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