Thesis of Dr. G. Santosh Babu - Department of Physics

Transcription

STRUCTURAL, LATTICE VIBRATIONAL AND MICROWAVE
DIELECTRIC STUDIES ON SOME RARE EARTH BASED
COMPLEX PEROVSKITES
A THESIS
submitted by
G. SANTOSH BABU
for the award of the degree
of
DOCTOR OF PHILOSOPHY
DEPARTMENT OF PHYSICS
INDIAN INSTITUTE OF TECHNOLOGY MADRAS
CHENNAI - 600 036, INDIA
MAY 2008
The imaginary numbers are a wonderful flight of God’s
spirit; they are almost an amphibian between being and
not being.
- Leibnitz
God made the laws only nearly symmetrical so that we
should not be jealous of his perfection.
-
Feynman
Each Soul is potentially Devine. The goal is to manifest
this divinity within; by controlling nature: external and
internal. Do this either by Work, or Worship, or Psychic
control, or Philosophy – by one, or more, and all of these
and be free. This is the whole of the Religion. Doctrines,
or dogmas, or rituals, or forms, or temples are but
secondary details.
- Swami Vivekananda
Dedicated to
My mother, father and teachers
CERTIFICATE
This is to certify that the thesis entitled “STRUCTURAL, LATTICE
VIBRATIONAL AND MICROWAVE DIELECTRIC STUDIES ON SOME
RARE
EARTH
BASED
COMPLEX
PEROVSKITES”
submitted
by
G.SANTOSH BABU to the Indian Institute of Technology Madras, Chennai for the
award of the degree of Doctor of Philosophy is a bonafide record of research work
carried out by him under our supervision. The contents of this thesis, in full or in
parts, have not been submitted to any other Institute or University for the award of
any degree or diploma.
Research guides
(Dr. V. SUBRAMANIAN)
Chennai 600 036
Date:
(Prof. V. R. K. MURTHY)
ACKNOWLEDGEMENTS
Last six years I have spent at Indian Institute of Technology Madras were days of
constant learning with the joy of exploring frontiers of science. I take this opportunity
to acknowledge all those who have helped me directly or indirectly.
Foremost, I would like to express my deep sense of gratitude to Dr. V. Subramanian
and Prof. V. R. K. Murthy for their unparalleled guidance, constructive and honest
criticism, constant encouragement and expert advice. I am greatly indebted to them
for providing me excellent freedom to work on what I was interested in.
I express my sincere thanks to the present and former Heads of the Department of
Physics Prof. P. C. Deshmukh and Prof. A. Subrahmanyam for their encouragement
and help throughout the course of my research work.
I am grateful to my doctoral committee members Prof. G. Markandeyulu, Prof. M. V.
Satyanarayana, Prof. K. Balasubramniam and Prof. Paramanand singh for their
constructive suggestions and fruitful discussions at various stages of my research
work.
I extend my heart-felt thanks to Dr. P. N. Santhosh, Department of Physics for
teaching me the techniques of Rietveld refinement and allowing me to use Crystal
Maker software.
I specially thank Dr. V. Sivasubramanian, IGCAAR, Kalpakam and Prof. K. V. Shiva
Kumar, Sri Krishnadevaraya University, Anantapur for all the discussions on sample
preparation and characterization.
I am greatly indebted to Dr. Patrick Woodward, Department of Chemistry, Ohio State
University, USA for answering all my e-mail queries and making me to understand
the structural aspects of perovskites through direct or indirect teaching. I also thank
him for providing all the informative papers.
i
I would like to thank Dr. R. L. Moreira, Departamento de Física, UFMG, Brazil for
lively discussions on spectroscopic analysis, recording IR spectra and helping me to
understand the concepts of lattice vibrations. I also thank Dr. R. P. S. M. Lobo,
Laboratoire Photons et Matière, Université Pierre et Marie Curie, France for careful
recording of IR spectra on some of the samples.
I express my sincere thanks Prof. I-Nan Lin, Department of Physics, Tamkang
University, Taiwan and Prof. Chia-Ta Chia and Prof. Hsiang-Lin Liu, Department
of Physics, National Taiwan Normal University, Taiwan for careful recording of
Raman and IR spectra on some of the samples.
I thank Dr. A.N. Salak, University of Aveiro, Portugal for all the discussions and
helping me to understand few concepts or rare earth based perovskites.
I thank Dr. D. De Sousa Meneses, CEMHTI, France for providing me Focus
software.
I thank Dr. Rick Ubic, Biose State University, USA for all the discussion on structure
and preparation of perovskites. I also thank him for sending necessary publications.
I thank many of my friends Dr. R. Sripad, Mr. P. Naresh, Mr. Sree Ram Mahesh, Dr.
K. Prabhakara Rao, Dr. Sufal Swaraj, Dr. N. Rajeev Kini, Mr. Ritesh Rawal, Mr.
G. A. Ravi and Mr. Sachidananda Mishra for sending me the necessary papers
whenever I needed.
I am extremely grateful to the staff members of Physics department and Machine shop
for their prompt service and cooperation.
I gratefully acknowledge my lab mates Dr. V. Radha Ramani, Dr. Dibyaranjan Rout,
Dr. E. D. V. Nagesh, Dr. Bibekananda Sundaray, Dr. D. V. B. Murthy, Dr. T.
Vishwam, Mr. V. Jagadeesh Babu, Mr. S. Roopas Kiran, Mr. N. Yogesh, Mr. G.
Ramesh, Mr. N. Raja Mohan, Mr. J. Magesh, Mr. Pradeep and Mr. Samiran
ii
Bhumik for their whole hearted help, discussions and cooperation. I also thank Mr.
N. Saravanan and Mr. P. Singaravelu for all their support and help.
I would like to thank all my Research Colleagues in the Physics department for their
help, cooperation and pleasant company during my stay in the campus.
I express my special thanks to Dr. S. Bhasker Reddy, Mr. C. Pedda Peraiah, Mr. P.
Koteswara Rao, Mr. Siva Nageswara Rao and Mr. Prasad Dudhgaaonker for all the
physics and non physics discussions, help and pleasant company during my stay in the
campus.
I thank my friend Mrs. Josephene Prabha for all the moral support. I also thank my
M.Sc class mates and friends for the moral support.
I would also like to sincerely thank Indian Institute of Technology Madras and
Council of Scientific and Industrial Research (CSIR), New Delhi for providing me
the financial support and facilities to carry out my research work.
I thank all my teachers who trained me to learn the science and made me to pursue
career in science. I specially thank my teacher Fr. Peter Daniel for all the moral
science classes during my high school.
Finally, I thank my wife for all the support, understanding and love during the final
stages of my work. I also thank all my family members for their love, affection and
moral support throughout the course of the work.
G. SANTOSH BABU
iii
ABSTRACT
KEYWORDS:
Complex Perovskites, Dielectric Resonators, Tolerance
Factor, Octahedral Tilting, B site Cation Ordering, Long
Range Order, Polarizability, Average Phonon Damping,
Raman Spectroscopy, Photonic Crystals
Dielectric Resonators (DRs) are miniature resonant devices in microwave integrated
circuits. They are dielectric materials with special characteristics. They are used in
telecommunications as frequency stabilizing elements in oscillator circuits and filters
etc. The principle of their operation is the ability of the dielectric/air interface to
reflect electromagnetic radiation, and thus the material can sustain a standing
electromagnetic wave within its body. A useful resonator should have ε ′r ≥ 20 for
suitable size reduction, quality factor (Q) > 3000 for low insertion loss and the
temperature coefficient of resonant frequency (τf) close to zero for temperature stable
operation.
Complex perovskite materials with AA'(BB')O3 chemical formula are best suitable
materials for microwave dielectric applications. Dielectric properties of these
materials depend upon the B site cation ordering, structure and phonon mode
characteristics. This thesis presents the investigation of crystal structure, estimation
of B site cation ordering, evaluation of phonon mode characteristics, determination of
intrinsic parameters and measurement of dielectric properties at microwave
frequencies. Thesis also presents study on one dimensional photonic crystals at
microwave frequencies using transfer matrix method and measurement of
transmittance.
iv
The thesis is divided into six chapters and the contents of each chapter are briefly
summarized below.
Chapter 1 begins with the general introduction of perovskite materials. A brief review
on dielectric resonator (DR) materials in the early days and their development, mode
characteristics of DRs and application of DRs are described. Literature survey on
alkaline earth materials with perovskite structure are discussed with respect to cation
ordering, structure, processing and dielectric properties. A brief review on rare earth
based perovskites concentrating on structure property relation and importance of
lattice vibrational study on DR materials is presented. Importance of photonic
crystals is also described briefly.
Chapter 2 deals with the synthesis and structure determination of rare earth based
complex perovskite DR compositions La(MgTi(1-x)Snx)0.5O3 (x = 0, 0.125, 0.25, 0.375
and 0.5), La(1-x)Ndx(MgSn)0.5O3 (x = 0, 0.25, 0.5, 0.75 and 1.0) and Nd(MgTi)0.5O3.
All the compositions are synthesized by solid state reaction method. The calcination
and sintering conditions are optimized to obtain the maximum density. The samples
are characterized by X-ray diffraction, indexing super lattice reflections and structure
determination. Rietveld refinement carried out on complex perovskites LMT, NMT
and La(1-x)Ndx(MgSn)0.5O3 is also presented.
Chapter 3 describes lattice vibrational studies on DR compositions. The results of
lattice mode characteristics obtained by the fit of the far infrared reflectance data to
four parameter model and fit of the Raman A1g mode to Lorentzian peak shape are
discussed. The frequencies of TO and LO modes, their damping coefficients, TO
v
mode strengths and intrinsic parameters are obtained from the fit of reflectance
spectra. FWHM of the A1g mode and Raman shift are obtained from the Raman
spectra. The variation of intrinsic Q.f with the long range order and average phonon
damping is discussed.
Chapter 4 describes the methods of microwave characterization techniques used in
this work on DR compositions. For measurement of the dielectric constant of DR
materials, Courtney’s method is employed. Dielectric constant is calculated by
measuring the frequency of TE011 mode. Q factor is measured using reflection
method. A cavity made of copper having the same aspect ratio of DR itself, but
diameter 2 to 3 times larger than DR is used for this purpose. The temperature
coefficient of resonant frequency is measured by an invar cavity. The results of the
measurement and discussion with respect to polarizability, structure and lattice
vibrations are presented.
Chapter 5 presents photonic band gap studies on one dimensional photonic crystals in
the frequency range of 10 to 20 GHz. Theoretically band gaps and defect modes are
analyzed by computation using transfer matrix method and experimentally by
measuring the transmittance through the photonic crystals constructed using glass and
ebonite dielectric materials. Computations carried out on photonic crystal constructed
with low dielectric loss and high dielectric constant material is also presented.
Chapter 6 presents the summary of the research work and the major conclusions
drawn from the results of individual chapters. A brief report on the scope for future
work is also presented.
vi
TABLE OF CONTENTS
Page No.
ACKNOWLEDGEMENTS …………………………………………………
i
ABSTRACT…………………………………………………………………..
iv
LIST OF TABLES…………………………………………………………...
x
LIST OF FIGURES………………………………………………………….
xii
ABBREVIATIONS…………………………………………………………..
xvi
NOTATIONS…………………………………………………………………
xvii
CHAPTER 1 INTRODUCTION
1.1 Dielectric Resonators…...………………………………………. ….......
4
1.1.1 Modes of Dielectric Resonator …...……………………………...
7
1.1.2 Applications of DRs.………...……………....................................
11
Perovskite Dielectric Resonators……………………………………......
12
1.2.1 Alkaline Earth Perovskites ………………….. …..……….………
12
1.2.2 Rare Earth Based Complex Perovskites…………………………..
15
1.2.3 Lattice Vibrational Aspects of Perovskites………………………..
17
1.3 Photonic Band Gap Structures…………………………………………...
20
1.4 Objective and Scope of the Present Work……………………………….
22
1.2
CHAPTER 2 PREPARATION AND STRUCTURAL
CHARACTERIZATION OF La(MgTi(1-x)Snx)0.5O3,
La(1-x)Ndx(MgSn)0.5O3 AND Nd(MgTi)0.5O3
2.1
Structural Aspects of Perovskites....………………………………….…
26
2.1.1 Tolerance Factor and Octahedral Tilting………………………….
26
2.1.2 Glazer Tilt Notation…..……………………………………..…….
26
2.1.3 B site Cation Ordering......………………….……………………..
27
2.1.4 Octahedral Tilting and X-ray Powder Diffraction ………………..
29
2.1.5 Effect of B site Cation Ordering on X-ray Powder Diffraction…...
30
vii
2.2
Structural Characterization……..……..………………... ………….......
32
2.2.1 Structural Study of La(Mg0.5Ti(0.5-x)Snx)O3……………………….
32
2.2.2 Structural Study of La(1-x)Ndx(MgSn)0.5O3 ….…………………..
37
2.2.2.1 X-ray Diffraction Patterns of La(1-x)Ndx(MgSn)0.5O3…….
37
2.2.2.2 Rietveld Refinement of La(1-x)Ndx(MgSn)0.5O3 ………….
39
2.2.3 Rietveld Refinement of La(MgTi)0.5O3 and Nd(MgTi)0.5O3………
52
2.3 Conclusions………………………………………………………………
56
CHAPTER 3 LATTICE VIBRATIONAL STUDIES ON
La(MgTi(1-x)Snx)0.5O3,
La(1-x)Ndx(MgSn)0.5O3 AND Nd(MgTi)0.5O3
3.1 Infrared Reflectance Studies…..…………………………………………
57
3.1.1 Four Parameter Model………………………………...………......
58
3.1.2 Experimental Details………………………………………………
59
3.1.3 Analysis of Infrared Reflectance Data……………….... …………
60
3.1.3.1 Mode Assignment and Data Treatment…………….........
60
3.1.3.2 IR Study on La(Mg0.5Ti(0.5-x)Snx)..............................….…
61
3.1.3.3 IR Study on La(1-x)Ndx(Mg0.5Sn0.5)O3 ................. ………..
67
3.2 Raman Scattering Studies………………………………………………..
72
3.2.1 Experimental Details………………………………………………
75
3.2.2 Raman Spectra of La(Mg0.5Ti(0.5-x)Snx)O3......................................
76
3.2.3 Raman Spectra of La(1-x)Ndx(MgSn)0.5O3…………………………
79
3.2.4 Raman Spectra of Nd(MgTi)0.5O3 and La(MgTi)0.5O3……………
81
3.3 Conclusions………………………………………………………………
84
CHAPTER 4: MICROWAVE DIELECTRIC CHARACTERIZATION
OF La(MgTi(1-x)Snx)0.5O3, La(1-x)Ndx(MgSn)0.5O3 AND
Nd(MgTi)0.5O3
4.1 Characterization Techniques of Microwave Dielectric Properties….…...
86
4.1.1 Measurement of Dielectric Constant ( ε ′r ) ……………………….
86
4.1.2 Measurement of Quality (Q) Factor…..……….……………….....
90
4.1.3 Measurement of Temperature Coefficient of Resonant Frequency
95
4.2 Results and Discussion. …………………..…………….... …………….
96
4.2.1 Microwave Dielectric Properties of La(Mg0.5Ti(0.5-x)Snx)O3 ……...
96
4.2.2 Microwave Dielectric Properties of La(1-x)Ndx(MgSn)0.5O3 and
viii
Nd(MgTi)0.5O3 …………………………………………………….
99
4.3 Conclusions…………. …………...……………………………………...
103
CHAPTER 5: PHOTONIC BAND GAP STUDIES ON ONE
DIMENSIONAL STRUCTURES
5.1 Computation Using Transfer Matrix Method …………………………….
105
5.1.1 Matrix for Photonic crystal (Transfer Matrix)..…………………...
105
5.1.2 Band Structure of Photonic Crystal………………………………..
106
5.1.3 Transmission Coefficient and Transmittance……………………...
107
5.1.4 Density of Modes…………………………………………………..
107
5.1.5 Method of Calculation....……..........................................................
107
5.2 Details of the Experiment ……………………………………………….
108
5.3 Analysis of One Dimensional Photonic Crystals…………………………
112
5.3.1 Glass and Ebonite Structures………………………………………
112
5.3.2 Double Periodic Structure…………………………………………
117
5.3.3 Low Loss and High Dielectric Constant Photonic Crystal………..
119
5.4 Conclusions………………………………………………………………..
121
CHAPTER 6 SUMMARY AND CONCLUSIONS
6.1 Scope for the Future Work……………………………………………….
126
127
REFERENCES……………………………………………………………….
139
LIST OF PUBLICATIONS……..…………………………………………...
ix
LIST OF TABLES
Table
Title
Page No.
Reflection coefficient for the plane wave incident from inside of
the dielectric material……………………………..………….………….
6
Space groups for all possible tilt systems both random B site cations
(random) and with 1:1 B-site cation ordering (ordered)…………………
31
2.2
Tolerance factor of La(Mg0.5Ti(0.5-x)Snx)O3……………………….……..
33
2.3
Tolerance factors, Rietveld discrepancy indices and long range
order parameter (LRO) of La(1-x)Ndx(MgSn)0.5O3 system…………………
43
Fractional atomic coordinates, thermal parameters and occupancies of
La(MgSn)0.5O3……………………………………………………………
43
La0.75Nd0.25(MgSn)0.5O3…………………………………………………..
44
La0.5Nd0.5(MgSn)0.5O3…………………………………………………..
44
La0.25Nd0.75(MgSn)0.5O3…………………………………………………..
45
Nd(MgSn)0.5O3……………………………………………………………
45
1.1
2.1
2.4
2.5
2.6
2.7
2.8
2.9
Fractional atomic coordinates, thermal parameters and occupancies
of cubic Nd2Sn2O7 pyrochlore with Fd 3m symmetry ………………….. 46
2.10
Lattice parameters and X-ray density of La(1-x)Ndx(MgSn)0.5O3 system….. 46
2.11
of La(MgTi)0.5O3………………………………………………………….. 54
2.12
of Nd(MgTi)0.5O3………………………………………………………….. 54
2.13
Lattice parameters and Rietveld discrepancy indices of
La(MgTi)0.5O3 and Nd(MgTi)0.5O3................................................................ 54
3.1
IR fit parameters and intrinsic dielectric constant (ε') obtained for
x = 0.0, x = 0.125 and x = 0.25 compositions of
La(Mg0.5Ti(0.5-x)Snx)O3..............……............................................................... 64
x
3.2
x = 0.375 and x = 0.5 compositions of La(Mg0.5Ti(0.5-x)Snx)O3……………. 65
3.3
x = 0.0, x = 0.25 and x = 0.5 compositions of La(1-x)Ndx(MgSn)0.5O3…….... 69
3.4
IR fit parameters and intrinsic dielectric constant (ε') obtained for x = 0.75
and x = 1.0 compositions of La(1-x)Ndx(MgSn)0.5O3………………………... 70
3.5
.
3.6
Atomic positions and Raman active modes for cubic crystal Fm3 m ……….....72
Raman shift of A1g mode and FWHM for La(Mg0.5Ti(0.5-x)Snx)O3 ceramics..... 79
3.7
Raman shift and FWHM of A1g mode for La(1-x)Ndx(MgSn)0.5O3 ceramics….... 81
4.1
Relative density (d), dielectric parameters determined at microwave
frequencies for the La(Mg0.5Ti(0.5-x)Snx)O3 ceramics………………………….. 98
4.2
Relative density, tolerance factor (t) and dielectric parameters determined
at microwave frequencies for the La(1-x)Ndx(MgSn)0.5O3 system………………101
5.1
Quality factor (Q) values of glass and ebonite photonic crystal………………..115
6.1 Relative density (d), dielectric characteristics extrapolated from infrared (IR)
data, Lorentzian fit parameters of A1g mode (Raman) and dielectric
parameters determined at microwave (MW) frequencies for the
La(Mg0.5Ti(0.5-x)Snx)O3 ceramics...........................................................................125
6.2 Relative density (d), dielectric characteristics extrapolated from infrared (IR)
data, Lorentzian fit parameters of A1g mode (Raman) and dielectric
La(1-x)Ndx(MgSn)0.5O3 system..............................................................................126
xi
LIST OF FIGURES
Figure
Title
Page No.
1.1
Ordered A(BB')X3 perovskite structure ( Fm3 m space group)
showing corner sharing of BX6 (light grey) and B'X6 (dark grey)
octahedra………. …………………..…………………………………………3
1.2
Electric field distribution in equatorial plane for TE01δ mode of a DR.………10
1.3
Magnetic field distribution in meridian plane for TE01δ mode of a DR………10
1.4
Dielectric resonator showing strongest electrics and magnetic field
lines for TE01δ mode…………………………………………………………..10
2.1
In phase tilting of BO6 octahedra ………………..............................................28
2.2
Out of phase tilting of BO6 octahedra ………………………………………...28
2.3
X-ray diffraction patterns of La(Mg0.5Ti(0.5-x)Snx)O3 indexed based on
a cubic perovskite cell. ………………………………………………............34
2.4
Evolution of ½(111) super lattice reflection with increase in Sn
concentration (x) for La(Mg0.5Ti(0.5-x)Snx)O3 ceramics ……………………....34
2.5
X-ray diffraction patterns of La(Mg0.5Ti(0.5-x)Snx)O3 indexed
based on a monoclinic P 21 / n unit cell………………………………….......35
2.6
The variation of (110) cubic or (200) monoclinic reflection with increase
in Sn concentration (x) of La(Mg0.5Ti(0.5-x)Snx)O3 ceramics............................35
2.7
The variation of lattice parameters a, b, c and β with increase
in Sn concentration (x) for La(Mg0.5Ti(0.5-x)Snx)O3 ceramics............................36
2.8
X-ray diffraction patterns of La(1-x)Ndx(MgSn)0.5O3 , x = 0.0 (bottom),
0.25, 0.5, 0.75 and 1.0 (top) ceramics…………………………………….…..38
2.9
Final observed (+ marks), calculated (solid line) and difference (below)
patterns, along with the calculated positions for La(MgSn)0.5O3.……….…….46
2.10
patterns, along with the calculated positions for La0.75Nd0.25(MgSn)0.5O3….…47
.
2.11
patterns, along with the calculated positions for La0.5Nd0.5(MgSn)0.5O3……...47
xii
2.12
patterns, along with the calculated positions for La0.25Nd0.75(MgSn)0.5O3…....48
2.13 Final observed (+ marks), calculated (solid line) and difference
(below) patterns, along with the calculated positions for Nd(MgSn)0.5O3.
Pyrochlore Nd2Sn2O7 positions are also shown……………………………..…48
2.14 a) Structure of La(MgSn)0.5O3. Unit cell is shown with dotted line
b) Structure of La(MgSn)0.5O3 in z direction………….…………………..……49
2.15 Structure of La(MgSn)0.5O3 :
a) x – direction, b) y- direction and c) z-direction………………………......…..50
2.16 Lattice parameters a, b, c and β of La(1-x)Ndx(MgSn)0.5O3 ceramics……..…….51
2.17 X-ray diffraction pattern of Nd(MgTi)0.5O3……………………………..……..53
(below) patterns, along with the calculated positions for La(MgTi)0.5O3...……..55
(below) patterns, along with the calculated positions for Nd(MgTi)0.5O3…...….55
3.1
.
3.2
IR reflectivity spectra of La(Mg0.5Ti(0.5-x)Snx)O3 perovskite system……...…….63
Imaginary part of dielectric constant (ε") obtained by fitting reflectivity
to four parameter model…………………………………………………………63
3.3
The variation of TO mode phonon strength of La(Mg0.5Ti(0.5-x)Snx)O3
as function of Sn concentration………………………………………………….66
3.4
Intrinsic Q.f values (circles) and average TO phonon damping (squares)
as function of Sn content, x…………………………………………………….. 66
3.5
IR reflectivity spectra of La(1-x)Ndx(MgSn)0.5O3 perovskite system…………….68
3.6
Imaginary part of dielectric constant (ε") obtained by fitting reflectivity to
four parameter model……………………………………………………………68
3.7
The variation of TO mode strength as function of Nd concentration, x…………71
3.8
Intrinsic Q.f values (open circles) and average TO phonon damping
(shaded squares) as functions of Nd content, x…………………………………..71
3.9
Vibration of O atom: a). A1g stretching mode
b). Eg antistretching mode and c). F2g bending mode…………..………………..74
3.10 Raman spectra of La(Mg0.5Ti(0.5-x)Snx)O3………………………………………..78
3.11 A1g mode of La(Mg0.5Ti(0.5-x)Snx)O3
xiii
a). A1g mode of x = 0.0 b). A1g mode of x = 0.125 to 0.5………………...……78
3.12 Raman spectra of La(1-x)Ndx(MgSn)0.5O3 ceramics...............................................80
3.13 Raman spectra of La(MgTi)0.5O3 and Nd(MgTi)0.5O3 ……………...…….…….83
3.14 Raman A1g mode of La(MgTi)0.5O3 and Nd(MgTi)0.5O3
a). A1g mode of La(MgTi)0.5O3. b). A1g mode of Nd(MgTi)0.5O3…...……….…..83
4.1 Schematic diagram of Courtney's method for measurement of the
dielectric constant of DRs………………………………………………………..88
4.2 Photograph of experimental arrangement to measure dielectric constant
of DRs………………………………………………………………………...…..88
.
4.3 The schematic diagram of cavity used to measure quality factor of DRs…...…...93
4.4 Photograph of the experimental arrangement for the measurement of
quality factor (Q) of DRs……………………………………………….…..…….93
4.5 Resonant frequency display of TE01δ mode for Nd(MgSn)0.5O3………...……….94
4.6 Smith chart display of TE01δ mode for Nd(MgSn)0.5O3………………...………..94
4.7
The schematic diagram for measurement temperature coefficient of
resonant frequency……………………………………………………………….97
4.8 Photograph of the experimental arrangement measure temperature
coefficient of resonant frequency………………………………………………...97
4.9
Tolerance factor (squares) and temperature coefficient of resonant
frequency (circles) of La(Mg0.5Ti(0.5-x)Snx)O3 ceramics as a function of Sn
concentration, x…………………………………………………………………..99
4.10 Polarizabilties of lanthanide ions versus radius cube…………………………...102
4.11 Classius-Mossotti and microwave dielectric constants of
La(1-x)Ndx(MgSn)0.5O3 system.............................................................................102
5.1
Schematics of the one-dimensional photonic band gap structure…….…………105
5.2
Variation of band gap width with increase in dielectric constant……….………109
5.3
Variation of band gap with increase in spacing between the glass sheets……....109
5.4
Experimental arrangement to observe the transmittance of the one
dimensional photonic crystal………………………………………………….....111
5.5
Photograph of transmittance measurement of one dimensional photonic
crystal…………………………………………………………………………....111
xiv
5.6
Variation of the gap width for glass ( ε ′r1 = 6.8, h1 = 0.3 cm) and ebonite
( ε ′r1 = 4, h1 = 0.27 cm) structures with thickness of air medium (h2) in the
range of 10 to 20 GHz…………………………………………………….…… 113
5.7
Band structure for photonic crystal with ε ′r1 = 6.8, h1 = 0.3 cm, ε ′r2 = 1
and h2 = 0.3 cm………………………………………………………………….113
5.8
Band structure for photonic crystal with ε ′r1 = 4, h1 = 0.27 cm, ε ′r2 = 1
and h2 = 0.4 cm………………………………………………………………….113
5.9
Schematics of the one-dimensional photonic crystal with defect…...…....….….114
5.10 Density of modes for the nine period glass photonic crystal without
and with defect…………………………………………………………....……..116
5.11 Density of modes for the nine period ebonite photonic crystal without
and with defect…………………………………………………………....……..116
.
5.12 Computed and measured transmittance through glass photonic crystal
without defect…………………………………………………………………....116
5.13 Computed and measured transmittance through ebonite photonic crystal
without defect…………………………………………………………………....116
5.14 Computed and measured transmittance through glass photonic crystal
with defect……………………………………………………………………....116
..
5.15 Computed and measured transmittance through ebonite photonic crystal
with defect…………………………………………………………………….…116
5.16 Schematics of double periodic one-dimensional photonic crystal……………....118
5.17 Band structure for double periodic photonic crystal ……………………….…...118
5.18 Density of modes for double periodic photonic crystal……………………….....118
5.19 Transmittance through double periodic photonic crystal……………….……….118
5.20 Transmittance through double periodic photonic crystal with defect….…….….118
5.21 Band structure for low loss and high dielectric constant photonic crystal……....120
5.22 Tranmsittance through low loss and high dielectric constant
photonic crystal……………………………………………………………….….120
5.23 Transmittance through low loss and high dielectric constant
photonic crystal with defect………………………………………………….…..120
xv
ABBREVIATIONS
DR
Dielectric Resonator
MW
Microwave
GHz
Gigahertz
BMT
Ba(Mg1/3Ta2/3)O3
BZT
Ba(Zn1/3Ta2/3)O3
LMT
La(MgTi)0.5O3
LZT
La(ZnTi)0.5O3
BT
BaTiO3
CT
CaTiO3
ST
SrTiO3
LMS
La(MgSn)0.5O3
NMT
Nd(MgTi)0.5O3
NMS
Nd(MgSn)0.5O3
FT-IR
Fourier transform infrared
XRD
X-ray diffraction
LRO
Long range order parameter
PVA
Poly vinyl alcohol
iso
Isotropic
ppm
parts per million
SPUDS
Structure prediction and diagnostics software
GSAS
General structural analysis system
EXPGUI
Graphical user interface for GSAS
TO
Transverse optic
LO
Longitudinal optic
CCD
Charge coupled device
VSWR
Voltage standing wave ratio
dB
decibel
FWHM
Full width at half maxima
IR
infrared
xvi
NOTATIONS
f
Frequency
λ0
Wavelength in free space
λd
Wavelength in dielectric
f0
Resonant frequency
Q
Quality factor
QL
Loaded quality factor
Q0
Unloaded quality factor
ε ′r
Dielectric constant (real part of relative permittivity)
ε ′r′
Dielectric loss (imaginary part of relative permittivity)
ε*
Complex dielectric constant
t
Tolerance factor
tanδ
Dielectric loss tangent
L
Thickness of the sample
D
Diameter of the sample
a, b & c
Lattice parameters
T
Temperature
U
Thermal parameter
o
odd
e
even
β
beta (the angle between a and c axes)
°C
degrees Celsius
p
Coupling coefficient
φ
phase
τf
The temperature coefficient of resonant frequency
τe
The temperature coefficient of dielectric constant
Jm
Bessel function of the order m
xvii
Km
Modified Bessel function of the order m
ν
Frequency expressed in wave numbers
ω
radian frequency (2πf)
γ
complex propagation constant
∆εj
Oscillator strength of TO mode
ΩjTO
Frequency of TO mode
ΩjLO
Frequency of LO mode
γjTO
Damping coefficient of TO mode
γjLO
Damping coefficient of LO mode
R
Reflectivity (or Reflectance)
K
Bloch wave vector
k
propagation wave vector
χ
goodness of fit
Rwp
RF
2
weighted profile R factor
F2 (structure factor)2 R factor
κ
filling factor
T
transmittance
t
transmission coefficinet
Y
admittance
ν
linear expansion coefficient
α
ionic polarizability
σ
sigma bond
π
pi bond
a
lattice constant of photonic crystal
d
relative density (ratio of experimental density and X-ray
density in percentage)
xviii
CHAPTER 1
INTRODUCTION
The perovskite family encompasses a variety of compounds ranging from
microwave dielectrics or dielectric resonators (BaMg1/3Ta2/3O3 and BaZn1/3Ta2/3O3),
ferroelectrics
(PbTiO3
and
BaTiO3),
piezoelectrics
(PbZr1-xTixO3),
relaxor
ferroelectrics (PbMg1/3Nb2/3O3), multiferroics (BiFeO3), colossal magneto resistance
materials (La1-xCaxMnO3), superconductors (Ba1-xKxBiO3) and non-linear optical
behavior materials (KNbO3). Originally the term perovskite was assigned to CaTiO3
and named after a Russian mineralogist Count Lev Aleksevich von Perovski.
Perovskites have the general formula ABX3, in which A represents a large
electropositive cation, B represents a small transition metal or main group ion and X
is commonly oxide or halide ion. The perovskite structure is very flexible and it can
accommodate most of the metallic elements of periodic table.
The ideal or aristotype perovskite structure with ABX3 stoichiometry belongs
to cubic space group Pm3 m and is composed of a three dimensional network of
vertex sharing BX6 octahedra. The A cation is surrounded by twelve X anions in a
dodecahedral environment, the B cation is octahedrally coordinated by six X anions
and the X ions are coordinated by two B cations and four A cations. The number of
possible compounds is greatly expanded when multiple ions are substituted for one or
more of original ions. In most cases, this substitution occurs on the cation site and
leads to large class of compounds known either as complex or double perovskites,
AA'BB'X6. Substituted ions can occupy the original cation in either a random or an
ordered fashion. If an ordered arrangement is adopted, the symmetry and in many
1
cases, the size of the unit cell are changed (Woodward, 1997). This structure type is
also called the elpasolite structure, named after the mineral K2NaAlF6. The B site
cation ordering is commonly seen in perovskites whereas A-site cation ordering is
rarely observed (Knapp and Woodward, 2006). The ideal cubic elpasolite structure
with rock salt ordering of B site cations is shown in Figure 1.1. The B site cation
ordering in perovskites gives rise to newer and interesting material properties. For
instance, realization of half-metallic ferrimagnetism in Sr2FeMO6 (M = Mo and Re),
relaxor ferroelectricity in Pb3MgNb2O9 and high dielectric constant with low
dielectric loss in Ba3ZnTa2O9 depends crucially on the ordering and nature of the B
site cations in the perovskite structure.
There exist three types of distortions in perovskites, viz. distortions of
octahedra, cation displacements within the octahedra and tilting of the octahedra. The
first two distortion mechanisms are driven by electronic instabilities of the octahedral
metal ion. The Jahn-Teller distortion in KCuF3 (Okazaki and Suemune, 1961) is an
example of electronic instability that leads to octahedral distortion. The ferroelectric
displacement of titanium in BaTiO3 (Shirane et al., 1957) is an example of electronic
instability that leads to cation displacements. The third and most common distortion,
the octahedral titling, can be realized by tilting the rigid BX6 octahedra while
maintaining their corner shared connectivity (Megaw and Darlington, 1975; Glazer,
1972; Glazer, 1975). This type of distortion is generally observed when the A-site
cation is too small for the cubic BX6 corner sharing octahedral network.
Perovskite microwave dielectrics (dielectric resonators) come into the class of
materials with octahedral tilting and cation ordering. Further detailed discussion on
tilting of octahedra and B-site cation ordering is dealt in chapter 2.
2
Fig. 1.1 Ordered A(BB')X3 perovskite structure ( Fm 3 m space group) showing corner
sharing of BX6 (light grey) and B'X6 (dark grey) octahedra. Small black shaded
spheres are X atoms and large hatched spheres are A site atoms.
3
1.1 DIELECTRIC RESONATORS
The dielectric resonator (DR) is a piece of high dielectric constant ceramic,
usually in the shape of cylinder that functions as a miniature microwave resonator.
Temperature-stable, medium dielectric constant ceramics have been used as
resonators in filters for microwave (MW) communications for several decades. The
growth of mobile phone market in the 1990s led to extensive research and
development in this area. The main driving forces were the greater utilization of
available bandwidth, which necessitates low dielectric loss (high quality factor, Q),
increase in dielectric constant so that smaller components could be fabricated with
cost reduction (Reaney and Iddles, 2006). The basic principle in the miniaturization is
that the wavelength of electromagnetic radiation λ in air gets reduced to the factor
1 / ε ′r inside the dielectric material ( ε ′r is the dielectric constant of the DR).
The term “Dielectric resonator” was first coined by Richtmyer, 1939, when he
showed that unmetallized dielectric objects (toroid) can function as microwave
resonators. However, his theoretical work failed to generate significant interest, and
practically nothing happened in this area for over 25 years. Later, a paper by Schlicke,
1953 reported on super high dielectric constant materials (~ 1000 or more) and their
applications as capacitors at relatively low RF frequencies. In the early 1960s,
researchers Okaya and Barash rediscovered DRs while working on high dielectric
materials (single crystal TiO2 rutile), paramagnetic resonance and masers. Their
papers (Okaya, 1960; Barash, 1962) provided the first analysis of modes and
resonator design. Nevertheless, the DR was still far from practical applications. High
dielectric constant materials such as rutile exhibited poor temperature stability,
correspondingly causing large resonant frequency changes. For this reason, in spite of
high Q-factor and small size, DRs were not considered in microwave devices.
4
In the mid 1960s, Cohn and his coworkers at Rantec Corporation performed
the extensive theoretical and experimental evaluation of the DR (Cohn, 1960). Rutile
ceramics were used for experiments that had an isotropic dielectric constant in the
order of 100. Again, poor temperature stability prevented development of practical
components. A real breakthrough in ceramic technology occurred in the early 1970s
when Masse et al., 1971, developed the first temperature stable low loss bariumtetratitanate ceramics. Temperature stable microwave DRs utilizing the composite
structure of positive and negative temperature coefficients of resonant frequency were
reported by Konishi, 1971. Later, a modified barium tetratitanate with improved
performance was reported by Bell Laboratories (Plourde et al., 1975). These positive
results led to actual implementations of DRs as microwave components. The
materials, however, were in scarce supply and, thus, were not commercially available.
The next major breakthrough came from Japan when the Murata Manufacturing
Company produced (ZrSn)TiO4 ceramics (Wakino et al., 1975; Wakino et al., 1977).
They offered adjustable compositions so that temperature coefficient could be varied
between +10 and -12 ppm/◦C. These devices became commercially available at
reasonable prices. Afterwards, the theoretical work and use of DRs expanded rapidly.
The ceramic element acts as a resonator due to multiple total internal reflections at
the high dielectric constant material and air boundary. The dielectric-air boundary will
act as perfect reflector of microwaves if the angle of incidence is greater than critical
(
)
angle θ c = sin −1 1/ ε ′r . For large values of ε ′r , the waves are internally reflected. The
values of reflection coefficient ( Γ ) calculated for DRs with various values of ε ′r
when the microwaves come out of the DR to free space (Kajfez, 1986a) are given in
Table 1.1
5
Table 1.1 Reflection coefficient for the plane wave incident from inside of the
dielectric material
ε ′r
2.5
10
37
100
Γ
0.225
0.519
0.717
0.818
It is seen from the Table 1.1 that for normal incidence, as the dielectric
constant increases, the value of reflection coefficient approaches unity. For this
reason, Kajfez, 1986a considered the dielectric-air boundary of a high dielectric
material is closer to a perfect magnetic conductor (PMC). This is a fictious material,
which requires the magnetic field tangential to its surface to be zero. Thus if ε ′r is
high, the electric and magnetic fields are confined in and near the resonator, resulting
in small radiation losses. To a first approximation, dielectric resonator is a dual of
metallic cavity. The radiation losses of the DRs with the commonly used dielectric
constant, however, are generally greater than the energy losses in the metallic cavities,
which necessitate proper shielding of the DR. But as a practical device, this leaking
field is useful, for it is this field that enables a DR to easily couple with strip lines and
other components in a microwave integrated circuit. The field confined inside the DR
is susceptible to intrinsic losses arising from the imaginary part of the dielectric
constant of the material ( ε ′r′ ). The unloaded quality factor Qu is thus limited by the
losses in the dielectric resonator. Hence only a low loss material can be used for DR
application.
The field inside a DR cannot be described as simple plane waves.
Nevertheless, any general electromagnetic field distribution can be considered as a
superposition of various plane waves incident under all possible angles. The boundary
6
conditions of a DR allow various field patterns, to be sustained at various frequencies
which are assigned as different modes of a DR.
1.1.1 Modes of Dielectric Resonator
Various modes supported by a DR can be approximately found out by
considering a DR as a truncated dielectric rod waveguide. The solution of the fields,
which satisfy Maxwell’s equation in the dielectric rod waveguide, leads to the
following eigen value equation (Kajfez, 1986b).
F1 (x)F2 (x) − F32 (x) = 0
(1.1)
F1 (x) , F2 (x) and F3 (x) are given by the following equations
F1 (x) =
J ′m (x) K ′m (y) J m (x)
+
x
ε ′r y K m (y)
(1.2)
F2 (x) =
J ′m (x) K ′m (y) J m (x)
+
x
y K m (y)
(1.3)
F3 (x) =
⎡1
1⎤
J m (x) ⎢ 2 + 2 ⎥
y ⎦
k 0 r ε ′r
⎣x
(1.4)
βrm
where Jm is the Bessel function of order m, Km is the modified Bessel function of the
same order and r is the radius of the dielectric rod. The values of x for which the
above equation becomes zero are called eigen values of the dielectric rod wave guide.
The relation between x and y is given below:
y = (k 0 r) 2 (ε ′r − 1) − x 2
(1.5)
In the above equation (1.5), x should not exceed a certain value, xmax given by
x max = k 0 r ε ′r − 1
(1.6)
otherwise y becomes purely imaginary thereby changing the modified Bessel function
Km into Hankel functions Hm representing outwardly traveling waves. There are,
therefore, only a finite number of eigen values for any specified ‘m’. The integer ‘m’
7
specifies the number of field variations in θ direction as in the case of hollow metallic
cylindrical waveguide. The eigen value equation (1.1), being transcendental equation,
needs to be solved by numerical procedures (Kajfez, 1983).
There are three distinct families of modes, which can be obtained from the
eigen value equation. For m = 0, the eigen equation splits into two separate equations.
One of them gives transverse electric family of modes TE0n and the other gives the
transverse magnetic family of modes TM0n. Modes with m = 0 do not show azimuthal
variation and their field patterns are circularly symmetric. For any m, different from
zero, the mode field is a mixture of both TE and TM kinds, and is called hybrid
electromagnetic (HEMmn) modes. The greater part of the power carried by a hybrid
electromagnetic wave inside the rod is either in the TE part of the HEM field or in the
TM part of the HEM field. As the value of x is gradually increased in search of
solutions of eigen value equation, quasi TE modes alternate the quasi TM modes.
Odd values of m produce hybrid modes of quasi TM type and even values of m result
in the hybrid modes of quasi-TE type.
The modes of a DR are indexed by the letters m, n and p. They are the number
of field variations in the azimuthal, radial and axial directions respectively. For
isolated DRs, the index p is designated as,
p=l+δ
(1.7)
where l = 0, 1, 2 etc. and δ signifies a non-integer number less than unity. To find out
the resonant frequencies of a DR, with a known value of dielectric constant ε ′r , having
a diameter to length ratio D/L (aspect ratio), the standard mode charts are available.
Kobayashi and Tanaka, 1980 reported a mode chart for a dielectric rod resonator short
circuited at both the ends. Mode chart graphically represents the variations of the
factor ε ′r (D/λ0)2 as a function of (D/L)2 for all the resonant modes. λ0 is the free space
8
wavelength corresponding to the resonant frequency of the modes. Mode chart is
obtained by solving the eigen value equation (1.1) for the resonant modes for various
aspect ratios and dielectric constants.
From the mode chart, the resonant frequencies of all the resonant modes of a
DR can be found out using its ε ′r , D and L values. Also one can find out the order in
which various modes appear in frequency spectrum using the mode chart. This is
important because the presence of some leaky modes can lead to degradation of its
performance. From the mode chart, one can find out the aspect ratio which must be
used for a DR with a given ε ′r value, to get maximum isolation for the mode of
interest. As an example, for TE011 mode, with ε ′r ≥ 10, the range of D/L values 1.0 1.3, 1.9 - 2.5, 3.0 - 3.3 etc. gives good isolation from the nearby low Q leaky modes
(Kobayashi and Katoh, 1985).
The mode that is widely used in material characterization is TE01δ mode.
Figures 1.2 and 1.3 display the electric and magnetic field for this mode (Glisson,
1986) respectively. The electric field is shown in the equatorial plane (Figure 1.2).
The magnetic field in the same plane is zero. The magnetic field in a meridian plane is
shown in Figure 1.3. Since this is an azimuthally symmetric mode, the plot of the
magnetic field will be the same in any meridian plane. The magnetic field is
perpendicular to the electric field and its maximum value occurs one-quarter period
later in time. Figure 1.4 gives the side view of a cylindrical DR showing strongest
electric and magnetic lines.
9
Fig. 1.2 Electric field distribution in equatorial plane for TE01δ mode of a DR.
Fig. 1.3 Magnetic field distribution in meridian plane for TE01δ mode of a DR.
Fig. 1.4 Dielectric resonator showing strongest electrics and magnetic field lines for
TE01δ mode.
10
Coupling to the TE01δ mode is often accomplished through the magnetic field
via a small horizontal loop placed in the equatorial plane or by placing the resonator
end face on a substrate near a microstrip line, so that the magnetic field lines link with
those of the loop or microstrip. Coupling to this mode via the electric field can also be
achieved using a small horizontal dipole, or a bent monopole. Knowledge of various
modes will be useful in selecting the coupling device, which is best suited for a
particular mode. A short electric probe would be most efficient if it is oriented along
E field lines, and placed at the location where the E field is strong. A small loop
should be effective if it is placed so that it couples with many H field lines.
1.1.2 Applications of DRs
Applications of dielectric materials in various microwave components are very
cost effective and lead to significant miniaturization, particularly when microwave
integrated circuit (MIC) or monolithic microwave integrated circuit (MMIC) are used.
Small size, light weight, low cost along with the features like high Q factor value,
temperature compensated performance, good coupling characteristics, frequency
tunability and a choice of modes with different characteristics make a DR an
important device in microwave integrated circuits. DRs are used as a resonating
element, feed back element or as radiating elements in various applications. The
microwave circuits, which employ them, are primarily oscillators, filters, duplexers
and miniature radiating elements. These circuits are key elements in systems used for
microwave communications, radar, navigation, electronic warfare systems, cellular
telephones, base stations, hand held radio transmitters, speed guns and automatic door
openers etc. DRs are nowadays widely used in the range of 1 to 30 GHz (Khanna,
1986). Additional applications include dielectric or superconductor testing and
antenna applications using radiating DRs. Miniature dielectric-filled coaxial
11
resonators are commonly used in wireless handsets such as cellular and personal
communication system (PCS) phones (Fiedziuszko et al., 2002).
High dielectric constant materials (80-100) have significant impact on lower
frequency microwave devices (1 GHz region). Such DRs are used in particularly all
cellular and PCS base stations (Fiedziuszko et al., 2002). Emerging working
frequencies from 900 MHz to 2.4 GHz, 2.8 GHz and even to 5.8 GHz require
moderate dielectric constant (around 20) and high quality factor materials. Low loss
dielectric materials with dielectric constant of 20 are being used in today’s global
positioning system (GPS) patch antennas, wireless local area network (WLAN) band
pass filters and even for 5.8 GHz industrial, scientific and medical applications (ISM)
band pass filters (Huang et al., 2008).
New applications for DRs are constantly emerging such as global positioning
systems (many of which use dielectric resonator antennas), low temperature co-fired
ceramics (LTCC) for embedded microwave circuitry, tunable filters and higher
frequency applications for advanced radar technology (Reaney and Iddles, 2006).
1.2 PEROVSKITE DIELECTRIC RESONATORS
1.2.1 Alkaline Earth Complex Perovskites
A large majority of microwave dielectric materials have perovskite or
perovskite related structure. One of the great advantages of the perovskite material is
that it can accommodate various kinds of equivalent ions at A and B positions,
thereby physical properties can be modified to suit the practical applications.
Materials based on Ba(Mg1/3Ta2/3)O3 (BMT) and Ba(Zn1/3Ta2/3)O3 (BZT) with 1:2 Bsite cation ratio was widely studied for the dielectric resonator applications (Nomura,
1983; Kawashima, 1983; Nomura and Kaneta, 1984; Desu and O’Bryan, 1985). In
12
specific, BaZrO3 doped BZT exhibits ε ′r = 30, Q.f = 1, 30, 000 GHz with zero τf and
is a commercial system sold by several companies (Reaney and Iddles, 2006). These
complex perovskites have a cubic structure ( Pm 3 m ) when disordered (Srinivas et al.,
1997; Srinivas et al., 2002) but often undergo a structural phase transition on cooling
from the sintering temperature to a trigonal ( P 3m1 ) ordered cell with ordering along
the (111) direction of the parent perovskite cubic lattice (Chai et al., 1997; Srinivas et
al., 2002). In complex perovskites ordering has shown to both increase Q factor and
decrease temperature coefficient of resonant frequency (Davies et al., 1997; Desu and
O’Bryan, 1985). Of all the perovskites with 1:2 cation perovskites, BMT and BZT
exhibit high quality factors. Achieving high percentage of ordering and thereby high
quality factors is not trivial in these materials. Typically, ordering is achieved by
annealing the ceramics for extended periods below the sintering temperature (Davies
et al., 1997).
Kawashima et al., 1983 reported the initial MW characterization of BZT for
which ε ′r = 30, Q = 6,500 at 12 GHz and τf ~ 0 ppm/°C and controlling the sintering
or annealing by heating at 1350°C for 120 hours, low loss BZT (Q =14,000 at 12
GHz) with perfect hexagonal ordered structure was obtained. At higher firing
temperatures of BZT, ZnO evaporation caused disordering by Ba replacement of Zn
on the B-site, and crystallographic distortion was observed. Another study concluded
that loss of ZnO from BZT by evaporation caused the Ba3Ta2O8 phase to form in the
matrix, which resulted in a decrease in the Q.f value (Desu and O’Bryan, 1985). Desu
and O’Bryan, 1985 and Tamura et al., 1984 conducted experiments on BZT by
addition of BaZrO3. They claimed that both sintering and crystallization in BZTBaZrO3 were accelerated by the addition of BaZrO3 and Q values were improved
accordingly.
13
BMT is another widely used material with near zero temperature coefficient of
resonant frequency and low dielectric loss. But dielectric constant of BMT (~24) is
less than that of BZT. A very high temperature (>1550°C) and long sintering time are
necessary to obtain both cationic ordering and a satisfactory density of BMT (Marinel
et al., 2003). The main hurdles in the synthesis of BMT include a) the high sintering
temperature above 1600˚C where volatisation of constituent MgO can occur; b)
formation of Mg free additional phases like BaTa2O6, Ba5Ta4O15 and Ba4Ta2O9 and c)
thermal destabilization of 1:2 to 1:1 order or disordered perovskite (Sebastian and
Surendran, 2005). Several studies such as addition of glass additives, addition of
lithium salts, BaTi4O9 and synthesis methods were carried out to improve the
sinterability and ordering of BMT ceramics (Surendran et al., 2004; Marinel et al.,
2003; Cheng, 2004; Surendran et al., 2007). Tien et al. reported that addition of
BaSnO3 to BMT decreased ordering and density but quality factors of BMT with 0.05
and 0.10 mole% of BaSnO3 are higher than that of those samples with 0.0 and 0.15
mole% of BaSnO3. The relation between microstructure and loss factor in BMT
ceramics is not well understood (Tien et al., 2000; Youn et al., 1996). Different
synthesis methods of BMT such as solution method, co-precipitation, citrate gel, sol
gel and solid state synthesis showed that solid state synthesis yields higher grain size
and low loss (Surendran et al., 2007). Theoretical calculations on Ba(B1/32+B2/35+)O3
found that greater the stability of the ordered 1:2 structure with respect to the
disordered phase, the higher the experimentally measured microwave Q factor
(Takahashi., 2000). First principle investigations on BMT ceramics concluded that
sintering at high temperature for a long time or prolonging the annealing should be
effective in enhancing the degree of cation order (Takahashi et. al., 2000).
14
Search of new materials with high dielectric constant and low loss, extensive
processing, high cost of Ta2O5, easy attainability of 1:1 order compared to 1:2
ordering (Setter and Cross, 1980) and thermal destabilization of cation order from 1:2
to 1:1 shifted the focus of research onto rare earth based complex perovskites with 1:1
ratio cations.
1.2.2 Rare Earth Based Complex Perovskites
Recently a lot of attention has been given to rare earth based complex
perovskites with the chemical formula Ln(BTi)0.5O3 (where Ln = La, Nd & Sm; B =
Mg, Co & Zn). Initially, Harshe et al., 1994 considered La(MgTi)0.5O3 (LMT) for
microwave dielectric resonator application. They reported that LMT crystallizes in a
cubic structure and do not exhibit B site cation ordering. Lee and his coworkers
determined the correct structure of LMT to be monoclinic P 21 / n symmetry with 1:1
B-site cation ordering (Lee et al., 2000). They also reported that LMT sintered at
1630˚C exhibits high quality factor (Q.f) of 63,000 GHz, moderate dielectric constant
( ε ′r ) of 28 and large negative temperature coefficient of resonant frequency (τf ) of 74 ppm/˚C. Cho et al. explored Ln(MgTi)0.5O3 (Ln = Dy, La, Nd, Pr, Sm and Y)
microwave dielectrics as substrates for high Tc superconductor thin films (Cho et al.,
1999). Their report concluded that cation ordering in Ln(MgTi)0.5O3 is easily
accomplished than in BMT. They also reported that with the exception of
Dy(MgTi)0.5O3, Ln(MgTi)0.5O3 perovskites possess large negative τf values. But Q
factor of Dy(MgTi)0.5O3 was found to be low (Q.f = 36,800 GHz).
Substitution of Zn in the place of Mg resulted a new material La(ZnTi)0.5O3
(LZT) (Yeo. et al., 1996). Kucheiko et al. prepared LZT by sol-gel process and they
showed that LZT exhibits ε ′r = 30 and Q.f = 60,000 GHz with τf equal to -79 ppm/˚C.
The crystal system was reported to be orthorhombic and formation of a second
15
orthorhombic phase La2TiO5 with ZnO evaporation at high sintering temperatures
above 1350˚C was observed (Kucheiko et al., 1996). Cho et al., 1997 made attempts
to analyze the influence of ZnO evaporation on microwave dielectric properties of
LZT. They reported that lattice distortion cannot explain the increase in quality factor
of LZT samples with ZnO evaporation and also reported the presence of (111)
reflection corresponding to ordering in the XRD pattern. Later, structural studies by
neutron diffraction experiments revealed that the true symmetry of LZT is monoclinic
P 21 / n with high degree of cation ordering but not orthorhombic Pbnm , which does
not support the B-site cation ordering (Ubic et al., 2006a; Ubic et al., 2006b).
In order to achieve near zero temperature coefficient of resonant frequency,
researchers synthesized solid solutions of negative τf materials (LMT and LZT) with
high positive τf materials such as BaTiO3 (BT), CaTiO3 (CT) and SrTiO3 (CT). Near
zero τf was obtained in these solid solutions (at a specific composition ratio) at the
cost of Q.f degradation together with the increase in dielectric constant values. In the
case of (1-x)LMT-xBT system, structural studies revealed that increase in BT induces
a series of structural transformations. These correspond to a progressive increase in
the average symmetry of the unit cell from monoclinic ( P 21 / n ; x ≤ 0.1) to
orthorhombic ( Pbnm ; x = 0.3), tetragonal ( I 4 / mcm ; x = 0.5) and cubic ( Pm3 m ; x >
0.5) (Avdeev et al., 2002a). The (1-x)LMT-xST (0 < x < 1) solid solution system
demonstrated gradual structural changes with increase of ST content [ P 21 / n
(monoclinic) → Pbnm (orthorhombic) → I mma (orthorhombic) → I 4 / mcm
(tetragonal)], which are related to the loss of B-site chemical ordering of (x > 0.1),
displacement of the A-site cation (x > 0.3), in-phase tilting (x > 0.5) and one axis antiphase tilting (x > 0.7), respectively (Avdeev et al., 2002b). Seabra et al., 2003a
reported that for higher concentrations of CT, B site ordering was disappeared in (116
x)LMT- xCT system and Pbnm symmetry was assigned. Cho et al., 1998 also
observed absence of super lattice reflections corresponding to cation ordering with the
higher concentration of ST in (1-x)LZT- xST system. In all the cases, cation ordering
was lost with the addition of high positive temperature coefficient materials with
LMT and LZT. Cho et al., 2001 concluded that cation ordering in B-sites was of
importance for the quality factor of A3+B3+O3 complex perovskites as in the case of
A2+B4+O3 complex perovskites such as BMT and BZT. Structural changes including
cation ordering was suggested as determining factor for the sign of τf in LZT based
perovskite systems.
1.2.3 Lattice Vibrational Aspects of Perovskites
Achieving a high dielectric constant, high quality factor and near zero
temperature coefficient of resonant frequency is a challenging task. Lattice vibrational
studies using far infrared spectroscopy and Raman spectroscopy have been used to
understand the microwave dielectric properties.
Microwave loss in ceramics has known to be caused by both extrinsic factors
(porosity, impurities, grain boundaries etc.) and intrinsic ones (lattice absorption due
to crystal anharmonicity). Usually, the extrinsic contribution can be minimized by
using proper processing conditions. Change of lattice anharmonicity and associated
dielectric loss in solid solutions depends on crystal symmetry, composition,
octahedral tilting, cation ordering etc. Thus there are various factors affecting the
behavior of the Q factor.
Mathematically, the Q factor is a ratio between real and imaginary parts of
complex dielectric constant, the real part being concerned with “resistance to an
electric field” and the imaginary part being concerned with losses. The real part of
permittivity is sensitive to the frequency at which given phonon oscillates. It is also
17
sensitive to the degree of polarity of a given phonon; the more polar it is the “wider”
the phonon will be. The imaginary part of the complex dielectric constant scales with
these parameters, but in addition will be affected by dampening. The sources of
dampening in mechanical systems are essentially friction and dissipation of energy
through heating of the spring due to its motion. In these systems, phonons become
scattered or absorbed as they propagate through material leading to extinction of
vibration. Factors that cause scattering or absorption of phonons include defects,
conduction related processes, and phonon-phonon interactions (Zheng et al., 2005).
Intrinsic dielectric properties can be estimated by using far infrared studies.
Infrared (IR) reflectivity may also be used as sensitive tool for revealing the degree of
B site cation order (Reaney et al., 1994a). IR reflectivity is not as sensitive to
processing as MW losses, provided sufficiently dense (> 95% of theoretical density)
materials are available and it may be used for a first estimate of intrinsic MW
properties of a new material (Petzelt et al., 1996). Further, in well processed ceramics
the room temperature losses are always proportional to frequency. There are two
different models to quantify the intrinsic parameters using IR reflectivity data. They
are classical oscillator model (Spitzer and Kleinmann, 1961) and four parameter
model (generalized oscillator model). The former is appropriate when the splitting
between transverse (TO) and longitudinal (LO) optic mode is weak i.e. the IR
reflection bands are narrow and symmetrical (Buixeradas, 2001) whereas later
describes more accurately broad reflectance bands and overlapping and interacting
modes (Luspin et al., 1980; Fontana et al., 1984). Further discussion on four
parameter model is dealt in Chapter 3.
Intrinsic loss steeply varies with the dielectric constant of DR. Petzelt et al.,
1996 concluded that at a fixed frequency, ε ′r′ α ε ′r 4 for Ba(B'B")0.5O3 system due to the
18
increase in anharmonicity with increasing permittivity. According to Tamura, by the
contribution of anharmonic terms in the crystal’s potential energy, the lower ε ′r
material always has higher Q.f value and disordered charge distribution in the crystal
contributes significant decrease in Q factor (Tamura, 2006). Investigation of far
infrared reflectance studies on (1-x)LMT-xBT shows that the decrease in quality
factor with increase in BT content is associated with increase in average TO phonon
damping (Salak et al., 2004). Increased average phonon damping is also observed
with increase in ST content for (1-x)LMT-xST ceramics (Seabra et al., 2004b).
The structural modifications, qualitative analysis of long range order and
distortions in the octahedra can be studied by using Raman spectroscopy (Moreira et
al., 2001). Zheng et al., 2003 observed that the F2g mode in complex perovskites was
present only when the compound contained long range order (coherence length of
ordering > 3 nm). Runka et al., 2005 concluded that the cubic F2g mode splits into
doublet and triplet with the lowering of the symmetry while the sharpness and
intensity of the F2g and A1g modes varied with the variation in the ordering. Reaney
and Iddles, 2006 concluded that the Q is optimized when the spread of tolerance
factor (∆t) is a minimum in solid solutions made of positive and negative temperature
coefficient perovskites and the short range order is detrimental, whereas the long
range order enhanced Q. The Raman spectroscopy technique is also highly sensitive
to short range order and offers a means of detection. The F2g mode is only sensitive to
long range order but A1g mode is either due to long range order or short range order
(Reaney et al., 2005). Short range order induces a distribution of unit cell parameters
of the order of a few nanometers; this may result in anharmonicity and phonon
damping, thereby reducing Q (Zheng et al., 2004). Low quality factors of 0.5LMT0.5CT, 0.5LMT-0.5BT and 0.5LMT-0.5ST solid solutions were due to the presence of
19
short range order (Reaney and Iddles, 2006; Zheng et al., 2004). Levin et al., 2005
observed asymmetric broadening in the A1g mode of LMT and stressed the need to
study detailed Raman analysis on solid solutions with varying order parameters to
clarify the origin of anomalous broadening.
1.3 PHOTONIC BAND GAP STRUCTURES
During the last fifteen years there is growing interest on periodic dielectric
structures, which attenuate electromagnetic waves in different frequency regions for
potential applications in one, two or three dimensions (Joannopoulos et al., 1995)
initiated by Yablonovitch, 1987. This periodic arrangement of dielectric or magnetic
structures does not allow certain regions of the electromagnetic spectrum to
propagate. This phenomenon is similar to the electronic band gaps formed in crystals.
Hence, these structures are called Photonic Band Gap structures (PBGs) or Photonic
Crystals (PCs). Unlike semiconductors, which possess electronic band gap, these
photonic crystals have to be artificially engineered. In these structures, the waves are
Bragg reflected and if these waves interfere destructively that gives rise to band gap.
Defects can also be created in an otherwise periodic structure as a result of
which a mode will appear with in the band gap. Defects can be created either by
changing the dielectric constant at a particular position or by changing the periodicity
locally. These band gap structures are useful in constructing efficient antennas, planar
wave guides, coupled cavity wave guides and resonant cavity enhanced detectors
(Thevenot et al., 1999; Gonzalo et al., 1999; Yang et al., 1999). Sirigiri et al, 2001
have constructed a gyrotron, which allows only a particular mode (TE041) using the
principle of Photonic Band Gap.
20
The unloaded Quality factor Q0 of a dielectric resonator consists of losses in
dielectric resonator and in the support structure and of losses in the metallic walls
shielding cavity (C) and is given by (Klien et al., 2001; Klien, 2005):
R
1
= κ tan δ + s ; κ =
Q0
G
ε r ∫ E 2 dV
DR
∫E
2
dV
ωµ 0
; G=
∫H
2
dV
DR + C
2
∫H
DR + C
dV
(1.8)
C
with Rs representing the surface resistance of the shielding cavity material, κ is the
filling factor corresponding to the fraction of electric energy stored in the dielectric
resonator with loss tangent (tanδ) and the geometric factor G represents the integral
squared amplitude of magnetic field of the inner surface of the shielding cavity.
In order to achieve ultimate Q0 values either the material or the mode and
geometry need to be optimized. The material issues are: 1) improving tanδ of
ceramics towards the intrinsic loss contribution by phonons, 2) device operation at
cryogenic temperature to reduce intrinsic dielectric losses and 3) high temperature
super conducting wall segments for reduction of Rs. On the other hand, the geometric
factor is strongly affected by the selected mode and geometry. In the case of,
cylindrically shaped dielectrics G increase staring from the HE11δ dual mode over the
TE01δ mono mode towards whispering gallery modes. For the majority of modes in
dielectric resonators κ is close to unity (κ of HE11δ = 0.8-0.9 and κ of TE01δ = 0.9
(Klien, 2005), i.e. the field concentration in the dielectric material is very high. The
periodically arranged dielectric structures (photonic crystals) provide a potential to
reduce κ significantly and provide high G values at the same time. Therefore photonic
crystals with low loss dielectrics are the potential structures for very high Q values.
21
1.4 OBJECTIVE AND SCOPE OF PRESENT WORK
Rare earth based complex perovskites attracted much attention due to the easy
attainability of 1:1 B-site cation ordering compared to 1:2 B- site cation ordering and
cost effective with the absence of expensive Ta. But efforts to achieve near zero
temperature coefficient of resonant frequency together with high Q factors were
unsuccessful. Solid solutions of LMT with high positive temperature coefficient or
resonant frequency materials resulted disappearance of long range order with
symmetry change, increase in average TO phonon damping and appearance of short
range order. The variation of temperature coefficient of resonant frequency with the
composition is also not understood very well.
This thesis work concentrates on analyzing the factors responsible for the
microwave dielectric properties of Ln(MgM)0.5O3 (where Ln = La and Nd; M = Ti and
Sn) complex perovskites by the study of structure determination and quantification of
long range order, estimation of phonon mode strengths and intrinsic parameters,
analysis of Raman modes corresponding to B-site cation ordering and measurement of
microwave dielectric properties. Smaller size Nd compared to La is chosen to analyze
the variation of temperature coefficient of resonant frequency. X-ray scattering length
difference between Mg2+ and Sn4+ is high, which enables accurate quantification of
cation ordering using Rietveld refinement of X-ray diffraction data. Further, main
group Sn4+ is less polarizable and high electronegative compared to transition metal
cation Ti4+. Sn4+ possesses fully occupied d orbitals whereas d orbitals of Ti4+ is
empty. The correlation of properties with combined study of structural
characterization, lattice vibrational analysis and measurement of microwave dielectric
properties is attempted in this work.
22
High quality factor values can be obtained using periodic dielectric structures
based on the photonic band gap concept. This work analyzes band gaps of one
dimensional photonic crystals at microwave frequencies using transfer matrix method.
Construction of photonic crystals using dielectric resonator materials is expensive and
equally difficult to make sheets or rods. Therefore, transfer matrix calculations are
performed on low loss structure. However, lossy periodic structures (glass and ebonite
photonic crystals) are analyzed both theoretically and experimentally.
23
CHAPTER 2
PREPARATION AND STRUCTURAL CHARACTERIZATION
OF La(MgTi(1-x)Snx)0.5O3, La(1-x)Ndx(MgSn)0.5O3 AND Nd(MgTi)0.5O3
In this chapter, the details of preparation and structural studies carried out
on La(MgTi(1-x)Snx)0.5O3 (x = 0, 0.125, 0.25, 0.375 and 0.5), La(1-x)Ndx(MgSn)0.5O3 (x
= 0, 0.25, 0.5, 0.75 and 1.0) and Nd(MgTi)0.5O3 dielectric resonators are described.
All the samples in this work were synthesized using conventional solid state
reaction method. The starting reagents were high pure oxides. Rare earth oxides
La2O3 and Nd2O3 are pre-fired at 1000 °C for 24 hours to remove the moisture content
and carbonates. Rare earth oxide powders were transferred into air tight bottles at 400
°C and then allowed powders to attain room temperature in desiccators. MgO was
also prefired at 800 °C for 6 hrs to remove the moisture.
The preparation method consists of different stages, viz. mixing, calcinations
and sintering. Preparation conditions were optimized to achieve higher density and
high percentage of perovskite phase. Stoichiometrically weighed oxides were initially
dry mixed using an agate mortar and pestle. Subsequently powders were wet mixed
with distilled water as medium. The amount of water used was just enough to form
slurry. The wet mixed reagents were dried in an oven at 150 °C for about 12 hours
and then ground in an agate mortar.
Calcination was carried out using alumina crucibles. Calcined powder with the
organic binder polyvinyl alcohol (PVA) was pressed into pellets using uniaxial press
and the binder was evaporated at 500 °C for 12 hrs. Prior to the sintering, the calcined
24
powders were well ground using an agate mortar. Sintering was carried out at high
temperature (1600 °C) by placing the pellet on a platinum foil. The heating rate used
was 8 °C/minute and cooling rate was 5 °C/minute. Sintered pellets were well ground
using agate mortar and pestle for structural studies. Densities of the samples were
measured by Archimedes method.
The La(MgTi(0.5-x)Snx)O3 ceramics were synthesized using La2O3 (Alfa Aesar,
99.99%), MgO (Alfa Aesar, 99.95%), TiO2 (Alfa Aesar, 99.9%) and SnO2 (Cerac,
99.9%) reagents. Calcination of x = 0.0 to x = 0.375 compositions was carried out at
1200 °C for 3 hours followed by mixing and subsequent heating at 1250 °C for 2
hours. For x = 0.5 composition, calcinations was done at 1200 °C for 3 hours followed
by mixing and subsequent heating at 1225 °C for 2 hours. Prior to uniaxial pressing,
calcined powders were remixed using agate mortar and pestle. Sintering temperature
for all the compositions was 1600 °C with the duration of 4 hours.
In the preparation of La(1-x)Ndx(MgSn)0.5O3 ceramics, starting reagents were
La2O3 (Alfa Aesar, 99.99%), MgO (Alfa Aesar, 99.95%), Nd2O3 (Alfa Aesar, 99.9%)
and SnO2 (Alfa Aesar, 99.9%). Calcination of x = 0.0, x = 0.75 and x = 1.0
compositions were carried out at 1200 °C for 3 hours. Calcination for x = 0.25 and x =
0.5 compositions was carried out at 1150 °C for 3 hours followed by mixing and
subsequent heating at 1200 °C. Calcined powder was remixed and pressed into pellets.
Sintering of all the compositions was carried out at 1600 °C for duration of 5 hours.
Nd(MgTi)0.5O3 was prepared using Nd2O3 (Alfa Aesar, 99.99%), MgO (Alfa
Aesar, 99.95%) and TiO2 (Alfa Aesar, 99.9%) reagents. Calcination was carried out at
1200 °C for 3 hours followed by mixing and subsequent heating at 1225 °C for 2
hours. Prior to uniaxial pressing, calcined powders were remixed using agate mortar
and pestle. Sintering was done at 1600 °C for duration of 5 hours.
25
2.1 STRUCTURAL ASPECTS OF PEROVSKITES
2.1.1 Tolerance Factor and Octahedral Tilting
Tolerance factor (t) is a parameter to predict the structural distortion in perovskites.
The tolerance factor of ABO3 perovskite is defined (Goldschmidt, 1926) as,
t=
R A+ R O
(2.1)
2 (R B + R O )
where RA, RB and RO are the ionic radii of A site cation with coordination number 12,
B site cation with coordination number 6 and oxygen anion with coordination number
6 respectively. This formula can be modified to determine the tolerance factor of a
complex perovskite by substituting the average radius of two or more different
cations.
The tolerance factor of an ideal cubic perovskite is equal to 1. If the tolerance
factor deviates from 1, structural distortions can occur. If t < 1, the A site cation is too
small for its site, results in BO6 octahedral tilting. Tilting of the BO6 octahedra within
the perovskite structure leads to lowering of its symmetry. Various types of possible
tilting in perovskites are discussed in the following section. If the tolerance factor is
too low, corner shared perovskite structure is not stable. The tolerance factor range in
which perovskite structure is preferred is 1.04 to 0.87.
2.1.2 Glazer Tilt Notation
Nomenclature describing the octahedral tilting distortions in perovskite was
developed by Glazer, 1972. This notation describes octahedral tilting occurring about
the x, y and z-axes. The magnitude of tilting along three axes is described by a set of
three letters, abc. If tilting along two or more axis is equal in magnitude, same letter is
repeated. For example, equal tilting about the x and y-axes is denoted by aab.
26
When tilting occurs along a particular axis, octahedra perpendicular to those
axes are forced to tilt in the opposite direction to maintain corner sharing connectivity.
Successive identical tilting of adjacent layers along a particular axis is known as in
phase tilting (Figure 2.1) and is denoted by a superscript “+”. If tilting of successive
octahedral layers along an axis is opposite to one another, then it is out of phase tilting
(Figure 2.2). This is denoted by superscript “-”. A superscript “0” denotes absence of
tilting.
Glazer initially defined 23 tilt systems and assigned them to 15 different space
groups (Glazer, 1972). Since then, the number of tilt systems and their corresponding
space group assignments has been revised (Leinenweber and Parise, 1995;
Woodward, 1997a). The Glazer tilt system and corresponding space groups for simple
(or disordered 1:1 B site) perovskites are presented in Table 2.1.
The lattice energy calculations by Woodward, 1997b shows that the a-a-c+ tilt
system and a-a-a- tilt system have more favorable lattice energies, while one and zero
tilt systems have are less stable. With decrease in tolerance factor, a-a-c+ tilt system is
more stable than a-a-a- tilt system.
2.1.3 B Site Cation Ordering
As mentioned in the previous chapter, complex perovskites can exhibit B-site
cation ordering. The arrangement of B site cations can be either completely ordered or
partially ordered or completely random (disordered). The charge difference between
the B site cations, difference in B site ionic radii and processing conditions influence
B site cation ordering (Anderson et al., 1993; Barnes, 2003). When there is a large
difference in size and/or charge, ordering can stabilize a structure by allowing each
cation to have its preferred environment rather than an average environment (Davies,
1999).
27
Fig. 2.1 in-phase tilting of BO6 octahedra (light grey shaded with unshaded oxygen
atoms at corners and dark grey shaded A atoms)
Fig. 2.2 out of phase tilting of BO6 octahedra (light grey shaded with unshaded
oxygen atoms at corners and black shaded A atoms)
28
The difference in charge between B site cations is the most important factor
influencing cation ordering (Anderson et al., 1993). If the difference in oxidation is
greater than two, highly ordered compounds are generally observed. The second most
influential factor is the size difference between B site cations (Anderson et al., 1993).
In general, larger differences in ionic radii give rise to higher degrees of cation
ordering. For example, Ba(MgW)0.5O3 is a completely ordered perovskite (Anderson
et al., 1993). The charge difference between Mg2+ and W6+ is four. It is not
electrostatically favorable for two W6+ ions to be nearer and ordering of the Mg2+ and
W6+ will ensure that each O2- ion is in contact with exactly one Mg2+ and one W6+.
Further, the size difference between Mg2+ and W6+ is high (0.1 Å), which makes O2ion between them to shift towards the W6+ cation to relieve the lattice strains arising
from the size mismatch (Knapp, 2006) and resulting the most favorable electrostatic
arrangement. In order to support the cation ordering, symmetry of an ideal perovskite
with a0a0a0 tilting transforms from Pm3 m to Fm3 m . The O2- ion sits on the special
Wyckoff position 24e (x, 0, 0) in Fm3 m space group, where x is a variable parameter.
In Pm3 m space group, anion sits on a general Wyckoff position 3d (0.5, 0, 0), which
is a fixed position. The x value of 24e position in Fm3 m symmetry adjusts (O2- ion
shifts) to relieve the strains developed by size mismatch of B-site cations. Therefore,
to accommodate the B-site cation ordering, perovskite structure transforms from
original symmetry to other. Woodward, 1997a determined symmetry of B-site cation
ordered complex perovskites with different tilt systems. Table 2.1 lists all the possible
tilt systems in complex perovskites and their symmetry.
2.1.4 Octahedral Tilting and X-ray Powder Diffraction
Octahedral tilting is the most important factor in determining the space group
symmetry for a given perovskite (Glazer, 1972). Tilting of the octahedra causes a
29
doubling of the unit cell axes and produces extra reflections corresponding to half
integral reciprocal lattice planes (super lattice reflections). In-phase, out-of-phase
tilting and antiparallel displacement of A site cations result to different types of
reflections. The in phase tilting results to odd-odd-even (ooe, oeo, eoo) type of
reflections, out of phase tilting results to odd-odd-odd (ooo, h+k+l > 3) type of
reflections and A site cation displacement results to odd-even-even (eoe, eeo, oee)
type of reflections (Glazer, 1975; Barnes et al., 2006).
2.1.5 Effect of B site Cation Ordering on X-ray Powder Diffraction
Evidences of cation ordering can be observed in powder X-ray diffraction
data. The odd-odd-odd reflection with h+k+l = 3 (i.e. (111)) corresponds to 1:1 B-site
cation ordering (Anderson et al., 1993; Avdeev et al., 2002a; Salak et al., 2003;
Seabra et al., 2004a). According to Howard et al., 2003, (111) super lattice reflection
is due to both B site cation ordering and out of phase tilting. But the contribution of
out of phase octahedral tilting to (111) is weak (Barnes et al., 2006). The cation
ordering contribution to the (111) reflection is strongly dependent upon the scattering
power contrast between B site ions. Therefore, X-ray diffraction can be powerful tool
for observing cation order if there is some difference in the number of electrons
between the two B site cations.
The degree of ordering within a given compound is determined by the Long
Range Order parameter (LRO):
LRO = [2 × (occ.) B − 1]× 100
(2.2)
where (occ.)B is the fractional occupancy of B site cation on the predominantly
occupied octahedral site (Woodward et al., 1994). This can be extracted from
diffraction data by refining the site occupancy using Rietveld refinement. For a 100%
ordered perovskite, the occupancy of B site cation on its crystallographic site is 1. If it
30
is completely disordered (LRO = 0%), the occupancy of the B-site cation on distinct
sites is ½.
Table 2.1 Space groups for all possible tilt systems both random B site cations
(random) and with 1:1 B-site cation ordering (ordered)
Tilt system
number
1
Tilt system symbol
a+b+c+
Space group
(random)
Immm
2
a+b+b+
Immm
Pnnn
3
a+a+a+
I m3
Pn 3
4
a+b+c-
Pmmn
P2/c
5
a+a+c-
P42/nmc
P42/n
6
a+b+b-
Pmmn
P2/c
7
a+a+a-
P42/nmc
P42/n
8
+ - -
a bc
P21/m
P1
9
a+a-c-
P21/m
P1
- - +
Space group
(ordered)
Pnnn
10
aac
Pnma
P21/n
11
a-a-a+
Pnma
P21/n
12
a-b-c-
F1
F1
13
- - -
abb
I2/a
F1
14
a-a-a-
R3c
R3
15
a0b+c+
Immm
Pnnn
16
a0b+b+
I4/mmm
P42/nnm
17
0 + -
Cmcm
C2/c
Cmcm
C2/c
abc
0 + -
18
abb
19
0 - -
a bc
I2/m
I1
20
a0b-b-
Imma
I2/m
21
22
23
0 0 +
aac
0 0 -
aac
0 0 0
aaa
31
P4/mbm
P4/mnc
I4/mcm
I4/m
Pm3 m
Fm3 m
2.2 STRUCTURAL CHARACTERIZATION
2.2.1 Structural Study of La(Mg0.5Ti(0.5-x)Snx)O3
X-ray diffraction data of powder samples of La(Mg0.5Ti(0.5-x)Snx)O3 (x = 0.0,
0.125, 0.25, 0.375 and 0.5) is collected using PANAlytical X’pert pro MPD in BraggBrentano geometry with an X’Celerator detector. The collection conditions were
CuKα radiation, 40 kV 30 mA, 0.017º step scan, 1.0º divergence slit and 0.02 rad.
incident and receiving soller slits. Figure 2.3 presents X-ray diffraction pattern of
La(Mg0.5Ti(0.5-x)Snx)O3 ceramics. An unidentified minor impurity peak (< 2 wt %) is
observed in all the patterns. All the reflections shift towards a lower angle, indicating
an increase in the unit cell dimensions with an increase in Sn concentration.
The tolerance factor (t) of La(Mg0.5Ti(0.5-x)Snx)O3 ceramics (Table 2.2) is less
than 1, which indicates that BO6 octahedra is tilted for all the compounds. In order to
identify the tilting system, X-ray diffraction patterns were indexed based on a cubic
perovskite unit cell. Initially, the lattice parameter of cubic cell was calculated using
highest intensity peak (110) of the perovskite and all the other reflections were
identified using the cubic lattice parameter. Super lattice reflections of all the patterns
with half integer Miller indices are shown in Figure 2.3. The main cubic reflections
are shown with integer Miller indices. Super lattice reflections corresponding to out of
phase tilting (½(311), ½(331) and ½(511)), in phase tilting (½(321)) and antiparallel
displacement of A site cations (½(210), ½(320), ½(410) and ½(432)) are observed in
all the compositions. Figure 2.4 shows the variation of ½(111) reflection intensity
with the increase in Sn concentration. The intensity improvement is assigned to
increase in the scattering length difference between B site cations. X-ray scattering
length difference between Mg2+ and Sn4+ is higher than that of Mg2+ and Ti4+. The
variation of (111) reflection intensity with increase in Sn concentration supports the
32
existence of cation ordering. Presence of out of phase tilting, in phase tilting,
antiparallel displacement of A site cation and B site cation ordering suggests that the
symmetry of these compounds is monoclinic P 21 / n with a-a-c+ tilting system. To
confirm the symmetry, La(Mg0.5Ti(0.5-x)Snx)O3 ceramics were indexed with a
monoclinic unit cell (Figure 2.5). The systematic absences of monoclinic P 21 / n
symmetry (h0l: h + l = 2n+1, 0kl: k = 2n+1 and 00l: l = 2n+1) are observed for all the
compositions.
The X-ray reflections show increase in the splitting with increase in Sn
concentration (Figure 2.3), indicating an increase in unit cell distortion. Figure 2.6
shows the evolution of splitting with increase in Sn concentration for (110) cubic or
(200) monoclinic reflection observed at 32 degrees. Splitting of (110) cubic reflection
gradually increases with increase in Sn concentration. The lattice constants of the
perovskites were calculated using CELREF software version 3 (Altermatt and Brown,
1987). The results show that lattice parameters a, b, c and β increase with increase in
Sn concentration (Figure 2.7). The difference between lattice parameters a and b also
increases indicating increase in the unit cell distortion. The tolerance factor of LMT is
0.946 and it decreases with increases in Sn concentration (Table 2.2).
Table 2.2 Tolerance factor of La(Mg0.5Ti(0.5-x)Snx)O3
x
Tolerance factor (t)
0.0
0.946
0.125
0.941
0.25
0.937
0.375
0.932
0.50
0.927
33
x = 0.5
*
Intensity (a.u.)
x = 0.375
*
x = 0.25
*
20
30
40
50
x = 0.0
(220)
1/2(432)
1/2(511)
(211)
(200)
1/2(410)
1/2(331)
(210)
1/2(421)
1/2(311)
(111)
1/2(320)
1/2(321)
*
1/2(300)
(110)
*
1/2(210)
(100)
1/2(111)
x = 0.125
60
70
2θ (deg)
Fig. 2.3 X-ray diffraction patterns of La(Mg0.5Ti(0.5-x)Snx)O3 indexed based on a cubic
perovskite cell. Half integer indices are super lattice reflections and integer indices are
cubic reflections. Impurity phase is shown by asterisk.
x = 0.5
Intensity (a.u.)
x = 0.375
x = 0.25
x = 0.125
x = 0.0
18.0
18.5
19.0
19.5
20.0
20.5
21.0
2θ (deg)
Fig. 2.4 Evolution of ½(111) super lattice reflection with increase in Sn concentration
(x) for La(Mg0.5Ti(0.5-x)Snx)O3 ceramics
34
x = 0.5
Intensity (a.u.)
x = 0.375
x = 0.25
20
30
40
50
(313)
(400)
(041)
x = 0.0
(231)
(024)
(222)
(311)
(004)
(221)
(121)
(022)
(113)
(021)
(200)
(002)
(101)
(111)
x = 0.125
60
70
2θ (deg)
Fig. 2.5 X-ray diffraction patterns of La(Mg0.5Ti(0.5-x)Snx)O3 indexed based on a
monoclinic P 21 / n unit cell
x = 0.5
Intensity (a.u.)
x = 0.375
x = 0.25
x = 0.125
x = 0.0
30.5
31.0
31.5
32.0
32.5
33.0
33.5
34.0
2θ (deg)
Fig. 2.6 The variation of (110) cubic or (200) monoclinic reflection with
increase in Sn concentration (x) of La(Mg0.5Ti(0.5-x)Snx)O3 ceramics
35
o
a,b,c (A )
8.0
7.8
a
b
c
5.7
5.6
90.45
90.40
β
β (deg)
90.35
90.30
90.25
90.20
90.15
90.10
90.05
0.0
0.1
0.2
0.3
0.4
0.5
Sn concentration (x)
Fig. 2.7 The variation of lattice parameters a, b, c and β with increase in Sn
concentration (x) for La(Mg0.5Ti(0.5-x)Snx)O3 ceramics. Error bars are also
indicated.
36
Decrease in tolerance factor results to increase in BO6 octahedral tilting and there by
lattice constants a, b and c should decrease. But the size of the Sn (ionic radius of Sn
= 0.69 Å) is greater than that of Ti (ionic radius of Ti = 0.61 Å). Therefore, volume of
the unit cell increases by substituting Sn in the place of Ti.
2.2.2 Structural Study of La(1-x)Ndx(MgSn)0.5O3
2.2.2.1 X-ray Diffraction Patterns of La(1-x)Ndx(MgSn)0.5O3
X-ray diffraction data of La(1-x)Ndx(MgSn)0.5O3 was collected using
PaNAlytical X’pert pro MPD in Bragg-Brentano geometry with X’Celerator detector.
The collection conditions were CuKα radiation, 40 kV 30 mA, 0.033º step scan, 1.0º
divergence slit and 0.02 rad incident and receiving soller slits.
X-ray diffraction patterns of La(1-x)Ndx(MgSn)0.5O3 (x = 0.0, 0.25, 0.5, 0.75
and 1.0) are presented in the Figure 2.8. X-ray patterns show an unidentified impurity
peak (< 2 wt %) for x = 0.0 and x = 0.25 compositions and Nd2Sn2O7 pyrochlore
phase (Kolekar et al., 2004) is observed for the x = 1.0 composition, Nd(MgSn)0.5O3
(NMS). X-ray reflections show increase in splitting with increase in Nd concentration.
Super lattice reflections corresponding to Mg/Sn ordering (ooo), in phase tilting (ooe,
oeo, eoo), out of phase tilting (ooo, h+k+l>3) and A site cation displacement (eoe,
eeo, oee) are observed in all the compositions. All the super lattice reflections are
indexed with half integer Miller indices (Figure 2.8). The ½(111) reflection indicates
the existence of 1:1 B site cation ordering in all the compositions. Presence of out-ofphase tilting, in-phase tilting and cation ordering suggest that symmetry of these
compounds is monoclinic P 21 / n with a-a-c+ tilting system.
37
1/2(511)
1/2(432)
1/2(331)
1/2(421)
1/2(410)
1/2(320)
1/2(321)
1/2(311)
1/2(300)
1/2(210)
Intensity (a.u.)
1/2(111)
Nd2Sn2O7
x = 1.0
x = 0.75
x = 0.5
x = 0.25
x = 0.0
20
30
40
2θ (deg)
50
60
70
Fig. 2.8 X-ray diffraction patterns of La(1-x)Ndx(MgSn)0.5O3, x = 0.0 (bottom), 0.25,
0.5, 0.75 and 1.0 (top) ceramics, indexed based on a cubic unit cell (unidentified
impurity peak of x = 0.0 and x = 0.25 compositions is shown by an arrow mark).
38
2.2.2.2 Rietveld Refinement of La(1-x)Ndx(MgSn)0.5O3
Starting atomic positions for the Rietveld refinement were generated by using
structure prediction and diagnostics software, SPUDS (Lufaso and Woodward, 2001;
Lufaso, 2002). SPUDS software predicts structure of perovskites using bond valence
concept (Brown, 1978; Brown and Altermatt, 1985; Brese and O’Keeffe, 1991;
Brown, 1992). SPUDS software was proven to predict the different types of
perovskite structures accurately (Lufaso et al., 2006).
Rietveld refinement on X-ray diffraction data of La(1-x)Ndx(MgSn)0.5O3 was
carried out using GSAS suite with EXPGUI frontend (Lorson and Von dreele, 2000;
Toby, 2001). Data collection and refinement strategy were according to the procedure
described in the Rietveld method (Young, 1993), Rietveld refinement guidelines
(McCusker et al., 1999) and Canadian powder workshop notes (Cranswick and
Swainson, 2007). The refinement process involved various stages: background fitting,
modeling profile shapes and refining atomic positions, thermal parameters and
occupancies. Initially, background was fitted manually and in later stages background
was allowed to fit with shifted Chebyschev polynomial. For all the compositions, nine
terms of Chebyschev polynomial were used (Function #1 in GSAS) for background
fitting. A pseudo-Voigt function was used to describe the peak shapes (Function #3 in
GSAS). Prior to the refinement of thermal parameters, atomic positions were refined.
The high X-ray scattering contrast between Mg2+ and Sn4+ enables an accurate
determination of the cation ordering. When the occupancies of the Mg2+ and Sn4+ sites
were refined, the unit cell content was constrained according to the chemical
composition. In order to avoid the unwanted correlation with the order parameter, the
isotropic thermal parameter (Uiso) values for the octahedral site were constrained to be
equal. Due to less number of electrons in O2- ion, X-ray scattering length of oxygen is
39
low. X-ray data analysis cannot determine the accurate position of oxygen in the
presence of high scattering atoms such as La and Nd. During the refinement, thermal
parameters of oxygen atoms were constrained to be equal.
Based on the analysis using superlattice reflections, symmetry
of the perovskites is monoclinic P 21 / n . Refinements of all the compositions
converge well with the monoclinic P 21 / n symmetry and did not converge with
orthorhombic Pbnm. The goodness of fit (χ2) is near to 1 for all the compositions and
Rietveld discrepancy indices are less than 7% (Table. 2.3). The final refinement plots
are presented in Figure 2.9 to Figure 2.13. The impurity pyrochlore phase Nd2Sn2O7
was also modeled along with NMS (Figure 2.13) and the fraction of the pyrochlore is
determined to be 1 wt% and perovskite is 99 wt%. Initial model for Nd2Sn2O7 (cubic
Fd3m symmetry) is obtained from previously reported atomic positions by Kolekar
et al., 2004. Fractional atomic coordinates, thermal parameters and occupancies of
La(1-x)Ndx(MgSn)0.5O3 are presented in Tables 2.4 to 2.8. Fractional atomic
coordinates and thermal parameters of Nd2Sn2O7 are presented in Table 2.9.
The structure of La(MgSn)0.5O3 (LMS) is drawn using CrystalMaker software.
It is shown in Figures 2.14 & 2.15. Figure 2.14a describes the unit cell and
coordination of La. Coordination number of A site ion is 8 for a-a-c+ tilt systems.
Figure 2.14b shows the structure in z-direction with unit cell. The unit cell of
monoclinic P 21 / n is nearly equal to
2a p ,
2a p , 2a p where ap is length of the
cubic Pm3 m perovskite cell. Figure 2.15 shows the structure of LMS describing out
of phase tilting in x-and y-directions and in phase tilting in z-direction. The number of
formula units is 4 and the number of atoms per unit cell is 20 for monoclinic P 21 / n
symmetry.
40
The z coordinate of the A site cation deviates very little from 0.25 (Table 2.4
to 2.9) for La(1-x)Ndx(MgSn)0.5O3 ceramics. The z coordinate of A site cation is 0.25
for orthorhombic Pbnm and A site lies on a mirror plane (Anderson et. al., 1993).
Lufaso et al., 2006 reported that the different identities and sizes of the B site cations
in monoclinic P 21 / n structure destroy the strict symmetry of the mirror plane, the
oxygen ions are still related by a pseudo-mirror plane unless the B site cations are
significantly different in size. Therefore, very slight deviation of z coordinate of the A
site cation in La(1-x)Ndx(MgSn)0.5O3 ceramics is due to small size difference between
the Mg2+ and Sn4+ ions (0.03 Å).
The tolerance factor of La(1-x)Ndx(MgSn)0.5O3 ceramics decreases with
increase in Nd concentration (Table 2.3). Decrease in tolerance factor increases
octahedral tilting, thereby it should result to decrease in lattice parameters a, b and c.
Refinement results reveal that the lattice parameters a and c decrease with increase in
Nd concentration (Figure 2.16). But the lattice parameter b is observed to slightly
increase with the increase in Nd concentration. Similar behavior of gradual increase in
lattice parameter b and decrease in lattice parameters a and c was also observed in the
La(1-x)Ndx(MgTi)0.5O3 system (Seabra et al., 2003b). In P 21 / n symmetry, lattice
parameters a and c are more sensitive to the tilting and even lattice parameter b
should decrease with increase in tilting. The slight increase in lattice parameter b may
be due to the distortion of BO6 octahedra. Decrease in lattice parameters a and c and
increase in lattice parameter b is also observed RTiO3 perovskites with a-a-c+ tilting
(Zhou and Goodenough, 2005). The lattice parameter β is observed to deviate more
from 90 degrees with increase in Nd concentration (decrease in tolerance factor),
indicating more distinct monoclinic unit cell.
41
The LRO of La(1-x)Ndx(MgSn)0.5O3 perovskites decreases from 92% to 88%
with increase in Nd concentration (Table 2.3). Neither the charge difference nor radii
difference of B site cations do not change, but LRO is observed to slightly decrease
with increase in Nd concentration. The small decrease in LRO is may be due to the
change in diffusion rates of B-site ions. By replacing La with Nd, diffusion rates at
the sintering temperature get affected due to the variation in melting points (melting
point of La = 920 °C and Nd = 1020 °C). Sintering conditions for all the samples of
La(1-x)Ndx(MgSn)0.5O3 were identical (1600 °C for 5 hours duration).
The size difference between the Mg2+ and Sn4+ is 0.03 Å, whereas the size
difference between Mg2+ and Ti4+ is 0.11 Å (Shannon, 1976). Even though size
difference between the B site cations (Mg2+ and Sn4+ ) is very small, the LRO of Mg2+
and Sn4+ is found to be high and comparable to LRO between Mg2+ and Ti4+ with
higher size difference. LRO values of LMT and NMT are 96% and 82% respectively
(discussed in Section 2.2.3). The high degree of LRO is may be due to the nature of
Sn4+ ion. The high electronegative Sn4+ always tries to form covalent bonds with
oxygen. But Sn4+ does not possess empty d orbitals that stabilize Sn-O-Sn linkage.
Therefore, Sn-O-Sn bonds are less favorable compared to Sn-O-Mg. Thus, the cation
order minimizes the adjacent Sn4+ interactions.
The high degree of LRO observed with main group ion Sn4+ is similar to the
high degree of LRO observed with main group ion Sb5+ compared to transition metal
ion Ta5+ (Woodward et al., 1994; Barnes, 2003). The high electronegative Sb5+ always
tries to form covalent bonds with O2- ion. But Sb5+ does not have empty d orbital.
Therefore, the Sb 5s orbital forms a strong σ bond with an oxygen 2p orbital and π
bonds are not possible. Sb-O-Sb linkage (oxygen bound to two Sb cations) is
favorable for each Sb to make bonds with different O 2p orbitals. This is possible
42
only for a 90 degree Sb-O-Sb bond, observed in ilmenite structure of NaSbO3
(Mizoguchi et al., 2004), but it is not possible in the perovskite structure where the
bonds are close to linear. Therefore, Sb-O-Sb bonds are less favorable in perovskite
structure and cation ordering is more favorable with Sb5+ compared to Ta5+.
Table 2.3 Tolerance factors, Rietveld discrepancy indices and long range order
parameter (LRO) of La(1-x)Ndx(MgSn)0.5O3 system
x
t
RF2
(%)
4.9
1.12
LRO
(%)
92
χ2
0
0.927
Rwp
(%)
6.8
0.25
0.920
5.8
4.4
1.05
92
0.5
0.912
5.7
4.3
1.10
90
0.75
0.905
5.4
4.2
1.08
88
1.0
0.897
4.9
4.4
1.04
88
Table 2.4 Fractional atomic coordinates, thermal parameters and occupancies of
La(MgSn)0.5O3
Site
x
0.4885(6)
y
La
4(e)
Mg
2(c)
0
0.5
Mg
2(d)
0.5
Sn
2(d)
Sn
z
0.5406(2)
0.2501(3)
Occupancy
Uiso(Å2)
1
0.0074(5)
0
0.962(3)
0.006(4)
0
0
0.038(3)
0.006(4)
0.5
0
0
0.962(3)
0.006(8)
2(c)
0
0.5
0
0.038(3)
0.006(8)
O
4(e)
0.280(3)
0.282(3)
0.057(3)
1
0.003(3)
O
4(e)
0.210(3)
0.808(3)
0.040(4)
1
0.003(3)
O
4(e)
0.589(2)
-0.024(2)
0.248(3)
1
0.003(3)
43
La0.75Nd0.25(MgSn)0.5O3
Site
x
y
z
Occupancy
Uiso(Å2)
La
4(e)
0.4883(5)
0.5443(2)
0.2504(2)
0.75
0.0093(1)
Nd
4(e)
0.4883(5)
0.5443(2)
0.2504(2)
0.25
0.0093(1)
Mg
2(c)
0
0.5
0
0.960(4)
0.008(5)
Mg
2(d)
0.5
0
0
0.040(4)
0.008(5)
Sn
2(d)
0.5
0
0
0.960(4)
0.008(1)
Sn
2(c)
0
0.5
0
0.040(4)
0.008(1)
O
4(e)
0.292(3)
0.294(3)
0.049(3)
1
0.018(3)
O
4(e)
0.210(3)
0.795(3)
0.045(4)
1
0.018(3)
O
4(e)
0.599(2)
-0.030(2)
0.247(3)
1
0.018(3)
La0.5Nd0.5(MgSn)0.5O3
Site
x
y
z
Occupancy
Uiso(Å2)
La
4(e)
0.4887(6)
0.5474(2)
0.2503(2)
0.5
0.010(11)
Nd
4(e)
0.4887(6)
0.5474(2)
0.2503(2)
0.5
0.010(11)
Mg
2(c)
0
0.5
0
0.953(4)
0.007(5)
Mg
2(d)
0.5
0
0
0.047(4)
0.007(5)
Sn
2(d)
0.5
0
0
0.953(4)
0.007(1)
Sn
2(c)
0
0.5
0
0.047(4)
0.007(1)
O
4(e)
0.297(3)
0.288(3)
0.040(4)
1
0.009(3)
O
4(e)
0.209(3)
0.800(3)
0.051(3)
1
0.009(3)
O
4(e)
0.607(2)
-0.028(2)
0.255(2)
1
0.009(3)
44
La0.25Nd0.75(MgSn)0.5O3
Site
x
y
z
Occupancy
Uiso(Å2)
La
4(e)
0.4865(5)
0.5510(2)
0.25001(23)
0.25
0.014(5)
Nd
4(e)
0.4865(5)
0.5510(2)
0.25001(23)
0.75
0.014(5)
Mg
2(c)
0
0.5
0
0.940(4)
0.017(7)
Mg
2(d)
0.5
0
0
0.060(4)
0.017(7)
Sn
2(d)
0.5
0
0
0.940(4)
0.017(3)
Sn
2(c)
0
0.5
0
0.060(4)
0.017(3)
O
4(e)
0.293(3)
0.295(3)
0.045(3)
1
0.014(3)
O
4(e)
0.201(3)
0.802(3)
0.052(3)
1
0.014(3)
O
4(e)
0.607(2)
-0.031(2)
0.262(2)
1
0.014(3)
Nd(MgSn)0.5O3
Site
x
Nd
4(e)
0.4859(6)
Mg
2(c)
0
Mg
2(d)
Sn
y
z
0.5535(2)
0.2502(3)
Occupancy
Uiso(Å2)
1
0.012(5)
0.5
0
0.936(4)
0.009(2)
0.5
0
0
0.054(4)
0.009(2)
2(d)
0.5
0
0
0.936(4)
0.009(4)
Sn
2(c)
0
0.5
0
0.054(4)
0.009(4)
O
4(e)
0.298(3)
0.295(3)
0.050(3)
1
0.013(4)
O
4(e)
0.199(3)
0.808(3)
0.061(3)
1
0.013(4)
O
4(e)
0.606(2)
-0.032(2)
0.257(3)
1
0.013(4)
45
cubic Nd2Sn2O7 pyrochlore with Fd 3m symmetry (Lattice parameter: 10.58 Å)
Uiso (Å2)
Site
x
y
z
Occupancy
Nd
16(d)
0.5
0.5
0. 5
1
0.01
Sn
16(c)
0
0
0
1
0.01
O
48(f)
0.29(2)
0.125
0.125
1
0.01
O
8(b)
0.375
0.375
0.375
1
0.01
Table 2.10 Lattice parameters and X-ray density of La(1-x)Ndx(MgSn)0.5O3 system
x
0
a (Å)
5.6387(4)
b (Å)
5.7250(4)
c (Å)
8.0219(6)
β (deg)
90.085(3)
Density (g/cc)
6.628
0.25
5.6140(4)
5.7255(4)
7.9989(5)
90.096(3)
6.710
0.5
5.5902(4)
5.7292(4)
7.9776(6)
90.123(2)
6.787
0.75
5.5652(4)
5.7298(4)
7.9533(6)
90.140(2)
6.872
1.0
5.5397(3)
5.7304(3)
7.9281(4)
90.155(2)
6.961
12000
Intensity (cps)
10000
8000
6000
50
4000
55
60
2000
0
20
40
60
80
100
2θ (deg)
Fig. 2.9 Final observed (+ marks), calculated (solid line) and difference (below)
patterns, along with the calculated positions for La(MgSn)0.5O3 (Inset shows goodness
of fit near 56 degree).
46
12000
Intensity (cps)
10000
8000
6000
4000
50
55
60
2000
0
20
40
60
80
100
2θ (deg)
patterns, along with the calculated positions for La0.75Nd0.25(MgSn)0.5O3 (Inset shows
goodness of fit near 56 degree).
12000
10000
Intensity (cps)
8000
6000
4000
50
55
60
2000
0
20
40
60
80
100
2θ (deg)
47
10000
Intensity (cps)
8000
6000
4000
50
55
60
2000
0
20
40
60
80
100
2θ (deg)
4
1.2x10
4
1.0x10
Intensity (cps)
3
8.0x10
3
6.0x10
3
4.0x10
50
55
60
3
2.0x10
0.0
20
40
60
80
100
2θ (deg)
patterns, along with the calculated positions for Nd(MgSn)0.5O3. Pyrochlore
Nd2Sn2O7 positions are also shown (Inset shows goodness of fit near 56 degree).
48
c
a
b
Fig. 2.14a
Fig. 2.14b
Fig. 2.14 a) Structure of La(MgSn)0.5O3. Unit cell is shown with dotted line.
b) Structure of La(MgSn)0.5O3 in z- direction (black shaded spheres: La atoms, dark
grey shaded octahedra: MgO6 octahedra and light grey shaded octahedra: SnO6
octahedra)
49
Fig. 2.15a
Fig. 2.15b
Fig. 2.15c
Fig. 2.15 Structure of La(MgSn)0.5O3 : a) x– direction b) y- direction and c) z-direction
(black shaded spheres: La atoms, dark grey shaded octahedra: MgO6 octahedra and
light grey shaded octahedra: SnO6 octahedra)
50
a
b
c
7.9
ο
a,b,c (Α )
8.0
5.7
5.6
5.5
90.16
90.15
β
β (deg)
90.14
90.13
90.12
90.11
90.10
90.09
90.08
0.0
0.2
0.4
0.6
0.8
Nd concentration (x)
1.0
Fig. 2.16 Lattice parameters a, b, c and β of La(1-x)Ndx(MgSn)0.5O3 ceramics.
(Error bars are indicated for all the parameters).
51
2.2.3 Rietveld Refinement of La(MgTi)0.5O3 and Nd(MgTi)0.5O3
Rietveld refinement of LMT and NMT were carried out using X-ray
diffraction data. Data was collected using PANAlytical X’pert pro MPD in BraggBrentano geometry with X’Celerator detector. The collection conditions were CuKα
radiation, 40 kV 30 mA, 0.017º step scan, 1.0º divergence slit and 0.02 rad. incident
and receiving soller slits. X-ray diffraction pattern of NMT is presented in Figure
2.17. An unidentified minor impurity peak (< 2 wt %) is observed. Tolerance factor of
LMT is 0.946 and that of NMT is 0.916. Super lattice reflections corresponding to out
of phase tilting, in phase tilting and A site cation displacements of LMT (discussed in
Section 2.2.1) are also observed in X-ray pattern of NMT. The clear splitting of
reflections is observed in NMT compared to LMT. This is due to the increase in
distortion of unit cell with decrease in tolerance factor.
Initial model for the refinements was obtained from previous refinements on
the same compounds (Levin et al., 2005; Greon et al., 1986). Fractional atomic
coordinates, thermal parameters and occupancies are presented in Tables 2.11 & 2.12.
Lattice parameters and Rietveld discrepancy indices are presented in Table 2.13. Final
refinement plots are presented in Figures 2.18 and 2.19. During the refinement of
occupancies of Mg2+ and Ti4+ ions, the unit cell content is constrained according to
the chemical composition. In order to avoid the unwanted correlation with the order
parameter, the thermal parameter (Uiso) values for the octahedral site are constrained
to be equal. The uncertainty values obtained for the occupancy factors of B-site ions
are high (Tables 2.11 & 2.12). This may be due to the low scattering length difference
between Mg2+ and Ti4+. The percentage of LRO obtained for LMT and NMT are 96%
and 82% respectively. Previous studies with the neutron refinement show that LRO of
LMT is 100% (Levin et al., 2005) and NMT exhibits 92% of LRO (Greon et al.,
52
1986). Very recent Rietveld refinement of LMT neutron data shows that LRO is 94%
(Salak et al., 2008).
Lower tolerance factor of NMT compared to LMT should result lower
lattice parameters a, b and c. But lattice parameters a and c of NMT are smaller than a
and c of LMT and lattice parameter b of NMT is slightly larger than b of LMT. The
trend of lattice parameters a, b and c variation is similar to that of the trend of lattice
parameter variation in La(1-x)Ndx(MgSn)0.5O3 perovskites (discussed in Section
*
20
30
40
50
60
(220)
1/2(432)
(211)
(210)
1/2(421)
1/2(300)
1/2(311)
(111)
1/2(320)
1/2(321)
(200)
1/2(410)
(110)
1/2(210)
1/2(111)
(100)
Intensity (a.u.)
2.2.2.3).
70
2θ (deg)
Fig. 2.17 X-ray diffraction pattern of Nd(MgTi)0.5O3 (Super lattice reflections are
shown with half integer indices and main cubic reflections are shown with integer
indices. Impurity peak is shown by asterisk).
53
Table 2.11 Fractional atomic coordinates, thermal parameters and occupancies
of La(MgTi)0.5O3
Site
La
4(e)
x
y
0.4964(8)
Occupancy
z
0.5299(1)
0.2542(4)
1
Uiso(Å2)
0.0053(4)
Mg 2(c)
0
0.5
0
0.98(2)
0.0023(5)
Mg 2(d)
0.5
0
0
0.02(2)
0.0023(5)
Ti
2(d)
0.5
0
0
0.98(2)
0.0023(5)
Ti
2(c)
0
0.5
0
0.02(2)
0.0023(5)
O
4(e)
0.307(3)
0.285(3)
0.059(3)
1
-0.008(3)
O
4(e)
0.234(3)
0.742(3)
0.028(4)
1
-0.008(3)
O
4(e)
0.571(2)
-0.010(2)
0.228(3)
1
-0.008(3)
Table 2.12 Fractional atomic coordinates, thermal parameters and occupancies
of Nd(MgTi)0.5O3
Site
Nd
4(e)
x
y
0.5110(3)
z
0.5473(1)
0.2512(5)
Occupancy
1
Uiso(Å2)
0.0048(4)
Mg 2(c)
0
0.5
0
0.91(3)
0.0039(7)
Mg 2(d)
0.5
0
0
0.09(3)
0.0039(7)
Ti
2(d)
0.5
0
0
0.91(3)
0.0039(7)
Ti
2(c)
0
0.5
0
0.09(3)
0.0039(7)
O
4(e)
0.215(3)
0.201(3)
-0.013(3)
1
-0.001(3)
O
4(e)
0.293(3)
0.708(3)
-0.072(2)
1
-0.001(3)
O
4(e)
0.405(1)
0.979(1)
0.229(3)
1
-0.001(3)
Table 2.13 Lattice parameters and Rietveld discrepancy indices of La(MgTi)0.5O3
and Nd(MgTi)0.5O3
a (Å)
b (Å)
c (Å)
β (deg)
LMT
5.5637(2)
5.5762(2)
7.8668(3)
89.945(3)
Rwp
(%)
6.6
NMT
5.4630(2)
5.5904(2)
7.7721(3)
89.991(3)
5.1
54
RF2
(%)
3.2
1.14
4.6
1.09
χ2
12000
Intensity (cps)
10000
8000
6000
50
4000
55
60
2000
0
20
40
60
80
100
2θ (deg)
patterns, along with the calculated positions for La(MgTi)0.5O3 (Inset shows goodness
of fit near 56 degree).
10000
Intensity (cps)
8000
6000
4000
48
52
56
60
2000
0
20
40
60
80
100
2θ (deg)
patterns, along with the calculated positions for Nd(MgTi)0.5O3 (Inset shows goodness
of fit in the range of 46 to 60 degree).
55
2.3 CONCLUSIONS
Complex perovskites La(Mg0.5Ti(0.5-x)Snx)O3 (x = 0.0, 0.125, 0.25, 0.375 and
0.5), La(1-x)Ndx(MgSn)0.5O3 (x = 0.0, 0.25, 0.5, 0.75 and 1.0) and Nd(MgTi)0.5O3
synthesized by solid state reaction method exhibit monoclinic P 21 / n symmetry
characteristic of a-a-c+ tilting and 1:1 B site cation ordering. The intensity of ½ (111)
super lattice reflection and lattice parameters a, b and c increase with increase in Sn
concentration for La(Mg0.5Ti(0.5-x)Snx)O3 ceramics. Splitting of the reflections and
increase in difference between a and b reveal that unit cell distorts with increase in Sn
concentration. X-ray diffraction patterns of La(1-x)Ndx(MgSn)0.5O3 show
½ (111)
super lattice reflection corresponding to 1:1 B site cation ordering. Rietveld
refinement of La(1-x)Ndx(MgSn)0.5O3 ceramics reveals that LRO decreases slightly
with increase in Nd concentration. Due to the main group Sn element with high
electronegativity and d10 orbitals, La(1-x)Ndx(MgSn)0.5O3 perovskites exhibit high
percentage of LRO. Lattice parameters a and c decrease with increase in Nd
concentration but lattice parameter b slightly increases with increase in Nd
concentration. LRO values obtained for LMT and NMT are 96% and 82%
respectively. Lattice parameter a and c of NMT are less than that of LMT but lattice
parameter b of NMT is higher than that of LMT. The slight increase in lattice
parameter b may be due to the distortion of BO6 octahedra.
56
CHAPTER 3
LATTICE VIBRATIONAL STUDIES ON
La(MgTi(1-x)Snx)0.5O3, La(1-x)Ndx(MgSn)0.5O3 AND Nd(MgTi)0.5O3
This chapter describes lattice vibrational studies using far infrared reflectance
studies and Raman spectroscopy studies. Number of lattice vibration modes of a
crystal with ‘n’ number atoms per unit cell is 3n; in which 3 are acoustic modes and
remaining 3n-3 are optic modes. These optic modes are categorized into infrared
active, Raman active and silent (neither Raman nor IR active) modes. The
fundamental requirement of a phonon to be Raman active is that the first derivative of
the polarizability with respect to the vibrational normal coordinate has a non-zero
value. For a mode to be IR active, dipole moment should change with the vibration.
According to the exclusion principle, no mode is both Raman and IR active for
centrosymmetric structures.
According to structural studies discussed in the previous chapter, symmetry of
the all compounds is monoclinic P 21 / n . Number of the atoms in P 21 / n unit cell is
20. Therefore, total number of the modes is 60, out of which 3 are acoustic. Factor
group analysis by Ayala et al., 2007 predicted that P 21 / n symmetry exhibits 24 (12
Ag + 12 Bg) Raman active modes and 33 (17 Au + 16 Bu) IR active modes.
3.1 INFRARED REFLECTANCE STUDIES
IR reflectance studies were carried out by fitting the experimental data to four
parameter model. In the infrared reflectance of solids, the experimental data provide
57
normalized reflectance R as a function of the frequency ω. The reflectance R(ω) is
related to the complex dielectric constant ε * (ω) = ε ' (ω) − i ε " (ω) through the
following equation.
R(ω) =
ε * (ω) − 1
(3.1)
ε * (ω) + 1
3.1.1 Four Parameter Model
The dielectric function as factorized form derived in stages by Cochran and
Cowley (1962), Barker (1964a), Berreman and Unterwald (1968), Lowndes (1970)
and Kukharski (1973). It is given by
ε * (ω) = ε ∞ ∏
j
ω 2jLO − ω 2 + iγ jLO ω
(3.2)
ω 2jTO − ω 2 + iγ jTO ω
where ωjTO is the frequency of the j-th transverse optic (TO) mode, ωjLO is the
frequency of the j-th longitudinal optic (LO) mode, γjTO and γjLO their respective
damping constants and ε ∞ is dielectric constant due to electronic polarization.
This model is useful for the spectra with broad or asymmetrical bands arising
from a TO-LO splitting (Luspin et al., 1980) and allows to attribute different damping
constants for TO and LO modes. The disadvantage of this model is that additional
conditions between the damping constants and frequencies should be taken into
account, otherwise non physical results can be obtained (for instance γLO > γTO and
(γ TO γ LO ) > (ω TO ω LO ) 2 should be fulfilled in the case of one oscillator fit). In
practical cases multiple oscillators are used for analyzing the data. In this case, a good
criterion is to maintain losses and optical conductivity always positive, otherwise the
model is valid in a limited range of frequencies.
58
The oscillator strength, ∆εj (contribution of each oscillator to the dielectric
constant) of the TO mode can be obtained from the frequencies of TO mode and LO
mode using the relation (Servoin et al., 1980a):
ω 2jLO − ω 2jTO
∆ε j = ε ∞
ω 2jTO
ω 2kLO − ω 2jTO
∏ω
k≠ j
2
kTO
(3.3)
− ω 2jTO
Equations for dielectric constant and dielectric loss at microwave frequencies
(ω2<<ω2TO) are given by (Wakino et al., 1986):
n
ε ′r = ε ∞ + ∑ ∆ε j
(3.4)
j=1
n
∆ε j γ jTO
j=1
ω 2jTO
ε ′r′ = ω∑
(3.5)
By using the above Equations 3.4 & 3.5
n
Q=
ε′
1
= r =
tanδ ε ′r′
ε ∞ + ∑ ∆ε j
j=1
n
∆ε j γ jTO
j=1
ω 2jTO
ω∑
(3.6)
From the above Equation 3.6, Q is inversely proportional to frequency. Therefore
Q.ω ~ constant
(3.7)
The average phonon damping or weighed sum of TO phonon damping (Salak et al.,
2004) can be estimated using the following expression:
n
γ(TO) =
∑γ
j=1
jTO
∆ε j
(3.8)
n
∑ ∆ε
j=1
j
3.1.2 Experimental Details
Far infrared spectra were recorded at two different places using Bruker IFS
66v FTIR spectrometer. IR reflectance data of La(Mg0.5Ti(0.5-x)Snx)O3 ceramics was
obtained from National Taiwan Normal University, Taipei, Taiwan. IR reflectance
59
data of La(1-x)Ndx(Mg0.5Sn0.5)O3 was obtained from Laboratoire Photons et Matière,
Université Pierre et Marie Curie, Paris, France.
During the experiment, different sources and detectors were used for far
IR and mid IR. The source for mid IR is Globar (SiC) source and KBr beam splitter
was used, whereas for far IR, Hg source and Ge coated mylar beam splitter were used.
The modulated light beam from the spectrometer was focused onto either the sample
or an Au-reference mirror. The reflected beam was directed onto a 4.2 K bolometer
detector (40-600 cm-1) and a B doped Si photoconductor (450-4000 cm-1). The
different sources, beam splitters, and detectors used in these studies provided
substantial spectral overlap, and the reflectance mismatch between adjacent spectral
ranges was less than 1%. For mid IR (350-5000 cm-1) the detector used in the case of
La(1-x)Ndx(Mg0.5Sn0.5)O3 ceramics was DTGS (Deuterated Triglycerine Sulfate): KBr
pyroelectric detector. The measurements were performed under low vacuum, with
beam width of 5 mm and in a reflection accessory with an internal reflection angle of
11 degrees. The samples for reflectance measurements were one-side polished using
0.25 µm diamond paste and subsequently annealed at 500 ºC for 8 hrs to remove the
residual stress. All the spectra were recorded at room temperature.
3.1.3 Analysis of Infrared Reflectance Data
3.1.3.1 Mode Assignment and Data Treatment
The reflectivity spectra studies on alkaline earth based perovskites suggest
three categories of modes. A–BO6 translation modes in the vicinity of 150 cm-1, B'-OB" bonding modes in the range of 200 to 400 cm-1 and B-O6 bending modes in the
range of 500-800 cm-1 (Furuya, 1999). Recent work on LZT suggests that modes
between 200 and 500 cm-1 are related to cation ordering and octahedral tilting (Kim et
al., 2001). Even though factor group analysis predicts 33 IR active modes, due to the
60
anisotropy averaging out in the poly crystalline samples, modes Au and Bu can not be
resolved and therefore, the number of effective modes would be 17.
Infrared data was fitted to four parameter model by using Focus curve fitting
software (Menses, 2005). The values of the mode parameters were varied until the
error between the calculated and measured reflectance was minimum. Initially the
frequencies of TO and LO modes can be approximately guessed from the reflectance
spectra itself (TO modes correspond to the peaks and LO modes to dips of the
reflectance spectra). Error in the intrinsic dielectric constant is 1% and error in the
intrinsic quality factor is 6%.
3.1.3.2 IR Study on La(Mg0.5Ti(0.5-x)Snx)O3
The infrared reflectivity spectra of La(Mg0.5Ti(0.5-x)Snx)O3 is shown in Figure 3.1. A
visual inspection of Figure 3.1 indicates presence of three categories of modes (ABO6, B'-O-B" and B-O6) in all the compositions and broadening of modes for
intermediate compositions. The four parameter model fitted data and experimental
data are shown in Figure 3.1 and imaginary part of dielectric constant (ε") is presented
in Figure 3.2. The TO and LO mode frequencies, damping coefficients and estimated
intrinsic dielectric constant values are presented in Table 3.1 and 3.2. Intermediate
compositions (x = 0.125, 0.25 and 0.375 compositions) are fitted with 15, 14 and 14
modes whereas the end compositions LMT and LMS are fitted with all the 17 modes.
The number of modes detected for intermediate compositions are less than 17, due to
the broadening of the modes.
Figure 3.3 presents TO mode frequencies and strengths as well as its variation
with Sn concentration (open circles denote strength of TO modes). The contribution
of the A-BO6 vibrations to the extrapolated dielectric constant is much higher,
followed by a mode in the vicinity of 350 cm-1. It is to be noted that the strength of the
61
modes corresponding to A-BO6 vibrations varies predominantly, even though there is
no substitution of ions at A site. The higher contribution of A-BO6 modes to the
dielectric constant is in accordance with the study of LMT-(NaNd)0.5TiO3 by Kim et
al., 2005. It is also observed that the contribution of the mode at 350 cm-1 decreases
with the increase in Sn concentration. This mode might have originated from O-B"-O
bending vibrations.
Figure 3.4 presents the variation of average phonon damping γ(TO) and
intrinsic Q as a function of Sn concentration. The intrinsic Q decreases up to x = 0.2
and then increases. The average phonon damping also shows a peak near x = 0.2.
Therefore, it may be inferred that the intrinsic Q correlates with the average phonon
damping closely.
According to structural refinement (described in Chapter 2), LRO of LMT is
98% and LMS is 92%. The low percentage of LRO should result in lower Q factor
(Tamura, 2006). But intrinsic Q factor of LMS is high compared to LMT. This is due
to the low dielectric constant of LMS. Increase in dielectric constant increases
anharmonicity and results to an increase in phonon damping. High phonon damping
gives rise to high dielectric loss (Petzelt et al, 1996; Tangastev et al., 1993). The loss
dependence of ε "r α ε 'r 4 was observed in perovskites Ba(B'1/2B"1/2)O3 studied by Petzelt
et al., 1996.
Even though dielectric constant of intermediate compositions (x = 0.125, 0.25
and 0.375) is less than that of LMT, intrinsic Q factors are low. The low Q and
correspondingly high phonon damping observed for intermediate compositions may
be due to low percentage of LRO, resulting from accommodating three different ions
at B site. The lowest Q is observed for x = 0.25 composition.
62
Reflectivity (a.u.)
x = 0 .5
x = 0 .3 7 5
x = 0 .2 5
x = 0 .1 2 5
x = 0 .0
200
400
600
800
-1
W a v e n u m b e r(c m )
1000
Fig. 3.1 IR reflectivity spectra of La(Mg0.5Ti(0.5-x)Snx)O3 perovskite system (open
circles represent the experimental data and continuous line represents the fitted
model)
120
80
40
0
x = 0.5
120
80
x = 0.375 40
0
120
80
40
0
ε"
x = 0.25
120
80
x = 0.125 40
0
120
80
40
0
x = 0.0
100
200
300
400
500
600
-1
Wave number (cm )
Fig. 3.2 Imaginary part of dielectric constant (ε") obtained by fitting reflectivity to
four parameter model.
63
Table 3.1 IR fit parameters and intrinsic dielectric constant (ε') obtained for x = 0.0, x = 0.125 and x = 0.25
compositions of La(Mg0.5Ti(0.5-x)Snx)O3
ωTO
(cm-1)
109.10
121.58
167.10
178.61
190.44
203.42
231.01
256.63
274.02
303.58
346.90
385.54
402.37
459.08
512.23
x = 0.0
γTO
(cm-1)
13.00
10.61
14.79
13.80
12.07
20.36
20.38
21.16
19.90
18.27
23.03
14.50
13.35
18.02
51.48
ωLO
(cm-1)
109.70
122.52
175.55
190.10
200.41
220.21
238.01
262.46
279.90
312.43
385.38
399.47
456.80
493.93
517.61
γLO
(cm-1)
8.66
7.74
16.82
10.00
20.27
5.89
12.22
25.37
15.60
18.74
15.56
14.97
20.03
29.52
24.51
545.30
30.54
553.61
22.38
ε' =
29.07
ωTO
(cm-1)
114.07
122.96
161.91
178.59
197.64
232.95
268.32
301.77
348.14
390.68
438.93
448.39
503.21
536.96
607.26
ε' =
x = 0.125
γTO
ωLO
-1
(cm-1)
(cm )
13.10
114.93
14.44
124.91
19.00
172.65
26.83
194.59
24.83
215.39
40.66
236.20
28.16
274.68
31.33
312.78
33.66
385.72
24.05
438.90
26.56
447.54
17.71
491.06
57.23
514.71
48.31
585.07
46.09
689.86
γLO
(cm-1)
5.02
10.92
23.11
27.69
20.17
25.26
27.07
26.80
32.07
26.52
21.59
32.90
39.40
71.29
24.77
26.97
4.48
ε∞ =
ε ∞ = 4.45
64
ωTO
(cm-1)
115.00
122.75
160.95
163.20
194.13
264.00
293.24
346.07
385.94
429.90
441.51
489.60
542.82
607.45
ε' =
x = 0.25
γTO
ωLO
-1
(cm-1)
(cm )
18.75
115.73
13.06
127.10
18.08
162.92
25.12
191.10
24.04
221.03
30.90
273.46
29.84
308.94
39.24
379.36
26.37
429.90
17.55
440.50
49.15
489.15
128.72
508.61
43.67
552.80
48.77
680.01
24.68
ε∞ =
γLO
(cm-1)
5.95
29.57
26.08
49.03
40.33
34.69
40.24
45.38
15.11
87.75
67.12
110.69
51.29
28.81
3.89
Table 3.2 IR fit parameters and intrinsic dielectric constant (ε') obtained
for x = 0.375 and x = 0.5 compositions of La(Mg0.5Ti(0.5-x)Snx)O3
ωTO
(cm-1)
115.43
125.95
156.26
159.10
171.02
204.50
269.00
283.56
343.10
369.71
392.92
435.39
603.50
ε' =
x = 0.375
ωLO
γTO
(cm-1)
(cm-1)
15.71
115.90
21.35
127.35
5.07
158.58
53.00
171.20
11.11
198.01
35.40
221.00
32.00
273.86
25.92
306.51
49.78
359.97
33.53
392.22
21.32
434.16
30.41
520.63
44.90
669.43
23.07
ε∞ =
x = 0.5
γLO
(cm-1)
9.94
15.57
14.69
16.12
14.47
47.90
23.33
26.23
44.90
25.81
36.93
74.80
21.60
3.77
ωTO
(cm-1)
112.99
123.18
156.29
162.03
170.92
240.05
272.22
284.36
314.38
332.22
356.95
387.18
412.60
440.35
γTO
(cm-1)
10.43
10.98
3.85
16.94
10.68
59.55
21.21
14.19
25.03
17.72
20.04
22.20
17.08
39.61
ωLO
(cm-1)
113.54
124.81
161.59
168.50
199.57
255.55
277.20
301.24
315.02
338.38
386.66
407.82
435.91
488.63
γLO
(cm-1)
5.95
6.36
4.09
9.91
4.02
49.11
32.57
18.68
25.52
15.04
38.03
24.12
41.76
40.09
517.46
580.53
610.88
ε' =
60.48
40.89
51.55
20.52
528.08
610.85
660.88
ε∞ =
26.17
71.09
25.21
3.79
65
10
x = 0 .5
5
0
10
Mode strength (∆ε)
x = 0 .3 7 5
5
0
10
x = 0 .2 5
5
0
10
x = 0 .1 2 5
5
0
10
x = 0 .0
5
0
100
200
300
400
500
600
-1
W a v e n u m b e r(c m )
Fig. 3.3 The variation of TO mode phonon strength of La(Mg0.5Ti(0.5-x)Snx)O3 as
function of Sn concentration (open circles represent TO modes).
22
110
21
20
100
19
17
80
16
70
Q.f(GHz)
<γTO>(cm-1)
90
18
15
60
14
13
50
0.0
0.1
0.2
0.3
Sn content, x
0.4
0.5
Fig. 3.4 Intrinsic Q.f values (circles) and average TO phonon damping (squares) as
function of Sn content, x.
66
3.1.3.3 IR Study on La(1-x)Ndx(Mg0.5Sn0.5)O3
Figure 3.5 presents the IR reflectivity spectra of the compositions studied.
Modes corresponding to A–BO6 translation modes (vicinity of 150 cm-1), B'-O-B"
stretching modes (200 to 500 cm-1) and B-O6 bending modes (500-800 cm-1) are
present in all the compositions. Imaginary part of dielectric constant (ε") obtained by
fitting the reflectivity data to four parameter model is presented in Figure 3.6. As
discussed in Section 3.1.1, condition of positive imaginary part is satisfied for all the
compositions. Modes between 200 and 500 cm-1 reconfirm the existence of B site
cation ordering and BO6 octahedral tilting in all the compositions in accordance with
the structural studies. It is also observed that the mode in the vicinity of 350 cm-1
broadens with the increase in Nd concentration suggesting a decrease in LRO.
All the compositions were fitted with 17 modes using four parameter model.
The TO and LO mode frequencies, damping coefficients and estimated intrinsic
values of dielectric constant are given in Tables 3.3 & 3.4. Figure 3.7 presents TO
mode wave numbers, strengths and its variation with Nd concentration (full squares
denote strength of TO modes). The intrinsic dielectric constant decreases slightly with
the increase in Nd concentration. The slight variation is due to the variation of mode
strengths in the range of 150-175 cm-1 and therefore the modes corresponding to ABO6 vibrations are responsible for the decrease in the dielectric constant. The strength
of A-BO6 modes is also high compared to B'-O-B" and B-O6 modes.
Figure 3.8 presents the variation of average phonon damping,
γ(TO) and
intrinsic Q as a function of Nd concentration. The intrinsic Q decreases with increase
in Nd concentration, whereas the average phonon damping is observed to increase.
The increase in phonon damping and decrease in intrinsic quality factor are attributed
to decrease in LRO.
67
x = 1.0
Reflectivity (a.u.)
x = 0.75
x = 0.5
x = 0.25
x = 0.0
200
400
600
800
1000
-1
Wave number (cm )
Fig. 3.5 IR reflectivity spectra of La(1-x)Ndx(MgSn)0.5O3 perovskite system (open
circles represent the experimental data and continuous line represents the fitted
model).
ε"
75
50
25
0
75
50
25
0
75
50
25
0
75
50
25
0
75
50
25
0
100
200
300
400
500
600
700
-1
Wave number (cm )
Fig. 3.6 Imaginary part of dielectric constant (ε") obtained by fitting reflectivity to
four parameter model.
68
Table 3.3 IR fit parameters obtained intrinsic dielectric constant (ε') for x = 0.0, x = 0.25 and x = 0.5
compositions of La(1-x)Ndx(MgSn)0.5O3
ωTO
(cm-1)
112.96
127.91
159.02
169.39
184.03
230.05
272.37
283.07
315.75
334.81
358.69
373.67
409.36
437.16
510.04
577.79
593.31
ε' =
x = 0.0
γTO
(cm-1)
8.87
8.90
10.07
10.07
7.19
32.95
15.14
12.11
13.16
17.31
15.05
13.36
10.97
56.24
56.41
34.02
17.82
19.07
ωLO
(cm-1)
113.96
128.29
164.41
183.97
198.99
235.64
277.81
301.32
317.57
340.66
372.44
407.51
436.91
488.35
524.57
591.42
657.66
ε∞ =
γLO
(cm-1)
3.74
6.01
6.52
5.04
3.22
29.59
16.40
12.62
13.45
11.22
12.03
11.24
56.96
36.76
26.74
20.19
17.56
3.93
ωTO
(cm-1)
112.92
127.59
158.97
171.71
186.25
230.07
271.91
282.09
318.06
335.69
362.15
376.82
410.52
461.64
515.36
579.33
410.52
ε' =
x = 0.25
γTO
ωLO
-1
(cm-1)
(cm )
7.06
112.97
7.57
128.00
14.06
168.27
9.07
186.06
4.97
199.22
25.06
234.48
12.34
276.62
12.24
301.45
11.05
319.34
15.04
339.96
18.07
375.44
13.59
407.91
13.21
461.03
32.28
491.11
52.97
526.36
39.19
596.23
13.21
461.03
18.91
ε∞ =
69
γLO
(cm-1)
3.43
5.20
7.89
5.53
5.55
26.49
14.16
15.77
10.84
11.97
14.35
13.80
19.50
33.81
25.94
24.98
34.09
3.98
ωTO
(cm-1)
111.93
127.62
157.92
172.55
186.07
225.29
272.80
283.07
316.95
336.35
363.69
378.07
411.49
460.37
514.46
579.08
596.17
ε' =
x = 0.50
γTO
ωLO
-1
(cm-1)
(cm )
4.72
112.89
12.00
127.65
14.53
168.32
10.06
185.85
5.69
199.60
25.72
227.78
16.08
277.49
14.02
302.88
12.74
318.60
13.63
339.49
21.18
376.29
16.16
408.52
14.08
460.35
34.60
493.57
49.33
527.11
40.03
596.08
19.71
596.07
18.47
ε∞ =
γLO
(cm-1)
4.02
10.77
11.93
5.79
6.26
35.46
21.12
15.04
14.08
13.35
18.55
14.14
34.46
33.87
27.02
24.94
18.24
3.94
Table 3.4 IR fit parameters intrinsic dielectric constant (ε') obtained for x = 0.75
and x = 1.0 compositions of La(1-x)Ndx(MgSn)0.5O3
x = 0.75
γTO
ωTO
(cm-1)
(cm-1)
112.83
8.50
127.28
8.07
159.99
14.66
173.94
11.09
186.19
5.54
224.60
29.48
273.09
20.27
284.73
16.58
316.98
16.17
337.82
12.66
367.16
23.95
381.77
19.33
413.38
17.08
460.38
34.46
515.82
45.79
579.01
40.32
595.17
20.62
ε' = 18.33
x = 1.0
ωLO
(cm-1)
113.76
127.98
169.05
186.11
199.12
229.28
278.86
303.33
319.70
340.25
378.80
409.54
460.21
496.22
528.43
594.44
659.58
ε∞ =
γLO
(cm-1)
6.07
6.95
10.13
5.72
8.51
35.98
24.52
16.62
19.44
13.24
22.45
16.27
34.17
32.87
27.34
26.03
18.94
4.00
ωTO
(cm-1)
113.03
127.07
160.36
175.07
186.87
222.83
270.81
288.66
318.07
340.89
368.63
384.43
415.04
460.56
520.06
579.09
593.99
ε' =
70
γTO
(cm-1)
5.97
10.11
17.28
9.94
7.83
29.46
17.06
17.19
12.22
10.05
25.01
22.21
17.31
34.25
39.19
37.57
16.00
17.97
ωLO
(cm-1)
114.15
127.40
170.22
186.56
199.45
227.39
281.15
304.59
320.32
342.74
379.68
410.47
460.35
500.13
529.58
593.00
659.79
ε∞ =
γLO
(cm-1)
4.51
10.27
11.64
8.49
8.53
35.65
31.41
14.86
16.28
10.26
28.15
13.97
30.02
23.52
21.31
17.38
31.94
4.00
Mode strength (∆εj)
6
4
2
0
x = 1.0
x = 0.75
6
4
2
0
6
4
2
0
x = 0.5
6
4
2
0
x = 0.25
6
4
2
0
x = 0.0
100
200
300
400
-1
Wave number (cm )
500
600
Fig. 3.7 The variation of TO mode strength as function of Nd concentration, x. (black
shaded squares represent TO modes).
125
18
120
-1
<γTO>( cm )
3
Q.f (x10 GHz)
17
115
110
16
105
15
100
14
95
0.0
0.2
0.4
0.6
0.8
1.0
Nd content, x
Fig. 3.8 Intrinsic Q.f values (open circles) and average TO phonon damping (shaded
squares) as functions of Nd content, x.
71
3.2 RAMAN SCATTERING STUDIES
The number of Raman active modes for monoclinic P 21 / n space group is 24
(12Ag+12Bg). First principle calculations by Prosandeev et al., 2005 suggest that 1:1
B site cation ordering yields F2g- Eg- and A1g- like modes for monoclinic
Ca(AlNb)0.5O3 with P 21 / n symmetry. In this work, mode analysis is carried out by
identifying modes responsible for cation ordering in an ideal perovskite. The ideal
cubic perovskite with Pm3 m symmetry do not support any Raman active mode
whereas B-site ordered cubic perovskite with Fm3 m symmetry exhibits 4
(A1g+2F2g+Eg) Raman active modes. In which, 3 modes (A1g+F2g+Eg) result from O
atom, and 1 mode (F2g) is due to A site atom (Gutter et al., 2003; Idink and White,
1994; Duyckaerts and Tarte, 1974; Rout et al., 2005; Ayala, 2007). Atomic positions,
site symmetry and Raman active modes of A(B'B")0.5O3 complex perovskite with
Fm3 m symmetry are listed in Table 3.5.
Table 3.5 Atomic positions and Raman active modes for cubic crystal Fm3 m
Atom
Wyckoff site
Site symmetry
Raman modes
A
8c
Td
F2g
B'
4b
Oh
-
B"
4a
Oh
-
O
24e
C4v
A1g, Eg & F2g
The A1g mode is a totally symmetric stretching mode of BO6 octahedra and Eg
mode is due to anti symmetric stretching of BO6 octahedra. F2g mode observed at
higher wave number is due to symmetric bending of octahedra combined with nonnegligible translation of A site cation (Liegeois-Duyckaerts and Tarte, 1974). During
the vibrations of the A1g and Eg symmetry modes, only the oxygen atom moves along
the B'-O-B" axis and all the cations are at rest. In this case, the corresponding
72
frequencies are determined by the B'-O and B"-O bonding forces (LiegeoisDuyckaerts and Tarte, 1974; Ayala et al., 2007). Higher frequencies in the Raman
spectrum are primarily related to A1g and Eg modes with frequency of A1g mode >
frequency of Eg mode (Ayala et al., 2007). The Triply degenerate F2g and doubly
degenerate Eg modes are sensitive to the symmetry changes. The cubic F2g mode splits
into doublet and triplet with the lowering of the symmetry (Runka et al., 2005).
Figure 3.9 shows O atom vibrations for non degenerate A1g, doubly degenerate Eg and
triply degenerate F2g modes.
The position of the F2g modes depends on the type of A site atom in perovskite
structure and A-O bonding force. Runka et al., 2005 has reported that sharpness or
intensity of F2g vibrational mode can be a key factor for determining the degree of
order in complex perovskites. The cubic F2g mode splits into doublet and triplet with
the lowering of the symmetry (Runka et al., 2005).
The non-degeneracy of the A1g mode forbids any splitting of the line, which
facilitates the qualitative estimation of cation ordering. The B-site cations can affect
the line shape of the A1g mode and A1g peak becomes narrower with increase in cation
ordering (Jiang et al., 2000, Zhao et al., 2005, Setter and Laulicht, 1987). If two
adjacent BO6 octahedra in complex perovskite are not equivalent, the constituent
oxygen atoms occur in the C4v local positions without an inversion centre, and their
vibrations become Raman active. In principle, if there is a statistically greater chance
of ions of different types occupying the adjacent octahedra, then the intensity of the
A1g band should increase and its width narrows (Zheng et al., 2003a; Zheng et al.,
2003b).
In this work, degree of B site cation ordering is studied by estimating the full
width at half maxima (FWHM) of A1g mode.
73
Eg1
Fig. 3.9a A1g stretching mode
F2g1
Eg2
Fig. 3.9b Eg antistretching mode
F2g2
F2g3
Fig. 3.9c F2g bending mode
Fig. 3.9 Vibration of O atom: a). A1g stretching mode b). Eg antistretching mode and
c). F2g bending mode. (A site atom vibration is not shown for F2g bending mode.
Shaded and unshaded big circles represent two different B-site atoms).
74
3.2.1 Experimental Details
Raman measurements were carried out using two different instruments.
Raman spectra of La(Mg0.5Ti(0.5-x)Snx)O3 ceramics (discussed in Section 3.2.2) was
obtained from National Taiwan Normal University, Taipei, Taiwan. Measurements
were carried out using a DILOR XY 800 triple-grating Raman spectrometer equipped
with a liquid-nitrogen-cooled CCD. The 514.5 nm line of an Ar+ ion laser with an
output 10 mW was used as the excitation source and an Olympus BH-2 microscope
with 100x objective was employed for micro-Raman detection. The resolution
obtained was 0.7 cm-1.
Raman
measurements
on
La(1-x)Ndx(MgSn)0.5O3,
La(MgTi)0.5O3
and
Nd(MgTi)0.5O3 (discussed in Sections 3.2.3. and 3.2.4.) were carried out using Horiba
Jobin Yvon HR 800 UV Raman spectrometer equipped with a thermoelectrically
cooled CCD. The 632.8 nm line of He-Ne laser with an output of 10 mW was used as
the excitation source and an Olympus BX-41 microscope with 100x objective was
employed for micro-Raman detection. The resolution obtained was 0.3 cm-1. Prior to
recording the spectra on dielectric samples, spectrometer was calibrated by using Si
standard.
The samples for Raman measurements were one-side polished using 0.25 µm
diamond paste and subsequently annealed at 500 ºC for 8 hrs to remove the residual
stress and measurements were carried out at room temperature. Raman data was
treated using Focus curve fitting software (Menses, 2005). The A1g mode is fitted to
Lorentzian peak shape and a baseline correction was applied prior to the fitting. The
error in the FWHM values is 1% and error in Raman shift values is 0.3%.
75
3.2.2 Raman Spectra of La(Mg0.5Ti(0.5-x)Snx)O3
Figure 3.10 presents Raman spectra of La(Mg0.5Ti(0.5-x)Snx)O3 ceramics. It is
observed that the modes shift (except modes near 177 and 286 cm-1) to low frequency
with increase in Sn concentration, attributed to increase in bond lengths, and thereby
decrease in force constants. The mode observed at the highest wave number (around
700 cm-1) is assigned to A1g vibrations. There exists some ambiguity with respect to
the identification of F2g mode and Eg mode in LMT. By analyzing LMT-LT solid
solutions, Levin et al., 2005 related modes at 139 cm-1 and 449 cm-1 to F2g like
vibrations, modes at 437 cm-1 and 454 cm-1 were attributed to octahedral tilting and
mode at 491 cm-1 due to Eg like vibrations, whereas Zheng et al. assigned mode at 353
cm-1 to the F2g vibrations in comparison with the spectra of Pb(ScTa)0.5O3 (Zheng et
al., 2004a).
The presence of F2g mode indicates the existence of long range order and the
FWHM of A1g mode gives the degree of long range order (Zheng et al., 2004; Reaney
et al., 2005). It is seen that the intensity of the mode at the vicinity of 353 cm-1
initially decreases slightly with the increase in Sn content and then increases (Figure
3.10). The intensity is minimum in the case of x = 0.25, indicating a lower percentage
of long range ordering, in accordance with IR analysis (Section 3.1.3.2). Therefore
present analysis supports that the mode at 353 cm-1 originates from F2g like vibrations.
Blasse et al., 1974 reported the absence of Eg mode in LMT and ascribed it to the d0
configuration of Ti. It is seen from Figure 3.10 that the intensity of the mode at 491
cm-1 gradually decreases with increase in Sn concentration, inferring that it is
dependent on Sn concentration. Therefore, this mode may not be due to Eg type
vibrations. Indeed, Eg mode was absent in many of the perovskites studied by
76
Liegeois-Duyckaerts and Tarte, 1974. The mode at 139 cm-1 is also assigned to F2g
type vibrations and its presence is observed in all the compositions.
A visual inspection of Figure 3.10 reveals that A1g mode of LMT is broader
with asymmetry compared to LMS. The fit of A1g mode of LMT and LMS to single
Lorentzian gave FWHM of 27.1 cm-1 and 25.1 cm-1 respectively. The broadness of
LMT (FWHM 27.1 cm-1) reveals that it has a lower percentage of LRO than LMS
(FWHM 25.1 cm-1). This also agrees with the earlier work on LMT and LMS by
Macke et al., 1976. But according to the structural studies with Rietveld refinement
(Section 2.2.2 & 2.2.3) LRO of LMS is less compared to LMT.
To analyze this further, asymmetric A1g mode of LMT is fitted to two
merging Lorentzians. Figure 3.11a shows a symmetric A1g mode at 722.2 cm-1
(FWHM 20.6 cm-1) followed by a weak satellite peak at higher frequency (737 cm-1).
The variation of FWHM of A1g mode with Sn concentration is presented in Table 3.6.
Since the FWHM of A1g mode of LMT (20.6 cm-1) is less than that of LMS (25.1 cm1
), it is clear that the LMT has more percentage of LRO as suggested by Rietveld
refinement. Figure 3.11b shows A1g mode for x = 0.125 to x = 0.5 compositions. A
very weak satellite peak is also observed for these compositions (inset of Figure 3.9b),
which is identified as a defect activated mode. The FWHM is maximum at x = 0.25
composition, which confirms the lowest percentage of long range order as suggested
by IR analysis. The FWHM of x = 0.125 and x = 0.375 is high compared to the end
compositions, indicating low percentage of the long range order compared to the end
compositions. These results are in accordance with IR studies.
Further discussion on asymmetry of the A1g mode and LRO in complex
perovskites is presented in the following Section 3.2.4.
77
x = 0.5
Intensity (a.u.)
x = 0.375
x = 0.25
F2g
F2g
A1g
x = 0.125
100
200
300
x=0
400
500
600
700
800
900
-1
Raman shift (cm )
Fig. 3.10 Raman spectra of La(Mg0.5Ti(0.5-x)Snx)O3
- x=0.125
-- x=0.25
.. x=0.375
-- peak 1
Intensity (a.u.)
. . peak 2
600
650
700
750
-1
Raman shift(cm )
800
Intensity (a.u.)
La(MgTi)0.5O3
- .x=0.5
600
850
700
700
750
800
850
800
-1
Raman shift(cm )
Fig. 3. 11a A1g mode of x = 0.0
Fig. 3.11b A1g mode of x = 0.125 to 0.5
Fig. 3.11 A1g mode of La(Mg0.5Ti(0.5-x)Snx)O3
78
Table 3.6 Raman shift of A1g mode and FWHM for La(Mg0.5Ti(0.5-x)Snx)O3
ceramics
Sn content, x
Raman shift (cm-1)
FWHM (cm-1)
0.0
722.2
20.6
0.125
713.8
33.0
0.25
701.2
36.0
0.375
685.1
33.1
0.5
668.6
25.1
By replacing Ti (LMT) with Sn (LMS), Raman shift of A1g mode decreases by
53 cm-1 (Table 3.6). The explanation for this large shift relies on the chemical nature
of tetravalent ion. The Sn4+ has a fully occupied d orbital, which avoids the formation
of π-type Sn-O bonds. On the other hand, Ti4+ has empty d-orbitals, which allows the
overlap of t2g orbital with oxygen p orbital, results in increase of B"-O6 bonding
energy (Duyckaerts, 1974). The low bonding energy in Sn based compound compared
to Ti, lowers the frequency. Similar behavior of large Raman shift in A1g mode was
previously observed in the case of Te6+ (d10 ion) and W6+ (d0 ion ) based complex
perovskites by Ayala et al., 2007.
3.2.3 Raman Spectra of La(1-x)Ndx(MgSn)0.5O3
The Raman spectra recorded for the La(1-x)Ndx(MgSn)0.5O3 solid solution
system are presented in Figure 3.12. The highest wave number mode (above 650 cm1
) is attributed to A1g-like mode corresponding to symmetric breathing of oxygen
octahedra. Modes between 300-370 cm-1 and other three in the range 130-140 cm-1
derive from F2g vibrations (A site cations). According to the structural studies in the
previous chapter, unit cell volume decreases with increasing Nd concentrations.
Decrease in bond lengths should increase force constants and it should result in
increase in frequencies.
79
x = 1.0
Intensity(a.u.)
x = 0.75
x = 0.5
100
A1g
F2g
F2g
x = 0.25
200
300
400
500
600
x = 0.0
700
800
900
-1
Raman shift(cm )
Fig.3.12 Raman spectra of La(1-x)Ndx(MgSn)0.5O3 ceramics (x = 0.0 (bottom) to 1.0
(top))
80
But with increasing Nd concentration (x), the modes do not present the same behavior
for frequencies, widths and intensities (Figure 3.12). This occurs because some bands
are more sensitive to the unit cell volume, other to the tolerance factor and other to B
site cation ordering, which have different effects on the bands. Differently, the F2glike modes in the vicinity of 330 cm-1 (associated to movements of A or O ions) tend
to merge with increasing x.
The Raman shift and FWHM of A1g mode are summarized in Table 3.7. The
Raman shift slightly decreases with the increase in Nd concentration, unlike the large
shift observed in La(Mg0.5Ti(0.5-x)Snx)O3 ceramics (Section 3.2.2). The FWHM
gradually increases with increasing Nd concentration, which indicates a decrease in
the long range order, also in agreement with the results of Rietveld refinement
presented in the previous chapter.
Table 3.7 Raman shift and FWHM of A1g mode for La(1-x)Ndx(MgSn)0.5O3
ceramics
Nd content, x
Raman shift (cm-1)
FWHM (cm-1)
0.0
667.7
24.0
0.25
665.0
24.4
0.5
663.2
24.9
0.75
661.1
25.2
1.0
658.5
25.4
3.2.4 Raman Spectra of Nd(MgTi)0.5O3 and La(MgTi)0.5O3
The Raman spectra recorded for NMT and LMT are presented in Figure 3.13.
By replacing La with Nd, it is observed that some modes split and some modes
merge. The mode observed at the highest wave number (near to 720 cm-1) is assigned
81
to A1g vibrations and the mode at 139 cm-1 and the modes in the vicinity of 350 cm-1
are assigned to F2g modes. F2g modes in the vicinity of 350 cm-1 tend to merge by
replacing La with Nd. This merging is similar to merging of the F2g modes observed
in La(1-x)Ndx(MgSn)0.5O3 system, which indicates the influence of A site cation on F2g
mode.
A visual inspection of Figure 3.13 reveals that there is asymmetry at high
frequency end of the A1g mode. In order to estimate the B-site cation ordering, A1g
mode is fitted to Lorentzian. The fit is not satisfactory with a single Lorentzian
whereas the improved fit is obtained by considering two merging Lorentzians. Figures
3.14a and 3.14b show the fit of A1g vibrations of LMT and NMT. Ubic et al., 2005
reported similar asymmetry in rare earth based perovskite LZT and splitting of the A1g
mode was observed in the case of NZT. Asymmetry or splitting was also observed in
other perovskites structures studied by Blasse and Corsmit, 1973; Blasse and Corsmit,
1974; Ratheesh et al., 2000; Fadini et al., 1978.
The splitting of A1g mode made to conclude that Ba(YNb)0.5O3(BYN) exhibits
low degree of B-site cation ordering (Blasse and Corsmit, 1973; Blasse and Corsmit,
1974). These conclusions were arrived with the lack of structural refinement results
by using neutron diffraction experiments. Due to the low scattering length difference
between Y3+ and Nb5+, X-ray diffraction pattern of Ba(YNb)0.5O3 do not show ½(111)
super lattice reflections corresponding to cation ordering. But ionic size difference
between Y3+ and Nb5 is 0.26 Å (Shannon, 1976). The large ionic size difference
between the B site cations should support high degree long range order. Recently,
Rietveld refinement on neutron diffraction data by Barnes et al., 2006 confirmed that
Ba(YNb)0.5O3 exhibits complete B site cation ordering (100% LRO).
82
F2g
A1g
F2g
Intensity(a.u.)
NMT
LMT
200
400
600
800
-1
Raman shift(cm )
Fig. 3.13 Raman spectra of La(MgTi)0.5O3 and Nd(MgTi)0.5O3.
- - peak 1
. . peak 2
Intensity (a.u.)
Intensity (a.u.)
A1g
650
700
750
800
-- peak 1
. . peak 2
650
-1
Raman shift(cm )
Fig. 3.14a A1g mode of La(MgTi)0.5O3.
A1g
700
750
-1
Raman shift(cm )
800
Fig. 3.14b A1g mode of Nd(MgTi)0.5O3.
Fig. 3.14 Raman A1g mode of La(MgTi)0.5O3 and Nd(MgTi)0.5O3 (two merging
Lorentzians are shown for both the cases).
83
Similarly, Fadini et al., 1978 reported splitting of A1g mode for Ba(YSb)0.5O3.
Later, Rietveld refinement of neutron diffraction data by Alonso et al., 1997 reveal
that it is completely ordered (100% LRO). The asymmetric feature of A1g mode was
also observed in LZT by Ubic et al., 2005, but neutron refinements reveal that it is
completely ordered (Ubic et al., 2006b). Structural studies on LMT and NMT
(Section 2.2.3) reveal that these perovskites exhibit high percentage of long range
order. Thus, asymmetry or splitting of A1g mode cannot be ascertained to low
percentage of LRO.
Raman spectra are sensitive to the local structure details. Gutter et al., 2003
reported that more number of peaks observed in the experimental spectra gives an
evidence for strong deviation of the local structure from the average structure. It was
also reported that distortions in B"O6 gives rise to number of peaks at A1g mode
(Fomichev, 1994). Therefore, to understand the physics behind the appearance of
extra peaks near the A1g mode, local structural studies using Extended X-ray
Absorption Fine Structure (EXAFS) or Pair Distribution Function (PDF) analysis and
phonon dispersion calculations are warranted.
3.3 CONCLUSIONS
Results of four parameter model fit with the IR reflectivity data reveals that
mode strengths corresponding to A-BO6 translational modes are high compared to
mode strengths of B'-O-B" stretching and B-O6 bending modes. Mode strengths of ABO6 modes in La(Mg0.5Ti(0.5-x)Snx)O3 ceramics vary with increase in Sn concentration.
Intermediate compositions of La(Mg0.5Ti(0.5-x)Snx)O3 exhibit high average phonon
damping (low Q factors) and average phonon damping is highest (lowest Q factor) for
x = 0.25 composition. Quality factor of LMS is determined to be highest of all the
84
compositions. Analysis on La(1-x)Ndx(MgSn)0.5O3 ceramics reveal that intrinsic
dielectric constant and Quality factors decrease with increase in Nd concentration.
Raman spectra analysis on La(Mg0.5Ti(0.5-x)Snx)O3 perovskites reveals
that long range order is low for the intermediate compositions and high for the end
compositions, confirming that average phonon damping is high with the low degree of
long range order. The difference in the Raman shift of the A1g mode for LMT and
LMS is determined to be 53 cm-1. The high value of difference in the shift is due to
low B"-O bonding energy with d10 configuration Sn.
Asymmetric feature of the A1g mode is observed in LMT and NMT and
analyses reveal existence of two merging Lorentzians. The asymmetric feature
observed for A1g mode is not due to low percentage of long range order and it may be
due to distortion in BO6 octahedra. In the case of La(1-x)Ndx(MgSn)0.5O3 ceramics,
FWHM of A1g mode is found to decrease with increase in Nd concentration,
confirming decrease in long range order in accordance with the structural and IR
studies.
85
CHAPTER 4
MICROWAVE DIELECTRIC CHARACTERIZATION
OF La(MgTi(1-x)Snx)0.5O3, La(1-x)Ndx(MgSn)0.5O3 AND Nd(MgTi)0.5O3
In this chapter, microwave dielectric characterization of DR samples is
discussed. All the measurements were carried out using vector network analyzer
(N5230A) by identifying the TE011/TE01δ mode. Dense and polished samples were
used for the measurements.
4.1 CHARACTERIZATION TECHNIQUES OF MICROWAVE DIELECTRIC
PROPERTIES
4.1.1 Measurement of Dielectric Constant ( ε ′r )
In order to measure the dielectric constant of dielectric resonator, the method
developed by Courtney, 1970 was employed. In this method, a dielectric resonator
(DR) whose dielectric constant to be measured was kept in between two metallic
circular plates made of copper. Two coupling antennas, which were used to propagate
and receive signals through DR, were kept near to the DR. The positions of the
antennas could be adjusted for optimum coupling. Network analyzer (N5230A)
sweeps the entire frequency range (10 MHz to 40 GHz) and displays all the possible
resonant modes of a DR. The detailed procedure to identify the resonant modes of a
DR was given by Wheles and Kajfez, 1985. By varying the vertical position of the
one of the antennas, the third subscript p can be identified. If p = 1, the field pattern
shows one maximum along the z-direction, and for p = 2, one can observe two
maxima, with a minimum at the position where the mode p = 1 had a maximum. The
86
second feature applicable to identify the modes is given by rotating the receiving loop
by 90o about its axis. The horizontal orientation is preferable for TEonp modes,
whereas vertical orientation is useful for TMonp family of modes. The third indicator
for the mode identification is azimuthal variation of the receiving antenna. The
receiving antenna can be placed at three different locations namely 90o, 135o and 180o
respect to transmitting antenna. The comparison of the signal amplitudes at different
azimuthal angles helps to identify the index m of the resonant mode. The circularly
symmetric mode (m = 0) should display the same signal amplitude for all the three
positions. A mode with index m = 1 should move from a maximum to minimum over
90o of azimuth change and modes with index m = 2 should show the same change
with 45o of change.
To calculate ε ′r , one has to use a particular mode, as mentioned in Chapter 1,
the widely used mode TE011 is used to calculate ε ′r . This mode was identified as the
second lowest mode in the mode spectrum displayed by network analyzer. However
this may be misleading due to the fact that the lowest HEM011 may not visible in
transmission spectrum as it is so leaky. When the movable plate is lowered, if the
stored magnetic energy is larger within the displaced volume than the stored electric
energy for a particular mode, then the resonant frequency will increase, otherwise it
will decrease (Kajfez, 1984). Therefore, when the movable plate is brought down to
touch the surface of DR, TE mode and predominantly TE hybrid modes will increase
in frequency while TM mode and predominantly TM hybrid modes will decrease in
frequency. Among the modes which increase in frequency, the mode which is having
lowest frequency when the top plate touches the DR is TE011 mode. A schematic
diagram of Courtney's method of measuring dielectric constant is shown in Figure 4.1.
The photograph of the set-up connected to network analyzer is shown in Figure 4.2.
87
Fig. 4.1 Schematic diagram of Courtney's method for measurement of the dielectric
constant of DRs.
Fig. 4.2 Photograph of experimental arrangement
to measure dielectric constant of DRs.
88
After identifying the TE011 mode, the value of ε 'r of DR can be calculated
using the frequency of this mode when the metallic plate touches the surface of DR
and the physical dimensions of DR. For TE011 mode, the electric field is always
tangential to the surface of the dielectric material, so that the frequency of TE011 mode
will not be affected by the presence of air gap between the sample and metallic plate
therefore the measurement of ε ′r will be accurate.
The characteristic equation for TEonl mode is
α
J 0 (α )
K (β )
= −β 0
J 1 (α )
K 1 (β )
(4.1)
where J0(α) and J1(α) are the Bessel functions of the first kind of orders zero and one
respectively, and K0(β) and K1(β) are the modified Bessel functions of second kind of
orders zero and one respectively. α and β are given by
2
πD ⎡
⎛ λ 0l ⎞ ⎤
α=
⎟ ⎥
⎢ε ′r − ⎜
λ 0 ⎣⎢
⎝ 2L ⎠ ⎦⎥
2
⎤
πD ⎡⎛ λ 0l ⎞
β=
− 1⎥
⎢⎜
⎟
λ 0 ⎢⎣⎝ 2L ⎠
⎥⎦
1/ 2
(4.2)
1/ 2
(4.3)
where λ0 is the free space wavelength. D is the diameter, L is the length of the DR and
l represents the number of half wave lengths along the axial direction of DR when
shorted at both ends, For TE011 mode l = 1.
For TE011 mode, from the equations for α and β (Equations 4.2 and 4.3), ε ′r can be
calculated as
⎛ c
ε ′r = 1 + ⎜⎜
⎝ πDf 0
⎞ 2
⎟⎟ α 1 + β12
⎠
(
)
(4.4)
where α1 and β1 are the first roots of the characteristic equation (4.1) with m = 0 and
l = 1 and f0 is the resonant frequency. β1 can be calculated from Equation 4.3 using D,
89
L and λ0. The corresponding value of α1 can be obtained from the graphical solution
of the characteristic equation for TEonl modes for the case n = 1. Hakki and Coleman,
1960 reported the graphical solution for TEonl modes for few 'n' values.
4.1.2 Measurement of Quality (Q) Factor
The figure of merit for dielectric resonator is Q factor. It is a measure of loss
or dissipation compared to the energy stored in the resonator. Q factor is given by
Q=
Maximum Energy Stored per cycle
Average Energy Dissipated per cycle
(4.5)
Q=
2π W0 ω 0 W0
=
PT
P
(4.6)
where W0 is the stored energy, P is power dissipation, ω0 is resonant frequency and
period T =
Q=
2π
. To a very good approximation, it can be shown that
ω0
ω0
f
= r
∆ω ∆f r
(4.7)
When the resonator is in actual circuit there arises the loaded quality factor QL
which includes both the internal and external losses. It is equal to
1
1
1
=
+
QL Qe Q0
(4.8)
where Qe is the external quality factor and Q0 is the unloaded quality factor. The
unloaded quality factor is due to internal losses which arise due to the interaction of
microwaves with phonons (dielectric loss), conduction losses and radiation losses. It
is given by
1
1
1
1
=
+
+
Q0 Qc Qd Qr
(4.9)
where Qc is conduction quality factor, Qd is the dielectric quality factor and Qr is the
radiation quality factor.
90
The Q factor was measured by reflection method (Hanson, 1986). In this
method, the sample is kept in a cylindrical metallic cavity having the same aspect
ratio as that of DR itself, but the diameter and height 2 to 3 times larger than DR. Dela
Balle et al., 1981 and Gureyev, 1988 showed that under these conditions, the
conducting losses due to metallic wall of the cavity can be minimum and neglected,
and consequently, the measured Q value is equal to that of DR, Qd. At the same time
radiation losses can also be prevented since the metallic cavity acts as a shield
(Kajfez, 1986).
A cavity made of copper satisfying the above conditions was fabricated with
the adjustable circular plates on the top and bottom. The inner side of the cavity was
well polished and gold plated. The position of the movable plunger (upper plate) can
be adjusted depending on the height of the pellet to meet required conditions. To
couple microwave to DR, a coupler with a single bent monopole oriented along
horizontal direction was inserted into the cavity at its centre. DR was placed at the
geometrical centre of the cavity using a styrofoam support whose dielectric constant is
much lower than that of the DR. the position of the bent monopole can be adjusted
properly to give an optimum coupling of the microwave power. A schematic diagram
of the Q factor cavity is given in Figure. 4.3. Figure 4.4 shows the photograph of the
set up used for measuring Q of DR materials.
The mode used for Q measurement of DR is TE01δ (The index 'δ' is generally
mentioned here because the condition inside the cavity is close to that of the isolated
DR). The identification of TE01δ is relatively easy in this set up, because once the ε ′r
value is known, the resonant frequency of the mode can be calculated for an isolated
condition. In the reflection mode, the network analyzer displays the magnitude and
the phase of S11 parameter (reflection coefficient). Sweeping across the calculated
91
value of the resonant frequency of TE01δ mode, a dip at the resonant frequency can be
seen on the network analyzer screen.
The measurement was performed by sweeping over a narrow frequency range
(20 MHz) around the region of resonant frequency. The single port of the network
analyzer was calibrated by the standard procedure before the measurement of QL, the
loaded Q. The unloaded Q value (Q0) is given by
Q0 = QL (1+p)
(4.10)
where 'p (= Q0 /Qe)' is the coupling coefficient. Depending on the type of coupling,
the coupling coefficient p can take the value > 1 (or) ≤ 1. For over coupled case p > 1,
for critically coupled case p = 1 and for weakly coupled p < 1. To get an accurate
value of Qu, it is preferable to use weakly coupled case (Aitken, 1976). The value of
coupling coefficient can be obtained from the measurement of Voltage Standing
Wave Ratio (VSWR). The coupling coefficient is related to the VSWR in the
following way. For critically coupled case, p = VSWR = 1, for weakly coupled case p
= 1/VSWR and for over coupled case, p = VSWR. However with the knowledge of
VSWR, it is not possible to identify the state of coupling. For a given value of
VSWR, the coupling coefficient p can correspond either to over coupled or weakly
coupled (Liao, 1991). Therefore to identify the state of coupling, Smith Chart was
used. In the reflection mode, the impedance circle will occupy the half circle of the
Smith Chart for critically coupled state and less than half circle for weakly coupled
state (Kajfez, 1994). The Q value was calculated from the above Equation 4.10.
Under these conditions of measurement Qu closely represents Qd, to a close
approximation (Kent, 1988). The resonant frequency of TE01δ mode in the reflection
mode and the corresponding Smith Chart display for the weakly coupled case for the
Nd(MgSn)0.5O3 dielectric resonator are shown in Figure 4.5 and Figure 4.6.
92
Copper cavity
Movable plate
Pellet
Styrofoam
support
Fig. 4.3 The schematic diagram of cavity used to measure quality factor of DRs.
93
Fig. 4.4 Photograph of the experimental arrangement for the measurement of quality
factor (Q) of DRs.
Fig. 4.5 Resonant frequency display of TE01δ mode for Nd(MgSn)0.5O3.
Fig. 4.6 Smith chart display corresponding to weakly coupled state of TE01δ mode for
Nd(MgSn)0.5O3. (corresponding to weakly coupled state)
94
4.1.3 Measurement of Temperature Coefficient of Resonant Frequency (τf)
The origin of τf is related to the linear thermal expansion coefficient (ν), which
effects the resonator’s dimensions and dielectric constant. The relationship can be
derived from universal fact that f.λ = c. As the wavelength of the standing wave
approximates to the diameter (D) of the resonator (λd ≈ D) in the simplest fundamental
mode, the frequency of the standing wave is:
f0 =
c
c
c
=
≈
λ 0 λ d ε ′r D ε ′r
(4.11)
If the temperature changes, then the resonant frequency f0 will change because of ε ′r
and D. Differentiating the above Equation 4.11 with respect to temperature gives:
1 ∂ f0
1 ∂D
1 ∂ ε ′r
=−
−
f0 ∂ T
D ∂ T 2ε ′r ∂ T
where
(4.12)
1 ∂ f0
is the temperature coefficient of resonant frequency, τf
f 0 ∂T
1 ∂D
is the linear expansion coefficient, ν and
D ∂T
1 ∂ ε ′r
is the temperature coefficient of dielectric constant, τε
ε ′r ∂ T
Substituting these equations into the above equation, the relationship can be expressed
more compactly as
τ ⎞
⎛
τ f = −⎜ ν + ε ⎟
2⎠
⎝
(4.13)
Measurement of τf requires a test cavity with the specifications given in the Section
4.1.2, to minimize the influence of metal wall on the measurement. To further reduce
the influence of thermal expansion of metallic cavity on τf, a cavity made of invar was
used which is having a linear thermal expansion coefficient of only 0.8 ppm/0C. τf
95
was calculated using TE01δ mode from 300C to 700C. The cavity was rigidly fixed to
a PID controlled hot plate (Model T-300 Step Electronics, USA). The τf value was
calculated from the following expression,
τf =
1 ∆f
×
f 0 ∆T
(4.14)
where f0 is the resonant frequency at 300C. ∆f is the change in the resonant frequency.
∆T is change in temperature. Figure 4.7 shows the schematic diagram of test holder
used for τf measurement and Figure 4.8 shows the photograph of the complete set-up.
The network analyzer has the stability of 1 KHz in GHz range and hence it can
measure τf value of less than 1 ppm/0C.
4.2 RESULTS AND DISCUSSION
All the measurements were carried out in the frequency range of 9-11 GHz.
All the techniques were verified on DR samples supplied by Trans Tech, USA and
found to agree with the values reported by them. Error in the measurement of
dielectric constant, quality factor and temperature coefficient of resonant frequency is
1%, 3% and 3% respectively.
4.2.1 Microwave Dielectric Properties of La(Mg0.5Ti(0.5-x)Snx)O3
Relative density, dielectric constant and microwave quality factors of
La(Mg0.5Ti(0.5-x)Snx)O3 system are presented in Table 4.1. Dielectric constant
decreases with increase in Sn concentration owing to the low ionic polarizability of
Sn compared to Ti. Ionic polarizability of Sn is 2.83 Å3 and that of Ti is 2.93 Å3
(Shannon, 1993). Quality factors are low for intermediate compositions and lowest
for x = 0.25 composition. Quality factors obtained are less than intrinsic values, but
the trend of microwave Q.f variation is similar to the variation of intrinsic Q.f values
for x = 0.25 composition. Quality factors obtained are less than intrinsic values, but
96
Invar cavity
Coaxial
Tuning
Dielectric
Low
loss
and low
ε′r
Fig. 4.7 The schematic diagram for measurement temperature coefficient of resonant
frequency.
Fig. 4.8 Photograph of the experimental arrangement measure temperature
coefficient of resonant frequency.
97
the trend of microwave Q.f variation is similar to the variation of intrinsic Q.f values
(discussed in Section 3.1.3.2). Low microwave quality factors are due to the
contribution of extrinsic losses. Q.f values of LMT were reported to be 48,000 GHz
by Kipkoech et al., 2005; 68,000 GHz by Kim et al., 2005 and 74,550 GHz by Seabra
and Ferriera, 2002. In the later case, powders were prepared by pechini method
(chemical route) and in the former cases ball milling was used for mixing, which
indicate that Q.f values depend upon method of preparation. In this study, quality
factor of LMT obtained is 55,000 GHz. Minor impurity phase of LMT may also be
responsible for lower Q compared to the values reported by Seabra and Ferriera, 2002
and Kim et al., 2005.
The variation of temperature coefficient of resonant frequency (τf) and
tolerance factor with Sn concentration is shown in Figure 4.9. Temperature coefficient
of resonant frequency is negative due to the presence of in-phase and anti-phase tilting
of octahedra and decreases with decrease in tolerance factor (Reaney et al., 2005).
The decrease in τf with decrease in tolerance factor is also reported for other alkaline
earth perovskites (Setter, 1993). This can be explained by an argument that in the
tilted region, increase in thermal energy is completely absorbed to recover the tilting
(Kim et al., 2005).
Table 4.1 Relative density (d), dielectric parameters determined at microwave
frequencies for the La(Mg0.5Ti(0.5-x)Snx)O3 ceramics
Sn concentration
(x)
0.0
Relative density
(%)
97.2
0.125
ε′r
28.4
Q.f
(GHz)
55,000
τf
(ppm/◦C)
-68
97.7
26.9
50,000
-74
0.25
97.6
24.4
46,000
-76
0.375
97.4
22.2
49,000
-80
0.5
97.8
19.7
63,000
-84
98
-64
0.950
0.945
-68
o
τf (ppm/ C)
0.935
-76
0.930
-80
0.925
toleance factor, t
0.940
-72
0.920
-84
0.915
-88
0.910
0.0
0.1
0.2
0.3
0.4
0.5
Sn content, x
Fig. 4.9 Tolerance factor (squares) and temperature coefficient of resonant frequency
(circles) of La(Mg0.5Ti(0.5-x)Snx)O3 ceramics as a function of Sn concentration, x.
(linear fit of the data are shown by continuous line).
4.2.2 Microwave
Nd(MgTi)0.5O3
Dielectric
Properties
of
La(1-x)Ndx(MgSn)0.5O3
and
Microwave dielectric properties, tolerance factor (t) and relative density of
La(1-x)Ndx(MgSn)0.5O3
are summarized in Table 4.2. The microwave dielectric
constant of NMT is 26.1, Q.f is 50,000 GHz, τf is -49 ppm/°C and relative density is
97%. The dielectric constant decreases slightly with the increase in Nd concentration
for La(1-x)Ndx(MgSn)0.5O3 ceramics. The intrinsic dielectric values presented in
Section 3.1.3.3 also show the same trend and intrinsic dielectric constants are slightly
lower than microwave dielectric constants. Dielectric constant of NMT (26.1) is less
than the dielectric constant of LMT (28.4). Decrease in dielectric constant with
increase in Nd concentration is observed for both the Sn based La(1-x)Ndx(MgSn)0.5O3
and the Ti based LMT and NMT.
Shannon, 1993 reported that polarizability of La is equal to 6.07 Å3 and that of
Nd is 5.01 Å3. But later, polarizability of the La was revised to 4.82 Å3 by Vineis,
99
1996. The revised polarizability of La is less than that of Nd. But trend of dielectric
constant variation with Nd substitution indicates that Nd polarizability is less than that
of La. To examine this further, dielectric constants are calculated using the reported
polarizability values. Delectric constants of DRs can be obtained by substituting
polarizability values in Classius-Mossotti equation. Clausius-Mossotti equation is
given as:
ε ′r =
3Vm + 8π α m
3Vm − 4π α m
(4.15)
where Vm and αm represent molar volume and polarizability respectively.
Structural studies (discussed in Chapter 2) revealed that symmetry of all the
compounds is monoclinic P 21 / n . Number of the formula units for P 21 / n symmetry
is 4. Therefore, to obtain polarizability of a compound, total polarizability of ions
should be multiplied with 4. For example, α(La(MgSn)1/2O3) = 4(α(La) +
0.5α(Mg+Sn) + 3α(O)).
By substituting α(La) = 6.07 Å3 in Clausius-Mossotti equation, dielectric
constants obtained for LMT and LMS are 123.2 and 33.7 respectively. Similarly, by
substituting α(La) = 4.82 Å3 dielectric constants obtained for LMT and LMS are 25.2
and 16.2 respectively. These results reveal that the value α(La) = 6.07 Å3 over
estimates dielectric constants and α(La) = 4.82 Å3 underestimates dielectric constants.
In order to recalculate the ionic polarizability of La3+, polarizability (α) of lanthanide
ions (other than La) was linearly fitted to r3 (ionic radius cube) values. For equal
valence ions polarizabiltiy is roughly proportional to r3 (Roberts, 1951). Radius (r)
values of lanthanides (coordination no 8) were obtained from tables reported by
Shannon, 1976 and Polarizabilities of ions were obtained from Shannon, 1993, except
for Ce3+ revised polarizability of 5.47 Å3 was used as reported in Paschoal, 2005.
100
Figure 4.10 shows the linear fit of α versus r3. By extrapolating the fitted straight
line, α(La3+) obtained is 5.71 Å3.
Dielectric constants of La(1-x)Ndx(MgSn)0.5O3
system obtained by substituting α(La3+) = 5.71 Å3 are presented in Figure 4.11. It is
found that Classius-Mossotti dielectric constant of LMS is 26.2 and dielectric constant
gradually decreases with increase in Nd concentration (dielectric constant is 21.6 for
NMS). Dielectric constants of LMT and NMT obtained by substituting α(La3+) = 5.71
Å3 in Classius-Mossotti equation are 59.6 and 40.2 respectively. The variation of
microwave dielectric constants is in qualitative aggrement with Classius-Mossotti
dielectric constants. But microwave dielectric constants are less than the ClassiusMossotti dielectric constants. The low microwave dielectric constants could partially
result from some micro structural defects, although the most probable situation is an
actual reduction of the ionic polarizabilities due to partial A–O covalent bonds (A =
La or Nd), similarly to ReTiTaO6 compounds (Paschoal, 2005).
Table 4.2 Relative density, tolerance factor (t) and dielectric parameters
determined at microwave frequencies for the La(1-x)Ndx(MgSn)0.5O3 system
Rel.
den. (%)
98.4
t
ε′r
0.927
19.8
Q.f
(GHz)
75,000
0.25
97.8
0.920
19.5
68,000
-75
0.5
97.6
0.912
19.4
64,000
-66
0.75
98.2
0.905
19.2
70,000
-60
1.0
98.5
0.897
19.1
68,000
-53
Nd concentration
(x)
0.0
101
τf
(ppm/◦C)
-82
6.0
27
26
dielectric constant (εr')
o3
polarizability, α(A )
5.5
5.0
y = 4.94x + 0.37
4.5
4.0
α
3.5
linear fit
0.6
0.7
0.8
0.9
3
25
24
23
22
Microwave
Clasius-Mosotti
21
20
19
1.0
1.1
1.0
o3
0.8
0.6
0.4
0.2
0.0
r (A )
La concentration, (1-x)
Fig. 4.10 Polarizabilties of lanthanide
ions versus radius cube. (polarizabilities
are shown with shaded squares and
continuous) represents linear fit.
Fig. 4.11 Classius-Mossotti and
microwave dielectric constants of
La(1-x)Ndx(MgSn)0.5O3 system.
The microwave quality factors obtained for La(1-x)Ndx(MgSn)0.5O3 are less
than the intrinsic values (presented in Section 3.1.3.3) due to extrinsic losses
contributed by structural defects. According to the trend predicted by intrinsic Q.f
values, microwave Q.f values of x = 0.75 and x = 1.0 composition are less than x =
0.0 composition. But microwave Q.f values obtained for x = 0.25 and 0.5
compositions are less than x = 0.75 and x = 1.0 compositions due to the lower
percentage of relative density of these samples compared to other compositions.
The temperature coefficient of resonant frequency of La(1-x)Ndx(MgSn)0.5O3
becomes less negative with the increase in Nd concentration (decrease in tolerance
factor). Temperature coefficient of resonant frequency of NMT (-49 ppm/°C) is less
negative than of LMT (-68 ppm/°C). But tolerance factor of NMT (0.916) is less than
that of LMT (0.946). These results show that decrease in tolerance factor do not
decrease temperature coefficient of resonant frequency for La(1-x)Ndx(MgSn)0.5O3,
LMT and NMT and suggest that temperature coefficient of resonant frequency
correlate with composition and ion type at A site. These results are in disagreement
102
with result that τf decreases with decrease in tolerance factor (discussed in Section
4.2.1).
However, there were very few attempts to relate bond valences of B site ions
(Lufaso, 2004), covalency of ions with τf (Paschol, 2005) and correlation of phonon
properties with τf by first principle calculations (Cockayne, 2001). Detailed studies,
viz. first principle calculations and bond valences obtained from accurate bond
lengths by neutron diffraction studies are required to understand the variation of τf.
4.3 CONCLUSIONS
Microwave
dielectric
constant
of
La(Mg0.5Ti(0.5-x)Snx)O3
decreases with increase in Sn concentration. Microwave Q.f values are low for
intermediate compositions of La(Mg0.5Ti(0.5-x)Snx)O3 and lowest for x = 0.25
composition and temperature coefficient of resonant frequency is found to decrease
with decrease in tolerance factor. Microwave dielectric constant of La(1x)Ndx(MgSn)0.5O3
decreases with the increase in Nd concentration. Microwave
dielectric constant obtained for NMT (26.1) is less than the dielectric constant of
LMT (28.4). Ionic polarizability of 6.07 Å3 for La ion overestimates dielectric
constants whereas polarizability of 4.82 Å3 underestimates dielectric constants. By
performing the liner fit of α to the r3, polarizability obtained for La is 5.71 Å3.
Microwave Q.f values of x = 0.25 and 0.5 are lower due to lesser percentage of
relative density compared to other compositions (x = 0.0, 0.75 and 1.0). Temperature
coefficient of resonant frequency of La(1-x)Ndx(MgSn)0.5O3 becomes less negative with
increase in Nd concentration (decrease in tolerance factor). Temperature coefficient of
resonant frequency of NMT (t = 0.916) is less negative than of LMT (t = 0.946). The
variation of temperature coefficient of resonant frequency suggests that it correlates
with the Nd concentration.
103
CHAPTER 5
PHOTONIC BAND GAP STUDIES ON ONE DIMENSIONAL
STRUCTURES
This chapter deals with photonic crystals (photonic band gap structures), one
of the possible applications of dielectric resonators. The analysis uses transfer matrix
method and experimental measurement of photonic band gaps in the frequency range
of 10 to 20 GHz. Transfer matrix relates the electric and magnetic fields of incident
plane electromagnetic wave and transmitted plane wave of the photonic crystal.
Theoretically photonic band gaps, density of modes and transmittance of the photonic
crystals are computed. Defects are created in these structures by removing the centre
dielectric material. Since it is expensive and not easy to fabricate sheets required for
one dimensional photonic crystal using dielectric resonator materials, computations
are only carried out for the low loss and high dielectric constant structure ( ε ′r = 20, ε ′r′
= 0.0025). These values correspond to the dielectric constant and loss of a typical
dielectric resonator. Theoretical results on photonic crystals constructed using lossy
dielectrics (glass and ebonite) are verified by performing transmission measurements.
All the photonic crystals constructed are nine-period structures. Commercially
available glass sheets and ebonite sheets are used to fabricate the photonic crystals.
Double periodic structure constructed by using glass sheets and ebonite sheets is also
analyzed.
104
5.1 COMPUTATION USING TRANSFER MATRIX METHOD
One-dimensional photonic crystal is a periodic stack of different materials of
alternating dielectric constants (εr1 and εr2) and is defined as
⎧ ε r1 , 0 < x < h 1
ε r (x) = ⎨
⎩ε r2 , h 1 < x < h 2
with ε r ( x + na ) = ε r ( x ) , where a = h1+ h2 is the lattice constant of the photonic
crystal, and n is an integer. Schematic diagram of one-dimensional photonic crystal is
shown in the Figure 5.1. It is periodic along z-axis and xy is the plane of incidence of
electromagnetic wave.
ε r1, h 1
x
ε r2, h 2
z
y
a
Fig 5.1 Schematics of the one-dimensional photonic band gap structure.
5.1.1 Matrix for Photonic crystal (Transfer Matrix)
The electric and magnetic fields of a n-layer photonic crystal are related by
E
⎡E 0 ⎤
n⎡ n ⎤
⎢H ⎥ = M ⎢H ⎥
⎣ n⎦
⎣ 0 ⎦
(5.1)
where M is a transfer matrix for one unit cell. E0 and H0 are incident electric and
magnetic fields. En and Hn are transmitted electric and magnetic fields.
The unit cell of photonic crystal consists of two dielectric materials with dielectric
constants εr1 and εr2. The transfer matrix for a unit cell was derived by Borns and
105
Wolf, 1980. Assuming the plane wave incident normally, i.e. with incident angle
equal to zero the transfer matrix is given by
⎛ m11 m12 ⎞
⎟⎟ =
M = ⎜⎜
⎝ m21 m22 ⎠
Y
⎛
cos γ1h1 cos γ2h 2 − 1 sin γ1h1sin γ2h 2
⎜
Y2
⎜
⎜ − i(Y sin γ h cos γ h + Y cosγ h sinγ h )
2
2 2
1 1
1
2 2
1 1
⎝
where γ1 =
⎛ 1
⎞⎞
1
− i⎜⎜ cos γ2h 2 sin γ1h1 + sin γ2h 2cos γ1h1 ⎟⎟ ⎟
Y2
⎠⎟
⎝ Y1
⎟
Y
cos γ1h1 cos γ2h 2 − 2 sin γ1h1sin γ2h 2 ⎟⎟
Y1
⎠
2π
2π
ε r 1 h1 , γ 2 =
ε r 2 h 2 , Y1 =
λ
λ
ε 0 ε r1 and Y2 =
(5.2)
ε 0 ε r2
5.1.2 Band Structure of Photonic Crystal
According to Bloch’s theorem, the electric and magnetic fields at the neighboring unit
cells of an infinitely extended photonic crystal are related through
⎡E m ⎤
⎡E m −1 ⎤
⎢H ⎥ = exp(i Ka) ⎢H ⎥
⎣ m⎦
⎣ m −1 ⎦
(5.3)
where K is Bloch wave number and 'a' is the lattice constant.
Using Equations 5.1 and 5.3, it follows that
⎡E m ⎤
⎡E m ⎤
M ⎢ ⎥ = exp(− i Ka) ⎢ ⎥
⎣H m ⎦
⎣H m ⎦
(5.4)
Above Equation 5.4 is an eigen value problem satisfied by electric and magnetic
fields of Bloch wave. The eigen values of the transfer matrix M are given by
exp(± i Ka) =
1
⎡1
⎤
(m11 + m 22 ) ± ⎢ (m11 + m 22 ) 2 − 1⎥
2
⎣2
⎦
1/2
(5.5)
Using the Equation 5.5, dispersion relation for one dimensional photonic crystal is
written as
K(f ) =
1
⎡1
⎤
cos −1 ⎢ (m11 + m 22 )⎥
a
⎣2
⎦
(5.6)
106
5.1.3 Transmission Coefficient and Transmittance
The transmission coefficient (t) and transmittance (T) are given by following
Equations
t=
T=
2γ 1
(m11 + m12 γ 2 ) γ1 + (m 21 + m 22 γ 2 )
γ2
t
γ1
(5.7)
2
(5.8)
5.1.4 Density of Modes
Equation for density of modes can be derived using the transmission
coefficient (Bendickson et al., 1996) of the photonic crystal
t = x + iy
(5.9)
where x and y are real and imaginary parts of transmission coefficient. If φ is the total
phase accumulated through the photonic crystal, then
tanφ =
y
x
(5.10)
But φ = kd
(5.11)
where k is the effective wave number and d is the physical thickness of the photonic
crystal.
Therefore,
tan(kd) =
y
x
(5.12)
Differentiating 5.12 and simplifying, equation for density of modes is written as:
ρ(ω) =
dk 1 y ′x − x ′y
=
df d x 2 + y 2
(5.13)
where y' and x' are derivatives of y and x with respect to frequency (f).
5.1.5 Method of Calculation
Band structures, transmittance and density of modes of photonic crystal were
computed using codes written in Matlab and C language. Photonic band gap depends
107
on the dielectric constant of the materials used, thickness of the dielectrics and lattice
constant of the photonic crystal. Typical variation of band gap with increase in
dielectric constant is shown in Figure 5.2. Unshaded regions are the regions in which
Bloch wave vector becomes imaginary (m11+m22 becomes greater than two in
equation 5.6) representing attenuated plane wave i.e. photonic band gap. Black shaded
regions are the regions in which Bloch wave vector is real. It is seen that the width of
the band gap increases with increase in dielectric constant and band gap shifts to
lower frequencies.
The variation of the band gap with increase in spacing (h2) between the
dielectrics ( ε ′r = 6.1, thickness (h1) = 0.31 cm) is shown in Figure 5.3 for the
frequency range of 1-20 GHz. Initially only one band gap opens up for the frequency
range (1-20 GHz) and at 0.8 cm spacing the second band gap opens up and
subsequent gaps open up at 1.6 and 2.4 cm spacing. These band gaps shift towards the
lower frequency region as the spacing increases. The band gap width increases
gradually and then decreases as the spacing increases between the sheets. It infers that
there exists particular geometry for which width of the band gap will be highest.
5.2 DETAILS OF THE EXPERIMENT
Schematic diagram to measure the transmittance of a photonic crystal is shown in
Figure 5.4 and photograph of the experiment is shown in Figure 5.5. A microwave
vector network analyzer (N 5230A) was used to get the transmitted power spectrum.
Two horn antennas (P-band (12-18 GHz)) were used to transmit and detect the
microwaves. Photonic crystals were constructed either using glass sheets ( ε ′r = 6.8,
ε ′r′ = 0.1) with thickness (h1) 0.3 cm or using ebonite sheets ( ε ′r = 4, ε ′r′ = 0.25) with
thickness 0.27 cm. Length of the sheets was 47 cm and the breadth was 28 cm. These
108
20
18
Frequency (GHz)
16
14
12
10
8
6
4
2
1
2
3
4
5
6
7
8
9
10
'
dielectric constant (εr )
Fig 5.2 Variation of band gap width with increase in dielectric constant (unshaded
region represents the band gap)
20
18
Frequency (GHz)
16
14
12
10
8
6
4
2
0.0
0.5
1.0
1.5
2.0
2.5
3.0
Spacing between the sheets (cm)
Fig 5.3 Variation of band gap with increase in spacing between the glass sheets
(unshaded region represents the band gap).
109
sheets were periodically arranged by separating them with a fixed distance using
styrofoam ( ε ′r = 1.03) supports.
Initially periodic structure was kept in between the antennas. The antennas
were adjusted such that periodic structure was in their field of view. The distance
from centre of lattice to the antennas was kept equal. Suitable reference level and
scale was selected in the network analyzer. Once these adjustments were made, the
periodic structure was removed. Without the structure, the transmittance spectrum
was saved and used as the reference for normalizing the spectrum. Once the
normalization procedure was followed, the periodic structure was kept again between
antenna and the transmittance was recorded.
The real and imaginary part of the dielectric constant of glass and ebonite used in
these experiments were measured by using cavity perturbation technique (Murthy and
Raman, 1989). In this technique, to determine the dielectric constant, the sample
under consideration is inserted in a cavity, which resonates at a particular frequency.
Measurements were carried out with a cavity resonating at 9 GHz. For measuring the
dielectric constant, the sample should be inserted at the position where the electric
field is maximum. If ε ′r and ε ′r′ are the real and imaginary parts of the dielectric
constant and if f0 is the resonant frequency without the sample and if f1 is the resonant
frequency of the cavity with the insertion of the sample and if Vc and Vs are the
volumes of the cavity and that of the sample respectively, then
ε ′r = 1 +
ε ′r′ =
Vc
4Vs
⎞
⎛ f 02
⎜ 2 − 1⎟
⎟
⎜f
⎠
⎝ 1
(5.14)
f 02 Vc ⎛ 1
1 ⎞
⎜⎜
⎟
−
2
f 1 4Vs ⎝ Q1 Q 0 ⎟⎠
(5.15)
110
N 5230A
Fig. 5.4 Experimental arrangement to observe the transmittance of the one
dimensional photonic crystal.
Fig. 5.5 Photograph of transmittance measurement of one dimensional photonic
crystal.
111
where Q0 and Q1 are the loaded quality factors of the cavity without the sample and
with sample respectively.
5.3 ANALYSIS OF ONE DIMENSIONAL PHOTONIC CRYSTALS
5.3.1 Glass and Ebonite Structures
In this section, one-dimensional photonic crystals are analyzed for nine period
glass and ebonite structures. These structures are analyzed by calculating band
structures followed by density of modes with and without defect for finite period
structures. Transmittances through the structures are measured to confirm the
predictions.
Initially, the variation of gap width with the air medium thickness (h2) is
computed for glass and ebonite photonic crystals (Figure 5.6) in the frequency range
of 10 to 20 GHz using Equation 5.6. It is seen from the graph that the band gap is
maximum with h2 = 0.3 cm (a = 0.6 cm) for the glass photonic crystal and with h2 =
0.4 cm (a = 0.67 cm) for the ebonite photonic crystal. Figure 5.7 shows band structure
for the periodic structure made of glass with a lattice constant, ‘a’ of 0.6 cm. Figure
5.8 shows the band structure for the periodic structure made of ebonite with a lattice
constant, ‘a’ of 0.67 cm. For convenience, only first three band gaps are shown for
both the structures. Band structure calculation reveals that first band gap for glass
photonic crystal exists between 10.4 and 16.5 GHz and for ebonite photonic crystal
between 12.6 and 19.3 GHz.
The density of modes for the first band gap of nine-unit cell glass and ebonite
structures are shown in Figures 5.10 and 5.11 (solid line). Density of modes confirms
the band gaps, as density of modes approaches zero in the band gap regions
112
Thicknes of the air medium (h2) for ebonite structure (c.m.)
0.4
65
0.5
0.6
0.2
0.3
Width of the gap (a.u.)
60
55
50
45
40
35
0.1
Thickness of the air medium (h2) for glass structure (c.m.)
Fig. 5.6 Variation of the gap width for glass ( ε ′r1 = 6.8, h1 = 0.3 cm) and ebonite ( ε ′r1 =
4, h1 = 0.27 cm) structures with thickness of air medium (h2) in the range of 10 to 20
GHz.
60
50
45
3
40
30
3
40
Frequency (GHz)
Frequency (GHz)
50
2
20
1
35
30
25
2
20
15
1
10
10
5
0.0
0.5
1.0
1.5
2.0
2.5
0.0
3.0
0.5
1.0
1.5
2.0
2.5
3.0
Ka
Ka
Fig. 5.8 Band structure for
photonic crystal with ε ′r1 = 4, h1 =
0.27 cm, ε ′r2 = 1 and h2 = 0.4 cm.
photonic crystal with ε ′r1 = 6.8, h1
= 0.3 cm, ε ′r2 = 1 and h2 = 0.3 cm.
113
predicted by band structures. The transmittance spectrum computed (solid line) and
experimentally measured (dotted line) for both nine-unit cell lossy glass and ebonite
structures are presented in Figures. 5.12 and 5.13. The experimental results confirm
the prediction of band gap. The nature of variation of the spectra also matches fairly
well with the computed transmittance.
In order to study the effect of defects, one dielectric sheet at the center of the
periodic structure is removed (schematically shown in Figure 5.9). The density of
modes computed for both the glass and ebonite defect structures are shown in Figures.
5.10 and 5.11 (dotted line), which predicts the existence of defect mode at 14.1 GHz
for glass structure and at 14.6 GHz for ebonite structure. The transmittance spectrum
computed (dotted line) and experimentally measured (solid line) for both of the above
defect structures are presented in Figures. 5.14 and 5.15. The defect mode for the
glass and ebonite structures using computed transmittance confirms the predictions
with quality factors of defect modes 108 and 32 respectively.
Fig. 5.9 Schematics of the one-dimensional photonic crystal with defect (defect is
created at the centre by removing a dielectric sheet).
It may be observed that defects resulted to widening of the band gap regions in
transmittance spectra, which is also seen from the profile of density of modes with
defect. Experimental defect modes for glass and ebonite structures are observed at
13.5 GHz and 14.3 GHz with quality factors 70 and 26 respectively.
114
The
experimental defect frequencies are with in the error of 5%. The experimentally
observed quality factors are less than the computed quality factors. This is due to
leakage loss of photonic crystal.
The total quality factor at the defect mode can be broken into two different quality
factors due to individual loss factors:
1
Q EXPT
=
1
Q COMP
+
1
(5.16)
Q LEAK
where QCOMP is the theoretically computed quality factor and QLEAK is due to the
leakage of electromagnetic field through the band gap structure. QLEAK is computed
for both the glass and ebonite structures and is given in the Table 5.1. In both the
cases, QLEAK is higher than the QEXPT. In order to obtain predicted Q values
experimentally, QLEAK has to be very high. QLEAK can be increased by increasing the
breadth and width of the sheets used in the experiment. According to the calculations,
dielectrics extend infinitely in x and y directions and the incident wave are plane
waves. But in experiment dielectric sheets are finite sheets and waves are not
perfectly plane waves.
Table 5.1 Quality factor (Q) values of glass and ebonite photonic crystal
Photonic crystal
QCOMPUTED
QEXPERIMENTAL
QLEAK
Glass
108
70
199
Ebonite
32
26
139
115
25
40
- without defect
.. with defect
20
- without defect
.. with defect
DOM (a.u.)
DOM (a.u.)
30
20
10
0
15
10
5
8
10
12
14
16
18
0
10
20
12
14
Frequency (GHz)
0
0
-10
-10
-20
-30
-40
10
- measured
.. computed
12
14
16
18
-30
-40
-50
10
20
12
0
-5
-10
-10
Transmittance (dB)
Transmittance (dB)
0
-15
-20
-25
-30
- measured
.. computed
12
14
16
16
18
20
Fig. 5.13 Computed and measured
transmittance through ebonite
photonic crystal without defect.
-5
-45
10
14
Frequency (GHz)
transmittance through glass photonic
crystal without defect.
-40
20
- measured
.. computed
-20
Frequency (GHz)
-35
18
Fig. 5.11 Density of modes for the
nine period ebonite photonic crystal
without and with defect.
Transmittance (dB)
Transmittance (dB)
Fig. 5.10 Density of modes for the
nine period glass photonic crystal
without and with defect.
-50
16
Frequency (GHz)
18
- measured
.. computed
-15
-20
-25
-30
-35
-40
10
20
Frequency (GHz)
12
14
16
18
20
Frequency (GHz)
transmittance through glass photonic
crystal with defect.
transmittance through ebonite
photonic crystal with defect.
116
5.3.2 Double Periodic Structure
The double periodic photonic crystal is constructed using glass sheets
( ε ′r1 = 6.8, ε ′r1′ = 0.1) with thickness (h1) 0.3 cm, and ebonite sheets ( ε ′r1 = 4, ε ′r1′ =
0.25) with thickness (h3) 0.27 cm of length 47 cm and breadth 28 cm. The structure
constructed
consists
of
total
nine
dielectrics
and
is
represented
as
{G → E → G → E → G → E → G → E → G} where 'G' represents glass and 'E'
represents ebonite. The separation distance between G to E (h2) is 0.4 cm and E to G
(h4) is 0.3 cm. Schematic diagram is shown in Figure 5.16. The unshaded region in the
figure represents the air gap. The structure is periodic in x direction and homogeneous
in y and z directions.
Figure 5.17 shows the band structure for the glass-ebonite photonic crystal. The first
band gap appears in between 6.0 to 7.7 GHz and second band gap in between 11.6 to
17.4 GHz. Densities of modes for the structure with and without defect are computed
(Figure 5.18). Density of modes approaches zero for the structure without defect
(solid line) in between 11.6 to 17.4 GHz, which confirms the existence of band gap.
By removing the center glass sheet, defect mode appears at 14.9 GHz by increase in
density of modes and the density of modes corresponding to band gap region (dotted
line) widens. The band gap is verified with measured and computed transmission
spectra (Figure 5.19). Figure 5.20 shows the transmittance of measured and computed
glass-ebonite photonic crystal with defect. The experimental defect mode is observed
at 14.8 GHz with a Q factor of 27 and computed defect mode is observed at 14.9 GHz
with Q factor of 33. The transmittance data also confirms the widening of the gap by
introducing the defect.
117
h1 h2 h3 h4
x
z
y
Fig. 5.16 Schematics of double periodic one-dimensional photonic crystal (glass
shaded in black and ebonite shaded in grey).
50
25
- without defect
.. with defect
6
20
40
30
DOM (a.u.)
Frequency (GHz)
5
4
3
20
15
10
2
5
10
1
0.0
0.5
1.0
1.5
2.0
2.5
0
10
3.0
12
Ka
double periodic photonic crystal.
0
-5
-10
-20
-25
-30
-35
-40
12
14
20
-20
-25
-30
-35
- measured
.. computed
-40
-45
10
18
-15
Trasmittance (dB)
Transmittance (dB)
- measured
.. computed
-15
16
Fig. 5.18 Density of modes
for double periodic photonic
crystal.
-5
-10
14
Frequency (GHz)
16
18
-45
10
20
Frequency (GHz)
12
14
16
18
Frequency (GHz)
Fig. 5.20 Transmittance through
double periodic photonic crystal
with defect.
double periodic photonic crystal.
118
20
Band gap of double periodic structure can be tuned more conveniently as the
gap depends on separation distance between glass to ebonite (h2) and ebonite to glass
(h4) for materials with thickness h1 and h3.
5.3.3 Low Loss and High Dielectric Constant Photonic Crystal
Photonic crystal with low loss dielectric is analyzed in this section. Band
structure calculations on one-dimensional structure with nine dielectric sheets ( ε ′r =
20, ε ′r′ = 0.0025) of thickness (h1) 0.4 cm and separation distance between the
dielectrics (h2) 0.2 cm (a = 0.6 cm) result photonic band in the frequency range of
12.3 to 16 GHz. Band structure showing the band gap is presented in Figure 5.22.
Figure 5.23 shows the computed transmittance through the structure. Defect mode is
created in this photonic crystal by removing the centre dielectric. Figure 5.24 shows
the computed transmittance of the photonic crystal with defect. Defect mode is
observed at 14.26 GHz with a quality factor of 1000.
Quality factor of the defect mode obtained with the low loss photonic crystal
(1000) is high compared to quality factors of defect mode with glass photonic crystal
(108) and ebonite photonic crystal (32). Therefore, photonic crystals constructed with
low loss structures exhibit high quality factor defect modes. Attenuation level in the
band gap region is high for the low loss photonic crystal (-70 dB) compared to glass
photonic crystal (-50 dB) and ebonite photonic crystal (-45 dB). High attenuation
level is due to the high dielectric constant of these materials. Transmission out side
the band gap region also attenuates due to the dielectric loss in photonic crystals with
lossy materials. Modeling periodic structures constructed with low loss (dielectric
resonator) materials in shielded environment is required to study their potentiality for
microwave applications.
119
18
Frequency (GHz)
16
Band gap region
14
12
10
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
Ka
Fig. 5.21 Band structure for low
loss and high dielectric constant
photonic crystal.
10
0
0
-10
12 to 16.3 GHz
-20
Transmittance (dB)
Transmittance (dB)
-10
-30
-40
-50
-60
Q = 1000
-20
-30
-40
-50
-60
-70
-80
-70
10
12
14
16
18
10
Frequency(GHz)
12
14
16
18
Frequency (GHz)
Fig. 5.22 Tranmsittance through
low loss and high dielectric
constant photonic crystal.
low loss and high dielectric
constant photonic crystal with
defect.
120
5.4 CONCLUSIONS
Photonic band gaps estimated by transfer matrix method closely agree with the
experimental band gaps for glass and ebonite structures. Defect modes predicted by
calculations are also observed experimentally. Experimentally obtained Q factors of
defect modes are lower than estimated Q factors. Studies also reveal that low loss and
high dielectric constant structures exhibit band gaps with high attenuation and high
quality factor defect modes.
121
CHAPTER 6
SUMMARY AND CONCLUSIONS
In this chapter, studies carried out on DR compositions and one dimensional
photonic crystals are summarized and the possible conclusions arrived at are
presented. Scope for the future work is also presented at the end of this chapter.
Complex perovskites La(MgTi(1-x)Snx)0.5O3, La(1-x)Ndx(MgSn)0.5O3 and
Nd(MgTi)0.5O3 were prepared by solid state reaction method. The preparation
conditions were optimized to obtain the maximum density and minimum impurity. Xray patterns of La(MgTi(1-x)Snx)0.5O3 and Nd(MgTi)0.5O3 show an unidentified
impurity peak (< 2 wt %). Minor pyrochlore Nd2Sn2O7 phase (1 wt %) is present in
NMS and an unidentified impurity peak (< 2 wt %) is present in x = 0.0 and x = 0.25
compositions of La(1-x)Ndx(MgSn)0.5O3. Indexing the X-ray patterns based on a cubic
perovskite reveal that all the patterns exhibit super lattice reflections corresponding to
out-of-phase tilting and in-phase tilting of BO6 octahedra, A site cation displacements
and ½(111) super lattice reflection corresponding to 1:1 B-site cation ordering. The
structure of all the compositions revealed to be monoclinic P 21 / n with a-a-c+ tilting.
Lattice parameters of La(MgTi(1-x)Snx)0.5O3 were calculated. Tolerance factor
decreases and the unit cell volume increases with increase in Sn concentration.
Difference between the lattice parameters a and b also increases with increase in Sn
concentration.
Rietveld refinement was carried using GSAS suite with EXPGUI frontend
and LRO of La(1-x)Ndx(MgSn)0.5O3, LMT and NMT is quantified. Rietveld refinement
studies on La(1-x)Ndx(MgSn)0.5O3 reveal that these materials exhibit high percentage of
122
LRO and LRO decreases with increase in Nd concentration. Lattice parameters a and
c decrease with increase in Nd concentration but lattice parameter b slightly increase
with increase in Nd concentration. LRO values obtained for LMT and NMT are 96%
and 82% respectively. Lattice parameter a and c of NMT are less than that of LMT
but lattice parameter b of NMT is slightly higher than that of LMT. The slight
increase in lattice parameter b with increase in Nd concentration (decrease in
tolerance factor) may be due to the distortion of BO6 octahedra.
The experimental IR reflectance spectra were fitted with four parameter model
to obtain lattice mode characteristics such as frequencies of TO and LO modes and
their damping coefficients. Intrinsic parameters obtained are presented in Table 6.1.
The strength of A-BO6 modes is high for all the compositions and found to vary with
Sn concentration. Intrinsic dielectric constant decreases with increase in Sn
concentration. Intrinsic Q.f decreases with increase in Sn concentration (from x = 0.0
to 0.25) and then increases with increase in Sn concentration (from x = 0.25 to x =
0.5). Average phonon damping is highest for x = 0.25 composition and lowest for x =
0.5 composition. Even though LRO of LMS is less than that of LMT, Q factor of
LMS is high. It is due to the low dielectric constant of LMS.
Infrared reflectance spectra of La(1-x)Ndx(MgSn)0.5O3 were fitted with 17
modes. Strength of A-BO6 modes is high for all the compositions. Intrinsic dielectric
constant decreases with increase in Nd concentration (Table 6.2). Intrinsic Q.f
decreases (Table 6.2) and average phonon damping increases with increase in Nd
concentration correlating with the decrease in LRO.
Raman spectra were analyzed by identifying F2g and A1g modes and fitting the
A1g mode to Lorentzian peak shape. LMT and NMT exhibit asymmetric feature of the
A1g mode and it is analyzed by fitting with two merging Lorentzians. The Raman shift
123
and FWHM of A1g mode for La(MgTi(1-x)Snx)0.5O3 is presented in Table 6.1. Lowest
FWHM is observed for LMT confirming highest percentage of LRO. High FWHM
for intermediate compositions reveal that these compositions exhibit low percentage
of long range order and it is responsible for high average phonon damping. The
difference in the Raman shift of the A1g mode for LMT and LMS is determined to be
53 cm-1. The high value of difference in the shift is due to low B"-O bonding energy
with d10 configuration Sn.
The FWHM of A1g mode for La(1-x)Ndx(MgSn)0.5O3 decreases with increase in
Nd concentration (Table 6.2), confirming that long range order decreases with
increase in Nd concentration. This result is also in agreement with the infrared studies
that average phonon damping increases with increase in Nd concentration. In addition
to this main study, Raman spectra analyses and Rietveld refinement of LMT and
NMT reveal that asymmetric feature of the A1g mode observed for some of the
complex perovskites should not be attributed to the low percentage of 1:1 B site
cation long range order.
The dielectric properties of these compositions were studied in the microwave
frequency range of 9 to 11 GHz. Dielectric properties of La(MgTi(1-x)Snx)0.5O3 are
presented in Table 6.1. The dielectric constant of La(MgTi(1-x)Snx)0.5O3 decreases with
increase in Sn concentration. The Q.f product decreases with increase in Sn
concentration (x = 0.0 to x = 0.25) and then increases with increase in Sn
concentration (x = 0.25 to x = 0.5). Temperature coefficient of resonant frequency
decreases with decrease in tolerance factor. The trend of dielectric constant and Q.f
variation follows the trend of intrinsic parameters.
Microwave dielectric properties of La(1-x)Ndx(MgSn)0.5O3 are presented in
Table 6.2. The dielectric constant of La(1-x)Ndx(MgSn)0.5O3 decreases with decrease in
124
Nd concentration and follow the trend of intrinsic dielectric constant. Q.f values
follow the trend intrinsic Q.f values except for x = 0.25 and x = 0.5 compositions. Q.f
values of x = 0.25 and x = 0.5 are low due to the lower percentage of relative density
compared to other compositions (x = 0.0, 0.75 and 1.0). Microwave Q.f values
obtained for these compositions are less than intrinsic values due to extrinsic
contributions from defects, impurities etc. Temperature coefficient of resonant
frequency becomes less negative with decrease in tolerance factor. The microwave
dielectric constant of NMT is 26.1, Q.f is 50,000 GHz, τf is -49 ppm/°C and relative
density is 97%. Dielectric constant and Q.f value obtained for NMT are less than the
dielectric constant and Q.f of LMT. Temperature coefficient of resonant frequency is
less negative compared to that of LMT.
Table 6.1 Relative density (d), dielectric characteristics extrapolated from
infrared (IR) data, Lorentzian fit parameters of A1g mode (Raman) and dielectric
La(Mg0.5Ti(0.5-x)Snx)O3 ceramics
97.2
A1g (Raman)
Shift
FWHM
-1
(cm ) (cm-1)
722.2
20.6
IR
Q.f
(GHz)
29.1
85,580
28.4
MW
Q.f
τf
(GHz) (ppm/◦C)
55,000
-68
0.125
97.7
713.8
33.0
27.0
60,520
26.9
50,000
-74
0.25
97.6
701.2
36.0
24.7
54,860
24.4
46,000
-76
0.375
97.4
685.1
33.1
23.1
70,550
22.2
49,000
-80
0.5
97.8
668.6
25.1
20.5
1,11,000
19.7
63,000
-84
Sn
conc.
(x)
0.0
d
(%)
ε ′r
ε ′r
Results on La(1-x)Ndx(MgSn)0.5O3, LMT and NMT show that temperature
coefficient of resonant frequency do not decrease with decrease in tolerance factor but
correlates with composition and type of the A site ion. Low dielectric constants with
high Nd concentration suggest that Nd polarizability is less than that of La. Analysis
125
with the Classius-Mossotti equation reveals that ionic polarizability of 6.07 Å3 for La
ion
overestimates
dielectric
constants
whereas
polarizability
of
4.82
Å3
underestimates dielectric constants. Ionic polarizability of lanthanides was linearly
fitted to r3 values. By extrapolating the linear fit, polarizability of La was obtained to
be 5.71 Å3.
Table 6.2 Relative density (d), dielectric characteristics extrapolated from infrared
(IR) data, Lorentzian fit parameters of A1g mode (Raman) and dielectric parameters
determined at microwave (MW) frequencies for the La(1-x)Ndx(MgSn)0.5O3 system.
Nd
d
A1g (Raman)
content (%) Shift FWHM
(x)
(cm-1) (cm-1)
0.0
98.4 667.7
24.0
IR
Q.f
ε ′r
ε ′r
(GHz)
19.1 1,23,600 19.8
MW
Q.f
τf
(GHz) (ppm/◦C)
75,000
-82
0.25
97.8 665.0
24.4
18.9
1,05,800 19.5
68,000
-75
0.5
97.6 663.2
24.9
18.5
1,02,900 19.4
64,000
-66
0.75
98.2 661.1
25.2
18.3
99,700
19.2
70,000
-60
1.0
98.5 658.5
25.4
18.0
93,900
19.1
68,000
-53
Photonic band gaps of one dimensional structures are analyzed by
computation of transmittance, band structure and density of modes using transfer
matrix method and experimental measurement of the transmission spectra.
Theoretically estimated band gaps and defect modes closely agree with the
experimental measurements for glass and ebonite structures. Studies also reveal that
low loss and high dielectric constant structures exhibit band gaps with higher
attenuation of transmittance and high quality factor defect modes.
SCOPE FOR THE FUTURE WORK
In order to obtain 100% perovskite phase (without any minor impurities) other
preparation methods such as chemical methods or ball milling the powders should be
126
tried. Isostatic pressing of the powders and sintering in oxygen atmosphere can be
tried to obtain better quality DR samples.
To get accurate bond valences, covalency of ions, distortion indices of BO6
octahedra and tilting angles of octahedra, structure of the perovskites should be
determined with the combined Rietveld refinement of X-ray diffraction data and high
resolution neutron diffraction data. Microstructure details such as anti phase
boundaries and twinning may be estimated using High Resolution Transmission
Electron Microscopy (HRTEM) and its relation to dielectric loss may be estimated.
To understand the details of local structure effect on Raman modes, detailed
studies are warranted. Local structural studies using Extended X-ray Absorption Fine
Structure (EXAFS) or Pair Distribution Function (PDF) analysis combined with
phonon dispersion calculations are required understand the asymmetry in A1g mode.
First principle calculations on phonon modes may be carried out to relate the
temperature coefficient of resonant frequency with the composition and structure.
Combined study with bond valences and first principle calculation may be carried out
to understand the factors responsible for the variation of temperature coefficient of
resonant frequency. To get an understanding of extrinsic loss mechanism on Q values,
microstructure studies using Electron Probe Microscopy Analysis (EPMA) may be
carried out.
Substitution of magnetic ion to achieve tunability while maintaining low
dielectric losses and preparing complex perovskites with simultaneous A and B site
cation ordering to examine the effect of A site cation ordering on dielectric losses can
also be attempted.
127
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138
LIST OF PUBLICATIONS BASED ON THE RESEARCH WORK
Publications in Refereed Journals:
1. G.Santosh Babu, V.Subramanian, V.Sivasubramanian and V.R.K.Murthy "Study
of One-dimensional Photonic Band Gaps for Microwave Filters" Ferroelectrics,
327, 19 (2005).
2. G.Santosh Babu, V.Subramanian and V.R.K.Murthy, “Structure Determination
and Microwave Dielectric properties of La(MgSn)0.5O3 Ceramics” J. Eur. Ceram.
Soc., 27, 2973 (2007).
3. G.Santosh Babu, V.Subramanian, V.R.K.Murthy, I-Nan Lin , Chia-Ta Chia and
Hsiang-Lin Liu “Far-Infrared, Raman spectroscopy and microwave dielectric
properties of La(Mg0.5Ti(0.5-x)Snx)O3 ceramics” J. Appl. Phys., 102, 064906
(2007).
4. G.Santosh Babu, V.Subramanian, V.R.K.Murthy, R.L.Moreira and R.P.S.M.
Lobo ”Crystal structure, Raman spectroscopy, far-infrared and microwave
dielectric properties of (1-x)La(MgSn)0.5O3 – xNd(MgSn)0.5O3 system” J. Appl.
Phys., 103, 084104 (2008).
Presentations in Conferences:
1. G.Santosh Babu, V.Subramanian, V.Sivasubramanian and V.R.K.Murthy
“Investigation of different Geometrical Structures for Microwave Band Gap
Studies” Presented at “Twelfth National Seminar on Ferroelectrics and
Dielectrics” held at IISc, Bangalore, India during 16th-18th December 2002.
2. G.Santosh Babu, V.Subramanian, V.Sivasubramanian and V.R.K.Murthy. "Band
Gap and Defect Mode in a Double Periodic One-dimensional Microwave Band
Gap Structure" Presented at "Thirteenth National Seminar on Ferroelectrics and
Dielectrics" held at New Delhi, India during 23rd-25th November 2004.
3. G.Santosh Babu, V.Subramanian, V.Sivasubramanian and V.R.K.Murthy.
"Microwave Propagation through One-dimensional Microwave Band Gap
Structures" Presented at "Asia-Pacific Microwave Conference - 4" held at New
Delhi, India during 15th-18th December 2004.
4. G.Santosh Babu, V.Subramanian and V.R.K.Murthy. “Preparation and
Characterization of La(MgTixSn(1-x))1/2O3 Ceramics” Presented at “Microwave
Materials and their Applications -2006” held at Oulu, Finland during 12th to 15th
June, 2006.
5. G.Santosh Babu, V.Subramanian and V.R.K.Murthy. “Structure Determination
of B-site Ordered Perovskites by Rietveld Analysis” Presented at "National
Conference on Ferroics" held at Hyderabad, India during 30th June-1st July, 2006.
139

Thesis of Dr. G. Santosh Babu - Department of Physics

Transcription

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