature during the deposition.
Fig. 20 The figures shown above are the simulated XRR patterns that are fitted to the experimental data.
For Tm/Ts values above 0.5, grain sizes reaches thin film thickness16. Regarding the
density values, both samples PTS0217 and PTS0318 present a density compatible
with anatase (3.78 g/cm3), whereas PSS0218 has a density half way between rutile
(4.23 g/cm3) and anatase.
5 CONCLUSIONS AND FUTURE WORK
A physical model is built with the intention of providing an alternative and
improved methodology for XRD analysis. This model converts the intended sample
into multilayers parallel to the sample surface, quantifying each layer by 3 physical
parameters: density, size and atomic composition, which are averaged over the two
directions parallel to the sample surface. This model was built in IMD software, and
the same model but with minor modifications is useful to fit XRR data.
The model is used to determine thin film fundamental parameters, as phase, c
lattice constant, density, grain size and layer thickness of TiO2 thin films (one pure
TiO2, and two doped with 1% and 0.5% Tantalum respectively) grown on STO
substrates. The values obtained from the model are compared (when possible) with
values obtained from traditional XRD analyses, and it is found that the utilization of
the model yields more information and more accurate than traditional analyses.
Plans for the future:
1. Several parameters, such as polarization and angular dispersion of the incident
beam need to be precisely determined. The precise knowledge of the latter
parameters yields in better fits.
2. The model need to be tested with other thin film and substrates that consist of
different materials, dopants and doping level and deposition conditions.
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APPENDIX 1
Intensity calculations for powder diffraction